Suppose that a firm\'s fixed proportion function is given by q = (min{k,l})^.5 I

Question

Suppose that a firm's fixed proportion function is given by q = (min{k,l})^.5

In the short run this firm must use 16 units of capital but can vary its amount of labor freely. The firm is a price taker both in the output and input markets. Denote the price of the output by p, the price of capital by v, and the price of labor by w.

a. write the short run maximimization problem and derive the 1st order condition.

b. If the wage is w=1 and the price of output is p=4, how much labor will the firm demand in the short run?

c. In the long run the firm can choose any amount of capital. Write down the long run total cost function for this firm.

d. Compute the firms contingent demand functions for capital and labor and find the firms long term supply function.

Explanation / Answer

a) Output is maximized when k=l. Thus k=16 implies l=16 in short run given the min production function. Employing any more labour does not increase the output. And if we employ lesser workforce the capital gets wasted.

b) k=16 implies l=16 in short run

c) TC= (w+r)*min(k,l)

d) q = (min{k,l})^.5

Thus, k=l= q^(1/5)

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