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152 Example Suppose effort is given by w  X t X J if w > X otherwise, X = (1  bu)Wa, (10.12) (10.13) where 0 < /3 < 1 and fo > 0. x is a measure of labormarket conditions. If fo equals 1, x is the wage paid at other firms multiphed by the fraction of workers who are employed. If fo is less than 1, workers put less weight on unemployment; this could occur if there are unemployment benefits or if workers value leisure. If fo is greater than 1, workers put more weight on unemployment; this might occur because workers who lose their jobs face unusually high chances of continued unemployment, or because of risk aversion. Finally, equation (10.12) states that for w > x, effort increases less than proportionately with w  x. Differentiation of (10.12) shows that for this functional form, the condition that the elasticity of effort with respect to the wage equals 1 (equation [10.11]) is w /w x\p" 1 [{wx)x] X ) x~ Straightforward algebra can be used to simplify (10.14) to (10.14) (1  bu) (10.15) Wa For small values of /3, 1/(1  /3) = 1  /3. Thus (10.15) implies that when p is small, the firm offers a premium of approximately fraction /3 over the index of labormarket opportunities, x. This example is based on Summers (1988). wage is w* and there is unemployment of amount I  M,*. And if M,* exceeds I, the wage is bid up and the labor market clears. This extended version of the model has promise for accounting for both the absence of any trend in unemployment over the long run and the fact that shifts in labor demand appear to have large effects on unemployment in the short run. This is most easily seen by means of an example.
Equilibrium requires that the representative firm wants to pay the prevailing wage, or that w = Wq. Imposing this cbndition in (10.15) yields (]  I3)wa =(lbu)Wa. (10.16) For this condition to be satisfied, the unemployment rate must be given by = "eq (10.17) As equation (10.15) shows, each firm wants to pay more than the prevailing wage if unemployment is less than ueq, and wants to pay less if unemployment is more than Ueq. Thus equilibrium requires that = Ueq. Substituting (10.17) and w = Wa into the effort function, (10.12), implies that equilibrium effort is given by Wq  (1  bUBQJWg (1  bUEQJWa L(l;6) 1;S (10.18) Finally, the equilibrium wage is determined by the condition that the marginal product of effective labor equals its cost (equation [10.10]): F"(el) = w/e. We can rewrite this condition as w = eF(eL). Since total employment is (1  iieq)L in equilibrium, each firm must hire (1  Ueq)I/N worlers. Thus the equilibrium wage is given by iQ = eEQFj. (10.19) Implications This analysis has three important implications. First, (10.17) implies that equilibrium unemployment depends only on the parameters of the effort function; the production function is irrelevant. Thus an upward trend in the production function does not produce a trend in unemployment. Second, relatively modest values of /3the elasticity of effort with respect to the premium firms pay over the index of ]abormarl<et conditionscan lead to nonnegligible unemployment. For example, either /3 = 0.06 and = 1 or /3 = 0.03 and b = 0.5 imply that equilibrium unemployment is 6%.
Cfixed = e{Wa,Wa,u) (10.20) Wg  il  bu)Wa (1  bu)Wa 1  bu bu ll the firm changes its wage, on the other hand, it sets it according to (10.15), and thus chooses w = x/(l  /3). In this case, the firms cost per unit of effective labor is Cadj = W  X x/(l;S) [x/(l  ;S)]  X x/(l;6) I I PHI pvp (10.21) (1  bu)Wa. Third, firms incentive to adjust wages or prices (or both) in response to changes in aggregate unemployment is likely to be small for reasonable cases. Suppose we embed this model of wages and effort in a model of pricesetting firms along the lines of Chapter 6. Consider a situation where the economy is initially in equilibnum, so that = ueq and marginal revenue and marginal cost are equal for the representative firm. Now suppose that the money supply falls and firms do not change their nominal wages or prices; as a result, unemployment rises above Ueq. We know from Chapter 6 that small barriers to wage and price adjustment can cause this to be an equilibrium only if the representative firms incentive to adjust is small. For concreteness, consider the incentive to adjust wages. Equation (10.15), w = (1  bu)Wa/{l  /3), shows that the costminimizing wage is decreasing in the unemployment rate. Thus the firm can reduce its costs, and hence raise its profits, by cutting its wage. The key issue is the size of the gain. Equation (10.12) for effort implies that if the firm leaves its wage equal to the prevaihng wage, Wa, its cost per unit of effective labor, w/e, is
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