Do labor markets provide enough short-hour jobs? An analysis of work hours and work incentives.

Rebitzer, James B. ; Taylor, Lowell J.

I. INTRODUCTION

Female labor force participation has increased dramatically over the past three decades. Much of this change is due to the increased participation of married women and women with children.(1) In households where both adults participate in the labor force, or where there is a single working parent, individuals will often have greater demands on their time at home and may therefore desire patterns of work hours that differ from other workers. Given the gender-based division of labor in most American households, many of the women entering the labor force may prefer shorter (and perhaps more flexible) work weeks.(2) Furthermore, as sex roles adjust to accommodate the changing work and career aspirations of women, it is reasonable to expect that increasing numbers of men will also prefer shorter work weeks. The prospects for equality of economic opportunity between men and women rest in large measure on how well and how rapidly labor markets accommodate the hours preferences of workers who desire this flexibility.

In this paper we ask whether labor markets will provide the optimal number of short-hour jobs in response to an increase in demand for short hours on the part of employees. According to the simple textbook model of the determinants of work hours, the answer to this question is clearly yes. Firms have an incentive to elicit information about their hours preferences because this allows them to offer labor contracts that minimize cost. Similarly, workers have an incentive to reveal their preferences to finns because this information is used to construct wage and hours packages in which workers are asked to work the utility-maximizing number of hours at the market-clearing wage.

Labor market outcomes may be considerably more complex, however, in a setting where firms rely on work incentives to regulate the effort exerted by employees.(3) We find that in a simple efficiency wage model (along the lines of Shapiro and Stiglitz [1984] and Bowles [1985], but with a heterogeneous work force) workers' hours preferences may provide an indicator of their responsiveness to the work incentives. In this setting employers will in general not be able to elicit accurate information about hours preferences from employees. We show that this market failure may lead in turn to labor market equilibria that are characterized by an underprovision of short-hour jobs. We find further that the shortage of short-hour jobs most likely occurs in high-wage labor markets.

Our model suggests that the simple textbook analysis of hours determination relies upon the wrong market metaphor. The conventional approach presumes that the determination of work hours is similar to the determination of car colors. Workers have an incentive to reveal their true hours preferences and employers have anincentive to solicit these preferences for the same reason that consumers have an incentive to reveal their color preferences and car makers have an incentive to solicit these preferences.

In our view, a more appropriate market metaphor for hours determination is the market for health insurance. Employees have an incentive to portray themselves as desiring long hours for the same reasons that purchasers of health insurance will want to portray themselves as having no health problems. Insurance providers must be concerned about the unobservable characteristics of individuals who are attracted to the various insurance contracts they offer. We suggest that employers face similar concerns when offering wage and hours packages.

The paper proceeds as follows. In the next section (section II), we analyze the determination of wages and work hours when finns use dismissal threats to motivate a homogeneous group of workers. Section III examines the case where workers have heterogeneous preferences with respect to hours of work. In section IV, we discuss the empirical plausibility of our model. Section V provides concluding remarks.

II. WAGES AND HOURS WITH HOMOGENEOUS WORKERS

A Simple Model of Work Incentives

In this section, we develop a model in which firms use dismissal threats to regulate the intensity with which their employees work. Models based on dismissal threats are analytically convenient and match nicely with the employment-at-will legal doctrine that governs labor law in the United States. However, the logic of our argument would not be appreciably altered if incentives revolved around promotion probabilities rather than separation probabilities.

The setup is quite simple. The economy is composed of a large number of firms each making use of the same concave production technology, such that the demand for labor is an inverse function of unit labor cost.

In each period there is a flow of identical workers into the labor market. These workers form queues at the firms, which in turn select workers from their own queue. Workers who are hired remain with the firm until they retire or are dismissed for working at low intensity. We assume that workers who are not hired when they first join the labor force will drop out of the market, and we similarly assume that workers who retire or are dismissed do not seek re-employment with other firms.(4) (One could alternatively suppose, as do Bulow and Summers [1986], that workers who are not selected from the queue or who lose their preferred job, accept employment in a market-clearing "secondary" sector comprised of firms that do not face the incentive issue we describe. Under this interpretation, the model we outline below applies only to the high-paying "primary" sector jobs.)

Workers in any period derive utility from income, the level of work effort, and leisure. Workers can adopt two levels of intensity, "high" or "low." A worker providing the lower level of effort is said to be "shirking." Let [U.sup.s](y, N) be the utility in any period of a worker who is shirking at a job paying income y with N hours of work. Similarly, let [U.sup.n](y, N) be the utility of a worker who is not shirking. Incentive problems arise because [U.sup.s](y, N) [greater than] [U.sup.n](y, N).

We assume that for the types of jobs relevant to our discussion, firms find it profitable to attempt to get employees to work at the high level of work intensity. They do this by dismissing those workers found to be providing the low level of intensity. These shirking workers are detected and dismissed in any period with probability D.(5) Under the assumption that employers learn about employee behavior by observing them on the job, D is a function of N, with D(0)=0 and D[prime] [greater than] 0.

For convenience, we abstract from issues relating to pensions and rising wage-tenure profiles by stipulating that workers are infinitely lived and that wages in each period are the same.(6) We also assume that the probability a worker retires, q, is the same each period. Under these conditions, the discounted present value of employment for a worker producing at the high level of work intensity is

(1) [V.sup.n] = [U.sup.n] + (1 - q)[V.sup.n]/(1 + r) + q[V.sup.u]/(1 + r),

where r is the discount rate and [V.sup.u] is the discounted present value of leaving the current job, i.e., [V.sup.u] is the present value of the flow of the reservation utility. We let this be the utility of full-time leisure - with y and N set to zero - though, as we indicate above, one could alternatively de-fine this to be the utility of employment in a "secondary" job.

In contrast to non-shirkers, shirking workers can be dismissed when they are discovered working at the low level of work intensity. The expected discounted present value of lifetime utility for a shirking worker is

(2) [V.sup.s] = [U.sup.s] + (1 - q)(1 - D)[V.sup.s] / (1 + r) + [1 - (1 - q)(1 - D)][V.sup.u]/(1 + r).

A rational worker will not shirk when [V.sup.n] [greater than or equal to] [V.sup.s]. Let [U.sup.u] be the utility derived from a single period of full-time leisure. Using equations (1) and (2) we notice that workers will not shirk as long as,(7)

[U.sup.n] - [U.sup.u] [greater than or equal to] [Gamma],

where

(3) [Gamma] = ([U.sup.s] - [U.sup.n])(r + q) / (1 - q)D.

