# Efficiency wages and equilibrium wages.

I. INTRODUCTION

Economists have long sought to explain unemployment in ways consistent with individual agents' optimizing behavior. A recent, although controversial, model in this vein is the efficiency wage paradigm, which focuses on the difficulty of monitoring worker performance when production is organized in teams. To discourage nonperformance by employees, firms monitor their workers and dismiss those who are caught shirking. A firm may further discourage nonfeasance by paying its workers higher wages than they could earn elsewhere. Then workers caught shirking not only become unemployed, but forfeit the rents associated with their jobs. (1) In this scenario, in order to maintain the incentives not to shirk, firms will not cut wages even though equally qualified, unemployed workers are willing to work for less. The result is disequilibrium in the labor market and possibly involuntary unemployment. (2)

A series of theoretical studies have demonstrated the conditions under which efficiency wages may occur. Standard references include Eaton and White [1982; 1983], Shapiro and Stiglitz [1984], Foster and Wan [1984], Bowles [1985], and Calvo [1985]. Yellen [1984] reviews the early literature while Katz [1986], Stiglitz [1987], and Carmichael [1990] discuss more recent work. But, as Carmichael emphasizes in his review, few papers have actually attempted to test the predictions of the efficiency wage models. Most of the existing empirical investigations examine the importance of industrial wage differentials. Krueger and Summers [1987; 1988], Dickens and Katz [1987], and Murphy and Topel [1987a] find that there are substantial unexplained industrial wage differentials, which Bulow and Summers [1986] and Krueger and Summers [1987; 1988] attribute to efficiency wages. However, the magnitude, as well as the interpretation, of these industrial wage differentials is disputed; see Carmichael [1990], Leonard [1987], and Murphy and Topel [1987a; 1987b] for differing views.

This paper extends the efficiency wage model in a straightforward way to address two questions: Will the labor market clear if firms can only imperfectly monitor worker effort, and do industrial wage differentials provide a test for the importance of efficiency wages? We offer two modifications to traditional efficiency wage models. First, we recognize that there may be an explicit cost to being fired. For instance, dismissed workers may find it more difficult to find future employment because of poor references from their previous employer, they may incur high psychic costs as a result of being fired, or they may have to suffer large search costs in locating another job. Second, we let the firm choose the performance standard. (3) It seems only natural that a firm wanting more effort from its employees would hold its workers to a more rigorous performance standard, just as a firm seeking a higher quality labor force may require more schooling of its applicants. While the firm must compensate the worker for the extra effort necessary to meet the higher performance standard, payment of efficiency wages could be avoided. Following the efficiency wage literature, we make the strong assumption that all implicit or explicit bonding mechanisms are prohibitively costly to implement. Our model of the worker's and firm's behavior is specified in section II.

The model includes as special cases a market paying equilibrium wages, one paying non-market-clearing efficiency wages, and a "dual" labor market that is a combination of the two. We demonstrate that the firm need not pay efficiency wages even in the presence of the agency problem. The use of the performance standard gives the firm another instrument (other than the wage) to control worker effort and welfare. There are limits to how high the performance standard will be adjusted, however, before the firm will offer an efficiency wage. We present the details of these arguments in section III.

In section IV, we discuss the empirical implications of our model. Since it includes equilibrium wages and disequilibrium efficiency wages as special cases, our model lends itself to generating empirically testable hypotheses that distinguish between the two. Industrial wage differentials are easily generated in our model, but they do not necessarily indicate the payment of efficiency wages. Wages may differ across industries if there are differences by industry in the choice of the performance standard or in the difficulty of accurately evaluating the worker's effort. However, the two cases may be distinguished by examining the relationship between wages and the dismissal rate. In Section V, we summarize the results and conclude the paper.

II. THE MODEL

In this section, we extend the basic efficiency wage model to incorporate a cost of dismissal and a choice of the performance standard. We consider both the worker's and the firm's optimization problems.

The Worker's Problem

Following the efficiency wage literature, we consider homogeneous, infinitely-lived workers whose utility at time t([V.sub.t]) is given by g([w.sub.t]) -h([e.sub.t]), where et is effort expended and [w.sub.t] is the wage paid in the period. Let the function g(*), which represents the worker's utility from income, be twice differentiable, monotonically increasing, and concave. We also assume that the function h(*), which measures the worker's disutility of effort, is twice differentiable, monotonically increasing, and convex. Thus, the worker's utility function is

(1) [V.sub.t] = g([w.sub.t]) - h([e.sub.t]) + D ([V.sup.0] - C)/(1+r)

+ (1-D) [V.sub.t+1]/(1+r),

where r is the discount rate. With probability D, the worker is dismissed from the firm and must find employment next period elsewhere. The expected utility in alternative employment is [V.sup.0], which we assume is exogenously given to the individual. If dismissed, the worker incurs a cost C (C [greater than or equal to] 0), which reflects any direct costs of dismissal due to search, relocation, or psychic costs. While this assumption is not found in the traditional efficiency wage literature, numerous studies recognize the potential importance of dismissal costs; see Bowles [1985], Sparks [1986], and Akerlof and Katz [19861. With probability l-D, the worker is not dismissed and obtains utility [V.sub.t+1].

The firm observes an unbiased but imperfect signal about the worker's effort. Let the signal, s, be given by

(2) s = [e.sub.t] + [u.sub.t],

where u is a normal random variable with zero mean and finite variance and is uncorrelated with effort, e. The signals are distributed independently and identically across workers. The firm sets a minimum reservation signal S, and if s < S, the firm fires the worker. The probability of dismissal then is

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [u.sup.*] [congruent to] S - [e.sub.t] and f(*) is the normal probability density function of u. We interpret the reservation signal level (S) as a performance standard: a firm with a higher reservation signal expects a greater level of effort. The signal s may be the result of numerous observations of the worker, but the signal entails only observations from the current period. (4)

We consider only contracts that allow for fixed wage payments, and, thus, we exclude the possibility that the wage payment is a function of the signal that the firm receives. While this is standard in the efficiency wage literature, MacLeod and Malcomson [1989] demonstrate how restrictive this assumption is. Even when the worker's productivity is not observable to third parties, MacLeod and Malcomson show that "piece rate" contracts are included in the class of incentive compatible contracts. (5) Similarly, we do not allow firms to require a bond to ensure worker performance. (6) For discussion of the use of such bonds, see Murphy and Topel [1987b], Carmichael's [1985] comment on Shapiro and Stiglitz [1984], and the reply by Shapiro and Stiglitz [1985].

In order for the worker to accept an employment offer, the firm's contract must provide utility at least as great as the worker's best alternative offer, or

(4) g([w.sub.t]) - h([e.sub.t]) + D([V.sup.0] - C)/(1+r)

+ (1-D) [V.sub.t+1]/(1+r) [greater than or equal to] [V.sup.0].

As in most of the efficiency wage literature we assume that the contract is stationary so that (w,S,[V.sup.0],[V.sub.t]) are constants; thus [V.sub.t+1] = [V.sub.t]. One may then reduce equation (4) to

(4') [g(w) - h(e)] (1+r) - D C [greater than or equal to] r [V.sup.0],

which is the reservation utility constraint for [V.sub.t] = [V.sup.0].

Given the stream of wages (w) and the performance standard (S), the worker selects the level of effort that maximizes his expected utility level. The necessary condition for an interior maximum (e > 0) is

(5) h'(e) =f([u.sup.*]) [V - ([V.sup.0]- C)]/(1+r),

where V is the (constant) expected utility of continued employment at the firm. The worker equates the marginal cost of additional effort with the marginal reduction in dismissal costs. Equation (5) requires that V > [V.sup.0] - C. (7) When there are no costs of dismissal (C = 0), then the firm must offer a level of utility that exceeds the worker's market alternative, which is the standard efficiency wage result. (8)

It is readily shown from equation (1) that

(6) [partial derivative]V/partial derivative]w = ([r+d).sup.-1] (1+r) g'(w) > 0

[partial derivative]V/[partial derivative]S = -[(r+D).sup.-1] f([u.sup.*]) (V + C - [V.sup.0]) [less than or equal to] 0.

Equation (6) is useful for evaluating the following comparative statics:

(7) [partial derivative]e/[partial derivative]w = f([u.sup.*]) [partial derivative]V/[partial derivative]w [DELTA],

(8) [partial derivative]e/[partial derivative]s = [f'([u.sup.*])(V+c-[V.sup.0]) ([partial derivative][u.sup.*]/ [partial derivative]S)

+f([u.sup.*]) ([partial derivative]V/[partial derivative]s)][DELTA],

where [DELTA] = [([-V.sub.ee]).sup.-1] > 0. Clearly, [partial derivative]e/ [partial derivative]w > 0; a higher wage improves the attractiveness of the job, thereby inducing the worker to work harder to avoid dismissal. The effect of an increase in the performance standard (S) on effort is ambiguous. A higher standard (when [u.sup.*] < 0) will increase the marginal probability of being discharged for a given level of effort because f'([u.sup.*])> 0. This effect induces more effort and is reflected by the first term of the right-hand side of equation (8). A higher S, however, lowers V, which makes the current job less valuable and lowers effort. The second term in the right-hand side of equation (8) represents this effect.

Using equation (6), equation (8) reduces to

(9) [partial derivative]e/[partial derivative]S = [f'([u.sup.*]) (r+D) -f [([u.sup.*]).sup.2]]

(V+C-[V.sup.0]) [DELTA][(r+D).sup.-1]

As u is distributed normally, the term f'([u.sup.*]) simplifies to f([u.sup.*])[-[u.sup.*]/ [[sigma].sup.2]], where [[sigma].sup.2] is the variance of u. Thus, equation (9) becomes

(9') [partial derivative]e/partial derivative]S = -f([u.sup.*])(V+C-[V.sup.0])

[([u.sup.*]/[sigma].sup.2])(r+D) + f([u.sup.*])]

[DELTA] [(r+D).sup.-1]

= -f([u.sup.*]) (V+C-[V.sup.0]) B [[DELTA][(r+D).sup.-1]

where B [congruent to] ([u.sup.*]/[[sigma].sup.2])(r+D) + f ([u.sup.*]). As

(10) [partial derivative]B/[partial derivative] [u.sup.*] = (r+D)/[[sigma].sup.2] + f([u.sup.*]) ([u.sup.*]/[sigma])

+f'([u.sup.*]) = (r+D)[[sigma].sup.2] > 0,

B is monotonically increasing and

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because B is monotonically increasing it must cross zero exactly once. Therefore, for a given interest rate, there exists exactly one [u.sup.*] such that [partial derivative]e/ [partial derivative]S = 0. Let this value of [u.sup.*] be denoted [u.sup.0] and note that [u.sup.0]<0.

