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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 9 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 Suppose a company has the production function: q = L k and has constant returns (not the same as constant returns to scale) (a) What is the short run production function? Answer: With capital fixed and labor variable, this is: q = f(L, k) = L k If there are constant returns, it means that (holding k constant) doubling L doubles q. Thus = 1 and: q = L k (b) What is the elasticity of output with respect to labor? Does your answer make sense given the company has constant returns? Answer: We have from part (a): q = L k 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain We want the elasticity of q with respect to L = % q / % L. q = L k log q = log L + log k d log q / d log L = 1 % q / % L = 1 Put simply: if L increases by x% then q will also increase by x%. This makes sense given that we have constant returns (increasing inputs by x% increases q by x% too). (c) Suppose the company doubles labor. Without using the equation for C(q) what is the impact on total variable cost? Average variable cost? Average cost? Answer: Total variable cost TVC(q) is: PL L. If the company doubles labor, then TVC(q) must also double. Now: AVC(q) = TVC(q)/q As the company doubles inputs because it has constant returns it doubles output. Thus, when the company doubles inputs, the ratio TVC(q)/q remains constant. On the other hand: AC(q) = C(q)/q AC(q) = {TFC + TVC(q)}/q AC(q) = TFC/q + TVC(q)/q AC(q) = AFC(q) + AVC(q) With constant returns, as labor is doubled, AVC(q) remains constant but AFC(q) declines with q. 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Thus, AC(q) declines. Question 2 In this question you will examine the properties of the short run cost, average cost, average variable, average fixed and marginal cost functions for the short run CobbDouglas production function: Q = L k. Note that capital is fixed at a level k. (a) When does this production function exhibit constant returns to labor? Answer: Remember that returns to scale is a long run concept: it is about the impact of changing all inputs on output. In contrast, returns is a short run concept: it is about the impact of changing all variable inputs on output. The only variable input here is labor and there are several ways of looking the impact of labor levels on output. You should do all to practice math techniques. First observe that with labor levels L, the output is: Initial Q = L k If we double labor levels, the new output is: New Q = (2L) k New Q = 2 L k New Q = 2 (Initial Output) From this: When > 1, New output = 2 (Initial Output) > 2*Initial Output, so that doubling labor has more than doubled output "increasing returns" When = 1, New output = 2 (Initial Output) = 2*Initial Output, so that doubling labor has doubled output "constant returns" 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain When < 1, New output = 2 (Initial Output) < 2*Initial Output, so that doubling labor has less than doubled output "decreasing (or diminishing) returns" Another way of seeing this is by elasticities: Q = L k log Q = log {L k} log Q = log L + log k log Q = log L + log k From this, the: Change in log Q = (change in log L) + (change in log k) d log Q = d log L + d log k But capital is fixed so that d log k = 0: d log Q = d log L Recall d log x = dx/x so that: dQ/Q = dL/L % in Q = (% in L) Note that: When > 1, % in Q > (% in L) so that a % change in labor leads to greater percentage change in Q (increasing returns) When = 1, % in Q = (% in L) so that a % change in labor leads to an equal percentage change in Q (constant returns) 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain When < 1, % in Q < (% in L) so that a % change in labor leads to lower percentage change in Q (decreasing returns) (b) Derive the short run demand for labor: Answer: There are two ways to solve for this. Let's purposely do both. The first technique is from recognizing that if a firm has target output q and fixed capital k, the "optimal" labor can be simply read off the isoquant: The equation of the isoquant is: q = L k L = q/k L = {q/k}1/ We'd get the same answer, if we did the Lagrangian: L = PL L [L k q] L = PL L + PK k L k + q The FOCs are: 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L/L = 0 PL L1 k = 0 PL = L1 k L/ = 0 L k = q The 2nd FOC is identical to the sans Lagrangian equation we used to solve for L: L k = q L = q/k L = {q/k} 1/ (c) Derive the short run total cost functions and examine the functional form when there are increasing, constant and decreasing returns. Answer: The short run total cost is: C = PL L + PK k C = PL L + PK k But L = {q/k} 1/ C = PL {q/k} 1/ + PK k C = PL {q/k} 1/ + TFC Let's examine the 3 cases: 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Returns Increasing ( > 1) Constant ( = 1) Decreasing ( < 1) Short Run Cost Function C = PL {q/k} 1/ + TFC C = PL {q/k} + TFC C = PL {q/k} 1/ + TFC Functional Form Nonlinear; concave Linear Nonlinear; convex Observe that because TFC is, by definition, a constant and therefore a horizontal line the shape of the total cost function (concave, linear, convex) must also be the shape of the total variable cost function: (d) Examine the functional form of the average fixed cost function. Answer: AFC is defined as: AFC = TFC/q AFC = (PK k)/q Thus, as q increases, AFC must fall intuitively the company is spreading fixed costs over volume. A simple example is a cable TV company. There is an enormous fixed cost of laying down the cable; but with every subscriber, the fixed cost per subscriber falls. 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (e) Examine the functional form of the average variable cost function. Answer: AVC is defined as: AVC = TVC/q AVC = (PL {q/k} 1/)/q AVC = PL {q1//k/ }/q AVC = PL {q1/ k/ }/q AVC = PL q(1/)1 k/ AVC = PL q(1 )/ k/ The functional form of the AVC function now depends on the parameter , representing "returns": Returns Increasing ( > 1) Constant ( = 1) Decreasing ( < 1) Short Run AVC Function AVC = PL q(1 )/ k/ AVC = PL q k/ AVC = PL q(1 )/ k/ Functional Form Nonlinear; falls with q Linear; constant Nonlinear; rises with q (f) Examine the functional form of the average cost function. Answer: C = TFC + TVC C/q = TFC/q + TVC/q AC = AFC + AVC We know that AFC always fall with output. On the other hand, AVC will fall, be constant, or rise 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain with increasing, constant, or decreasing returns. The shape of the AC function depends on the interaction between AFC and AVC. The general shapes are as follows: Notice a few things: with increasing returns, AC falls with volume; with constant returns, AC falls up to a point and then flattens out this is because the constant AVC term dominates the ever shrinking AFC; with decreasing returns, AC is (possibly) Ushaped: at first the declining AFC may dominate the growing AVC, but eventually the AVC dominates. (g) Give a returns to scale explanation for the long run Ushaped AC curve and contrast this with a returns explanation for the short run Ushaped AC curve. Restrict the discussion to the Cobb Douglas production function. Answer: Restricting ourselves to a returns to scale explanation for the Ushaped long run AC curve (i.e. without invoking inputs becoming cheaper and then more expensive), we saw that with a Cobb Douglas production function, the only way you can have a Ushaped long run AC curve is if, as volume expands, there are increasing returns to scale, followed by constant returns to scale, followed by decreasing returns to scale. This would mean that ( + ) is initially ( + ) > 1 then ( + ) = 1 and finally ( + ) < 1. Turning to a returns explanation for the Ushaped short run AC curve (i.e. without invoking inputs becoming cheaper and then more expensive), we see that with a CobbDouglas production function, the only way you can have a Ushaped long run AC curve is if there are decreasing returns or that < 1 regardless of the returns to scale (i.e. whether ( + ) is > 1, = 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 1, or < 1). In contrast to the long run case, for the short run AC to have a Ushape, you do not need to have the "returns" changing with volume. Now you can see why ECO 100 tends to give you Ushaped AC curves: as we'll see, real life companies are best modeled through decreasing returns. Hence, these companies can be modeled by Ushaped AC curves. But be warned, the AC curve in general can have any shape. (h) Examine the functional form of the marginal cost function. In particular, when is the MC curve above the AVC curve? When is the MC curve coincidental