eco204_summer_2009_practice_problem_7

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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 7 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 Ajax Corporation has production function q = L1/3 K2/3 and has target output q. Currently, PL = $5 and PK = $10. Please note that PK can be either the price of leasing capital or using capital (aka "user cost of capital"). (a) Calculate the MRTS and the slope of the isocost curve. (b) Solve for the optimal L and K. You may want to check your answers with the "formulas" derived in lectures for q = L K the CMP yielded: L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) (c) What is the elasticity of labor demand with respect to q, PL and PK? Interpret your answers. (d) What is the change in labor demand with respect to q, PL and PK? Interpret your answers. (e) What is Ajax's long run cost function C(q)? (f) What is the elasticity of Ajax's long run cost with respect to target output? (g) What is the long run average cost? 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (h) What is the long run marginal cost? Question 2 Bob Sinclair Corporation has production function q = (1/5)L + (3/5)K and has target output q. Currently, PL = $5 and PK = $10. (a) Solve for the optimal demand for labor and capital. (b) What is Bob Sinclair's long run cost function C(q)? (c) What is the elasticity of Bob Sinclair long run cost with respect to target output (d) What is the long run average cost? (e) What is the long run marginal cost? Question 3 Guetta Corporation has production function q = min((1/3)L , (2/3)K) and has target output q. Currently, PL = $5 and PK = $10 (a) Solve for the optimal labor and capital demands: (b) What is Guetta's long run cost function C(q)? (c) What is the elasticity of Guetta's long run cost with respect to target output? (d) What is the long run average cost? (e) What is the long run marginal cost? About Questions 4 6 Suppose we analyze a company with 2 inputs capital (K) and labor (L). The production function is Q = f(K, L). We've discussed these production technologies: CobbDouglas Technology: Q = L K Perfect Substitutes Technology: Q = L + K Complements Inputs Technology: Q = min(L, K) 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now you'll look at the CES production function: Q = [ L( 1)/ + K( 1)/ ] /( 1) It turns out that the isoquants of the CES production function approximate the isoquants of: CobbDouglas Technology: when lim 1 of Q = [ L( 1)/ + K( 1)/ ] /( 1) Perfect Substitutes Inputs Technology: when lim of Q = [ L( 1)/ + K( 1)/ ] /( 1) Complements Inputs Technology: Q = min(L, K): when lim 0 of Q = [ L( 1)/ + K( 1)/ ] /( 1) Question 4 Assume 0 and derive the MRTS = (dK/dL) = (dQ/dL)/(dQ/dK) of the CES production function. Question 5 Consider a company using K and L as imperfect substitute: Q = L K. (a) Derive the MPK = dQ/dK and interpret it. (b) Derive the MPL = dQ/dL and interpret it. (c) Derive the MRTS = dK/dL = (dQ/dL)/(dQ/dK) = MPL/MPK. Interpret this. (d) Substitute = 1 in the MRTS of the CES production function: do you get MRTS of Q = L K? (e) In consumer theory, we saw the utility function U = Q11/3 Q22/3 represents the same preferences as (say) U = Q12/3 Q24/3. By the same token, does the production function Q = L1/3 K2/3 represent the same technology as Q = L2/3 K4/3? Question 6 Consider a company using K and L as perfect substitute: Q = L + K. 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (a) Derive the MPK = dQ/dK and interpret it. (b) Derive the MPL = dQ/dL and interpret it. (c) Derive the MRTS = dK/dL = (dQ/dL)/(dQ/dK) = MPL/MPK. Interpret this. (d) Substitute = in the MRTS of the CES production function: do you get MRTS of Q = L + K? Question 7 Consider a company using K and L as complements: Q = min(L, K). Solve for the optimal demand for labor and capital. Question 8 (2008 Test 2 Question) Edison Chang has the production function Q = L K + L + K. Suppose Edison Chang is in the long run. Calculate Edison Chang's MRTS. Question 9 (2008 Test 2 Question) ETorre Enterprises used labor (L) and capital (K) to produce its target output q = 80 according to the production function q = L + K where = 2 and = 8. ETorre procures labor and leases capital in competitive markets. Currently, PL = $100 and PK = $10. (a) If ETorre is in the long run, how much labor and capital does it use? Show your calculations below. (b) Plot your answer to part (a) in on a graph with labor on xaxis and wages on yaxis. Now suppose PL decreases: graph the resulting demand curve for labor in the figure. Question 10 (Summer 2008 Final Exam Question) Etorre manufactures Italian gelato using machines and workers as complements in a 2:1 ratio. It leases machines at a price of $20/machine and hires workers at a wage of $10/worker. If E torre sells gelato into a competitive market and currently has 0 profits, what is the price of gelato? Show all steps and calculations clearly. Question 11 Ever wonder how a particular brand of milk has the same "taste"? It's because the dairy and animal husbandry industry uses scientific methods to design animal specific diet. There are methods which change diets according to the age, health, and condition etc of animals. In this question you will use actual figures to solve a simple CMP. 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain A 200 pound steer can be sustained on a diet with the following combinations of grass and grain (pounds/day): Pounds of Grass Pounds of Grain 50 80 56 70 60 65 68 60 80 54 88 52 Currently, grass is $0.10/lb and grain is $0.07/lb. Suppose a farmer currently uses a feed mix of 68 lbs grass and 60 lbs grain is this optimal? 5 ...
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