3. (14 points) Let the production function be Q(K,L)=2K L. The cost of one unit of labor is w, the cost of one unit of capital is r, and the quanitity to produce is y. (a) (4 points) Set up the firm's cost-minimization problem, and clearly write down the con- Page 6 straint the firm faces. The cost minimization problem of the firm is min(KL) WL+rK. The constraint: y=2K!L!. anscribed Text (b) (5 points) What are two conditions that the optimal bundle L and K need to satisfy? Write down their expressions. Tangency: -=-; production: y = 2(K)! (L)!. (c) (5 points) Solve for the optimal bundle of L. and K that minimizes the firm's cost at y=144 by finding L and K", assuming that w $16 and r $81. By solving the system of two conditions in Part (b), it should be not that hard to get L = 162 and K* = 32.
3. (14 points) Let the production function be Q(K,L)=2K L. The cost of one unit of labor is w, the cost of one unit of capital is r, and the quanitity to produce is y. (a) (4 points) Set up the firm's cost-minimization problem, and clearly write down the con- Page 6 straint the firm faces. The cost minimization problem of the firm is min(KL) WL+rK. The constraint: y=2K!L!. anscribed Text (b) (5 points) What are two conditions that the optimal bundle L and K need to satisfy? Write down their expressions. Tangency: -=-; production: y = 2(K)! (L)!. (c) (5 points) Solve for the optimal bundle of L. and K that minimizes the firm's cost at y=144 by finding L and K", assuming that w $16 and r $81. By solving the system of two conditions in Part (b), it should be not that hard to get L = 162 and K* = 32.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Transcribed Image Text:3. (14 points) Let the production function be Q(K,L)=2K L. The cost of one unit of labor
is w, the cost of one unit of capital is r, and the quanitity to produce is y.
(a) (4 points) Set up the firm's cost-minimization problem, and clearly write down the con-
Page 6
straint the firm faces.
The cost minimization problem of the firm is min(KL) WL+rK. The constraint:
y=2K!L!.
anscribed Text
(b) (5 points) What are two conditions that the optimal bundle L and K need to satisfy?
Write down their expressions.
Tangency: -=-; production: y = 2(K)! (L)!.
(c) (5 points) Solve for the optimal bundle of L. and K that minimizes the firm's cost at
y=144 by finding L and K", assuming that w $16 and r $81.
By solving the system of two conditions in Part (b), it should be not that hard to get
L = 162 and K* = 32.
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