eco204_HW_6 - University of Toronto, Department of...

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain ECO 204 2008‐2009 Ajaz Hussain HW 6 In lecture 7, we derived the optimal labor, capital and the long run average cost functions for imperfect substitutes, perfect substitutes and complements technologies for a company that has a target output q and procures labor and capital at prices PL and PK respectively. For your convenience these are summarized below: Imperfect Substitutes Technology: q = Lα Kβ L = q1/(α + β) [(α/β)(PK/PL)]β/(α + β) K = q1/(α + β) [(β/α)(PL/PK)]α/(α + β) C(q) = q1/(α + β) PLα /(α + β) PKβ/(α + β) [(α/β)β + (β/α)α]1/(α + β) Perfect Substitutes Technology: q = αL + βK Case 1: If PL/PK < MRTS L = q/α K = 0 C(q) = PL q/α Case 2: If PL/PK > MRTS L = 0 1 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain K = q/β C(q) = PK q/β Case 3: If PL/PK = MRTS Any L and K satisfying: q = αL + βK C(q) = PLL + PK K Complements Technology: q = min(αL, βK) L = q/α K = q/β C(q) = q (PL/α + PK/β) Question 1 Ajax Corporation has production function q = L1/3 K2/3 and has target output q. Currently, PL = $5 and PK = $10 a) What is Ajax’s returns to scale: increasing, constant or decreasing? b) From graphical analysis what must be true about the iso‐cost and iso‐quant slopes? c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. d) Now use the formulas above and check if you get the same answer. e) What is Ajax’s long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above? f) What is the elasticity of Ajax’s long run cost with respect to target output? Interpret this result given your answer in part (a). g) What is the long run average cost? Interpret this result given your answer in part (a). h) What is the long run marginal cost? Interpret this result given your answer in part (a). 2 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain Question 2 Tiesto Corporation has production function q = L1/5 K3/5 and has target output q. Currently, PL = $5 and PK = $10 a) What is Tiesto’s returns to scale: increasing, constant or decreasing? b) From graphical analysis what must be true about the iso‐cost and iso‐quant slopes? c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. d) Now use the formulas above and check if you get the same answer. e) What is Tiesto’s long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above. f) What is the elasticity of Tiesto’s long run cost with respect to target output? Interpret this result given your answer in part (a). g) What is the long run average cost? Interpret this result given your answer in part (a). h) What is the long run marginal cost? Interpret this result given your answer in part (a). Question 3 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) What is Bob Sinclair’s returns to scale: increasing, constant or decreasing? b) From graphical analysis what must be true about the iso‐cost and iso‐quant slopes? c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. d) Now use the formulas above and check if you get the same answer. 3 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain e) What is Bob Sinclair’s long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above. f) What is the elasticity of Bob Sinclair long run cost with respect to target output? Interpret this result given your answer in part (a). g) What is the long run average cost? Interpret this result given your answer in part (a). h) What is the long run marginal cost? Interpret this result given your answer in part (a). Question 4 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) What is Guetta’s returns to scale: increasing, constant or decreasing? b) From graphical analysis what must be true about Guetta’s optimal choice of labor and capital? c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. d) Now use the formulas above and check if you get the same answer. e) What is Guetta’s long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above? f) What is the elasticity of Guetta’s long run cost with respect to target output? Interpret this result given your answer in part (a). g) What is the long run average cost? Interpret this result given your answer in part (a). h) What is the long run marginal cost? Interpret this result given your answer in part (a). 4 ...
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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.

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