
Romer D. Advanced Macroeconomics
.pdf9.2 A Generic ERiciency-Wage Model 413
of the applicant pool, and thus raises the average ability of the workers the firm hires.'
Finally, a hlgh wage can build loyalty among workers and hence induce high effort; conversely, a low wage can cause anger and desire for revenge, and thereby lead to shirking or sabotage. Akerlof and Yellcn (1990)present extensive evidence that workers' effort is affected by such forces as anger, jealousy, and gratitude. For example, they describe studies showing thar workers who believc they are underpaid sometimes perform their work in ways that are harder for them in ol'der to rcducc their employers' profits.3
Other Compensation Schemes
This discussion implicitly assumes that a firm's financial arrangements with its workcrs take the form of some %'age per unit of time. An important question is whether there are more complicated ways for the firm to compensate its workers thar allow it to obtain the benefits of a higher wagc less expensively. The nutritional advantages of a higher wage, for example, can be obtained by compensating workers partly in kind (such as by feeding them at work). 'So give another example, firms can givc workers an incentive to exert effort by requiring them to post a bond that they lose if they are caught shirking.
If there are cheaper ways for firms to obtain the benefits of a higher wage, then these bencfits lead not lo a higher wage but just to complicated compensation policies. Whether the benefits can be obtained in such ways dcpends on the specific reason that a higher wage is advantageous. We will therefore not attempt a general treatment. The end of Section 9.4 discusses this issue in the context of efficiency-wagctheories based on imperfect monitoring of workers' effort. In this section and the next, however, wc simply assume that compensation takes the form of a conventional wage, and investigate the effects of efficiency wages under this assumption.
Assumptions
We now turn to a model of efficiencywages. There is a large number, N,of identical competilive firms..' The reprcscntative firm seeks to maximize its
\\.hen ability is obsen'able. the firm can pay higher wajirs to more able workers; thus ubsrrvablc ability differences do nu1 lead lo any drpartures fronl the Walrasian case.
: See Problem 9.5 fur a formalization of this idea. Thrpe other pulenlial advantages of a higher wage art. that it ran reduce turnover (and hence recruilmt'nt and training costs, if they are borne by lhc hrm); that it can lower the likehhoud that ihr workcrs will unionize; and that it can raisr thc utility of managers who ha\ t. sunlr abiliry to pursue objectives other than maAnli7ing profits.
\Vc can think of the number ul firms as bring drternlined by the amount of capilal in thr rconomy, which is furd in lhc short run.
414 Chapter 9 UNEMPLOYMENT
real profits, which are given by
lT=Y - wL, |
(9.11 |
where Y is the firm's output, w is the real wage that it pays, and L is the amount of labor it hires.
A firm's output depends both on the number of workers it employs and on their effort. For simplicity we neglect other inputs and assume that labor and effort enter the production function multiplicatively. Thus the representative firm's output is
where e denotes workers' effort. The crucial assumption of efficiency-wage models is that effort depends positively on the wage the firm pays. In this section we consider thc simple case (due to Solow, 1979)where the wage is the only determinant of effort. Thus,
c = e(w), e'(.) > 0. |
(9.31 |
Finally, there are iidentical workers, each of whom supplies 1unit of labor inelasticallv.
L
Analyzing the Model
The problem facing thc representative firm is |
|
max F(e(w)L)- wL. |
(9.4) |
L. W |
|
If there arc unemployed workers, the firm can choose the wage freely. If unemployment is zero, on the other hand, the firm must pay at least the wage paid by other firms.
Whcn the firm is unconstrained, the first-order conditions for L and w
are5 |
|
F'(e(w)L)Le'(w)- L = O. |
(9.6) |
We can rewrite (9.5) as |
|
Substituting (9.7)into (9.6)and dividing by L yieIds
We assurnc that the second~orderconditions are satisfied.
9.2 A Generic Efficiency-Wage Model 41 5
Equation (9.8) 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 firm therefore wants to hire effective labor as cheaply as possible. Whenthe firm hires a worker, it obtains e(w)units of effectivelabor at a cost of w; thus the cost per unit of effective labor is w/e(w) .W-hen 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 first-order condition for the problem of choosing w to minimize the cost of effective labor. 'l'hc wage satisfying (9.8) is !&own as the eficiency wagc.
