Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 4 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. Question 1 (ECO 204 2008 Final Exam Question) If Ajax if doesn't have at least 5 pounds of food a day, he will die. In fact, with less than 5 pounds of food a day, he doesn't care about anything else. Let food be on the xaxis and everything else on the y axis. Answer the following questions: suppose that once he has the threshold level of food: (a) He prefers food and everything else as perfect complements. Graph Ajax's indifference curves. Answer: Assume that once Ajax has 5 lbs of food, that he prefers food and everything else as complements in a 1:1 ratio. In this case, his indifference curves are: 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Even though the question doesn't ask for it, you may want to think about the equation of the utility function corresponding to the indifference curves above. Suppose Q1 is food and Q2 is everything else. Then the indifference curves above have the utility function: U = Q1 for Q1 5 U = min(Q1, Q2) for Q1 > 5 (b) He prefers food and everything else as imperfect substitutes with decreasing MRS. Graph Ajax's indifference curves. Answer: Assume that once Ajax has 5 lbs of food, that he prefers food and everything else as imperfect substitutes. Recall that imperfect substitute indifference curves are one example of indifference curves with decreasing MRS. Using these, Ajax's indifference curves are: Even though the question doesn't ask for it, you may want to think about the equation of the utility function corresponding to the indifference curves above. Suppose Q1 is food and Q2 is everything else. Then the indifference curves above have the utility function: 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain U = Q1 for Q1 5 U = Q1Q2 for Q1 > 5 where and are some (positive) unknown parameters. (c) He prefers food and everything else as perfect substitutes with MRS of 1. Graph Ajax's indifference curves. Answer: Assume that once Ajax has 5 lbs of food, that he prefers food and everything else as perfect substitutes in 1:1 ratio. Recall that perfect substitutes are so called because the MRS is constant (and may very well be 1). In this case, after Ajax has had 5 lbs of food, the MRS is 1 and so Ajax's indifference curves are: Even though the question doesn't ask for it, you may want to think about the equation of the utility function corresponding to the indifference curves above. Suppose Q1 is food and Q2 is everything else. Then the indifference curves above have the utility function: U = Q1 for Q1 5 U = Q1 + Q2 for Q1 > 5 (d) He prefers food and everything else as perfect substitutes with MRS of 1/2. Graph Ajax's indifference curves. Answer: Assume that once Ajax has 5 lbs of food, that he prefers food and everything else as perfect substitutes: for an additional unit of food, he's willing to give up a unit of everything else to 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain remain on the same indifference curve. In this case, after Ajax has had 5 lbs of food, the MRS is 1/2 and so Ajax's indifference curves are: Even though the question doesn't ask for it, you may want to think about the equation of the utility function corresponding to the indifference curves above. Suppose Q1 is food and Q2 is everything else. Recall that the utility function for perfect substitutes is: U = Q1 +Q2 which has MRS = /. Thus: / = 1/2 and / = which implies that any ratio of / = will generate indifference curves with constant MRS = 1/2. Here is one: U = Q1 for Q1 5 U = Q1 + 2Q2 for Q1 > 5 Here is another: U = Q1 for Q1 5 U = 2Q1 + 4Q2 for Q1 > 5 (e) He only cares about everything else. Graph Ajax's indifference curves. Answer: Assume that once Ajax has 5 lbs of food that he does not care about food anymore. Ajax's indifference