The term [Gamma] is the minimum value of the difference between the current utility of employment and unemployment that is required to assure no shirking. Since this "employment rent" is always positive, the utility of employment at the high level of work intensity exceeds the utility of not working for the marginal worker. Assuming that the no-shirking condition represents a binding constraint in the market, there will be workers who would like to work but who nonetheless cannot find employment.(8)

Determining Hours of Work

Under the assumption of constant returns to scale in hours, and in the absence of fixed employment costs, cost-minimizing firms will set hours, N, so as to minimize the wage that assures no shirking. Consider the case where utility from working N hours at wage w for a shirking and non-shirking worker is given, respectively, by the following simple quasi-linear form:

[U.sup.s] = wN - [e.sub.s](N) + [Mu]g(N),

and

(4) [U.sup.n] = wN - [e.sub.n](N) + [Mu]g(N),

where wN is the utility of income, [Mu]g(N) is the utility derived from leisure with g being a function of N (g[prime] [less than] 0, g[double prime] [less than] 0) and [Mu] a positive parameter, [e.sub.s](N) is the disutility of work effort for a shirking worker, and [e.sub.n](N) is the corresponding disutility for a non-shirking worker. For both shirking and non-shirking workers, e(0) = 0, e[prime](N) [greater than] 0 and e[double prime](N) [greater than or equal to] 0, and since working at the higher level of effort is distasteful, [e.sub.n](N) [greater than] [e.sub.s](N). With this utility function, the no-shirking condition becomes

(5) wN - [e.sub.n](N) + [Mu]g(N) - [Mu]g(0) [greater than or equal to] [Gamma], with

[Gamma] [equivalent to] (r + q)E/(1 - q)D,

and

E = [e.sub.n](N) - [e.sub.s](N).

By examining the derivative of equation (5) we find that firms will be paying the minimum wage consistent with no shirking when they set hours such that

(6) w - [e.sub.n][prime] + [Mu]g[prime] = [[Gamma].sub.N],

where

[[Gamma].sub.N] = [(r + q)E'] / [(1 - q)D]

[1 - (D[prime] / D) / (E[prime] / E)]

The terms on the left-hand side of equation (6) represent the utility of income generated from an additional hour of work (w) net of the disutility of work effort (-[e.sub.n][prime]), and the utility lost from giving up an hour of leisure ([Mu]g[prime]).

For any given wage, the worker will be maximizing utility when he or she works the number of hours for which the sum of the terms on the left-hand side of (6) is zero. Thus, the wage and hours package offered by firms will entail the utility maximizing number of hours only when the employment rent is invariant with respect to hours, i.e., when [[Gamma].sub.N] = 0.(9) While it is in principle possible that [[Gamma].sub.N] will be zero, it also possible for [[Gamma].sub.N] [greater than] 0.(10) When this happens, the employment rent increases with work hours and firms offer a wage and hours package such that the effort-adjusted marginal utility of income exceeds the marginal loss of utility from giving up leisure. Employees accepting this wage-hours package will therefore perceive themselves to be hours constrained.

For our purposes, the central question is how labor markets will respond when worker preferences shift in favor of shorter working hours. Let ([w.sub.0], [N.sub.0]) be the cost-minimizing wage and hours package. Differentiating the no-shirking condition (5) and first-order condition for cost minimization (6), and assuming that the second-order condition holds for the minimization problem, it is easy to demonstrate that

(7) d[w.sub.0]/d[Mu] = [g(0) - g([N.sub.0])]/[N.sub.0] [greater than] 0,

and

(8) d[N.sub.0]/d[Mu] = -[([d.sup.2]w/d[N.sup.2]).sup.-1].

(1/N)[g[prime] + N(d[w.sub.0]/d[Mu])]

The result that d[w.sub.0]/d[Mu] is positive (equation 7) has a strong intuitive appeal. The effectiveness of dismissal threats rests on denying shirking workers access to high future income streams. Since an increase in [Mu] corresponds to an increase in the value workers place on leisure relative to income, a higher income stream (and therefore a higher wage) is required to assure no shirking.(11) One might expect expression (8) to be negative. As [Mu] increases so does employees' marginal utility of leisure relative to income. Since firms prevent shirking by offering a wage-hours package that is attractive to workers, firms would want to respond to this shift in preferences by offering jobs entailing shorter hours, all else equal. All else is not equal, however. As indicated by (7), the wage must be increasing with [Mu], and this induces workers to seek longer hours. The net effect is ambiguous.

Our findings suggest that when the work force is homogeneous, firms using efficiency wage incentive schemes will respond to a change in employee preferences favoring more leisure by offering wage-hours packages entailing higher wages. Firms will also adjust hours to reflect worker preferences. Nonetheless, jobs may be characterized by hours constraints, and individual workers may not perceive themselves to be working optimal hours.

III. WAGES AND HOURS WITH HETEROGENEOUS WORKERS

Work Hours as an Indicator

We turn next to the analysis of incentive schemes firms use when employees have heterogeneous preferences. We focus our attention on the wage-hours packages firms offer when individual workers have different values of [Mu], i.e., differing marginal rates of substitution between leisure and income.

In the preceding section, we notice that workers desiring short hours (i.e., workers with high values of [Mu]) will require a higher wage to assure no shirking. If worker preferences are known to firms, firms will clearly choose not to hire the more expensive short-hour workers. If worker preferences are not apparent to firms, however, this simple outcome may not be possible. Employees, fearing the consequences of the signal they are sending by asking to work a given number of hours, will not reveal their true hours preferences to their employers.(12)

In the sections below we explore the implications this market failure has for the ability of markets to respond to the preferences of workers desiring short hours. To highlight the role played by the unobservable (to the firm) worker heterogeneity, we assume in what follows that the probability of dismissal, D, and the single-period utility gain from shirking, E, are linear functions of hours, N, such that D = dN and E = eN. As noted in footnote 9, [[Gamma].sub.n] = 0 in this case, and thus no hours restrictions would emerge if the work force were homogeneous.

Pooling and Separating Contracts

We consider a labor force with two types of workers, type S and type L. These workers are identical in all respects except that [[Mu].sub.S] [greater than] [[Mu].sub.L]. Note that type S workers will, at any wage, prefer shorter hours than type L workers; we refer to type S and L workers as, respectively, short- and long-hour workers. Let [Theta] represent the proportion of short-hour workers in the population.