For values of [u.sup.*] < [u.sup.0], increases in the performance standard increase worker effort. If the standard is increased sufficiently ([u.sup.*] > [u.sup.0]), however, the worker reduces his or her level of effort because the increase in the performance standard has reduced the value of continued employment.

The Firm's Problem

We now examine the firm's selection of the wage payment, the level of employment, and the performance standard. In contrast, most of the efficiency wage literature, with the exception of Bulow and Summers [1986], Sparks [1986], and Akerlof and Katz [1986], assumes that the performance standard is exogenous. The firm selects the wage payment, the performance standard, and the level of employment to maximize profits. In a steady state, this is equivalent to maximizing per period profits, which are given by

(12) [pi] = F(N,e) - wN,

where F(*) is a concave production function with [F.sub.N](*) > 0 and [F.sub.e](*) > 0, and N is the level of employment of the homogeneous workers. The firm also must guarantee that workers receive a utility level of at least [V.sup.0], which is the constraint described by equation (4').

The first-order conditions for this problem are

(13) [partial derivative][pi]/[partial derivative]w = [F.sub.e](N,e) [partial derivative]e/[partial derivative]w

- N + [lambda][g'(w)] (1 + r) = 0

(14) [partial derivative][pi]/[partial derivative]S = [F.sub.e](N, e) [partial derivative]e/[partial derivative]S -[lambda] [C [partial derivative] D/[partial derivative]D/[partial derivative]S] = 0,

(15) [partial derivative][pi]/ [partial derivative]N = [F.sub.N](N, e) - w = 0

where [lambda] is the Kuhn-Tucker multiplier associated with the utility constraint given in equation (4'). The first-order conditions have the usual marginal benefit-marginal cost interpretation. The terms involving [lambda] reflect the influence of the firm's choices on the workers' welfare. Equation (14) implies that [partial derivative]e/[partial derivative]S [greater than or equal to] 0, which in turn implies that [u.sup.*] [less than or equal to] [u.sup.0].

If the firm chooses w and S such that the utility constraint is not binding, then [lambda], = 0, and the first-order conditions reduce to

(13') [F.sub.e](N, e) [partial derivative]e/[partial derivative]w - N = 0,

(14') [F.sub.e](N, e) [partial derivative]e/[partial derivative]S = 0,

(15') [F.sub.N](N, e) - w = 0.

III. EQUILIBRIUM WAGES OR EFFICIENCY WAGES?

In this section we consider how firms' choices of wages and performance standards may result in a labor market in equilibrium with utility equalized across all jobs, in a disequilibrium market paying efficiency wages, or in a "dual" labor market that is a combination of the two. Because of differences in technology, firms may choose different levels of the performance standard and wages, which cause workers to provide different levels of effort.

The introduction of an endogenous performance standard provides the firm with a method of increasing the worker's effort without increasing the worker's utility level. Thus, for some values of S, if the firm desires more effort from the worker, it can raise the performance standard and fire the worker if the standard is not met. The firm must pay a higher wage to compensate the worker for increased effort, but the firm need not raise the worker's utility above the reservation level, [V.sup.0]. In this scenario, the firm chooses w and S along the indifference curve [V.sup.0], which is depicted in Figure 1. As utility is equalized across jobs, the labor market clears. The equilibrium is much like a hedonic wage equilibrium, where the hedonic wage locus is coincident with the worker's indifference curve.

[FIGURE 1 OMITTED]

The slope of the indifference curve is positive and, from equation (6), is given by

(16) [partial derivative]w/[partial derivative]S = f ([u.sup.*]) C [rho]

with V = [V.sup.0] and where [rho] = [[(1+r)g'(w)].sup.-1]. The firm's selection of w and S will determine the worker's supply of effort. In (w,S) space, the slope of an isoeffort curve is

(17) [[partial derivative]w/[partial derivative]s |.sub.e] = (-[partial derivative]e/[partial derivative]SV([partial derivative]e/ [partial derivative]w)

= {[(r+D)([u.sup.*]/[[sigma].sup.2]) +f([u.sup.*])](V+C-[V.sup.0])} [rho].

For values of S such that [u.sup.*] < [u.sup.0], the expression is negative: an increase in the performance standard will raise the worker's effort and the firm must lower the wage to leave the worker's effort unchanged. At the point -(r+D)([u.sup.0]/[[sigma].sup.2]) = f([u.sup.0]), the expression obtains a minimum. For values of u > [u.sup.0], the isoeffort curve will slope upward. Several isoeffort curves are depicted in Figure 1.

In Figure 1, point A represents the contract ([w.sup.0], [S.sup.0]) where the firm guarantees the worker a utility level [V.sup.0] and receives an effort level [e.sup.0] in return. If the firm wishes to increase the effort level to [e.sup.1], it could do so by maintaining the performance standard at [S.sup.0] and increasing the wage to [w.sup.*], which would raise the worker's utility level. This is depicted by point B in Figure 1. The employer, however, will not choose this contract. Instead, the firm will increase the performance standard to [S.sup.1] and increase the wage to [w.sup.1] at point C. As point C provides the same level of effort at a lower wage than point B, point C is clearly the contract that the firm prefers.

By allowing the firm to increase the performance standard and the wage simultaneously we have increased the number of instruments that employers can use to induce the worker to provide more effort. (9) A natural question to ask is: Will we ever observe efficiency wages in the labor market? An efficiency wage contract occurs only when the utility constraint is nonbinding on firms, or when [lambda] = 0. From equation (14'), this results in a choice of S at the minimum of an isoeffort curve ([partial derivative]e/[partial derivative]S = 0). The set of minimum points, defined as the solutions to equation (14') for various levels of effort, define an upward sloping locus of points in (w,S) space. A set of possible efficiency wage contracts is depicted as the locus [E.sup.0] in Figure 2.

[FIGURE 2 OMITTED]

The part of this set that lies below the indifference curve V = [V.sup.0], however, is not feasible: the utility constraint is not met. (10) In the appendix we show that for some effort levels the firm prefers an efficiency wage contract to an equilibrium wage contract, and that there exists an effort level, depicted as [e.sup.2] in Figure 2, such that for e > [e.sup.2] the firm chooses an efficiency wage contract.

In Figure 2, point A represents the contract that is both an equilibrium and an efficiency wage contract. If the firm desires the level of effort [e.sup.3], it could elect to keep the worker on indifference curve [V.sup.0] and move to point B. The firm may reduce the performance standard and the wage, but still retain the same level of effort at point C. This efficiency wage contract is less expensive than the equilibrium wage contract in obtaining the level of effort [e.sup.3]. For higher levels of effort the efficiency wage contracts remain the preferred alternative: the firm will move along the locus [E.sup.0] rather than the locus [V.sup.0] in Figure 2.

Our model also admits "dual" labor markets as in Jones [1985] and Bulow and Summers [1986]. One "sector" may be used to define the reservation level of utility [V.sup.0]. Because we assume losing a job is costly to workers, we need not require that this "sector" be able to monitor perfectly employee effort, as in Jones and Bulow and Summers. Workers in this sector will receive compensating wage differentials for increases in the performance standard as in our equilibrium model. Firms in the "primary" sector, i.e., those wanting higher levels of effort, must pay efficiency wages. Workers clearly would prefer to be employed in the primary sector and earn the rents associated with efficiency wages.

Finally, recall that all of our analysis is predicated on the assumption that employers find it prohibitively costly to use bonds. Yet, Carmichael [1985] has cogently argued that such bonds should be a robust feature of labor markets with worker shirking. Our model suggests a reason why such bonds may not be necessary: if the cost of being fired is sufficiently high, the labor market will be in hedonic equilibrium, and there is no need for such bonds. As Carmichael [1989] notes, bonds are extremely rare. Thus, our model suggests that the absence of bonding implies that labor markets are in equilibrium and observed wage differentials are simply compensatory. (11)

IV. EMPIRICAL IMPLICATIONS

As our model allows for the possibility of a market-clearing equilibrium, or the payment of efficiency wages, or a combination of the two, we must find ways to distinguish between these settings. In part A, we show how our model generates industrial wage differentials and why they do not indicate the presence of efficiency wages. In part B, however, we demonstrate that one can use the relationship between wages and dismissals to distinguish between the efficiency wage and equilibrium settings.

A. Industrial Wage Differentials

After controlling for a variety of individual and workplace characteristics, several recent studies conclude that there are substantial wage differences across industries; see Krueger and Summers [1987; 1988] and Dickens and Katz [1987]. For some, these unexplained differentials suggest the presence of efficiency wages. Murphy and Topel [1987a; 1987b], however, argue that unobserved differences in workers account for a substantial part of the wage differentials. They examine wage changes for individuals who move between industries and document that the changes imply a much smaller wage differential than do those from cross-section estimates. They conclude that inter-industry wage differences are largely due to individual-specific differences in productivity, not efficiency wages. Krueger and Summers [1988], however, dispute this claim.

Our model suggests that even among a homogeneous work force the presence industrial wage differentials does not demonstrate the payment of efficiency wages. To illustrate, consider Figure 3. The loci [V.sup.0] and [E.sup.0] are the same as in Figure 2. Recall that the firm will choose a (w,S) combination along [V.sup.0] until point C, and then move along [E.sup.0]. Firms with production technologies exhibiting a higher marginal product of effort will choose a (w,S) combination further to the northeast in Figure 3, as effort increases in this direction. Thus, we may find one firm located at point A and another at point B. We observe a wage differential between these two firms, but it is an equalizing difference. Firms that choose a higher S demand more effort from workers and must compensate them for it. Other firms may have production functions with an even higher marginal productivity of effort and locate at point D. Here, the higher wage does indicate an efficiency wage payment. Simply examining wage differentials, however, does not reveal which is and which is not an efficiency wage. Thus, if industries tend to have common technologies, and firms in one industry cluster about point A, those in another cluster about B, and those in a third cluster around D, the industrial wage differentials do not distinguish between equalizing differences and efficiency wages. Also note that the relationship between higher effort and higher wages reported in Krueger and Summers [1986] does not reflect the payment of efficiency wages; we expect that relationship to hold in the compensating differentials equilibrium.