curves are: 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Even though the question doesn't ask for it, you may want to think about the equation of the utility function corresponding to the indifference curves above. Suppose Q1 is food and Q2 is everything else. Then the indifference curves above have the utility function: U = Q1 for Q1 5 U = Q2 for Q1 > 5 Question 2 After giving really hard tests in his ECO 200 class, Carlos loves to go out and celebrate over dinner where he likes to have Porterhouse steaks with glasses of Chteau Mouton Rothschild 1986 wine as complements. Let wine be on the xaxis and steak on the yaxis. Draw Carlos' indifference curves given that as he has more steaks, he prefers to have more glasses of wine per steak at an increasing rate. For example, he may have a glass of wine with the first steak; more than 2 glasses of wine with the second steak and so on. There is no one right answer depending on the assumptions you make, there can be a range of correct answers. Answer: Carlos's indifference curves are depicted above. One utility function that may generate these indifference curves is: 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain U = min({Q1}1/2 , Q2 ) Question 3 In this question you will practice the CobbDouglas utility function. Ajax has utility function U = Q11/5Q24/5. (a) What are and in U = Q1Q2? Answer: = 1/5 and = 4/5. (b) Does Ajax's utility function U = Q11/5Q24/5 represent monotone "more is better" preferences over Q1 and Q2? Answer: Yes. There are two ways of seeing this. Note that as either Q1 and/or Q2 increase, utility increases. More formally, the marginal utility of goods 1 and 2 for positive amounts of goods 1 and 2 are positive: MU1 = dU/dQ1 = (1/5)(Q2/Q1)4/5 MU2 = dU/dQ2 = (4/5)(Q1/Q2)1/5 Observe the utility function has diminishing marginal utility: as Q1 rises, MU1 increases, but at a diminishing rate. The same is true for MU2: as Q2 rises, MU2 increases, but at a diminishing rate. (b) Why does the utility function U = (1/5) log (Q1) + (4/5) log (Q2) also represent Ajax's preferences over Q1 and Q2? Answer: Recall that a monotonic transformation of a utility function represents the same preferences. Intuitively, any scaling up or down of the utility function preserves the ordering of preferences. Let's see if the utility function U = (1/5) log (Q1) + (4/5) log (Q2) is a monotonic transformation of U = Q11/5Q24/5. Now: U = Q11/5Q24/5 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain log U = log (Q11/5Q24/5) log U = log Q11/5 + log Q24/5 log U = (1/5) log (Q1) + (4/5) log (Q2) Now if A > B, then log (A) > log (B). Thus if bundle A consisting of goods 1 and good 2 is preferred to bundle B consisting of goods 1 and 2, then both U = (1/5) ln(Q1) + (4/5)ln(Q2) and U = Q11/5Q24/5 will assign a higher value to A than B. As an example, suppose bundle A has Q1 = 2 and Q2 = 2 while bundle B has Q1 = 1 and Q2 = 1. Since preferences are monotone, A is preferred to B. In fact, both utility functions give this result: U = Q11/5Q24/5 U(A) = (2)1/5(2)4/5 > (1)1/5(1)4/5 = U(B) and: U = (1/5) ln(Q1) + (4/5)ln(Q2) U(A) = (1/5)ln(2) + (4/5)ln(2) > (1/5) ln(1) + (4/5)ln(1) = U(B) (c) Given an arbitrary level of utility, what is the equation of Ajax's indifference curve over Q1 and Q2? Answer: If Ajax has utility function U = Q11/5Q24/5 then: U = Q11/5Q24/5 Q24/5 = U/Q11/5 Q2 = U5/4/Q11/4 In a Q1, Q2 plot this will have the form of an "imperfect substitutes" indifference curve. (d) Suppose all good are sold at uniform prices. Assume Ajax is a price taker. Show that Ajax will always spend 20% of his income Y on good 1 and the remainder on good 2. 