# Labor Economics & Market Marginal Costs & Profit Maximization

## Labor Economics & Market Marginal Costs & Profit Maximization

1. (25 points total) Suppose a firm uses only one input (*L*) to produce output *y*, with the production function *y = L** ^{1/2}*. Suppose the firm sells its output in a competitive market at price

*p*, and buys labor in a competitive market at price

*w*.

a. (5) What is the marginal cost of a unit of labor for the firm? What is the marginal revenue product of a unit of labor for the firm? Graph the marginal cost and marginal revenue product against *L*.

b. (5) If the firm maximizes profits, what is its labor demand function? Show the profit-maximizing quantity of labor employed on the graph. If *w* = 2 and *p *= 8, what is the quantity of labor demanded?

c. (5) What happens to your graph in parts a and b if the wage increases? Use the graph to explain how and why the profit-maximizing labor demand choice changes.

d. (10) Now assume that the firm is a monopolist, and faces the product demand curve *a – b∙y*. (Everything else remains the same). If the monopolist maximizes profits, what is its labor demand function? After you derive the labor demand function, discuss whether the effects of the parameters or variables that determine labor demand have the expected signs. (Hint: You should be able to show that if *a = p* and *b = *0, you get the labor demand function that you derived in part b.)

2. (15 points total) Suppose a profit-maximizing firm in a competitive product market and a competitive labor market has production function *Y* = *L*^{1/3}*K*^{1/3}. Input prices are *w* and *r* (for *L* and *K*, respectively), and output price is *p*.

a. (5) Draw isoquants for this firm for *Y *= 1, *Y *= 4, and *Y *= 9, labeling at least one point

(*L*,*K*) on each isoquant.

b. (5) Solve for the conditional labor demand curve for the firm (i.e., when it cost minimizes)? What variables does this depend on, and why?

c. (5) Solve for the unconditional labor demand curve for this firm. What variables does this depend on, and why?