[FIGURE 3 OMITTED]

Thus, heterogeneity of production technologies among firms and industries leads to industrial wage differentials that need not reflect efficiency wages. The wage differences may be accounted for by a compensating differential paid for a higher performance standard that requires more worker effort. Finding wage changes when workers change industries is not inconsistent with equilibrium compensating wage differentials. If a worker moves from an industry at A to one at B, wages increase but utility does not. Essentially, the difficulty with this type of empirical analysis is the failure to take into account the variation in S across industries.

If industries differ in their marginal productivity of effort, they may also differ in their ability to monitor their employees. Heterogeneity in the nature of production processes is likely to result in some industries having less accurate measures of worker effort than other industries. This generates dispersion in [[sigma].sup.2], the variance of the signal. (12) A shift in [sigma] has two effects. First, the [V.sup.0] curve shifts left. An increase in the variance of the signal makes workers worse off because, with [u.sup.*] < 0, a higher [sigma] raises the probability of discharge for a given level of effort. Thus, for a given value of S, a larger [sigma] must be accompanied by a higher wage to keep the worker as well off. In Figure 3, we depict the new position of the indifference curve as [V.sup.0]'. Second, the locus [E.sup.0] shifts inward to the left. For a firm located on [E.sup.0], an increase in the variance of the signal reduces the utility of holding the job and so reduces effort. Thus, to remain on this locus, a firm must increase wage payments to maintain the same level of effort, implying that [E.sup.0] shifts inward to the left, as [E.sup.0]' is depicted in Figure 3. These results are formally derived in the appendix. It can also be shown (see the appendix) that the wage at which the [V.sup.0]' and the [E.sup.0]' loci intersect is lower than the wage where the [V.sup.0] and [E.sup.0] curves intersect. Thus, efficiency wages start at a lower wage when the variance is higher, which corresponds to the intuitive notion that we are more likely to observe efficiency wages in firms with greater monitoring difficulties. (13)

Firms (or industries) with this larger a now locate along [V.sup.0]' until point G, and then along [E.sup.0]'. Again, it is possible that higher wages are merely compensating for a higher S. It is also possible, however, that compensating wages are paid for a larger [sigma], S constant. Consider an industry whose firms have a larger value of [sigma] and locate at point F, and another industry where the firms have a smaller [sigma] and locate at I. The performance standard is the same for both, but wages are higher in the former industry. This does not indicate an efficiency wage, but rather compensation for the higher o. Another possibility is for an industry to have a lower performance standard and a higher wage and still not be paying efficiency wages. Compare industries located at A and F. At F, S is lower, but not low enough to fully compensate for the higher [sigma], so the wage is higher. (14)

In addition, consider an industry whose firms are located at point B, and another whose firms are at point H. The latter is paying efficiency wages while the former is not, but the efficiency wage is actually lower than the equalizing difference wage. The criterion of higher wages yields a perverse prediction of which is an efficiency wage. We obtain equalizing differences for fairly subtle (given current data) differences in the nature of firms--differences in performance standards and the accuracy of monitoring. Well-functioning, competitive labor markets will generate wage differentials on these bases, but they are unmeasured by the investigator and the consequent wage differences may be mistaken for the payment of efficiency wages. Our analysis allows for the possibility of efficiency wages, but we find that simply looking for industrial wage differentials (using either wage level or wage change data) cannot detect the payment of efficiency wages.

Several investigators have argued that heterogeneity among workers may generate industrial wage differentials. Murphy and Topel [1987b] demonstrate that a substantial part of observed industrial wage differentials may be attributable to unobserved individual differences in productivity. Although our model does not include differences in abilities across workers, we might expect variations across occupations in the cost to workers of being discharged. This may affect the incidence of efficiency wages. In many efficiency wage models--such as Shapiro and Stiglitz [1984] for instance--the worker faces no cost from being dismissed. To the extent, however, that employers perceive that previous discharges are a good predictor of future performance, dismissals will damage the reputation of the worker. While an unskilled, temporary laborer may find it easy to obtain employment without references from his past employer, for many skilled and professional workers this is unlikely. Thus, it would be very costly for skilled and professional workers to be dismissed for shirking, but not so costly for unskilled, temporary workers. Efficiency wages, therefore, may be concentrated among the low-paying, rather than higher-paying occupations.

This "reputation" effect makes dismissal costly to the worker. In Figure 4, we depict the impact of an increase in dismissal costs (C). When C increases, at a given performance standard, the wage necessary to ensure the utility level [V.sup.0] increases, shifting the isoquant to the left ([V.sup.0]'). It is straightforward to demonstrate that the efficiency wage locus shifts to the right ([E.sup.0]'), reflecting the higher effort that may be obtained for a given wage and standard. As a result, the point of intersection between the isoquant and efficiency wage locus is at a higher wage. Occupations in which the worker's reputation is valuable are less likely to rely on efficiency wages to prevent shirking.

[FIGURE 4 OMITTED]

B. Wages and Dismissals

Although industrial wage differentials do not enable us to differentiate between efficiency and equilibrium wages, the relationship between wages and dismissals does allow us to distinguish between the two models. For firms paying equalizing differences, as the performance standard (and the wage) is raised, the worker's effort increases, but the probability of dismissal also increases. The worker reacts to the increase in the performance standard by increasing effort, but by less than the increase in the performance standard. From equation (3), and recalling that [u.sup.*] = S - e, we obtain

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the second equality is derived from substituting for the partial derivatives given in equations (7), (8), and (16). The wage rate increases as the firm increases the performance standard in an equalizing differences setting, so we will find a positive relationship between wages and the dismissal rate.

With an efficiency wage contract, this is not the case: the rate of dismissals at the firm is independent of the wage. In the efficiency wage region, [partial derivative] e/[partial derivative]S = 0, which, from equation (9'), implies ([u.sup.*]/[[sigma].sup.2])(r+D) +f([u.sup.*])=0. Define [x.sup.*] to be the standard normal random variable x* = [u.sup.*]/[sigma], and the above expression can be rewritten as

(19) [x.sup.*] [r + [PHI]([x.sup.*])] + [phi]([x.sup.*]) = 0,

where [phi] and [PHI] are the standard normal density and cumulative distribution functions. The solution to this expression implies a unique [x.sup.*]. Therefore, for an efficiency wage contract, d[x.sup.*] = 0. The discharge rate, D, is equal to [PHI]([x.sup.*]), and so dD = [PHI]' d[x.sup.*] = 0. The dismissal rate is invariant in the efficiency wage region. Wages still increase with the performance standard, but at a faster rate than in the equilibrium setting, inducing enough additional effort to offset fully the direct, positive effect of higher standards on dismissals. Changes in o also have no effect on D, and, thus, the wage rates for industries utilizing efficiency wage contracts should be uncorrelated with their rates of dismissals.

While this result enables us to distinguish between industries that pay efficiency wages and those that do not, the prediction does depend crucially on the assumption of a normal distribution for the errors in the signal. In addition, there are likely to be difficulties in the empirical implementation of this test. The prediction derived implicitly holds constant the characteristics of the workers. As many dismissals may be attributable to unobserved differences in ability, rather than effort, it may prove quite difficult in practice to control for these other influences on discharges.

V. SUMMARY AND CONCLUSION

In this paper we present a model that contains as special cases a labor market with a compensating differentials equilibrium, a labor market with efficiency wages, and a "dual" labor market with both. Our approach differs from the traditional efficiency wage model in two important ways. First, unlike most other efficiency wage models, we allow for a direct cost of dismissal. This seems quite plausible as it may simply reflect the search costs that a worker may incur when seeking alternative employment or the damage to the worker's reputation a dismissal might cause. Second, our approach allows the firm to choose the performance standard.

We demonstrate that even in the absence of a bonding mechanism and wage payments based on output, payment of above-market wages need not occur when there is a cost of dismissal and an endogenous performance standard. Our analysis also indicates that recent attempts to detect efficiency wages by examining industrial wage differentials fail to distinguish an equilibrium model from an efficiency wage model. We are able however, to distinguish, between the labor market equilibrium and the efficiency wage settings by examining the relationship between wages and turnover.

APPENDIX

I. The Uniqueness of the Optimal Level of Effort.

We now demonstrate that if V(0) < [V.sup.0] and if h"(e)/h'(e) is a nondecreasing function, then the worker's problem will yield a unique optimum. The second-order condition is

(A1) [V.sub.ee] = - h"(e)(1 + r) + [u.sup.*]f ([u.sup.*]) [V - ([V.sup.0]-C)].

For e > S, clearly the second derivative is negative. At any critical point ([V.sub.e] = 0), we may use the first-order condition (5) to obtain

(A2) sgn{[V.sub.ee]} = sgn { - h"(e)/h'(e) + (S-e)}.

Let [e.sup.*] be a critical point such that [V.sub.ee] < 0. Using equation (A2) we will now show that any other critical point must be an inflection point or a local minimum. By way of contradiction, assume that there is a critical point [e.sup.l] > [e.sup.*]. As h"(e)/h'(e) is nondecreasing, any critical point [e.sup.1] > [e.sup.*] must be a local maximum. But as V is a continuous function, there must exists a local minimum between two local maxima, which is a contradiction. For any critical point [e.sup.2] < [e.sup.*], equation (A2) ensures it must be a local minimum because h"(e)/h'(e) is a nondecreasing function. As V([e.sup.*]) [greater than or equal to] [V.sup.0], we know that V(0)< V([e.sup.*]). Thus, there is a unique optimum.

II. The Use of Efficiency Wage Contracts.

To demonstrate that for some levels of effort the firm prefers an efficiency wage contract to an equilibrium wage contract, consider the contract (w,S), which corresponds to the effort level [E.sup.[dagger]], the highest effort level obtainable under the equilibrium contract. Rewriting equation (5), this point is defined by

h'([e.sup.[dagger]])- f(0) C[(1+r).sup.-1],

where f(0) is the maximum value of the probability density function f(*). But at this point

(A3) [dw/dS|.sub.e][dagger] = C [rho] f(0) [(r + D).sup.-1].

As [dw/dS|.sub.e][dagger]] > 0, the firm may lower the performance standard and the wage, and maintain the same level of effort. As the level of effort is maintained and wages are lower, (w,S) cannot be a profit-maximizing contract. The profit-maximizing contract, of course, simply satisfies

(A4) (1+r) h'(e) =f(u) [V(e) - [V.sup.0] + C],

which is the efficiency wage contract.

