1.) A firm produces an output with the production function Q = KL, where Q is th

Question

1.) A firm produces an output with the production function Q = KL, where Q is the number of units of output per hour when the firm uses K machines and hires L workers each hour. The factor price of K is 4 and the factor price of L is 2. The firm is currently using K = 16.

a. Please write out the expression for the short-run total cost curve, the average total cost curve and the marginal cost curve. Draw the average total cost curve and the marginal cost curve in one graph. Explain the relative position of these two curves.

b. In the short-run, the firm uses just enough L to produce Q = 32. How much could the firm save if it were to adjust K and L to produce 32 units in the least costly way possible

2.)True or False? Please explain why.

a. When the marginal product decreases, the average product may or may not be decreasing.

b. When a firm acquires more of a fixed factor of production, its marginal cost curve shifts down and to the right.

c. Suppose the production function is F(L,K)=min{L,K}, then the cost minimizing quantity of labor equals the cost minimizing quantity of capital, independent of wage rate w and capital rent r.

Explanation / Answer

Currently the firm must be using L = Q/K = 32/16 = 2 units of labor. Let the factor prices of capital and labor be, respectively, r and w.

Its total expenditure is C = wL + rK = 2(2) + 4(16) = 68.

If it were to minimize cost, it would hire L and K so that (1) MPK/r = MPL/w, or L/4 = K/2, or L = 2K and (2) Q = LK.

(1) and (2) imply that Q = 2K2, or 32 = 2K2, and thus K = 4 and L = 8.

So Q = 32 can be produced efficiently with a cost of C = wL + rK = 2(8) + 4(4) = 32.

The firm could save 68

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