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Unformatted text preview: UNIVERSITY OF WATERLOO DEPARTMENT OF ECONOMICS ECONOMICS 301 MICROECONOMIC THEORY II C. H. VAN DE WAAL The learning materials contained in this booklet have been reproduced for the exclusive use of the students registered in the above course at the University of Waterloo. They may not be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopied, recorded or otherwise, either in whole or in part, without obtaining the written permission of the course author. TABLE OF CONTENTS LECTURE 1 LECTURE 2 LECTURE 3 LECTURE 4 LECTURE 5 LECTURE 6 LECTURE 7 LECTURE 8 LECTURE 9 LECTURE 10 LECTURE 11 LECTURE 12 LECTURE 13 LECTURE 14 LECTURE 15 LECTURE 16 LECTURE 17 LECTURE 18 LECTURE 19 1 14 39 54 71 89 109 129 145 166 180 197 220 236 260 281 294 309 325 i ECON 301 LECTURE #1 REVIEW: Consumer Equilibrium (Utility Maximization) The consumer desires to maximize their Total Utility subject to their budget constraint. Let's consider the case where the consumer has the choice between 2 goods, X and Y. Max U(X,Y) subject to PX X + PY Y = M X,Y Using ECON 201 logic, we know that for the consumer to reach an equilibrium he/she must have no further incentives to shift consumption among the two goods (in terms of Utility). This implies that the last unit purchased of X and the last unit purchased of Y must give the consumer the same "satisfaction". Suppose, for example, that PX = 2 and PY = 1. We should notice the relationship between PX and PY is simply PX = 2PY. Our ECON 201 logic is that if X costs twice as much as Y ... to be worth paying double for X, its marginal utility (utility per additional unit) must be twice the marginal utility of Y in equilibrium. This implies that; Rearranging, or simply, MUX = MUY PX PY Of course, this condition extends to the case of multiple (more than two) goods as follows: MU1 = MU2 = ... = MUn P1 P2 Pn Now that we have taken ECON 211, we can arrive at this conclusion quantitatively. Using the Lagrangian technique learned in ECON 211, we can set up the consumer's problem in general as: 1 MUX = 2MUY MUX = MUY 2 1 L: Max U(X,Y) + (M PX X PY Y) X,Y Taking the First Order Conditions, we get: L / X = FOCX L / Y = FOCY UX PX = 0 UY PY = 0 (1) (2) Where, UX = Marginal Utility of good X. UY = Marginal Utility of good Y. Rearrange (1) and (2) to isolate on the LHS... = UX / PX = UY / PY Equate (3) and (4) to get, UX = UY PX PY MUX = MUY PY PX which is the consumer's equilibrium condition that we arrived at above using our ECON 201 logic and intuition. This condition is central to microeconomics and this course. We will often use variations of the equilibrium condition above, for example, MUX = PX MUY PY which says that the ratio of the marginal utilities of two goods is proportional to the ratio of the prices of the two goods. Also, this says that the MRSXY is equal to the relative prices of X and Y (the price ratio). from (1) from (2) (3) (4) or simply, 2 Some popular utility functions used in economics are of the forms: U(x,y) = aXY Cobb-Douglas Preferences U(x,y) = aX + bY Perfect Substitutes Utility U(x,y) = min{aX , bY) Liontief Preferences Perfect Compliments U(x,y) = X + aY1/2 Quasi-linear Preferences Quasi-linear preferences are useful in economic applications because they allow us to ignore wealth effects. These have a similar shape to CobbDouglas preferences. Expectations for Diagrams, Graphs and Charts in this Course We will continue to rely quite heavily on the intuition provided by diagrams in this course. It is important to take the production of these diagrams very seriously and to put sufficient care into drawing them since they can provide an excellent basis to guide our economic intuition. You should attempt to produce good diagrams and graphs that are large, properly (and completely) labeled, with curves that curve and straight lines that are straight. You should also take brief yet detailed notes concerning the key features of the diagram since at test time this will refresh your memory about the point that the graph is intended to illustrate when you study. The intended result of a good diagram is that it be self-contained and selfexplanatory (without long paragraphs of explanatory text). Partial vs. General Equilibrium Analysis In ECON 201, we restricted our analysis to one individual object at a time whether it was a market, a consumer, or a producer while we assumed away the rest of the world. As an example, when we considered the market for air travel in ECON 201, we did not investigate the effects on the market for train travel or other travel services. By doing that, we essentially assume that only the market for air travel is affected by our investigation and the rest of the travel industry is unaffected and will not respond to activities in the market for air travel. 3 This method of investigation is called Partial Equilibrium Analysis. We sometimes use the phrase "ceteris paribus" in this line of investigation, which is Latin for Everything Else Being the Same. In this course, we begin our training in General Equilibrium Analysis. We no longer restrict our attention to a single market, consumer, or producer in isolation. As a result, we have the opportunity to explore the inter-relationship and interdependence among the various components of the economy. As an example, when considering issues of free-trade (markets for goods and services) we want to investigate the possible impacts on employment (market for labour) and maybe even on exchange rates (the market for money). The multiple market feature of General Equilibrium analysis is described by the assumption of "mutual inter-dependence" or simply as Everything Depends on Everything Else. Now that we know where we are headed, let's begin to build the necessary toolbox of skills to deal with the challenges of this type of analysis. Integral Exponential Powers This can be thought of as a shortcut notation for repeated multiplication of a number by itself. Normal Notation Multiply X by itself once Multiply X by itself twice Multiply X by itself thrice Multiply X by itself four times Multiply X by itself n times X XX XXX XXXX XXXX...X { n times } Exponential Notation X1 X2 X3 X4 Xn Thus, given a number X and an integer n, the exponential power n of a number X can be defined as the product of multiplying X by itself n times. This can be said in a number of ways, each meaning the same thing... "exponential power n of X" "nth exponential power of X" "X raised to the power of n" Let's consider the functional forms for n=1,2,3. First, n=1 means we are multiplying X by itself once which is simply X 4 Secondly, n=2 is the special case of the square function. This is X2 and produces a parabola. FIGURE 1: SQUARE FUNCTION 100 X -10 0 10 FIGURE 2: CUBIC FUNCTION If n=3 is the special case of the cubic function. This is X3 and produces an inflection point at 0. 64 -4 -2 8 0 -8 -64 2 4 X Now, what if n=0? There is not an intuitive way to evaluate the result of multiplying a number by itself zero times. It is generally accepted that X0 = 1. It is important to notice that the zeroth exponent is a definition and not a calculation. As an example, notice that we always have the following result: 10 20 15540 0 (-2.81)0 00 = = = = = = 1 1 1 1 1 1 Even negative numbers, or zero, have their zeroth exponent defined to be one. 5 Fractional Exponential Powers The concept of exponential powers can easily be extended to the case where the power is a fraction of the form 1/n or m/n. How do we define X to the power of 1/n? The exponential power X1/n means that if we repeatedly multiply X1/n by itself n times we get the original number X. X1/n X1/n X1/n X1/n ... X1/n = X { n times } or simply, [X1/n]n = X There is a close relationship between the integral exponential Xn and the fractional exponential X1/n. To see this, let's define a new variable Y = X1/n and then by raising Y to the exponent n (or multiplying Y by itself n times) we get the original variable X back. Yn = X1/n X1/n X1/n X1/n ... X1/n = X { n times } Let's look at a couple of common fractional exponents used in economics. First, X1/2, which means if we square it we'll get X back. This can also be thought of as the square root of X (X). FIGURE 3: SQUARE ROOT FUNCTION 2 1 0 1 4 X 6 Similarly, the exponential power X1/3 means that if we cube it we will get X back. This is also referred to as the cube root of X (3X). The graph is qualitatively the same as the square root function displayed above. Okay. Now let's calculate one of these things as an example. What is 253/2? 253/2 = 251/2 251/2 251/2 =555 = 125 How do we get back to 25 from 125? We want to raise 125 to the exponent 2/3. To see this, consider... 1252/3 = [125 125]1/3 = [15625]1/3 = [25 25 25]1/3 = 25 So we have the result that 253/2 = 125 and 1252/3 = 25. EXPONENTIAL POWER ARITHMETIC RULES Exponential powers provide a convenient way for us to perform multiplication and division of numbers by performing simple addition and subtraction on their exponential powers. Multiplication: To multiply two exponential powers Xm and Xn of a given number X, we can simply add the powers m and n together. Xm Xn = Xm+n This can be seen as follows, Xm Xn = (X X X X ... X) (X X X X ... X) { m times } { n times } = (X X X X ... X) { m+n times } = Xm+n 7 Example, X3 X4 = (X X X) (X X X X) =XXXXXXX = X7 = X3+4 Division: To divide two exponential powers Xm and Xn of a given positive number X, we subtract the powers m and n. Xm / Xn = Xm-n This can be seen as follows: Xm / Xn = (X X X X ... X) / (X X X X ... X) { m times } { n times } = (X X X X ... X) { m-n times } = Xm-n Example, X6 / X4 = (X X X X X X) / (X X X X) =XXXXXX XXXX = X2 = X(6-4) Okay, but what if m=0? If m=0, then m - n = -n. We'll use the fact that the power 0 of a number is defined to be 1, to get... X-n = X0-n = X0 Xn =1 Xn 8 That is, raising a number to a negative power is equivalent to taking its reciprocal. For example, X-2 = 1 X2 X-6 = 1 X6 X-1/2 = 1 X1/2 =1 X X-2/3 = 1 X2/3 Finally, to take the exponential power m of an exponential power Xn, we multiply the powers m and n together. [Xn]m = Xn Xn Xn Xn Xn ... Xn ( m times ) = (X X ... X) (X X ... X) ... (X X ... X) ( n times ) ( n times ) ( n times ) { m times } =XX...X (nm times) = Xnm For example, [X3]2 = X3 X3 = (X X X) (X X X) =XXXXXX = X6 = X32 = Xnm 9 Let's summarize, PROPERTIES OF EXPONENTIAL POWERS X0 = 1 Xn = X X X X ... X ( n times ) X-n = 1 Xn Xm Xn = Xm+n Xm / Xn = Xm-n [Xn]m = Xnm Exponential powers are used widely in economic functional forms. Some examples include, SQUARE ROOT UTILITY U(X,Y) = XY = XY = X1/2 Y1/2 COBB-DOUGLAS UTILITY U(X,Y) = X Y where , are parameters for shares and is for scaling. CES UTILITY U(X,Y) = [X- + Y-]-1/ where , are parameters for shares, is for curvature and is for scaling. The production side of the economy can also be represented equivalently in these functional forms. To do this, simply make all of the Xs into Ks and all of the Ys into Ls. Let's go through a couple of examples where we will find the consumer demands using the Lagrange technique and our new found knowledge about exponents... 10 1. U(X,Y) = (X1/3)(X1/3)[(Y2/3)/(Y1/3)] subject to 9X + 3Y = 1500. First we need to set up the Lagrange...but what exponents do we have on X and Y? L: Max X2/3Y1/3 + (1500 - 9X - 3Y) X,Y Taking the First Order Conditions, we get: L / X = FOCx L / Y = FOCy 2Y1/3 / 3 X1/3 - 9 = 0 X2/3 / 3 Y2/3 3 = 0 (1) (2) Rearrange (1) and (2) to isolate on the LHS... = 2Y1/3 / 27 X1/3 = X2/3 / 9 Y2/3 Equate (3) and (4) to get, X2/3 / 9 Y2/3 = 2 Y1/3 / 27 X1/3 or simply, 27X = 18Y 2Y = 3X Y = 3/2 X X = 2/3 Y (5) (6) from (1) from (2) (3) (4) or Substitute (5) into the Budget Constraint to get X: 3(3/2 X) + 9X = 1500 27/2 X = 1500 27X = 3000 X = 111.111 Substitute (6) into the Budget Constraint to get Y: 3Y + 9(2/3 Y) = 1500 27/3 Y = 1500 9 Y = 1500 Y = 166.6667 Note: The share of income spent on good X is: Px X = 9(111.111) = 1000 = 2 = exponent on X in the Utility Function M 1500 1500 3 11 Similarly, the share of income spent on good Y is: Py Y = 3(166.666) = 500 = 1 = exponent on Y in the Utility Function M 1500 1500 3 2. U(x,y) = [(X1/5) / (X2/5)]-5 + 2 {[(Y2/3) (Y1/3)] / (Y3/4)}2 subject to 5X + 2Y = 40. Once again, we need to set up the Lagrange...but what exponents do we have on X and Y? L: Max X + 2Y1/2 + (40 - 5X - 2Y) X,Y Taking the First Order Conditions, we get: L / X = FOCx L / Y = FOCy 1 - 5 = 0 1 / Y1/2 2 = 0 (1) (2) Rearrange (1) and (2) to isolate on the LHS... = 1/5 = 1 / 2Y1/2 Equate (3) and (4) to get, 1 / 2Y1/2 = 1/5 or simply, 2/5 Y1/2 = 1 Y1/2 = 5/2 Y = 25/4 (or 6.25) from (1) from (2) (3) (4) (5) Substitute (5) into the Budget Constraint to get X: 5X + 2(25/4) = 40 5 X = 110/4 X = 110/20 X=5 12 HOMEWORK: Find the consumer equilibrium (i.e. demands for X and Y) for the following situations: 1. U(x,y) = X + Y subject to 2X + 3Y = 150 ... (HINT: draw the graph first). 2. U(x,y) = X1/3Y2/3 subject to 3X + 9Y = 3000. What can you say about your result in terms of the shares of income spent on each good? 3. U(x,y) = X + 2 Y1/2 subject to 5X + 2Y = 40. 4. Evaluate the following expressions. a) b) c) d) e) (x+y)2 (x3 + y2)2 (z y4)2 (c1/2 + b5)0 [(c1/2)8]1/2 5. Verify that the CES production function, f(K,L) = [K- + L-]-1/, can be written equivalently as... f(K,L) = _____________ [ / K + / L]1/ Find the consumer equilibrium (i.e. demands for X and Y) for the following situations: 6. U(x,y) = 7(X1/6)3 (Y2/3)1/2 subject to 7X + Y = 800. 7. U(x,y) = [X / (X1/2)] + 6 Y1/2 subject to 3X + 2Y = 330. 13 ECON 301 LECTURE #2 Slopes A slope is the measure of steepness of a line. That is, the bigger the slope, the steeper the line. Let's consider the following linear equation of a straight line: Y = aX + b In this case the slope is determined by the coefficient "a" of the independent variable X. The vertical intercept is determined by the constant "b" and the horizontal intercept is determined by the ratio "-b/a". Since we are investigating a straight line we know that the slope is constant and as such it does not matter which point on the line we evaluate the slope (we should always get the same answer). Slope = a b -b/a 0 X In the special case where b = 0, we have both the vertical intercept and horizontal intercept at the origin. Slopes of Curves A tangent is a straight line that touches a curve at a single point. The line is referred to as a tangent line and the point on the curve that is touching the line is called the point of tangency. At this point of tangency, both the line and the curve will have the same value of the independent variable. 14 Y Curve Y* Tangent line 0 X* X For example, if the straight line Yline = f(X) and the curve Ycurve = g(X) are tangent to each other at the point of tangency (X*, Y*), then we have... Y*line = Y*curve = Y* f(X*) = g(X*) So, at each point on a curve, the slope of the curve at that point is defined as the slope of the tangent line at the point of tangency. Why do we care so much about slopes and tangencies in economics? We care because they are very closely related to various marginal measures that we find interesting and useful in economic analysis. 15 Curves / Functions Utility function Production function Total cost Total revenue Total profit Indifference curve Isoquant Differentiation Rules Slopes / Tangents Marginal utility Marginal product Marginal cost Marginal revenue Marginal profit Marginal rate of Substitution Marginal rate of Technical Substitution To calculate the slope of a curve we take the derivative of the function with respect to the independent variable. For example, Curves / Functions Y=a Y = aX + b Y = aX2 + bX + c Y = aX3 + bX2 + cX + d Y = Xn Slopes / Tangents Y / X = 0 (Constant) Y / X = a (Linear) Y / X = 2aX + b (Quadratic) Y / X = 3aX2 + 2bX + c (Cubic) Y / X = nXn-1 (Exponential) In addition, we can calculate the slope of a combination of two curves, f(X) and g(X). Curves / Functions Y = f(X) + g(X) Y = f(X) - g(X) Y = f(X) g(X) Y = f(X) / g(X) Slopes / Tangents Y / X = [f / X] + [g / X] Y / X = [f / X] - [g / X] Y / X = [f / X] g(X) + [g / X] f(X) Y / X = [f / X] g(X) - [g / X] f(X) [g(X)]2 (Addition) (Subtraction) (Multiplication) (Division) 16 Let's use each of these rules for the example of f(X) = 2X3 and g(X) = 7X2. Y = f(X) + g(X) Y / X = [f / X] + [g / X] 6X2 + 14X Y = f(X) - g(X) Y / X = [f / X] - [g / X] 6X2 - 14X Y = f(X) g(X) Y / X = [f / X] g(X) + [g / X] f(X) (6X2)( 7X2) + (14X)( 2X3) 42X4 + 28X4 70X4 Y / X = [f / X] g(X) - [g / X] f(X) [g(X)]2 (6X2)( 7X2) - (14X)( 2X3) ( 7X2)2 42X4 - 28X4 49X4 14X4 49X4 Envelopes A family of curves is a collection of curves which are related to each other by a common variable (parameter). A typical example is a family of short run cost curves, CSR each of which is associated with a plant size, . CSR = CSR (Q,) Figure #3: Family of SR Cost Curves $ Y = f(X) / g(X) 2 7 CSR (1) CSR (2) CSR (3) CSR (4) CLR Q 17 In this example, the long run cost curve is the envelope (a curve that is tangent to each member curve in the family of curves) of the family of short run cost curves. We can calculate the envelope of a family of curves using the following steps... 1. Get the slope with respect to the parameter , 2. Set this slope equal to zero and solve for the optimal parameter *, 3. Substitute the optimal parameter * into the equation of the curve. Let's do an example, consider the following family of short run cost curves: CSR (Q,) = - 2Q2 + (15 )Q + 2 where Q denotes the output level and denotes the plant size. STEP 1: Get the slope with respect to the parameter . CSR (Q,) / = 2 Q STEP 2: Set this slope equal to zero and solve for the optimal parameter *. CSR (Q,) / = 0 2 Q = 0 or * = Q / 2 STEP 3: Substitute the optimal parameter * into the equation of the curve. CLR (Q) = - 2Q2 + (15 *)Q + *2 = - 2Q2 + (15 Q / 2)Q + (Q / 2)2 = 8/4Q2 2/4Q2 + 1/4Q2 + 15Q = 9/4Q2 + 15Q This is the equation of the envelope of this family of short run cost curves. Notice that there is no fixed cost component in the equation in the long run cost curve. This is consistent with the notion that in the long run all costs are variable. Review: MRS and MRTS MRS or Marginal Rate of Substitution is the rate at which one good can be substituted for another good along the same indifference curve. MRS = -Y (along the same IC or holding utility constant) X 18 This is equivalent to saying that MRS is equal to the (negative) slope of the indifference curve. We should also remember from ECON 201 that the MRS is the ratio of the two marginal utilities. MRS = MUX / MUY MRTS or Marginal Rate of Technical Substitution is the rate at which one factor can be substituted for another factor along the same isoquant. MRTS = -L (along the same IQ or holding output constant) K This is equivalent to saying that MRTS is equal to the (negative) slope of the isoquant. We should also remember from ECON 201 that the MRTS is the ratio of the two marginal products. MRTS = MPK / MPL Marginal Rate of Transformation The marginal rate of transformation (MRT) refers to the rate that one good can be substituted for another good in the production process of an economy without changing the production technology and resources of the economy. Essentially, MRT measures the trade-off between one good and another along the same PPF (production possibility frontier). Recall from ECON 201 that a PPF refers to all possible combinations of goods that can be produced efficiently with given levels of factor inputs and production technology. Figure #4: Production Possibility Frontier Jalapeno Chips Bubble Gum 19 A simple example is the trade-off between Chips and Gum. With fixed levels of factor inputs and production technology, the economy can either produce more gum and less chips OR more chips and less gum, NOT BOTH. To produce more gum, capital and labour must be moved from the chip sector to the gum sector, and hence less chips will be produced. So, we can see that MRT differs from MRS and MRTS mainly in the type of commodities being substituted. MRT vs. MRS MRS refers to the consumption side while MRT refers to the production side. MRS measures the substitution of outputs consumed while MRT measures the substitution of outputs produced. MRT vs. MRTS MRTS refers to the input side while MRT refers to the output side. MRTS measures the substitution of factor inputs while MRT measures the substitution of outputs produced. Similar to the relationship between MRS and an indifference curve, the MRT is defined as the (negative) slope of the PPF. MRT is the rate at which one good can be substituted for another good along the same production possibility frontier. MRT = -Chips (along the same PPF or holding factors and technology constant) Gum We can also show that the MRT is the ratio of the two marginal costs. MRT = MCGum / MCChips Take the simple case of a single factor input (say, labour) which is used to produce the two outputs, Chips and Gum. Assume the general production functions that follow: Gum = f(LGum) Chips = g(LChips) LGum + LChips = LTotal 20 where, LGum = the amount of labour used in the production of Gum. LChips = the amount of labour used in the production of Chips. LTotal = the fixed total amount of labour in the economy. We can generalize the equation of the PPF as: LTotal = L(Gum,Chips) and we know that any movement along the PPF does not change the total amount of labour, LTotal. Thus, LTotal = 0. The total change in labour can be broken down into the change due to labour movements in the Gum sector and changes due to labour movements in the Chips sector. (L / Gum) Gum + (L / Chips) Chips = 0 Observe that (L / Gum) refers to the change in labour as a result of a change in the output level of Gum. This is essentially the marginal cost of Gum (technically, not exactly correct). Similarly, (L / Chips) can be interpreted as the marginal cost of Chips. We re-write in terms of marginal costs and get: MCGum Gum + MCChips Chips = 0 MCGum Gum = - MCChips Chips MCGum / MCChips = -Chips Gum Recall that we defined MRT above (just before the squiggly line) as Chips !!!!! Gum Thus, MCGum / MCChips = MRT. Q.E.D. Now let's turn our attention to three popular economic functions Cobb-Douglas, Liontief, and CES (Constant Elasticity of Substitution). As we know, these functions can be applied to both utility functions (theory of consumption) and production functions (theory of production). 21 COBB-DOUGLAS FUNCTIONS The general form of a Cobb-Douglas utility function is defined as: U(x,y) = XY with positive constants, > 0 expenditure share of good X > 0 expenditure share of good Y > 0 scale constant Cobb-Douglas utility functions have the following marginal utilities: MUX = U X MUY = U Y Don't take my word for it...let's derive the Cobb-Douglas marginal utilities. To do this we must take the partial derivatives of the utility function with respect to X and Y, respectively. MUX = U X MUX = Y(X-1) MUX = Y(X) X MUX = (XY) X MUX = U X and... MUY = U Y MUY = X(Y-1) MUY = X(Y) Y 22 MUY = (XY) Y MUY = U Y Now we can calculate the general form of the MRS for a Cobb-Douglas utility function. Recall that MRS is defined as: MRS = MUX MUY Using the marginal utilities we just derived for the general form of the CobbDouglas utility function we have: MRS = MUX MUY MRS = U __X__ U Y Y X MRS = Let's do an example, assume you are given that = 1,969, = 0.35, and = 0.65. Then we have the following Cobb-Douglas utility function: U(x,y) = XY = 1,969X0.35Y0.65 and the marginal rate of substitution (using our shortcut) is... MRS = MRS = MRS = Y X 0.35Y 0.65X 7Y 13X Let's check that this is correct using the long way around... 23 MRS = MUX MUY MRS = 1969(0.35)X-0.65Y0.65 1969(0.65) X0.35Y-0.35 MRS = (0.35)Y0.35Y0.65 (0.65) X0.35X0.65 MRS = 0.35Y 0.65X MRS = 7Y 13X Does this result hold if the Cobb-Douglas utility function is NOT constant returns to scale? Assume you are given that = 1,969, = 0.65, and = 0.65. Then we have the following Cobb-Douglas utility function: U(x,y) = XY = 1,969X0.65Y0.65 and the marginal rate of substitution (using our shortcut) is... MRS = MRS = MRS = Y X 0.65Y 0.65X Y X Let's check that this is correct using the long way around... MRS = MUX MUY MRS = 1969(0.65)X-0.35Y0.65 1969(0.65) X0.65Y-0.35 24 MRS = (0.65)Y0.35Y0.65 (0.65) X0.65X0.35 MRS = 0.65Y 0.65X MRS = Y X This result DOES, in fact, hold for IRS functions (or DRS functions). LIONTIEF FUNCTIONS The general form of a Liontief utility function is defined as: U(x,y) = min{X , Y} meaning, U(x,y) = {X {Y if X < Y if Y < X This is interpreted as meaning that the utility is equal to the smaller of the two values X and Y. In the special case where X = Y, the utility is equal to either one of the values (since they are the same). We know what a Liontief indifference curve looks like, but let's investigate why this is. As an example, let's draw the unit indifference curve (the indifference curve that gives the consumer one unit of happiness) of the following Liontief utility function: U(x,y) = min{2X , 3Y} meaning, U(x,y) = {2X {3Y if 2X < 3Y if 3Y < 2X so we need to find some combinations of goods X and Y that result in U(x,y) = 1. Consider the following 3 points: X 1/2 1/2 1 Y 1/3 1 1/3 2X 2(1/2) = 1 2(1/2) = 1 2(1) = 2 3Y 3(1/3) = 1 3(1) = 3 3(1/3) = 1 U(x,y) = min{2X , 3Y} min {1 , 1} = 1 min {1 , 3} = 1 min {2 , 1} = 1 25 Each of these points yields exactly one unit of utility as defined by our Liontief unit indifference curve. The rigid preference structure for a Liontief consumer gives us the L-shaped indifference curve with a kink at the corner point. The corner point is where both 2X and 3Y give us one unit of utility (in general, where X = Y). This occurs in our example when X = 1/2 (or generally, when X = 1/) and Y = 1/3 (or generally, when Y = 1/). The straight line (or kink line) joining the point of origin and the corner points for this family of indifference curves has a slope determined by 2X = 3Y or Y = 2/3X (or generally, /). Y 1 A Slope = 2/3 = / 1/3 Kink B Indifference curve X 1/2 1 The kink point is the only real point of interest to us when dealing with Liontief preferences (or production). If we compare the kink point (1/2 , 1/3) with point A (1/2 , 1) on the vertical segment and point B (1 , 1/3) on the horizontal segment, it should be clear that while these three points lead to the same utility level for the consumer, point A requires more of good Y and point B requires more of good X than the kink point does to achieve the same utility level. This means that the kink point gives this consumer the "best" (most efficient in terms of resources) combination of goods X and Y according to a fixed proportion (i.e. X = Y). Thus, this type of function has a rather rigid structure in the sense that extra amounts of the goods beyond these proportions are completely ruled out because they do not add any extra utility to the consumer (and with strictly positive prices, these superfluous goods have a cost to the consumer). As a result of the argument above, the indifference curve for these preferences has an L-shape with the vertical segment representing all of the combinations of extra good Y and the horizontal segment representing all of the combinations of 26 extra good X. For this reason, Liontief functions are also referred to as "fixedcoefficient" functions. So now let's consider the MRS of a Liontief function. The marginal rate of substitution, as we saw in the last class, is the slope of the tangent line at the point of tangency. We just discussed that the real point of interest to us when dealing with a Liontief function is the kink point. If we want to have the point of tangency at the kink point, do we get a unique solution? NO!!! There are infinitely many tangent lines, with infinitely many different slopes, that are tangent to the kink point. For this reason, Liontief functions are not differentiable at the kink and the marginal utility (or marginal product) concept does not apply to these functions. As such, the concept of the marginal rate of substitution is undefined for the Liontief function (more later). CONSTANT ELASTICITY OF SUBSTITUTION (CES) FUNCTIONS The general form of a CES production function is defined as: Q = f(K,L) = (K- + L-)-1/ or Q = f(K,L) = _____________ [ / K + / L]1/ So what are the marginal products? MPK = Q K = / (Q / K)1 + (holding L constant) MPL = Q L = / (Q / L)1 + (holding K constant) If we take the ratio of these two marginal products, we get the MRTS as follows: MRTS = MPK MPL 27 = / (Q / K)1 + / (Q / L)1 + = (QL)1 + (QK)1 + = [L/K]1 + So let's compute an example using = 1, = = 0.5, and = 1. We get the following CES production function. Q = f(K,L) = _____________ [ / K + / L]1/ Q = f(K,L) = _____1________ [0.5 / K + 0.5 / L] Q = f(K,L) = _____2KL_____ [KL / K + KL / L] Q = f(K,L) = _____2KL_____ L+K with marginal products of: MPK = / (Q / K)1 + = 0.5 (Q / K)2 MPL = / (Q / L)1 + = 0.5 (Q / L)2 and the marginal rate of technical substitution of: MRTS = / (Q / K)1 + / (Q / L)1 + = 0.5 (Q / K)2 0.5 (Q / L)2 = (L / K)2 28 THE IMPORTANCE OF The parameter > 0 measures the degree of flexibility of input (or factor) substitution in the production process. It can also be defined in terms of the elasticity of substitution, , as follows: = ___1___ 1+ or, =1-1 THE IMPORTANCE OF The parameter indicates how easily inputs can be substituted among each other along the same isoquant. The higher the value of , the more flexibility there is in the substitution between capital and labour in the production process. For example, = 0 allows no substitution between capital and labour (as in the case of a Liontief fixed-coefficient isoquant). = allows perfect substitution between capital and labour (as in the case of a linear isoquant). A special case of the CES function is when = 0 (or = 1). This gives us the Cobb-Douglas function!!!!!! To see this consider the marginal products of a Cobb-Douglas production function: COBB DOUGLAS MPK = Q K MPL =Q L MRTS = L K CES MPK = / (Q / K)1 + MPL = / (Q / L)1 + MRTS = [L/K]1 + So we can see that there is some relationship between Cobb-Douglas, Liontief, Linear, and CES functions. What values for and do we need to make a CES function into a Cobb-Douglas, Liontief, or a Linear function? 29 Elasticity of Substitution =1 =0 = 0<< Value for =0 = = -1 -1 < < Resulting Function Cobb-Douglas Liontief Linear CES Finally, let's look at these relationships in terms of a graph. Elasticity of Substitution L =0 lower higher = K Geometrically, refers to the curvature of the isoquant such that the higher the value has, the flatter the isoquant is. At the two extremes (as seen in the graph), = 0 implies an L-shaped isoquant with no substitutability between capital and labour while = implies a straight line isoquant with perfect substitutability. Rational Behaviour Consumer Equilibrium A consumer equilibrium is the solution to the consumer's basic problem of selecting the best possible collection of commodities given all of the influencing economic factors. The choice problem is essentially to balance the following two factors: 30 Wants / Preferences What the consumer wants is represented by preferences, utility, and indifference curves. The consumer wants to go as high as possible in this direction. Affordability / Budget Constraint What the consumer can afford is represented by income, prices and thus, the budget lines. The budget constraint holds the consumer back, limiting them. The basic theoretical premise of "rational behaviour" is that the consumer solves their choice problem within the context of a constrained optimization problem (i.e. Lagrangians). We call this the "utility maximization problem". The economic intuition behind the utility maximization problem is to balance the opposing forces represented by the indifference curve and the budget line. This means that consumers are choosing X and Y to achieve the highest indifference curve that is affordable given prices and income. Y Indifference curve Consumer Equilibrium Budget Line X The slope at the equilibrium point where the indifference curve and the budget line are tangent to each other will correspond to the condition: slope of the budget line = slope of the curve at the point of tangency slope of the budget line = the slope of the indifference curve 31 The slope of the budget line is: PXX + PYY = M Y = M - PX X PY PY Thus, the slope of the budget line is - PX . PY The slope of the indifference curve is MRS, as we've seen before, so... MRS = PX PY This gives us a new element to our definition of MRS, namely that MRS is simply the ratio of prices. In addition, the consumer equilibrium point must lie on the budget line to ensure the affordability requirement. This means that the budget constraint must be exactly satisfied at the consumer equilibrium point. Why do we need both equations for consumer equilibrium to occur? One equation is not enough to describe the concept of a consumer equilibrium point because: 1. MRS = PX / PY only ensures that the indifference curve and the budget line have the same slope (regardless of whether the budget constraint is satisfied or not). 2. PXX + PYY = M only ensures that the budget constraint is satisfied (regardless of whether the budget line has the same slope as the indifference curve or not). Only when both of these equations are satisfied do we have a consumer equilibrium corresponding to constrained utility maximization. Let's do an example, Suppose you have a square root utility function and the following data on prices and income: U(X,Y) = X0.5Y0.5 PX = 1 PY = 1 M = 10 32 We calculate the MRS as: MRS = Y X = 0.5Y 0.5X =Y X Now we can say, MRS = PX so, PY Y=1 X 1 or, X=Y and, X + Y = 10 thus, X + X = 10 2X = 10 X=5 Y=5 Thus, the optimal solution for this problem is to choose X*=Y*=5 for a maximum utility of (X0.5Y0.5) = (50.550.5) = 51 = 5. We also have an exactly satisfied budget constraint of: X + Y = 10 5 + 5 = 10 Consumer Demand Functions Let's take this a step further and generate a consumer's demand function. To do this we need to calculate the consumer equilibrium for every value of prices PX and PY. This can be done by solving for the quantities of goods demanded in terms of the prices alone. 33 So now we change our example above as follows: U(X,Y) = X0.5Y0.5 PX = ? PY = ? M = 10 We calculate the MRS as: MRS = Y X = 0.5Y 0.5X =Y X Now we can say, MRS = PX so, PY Y = PX X PY PXX + PYY = 10 Now, given the two relationships above, we can solve the optimal quantities of X and Y as before (this time, however, will yield a demand function not just a single consumer equilibrium). Y = PX X PY PXX = PYY PXX + PXX = 10 2PXX = 10 X = _5__ PX This gives us the demand for good X as a function of price PX. Similarly, we can get the demand function for good Y as a function of price PY. Using these functions, and varying the prices for each, one can trace out the demand curve for each good. and, leads to... so, 34 Rational Behaviour Producer Equilibrium Since the theory of producer equilibrium does not differ significantly from that of the theory of consumer equilibrium, we will examine producer theory in the context of a comparison with consumer theory. CONSUMER EQUILIBRIUM goods utility given prices given income indifference curve budget constraint X,Y U = U(X,Y) PX , PY MRS = -slope PX X+ PY Y = PRODUCER EQUILIBRIUM inputs K,L output Q = f(K,L) given prices PX r , PY w given cost C ( C-bar) isoquant MRTS = -slope cost constraint rK + wL = C The choice problem is to get the highest output level out of the pre-determined expenditures to be spent on inputs. Choose K & L to maximize output f(K,L) subject to rK + wL = C At the point of producer equilib. the isoquant should be tangent to the isocost line. In other words, these two curves should have the same slope. L Isoquant The choice problem is to get the highest utility level out of the pre-determined expenditures to be spent on goods. Choose X & Y to maximize utility U(X,Y) subject to PX X+ PY Y = At the point of consumer equilibrium, the indifference curve should be tangent to the budget line. In other words, these two curves should have the same slope. Indifference curve Y Consumer Equilibrium Producer Equilibrium Budget Line X Isocost Line K 35 CONSUMER EQUILIBRIUM Analytically, the following two equations of consumer equilibrium must be satisfied: MRS = PX PY PX X+ PY Y = Solving these two equations for quantities of goods X and Y demanded, we obtain consumer demands as functions of prices PX , PY, and income, . X = X(PX , PY , ) Y = Y(PX , PY , ) PRODUCER EQUILIBRIUM Analytically, these two equations of producer equilibrium must be satisfied: MRTS = _r_ w rK + wL = C Solving these two equations for quantities of inputs K and L demanded, we get producer demands as functions of prices r , w , and cost, C. K = K(r , w , C) L = L(r , w , C) Now, let's take an example using the simple square root function for both the consumer and the producer. SQUARE ROOT UTILITY On the consumer side, we have the following utility maximization problem: Choose X & Y to maximize utility X0.5Y0.5 subject to PX X+ PY Y = The following two equations of consumer equilibrium must be satisfied: MRS = PX PY PX X+ PY Y = Specifically for the square root utility function, we have: SQUARE ROOT PRODUCTION On the consumer side, we have the following output maximization problem: Choose K & L to maximize output K0.5L0.5 subject to rK + wL = C These two equations of producer equilibrium must be satisfied: MRTS = _r_ w rK + wL = C Specifically for the square root production function, we have: 36 SQUARE ROOT UTILITY Y = PX X PY PX X+ PY Y = Solving these two equations for quantities of goods X and Y demanded, we obtain consumer demands as functions of prices PX , PY, and income, . X = __ 2PX Y = __ 2PY If we substitute these demands into the objective function, we get the optimal value of utility. U = X0.5Y0.5 U = [ __ . __ ]1/2 2PX 2PY U = _________ 2(PX PY)1/2 If we rearrange this result, we get the consumer expenditure function as a function of prices PX , PY , and utility level, U. = 2(PX PY)1/2 U SQUARE ROOT PRODUCTION L = _r_ K w rK + wL = C Solving these two equations for quantities of inputs K and L demanded, we get producer demands as functions of prices r , w , and cost, C. K = _C_ 2r L = _C_ 2w If we substitute these demands into the objective function, we get the optimal value of output. Q = K0.5L0.5 Q = [ _C_ . _C_ ] 1/2 2r 2w Q = ____C____ 2(r w)1/2 If we rearrange this result, we get the producer's cost function as a function of prices r , w, and output level, Q. C = 2(r w)1/2 Q 37 HOMEWORK: 1. Find the MRS for each of the following utility functions: a) b) c) d) e) U(X,Y) = 3X0.3Y0.7 U(X,Y) = 3X0.7Y0.3 U(X,Y) = 0.3X + 0.7Y U(X,Y) = 3X + Y0.7 U(X,Y) = min{3X , 2Y} 2. Show that a Cobb-Douglas production function defined by Q = f(K,L) = KL with positive constants, > 0 capital share, > 0 labour share, > 0 scale constant, has a marginal rate of technical substitution of: MRTS = L. K 3. Consider the Cobb-Douglas utility function U = U(X, Y, Z) for the case of three goods X, Y, and Z defined by: U = U(X, Y, Z) = X1/3Y1/3Z1/3 What are the marginal utilities and marginal rates of substitution for this utility function? 4. Show that the MPK of Q = f(K,L) = (K- + L-)-1/ is MPK = (Q / K)1 + . 38 ECON 301 LECTURE #3 DUALITY OF PRODUCTION AND COST What do we mean by the duality of production and cost? Well, essentially there are many distinct similarities in the logical structure of the theories of production and cost. In reality, production and cost are two sides of the same coin. That is, an efficient producer must consider the following twin problems: On one hand, in terms of production, a producer needs to consider the primal problem of producing the highest level of output for a fixed cost constraint (output maximization problem). On the other hand, in terms of cost, the same producer needs to consider the dual problem of keeping the lowest level of cost for a fixed production level (cost minimization problem). OUTPUT MAXIMIZATION The primal problem is to get the highest output possible from a given cost level. That is, for a given level of cost, C, the producer must choose K & L to maximize output, Q. Choose K & L to maximize output f(K,L) subject to rK + wL = C L COST MINIMIZATION The dual problem is to get the lowest factor cost required to produce a given output level. For a given level of output, Q , the producer must choose K & L to minimize cost, C. Choose K & L to minimize cost rK + wL = C subject to f(K,L) = Q L Highest isoquant for a given isocost Lowest isocost for a given isoquant K K Objective: Highest isoquant Constraint: Fixed isocost Objective: Lowest isocost Constraint: Fixed isoquant 39 As before, we need to satisfy two conditions of producer equilibrium. MRTS = _r_ w rK + wL = C Solving these two equations, we get the producer demands for K & L as functions of factor prices r , w, and cost level, C. K = K(r , w , C) L = L(r , w , C) If we substitute these demands into the objective function, we get the maximum output. Q = f(K(r , w , C) , L(r , w , C)) Q = Q(r , w , C) Q = Q(C) Quantity can now be expressed as a function of the given cost level. PRODUCER'S DUALITY EXAMPLE As before, we need to satisfy two conditions of producer equilibrium. MRTS = _r_ w f(K , L) = Q Solving these two equations, we get producer demands for K & L as functions of r,w, and output level, Q . K = K(r , w , Q ) L = L(r , w , Q ) If we substitute these demands into the objective function, we get the minimum cost. C = r K(r , w , Q ) + w L(r , w , Q ) C = C(r , w , Q ) C = C(Q ) Cost can be expressed as a function of the given output level. Duality theory provides us with a procedure to construct a cost function from only two pieces of information, namely, a production function and factor prices. For example, suppose we have unit factor prices r=w=1 and the following Cobb-Douglas production function Q = K4/5L1/5 Q = K0.8L0.2 40 Using our "short cut" from last class, we get the marginal rate of technical substitution MRTS = 0.8L 0.2K = 4L K We solve the problem using both the primal and dual formulations as follows: Output Maximization (Primal) Using the primal formulation of output maximization, we have the following two equations of producer equilibrium: MRTS = _r_ w rK + wL = C 4L K K+L=C Solving these two equations, we get the factor demands for K & L 4L = _1_ K 1 4L = _1_ K 1 4L = K 4L = K 5L = C L = 0.2C K = 0.8C L = 0.25K 5/4 K = C Substituting these factor demands into the production function, we get the optimal output as a function of the cost term Q = K0.8L0.2 = (0.8C)0.8(0.2C)0.2 = (0.8)0.8(0.2)0.2C0.8 + 0.2 = (0.836511642)(0.724779663)C = 0.606286626C 41 Rearranging the result so that the cost term is on the left hand side, we get the cost term as a function of the output level. C = _____Q_____ 0.606286626 = 1.64938489Q Which is the cost function! Cost Minimization (Dual) Using the dual formulation of cost minimization, we have the following two equations of producer equilibrium: MRTS = _r_ w f(K,L) = Q 4L K K0.8L0.2 = Q Solving these two equations, we get the factor demands for K & L 4L = _1_ 4L = K (4L)0.8L0.2 = Q 40.8 L0.8 L0.2 = Q 3.031433133L = Q K 1 So, L = _____Q _____ 3.031433133 or simply, L = 0.329876977Q 4L = _1_ 4L = K L = 0.25K K0.8(0.25K)0.2 = Q (0.25)0.2 K0.8 K0.2 = Q K 1 0.757858283K = Q So, K = _____Q _____ 0.757858283K ASIDE: Once we had the demand for labour, L = 0.329876977Q , we could have simply taken the relationship K = 4L and calculated: K = 4(0.329876977Q ) = 1.319507908Q (difference in 9th decimal place is due to machine imprecision). BACK TO OUR REGULARLY SCHEDULED PROGRAMMING!! or simply, K = 1.319507911Q Substituting these factor demands into the cost constraint, we get the cost term as a function of the output level. 42 C = rK + wL =K+L = 1.319507911Q + 0.329876977Q = 1.64938489Q Rearranging the result so that the output term is on the left hand side, we get the output level as a function of the cost term. Q = _____C_____ 1.64938489 = 0.606286626C SAME AS THE RESULTS FROM THE OUTPUT MAXIMIZATION PROBLEM!!!! DUALITY OF UTILITY AND EXPENDITURE As in the investigation of production and cost, there are also many similarities between the theory of utility maximization and expenditure minimization. That is, an efficient consumer must consider the following twin problems: On one hand, in terms of utility, a consumer needs to consider the primal problem of producing the highest level of utility for a fixed budget constraint (utility maximization problem). On the other hand, in terms of expenditure, the same consumer needs to consider the dual problem of keeping the lowest level of expenditure for a fixed utility level (expenditure minimization problem). UTILITY MAXIMIZATION EXPENDITURE MINIMIZATION The primal problem is to get the highest The dual problem is to get the utility possible from a given expenditure level. lowest expenditure level required to produce a given utility level. That is, for a given level of income, , the consumer must choose X & Y to maximize utility, U. For a given level of utility, , the consumer must choose X & Y to minimize expenditure, I. 43 Choose X & Y to maximize utility U(X,Y) subject to PXX + PYY = Y Choose X & Y to minimize expenditure PXX + PYY = I subject to U(X,Y) = Y Highest IC for a given budget line Lowest BL for a given indifference curve X X Objective: Highest indifference curve Constraint: Fixed budget line As before, we need to satisfy two conditions of consumer equilibrium. MRS = _PX_ PY PXX + PYY = Solving these two equations, we get the consumer demands for X & Y as functions of prices PX , PY , and income level, . X = X(PX , PY , ) Y = Y(PX , PY , ) If we substitute these demands into the objective function, we get the indirect utility function as a function of output prices PX , PY , and income level, . Objective: Lowest budget line Constraint: Fixed indifference curve As before, we need to satisfy two conditions of consumer equilibrium. MRS = _PX_ PY U(X , Y) = Solving these two equations, we get consumer demands for X & Y as functions of prices PX , PY , & utility level, . X = X(PX , PY , ) Y = Y(PX , PY , ) If we substitute these demands into the objective function, we get the expenditure function as a function of output prices PX , PY , and utility level, . 44 V = U(X(PX , PY , ), Y(PX , PY , )) V = V(PX , PY , ) CONSUMER'S DUALITY EXAMPLE E = PXX(PX,PY,) + PYY(PX,PY,) E = E(PX , PY , ) As we mentioned above, duality theory provides us with a procedure to construct an expenditure function from only two pieces of information, namely, a utility function and the prices of the goods. For example, suppose we have prices PX = 1 and PY = 2 and the following Cobb-Douglas utility function U = X3/4Y1/4 U = X0.75Y0.25 Using our "short cut" from last class, we get the marginal rate of substitution MRS = 0.75Y 0.25X = 3Y X We solve the problem using both the primal and dual formulations as follows: Utility Maximization (Primal) Using the primal formulation of utility maximization, we have the following two equations of consumer equilibrium: MRS = _PX_ = 3Y PY X PXX + PYY = Solving these two equations, we get the demands for X & Y 3Y = _1_ X 2 3Y = _1_ X 2 6Y = X 6Y = X Y = 1/6 X 8Y = 4/3 X = Y = 0.125 X = 0.75 X + 2Y = 45 Substituting these demands into the utility function, we get the optimal utility as a function of income (expenditure). U = X0.75Y0.25 = (0.75)0.75(0.125)0.25 = (0.75)0.75(0.125)0.25 0.75 + 0.25 = (0.805927448)(0.594603557) = 0.479207327 Rearranging the result so that the expenditure term is on the left hand side, we get the expenditure term as a function of the utility level. = _____U_____ 0.479207327 = 2.086779445 U Which is the expenditure function! Expenditure Minimization (Dual) Using the dual formulation of expenditure minimization, we have the following two equations of consumer equilibrium: MRS = _PX_ = 3Y PY X U(X,Y) = X0.75Y0.25 = Solving these two equations, we get the demands for X & Y 3Y = _1_ 6Y = X (6Y)0.75Y0.2 = 60.75 Y0.75 Y0.25 = 3.833658625Y= X 2 So, Y = __________ 3.833658625 or simply, Y = 0.26084743 3Y = _1_ 6Y = X Y = 0.1666X X0.75(0.1666X)0.25 = X 2 (0.1666)0.25 X0.75 X0.25 = 0.638943104X = 46 So, X = __________ 0.638943104X ASIDE: or simply, X = 1.565084581 Once we had the demand for good Y, Y = 0.26084743 , we could have simply taken the relationship X = 6Y and calculated: X = 6(0.26084743 ) = 1.565084580 (difference in 9th decimal place is due to machine imprecision). BACK TO OUR REGULARLY SCHEDULED PROGRAMMING!! Substituting these factor demands into the budget constraint, we get the expenditure term as a function of the utility level. I = PXX + PYY = X + 2Y = 1.565084581 + 2(0.26084743 ) = 1.565084581 + 0.52169486 = 2.086779441 Rearranging the result so that the utility term is on the left hand side, we get the utility level as a function of the income(expenditure) term. = _____I_____ 2.086779441 = 0.479207328 I SAME AS THE RESULTS FROM THE UTILITY MAXIMIZATION PROBLEM!!!! FACTOR MARKETS Factor markets, especially labour and human capital, have some peculiar features that deserve some special attention. We will consider factor market pricing in the labour market under perfect competition and imperfect competition. So how do we price labour in a competitive market? 