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: Recall some important properties of the CobbDouglas utility function: the price elasticity is 1, the income elasticity is 1 and the consumer (obviously why?) spends a constant fraction of income on both goods which implies that her expenditure on each good is a constant. In fact, you should again verify that if: U = Q1Q2 then P1Q1 = {/( + )} Y and P2Q2 = {/( + )}Y. Thus, with = 1/5 and = 4/5 we have: P1Q1 = {/( + )}Y = Y/5, or 20% of income P2Q2 = {/( + )}Y = 4Y/5, or 80% of income Here is a "proof" similar to the one in the lectures: since these are imperfect substitute indifference curves given uniform prices at the optimal choice, the budget line will be tangent to the indifference curve. Thus for Ajax's UMP: MU1/MU2 = P1/P2 {(1/5)(Q2/Q1)4/5}/{(4/5)(Q1/Q2)1/5} = P1/P2 (1/4)(Q2/Q1) = P1/P2 (1/4)(P2Q2) = P1Q1 This says, the expenditure on good 1 is 25% of the expenditure on good 2. Now, because these are more is better preferences, we know Ajax will spend his entire income. Thus: P1Q1 + P2Q2 = Y Substituting (1/4)(P2Q2) = P1Q1 above, we have: (1/4)(P2Q2) + P2Q2 = Y (5/4)(P2Q2) = Y 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain P2Q2 = (4/5)Y Ajax spends 80% of his income on good 1 and therefore, he spends 20% on good 1: P1Q1 = (1/5)Y. (e) Derive Ajax's demand equation for good 1. Answer: The demand equation is a function that gives quantity in terms of a given price. Mathematically it is: Q1 = f(P1). From above, we have: P1Q1 = (1/5)Y Q1 = (1/5)Y/P1 Q1 = (1/5) Y P11 With Y ceteris paribus this is the (constant elasticity) demand equation for good 1. (f) What is Ajax's price elasticity of good 1? Answer: From above, we had: Q1 = (1/5) Y P11 Thus, the price elasticity is 1. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing E = (d log Q1)/(d log P1) or (iii) E = (dQ1/dP1)(P1/Q1). I'll do (ii) and (iii) below instead of reading, you should do these: (ii) Start with Q1 = (1/5) Y P11 : log Q1 = log {(1/5) Y P11 } log Q1 = log (1/5) + log Y + log P11 log Q1 = log (1/5) + log Y log P1 Now: E = (d ln Q1)/(d ln P1). Thus: 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain E = (d log Q1)/(d log P1) = 1 (You're taking log Q1 as the "y" variable and log P1 as the "x" variable and doing dy/dx). (iii) Start with Q1 = (1/5) Y P11 : Now: E = (dQ1/dP1)(P1/Q1) E = ( (1/5) Y P12 ) (P1/Q1) E = (1/5) Y P11 /Q1 Substituting Q1 = (1/5) Y P11 above we get: E = {(1/5) Y P11 }/{(1/5) Y P11 } E = 1 Given that Ajax always spends a constant fraction of income on each good, you should lend some thought as to why the price elasticity must be 1. (g) What is Ajax's income elasticity of good 1? Answer: From above, we had: Q1 = (1/5) Y P11 Thus, the income elasticity is 1. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d log Q1)/(d log Y) or (iii) (dQ1/dY)(Y/Q1). I'll do (ii) and (iii) below instead of reading, you should do these: (ii) Start with Q1 = (1/5) Y P11 : log Q1 = log {(1/5) Y P11 } log Q1 = log (1/5) + log Y + log P11 log Q1 = log (1/5) + log Y log P1 Now: E = (d log Q1)/(d log Y). Thus: 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain E = (d log Q1)/(d log Y) = 1 (You're taking log Q1 as the "y" variable and log Y as the "x" variable and doing dy/dx). (iii) Start with Q1 = (1/5) Y P11 : Now: E = (dQ1/dY)(Y/Q1) E = ((1/5)P11 ) (Y/Q1) E = (1/5) P11 Y /Q1 Substituting Q1 = (1/5) Y P11 above we get: E = {(1/5) Y P11 }/{(1/5) Y P11 } E = 1 Given that Ajax always spends a constant fraction of income on each good, you should lend some thought as to why the income elasticity must be 1. (h) What is Ajax's crossprice elasticity for good 1? Answer: From above, we had: Q1 = (1/5) Y P11 Note that demand for good 1 does not depend on price of good 2: so we know the cross price elasticity must be zero. If you wanted to show this mathematically then observe that the equation above can be written as: Q1 = (1/5) Y P11 P20 Thus, the crossprice elasticity of good 1 (with respect to good 2) is 0. You can justify this three