Question

A profit-maximizing competitive firm has the production function ? = ?^.25?^.25 , where output ? is

produced using capital (?) and labor (?). The price of labor is $2 and the price of capital is $6. The price of the firm’s output is $100.

a. Write the firm’s profit function and calculate the first-order conditions. Provide an economic interpretation for each of the first-order conditions.

b. Find the profit-maximizing quantities ?* and ?*.

c. Find the firm’s total costs.

d. Find the firm’s maximized profit. Is this profit a long-run outcome for the firm? Why or why not?

Answer 1

a) ? = ?^.25?^.25

Revenue, R = PQ = 100*?^.25?^.25 .... [ As price of the firm’s output, P = $100]

Total Cost, TC = wL +rK = 2L + 6K ... [ As price of labor, w = $2 and the price of capital, r = $6]

Profit, Z = Revenue - Cost

or, Z = 100*?^.25?^.25 - 2L - 6K

F.O.Cs for profit maximization are-

i)

or, 25?^.25?^(-0.75) - 2 = 0

or, 25?^.25?^(-0.75) = 2.... eqn (1)

ii)

or, 25?^(-0.75)?^.25 - 6 = 0

or, 25?^(-0.75)?^.25 = 6... eqn (2)

These F.O.C s indicate that the optimal condition at which if we increase the input, profit will not increase anymore. The level of L or K at which derivative of profit is zero, indicates the optimal level of input. At this point, marginal revenue from each input is same as the cost of that input. So, increase in revenue due to one unit increase in input will be same as the increase in cost due to that extra unit of input. hence, the increase in profit will be zero.

Eqn (1) shows the level of labour at which the change in profit due to an unit increase in labour is zero.

Eqn (2) shows the level of capital at which the change in profit due to an unit increase in capital is zero.

This is a case of long run profit.

In short run, we caonsider capital as a fixed input but in the long run, both labour and capital are variable. Here, we calculate both optimum labour and capital to maximize profit. Hence, this profit is the long run outcome.