47 Just as in any other market, pricing in a competitive labour market is a result of balancing two opposing forces...the demand side (producers/firms that wish to employ labour) and the supply side (consumers/workers that wish to offer labour services). [Demand for Labour by Firms] [Supply of Labour by Consumers] Labour Market Let's look at each side separately in order to derive the demand for labour by firms and the supply of labour by workers. Demand for Labour The demand for labour can be derived from the usual balance of costs and benefits by a producer. For example, consider a competitive firm having the following simple production function which takes a single variable factor input...say, labour: Q = f (L) where, Q denotes the level of output produced, and L denotes the amount of labour input required. PHYSICAL UNITS Marginal (Physical) Product The marginal (physical) product of labour is the change in the output level as a result of a change in the quantity of labour input. MP = Q / L MP refers to the change in the output measured in physical units of output as a result of a marginal change in labour. On the benefit side, MP measures the marginal benefit to the firm in terms of physical units. DOLLAR VALUES Marginal Revenue Product The marginal revenue product of labour is defined as the product of the output price and the marginal (physical) product of labour. MRP = MP P MRP refers to the change in revenue measured in dollar values as a result of a marginal change in labour. On the benefit side, MRP measures the marginal benefit to the firm in dollar values. 48 COST CALCULATION In a competitive labour market, the labour wage is taken as given by the firm. For each additional unit of labour hired, the firm has to pay the labour wage in physical units (w / P). This is the real wage. COST CALCULATION In a competitive labour market, the labour wage is taken as given by the firm. For each additional unit of labour hired, the firm has to pay the Labour wage in dollar terms (w). This is the nominal wage. On the cost side, w measures the marginal cost to the firm in terms of dollar values. COST VS BENEFIT The optimal decision for the firm is to hire labour up to the point where MRP = w. Where did the (w) come from? On the cost side, (w / P) measures the marginal cost to the firm in terms of physical units. COST VS BENEFIT The optimal decision for the firm is to hire labour up to the point where MP = (w / P). Where did the (w / P) come from? In perfectly competitive labour markets, the key implication is that firms are price takers in both the output and the input markets (sell at P determined in the market, pay labour w determined in the market). Recall that we have only one variable input, labour, in our investigation and the production function is: Q = f (L) It is important to remember the goal of the producer under perfect competition is to maximize profits as follows: f (P,w) = max PQ wL subject to Q = f (L) Q,L f (P,w) = max P f (L) wL L FOC: (P,w) = 0 L [1] FOC: Pf(L) w = 0 This characterizes the optimal solution. Pf(L) w = 0 Pf(L) = w 49 or, MP P = w MRP = w MRP = MCL [2] FOC: Pf(L) w = 0 Pf(L) = w f(L) = __w__ P MP = __w__ P MP = real wage [3] FOC: Pf(L) w = 0 Pf(L) = w P = __w__ f(L) MRQ = MCQ The decision rule based on dollar values turns out to be exactly the same as the decision rule based on physical units (because they are variations on the same expression). In [3] above, we can see that price is the ratio of w and MPL (which is the marginal cost of producing one additional unit of Q). We can also see, under perfect competition, P = MR so we get the familiar optimal condition under perfect competition where P = MC. or, or, 50 Slope = w / P Q f (L) w / P = f (L) = PQ wL Q = / P + (w / P) L L Let's do a simple example. Consider the square root production function with only the labour input. Q = L1/2 The MPL is: MPL = __1__ 2 L1/2 Now, suppose that we are in a competitive setting and the price of output is P = 24 and the price of labour is w = 4. We can derive the optimal amount of labour demanded by the firm using the decision rule either in terms of physical units or in terms of dollar values as: PHYSICAL UNITS Optimal Decision Rule MPL = __w__ P __4__ = __1__ 24 2 L1/2 8 L1/2 = 24 L1/2 = 3 L=9 51 DOLLAR VALUES Optimal Decision Rule MRP = w __1__ 24 = 4 2 L1/2 8 L1/2 = 24 L1/2 = 3 L=9 MP Physical Units $ MR w/P w L 9 9 L The Demand Curve for Labour The example above gives us the quantity of labour demanded at a market wage of $4. If we vary the values of the market wage, w, and calculate the corresponding labour quantities demanded by the firm, we get the derived demand curve for labour by the firm. Using the above example, we get the labour demand equation as: PHYSICAL UNITS Optimal Decision Rule MPL = __w__ P __w__ = __1__ 24 2 L1/2 2w L1/2 = 24 L1/2 = 12 / w L = 144 / w2 DOLLAR VALUES Optimal Decision Rule MRP = w __1__ 24 = w 2 L1/2 2w L1/2 = 24 L1/2 = 12 / w L = 144 / w2 52 w w L = 144 / w2 L = 144 / w2 L L HOMEWORK 1. Formulate and solve the utility maximization problem for the general CobbDouglas utility function U(X,Y) = XY And confirm that the consumer demands are: X = ______ [I / PX] + and Y = ______ [I / PY] + 2. Formulate and solve the output maximization problem for the general CobbDouglas production function f(X,Y) = KL And confirm that the factor demands are: K = ______ [C / r] + and L = ______ [C / w] + 3. Using your results from question 2, verify that the cost function is if + = 1. C = _Q_ [r / ] [w / ] 4. Consider a production function with only the labour input. If the wage rate is $17 per hour and the price of output is $153, what is the demand for labour if we assume that all markets are competitive. Find the demand for labour using the method shown for physical quantities and then confirm your answer by finding demand for labour using the method shown for dollar values if the production function is defined as Q = L1/5. 53 ECON 301 LECTURE #4 Market Demand for Labour The market demand curve for labour is obtained by the usual horizontal summation technique. This involves adding up all of the individual firm's demands for labour along the horizontal axis. $ Market Demand Curve for Labour d1 d2 d4 d3 L Now, suppose we have a production function f (L) = ln(1+L). What is the firm's demand for labour, in general? Using the firm's profit function we can find the profit maximizing labour input... Max {P ln(1 + L) wL} FOC __P__ = w 1+L P/w=1+L L = (P w) w In general, we can say that for any number of inputs i = 1, 2, ... , n P MPi = wi P MPL = w P MPK = r P MPLAND = t (rental rate of land), etc. 54 Labour Supply The supply of labour is derived from the consumer's problem of choosing between work and PLAY!!! Work (BOOOO!!!!) yields income, while play (OH YEAHHHHH!!!) gives us utility in the form of leisure. We formulate this problem of the consumer's income-leisure choice as follows: [1] The consumer has a utility function defined on income, M, and leisure, (where leisure, = 24 L in a day). U = U ( , M) The indifference curve corresponding to a given utility level connects all combinations of income and leisure (M , ) that satisfy the equation U ( , M) = Within this income-leisure choice framework, the marginal rate of substitution refers to the change in income as a result of a marginal change in leisure MRS = M M Indifference Curve U ( , M) = 55 [2] Suppose the consumer has a time endowment of Tbar = 24 hours per day to spend either on work to produce income or on play to produce pleasure. Given the market wage, w, if she allocates a time amount for play then the value of her leisure would be w (which is the income lost by playing around) or the opportunity cost of choosing not to work. The income earned from work would be M = w (Tbar ) The budget constraint for the income-leisure choice can be written as: w + M = w Tbar M wTbar Budget Line w + M = wTbar Tbar The budget line is thus a straight line with slope w as described by the equation M = w Tbar w [3] The consumer choice problem can now be formulated as the standard constrained utility maximization problem: Maximize U ( , M) Subject to w + M = w Tbar 56 M Indifference Curve wTbar M* Consumer's Equilibrium Income-Leisure Choice Budget Line * Tbar Remember, at the consumer equilibrium point, the indifference curve must be tangent to the budget line. [4] Solving for the optimal quantities for leisure, *, and income, M*, we can derive the optimal amount of labour, L*, supplied by the consumer at a given market wage. L* = Tbar * In general, if we can write the amount of labour supply as a function of the market wage, w, then we have the following individual supply of labour by the consumer: L = L(w) How do we go about this? Let's use an example to illustrate... Recall that consumers are endowed with time. Essentially, there are two uses of time in this model. Labour which produces income that, in turn, allows the consumer to purchase consumption (this gives the consumer utility). Leisure This produces utility for the consumer at the opportunity cost of foregone work (and thus, foregone consumption). For our model, we will consider 2 goods consumption, c, and leisure, , with consumer tastes represented by a utility function U = U(c , ) 57 Now, we have a utility function with the consumer's preferences over consumption and leisure defined as U = U(c , ). But what is the constraint in this case? The consumer can purchase consumption at a price, P, and receives a given wage, w, for each unit of labour they supply to the labour market. Thus, the consumer's consumption is constrained by their wage income... Pc = w (Tbar ) Pc + w = w Tbar So generally, Max U(c , ) + (w Tbar Pc w) c, MRS = MU MUc FOC FOCc U () w = 0 Uc () P = 0 (1) (2) Now we isolate for (1) and (2) and equate these expressions (3) = (4): U () = w Uc () = P U () = Uc () w P U () = __w__ Uc () P So, the MRS is the real wage! Okay, now that we've established that we can take an example of how to find the labour supply function for a consumer. (3) (4) 58 Suppose a consumer has preferences over consumption and leisure that are represented by the following utility function: U(c , ) = ln c + ln Is this a Cobb-Douglas form of the utility function? Yes! Why? ln c + ln is equivalent to c1 1/2 So, our constraint is still income = cost of consumption... Pc = w (Tbar ) Pc + w = w Tbar and we form the Lagrangian, Max ln c + ln + (w Tbar Pc w) c, MRS = MU MUc FOC FOCc __1__ w = 0 2 __1__ P = 0 c (1) (2) Now we isolate for (1) and (2) and equate these expressions (3) = (4): __1__ = 2w __1__ = Pc __1__ = __1__ 2w Pc __c__ = __w__ 2 P 59 (3) (4) So, we can now find the demand functions for consumption and leisure in terms of P, w, and Tbar. c = __2w__ P (5) Now, we sub (5) into our resource constraint to get our demand function for leisure. P 2w + w = wTbar P 3w = wTbar * = Tbar 3 (6) Now, we can sub * into (5) to get the consumer's demand function for consumption. c* = __2w Tbar __ 3P (7) If this is truly a Cobb-Douglas form of the utility function we should be able to find proportions of income "spent" on leisure and consumption... (w , P , Tbar) = Tbar 3 c(w , P , Tbar) = 2wTbar 3P wTbar 3w 2M 3P 1/3 M w 2/3 M P How is this the proportion of income spent on consumption and leisure? Recall, the proportion spent on a generic good X is PXX ... M So, the proportion spent on consumption is: P c(w , P , Tbar) M _P_ 2/3 M = 2/3 M P proportion of income to consume goods! 60 So, the proportion spent on leisure is: w (w , P , Tbar) M _w_ 1/3 M = 1/3 M w proportion of income to consume leisure! Labour Market Supply If there is more than one consumer, we can derive the market labour supply curve by the usual horizontal summation technique. Labour Market Equilibrium To determine the labour market equilibrium, we simply find the point of intersection of the market supply curve of labour by consumers and the market demand curve for labour by producers. Ld (w) = Ls (w) This will show us the equilibrium market wage, w*, and quantity of labour, L*, where labour demanded by firms is equal to labour supplied by workers. w Market Supply w* Market Demand L* IMPERFECT COMPETITION IN THE LABOUR MARKET Like any other market, perfect competition is not the only market structure possible for the labour market. For example, we could have the following two extremes of labour market imperfections: 61 L [1] Monopoly: there is a single seller that controls the supply of labour in the market. [2] Monopsony: there is a single buyer that controls the demand for labour in the market. A firm can be a monopoly or a monopsony in the labour market and still operate in a perfectly competitive output market. Similarly, a consumer (or group of consumers) can be a monopoly or a monopsony in the labour market and still face perfect competition in the output market. The following table lists some of the possible combinations of market structures in both the goods and factor markets with the following economic agents: [1] consumers play the role of buyers in the goods market and, at the same time, the role of sellers in the labour market; [2] producers play the role of sellers in the goods market and, at the same time, the role of buyers in the labour market. Case 1 2 3 4 5 6 7 Consumer Producer Consumer Producer Consumer Producer Consumer Producer Consumer Producer Consumer Producer Consumer Producer Goods Market Competitive Competitive Competitive Monopoly Monopsony Competitive Monopsony Monopoly Competitive Competitive Competitive Competitive Competitive Competitive Labour Market Competitive Competitive Competitive Competitive Competitive Competitive Competitive Competitive Competitive Monopsony Monopoly Competitive Monopoly Monopsony While competitive firms must take the market price (price-takers), P in the goods market and w in the labour market, monopolists and monopsonists operate with different rules for optimality. Namely, downward sloping demand curves for monopolists and upward sloping supply curves for monopsonists. As a consequence, we need to incorporate the corresponding marginal curve into the optimal decision of these non-competitive agents accordingly. For example, 62 [1] Monopolist in the goods market... In case 2 above, the firm is a monopolist in the goods market (the only seller). The relevant variable in the optimal decision is the marginal revenue, MR, instead of the exogenously fixed output price, P, when dealing with PC markets. As a result, the marginal revenue product is defined as MRP = MP MR instead of MRP = MP P and the optimal decision rule of the firm in the competitive labour market is: MRP = w or, MP MR = w So, the firm wants to maximize TR TC. Perfect Competition P Monopoly P P* D = AR = MR MR Q D = AR Q Max {P(Q) Q C(Q)} Q FOCQ P(Q) + Q P(Q) = C(Q) [MRQ] = [MCQ] Rearranging, P(Q) - C(Q) = - Q P(Q) 63 and dividing both sides by P(Q) gives us the percentage markup on the left hand side: P(Q) - C(Q) = - Q P(Q) P(Q) P(Q) But what is the right hand side? Remember, P(Q) is simply the P / Q P(Q) - C(Q) = - __Q__ P(Q) P(Q) P P(Q) - C(Q) = - __Q__ P P(Q) P Q and the right hand side is simply, 1 / or the Lerner condition we learned in ECON 201. How is the RHS = 1 / ? _____1______ = - __Q__ P P Q - _P_ Q Q P So, the monopolist has a percentage markup that is equal to 1 / , as follows: P(Q) - C(Q) = __1__ P(Q) [% markup] = __1__ [2] Monopsonist in the labour market... If the firm is a monopsonist in the labour market, as in Case 5, the relevant variable in the optimal decision is the marginal expenditure on labour, MEL, instead of the exogenously fixed market wage, w, when dealing with PC markets. Given an upward sloping supply curve for labour, L = L(w) we have the following expression for the total expenditure on labour: EL = w L(w) (Lerner Condition) 64 and the marginal expenditure on labour, MEL, (i.e. the change in the total expenditure on labour as a result of an additional unit of labour): MEL = w + L w L where, w / L is the slope of the labour supply curve (w / L > 0). Notice that there is a markup in terms of the marginal expenditure on labour over the perfectly competitive wage, w. The monopsony employer now sets their demand for labour according to MEL (also thought of as marginal cost of labour by some) = MRP. The optimal decision rule of the firm in the competitive goods market remains as: MRP = w or, MP P = w This type of situation leads to what is called Monopsonistic Exploitation. So you might want to ask me at this point, "What in the HECK are you talking about?" Let me attempt to clarify using a more familiar formulation of the problem and using a graphical representation... Monopsonistic Exploitation is when the firm is the only buyer of labour in the labour market and the consumer supplies their work effort in a competitive labour market. In this case, the firm can induce a w* < MRP and exploit the worker! Let's see how this works by considering the firm's maximization problem. The firm wants to: Max {P f(L) w(L) L} L FOCL P f (L) w(L) L w(L) = 0 P f (L) = w(L) + L w(L) [MRP] = [MCL in PC] + [Premium] 65 so, [MRP] = [MCL in Monopsony or MEL] The graph shows that the firms demand for labour is equal to the MRP and their optimal decision is to set MRP equal to MEL (not the actual supply of labour!). This means that w* < MRP and the worker receives a lower wage than they would if the firm was not the only buyer of labour in the market...thus the terminology "Monopolistic Exploitation". w MEL Deadweight Loss S = ACL w*PC w*M D = MRP = P MPL L L*M L*PC We said that while competitive firms must take the market price (price-takers), P in the goods market and w in the labour market, monopolists and monopsonists operate with different rules for optimality. Namely, downward sloping demand curves for monopolists and upward sloping supply curves for monopsonists. As a consequence, we need to incorporate the corresponding marginal curve into the optimal decision of these non-competitive agents accordingly. We just looked at the case where the producer has a monopoly in the goods market, as well as, the case where the firm is a monopsonist in the labour market. Now, we will consider the case where we have both a monopolist and a monopsonist in the labour market. 66 This scenario is case 7 in the table above and it refers to what is often called a bilateral monopoly with imperfect competition on both sides of the market: [1] The selling side of the labour market has a single seller (i.e. a monopoly) played by a labour union that controls the supply of labour, [2] The buying side of the labour market has a single buyer (i.e. a monopsonist) played by a firm contolling the demand for labour. This situation is quite interesting. The countervailing power between big business and big labour can be analyzed as follows: [a] In the absence of market imperfection, the market equilibrium is determined at the intersection of the market demand for labour by the firm and the market supply of labour by the union (point C in the diagram below). The equilibrium market wage is wc and the equilibrium quantity of labour is Lc. w S wc C D Lc L [b] The labour union, acting as a monopolist in the labour market faces a downward sloping demand curve for labour and the corresponding marginal revenue curve, MR. The relevant decision variables for this monopolist are the marginal revenue on the benefit side and the labour supply (viewed in terms of its marginal cost) on the cost side. The optimal decision for the labour union is the intersection point (point 1) of the marginal revenue and marginal cost curves. The optimal wage required by the labour union is wU and the corresponding labour supplied is LU (point U). 67 w S = MC U wU wc 1 MR LU C D Lc L [c] On the other hand, the firm acting as a monopsonist in the labour market faces an upward sloping supply curve of labour and the corresponding marginal expenditure on labour curve, MEL. The relevant decision variables for this monopsonist are the marginal expenditure on labour (cost side) and the labour demand (viewed in terms of its MRP) on the benefit side. The optimal decision for the firm is the intersection point (point 2) of the marginal expenditure on labour, MEL, and the MRP curve. The optimal wage offered by the firm is wF and the corresponding quantity of labour demanded is LF (point F). w MEL S 2 wU U wc F C wF D = MRP LF LU Lc L 68 [d] The optimal decisions for the monopolist (labour union) and the monopsonist (firm) are not compatible with each other. While the labour union requires a higher wage, wU, with a higher employment level, LU (see point U), the firm is offering a lower wage, wF, with a lower employment level, LF (see point F). We should notice that the competitive equilibrium wage, wc (point C) lies between these two extreme wages determined by the labour union and the firm: wF [firm monopsonist] wc [competitive equilibrium] wU [union monopolist] As well, we should notice that both of the employment levels determined by the firm and the labour union are below the competitive employment level, Lc. LF [firm monopsonist] LU [union monopolist] Lc [competitive equilibrium] So, the bilateral monopoly solution is indeterminate in the sense that the wage rate, wF, and employment level, LF, offered by the firm are not matched by the wage rate, wU, and employment level, LU, required by the labour union. The discrepancy in the wage rates determined by the labour union and the firm can be eliminated (?) through the process of collective bargaining negotiations. The outcomes of this negotiation process can best be analyzed by using game theory...if we have time at the end of the course, we may return to this issue in the context of game theory. For now, we just want to notice the conditions that give rise to the need for labour negotiations. w MEL S = MC 2 wU U wc F 1 MR LF LU C wF D = MRP Lc L 69 HOMEWORK 1. Suppose a consumer has preferences over consumption and leisure that are represented by the following utility function: U(c , ) = 1/2ln c + 1/3 ln Find the consumer's demand functions for consumption, c*, and leisure, * in terms of P, w, and Tbar. 70 ECON 301 LECTURE #5 Minimum Wages Another labour market imperfection can arise through the imposition of minimum wages into an otherwise competitive labour market. In the early proliferation of minimum wages as a conceptual framework, it was thought of and intended as an appropriate policy instrument to alleviate poverty and to protect workers from exploitation. In the past few decades, researchers have determined that theoretically, at least, - minimum wages reduce employment (or create unemployment) if the minimum wage floor is set above the competitive equilibrium wage. As for minimum wages reducing poverty, at least one researcher has determined that minimum wages in highly developed countries, like Canada, do not necessarily reduce the proportion of families in poverty rather they increase the proportion. If any of you are interested in my methodology and results, please come to my office and I'll be happy to discuss my working paper with you in detail. For now, let's focus on the low wage labour market and see how minimum wages can actually decrease employment (create unemployment)... w unemployment wmin S = MC w* LDM L* LSM D = MRP L So, in the simple diagram above, we can see that at the competitive equilibrium the wage rate is w* and the employment level is L*...the market clears. 71 Suppose the provincial government introduces minimum wage legislation and imposes a minimum wage of wmin. Now, in this low wage market all firms must pay their workers wmin > w*. This induces the firms to demand a lower quantity of labour (movement along the labour demand curve). Meanwhile, this policy induces workers to supply more of their labour into the market because of the higher wage (movement along the labour supply curve). As we can see in the diagram, firms will now demand LDM at wmin and workers will supply LSM at wmin. We know that only the labour demanded will be employed at wmin so we have an excess supply of labour in this low wage market. The unemployment that the minimum wage policy creates is LSM LDM. Slutsky Equation Hopefully, you were exposed to the Slutsky equation in ECON 201. For those who were not, we will do a brief review of the salient points of the formulation before we construct the equation using calculus. We want to investigate how a consumer's choice of a good responds to a change in the price of the good. This response can be broken down into two effects that arise from a price change in the good, namely the substitution effect and the income effect. So, when the price of a good changes, there are two sorts of effects: The rate at which you can exchange one good for another changes, and the total purchasing power of your income is altered as a result of the price change. So let's take the example of two goods X and Y. A consumer has some preferences over X and Y represented by a utility function, U(X,Y), and a budget constraint, PXX + PYY = M. Suppose the price of X falls, PX . Then, we can see directly, that the consumer can buy more of good X with his income, M. In turn, this will affect the consumer's utility level (since we assume that MUX > 0). How do we represent this scenario? 72 Y M / PY M / PY B A C Shift BLO s.e i.e Pivot BLF X BLC So, in order to determine the substitution effect we need to use the consumer's demand function to calculate the optimal choices at the new price and compensated income X(PX, M) and the consumer demand at the original point X(PX, M). So the change in X due to the substitution effect is: Xs = X(PX, PY, M) - X(PX, PY, M) The change in demand for X may be large or small, depending on the shape of the consumer's indifference curves. But given the demand function, it is straight forward to plug in the numbers and calculate the substitution effect (holding PY constant). The substitution effect is often described as the change in compensated demand. The idea is that the consumer is being compensated for a price rise (or penalized for a price decrease) by having enough income given back to them to purchase their old demand bundle. Of course, as I just mentioned, if the price goes down the consumer is "compensated" by having income taken away from them. Let's use an example to illustrate how we calculate the substitution effect. Suppose that Ricky has a demand function for pepperoni given by: X = 10 + __M__ 10PP 73 Originally, the government of Nova Scotia pays Ricky $120 per week in social assistance and the price of pepperoni is $3 per pound. This means that Ricky's original demand for pepperoni is 10 + 120/ (10 3) = 14 pounds per week. Now, let's suppose that the price of pepperoni falls to $2 per pound. Then his demand for pepperoni at the new price will be 10 + 120/ (10 2) = 16 pounds per week. So, as a result of the decrease in the price of pepperoni, Ricky now demands 2 pounds per week more of it. To calculate how much of this 2 pounds per week increase is due to the substitution effect, we need to calculate how much Ricky's income has to change in order to make the original consumption level of 14 pounds of pepperoni just affordable to Ricky at the new price of pepperoni ($2 per pound). M = X PP = 14 (2 3) = -14 So the level of income needed to keep Ricky's purchasing power constant is: M = M + M = 120 14 = 106 So, now we have Ricky's compensated demand for pepperoni at the new price and compensated income level. X(PX, PY, M) = X(2, PY, 106) = 10 + _106__ = 15.3 (10 2) and we can figure out how much of Ricky's demand for pepperoni changed due to the substitution effect as follows: Xs = X(PX, PY, M) - X(PX, PY, M) Xs = 15.3 14 = 1.3 pounds per week Now, we want to investigate the income effect. Remember that the income effect is the parallel shift of the budget line due to income changes once the change in the relative prices is accounted for. We simply change the consumer's income from M to M, keeping prices constant at (PX, PY). In the diagram, this is the shift from BLC to BLF and, more specifically, from B to C. 74 Y M / PY M / PY B A C Shift BLO s.e i.e Pivot BLF X BLC It is natural to call this effect the income (or wealth) effect, since all we are doing is changing income while keeping the prices fixed at the new price vector. More precisely, the income effect, XI, is the change in the demand for good X when we change income from M to M, keeping the price of good X fixed at PX: XI = X(PX, PY, M) X(PX, PY, M) From Intermediate Micro I, we know that the wealth effect can operate in either direction. It will tend to increase or decrease the demand for good X depending on whether X is a normal or an inferior good. When the price of the good decreases (as in our example), we need to decrease income in order to keep purchasing power constant. If the good is a normal good, then this decrease in income will lead to a decrease in demand (we are moving from BLF to BLC). If the good is an inferior good, then this decrease in income will lead to an increase in demand (again, we are moving from BLF to BLC). Now, let's complete our example... Remember, that Ricky had a demand function for pepperoni given by: X = 10 + __M__ 10PP 75 The government of Nova Scotia pays Ricky $120 per week and the price of pepperoni was $3 per pound and it falls to $2 per pound...giving us Ricky's original demand for pepperoni as X(3, PY, 120) = 10 + 120/ (10 3) = 14 pounds per week. and his final demand for pepperoni at the new price of $2 as: X(2, PY, 120) = 10 + 120/ (10 2) = 16 pounds per week. So, as a result of the decrease in the price of pepperoni, Ricky now demands 2 pounds per week more of it. We calculated the amount of the 2 pound per week increase in Ricky's demand for pepperoni that was due to the substitution effect by finding his compensated demand: M = X PP = 14 (2 3) = -14 So the level of income needed to keep Ricky's purchasing power constant is: M = M + M = 120 14 = 106 So, now we have Ricky's compensated demand for pepperoni at the new price and compensated income level. X(PX, PY, M) = X(2, PY, 106) = 10 + _106__ = 15.3 (10 2) Now, we can find the magnitude of the wealth effect on Ricky's demand for pepperoni as follows: XI = X(PX, PY, M) X(PX, PY, M) XI = X(2, PY, 120) X(2, PY, 106) XI = 16 15.3 XI = 0.7 pounds per week So we can infer that pepperoni is a normal good for Ricky, since as his income increases (we are moving from BLC to BLF) his demand for pepperoni increases. Let's put it all together now... 76 X = Xs + XI X = 1.3 + 0.7 X = 2 It is convenient to write our expression above in terms of the rate of change and also to redefine -XI = X(PX, PY, M) X(PX, PY, M) = XM...so, the relationship becomes, X = Xs XM PX PX PX (1) We can finish this formulation by noticing that the income change, M, is related to the price change, PX, as follows: M = x PX. Solving for PX... PX = M X and subbing this into equation (1) above: X = Xs XM x PX PX M (2) We will see that this is exactly the form of the Slutsky equation derived using calculus. Deriving the Slutsky Equation Consider the Slutsky definition of the substitution effect, in which income is adjusted so as to give the consumer just enough to buy the original consumption bundle. Let's denote this bundle as (x', y'). If the prices are (PX, PY), then the consumer's actual choice with the income adjustment will depend on (PX, PY) and (x', y'). Let's call this relationship the Slutsky demand function for good X and write it as XS(PX, PY, x', y'). Suppose the original demanded bundle is (x', y') at prices (P'X, P'Y) and income, M'. The Slutsky demand function tells us what the consumer would demand facing some different prices (PX, PY) and having income PXx' + PYy'. Thus the Slutsky demand function at (PX, PY, x', y') is the ordinary demand at (PX, PY) and income PXx' + PYy'. Meaning, 77 XS(PX, PY, x', y') X(PX, PY, PX x' + PY y') This equation says that the Slutsky demand at prices (PX, PY) is the amount that the consumer would demand if they had enough income to buy their original bundle of goods (x', y'). If we differentiate the identity with respect to PX, we get XS(PX, PY, x', y') X(PX, PY, M') + X(PX, PY, M') x' PX PX M Rearranging, we have... X(PX, PY, M') = XS(PX, PY, x', y') - X(PX, PY, M') x' PX M PX This is the derivative form of the Slutsky equation. It says that the total effect of a price change is composed of a substitution effect (where income is adjusted to keep the bundle (x', y') feasible) and an income effect. We know from ECON 201 that the substitution effect is always negatively related to the price change and that the sign of the income effect will depend on whether the good is a normal good or an inferior good. As you can see, this is simply the form of the Slutsky equation we considered without calculus, except we have replaced the 's with derivative signs. Let's summarize what we have so far: X = Xs XM x PX PX M Let's interpret each term! X = X(PX, PY, M) - X(PX, PY, M) PX PX Meaning, the (demand with the new price) (demand with the old price) (change in the price from old to new) Our example showed that this was (16 14) = 2 1 (2) 78 XS = X(PX, PY, M) - X(PX, PY, M) PX PX (demand with new price & compensated income) (demand with old price & income) (change in the price from old to new) Our example showed that this was (15.3 14) = 1.3 1 XM x = X(PX, PY, M) - X(PX, PY, M) X(PX, PY, M) M M - M (demand with new price & compensated income) (demand with new price & old income) original demand of x (change in income needed to compensate) Our example showed that this was (15.3 16) 14 = 0.7. -14 Okay, now let's move on to Walrasian General Equilibrium Theory... WALRASIAN THEORY General equilibrium theory studies the interdependence among various components of an economy. Liontief presented an input-output technique that won him a Nobel Prize that was essentially a special case of general equilibrium analysis where all of the functional forms are assumed to be linear with fixed coefficients. We will consider this model, which uses matrix algebra in its formulation, later in the course. The economic theory that you have been exposed to until now has been limited to partial equilibrium analysis. Partial equilibrium analysis focuses on a single market or sector of the economy at a time while ignoring outside influences. This means that we have been studying the economics of a single situation holding all other things the same (ceteris paribus). On the other hand, general equilibrium analysis considers the economy as a whole with everything being fully accounted for. We use the phrase, "everything depends on everything else" to describe general equilibrium theory. While general equilibrium theory is presented in the microeconomics stream of courses, it is general enough to be applicable in a number of other areas other than microeconomics. As an example, international economics provides many natural examples of general equilibrium analysis pertaining to the interdependence among many countries that produce and exchange commodities with each other. 79 Another example is the more modern treatment of macroeconomics which is primarily concerned with a general equilibrium system of at least four different aggregate markets, namely, the commodity market, the labour market, the money market, and the bonds market. In fact, the so-called modern theory of "fixed-price equilibria" attempts to validate macroeconomic theory by providing a micro foundation to macroeconomics. This attempt to bridge the two main components of economics has lent some credibility to economic analysis in the face of outside criticism that the methodologies of micro and macroeconomics have been too far apart. The heading of this section is Walrasian Equilibrium, yet we have been talking about general equilibrium throughout this section. General equilibrium analysis is named after the French economist Marie Esprit Leon Walras (1834 1910) who was born in Evreux, France. He studied literature and science but twice failed the entrance exam to the prestigious Ecole Polytechnique a Paris as a result of his poor mathematics. He had no formal training in economics and so his perspective and economic intuition was self-taught and commonsensical. Despite his lack of formal training, Walras was appointed the chair of professor of political economy at the Universite de Lausanne (Switzerland) in 1870 and remained there until his death in 1910. He pioneered the analysis of multiple markets in a general equilibrium framework and is considered one of the top five greatest economists of all time (Adam Smith, John Maynard Keynes, Leon Walras, and arguments still rage about who fills the four and five spots). The basic objective of general equilibrium theory is to analyze the economy as a whole with everything being fully accounted for. Thus, all components of the economy must be considered in the analysis. For example, in a simple economy, there are three basic components: [1] Consumers make buying decisions on goods according to their individual rational behaviour. This results in a "consumer equilibrium" for each and every individual consumer. Y Consumer Equilibrium IC BL X 80 [2] Producers make buying decisions on capital and labour according to their individual rational behavior. This results in a "producer equilibrium" for each and every individual producer. L Producer Equilibrium IQ Isocost K [3] Markets balance between quantities demanded and quantities supplied for each good and factor input. This results in a "market equilibrium" for each and every individual market. P S Market Equilibrium D Q General equilibrium theory studies the interdependence among these three basic components of the economy. We will look at the notion of general equilibrium at the individual level (i.e. consumer and producer equilibrium) and at the aggregate level (i.e. market equilibrium). So, how do we go from individual levels to aggregate levels? 81 The process of going from individual levels to aggregate levels is called aggregation or summation. We will use the technique called horizontal summation in order to aggregate the various individual levels to market levels. This process is called horizontal summation because we fix the common independent variable at some level on the vertical axis and then add all of the individual dependent variables along the horizontal axis. P DB DA Aggregate Demand Curve PX X X A X B X +X A B For example, suppose that we have two consumers, A and B, with the following individual demands for good X: XA = XA(PX, PY, MA) XB = XB(PX, PY, MB) These individual demand functions could have been derived from the usual utility maximization problem with given market prices, PX, PY, and individual incomes MA and MB as follows: CONSUMER A Utility Maximization problem: Maximize UA(XA, YA) Subject to: PXXA + PYYA = MA CONSUMER B Utility Maximization problem: Maximize UB(XB, YB) Subject to: PXXB + PYYB = MB 82 Consumer equilibrium condition: MRSA = PX PY PXXA + PYYA = MA Individual demand functions: XA = XA(PX, PY, MA) YA = YA(PX, PY, MA) Consumer equilibrium condition: MRSB = PX PY PXXB + PYYB = MB Individual demand functions: XB = XB(PX, PY, MB) YB = YB(PX, PY, MB) Now the aggregate demand can be constructed from these two individual demands by the technique of horizontal summation which basically adds the quantities demanded by both A and B as follows: X = XA(PX, PY, MA) + XB(PX, PY, MB) X = X(PX, PY, MA, MB) Note that while each individual demand is a function of incomes MA and MB separately, the aggregate demand is a function of both MA and MB jointly. We just said that general equilibrium theory studies the interdependence among the three basic components of the economy (consumer equilibrium, producer equilibrium, and market equilibrium). We showed how we go from individual levels to aggregate levels using aggregation or horizontal summation. Remember that we said this process is called horizontal summation because we fix the common independent variable at some level on the vertical axis and then add all of the individual dependent variables along the horizontal axis. 83 P DB DA Aggregate Demand Curve PX X X A X B X +X A B For example, suppose that we have two consumers, Ross and Rachel, with the following individual demands for good X and good Y XRoss = XRoss(PX, PY, MRoss) = MRoss / (2 PX) YRoss = YRoss(PX, PY, MRoss) = MRoss / (2 PY) XRachel = XRachel(PX, PY, MRachel) = MRachel / (3 PX) YRachel = YRachel(PX, PY, MRachel) = 2MRachel / (3 PY) To calculate the aggregate (market demand in this tiny market of two people) demand, we simply add up their individual demands for each good. The market for X would have an aggregate demand of: XTOTAL = XRoss + XRachel = MRoss / (2 PX) + MRachel / (3 PX) = 3MRoss + 2MRachel 6 PX The market for Y would have an aggregate demand of: YTOTAL = YRoss + YRachel = MRoss / (2 PY) + 2MRachel / (3 PY) 84 = 3MRoss + 4MRachel 6 PY This illustrates our claim that the aggregate demands are a function of both individuals' incomes (unlike their individual demands where they optimize with only their own incomes in mind). Obviously, this is a very simple example with only two people in the economy. Of course, the horizontal summation technique can be extended to any number of consumers in the particular market by summing XTOTAL = XRoss + XRachel + XJoey + XChandler + ... + XMonica where there are any number of individual demands represented in the "..." portion of the summation. This can be represented generally as: XTOTAL = n Xi(PX, PY, Mi) where, n represents the summation for i = 1,...,n XTOTAL = Xi(PX, PY, M1, M2,..., Mi) So, for instance if n = 154 we have an aggregate demand as a function of two prices, PX & PY, and 154 individual incomes, M1, M2,..., M154. One of the key considerations of general equilibrium theory can be illustrated by considering an economy with a specified number of consumers and producers making individual optimizing decisions on a specified number of goods and factors. The consumers, such as they are fully described by their utility functions and incomes and the producers such as they are fully described by their production functions. Now the fundamental question of general equilibrium theory is to find a set of prices for goods and factors which simultaneously satisfy all the following conditions: [1] every consumer is in equilibrium (their indifference curve is tangent to their budget line), [2] every producer is in equilibrium (their isoquant is tangent to their isocost line), [3] every market is in equilibrium (the quantity demanded in every market is equal to the quantity supplied in every market). 85 Such a set of prices, if it exists, is called an equilibrium price vector. So the most basic and central question of general equilibrium theory can be stated simply as: Is there such an equilibrium price vector? How do we prove that it exists? Why are we concerned mainly with equilibrium prices and not the quantities or other variables? Well, this is because once we know the equilibrium price vector; we can calculate all other economic variables accordingly since we have defined demands in terms of prices. To illustrate this point... [1] if we know equilibrium prices, we can use consumer demand functions to calculate all of the individual quantities of goods demanded; [2] we can also use producer demand functions (derived from their production functions) to calculate all individual quantities of factors demanded; [3] from all individual quantities demanded, we can obtain the corresponding aggregate quantities by the horizontal summation technique; [4] finally, we can use the market equilibrium conditions to calculate the aggregate supplies from the aggregate quantities of goods or factors demanded. To summarize, once we know the equilibrium price vector, we can determine all of the other relevant economic variables in the economy. There are many general equilibrium models with various special features such as corporate income taxes, personal income taxes, international trade, economic development and even money markets. We will concentrate on the following basic GE models of a simple economy. While these models are simple and clearly not overly realistic, they are quite useful in demonstrating the salient features of general equilibrium theory without dwelling too heavily on technical details or being encumbered by performing thousands of calculations. [1] The pure exchange economy is an economy with two consumers, A and B, exchanging two goods, X and Y. This is essentially a simple barter economy. [2] The simple production economy is an economy with one producer using labour to produce one good, C. On the consumption side there is one aggregate 86 consumer. This is often referred to as a Robinson Crusoe economy and is a special (or strange) case of a production economy. [3] The production economy is an economy with two producers using capital and labour to produce two goods, X and Y. On the consumption side there are two consumers exchanging the two goods produced, X and Y. Okay, so we now understand where we are going...but how do we get there? Don't we need to consider market structures and all of the other factors that affect the determination of these magical equilibrium price vectors? I'm really glad that you asked me that!!!! For now, not really...meaning that we won't need to concern ourselves with all of the potential market structures in all of the various markets if we assume that all markets under consideration are perfectly competitive. Remember, that simplicity is instructive and we are trying to get the basic idea of how to approach the general equilibrium problem. For example, when you first start training for a marathon you certainly don't start off sprinting for 6 miles... Now, since we are assuming a perfectly competitive market structure for every market, I feel it is imperative that we very briefly review the theoretical features of perfectly competitive markets. Perfect Competition Perfect competition is the most common theoretical model of market structure used in microeconomic analysis. Its definition consists of the following requirements: [1] Price Takers There are so many agents in the market (both buyers and sellers) that no single agent can have any influence on the market price. The key point is thus not the exact number of agents but the fact that none of them can have any effect on price. In other words, this feature requires that all agents are price takers (they take the market price as given in their optimizations). [2] Free Entry and Exit Agents can freely enter or leave the market without any type of restrictive barriers (such as regulations, certifications, unions, professional associations, etc.). Firms must be able to enter the market if there are profits to be earned or leave the market if there are losses to incur. Markets must be accessible without costs or restrictions. In other words, this feature requires that free entry and exit prevails in the market without costs. 