We now show that there is exactly one contract ([w.sup.2],[S.sup.2]), depicted as point A in Figure 2, that is both an equilibrium and an efficiency wage contract. This contract satisfies

(A5) (l+r)h'([e.sup.2]) = f([u.sup.0]) C.

At this point, V = [V.sup.0] and u = [u.sup.0]. To establish that this contract is unique, we will show that the locus of possible efficiency wage contracts cuts the isoquant V = [V.sup.0] from below. Consider first the slope of the efficiency wage locus. As we are dealing only in variations in w and S along [E.sup.0], [u.sup.0]= S-e is constant. Thus, du = dS -([e.sub.w], dw + [e.sub.s] dS) = dS - [e.sub.w] dw - 0; the second equality is because [e.sub.s] = 0. Therefore, dw/dS [ [u.sup.0] = 1/[e.sub.w]. From equation (7), we obtain

(A6) dw/dS|[u.sup.0] = 1/[e.sub.w] = ([[DELTA].sup.-1] [rho])/f(u),

where again [DELTA] = [(-[V.sub.ee]).sup.-1] and [rho] = [[(1+r)g'(w)].sup.-1]. The slope of the isoquant is

(A7) dw/dS|[V.sup.0] = -[V.sub.S]/[V.sub.w]

= f([u.sup.*]) (V+C-[V.sup.0]) [rho].

At the point ([w.sup.2],[S.sup.2]) we have

(A8) dw/dS|[u.sup.0] - dw/dS|[V.sup.0]

= [(r+D)(1+r)h"([e.sup.2]) [rho]]/f([u.sup.0]) > 0,

where we make use of the condition that [e.sub.s] = 0 at this point. The continuity of this difference implies that in the neighborhood around the point ([w.sup.2],[S.sup.2]) this difference is positive as well. Thus the locus of efficiency wage contracts cuts the indifference curve V = [V.sup.0] from below and the intersection point is locally unique.

To establish that this intersection point is globally unique, we need only note that the difference given in equation (A7) holds any time the two curves intersect. As the locus of efficiency wage contracts must cut the indifference curve V = [V.sup.0] from below, there must be no more than one such intersection point. As the set of feasible efficiency wage contracts is nonempty, there must be exactly one intersection point. Thus, for e > [e.sup.2] the firm chooses an efficiency wage contract.

III. Change in Variance of the Signal.

We now show that the effect of an increase in the variance of the signal shifts the isoquant to the left. To see this, consider the following:

(A9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the second equality comes from (6). This term is negative for [u.sup.*] < 0. Now, consider the effect of [sigma] on effort. With some algebra and using the properties of the normal distribution, we find

(A10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the expressions in (9') and (6), this becomes

(A11) [partial derivative]e/[partial derivative] [sigma] = ([u.sup.*]/[sigma]) [partial derivative]e/[partial derivative]S

+ ([partial derivative]V/[partial derivative]S)[DELTA]/[sigma].

Along the locus [E.sup.0], [partial derivative]e/ [partial derivative]S = 0, so [partial derivative]e/[partial derivative][sigma] = ([DELTA]/[sigma]) [partial derivative]V/ [partial derivative]S < 0. Also, along [E.sup.0], equation (19) holds, implying a unique [x.sup.*] = [u.sup.*]/[sigma]. Thus, d[x.sup.*] = 0, and we obtain

(A12) d[x.sup.*] = 0 = {[dS ([e.sub.w] dw + [e.sub.s] dS

+ [e.sub.[sigma]] d[sigma])][sigma] - [u.sup.*] d[sigma]}/[[sigma].sup.2].

Noting that [e.sub.s] = 0, this simplifies to

(A13) dS - [e.sub.w] dw - ([x.sup.*] + [e.sub.[sigma]]) d[sigma].

If [sigma] increases (d[sigma] > 0, the right-hand side of (A13) is negative, so the left-hand side also must be negative. This is accomplished by either a decline in S, an increase in w, or a combination of the two.

Efficiency wages do result at a lower value of w when [sigma] is larger. This implies that the crossing point of [V.sup.0,] and [E.sup.0,] is at a lower wage than previously. For this to occur, holding w constant, the shift inward in indifference curve ([V.sup.0]) when [sigma] rises must be smaller than the shift in the E locus. To see this, first consider the indirect utility function [V.sup.0] = V(w,S,[sigma]). For utility to be constant, d[V.sup.0] = 0. With w unchanged, this implies [V.sub.s]dS + [V.sub.[sigma]]d[sigma] = 0; or

(A14)dS = - ([V.sub.[sigma]]/[V.sub.s]) d[sigma] = [x.sup.*] d[sigma]

where the second equality comes from (A8). Now, along [E.sup.0], recall that d[x.sup.*] = 0. With w constant, from (A13), we see this results in

(A15) dS = ([x.sup.*] + [e.sub.[sigma]])d[sigma].

Now, [x.sup.*] < 0, and along the E locus, [e.sub.s] = 0, so [e.sub.[sigma]] < 0. Comparing (A14) to (A15) shows that for a given increase in [sigma], remaining on the E locus calls for a larger decline in S than to remain on the indifference curve ([V.sup.0]). This verifies that the crossing point of [E.sup.0] and [V.sup.0] occurs at a lower wage than that of [V.sup.0] and [E.sup.0]. This is as it is depicted in Figure 3.

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Sparks, Roger. "A Model of Involuntary Unemployment and Wage Rigidity: Worker Incentives and the Threat of Dismissal." Journal of Labor Economics, October 1986, 560-81.

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* Associate Professors, Department of Economics, University of Kentucky. We thank Richard Jensen, Mark Loewenstein, Timothy Perri, Roger Sparks, two anonymous reviewers, and members of the Macroeconomics Workshop at the University of Kentucky for comments on earlier drafts, but we remain responsible for any errors. We gratefully acknowledge financial support from the National Science Foundation (Grant No. RII-8610671) and the Commonwealth of Kentucky through the EPSCoR program.

(1.) There are numerous possible reasons for the payment of wages above the market-clearing level, but in this paper we focus on the monitoring-shirking paradigm. See Yellen [1984] for a good review of the various types of efficiency wage models.

(2.) This conclusion has been challenged by some; see Carmichael [1985], Oi [1986], and Murphy and Topel [1987b], for example. A common argument is that some type of implicit or explicit bonding scheme as proposed by Becker and Stigler [1974] can deter shirking and equalize the value of jobs.

(3.) Others have considered variation in the performance standard. See Akerlof and Katz [1986], Bulow and Summers [1986], and Sparks [1986].

(4.) Unlike the work of Radner [1981; 1985] and Rubenstein and Yaari [1983], we do not allow the firm to use observations front past periods when deciding whether or not to retain the worker. The use of information from only one period may not be optimal when there is a great deal of noise in the signal. There may exist gains to using "review strategies" in which the firm uses past information about the worker's performance to decide whether or not the worker should be retained. For a further discussion, see footnote 13.

(5.) See Malcomson [1981] for a good discussion of why such conditional contracts may not be offered.

(6.) This assumption means that the loss that the worker suffers from dismissal is simply the cost of being dismissed. Thus, we implicitly rule out forcing contracts that provide for very large penalties if the worker is apprehended.

(7.) If we allow firms to offer experienced workers contracts different from those offered new workers and we interpret C as a cost of changing jobs rather than as a cost of dismissal, it would be possible that the utility of experienced employees would be below that of their market alternative, [V.sup.0]. We rule out such contracts, assuming that firms will not take advantage of experienced workers for fear of damaging their reputations.

(8.) To ensure that we may use the first-order approach to this problem, we assume that h"(e)/h'(e) is a nondecreasing function of effort, and that V(0) < [V.sup.0]. The latter assumption simply implies that the firm can readily detect and dismiss workers that provide no effort-the worker must at least show up if he wishes to deceive the firm. The first assumption is satisfied by a wide class of functions (e.g., h(e) = [gamma]e, or h(e) = exp[[gamma]e]). While clearly restrictive, these assumptions allow us to use the first-order approach to this principal-agent problem, which vastly simplifies the description of the optimal contract. Alternatively, we could restrict the distribution of uncertainty in the model. Until recently, the sufficient conditions generally invoked required that the probability density function be nondecreasing. Jewitt [1988], however, has found more general sufficient conditions that allow for a wide variety of distribution functions. Unfortunately, Jewitt's conditions do not admit the normal distribution, and thus we choose to impose restrictions on the preferences of workers rather than the distribution function. In the appendix, we demonstrate that these restrictions are sufficient to ensure that there is a unique optimum to the worker's problem.

(9.) In this discussion, we abstract from the firm's decision of how many workers to employ. Oi [1983] notes, however, that larger firms with high costs of monitoring may economize on these costs by hiring more productive labor and using more capital. Barron, Black, and Loewenstein [1987] find that large firms do indeed use more capital per worker, train their workers more, and search more intensively for workers than do small firms.

(10.) The efficiency wages below the [V.sup.0] locus are well defined as long as h'(e) -f(u)[V+C-[V.sup.0]], which can occur when the cost of changing jobs is strictly positive. The only difficulty with this contract is that firms cannot recruit new labor. Note that our geometric exposition of this model is similar to that of Oi [1986].

11. We are grateful to an anonymous referee for this interpretation.

(12.) Presumably, firms have some control over [[sigma].sup.2] by devoting more or less resources to monitoring workers. There is, however, likely to be considerable exogenous variation in the ability to obtain an accurate signal. Our focus is on this exogenous component.

(13.) The work of Radner [1981; 1985] and Rubenstein and Yaari [1983] suggests another margin on which firms may respond. In our model we force firms to decide whether or not to retain the worker based solely on observations from the current period. But Radner (1985) demonstrates that such a strategy may not be optimal. Rather, the firm specifies a review period of, say, R periods. Defining the cumulative signal as [zeta] = [s.sub.1] + [s.sub.2],+ ... +[s.sub.R], the firm retains the worker if [zeta] [greater than or equal to] [R.sup.*], where [R.sup.*] is the "review performance standard." If the worker passes the review, the process starts again. This allows the firm to "average" the worker's signals over several periods, which reduces the variance of the signal. As the variance of the signal has been reduced, the firm is less likely to rely on efficiency wages. Clearly, our monitoring technology is a special case of Radner's, with R = 1. Indeed, one can interpret Radner's monitoring technology as determining the optimal length of a period in our model.

