1. Consider a firm in the short run, when capital is fixed and the only variable input is labor. For simplicity, we will simply ignore capital. In this situation, suppose that the firm’s production function is given by Q = f(L) = ?L – (1/2)L2 , where Q represents the quantity of output produced, L represents the amount of labor employed, and the parameter ? is a positive constant. a. Derive this firm’s marginal product of labor function? Under what conditions will the marginal product of labor be positive? Under what conditions will the marginal product of labor be diminishing?

b. This firm is perfectly competitive (i.e., a price taker) in both input and output markets. It must pay the wage w for each unit of labor it employs, and it receives the price p for each unit of output it sells. Write down the firm’s profits, and solve for the firm’s profit-maximizing choice of labor (i.e., the firm’s labor demand function). Show both the first-order condition and the second-order condition associated with the firm’s profit maximization problem. Is the second-order condition satisfied?

c. How will the firm’s profit-maximizing choice of labor respond to an increase in ? (holding w and p fixed)? How will it respond to an increase in w (holding ? and p fixed)? How will it respond to an increase in p (holding ? and w fixed)? How would it respond if, at the same time, both w and p were to double (holding ? fixed)? Explain the economic intuition behind each of these results.

d. Draw a graph of the firm’s labor demand curve (remember that we draw labor demand curves with w on the vertical axis and L on the horizontal axis). Label the intercepts and slope of this curve as precisely as you can. How does the labor demand curve change when ? increases? How does the labor demand curve change when p increases? Why do these changes make sense?

e. So far, we have been considering labor demand by a single firm. Suppose that the relevant market is made up of n firms identical to the firm we have described, where n is a positive integer greater than one. In this situation, derive the market labor demand function (i.e., the aggregate amount of labor employed by all n firms combined). Graph the market labor demand curve, labeling the intercepts and slope. How does the market curve compare with the curve for an individual firm? Why does this make sense?

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a To derive the marginal product of labor function we can take the derivative of the production function with respect to labor MPL QL fL L The marginal product of labor is positive when L and it is di... View the full answer