We suppose that firms independently offer contracts that they always proceed to honor. These contracts specify wages and hours available to employees, and always stipulate that workers who provide the low level of effort will be dismissed if detected. As we proceed we will assume, for simplicity, that shirking workers produce zero output.

The presence of two types of workers leads firms to offer one of two types of employment contracts, "pooling contracts" or "separating contracts." Under a pooling contract, a firm offers a single wage-hours package to all workers. In principle, this contract could be a "long-hour" or "short-hour" pooling contract - formed to meet the no-shirking conditions of either long- or short-hour workers. In the long-hour pooling contract, a firm would offer to all employees the minimum cost wage-hours package necessary to elicit the high level of work effort from long-hour workers. In the short-hour pooling contract, a firm would offer employees the minimum cost wage-hours package necessary to elicit the high level of work effort from short-hour workers. However, short-hour pooling contracts will always be at least as expensive as the separating contracts we discuss in the following paragraph. In analyzing pooling employment contracts we therefore need only concern ourselves with long-hour pooling contracts.

Under long-hour pooling contracts (henceforth, simply "pooling contracts"), long-hour workers provide high levels of work effort, while short-hour workers shirk. For a firm that wants to prevent shirking among both types of workers, the cost-minimizing strategy is to specify a contract allowing workers to select either a long- or short-hour option. We refer to such contracts as "separating contracts." Under a separating contract, the firm's objective is to offer the minimum-cost wage-hours packages subject to the constraints imposed by the no-shirking conditions.

A final type of contract that could arise might be termed a "screening contract." Under this contract a firm would offer a wage and hours package with very long hours, so that short-hour workers would find the positions less attractive than the alternative of unemployment. These workers would thus be effectively screened out of the labor force. For a screening contract to be effective, the wage and hours package must not only be sufficiently unattractive to short-hour workers, but must also meet the no shirking condition for long-hour workers. It is fairly easy to demonstrate that screening contracts will not always be feasible - here may be cases for which there is no wage and hour package that induces no shirking for long-hour workers while at the same time screens out short-hour workers. Moreover, when such contracts are feasible they may be more expensive than the pooling or separating contracts. In the present analysis we will assume that screening contracts are not a viable option to firms.(13) Thus firms will restrict attention to two types of contract - pooling contracts and separating contracts. We proceed by describing first the pooling contracts, then the separating contracts.

Pooling Contracts. With a pooling contract, an employer offers all workers the wage-hours package designed to prevent long-hour workers from shirking. Short-hour workers hired under this contract shirk, and the resulting dismissals increase the exit rate of short-hour workers. Thus, the proportion of short-hour workers in the population, [Theta], exceeds the proportion of short-hour workers who are employed in firms offering a pooling contract, [[Theta].sub.p].

It is straightforward to demonstrate that [[Theta].sub.p] [less than or equal to] [Theta] in a steady state the following relationship holds:

(9) [[Theta].sub.p] = q[Theta]/[q + (1 - [Theta])(1 - q)D].

It is clear that [[Theta].sub.p] [less than or equal to] 0, with the equality holding only when [Theta] equals zero or one. For future reference it is also worth noting that [[Theta].sub.p] is a monotonically increasing function of [Theta].

Due to shirking, the average productivity of the short-hour workers is less than the long-hour workers. Here we are assuming shirkers produce zero output. Thus, in an equilibrium in which all firms offer pooling contracts, the per unit labor cost will be [w.sub.0]/(1-[[Theta].sub.p]), where [w.sub.0] is the minimum wage consistent with no shirking on the part of long-hour workers. Using (9) we can express unit costs in this case, [C.sup.p], as a function of the proportion of short-hour workers in the population, [Theta]:

(10) [C.sup.p]([Theta]) = [w.sub.0]{1 - q[Theta]/[q + (1 - [Theta])(1 - q)D]}s.

Notice that unit labor cost in the pooling equilibrium is an increasing function of [Theta].

Separating Contracts. Firms may alternatively offer a separating contract in response to worker heterogeneity. The problem for the firm is to construct wage-hours packages, ([w.sub.S], [N.sub.S]) and ([w.sub.L], [N.sub.L]), such that the package taken by any worker will meet that worker's no-shirking condition. Suppose that for a firm offering this separating contract, [[Theta].sub.S] is the proportion of the short-hour workers. Then the formal cost-minimization problem is

(11) min [[[Theta].sub.S][w.sub.S][N.sub.S] +(1 - [[Theta].sub.S])[w.sub.L][N.sub.L]]/[[[Theta].sub.S][N.sub.S] + (1 - [[Theta].sub.S])[N.sub.L]],

subject to the binding no-shirking condition for the type S worker,(14)

(12) [w.sub.S][N.sub.S] - e[N.sub.S] + [[Mu].sub.S]g([N.sub.S]) [greater than or equal to] [Gamma] + [[Mu].sub.S]g(0),

and the constraint that type L workers prefer the ([w.sub.L], [N.sub.L]) package,

(13) [w.sub.L][N.sub.L] - e[N.sub.L] + [[Mu].sub.L]g([N.sub.L]) [greater than or equal to][w.sub.S][N.sub.S] - e[N.sub.S] + [[Mu].sub.L]g([N.sub.S]).

For a concrete example, suppose that the utility for leisure is a simple quadratic term, g(N)= -[N.sup.2]. For ease of notation we let [[Mu].sub.L] = 1 and note that [[Mu].sub.S] must then exceed unity. After several algebraic steps, we find that a firm's minimum unit labor cost when offering a separating contract is

(14) [C.sup.s]([[Theta].sub.S] = e + [{2[Gamma][[[Mu].sub.S] - (1 - [[Theta].sub.S])]/[[[Theta].sub.S] + (1 - [[Theta].sub.S])([[Mu].sub.S] - 1)]}.sup.1/2].

The hours offered to short- and long-hours workers respectively are

[N.sub.S] = ([C.sup.s] - e)/{2[1 + ([[Mu].sub.S] - 1)/[[Theta].sub.S]]}

and

(15) [N.sub.L] = ([C.sup.s] - e)/2.

