eco204_summer_2009_HW_4

# eco204_summer_2009_HW_4 - University of Toronto Department...

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain HW 4 Please help improve the course by sending me an email about typos or suggestions for improvements Question 1 Consider the (general forms) of the following typical production functions without technological progress A: CobbDouglas production function: Q = L K Complements production function: Q = min(L, K) Perfect Substitutes production function: Q = L + K CES production function (assume 0 ) Q = [ L( 1)/ + K( 1)/ ] /( 1) Do note that the CES production function is also sometimes written as: Q = [ L + K ] 1/ (a) Derive the MRTS for each production function and interpret it for general values of and , and = = . (b) Derive the MRTS for each production function and interpret it for general values of and , assuming there is technological progress A of the type where Q = A f(K, L). How does technological progress affect the isoquants? Question 2 Suppose a company has the production function Q = L1/2 K1/2. Currently PL = PK = \$10. (a) Interpret PL = PK. (b) Solve the optimal labor and capital using the Lagrangian method. 1 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain (c) Use the envelope theorem to calculate the impact of higher target output on total cost, subject to the constraint that the firm produce the target output q. (d) Use the (long run) cost function to calculate the impact of higher target output on total cost, subject to the constraint that the firm produce the target output q. (e) What is the impact ceteris paribus of a 1% increase in q, PL and PK on the optimal demand for labor? Hint: you may need to use the expressions for the CobbDouglas production function: Q = L K L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) C(q) = q1/( + ) PL /( + ) PK/( + ) [(/)/( + ) + (/)/( + )] (f) What is the impact ceteris paribus of a 1% increase in q, PL and PK on the total cost? Hint: you may need to use the expressions for the CobbDouglas production function: Q = L K L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) C(q) = q1/( + ) PL /( + ) PK/( + ) [(/)/( + ) + (/)/( + )] (g) What kind of returns to scale does the production function Q = L1/2 K1/2 possess? Does the cost function reflect this? (h) Now suppose the company is the short run with capital level fixed at k (note small "k" not large "K"). Solve for the optimal amount of labor. (i) What is the impact ceteris paribus of a 1% increase in q, PL and PK on the optimal demand for labor in the short run? (j) Compare the short vs. long run effect of a ceteris paribus of a 1% increase in q, PL and PK on the optimal demand for labor. What do you notice? 2 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain (k) Derive the total short run cost and average cost functions. Contrast the result with the long run cost and average cost function. (l) Now suppose the company is the short run with capital level fixed at k (note small "k" not large "K"). Solve for the optimal amount of labor assuming there is technological progress impacting both labor and capital. (m) Suppose there is a 1% increase in technological progress that impacts both K and L, i.e. Q = A f(K, L). What is the % impact on demand for labor in the short vs. the long run? Question 3 Ajax Corporation has production function q = L K1/2 and has target output q. Currently, PL = \$1 and PK = \$1. Long Run (a) Under what conditions will Ajax have constant returns to scale? Interpret constant returns to scale. (b) For a given value of , what is Ajax's long run demand for labor and capital? (c) For a given value of , what is Ajax's long run cost function? (d) Will Ajax's long run AC decline, remain constant, or increase with q? Short run: Suppose k = 100 (e) Under what conditions will Ajax have constant returns? Interpret constant returns. It is important not to confuse RTS with returns. These are two entirely different concepts. More crucially, even if there are constant returns to scale, that does not mean there are constant returns. (f) For a given value of , what is Ajax's short run demand for labor? (g) For a given value of , what is Ajax's short run cost function? (h) Will Ajax's short run AC(q) decline, remain constant, or increase with q? 3 ...
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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.

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