(14.) Effort may be lower at point F as effort does vary with [sigma].

Economists have long sought to explain unemployment in ways consistent with individual agents' optimizing behavior. A recent, although controversial, model in this vein is the efficiency wage paradigm, which focuses on the difficulty of monitoring worker performance when production is organized in teams. To discourage nonperformance by employees, firms monitor their workers and dismiss those who are caught shirking. A firm may further discourage nonfeasance by paying its workers higher wages than they could earn elsewhere. Then workers caught shirking not only become unemployed, but forfeit the rents associated with their jobs. (1) In this scenario, in order to maintain the incentives not to shirk, firms will not cut wages even though equally qualified, unemployed workers are willing to work for less. The result is disequilibrium in the labor market and possibly involuntary unemployment. (2)

A series of theoretical studies have demonstrated the conditions under which efficiency wages may occur. Standard references include Eaton and White [1982; 1983], Shapiro and Stiglitz [1984], Foster and Wan [1984], Bowles [1985], and Calvo [1985]. Yellen [1984] reviews the early literature while Katz [1986], Stiglitz [1987], and Carmichael [1990] discuss more recent work. But, as Carmichael emphasizes in his review, few papers have actually attempted to test the predictions of the efficiency wage models. Most of the existing empirical investigations examine the importance of industrial wage differentials. Krueger and Summers [1987; 1988], Dickens and Katz [1987], and Murphy and Topel [1987a] find that there are substantial unexplained industrial wage differentials, which Bulow and Summers [1986] and Krueger and Summers [1987; 1988] attribute to efficiency wages. However, the magnitude, as well as the interpretation, of these industrial wage differentials is disputed; see Carmichael [1990], Leonard [1987], and Murphy and Topel [1987a; 1987b] for differing views.

This paper extends the efficiency wage model in a straightforward way to address two questions: Will the labor market clear if firms can only imperfectly monitor worker effort, and do industrial wage differentials provide a test for the importance of efficiency wages? We offer two modifications to traditional efficiency wage models. First, we recognize that there may be an explicit cost to being fired. For instance, dismissed workers may find it more difficult to find future employment because of poor references from their previous employer, they may incur high psychic costs as a result of being fired, or they may have to suffer large search costs in locating another job. Second, we let the firm choose the performance standard. (3) It seems only natural that a firm wanting more effort from its employees would hold its workers to a more rigorous performance standard, just as a firm seeking a higher quality labor force may require more schooling of its applicants. While the firm must compensate the worker for the extra effort necessary to meet the higher performance standard, payment of efficiency wages could be avoided. Following the efficiency wage literature, we make the strong assumption that all implicit or explicit bonding mechanisms are prohibitively costly to implement. Our model of the worker's and firm's behavior is specified in section II.

The model includes as special cases a market paying equilibrium wages, one paying non-market-clearing efficiency wages, and a "dual" labor market that is a combination of the two. We demonstrate that the firm need not pay efficiency wages even in the presence of the agency problem. The use of the performance standard gives the firm another instrument (other than the wage) to control worker effort and welfare. There are limits to how high the performance standard will be adjusted, however, before the firm will offer an efficiency wage. We present the details of these arguments in section III.

In section IV, we discuss the empirical implications of our model. Since it includes equilibrium wages and disequilibrium efficiency wages as special cases, our model lends itself to generating empirically testable hypotheses that distinguish between the two. Industrial wage differentials are easily generated in our model, but they do not necessarily indicate the payment of efficiency wages. Wages may differ across industries if there are differences by industry in the choice of the performance standard or in the difficulty of accurately evaluating the worker's effort. However, the two cases may be distinguished by examining the relationship between wages and the dismissal rate. In Section V, we summarize the results and conclude the paper.

II. THE MODEL

In this section, we extend the basic efficiency wage model to incorporate a cost of dismissal and a choice of the performance standard. We consider both the worker's and the firm's optimization problems.

The Worker's Problem

Following the efficiency wage literature, we consider homogeneous, infinitely-lived workers whose utility at time t([V.sub.t]) is given by g([w.sub.t]) -h([e.sub.t]), where et is effort expended and [w.sub.t] is the wage paid in the period. Let the function g(*), which represents the worker's utility from income, be twice differentiable, monotonically increasing, and concave. We also assume that the function h(*), which measures the worker's disutility of effort, is twice differentiable, monotonically increasing, and convex. Thus, the worker's utility function is

(1) [V.sub.t] = g([w.sub.t]) - h([e.sub.t]) + D ([V.sup.0] - C)/(1+r)

+ (1-D) [V.sub.t+1]/(1+r),

where r is the discount rate. With probability D, the worker is dismissed from the firm and must find employment next period elsewhere. The expected utility in alternative employment is [V.sup.0], which we assume is exogenously given to the individual. If dismissed, the worker incurs a cost C (C [greater than or equal to] 0), which reflects any direct costs of dismissal due to search, relocation, or psychic costs. While this assumption is not found in the traditional efficiency wage literature, numerous studies recognize the potential importance of dismissal costs; see Bowles [1985], Sparks [1986], and Akerlof and Katz [19861. With probability l-D, the worker is not dismissed and obtains utility [V.sub.t+1].

The firm observes an unbiased but imperfect signal about the worker's effort. Let the signal, s, be given by

(2) s = [e.sub.t] + [u.sub.t],

where u is a normal random variable with zero mean and finite variance and is uncorrelated with effort, e. The signals are distributed independently and identically across workers. The firm sets a minimum reservation signal S, and if s < S, the firm fires the worker. The probability of dismissal then is

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [u.sup.*] [congruent to] S - [e.sub.t] and f(*) is the normal probability density function of u. We interpret the reservation signal level (S) as a performance standard: a firm with a higher reservation signal expects a greater level of effort. The signal s may be the result of numerous observations of the worker, but the signal entails only observations from the current period. (4)

We consider only contracts that allow for fixed wage payments, and, thus, we exclude the possibility that the wage payment is a function of the signal that the firm receives. While this is standard in the efficiency wage literature, MacLeod and Malcomson [1989] demonstrate how restrictive this assumption is. Even when the worker's productivity is not observable to third parties, MacLeod and Malcomson show that "piece rate" contracts are included in the class of incentive compatible contracts. (5) Similarly, we do not allow firms to require a bond to ensure worker performance. (6) For discussion of the use of such bonds, see Murphy and Topel [1987b], Carmichael's [1985] comment on Shapiro and Stiglitz [1984], and the reply by Shapiro and Stiglitz [1985].

In order for the worker to accept an employment offer, the firm's contract must provide utility at least as great as the worker's best alternative offer, or

(4) g([w.sub.t]) - h([e.sub.t]) + D([V.sup.0] - C)/(1+r)

+ (1-D) [V.sub.t+1]/(1+r) [greater than or equal to] [V.sup.0].

As in most of the efficiency wage literature we assume that the contract is stationary so that (w,S,[V.sup.0],[V.sub.t]) are constants; thus [V.sub.t+1] = [V.sub.t]. One may then reduce equation (4) to

(4') [g(w) - h(e)] (1+r) - D C [greater than or equal to] r [V.sup.0],

which is the reservation utility constraint for [V.sub.t] = [V.sup.0].

Given the stream of wages (w) and the performance standard (S), the worker selects the level of effort that maximizes his expected utility level. The necessary condition for an interior maximum (e > 0) is

(5) h'(e) =f([u.sup.*]) [V - ([V.sup.0]- C)]/(1+r),

where V is the (constant) expected utility of continued employment at the firm. The worker equates the marginal cost of additional effort with the marginal reduction in dismissal costs. Equation (5) requires that V > [V.sup.0] - C. (7) When there are no costs of dismissal (C = 0), then the firm must offer a level of utility that exceeds the worker's market alternative, which is the standard efficiency wage result. (8)

It is readily shown from equation (1) that

(6) [partial derivative]V/partial derivative]w = ([r+d).sup.-1] (1+r) g'(w) > 0

[partial derivative]V/[partial derivative]S = -[(r+D).sup.-1] f([u.sup.*]) (V + C - [V.sup.0]) [less than or equal to] 0.

Equation (6) is useful for evaluating the following comparative statics:

(7) [partial derivative]e/[partial derivative]w = f([u.sup.*]) [partial derivative]V/[partial derivative]w [DELTA],

(8) [partial derivative]e/[partial derivative]s = [f'([u.sup.*])(V+c-[V.sup.0]) ([partial derivative][u.sup.*]/ [partial derivative]S)

+f([u.sup.*]) ([partial derivative]V/[partial derivative]s)][DELTA],

where [DELTA] = [([-V.sub.ee]).sup.-1] > 0. Clearly, [partial derivative]e/ [partial derivative]w > 0; a higher wage improves the attractiveness of the job, thereby inducing the worker to work harder to avoid dismissal. The effect of an increase in the performance standard (S) on effort is ambiguous. A higher standard (when [u.sup.*] < 0) will increase the marginal probability of being discharged for a given level of effort because f'([u.sup.*])> 0. This effect induces more effort and is reflected by the first term of the right-hand side of equation (8). A higher S, however, lowers V, which makes the current job less valuable and lowers effort. The second term in the right-hand side of equation (8) represents this effect.

Using equation (6), equation (8) reduces to

(9) [partial derivative]e/[partial derivative]S = [f'([u.sup.*]) (r+D) -f [([u.sup.*]).sup.2]]

(V+C-[V.sup.0]) [DELTA][(r+D).sup.-1]

As u is distributed normally, the term f'([u.sup.*]) simplifies to f([u.sup.*])[-[u.sup.*]/ [[sigma].sup.2]], where [[sigma].sup.2] is the variance of u. Thus, equation (9) becomes

(9') [partial derivative]e/partial derivative]S = -f([u.sup.*])(V+C-[V.sup.0])

[([u.sup.*]/[sigma].sup.2])(r+D) + f([u.sup.*])]

[DELTA] [(r+D).sup.-1]

= -f([u.sup.*]) (V+C-[V.sup.0]) B [[DELTA][(r+D).sup.-1]

where B [congruent to] ([u.sup.*]/[[sigma].sup.2])(r+D) + f ([u.sup.*]). As

(10) [partial derivative]B/[partial derivative] [u.sup.*] = (r+D)/[[sigma].sup.2] + f([u.sup.*]) ([u.sup.*]/[sigma])

+f'([u.sup.*]) = (r+D)[[sigma].sup.2] > 0,

B is monotonically increasing and

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because B is monotonically increasing it must cross zero exactly once. Therefore, for a given interest rate, there exists exactly one [u.sup.*] such that [partial derivative]e/ [partial derivative]S = 0. Let this value of [u.sup.*] be denoted [u.sup.0] and note that [u.sup.0]<0.

