A firm that only employs labor L has the following productio

A firm that only employs labor (L) has the following production function: f(L) = 20L - L2

Let the price of output be normalized to one and the price of labor (relative to output price) is w.

a. Write out the profit function for this firm as a function of labor, L.

b. What is the necessary first-order condition for the firm to maximize profit when L > 0?

c. Compute the profit maximizing amount of labor as a function of the wage. What is the effect of an increase in wage on the firms optimal employment level? Use calculus to solve this.

Solution

a. Profit(P) = f(L) - wL = 20L - L2- wL

b. The first-order condition is dP/dL=0 ==> 20 - 2L^m - w = 0

c. Solving for L==>L^m(w) = 10 - w/2

An increase in wage will decrease the level of employment by half the rate of change of wage.