with the AVC curve? When is the MC curve below the AVC curve? Hint: derive MC and express it terms of AVC. Answer: It is important to understand the difference between AVC and MC: students tend to get confused. AVC is the variable cost per unit; MC is the cost of the next unit. Graphically, AVC is the slope of the ray from the origin to the cost or TVC function. MC is the slope of the tangent on a point on the cost or TVC function. Now: MC is defined as: MC = dC/dq MC = d{TFC + TVC}/dq MC = dTFC/dq + dTVC/dq But TFC is a constant MC = dTVC/dq MC = d(PL {q/k} 1/)/dq MC = d{PL q1/ k/ }/dq MC = (1/) PL q(1/)1 k/ MC = (1/) PL q(1 )/ k/ If you look at this carefully you'll notice that: MC = (1/) AVC 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Hence, the functional form of the MC function is the same as the AVC function. So: if there are increasing returns, AVC falls with volume and so does MC if there are constant returns, AVC is constant with volume and so is MC if there are decreasing returns, AVC rises with volume and so does MC Short Run MC Function MC = (1/) PL q(1 )/ k/ MC = PL q k/ MC = (1/) PL q(1 )/ k/ Functional Form Nonlinear; falls with q Linear; constant Nonlinear; rises with q Returns Increasing ( > 1) Constant ( = 1) Decreasing ( < 1) In fact, you can confirm something that ECO 100 tells you: When AVC falls, MC also falls, and the MC curve lies below the AVC curve When AVC is constant, MC is also constant, and the MC curve coincides with the AVC curve When AVC rises, MC also rises, and the MC curve lies above the AVC curve Can we confirm this from the equations? Yes. Look at: MC = (1/) AVC Observe that if: > 1 (1/) < 1 = 1 (1/) = 1 < 1 (1/) > 1 Hence: 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Returns Functional Form of MC and AVC Nonlinear; falls with q MC vs. AVC MC < AVC (i.e. MC lies below AVC) MC = AVC (i.e. MC coincides with AVC) MC > AVC (i.e. MC lies above AVC) Increasing ( > 1) Constant ( = 1) Linear; constant Decreasing ( < 1) Nonlinear; rises with q Hopefully, you now have a deep understanding of cost curves. Question 3 Ajax Inc. produces cranberry juice and has the short run cost function C(q) = 5 + Qc. Don Damiano Inc. produces grape juice and has the short run cost function C(q) = 5 + Qg. If Ajax and Don Daminao merge, the short run cost function for both cranberry and grape juice is: C(q) = 5 + Qc + Qg. Are there economies of scope from a Ajax and Don Damiano merger? Answer: To see if there economies of scope, we simply need to check if cost of cranberry and grape juices is lower than the sum of cranberry plus grape juice. Now: C(Qc ,Qg) = 5 + Qc + Qg Whereas C(Qc) + C(Qg) = 5 + Qc +5 + Qg.= 10 + Qc + Qg Observe: C(Qc ,Qg) < C(Qc) + C(Qg) This shows that it is cheaper to produce the juices together than separately. Therefore, there are economies of scope. 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 4 (Summer 2008 Test Question) The faculty club bar at the University of Toronto employs labor L (labor hours) to serve customers Q. The production function is: Q = 60L L2. Currently, PL = $16, P = $2 and L = 1. The club hires L and sells Q in competitive markets. (a) Currently, at L = 1, what is the MPL = dQ/dL? Show all calculations clearly and interpret your answer. Answer: The MPL, or dQ/dL, is how much output an additional worker would produce. Now: Q = 60L L2 dQ/dL = 60 2L Note how the output from the next workers depends on how many workers there are: as L then MPL . In a sense, there is a case of "too many cooks in the kitchen, spoil the broth" story. At L = 1: MPL = 60 2(1) = 58 This says that starting with 1 laborhour, if the club hires another unit of laborhour, an additional 58 customers can be served. (b) Does the faculty club have decreasing, increasing or constant marginal returns? Show all calculations clearly. Answer: Note how MPL = 60 2L decreases with L. Hence the club has decreasing marginal returns each successive (marginal) worker is less productive than his/her predecessor. (c) What is the profit maximizing number of customers and labor? Show all calculations clearly. Hint: this is not the typical producer CMP. Maybe you want to maximize profits. Answer: This problem can be solved in several ways. Let's do the Lagrangian first. In fact, you don't want to do the CMP as that only works if you know the target output. In this