ways: (i) by recognizing this is a constant elasticity demand function (ii) doing (d log Q1)/(d log P2) or (iii) (dQ1/dP2)(P2/Q1). I'll do (ii) and (iii) below instead of reading, you should do these: (ii) Start with Q1 = (1/5) Y P11 P20: log Q1 = log {(1/5) Y P11 P20} log Q1 = log (1/5) + log Y + log P11 + log P20 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain log Q1 = log (1/5) + log Y log P1 + 0 Now: E = (d log Q1)/(d log P2). Thus: E = (d log Q1)/(d log P2) = 0 (You're taking log Q1 as the "y" variable and log Y as the "x" variable and doing dy/dx). (iii) Start with Q1 = (1/5) Y P11 P20 Now: E = (dQ1/ P2)( P2/Q1) E = (0)(P2/Q1) E = 0 Question 4 Jenn has utility function U = Q11/5. (a) What are and in U = Q1Q2? Answer: = 1/5 and = 0. Right away, from the CobbDouglas utility function above, you should recognize that Jenn will spend all her income on good 1 and none on good 2 since: P1Q1 = {/( + )}Y = (5/5)Y, or 100% of income P2Q2 = {/( + )}Y = (0/5)Y, or 0% of income. (b) Given an arbitrary level of utility, what is the equation of Jenn's indifference curve over Q1 and Q2? Answer: If Jenn has utility function U = Q11/5 then: U = Q11/5 Q11/5 = U Q1 = U5 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain In a Q1, Q2 plot this will have the form of "vertical" indifference curves: (c) Suppose all goods are sold at uniform prices. Assume Jenn is a price taker..Show that Jenn will always spend 100% of his income Y on good 1 and the nothing on good 2. Answer: Notice that you cannot show this by appealing to a tangency argument: since these are vertical indifference curves given uniform prices at the optimal choice, the budget line will be not be tangent to the indifference curve: Thus for Jenn's UMP: MU1/MU2 P1/P2 In fact, it gets worse: you can't even solve this problem using the Lagrangian method. The technical explanation for the Lagrangian to "work" is that the indifference curves must be tangent to the budget line. This is not the case here. Hence, if you tried using the Lagrangian you'd get: 13 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Choose Q1 and Q2 to maximize L = Q1 (P1Q1 + P2Q2 Y) Take first order conditions: L/Q1 = 0 1 P1 = 0 L/Q2 = 0 0 P2 = 0 L/ = 0 P1Q1 + P2Q2 = Y Now, remember that prices are positive: P1 > 0 and P2 > 0. The L/Q2 = 0 equation implies therefore that = 0. But the L/Q1 = 0 implies that > 0, which is a contradiction. Hence, you can't use the Lagrangian to solve it. Instead, appeal to the graph above: there, observe that Jenn will spend her entire income on good 1. Since Jenn has more is better preferences, she will spend her entire income and thus: P1Q1 = Y Q1 = Y/P1 Question 5 GIndart has utility function U = min(2Q1 , 4Q2) (a) What are and in U = min(Q1,Q2)? Answer: = 2, = 4. (b) Does GIndart's utility function U = min(2Q1 , 4Q2) represent monotone "more is better" preferences over Q1 and Q2? Answer: Yes. You could show it mathematically that if bundle A >> bundle B and A is preferred to B then these are monotone preferences. For example, suppose: 14 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Bundle A (Q1, Q2) = (2, 2) Bundle B has (Q1, Q2) = (1, 1). Note how bundle A has "more" of good 1 and good 2 than bundle B. Observe: U(A) = min(2(2) , 4(2) ) = min(4,8) = 4 and: U(B) = min(2(1) , 4(1) ) = min(2,4) = 2 Thus U(A) > U(B) and thus preferences are monotone. (c) Suppose GIndart purchases goods 1 and 2 for (uniform) prices P1 and P2. Assume GIndart is a price taker. Derive an expression for GIndart's expenditure on goods 1 and 2. Answer: From the discussion of complements, recall that GIndart must spend his entire income and that the optimal choice must be on the line Q2 = (1/2)Q1. Substitute this in the budget constraint: P1Q1 + P2Q2 = Y P1Q1 + P2 (1/2)Q1 = Y Q1(P1 + P2/2) = Y Q1(2P1 + P2)/2 = Y Q1 = 2Y/(2P1 + P2) This is the demand equation for good 1. To obtain an expression for the expenditure on good 1, simply do: P1Q1 = 2YP1/(2P1 + P2) (d) Derive GIndart's demand equation for good 1. 