Figure 9.1 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. T h s occurs where the e ( w )function is just tangent to one of the rays-that is, where the elasticity of e with respect to w is I. Panel (a) shows a case where effortis 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 age.
Finally,equation (9.7)states that the firm hires workers until the margin- al product of effective labor equals its cost. This is analogous to the condition in a slandard labor-demand problem that the firm hires labor up to the point where the marginal product equals the wagc.
Equations (9.7)and (9.8)describe the behavior of a single firm. Ucscribing the economy-wideequilibrium is straightforward. Let w" and L* denote the values of w and L that satisfy (9.7) and (9.8). Since firms are identical, each firm chooses these same values of w and L. Total labor dcmand is therefore NL". If labor supply, 1, exceeds this amount, firms are unconstrained in their choice of w. In this case the wage is w*, employment is NL', and there is unemploy-mcnt of amounl i- NL*. 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 arc in balance, and there is no uncmployment.
Implications
This model shows how elficiency wages can give rise to unemployment. In addition, the model implies that the real wage is unresponsive to demand shifts. Suppose the dcmand for labor increases. Since the efficiencywage. w*, is determined enlirely by the properties of the effort function, e ( * ) , there is no reason for firms to adjust their \$ages. Thus the modcl provides a candidate explanation of hhy shifts in labor dcmand lead lo large movemenrs in employment and small changes in the real wage. In addition, the fact that the real wagc and effort do not change implies that firms' labor costs do not change. As a result, in a modcl hith price-setting firms, the incentive to adiusl prices is small.
9.3 A More General Version 417
bor ourward, the real wag? remains unchanged and unemployment trends down%rard.Eventually, uncmplo~~mentreaches zero, at which point further increases in demand lead to increases in the real wagc. In practice, however. we observe no clear trend in unemployment over kxlended 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 mainly on employment in the short run; it is also that they fall almost entirely on the real wage in the long run. Our model does nol explain t h s pattern.
9.3A More General Version
Introduction
With many of the potential sources of efficiencywages, the wage is unlikely to be the only determinant of effort. Suppose, for example, that the wage affects cfiort becausz firms cannot monitor workers perfectly and workers are concerned about the possibility of losing their jobs if the firm catches them shn-king. In such a situation, the cost to a worker of being fired depends not just on the wagc 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 effortat a given wage when unemployment is higher, and to exert less effort when the wagc paid by other firms is higher. Similar arguments apply to situations where ihe wage affects effort because of unobsenred ability or feelings of gratitude or anger.
Thus a natural generalization of the effortfunction, (9.31,is
where w, is the wage paid by other firms and u is the unemployment rate, and where subscripts denote partial derivati\,es.
Each firm is small relative to the economy, and therefore takes w , and u as given. The representatiw firm's problem is the same as before, except that w, and u now affect the cffortfunction. The first-order conditions can therefore be rearranged to obtain
These conditions are analogous to (9.7) and (9.8)in the simpler version of the model.
Assume lhat the e ( . ) function is sufficientlywell behaved that there is a unique optimal w for a given w , and u. G i ~ e nthis assumption, equihbrium
418 Chapter 9 UNEMPLOYMENT
requires w = w,; if not, each firm wants to pay a wage &fferent from the prevading wage. Let w * and L* denote the values of w and L satisfying (9.10)-(9.11)with w = w,. As before, if NL* is less than E, the equilibrium wage is w* and there is unemployment of amount i - NL*. And if NL' 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 uncmployrnent in the short run. This is most easily seen by means of an e ~ a m p l e . ~
Example
Suppose effort is given by
L O otherwise,
where 0 < b < 1 and b > 0. x is a measure of labor-market conditions. If b equals 1,x is the wage paid at other firms multiplied by the fraction of workers who are employed. If b is less than 1, workers put less weight on uncmploynent; this could occur if there are unemployment benefits or if workers value leisure. If b 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 (9.12)states that for w > x, effort increases less than proportionately with w - x.