Figure 1 illustrates such a separating contract. In this figure, NS[C.sub.S] and NS[C.sub.L] represent the no-shirking conditions for short- and long-hour workers, respectively. Firms offer a wage-hours package, ([w.sub.S], [N.sub.S]), that attracts short-hour workers and meets their no-shirking constraint. Also offered is a package, ([w.sub.L], [N.sub.L]), with hours and wages that are as attractive to long-hour workers as the short-hour package.(15)

An important feature of the separating contract is that the labor cost for a firm adopting the separating equilibrium, [C.sup.s]([[Theta].sub.S]), depends crucially on the mix of the two types of workers. As the proportion of type S workers ([Theta]) approaches zero, the labor cost approaches

(16) [C.sup.s](0) = e + 2[[Gamma].sup.1/2],

which, it can be shown, is simply the minimum no-shirking wage for the long-hour workers, [w.sub.0]. Unit labor cost is strictly increasing in [[Theta].sub.S], and as [[Theta].sub.S] approaches 1, labor cost converges to the minimum wage that solves the no-shirking constraint for short-hour workers,

(17) [C.sup.s](1) = e + 2[([[Mu].sub.S][Gamma]).sup.1/2].

The Provision of Short-Hour Jobs

Our central concern in this paper is analyzing the response of labor markets to an influx of workers desiring short hours. Beginning with a labor market composed entirely of long-hour workers, we explore how the wage-hours packages offered by firms change as we increase the number of short-hour workers in the population. Our focus will be on steady states.

As outlined above, all firms must choose between offering pooling contracts in which some fraction of employees will shirk, or separating contracts in which all workers are given wage-hours packages that induce no-shirking. In deciding which strategy to pursue, firms will compare the costs of pooling and separating contracts. The cost of each contract for any firm depends in turn on the composition of workers in that firm's job queue.

Recall that we have assumed that workers who enter the labor force form queues at each firm, and when hiring, a firm selects at random workers from its own queue. A worker cannot at any time be in more than one queue. We assume that workers choose which queue to join based on the contracts named by the prospective employers. More precisely, under our assumption that individuals maximize expected utility, a worker's decision to join a job queue is based upon the expected utility of the job and the probability of being hired into the job out of the queue. This latter probability is determined by the number of vacancies to be filled and the number of workers in the queue.

With this in mind, consider an initial equilibrium in which all firms are offering the pooling contracts, i.e., offering exclusively jobs with wages and hours designed to prevent shirking among long-hour workers. In response to an influx of short-hour workers, suppose a firm is considering a unilateral switch from offering the pooling labor contract to a separating labor contract, i.e., a contract that also accommodates short-hour workers. As we have seen, the cost of labor to a firm offering a separating contract will hinge on the mix of workers the firm attracts, [[Theta].sub.S]. In turn, the proportion of short-hour workers in any firm depends on the actions of other firms in the market. In a market in which all firms are offering only long-hour jobs, a firm that deviates from the market by offering a mix of short- and long-hours jobs will generally attract a disproportionate number of workers who prefer short-hour work. Indeed, such a firm may well attract only short-hour workers.

The process we describe can easily lead to a sub-optimal market equilibrium. In particular, an equilibrium can persist in which individually optimizing firms offer pooling contracts, even though a switch by all firms to separating contracts would (i) reduce unit labor cost and thus lead to higher labor utilization, and (ii) increase wages and utility of employment for all workers.

To establish this result, we examine the decision of the first firm considering the shift from a pooling contract to a separating contract. The firm will pursue this option only if the shift reduces unit labor cost, i.e., if [C.sup.s]([[Theta].sub.S]) [less than] [C.sup.p]([Theta]). It is clear that a firm offering a separating contract will attract a disproportionately high fraction of short-hour workers to its queue. For simplicity we proceed with the case in which a firm offering a separating contract attracts exclusively short-hour workers.(16) The firm will prefer the separating contract if

(18) [C.sup.s](1) [less than] [C.sup.p]([Theta]).

Using (10) and (17), expression (18) can be rewritten to show that the firm will offer the separating contract only when [Theta] is such that

(19) e + 2[([[Mu].sub.S][Gamma]).sup.1/2] [less than] (e + 2[[Gamma].sup.1/2])

[{1 - q[Theta]/[q + (1 - [Theta])(1 - q)D]}.sup.-1].

The right-hand side of (19), the unit cost of offering a pooling contract, is increasing in [Theta] and approaches infinity as [Theta] tends toward one. Thus, when the workers who prefer short hours constitute a large enough group in the labor force, the firm will always find it profitable to offer short-hour positions if all other firms are offering long-hour jobs. On the other hand, if the fraction of type S workers in the population is small enough (i.e., [Theta] close to zero), inequality (19) cannot hold. In this case, no firm will be inclined to deviate from the norm of offering only long-hour positions.

Notice, however, that if all firms in the market were to switch from offering pooling to separating contracts, each would attract the population proportion of short-hour workers to its queue. Such a switch would reduce labor cost if [Theta] were such that

(20) [C.sup.s]([Theta]) [less than] [C.sup.p]([Theta]),

or, using (10) and (14),

e + [{2[Gamma][[[Mu].sub.S] - (1 - [Theta])]/[[Theta] + (1 - [Theta])([[Mu].sub.S] - 1)]}.sup.1/2] [less than]

(21) (e + 2[[Gamma].sup.1/2])[{1 - q[Theta]/[q + (1 - [Theta])(1 - q)D]}.sup.-1].

The term in brackets on the left-hand side of (21) is always less than [[Mu].sub.S]. Thus by comparing inequalities (19) and (21), we observe a key feature of our model - that there is a range for the value of [Theta] for which unit labor cost declines when all firms switch to separating contracts, but for which any individually maximizing firm nonetheless continues to offer the pooling contracts. That is, there is a range of values of [Theta] for which

(22) [C.sup.s]([Theta]) [less than] [C.sup.p]([Theta]) [less than] [C.sup.s](1).

In particular, let [Theta][prime] be the value of [Theta] such that (21) holds with equality. When [Theta] exceeds [Theta][prime], labor cost will be lower if all firms offer separating contracts than if all firms offer pooling contracts. However, no firm will have the incentive to offer a separating contract unless [Theta] exceeds [Theta][double prime], where [Theta][double prime] is the value of [Theta] that solves (18) with equality. For any mix of workers such that (22) holds, unit labor cost will decline and output will increase if firms collectively adopt separating contracts. Nonetheless, no one firm finds it advantageous to abandon its practice of offering pooling contracts. As compared with the pooling equilibrium, the separating equilibrium is characterized by lower labor cost, even though the wages received by both types of workers are higher. In the separating equilibrium, the quantity of labor utilized and market output are higher. Both short-hour and long-hour workers gain higher utility from employment in the separating equilibrium than in the pooling equilibrium.