question, you need to 13 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain determine the output and labor. You can instead do the profit maximization problem: Choose L and q to maximize: L = Pq {PLL [f(L) q]} This is revenues minus cost and the constraint that output should be equal to target output. P is the price of output this being a competitive market, take P as given and P = $2 (given in question). Note how the production function has been expressed only in terms of L: there is no capital here. Now PL = $16 and Q = 60L L2 so that: L = 2q {16L [60L L2 q]} L = 2q 16 L + [60L L2 q] L = 2q 16 L + 60L L2 q The first order conditions are: dL/dq = 0 2 = 0 = 2 This is nice: we've managed to solve for ; but what does it mean? Look at the Lagrangian: note how dL/dq = . Thus is the impact on profits from having another customer: in this case, each additional customer results in profits of $2. dL/dL = 0 16 + 60 2 L = 0 Substitute: = 2 16 + 60(2) 2(2) L = 0 16 + 120 4 L = 0 104 = 4 L 14 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L = 26 Hence, we need to hire 26 labor hours. But what about output? From the last FOC: dL/d = 0 60L L2 q = 0 But L = 26 60(26) (26)2 q = 0 q = 884 The decision: serve 884 customers by hiring 26 labor hours, provided the wage/laborhour is $16 and the price per drink is $2. By the way, here is another way to solve the question: since there is no demand curve in this question you could express revenues and costs in terms of labor, and solve for the optimal labor: (L) = R(L) C(L) (L) = PQ(L) TFC TVC(L) (L) = PQ(L) TFC PLL (L) = 2 Q(L) TFC 16L (L) = 2[60L L2] TFC 16L (L) = 120L 2L2 TFC 16L (L) = 104L 2L2 TFC Maximizing profits with respect to L: d (L)/dL = 0 L = 26 15 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Substitute back in the production function: Q = 60L L2 Q = 60(26) (26)2 Q = 884 Same answers. Nice. (d) What is the impact on the club's profits from, ceteris paribus, an increase in the price of output and wages? Answer: Apply the envelope theorem! Express the Lagrangian in variable/parameter form: L = Pq PL L + [60L L2 q] Profits will be impacted by a higher output price by: dL/dP = q If the club is selling at least one drink, so that q > 0, as price of drinks rises, profits will increase in fact, by the amount of drinks sold. Profits will be impacted by higher wages by: dL/d PL = L Each dollar increase in wages, results in lower profits and in the amount of L employed at faculty club. Question 5 (Summer 2008 Test question) Jen Burgers (JB) uses labor L and capital K to "produce" burgers. Her production function is: q = K 1/3L2/3. (a) Does JB have increasing, decreasing or constant returns to scale? Show all calculations. 16 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: This is a CobbDouglas production function where the powers add up to 1, which is a constant returns to scale production function. (b) Suppose JB always produces a fixed target output q. Show that if JB uses the optimal amount of capital and labor, then 2/3 of her cost will be due to labor costs. Answer: For this question, I am going to exploit the fact that at the optimal choice, the MRTS is equal to the slope of the isocost line. You can, if you want, do the Lagrangian. If MRTS is tangent to the isocost line: MPL/MPK = PL/PK {2/3 (K 1/3L1/3 )}/{1/3 (K2/3L2/3 )} = PL/PK 2 PK K = PL L PK K = (PL L )/2 The cost is: C = PL L + PK K Using the condition PK K = (PL L)/2 in the cost equation yields: C = (3/2) PL L PL L = (2/3)C That is, JB's labor cost is 2/3 of total cost. (c) Now suppose capital is a fixed input (say k = 1). Does JB have a concave, convex, or linear short run production function? Show all calculations and graph the production function below. Answer: Say k = 1 then: 17 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain q(K, L) = K 1/3L2/3 q( L) = L2/3 This is a concave short run production function: Output Labor (d) Based on the answer in part (c), derive JB's TVC(q) equation. Please plot this equation below and indicate whether it is concave, convex, or linear. Answer: Answer: The TVC is simply: TVC(Q) = PL L = PLQ3/2 which is convex (i.e. MC increases with volume): TVC(Q) 18 Output Labor ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto Toronto.
 Fall '08
 HUSSEIN
 Economics, Microeconomics

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