15 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: From above, this is: Q1 = 2Y/(2P1 + P2) Note that unlike the CobbDouglas UMP in questions 1 and 2, demand for good 1 here depends on the price of goods 1 and 2. This is obvious since GIndart prefers the two goods as complements and so the price of both goods will determine his demand for good 1 (and good 2). Even though the question does not ask for it, you can derive good 2's demand equation by noting that since: Q2 = (1/2)Q1 Q2 = (1/2) {2Y/(2P1 + P2)} = Y/(2P1 + P2) (e) What is GIndart's price elasticity of good 1? Answer: It's best to use the formula E = (dQ/dP)(P/Q) for this question. First rewrite the demand function as: Q1 = 2Y(2P1 + P2)1 Now: dQ1/dP1 = 4Y/(2P1 + P2)2 Thus: E = (dQ1/dP1)(P1/Q1) = 4(Y/(2P1 + P2)2 )(P1/Q1) E = 4(Y/(2P1 + P2)2 )(P1/{2Y/(2P1 + P2)}) E = 2P1/(2P1 + P2) Which is not a constant elasticity. (f) What is GIndart's income elasticity of good 1? 16 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: Again, start with: Q1 = 2Y/(2P1 + P2) Use the formula E = (dQ/dY)(Y/Q): Now: dQ1/dY = 2/(2P1 + P2) Thus: E = (dQ1/dY)(Y/Q1) = (2/(2P1 + P2)) (Y/Q1) E = (2/(2P1 + P2)) (Y/{2Y/(2P1 + P2)}) E = 1 which is a constant elasticity. Question 6 (ECO 204 2008 Final Exam Question) Alma's preferences over goods 1 and 2 are given by the utility function: U = Q1 Q2 where Q1 and Q2 are units of goods 1 and 2 respectively. Suppose + = 1. Denote Alma's weekly income by Y. Denote the price of good 1 by P1 and the price of good 2 by P2. (a) Derive Alma's demand functions for goods 1 and 2. Show your calculations. Answer: Set up the Lagrangian: Choose Q1 , Q2 to max L = Q1 Q2 (P1Q1 + P2Q2 Y) The first order conditions are: L/Q1 = 0 Q11 Q2 P1 = 0 17 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L/Q2 = 0 Q1 Q21 P2 = 0 L/ = 0 P1Q1 + P2Q2 = Y Dividing first and second equations: { Q1 1 Q2} / { Q1 Q2 1 } = P1 / P2 ( / ) Q2 / Q1 = P1 / P2 Q2 = Q1 (/) (P1 / P2) Next turn to the budget constraint: P1Q1 + P2Q2 = Y P1Q1 + P2 Q1 (/) (P1 / P2) = Y P1Q1 + P1Q1 (/) = Y P1Q1 (1 + (/)) = Y P1Q1 ( + )/ = Y Q1 = {/( + )} Y/ P1 But + = 1. Thus: Q1 = Y/ P1 This implies that: Q2 = Y/ P2 (b) True or false: for a company selling good 1 to Alma any price will maximize revenues. Show your reasoning. Answer: True. Alma's demand function for good 1 is: Q1 = Y/ P1 18 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Rewrite this as: Q1 = Y P11 This is a constant elasticity demand function with E = 1. We know that revenue maximization is the same as E = 1. Thus, any price of good 1 will maximize revenues. Here is another explanation. From the demand curve: Q1 = Y/ P1 P1 Q1 = Y R1 = Y Note how revenue from good 1 is always a constant fraction of income. Thus, if price increases by x% demand must fall by x%. Thus E = 1. (c) True or false: a 10% increase in all prices and Alma's income will not change her optimal consumption of goods 1 and 2. Briefly explain your reasoning. Answer: The statement is correct. From the demand functions of goods 1 and 2: Q1 = Y P11 Q2 = Y P21 Observe that as income and prices increase by 10% the ratio of Y to price remains constant so that Alma will not change her consumption of goods 1 and 2. You can also show this result graphically by showing that if income and prices increase by 10%, Alma's budget set will not change so that, given her preferences, her consumption of goods 1 and 2 does not change. (d) Suppose = 0.25 and = 0.75. Evaluate the statement: "a 100% increase in Alma's income leads to a 25% increase in the consumption of good 1 and a 75% increase in the consumption of good 2". Answer: The statement is incorrect. From the