87 [3] Homogeneous Goods The product in the market is homogeneous or identical in every aspect except pricing. In other words, this feature requires that price is the only criterion to be considered in the economic decision process. [4] Perfect Knowledge Information about price, quantity, the future, etc. must be available to allow agents to make informed decisions in their economic activities. In other words, perfect knowledge must be available for agents to make sensible (or rational) economic decisions. So having described the features of perfect competition, one might wonder how realistic our assumption that all markets structures are perfectly competitive is. Does a perfectly competitive market actually really exist? The short answer is no, not really... Perfect competition is a theoretical construct to facilitate the economic analysis of complicated market activities that occur in "real life". The pure concept of perfect competition to economists is just like the concept of infinities to mathematicians or galaxies to astronomers. I certainly would never claim that perfect competition occurs in "real life" nor would I want you walking around thinking that it does! The concept, however, does assist us to think about the analysis in terms of what to expect in the most perfect idealistic situation. What you are learning is a thought process or way of thinking analytically about a problem...and that's what higher learning is essentially all about. In reality, some markets are closer to perfect competition than others. HOMEWORK 1. Suppose that Julian has a demand function for cola given by: QCOLA = 17 + __M__ 54PCOLA [a] Originally, Julian makes $216 per week in income and the price of cola is $2 per bottle. Suppose the price of cola falls to $1 per bottle. [i] Find Julian's total demand response to the price change. [ii] Find Julian's response to the price change due to the substitution effect. [iii] Finally, find Julian's response to the price change due to the income effect. [b] Is cola a normal good or an inferior good to Julian? [c] Draw the diagram to explain the Slutsky decomposition of the two effects on Julian's overall demand for cola. 88 ECON 301 LECTURE #6 ENDOWMENTS AND INCOMES Let's look at how we define the budget line in general equilibrium theory. Recall that given prices, PX and PY, and income, M, the budget constraint of the consumer is written as: PXX + PYY = M We can rewrite this equation as Y = M PXX PY = _M_ - _PX_ X PY PY so that the budget line is a straight line with... slope = - _PX_ 0 PY and vertical intercept = _M_ PY As the price of good X goes up (everything else remaining the same), the budget line rotates around its vertical intercept while the consumer income remains the same. Y M / PY Y = M / PY (PX / PY)X BL2 BL1 X 89 In general equilibrium theory, the consumer income, M, is no longer exogenously fixed. Instead, we go one step further and define the consumer income, M, in terms of the concept of a consumer's endowment income as follows: Me = PX X + PY Y where X and Y denote the amounts of goods X and Y originally owned by the consumer, respectively. That is, [1] PX X is the market value of endowments in good X evaluated at price PX. [2] PY Y is the market value of endowments in good Y evaluated at price PY. As a result, instead of being fixed, the consumer endowment income Me fluctuates with market prices. So now we will have a new budget line. Using the consumer endowment income, Me, instead of the fixed consumer income, M, we can write the equation of the budget constraint as follows: PXX + PYY = Me PXX + PYY = PX X + PY Y and we can rewrite this equation as: Y = PX X + PY Y - PXX PY = _ PX X + PY Y _ - _PX_ X PY PY = _PX_ X + Y - PX X PY PY so that the new budget line is a straight line with... slope = - _PX_ 0 PY and vertical intercept = __PX_ X + Y PY When the consumer's income is defined in terms of endowments, as the price of good X goes up, ceteris paribus, the budget line rotates around the endowment point (X , Y) instead of the vertical intercept. 90 Y (PX / PY) X + Y Y Endowment Point Y = (PX / PY) X + Y - (PX / PY) X BL2 X BL1 X So let me explain the difference between the concept of exogenous income, M, and the endowment income, Me. Before, we were studying the theory of consumer equilibrium in a selfcontained framework where the consumer's income was exogenously fixed (exogenous = determined from outside the modeling process). In that case, we were concerned only with the optimizing behavior of a single consumer without any relationship to other consumers or producers in the economy. Now, in general equilibrium theory, we are studying the optimizing behavior of a consumer in the context of the inter relationship with other consumers and producers. We go one step further to endogenize the consumer income (endogenous = determined within the modeling process). That is, the endowment income now depends on both the market prices (PX and PY) and the original endowment points (X , Y). When prices change, the endowment income also changes and hence, the consumer decisions also must change accordingly. In other words, the concept of endowment income provides an added dimension to the interaction among agents in the economy. RELATIVE PRICES Microeconomics uses the notion of a price ratio to express the relative magnitude of the prices of two or more goods. For example, the theory of consumer equilibrium equates the marginal rate of substitution with the price ratio of two goods: 91 MRS = PX PY while the theory of producer equilibrium equates the marginal rate of technical substitution with the price ratio of the two factors: MRTS = r / w Thus, given a set of prices (PX , PY , r , w) we can choose the price of any of the following goods or factors (X , Y , K , L) as a unit of value (called a numeraire) and express the prices of the remaining goods and factors in terms of this unit of value. These prices, when expressed in terms of this unit of value, are called the relative prices. So, a numeraire is an archaic French term originally used by Walras to refer to the good or factor which has been selected as the unit of value. Once a good or factor has been selected as a numeraire, the prices of all other goods and factors are expressed in terms of their relative value to the numeraire. The numeraire itself is simply the good or factor that has a relative price (its price relative to its own price) equal to one. How do you choose and calculate the numeraire? We can illustrate this process quite easily. Since the numeraire is just an arbitrary designation; any good or factor can be a numeraire. Consider the price vector (PX , PY , r , w) = (2 , 3 , 4 , )... Good / Factor Selected to be the Numeraire GOOD X (divide all by PX = 2) GOOD Y (divide all by PY = 3) CAPITAL (divide all by r = 4) LABOUR (divide all by w = ) GOOD X PX = 2 1 2/3 4 GOOD Y PY = 3 3/2 1 6 CAPITAL LABOUR r=4 w= 2 4/3 1/6 1 1/8 8 1 Now, of course, this is not the only way to get relative prices... Instead of choosing a numeraire and dividing all the remaining prices by the price of the numeraire good / factor, we can get normalized relative prices in the following two steps: [1] First, we need to get the sum of all the prices: = 2 + 3 + 4 + = 19/2 [2] Then we divide all prices by this sum 92 PX = _2_ = _4_ 19 PY = _3_ = _6_ 19 r = _4_ = _8_ 19 w = _1/2_ = _1_ 19 This process of dividing all prices by the sum is called normalization and the prices calculated are called normalized prices. We should notice that the sum of the normalized prices is equal to one. PX + PY + r + w = 4/19 + 6/19 + 8/19 + 1/19 = 19/19 =1 The reason for this is simply that we have divided all prices by (which is the sum of all prices). Now, notice that we can obtain the relative prices that we calculated in the chart before by taking the ratios of the normalized prices. For example, when we used labour as the numeraire we had the relative price vector (PX , PY , r , w)Rel. to L = (4 , 6 , 8 , 1) and if we use normalized prices and take their ratio to w we get: PX = _4/19_ = _4_ w 1/19 1 PY = _6/19_ = _6_ w 1/19 1 r = _8/19_ = _8_ w 1/19 1 w = _1/19_ = _1_ w 1/19 1 resulting in the relative price vector (PX , PY , r , w)Rel. to L = (4 , 6 , 8 , 1). This works for all of the relative prices found using the numeraire method (shown in the table above). 93 So, as you can see, we can construct a lot of prices for a given set of four commodities! How many? Well, for a given set of four commodities (X , Y , K , L), we can have up to four sets of relative prices (one set for each choice of the numeraire) plus one set of normalized prices. In total, there are 4 commodities and 5 sets of prices (4 times 5 = 20) or 20 individual prices altogether. Does it matter if we change from one set of relative prices to another set of relative prices? That depends on the type of variables under consideration. Some economic variables change with prices while others do not. This is the distinction between nominal and real variables: [1] Nominal variables Nominal variables are variables which will change when we change from one set of relative prices to another. For example, if we increase or decrease prices, the following nominal variables will change accordingly: Endowment income Production Cost Profit [2] Real variables Real variables are variables which will not change when we change from one set of relative prices to another set of relative prices. For example, the following real variables will not change for the very simple reason that we divide both the numerator and denominator by the same number (i.e. the price of the numeraire): Demand for good X Demand for good Y MRS X = M / PX Y = M / PY MRS = Y / X K = C / r L = C / w MRTS = L / K 94 Me = PX X + PY Y C = rK + wL = PQ (rK + wL) Demand for Capital Demand for Labour MRTS The choice of the numeraire will also not affect the budget constraint equation: PXX + PYY = PX X + PY Y Which has prices PX and PY on both sides, it will not change if we divide both sides of the price of the numeraire. Consequently, the following consumer demand functions derived from the solution of the constrained utility maximization problem: X = X(PX , PY) Y = Y(PX , PY) remain invariant to a change of all prices PX , PY. We describe this property by saying that the consumer demand functions are homogeneous of degree zero to prices. PURE EXCHANGE ECONOMY Now let's set up the logical structure of the pure exchange economy general equilibrium model (two consumers and two goods). As simple as it is, the pure exchange economy is an excellent introduction to various techniques and tools used in general equilibrium theory. This pure exchange economy focuses our attention on the consumption side of the economy while leaving the production side mysteriously unaccounted for. Basically, there are two consumers, A and B, making consumption and exchange decisions (barter) on two goods, X and Y. How do we get around this simplification? We assume that, since there is no production in the model, the consumers are endowed with some distribution of the two goods. This makes the supply side quite simple with the following endowment distribution: Consumer A Consumer B Total GOOD X XA XB X GOOD Y YA YB Y XA , YA denote the amounts of the goods originally owned by consumer A and XB , YB denote the amounts of the goods originally owned by consumer B. 95 The total (or market supply) of goods owned by both consumers are: X = XA + XB Y = YA + YB This endowment distribution completely specifies the supply side of the economy. There is no production of goods X, Y and the exogenously fixed amounts X and Y (of goods X and Y) are divided between consumers A and B. In other words, the supply curves are vertical for both goods, X and Y. PX "SX" PY "SY" X X Y Y Of course, the treatment of the demand side is more complex than that of the supply side. The consumers are equipped with individual utility functions, UA and UB, and endowments XA , YA , XB , YB. These consumers will make optimal decisions on demands XA , YA , XB , YB as follows: CONSUMER A decisions utility XA , YA UA = UA(XA,YA) CONSUMER B decisions utility XB , YB UB = UB(XB,YB) endowments XA , YA prices income PX , PY MA = PX XA + PY YA endowments XB , YB prices income PX , PY MB = PX XB + PY YB Note that the endowment income, MA, is also a function of prices, PX , PY. Note that endowment income, MB, is also a function of prices, PX , PY. 96 Utility Maximization maximize UA(XA,YA) subject to PX XA + PY YA = MA Consumer Equilibrium Analytically, the two conditions for consumer equilibrium must be satisfied: MRSA = PX PY PX XA + PY YA = MA Solving these two equations for XA & YA we get the demands by consumer A. XA = XA(PX, PY, MA) YA = YA(PX, PY, MA) Since the endowment income, MA, is also a function of prices, PX, PY, we can eliminate MA and express the consumer demands in terms of prices PX, PY, alone. XA = XA(PX, PY) YA = YA(PX, PY) Utility Maximization maximize UB(XB,YB) subject to PX XB + PY YB = MB Consumer Equillibrium Analytically, the two conditions for consumer equilibrium must be satisfied: MRSB = PX PY PX XB + PY YB = MB Solving these two equations for XB & YB we get the demands by consumer B. XB = XB(PX, PY, MB) YB = YB(PX, PY, MB) Since the endowment income, MB, is also a function of prices, PX, PY, we can eliminate MB and express the consumer demands in terms of prices PX, PY, alone. XB = XB(PX, PY) YB = YB(PX, PY) Consumers A and B make their own optimal decisions on their quantities (XA and YA or XB and YB respectively), independent of the other consumer, yet they are linked to each other through market prices (PX, PY). Now, on one hand we have the individual consumer demands XA, YA, XB, YB and on the other hand we have fixed quantities of goods supplied represented by the total endowment of the good in the economy, X , Y. Now, for equilibrium to occur, we need to put these hands together! 97 MARKET DEMAND FOR X On the demand side, using the horizontal summation technique, we obtain the aggregate demand function for good X. X = XA(PX, PY) + XB(PX, PY) = X(PX, PY) MARKET DEMAND FOR Y On the demand side, using the horizontal summation technique, we obtain the aggregate demand function for good Y. Y = YA(PX, PY) + YB(PX, PY) = Y(PX, PY) On the supply side, we already know that the market supply of good X is fixed at X. supply = X At market equilibrium, the aggregate demand for good X must be equal to the aggregate supply of good X. X(PX, PY) = X Graphically, the market equilibrium is defined as the intersection of the aggregate demand and supply curves. On the supply side, we already know that the market supply of good Y is fixed at Y. supply = Y At market equilibrium, the aggregate demand for good Y must be equal to the aggregate supply of good Y. Y(PX, PY) = Y Graphically, the market equilibrium is defined as the intersection of the aggregate demand and supply curves. PY "SY" PX "SX" PY* Market Equilibrium Market Equilibrium PX* X(PX , PY) X X Y Y(PX , PY) Y 98 To get the equilibrium price vector, we solve the market equilibrium equations: X(PX, PY) = X Y(PX, PY) = Y We denote these equilibrium price solutions as (PX*, PY*) to emphasize the fact that they are the prices that give rise to both the individual and market equilibria at the same time. Once these equilibrium prices are known, all other variables can be calculated accordingly. For example, we can calculate the quantity of goods demanded by consumer A and consumer B: XA* = XA(PX*, PY*) YA* = YA(PX*, PY*) PURE EXCHANGE ECONOMY EXAMPLE Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has a square root utility function while Consumer B has a CobbDouglas utility function with = 0.25, = 0.75, and = 1. There is one unit of each good allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = XB = X = + = 1 GOOD Y YA = YB = Y = + = 1 XB* = XB(PX*, PY*) YB* = YB(PX*, PY*) We solve this particular pure exchange economy as follows: Let's start with Consumer A. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: UA = UA(XA,YA) = X1/2Y1/2 MA = PX XA + PY YA = PX + PY 99 We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = 0.5 X-1/2Y1/2 0.5 X1/2Y-1/2 = YA XA At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = YA = PX XA PY Rearranging (1) we get: PX XA + PY YA = MA XA PX = YA PY (1) (2) (3) Meaning we can get demands for XA and YA by subbing (3) into (2) as follows: PX XA + PX XA = MA 2 PX XA = MA XA = M A 2 PX (4) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = PX + PY 2 PX XA = + _PY__ 4 PX (5) PY YA + PY YA = MA 2 PY YA = MA YA = M A 2 PY 100 (6) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (6) to get: YA = PX + PY 2 PY YA = + _PX__ 4 PY (7) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: UB = UB(XB,YB) = X1/4Y3/4 MB = PX XB + PY YB = PX + PY We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.25 X-3/4Y3/4 0.75 X1/4Y-1/4 = YB 3XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB = PX 3XB PY Rearranging (1B) we get: PX XB + PY YB = MB 3 XB PX = YB PY (1B) (2B) (3B) Meaning we can get demands for XB and YB by subbing (3B) into (2B) as follows: PX XB + 3 PX XB = MB 4 PX XB = MB 101 XB = MB 4 PX (4B) and we know that MB = PX + PY is the endowment income of consumer B, so we sub this in for the MB in (4B) to get: XB = PX + PY 4PX XB = 1/8 + _PY__ 8 PX (5B) 1/3 PY YB + PY YB = MB 4/3 PY YB = MB YB = 3MB 4 PY (6B) and we know that MB = PX + PY is the endowment income of consumer B, so we sub this in for the MB in (6B) to get: YB = 3 ( PX + PY) 4 PY YB = 3/8 + _3PX__ 8 PY (7B) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, X = XA + XB = + _PY__ + 1/8 + _PY__ 4 PX 8 PX = 3/8 + _3PY__ 8 PX and we know that the fixed supply of X in the economy is the total endowment of X... X = X 102 so, 3/8 + _3PY__ = 1 8 PX and solving the market equilibrium, we get the following equilibrium price ratio... _3PY__ = 5/8 8 PX or, 3PY = 5 PX _PX__ = __3__ 5 PY (8) We can do the same thing in the market for good Y. Okay, let's do it! Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, Y = YA + YB = + _PX__ + 3/8 + _3PX__ 8 PY 4 PY = 5/8 + _5PX__ 8 PY and we know that the fixed supply of Y in the economy is the total endowment of Y... Y = Y so, 5/8 + _5PX__ = 1 8 PY and solving the market equilibrium, we get the following equilibrium price ratio... _5PX__ = 3/8 8 PY or, 5PX = 3 PY 103 _PX__ = __3__ PY 5 (8B) So now that we have the equilibrium price ratio and the individual consumer demands (and the market demands as a result), we can find the equilibrium quantities demanded by the individuals, A and B, by subbing the price ratio into the individual demand functions. XA = + _PY__ 4 PX = + (5/3) XA* = 2/3 YA = + _PX__ 4 PY = + (3/5) YA* = 2/5 XB = 1/8 + _PY__ 8 PX = 1/8 + 1/8 (5/3) XB* = 1/3 YB = 3/8 + _3PX__ 8 PY = 3/8 + 3/8 (3/5) YB* = 3/5 As a verification (or check) of the market equilibrium condition, we add the individual consumer demands (X = XA + XB and Y = YA + YB) and they should sum to the fixed supply in terms of endowments of the goods. 104 XA + XB = 2/3 + 1/3 XA + XB = 1 XA + XB = X YA + YB = 2/5 + 3/5 YA + YB = 1 YA + YB = Y So, in both the market for X and the market for Y, demands are equal to supplies and we have market clearing (equilibrium). What if we specifically chose a particular good as the numeraire? The general equilibrium that we solved above only gives us the ratio of the two prices... _PX__ = __3__ PY 5 (8) What if we choose X as the numeraire? Then PX =1 and expression (8) above becomes... __1__ = __3__ PY 5 or, PY = 5/3 What if we choose Y as the numeraire? Then PY =1 and expression (8) above becomes... _ PX___ = __3__ 1 5 or, PX = 3/5 What if we normalized the prices? Then PX + PY = 1 and using expression (8), we get 105 and, Rearrange (9), to get... _ PX___ = __3__ PY 5 PX + PY = 1 (9) Sub (9.1) into (8) to get PX = 1 PY PY = 1 PX _ 1 PY___ = __3__ PY 5 _ 1__ = __8__ PY 5 PY = 5/8 (9.1) (9.2) Sub (9.2) into (8) to get _ PX___ = __3__ 1 PX 5 5PX = 3 3PX PX = 3/8 Are these 3 sets of prices the same or are they different? Recall, when we used X as the numeraire good, we had PX = 1, PY = 5/3. Recall, when we used Y as the numeraire good, we had PX = 3/5, PY = 1. Recall, when we used normalization, we had PX = 3/8, PY = 5/8. These are the same things! To see this consider the price ratio _ PX__ in each of PY the three situations. [1] When we used X as the numeraire good, we had PX = 1, PY = 5/3 so... _ PX__ = __1__ = 3/5 5/3 PY [2] When we used Y as the numeraire good, we had PX = 3/5, PY = 1 so... _ PX__ = 3/5 PY 106 [3] When we used normalization, we had PX = 3/8, PY = 5/8 so... _ PX__ = 3/8 = 3/5 PY 5/8 The point is that in general equilibrium theory the actual prices themselves do not matter, only the ratio of the prices matter. As we discussed before, nominal variables may change in value depending upon which set of prices we use (i.e. numeraires or normalized) but the real variables will not change in value and the nominal variables are not usually the variables that are of central interest to us anyway. 107 HOMEWORK 1. Calculate all of the possible relative and normalized prices for the following price vector (and fill on the chart below with your answers): (PX , PY , r , w) = (7 , 9 , 23 , 18) Good / Factor Selected to be the Numeraire GOOD X GOOD Y CAPITAL LABOUR GOOD X PX = 7 GOOD Y PY = 9 CAPITAL LABOUR r = 23 w = 18 2. A price simplex of dimension 1 is a set of all non-negative price vectors (PX , PY) that sum to one, S = {(PX , PY) 0 | PX + PY = 1} What does this price simplex look like in a two dimensional diagram? 3. Ross, Phoebe, Joey, Monica, Rachel, and Chandler have the following individual demands for cappuccino, C: CRoss = CRoss(PC , MRoss) = MRoss / (2 PC) CPhoebe = CPhoebe(PC , MPhoebe) = MPhoebe / (7 PC) CJoey = CJoey(PC , MJoey) = 5MJoey / (14 PC) CMonica = CMonica(PC , MMonica) = 9MMonica / (28 PC) CRachel = CRachel(PC , MRachel) = 10MRachel / ( 21PC) CChandler = CChandler(PC , MChandler) = 21MChandler / (56 PC) Using horizontal summation, find the market demand for cappuccino if these 6 "friends" are the only consumers in the cappuccino market. Also, draw a diagram representing the horizontal summation that you performed (not to scale, but label the diagram completely). 108 ECON 301 LECTURE #7 WALRAS LAW Is there a particular reason that we can solve the equilibrium price ratio from the market equilibrium condition of either good X or good Y? Remember, in our example we got the same value for _ PX__ = 3/5 from the PY market equilibrium condition of both good X and good Y. This is not a coincidence! Indeed, it comes from a general theoretical result, namely, Walras Law. Walras Law can be stated in the context of a pure exchange economy as follows: If the market for one good (say, good Y) is in equilibrium, then the market for the other good (i.e. good X) is also in equilibrium. Walras Law states that if the market for good X is already in equilibrium, then so is the market for good Y. Therefore, if we have already solved for the equilibrium price ratio from the market equilibrium equation of good X, then we do not need to solve the other market equilibrium equation since it will be in equilibrium as well (and give us the same answer in terms of the equilibrium price ratio...needless suffering). The logic behind Walras Law lies in the close relationship between the two concepts of consumer equilibrium and market equilibrium. It is not difficult to show (prove) that Walras Law works within the context of the pure exchange economy. To do so, we start with the individual budget constraints PX XA + PY YA = PX XA + PY YA PX XB + PY YB = PX XB + PY YB If we sum these two budget constraints together, we get... PX (XA + XB) + PY (YA + YB) = PX (XA + XB) + PY (YA + YB) [market X] [market Y] [endowment X] [endowment Y] PX X + PY Y = PX X + PY Y PX (X X) + PY (Y Y) = 0 (10) Now, let's suppose the market for good X is in equilibrium already. That is the aggregate demand = the aggregate supply of X. 109 XA + XB = X [market X] X = X X X = 0 Thus, PX (X X) = 0 "excess demand" = 0 "value of excess demand" = 0 Substituting this result into (10), we get: PX (X X) + PY (Y Y) = 0 PY (Y Y) = 0 Y Y = 0 Y = Y (10) "value of excess demand" = 0 "excess demand" = 0 "demand" = "supply" And the market for Y is in equilibrium! What we have done here is to show that if we know the market for X is in equilibrium then we can derive that the market for Y is also in equilibrium. What do I mean by excess demands? We can define excess demands for both goods by saying that "market demand market supply = excess demand"1. ZX = X X ZY = Y Y and Walras Law can be written as PX ZX + PY ZY = 0 In this context, we can state Walras Law as: The value of all market excess demands must sum to zero. (11) For those of you who are wondering, Walras Law works for any number of markets (not just two). The statement of Walras Law for the general case of multiple markets is: 1 Of course, if the expression is negative we have excess supply. 110 In an economy with n markets, if (n 1) markets are in equilibrium, then the nth market is also in equilibrium. This implies that we only need to solve for (n 1) equations. Why? Suppose that P1 Z1 + P2 Z2 + P3 Z3 + P4 Z4 + ... + Pn Zn = 0 and further, suppose the first (n 1) markets are in equilibrium. This means that... Z1 = Z2 = Z3 = Z4 = ... = Zn-1 = 0 0 + 0 + 0 + 0 + ... + Pn Zn = 0 so either Pn = 0 or Zn = 0. So the market for good n either results in n being a free good or the market for good n is in equilibrium. PURE EXCHANGE EXAMPLE (2 COBB-DOUGLAS CONSUMERS...again) As promised, we will be going through a couple more examples of how to solve a pure exchange economy using a variety of functional forms in combination. Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has a Cobb-Douglas utility function with = 0.4, = 0.6, and = 1while Consumer B has a Cobb-Douglas utility function with = 0.65, = 0.35, and = 1. There are 12 units of each good allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = 7 XB = 5 X = 7 + 5 = 12 GOOD Y YA = 3 YB = 9 Y = 3 + 9 = 12 We solve this particular pure exchange economy as follows: Let's start with Consumer A. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: 111 UA = UA(XA,YA) = X0.4Y0.6 MA = PX XA + PY YA = 7 PX + 3 PY We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = 0.4 X-0.6Y0.6 0.6 X0.4Y-0.4 = 2YA 3XA At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = 2YA = PX 3XA PY Rearranging (1) we get: PX XA + PY YA = MA 3XA PX = 2YA PY (1) (2) (3) Meaning we can get demands for XA and YA by subbing (3) into (2) as follows: PX XA + (3/2) PX XA = MA (5/2) PX XA = MA XA = 2MA 5 PX (4) and we know that MA = 7 PX + 3 PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = 14 PX + 6 PY 5 PX XA = 14/5 + _6PY__ 5 PX (5) (2/3) PY YA + PY YA = MA 112 (5/3) PY YA = MA YA = 3MA (6) 5 PY and we know that MA = 7 PX + 3 PY is the endowment income of consumer A, so we sub this in for the MA in (6) to get: YA = 21 PX + 9 PY 5 PY YA = 9/5 + _21PX__ 5PY (7) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: UB = UB(XB,YB) = X0.65Y0.35 MB = PX XB + PY YB = 5 PX + 9 PY We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.65 X-0.35Y0.35 0.35 X0.65Y-0.65 = 13YB 7XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = 13YB = PX 7XB PY Rearranging (1B) we get: PX XB + PY YB = MB 7 XB PX = 13 YB PY (1B) (2B) (3B) Meaning we can get demands for XB and YB by subbing (3B) into (2B) as follows: PX XB + (7/13) PX XB = MB 113 (20/13) PX XB = MB XB = 13MB 20 PX (4B) and we know that MB = 5 PX + 9 PY is the endowment income of consumer B, so we sub this in for the MB in (4B) to get: XB = 65 PX + 117 PY 20PX XB = 13/4 + _117PY__ 20 PX (5B) (13/7) PY YB + PY YB = MB (20/7) PY YB = MB YB = 7MB 20 PY (6B) and we know that MB = 5 PX + 9 PY is the endowment income of consumer B, so we sub this in for the MB in (6B) to get: YB = 35 PX + 63 PY 20 PY YB = 63/20 + _7PX__ 4PY (7B) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, X = XA + XB = 14/5 + _6PY__ + 13/4 + _117PY__ 5PX 20PX = 121/20 + _141PY__ 20PX 114 and we know that the fixed supply of X in the economy is the total endowment of X... X = X so, 121/20 + _141PY__ = 12 20PX and solving the market equilibrium, we get the following equilibrium price ratio... _141PY__ = 119/20 20PX or, 141PY = 119 PX _PX__ = __141__ PY 119 (8) We can do the same thing in the market for good Y. Okay, let's do it! Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, Y = YA + YB = 9/5 + _21PX__ + 63/20 + _7PX__ 5PY 4PY = 99/20 + _119PX__ 20PY and we know that the fixed supply of Y in the economy is the total endowment of Y... Y = Y so, 99/20 + _119PX__ = 12 20PY and solving the market equilibrium, we get the following equilibrium price ratio... _119PX__ = 141/20 20PY 115 or, 119 PX = 141 PY _PX__ = __141__ 119 PY (8B) So now that we have the equilibrium price ratio and the individual consumer demands (and the market demands as a result), we can find the equilibrium quantities demanded by the individuals, A and B, by subbing the price ratio into the individual demand functions. XA = 14/5 + _6PY__ 5PX = 14/5 + 6/5 (119/141) = 1974/705 + 714/705 = 2688/705 XA* = 3.8128 YA = 9/5 + _21PX__ 5PY = 9/5 + 21/5 (141/119) = 1071/595 + 2961/595 = 4032/595 YA* = 6.7765 XB = 13/4 + _117PY__ 20 PX = 13/4 + 117/20 (119/141) = 9165/2820 + 13923/2820 = 23088/2820 or 5772/705 XB* = 8.1872 116 YB = 63/20 + _7PX__ 4 PY = 63/20 + 7/4 (141/119) = 7497/2380 + 4935/2380 = 12432/2380 or 3108 / 595 YB* = 5.2235 As a verification (or check) of the market equilibrium condition, we add the individual consumer demands (X = XA + XB and Y = YA + YB) and they should sum to the fixed supply in terms of endowments of the goods. XA + XB = 3.8128 + 8.1872 XA + XB = 12 XA + XB = X YA + YB = 6.7765 + 5.2235 YA + YB = 12 YA + YB = Y So, in both the market for X and the market for Y, demands are equal to supplies and we have market clearing (equilibrium). PURE EXCHANGE EXAMPLE (1 QUASI-LINEAR & 1 LINEAR CONSUMER) Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. UA = 2X + 4Y1/2 A = (XA , YA) = (6,4) 117 UB = X + Y B = (XB , YB) = (4,6) Let's clear up some terminology before we continue... Above, we have the consumers' initial endowments given by: A = (XA , YA) and B = (XB , YB) and their utility functions. What we are trying to find is the equilibrium price ratio and to do this we need the consumers' consumption vectors represented as: CA = (XA , YA) and CB = (XB , YB) to obtain total market demands represented as: X = (XA + XB) and Y = (YA , YB) Definition: An allocation is a specification of a consumption vector for each consumer and CB = (XB , YB) CA = (XA , YA) Definition: A feasible allocation is an allocation which is physically possible given the resources of the economy. An allocation is feasible if XA + XB XA + XB YA + YB YA + YB Now, back to our example... We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = ___2___ 2Y-1/2 118 = YA 1/2 At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = YA 1/2 = PX PY YA = PX2 PY2 Subbing (1) into (2) we get: PX XA + PY YA = MA PX XA + PX2 = MA PY XA = MA - PX PX PY (3) (1) (2) and we know that MA = 6 PX + 4 PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = 6 PX + 4 PY - PX PX PY XA = 6 + _4PY__ - PX PX PY (4) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment that constrains their utility as follows: UB = X + Y B = (XB , YB) = (4,6) We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = __1__ 1 119 At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = _1_ = PX 1 PY (1B) So now that we have the equilibrium price ratio. Since only the ratio matters we can use the relationship PX = PY in consumer A's demands above (we can derive consumer B's demands from consumer A's...more in a second) XA = 6 + _4PY__ - PX PX PY = 6 + 4 (PX) 1 PX XA* = 9 YA = PX2 PY2 YA = 12 12 YA* = 1 XB = X XA* XB* = 1 YB = Y YA* YB* = 9 This example illustrates that the individual with a linear indifference curve (consumer B) will dictate the price ratio, while the individual with the quasilinear indifference curve will dictate the final demands in the pure exchange economy. 120 PURE EXCHANGE EXAMPLE (1 LIONTIEF & 1 LINEAR CONSUMER) Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. UA = min {2X , Y} A = (XA , YA) = (2,3) UB = X + 3Y B = (XB , YB) = (3,2) We can figure out consumer A's marginal rate of substitution as: MRSA = ? Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment that constrains their utility as follows: UB = X + 3Y B = (XB , YB) = (3,2) We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = __1__ 3 At the consumer equilibrium, the price ratio is dictated by consumer B and is simply: MRSB = _1_ = PX 3 PY This gives us the relationship between PX & PY as: 3PX = PY Back to Consumer A... At the consumer equilibrium, the equations consumer A needs to satisfy are: 2XA = YA (1) (1a) (1b) 121 PX XA + PY YA = MA Subbing (1) into (2) we get: PX XA + 2 PY XA = MA PX XA + 2 PY XA = 2 PX + 3 PY and we know that 3 PX = PY must hold in equilibrium, so... PX XA + 6 PX XA = 2 PX + 9 PX 7 PX XA = 11 PX XA = 11 / 7 (2) Now let's go back and find YA, remember that the equations consumer A needs to satisfy are: 2XA = YA Subbing (1) into (2) we get: PX XA + PY YA = MA PX YA + PY YA = MA PX YA + PY YA = 2 PX + 3 PY and we know that PX = 1/3 PY must hold in equilibrium, so... 1/6 PY YA + PY YA = 2/3 PY + 3 PY PY YA + 6 PY YA = 4 PY + 18 PY 7 PY YA = 22 PY YA = 22 / 7 We should notice that Consumer A's demands lie on both the equilibrium price vector (common budget line) and on the kink line defined by their perfect complement preferences. (1) (2) 122 It is now elementary to find Consumer B's equilibrium allocation... XB = X - XA = 5 11 / 7 = 24 / 7 YB = Y - YA = 5 22 / 7 = 13 / 7 PURE EXCHANGE EXAMPLE (1 COBB-DOUGLAS & 1 QUASI-LINEAR CONSUMER) Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. UA = 2X + 4Y1/2 A = (XA , YA) = (1,19) UB = X0.5Y0.5 B = (XB , YB) = (19,1) We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = ___2___ 2Y-1/2 = YA 1/2 At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = YA 1/2 = PX PY YA = PX2 PY2 Subbing (1) into (2) we get: PX XA + PY YA = MA PX XA + PX2 = MA PY XA = MA - PX PX PY 123 (3) (1) (2) and we know that MA = PX + 19 PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = PX + 19 PY - PX PX PY XA = 1 + _19PY__ - PX PX PY (4) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment that constrains their utility as follows: UB = X0.5Y0.5 B = (XB , YB) = (19,1) We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.5 X-0.5Y0.5 0.5 X0.5Y-0.5 = YB XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB = PX XB PY Rearranging (1B) we get: PX XB + PY YB = MB XB PX = YB PY (1B) (2B) (3B) Meaning we can get demands for XB and YB by subbing (3B) into (2B) as follows: PX XB + PX XB = MB 2 PX XB = MB 124 XB = MB 2PX (4B) and we know that MB = 19 PX + PY is the endowment income of consumer B, so we sub this in for the MB in (4B) to get: XB = 19 PX + PY 2PX XB = 19/2 + _PY__ 2 PX (5B) PY YB + PY YB = MB 2 PY YB = MB YB = MB 2 PY (6B) and we know that MB = 19 PX + PY is the endowment income of consumer B, so we sub this in for the MB in (6B) to get: YB = 19 PX + PY 2 PY YB = 1/2 + _19PX__ 2 PY (7B) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, X = XA + XB = 1 + _19PY_ - PX + 19/2 + _PY__ PX PY 2 PX = 21/2 + _39PY__ - PX 2PX PY and we know that the fixed supply of X in the economy is the total endowment of X... X = X 125 so, 21/2 + _39PY__ - PX = 20 2PX PY _39PY__ - PX = 19/2 2PX PY 39PY2 19PXPY - 2PX2 = 0 2PX2 + 19PXPY - 39PY2 = 0 So, we need to crack out our quadratic formula... -b + (b2 4ac)1/2 2a -19 + (361 4(2)(-39))1/2 4 -19 + (673)1/2 4 -19 + 25.94224354 4 1.735560886 = PX and so, YA = PX2 PY2 YA = 1.7355608862 YA = 3.012171587 So now that we have the equilibrium price ratio and the individual consumer demands (and the market demands as a result), we can find the equilibrium quantities demanded by the individuals, A and B, by subbing the price ratio into the individual demand functions. XA = 1 + _19PY__ - PX PX PY = 1 + 19 / 1.735560886 - (1.735560886) = 1 + 10.94746958 1.735560886 126 when PY = 1 The other part is irrelevant since we assume PX > 0 XA* = 10.21190869 XB = 19/2 + _PY__ 2 PX = 19/2 + 1/(2 1.735560886) = 9.5 + 1 / 3.471121772 XB* = 9.788091304 YB = 1/2 + _19PX__ 2PY = 1/2 + 19 (1.735560886) / 2 = 0.5 + 32.97565683/2 YB* = 16.98782842 As a verification (or check) of the market equilibrium condition, we add the individual consumer demands (X = XA + XB and Y = YA + YB) and they should sum to the fixed supply in terms of endowments of the goods. XA + XB = 10.21190869 + 9.788091304 XA + XB = 20 XA + XB = X YA + YB = 3.012171587 + 16.98782842 YA + YB = 20 YA + YB = Y 127 So, in both the market for X and the market for Y, demands are equal to supplies and we have market clearing (equilibrium). HOMEWORK 1. Consider a pure exchange economy with two goods, Xylophones (X) and Yarn (Y), and two consumers, Anne (A) and Betty (B). Anne has a Cobb-Douglas utility function with = 0.2, = 0.8, and = 4 while Betty has a Cobb-Douglas utility function with = 0.35, = 0.65, and = 1. There are 12 Xylophones and 8 balls of Yarn allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = 7 XB = 5 X = 7 + 5 = 12 GOOD Y YA = 2 YB = 6 Y = 2 + 6 = 8 Solve for the general equilibrium price ratio, and report the equilibrium quantities demanded for each consumer. 2. Consider a pure exchange economy with two goods, aXes (X) and knYves (Y), and two consumers, Andy (A) and Bill (B). UA = 4X1/2 + 2Y UB = X + Y A = (XA , YA) = (2,1) B = (XB , YB) = (1,2) Solve for the general equilibrium price ratio, and report the equilibrium quantities demanded for each consumer. 3. Consider a pure exchange economy with three goods,(X, Y, and Z), and two consumers, A and B. UA = X1/2Y1/4Z1/4 A = (XA , YA, ZA) = (1,1,1) UB = X1/4Y1/2Z1/4 B = (XB , YB, YB) = (1,1,1) Solve for the general equilibrium price ratios, and report the equilibrium quantities demanded for each consumer. 128 ECON 301 LECTURE #8 PRODUCTION ECONOMY The production economy introduces, you guessed it, production into the pure exchange general equilibrium model. This analysis is considerably more complicated than the pure exchange model without production since now we must trace the interaction between consumers and producers in both the goods markets and the factor markets. So now we have two goods, X and Y, two factors, K and L, and two consumers, A and B. The consumers play the dual role of buyers in the goods markets and sellers in the factor markets while the opposite is true for the producers. Using this simple structure, we will be able to trace the inter-relationship between three basic components of the economy: consumers, producers and markets. Here are the details: Commodities Prices Buyers Sellers Good X PX Consumer A Consumer B Producer X Good Y PY Consumer A Consumer B Producer Y Capital (K) r Producer X Producer Y Consumer A Consumer B Labour (L) w Producer X Producer Y Consumer A Consumer B In this case, we are using the terminology of "producer" to mean a firm, industry, or a sector. Additionally, we will assume that each producer produces only one good at a time (i.e. producer X produces good X while producer Y produces good Y). That is, we will not consider the very complex case of joint production where all producers can make several different types of products. We will approach the distribution of endowments differently in this model as well. Unlike the pure exchange model which has initial endowments defined in terms of goods, the production economy takes a step back and defines the initial endowment distribution in terms of factors as follows: Consumer A Consumer B Total Capital (K) KA KB KT Labour (L) LA LB LT where KA , KA denote the amounts of capital and labour originally owned by consumer A and KB , LB denote the amounts of capital and labour originally owned by consumer B. 129 The total (or market supply) of capital and labour owned by both consumers are: KT = KA + KB LT = LA + LB How does this factor endowment distribution affect the consumer optimizing behavior? The interesting part of the factor endowment distribution is that the endowments of each consumer need to be determined at market prices. They can fluctuate along with market prices and hence, influence each consumer's decision in other markets. For example, if wage increases then the consumers' endowment incomes will increase accordingly. Under normal circumstances we would expect that, with more income in their hands, consumers will increase their demands for the goods. This triggers an increase in the production of goods which, on turn, feeds back to the demand for factors and ultimately back to the income side. Let's look at how the factor endowments enter the utility maximization problems of both consumers A and B. CONSUMER A decisions utility XA , YA UA = UA(XA,YA) CONSUMER B decisions utility XB , YB UB = UB(XB,YB) endowments KA , LA prices income PX , PY MA = r KA + w LA endowments KB , LB prices income PX , PY MB = r KB + w LB Note that the endowment income, MA, is also a function of prices, r , w. Utility Maximization maximize UA(XA,YA) subject to PX XA + PY YA = MA Note that endowment income, MB, is also a function of prices, r , w. Utility Maximization maximize UB(XB,YB) subject to PX XB + PY YB = MB 130 Consumer Equilibrium Analytically, the two conditions for consumer equilibrium must be satisfied: MRSA = PX PY PX XA + PY YA = MA Solving these two equations for XA & YA we get the demands by consumer A. XA = XA(PX, PY, MA) YA = YA(PX, PY, MA) Since the endowment income, MA, is a function of factor prices, r, w, we can eliminate MA and express the consumer demands in terms of prices (PX, PY, r, w) alone. XA = XA(PX, PY, r, w) YA = YA(PX, PY, r, w) Consumer Equillibrium Analytically, the two conditions for consumer equilibrium must be satisfied: MRSB = PX PY PX XB + PY YB = MB Solving these two equations for XB & YB we get the demands by consumer B. XB = XB(PX, PY, MB) YB = YB(PX, PY, MB) Since the endowment income, MB, is a function of factor prices, r, w, we can eliminate MB and express the consumer demands in terms of prices (PX, PY, r, w) alone. XB = XB(PX, PY, r, w) YB = YB(PX, PY, r, w) Consumers A and B make their own optimal decisions on their quantities (XA and YA or XB and YB respectively), independent of the other consumer, yet they are linked to each other through market prices (PX, PY, r, w). The treatment of production is the key difference between the pure exchange economy and this production economy model. Each producer solves its own individual optimizing decision to get its demands for factors. PRODUCER OF GOOD X decisions KX , LX PRODUCER OF GOOD Y decisions K Y , LY production function QX = f(KX,LX) Assume CRS production function. 131 production function QY = g(KY,LY) Assume CRS production fnc. prices r,w prices r,w output level QX Cost Minimization minimize cost r KX + w LX subject to f(KX,LX) = QX Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: MRTSX = _r_ w f(KX,LX) = QX Solving these two equations for KX & LX we get the demands by producer X. KX = KX(r, w, QX) LX = LX(r, w, QX) Constant Returns to Scale The factor demands by producer X are thus functions of factor prices and its prices output level QX. Since f(KX,LX) is a constant returns to scale production function, we can divide factor demands KX,LX by the total output level QX in order to get the factor demands on a per unit of output basis as follows: KX = kX(r, w) QX output level QY Cost Minimization minimize cost r KY + w LY subject to g(KY,LY) = QY Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: MRTSY = _r_ w g(KY,LY) = QY Solving these two equations for KY & LY we get the demands by producer Y. KY = KY(r, w, QY) LY = LY(r, w, QY) Constant Returns to Scale The factor demands by producer Y are thus functions of factor and its output level QY. Since g(KY,LY) is a constant returns to scale production function, we can divide factor demands KY,LY by the total output level QY in order to get the factor demands on a per unit of output basis as follows: KY = kY(r, w) QY 132 LX = lX(r, w) QX KX and LX refer to the quantities of factors required by producer X to produce QX units of output. On the other hand, kX and lX refer to the quantities of factors required by producer X to produce 1 unit of output. Also, note that kX and lX are functions of factor prices (r,w) alone while KX and LX are functions of both factor prices and the total output QX. Marginal Cost We know that the marginal cost of good X is the cost of producing one more unit of the good using K and L. MCX = r kX + w lX and this is a function of only r and w. Perfect Competition Under perfect competition, producer X must satisfy the zero profit condition: PX = MCX (r,w) PX = PX (r,w) LY = lY(r, w) QY KY and LY refer to the quantities of factors required by producer Y to produce QY units of output. On the other hand, kY and lY refer to the quantities of factors required by producer Y to produce 1 unit of output. Also, note that kY , lY are functions of factor prices (r,w) alone while KY and LY are functions of both factor prices and the total output QY. Marginal Cost We know that the marginal cost of good Y is the cost of producing one more unit of the good using K and L. MCY = r kY + w lY and this is a function of only r and w. Perfect Competition Under perfect competition, producer Y must satisfy the zero profit condition: PY = MCY (r,w) PY = PY (r,w) Of course, we wouldn't go to all this trouble unless there was some significance to the fact that output prices PX and PY are functions of the factor prices (r,w) alone. The fact that the output prices PX and PY are functions of the factor prices (r,w) has a very important implication. 133 From now on, we only need to focus our attention on the factor prices (r,w) because once we have the equilibrium factor prices (r,w), we can always calculate the equilibrium output prices PX and PY accordingly. In other words, instead of solving for four equilibrium prices (PX, PY, r, w), we need only solve for two equilibrium factor prices (r,w). Alright, now we have considered the consumer and the producer but what about the goods market? On the demand side, we have individual consumer demands XA , YA , XB and YB while on the supply side we have output supplies produced by the two producers. All we need to do now is to equate the demand side with the supply side... MARKET FOR GOOD X On the demand side, we have individual consumer demands for good X. XA = XA(PX, PY, r, w) XB = XB(PX, PY, r, w) and aggregate demand for good X X = XA(r, w) + XB(r, w) X = X(r , w) On the supply side, producer X provides QX as the market supply of good X. At market equilibrium, the output supply of good X must equal the aggregate demand of good X. QX = X(r , w) That is, producer X must produce just enough good X to satisfy the market demands (i.e. leaving neither surplus nor shortage). In other words, the market equilibrium MARKET FOR GOOD Y On the demand side, we have individual consumer demands for good Y. YA = YA(PX, PY, r, w) YB = YB(PX, PY, r, w) and aggregate demand for good Y Y = YA(r, w) + YB(r, w) Y = Y(r, w) On the supply side, producer Y provides QY as the market supply of good Y. At market equilibrium, the supply of good Y must equal the aggregate demand of good Y. QY = Y(r , w) That is, producer Y must make just enough good Y to satisfy the market demands (i.e. leaving neither surplus nor shortage). In other words, the market 134 condition for good X allows us to determine the output supply QX. equilibrium condition for good Y allows us to determine the output supply QY. We find the factor market equilibrium using the same basic principle of demand equal to supply in each market. MARKET FOR CAPITAL On the demand side, we have the individual producer demands for capital KX(r, w) = kX QX KY(r, w) = kY QY and aggregate demand for capital K = KX(r, w) + KY(r, w) K = K(r , w) On the supply side, we have the aggregate capital endowment as determined by K = KA + KB At capital market equilibrium, the Aggregate capital demand must match the aggregate capital supply. K(r , w) = KA + KB = K MARKET FOR LABOUR On the demand side, we have the individual producer demands for labour LX(r, w) = lX QX LY(r, w) = lY QY and aggregate demand for labour L = LX(r, w) + LY(r, w) L = L(r, w) On the supply side, we have the aggregate labour endowment as determined by L = LA + LB At labour market equilibrium, the the aggregate labour demand must match the aggregate labour supply. L(r , w) = LA + LB = L We now have a system of two simultaneous factor market equilibrium equations in two unknown factor prices: K(r , w) = K L(r , w) = L (1) (2) which must be solved for the equilibrium factor prices (r*, w*). As we know, the equilibrium factor prices are defined as those factor prices which match demand and supply in each factor market. 