HW9_Sol - Problem Set 9 Solutions ECON 401, Winter 2008...

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Unformatted text preview: Problem Set 9 Solutions ECON 401, Winter 2008 Perloff 6.25 In the short run, a firm cannot vary its capital, K = 2 , but it can vary its labor, L . It produces output q . Explain why the firm will or will not experience diminishing marginal returns to labor in the short run if its production function is: a) q = 10 L + K 2 q L 2 = 0 so the production function exhibits constant marginal returns to labor in the short run. b) q = L 1 2 K 1 2 q L = . 5 L- . 5 K . 5 and 2 q L 2 =- . 25 L- 1 . 5 K . 5 < 0 so the production function exhibits dimin- ishing marginal returns to labor in the short run. Perloff 6.26 Under what conditions do the following production functions exhibit decreas- ing, constant, or increasing returns to scale? a) q = L + K Always constant returns to scale: If you double L and K , then output will double. b) q = AL a K b If we double L and K , then output changes from f ( L,K ) = AL a K b to f (2 L, 2 K ) = A (2 L ) a (2 K ) b = A 2 a + b L a K b . Thus: If a + b < 1, the f (2 L, 2 K ) < 2 f ( L,K ), so f ( L,K ) exhibits decreasing returns to scale. If a + b = 1, the f (2 L, 2 K ) = 2 f ( L,K ), so f ( L,K ) exhibits constant returns to scale. If a + b > 1, the f (2 L, 2 K ) > 2 f ( L,K ), so f ( L,K ) exhibits increasing returns to scale. c) q = L + L a K b + K The answer is the same as part b). The linear part L + K has constant returns to scale, so the returns to scale will depend on the middle term. Perloff 7.21 Give the formulas and plot AFC, MC, AVC, and AC if the cost function is a) C = 10 + 10 q ; AFC = 10 q ; MC = 10; AV C = 10; AC = 10 q + 10 1 2 b) C = 10 + q 2 ; AFC = 10 q ; MC = 2 q ; AV C = q AC = 10 q + q c) C = 10 + 10 q- 4 q 2 + q 3 ; AFC = 10 q ; MC = 10- 8 q + 3 q 2 ; AV C = 10- 4 q + q 2 ; AC = 10 q + 10- 4 q + q 2 . Perloff 7.24 Gail works in a flower shop, where she produces 10 floral arrangements per hour. She is paid $ 10 an hour for the first eight hours she works and $ 15 an hour for each additional hour she works. What is the firms cost function? What are its AC, AVC, and MC functions? Draw the AC, AVC, and MC curves. 3 Let L be the number of labor hours and q the number of floral arrangements. The produc- tion function is: q = 10 * L so to produce q , we need L = 1 10 q units of labor. Thus, for L 8, and thus q 80, the cost is wL = 10 * 1 10 q = q . For q > 80, costs equal the $80 required to produce 80 units plus 15 * 1 10 = $1 . 5 required to produce every unit above 80. Summarizing: C ( q ) = q if q 80; 80 + 3 2 ( q- 80) = 3 2 q- 40 if q > 80 . Divide by q to get AC , which are the same as AV C because FC = 0. AC = AV C = 1 if q 80; 3 2- 40 q if q > 80 . Take the derivative of C ( q ) to get marginal costs: AC = AV C = 1 if q 80; 3 2 if q > 80 ....
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HW9_Sol - Problem Set 9 Solutions ECON 401, Winter 2008...

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