For values of [u.sup.*] < [u.sup.0], increases in the performance standard increase worker effort. If the standard is increased sufficiently ([u.sup.*] > [u.sup.0]), however, the worker reduces his or her level of effort because the increase in the performance standard has reduced the value of continued employment.

The Firm's Problem

We now examine the firm's selection of the wage payment, the level of employment, and the performance standard. In contrast, most of the efficiency wage literature, with the exception of Bulow and Summers [1986], Sparks [1986], and Akerlof and Katz [1986], assumes that the performance standard is exogenous. The firm selects the wage payment, the performance standard, and the level of employment to maximize profits. In a steady state, this is equivalent to maximizing per period profits, which are given by

(12) [pi] = F(N,e) - wN,

where F(*) is a concave production function with [F.sub.N](*) > 0 and [F.sub.e](*) > 0, and N is the level of employment of the homogeneous workers. The firm also must guarantee that workers receive a utility level of at least [V.sup.0], which is the constraint described by equation (4').

The first-order conditions for this problem are

(13) [partial derivative][pi]/[partial derivative]w = [F.sub.e](N,e) [partial derivative]e/[partial derivative]w

- N + [lambda][g'(w)] (1 + r) = 0

(14) [partial derivative][pi]/[partial derivative]S = [F.sub.e](N, e) [partial derivative]e/[partial derivative]S -[lambda] [C [partial derivative] D/[partial derivative]D/[partial derivative]S] = 0,

(15) [partial derivative][pi]/ [partial derivative]N = [F.sub.N](N, e) - w = 0

where [lambda] is the Kuhn-Tucker multiplier associated with the utility constraint given in equation (4'). The first-order conditions have the usual marginal benefit-marginal cost interpretation. The terms involving [lambda] reflect the influence of the firm's choices on the workers' welfare. Equation (14) implies that [partial derivative]e/[partial derivative]S [greater than or equal to] 0, which in turn implies that [u.sup.*] [less than or equal to] [u.sup.0].

If the firm chooses w and S such that the utility constraint is not binding, then [lambda], = 0, and the first-order conditions reduce to

(13') [F.sub.e](N, e) [partial derivative]e/[partial derivative]w - N = 0,

(14') [F.sub.e](N, e) [partial derivative]e/[partial derivative]S = 0,

(15') [F.sub.N](N, e) - w = 0.

III. EQUILIBRIUM WAGES OR EFFICIENCY WAGES?

In this section we consider how firms' choices of wages and performance standards may result in a labor market in equilibrium with utility equalized across all jobs, in a disequilibrium market paying efficiency wages, or in a "dual" labor market that is a combination of the two. Because of differences in technology, firms may choose different levels of the performance standard and wages, which cause workers to provide different levels of effort.

The introduction of an endogenous performance standard provides the firm with a method of increasing the worker's effort without increasing the worker's utility level. Thus, for some values of S, if the firm desires more effort from the worker, it can raise the performance standard and fire the worker if the standard is not met. The firm must pay a higher wage to compensate the worker for increased effort, but the firm need not raise the worker's utility above the reservation level, [V.sup.0]. In this scenario, the firm chooses w and S along the indifference curve [V.sup.0], which is depicted in Figure 1. As utility is equalized across jobs, the labor market clears. The equilibrium is much like a hedonic wage equilibrium, where the hedonic wage locus is coincident with the worker's indifference curve.

[FIGURE 1 OMITTED]

The slope of the indifference curve is positive and, from equation (6), is given by

(16) [partial derivative]w/[partial derivative]S = f ([u.sup.*]) C [rho]

with V = [V.sup.0] and where [rho] = [[(1+r)g'(w)].sup.-1]. The firm's selection of w and S will determine the worker's supply of effort. In (w,S) space, the slope of an isoeffort curve is

(17) [[partial derivative]w/[partial derivative]s |.sub.e] = (-[partial derivative]e/[partial derivative]SV([partial derivative]e/ [partial derivative]w)

= {[(r+D)([u.sup.*]/[[sigma].sup.2]) +f([u.sup.*])](V+C-[V.sup.0])} [rho].

For values of S such that [u.sup.*] < [u.sup.0], the expression is negative: an increase in the performance standard will raise the worker's effort and the firm must lower the wage to leave the worker's effort unchanged. At the point -(r+D)([u.sup.0]/[[sigma].sup.2]) = f([u.sup.0]), the expression obtains a minimum. For values of u > [u.sup.0], the isoeffort curve will slope upward. Several isoeffort curves are depicted in Figure 1.

In Figure 1, point A represents the contract ([w.sup.0], [S.sup.0]) where the firm guarantees the worker a utility level [V.sup.0] and receives an effort level [e.sup.0] in return. If the firm wishes to increase the effort level to [e.sup.1], it could do so by maintaining the performance standard at [S.sup.0] and increasing the wage to [w.sup.*], which would raise the worker's utility level. This is depicted by point B in Figure 1. The employer, however, will not choose this contract. Instead, the firm will increase the performance standard to [S.sup.1] and increase the wage to [w.sup.1] at point C. As point C provides the same level of effort at a lower wage than point B, point C is clearly the contract that the firm prefers.

By allowing the firm to increase the performance standard and the wage simultaneously we have increased the number of instruments that employers can use to induce the worker to provide more effort. (9) A natural question to ask is: Will we ever observe efficiency wages in the labor market? An efficiency wage contract occurs only when the utility constraint is nonbinding on firms, or when [lambda] = 0. From equation (14'), this results in a choice of S at the minimum of an isoeffort curve ([partial derivative]e/[partial derivative]S = 0). The set of minimum points, defined as the solutions to equation (14') for various levels of effort, define an upward sloping locus of points in (w,S) space. A set of possible efficiency wage contracts is depicted as the locus [E.sup.0] in Figure 2.

[FIGURE 2 OMITTED]

The part of this set that lies below the indifference curve V = [V.sup.0], however, is not feasible: the utility constraint is not met. (10) In the appendix we show that for some effort levels the firm prefers an efficiency wage contract to an equilibrium wage contract, and that there exists an effort level, depicted as [e.sup.2] in Figure 2, such that for e > [e.sup.2] the firm chooses an efficiency wage contract.

In Figure 2, point A represents the contract that is both an equilibrium and an efficiency wage contract. If the firm desires the level of effort [e.sup.3], it could elect to keep the worker on indifference curve [V.sup.0] and move to point B. The firm may reduce the performance standard and the wage, but still retain the same level of effort at point C. This efficiency wage contract is less expensive than the equilibrium wage contract in obtaining the level of effort [e.sup.3]. For higher levels of effort the efficiency wage contracts remain the preferred alternative: the firm will move along the locus [E.sup.0] rather than the locus [V.sup.0] in Figure 2.

Our model also admits "dual" labor markets as in Jones [1985] and Bulow and Summers [1986]. One "sector" may be used to define the reservation level of utility [V.sup.0]. Because we assume losing a job is costly to workers, we need not require that this "sector" be able to monitor perfectly employee effort, as in Jones and Bulow and Summers. Workers in this sector will receive compensating wage differentials for increases in the performance standard as in our equilibrium model. Firms in the "primary" sector, i.e., those wanting higher levels of effort, must pay efficiency wages. Workers clearly would prefer to be employed in the primary sector and earn the rents associated with efficiency wages.

Finally, recall that all of our analysis is predicated on the assumption that employers find it prohibitively costly to use bonds. Yet, Carmichael [1985] has cogently argued that such bonds should be a robust feature of labor markets with worker shirking. Our model suggests a reason why such bonds may not be necessary: if the cost of being fired is sufficiently high, the labor market will be in hedonic equilibrium, and there is no need for such bonds. As Carmichael [1989] notes, bonds are extremely rare. Thus, our model suggests that the absence of bonding implies that labor markets are in equilibrium and observed wage differentials are simply compensatory. (11)

IV. EMPIRICAL IMPLICATIONS

As our model allows for the possibility of a market-clearing equilibrium, or the payment of efficiency wages, or a combination of the two, we must find ways to distinguish between these settings. In part A, we show how our model generates industrial wage differentials and why they do not indicate the presence of efficiency wages. In part B, however, we demonstrate that one can use the relationship between wages and dismissals to distinguish between the efficiency wage and equilibrium settings.

A. Industrial Wage Differentials

After controlling for a variety of individual and workplace characteristics, several recent studies conclude that there are substantial wage differences across industries; see Krueger and Summers [1987; 1988] and Dickens and Katz [1987]. For some, these unexplained differentials suggest the presence of efficiency wages. Murphy and Topel [1987a; 1987b], however, argue that unobserved differences in workers account for a substantial part of the wage differentials. They examine wage changes for individuals who move between industries and document that the changes imply a much smaller wage differential than do those from cross-section estimates. They conclude that inter-industry wage differences are largely due to individual-specific differences in productivity, not efficiency wages. Krueger and Summers [1988], however, dispute this claim.

Our model suggests that even among a homogeneous work force the presence industrial wage differentials does not demonstrate the payment of efficiency wages. To illustrate, consider Figure 3. The loci [V.sup.0] and [E.sup.0] are the same as in Figure 2. Recall that the firm will choose a (w,S) combination along [V.sup.0] until point C, and then move along [E.sup.0]. Firms with production technologies exhibiting a higher marginal product of effort will choose a (w,S) combination further to the northeast in Figure 3, as effort increases in this direction. Thus, we may find one firm located at point A and another at point B. We observe a wage differential between these two firms, but it is an equalizing difference. Firms that choose a higher S demand more effort from workers and must compensate them for it. Other firms may have production functions with an even higher marginal productivity of effort and locate at point D. Here, the higher wage does indicate an efficiency wage payment. Simply examining wage differentials, however, does not reveal which is and which is not an efficiency wage. Thus, if industries tend to have common technologies, and firms in one industry cluster about point A, those in another cluster about B, and those in a third cluster around D, the industrial wage differentials do not distinguish between equalizing differences and efficiency wages. Also note that the relationship between higher effort and higher wages reported in Krueger and Summers [1986] does not reflect the payment of efficiency wages; we expect that relationship to hold in the compensating differentials equilibrium.