135 K(r* , w*) = K L(r* , w*) = L Once (r*, w*) are known, we can further calculate equilibrium output prices PX, PY, and all remaining equilibrium quantities. Now we can reduce our problem even further by using our knowledge of Walras Law and the numeraire. RELATIVE PRICES If we specify a good or a factor (say, labour) as the numeraire then we can express all other prices in terms of the numeraire. PX = PX (r, w) PY = PY (r, w) r = ? (to be solved) w = 1 (chosen as numeraire) That is, we have reduced the number of unknown prices to solve from four prices (PX, PY, r, w) down to three prices (PX, PY, r) since we automatically have w = 1 from choosing labour as numeraire. WALRAS LAW If the markets for good X, good Y, & capital are all in equilibrium, then Walras Law ensures that the market for labour is also in equilibrium. X(r , w) = QX Y(r , w) = QY K(r , w) = K L(r , w) = L That is, we have reduced the number of market equilibrium conditions to solve from four markets (X, Y, K, L) to three markets (X, Y, K) since we automatically have the labour market in equilibrium from Walras Law (once all other markets are in equilibrium). In other words, we can eliminate one equation from the solution by using Walras Law. In other words, we can eliminate one unknown from the solution by using relative prices (numeraire). Thus, using both Walras Law and labour as the numeraire (w 1), we can further reduce the two equations (1), (2) to one equation of capital market equilibrium in one unknown capital price K(r, w) = K 136 This is the simplest reduction of the problem: solving one equation for one unknown. We cannot simplify the production economy any further. How about an example... Consider the "square root economy" with square root functions for both consumers and producers UA = XA1/2YA1/2 UB = XB1/2YB1/2 QX = KX1/2LX1/2 QY = KY1/2LY1/2 and the following initial factor endowment distribution: Consumer A Consumer B Total Capital (K) KA = 0.2 KB = 0.8 KT = 1 Labour (L) LA = 0.6 LB = 0.4 LT = 1 We want to find the equilibrium prices (PX, PY, r, w) that clear all markets simultaneously. CONSUMER A Consumer A has the following data on utility function and endowment income: MRSA = YA XA MA = r KA + w LA = 0.2 r + 0.6 w Utility Maximization maximize UA = XA1/2YA1/2 subject to PX XA + PY YA = MA Consumer Equilibrium Analytically, the two conditions for consumer equilibrium must be satisfied: CONSUMER B Consumer B has the following data on utility function and endowment income: MRSB = YB XB MB = r KB + w LB = 0.8 r + 0.4 w Utility Maximization maximize UB = XB1/2YB1/2 subject to PX XB + PY YB = MB Consumer Equillibrium Analytically, the two conditions for consumer equilibrium must be satisfied: 137 MRSA = PX PY PX XA + PY YA = MA Solving these two equations for XA & YA we get the demands by consumer A. XA = M A 2PX XA = 0.2r + 0.6w 2PX XA = r + 3w 10PX YA = M A 2PY YA = 0.2r + 0.6w 2PY YA = r + 3w 10PY PRODUCER OF GOOD X The producer of good X has the following MRTS: MRTSX = LX KX Cost Minimization minimize cost r KX + w LX subject to KX1/2LX1/2 = QX Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: MRSB = PX PY PX XB + PY YB = MB Solving these two equations for XB & YB we get the demands by consumer B. XB = MB 2PX XB = 0.8r + 0.4w 2PX XB = 4r + 2w 10PX YB = MB 2PY YB = 0.8r + 0.4w 2PY YB = 4r + 2w 10PY PRODUCER OF GOOD Y The producer of good Y has the following MRTS: MRTSY = LY KY Cost Minimization minimize cost r KY + w LY subject to KY1/2LY1/2 = QY Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: 138 MRTSX = _r_ w KX1/2LX1/2 = QX Solving these two equations for KX & LX we get the demands by producer X. KX-1/2LX1/2 = _r_ KX1/2LX1/2 w LX = _r_ KX w KX = LXw r LX = KXr w sub KX into production function: QX = [LX(w / r)]1/2LX1/2 QX = LX(w / r)1/2 LX = QX(r / w)1/2 Similarly, we can find that... KX = QX(w / r)1/2 Constant Returns to Scale Since f(KX,LX) is a constant returns to returns scale production function, we can candivide factor demands KX,LX by the total output level QX in order to get the factor demands on a per unit of output basis as follows: KX = kX = (w / r)1/2 QX MRTSY = _r_ w KY1/2LY1/2 = QY Solving these two equations for KY & LY we get the demands by producer Y. KY-1/2LY1/2 = _r_ KY1/2LY1/2 w LY = _r_ KY w KY = LYw r LY = KYr w sub KY into production function: QY = [LY(w / r)]1/2LY1/2 QY = LY(w / r)1/2 LY = QY(r / w)1/2 Similarly, we can find that... KY = QY(w / r)1/2 Constant Returns to Scale Since g(KY,LY) is a constant to scale production function, we can divide factor demands KY,LY by the total output level QY in order to get the factor demands on a per unit of output basis as follows: KY = kY = (w / r)1/2 QY 139 LX = lX = (r / w)1/2 QX Marginal Cost MCX = r kX + w lX Perfect Competition Under perfect competition, producer X must satisfy the zero profit condition: PX = MCX (r,w) PX = r kX + w lX PX = r (w / r)1/2 + w (r / w)1/2 PX = (wr2 / r)1/2 + (rw2 / w)1/2 PX = 2(wr)1/2 MARKET FOR GOOD X Substituting these output prices into individual consumer demands... XA = r + 3w = r + 3w 10PX 20(wr)1/2 XB = 4r + 2w = 4r + 2w 20(wr)1/2 10PX and aggregate demand for good X X = r + 3w + 20(wr)1/2 X = 5r + 5w 20(wr)1/2 X=r+w 4(wr)1/2 140 4r + 2w 20(wr)1/2 LY = lY = (r / w)1/2 QY Marginal Cost MCY = r kY + w lY Perfect Competition Under perfect competition, producer Y must satisfy the zero profit condition: PY = MCY (r,w) PY = r kY + w lY PY = r (w / r)1/2 + w (r / w)1/2 PY = (wr2 / r)1/2 + w (rw2 / w)1/2 PY = 2(wr)1/2 MARKET FOR GOOD Y Substituting these output prices into individual consumer demands... YA = r + 3w = r + 3w 10PY 20(wr)1/2 YB = 4r + 2w = 4r + 2w 10PY 20(wr)1/2 and aggregate demand for good Y Y = r + 3w + 20(wr)1/2 Y = 5r + 5w 20(wr)1/2 Y=r+w 4(wr)1/2 4r + 2w 20(wr)1/2 X = X(r , w) On the supply side, producer X provides QX as the market supply of good X. At market equilibrium, the output supply of good X must equal the aggregate demand of good X. QX = X(r , w) = r + w 4(wr)1/2 MARKET FOR CAPITAL Substituting this output supply, QX, into the producer demands for capital, we get... KX = QX(w / r)1/2 KX = r + w 4(wr)1/2 KX = r + w 4r KY = QY(w / r)1/2 KY = r + w w1/2 4(wr)1/2 r1/2 KY = r + w 4r and aggregate demand for capital K=r+w 4r K=r+w 2r At capital market equilibrium, the Aggregate capital demand must match the aggregate capital supply. +r+w 4r w1/2 r1/2 Y = Y(r, w) On the supply side, producer Y provides QY as the market supply of good Y. At market equilibrium, the supply of good Y must equal the aggregate demand of good Y. QY = Y(r , w) = r + w 4(wr)1/2 MARKET FOR LABOUR Substituting this output supply, QY, into the producer demands for labour, we get... LX = QX(r / w)1/2 LX = r + w r1/2 1/2 4(wr) w1/2 LX = r + w 4w LY(r, w) = lY QY LY = r + w r1/2 4(wr)1/2 w1/2 LY = r + w 4w and aggregate demand for labour L=r+w +r+w 4w 4w L=r+w 2w At labour market equilibrium, the the aggregate labour demand must match the aggregate labour supply. 141 r+w 2r =1 r+w =1 2w If we solve for capital market equilibrium condition, we get the relationship between the price of labour and the price of capital. 2w = r + w w=r r/w=1 This is the equilibrium price solution that we need to figure out everything else. If we solve for capital market equilibrium condition, we get the following relationship between the price of labour and the price of capital. 2r = r + w r=w w/r=1 This is the equilibrium price solution that we need to figure out everything else. Let's figure out the rest of our equilibrium price vector... If we choose capital as the numeraire then r=1 r=w=1 PX = 2(wr)1/2 = 2r = 2 PY = 2(wr)1/2 = 2r = 2 Now let's figure out the consumer demands in equilibrium... XA = r + 3w = r + 3w 10PX 20(wr)1/2 X*A = 4 / 20 = 1 / 5 XB = 4r + 2w = 4r + 2w 10PX 20(wr)1/2 X*B = 6 / 20 = 3 / 10 So goods market demands are... X* = X*A + X*B = Y* = Y*A + Y*B = YA = r + 3w = r + 3w 10PY 20(wr)1/2 Y*A = 4 / 20 = 1 / 5 YB = 4r + 2w = 4r + 2w 10PY 20(wr)1/2 Y*B = 6 / 20 = 3 / 10 142 and just to check market clearing, this should be the market supply (i.e. amount produced by the producers) KX = r + w 4r KY = r + w 4r K=r+w 2r = = =1 LX = r + w = 4w LY = r + w = 4w L=r+w=1 2w (notice that this implies factor market clearing since factor endowments = 1 for each of L and K) QX = KX1/2LX1/2 QX = (1/2)1/2(1/2)1/2 QX = (notice goods market clearing, here!!!!) So all markets clear at the equilibrium price vector (PX, PY, r, w) = (2, 2, 1, 1) and the resulting demands for goods and factors are: (XA, YA) = (2/10 , 2/10) (KX, KY) = (1/2 , 1/2) (XB, YB) = (3/10 , 3/10) (LX, LY) = (1/2 , 1/2) QY = KY1/2LY1/2 QY = (1/2)1/2(1/2)1/2 QY = There are models where there is only one consumer in the production economy with two goods, two factors and one (representative) consumer. In such a case, there is no need to use the horizontal summation technique to sum over individual consumer demands to get the market demands for goods, since there is one consumer and their demands are the market demands in the goods market. The same applies to the aggregate supply of factors, since the one consumer is the owner of all the factor endowments of the economy. Otherwise, the model for one consumer, two factors, and two goods is very similar to the logical formulation of the model we just went through (two consumers, two factors, two goods production economy). ROBINSON CRUSOE ECONOMY Next time... 143 HOMEWORK 1. Find the equilibrium prices and quantities for the production economy with: UA = XA1/4YA3/4 UB = XB0.6YB0.4 QX = KX1/2LX1/2 QY = KY1/2LY1/2 and the following initial factor endowment distribution: Consumer A Consumer B Total Capital (K) KA = 1 KB = 1 KT = 2 Labour (L) LA = 1 LB = 2 LT = 3 144 ECON 301 LECTURE #9 ROBINSON CRUSOE ECONOMY In this economy, Robinson Crusoe plays a dual role: he is both the only producer and the only consumer. He can choose to spend his time getting a tan on the beach (consuming leisure) or he can spend it gathering coconuts. The more coconuts he finds, the more he has to eat but the less time he has to work on his tan. We know that Rob will have preferences over coconuts and leisure represented by some utility function. He will also have some production function that represents the relationship between how much he works and how many coconuts he can get. This production function is likely to be one with diminishing marginal returns to labour. Given these two bits of information, we can find out how much Robin works and how many coconuts he will have (his general equilibrium). This is simply the tangency between his production function and his family of utility curves, point (C*,L*) below. Indifference Curves Coconuts Production Function C* Labour L* So, at (C*,L*) the slope of the indifference curve must be equal to the slope of the production function by the standard logic that if they crossed, there would be some other feasible point that is preferred. This means that the marginal product of labour must equal the marginal rate of substitution between leisure and coconuts. 145 Great! But now the story takes a little twist... Since Robby is stranded on an uncharted island, he might get a little...eccentric (insane). So suppose he decides that he is now not simply one person that just happens to be both the producer and consumer for the economy, but rather he is two people (split personality?) one of which is the consumer and the other being the producer. In order to facilitate transactions Bob decides to set up a labour market and a market for coconuts. He sets up a firm, "Bobby C's Inc." and becomes the sole shareholder. This firm considers the prices for labour and for coconuts when deciding how much labour to hire and how many coconuts to produce. The firm is guided by the principles of profit maximization when making these decisions. Robinson, in his role as the worker, will collect income from working which he will use as his constraint as the sole consumer when he decides how many coconuts to buy. Bobby C, in his role as the firm's sole shareholder will collect the firm's profits. In order to keep track of the transactions, Bobby C and Robinson come to an agreement on a currency called sand-dollars, or dollars for short. They also agree that one coconut will be worth one dollar (numeraire good = coconuts). Now all they need to agree on is a wage rate... We want to consider this "economy" after it has been established and has been operating for a period of time and everything has settled into an equilibrium. In this equilibrium state, the demand for coconuts equals the supply of coconuts and the demand for labour equals the supply of labour. Both Bobby C's Inc. and Robinson the Consumer will be making optimal choices given the constraints that they face. THE POINT OF VIEW OF BOBBY C's INC. Each night, Bobby C's Inc. decides how much labour it wants to hire for the next day and how many coconuts it wants to produce. Given a price of $1 for coconuts and a wage rate of labour, w, we can solve the firm's profit maximization problem. We first consider all of the combinations of coconuts and labour that yield a constant level of profits (isoprofit line), . This means that = C wL 146 solving for C, we get C = + wL This formula describes a family of isoprofit lines all combinations of coconuts and labour that yield profits of . Bobby C's Inc. will choose the point where profits are maximized. As always, this means they will choose the point of tangency where the slope of the production function (marginal product of labour) is equal to the slope of the isoprofit line, which is simply w. Coconuts Isoprofit Line Production Function C* Profit = * Labour L* Thus, the vertical intercept of the isoprofit line measures the maximal level of profits measured in units of coconuts. If Bobby C's Inc. generates * dollars in profit, the money can buy * coconuts, since the price of coconuts is $1. Okay, Bobby C's Inc. has done their job. Given the wage, w, it has determined how much labour it wants to hire, how many coconuts it wants to produce, and what level of profits it will generate using profit maximization as an overriding philosophy. Now, Bobby C's Inc. mails out a dividend in the amount of * to its sole shareholder Bobby/Robinson. 147 THE POINT OF VIEW OF ROBINSON THE CONSUMER The very next morning Robinson awakens to find his dividend of * dollars. While eating his shredded coconut cereal, he considers how much he wants to work today and how much he wants to consume. He might just consume his endowment by spending his dividend of * on coconuts and take it easy or he may decide to work a bit today. So he heads down the beach into Bobby C's Inc. for a bit of work. We can describe Robinson the consumer's leisure-consumption choice using the familiar indifference curve analysis. But first, let's figure out Robinson the Consumer's budget constraint. He has his dividend of * without working and he can get w for every unit of labour he offers into the labour market. This income amount is equal to what he can consume in terms of coconuts... C = * + wL Now, Robinson the Consumer will try to achieve the highest indifference curve that he can taking into account his budget constraint... Coconuts Indifference Curve C* Profit = * Labour L* Since labour is a "bad" and coconuts are a good, the indifference curve has a positive slope. If we were to denote the maximum labour as TBAR as we did before, then the distance between TBAR and L* would be the demand for leisure. This is just like the leisure consumption choice except the horizontal axis has the origin reversed. 148 Robinson the Consumer's budget line has a slope of w and passes through the endowment point (0, *). Given the wage rate, Robinson the Consumer chooses optimally how much he wants to work and how many coconuts he wants to consume. At this optimal choice, the marginal rate of substitution between consumption and leisure must equal the wage rate, just as in the standard consumer choice problem. PUTTING THE SPLIT PERSONALITY TOGETHER AGAIN Now if we superimpose the Bobby C's Inc. profit maximization picture onto the Robinson the Consumer picture we get... Indifference Curve Coconuts Budget Line / Isoprofit Line Production Function C* Consumption Optimum Production Optimum Profit = * Labour L* It turns out that the schizophrenic Robinson did exactly what the "sane" Robinson did. This optimal solution is the same one as we started with before when Robinson made all the decisions at once. Using the market system results in the same outcome as choosing the consumption and production plans directly. This is directly due to the fact that the MRS between consumption and leisure and the MPL are equal to w. This assures us that the MRS = the MPL and thus, the slope of the indifference curve and the slope of the production function are equal. 149 Let's do a simple example... Suppose Robinson Crusoe has a utility function and a production function defined as: URC = URC (C , ) = C1/2(24 - L)1/2 C = f(L) = L We can find Robinson's optimal consumption-leisure choice by equating his MRS between consumption and leisure with his MP of leisure... First, let (24 L) = , where represents leisure MRS = C-1/21/2 = L C1/2-1/2 C MPL = 1 So, (24 - L) = 1 C (24 - L) = C (24 - C) = C 2C = 24 (24 - L) C C = 12 = L and = (24 - L) = 12 Now let's do one that is a little more challenging. Suppose Robinson Crusoe has a utility function and a production function defined as: URC = URC (C , ) = C1/2(18 - L)1/2 C = f(L) = 4 L1/2 Now remember Robinson wants to maximize his leisure subject to his maximum profit First, let (18 ) = L or alternatively, (18 L) = (meaning that Robinson sleeps 6 hours per night and this sleep time is counted as neither labour nor leisure). 150 Max PC C + w since w = the opportunity cost for Robinson. Max 4 PC (L1/2) + w (18 - L) FOCL 2 PC L-1/2 - w = 0 L-1/2 = w 2 PC L1/2 = 2 PC w L = 4 PC2 w2 Now, profit is maximized at the point where total revenue is equal to total cost (i.e. marginal profit is zero). PC C = w PC C = w (18 L) Let w = 1 (Labour = numeraire) PC (4 L1/2) = 18 4 PC2 4PC (2PC) = 18 4 PC2 8PC2 + 4 PC2 = 18 12PC2 = 18 PC2 = 3 / 2 PC = 3 / 2 Now, remember that we had L* = 4 PC2 = 4(3 / 2)2 = 6 w2 (1)2 C = 4L C* = 46 = (18 L) * = (18 6) = 12 How do we know that the Consumption / Leisure choice is, in fact, the one which maximized Robinson's utility? 151 Let's see how! Recall, Robinson's utility was defined as... URC = URC (C , ) = C1/2(18 - L)1/2 C = f(L) = 4 L1/2 1. First, we figure out the supposed maximum utility level based on the optimal quantities of leisure and consumption that we determined above. We had... C* = 46 * = (18 6) = 12 URC* = (46)1/2(12)1/2 URC* = (3.13016916)(3.464101615) URC* = 10.84322404 2. Next, we choose a value for L that is slightly higher than the supposed optimum, say 6.1, (and calculate the associated C and ). We then use this point on the production function to see if utility is higher or lower. When L = 6.1 we get... C* = 46.1 and using this point... URC1 = (46.1)1/2(11.9)1/2 URC1 = (3.1431308)(3.449637662) URC1 = 10.84266238 which is less than URC* = 10.84322404. 3. Next, we choose a value for L that is slightly lower than the supposed optimum, say 5.9, (and calculate the associated C and ). We then use this point on the production function to see if utility is higher or lower. When L = 5.9 we get... C* = 45.9 and using this point... 152 * = (18 5.9) = 12.1 * = (18 6.1) = 11.9 URC2 = (45.9)1/2(12.1)1/2 URC2 = (3.117044472)(3.478505426) URC2 = 10.84265611 which is less than URC* = 10.84322404. Now let's do one that is even more challenging. Suppose Robinson Crusoe has a utility function and a production function defined as: URC = URC (C , ) = C1/2(16 - L)1/2 C = f(L) = 8 L1/2 Now remember Robinson wants to maximize his leisure subject to his maximum profit. First, let (16 ) = L or alternatively, (16 L) = (meaning that Robinson sleeps 8 hours per night and this sleep time is counted as neither labour nor leisure). Max PC C w since w = the opportunity cost for Robinson. Max 8 PC (L1/2) w (16 - L) FOCL 4 PC L-1/2 + w = 0 L-1/2 = - w__ 4 PC L = __ 16PC2___ w2 Now, profit is maximized at the point where total revenue is equal to total cost (i.e. marginal profit is zero). PC C = w PC C = w (16 L) 153 Let w = 1 (labour = numeraire) PC (8 L1/2) = 16 16 PC2 8 PC (4PC) = 16 16 PC2 48 PC2 = 16 PC2 = 1 / 3 PC = 1 / Now, remember that we had L = __ 16PC2__ = 16 / 3 w2 C = 8 L1/2 C* = 8 (16 / 3)1/2 C* = 18.47520861 * = (16 16 / 3) = 32 / 3 Homework #3: How do we know that the Consumption / Leisure choice is, in fact, the one which maximized Robinson's utility? (Use L = 5.4 and L = 5.3 in your higher/lower calculations). PARETO OPTIMALITY THEORY EQUILIBRIUM VS. OPTIMALITY Vilfredo Pareto (1848 1923) was born in Paris to an Italian father and a French mother. His professional background was in engineering. He became interested in Economics after a successful career in industry. In 1893, he was appointed as a professor of political economy (the post previously held by Leon Walras) at the Universite de Lausanne in Switzerland. In 1906 he retired to study sociology. We are concerned with Pareto Theory in the context of what it adds to Walrasian General Equilibrium analysis. Recall that Walrasian General Equilibrium theory incorporates three equilibrium concepts, namely, consumer equilibrium, producer equilibrium, and market 154 equilibrium. The key that holds these three equilibrium concepts together is the fundamental concept of market prices. The question of Walrasian Theory is [1] Existence of General Equilibrium: how do we ascertain that there exists an equilibrium price vector which can simultaneously satisfy the following conditions: all consumers are in equilibrium, all producers are in equilibrium, all markets are in equilibrium? [2] Computation of General Equilibrium: if we can ascertain that there exists such an equilibrium price, then how do we proceed to find it? The question of Pareto Theory is ... after we get an equilibrium price vector ... [1] Is this equilibrium price vector the "best" one for the economy? [2] Is the Walrasian general equilibrium solution the "best" way to allocate resources in the economy? [3] Can we get the "best" allocation of resources in the economy without going through a general equilibrium solution? [4] Can we find all of the "best" allocations of resources in the economy? Now, to answer any of these questions, we need to understand what is meant be the word "best" and this is exactly the question which Pareto Optimal Theory is trying to clarify. The theory suggests the notion of "Pareto Optimality" or "Pareto Efficiency" which can be stated as follows: An allocation is Pareto Optimal if no one can be made better off without making someone worse off. So this is an awkward statement, isn't it? To clarify, if someone can be made better off while no one else is worse off then the allocation can be improved. Of course, if the allocation can be improved then it cannot be the best allocation, and hence, is not Pareto Optimal. 155 On the other hand, if someone can be made better off only when someone else has to be made worse off then the allocation cannot be further improved. Of course, if the allocation cannot be further improved then, by definition, it is the best allocation i.e. it is Pareto Optimal. So let's incorporate the concept of Pareto Optimality into the pure exchange economy. It is essentially the same problem that we had before except now we need to satisfy the following two conditions: [1] The quantities of goods allocated must equal the quantity of goods available (feasibility). XA + XB = X YA + YB = Y (1) (2) [2] The allocation must be Pareto Optimal in the sense that no one can be made better off without making someone else worse off. EDGEWORTH BOX First, we need to know what an Edgeworth Box is... We will do this a little differently than most by using a three step graphical representation. This is so you will understand the configuration of the box, as well as, the dimensions and labelling features. The first step is to imagine our two consumers, A and B, with their indifference curves in our normal depiction of a quadrant space. Next, we take the axes for consumer B and flip them around the point Z in the first diagram. This results in the second diagram. In the second diagram, we flip consumer B's axes again but this time upward (around the point S in diagram 2 below. This results in diagram three which is the Edgeworth Box. Y A YB Imagine a scenario where we have the two consumers in a similarly configured quadrant space, as below. ICB IC A Z XB XA 156 S YB YA ICB IC XB XA A Then we turn consumer B's axes up, flipping upwards around the point S. In the box below we have the finished product...an Edgeworth box. YA XB ICB OB ICA OA Y B XA This box must be of the dimensions described in equations (1) and (2) above to represent the feasible allocations in the economy. Let's consider what this means... 157 Y = YA + YB XB OB YB Endowment Point YA OA XA X = XA + XB The box must have the following properties: [1] The total dimension of the box is X by Y. The size of the box is determined by the total amounts of both goods available in the economy. As a result, we cannot have a "feasible allocation" outside the boundary of the box. This does not say that consumer demands are not well defined outside the boundary of the box...they are well defined just not feasible given the economy's resources. [2] One person, say A, has their origin at the southwest corner (bottom left) of the box while the other person, say B, has their origin at the northeast corner (top right) of the box. This is what we showed above graphically. [3] At any point of the box, either on the interior or on the boundary, now represents a "feasible allocation" (XA, XB, YA, YB) which satisfy equations (1) and (2) above. Note that corner OA represents the extreme allocation which results in everything in the economy belonging to consumer B. Similarly, OB represents the extreme allocation which results in everything in the economy belonging to consumer A. Further, note that the initial endowment point (XA, YA, XB, YB) defines the dimensions of the box and serves as the starting point for trade to occur. So, we can formulate the allocation problem in terms of the Edgeworth Box as: 158 To find a point of the Edgeworth Box such that no one can be made better off without making someone else worse off. The way we want to think about "better off" and "worse off" in our analytical formulation of this problem is by using the basic tools of microeconomic analysis, namely, the utility function and indifference curve analysis. BETTER OFF "Better off" means a higher utility level or a higher indifference curve. "No one can be made better off" means that the indifference curve cannot go any higher... YA IC A WORSE OFF "Worse off" means a lower utility level or a lower indifference curve. "No one can be made worse off" means that the indifference curve cannot go any lower... YB IC B XA XB Putting these two notions of "better off" and "worse off" together implies that one indifference curve cannot go any higher while the other indifference curve cannot go any lower (in the opposite direction). Geometrically, this implies that the two indifference curves must be positioned such that they are tangent to each other. The allocation problem can now be restated as: To find a point of the Edgeworth Box such that the two indifference curves are tangent to each other. 159 So how do we draw this thing? As we said above, we begin with consumer A's origin at the southwest corner and person B's origin at the northeast corner. Then we draw the two indifference curves ICA, ICB, in opposite directions (for Cobb-Douglas utility) because one set of coordinates is right side up and the other is right side down. YA XB OB ICB ICA OA YB XA Now, there are a set of points in the CORE that give rise to a final Pareto Optimal allocation. The core is the set of Pareto Optimal points between the original indifference curve positionings. We may eventually arrive at a Pareto Optimal allocation where the indifference curves are tangent to one another... YA XB Price Ratio OB Pareto Optimal solution ICB OA ICA YB XA 160 All right, but how do we get this tangency condition in terms of the slopes of the indifference curves? The tangency condition of the two indifference curves implies that they must have the same slopes, and hence, the same marginal rates of substitution. Indifference Curve of is tangent to Consumer A Consumer B Indifference Curve of Slope of Indifference Curve = of Consumer A Slope of Indifference Curve of Consumer B Simply means that: MRSA = MRSB Now, we can finally rephrase our allocation problem in terms of the basic concept of marginal rates of substitution as follows: To find a point in the Edgeworth Box such that MRSA = MRSB Is this going to give us a unique allocation? In other words, is there only one Pareto Optimal allocation? Not necessarily! There may be more than one Pareto Optimal allocation in the Edgeworth Box. That is, it is possible (indeed, likely) that there are several pairs of indifference curves tangent to each other at various points. In this case, the curve that connects all of these Pareto Optimal points is called a contract curve. 161 YA XB ICB1 Price Ratio Pareto Optimal solution OB Pareto Optimal solution ICA1 Price Ratio ICA2 ICB2 OA Y B XA So the curve connecting the origins and all points of tangency between the two families of indifference curves is the Contract Curve. CONTRACT CURVE How do we derive the contract curve? Let's suppose that consumer A and consumer B have utility functions that are both Cobb-Douglas form as follows: UA = XAYA1- MRSA = __YA__ (1-) XA UB = XBYB1- MRSB = __YB__ (1-) XB And we know that the Pareto Optimal allocations must satisfy MRSA = MRSB, so... __YA__ = __YB__ (1-) XA (1-) XB and we want our contract curve coordinates in terms of consumer A so we can convert consumer B's demands into: YB = Y YA XB = X XA 162 So... ___ YA__ = __ Y YA __ (1-) XA (1-) X XA YA__ = __(1-) XA (1-) Let Z = (1-) (1-) YA (X XA) = Z (Y YA ) XA YA X YA XA + Z YA XA = Z Y YA [X (1 Z) XA] = Z Y YA = _____Z Y________ X (1 Z) XA Now the "bow" is either the second derivative of this mess above or we can compare on the main diagonal to estimate the shape of the contract curve. How do we do that? We compare the ______ and the ______ terms. (1 ) (1 ) Suppose that the ______ > ______ then the indifference curve for A is steeper than (1 ) (1 ) the indifference curve for B on the main diagonal, this leads to the slopes being equal below the main diagonal. In the reverse case, the opposite is true. Y YA __ X XA 163 YA XB OB ICA ICB OA YB XA We can draw the utility levels of the contract curve. By definition, the contract curve is a collection of all of the possible Pareto Optimal allocations in the economy. Each point on the contract curve is a Pareto optimal allocation from which we can calculate the corresponding utility levels UA and UB. If we plot these pairs of Pareto optimal utility levels, we have the utility possibility frontier for the economy (UPF). U B UPF UA The difference between the contract curve and the UPF is that while both refer to Pareto optimal allocations, they use different units of measurement. 164 [1] A contract curve yields allocations (XA, XB, YA, YB) in terms of goods. [2] A UPF yields allocations (UA, UB) in terms of utility levels. HOMEWORK [1] Draw the Edgeworth Box representations of your homework solutions to the pure exchange questions in Lecture #8. [2] Find the contract curve and draw it for the following two consumers in a pure exchange economy: UA = XA + YA1/2 Don't forget... Homework #3: How do we know that the Consumption / Leisure choice is, in fact, the one which maximized Robinson's utility? (Use L = 5.4 and L = 5.3 in your higher/lower calculations). SEE PAGE 154 . UB = XB + YB 165 ECON 301 LECTURE #10 PARETO OPTIMAL THEORY PURE EXCHANGE ECONOMY EXAMPLE Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has a square root utility function while Consumer B has a CobbDouglas utility function with = 0.25, = 0.75, and = 1. There is one unit of each good allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = XB = X = + = 1 GOOD Y YA = YB = Y = + = 1 We solve this particular pure exchange economy as follows: Let's start with Consumer A. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: UA = UA(XA,YA) = X1/2Y1/2 MA = PX XA + PY YA = PX + PY We can figure out consumer A's marginal rate of substitution as: MRSA = MUXA MUYA = 0.5 X-1/2Y1/2 0.5 X1/2Y-1/2 = YA XA At the consumer equilibrium, the two equations consumer A needs to satisfy are: MRSA = YA = PX XA PY PX XA + PY YA = MA 166 (1) (2) Rearranging (1) we get: XA PX = YA PY (3) Meaning we can get demands for XA and YA by subbing (3) into (2) as follows: PX XA + PX XA = MA 2 PX XA = MA XA = M A 2 PX (4) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (4) to get: XA = PX + PY 2 PX XA = 1/8 + _3PY__ 8 PX PY YA + PY YA = MA 2 PY YA = MA YA = M A 2 PY (6) (5) and we know that MA = PX + PY is the endowment income of consumer A, so we sub this in for the MA in (6) to get: YA = PX + PY 2 PY YA = 3/8 + _PX__ 8 PY (7) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment income that constrains their utility as follows: UB = UB(XB,YB) = X1/4Y3/4 167 MB = PX XB + PY YB = PX + PY We can figure out consumer B's marginal rate of substitution as: MRSB = MUXB MUYB = 0.25 X-3/4Y3/4 0.75 X1/4Y-1/4 = YB 3XB At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB = PX 3XB PY Rearranging (1B) we get: PX XB + PY YB = MB 3 XB PX = YB PY (1B) (2B) (3B) Meaning we can get demands for XB and YB by subbing (3B) into (2B) as follows: PX XB + 3 PX XB = MB 4 PX XB = MB XB = MB 4 PX (4B) and we know that MB = PX + PY is the endowment income of consumer B, so we sub this in for the MB in (4B) to get: XB = PX + PY 4PX XB = 3/16 + _PY__ 16 PX 1/3 PY YB + PY YB = MB 4/3 PY YB = MB 168 (5B) YB = 3MB 4 PY (6B) and we know that MB = PX + PY is the endowment income of consumer B, so we sub this in for the MB in (6B) to get: YB = 3 ( PX + PY) 4 PY YB = 3/16 + _9PX__ 16 PY (7B) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, X = XA + XB = 1/8 + _3PY__ + 3/16 + _PY__ 8 PX 16 PX = 5/16 + _7PY__ 16 PX and we know that the fixed supply of X in the economy is the total endowment of X... X = X so, 5/16 + _7PY__ = 1 16 PX and solving the market equilibrium, we get the following equilibrium price ratio... _7PY__ = 11/16 16 PX or, 7PY = 11PX _PX__ = __7__ PY 11 We can do the same thing in the market for good Y. Okay, let's do it! 169 (8) Now that we have the individual demands for each consumer for both goods, we can do our horizontal summation to figure out the market demand. Recall, Y = YA + YB = 3/8 + _PX__ + 3/16 + _9PX__ 8 PY 16 PY = 9/16 + 11PX__ 16 PY and we know that the fixed supply of Y in the economy is the total endowment of Y... Y = Y so, 9/16 + _11PX__ = 1 16 PY and solving the market equilibrium, we get the following equilibrium price ratio... _11PX__ = 7/16 16 PY or, 11PX = 7PY _PX__ = __7__ PY 11 (8B) So now that we have the equilibrium price ratio and the individual consumer demands (and the market demands as a result), we can find the equilibrium quantities demanded by the individuals, A and B, by subbing the price ratio into the individual demand functions. XA = 1/8 + _3PY__ 8 PX = 1/8 + 3/8 (11/7) XA* = 5/7 170 YA = 3/8 + _PX__ 8 PY = 3/8 + 1/8 (7/11) YA* = 5/11 XB = 3/16 + _PY__ 16 PX = 3/16 + 1/16 (11/7) XB* = 2/7 YB = 3/16 + 9PX__ 16 PY = 3/16 + 9/16 (7/11) YB* = 6/11 As a verification (or check) of the market equilibrium condition, we add the individual consumer demands (X = XA + XB and Y = YA + YB) and they should sum to the fixed supply in terms of endowments of the goods. XA + XB = 5/7 + 2/7 XA + XB = 1 XA + XB = X YA + YB = 5/11 + 6/11 YA + YB = 1 YA + YB = Y So, in both the market for X and the market for Y, demands are equal to supplies and we have market clearing (equilibrium). 171 Now, let's look at this in an Edgeworth Box diagram... Y=1 O B Core ICA IC B Here we have our original situation...Person A has their endowment of (1/4, 3/4) and Person B has their endowment of (3/4, 1/4). Both consumers have an indifference curve through the endowment point. This means that along these indifference curves each consumer is just as happy without trade. X=1 OA Now let's consider how we get from this original situation to our final Pareto optimal competitive equilibrium... Y=1 ICA 2/7 O B Slope = -7/11 5/11 6/11 Now we can see that from the original endowment these consumers (guided by their preferences, i.e. indifference curves) trade goods X and Y such that the prevailing price ratio, PX / PY, is 7/11 and the tangency line separates both indifference curves at the point of tangency (i.e. where PX / PY = MRSA = MRSB). This point of tangency turns out to be where A A B B (X , Y , X , Y ) = (5/7, 5/11, 2/7, 6/11). ICB OA 5/7 X=1 172 Alright, now let's see if the contract curve is bowed in the right direction... We know that consumer A and consumer B have utility functions that are both Cobb-Douglas form as follows: UA = XA1/2YA1/2 MRSA = __1/2YA__ 1/2 XA UB = XB1/4YB3/4 MRSB = __1/4YB__ 3/4 XB And we know that the Pareto Optimal allocations must satisfy MRSA = MRSB, so... __1/2YA__ = __1/4YB__ 1/2 XA 3/4 XB and we want our contract curve coordinates in terms of consumer A so we can convert consumer B's demands into: YB = Y YA So... XB = X XA _1/2__ YA__ = __1/4 1/2 XA 3/4 YA__ = __1/8 XA 3/8 Y YA __ X XA Y YA __ X XA YA (X XA) = 1/3 (Y YA ) XA YA X YA XA + 1/3 YA XA = 1/3 Y YA [X 2/3 XA] = 1/3 Y YA = _____1/3 Y________ X (2/3) XA Now the "bow" is either the second derivative of this mess above or we can compare on the main diagonal to estimate the shape of the contract curve. How do we do that? We compare the ______ and the ______ terms. (1 ) (1 ) 173 We know that ___1/2___ > ___1/4___ thus the indifference curve for A is 1/2 3/4 steeper than the indifference curve for B on the main diagonal, this leads to the slopes being equal below the main diagonal. In the reverse case, the opposite is true. YA XB OB ICA ICB OA Y B XA So now we can add the contract curve to our Edgeworth Box diagram that represents the Pareto Optimal competitive equilibrium solution to our example... 174 Y=1 ICA 2/7 OB Now we can see that from the main A diagonal, IC is steeper as it crosses the main diagonal than ICB is. This means that the Pareto Optimal set (contract curve) is bowed down from the main diagonal and connects the origins of the consumers. Very nice! Now we know that our solution is in the right range of the box to be on the contract curve (MRSA = B MRS ). This completes our Edgeworth Box representation of this example. Slope = -7/11 5/11 6/11 ICB OA 5/7 X=1 Now, what if there is a quasi-linear consumer? They don't have any wealth effects so wouldn't the Contract Curve look different if we had a quasi-linear agent? I'm very glad that someone asked me that!!!! PARETO OPTIMAL THEORY PURE EXCHANGE ECONOMY EXAMPLE: QUASI-LINEAR PREFERENCES Consider a simple pure exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has a Perfect Substitutes utility function defined by U(X,Y) = 5X +2Y while Consumer B has Quasi-linear preferences defined by U(X,Y) = 2X + 4Y1/2 . Each good is allocated between the two consumers according to the following endowment distribution: Consumer A Consumer B Total GOOD X XA = 12 XB = 3 X = 15 175 GOOD Y YA = 1 YB = 14 Y = 15 We want to solve for the general equilibrium price ratio, and report the equilibrium quantities of both goods demanded for each consumer. We also wish to draw the final Edgeworth Box diagram for this economy. UA = 5X + 2Y A = (XA , YA) = (12,1) We can figure out consumer B's marginal rate of substitution as: MRSA = MUXA MUYA = __5__ 2 At the consumer equilibrium, the equation consumer A needs to satisfy is: MRSA = _5_ = PX 2 PY (1B) So now that we have the equilibrium price ratio. Since only the ratio matters we can use the relationship PX = PY in consumer B's demands below (we can derive consumer A's demands from consumer B's...more in a second) Now let's turn our attention to Consumer B. This consumer has a utility function that allows us to determine their marginal rate of substitution and an endowment that constrains their utility as follows: UB = 2X + 4Y1/2 B = (XB , YB) = (3,14) We can figure out consumer A's marginal rate of substitution as: MRSB = MUXB MUYB = ___2___ 2Y-1/2 = YB 176 At the consumer equilibrium, the two equations consumer B needs to satisfy are: MRSB = YB 1/2 = PX PY YB = PX2 PY2 Subbing (1) into (2) we get: PX XB + PY YB = MB PX XB + PX2 = MB PY XB = MB - PX PX PY (3) (1) (2) and we know that MB = 3 PX + 14 PY is the endowment income of consumer B, so we sub this in for the MB in (4) to get: XB = 3 PX +14 PY - PX PX PY XB = 3 + _14 PY__ - PX PY PX XB = 3 + _14PY__ - PX PY PX = 3 + 28_ 5 / 2 5 = 3 + 56 / 10 25 / 10 XB* = 6.1 YA = PX2 PY2 YA = 52 22 YB* = 6.25 177 (4) XA = X XB* XA* = 8.9 YA = Y YB* YA* = 8.75 This example illustrates that the individual with a linear indifference curve (consumer A) will dictate the price ratio, while the individual with the non-linear indifference curve will dictate the final demands in the pure exchange economy. Now, what about the Edgeworth Box for this problem? Recall, in the first example (2 Cobb-Douglas consumers) we derived the contract curve in terms of YA mostly because this gives us a familiar reference point in our usual north-east quadrant space. With quasi-linear preferences, the contract curve will be the level of consumption of the quasi-linear good for the individual that has the quasi-linear preferences. So, in the example above consumer B is quasi-linear in good Y. Thus, the value of YB = 6.25 is the contract curve on the interior of the box. We have said in the past that the contract curve connects the origins...and this case is no exception. Thus, the contract curve for this example has three components to it. The first is from the origin for Person B along the Y-axis for Person B up to the point where YB = 6.25. The second section moves horizontally across the box until reaching the Y-axis for Person A and the final section of the contract curve is along the Y-axis for Person A all the way to the origin for Person A. 178 YA XB ICA 6.1 3 OB 8.75 Contract Curve 6.25 Price Ratio = 2.5 ICB 1 OA 8.9 12 Y B 14 XA HOMEWORK [1] Create your own (different) example of a pure exchange economy by selecting the parameters , (1- ), , (1-), XA, YA, XB, YB, such that: Consumer A Consumer B Total GOOD X XA = ? XB = ? X = XA + XB = 1 GOOD Y YA = ? YB = ? Y = YA + YB = 1 And the consumer's utility functions are Cobb-Douglas constant returns to scale form... [a] Solve the competitive equilibrium for this pure exchange economy (i.e. find the equilibrium price vector and the equilibrium demands for each good by each consumer). [b] Represent your equilibrium solution in an Edgeworth Box diagram similar to the one presented in class. [c] Find the expression for the contract curve and (in a new Edgeworth Box) draw a diagram showing your Pareto Optimal competitive equilibrium solution. 179 ECON 301 LECTURE #11 PARETO OPTIMAL THEORY PRODUCTION ECONOMY Now let's extend our Pareto Optimal concept to a production economy... Two Goods Two Factors Two Consumers Good X Capital (K) Consumer A Good Y Labour (L) Consumer B We need to consider both the exchange efficiencies and the production efficiencies in this framework. EXCHANGE EFFICIENCY Given exogenously fixed quantities of aggregate goods endowments X and Y, the pure exchange economy provides Pareto Optimal allocations of goods X between consumers A and B. PRODUCTION EFFICIENCY Given exogenously fixed quantities of aggregate factor endowments K and L, the production side will provide Pareto Optimal allocations of factors K and L between two producers, Producer X & Producer Y To combine these efficiencies into our Production Economy we use the following general strategy: [1] Take the quantities of aggregate factor endowments K and L as exogenously given. [2] Solve the production side for Pareto Optimal factor allocations, and hence the corresponding output levels X and Y on the PPF (production possibility frontier). This ensures the production efficiency in the economy. [3] Using the output levels of X and Y produced, we then solve the exchange economy to generate Pareto Optimal goods allocations, and hence the corresponding utility possibility frontier (UPF). This ensures the consumption (or exchange) efficiency in the economy. [4] We then impose additional conditions so that the production allocations are compatible with the exchange allocations. This will ensure the overall efficiency in the economy. Okay. 180 Now, analytically, we can describe the Pareto Optimal Production economy using 9 equations outlining the Pareto Optimal conditions of the economy as follows: Given aggregate factor endowments, K and L, we want to find... allocations (XA, XB, YA, YB) of goods allocations (KX, KY, LX, LY) of factors which satisfy the following conditions. KX + KY = K LX + LY = L MRTSX = MRTSY (= MRTS) (1) (2) (3) These first three equations (1), (2), and (3) make up the production side efficiencies. This gives us our PPF. XA + XB = X YA + YB = Y MRSA = MRSB (= MRS) (4) (5) (6) These three equations (4), (5), and (6) make up the consumption side efficiencies. This gives us our UPF. And, of course we need the production functions... X = f(KX, LX) Y = g(KY, LY) (7) (8) As well, we need the mysterious 9th equation that ensures the Overall Efficiency Requirement... MRS = MRT (9) While the consumption side uses MRS and the production side uses MRTS to determine efficient allocations, we cannot equate these measures to ensure overall efficiency. This stems from the fact that we cannot compare the two different object categories, namely MRS which is measured in terms of goods 181 and MRTS which is measured in terms of factors. Instead we must equate MRS = MRT which are both measured in terms of goods. So let's investigate the logic behind the overall efficiency requirement that MRS = MRT... This requirement means that the consumption allocations must be compatible with the production allocations. MRS (marginal rate of substitution) specifies the amount of one good, say good Y, which can be substituted for another good, say good X, along the same indifference curve. For example, MRS = 1 means that to reduce one unit of good X, the consumer requires 1 additional unit of good Y in order to be on the same indifference curve. MRT (marginal rate of transformation) specifies the amount of one good, say good Y, which can be substituted for another good, say X, along the same PPF. For example, MRT = 2 means that to reduce one unit of X, the economy has the resources to produce 2 more units of good Y and is still on the same PPF. The consumer is thus better off since the consumer requires only 1 additional unit of good Y and yet the economy can produce 2 additional units of good Y. In other words, the allocation associated with MRS MRT is NOT Pareto Optimal. 182 THE PRODUCTION SIDE OF THE MODEL First we solve the equations that make up the production side efficiencies: KX + KY = K LX + LY = L MRTSX = MRTSY (= MRTS) (1) (2) (3) These solutions will give us all of the Pareto Optimal allocations (KX, KY, LX, LY) of factors. Next, we substitute these factor allocations into the production functions... X = f(KX, LX) Y = g(KY, LY) (7) (8) and we obtain all possible corresponding output levels X and Y. That is, we first derive the factor contact curve and then construct the corresponding PPF. In other words, the production possibility frontier represents the information captured in the optimal solutions of the five equations (1), (2), (3), (7), and (8) relating to the productions side of the economy. Y PPF X So this is the same PPF you heard about in your introductory economics courses except for the following difference...previously, you were asked to accept the PPF as exogenously given while here, the PPF is analytically derived from the solution of the Pareto Optimal allocation problem. We have reached the point 183 now where we do not need to take this result (the PPF) as a primitive or primal concept but can use the basic concept of Pareto Optimality and derive the PPF as a consequence. Pretty cool, huh? Now that we have the production side of the economy all solved and summarized in the PPF, we can proceed to solve the remaining four equations to get the consumption efficiency and overall efficiency. XA + XB = X YA + YB = Y MRSA = MRSB MRS = MRT (= MRS) (4) (5) (6) (9) Solving these we find all possible Pareto optimal allocations (XA, XB, YA, YB) of goods. That is, we solve the pure exchange economy for every point on the PPF while making sure that the overall efficiency condition is also satisfied: [1] We move along the PPF to maintain the production efficiency. [2] For each point on the PPF, we can solve the pure exchange economy to obtain consumption efficiency at that point. Since the output levels X and Y vary as we move along the PPF, the dimensions of the Edgeworth box will vary accordingly. Each point on the PPF has a different Edgeworth box having different dimensions. Of course, each associated pure exchange solution provides a contract curve of Pareto Optimal allocations of goods (and the corresponding UPF). [3] To ensure the overall efficiency condition, we need to find a point on the contract curve at which the common MRS (slope of the indifference curves) is equal to the slope of the PPF (MRT). 