It is tempting to suggest that the coordination problem highlighted above might be circumvented if, by collective agreement, all firms were to agree to offer separating employment contracts. However, each firm will have an incentive to defect from this agreement by offering a pooling contract. For instance, a defecting firm could restrict its offers to the long-hour lower-wage positions offered by other firms. This firm would thus attract a disproportionately large number of relatively inexpensive long-hour workers. If other firms were to follow suit, the separating equilibrium would unravel and the market would once again be characterized by a sub-optimal pooling equilibrium.

Of course when there are a very large number of short-hour workers in the economy, so that [Theta] exceeds [Theta][double prime], firms will no longer offer exclusively pooling contracts; if they were all offering such contracts any single firm could reduce labor cost (from [C.sup.p]([Theta]) to [C.sup.s](1)) by offering a separating contract. Nonetheless, it is unlikely that an equilibrium can prevail in which all firms offer separating contracts, because, as we argue in the preceding paragraph, in this latter equilibrium any firm can always profit by restricting its offers to exclusively long-hour jobs which would then attract the relatively lower-cost long-hour jobs. Thus, even when [Theta] is large - when there are many short-hour workers in the economy - the market will not be characterized by the simple separating equilibrium which would minimize labor cost.(17)

An important implication of our model is that extra-market intervention can in principle improve welfare in this labor market. As an intellectual exercise, consider, for instance, a law that eliminated the pooling equilibrium by mandating that (i) all employees must be offered the option of working at short or long hours, and (ii) dismissals must be for just cause. If the dismissal of short-hour workers in high proportion were taken as evidence of violation of the law, firms would be induced to offer short-hour jobs by providing separating contracts. As we have demonstrated, it is possible that such a law will result in lower labor cost to firms and higher wages to workers. While it may seem paradoxical that by restricting the actions of parties to an exchange one could improve the welfare of the parties, this conclusion has been reached in a number of other models where information is imperfect.(18)

IV. RECONCILING THE THEORETICAL MODEL WITH THE EVIDENCE

The central conclusions of the preceding sections can be briefly summarized. Work incentives of the sort described by efficiency wage models can produce labor market equilibria in which labor markets will not clear and employers will use desired work hours as an indicator for screening potential employees. Under these conditions we derive three propositions:

1. Job offers will specify both wages and work hours, and many individuals will not be able to work the number of hours they perceive to be optimal given their wage.

2. Adverse selection problems may lead employers to offer an insufficient number of short-hour jobs.

3. As the number of individuals who would prefer short-hour jobs increase, markets will be inefficiently slow in providing increasing numbers of short-hour jobs.

We make no effort here to provide an empirical test of these propositions. Our goal in the following discussion is instead to make a case for the empirical plausibility of the three propositions and to describe what we see as promising future directions for empirical research. In doing so, we hope to encourage more empirical and theoretical investigations of the issues raised in this paper.

Evidence Concerning Proposition 1

A series of papers by Altonji and Paxson offer strong empirical evidence that work hours are tied to jobs. In their first paper, Altonji and Paxson [1986] use longitudinal data to study the variance of the change in work hours over time. They find that the variance of the change in hours is much larger for job changers than job stayers. This result suggests that, in many cases, hours are tied to job offers so that changing hours requires a change in jobs. In a related longitudinal study of married women, Altonji and Paxson [1988] present additional evidence that hours are tied to jobs. They find that changes in variables related to labor supply (e.g., the birth of a new child) lead to bigger hours adjustments for women who change jobs than for women who remain at the same job. This result holds even after controls for individual quit propensities are introduced into the analysis.

The finding that hours are tied to jobs would be of little consequence if employees could easily find optimum work hours by switching jobs. To investigate the role of job mobility in hours adjustment, Altonji and Paxson [1986] compare the variance of work hours for job quitters and those who experience layoffs. Since quits are voluntary, the group of job quitters should include some people who voluntarily leave their old job in order to accommodate a change in hours preferences. Layoffs, in contrast, are involuntary separations and most laid-off workers should return to work at jobs offering their old, and still optimal, level of work hours. Thus, if workers can easily adjust hours by changing jobs, quitters should have a larger variance of the change in work hours than those who are laid off.(19) Contrary to this prediction, Altonji and Paxson observe that the variance of the change in work hours is at least as great among those who are laid off as among those who quit.

The proposition that individuals work the utility-maximizing number of hours (either within jobs or by moving across jobs) can be tested directly by examining an individual's stated hours preferences given the current structure of their compensation. If employees are working the utility-maximizing number of hours then, conditional on their current hourly compensation, they should not want to adjust their work hours if they are given the opportunity. A number of microeconomic studies have found that substantial proportions of the work force are, by this criterion, not working the optimal number of work hours (see Kahn and Lang [1988; 1991; 1992]; Best [1978], and Shank [1986]). In a study of Canadian employees, based on responses to questions that were especially carefully designed to examine the hours preferences of individuals, it was found that 48 percent of respondents were working "too few" hours, 16 percent were working "too many" hours and 36 percent were working optimal hours (see Kahn and Lang [1991]). Kahn and Lang [1992], using data from the Panel Study of Income Dynamics (PSID) and the Current Population Survey, analyzed the hours preferences of non-self employed, male heads of households, aged twenty-five to fifty-four. They found that between 35 and 41 percent of the workers in their samples would prefer to work more if more work were available. On the other hand, between 4 and 8 percent of their sample wanted to work less.(20)

At first glance it might seem that the high incidence of reported underwork relative to overwork is evidence against our prediction of a shortage of short-hour jobs. Our model, however, makes no clear prediction about which type of hours constraints will predominate.

In the case of a homogeneous work force, described in section II, we demonstrate that the incidence of overwork depends on how the probability of detection and dismissal (D) and the disutility of not shirking (E) respond to changes in work hours. Only by making assumptions$about the shape of the D and E functions can we make clear predictions about the types of hours constraints appearing in the homogenous model. For example, if D and E are both linear functions of work hours, then the wage-hours packages firms offer result in employees working the utility-maximizing number of hours.

In the heterogeneous case, discussed in section III, workers may perceive themselves to be underworked or overworked even when D and E are linear. In this case the pooling contract will specify wages and hours so that w - [e.sub.n] - [[Mu].sub.L]g[prime] = 0 for long-hour workers. (This means that the marginal utility of work just equals the marginal utility of income for non-shirking long-hour workers.) Short-hour workers will shirk under the pooling equilibrium, and given that they are shirking, the marginal effect on utility of a small change in work hours will be w - [e.sub.s] - [[Mu].sub.S]g[prime], which can be positive or negative; we cannot say whether these workers would prefer to work more or fewer hours at the wage offered by firms. Our model therefore makes no clear prediction how short-hour workers will respond to a question about hours constraints.