[FIGURE 3 OMITTED]

Thus, heterogeneity of production technologies among firms and industries leads to industrial wage differentials that need not reflect efficiency wages. The wage differences may be accounted for by a compensating differential paid for a higher performance standard that requires more worker effort. Finding wage changes when workers change industries is not inconsistent with equilibrium compensating wage differentials. If a worker moves from an industry at A to one at B, wages increase but utility does not. Essentially, the difficulty with this type of empirical analysis is the failure to take into account the variation in S across industries.

If industries differ in their marginal productivity of effort, they may also differ in their ability to monitor their employees. Heterogeneity in the nature of production processes is likely to result in some industries having less accurate measures of worker effort than other industries. This generates dispersion in [[sigma].sup.2], the variance of the signal. (12) A shift in [sigma] has two effects. First, the [V.sup.0] curve shifts left. An increase in the variance of the signal makes workers worse off because, with [u.sup.*] < 0, a higher [sigma] raises the probability of discharge for a given level of effort. Thus, for a given value of S, a larger [sigma] must be accompanied by a higher wage to keep the worker as well off. In Figure 3, we depict the new position of the indifference curve as [V.sup.0]'. Second, the locus [E.sup.0] shifts inward to the left. For a firm located on [E.sup.0], an increase in the variance of the signal reduces the utility of holding the job and so reduces effort. Thus, to remain on this locus, a firm must increase wage payments to maintain the same level of effort, implying that [E.sup.0] shifts inward to the left, as [E.sup.0]' is depicted in Figure 3. These results are formally derived in the appendix. It can also be shown (see the appendix) that the wage at which the [V.sup.0]' and the [E.sup.0]' loci intersect is lower than the wage where the [V.sup.0] and [E.sup.0] curves intersect. Thus, efficiency wages start at a lower wage when the variance is higher, which corresponds to the intuitive notion that we are more likely to observe efficiency wages in firms with greater monitoring difficulties. (13)

Firms (or industries) with this larger a now locate along [V.sup.0]' until point G, and then along [E.sup.0]'. Again, it is possible that higher wages are merely compensating for a higher S. It is also possible, however, that compensating wages are paid for a larger [sigma], S constant. Consider an industry whose firms have a larger value of [sigma] and locate at point F, and another industry where the firms have a smaller [sigma] and locate at I. The performance standard is the same for both, but wages are higher in the former industry. This does not indicate an efficiency wage, but rather compensation for the higher o. Another possibility is for an industry to have a lower performance standard and a higher wage and still not be paying efficiency wages. Compare industries located at A and F. At F, S is lower, but not low enough to fully compensate for the higher [sigma], so the wage is higher. (14)

In addition, consider an industry whose firms are located at point B, and another whose firms are at point H. The latter is paying efficiency wages while the former is not, but the efficiency wage is actually lower than the equalizing difference wage. The criterion of higher wages yields a perverse prediction of which is an efficiency wage. We obtain equalizing differences for fairly subtle (given current data) differences in the nature of firms--differences in performance standards and the accuracy of monitoring. Well-functioning, competitive labor markets will generate wage differentials on these bases, but they are unmeasured by the investigator and the consequent wage differences may be mistaken for the payment of efficiency wages. Our analysis allows for the possibility of efficiency wages, but we find that simply looking for industrial wage differentials (using either wage level or wage change data) cannot detect the payment of efficiency wages.

Several investigators have argued that heterogeneity among workers may generate industrial wage differentials. Murphy and Topel [1987b] demonstrate that a substantial part of observed industrial wage differentials may be attributable to unobserved individual differences in productivity. Although our model does not include differences in abilities across workers, we might expect variations across occupations in the cost to workers of being discharged. This may affect the incidence of efficiency wages. In many efficiency wage models--such as Shapiro and Stiglitz [1984] for instance--the worker faces no cost from being dismissed. To the extent, however, that employers perceive that previous discharges are a good predictor of future performance, dismissals will damage the reputation of the worker. While an unskilled, temporary laborer may find it easy to obtain employment without references from his past employer, for many skilled and professional workers this is unlikely. Thus, it would be very costly for skilled and professional workers to be dismissed for shirking, but not so costly for unskilled, temporary workers. Efficiency wages, therefore, may be concentrated among the low-paying, rather than higher-paying occupations.

This "reputation" effect makes dismissal costly to the worker. In Figure 4, we depict the impact of an increase in dismissal costs (C). When C increases, at a given performance standard, the wage necessary to ensure the utility level [V.sup.0] increases, shifting the isoquant to the left ([V.sup.0]'). It is straightforward to demonstrate that the efficiency wage locus shifts to the right ([E.sup.0]'), reflecting the higher effort that may be obtained for a given wage and standard. As a result, the point of intersection between the isoquant and efficiency wage locus is at a higher wage. Occupations in which the worker's reputation is valuable are less likely to rely on efficiency wages to prevent shirking.

[FIGURE 4 OMITTED]

B. Wages and Dismissals

Although industrial wage differentials do not enable us to differentiate between efficiency and equilibrium wages, the relationship between wages and dismissals does allow us to distinguish between the two models. For firms paying equalizing differences, as the performance standard (and the wage) is raised, the worker's effort increases, but the probability of dismissal also increases. The worker reacts to the increase in the performance standard by increasing effort, but by less than the increase in the performance standard. From equation (3), and recalling that [u.sup.*] = S - e, we obtain

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the second equality is derived from substituting for the partial derivatives given in equations (7), (8), and (16). The wage rate increases as the firm increases the performance standard in an equalizing differences setting, so we will find a positive relationship between wages and the dismissal rate.

With an efficiency wage contract, this is not the case: the rate of dismissals at the firm is independent of the wage. In the efficiency wage region, [partial derivative] e/[partial derivative]S = 0, which, from equation (9'), implies ([u.sup.*]/[[sigma].sup.2])(r+D) +f([u.sup.*])=0. Define [x.sup.*] to be the standard normal random variable x* = [u.sup.*]/[sigma], and the above expression can be rewritten as

(19) [x.sup.*] [r + [PHI]([x.sup.*])] + [phi]([x.sup.*]) = 0,

where [phi] and [PHI] are the standard normal density and cumulative distribution functions. The solution to this expression implies a unique [x.sup.*]. Therefore, for an efficiency wage contract, d[x.sup.*] = 0. The discharge rate, D, is equal to [PHI]([x.sup.*]), and so dD = [PHI]' d[x.sup.*] = 0. The dismissal rate is invariant in the efficiency wage region. Wages still increase with the performance standard, but at a faster rate than in the equilibrium setting, inducing enough additional effort to offset fully the direct, positive effect of higher standards on dismissals. Changes in o also have no effect on D, and, thus, the wage rates for industries utilizing efficiency wage contracts should be uncorrelated with their rates of dismissals.

While this result enables us to distinguish between industries that pay efficiency wages and those that do not, the prediction does depend crucially on the assumption of a normal distribution for the errors in the signal. In addition, there are likely to be difficulties in the empirical implementation of this test. The prediction derived implicitly holds constant the characteristics of the workers. As many dismissals may be attributable to unobserved differences in ability, rather than effort, it may prove quite difficult in practice to control for these other influences on discharges.

V. SUMMARY AND CONCLUSION

In this paper we present a model that contains as special cases a labor market with a compensating differentials equilibrium, a labor market with efficiency wages, and a "dual" labor market with both. Our approach differs from the traditional efficiency wage model in two important ways. First, unlike most other efficiency wage models, we allow for a direct cost of dismissal. This seems quite plausible as it may simply reflect the search costs that a worker may incur when seeking alternative employment or the damage to the worker's reputation a dismissal might cause. Second, our approach allows the firm to choose the performance standard.

We demonstrate that even in the absence of a bonding mechanism and wage payments based on output, payment of above-market wages need not occur when there is a cost of dismissal and an endogenous performance standard. Our analysis also indicates that recent attempts to detect efficiency wages by examining industrial wage differentials fail to distinguish an equilibrium model from an efficiency wage model. We are able however, to distinguish, between the labor market equilibrium and the efficiency wage settings by examining the relationship between wages and turnover.

APPENDIX

I. The Uniqueness of the Optimal Level of Effort.

We now demonstrate that if V(0) < [V.sup.0] and if h"(e)/h'(e) is a nondecreasing function, then the worker's problem will yield a unique optimum. The second-order condition is

(A1) [V.sub.ee] = - h"(e)(1 + r) + [u.sup.*]f ([u.sup.*]) [V - ([V.sup.0]-C)].

For e > S, clearly the second derivative is negative. At any critical point ([V.sub.e] = 0), we may use the first-order condition (5) to obtain

(A2) sgn{[V.sub.ee]} = sgn { - h"(e)/h'(e) + (S-e)}.

Let [e.sup.*] be a critical point such that [V.sub.ee] < 0. Using equation (A2) we will now show that any other critical point must be an inflection point or a local minimum. By way of contradiction, assume that there is a critical point [e.sup.l] > [e.sup.*]. As h"(e)/h'(e) is nondecreasing, any critical point [e.sup.1] > [e.sup.*] must be a local maximum. But as V is a continuous function, there must exists a local minimum between two local maxima, which is a contradiction. For any critical point [e.sup.2] < [e.sup.*], equation (A2) ensures it must be a local minimum because h"(e)/h'(e) is a nondecreasing function. As V([e.sup.*]) [greater than or equal to] [V.sup.0], we know that V(0)< V([e.sup.*]). Thus, there is a unique optimum.

II. The Use of Efficiency Wage Contracts.

To demonstrate that for some levels of effort the firm prefers an efficiency wage contract to an equilibrium wage contract, consider the contract (w,S), which corresponds to the effort level [E.sup.[dagger]], the highest effort level obtainable under the equilibrium contract. Rewriting equation (5), this point is defined by

h'([e.sup.[dagger]])- f(0) C[(1+r).sup.-1],

where f(0) is the maximum value of the probability density function f(*). But at this point

(A3) [dw/dS|.sub.e][dagger] = C [rho] f(0) [(r + D).sup.-1].

As [dw/dS|.sub.e][dagger]] > 0, the firm may lower the performance standard and the wage, and maintain the same level of effort. As the level of effort is maintained and wages are lower, (w,S) cannot be a profit-maximizing contract. The profit-maximizing contract, of course, simply satisfies

(A4) (1+r) h'(e) =f(u) [V(e) - [V.sup.0] + C],

which is the efficiency wage contract.