demand function of good 1: Q1 = Y P11 19 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain We can derive the income elasticity. Either using (dQ/dY)(Y/Q) or d log Q / d log Y we see that the income elasticity for good 1 is: EY = 1 Thus, a 100% increase in Y will lead a 100% increase in the consumption of good 1. The same result holds for good 2 since its demand function also has EY = 1. Question 7 (ECO 204 2008 Final Exam Question) Your utility is a function of three goods: U = min(Q1 , Q2 , Q3 ) Denote income by Y and the price of good 1 by P1, the price of good 2 by P2 and the price of good 3 by P3. (a) Derive the demand equations for goods 1, 2 and 3. Explain your reasoning and show all calculations. Answer: The easiest way to solve this equation is to remember how you did the two good case. There: U = min(Q1 , Q2 ) And the budget constraint was: P1Q1 + P2Q2 = Y We argued that the optimal solution must be at the "corner" of the indifference curves. Thus at the optimum: Q1 = Q2 Inserting into budget constraint gives: P1Q1 + P2Q1 = Y Q1 (P1 + P2) = Y Q1 = Y/(P1 + P2) Which implied that: Q2 = Y/(P1 + P2) 20 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain In the three goods case it be true that at the optimum, the consumption of all three goods must be equal or that: Q1 = Q2 = Q3 Inserting into budget constraint: P1Q1 + P2Q2 + P3Q3 = Y Q1 (P1 + P2 + P3) = Y Q1 = Y/(P1 + P2 + P3) And therefore: Q1 = Q2 = Q3 = Y/(P1 + P2 + P3) (b) What is the income elasticity for good 1? Show your calculations. Answer: The demand function for good 1 is: Q1 = Y/(P1 + P2 + P3) Now: log Q1 = log Y log (P1 + P2 + P3) EY = d log Q1 /d log Y = 1 The income elasticity is 1. Question 8 In this question you will do a variation of a utility maximization problem (UMP) that we've seen before. Suppose a consumer's utility over 3 goods is given by: 21 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain U = Q1 Q2 Q3 Denote the consumer's income by Y. Solve for the optimal demands for goods 1, 2 and 3. Answer: Form the Lagrangian: Choose Q1 , Q2 , Q3 to max L = Q1 Q2 Q3 (P1Q1 + P2Q2 + P3Q3 Y) The first order conditions (FOCs) are: L/Q1 = 0 Q11 Q2 Q3 P1 = 0 L/Q2 = 0 Q1 Q21 Q3 P2 = 0 L/Q3 = 0 Q1 Q21 Q31 P3 = 0 L/ = 0 P1Q1 + P2Q2 + P3Q3= Y Dividing first and second FOCs: Q11 Q2 Q3 / Q1 Q21 Q3 = P1/P2 Q2 / Q1= P1/P2 .. (1) Dividing second and third FOCs: Q1 Q21 Q3 / Q1 Q2 Q31 = P2/P3 Q3 / Q2 = P2/P3 .. (2) Dividing first and third FOCs: Q11 Q2 Q3 / Q1 Q2 Q31 = P1/P3 Q3 / Q1 = P1/P3 .. (3) Now the budget constraint is: P1Q1 + P2Q2 + P3Q3 = Y. Substitute Q2 in terms of Q3 from equation (2) and Q3 in terms of Q1 from equation (3): 22 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain P1Q1 + P2Q2 + P3Q3 = Y P1Q1 + P2 (/)(P3/P2)Q3 + P3 (/)(P1/P3)Q1 = Y P1Q1 + (/) P3 Q3 + (/) P1Q1 = Y Now substitute Q3 in terms of Q1 from equation (3): P1Q1 + (/) P3 (P1/P3) Q1 (/) + (/) P1Q1 = Y P1Q1 + (/) P1 Q1 + (/) P1Q1 = Y P1Q1 [1 + (/) + (/)] = Y P1Q1 [ + + ]/ = Y P1Q1 = {/[ + + ]} Y This says, the expenditure of good 1 is a constant fraction of income. Now: Q1 = {/[ + + ]} Y/ P1 which yields the optimal demand for good 1. Now use equations (2) and (3) to get the optimal demands for goods 2 and 3: Q2 = {/[ + + ]} Y/ P2 Q3 = {/[ + + ]} Y/ P3 Notice these are each constant elasticity demand functions with income elasticity 1 and price elasticity 1. In particular, if + + = 1 then: Q1 = Y/ P1 Q2 = Y/ P2 Q3 = Y/ P3 Contrast this with the two goods case and you'll see a pattern emerge. If you see the pattern, solving for the 4 good or 5 goods case should be really easy. Try it. 23 ...
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 Fall '08
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 Economics, Microeconomics, Utility, S. Ajaz Hussain, Ajaz Hussain

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