184 Y MRT MRT PPF X {Common slope of ICs} = {Slope of the production possibility frontier (PPF)} MRS = MRT [4] Each point of the PPF produces an individual utility possibility frontier (UPF) since the Pareto Optimal solutions of the pure exchange economy (contract curve) can be converted to utility levels. Since there are infinitely many points on the PPF, there will be infinitely many corresponding UPFs. UB Grand UPF UA 185 It can be shown that the Pareto optimal solution to the Production economy results in the overall utility possibility frontier which is the envelope of all of these individual UPFs. We call this envelope the Grand UPF or Grand utility possibility frontier. All right then...what about an example. Let's consider the square root economy with unit aggregate factor endowments (K = L = 1). We solve the Pareto Optimal allocation problem for this simple economy as follows: PRODUCTION EFFICIENCY The problem is to find factor allocations (KX, KY, LX, LY) such that KX + KY = K LX + LY = L MRTSX = MRTSY (= MRTS) (1) (2) (3) Applying these general conditions to the specific problem in our square root example, we have KX + KY = 1 LX + LY = 1 LX = LY KX KY which can be solved for the following production contract curve: LX = KX L Y = KY To construct the production possibility frontier, we first substitute Pareto Optimal allocations (KX, KY, LX, LY) into the production function for good X... X = (LX KX)1/2 = (KX KX)1/2 = KX (1) (2) (3) 186 Similarly for good Y... Y = (LY KY)1/2 = (KY KY)1/2 = KY Then we can extract the following relationship between X and Y from the above information: X + Y = KX + KY =K =1 The production side of the model is thus solved with linear upward sloping contract curve (slope = 1) and a linear downward sloping PPF (MRT = slope = 1). L 1 Production Contract Curve K 1 187 CONSUMPTION EFFICIENCY The problem is to find allocations (XA, XB, YA, YB) of goods such that XA + XB = X YA + YB = Y MRSA = MRSB (= MRS) (4) (5) (6) Applying these general conditions to our specific square root example, we get XA + XB = X YA + YB = Y YA = YB XA XB (4) (5) (6) for each point on the PPF above. For example, we will solve the pure exchange model at the following three points on the PPF: Point A Point B Point C X = 1/3 X = 1/2 X = 3/4 Y = 1 1/3 = 2/3 Y = 1 1/2 = 1/2 Y = 1 3/4 = 1/4 For each point, we obtain one contract curve and one individual utility possibility frontier (UPF) as follows : [a] At point A on the PPF, we have the following equations: XA + XB = 1/3 YA + YB = 2/3 YA = YB XA XB which can be solved for the equations of the contract curve YA = 2XA YB = 2XB 188 To find the equation of the UPF at point A, we substitute Pareto Optimal allocations (XA, XB, YA, YB) into the utility functions UA = (XA YA)1/2 = (XA 2XA)1/2 = XA (21/2) UB = (XB YB)1/2 = (XB 2XB)1/2 = XB (21/2) and then extract the following functional relationship between utility levels UA and UB: UA + UB = XA (21/2) + XB (21/2) UA + UB = (XA + XB)(21/2) UA + UB = (1/3)(21/2) UA + UB = (21/2) 3 [b] At point B on the PPF, we have the following equations: XA + XB = 1/2 YA + YB = 1/2 YA = YB XA XB which can be solved for the equations of the contract curve YA = XA YB = XB To find the equation of the UPF at point B, we substitute Pareto Optimal allocations (XA, XB, YA, YB) into the utility functions UA = (XA YA)1/2 = (XA XA)1/2 = XA UB = (XB YB)1/2 = (XB XB)1/2 = XB and then extract the following functional relationship between utility levels UA and UB: UA + UB = XA + XB 189 UA + UB = X U A + UB = [c] At point C on the PPF, we have the following equations: XA + XB = 3/4 YA + YB = 1/4 YA = YB XA XB which can be solved for the equations of the contract curve YA = 1/3XA YB = 1/3XB To find the equation of the UPF at point C, we substitute Pareto Optimal allocations (XA, XB, YA, YB) into the utility functions UA = (XA YA)1/2 = (XA 1/3XA)1/2 = XA (1/3)1/2 UB = (XB YB)1/2 = (XB 1/3XB)1/2 = XB (1/3)1/2 and then extract the following functional relationship between utility levels UA and UB: UA + UB = XA (1/3)1/2 + XB (1/3)1/2 UA + UB = (XA + XB)(1/3)1/2 UA + UB = (3/4)(1/3)1/2 UA + UB = (3)1/2 4 190 Y 1 MRS = 2 2/3 MRS = 1 MRS = 1/3 PPF (slope = -1) 1/3 1 X OVERALL EFFICIENCY Among the three points A, B, and C on the PPF, only point B satisfies the overall efficiency condition. Point A Point B Point C MRS = 2 MRS = 1 MRT = 1 MRT = 1 MRS = 1 MRT = 1 3 That is, while production efficiency and consumption efficiency hold separately at all three of these points, only at point B are production and consumption efficiency compatible with each other (i.e. MRS = MRT = 1). In the diagram below, this overall efficiency condition can be seen as the UPF corresponding to point B is higher than that of the other two points. In fact, it can be shown that in this special case, the UPF at point B is also the Grand UPF. 191 UB 2/3 UPF (point B) = GRAND UPF UPF (point A) 3/4 UPF (point C) 3/4 2/3 UA SOCIAL WELFARE FUNCTIONS So how many solutions are there for the Pareto Optimal Production Economy? For each point on the PPF, we have to solve one pure exchange economy to get [1] one Edgeworth box, [2] one contract curve, [3] one utility possibility frontier (UPF). There will be at least one point on the contract curve at which the overall efficiency condition holds (MRS = MRT). Consequently, there will be at least one point on the UPF which will also be on the Grand UPF. Since there are infinitely many points on the PPF, there will be [1] infinitely many corresponding UPFs, [2] infinitely many points satisfying the overall efficiency condition, [3] infinitely many points on the Grand UPF (all of which are equally good). Thus, there are infinitely many solutions for the Pareto Optimal Production Economy. Furthermore, all of these solutions are Pareto Optimal. In other words, the solution to the Pareto Optimal Production Economy problem is indeterminate. 192 Since there are infinitely many solutions to this problem...which one should we choose? Since this model has infinitely many solutions, all of which are equally efficient, the selection of a specific solution will depend on other criteria, such as social values and judgments... How do economists formulate an analytical framework that includes social values and judgments? We use a conceptual framework that incorporates a social welfare function SWF = W(UA, UB) to summarize criteria such as social values and judgments. A social welfare function is thus a social choice on individual welfare (policy decision). This distinction differs somewhat from the concept of our usual utility function. The concept of a utility function defines individual choices over commodities UA = UA(XA, YA) UB = UB(XB, YB) while the concept of a social welfare function defines social choices over individual utility levels SWF = W(UA, UB) This is the idea that differentiates the desirability of utility distributions. For instance, one of the Pareto Optimal solutions to the problem may be that consumer A has almost everything in terms of utility while consumer B has next to nothing. This is a Pareto efficient solution that may not be socially desirable. A social welfare contour is akin to the usual concept of an indifference curve. That is, a SWF contour is a curve that connects all of the combinations of individual welfare UA, UB having the same SWF level. W(UA, UB) = Thus, from a social point of view, every point on a social welfare contour is equally desirable. Society as a whole will be indifferent among all of the points on the same SWF contour. 193 Let's have a look at a few of the more famous social welfare functions. Just like individual utility functions, SWF can have any functional form such as square root, linear, Cobb-Douglas, Liontief, or CES. [1] The following SWF SWF = UA + UB indicates that society, as a whole, weigh consumers A and B equally by giving them the same weight of . This is a simple form of egalitarianism. UB 2 1 1 2 UA [2] On the other hand, the following SWF SWF = 2/3 UA + 1/3 UB indicates that society, as a whole, gives a larger weight to consumer A than consumer B. In other words, consumer A (say, the rich) is viewed as more important to society than consumer B (say, the poor). 194 UB 3 1 1 3/2 UA [3] Finally, the following SWF SWF = min{UA, UB} indicates that society, as a whole, identifies themselves as those having lower utility levels (say, the poor). This implies that society cares exclusively about a base minimum of utility for the poorest members of society. This is a simple form of the so-called Rawlsian Social welfare function. UB SWF contour for min {UA, UB} = 1 2 1 1 2 UA Fine. But does the SWF help us to break the indeterminacy of the Pareto Optimal Production model? 195 We'll use a diagram to show how the concept of a social welfare function can be used to break the indeterminacy of the problem. The choice problem of society as a whole is to choose one point out of infinitely many points on the Grand UPF. The analytical tools for this social choice problem consist of the following two curves: [1] an upward moving SWF contour representing social equity [2] a stationary Grand UPF representing economic efficiency The optimal choice for society is thus the point where the social welfare function contour is tangent to the Grand UPF (i.e. point S on the diagram below). Both points S and N are Pareto efficient because they are both on the Grand UPF. However, point S is more socially desirable because it corresponds to a higher SWF contour than point N. Point S provides both economic efficiency (on the Grand UPF) and social equity (on the highest attainable SWF contour). UB N S SWF1 SWF2 Grand UPF UA HOMEWORK Construct the individual UPFs for the following additional points on the PPF of the square root economy example we did in today's lecture: Point D Point E Point F X = 0.4 X = 0.2 X = 0.1 196 Y = 1 0.4 = 0.6 Y = 1 0.2 = 0.8 Y = 1 0.1 = 0.9 ECON 301 LECTURE #12 WELFARE THEOREMS STATEMENT: "A competitive equilibrium is Pareto Optimal". PROOF: (by contradiction) Suppose that an allocation bundle x (i.e. x = (x1 A, x2A, x1B, x2 B)) is a competitive equilibrium that is not Pareto Optimal. Thus, there exists an allocation bundle y (i.e. y = (y1 A, y2A, y1B, y2 B)) such that y is feasible... y1 A + y1 B = 1 A + 1 B y2 A + y2 B = 2 A + 2 B (1) (2) and UA (y) UA (x) but UB (y) > UB (x) However, since x was the allocation bundle chosen for utility maximization at the equilibrium price vector, we have... P1x1 A + P2x2A = P11A + P22A P1x1 + P2x2 = P11 + P22 B B B B Budget Constraints are satisfied. meaning that for allocation bundle y to be "better" than allocation bundle x the following must be true... P1y1 A + P2y2A P11A + P22A P1y1B + P2y2 B > P11B + P22 B which implies, P1 (y1 A + y1B ) + P2 (y2A + y2B) > P1 (1 A + 1B ) + P2 (2A + 2B) (3) but the allocation bundle y must be feasible...subbing (1) and (2) into (3), we get P1 (1 A + 1B) + P2 (2A + 2B) > P1 (1 A + 1B ) + P2 (2A + 2B) the above says that, at allocation bundle y, the value of the individual endowments for good 1 and good 2 exceeds the value of the individual endowments of goods 1 and 2 in the economy. So at this equilibrium price vector the allocation bundle y is not feasible...we have our contradiction! 197 We derived this contradiction by assuming that the competitive equilibrium solution (allocation bundle x) was not Pareto efficient. Therefore, this assumption must be wrong. Thus, if an allocation bundle x is a competitive equilibrium then the allocation bundle x is Pareto Optimal. This result is known as the First Fundamental Theorem of Welfare Economics. THE FIRST FUNDAMENTAL THEOREM The First Fundamental Theorem of Welfare Economics provides the link which connects the Walrasian General Equilibrium Theory to the Pareto Optimal Theory. On one hand, Walrasian Theory deals with the various concepts of equilibrium. On the other hand, Pareto Theory deals with the various concepts of efficiency. The First Fundamental Theorem of Welfare Economics essentially says that "the Walrasian concept of general equilibrium is consistent with the Pareto concept of efficiency". So what does this mean in the context of our Pure Exchange Economy? We have studied both the Walrasian general equilibrium pure exchange economy and the Pareto optimal pure exchange economy with two goods (X,Y) and two consumers (A,B). Let's summarize what we know so far... WALRASIAN PURE EXCHANGE The Walrasian pure exchange model provides equilibrium prices (PX, PY) and allocation (XA, XB, YA, YB) such that (a) both consumers are in equilibrium (indifference curves are tangent to budget lines) (b) both markets are in equilibrium (demands equal supplies) PARETO OPTIMAL PURE EXCHANGE The Pareto Optimal pure exchange model provides optimal allocations (XA, XB, YA, YB) such that (a) no one can be made better off without making the other worse off (indifference curves are tangent to each other) (b) both goods are fully allocated (quantities distributed equal the quantities available) 198 In general, the First Fundamental Theorem of Welfare Economics provides a bridge from the Walrasian theory to the Pareto theory. For our exchange economy, we have the following specific statement of the First Fundamental Theorem of Welfare Economics: If the general equilibrium conditions of the Walrasian pure exchange economy are satisfied then the optimal conditions of the Pareto Optimal pure exchange economy are also satisfied. Let's look at the equilibrium conditions of the Walrasian GE pure exchange model and the optimal conditions of the Pareto Optimal pure exchange economy side-by-side: WALRASIAN PURE EXCHANGE PARETO OPTIMAL PURE EXCHANGE (EQUILIBRIUM CONDITIONS) Consumer Equilibrium for Person A MRSA = PX PY PXXA + PYYA = MA Consumer Equilibrium for Person B MRSB = PX PY PXXB + PYYB = MB Market equilibrium for good X XA + XB = X Market equilibrium for good Y YA + YB = Y (OPTIMAL CONDITIONS) Pareto optimality for both Person A and B MRSA = MRSB Inside the Edgeworth Box XA + XB = X Inside the Edgeworth Box YA + YB = Y We can clearly see that the Walrasian pure exchange solution (competitive equilibrium) does indeed satisfy the optimality condition MRSA = MRSB of the Pareto optimal pure exchange model since at the competitive equilibrium, both of the consumers equate their marginal rates of substitution to the common price ratio 199 MRSA = MRSB = PX PY In other words, the general equilibrium solution provided by solving the Walrasian competitive equilibrium is also an optimal solution (along with many other possible solutions) of the Pareto Optimal pure exchange model. The key link is the common price ratio, PX. PY So I bet you're wondering what the First Fundamental Theorem of Welfare Economics has to say about the Production Economy... For a Production Economy with 2 goods (X,Y) and two factors (K,L) and two consumers (A,B) we have studied both the Walrasian general equilibrium model and the Pareto optimal model. Let's compare the logical structure of the two models side-by-side: WALRASIAN PRODUCTION MODEL PARETO OPTIMAL PRODUCTION MODEL (EQUILIBRIUM CONDITIONS) Consumer Equilibrium for Person A MRSA = PX PY PXXA + PYYA = MA Consumer Equilibrium for Person B MRSB = PX PY PXXB + PYYB = MB Producer Equilibrium for Producer X MRTSX = r w X = f(KX, LX) Producer Equilibrium for Producer Y MRTSY = r w Y = g(KY, LY) (OPTIMAL CONDITIONS) Pareto optimality for both Person A and B MRSA = MRSB Production efficiency MRTSX = MRTSY X = f(KX, LX) Y = g(KY, LY) 200 Zero Profit Conditions PX = rkX + wlX PY = rkY + wlY Market equilibrium for goods XA + XB = X YA + YB = Y Market equilibrium for factors KX + KY = K LX + LY = L Inside the Edgeworth Box for goods XA + XB = X YA + YB = Y Inside the Edgeworth Box for factors KX + KY = K LX + LY = L Overall efficiency MRS = MRT We can show that the equilibrium conditions of the Walrasian Production economy will automatically satisfy the three optimal conditions required by the Pareto optimal production economy (namely, exchange efficiency, production efficiency, and overall efficiency) as follows: MRSA = MRSB = PX PY MRTSX = MRTSY = r w (exchange efficiency) (production efficiency) For overall efficiency, the proof hinges on the following property: under perfect competition, zero profit conditions allow the ratio of output prices to be expressed as a ratio of marginal costs. By definition, this ratio of marginal costs is nothing more than the marginal rate of transformation. MRS = PX PY = rkX + wlX rkY + wlY (consumer equilibrium) (zero profit condition) 201 = MCX MCY MRS = MRT (ratio of marginal costs) (overall efficiency) In summary, we have the following First Fundamental Theorem of Welfare Economics for a production economy: If the general equilibrium conditions of the Walrasian Production Economy are satisfied then the optimal conditions of the Pareto Optimal Production Economy are also satisfied. In general, we can say that under perfect competition general equilibrium implies Pareto optimality. [General Equilibrium] Perfect Competition => [Pareto Optimality] Note the critical role of the requirement of perfect competition in this statement of the theorem. THE SECOND FUNDAMENTAL THEOREM The Second Fundamental Theorem of Welfare Economics provides a second link which runs from Pareto Optimal Theory back to Walrasian General Equilibrium Theory. The key question of the Second Fundamental Theorem of Welfare Economics is: Given a specific Pareto Optimal allocation, is there a set of prices which makes that particular Pareto Optimal allocation a general equilibrium (competitive equilibrium) solution? Wait just a minute! Why does this matter to anyone?!? We are interested in the Second Fundamental Theorem of Welfare Economics because once we have established it, it implies the following important feature of a market economy: It is possible to use the market pricing mechanism to achieve a given Pareto Optimal allocation. That is, there is no need to rely on other methods (such as dictatorship) to make allocations of resources in the economy. The market process itself is Pareto efficient. So to answer the key question above: 202 Yes, it is possible to get a set of prices which makes a given Pareto optimal allocation a general equilibrium solution. However, this answer only works under the additional assumption of convex preferences (i.e. all indifference curves are convex). Thus, the Second Fundamental Theorem of Welfare Economics can be stated as follows: Under perfect competition and the additional assumption of convex preferences Pareto optimality can be realized by general equilibrium solution. [Pareto Optimality] Perfect Competition Convex Preferences => [General Equilibrium] Now, what if we don't have the additional assumption of convex preferences? It is possible that the Second Fundamental Theorem of Welfare Economics will break down!!!! For example, in the diagram below we have consumer A with non-convex preferences where point 1 represents a Pareto optimal allocation (since the two indifference curves are tangent to one another at point 1) but not a general equilibrium solution (since the two consumer equilibrium points do not match up...consumer B is on their highest indifference curve at point 1 but consumer A is on a higher indifference curve at point 2). YA XB Price Ratio OB 1 Pareto Optimal solution 2 ICA2 ICA1 O A Y B ICB XA 203 WELFARE MEASURES We will look at two commonly used measures in applied welfare economics: Compensating Variations (CV) and Equivalent Variations. We need these welfare measures to coordinate our theoretical results with "the Real World". We already know that the two Fundamental Theorems (taken together) establish a one-to-one correspondence between economic efficiency and optimality for a market economy under perfect competition. The "Real World" is, however, not perfect and hence, we would not really expect Pareto optimality to prevail in "reality". This generates the need for analytical measures that can tell us just how far (or how close) we are from the ideal theoretical world of perfect competition. [Perfect Competition] Welfare Measures ["Real World"] Two of the most popular welfare measures are compensating variations (CV) and equivalent variations (EV). These measures are closely related to one another indeed, nearly identical, at times, in the order of magnitude. COMPENSATING VARIATIONS (CV) Consider a consumer with a utility function U = U(X,Y) and income M, and let the original prices of the goods be PX, PY. [1] At the initial consumer equilibrium (point E1), the consumer maximizes utility subject to the budget constraint according to the following conditions: MRS = PX PY PXX + PYY = M which can be solved for the quantities demanded of goods X and Y. (2) (3) (1) 204 Y E1 X [2] Suppose that PX increases to PX while everything else remains the same. The budget line rotates around the vertical intercept (inward) and the consumer revises their quantities demanded accordingly. Y E2 E1 X The new consumer equilibrium is now at point E2 where the consumer still maximizes utility but now subject to their new budget constraint according to the following conditions: MRS = PX PY PXX + PYY = M 205 (4) (5) which can be solved for the new quantities of goods demanded. The consumer has moved from point E1 to point E2 to respond to the change in prices. We want to measure the change in the consumer welfare as a result of such a move. [3] To measure the welfare change from E1 to E2, CV uses E1 as the basis for comparison by asking the following question: A price change brings the consumer from E1 to E2. How much income should we give the consumer in order to keep him equally well off as before the price change? That is, we want to offset the effect of the price change (which brings the consumer from E1 to E2) by an income change (which brings the consumer from E2 to CV) so that the consumer gets back to the original indifference curve. Y CV E2 E1 X Therefore, CV is the amount of income change which makes a parallel movement of the budget line from E2 (at new prices, new utility level) to CV (new prices, old utility level). In other words, CV is the amount of income change to "compensate" for the welfare effect of a price change so that the consumer is back to the old indifference curve. 206 EQUIVALENT VARIATIONS (EV) Unlike the compensating variation (CV) which is based on the initial point E1 to measure the welfare change from E1 to E2, EV uses E2 as the basis for comparison by asking the following question: A price change brings the consumer from E1 to E2. How much income should we take away from the consumer in order to keep them equally worse-off as after the price change? That is, we want to simulate the effect of a price change (which brings the consumer from E1 to E2) by an income change (which brings the consumer from E1 to EV) so that the consumer has the same effect as moving to the new indifference curve. Y E2 EV E1 X Therefore, EV is the amount of income change which makes a parallel movement of the budget line from E1 (at old prices, old utility level) to EV (old prices, new utility level). In other words, EV is the amount of income change which is "equivalent" to the welfare effect of the price change so that the consumer feels the same as moving to the new indifference curve. So clearly, the main difference between CV and EV is the reference to which the welfare comparison is being made: [1] CV uses E1, new prices and old utility level as the basis for comparison, 207 [2] EV uses E2, old prices and new utility level as the basis for comparison. The differences between the two measures are summarized in the following table and diagram: Price of Good X Price of Good Y Utility Level Income E1 PX PY U M E2 PX PY U M CV PX PY U M + CV EV PX PY U M + EV A positive value for CV indicates that the consumer has been made worse off and hence, needs an increase in their income to "compensate" for the welfare loss. A negative value of CV indicates that the consumer has been made better off and hence, needs an income reduction to "compensate" for the welfare gain. On the other hand, a negative value for the EV indicates that the consumer has been made worse off since it is "equivalent" to a welfare loss. However, a positive value for the EV measure indicates that the consumer has been made better off since it is "equivalent" to a welfare gain. Welfare Position Worse off Better off CV Positive Negative EV Negative Positive Alright! Let's do a numerical example to see how this really works! Consider a consumer with a square root utility function, income of $10 (M = 10), and unit prices (PX = PY = 1). [1] At the initial equilibrium E1, we have the following general consumer equilibrium conditions: MRS = PX PY PX X + PY Y = M 208 Applying these general conditions to the problem at hand, we get... Y=1 X X + Y = 10 Solving for our consumer's initial equilibrium X=5 Y=5 U = XY = 25 = 5 (denoted as U1) [2] Now suppose that PX increases by 25% to PX = 1.25, while everything else remains the same. At the new consumer equilibrium E2, we have the following general consumer equilibrium conditions: MRS = PX PY PX X + PY Y = M Applying these general conditions to the problem at hand, we get... Y = 1.25 X 1.25 X + Y = 10 Solving for our consumer's new equilibrium X=4 Y=5 U = XY = 20 = 4.47 (denoted as U2) Thus, a 25% increase in the price of good X, ceteris paribus, will make the consumer worse off since their utility level falls from U1 = 5 to U2 = 4.47. The consumer's demand for good X falls from X = 5 to X = 4 and their demand for good Y remains the same at Y = Y = 5. [3] At point CV (using new prices and old utility), we have... MRS = PX PY PX X + PY Y = M + CV 209 U (X,Y) = U1 Applying these general conditions to the problem at hand, we get... Y = 1.25 X 1.25 X + Y = 10 + CV XY = 25 = 5 Solving for our consumer's equilibrium at CV 1.25 X = Y X = 4Y 5 Subbing (1) into the budget constraint... 2.5 X = 10 + CV X = 4 + 2CV 5 Subbing (2) into the budget constraint... 1.25 (4/5 Y) + Y = 10 + CV 2Y = 10 + CV Y = 5 + CV 2 Now we use the old utility level to determine CV... XY = 25 = 5 [(4 + 2CV)( 5 + CV)]1/2 = 25 5 2 (4 + 2CV)( 5 + CV) = 25 5 2 20 + 2CV + 2CV + 2CV2 = 25 10 210 (4) (3) (1) (2) 40CV + 2 CV2 = 50 CV2 + 20 CV 25 = 0 -b + (b2 4ac)1/2 2a -20 + (400 + 100)1/2 2 -20 + 22.36067977 2 2.36067977 2 1.180339887 = CV (5) Why don't we do the "minus" part of the "plus or minus" in the quadratic formula? Since we know that the consumer is made worse off and is being "compensated" for a welfare loss, we know that the CV measure must be positive...so we don't need the negative solution from the quadratic formula! So now that we know that CV = 1.18, we can figure out the demands at point CV by subbing (5) into both (3) and (4) respectively. This gives us the point CV as follows: X = 4 + 2CV 5 X = 4 + 2.36 5 X = 4.472 Y = 5 + CV 2 Y = 5 + 1.18 2 Y = 5.59 Just to be sure, let's check the old utility level to see if the point CV is on the old indifference curve... XY = 25 = 5 [(4.472)(5.59)]1/2 = 24.99848 25 (4) (3) 211 So, to summarize the solution at point CV X = 4.472 Y = 5.59 U=5 CV = 1.18 (same as U1) Thus, an income increase of CV = 1.18 will "compensate" for the welfare loss due to a 25% increase in PX, ceteris paribus. [4] At point EV (using old prices and new utility), we have... MRS = PX PY PX X + PY Y = M + EV U (X,Y) = U2 Applying these general conditions to the problem at hand, we get... Y=1 X X + Y = 10 + EV XY = 20 = 4.47 Solving for our consumer's equilibrium at EV X=Y Subbing (1) into the budget constraint... 2 X = 10 + EV X = 5 + EV 2 Now we use the new utility level to determine EV... XY = 20 = 4.47 XX = X = 20 = 4.47 Thus, X = Y = 4.47 212 (2) (1) So now that we know that X = Y = 4.47, we can figure out the EV by solving (2) for EV as follows: X = 5 + EV 2 4.47 = 5 + EV 2 EV = -0.53 2 So, to summarize the solution at point EV X = 4.47 Y = 4.47 U = 4.47 EV = -1.06 EV = -1.06 (2) (same as U2) Thus, an income decrease of EV = -1.06 will be "equivalent" to the welfare loss due to a 25% increase in PX, ceteris paribus. BOADWAY'S PARADOX If we can calculate welfare measures for all of the individual consumers, then the usual procedure to get a welfare measure for the entire economy is to sum all individual welfare as follows: Consumer 1 Consumer 2 Consumer 3 . . . . Consumer n Entire Economy CV Calculation CV1 CV2 CV3 . . . . CVn CVsum = in CVi EV Calculation EV1 EV2 EV3 . . . . EVn EVsum = in EVi Do you think that there are any problems in this method of adding (aggregating) welfare like this? The answer is, perhaps, both yes and no! 213 On one hand, welfare measures are essentially income changes measured in dollar values, and hence, can be summed over individual consumers in the economy. For example, if we want to evaluate the welfare effect of a tax reform policy, we can calculate all the individual welfare effects and then add them up. On the other hand, if we have more than one consumer in the economy (and we do!), there will be those who gain in terms of welfare from the policy and those who will lose in terms of welfare from the policy. This makes it difficult to interpret the aggregation results. So, how do we know whether a policy change generates a society wide welfare improvement? To assert the welfare effect of a policy change, we can use the following compensation test: A policy change is a "potential Pareto improvement" if gainers can hypothetically compensate losers and yet on the whole remain better off. The change is "potential" in nature because the test is hypothetical (gainers are not actually compensating the losers for real) and it is a "Pareto improvement" because once "compensated" the losers are indifferent while the gainers are better off. [Potential Pareto Improvement] Compensation Test => [Aggregate Welfare Change] As a result, a potential Pareto improvement implies a net aggregate welfare change CVsum < 0 EVsum > 0 The heading for this section is "Boadway's Paradox" so what is this paradox all about? While it's true that a potential Pareto improvement will lead to a net aggregate welfare change, Boadway's paradox warns about the potential danger in believing that the reverse will be true. The reverse is not necessarily always true. [Aggregate Welfare Change] Boadway's Paradox xxxxxxxx [Potential Pareto Improvement] 214 Boadway sets up his argument within the context of a simple exchange economy with two people and two goods. Let E1 and E2 be two points on the contract curve so that there will be no potential Pareto improvements from moving from one point to the other (since both of the points are already Pareto Optimal by definition). He then proceeds to calculate individual welfare measures in terms of CV and EV and shows that there is a net non-zero aggregate welfare change. This is a paradox because we would expect a zero net aggregate welfare change from such a move (between two Pareto Optimal points on the contract curve). HOW COULD THIS BE? [1] Since both E1 and E2 are on the contract curve by construction, they are equally efficient and there will not be any potential Pareto improvement in moving from one point to the other. ICB2 YA XB Price Ratio E2 OB E1 ICA2 Price Ratio ICA1 ICB1 OA Y B XA [2] If we move from E1 to E2, consumer B will be worse off (with a lower indifference curve and a positive value CVB > 0). This means we have to give B an income supplement of the amount CVB in order to bring her back to the old indifference curve ICB1 at E1. 215 ICB2 YA XB OB Price Ratio E2 E1 Price Ratio ICA2 ICA1 ICB1 CVB OA Y B XA [3] On the other hand, if we move from E1 to E2, consumer A will be better off (on a higher indifference curve and having a negative value for CVA < 0). This means we have to take away an income amount CVA in order to bring him back to the old indifference curve ICA1 at E1. ICB2 YA XB ICB1 E1 Price Ratio OB Price Ratio E2 ICA2 ICA1 CVA CVB OA Y B XA [4] Since, by construction, the absolute value of the welfare change CVA < 0 of consumer A is greater than the absolute value of the welfare change CVB > 0 of consumer B, we have a net aggregate welfare change when moving from E1 to E2... 216 CVsum = CVA + CVB <0 We would presumably conclude that E2 is "better" than E1 since we get a net welfare change in CV by moving from E1 to E2. This is, however, a contradiction since both E1 and E2 are on the same contract curve and, as such, are both Pareto Optimal equilibrium points (i.e. no potential Pareto improvement exists). So Boadway's Paradox shows us that there are no potential Pareto improvements in moving from E1 to E2 and yet, we found a net aggregate welfare effect where CVsum < 0. The lesson is that aggregate welfare calculations could be misleading if we are not careful about possible (re)distributional issues. Okay. Let's take an example...Consider a simple exchange economy with two goods with unit aggregate endowments X=Y=1 Consumer A has a square root utility function while Consumer B has a CobbDouglas utility function with = 2/3 and = 1/3. MRSA = YA XA MRSB = 2YB XB Without any loss of generality, we choose good Y as the numeraire so that PY = 1. [1] We first construct the contract curve from the Pareto Optimal conditions MRSA = MRSB XA + XB = X YA + YB = Y Applying these general equations to the specific problem at hand YA = 2YB XB XA XA + XB = 1 YA + YB = 1 217 Solving for the contract curve, we get YA = 2XA 1 + XA [2] Now we can select two points E1 and E2 on the same contract curve (defined above). Suppose that we choose E1 at XA = 0.25 and E2 at XA = 0.50. Substitute these values into the equation of the contract curve and we get... XA = 0.25 XA = 0.50 YA = 0.40 YA = 2/3 (at point E1) (at point E2) From these values, we can calculate various variables such as MRS, PX, incomes and utility levels that are needed for the welfare calculations. The results of the calculations are: Consumer A XA YA MRSA PX PY (numeraire) MA UA CVA Consumer B XB YB MRSB PX PY (numeraire) MB UB CVB Point E1 0.25 0.40 1.60 1.60 1.00 0.80 0.316228 Point E1 0.75 0.60 1.60 1.60 1.00 1.80 0.696238 Point E2 0.50 0.666667 1.333333 1.333333 1.00 1.333333 0.577350 -0.603037 Point E2 0.50 0.333333 1.333333 1.333333 1.00 1.00 0.436790 0.593988 218 The calculation shows that A gains and B loses during a move from E1 to E2. Moreover, A can compensate B and still remains better off with a net gain of 0.009049 in CV. This welfare result is indeed misleading since it is contrary to the fact that we have pre-selected E1 and E2 to be equally optimal. Consumer A (better off) Consumer B (worse off) Total HOMEWORK [1] Re-calculate CV and EV for our example where a consumer with a square root utility function, income of $10 (M = 10), and unit prices (PX = PY = 1) but both PX and PY increase by 25% with everything else remaining the same. [2] Verify the welfare calculations for the Boadway's Paradox example above (i.e. solve the example and see if you can get the same results as are in the table provided above). CVA = -0.603037 CVB = 0.593988 CVsum = -0.009049 219 ECON 301 LECTURE #13 Liontief Input Output Models Recall that throughout the course we have discussed the perfect compliments production function. This functional form is often referred to as a fixed coefficients or fixed proportions function and was named after Professor Wassily Liontief and his seminal work from the early 1950s. In its "static" version, Liontief's input output2 analysis deals with the following question: What level of output should each of the n industries in an economy produce such that that output will be just enough to satisfy the total demand for that product? The reason for the term input output analysis now becomes quite clear. The output of any given industry (let's say the steel industry) is needed as an input in many other industries, or even as an input into its own industry; thus the correct level of steel output (i.e. shortage-free and surplus-free) will depend on the input requirements of all of the n industries in the economy. In turn, the output of many other industries will enter into the steel industry as inputs, and consequently the "correct levels of these other products will depend partly upon the input requirements of the steel industry. In light of this inter-industry dependence, any set of "correct" output levels for the n industries must be one that is consistent with all of the input requirements in the economy (so that no bottlenecks will arise anywhere). So it becomes abundantly clear that this input output analysis should be of paramount use in production planning applications, such as planning for the economic development of a nation or even for a domestic defense program. CAVEAT: Strictly speaking, input output analysis is not really a form of general equilibrium analysis. Although the interdependence of the various industries is emphasized, the "correct" output levels are those which satisfy a set of technical input output relationships rather than a set of market equilibrium conditions. Nevertheless, the problem posed by the input output analysis does boil down to solving a system of simultaneous equations and this gives us a chance to investigate the interdependence among markets using matrix algebra. 2 Liontief, Wassily W. The Structure of American Economy 1919-1939, 2d ed., Oxford University Press, Fair Lawn, N.J., 1951. 220 The Structure of an Input Output Model Since an input output model normally considers a large number of industries, its framework is by necessity rather involved. To simplify the problem, we will make the following standard assumptions: 1. Each industry produces only one homogeneous good (broadly interpreted, this implies that they could produce two or more jointly produced goods provided that they are made in fixed proportions to one another). 2. Each industry uses a fixed input ratio or factor combination for the production of its output. 3. Production in every industry is subject to constant returns to scale so that a k-fold change in every input will result in a k-fold change in the output. Of course, these assumptions are quite unrealistic. Even so, we can remedy at least some of these issues conceptually. For example, even if an industry does produce two different goods or uses two different possible factor combinations in production we can (at least conceptually) break the industry down into two separate industries. From the above assumptions we see that, in order to produce each unit of the jth commodity, the input need for the ith commodity must be a fixed amount that we will denote as aij. To clarify, the production of each unit of the jth commodity will require a1j of the first commodity, a2j of the second commodity,..., and anj of the nth commodity. The order of the subscripts in aij is easy to remember: the first subscript refers to the input and the second refers to the output so that aij indicates how much of the ith commodity is used for the production of each unit of the jth commodity. For our purposes, let's assume that prices are given and this allows us to adopt "a dollar's worth" of each commodity as its unit of measurement. Then the statement a32 = 0.35 will mean that 35 worth of the third commodity is required as an input for producing a dollar's worth of the second commodity. This means that the aij symbol can be referred to as the input coefficient. For an n-industry economy, the input coefficients can be arranged into a matrix A = [aij], as in the table on the next page. Each column specifies the input requirements for the production of one unit of the output for a particular industry. The second column, for example, states that to produce a unit (one dollar's worth) of commodity 2, the inputs needed are: 221 a12 units of commodit 1, a22 un of com ty nits mmodity 2,... and an2 units of ..., com mmodity n. . ses s then the ele ements in th principal he l If no industry us its own product as an input, t diago onal of mat A will all be zero. trix del The Open Mod esides the n industries the model also conta s, ains an "ope sector ( en" (say, If, be hous seholds) wh hich exogen nously dete ermines a fin demand (i.e. a non nal d n-input dema and) for the product of each industry and wh e f hich supplie a primar input es ry (say, labour) no produced by the n in , ot d ndustries th hemselves, then the model is an open model. n With the case of the prese o ence of an o open sector the sum o the eleme r, of ents in eac ch colum of the in mn nput coeffic cient matrix A must be less than 1 Each column sum 1. repre esents the partial input cost (not i p including th cost of th primary input) he he incur rred in prod ducing a dollar's worth of some co ommodity; if this sum is greater than or equal to 1 then pro o oduction wo ould not be economica justifiable. ally Symbolically, th fact can be stated a follows: his as 222 re mation is ov i, that is over the e ver s, elements appearing in the variou n us wher the summ rows of a specif column j s fic j. Takin this line of thought a step furth it may also be sta ng her, ated that, sin the nce value of output ($1) must b fully abs e be sorbed by th payment to all fact he ts tors of production, the amount by which the column sum falls shor of $1 mus be the y m rt st value of the pay e yment to the primary in e nput (labou of the op sector. ur) pen Thus the value of the prim s, mary input n needed in p producing a unit of the jth comm modity shou be: uld If ind dustry 1 is to produce a output ju sufficien to meet the input req o an ust nt quirements s of the n industries as well a the final demand of the open s e as f sector, its o output level l x1 must satisfy the followin equation: t ng x1 = a11x1 + a12x2 + a13x3 +... a1nxn + d1 ...+ or (1 a11) x1 a12x2 a13x3 - ... - a1nxn = d1 re es or t epresents t input the wher d1 denote the final demand fo its output and a1jxj re 3 dema from th jth industry. Notice that, exce for the f and he e ept first coefficie (1 ent a11), the remain ning coeffici ients in the last equation are exac the first row of the ctly t e matrix on page 3 above ex xcept they a all nega are ative. Similarly, the corre esponding equation for industry 2 will have t same co e the oefficients a in the as seco row of the matrix o page 3 a ond on above, exce that the variable x2 will have ept the c coefficient (1 a22) ins stead of a22. For t entire se of n indu the et ustries, the "correct" ou utput levels can be sum s mmarized by th following system of n linear eq he g quations: 3 Do n add up the input coeffic not e cients across a row; such a sum say, a11 + a12 + a1 +...+ a1n is 13 devoid of economic meaning. T sum of th products a1 x1 + a12x2 + a13x3 +...+ a1nxn on the d The he 11 other hand, does have an econo h omic meaning it represents the total am g; mount of x1 ne eeded as an input for all the n in ndustries. 223 (1 a11) x1 a21 x1 a12x2 (1 a22) x2 2 a13x3 - ... a23x3 - ... - a1nxn = d1 a2nxn = d2 an1 x1 an2 x2 2 an3x3 - ... - (1 ann) xn = dn In ma atrix notatio this can be written as: on, n e p agonal of th matrix on the left ar ignored, then the he n re If the 1s in the principal dia matrix is simply A = [-aij]. As it is, o the other hand, the matrix is th sum of y on r he the id dentity matrix In (with 1 in the pr 1s rincipal diag gonal and z zeros every ywhere else e) and t matrix A. the Ther refore, we can write the above se of matrice as: c et es (I A)x = d re a variable vec and the final dema ctor e and wher x and d are, respectively, the v (cons stant-term) vector. Th matrix (I A) is called the tech he hnology ma atrix, and we e can d denote it by T. y Ther refore, we can also write this syst c tem as: Tx = d As lo as T is non-singula and the is no re ong ar ere eason why i should no be we it ot -1 shall be able to find its inve erse T , an obtain th unique solution of th system nd he he from the equatio on: _ 1 x = T-1 d = (I A)-1 d ) 224 del merical Exa ample The Open Mod A Num For t purpose of illustra the es ation, suppo that the are only three indu ose ere y ustries in the econ nomy and th the inpu co-efficient matrix is as follows: hat ut s e e n ss as Note that in A each column sum is les than 1, a it should be for the open mode Further, if we deno by a0j th dollar am el. ote he mount of the primary in e nput (labour r) th used in producing a dollar' worth of t j comm d 's the modity, we can write (by subtr racting each column s sum in the a above matri from 1): ix a01 = 0.3 0 a02 = 0.3 a03 = 0.4 With the matrix A above, th open inp output system ca be expre he put an essed in the e form we derived earlier...T = (I A)x = d as fol d Tx x llows: e d any c emands d1, Here we have deliberately not given a specific values to the final de d2, and d3. In th way, by keeping th vector d in parametric form, ou solution his he ur appear as a "formula" into which w can feed various s we specific d ve ectors to will a obtai various corresponding solution (i.e. like a demand f in c ns function in consumer theor ry). 225 Now, by invertin the 3 3 technolog matrix T, the solutio of this sy ng gy on ystem can be fo ound (appro oximately, b because of rounding) t be: to Now, just how did we get t d this inverted technolog matrix? Recall, from your d gy m ECO 211 clas that inver ON ss rting a 3 3 matrix is a simple a the follow as as wing rule: A-1 = __1__ adj A A to nverse of th technolo matrix T he ogy T... So, t find the in we c find the cofactor m can matrix (we kn now that the determinant of T is n zero, so not o T is i invertible). The cofact matrix is tor s... 226 This, of course leads us to the adj T ( , o (i.e. row 1 b becomes co olumn 1, row 2 beco omes colum 2, and ro 3 becom column 3) as follow mn ow mes n ws: Now all we need to do is find the dete erminant of the techno f ology matrix to x comp plete the inversion! So let's do it o, t! member, from ECON 211, that the determina of a 3 3 matrix is as simple m e ant Rem as th following rule: he g or sim mply, A = a11a22a33 a11a23a32 + a12a23a3 a12a21a33 + a13a21a32 a13a22a31 3 31 2 For hnology ma atrix, the de eterminant w would be: which is simply a scalar! F our tech T= (0.8)(0.9) = )(0.8) (0.8 8)(-0.2)(-0.3 + (-0.3)(-0 3) 0.2)(-0.1) (-0.3)(-0.4 4)(0.8) + (-0.2)(-0.4 4)(-0.3) (-0.2)(0.9)(-0 0.1) T= 0.576 0.048 0.00 0.096 0.024 0 = 0 06 0.018 T= 0.384 = quired parts to get the matrix inve s ersion solution shown Now we have all of the req he age on th top of pa 226. 