Hours constraints can be generated by a number of models with quite different predictions than the model developed in this paper (for discussion see Kahn and Lang [1992]). The more distinctive claim of our model concerns propositions 2 and 3.

Evidence Concerning Proposition 2

Proposition 2 concerns the underprovision of short-hour jobs as a result of adverse selection. It is not possible to directly observe the absence of "enough" short-hour jobs. To make the case for the plausibility of our arguments, we appeal to evidence about a similar case in which adverse selection can play a key role, the insurance coverage for childbirth.

Gruber [1992] analyzes the response of labor markets to the imposition of state laws mandating comprehensive coverage for childbirth in health insurance policies. Prior to the mid-1970s health insurance policies commonly offered little or no coverage for childbirth. Given this backdrop, Gruber studies the effect of the imposition of laws prohibiting insurance policies from treating pregnancy differently than "comparable illnesses." In the standard model the result of these laws should be a reduction in wages and employment.(21) If, however, adverse selection prevented optimal provision of maternity benefits in the unregulated markets, there need not be any reduction in employment resulting from passage of laws mandating provision of benefits (for a discussion of this point see Summers [1989]). Consistent with the predictions of the adverse-selection model, Gruber finds that the total employment of women "at risk" of having children was not changed by the passage of laws mandating the provision of health insurance benefits.

We argue that an analogous mechanism may be at work in the provision of short-hour jobs. That is, for good profit-maximizing reasons in the face of imperfect information, the labor market does not offer jobs that it "should." In our case the result is too few short-hour jobs. It is worth noting, moreover, that if our argument is correct, adverse selection would induce firms to limit the provision of any benefit that is likely to be valued more by short-hour than long-hour workers - including perhaps insurance coverage for child bearing.

Evidence Concerning Proposition 3

Aggregate studies of the determination of work hours indicate that median work hours have remained roughly constant for men since 1940. Recent disaggregated analysis, however, suggests a different pattern - one that offers some indirect support for proposition 3.

Before turning to this evidence we find it useful to generalize our model by introducing a modification mentioned briefly in the introduction - that there are two different labor markets, primary and secondary. The primary labor market is composed of firms making use of the incentive schemes described in the preceding sections. Firms in the secondary labor market use technology that makes it easy to observe the work activities of employees. It follows then that secondary labor markets pay a market-clearing wage. Workers who are not fortunate enough to get selected out of the queue for primary jobs, and workers who are dismissed from primary jobs, can always secure employment in the lower-paying secondary sector.(22)

In the dual labor market context, our model predicts that the shortage of short-hour jobs is most likely to occur in primary labor markets. A reasonable first step in investigating proposition 3 is to examine the historical trend in work hours in primary versus secondary jobs.

A recent study by Coleman and Pencavel [1993a, Table 14, 280] found that white men with sixteen or more years of schooling (a reasonable proxy for primary employment) increased both their weekly and annual work hours considerably between 1940 and 1988. In contrast, the weekly and annual work hours of white men with a high-school education or less followed a downward trend over this same period.(23) In a separate study of female work hours, Coleman and Pencavel [1993b] also find that weekly and annual hours have increased for highly educated women between 1940 and 1988, but have decreased for less-educated women. This pattern is broadly consistent with a model in which there is a general increase in the number of workers preferring shorter hours, but in which primary labor markets are slow to adjust.

Alternative explanations for Coleman and Pencavel's results are, of course, possible, and we make no attempt to dismiss them here. Our claim is simply that proposition 3 is plausibly consistent with the historical record.

Turning to the issue of part-time work, we note that prior to 1950 part-time work was virtually non-existent in the United States economy. During the 1950s employers began offering jobs having less than thirty-five hours per week as a means of attracting older, married women into the labor force (Goldin [1990, 180-83]). Over the past twenty years, the percentage of women working part-time has remained roughly constant at about 25 percent (Blank [1990]). Considering the rapid growth of the female labor force, this figure suggests a significant rate of growth of part-time employment.

Can the rapid growth of part-time work be reconciled with our claim that work incentives inhibit the introduction of short-hour jobs? To answer this question, we return to the dual labor market model discussed above. In this model, wages in the secondary labor market are determined competitively; firms do not pay efficiency wages. Employers in the secondary market will therefore allow employees to choose the hours of work that maximize utility. This stands in contrast to the primary labor market where firms offer the minimum-cost wage and hours package that satisfies the no-shirking condition. On the basis of our model, we would expect that most part-time jobs are found in the secondary sector.

Consistent with the hypothesis that part-time jobs are generally found in the secondary labor market, Rebitzer and Taylor [1991] find that industries with high concentrations of part-time workers tend to be low-wage industries. In a study of labor market segmentation among nonunion men, Rebitzer and Robinson [1991] find that 4.9 percent of the men in the primary sector work part time. In contrast, 31 percent of the men in the secondary sector are part-timers. Similarly, Blank [1990] offers an extensive analysis of wage differentials between part-time and full-time workers. The data she presents indicate that 71 percent of part-time workers are found in the generally low-wage sales, clerical and service occupations, and only 22 percent are found in the relatively high-wage professional, managerial and technical occupations (Blank [1990, 129]). The comparable figures for women generally are 30 percent and 57.2 percent respectively.

Of course, none of this evidence tests directly the behavioral mechanisms posited in our theory. A direction for future research is empirical examination of the behavior that forms the basis of our model of hours determination. In particular, it might prove useful to study two fundamental propositions in our model that are, in principle, subject to direct empirical testing. First, employers use work hours (and possibly other attributes that are linked to work hours) as an indicator of a desirable and hard-to-observe attribute of employees. Second, awareness of the signaling value of work hours influences the process by which work hours are determined. Evidence favoring these propositions will not in and of itself constitute proof that labor markets provide an insufficient number of short-hour jobs. Such evidence would, however, give added weight to the possibility of market failures of the sort we describe in this paper.

Empirical investigation of these behavioral propositions requires the development of data sources that are quite different from the microeconomic data sets typically used by labor economists. In addition to information on work hours and compensation, it would be necessary to collect detailed information about the incentive systems used in organizations, managers' perception of the value of work hours as indicators, employee beliefs about managers' perceptions of work hours, and employee preferences regarding trade-offs of income for non-work time. In order for data of this sort to be meaningful, empirical research will necessarily have to focus on narrowly defined organizations and labor markets.

