Question

# 5. A competitive firm has a production function described as follows. “Weekly output is the square...

5. A competitive firm has a production function described as follows. “Weekly output is the square root of the minimum of the number of units of capital and the number of units of labor employed per week.” Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely.

a) Write down a formula that describes the marginal product of labor in the short run as a function of the amount of labor used. (Be careful at the boundaries.)

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) What if w=\$1 and p=\$10?

Part A

The long run formula for this firm is,

So, in the short run, labor greater than 16 unit have no effect on total output of the firm. Hence, the short run production function is,

The marginal product of labor would be,

Part B

Here wage is \$1 and price of output is \$10.

The optimal demand for labor is,

. Here,

or,