227 So, if the specif final dem f fic mand vector (say, the f r final output target of a deve elopment pr rogram) hap ppens to be e: in billions of dollars then the s s, follow wing specif solution v fic values will emerge fro our input output m om model (again in billions of dollars): s _ x1 = ___1___ [0 0.66 (10) + 0.30 (5) + 0 0.24 (6)] = 9.54 = 24.8 84 0.384 0.384 0 _ x2 = ___1___ [0 0.34 (10) + 0.62 (5) + 0 0.24 (6)] = 7.94 = 20.6 68 0.384 0.384 0 _ 0.21 (10) + 0.27 (5) + 0 0.60 (6)] = 7.05 = 18.3 36 x3 = ___1___ [0 0.384 0.384 An im mportant qu uestion now arises. Th producti of the o w he ion output mix w have we solve must ent a definit required amount of the primary input (lab ed tail te y bour). Would the amou required be consis unt d stent with w what is avail lable in the economy? ? On th basis of our calcula he ations of a01, a02, and a03, the req quired prima input ary can b calculate as follow be ed ws: 228 Ther refore, the specific fina demands s al s wi be feasib if and on if the av ill ble nly vailable amo ount of the prima input (la ary abour) is at least $21 b t billion worth If the am h. mount availa able falls short then that particular p t, production t target will, of course, h have to be revised down nward acco ordingly. One important feature of th above analysis is th as long as the inp f he hat, g put -1 -1 coeff ficients rem main the sam the inve me, erse T = ( A) will not change (I e. Ther refore, only one matrix inversion n x needs to be performed even if w are to e d, we cons sider a hund dred or eve a thousand different final dema vectors (which en t and s could occur if we were ask to analy a full sp d w ked yse pectrum of alternative deve elopment ta argets). This means tha we can sa a pile o computat at ave of tional effort when compared to r ( mination-of f-variables m method), es specially if large other methods (like the elim equa ation system are invo ms olved. Note that this ad e dvantage is not shared by s d Cram mer's Rule. By C Cramer's Ru the solu ule, ution will be calculated according to the form e d g mula: xj = Tj / T each time a different final demand vector is u d used, we m must re-evaluate the but e deter rminants Tj. This w T would be mu more time consum uch ming than th he multi iplication of a known T-1 by a new vector d. f w The Closed Mo odel e us nput outp model is absorbed into the put s If the exogenou sector of the open in syste as just another industry, then the model becomes a closed mo em a odel. In su a model, final dem uch mands and p primary inpu do not a uts appear; in t their place will b the input requireme be t ents and the output of t newly c e the conceived industry. All good become intermediat in nature because everything that is prod ds te e, duced is produced only for the sake of satisfyin the inpu requireme f e ng ut ents of the (n+1) indus stries in the model. e At fir glance, the convers rst t sion of the o open sector into an ad dditional ind dustry would d not s seem to cre eate any sig gnificant cha ange to the analysis. Actually, ho e owever, 229 since the new in e ndustry is a assumed to have a fixe input rat as does every other ed tio indus stry, the supply of wha used to b the prima input (la at be ary abour) mus now bear st a fixe proportio to what used to be called the f ed on final deman nd. More concretely this may mean, for e e y, example, th househo hat olds will consume each commodity in a fixed proportion to the labo services they provi h our s ide. This certa ainly constit tutes a sign nificant theo oretical change in the a analytical fr ramework comp pared to the open mod e del. Math hematically speaking, t disappe the earance of final deman means that we will nds now have a hom mogeneous s-equation s system. As ssuming fou industries only ur (inclu uding the newly create one de ed esignated b the subsc by cript 0), the "correct" e outpu levels will be those that satisfy the followi system of equation ut y ing ns: Beca ause this eq quation sys stem is hom mogeneous, it can have a non-triv solution e vial n if and only if the 4 4 tech d e hnology ma atrix (I A) has a vanis shing deter rminant. Of co ourse, this condition is always sa c s atisfied. In a closed mo odel, there is no more e prima input (la ary abour) and hence, eac column s ch sum in the input outp put coeff ficient matr A must n rix now be exac equal to (instead o less than 1. That ctly of n) mean that ns a0j + a1 + a2j + a3j = 1 1j or a0j = 1 a1j a2j a3j Yet, this implies that in eve column of the matr (I A) above, the to element s ery rix op t ways equal to the nega ative sum o the other three elem of r ments. As a is alw cons sequence, the four row are linea dependent and we must find t ws arly e that I A= 0. This guarantees that the system poss s sesses non n-trivial solu utions. In fa it has an act, al s. h infinite number of non-trivia solutions This implies that in the closed model with a hom mogeneous equation system the is no unique "correct" output m s ere mix. 230 _ _ _ _ We c determine the outp levels x0, x1, x2, an x3 in proportion to o another can put nd one r, but w cannot fix their abs we f solute levels unless ad s dditional res strictions ar imposed re on th model. We will see that this is much less useful (no really very useful at he W e s s ot y all!) t than our res sults from t open model since we cannot invert the t the technology matrix in the clo osed model. odel A Nu umerical E Example The Closed Mo Supp pose that we have the same exam w mple as we solved in t open m e the model above e, but w convert the primary input into "just another industry" and appro we y " oach the problem as a closed input output m model. all s e efficient ma atrix for the open mode is as el Reca that we said that the input co-e follow ws: and i we denote by a0j the dollar amo if e ount of the primary inp (labour) used in put producing a dollar's worth of the jth co ommodity, w can writ (by subtr we te racting each h colum sum in the above m mn t matrix from 1): m a01 = 0.3 0 a02 = 0.3 a03 = 0.4 ow e nput into "ju another industry" by adding ust y so no we can convert the primary in these elements to the inpu co-efficient matrix as follows: e ut s With the matrix A above, th closed in he nput outp system c be exp put can pressed in the fo we der orm rived earlier...Tx = (I A)x = d as follows: 231 Or sp pecifically, we ove get ) o em So, w said abo that to g a specific (unique) solution to the proble the techn nology matrix must be invertible.. e .. We k know that fo the close model th it won't b invertible but let's confirm this or ed hat be e, s by fin nding the determinant of the tech hnology mat trix. We wi use the fi row to ill irst expa along to evaluate t determi and o the inant... f minants is si imply the de eterminant that we Now, the first of the four 3 3 determ figure out in th open ver ed he rsion of the model. Th first one works out to be 0.384 his 4. 232 The second term of this de m eterminant c be foun as follow can nd ws: The t third term of this deter o rminant can be found as follows: n 233 The f fourth term of this dete erminant ca be found similarly a follows: an d as o s we d minant of the 4 4 So bringing all of the terms together w can find the determ nology matrix for the c closed input output m t model as: techn T = 0.384 0.1152 0.1152 0.1536 0 T = 0 s, x ot s ertible). This implies Thus the matrix T does no have rank (and thus is not inve (as e expected) th there is no unique solution to the closed input out hat tput model, , simp output ra ply atios. 234 mework: Hom 1. Using th open mo he odel example shown in the class notes, if the final n e demand vector for t three se the ectors is: what will be the solution output lev w vels for the three indus stries? (Round your answers to two decima places). t al 2. Recall th input co-efficient ma he atrix from our open mo odel examp ple... a) What is the econom meaning of the elements a22 = 0.1, a23 = 0.2, and t mic g a31 = 0.1? b) What is the econom meaning (if any) of the first co t mic g f olumn sum? ? c) What is the econom meaning (if any) of the first row sum? t mic g f w 3. In a two industry ec conomy it is known tha Industry 1 uses 10 of its own s at product and 60 of commodity 2 to produ a dollar worth of c a y uce rs commodity 1. It is also known t that Industr 2 uses none of its o ry own product but uses t 50 of co ommodity 1 to produce a dollars w e worth of co ommodity 2. The open n sector de emands are $1000 billion of comm e modity 1 an $2000 billion of nd commodity 2. a) Write out the inp co-efficient matrix, the techno put ology matrix and the x, specific input o output matr for this o rix open econo omy. b) Find the solution to the output levels u t n using the m method show in class. wn . 235 ECON 301 LECTURE #14 PUBLIC GOODS We want to introduce the concept of public goods into the theory of welfare economics. The economic theory that we have studied so far deals exclusively with private goods (i.e. goods that are privately owned by agents in the economy. The existence of public goods in the real world requires important modifications with respect to the way we conduct our economic analysis and with respect to the final welfare results. PUBLIC VS PRIVATE So what is the difference between private and public goods? Pure public goods (or just public goods for short) and private goods are two ends of the entire spectrum of goods. [Regular private goods] PRIVATE GOODS Private goods are characterized by private ownership and consumption which basically exclude the joint ownership and consumption of the good (i.e. exludability, rivalrous). [Pure public goods] PUBLIC GOODS Public goods are characterized by public ownership and consumption which do not exclude the joint ownership and consumption of the good (i.e. non-exludability, nonrivalrous). Public goods can be owned or consumed jointly and equally by all in the economy. (i.e. national security, air quality, community fireworks displays, etc.). When private goods are owned or consumed by their owners, everyone else is barred from owning or consuming those goods. Private goods are the only type of goods we have studied thus far. Of course, there are goods that are categorized somewhere in the middle of the spectrum. These might be referred to as "local public goods" and these are characterized by some combination of features that make them "sort of public, sort of private". [Regular private goods] Local Public Goods [Pure public goods] Although these intermediate goods are not publicly owned (i.e. it costs money to gain entitlement, entrance or membership in order to consume them), they still 236 allow for joint consumption (i.e. every fee paying member can equally and jointly consume the goods). Some examples of these local public goods might include private club membership, local environmental quality, tickets to a concert stadium, etc. There are three basic economic issues involving public goods: [1] Efficient allocation of public goods We already know that in an economy with private goods only, Pareto Optimal conditions such as, MRSA = MRSB allow us to determine efficient allocations of goods and factors. Now, the issue is: Do these Pareto Optimal conditions still hold in the presence of public goods? [2] Pricing of public goods We also know that in an economy with only private goods, market equilibrium conditions such as Qdemanded = Qsupplied allow us to determine equilibrium prices and quantities in the markets. Again, the issue at hand is: Do these market equilibrium conditions still hold in the presence of public goods? [3] The free rider problem The non-excludability feature of public goods suggests that it is possible for people to enjoy the full benefits of public goods through joint consumption, without contributing anything to the costs of public goods. So the policy issue is: How do we encourage people to voluntarily contribute to the costs of public goods? SAMUELSON'S ALLOCATION OF PUBLIC GOODS Our first question is answered using Samuelson's theory on the allocation of public goods. Paul Samuelson's contribution to the theory of public goods lies in his innovative formulation of the Pareto Optimal allocation problem of an economy having both public and private goods. Consider an economy with two people, A and B, and two goods, X and Y. Let X be a public good and let Y be a private good. 237 [a] The fact that X is a public good means that we have non-excludability and joint consumption (non-rivalrous consumption) for X which can be formulated analytically as follows: XA = XB = X That is, there are no restrictions on the consumption of X. Not only can person A consume as much of good X as person B but also both people can consume as much of good X as can be produced. [b] On the other hand, the fact that good Y is a private good means that the total amount of good Y allocated to both people A and B must match the total amount of good Y. That is, YA + YB = Y [c] The usual Pareto optimality requirement that no one can be made better off without making someone else worse off can be analytically formulated as the following welfare maximization problem: maximize UB (XB, YB) subject to UA (XA, YA) = A where A is some arbitrary level of utility for consumer A. That is, one person (person B, in this case) can attain the highest level of utility without affecting the utility level of the other person (we have constrained consumer A's utility to be a fixed level). The optimal solution to this formulation cannot be improved upon (i.e. Pareto optimal allocation). [d] The production side of the economy can be succinctly described by a PPF. We write the equation of the PPF as F(X, Y) = 0 which summarizes all Pareto optimal factor allocations (KX, KY, LX, LY) satisfying the optimal conditions of the production side of the economy. MRTSX = MRTSY KX + KY = K LX + LY = L X = f(KX, LX) Y = g(KY, LY) 238 That is, as we said previously in this course, the PPF equation alone is functionally equivalent to the system of five optimal equations that characterize the production side of the economy. Private Good Y PPF F (X, Y) = 0 Public Good X In terms of production, there is no real difference between public and private goods: both of them require the same capital and labour to be input into the same production processes of the economy. For example, we use the same manufacturing process to produce buses whether they are to transport children to public school (public good) or for private (fee per trip) transportation (i.e. Greyhound). [e] In summary, the model consists of the following components: [i] non-excludability and joint consumption (non-rivalrous consumption) of public goods (equation (3) below). [ii] excludability and rivalrous consumption of private goods (equation (4) below). [iii] Pareto optimality (equations (1) and (2) below). [iv] and a PPF (equation (5) below) Samuelson's theory of efficient allocation of public goods can now be re-stated as the following constrained welfare maximization problem: maximize UB (XB, YB) subject to UA (XA, YA) = A 239 (1) (2) XA = XB = X YA + YB = Y F(X, Y) = 0 (3) (4) (5) Theoretically, we can use the calculus techniques of classical optimization to solve this complicated problem of multiple constraint welfare maximization. You will be relieved to know that this is not the route we are going to take. Instead, we will use a simple diagrammatic approach to investigate this problem. [1] First, we describe the production side of the economy by using the PPF from equation (5). Private Good Y PPF F (X, Y) = 0 Public Good X [2] Then we draw an indifference curve for consumer A which corresponds to the given utility level A as in equation (2). Private Good Y ICA = A PPF Public Good X 240 [3] Since good X is a public good, the amount of the good X produced by the PPF must be the same as the amount consumed by EACH person... XA = XB = X On the other hand, since good Y is a private good, the amount of good Y produced by the PPF must be allocated BETWEEN both consumers... YA + YB = Y For each level of good X on the horizontal axis, the total amount of good Y produced must first be allocated to YA required by consumer A so that his welfare level is unchanged (as required by the constraint). Whatever amount of good Y is left over is then allocated to person B. YB = Y YA In other words, the vertical difference between the PPF and the indifference curve of A represents all the options available to B in order to maintain the same welfare level for person A. [Options Available to B Without Hurting A] = PPF ICA Private Good Y ICA = A Options Available to B PPF Public Good X [4] With the choice set of all options available to B (without hurting A) being fully described, the remaining task is to pick the point on that option curve which corresponds to the highest indifference curve for B. This amounts to choosing the highest indifference curve ICB which is tangent to the option curve (point S in the diagram below). 241 Private Good Y S ICB PPF ICA = A Public Good X [5] At point S of the optimal solution to the constrained welfare maximization problem, the curve of options available to B must be tangent to the indifference curve of B. In other words, these two curves must have the same slope at the point of tangency, point S. [Slope of Options Curve for B] = [Slope of PPF] MRT So... MRSA + MRSB = MRT [Slope of ICA] MRSA = = [Slope of ICB] [Slope of ICB] MRSB So, we can see the difference between Samuelson's theory and the regular Pareto optimal allocation rules... Recall that the usual Pareto optimal allocation rules equate all individual marginal rates of substitution with the common marginal rate of transformation. That is, in a two consumer, two good world the following equation must be satisfied: MRSA = MRSB = MRT Samuelson's theory basically shows that, in the presence of public goods, this usual Pareto optimal allocation rule does not hold anymore. Instead, we must replace it with the Samuelson allocation rule... MRSA + MRSB 242 = MRT which "adds" (not "equates") all individual marginal rates of substitution and then matches them with the common marginal rate of transformation. EXAMPLE Consider the square root economy with unit factor endowments. We have done this example before and found that the equation of the PPF is X+Y=1 with a marginal rate of transformation of MRT = 1 Let's consider the optimal solutions of the Production economy for the usual case of private goods side-by-side with the case of one private and one public good: PRIVATE 222 Model data Good X private good Good Y private good Pareto Optimal Allocation Rules MRSA = MRSB = MRT XA + XB = X YA + YB = Y X+Y=1 Applying to the square root economy YA = YB = 1 XA XB XA + XB = X YA + YB = Y X+Y=1 PUBLIC 222 Model data Good X public good Good Y private good Samuelson Optimal Allocation Rules MRSA + MRSB = MRT XA = XB = X YA + YB = Y X+Y=1 Applying to the square root economy YA + YB = 1 XA XB XA = XB = X YA + YB = Y X+Y=1 243 Solving for the optimal solution YA = XA YB = XB X = XA + XB = Y = YA + YB = Utility Possibility Frontier (UPF) UA + UB = UB Solving for the optimal solution X = XA = XB = Y = YA + YB = Utility Possibility Frontier (UPF) [UA]2 + [UB]2 = UB UA UA LINDAHL'S PRICING OF PUBLIC GOODS Now that Samuelson has figured out how we should allocate public goods, we need a process for dealing with the problem of pricing them. So while Samuelson's theory addresses the issue of Pareto optimal allocations of public goods, Lindahl's theory attempts to answer the following question: Does the usual concept of market equilibrium "Demand = Supply" still apply in the presence of public goods? This is an important question because if we can answer "yes", then we can use our familiar market equilibrium condition D = S to solve the problem of public good pricing. Since there are no differences in the production of public goods relative to private goods (supply side), the main problem is to investigate whether it makes any sense to have the notion of market demand for public goods. 244 PUBLIC GOODS If good X is a public good, then everyBody consumes the same amount of good X. As a result, the difference is the price each person is willing to pay for the good. Given the common amount of X A pays the price PXA B pays the price PXB A and B pay the price PXA + PXB The market demand for a public good X is the vertical summation of all the individual "willingness to pays". Price PRIVATE GOODS If good Y is a private good, then everybody pays the same price for good Y. As a result, the difference is the amount of good Y each person is willing to buy. Given the common price PY A buys the amount YA B buys the amount YB A and B buy the amount YA + YB The market demand for private good Y is the horizontal summation of all individual demands. Price PX + PX A B PX PX B A PY Public Good X X Y A Private Good Y Y B Y +Y A B Once both the market demand and supply curves for the public good X have been determined, we can solve for the market equilibrium by equating demand and supply. In this final step, there is no difference between a market for a public good and a market for a private good. Once both the market demand and supply curves for the private good Y have been determined, we can solve for the market equilibrium by equating demand and supply. In this final step, there is no difference between a market for a public good and a market for a private good. 245 Price Price S S PX PY D Public Good X X Y D Private Good Y EXAMPLE Again, consider the square root economy (example used from a previous lecture) with unit aggregate endowments allocated between two consumers A and B as follows: Consumer A Consumer B Total Capital (K) KA = 0.2 KB = 0.8 KT = 1 Labour (L) LA = 0.6 LB = 0.4 LT = 1 Let's look at the example side-by-side for the usual case of private goods and the case of public goods (using Lindahl pricing): PRIVATE 222 Solving the cost minimization problem faced by the producer of each good we have the following factor demands on a per unit of output basis: kX = kY = (w / r)1/2 lX = lY = (r / w)1/2 using the zero profit condition of perfect competition, we can express output prices in terms of factor prices PX = 2(wr)1/2 PY = 2(wr)1/2 246 PUBLIC 222 Solving a cost minimization problem faced by the producer of each good we have the following factor demand on a per unit of output basis: kX = kY = (w / r)1/2 lX = lY = (r / w)1/2 using the zero profit condition of perfect competition, we can express output prices in terms of factor prices PX = 2(wr)1/2 PY = 2(wr)1/2 Solving for individual demand functions from the usual constrained utility maximization problem XA = 0.2r + 0.6w 2PX XB = 0.8r + 0.4w 2PX Solving individual demand functions from the usual constrained utility maximization problem. XA = 0.2r + 0.6w 2PXA XB = 0.8r + 0.4w 2PXB For the public good model, we do one extra step to invert these individual demand functions so that prices are now expressed in terms of quantities PXA = 0.2r + 0.6w 2 XA PXB = 0.8r + 0.4w 2 XB Now, remember that for public goods XA = XB = X so the above expressions become... PXA = 0.2r + 0.6w 2X PXB = 0.8r + 0.4w 2X For private good X, we use horizontal summation to obtain the market demand for good X from the two individual demand functions X = XA + XB =r+w 2PX For public good X, we use vertical summation to obtain the market demand for good X from the two individual willingness to pays PX = PXA + PXB =r+w 2X Again we do an extra step to convert the market willingness to 247 pay into the usual form of demand functions where quantities are expressed in terms of prices X=r+w 2PX Note that we just happen to have the same market demand for good X in this example for both "X as a private good" and "X as a public good"... Substituting the zero profit condition above so that the market demand for good X is expressed in terms of factor prices only X=r+w 4(wr)1/2 Subbing the zero profit condition above so that the market demand for good X is expressed in terms of factor prices only X=r+w 4(wr)1/2 Since good Y is private, the derivation of the market demand for good Y should be the same for both models Y=r+w 4(wr)1/2 Using these market demands for goods in the factor market equilibrium conditions kX X + kY Y = K lX X + lY Y = L and solving for the equilibrium prices r=w=1 PX = PY = 2 Y=r+w 4(wr)1/2 Using these market demands for goods in the factor market equilibrium conditions kX X + kY Y = K lX X + lY Y = L and solving for the equilibrium prices r=w=1 PX = PY = 2 248 Solving for equilibrium quantities XA = 0.2 XB = 0.3 X = 0.5 YA = 0.2 YB = 0.3 Y = 0.5 Solving for equilibrium quantities XA = 0.5 XB = 0.5 X = 0.5 YA = 0.2 YB = 0.3 Y = 0.5 And the prices that consumer A and consumer B are willing to pay for the same amount of public good X PXA = 0.2r + 0.6w 2X PXB = 0.8r + 0.4w 2X = 0.8 = 1.2 INCENTIVE MECHANISM So we have addressed the issue of optimal allocation of public goods (Samuelson) and the issue of pricing public goods (Lindahl) but if you can recall the beginning of the class, we said there were three issues surrounding public goods. The third issue is the free rider problem...how do we get people to reveal their willingness to pay for a public good? For that matter, how do we get them to be willing to pay anything at all? So to have an appropriate solution to the Lindahl pricing method above, we need a truthful reporting of individual's willingness to pay for the public good. If people do not truthfully reveal the prices that they are willing to pay for the public good, X, the method of vertical summation will yield an incorrect market demand for the public good. Consequently, the market equilibrium condition Demand = Supply will give us an incorrect equilibrium quantity of the public good in the economy. Presumably, with an under-reporting of WTP there will be an under-provision of the public good in the market. Why might people lie about or otherwise mis-report their willingness to pay for public goods? 249 By their very nature, public goods are available to all consumers in the economy regardless of whether they contribute to the cost of their production or not. We cannot exclude those who do not pay nor can we distribute the public good to only those that pay. Consumers generally learn quickly that if they do not reveal their demands for public goods, they can get away with paying less than what they should have paid for their consumption of the public goods (or even not pay at all). On the other hand, they cannot be prevented from enjoying the public goods and so there are economic incentives for consumers to be a free rider in an economy that includes public goods. How do we solve this free rider problem? Since the source of the free rider problem comes from the fact that there are economic incentives for consumers to hide their true valuations of public goods, the solution concept is to create an economic incentive mechanism that will give consumers the incentive to report their true valuations (such as some sort of penalty for mis-reporting). In the presence of this incentive mechanism, there would be a cost to consumers for not revealing their true valuations of public goods. The consumer decision to lie or reveal their true valuation becomes a cost/benefit problem of the individuals' economic welfare calculations. The design of incentive mechanisms is a technically difficult subject in microeconomic theory and is of particular interest to public economists. We will not explore this theoretical construct in any further detail (leaving this for ECON 341). For the purposes of this course, what we need to understand is that the free rider problem exists, what it is, and that there is some advanced theoretical models designed to induce consumers to truthfully report their valuations of public goods. 250 Supplement to Lecture #14 Detailed Lindahl Pricing Example Consider the "square root economy" with square root functions for both consumers and producers UA = XA1/2YA1/2 UB = XB1/2YB1/2 QX = KX1/2LX1/2 QY = KY1/2LY1/2 and the following initial factor endowment distribution: Consumer A Consumer B Total Capital (K) KA = 0.2 KB = 0.8 KT = 1 Labour (L) LA = 0.6 LB = 0.4 LT = 1 We want to find the equilibrium prices (PX, PY, r, w) that clear all markets simultaneously. CONSUMER A Consumer A has the following data on utility function and endowment income: MRSA = YA XA MA = r KA + w LA = 0.2 r + 0.6 w Utility Maximization maximize UA = XA1/2YA1/2 subject to PX XA + PY YA = MA Consumer Equilibrium Analytically, the two conditions for consumer equilibrium must be satisfied: CONSUMER B Consumer B has the following data on utility function and endowment income: MRSB = YB XB MB = r KB + w LB = 0.8 r + 0.4 w Utility Maximization maximize UB = XB1/2YB1/2 subject to PX XB + PY YB = MB Consumer Equillibrium Analytically, the two conditions for consumer equilibrium must be satisfied: 251 MRSA = PX PY PX XA + PY YA = MA Solving these two equations for XA & YA we get the demands by consumer A. XA = M A 2PX XA = 0.2r + 0.6w 2PX Additional Step (notice ... but ignore for now) MRSB = PX PY PX XB + PY YB = MB Solving these two equations for XB & YB we get the demands by consumer B. XB = MB 2PX XB = 0.8r + 0.4w 2PX Since good X is our Public Good we need one extra step to find the individual consumer's willingness to pay for the public good. We can derive this from their individual demand functions as follows: XA = 0.2r + 0.6w 2PXA PXA = 0.2r + 0.6w 2 XA XB = 0.8r + 0.4w 2PXB PXB = 0.8r + 0.4 2 XB Now, remember that for public goods XA = XB = X so the above expressions become... PXA = 0.2r + 0.6w 2X PXB = 0.8r + 0.4 2X Now back to our original problem strategy. XA = r + 3w 10PX YA = M A 2PY YA = 0.2r + 0.6w 2PY YA = r + 3w 10PY 252 XB = 4r + 2w 10PX YB = MB 2PY YB = 0.8r + 0.4w 2PY YB = 4r + 2w 10PY PRODUCER OF GOOD X The producer of good X has the following MRTS: MRTSX = LX KX Cost Minimization minimize cost r KX + w LX subject to KX1/2LX1/2 = QX Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: MRTSX = _r_ w KX1/2LX1/2 = QX Solving these two equations for KX & LX we get the demands by producer X. KX-1/2LX1/2 = _r_ w KX1/2LX1/2 LX = _r_ KX w KX = LXw r LX = KXr w sub KX into production function: QX = [LX(w / r)]1/2LX1/2 QX = LX(w / r)1/2 253 PRODUCER OF GOOD Y The producer of good Y has the following MRTS: MRTSY = LY KY Cost Minimization minimize cost r KY + w LY subject to KY1/2LY1/2 = QY Producer Equilibrium Analytically, the two conditions for producer equilibrium must be satisfied: MRTSY = _r_ w KY1/2LY1/2 = QY Solving these two equations for KY & LY we get the demands by producer Y. KY-1/2LY1/2 = _r_ KY1/2LY1/2 w LY = _r_ KY w KY = LYw r LY = KYr w sub KY into production function: QY = [LY(w / r)]1/2LY1/2 QY = LY(w / r)1/2 LX = QX(r / w)1/2 Similarly, we can find that... KX = QX(w / r)1/2 Constant Returns to Scale Since f(KX,LX) is a constant returns to scale production function, we can divide factor demands KX,LX by the total output level QX in order to get the factor demands on a per unit of output basis as follows: KX = kX = (w / r)1/2 QX LX = lX = (r / w)1/2 QX Marginal Cost MCX = r kX + w lX Perfect Competition Under perfect competition, producer X must satisfy the zero profit condition: PX = MCX (r,w) PX = r kX + w lX PX = r (w / r)1/2 + w (r / w)1/2 PX = (wr2 / r)1/2 + (rw2 / w)1/2 PX = 2(wr)1/2 MARKET FOR GOOD X Substituting these output prices into individual consumer demands... 254 LY = QY(r / w)1/2 Similarly, we can find that... KY = QY(w / r)1/2 Constant Returns to Scale Since g(KY,LY) is a constant returns to scale production function, we can divide factor demands KY,LY by the total output level QY in order to get the factor demands on a per unit of output basis as follows: KY = kY = (w / r)1/2 QY LY = lY = (r / w)1/2 QY Marginal Cost MCY = r kY + w lY Perfect Competition Under perfect competition, producer Y must satisfy the zero profit condition: PY = MCY (r,w) PY = r kY + w lY PY = r (w / r)1/2 + w (r / w)1/2 PY = (wr2 / r)1/2 + w (rw2 / w)1/2 PY = 2(wr)1/2 MARKET FOR GOOD Y Substituting these output prices into individual consumer demands... XA = r + 3w = r + 3w 10PX 20(wr)1/2 XB = 4r + 2w = 4r + 2w 10PX 20(wr)1/2 and aggregate demand for good X X = r + 3w + 20(wr)1/2 X = 5r + 5w 20(wr)1/2 X=r+w 4(wr)1/2 X = X(r , w) On the supply side, producer X provides QX as the market supply of good X. At market equilibrium, the output supply of good X must equal the aggregate demand of good X. QX = X(r , w) = r + w 4(wr)1/2 MARKET FOR CAPITAL Substituting this output supply, QX, into the producer demands for capital, we get... KX = QX(w / r)1/2 w1/2 KX = r + w 1/2 4(wr) r1/2 KX = r + w 4r 255 4r + 2w 20(wr)1/2 YA = r + 3w = r + 3w 10PY 20(wr)1/2 YB = 4r + 2w = 4r + 2w 10PY 20(wr)1/2 and aggregate demand for good Y Y = r + 3w + 20(wr)1/2 Y = 5r + 5w 20(wr)1/2 Y=r+w 4(wr)1/2 Y = Y(r, w) On the supply side, producer Y provides QY as the market supply of good Y. At market equilibrium, the supply of good Y must equal the aggregate demand of good Y. QY = Y(r , w) = r + w 4(wr)1/2 MARKET FOR LABOUR Substituting this output supply, QY, into the producer demands for labour, we get... LX = QX(r / w)1/2 LX = r + w r1/2 1/2 4(wr) w1/2 LX = r + w 4w 4r + 2w 20(wr)1/2 KY = QY(w / r)1/2 w1/2 KY = r + w 4(wr)1/2 r1/2 KY = r + w 4r and aggregate demand for capital K=r+w 4r K=r+w 2r At capital market equilibrium, the Aggregate capital demand must match the aggregate capital supply. r+w 2r =1 +r+w 4r LY(r, w) = lY QY LY = r + w r1/2 4(wr)1/2 w1/2 LY = r + w 4w and aggregate demand for labour L=r+w +r+w 4w 4w L=r+w 2w At labour market equilibrium, the the aggregate labour demand must match the aggregate labour supply. r+w =1 2w If we solve for capital market equilibrium condition, we get the relationship between the price of labour and the price of capital. 2w = r + w w=r r/w=1 This is the equilibrium price solution that we need to figure out everything else. If we solve for capital market equilibrium condition, we get the following relationship between the price of labour and the price of capital. 2r = r + w r=w w/r=1 This is the equilibrium price solution that we need to figure out everything else. Let's figure out the rest of our equilibrium price vector... If we choose capital as the numeraire then r=1 256 r=w=1 PX = 2(wr)1/2 = 2r = 2 PY = 2(wr)1/2 = 2r = 2 Now let's figure out the consumer demands in equilibrium (both private goods for now)... XA = r + 3w = r + 3w 10PX 20(wr)1/2 X*A = 4 / 20 = 1 / 5 XB = 4r + 2w = 4r + 2w 10PX 20(wr)1/2 X*B = 6 / 20 = 3 / 10 So goods market demands are... X* = X*A + X*B = Y* = Y*A + Y*B = YA = r + 3w = r + 3w 10PY 20(wr)1/2 Y*A = 4 / 20 = 1 / 5 YB = 4r + 2w = 4r + 2w 10PY 20(wr)1/2 Y*B = 6 / 20 = 3 / 10 and just to check market clearing, this should be the market supply (i.e. amount produced by the producers) KX = r + w 4r KY = r + w 4r K=r+w 2r = = =1 LX = r + w = 4w LY = r + w = 4w L=r+w=1 2w (notice that this implies factor market clearing since factor endowments = 1 for each of L and K) QX = KX1/2LX1/2 QX = (1/2)1/2(1/2)1/2 QX = (notice goods market clearing, here!!!!) 257 QY = KY1/2LY1/2 QY = (1/2)1/2(1/2)1/2 QY = So all markets clear at the equilibrium price vector (PX, PY, r, w) = (2, 2, 1, 1) and the resulting demands for goods and factors are: (XA, YA) = (2/10 , 2/10) (KX, KY) = (1/2 , 1/2) (XB, YB) = (3/10 , 3/10) (LX, LY) = (1/2 , 1/2) However, if good X is our public good, then ... For public good X, we use vertical summation to obtain the market demand for good X from the two individual willingness to pays PX = PXA + PXB =r+w 2X Again we do an extra step to convert the market willingness to pay into the usual form of demand functions where quantities are expressed in terms of prices X=r+w 2PX Note that we just happen to have the same market demand for good X in this example for both "X as a private good" and "X as a public good"...this is not usually the case! Substituting the zero profit condition above so that the market demand for good X is expressed in terms of factor prices only... X=r+w 4(wr)1/2 Since good Y is private, the derivation of the market demand for good Y should be the same for both models Y=r+w 4(wr)1/2 Using these market demands for goods in the factor market equilibrium conditions kX X + kY Y = K lX X + lY Y = L 258 and solving for the equilibrium price (not any different from the 2 private good case above) the resultant price vector is as it was in the two good case. r=w=1 PX = PY = 2 r=w=1 PX = PY = 2 Solving for equilibrium quantities using the Public Good model... XA = 0.2r + 0.6w 2PXA PXA = 0.2r + 0.6w 2 XA XB = 0.8r + 0.4w 2PXB PXB = 0.8r + 0.4 2 XB Now, remember that for public goods XA = XB = X so the above expressions become... PXA = 0.2r + 0.6w 2X PXB = 0.8r + 0.4 2X And the prices that consumer A and consumer B are willing to pay for the same amount of public good X PXA = 0.2r + 0.6w 2X PXA = 0.2 + 0.6 1 PXB = 0.8r + 0.4w 2X PXB = 0.8 + 0.4 1 = 1.2 = 0.8 Solving equilibrium quantities using the equilibrium price vector (PXA, PXB, PX, PY, r, w) = (0.8, 1.2, 2, 2, 1, 1) we get... XA = 0.5 YA = 0.2 XB = 0.5 YB = 0.3 X = 0.5 Y = 0.5 And the capital and labour markets clear exactly as they did in the 2 Private good case. 259 ECON 301 LECTURE #15 EXTERNALITIES As we have been discussing, the existence of public goods has a serious impact on both welfare and the resource allocation in the economy. Aside from public goods, other phenomena such as externalities have far-reaching effects, both positive and negative, on the efficiency of the market mechanism. The subject of externalities is, however, quite conceptually difficult and technically demanding. POSITIVE VS NEGATIVE EXTERNALITIES Individuals in a market pricing economy are normally independent and unrelated to each other. The only indirect link among them is the pricing system in the market. [Economic Activities of Individual A] Prices (Indirect) [Economic Activities of Individual B] Externalities refer to the situation where the economic activities of one individual (or group of individuals) directly affect those of the other individuals. Common examples of externalities include second-hand smoke, drunk driving and environmental pollution (to name just a few). For instance, the economic activities of a drunk driver have a direct effect on car accident victims. This direct effect is completely outside the normal market pricing system. [Economic Activities of Individual A] Externalities (Direct) [Economic Activities of Individual B] The examples of externalities presented so far have all had negative effects on others. Are all externalities negative? No! Externalities are not necessarily always bad. The effect of the economic activities of one individual on another individual can be either positive (in which case we call these positive externalities) or negative (in which case, you guessed it, we call these negative externalities). Examples of positive externalities include the happy event that bees owned by a bee keeper pollinate the apple trees of a neighboring apple orchard. Examples of a negative externality include the unhappy event that the babies of drug addicted mothers become addicted and incur severe health problems (physical and developmental) before they are even born. We can classify externalities in ways other than simply positive or negative. The positive or negative classification refers only to the sign or magnitude of the 260 externality without revealing the source of the externality or who receives the consequences of the externality. To help clarify, the following table might help... Consumer Producer Consumer (a) (c) Producer (b) (d) The example of the bee keeper and the apple grower falls under category (d) where there are "producer producer" externalities whereas the unborn drug addict example falls under category (a) where there are "consumer consumer" externalities. A typical example of a category (c) externality or "producer consumer" externalities is the case of chemical dumping (i.e. a factory pouring chemical pollutants into a river basin that turns out to be the source of drinking water for local residents...(see Erin Brocovich for details)). Finally, an example of a category (b) externality or "consumer producer" externalities is the case where the popularity of home computer use gives rise to increases in computer literacy in the labour force which in turn increases productivity in many sectors of the economy. SOCIAL VS PRIVATE COSTS Of course, negative externalities impose costs to society and positive externalities accrue some benefit to society. Being economists, we are interested primarily with the costs incurred by negative externalities (economics being the "dismal science"). In the absence of externalities, there are no differences between social and private costs since everything is fully accounted for in our calculations of costs and benefits. Optimal decisions are characterized by the balance between the marginal private costs incurred by individuals and the marginal private benefits accrued to individuals. This condition (optimal cost/benefit analysis) applies to both individuals and society. [Marginal Private Costs] = [Marginal Private Benefits] In the presence of externalities, social costs are, however, different than private costs in the magnitude of the effect of the externality. Individual optimal decisions are characterized by the balance between marginal private costs and marginal private benefits. On the other hand, social optimal decisions must account for the effects of the externalities (which have been ignored by the individuals). As a result, social optimal decisions are characterized by the balance between marginal private costs and benefits "net of externalities". 261 [Marginal Private Costs] including externalities = [Marginal Private Benefits] including externalities In other words, we need to have a balance between marginal social costs and marginal social benefits as follows: [Marginal Social Costs] = [Marginal Social Benefits] For example, one of the key economic issues in environmental pollution is that a polluter is making an individual optimal decision (without having to account for its polluting activities) while society has to bear the costs of environmental clean-up. The situation is sub-optimal because we have individual optimality but a suboptimal condition for society. Many policy analysts have studied this particular issue and governments have attempted pollution fines, pollution taxes, and most recently there has been some promise in the theory of pollution permits to limit the aggregate pollution in society to an "optimal (?)" pollution level. Any sort of regulation as it pertains to pollution reduction strategies must be enforceable and needs to monitored. Perhaps, in today's generation where the environment has overtaken the economy as the foremost issue in voters' minds, government policy will find new ways (or re-address old methods) of bringing the marginal private costs of pollution in line with the marginal social costs of pollution (i.e. make industry accountable). INEFFICIENCY OF EXTERNALITIES Where does the inefficiency of externalities come from? In the presence of externalities, the usual market pricing mechanism fails to send a correct market signal to both the culprit and the victims. For example, polluters keep polluting because clean-up costs are not included in their optimal decisions. At the same time, the victims keep getting hurt by the pollution in the environment because compensation remedies are not included in their optimal decisions as well. In other words, the economic inefficiency of externalities comes from the fact that the effects of externalities are not incorporated into the optimal economic decisions of individuals in the economy. So, how can we correct this situation? Since the main problem of externalities is that externalities are not incorporated into the optimal decisions of individuals, the solution concept is, in principle, quite simple: 262 If individuals do not incorporate externalities into their optimal decisions then we will have to make them accountable for the externalities caused by their actions. If the problem was caused by the fact that externalities have remained to be "outside" the optimal decisions of individuals, then a solution to the problem requires us to move them "inside" the individual decision. In other words, the solution concept boils down to ... Internalizing Externalities EXAMPLE Consider a simple example economy with unit aggregate endowments of both goods X and Y. X=Y=1 unit initial endowment incomes M A = MB = 1 and the following utility functions for person A and person B: UA = XAYA XB UB = XBYB The interesting thing about this example is that the quantity (XB) of good X consumed by B now enters the utility function of A. Everything else will be the same, the more good X that B consumes, the less utility that A will get. This is a simple example of a negative externality of the "consumer-consumer" type. Let's use the contract curve to illustrate...first, we need to solve for the contract curve! To derive the contract curve for this externality economy, we need to solve the Pareto problem. A unit of aggregate endowment for both goods means that inside the Edgeworth Box, we have the following equations: XA + XB = 1 YA +YB = 1 First, we need to use the fact that XB = 1 XA to eliminate the XB term from the utility function of A. 263 UA = XAYA XB UA = XAYA (1 XA) UA = (YA + ) XA And then calculating the marginal rate of substitution for person A MRSA = (YA + ) XA and then calculating the marginal rate of substitution for person B MRSB = YB XB Along the contract curve, we have the following Pareto optimal conditions: MRSA = MRSB XA + XB = 1 YA +YB = 1 Applying these conditions to the specific problem at hand (YA + ) = YB XA XB (YA + ) = 1 YA 1 XA XA And we get the following contract curve (convince yourself that this is correct!): YA = 5 XA 4 which is a straight line passing through the horizontal intercept (1/5, 0) instead of the point of origin for person A (convince yourself that this is correct!). 264 YA XB OB Contract Curve OA 1/5 Y B XA What might the Walrasian General Equilibrium solution look like? The following analysis shows the general equilibrium solution for this simple economy where externalities are not corrected for by any form of taxes on the polluter (person B, in this case) or subsidies to the victim (person A, in this case). VICTIM Utility Maximization Max XAYA XB subject to PX XA + PY YA = MA with XB being taken as given. At consumer equilibrium, we have the following conditions (ignoring the externality): YA = PX XA PY PX XA + PY YA = MA Solving these two equations, we have the following consumer demand functions for the victim: 265 At consumer equilibrium, we have the following conditions: YB = PX XB PY PX XB + PY YB = MB Solving these two equations, we get the following consumer demand functions for the polluter: POLLUTER Utility Maximization Max XBYB subject to PX XB + PY YB = MB XA = M A 2PX YA = M A 2PY XB = MB 2PX YB = MB 2PY At the market equilibrium for both goods, we have the following equations: XA + XB = X YA + YB = Y which can be solved for equilibrium prices and quantities PX = PY = 1 XA = XB = YA = YB = YA XB OB General Equilibrium Solution (no externality) Contract curve (with externality) OA 1/5 Y B XA The diagram shows that the uncorrected general equilibrium solution XA = XB = YA = YB = is not on the contract curve derived when accounting for the negative externality. That is, in the presence of externalities, the uncorrected general equilibrium solution is no longer on the contract curve of Pareto Optimal allocations. In other 266 words, there is no relationship between the two inter-twined concepts of Pareto optimality and general equilibrium: in the presence of externalities, the first fundamental theorem of welfare economics breaks down! How do externalities affect the production side? To examine the effects of externalities on the production side, we consider the following simple example of an economy with and without externalities. Labour is the only factor in the economy consisting of a fixed aggregate endowment of L=8 WITHOUT EXTERNALITIES Given production functions X = 2.5 LX Y = 50 LY WITH EXTERNALITIES Given production functions X = 2.5 LX Y = (50 2.5X) LY Note that in this externality case, the production of good X has a detrimental effect on the production of good Y. To derive the PPF for this simple economy, we substitute the production functions into the factor allocation condition LX + LY = L _X_ + _Y_ = 8 2.5 50 Solving for X and Y, we have Y + 20X = 400 This is a linear equation which means that the PPF is a straight line. To derive the PPF for this economy, we substitute production functions into the factor allocation condition LX + LY = L _X_ + ____Y____ = 8 2.5 (50 2.5X) Solving for X and Y, we have Y + 40X X2 = 400 This is a quadratic equation which means that the PPF is a parabolic curve which we refer to as a non-convex PPF. 267 Y 400 Y 400 X 20 20 X So, how does this non-convex PPF compare to the usual PPF? The key difference is the direction of the curvature as shown below (this will have consequences for our maximization problems). Y 400 Y 400 Normal PPF Non-convex PPF X 20 20 X A normal PPF does not go below the straight line. Okay. A non-convex PPF does not go above the straight line. So what's wrong with a non-convex PPF? The main issue with a non-convex PPF is that it does not give us a correct solution to the welfare maximization process. For example, when we use the concept of a social welfare function (SWF) to break the indeterminacy of the Pareto optimal solution, a non-convex PPF can produce more than one point of tangency (see the diagrams below). 268 Y 400 SWF Contour Non-convex PPF 20 X or we might get the point of tangency at a minimum instead of a maximum welfare solution! Y 400 Non-convex PPF Minimum Welfare 20 INSTITUTIONAL CONSTRAINTS X We have been considering some sources of economic inefficiency. We will continue in this spirit by thinking about some additional sources of economic inefficiency arising from institutional constraints such as monopoly power, government policies, and regulations. With this context in mind, we will look at some of the potential pitfalls when conducting economic policy by examining the "theory of second best". 