As a first step in investigating the behavioral assumptions underpinning our model, we are currently studying data we have collected from associates and partners working in large corporate law firms.(24) Law firms are a convenient vehicle for such an investigation because they have a simple and uniform incentive scheme that applies to nearly all the associates in the firm. In addition, the legal profession has experienced an unprecedented influx of women (and men married to women with careers) while work hours have remained relatively fixed at high levels (Rosen [1992]).

V. CONCLUSION

Firms look for workers whose attitudes and preferences make them responsive to the work incentives prevalent in the firm. These incentives often involve promises to provide (and threats not to provide) income in the future. Thus, preferences towards income and leisure will be important to firms in deciding whom to hire.

In this paper we have presented a model of wage and hours determination in which firms use dismissal threats to elicit high levels of work effort. In our model workers who prefer long hours will be more responsive to dismissal-based incentive schemes than other workers. Whenever possible, employers will therefore put job seekers with preferences for short hours at the bottom of the queue of workers seeking jobs.

In order to avoid unemployment or employment in low-wage positions, job seekers will not reveal their true preferences for income and leisure. We demonstrate that the response by employers may result in the provision of fewer short-hour jobs than is optimal. In the context of a model of labor market segmentation, the shortage of short-hour jobs will occur in the high-wage, primary sector.

The logic of the model we present suggests that labor markets will not adjust smoothly to the changes brought about by the rise in female labor force participation. In the absence of some intervention in the market, firms will find it difficult to provide the optimum number of short-hour jobs in response to the increasing numbers of female and male workers seeking to balance job and family responsibilities.

1. In 1950 the labor force participation rate of married women was 29.5 percent compared to 51.1 percent in 1980 (Goldin [1990, 17]). The 1950 figure includes women over age fourteen, while the 1980 figure includes only women over fifteen years old. In 1970 the labor force participation rate of women with children under eighteen was 42.1 percent. By 1985 this figure was 62.1 percent (Bergmann [1986, 2]).

2. Fuchs [1986] estimates that in 1979 women spent 1,497 hours per year on non-market work compared to 595 hours for men. Non-market work includes activities like shopping, yard work and child care. Leete-Guy and Schor [1990] estimate that on average women in 1979 spent 1514 hours per year on non-market work. This figure declined to 1442 in 1987. The comparable figures for men were 860 and 853 respectively (Leete-Guy and Schor [1990, Table 2]).

3. A number of previous studies have concluded that work incentives may influence the determination of hours of work. Lang [1989] and Lazear [1981] argue that work incentives may cause employers to offer wage-hours packages under which employees do not perceive themselves to be working optimal hours. Bulow and Summers [1986] argue that reliance upon work incentives may cause employers to seek to avoid hiring employees having preferences for short hours. Bulow and Summers' consideration of heterogeneous workers focuses primarily on the case where different hours preferences (and turnover propensities) are known to the employer because they vary by an observable characteristic, gender. For a discussion of the effects of worker heterogeneity on unemployment and wages, see Weiss [1990].

4. In an earlier version of this paper, we set up our model using an alternative assumption that workers who quit or are fired may subsequently re-enter the labor force. This alternative set-up leads to similar results as those presented here, but the derivations are much more cumbersome.

5. We assume that firms always correctly identify shirkers. It is straightforward, however, to introduce erroneous dismissals into the model. For a discussion of the implications that erroneous dismissals have on labor market outcomes see Levine [1989].

6. For a discussion of these issues in a closely related model see Lazear [1981] and Akerlof and Katz [1990].

7. To highlight the central point of the model we do not allow workers to post performance bonds. (Dickens, Katz, Lang and Summers [1989] and Ritter and Taylor [1994] provide discussions of this issue.) Further, we stipulate that workers in each period are paid prior to the observation of their work activities in the period. Thus the discipline effect of dismissal is derived entirely from lost future earnings.

8. The finding that labor market equilibria may be characterized by unemployment is common to many effort regulation models. See Bowles [1985] and Stiglitz [1987].

9. For example, [[Gamma].sub.N] will be equal to zero when both E and D are linear functions of N. In this case D[prime]/D = E[prime]/E = 1 / N, since D(0) = E(0) = 0. [[Gamma].sub.N] may also be zero if E[prime] = 0.

10. This is the case analyzed in Lang [1989]. Lang ensures that [[Gamma].sub.N] [greater than] 0 by assuming D to be concave and [e.sub.s] to be zero. In theory, it is also possible for [[Gamma].sub.N] to be negative. In this situation workers will be required to work more than the optimal number of hours. As we discuss in section IV, however, the literature indicates that the incidence of "excess hours" is small compared to the number of workers reporting they worked optimal hours or wished to work more hours.

11. Bulow and Summers [1986] express this point vividly by noting, "Firms prefer to give jobs to workers who 'really need them' than to workers who gain less surplus from holding them" (p. 400).

12. In order to eliminate short-hour workers from their job queues, firms may discriminate in hiring against groups known to have, on average, a preference for shorter hours. Such discrimination may be important in many labor markets. In this paper we abstract from statistical discrimination because it not essential to our argument.

13. The central point of our paper - that labor market processes may produce outcomes entailing a shortage of short-hour jobs - would only be strengthened if we were to assume that some firms do find it profitable to use screening contracts. Further, the use of screening contracts by some firms is a likely outcome in a more realistic setting in which firms differ with respect to monitoring difficulty and cost to the firm of shirking.

14. Note that because the no-shirking wage is lower for long-hour than for short-hour workers at any N, only the no-shirking condition for short-hour workers will be binding in this problem.

15. As shown in Figure 1, hours constraints can arise due to the nature of the separating contract. For instance, short-hour workers are asked to work fewer hours than they would choose given their wage, [w.sub.S]. This occurs because firms do not find it optimal to provide short-hour jobs lying at the minimum point of these workers' no-shirking curve, NS[C.sub.S]. In comparison with this minimum point, a move by the firm to the left along the no-shirking curve has only a small adverse effect on the wage, [w.sub.S]. This effect is more than offset by the reduction in cost associated with the decline in the use of the relatively expensive type S workers and the drop in the wage that must be paid type L workers.

16. All we need to establish the results that follow is that there be some adverse selection at work, i.e., that the proportion of type S workers the firm attracts be greater than the population [Theta]; the assumption that [[Theta].sub.S] = 1 makes for clearer exposition.

