## Demand for a Factor Input When Only One Input Is Variable

Demand curves for factors of production are downward sloping, just like demand curves for the final goods that result from the production process. Unlike consumers' demands for goods and services, however, factor demands are derived demands-they depend on, and are derived from, the firm's level of output and the costs of inputs. For example, the demand of Microsoft Corporation for computer programmers is a derived demand that depends not only on the current salaries of programmers, but also on how much software Microsoft expects to sell.

To analyze factor demands, we will use the material from Chapter 7 that shows how a firm chooses its production inputs. We will assume that the firm produces its output using two inputs, capital K and labor L, that can be purchased at the prices r (the rental cost of capital) and w (the wage rate), respectively.1 We will also assume that the firm has its plant and equipment in place (as in a short-run analysis), and must decide how much labor to hire.

Suppose that the firm has hired a certain number of workers and wants to know whether it is profitable to hire one additional worker. This will be profitable if the additional revenue from the output of the worker's labor is greater than the cost of his or her labor. The additional revenue from an incremental unit of labor, the marginal revenue product of labor, is denoted MRP/ . We know that the firm should hire more labor if the MRPl is at least as large as the wage rate w.

How do we measure the MRPl? It's the additional output obtained from the additional unit of labor, multiplied by the additional revenue from an extra unit of output. The additional output is given by the marginal product of labor MPl and the additional revenue by the marginal revenue MR. Thus 2

This important result holds for any competitive factor market, whether the output market is competitive or not. However, to examine the characteristics of the MRPl, let's begin with the case of a perfectly competitive output (and input) market. In a competitive output market, a firm will sell all its output at the market price P. The marginal revenue from the sale of an additional unit of output is then equal to P. In this case the marginal revenue product of labor is equal to the marginal product of labor times the price of the product:

The higher of the two curves in Figure 14.1 represents the MRPl curve for a firm in a competitive output market. Note that the marginal product of la-

1 We implicitly assume that all inputs to production are identical in quality. Differences in workers' skills and abilities are discussed in Chapter 17.

The marginal revenue product is AR/AL, where L is the number of units of labor input and R is revenue. Note that MPL = AQ/AL, and MR = AR/AQ, where Q is output. Therefore, MRP, = AR/AL = (AR/AQ)(AQ/AL) = (MR)(MPt).

MRP,. = MPl • P

Hours of Work

FIGURE 14.1 Marginal Revenue Product In a Competitive factor market in which the producer of the product is a price taker, the buyer's demand for an input is given by the marginal revenue product curve. The.MRP curve falls because the marginal product of labor falls as hours of work increases. When the producer of the product has monopoly power, the demand for the input is also given by the MRP curve, but the MRP curve falls because both the marginal product of labor and marginal revenue fall.

bor falls as the number of hours of labor increases because there are diminishing returns to labor. The marginal revenue product curve thus slopes downward, even though the price of the product is constant.

The lower curve in Figure 14.1 is the MRPl curve when the firm has monopoly power in the output market. When firms have monopoly power, they must lower the price of all units, of the product to sell more of it. As a result, marginal revenue is always less than price (MR < P), and marginal revenue falls as output increases. Thus, the marginal revenue product curve slopes downward in this case because the marginal revenue curve and the marginal product curve slope downward.

The concept of marginal revenue product can be applied to firms' hiring of workers; the marginal revenue product tells us how much the firm will pay to hire an additional unit of labor. As long as the MRPl is greater than the wage rate, the firm should hire an additional unit of labor. If the marginal revenue product is less than the wage rate, the firm should lay off workers. Only when the marginal revenue product is equal to the wage rate will the firm have hired the profit-maximizing amount of labor. So the profit-maximizing condition is

Price of Labor

Quantity of Labor

FIGURE 142 Hiring by a Firm in the Labor Market (with Capital Fixed). In a competitive labor market, a firm faces a perfectly elastic supply of labor Sl and can hire as many workers as it wants at a wage rate w*. The firm's demand for labor Dl is given by its marginal revenue product of labor MRPl. The profit-maximizing firm will hire L* units of labor at the point where the marginal revenue product of labor is equal to the wage rate.

Figure 14.2 illustrates this condition. The demand for labor curve Dl is the MRPL.Note that the quantity of labor demanded increases as the wage rate falls. Since the labor market is perfectly competitive, the firm can hire as many workers as it wants at the market wage w* so that the supply of labor curve facing the firm, Sl, is a horizontal line. The profit-maximizing amount of labor that the firm hires, L*, is at the intersection of the supply and demand curves.

Figure 14.3 shows how the quantity of labor demanded changes in response to a drop in the market wage rate from wi to W2. The wage rate might decrease if more people entering the labor force are looking for jobs for the first time (as happened, for example, when all the baby boomers came of age). The quantity of labor demanded by the firm is initially Li, at the intersection of MRPl and Si. However, when the supply of labor curve shifts from Si to S2 the wage falls from w 1 to W2 and the quantity of labor demanded increases from Li to Li.

Factor markets are similar to output markets in many ways. For example, the factor market profit-maximizing condition that the marginal revenue product of labor be equal to the wage rate is analogous to the output market condition that marginal revenue be equal to marginal cost. To see why this is true, recall that MRPl = (MPl)(MR) and divide both sides of equation (14.3) by the marginal product of labor. Then,

Since MTl measures the additional output per unit of input, the right-hand side of equation (14.4) measures the cost of an additional unit of output (the wage rate multiplied by the labor needed to produce one unit of output), i.e.,

Price of Labor

Quantity of Labor

FIGURE 143 A Shift in the Supply of Labor. When the supply of labor facing the firm is Si, the farm hires Li units of labor at wage wi. But when the market wage rate decreases and the supply of labor shifts to S2, the firm maximizes its profit by moving along the demand for labor curve until the new wage rate wi is equal to the marginal revenue product of labor, and L2 units of labor are hired.

the marginal cost of production. Equation (14.4) shows that both the hiring and output choices of the firm follow the same rule-inputs or outputs are chosen so that the marginal revenue (from the sale of output) is equal to marginal cost (from the purchase of inputs). This result holds in both competitive and noncompetitive markets.