269 MONOPOLY We already know that monopoly and perfect competition are the two ends of the spectrum of market structures. [Perfect Competition] Market Structure [Monopoly] Let's refresh our memory by comparing the salient features of these two extreme theoretical models of market structure side-by-side. PERFECT COMPETITION Many individual firms make up the supply side of the market. None of these firms has any significant influence on the market. Both free entry and exit are allowed. All competitive firms face horizontal demand curves. MONOPOLY One single firm controls the supply side of the market. This single firm (called the monopolist) has a large influence on the market. There is no free entry, but free exit is sometimes still allowable. The monopolist faces a downward sloping demand curve and a corresponding marginal revenue curve. Profits can be positive, negative or zero depending on how high or low the demand curve is. The optimal level of output for a monopolist is determined by the rule MR = MC Which is the intersection point of (a) marginal revenue curve MR (b) marginal cost curve MC There are zero profits in the long run due to free entry and exit. The optimal output level of a competitive firm is determined by the rule P = MC Which is the intersection point of (a) horizontal price line PC (b) marginal cost curve MC (c) average cost curve AC 270 P MC AC P MC PM PC D MR QM Q QC Q There is a very close relationship between the firm's demand curve and its marginal revenue curve. Consider the following demand curve: P = f(Q) where P and Q denote the price and quantity, respectively. The total revenue is defined as the product of price and quantity TR = P Q and the corresponding marginal revenue is defined as the change in the total revenue as a result of a change in the quantity, Q. MR = TR Q = PQ + QP Q = P + Q(P/Q) Multiplying and dividing the last term on the RHS by one (P/P), we get... = P + P Q P P Q = P + P (P/P) (Q/Q) = P + P (%P) (%Q) 271 = P + P ___1__ (%Q) (%P) = P + P __-1__ - (%Q) (%P) Noticing, as I'm sure all of you have, that the ratio of the two percentage changes in the denominator is simply the price elasticity of demand = - (%Q) (%P) we can, therefore, rewrite our MR expression as... MR = P + P __-1__ - (%Q) (%P) MR = P + P __-1__ MR = P [1 1/] In other words, the marginal revenue (MR) is linked to the price (P) by the price elasticity of demand () according to the formula: MR = P [1 1/] In the limiting case of a horizontal demand curve (perfectly competitive firm demands), the price elasticity of demand is approaching infinity and the marginal revenue is equal to the price MR = P This is the case of the perfectly competitive market structure. How does monopoly compare to this result? Well, we know that under perfect competition output is determined by P = MC where the horizontal demand curve PC intersects the marginal cost curve MC at the minimum point of the average cost curve AC. 272 On the other hand, with a monopoly, the output is determined by the rule MR = MC where the marginal revenue curve MR intersects the marginal cost curve MC at a point to the left of the minimum point of the average cost curve AC. P AC MC PM PC = MRC DM MR QM QC Q Since the marginal revenue curve is below the monopolist's market demand curve, the price charged by the monopolist is higher than the corresponding price under perfect competition. There will be a higher price and lower quantity under monopoly than under perfect competition. In other words, we have the following result: Monopoly is less efficient than Perfect Competition. Does this result hold under general equilibrium conditions? Our analysis above obtains the result that monopoly is less efficient than perfect competition using partial equilibrium analysis. To see whether this statement holds in general equilibrium, we will need to investigate whether the monopoly creates any departure from the usual Pareto optimal conditions of an economy under perfect competitive forces. For example, consider an economy of two sectors with different market structures as follows: 273 [1] Monopoly in sector X Sector X has a monopolist operating according to the profit maximization rule MRX = MCX [2] Perfect Competition in sector Y Sector Y has competitive firms operating according to the zero profit condition PY = MCY The factor markets for capital and labour are still under perfect competition, and hence, we can still pick a point on the production possibility frontier of the economy. In terms of consumption, we have the following marginal rate of substitution: MRS = PX PY On the other hand, in terms of production, we have the following marginal rate of transformation: MRT = MCX MCY MRT = MCX PY And since MRX = MCX in equilibrium... MRT = MRX PY Applying the relationship between marginal revenue and market price calculation that we did previously (i.e. MCX = PX [ 1 1/]), we get MRT = PX [ 1 1/] PY MRT = PX [ 1 1/] PY MRT = MRS [ 1 1/] 274 The presence of the price elasticity of demand creates a discrepancy between the marginal rate of substitution on the consumption side and the marginal rate of transformation on the production side MRS MRT This departure from the usual Pareto optimal conditions indicates the lack of Pareto efficiency under monopoly. In other words, the result Monopoly is less efficient than Perfect Competition. still holds in general equilibrium! THEORY OF SECOND BEST The theory of Second Best studies the welfare effects of an economy having various institutional constraints such as market imperfections, government policies and regulations. These institutional constraints result in a departure from the usual optimal marginal cost pricing rule P = MC For example, market imperfection (such as a monopoly) and market interventions through policy and regulation (such as a tax or subsidy) create a divide between the price and the marginal cost Monopoly Tax Subsidy MR = MC (1 t)P = MC (1 + s)P = MC => => => P MC P MC P MC In general, these institutional constraints can be represented by an appropriate scale factor in the marginal cost pricing rule as follows: P = MC where = 1 (without any institutional constraints) 1 (when institutional constraints exist) The theory of Second Best is aptly named because it deals with the welfare analysis of an economy in the presence of unavoidable institutional constraints. The First Best refers to the idealistic scenario in the absence of any institutional constraints (i.e. perfect competition). 275 FIRST BEST Optimal solution to the welfare maximization max. in the absence of institutional constraints. For example, given the PPF equation F (X, Y) = 0 SECOND BEST Optimal solution to the welfare in the presence of institutional constraints. For example, given the PPF equation F (X, Y) = 0 and the institutional constraint P = MC the first best solution solves the welfare maximization problem Maximize U(X, Y) Subject to F (X, Y) = 0 We need only one constraint in the welfare maximization problem, namely, the PPF which represents the structure of the economy. the second best solution solves the welfare maximization problem Maximize U(X, Y) Subject to F (X, Y) = 0 PX = MCX We need two constraints in the welfare max. problem, namely, the PPF which represents the structure of the economy and the added constraint for other institutional features. Note that all added institutional constraints must be fully incorporated into the welfare maximization problem. EXAMPLE For example, consider the following simple welfare maximization problem of a two sector economy with a square root welfare (utility) function: U = XY and a linear production possibility frontier X+Y=1 276 We can compare three potential methods of finding a welfare maximizing solution: [1] FIRST BEST SOLUTION The first best solution maximizes the welfare function subject to the constraint given by the production possibility frontier Maximize U(X, Y) = XY Subject to X + Y = 1 At the point of maximum welfare (see point 1 in the diagram below), the welfare contour is tangent to the PPF, and hence, MRS = MRT Y=1 X X+Y=1 Solving for the first best solution X= Y= U= Y 1 PPF X 277 [2] SECOND BEST SOLUTION The second best solution incorporates all institutional constraints directly in the welfare maximization problem. It thus maximizes the welfare function subject to the constraint given by the PPF plus the additional institutional constraint. Maximize U(X, Y) = XY Subject to X + Y = 1 Y=X At the point of maximum welfare (see point 2 in the diagram below), the welfare contour must intersect both the PPF and the institutional constraint line X+Y=1 Y=X Solving for the second best solution X = 2/3 = 0.66667 Y = 1/3 = 0.33333 U = 2/3 = 0.4714 Y 1/3 1 2 Constraint PPF 2/3 X It is clear that the second best solution results in a lower welfare level (U = 0.47) compared to the first best solution (U = 0.5). In the diagram, point 2 (the second best solution) is on a lower social welfare contour than point 1 (the first best solution). 278 [3] PIECEMEAL SOLUTION In addition to the first best and second best solutions, let's consider what is often called the "piecemeal" solution (see point 3 in the diagram below). This applies the institutional constraint to the first best solution. This approach is half-hearted since it attempts to satisfy the institutional constraint without incorporating it into the welfare maximization problem (as in the second best solution). Taking the first best solution for X X= and satisfying the institutional constrant Y=X gives us the "piecemeal" solution (which is somewhat ridiculous, don't you think?) X= Y= U = 1/8 = 0.3536 Y PPF 1/3 1 2 Constraint 3 X 2/3 The diagram shows that the "piecemeal" solution results in the lowest welfare level (U = 0.35) compared to the second best solution (U = 0.47) and compared to the first best solution (U = 0.5). So, the message that the theory of Second Best is trying to deliver is that with the presence of institutional constraints it may not be possible to satisfy the Pareto efficiency conditions in practice. The theory of Second Best has an important policy implication, that is 279 If the full set of Pareto optimality conditions cannot be satisfied, then any subset of these Pareto optimality conditions will not do any good in the presence of institutional constraints. That is, either we meet all of the conditions of Pareto optimality (and have a Pareto optimal solution) or we do not meet them all (and do not have a Pareto optimal solution). There is no middle choice. The theory of Second Best suggests to us that the "piecemeal" approach to economic policy (that is, to remove part of the distortions in the economy) might end up resulting in a welfare reducing situation. 280 ECON 301 LECTURE #16 Oligopoly The term oligopoly covers a variety of market structures and market behaviours. As a result, there does not exist a definitive model of oligopoly. However, the following assumptions capture the essential elements of this market structure. ASSUMPTION #1: There are few sellers of the product. In some cases, all of these "few" sellers are large firms (like in car manufacturing). However, more often there are a couple of large firms and some small firms in the industry (like in the brewing industry). One of the most studied special cases is that of a single, large, low-cost "dominant firm" that acts as a leader to the group of follower firms (Stackelberg leadership model). ASSUMPTION #2: There are significant barriers to entry into the industry. The presence of these significant barriers to entry accounts for the small number of firms in oligopolistic industries. Entry prevention is often made possible by the cost advantages realized by the existing firms. C,P ATC (potential entrants) MC P0 P1 ATC (existing firms) MR X0 D X First, let's consider the case of a well established firm that has lower costs per unit (ATC) than potential entrants. By charging a price such as P1 the established firm would more than cover its own costs while preventing potential entrants from operating at even normal profits. 281 This sort of barrier to entry is difficult to continue in the long-run unless the existing firms have considerable monopoly-like power over resources and/or technology. Sometimes the presence of large economies of scale provides a major barrier to entry even if the existing firms do not actively engage in entry preventing behaviour. C C LRAC (with advertising) LRAC df X Car Manufacturing LRAC X Toothpaste Manufacturing df Large Economies of Scale It takes a large market share (df) to attain an output level at which costs per unit are low enough to compete with existing firms. (Large Scale Barriers to Entry). Small Economies of Scale Technology is so simple by comparison that most reductions in costs per unit can be realized with a small market share. (Weak Scale Barriers to Entry). Obviously, it takes relatively little investment and knowledge to enter the toothpaste industry than it does to enter the auto industry. For this reason, existing firms in such industries often protect themselves against entry by artificially raising costs. Heavy advertising, for example, effectively raises the LRAC curve. Another way to discourage entry is for the existing firm to create many different brands of toothpaste that cater to different tastes of consumers, hence making it difficult for new entrants to get into niche markets without considerable expense. These (and other) techniques are referred to as created barriers to entry and they are most often used in markets that are not naturally protected by the existence of large economies of scale. ASSUMPTION #3: Firms in oligopolistic markets are aware of their own interdependence with competing firms and act accordingly. 282 Oligopolistic firms often confront large competitors that are not tightly controlled by the discipline of competitive forces and whose actions can significantly affect prices and market shares. For this reason, they must be constantly on guard against their competitors' hostile actions and must weigh the potential costs and benefits of their own market strategies in terms of their competitors' responses. One of the most crucial decisions facing oligopolistic firms is whether to act on their own or to collude with potential competitors against consumers. This choice provides the basis for the theoretical and practical distinction between noncollusive and collusive oligopolies. Non-collusive Oligopoly: Cournot's Duopoly Model (MC=0) Cournot considered the simple case of two identical mineral water springs that stand side by side and are owned by two firms, A and B. In this model, production costs are limited to the fixed costs of digging the wells, since it is assumed that the consumers fetch the water themselves and provide their own containers. It follows that both variable costs and marginal costs are zero, since the production of one extra unit of water requires no additional expense beyond digging the well. Cournot's analysis of duopoly is built on the assumption that both firms aim at profit maximization so that they will both produce the level of output for which MR=MC. Since in the case of the two wells we have a MC=0, the profit maximization condition becomes MR=0. We should recall that this condition is fulfilled at the midpoint of a straight line demand curve, where the quantity sold equals one half of the market. P D' MRA 0 QA = 0.5 A MRB B QB = 0.25 D 1 X 283 It is assumed that initially firm A is the only producer and seller of mineral water. Firm A maximizes profits where MRA = 0, by producing OA = 0.5 (one half of the quantity that consumers would be willing to drink...according to their demand curve, if the price was zero). A second behavioural assumption of Cournot's model is that each producer believes that the other will not change its output and therefore takes the other producer's output as given. Accordingly, when producer B enters the market, it assumes that its effective demand curve is D'D and that the corresponding MR curve is MRB. Producer B thus maximizes profits where MRB = 0 or AB = 0.25 (1/4 of the market demand when P = 0). Is this a stable final equilibrium point? This situation, where A takes half of the market and B takes one quarter of the market, can be shown to be unstable based on Cournot's behavioural assumptions. When B enters the market and takes one quarter, A takes B's production as given and thus believes that its own market has shrunk to of its previous size. Accordingly, A reduces their production from of 1 to of , or 3/8. In the next round, B takes A's output reduction as a sign to increase their production to of 5/8 (1 3/8 leaves 5/8), or 5/16. This process continues until A's output falls to 1/3 and B's output increases to 1/3, as we can see graphically below. P D' D 0 QA = 0.5 QA = 3/8 QA = 11/32 A QB = 0.25 QB = 5/16 and so on... B 1 X 284 And so on. It can be proven that this process will stop when each of the two producers takes one third of the market since only at this point will there be no further reaction by either firm. For clarity, if B is taking 1/3 of the market then A will be happy to produce half of the remaining 2/3, or 1/3 of the market. The final (stable) equilibrium in our development of the simple Cournot Duopoly Model looks like this: P PM PD A 1/3 QA = 0.3333 TR max B D 1/2 QB = 0.3333 2/3 1 X 0 The two firms A and B jointly sell 2/3 at PD and get a total revenue that is equal to the shaded area. We have noted that this is a stable equilibrium since each producer maximizes profits by taking of what it believes to be the available market. However, at this equilibrium joint profits are not maximized. The duopolists would be better off by colluding and acting as a monopoly, selling at PM and sharing the maximum total revenue at the midpoint of the demand curve. This analysis is FAR too simplistic, but most of its weaknesses can be overcome using more complex models (i.e. where MC 0 and/or is not equal for the two firms, etc.). We will consider an example that develops reaction functions for a more complex model soon, but before we do...we should be quite critical of the above model. Our criticisms would include the following: 1. The assumption of costless production is unrealistic in most, if not all, applications. 2. The model is limited in scope because it does not consider the possibility of new firms entering the industry. 3. The assumption that firms take each other's output as given and do not learn from past experiences is nave. 285 4. The model focuses exclusively on quantity competition, with no attention paid to price competition whatsoever. 5. It is difficult to believe that a small number of firms would not collude against consumers in one way or another. Let's consider a Cournot Duopoly Quantity competition in a simultaneous game theory setting. Consider 2 firms, firm A and B, producing identical products so that they are forced to charge the same price. This leaves the sole strategic choice as the amount the firms choose to produce, QA and QB. Once the firms select their quantities, the resulting price is whatever price is required to "clear the market". This is the price where consumers are willing to buy QA + QB (the total production). Suppose the firms have no marginal cost of production (for now). Let the market demand be given by: where, QA + QB = QTOTAL QA + QB = 600 P Now, we are interested in price so we rewrite market demand as: P = 600 - QA - QB If these firms make their output decisions simultaneously, how much will each firm produce? To reach an equilibrium, both firms must be choosing the output level that is a "best response" to the choice of the other firm. Consider the output decision of Firm A. For QA to be an equilibrium output for firm A, it needs to maximize profits for firm A given firm B's choice, QB. So, suppose that firm A believes that firm B is going to produce QB. Then firm A estimates that if it produces QA units of output its profit will be: A = P QA which is, A = (600 - QA - QB) QA A = 600QA - QA2 - QAQB To maximize this profit function we need to take the partial of with respect to QA and set the resulting expression equal to zero (i.e. marginal profit =0). 286 FOCQA 600 QB - 2QA = 0 2QA = 600 QB QA = 300 0.5QB (1) This profit maximizing output equation for QA is called firm A's best response to firm B (firm A's reaction function). We should notice that QA is a decreasing function of QB. This means if firm A expects firm B to increase output they will reduce QA. Now, we can find firm B's reaction function in a similar manner. B = P QB B = (600 - QA - QB) QB B = 600QB QB2 - QAQB FOCQB 600 QA - 2QB = 0 2QB = 600 QA QB = 300 0.5QA (2) We now have both firm's best responses to each other's output choices. This means that we know what each firm will produce given the output choices of the other firm. For this to be in equilibrium, we must have both firms playing best responses, otherwise one of the firms will have an incentive to change their output. So, let's re-write (1) as QB = 600 2QA and since (2) = (3) we can set them equal and solve for QA. 600 2QA = 300 0.5QA 3/2QA = 300 QA* = 200 We can similarly proceed to solve for QB*, using (1) and re-writing (2) as: QA = 600 2QB (4) (3) 287 and since (1) = (4) we can set them equal and solve for QB. 600 2QB = 300 0.5QB 3/2QB = 300 QB* = 200 and we can find price in equilibrium by subbing QA* = QB* = 200 into the demand equation. P = 600 - QA - QB P* = 200 and further we can solve the firm's profit function in equilibrium. A* = P* QA* = 200 200 = 40,000 B* = P* QB* = 200 200 = 40,000 For the next example, we will consider a Cournot Duopoly Quantity competition with ATC 0 and a situation where one of the firms has a cost advantage. Consider the same two firms as in the example above but this time firm A has a marginal cost of $6 per unit and firm B has a marginal cost of $8 per unit. Let the market demand remain the same... P = 600 - QA - QB The reaction function for firm A is: A = P QA - 6 QA A = (600 - QA - QB) QA 6 QA A = 600QA - QA2 - QAQB 6 QA FOCQA 600 QB - 2QA 6 = 0 2QA = 594 QB QA = 297 0.5QB (1) 288 The reaction function for firm B is: B = P QB - 8 QB B = (600 - QA - QB) QB 8 QB B = 600QB QB2 - QAQB 8 QB FOCQB 600 QA - 2QB 8 = 0 2QB = 592 QA QB = 296 0.5QA Re-write (1) as QB = 594 2QA Re-write (2) as QA = 592 2QB Now, equate (1) and (4) to find QB*... 592 2QB = 297 0.5QB 3/2 QB = 295 QB* = 196.666666 Now, equate (2) and (3) to find QA*... 594 2QA = 296 0.5QA 3/2 QA = 298 QA* = 198.666666 Now, we can find P* using QA* = 198.666666 and QB* = 196.666666 in the demand equation. P* = 600 - QA* - QB* P* = 600 198.666666 196.666666 P* = 204.666666 (2) (3) (4) 289 This allows us to calculate the firm's equilibrium profits... A* = P* QA* - 6 QA* = (204.6666 6) 198.6666 = $39,468.44 B* = P* QB* - 8 QB* = (204.6666 8) 196.6666 = $38,677.77 The point of this example is to show that if a firm, like firm A, has a cost advantage over another firm, like firm B, they will be able to get more of the market in equilibrium (cost advantages create market power in this framework). STACKELBERG QUANTITY COMPETITION The Cournot model has been criticized as somewhat unrealistic in the sense that it is not very likely that the two firms would actually be forced to choose quantity simultaneously. In reality, one firm chooses a production quantity and the other firm would, observing the first firm's decision, follow with a quantity decision of their own. Would this sequential move game result in a different outcome than Cournot competition? The answer is a resounding YES!!! But is it better to go first or to wait, see what the leader chooses and then respond (going second)? The intuitive answer for some people is to wait and go second, since by observing what the leader decides we can gather more useful information with which to make our output decision. However, we will soon see, the first mover (leader) has the advantage. Let's use an example to illustrate, Consider a two-step model. The first step has firm A selecting its output and the second step has firm B observing firm A's decision and then selecting their output accordingly. Let's try to think why firm A will be better off. The only perceived advantage to going second is that firm B can observe what firm A chose before selecting their profit maximizing output. But, firm A can calculate firm B's reaction to their move ahead of time and take it into account. This gives firm A more power in the market. Using the same demand curve as before... P = 600 - QA - QB 290 and we will assume no marginal costs once again, so we can focus on only the difference in quantity announcement timing. When firm A calculates their decision they take firm B's reaction to their decision into account, thus they calculate firm B's reaction function... B = P QB B = (600 - QA - QB) QB B = 600QB QB2 - QAQB FOCQB 600 QA - 2QB = 0 2QB = 600 QA QB = 300 0.5QA (1) Now, firm A will anticipate firm B's response when they maximize profits. A = P QA A = (600 - QA - QB) QA and firm A knows what firm B will do based on (1)... subbing (1) in for QB, gives us A = [600 - QA (300 0.5QA)] QA = 600 QA - QA2 - 300 QA + 0.5 QA2 = 300 QA 0.5 QA2 and we can now find the profit maximizing output for the leading firm, firm A, as: A = 0 QA or, 300 - QA = 0 QA* = 300 It follows (by using firm B's reaction function) from (1) that if firm A produces QA* = 300, then firm B produces... 291 QB = 300 0.5QA = 300 0.5(300) QB* = 150 We can find the market price, P*, using QA* = 300 and QB* = 150 in the demand equation, P* = 600 - QA* - QB* = 600 300 150 = 150 and thus, the firms profits are, A* = P* QA* = 150 300 = 45,000 B* = P* QB* = 150 150 = 22,500 Remember, in the first example, the Cournot (simultaneous decision game)gave us a result of P* = 200, QA* = 200, and QB* = 200 with profits for the firms as follows: A* = P* QA* = 200 200 = 40,000 B* = P* QB* = 200 200 = 40,000 Clearly, the first mover advantage benefits firm A. In the sequential model, when firm A is a Stackelberg leader, firm A is able to commit to a higher quantity than in the Cournot scenario. This forces firm B to cut back their "best response" production. This enables firm A to take a 2/3 share of the market as opposed to the share they had in the simultaneous move game. We should note that while the Stackelberg leadership model results in a higher firm A profit, the joint profits in the market fall from $80,000 to $67,500. 292 HOMEWORK 1. Consider the market for automobiles. Let's assume there are two auto manufacturers, Toyota and Honda. Suppose Honda has an ATC of 9,000 per unit and Toyota has an ATC of $11,000 per unit. Assume the following market demand in this automobile market: P = 54,000 QHONDA QTOYOTA Solve the Cournot quantity competition to get Q*HONDA, Q*TOYOTA, and P*. Also, find the equilibrium profits for each firm as *HONDA and *TOYOTA. 2. . Consider the same market for automobiles. Suppose Honda has the same ATC of 9,000 per unit and Toyota has the same ATC of $11,000 per unit as in question 1. Assume the same market demand of: P = 54,000 QHONDA QTOYOTA Solve the Stackelberg leadership quantity competition to get Q*HONDA, Q*TOYOTA, and P* if Honda is the first mover. Also, find the equilibrium profits for each firm as *HONDA and *TOYOTA and compare this to the Cournot results. 3. Consider the same market for automobiles. Suppose Honda has the same ATC of 9,000 per unit and Toyota has the same ATC of $11,000 per unit as in question 1. Assume the same market demand of: P = 54,000 QHONDA QTOYOTA Solve the Stackelberg leadership quantity competition to get Q*HONDA, Q*TOYOTA, and P* if Toyota is the first mover. Also, find the equilibrium profits for each firm as *HONDA and *TOYOTA and compare this to the Cournot results and the results from question 2. 293 ECON 301 LECTURE #17 GAME THEORY NON-COOPERATIVE GAME THEORY There are two broad approaches to decision problem analysis using game theory. The distinction lies between cooperative and non-cooperative games. This distinction does not mean that agents necessarily cooperate in cooperative game theory nor does it mean that agents necessarily do not cooperate in noncooperative game theory. What is the difference then? "Cooperative" means that the players can write binding contracts on their actions. For example, in cooperative game theory, players can commit to actions, which when they have to be taken, possibly end up being against that players interests at that point in time. For instance, a player may commit to becoming a member of a coalition that is bound via contract to behave a certain way in a given state of the world...once that state of the world presents itself (if it ever does) the coalition is obligated to act as they said they would, even if there are better payoffs from another course of action. By contrast, non-cooperative game theory promises no such commitment to future actions. When a set of moves results in a player's turn to decide upon a move going forward, the player is not constrained by any prior commitments and will decide their move based on their best personal outcome. We will only concern ourselves with non-cooperative game theory in this course. There are two distinct ways of representing non-cooperative games. EXTENSIVE FORM GAMES A game in extensive form provides us with a way of describing: [1] Who is involved? [2] The rules of the game (i.e. who moves when? what do they know? what moves are possible?), [3] The outcomes (i.e. what happens? (a physical outcome)), [4] The payoffs. 294 Defin nition: A game tree, , is a finite c , collection o nodes, ca of alled vertice connect by lines es, ted s, a at cted (i.e. the exists a set of arcs ere s called arcs, so as to form a figure tha is connec conn necting any one vertex to another and conta x r) ains no sim mple closed curves (i.e. there does not exist a set o arcs connecting a v e e of vertex to itse elf). We c further assert that a game tre has a dis can ee stinguished node or ro this is d oot; wher the game starts. No let's co re e ow, onsider som examples to illustra the me ate conc cepts of info ormation se and perf ets fect recall. 295 In thi case, Ch is handler knows whether Joey chos to bring his umbrella or not but se he do oesn't know whether it is raining or not raining. w ORMATION SETS N INFO Joey has one in y nformation s (Joey1) and one m set move or dec cision to ma (bring ake an um mbrella or not). n ndler has tw informat wo tion sets. If it is true th Joey ha his umbr f hat as rella then Chan 1 Chan ndler is at in nformation set Chandl . If Joey forgot his umbrella then ler y Chan ndler is at in nformation set Chandl 2. He als only has one move or decision ler so s n to ma (a or b) yet he do not know if it is rain ake ), oes ning or sun (i.e. he doesn't nny know which ver w rtex he is at in each inf t formation s set). Now, what does this game tree look like if Chand does n know if J s e dler not Joey has umbrella but he does k know wheth it's rainin or not ra her ng aining? his u 296 In thi case, Ch is handler knows if it is ra aining or no (rain is pa of inform ot art mation set 1 and n rain is part of inform no p mation set 2 but he is sure wh 2) sn't hether Joey has his y umbr rella today (information set 1 is w where it's ra aining but he doesn't k know which node he is at in the informa e ation set, th he doesn't know if Joe has his umbrella hus f a or no ot). Defin nition: An extensive fo game is said to be a game of perfect rec if all pla orm s e f call ayers never r forge informatio once it is known an they nev forget th own prior moves. et on s nd ver heir For e example, th following game tree is not a ga he g e ame of perf fect recall... ...can you see w why? is onica does not know (i she can remembe if she's starting the i.e. n't er) e In thi case, Mo game or ending it (i.e. mak e g king a choic that lead to a term ce ds minal node... ...and payo offs). Essen ntially, she has forgott whether she has a ten r already turned left or turne right or, in fact, has not moved yet. ed s d ow omes very problematic in the gam theoretic c me c You can see ho this beco , t, want to consider game of perfec recall. es ct framework and, as a result we only w PER RFECT VS IMPERFEC INFORM CT MATION nition: Defin An extensive fo game is said to be a game of perfect info orm s e f ormation if all mation sets are single s etons (i.e. players can identify exa actly which node they inform are a when ma at aking a mov ve/decision) ). If the informatio sets are not all singletons, it is a game of imperfect information. e on 297 STRA ATEGIES Defin nition: A pure strategy for each player i {1,2,...,n} is a function i that associates ever ry inform mation set with one of the availab choices f ble s. Let's call the pu strategy sets i. Fo the follow s ure y or wing game (in extensive form) the pure strategy se for playe 1 and pla ets er ayer 2 are: 1 = {(L, 5), (L, 10), (R, 5), (R, 10)} 2 = {(u, a, b), (u, a, a), (u, b, b), (u, b a), (d, a, b), (d, a, a) (d, b, b), ( b, a)} b, ), (d, represent all of the pot tential selec ctions that e either The pure strategy sets i r er ce ame. playe could potentially fac in the ga For n now, think about one likely outcom to this g a me game. We'll investigat this later te r when we discus Pure stra n ss ategy Nash equilibria, in the norm or "strat h mal tegic" form game es. STRA ATEGIC OR NORMAL FORM GAMES We s should notic that the extensive f ce form of a ga ame is aptly named. T y The inform mation that is given in a game tre is "exten t ee nsive". Do w need this much we inform mation? Sometimes n not! Often games of this sort ar summari n f re ized in a "payoff matrix or what w call the x" we strate egic form of the game o e. 298 Let's put the above extens s sive form ga ame into str rategic form (for practi m ice). We shou notice th this gam is relativ uld hat me vely large in its strategic form. n In thi "payoff matrix", play 1's pure strategies are the row and play 2's pure is m yer e ws yer e strate egies are th columns he s. The payoffs are recorded i the form (row, colum e in mn)...so a p payoff pair in this matrix such as (1, 7) mean that play 1 gets 1 util of happ ( ns yer piness and player 2 gets 7 utils of ha appiness fr rom the gam me's outcom if those pure strate me egies are playe Keep th ed. hinking abo a likely s out solution to t this game. ONCEPTS SOLUTION CO THE ELIMINAT TION OF (W WEAKLY) D DOMINATED STRATE D EGIES A particular pure strategy i strictly do is ominated if the payoffs to a playe from s er playing that pur strategy a always less than if the player chose an alternative re are f r egy. In our example a r above there are many strictly dom e minated stra ategies. strate Can y see any? you 299 We c cannot redu the pure strategies any furthe So the p uce e s er. pure strateg gies that resul are: lt ((R 5), (d, b, b)) R, , ((R 5), (d, b, a)) R, , ((R, 10), (d, b, b)) ( ((R, 10), (d, b, a)) ( So, s successive elimination of (weakly dominate strategie n y) ed es... [i] ma not yield a unique o ay d outcome, [ii] re equires com mmon know wledge, and s for dominated strategies ( (after all, th player is he [iii] is particularly dubious f weakly d indiff ferent). Now let's do thr rough anoth exercise using the successive eliminatio of her e e on (wea akly) domina ated strateg gies... 300 Cons sider a two player gam in strateg form giv as: me gic ven Now, let's illustr rate the "inf formation lo oss" we exp perience by analysing a game in y the s strategic for rm. Take a few minutes to draw an extensive form re e w epresentati of the tw player ion wo strate egic form game above Is your representati the only possible e g e. ion y extensive form game for this payoff m matrix? BLE TEGIES AN NASH E ND EQUILIBRIA A RATIONALIZAB STRAT Defin nition: A str rategy, s, is a "best res s sponse" for player i to his rival's s r strategies if the payoff f f to pla ayer i from playing s is greater th the payoff to playe i from any other s han er y strate (taking the rival's strategies a given). egy as 301 Defin nition: A Na Equilibr ash rium in pure strategies is a set of strategies, one for ea player, e s ach such that each player's str h rategy maxi imizes their payoff tak r king the othe player's er strate egies as giv (i.e. mu ven utual best r response). Let's see how th works... s his ...consider th following two perso game in strategic he g on form: How do we find these "mutual best re esponses"? First, let's take as given that player 1 will choose his pure s a e strategy T. What is playe 2's best response to this inform er o mation? en payoffs When player 1 plays T, the player 2 can choose L, C, or R with his p g being 4 for L, 2 for C, and 3 for R. Clearly, when player 1 play T the be response from play 2 is to p p ys est yer play L. Now, we need to check if T is player 1's best res t sponse if pl layer 2 play L. ys en B, offs When player 2 plays L, the player 1 can choose T, M, or B with payo being 4 for T 2 for M, and 3 for B. So when p T, a player 2 pla L the be respons from ays est se playe 1 is to pla T. er ay To su ummarize, ayer 1 plays T then pla s ayer 2's bes response is to play L st e L...(T, L). if pla if pla ayer 2 plays L then pla s ayer 1's bes response is to play T st T...(T, L). So th pure stra he ategy pair (T, L) is a m mutual best response a as such is a pure and h strate Nash equilibrium. egy e ONE DOW ONE T GO! WN, TO Now take as giv that pla ven ayer 1 will c choose his p pure strateg as M. W gy What is playe 2's best response to this? er o 302 Hope efully, we can see that when play 1 choos M then player 2's b t yer ses best respo onse is C. Now we ne to chec to see if M is a best response t Player 2 eed ck t to choo osing C...an it is! nd So th pure stra he ategy pair (M, C) is a m mutual best response and as suc is a pure t ch e strate Nash Equilibrium. egy E . WN, E TWO DOW NONE TO GO! Take a couple of seconds to try to find a mutual best respo e o onse to Play 1's yer choic of B and let me kno what you find... ce d ow u TCHING PE ENNIES MAT This game sets up as follo ows: Two friends take a penny i their han and place it either "h in nd e heads up" o "tails up" or ". They have agre to payo as follow y eed offs ws: If, wh they op their ha hen pen ands, the pe ennies are the same s side up then player 1 n gets a dollar fro player 2. om If, wh they op their ha hen pen ands, the pe ennies show opposite sides up th player 2 w hen gets a dollar fro player 1. om s form and th normal fo he orm... Let's represent this game in both the extensive f BAT TTLE OF TH SEXES HE meo arated by m many miles with no way of commu unicating Rom and Juliet are sepa (ther were no cell phones or Blackbe re c s erries in the 18th century!). e The last time that they saw each othe they agre to go out on Satur w er eed rday night but th could not agree to which eve they would attend before they were hey n o ent y sepa arated. 303 Juliet wants to go to the Manchester United gam (soccer = sports, de g me enoted as S) bu she know that Rom wants t go to see the Londo Philharm ut ws meo to e on monic Orch hestra (mus denoted as M). sic, d Of co ourse, Rom and Juliet are quite taken with each othe and would prefer to meo e er o be to ogether than to be sep parate, how wever, each of them wo ould derive more pleas sure by being together at the eve that they prefer. ent y For e example, Romeo want to be with Juliet liste R ts h ening to mu usic (his big ggest payo and Juliet wants to be with Ro off) o omeo watch hing the soc ccer match (her bigge payoff). est meo multaneous choose ( sly (without hav ving agreed upon d Rom and Juliet must sim wher to meet) which venu to arrive at. re ue e s form and th normal fo he orm... Let's represent this game in both the extensive f If Ro omeo choos M then Juliet's bes response is M. Sinc M is also Romeo's ses st e ce o best response when Juliet chooses M we have a mutual be respons Thus, w est se. M) e Nash Equilib brium. (M, M is a pure strategy N If Ro omeo choos S then J ses Juliet's bes response is S and by inspection we can st see t this is also Romeo's b best respons to Juliet picking S. Thus, (S, S) is also a se pure strategy Nash Equilib brium. Grea Now the have no problem, right? at! ey WRONG! W ne wo decides to g to the go The Nash equilibria only happen if on of the tw players d nt even that they don't want to go to. pose that Romeo decid that he simply can R des e nnot miss th orchestr and he ra Supp belie eves that Ju uliet will sett for musi just so th they can be togethe tle ic hat n er... 304 At the same time, suppose that Juliet figures that Romeo is a sensitive fellow and because he knows how much she wants to see the soccer match so she goes to the sports venue expecting that he will meet her there... If this scenario played itself out we would have the outcome (M, S) and the payoffs would be (5, 5). Both of the players are where they want to be...but without each other's company. "SELFISH OUTCOME" Interestingly, the "selfish outcome" described above is not the worst outcome... Suppose that Romeo IS the sensitive and considerate fellow that Juliet believed him to be and he decides to go to the soccer match simply to be with her (i.e. sacrifices his musical experience). At the same time, suppose Juliet DID decide to settle for a night of music because she knows how much it means to Romeo... If this scenario played itself out we would have the outcome (S, M) and the payoffs would be (0, 0). Both of the players are where they do not want to be...AND without each other's company. The "SELFLESS OUTCOME" ends up resulting in the worst case scenario. The moral of the story is... (I'll let you decide!) PRISONER'S DILEMMA Two criminals are picked up for shoplifting and immediately separated by the police (i.e. put into different interrogation rooms) before they can communicate. The police "know" but cannot prove that these are the same two crooks that have robbed seven area banks in the last two months. Now, the police need a confession from at least one of these "gentlemen" to close the bank robbery case...so... The prosecutor goes to each interrogation room and tells each of the crooks that he is offering both of them the following deal: "If one of you confesses to the bank jobs and the other does not, then the confessor will get one year in jail and the other will do a twenty-five year term." "If you both confess to the bank jobs then it's ten years for each of you." "and, if both of you refuse to confess the evidence that the police have on you for the shoplifting incident will get you both two year jail terms...but Constable 305 O'Ma alley tells me that it loo like you partner is about ready to `spill t beans' m oks ur s the and c confess." come? We each of t crimina will confe and end up ell, the als ess What is the outc servi 10 year apiece! ing rs If this doesn't make any se s m ense to you you are n alone... u, not Let's use our kn s nowledge o game theory to see w this "su of why ub-optimal" outcome " occu urs. So f nfesses, the Nuckles en [1] S let's consider Nuckles' strategies first...If Snake con is be etter off to confess as w (i.e. -10 is much b c well 0 better than -25). If Sna does ake not c confess, the Nuckles is better of to confess since -1 is better tha -2. en ff s s an This means tha Nuckles' p at pure strateg to confess strictly d gy dominates t pure the strate not to confess! egy c So s Snake's str rategies...If Nuckles co f onfesses, then Snake [2] S now let's consider S is be etter off to confess as w (i.e. -10 is much b c well 0 better than -25). If Nuc ckles does not c confess, the Snake is better off t confess since -1 is better than -2. en s to This means tha Snake's p at pure strateg to confes strictly dominates th pure gy ss he strate not to confess! egy c 306 So w end up with... we 307 [2] D Draw both th extensive form and normal for represen he rm ntations of t the follow wing game: : ROC PAPER SCISSOR CK, R, RS ngelina and Jennifer, a trying to decide wh get's to d are o ho date Brad Two people, An . ee he per, rs on Friday night. They agre to use th rock, pap scissor game to decide h r e (with the winner getting the date). For c convenienc represen a win with a payoff o 1 and a loss with a payoff of -1 ce, nt of 1. If the is a tie the payoff is (0, 0). Ar there any pure strat ere t s re y tegy Nash e equilibria? Here are the rul of the game: e les Rock smashes Scissors k Pape covers Rock er R Sciss sors cuts Paper Roc wins ove Scissors ck er Pap wins ov Rock per ver Scissors wins over Paper Deno Rock = "R", Paper = "P", Scis ote ssors = "S" Both players rev veal their choice simultaneously. Note that the normal form game l e three by thr matrix a the exte ree and ensive form game has nine termi m s inal nodes is a t (i.e. p payoffs). 308 ECON 301 LECTUR #18 1 RE Reca the matc all ching pennie game fro last time... es om TCHING PE ENNIES MAT Two friends take a penny i their han and place it either "h in nd e heads up" o "tails up" or ". y eed offs ws: They have agre to payo as follow If, wh they op their ha hen pen ands, the pe ennies are the same s side up then player 1 n gets a dollar fro player 2. om hen pen ands, the pe ennies show opposite sides up th player 2 w hen If, wh they op their ha gets a dollar fro player 1. om Let's represent this game in both the extensive f s form and th normal fo he orm... Rem member that we said th this gam has no p t hat me pure strateg Nash equ gy uilibrium. However, if thes players mix strategies (and the game is f se finite) there will exist a e mixe strategy Nash equilibrium. ed MIXE STRATEGY NASH EQUILIBRIA ED H If a p player is going to mix o a set of pure strategies then a of the stra on all ategies in the s must ha the sam payoff. H set ave me Hence, the player's pr robability dis stribution over these strat tegies is "ar rbitrary". Of co ourse, the other player's payoffs do depend upon the p o probability d distribution and h hence, a player's mixe strategy is complet ed tely determi ined by the other playe ers. Let's clarify how this works by solving the mixed strategy N s w s g d Nash equilib brium for the e matc ching pennies game an then the (slightly) m nd e more compl Battle o the Sexes lex of s game e. 309 ote 1 ng that Player 2 mixes as 1 (H, ) r s Deno Player 1's payoffs from playin H given t and a also denote Player 1's payoffs fro playing T given tha Player 2 mixes as e s om at 1 (T ). T, Now, let's consi ider Player 1's payoff from playin H... ng 1 (H, ) = (1) + (-1) (1 ) ) 1 (H, ) = 1 + 1 (H ) = 2 1 H, and w what about Player 1's payoff from playing T t m T... 1 (T, ) = (- + (1) (1 ) -1) ) 1 (T, = - + 1 ) 1 (T ) = 1 2 T, 2 Now, for Player 2 to make Player 1 in r ndifferent be etween play ying H or T they must ose Player 1's p payoffs the s same for H and T (i.e. 1 (H, ) = choo the that makes P 1 (T )). T, So... ... 1 (H, ) = 1 (T, ) , 2 1 = 1 2 4 = 2 = 310 Denote Player 2's payoffs from playing H given that Player 1 mixes as 2 (, H) and also denote Player 2's payoffs from playing T given that Player 1 mixes as 2 (, T). Now, let's consider Player 2's payoff from playing H... 2 (, H) = (-1) + (1) (1 ) 2 (, H) = - + 1 2 (, H) = 1 - 2 and what about Player 2's payoff from playing T... 2 (, T) = (1) + (- 1) (1 ) 2 (, T) = 1 + 2 (, T) = 2 1 Now, for Player 1 to make Player 2 indifferent between playing H or T they must choose the that makes Player 2's payoffs the same for H and T (i.e. 2 (, H) = 2 (, T)). So... 2 (, H) = 2 (, T) 1 2 = 2 1 4 = 2 = Person 1's choice to play a mixed strategy depends on Person 2's choice of B. For example, if Person 2 chooses to mix using = as above, then Player 1 will mix on his pure strategies since he is indifferent between them (moreover, Person 1 will mix such that Person 2 is indifferent between the pure strategies available to her). This results in a Mixed Strategy Nash Equilibrium of the form: [(, (1 )), (, (1 ))] or in this case... (( , ), ( , )) where the expected payoffs from these strategies are: 1 (H, ) = 2 1 1 (T, ) = 1 2 2 (, H) = 1 - 2 2 (, T) = 2 1 2( ) 1 = 0 1 2( ) = 0 1 2( ) = 0 2( ) 1= 0 311 Here of course, the individ e, dual outcom of each individual p me play is still u unknown but b mixing st by trategies as above, ea player is ensuring themselves the best s ach s payo in a large number o repeated trials. off e of TTLE OF TH SEXES HE BAT Let's quickly ref s fresh our m memory with respect to Romeo an Juliet's s h o nd situation. They are separa y ated by ma miles w no way of commun any with nicating. Th last time he e that t they saw ea other th agreed to go out o Saturday night but they could ach hey d on y not a agree to wh hich event th would a hey attend befo they were separate ore ed. Juliet wants to go to the Manchester United gam (soccer = sports, de g me enoted as S) bu she know that Rom wants t go to see the Londo Philharm ut ws meo to e on monic Orch hestra (mus denoted as M). sic, d Of co ourse, Rom and Juliet are quite taken with each othe and would prefer to meo e er o be to ogether than to be sep parate, how wever, each of them wo ould derive more pleas sure by being together at the eve that they prefer. ent y For e example, Romeo want to be with Juliet liste R ts h ening to mu usic (his big ggest payo and Juliet wants to be with Ro off) o omeo watch hing the soc ccer match (her bigge payoff). est Rom and Juliet must sim meo multaneous choose ( sly (without hav ving agreed upon d wher to meet) which venu to arrive at. re ue e We r represented this game in both the extensive form and t normal form as d e e e the follow ws: at! ey Grea Now the have no problem, right? ONG! The Nash equilibria only happen if on of the tw players d ne wo decides to WRO go to the event that they don't want to go to. o o 312 So w we've still go a problem Now, let's suppose that they a both un ot m! e are nable to learn from their past exper n riences (a r realistic pro oblem in som couples and this me s) same situation presents its every s e self single time t that they ag gree to mee et. ote 's g hat mixes as R (M, ) and d Deno Romeo' payoffs from playing M given th Juliet m also denote Romeo's payo from pla offs aying S giv that Juli mixes as R (S, ). ven iet s . Now, let's consi ider Romeo payoff fr o's rom playing M... g R (M, ) = (60) + (5 (1 ) 5) R (M, ) = 60 + 5 5 R (M, ) = 55 + 5 and w what about Romeo's p t payoff from playing S... ... R (S, ) = (0 + (40) (1 ) 0) R (S, ) = 40 4 40 t omeo indiffe erent betwe playing M or S she must een e Now, for Juliet to make Ro choo the that makes R ose Romeo's pa ayoffs the same for M and S (i.e. R (M, ) = R (S )). S, 313 So... ... R (M, ) = R (S ) S, 55 + 5 = 40 40 95 9 = 35 (1 ) = 12 / 19 = 7 / 19 Deno Juliet's payoffs from playing M given tha Romeo m ote at mixes as J (, M) and also denote Juliet's payoffs from play s ying S given that Rome mixes as J (, S). n eo s Now, let's consi ider Juliet's payoff from playing M s m M... J (, M) = (2 + (0) (1 ) 20) J ( M) = 20 , and w what about Juliet's payoff from playing S... t J (, S) = (5 + (80) (1 ) 5) ) J (, S) = 5 + 80 - 80 75 J (, S) = 80 - 7 Now, for Romeo to make J o Juliet indiffe erent betwe playing M or S he must een ose Juliet's payo the sam for M an S (i.e. J (, M) = J offs me nd choo the that makes J (, S S)). ... So... J (, M) = J (, S) , 20 = 80 - 75 95 9 = 80 = 16 / 19 (1 = 3 / 19 ) This results in a Mixed Stra ategy Nash Equilibrium of the for h m rm: [(, (1 )), (, (1 ))] or in this case... ... ((16/19 , 3/19), (7/19 , 12/19)) ( 314 wher the expected payoff from thes strategie are: re fs se es R (M, ) = 55 + 5 0 R (S, ) = 40 40 J (, M) = 20 M J (, S) = 80 75 S 5 55 5(7/19) + 5 = 480/19 40 40(7/19 = 480/19 0 9) 20 0(16/19) = 3 320/19 80 75(16/19) = 320/19 0 9 (or 25.2 263) (or 25.2 263) (or 16.8 842) (or 16.8 842) Why does Romeo get a hig gher expec cted payoff than Juliet? ? Why can Romeo mix with = 16 / 19 on music a only 3 / 19 on spo and orts? Does Romeo ha some a s ave advantage o power in this game? or n ? If Ro omeo and Juliet mix strategies the their pay en yoffs will be in the inte e erior of their r utility possibility frontier. y y Cons sider the sit tuation whe they alw ere ways both g to the orc go chestra (M, M). We'll , call t this point A and the pa ayoffs are (6 20). 