We now show that there is exactly one contract ([w.sup.2],[S.sup.2]), depicted as point A in Figure 2, that is both an equilibrium and an efficiency wage contract. This contract satisfies

(A5) (l+r)h'([e.sup.2]) = f([u.sup.0]) C.

At this point, V = [V.sup.0] and u = [u.sup.0]. To establish that this contract is unique, we will show that the locus of possible efficiency wage contracts cuts the isoquant V = [V.sup.0] from below. Consider first the slope of the efficiency wage locus. As we are dealing only in variations in w and S along [E.sup.0], [u.sup.0]= S-e is constant. Thus, du = dS -([e.sub.w], dw + [e.sub.s] dS) = dS - [e.sub.w] dw - 0; the second equality is because [e.sub.s] = 0. Therefore, dw/dS [ [u.sup.0] = 1/[e.sub.w]. From equation (7), we obtain

(A6) dw/dS|[u.sup.0] = 1/[e.sub.w] = ([[DELTA].sup.-1] [rho])/f(u),

where again [DELTA] = [(-[V.sub.ee]).sup.-1] and [rho] = [[(1+r)g'(w)].sup.-1]. The slope of the isoquant is

(A7) dw/dS|[V.sup.0] = -[V.sub.S]/[V.sub.w]

= f([u.sup.*]) (V+C-[V.sup.0]) [rho].

At the point ([w.sup.2],[S.sup.2]) we have

(A8) dw/dS|[u.sup.0] - dw/dS|[V.sup.0]

= [(r+D)(1+r)h"([e.sup.2]) [rho]]/f([u.sup.0]) > 0,

where we make use of the condition that [e.sub.s] = 0 at this point. The continuity of this difference implies that in the neighborhood around the point ([w.sup.2],[S.sup.2]) this difference is positive as well. Thus the locus of efficiency wage contracts cuts the indifference curve V = [V.sup.0] from below and the intersection point is locally unique.

To establish that this intersection point is globally unique, we need only note that the difference given in equation (A7) holds any time the two curves intersect. As the locus of efficiency wage contracts must cut the indifference curve V = [V.sup.0] from below, there must be no more than one such intersection point. As the set of feasible efficiency wage contracts is nonempty, there must be exactly one intersection point. Thus, for e > [e.sup.2] the firm chooses an efficiency wage contract.

III. Change in Variance of the Signal.

We now show that the effect of an increase in the variance of the signal shifts the isoquant to the left. To see this, consider the following:

(A9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the second equality comes from (6). This term is negative for [u.sup.*] < 0. Now, consider the effect of [sigma] on effort. With some algebra and using the properties of the normal distribution, we find

(A10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the expressions in (9') and (6), this becomes

(A11) [partial derivative]e/[partial derivative] [sigma] = ([u.sup.*]/[sigma]) [partial derivative]e/[partial derivative]S

+ ([partial derivative]V/[partial derivative]S)[DELTA]/[sigma].

Along the locus [E.sup.0], [partial derivative]e/ [partial derivative]S = 0, so [partial derivative]e/[partial derivative][sigma] = ([DELTA]/[sigma]) [partial derivative]V/ [partial derivative]S < 0. Also, along [E.sup.0], equation (19) holds, implying a unique [x.sup.*] = [u.sup.*]/[sigma]. Thus, d[x.sup.*] = 0, and we obtain

(A12) d[x.sup.*] = 0 = {[dS ([e.sub.w] dw + [e.sub.s] dS

+ [e.sub.[sigma]] d[sigma])][sigma] - [u.sup.*] d[sigma]}/[[sigma].sup.2].

Noting that [e.sub.s] = 0, this simplifies to

(A13) dS - [e.sub.w] dw - ([x.sup.*] + [e.sub.[sigma]]) d[sigma].

If [sigma] increases (d[sigma] > 0, the right-hand side of (A13) is negative, so the left-hand side also must be negative. This is accomplished by either a decline in S, an increase in w, or a combination of the two.

Efficiency wages do result at a lower value of w when [sigma] is larger. This implies that the crossing point of [V.sup.0,] and [E.sup.0,] is at a lower wage than previously. For this to occur, holding w constant, the shift inward in indifference curve ([V.sup.0]) when [sigma] rises must be smaller than the shift in the E locus. To see this, first consider the indirect utility function [V.sup.0] = V(w,S,[sigma]). For utility to be constant, d[V.sup.0] = 0. With w unchanged, this implies [V.sub.s]dS + [V.sub.[sigma]]d[sigma] = 0; or

(A14)dS = - ([V.sub.[sigma]]/[V.sub.s]) d[sigma] = [x.sup.*] d[sigma]

where the second equality comes from (A8). Now, along [E.sup.0], recall that d[x.sup.*] = 0. With w constant, from (A13), we see this results in

(A15) dS = ([x.sup.*] + [e.sub.[sigma]])d[sigma].

Now, [x.sup.*] < 0, and along the E locus, [e.sub.s] = 0, so [e.sub.[sigma]] < 0. Comparing (A14) to (A15) shows that for a given increase in [sigma], remaining on the E locus calls for a larger decline in S than to remain on the indifference curve ([V.sup.0]). This verifies that the crossing point of [E.sup.0] and [V.sup.0] occurs at a lower wage than that of [V.sup.0] and [E.sup.0]. This is as it is depicted in Figure 3.

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* Associate Professors, Department of Economics, University of Kentucky. We thank Richard Jensen, Mark Loewenstein, Timothy Perri, Roger Sparks, two anonymous reviewers, and members of the Macroeconomics Workshop at the University of Kentucky for comments on earlier drafts, but we remain responsible for any errors. We gratefully acknowledge financial support from the National Science Foundation (Grant No. RII-8610671) and the Commonwealth of Kentucky through the EPSCoR program.

(1.) There are numerous possible reasons for the payment of wages above the market-clearing level, but in this paper we focus on the monitoring-shirking paradigm. See Yellen [1984] for a good review of the various types of efficiency wage models.

(2.) This conclusion has been challenged by some; see Carmichael [1985], Oi [1986], and Murphy and Topel [1987b], for example. A common argument is that some type of implicit or explicit bonding scheme as proposed by Becker and Stigler [1974] can deter shirking and equalize the value of jobs.

(3.) Others have considered variation in the performance standard. See Akerlof and Katz [1986], Bulow and Summers [1986], and Sparks [1986].

(4.) Unlike the work of Radner [1981; 1985] and Rubenstein and Yaari [1983], we do not allow the firm to use observations front past periods when deciding whether or not to retain the worker. The use of information from only one period may not be optimal when there is a great deal of noise in the signal. There may exist gains to using "review strategies" in which the firm uses past information about the worker's performance to decide whether or not the worker should be retained. For a further discussion, see footnote 13.

(5.) See Malcomson [1981] for a good discussion of why such conditional contracts may not be offered.

(6.) This assumption means that the loss that the worker suffers from dismissal is simply the cost of being dismissed. Thus, we implicitly rule out forcing contracts that provide for very large penalties if the worker is apprehended.

(7.) If we allow firms to offer experienced workers contracts different from those offered new workers and we interpret C as a cost of changing jobs rather than as a cost of dismissal, it would be possible that the utility of experienced employees would be below that of their market alternative, [V.sup.0]. We rule out such contracts, assuming that firms will not take advantage of experienced workers for fear of damaging their reputations.

(8.) To ensure that we may use the first-order approach to this problem, we assume that h"(e)/h'(e) is a nondecreasing function of effort, and that V(0) < [V.sup.0]. The latter assumption simply implies that the firm can readily detect and dismiss workers that provide no effort-the worker must at least show up if he wishes to deceive the firm. The first assumption is satisfied by a wide class of functions (e.g., h(e) = [gamma]e, or h(e) = exp[[gamma]e]). While clearly restrictive, these assumptions allow us to use the first-order approach to this principal-agent problem, which vastly simplifies the description of the optimal contract. Alternatively, we could restrict the distribution of uncertainty in the model. Until recently, the sufficient conditions generally invoked required that the probability density function be nondecreasing. Jewitt [1988], however, has found more general sufficient conditions that allow for a wide variety of distribution functions. Unfortunately, Jewitt's conditions do not admit the normal distribution, and thus we choose to impose restrictions on the preferences of workers rather than the distribution function. In the appendix, we demonstrate that these restrictions are sufficient to ensure that there is a unique optimum to the worker's problem.

(9.) In this discussion, we abstract from the firm's decision of how many workers to employ. Oi [1983] notes, however, that larger firms with high costs of monitoring may economize on these costs by hiring more productive labor and using more capital. Barron, Black, and Loewenstein [1987] find that large firms do indeed use more capital per worker, train their workers more, and search more intensively for workers than do small firms.

(10.) The efficiency wages below the [V.sup.0] locus are well defined as long as h'(e) -f(u)[V+C-[V.sup.0]], which can occur when the cost of changing jobs is strictly positive. The only difficulty with this contract is that firms cannot recruit new labor. Note that our geometric exposition of this model is similar to that of Oi [1986].

11. We are grateful to an anonymous referee for this interpretation.

(12.) Presumably, firms have some control over [[sigma].sup.2] by devoting more or less resources to monitoring workers. There is, however, likely to be considerable exogenous variation in the ability to obtain an accurate signal. Our focus is on this exogenous component.

(13.) The work of Radner [1981; 1985] and Rubenstein and Yaari [1983] suggests another margin on which firms may respond. In our model we force firms to decide whether or not to retain the worker based solely on observations from the current period. But Radner (1985) demonstrates that such a strategy may not be optimal. Rather, the firm specifies a review period of, say, R periods. Defining the cumulative signal as [zeta] = [s.sub.1] + [s.sub.2],+ ... +[s.sub.R], the firm retains the worker if [zeta] [greater than or equal to] [R.sup.*], where [R.sup.*] is the "review performance standard." If the worker passes the review, the process starts again. This allows the firm to "average" the worker's signals over several periods, which reduces the variance of the signal. As the variance of the signal has been reduced, the firm is less likely to rely on efficiency wages. Clearly, our monitoring technology is a special case of Radner's, with R = 1. Indeed, one can interpret Radner's monitoring technology as determining the optimal length of a period in our model.

(14.) Effort may be lower at point F as effort does vary with [sigma].

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Author: | Black, Dan A.; Garen, John E. |
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Publication: | Economic Inquiry |

Date: | Jul 1, 1991 |

Words: | 8559 |

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