3. Suppose an individual's utility function over income (M) and leisure (L) is U(M,L)·L, and the individual has non-labor income Y and earns wage w. The total amount of time available is 16 hours.
a. What is the equation for the budget constraint? Graph the budget constraint.
b. Suppose Y = 0 and w = 2. What is the individual's utility-maximizing choice of M and L?
c. Now suppose Y = 10. If the individual works, what is the utility-maximizing choice of M and L? What is the individual's utility if she does not work? Does she choose to participate in the labor force or not?
d. Now suppose there are fixed costs of working, such as expenditures on work clothes and transportation, and these costs are F = 5. Draw the budget constraint. Answer the questions in part c in this case. Do your answers change? Why?
4. Suppose a firm uses labor L and capital K to product output y, with the production function Suppose the firm sells its output in a competitive labor market at price p, and buys labor in a competitive market at price w, and assume the firm maximizes profits.
a. Let the level of capital be fixed at in the short-run. Provide an expression for the demand curve for labor.
b. Using your answer for part a, how does L change with w, p, and ? Explain why each of these changes makes sense.
5. Suppose a firm uses only one input (L) to produce output y, with the production function . Suppose the firm sells its output in a competitive market at price p, and buys labor in a competitive market at price w.
a. Write an expression for the profits of the firm as a function of w, p, and L.
b. What is the marginal cost of hiring an additional unit of labor? Graph the marginal cost of labor curve.
c. What is the marginal revenue from hiring an additional unit of labor? Graph the marginal revenue curve on the same graph as in part b.
d. Assume the firm maximizes profits. How much labor should it hire as a function of the real wage w/p? Find the solution in terms of L, and also display it on the graph.
e. Does the firm hire more or less labor as p increases? Why?
6. Suppose a firm's production function is Q = min(L,K). (This means the level of Q produced is the smaller of L and K.)
a. Graph some isoquants for this firm.
b. Let w = 2, r = 1, and suppose the firm's expenditures are C = 12. What are the firm's demands for L and K? What is the share of labor in the cost of output?
c. Now let w rise to 3. What are the firm's new demands for L and K?
d. Now suppose instead that the production function is Q = min(L,2K). Draw some isoquants for this firm.
e. Again, let w = 2 and r = 1, and suppose the firm's expenditures are C = 12. What are the firm's demands for L and K? What is the share of labor in the cost of output?
f. Now let w rise to 3. What are the firm's new demands for L and K?
g. Why does labor demand fall more in part f than in part c? (Hint: Use one of Marshall's Laws.)