60, Also, consider the situation where the always g to the so , t n ey go occer match (S, S). h We'll call this po B and t payoffs are (40, 80 l oint the 0). Furth conside the mixe strategy Nash Equilibrium that we just figu her, er ed ured out ((16/ , 3/19), (7/19 , 12/1 /19 19)). We'll call this point C and th payoffs a (25.263 he are 3, 16.84 42). should be able to see (even witho the diag a out gram that fo ollows) that point C is We s interi to the ut ior tility possibilities create by conn ed necting poin A and point B with a nt line s segment. pose that th agree th if they w hey hat went to spo last time then the n orts e next time Supp they would go to the music concert. T o c Then over r repeated tr rials they wo ould go to each venue half of the time and be gu h f e uaranteed t meet. to 315 Their expected utility would then beco r d ome: E ( ROMEO) = (60) + (40) = 50 (U E ( JULIET) = (20) + (80) = 50 (U Let's call this po D wher the payof are (50, 50). s oint re ffs Now, just to be clear, it doe esn't need to be a fair split to be efficient (i.e on the r e. UPF). Supp pose that th agree th they wo hey hat ould go to s sports twice in a row and then to e musi once (in an exactly repeated pa ic a attern). Th hen, over tim they wo me, ould go to sport 2/3 of the time and music conc ts e certs 1/3 of the time. Their expected utility would then beco r d ome: E (UROM ) = 1/3 (60) + 2/3 (4 = 46.66 40) 6666 MEO E (UJUL ) = 1/3 (2 + 2/3 (8 = 66.66666 20) 80) LIET s oint re ffs 66666, 66.6 66666). Let's call this po E wher the payof are (46.6 The key thing to recognize is that points A, B, D, E constitu situation where o e ute ns Rom and Juliet commun meo nicate and a agree on a pattern of meeting be ehaviours wher reas point C is the mix strategy Nash outc xed y come in a s situation of uncertainty y abou their mee ut eting behaviours. The f following diagram sug ggests that w when they can commu unicate and agree, d they can reach the bounda of their utility possi ary ibility set...w when there e's unce ertainty they can't. 316 This results in three pure s strategy Na equilibri They ar (t, T), (b, T), and (b, ash ia. re , B). We w want to dev velop a way to eliminat the "risky pure stra y te y" ategy Nash equilibria from this game. Can you s which t . see two pure strategy Nas equilibria are sh a "risky y"? Of co ourse, the "risky" Nash equilibria are (t, T) and (b, B). If one of the players " h e misc calculates or misreport their strat o ts tegy then th worst ca scenario may arise he ase e wher both play re yers get -50 as their pa 0 ayoff. can roblem by re efining our pure strate equilibria using the egy e We c fix this possible pr conc cept of Perfe Nash E ect Equilibrium. Defin nition: A completely mixed strateg for a pla gy ayer is one t that attache a strictly positive es y probability to ev very pure st trategy for t that player. Defin nition: An epsilon-perfe equilibr ect rium of a no ormal form g game is the equilibrium that e m resul from com lts mpletely mixed strateg gies for all p players whe the prob en bability of playing poor str rategies is s to less t set than in the mix of pure strategie ( > 0). e es, nition: Defin A perfect equilib brium is the limit point of an -per e rfect equilib brium as (no 0. uniqu ueness of equilibrium is assured) e ). his ee... So how does th work for an example? Let's se 317 So, if Player 2 mixes again then 2 (, B) < 2 (, B) for all > 0. f m nst at will r y g n This means tha Player 2 w set their probability of playing B to less than when st 1 they mix agains . i.e. (1 ) < Now, consider Player 1's p P payoffs from mixing ag m gainst . 1 (t, ) = (100) + (-50 (1 ) 0) 1 (t, = 150 50 ) what about Player 1's payoff from playing b... t m and w (b, 00) 00) ) 1 ( ) = (10 + (10 (1 ) 1 ( ) = 100 (b, 0 f m nst So, if Player 1 mixes again then 1 (t, ) < 1 (b, ) for all < 1. This means tha Player 1 w set their probability of playing t to less th when at will r y g han they mix agains . i.e. < st In the limit, as 0, the pu strategy T for playe 2 emerge (i.e. 1) and the e ure y er es pure strategy b for player 1 emerges (i.e. (1 ) )1). This gives us th Perfect he Nash Equilibrium at (b, T). h Hope efully, we notice that th success he sive elimina ation of wea akly domina ated strate egies would have resu d ulted in the same equilibrium with hout all of th effort. his This is only coin ncidental since the exa ample is de esigned to b simple a be and illustrative. 318 RE S Y QUILIBRIA A MOR MIXED STRATEGY NASH EQ Rem member, the were no pure strate Nash E ere o egy Equilibria for the "Rock Paper, k, Sciss sors" game e. We p proceed jus like we did with a 2X matrix. F st X2 First we find Player 1's payoffs d s from playing each of their o options aga ainst the d distribution. 1 (R, ) = (0) 1 + (-1) (2) + (1) (1- 1 - 2) 1 (R, ) = - 2 + (1 - 1 - 2) 1 (R, ) = 1 - 1 - 22 (1) 1 (P, ) = (1) 1 + (0) (2) + (-1) (1- 1 - 2) 1 (P, ) = 1 - 1 + 1 + 2 1 (P, ) = 21 + 2 - 1 (2) 1 (S, ) = (-1) 1 + (1) (2) + (0) (1- 1 - 2) 1 (S, ) = - 1 + 2 1 (P, ) = 2 1 (3) column play yer's choice of 's tha will make the row e at Now we can solve for the c er nt ons s h e playe indifferen among his/her optio (payoffs from each alternative equal). 319 Setting (1) and (2) equal, we get 1 - 1 - 22 = 21 + 2 - 1 2 - 32 = 31 1 = 2/3 - 2 Setting (1) and (3) equal, we get 1 - 1 - 22 = 2 1 1 = 32 2 = 1/3 So 1 = 2/3 - 2 results in 1 = 1/3 and, of course, 1 1 2 = 1/3 Now we can solve for the row player's choice of 's that will make the column player indifferent among his/her options (payoffs from each alternative equal). First we find Player 2's payoffs from playing each of their options against the distribution. 2 (, R) = (0) 1 + (-1) (2) + (1) (1 1 2) 2 (, R) = - 2 + 1 1 2 (1) 2 (, R) = 1 1 22 2 (, P) = (1) 1 + (0) (2) + (-1) (1 1 2) 2 (, P) = 1 1 + 1 + 2 (2) 2 (, P) = 21 + 2 - 1 2 (, S) = (-1) 1 + (1) (2) + (0) (1 1 2) 2 (, S) = - 1 + 2 2 (, S) = 2 1 (3) Now we can solve for the row player's choice of 's that will make the column player indifferent among his/her options (payoffs from each alternative equal). Setting (1) and (2) equal, we get 1 1 22 = 21 + 2 1 2 - 32 = 31 1 = 2/3 - 2 Setting (1) and (3) equal, we get 1 1 22 = 2 1 1 = 32 2 = 1/3 320 So 1 = 2/3 - 2 results in 1 = 1/3 and of course 1 1 2 = 1/3 d, e, We w write the Mi ixed Strateg Nash Eq gy quilibrium fo the "Rock, Paper, S or Scissors" game in the following form e m: [(1, 2, (1 1 2)), (1, 2, (1 1 2))] or in this case [( (1/3 , 1/3, 1 1/3), (1/3 , 1 1/3)] 1/3, We p proceed jus as before First we f st e. find Player 1's payoffs from playing each of s their options against the distribution n. 1 (F, ) = (3) 1 + (0) (2) + (5) (1- 1 - 2) 1 (F, ) = 31 + 5 - 51 - 52 1 (F, ) = 5 - 21 - 52 (1) 1 (G, ) = (5) 1 + (3) (2) + (0) (1- 1 - 2) 1 (G, ) = 51 + 32 (2) 1 (H, ) = (0) 1 + (5) (2) + (3) (1- 1 - 2) 1 (H, ) = 52 + 3 - 3 1 - 32 3 1 (H, ) = 3 + 22 3 1 (3) 3 Now we can solve for the c column play yer's choice of 's tha will make the row e at playe indifferen among his/her optio (payoffs from each alternative equal). er nt ons s h e Setting (1) and (2) equal, w get we 321 5 - 21 - 52 = 51 + 32 5 - 82 = 71 1 = 5/7 8/7 2 Setting (1) and (3) equal, w get we 5 - 21 - 52 = 3 + 22 31 2 + 1= 72 1 = 72 2 Setting 1 = 5/7 8/7 2 e 7 equal to 1 = 72 - 2 gives us... 5/7 8 2 = 72 - 2 8/7 57/7 2 = 19/57 7 7 2 = 1/3 1 = 1/3 we olved for 1 & 2. Now we can get 1 - 1 2 by using the values w have so (1 - 1 2) = 1 1/3 1/3 3 (1 - 1 2) = 1/ /3 This leads to pa ayoffs for pl layer 1 as f follows: 1 (F, ) = 5 - 21 - 52 = 5 2(1/3) 5(1/3) = ( 2 5) / 3 2 (15 ) 1 (F, ) = 8 / 3 G, 1/3) 3) 1 (G ) = 51 + 32 = 5(1 + 3(1/3 = (5 + 3) / 3 1 (G ) = 8 / 3 G, H, (9 1 (H ) = 3 + 22 31= 3 + 2(1/3) 3(1/3) = ( + 2 3) / 3 1 (H ) = 8 / 3 H, First we find Pla ayer 2's pay yoffs from p playing eac of their o ch options agai inst the ibution. distri 322 2 (, X) = (2) 1 + (1) (2) + (5) (1 1 2) 2 (, X) = 21 + 2 + 5 51 52 2 (, X) = 5 31 42 (1) 2 (, Y) = (4) 1 + (2) (2) + (0) (1 1 2) 2 (, Y) = 41 + 22 (2) 2 (, Z) = (0) 1 + (5) (2) + (3) (1 1 2) 2 (, Z) = 52 + 3 31 32 2 (, Z) = 3 + 22 31 (3) Now we can solve for the row player's choice of 's that will make the column player indifferent among his/her options (payoffs from each alternative equal). Setting (1) and (2) equal, we get 5 31 42 = 41 + 22 5 62 = 71 Setting (1) and (3) equal, we get 5 31 42 = 3 + 22 31 62 = 2 2 = 1/3 Subbing 2 = 1/3 into 71 = 5 62 gives us... 71 = 5 6(1/3) 7 1 = 3 1 = 3/7 Now we can get 1 1 2 by using the values we have solved for 1 & 2. (1 1 2 ) = 1 3/7 1/3 (1 1 2 ) = 21/21 9/21 7/21 (1 1 2 ) = 5/21 This leads to payoffs for player 2 as follows: 2 (, X) = 5 31 42 = 5 3(3/7) 4(1/3) = (105 27 28) / 21 2 (, X) = 50 / 21 2 (, Y) = 41 + 22 = 4(3/7) + 2(1/3) = (36 + 14) / 21 2 (, Y) = 50 / 21 323 2 (, Z) = 3 + 22 31 = 3 + 2(1/3) 3(3/7) = (63 + 14 27) / 21 2 (, Z) = 50 / 21 We write the Mixed Strategy Nash Equilibrium for the game above in the following form: [(1, 2, (1 1 2)), (1, 2, (1 1 2))] or in this case [(1/3 , 1/3, 1/3), (3/7 , 1/3, 5/21)] 324 ECON 301 LECTURE #19 INFORMATION ECONOMICS ASYMMETRIC INFORMATION So far in our studies we have neglected to examine the issues raised by differences in information. Up until now, we assumed that both buyers and sellers were perfectly informed about the quality of the goods being sold in the market. This assumption can be appropriate if it is easy to verify the quality of an item. We can argue that if it is not costly to determine which goods are high quality goods and which goods are lower quality goods then the pricing mechanism will simply adjust prices to reflect any quality differences. However, if getting the information about the quality of the good is costly then it is no longer plausible to believe that both the buyer and the seller have the same information about the good involved in the transaction. There are most certainly many markets in the "real world" in which it may be very costly or even impossible to get accurate information about the quality of the goods being sold. One example is the labour market. In any model that we have discussed so far, labour was a homogeneous product meaning that everyone had the same "kind" of labour and supplied the same amount of effort per hour worked. This is clearly not the case in the "real world". For practical purposes, it may be very difficult for firms to determine how productive its employees are (or will be once hired). Costly information is not just an issue for labour markets. Similar issues arise in markets for consumer products. When a consumer buys a used car it is often very difficult for them to ascertain whether or not the car is a good car or a "lemon". On the other hand, the seller of the used car likely has a pretty good idea about the quality of the car. We will see that this asymmetric information is apt to cause a substantial problem with the efficient operation of a market. THE MARKET FOR LEMONS Let's consider a model of a market where the buyers and sellers have different information about the quality of the goods being sold (see Akerlof)4. Consider a market with 100 people that want to sell their used cars and 100 people that want to buy a used car. Suppose that everyone knows that 50 of these cars are "plums" (i.e. good cars) and 50 of these cars are "lemons" (i.e. Akerlof, George "The Market for Lemons: Quality Uncertainty and the Market Mechanism," The Quarterly Journal of Economics, 84, 1970, pp.488-500. 4 325 bad cars). The current owner (seller) of the car knows the quality but the prospective buyers don't know whether any given car is a plum or a lemon. Further, the owner of a lemon will be willing to part with it at a price of $1000 and the owner of a plum will be willing to part with it at a price of $2000. The buyers will be willing to pay $1200 for a lemon and $2400 for a plum. This market has no problems if it is easy to verify the quality type of each car. The lemons would sell for somewhere between $1000 and $1200 and the plums would sell for somewhere between $2000 and $2400. But what happens when the buyers can't observe the quality of the car? In this case, the buyer is at an informational disadvantage and must "guess" about how much a particular car is worth. We will make a simple assumption about the form of the buyer's guesses, that is, we will assume that if a car is equally likely to be a plum or a lemon then a typical buyer would be willing to pay the expected value of the car. Using the numbers from the example this means that the buyer would be wiling to pay: ($1200) + ($2400) = $1800. Wait a minute! Now who are the sellers that are willing to sell at such a price? The owners of lemons would, but the owners of plums would not want to sell their cars (they require at least $2000 by assumption). So the price the buyers are willing to pay for an "average" car is less than the price that the sellers of the plum cars want in order to part with their cars. At a price of $1800 only lemons would be offered for sale!! But if the buyer was certain that he would get a lemon then he wouldn't pay $1800 for it. In fact, the equilibrium price in this market would need to be somewhere between $1000 and $1200. For prices in this range we know only owners of lemons would offer their cars for sale, and buyers would expect (correctly) to get a lemon. In this market, none of the plums ever get sold even though the price that buyers are willing to pay exceeds the price the sellers are willing to accept (i.e. $2400 > $2000). This is due to the fact that the buyer and seller have asymmetric information about the quality of the cars. Let's think about the source of this market failure... The problem is that there's an externality between the sellers of good cars and the sellers of bad cars; when a seller decides to sell a bad car, he affects the buyer's perception of the quality of the average car on the market. 326 This lowers the price that the buyers are willing to pay for the average car, and thus hurts the people who are trying to sell good cars. It is this externality that results in the market failure. Let's demonstrate with another (similar) example... Suppose now that there are 100 sellers and 100 buyers, as before, but now there are 80 plums and only 20 lemons. The expected value of the average car to the buyers then becomes: 0.2 ($1200) + 0.8 ($2400) = $2160. With a lower proportion of bad cars on the market we have a situation where the average car is worth $2160 to the prospective buyers and plum cars will now be sold! The caveat is that the cars that are most likely to be offered for sale are the ones that people most want to get rid of. The very act of offering to sell something sends a signal to the prospective buyer about its quality level. If too many low quality items are offered for sale (as we see above) it makes it more and more difficult for the owners of high quality items to sell their products. The "Lemon's Problem" is the classic example of what is called adverse selection. The low quality items in the model served to "crowd-out" the high quality items because of the high cost of acquiring information. This adverse selection problem may become so severe that it can completely destroy the market. Another example might serve to clarify what we mean by adverse selection. Consider the insurance industry. Suppose that an insurance company wants to offer insurance for bicycle theft. They do a careful market survey and find that the incidence of theft varies widely across communities. In some areas there is a high probability that a bicycle will be stolen, and in other areas bike thefts are quite rare. Suppose that the insurance company decides to offer the insurance based on the average theft rate. What do you think will happen? ANSWER: The insurance company is likely to go broke quite quickly! Who is going to buy the insurance at the average theft rate? Not the people in the safer communities they don't need much insurance anyway. Instead, the people in the communities with a high incidence of theft will want the insurance they are the ones who need it. 327 But this means that the insurance claims will be made mostly in the communities with a higher theft rate than the average. Insurance prices that are based on the average theft rate will be a misleading indicator of the actual claims experience for the insurance company. The insurance company will not get an unbiased selection of customers rather they will get an adverse selection. In fact, the term "adverse selection" was first used in the insurance industry to describe this sort of problem. It follows that insurance pricing uses a "worst-case" forecast based on the fact that low risk customers will not purchase the policy. There is a very interesting literature in this field of economics that attempts to discuss the ideas of separating equilibria, pooling equilibria, and market signalling. Staying with the insurance industry, we can have another type of information problem known as moral hazard. While the terminology is somewhat strange the problem is not difficult to describe. Let's consider the bike theft issue again and suppose, for simplicity, that all of the consumers live in areas with identical theft rates, so there is no problem of adverse selection. On the other hand, the probability of theft may be affected by the actions of the bicycle owners. For example, if the bike owner do not bother to lock their bikes then the bicycle is much more likely to be stolen than if they use a sturdy lock. Let's refer to actions that affect the probability that some event occurs as "taking care". When it sets the insurance rates the insurance company has to take into account the incentives that the consumers have to take an appropriate amount of care. If no insurance is available then consumers have an incentive to take the maximum possible amount of care. For example, if it were impossible to get bike insurance then all bike owners would use large, expensive, and secure locks. In this case, the individual bears the full cost of his actions and wants to "invest" in taking care until the marginal benefit of taking care just equals the marginal cost of doing so. However, if a consumer can buy bike insurance then the cost inflicted on the individual from having their bike stolen is much less. After all, if the bike is stolen then the person simply needs to report the theft to the insurance company and will receive insurance money to replace it. In the extreme case, where the insurance company completely reimburses the individual for the theft, the individual has no incentive to take care at all. This lack of incentive to take care is called moral hazard. Notice the trade-off involved in this scenario... 328 Too little insurance means that people bear a lot of risk, too much insurance means that people will bear too little risk and will not take appropriate care. If the amount of care taken were observable by the insurance company then there would be no problem. The insurance company could just base their rates on the amount of care taken by each individual. In reality, it is common for insurance companies to give different rates to businesses that have a fire sprinkler system in their building, or to give a better rate on home content insurance to homeowners that have a home security system. It is also quite common for smokers to pay a higher rate for life and health insurance than nonsmokers. Yet, in our example of the bike insurance we can't expect the insurance company to be able to observe the actions of the consumer...and we will still have the trade-off. What does this imply about the types of insurance contracts that will be offered? In general, insurance companies will not want to offer the consumer "complete insurance". They will always want the consumer to face some of the risk. This is why most insurance contracts include a "deductible", which is an amount that the insured party must pay in any claim. By making the individual pay some part of the claim, the insurance company can make sure that the consumer has some incentive to take care. This differs from our previous market analysis where the amount of a good traded in a competitive market is determined by the condition where demand = supply. In the case of moral hazard, a market equilibrium has the property that: [1] each individual would like to buy more insurance (i.e. fully insure themselves against loss), and [2] the insurance company would like to sell more insurance if they could know that the consumer was taking care, but, the trade does not occur because if the consumer could fully insure then they would rationally choose to take less care. SUMMARIZING ADVERSE SELECTION AND MORAL HAZARD Adverse selection refers to situations where one side of the market cannot observe the "type" or quality of the goods on the other side of the market. For this reason it is sometimes called a hidden information problem. Equilibrium in a market involving hidden information will typically involve too little trade taking place because of the externality between the "good" and the "bad" types of goods. 329 Moral hazard refers to situations where one side of the market cannot observe the actions of the other side of the market. For this reason it is sometimes called a hidden action problem. Equilibrium in a market involving hidden action typically involves some form of rationing firms would like to provide more than they do, but they are unwilling to do so since it will change the incentives of their consumers. STEPPING INTO "THE REAL WORLD?" Equilibrium outcomes in these markets will appear to be inefficient, but one has to be careful when making this assertion in "the real world". In reality, we must ask the question, "inefficient relative to what?" The equilibrium in these markets will always be inefficient relative to the equilibrium with full information. However, this is of little (or no) help when we need to make policy decisions: if the participants in the industry find it too costly to collect more information then the government would likely find it too costly as well. The real question is to ask whether some sort of government intervention in the market could improve efficiency. In the case of hidden action, the answer is usually "no". If the government cannot observe the care taken by individuals, then it can do no better than the insurance companies. Of course, the government may have other tools at its disposal that are not available to the insurance company it could compel a particular level of care, and it could set criminal punishments for those that did not take due care. However, in most cases, the government can do no better than insurance companies can in markets with hidden action. Similar issues arise in the case of hidden information. We can imagine that if the government can compel people of all risk classes to buy insurance, it is possible for everyone to be made better off. That is, on the surface, this seems to justify government intervention here. On the other hand, there are costs to government intervention as well. Economic decisions made by governmental decree may not be as cost effective as those made by private firms. Just because there are government actions that can improve social welfare does not guarantee that those actions will be taken. Furthermore, there may be private solutions to the adverse selection problem. For example, if the seller of a good car can signal to the buyer that their car is a plum (by offering a 1 year warranty, or a money back guarantee if anything goes wrong within the first 6 months) then there are incentives for the buyer to believe the car is not a lemon. 330 These solutions are often referred to in the literature as signals (the literature on signalling is becoming quite vast in areas of concern from labour signalling models to signalling product types). PREFERENCE RELATIONS AND CHOICE RULES5 In this section, we will begin our exploration of the theory of individual decision making. Our approach will be to consider decision making behaviour in a completely abstract setting. As a point of origin for our discussion we begin by thinking about any individual decision problem as a set of possible (mutually exclusive) alternatives from which the consumer must choose. In the following analysis, we will denote the set of alternatives simply by X. For now, this set can be anything. For example, when an individual faces a decision of which career path to follow then the set of possible alternatives might include: {attend medical school, go to graduate school for economics, take over the family business, ... , become a taxi driver}. There are two separate approaches to modeling individual choice behaviour. The first method treats the decision maker's tastes (as summarized in their preference relation), as the primitive characteristic of the individual. The theory begins by developing rationality axioms on the decision maker's preferences and then analysing the effects of these preferences for their choice behaviour (i.e. looking at the results with respect to the choices made). This preference based approach is the more traditional of the two general approaches. The second approach treats the individual's choice behaviour as the primitive feature and goes forward with the aid of assumptions that help to define this behaviour. The assumption that is of paramount importance in this line of investigation is referred to as the Weak Axiom of Revealed Preference (WARP). This axiom of individual behaviour will define behavioural consistency on an individual's choices. This mirrors the idea from the traditional preference based approach that the consumer will behave rationally. The choice based approach offers up some very appealing properties. The scope of the theory is broader and opens up wider possibilities for general behaviour than is feasible under the preference based approach to decision making. The choice based approach also appeals more to our human sensibilities. The assumptions that we make about choice behaviour are applied to consumer choice behaviours that are readily observable rather than 5 Paraphrased from "Microeconomic Theory" by Mas-Colell, Whinston and Green, Chapter 1 331 assu umptions ab bout intangible or unob bservable p phenomena such as pr a references (i.e. u utility functi ions). haps the mo attractiv feature o the choice based ap ost ve of e pproach is th it tells hat Perh us th individua decision processes need not n hat al necessarily be based o a on proce of "gue ess ess-timation (or induce and/or p n" ed purely theor retical cons structs such h as ut tility functio ons). What the choice based app proach says is that we can s analy individu decision making on an entirely behaviou or obse yse ual n n ural ervational basis s. ENCE BASED APPRO OACH THE PREFERE PREFERENCE RELATIONS The preference based app proach sum mmarizes the behavior of the decis e sion maker r preference relation wh hich we den note by . is a binary in a p Technically, relati on the set of altern ion s natives X th allows u to compa pairs of alternative hat us are es x,y X. example, "x x For e y" m means that "x is at leas as good a y" st as we w can der rive two other importan relations nt , define by ed y x y but not y x (1) m rence relatio , on From the prefer on X X. p First, the strict preference relation, x wher (1) is rea as "x is p re ad preferred to y". o ond, the ind difference re elation, ~, d defined by Seco x~y x wher (2) is rea as "x is in re ad ndifferent to y". o As w have disc we cussed pre eviously, microeconom theory uses the bas mic sic assu umption that consumer preference are ratio r es onal. This a assertion ca be an summ marized in the followin two basic assumptions placed on the pre ng c eference relati ; ion these two a assumption are comp ns pleteness a transitiv and vity. RAT TIONAL PR REFERENCES Our p preference relation and t transitive. is rational if it has the proper rties of bein complete ng e y and y x (2) 332 Com mpleteness: for all x,y X, we ha that x ave Transitivity: for all x,y,z X, if x r y and y y or y x (or both). z then x z. Our a assumption that n is complete has a po owerful implication. Th is, if the hat e prefe erence relat tion is complete then t individu has a cle the ual early define ed prefe erence betw ween ANY t two possibl alternativ le ves. Let's pause to consider wh this assumption me s c hat eans in pra actical terms This s. mean that the consumer has taken t time and effort to r ns the reflect on ev very poss sible combin nation of ch hoices they could compare and decided betw ween them in so ome comple ordered way (i.e. in terms of in ete n ndifference or preference). The e comp pleteness assumption allows us t move for a to rward with t assuran that this the nce s arduous task ha been performed by the individual decision maker. as n assumption that preferences are transitive is also a po n owerful assu umption and d Our a it ens sures the concept of ra ationality. T transiti The ivity proper suggests that it is rty s not p possible for the individual to make a series o pairwise choices tha result in a r e of at circu flow of preference. For exam ular p . mple, transit tivity implies that a con s nsumer cann report th they believe that dr not hat riving is at l least as good as taking the bus and t that taking the bus is a least as g at good as rid ding a horse but then a e also prefe erring riding a horse to driving. g o The t transitivity property an the comp nd pleteness p property can be difficult to satisfy n when the decisi maker is facing choices that a well rem n ion are moved from their m comm experiences. But, compared to the com mon d mpleteness property, t s the trans sitivity prope is more fundamen in the s erty e ntal sense that m much of the economic e c theor we learn would not survive if e ry economic ag gents aren' assumed to display 't trans sitive prefer rences. The assumption that the preference r n relation is complete and tra ansitive (i.e. nal mplications for the stric preferenc and ct ce ration preferences) has important im indiff ference rela ations If 1) (1 and , and ~. is rational (comple and tran ete nsitive) then n: is both irrefl s lexive (x x never holds) and transitive ( x d (if y z, then we must have x z). y (2 2) ~ is reflexive (x ~ x for alll x), transitive (if x ~ y and y ~ z then x ~ x z, s en z), and symmetric (if x ~ y, the y ~ x). 333 (3 If x 3) y z, z then x z. What does it all mean? Well, the first st , tatement sa that if w have a p ays we preference r relation that is ration then th strict preference rela nal, he ation will be irreflexive and transi e e itive. This simp means th I can ra my pref ply hat ank ference for cantaloupe at least as good as e s canta aloupe but I cannot rank my prefe erence for c cantaloupe as strictly preferred to e o 6 canta aloupe . The t transitive property of t first stat p the tement say that if I st ys trictly prefer steak to cat fo and I strictly prefe cat food t cantaloupe, then I m ood s er to must strictly prefer y steak to cantalo k oupe. This is sensible. The second sta atement is s simply point ting out that the indiffe erence relat tion is reflex xive, transit tive, and sy ymmetric. R Reflexivity m means that I am indiffe t erent betw ween one ca antaloupe a another cantaloupe AND I ca switch th order and r an heir and s remain indifferent. The trans still . sitivity prope says th if I am in erty hat ndifferent betw ween mango and apples and I am indifferen between apples and cherries, os m nt d then I must be indifferent b between ma angos and cherries. F Finally, the symmetric property says th I can sw hat witch goods around in this indiffer s rence ranking. For exam mple: Waterme elon ~ Lem mons can be equ uivalently st tated as Lemons ~ Waterm melon third conse equence of g s ve-like property that The t being rational is a transitiv holds for the str preferen relation when coupled with th at least a good as s rict nce n he as relati ion. This means that if I strictly p m prefer Conv verse All-Sta to Nike shoes and ars Nike shoes are at least as good as Ad didas shoes, then I must strictly p prefer Conv verse All-St tars to Adid shoes. Again, most of this is sensible in this das n conte ext. Yet, there are a couple of c contexts in which thes statemen are not so clear se nts ider a situation where you are choosing a pa colour for a room aint cut. Let's consi in yo house. Suppose you are give the choic between two shade of purple our en ce n es e that a imperce are eptibly clos For exa se. ample, 6 Cantaloupe is not "at least as good as" any ything. It is wh evil must taste like! hat 334 This shade compared to this shade. s suppose that you are indifferent b between these two shades. Nex suppose xt, and s you a given th choice b are he between the slightly lig e ghter shade (on the right) and an ever so slightly lighter shad still. For example, de r This shade compared to this shade. s and y expres your indif you ss fference be etween thes two shad se des. Repea ated comp parisons ar made comparing the lighter (rig hand side) shade t a slightly re e ght to y lighte shade an indifference is expr er nd ressed throu ughout. Un we arriv at the las ntil ve st comp parison... s This shade compared to this shade. and y expres your indif you ss fference be etween thes two shad se des. So we would e have the relatio e onships as f follows: ~ ~ ~...~ ~ Now, suppose you were given the choice between the first shade and the last y d at er ple purple. The en... one. Further, suppose tha you prefe dark purp to light p This shade compared to this shade. s violating the e and we would be v would result in sitivity prope erty. This r result stems from the f s fact that the pair wise e trans comp parisons wh here we rep ported our i indifference compared shades th were e d hat indistinguishable (barely different) wh hereas the f final compa arison is distin nguishable and a prefe erence can be reporte ed. Anot ther potential problem arises whe choices a present to us in certain en are ted ways This is re s. eferred to a the "fram as ming problem m". Supp pose that yo are abou to buy a laptop for $ ou ut $1000 and a software program fo or $100 The sale clerk tells you that th software is on sale for $20 off at another 0. es s he e e f r branch of the st tore, which is a 20 min nute drive a away. The laptop is th same he e er price at the othe store. t ses...you are about to buy a lapto for o op Now, suppose the same situation aris $100 and a so 00 oftware prog gram for $100. The sa ales clerk te you tha the laptop ells at p 335 is 2% off at ano % other branch of the store, which is a 20 minu drive aw h s ute way. The softw ware is the same price at the othe store. s e er of ents that rep that the would dr port ey rive to the Interestingly, the fraction o responde other store is much higher for the $20 discount o the softw r m r 0 on ware than fo the 2% or 7 savin on the laptop. ngs This is the findin even tho ng ough the inconvenienc of travel and the sa ce avings to the e 8 cons sumer is the same in b e both cases. Okay but how does this le to a violation of the transitivity property? y, d ead e y ? , o nsumer's re esponse if t the Well, we would expect indifference to be the con sales sperson said that the s store was o of stock on both the laptop an the out e nd softw ware and the consume would have to go to the other s er store for bot goods th (rece eiving $20 savings on the purchase of the tw goods co s wo ombined). s the ual vity ty. If this were the case then t individu violates the transitiv propert To see this, denote Travel to the other store to get $2 off the so 20 oftware. X=T Y=T Travel to the other store to get 2% off the laptop. % Z = b both ite buy ems at the f first store. first two choices (comparing the savings wit the optio of buying from the th on g The f first s store) say that... t X Z and Z Y, b the last scenario (w but where we c cannot buy the goods from the first sto says th X ore) hat GRO OUP DECIS SION MAKIN NG ~ Y. What problems can arise in the conte of group decision m ext p making? Supp pose we ha a house ave ehold comp prised of 3 p people Sp panky, Alfal and lfa, Buck kwheat. Fu urther suppo that the gentlem are tryi to decid upon a ose ese men ing de Frida night act ay tivity. The c choices are going to see a movie (M), going to the Pub e e g b (P), o going to Dana Porte Library to study ECON 301 (L) or er o ). Kahneman and Tversky (1984 T 4) Kahneman and Tversky attribu this to individuals keep T ute ping "mental a accounts" in w which the saving are compa gs ared to the pri of the item on which they are receiv ice m ved. 8 7 336 It is a agreed that the household is to o t operate by m majority vot Now, th te. he prefe erences of these three "Little Ras t e scals" are as follows: Alfalf fa Buck kwheat Span nky M L P P M L P M L Ther is no problem with these individ re dual prefere ences, they are transit y tive and comp plete (ration nal). But in the contex of a majo n xt ority vote...a they? are pose that we take a pa wise vote between M and L... w air e Supp Alfalf and Spanky both pr fa refer M to L while Buckwheat pre efers L to M. Majority vote will be M and they wo a ould go to th movies. he M L Supp pose that we take a pa wise vote between L and P... w air e Alfalf and Buckwheat both prefer L t P while S fa to Spanky pref fers P to L. Majority vote will be L an they wou go to th library. nd uld he L P Supp pose that we take a pa wise vote between P and M... w air e Span and Buckwheat bo prefer P to M while Alfalfa pre nky oth e efers M to P Majority P. vote will be P and they would go to th pub. he P M given these pair wise v e votes the group decision preferen nces are as follows: s So, g M L P M and t househ the hold prefere ences have taken on a intransitiv form. Th an ve he intran nsitivity illus strated in th example is known as the Con his e ndorcet para adox and it t is the central pr e roblem for the theory o group decision making. of 337 OM RENCE RE ELATIONS TO UTILIT FUNCTIO TY ONS FRO PREFER In ec conomics, we like to us utility fun w se nctions to e express pre eference relations. A utility function u(x) assigns a numeric value to each element in X, es y u s cal ssentially ranki the elem ing ments of X according t the consumer's preferences. to DEFINITION: A fun nction u: X R is a ut tility functio represen on nting prefere ence relatio on for all x, y X, x y u(x) u(y). if, s nce is no unique. ot Now, a utility function that represents a preferen relation any x) s lity n For a strictly increasing function f: R R, v(x = f(u(x)) is a new util function repre esenting the same pre e eferences as u(). It is only the ra anking of alt ternatives that m matters. As we know, from ECO 201 utilit functions with prope , ON ty s erties that are in nvariant to any strictly positive tra y ansformatio are calle ordinal. Of course, on ed the c cardinal pro operties of these utility functions a not pres are served under such asso trans sformations so we can see that th preferen relation s n he nce ociated with h a util function is an ordin property (only pres lity n nal y serves the o order or ran nking of prefe erences). Our a ability to represent con nsumer pre eferences b a utility fu by unction is very closely tied t the assumption that preference are ratio to t es onal. TEMENT: A preference relation STAT funct tion only if it is rational. i PRO OOF: tatement, w need to s we show that if there is a utility function that f To prove this st esents prefe erences , then must be comple and tran t ete nsitive. repre mpleteness: Since u() is a real-va alued function defined on X, it mu be that d ust Com for any x, y X, either u(x) u(y) or u u(x). But becau u() is a utility X u(y) use tion represe enting , this imp plies either t that x y or that y x. funct Thus s, mu be comp ust plete. can be represe ented by a u utility Tran nsitivity: Suppose that x ya y and z. Becaus u() repr se resents , we must have u(x) u(y) and u(y) u(z). Thus u(x) u(z Because u() s, z). this impli x y and y esents ies z. So, we have show that x wn repre z im mply x z, and so transitivity is establish hed. a but ht whether any rational pr y reference This is all well and good, b we migh wonder w relati ion can be repre c esented by a utility fun nction. As i turns out, the answe it er is no But I'll lea that inv o. ave vestigation to your furt ther studies s... 338 B PPROACH THE CHOICE BASED AP OICE RULE ES CHO In the second approach to the theory of decision making, c e n choice beha avior itself is s taken to be the primitive ob n bject of the theoretical construct. Formally, choice l beha avior is repr resented by means of a choice st y tructure. A choice stru ucture (B, C()) consists of two bits: f 1. B is a family (a set) of n y nonempty s subsets of X that is, every eleme of B is a X; ent set B X. This simp means th the elem ply hat ments B B are budget sets. The T budget sets in B ca be thoug of as a complete listing of all the choice an ght experiments that the re s estricted (institutionally physically socially, o y, y, or s n bly t otherwise) situation can conceivab present to the decision maker. A warning, tho w ough, it nee not includ all of the elements of X (the se of all ed de e et possible alte ernatives). C() oice rule tha assigns a nonempty set of cho at y osen elemen C(B) nts 2. C is a cho B for every budget set B B. When C(B) co b ontains a sin ngle elemen that nt, element is th individua choice from among the altern he al's natives in B The set B. C(B) may, however, co C ontain more than one e e element. W When it does the s, elements in B are the a alternatives in B that th decision maker mig choose. he ght Hunh?!? H These are his/her acce T eptable alternatives fro B. In this case, the set C(B) om e can be thoug of as co ght ontaining th hose alterna atives that w would a we actually see e chosen if the decision m e maker were to repeate e edly face th problem of choosing he g an alternativ from set B. ve EXAMPLE: pose that X = {x, y, z} and B = {{x y}, {x, y, z One po x, z}}. ossible choi structur ice re Supp is (B, C1 ()), wh here the cho oice rule C1() is: C1({x y}) = {x} a C1({x, y z}) = {x}. x, and y, In thi case, we see that x is chosen no matter w is e what budge the decisi maker et ion faces s. Anot ther possibl choice st le tructure is ( C2 ()), w (B, where the c choice rule C2() is: C2({x y}) = {x} and C2({x, y z}) = {x, y In this c x, a y, y}. case, we se that x is c ee chosen when never the decision ma aker faces b budget {x, y but they may choos x or y y}, se when they face the budget set {x, y, z n t z}. oice structu ures to mod individua consume behavior, we need to del al er , When using cho impo some re ose easonable r restrictions on the indi ividual's choice behav vior. A key assu umption, called the wea axiom of revealed p ak preference or "WARP" for short, " reflec the expectation tha a person choices w be obse cts at n's will erved to show a certa amount of consiste ain ency. 339 For e example, if an individu chooses x when fac with a c ual s ced choice betw ween x and d y it w would be qu surprising if we observed him uite m/her choos sing y when faced with h a cho oice betwee x, y, and a third alte en d ernative z. The point is that the c choice of x when facing the alternative {x, y} sug n e es ggests a tendency to c choose x ov y that ver we should see when the in w ndividual is faced with the alterna atives {x, y, z}. THE WEAK AX XIOM OF RE EVEALED PREFEREN NCE (WAR RP) DEFINITION #1 1: The choice str ructure (B, C ()) satis sfies the w weak axiom of reveale m ed prefe erence if th followin property holds: he ng y r have x C C(B), then f any B' B with x, for , If for some B B with x, y B we h y B and y C(B'), we must also have x C B' C(B'). Translation: the weak axio says tha if x is eve chosen w e om at er when y is av vailable, then there can be no budg set containing both x and y wh b get h here y is ch hosen and x is no ot. should notic that the assumption that indivi ce n idual choice behavior satisfies the e We s weak axiom imp k plies consis stency of ch hoices... If C({ y}) = {x}, then the w {x, weak axiom says that w cannot have C({x, y, z}) = {y}. m we . can he iom in a sim mpler way b defining a revealed preference by e We c state th weak axi relation fr rom the obs served choice behavio in C(). or DEFINITION #2 2: Give the choi structu (B, C () the revea en ice ure )) aled prefer rence relat tion is de efined by x y th here is som B B su that x, y B and x C(B). me uch d hat tuitive. It sa that "If x is reveale at least ays ed This translation is somewh more int as go as y, th y canno be revea ood hen ot aled preferre to x." ed We r read x y as "x is revealed a least as g s at goos as y." need no be either complete o ot or Notic that the revealed pr ce reference re elation trans sitive. Particularly, for any pair of alternative x, y to be comparab all we ne is that f es e ble eed for so ome B B we have x, y B and either x C(B) or y C(B) or both. d 340 rly hat red there is some B B we e Clear then, we would say th "x is revealed preferr to y" if t have x, y B and x C(B) a y C(B), that is, if x is ever cho d and osen over y w when both are fe easible. EXAMPLE: Rem member our previous example? D the two c Do choice struc ctures satis the weak sfy axiom m? Cons sider the ch hoice structure (B, C1 ( ()). With th choice s his structure we can infer e that... ... x y and x z but the is no re ere evealed pre eference relation that can b inferred b n be between y and z. This choice stru ucture satisfies the weak axiom since y and z are never chosen. e ). e z}) Now, let's think about choice structure (B, C2 ()) Because C2({x, y, z = {x, y} we h have these relationship that can be inferred r ps d: y x x y x z y z use y}) is to efore, the However, becau C2({x, y = {x}, x i revealed preferred t y. There choic structure (B, C2 ()) violates the weak axiom. ce e weak axiom (WARP) restricts the choice be m e ehavior of th consume in much he er The w the s same mann as the ra ner ationality as ssumption f preferen relation This for nce ns. begs the question: What is the relatio s onship betw ween the tw approaches? wo eved to hea that this is a question for furthe study (EC ar s er CON 401, You will be relie ON ECO 601). 341 ...
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This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.

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