Differentiation of (9.12 ) shows that for this functional form, thc condition that the elasticity of effort with respect to the wage equals 1 (equation [9.11])is
Straightforward algebra can be used to simplify (9.14)to
|
(9.15) |
- l b u |
|
- |
1wo . |
For small values of 8, l / ( l- f i ) - |
1 + fi. Thus (9.15)implies that when 6 is |
small, the firm offers a premium of approxi~llatelyfraction 8 over thc index of labor-market opportunities, x.
(his examjllr i s hased on Sunnners (1988).
9.3 A More General Version 419
Equilibrium requires rhdl tile representative firm wants to pay the pre- v a h g wage, or that N = lmposing tlns condition in (9.15) yields
For this condition to be sdt~hhed,the unemployment rate must be given by
As equation (9.15) shows, each firm wants to pay more than the prevailing wagc if unemployment is less than UEQ, and wants to pay less if unemployment is more than UEQ. Thus equilibrium requires that u = UEQ
Substituting (9.17) and w = w, |
into thc effort function, (9.12). implies |
that equilibrium efforr is given by |
|
w |
, (1-bulQ)wo |
Finally, the equilibrium wagc is dctcrmined by the condition that the marginal product of effcctivc labor equals its cost (equation [9.101):F'ieL) = w/e. We can rewrite this condition as w = eY'(cL). Since total employment is (1- uEa)Tin equilibrium, each firm n u s t hire (1- UFQ ) T I Nworkers. Thus the equilibrium wage is given by
Implications
This analysis has thrcc important implications. First, (9.17) implies that equilibrium unemployment depends only on the parameters 01the effort funcrion; the production function is irrelevant. Thus an upward trcild in the production function does not produce a trend in unemployment.
Second, relatively modest valucs of &the elasticity of efforr with rcspcct to the premium firms pay over the index of labor-market conditionscan lead to nonnegligihle unemploynent. kor cxample, either fi = 0.06 and b = 1 or 6 = 0.03 and b = 0.5 imp1~thatequilibrium unemployment is 6 percent. This result is not as strong as it ma!- appear, howcvcr: while these parameter valucs imply a low elasticit\ of efforr with respect to (w - x ) / x ,
420 Chapter 9 UNEMPLOYMENT
they also imply that workers exert no effort at all until the wage is quite high. For example, if b is 0.5 and unemployment is at its equilibrium l e d of 6 percent, effort is zero until a firm's wage reaches 97 percent of the prevailing wage. In that sense, efficiency-wage forces are quite strong for these parameter values.
Third, firms' incentive to adjust wages or prices (or both) in response to changes in aggregate unemploymcnt is likely to be small for reasonable cases. Suppose we cmbed 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 equilibrium, so that u = uEaand marginal revenue and marginal cost are equal for the representative firm. Now suppose thar the money supply falls and firms do not change their nominal wages or prices; as a rcsult, uncmployment rises above utu. We know from Chapter 6 that small barriers lo wage and price adjustment can cause this to be an equilibrium only if the representative firm's incentive to adjust is small.
For concreteness, consider the incentive to adjust wages. Equation (9.151 w = ( 1b u ) w , / ( l - 81, shows thar the cost-minimizing wage is decreasing in the unemploymcnt 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 (9.12)for effort implics that if the firm leaves its wage equal to the prevailing wage, w,, its cost per unit of effective labor, wje, is
wo
CIIX~I)= e ( ~ , , w ,U~) .
-w',
-(7)"
-
-
(9.20,
w,
w, - ( 1 - bu)w, 1 ( 1 - bu)w,
If the firm changes its wage, on the bther hand, it sets it according to (9.151,and thus chooses w = x / ( l In this case, the firm's cost per unit of effective labor is
9.4 The Shapiro-Stiglitz Model 421
Supposc that fi = 0.06and h = 1, so that ura = (5%. Suppose, however, that unemployment risps to 9 percent and that other firms do not change their wages. Equations (9.20)and (9.21)imply that this rise lowers CFIXEDby 2.6 percent and CW, by- 3 . 2 percent. Thus the firm can save only 0.6 percent
or costs by cutting its M-ages.For fi = 0.03 and b = 0.5, the declines in C F ~ and CAD,are 1.3 percent and 1.5 percent; thus in thls casc the incentive to
cut wages is even smaller.'
