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151 Implications This model shows how efficiency wages can give rise to unemployment. In addition, the model imphes that the real wage is unresponsive to demand shifts. Suppose the demand for labor increases. Since the efficiency wage, w*, is determined entirely by the properties of the effort function, e(*), there is no reason for firms to adjust their wages. Thus the model provides a candidate explanation of why shifts in labor demand lead to large movements in employment and small changes in the real wage. In addition, the fact that the real wage and effort do not change implies that firms labor costs do not change. As a result, in a model with pricesetting firms, the incentive to adjust prices is small. Equation (10.8) states that at the optimum, the elasticity of effort with respect to the wage is 1. To understand this condition, note that output is a function of the quantity of effective labor, el. The ftrm therefore wants to hire effective labor as cheaply as possible. When the firm hires a worker, it obtains e(w) units of effective labor at a cost of w; thus the cost per unit of effective labor is w/e(w). When the elasticity of e with respect to w is 1, a marginal change in w has no effect on this ratio; thus this is the firstorder condition for the problem of choosing w to minimize the cost of effective labor. The wage satisfying (10.8) is known as the efficiency wage. Figure 10.1 depicts the choice of w graphically in (w, e) space. The rays coming out from the origin are lines where the ratio of e to w is constant; the ratio is larger on the higher rays. Thus the firm wants to choose w to attain as high a ray as possible. This occurs where the e(w) function is just tangent to one of the raysthat is, where the elasticity of e with respect to w is 1. Pane] (a) shows a case where effort is sufficiently responsive to the wage that over some range the firm prefers a higher wage. Panel (b) shows a case where the firm always prefers a lower wage. Finally, equation (10.7) states that the firm hires workers until the marginal product of effective labor equals its cost. This is analogous to the condition in a standard labordemand problem that the firm hires labor up to the point where the marginal product equals the wage. Equations (10.7) and (10.8) describe the behavior of a single firm. Describing the economywide equilibrium is straightforward. Let w* and L* denote the values of w and I that satisfy (10.7) and (10.8). Since firms are identical, each firm chooses these same values of w and I. Total labor demand is therefore M,*. If labor supply, I, exceeds this amount, firms are unconstrained in their choice of w. In this case the wage is w*, employment is NL*, and there is unemployment of amount I  M,*. If NL* exceeds I, on the other hand, firms are constrained. In this case, the wage is bid up to the point where demand and supply are in balance, and there is no unemployment.
FIGURE 10.1 The determination of the efficiency wage Unfortunately, these results are less promising than they may appear. The difhculty is that they apply not just to the short run but to the long run: the model imphes that as economic growth shifts out the demand for labor, the real wage remains unchanged and unemployment trends downward. Eventually, unemployment reaches zero, at which pomt further mcreases m demand lead to increases in the real wage. In practice, however, we observe
no clear trend in unemployment over extended periods. In other words, the basic fact about the labor market that we need to understand is not just that shifts in labor demand appear to have little impact on the real wage and fall almost entirely on employment in the short run; it is also that they faU almost entirely on the real wage in the long run. Our model does not explain this pattern. 10.3 A More General Version Introduction With many of the potential sources of efficiency wages, the wage is unlikely to be the only determinant of effort. Suppose, for example, that the wage affects effort because firms cannot monitor workers perfectly and workers are concerned about the possibility of losing their jobs if the firm catches them shirking. In such a situation, the cost to a worker of being fired depends not just on the wage the job pays, but also on how easy it is to obtain other jobs and on the wages those jobs pay. Thus workers are likely to exert more effort at a given wage when unemployment is higher, and to exert less effort when the wage paid by other firms is higher. Similar arguments apply to situations where the wage affects effort because of unobserved ability or feelings of gratitude or anger. Thus a natural generalization of the effort function, (10.3), is e = e(w,wa,u), ei()>0, e2(*) < 0, () > 0, (10.9) where Wa is the wage paid by other firms and is the unemployment rate, and where subscripts denote partial derivatives. Each firm is small relative to the economy, and therefore takes Wa and as given. The representative firms problem is the same as before, except that Wa and now affect the effort function. The firstorder conditions can therefore be rearranged to obtain F{e(w,Wa,u)L)= (10.10) e{w,Wa,u) /" = 1. (10.11) These conditions are analogous to (10.7) and (10.8) in the simpler version of the model. Assume that the e(*) function is sufficiently well behaved that there is a unique optimal w for a given Wa and . Given this assumption, equilibrium requires w = Wq ; if not, each firm wants to pay a wage different from the prevailing wage. Let w* and I* denote the values of w and L satisfying (lO.lO)(lO.l 1) with w = Wq . As before, if NL* is less than L, the equilibrium
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