A profit-maximizing firm that is producing q 0 units of output obviously wants to pro- duce these units at the lowest possible cost. Figure 3-8 illustrates the solution to this cost- minimization problem. In particular, the firm chooses the combination of labor and FIGURE 3-7 Isocost Lines
All capital-labor combinations that lie along a single isocost curve are equally costly. Capital-labor combinations that lie on a higher isocost curve are more costly. The slope of an isoquant equals the ratio of input prices (w/r).
Isocost with Cost Outlay C1
Isocost with Cost Outlay C0
Capital
Employment C1/w
C0 /w C0 /r
C1 /r
capital (100 workers and 175 machines) given by point P, where the isocost is tangent to the isoquant. At point P, the firm produces q 0 units of output at the lowest possible cost because it uses a capital-labor combination that lies on the lowest possible isocost. The firm can produce q 0 units of output using other capital-labor combinations, such as point A or B on the isoquant. This choice, however, would be more costly because it places the firm on a higher isocost line (with a cost outlay of C 1 dollars).
At the cost-minimizing solution P, the slope of the isocost equals the slope of the iso- quant, or
MPE MPK = w
r (3-13)
Cost minimization, therefore, requires that the marginal rate of technical substitution equal the ratio of prices. The intuition behind this condition is easily grasped if we rewrite it as
MPE
w = MPK
r (3-14)
The last worker hired produces MP E units of output for the firm at a cost of w dollars. If the marginal product of labor is 20 units and the wage is $10, the ratio MP E / w implies that the last dollar spent on labor yields two units of output. Similarly, the ratio MP K / r gives the output
A
P
B
q0
Capital
Employment 100
C0/r
175 C1/r
FIGURE 3-8 The Firm’s Optimal Combination of Inputs
A firm minimizes the costs of producing q0 units of output by using the capital-labor combination at point P, where the isoquant is tangent to the isocost. All other capital-labor combinations (such as those given by points A and B) lie on a higher isocost curve.
yield of the last dollar spent on capital. Cost-minimization requires that the last dollar spent on labor yield as much output as the last dollar spent on capital. In other words, the last dollar spent on each input gives the same “bang for the buck.”
The hypothesis that firms minimize the cost of producing a particular level of output is often confused with the hypothesis that firms maximize profits. It should be clear that if we constrain the firm to produce q 0 units of output, the firm must produce this level of out- put in a cost-minimizing way in order to maximize profits. Profit-maximizing firms, there- fore, will always use the combination of labor and capital that equates the ratio of marginal products to the ratio of input prices. This condition alone, however, does not describe the behavior of profit-maximizing firms. After all, the equality of ratios in equation (3-13) was derived by assuming that the firm was going to produce q 0 units of output, regardless of any other considerations. A profit-maximizing firm will not choose to produce just any level of output. Rather, a profit-maximizing firm will choose to produce the optimal level of output—that is, the level of output that maximizes profits, where the marginal cost of production equals the price of the output (or q * units in Figure 3-5 ).
Therefore, the condition that the ratio of marginal products equals the ratio of prices does not tell us everything we need to know about the behavior of profit-maximizing firms in the long run. We saw earlier that for a given level of capital— including the optimal level of capital —the firm’s employment is determined by equating the wage with the value of marginal product of labor. By analogy, the profit-maximizing condition that tells the firm how much capital to hire is obtained by equating the price of capital ( r ) and the value of marginal product of capital VMP K . Therefore, long-run profit maximization also requires that labor and capital be hired up to the point where
w = p * MPE and r = p * MPK (3-15) These profit-maximizing conditions imply cost minimization. Note that the ratio of the two marginal productivity conditions in equation (3-15) implies that the ratio of input prices equals the ratio of marginal products. 3
3-4 The Long-Run Demand Curve for Labor
We can now determine what happens to the firm’s long-run demand for labor when the wage changes. We initially consider a firm that produces q 0 units of output. We assume that this output is the profit-maximizing level of output, in the sense that, at that level of production, output price equals marginal cost. A profit-maximizing firm will produce this output at the lowest cost possible, so it uses a mix of labor and capital where the ratio of marginal products equals the ratio of input prices. The wage is initially equal to w 0 . The optimal combination of inputs for this firm is illustrated in Figure 3-9 , where the firm uses 75 units of capital and 25 workers to produce the q 0 units of output. Note that the cost out- lay associated with producing this level of output equals C 0 dollars.
Suppose the market wage falls to w 1 ; how will the firm respond? The absolute value of the slope of the isocost line is equal to the ratio of input prices (or w 1 / r ), so the isocost line
3 To restate the point, profit maximization implies cost minimization, but cost minimization need not imply profit maximization.
will be flattened by the wage cut. Because of the resemblance between the wage change in Figure 3-9 and the wage change in the neoclassical model of labor-leisure choice that we discussed in Chapter 2, there is a strong inclination to duplicate the various steps of our earlier geometric analysis.
We have to be extremely careful when drawing the new isocost line, however, because the obvious way of shifting the isocost line is also the wrong way of shifting it. As illus- trated in Figure 3-9 , we may want to shift the isocost by rotating it around the original intercept C 0 / r. If this rotation of the isocost line were “legal,” the firm would move from point P to point R. The wage reduction increases the firm’s employment from 25 to 40 workers and increases output from q 0 to q 0 units.
Although we are tempted to draw Figure 3-9 , the analysis is simply wrong! The rota- tion of the isocost around the original intercept C 0 / r implies that the firm’s cost outlay is being held constant, at C 0 dollars. There is nothing in the theory of profit maximization to require that the firm incur the same costs before and after the wage change. The long-run constraints of the firm are given by the technology (as summarized by the production func- tion) and by the constant price of the output and other inputs ( p and r ). In general, the firm will not maximize its profits by constraining itself to incur the same costs before and after a wage change.
FIGURE 3-9 The Impact of a Wage Reduction, Holding Constant Initial Cost Outlay at C0
A wage reduction flattens the isocost curve. If the firm were to hold the initial cost outlay constant at C0 dollars, the isocost would rotate around C0 and the firm would move from point P to point R. A profit-maximizing firm, however, will not generally want to hold the cost outlay constant when the wage changes.
R P
Wage Is w0
Wage Is w1
q'0
Capital
Employment 40
25 C0/r
75