Ans_HW7 - Answer to Practice Homework 7 8.9 a)...

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Answer to Practice Homework 7 8.9 a) Cost-minimizing quantities of inputs are equal to L = Q ( r / w ) and K = Q / ( r / w ). Hence, in the long-run the total cost of producing Q units of output is equal to TC ( Q ) = 10 + 2 ( Qrw ). For w = 1 and r = 1 we have TC ( Q ) = 2 Q . b) When capital is fixed at a quantity of 5 units (i.e., K * = 5) we have Q = K * L = 5 L . Hence, in the short-run the total cost of producing Q units of output is equal to S TC ( Q ) = 5 + Q /5. c) We have L = Q ( r / w ) and K = Q / ( r / w ). Hence, TC ( Q ) = 2 ( Qrw ) and S TC ( Q ) = 5 r + wQ /5. When w = 1 and r = 4 we have TC ( Q ) = 4 Q and S TC ( Q ) = 20 + Q /5. TC Q 25 5 STC ( Q ) TC ( Q )
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d) When w = 4 and r = 1 we have TC ( Q ) = 4 Q and S TC ( Q ) = 4 Q /5. TC Q 25 5 STC ( Q ) TC ( Q ) 100 20 STC ( Q ), w =1, r = 4 TC ( Q ), w =1, r = 4 TC Q 25 10 STC ( Q ) TC ( Q ) 25/4 STC ( Q ), w = 4, r = 1 TC ( Q ), w = 4, r = 1
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8.11 a) With a linear production function, the firm operates at a corner point depending on whether w < r or w > r . If w < r , the firm uses only labor and thus sets L = Q . In this case, the total cost (including the fixed cost) is wQ . If w > r , the firm uses only capital and thus sets K = Q . in this case, the total cost is rQ . When w = r = 1, the firm is indifferent among combination of L and K that make L + K = 10. Thus, we have TC ( Q ) = Q . b) When capital is fixed at 5 units, the firm’s output would be given by Q = 5 + L . If the firm wants to produce Q < 5 units of output, it must produce 5 units and throw away 5 – Q of them. The total cost of producing fewer than 5 units is constant and equal to $5, the cost of the fixed capital. For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, to produce Q units of output, the firm uses Q – 5 units of labor, for a cost of Q 5, and 5 units of capital, for a cost of 5. Thus, STC ( Q ) = Q – 5 + 5 = Q
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c) In the long run, since w < r , the firm produces its output entirely with labor. Thus,
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Ans_HW7 - Answer to Practice Homework 7 8.9 a)...

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