In a compctitive labor market, in contrast, the equilibrium wagc falls by the percentage fall in employment divided by the elasticity of labor supply. For a 3 percent fall in employment and a labor supply elasticity of 0.2, for example, the equilibrium wage falls by 1 5 percent. And without cndogenous effort,a 1 5 percent fall in wages translates dircctly into a 15 percent fall in costs. Firms therefore have an overwhelming incentive to cut wagcs and prices in this case.8
Thus efficiencywages have a potentially large impact on the incentive to adjust wages in the face of fluctuations in aggregate output. As a result, they have the potential to cxplain why shifts in labor demand mainly affect employment in the short run. Intuitively, in a competitive market firms are initially at a corner solution with respect to u7agcs: firms pay the lowest possible wage at which thcy can hire workers. Thus wage reductions, if possible, are unambiguously beneficial. With efficiency wagcs, in contrast, firms are initially at an interior optimum whcre the marginal benefits and costs of wage cuts are equal.
The Shapiro-Stiglitz Model
One source of efficiency wages that has rcccivcd a great deal of attention is the possibility that firms' limited monitoring abilities force them to
Onr can also show that if firms do not change their wagcs, for reasonable cases their inrrnrivc to adjust the& prices is ~ S smallU . 1T Maps arc romplctcly tlexible, however, the incentive to adjust prices is nut small. With u grratcr than UEI), each firm r a n t s to pay less than other firms are paying (scc [0.151). Thus if wages are completely flexible, 1ht.y must fall O-or, i l workcrs have a positive resewation wage, to this reservation wag? As a result, firms' labor costs are extremely lory, and thus their incenlive to rut prircs and incrvasr ourput is high. Thus in the absence of any barriers lu changlr~gwages, snlall costs to changing prices are not enough lu prwenl prirr adjusrmrnt in this model.
In fact,in a cumpelilivr labor markrr, an inditidual firm's incentive to reduce wages if other firms do nu1 is evcn largrr than the fall h the equilibrium wage. If other firms du no1 cut wages, some workrrs arc unemployed. Thus the firm can hire workers at an arbitrarily small wage (ur at workrrs' reservation wage).
422 Chapter 9 UNEMPLOYMENT
provide their workers with an incentive to exert effort. This section presents a specific model, due to Shapiro and Sriglitz (19841,of this possibility."
Presenting a formal model of imperfect monitoring serves three purposes. First, it allows us to investigate whether this idea holds up under scrutiny. Second, it permits us to analyze additional questions; for example, only with a formal model can we ask whether government policies cz improve welfare. Third, the mathematical tools the model employs are useful in other settings.
Assumptions
The economy consists of a large number of workers, i,and a large number of firms, N. 'The workers maximize their expected discounted utilities and firms maximize their expected discounted profits. The model is set continuous time. For simplicity, the analysis focuses on steady states.
Consider workers first. The representative worker's lifetime utility is
u ( t ) is instantaneous utility at time t, and p is the discount rate. Instantaneous utility is
u(t) = |
w ( t ) - e(r) if employed |
|
if unemployed. |
||
|
w is the wage and e is the worker's effort. There are only two possible effort levels, e = 0 and e = 2. Thus at any moment a worker must be in one of three states: employed and exerting elfort (denoted E ) , employed and nor exerting effort (denoted S, for shirking), or unemployed (denoted U ) .
A key ingredient of the model is its assumptions concerning workers' transitions among the thrce states. First, there is an exogenous rare at which jobs end. Specifically, if a worker begins working in a job at some time. to (and if the worker exerts effort), the probability that the worker is stili employed in the job at some later time, r, is
(9.24)impliesthat P ( t + T ) / P ( ~equals) chi,and thus that it is independent of t: if a worker is employed at some time, the probability that he or she is still employed time T later is e-bTregardless of how long the worker has already been employed. THis lack of time dependence simplifies the analysis greatly, because it implies that there is no need to keep track of how long
"ick~na, Katz. Lang, and Summers ( l i l R ' 1 ) docu~nentthe impurlanrc of worker thrfl and shirking in the 1Jnired Statrs and argur that these phenom~naar? essential to undrrslanding the labor market.