17. We make no effort to formally describe what equilibrium would look like for this extreme case, but it is clear that it would necessarily be characterized by a mix of firm strategies, with many firms continuing to offer contracts with the idea of excluding short-hour workers.

18. See, for example, Levine [1990] and Aghion and Hermalin [1990].

19. This of course assumes that the probability of layoff is unrelated to heterogeneity across individuals in the change in desired hours.

20. The reliability of the PSID data is supported by the results of Altonji and Paxson [1988]. They examined whether "hours constraints" and "overwork" on the initial job affects the relations between hours changes and wage changes for quitters. Job changers who were initially hours constrained required an additional wage increase to go to a new job at which they worked fewer hours. Conversely, job changers who were initially overworked required an additional wage increase to induce them to accept a job entailing still longer hours.

21. This assumes, of course, that the labor supply curve is not perfectly inelastic.

22. This dual labor market model is essentially the same as that presented in Bulow and Summers [1986]. The theory of dual labor markets has generated a large body of qualitative and quantitative research. For surveys see Rebitzer [1989] and Dickens and Lang [1988].

23. These results were found in an equation that controls for changes in GNP, cohort size, experience and real hourly earnings.

24. See Landers, Rebitzer and Taylor [1994].

REFERENCES

Aghion, Philippe, and Benjamin Hermalin. "Legal Restrictions on Private Contracts Can Enhance Efficiency." Journal of Law, Economics and Organization, Fall 1990, 381-410.

Akerlof, George A., and Lawrence F. Katz. "Workers' Trust Funds and the Logic of Wage Profiles." Quarterly Journal of Economics, August 1989, 525-36.

Altonji, Joseph G., and Christina H. Paxson. "Labor Supply Preferences, Hours Constraints, and Hours-Wage Trade-offs." Journal of Labor Economics, April 1988, 229-53.

-----. "Job Characteristics and Hours of Work," in Research in Labor Economics, edited by Ronald G. Ehrenberg. JAI Press, 1986, part A, 1-55.

Bergmann, Barbara. The Economic Emergence of Women. New York: Basic Books, 1986.

Best, Fred. "Preferences on Worklife Scheduling and Work-Leisure Tradeoffs." Monthly Labor Review, June 1978, 31-37.

Blank, Rebecca. "Are Part-Time Jobs Bad Jobs?" in A Future of Lousy Jobs? edited by Gary Burtless. Washington, D.C.: Brookings Institution, 1990.

Bowles, Samuel. "The Production Process in a Competitive Economy: Walrasian, Neo-Hobbesian, and Marxian Models." American Economic Review, March 1985, 16-36.

Bulow, Jeremy I., and Lawrence H. Summers. "A Theory of Dual Labor Markets with Application to Industrial Policy, Discrimination, and Keynesian Unemployment." Journal of Labor Economics, July 1986, part 1, 376-414.

Coleman, Mary T., and John Pencavel. "Trends in the Market Work Behavior of Women Since 1940." Industrial and Labor Relations Review, July 1993a, 653-76.

-----. "Changes in the Work Hours of Male Employees Since 1940." Industrial and Labor Relations Review, January 1993b, 653-76.

Dickens, William T., Lawrence Katz, Kevin Lang, and Lawrence Summers. "Employee Crime and the Monitoring Puzzle." Journal of Labor Economics, July 1989, 331-47.

Dickens, William, and Kevin Lang. "Neo-Classical and Sociological Perspectives on Segmented Labor Markets," in Industries, Firms and Jobs: Sociological and Economic Approaches, edited by George Farkas and Paula England. New York: Plenum Press, 1988.

Dickens, William T., and Shelly J. Lundberg. "Hours Restrictions and Labor Supply." NBER Working Paper No. 1638, 1985.

Fuchs, Victor. "His and Hers: Gender Differences in Work and Income, 1969-1979." Journal of Labor Economics, July 1986, pt. 2, S245-S272.

Goldin, Claudia. Understanding the Gender Gap: An Economic History of American Women. New York: Oxford University Press, 1990.

Gruber, Jonathan. "The Efficiency of a Group-Specific Mandated Benefit: Evidence from Health Insurance Benefits for Maternity." NBER Working Paper No. 4157, 1992.

Kahn, Shulamit, and Kevin Lang. "The Causes of Hours Constraints: Evidence from Canada." Working Paper, 1988.

-----. "The Effects of Hours Constraints on Labor Supply Estimates." Review of Economics and Statistics, November 1991, 605-11.

-----. "Constraints on the Choice of Work Hours: Agency versus Specific-Capital." Journal of Human Resources, Fall 1992, 661-78.

Landers, Renee, James Rebitzer, and Lowell Taylor. "Rat Race Redux: Adverse Selection in the Determination of Work Hours." Working Paper, Massachusetts Institute of Technology, 1994.

Lang, Kevin. "Why Was There Mandatory Retirement?" Journal of Public Economics, June 1989, 127-36.

Lazear, Edward P. "Agency, Earnings Profiles, Productivity, and Hours Restrictions." American Economic Review, September 1981, 606-20.

Leet-Guy, Laura, and Juliet Schor. "Assessing the Time Squeeze Hypothesis: Estimates of Market and Non-Market Hours in the United States, 1969-1987." Working Paper, Harvard Department of Economics, 1990.

Levine, David I. "Just-Cause Employment Policies When Unemployment is a Worker Discipline Device." American Economic Review, September 1989, 902-05.

-----. "Just-Cause Employment Policies in the Presence of Worker Adverse Selection." Working Paper, University of California, 1990.

Rebitzer, James. "Efficiency Wages and Implicit Contracts: An Institutional Evaluation," in Microeconomic Issues in Labor Economics, edited by Robert Drago and Richard Perlman. Sussex, U.K.: Wheatsheaf Books, 1989.

Rebitzer, James, and Michael Robinson. "Plant Size and Dual Labor Markets." Review of Economics and Statistics, November 1991, 710-15.

Rebitzer, James, and Lowell Taylor. "Work Incentives and the Demand for Primary and Contingent Labor." National Bureau of Economic Research Working Paper No. 3647, 1991.

Ritter, Joseph, and Lowell Taylor. "Workers as Creditors: Performance Bonds and Efficiency Wages." American Economic Review, June 1994, 694-704.

Rosen, Sherwin. "The Market For Lawyers." Journal of Law and Economics, October 1992, 215-46.

Schor, Juliet B. The Overworked American: The Unexpected Decline of Leisure. New York: Basic Books, 1991.

Shank, Susan E. "Preferred Hours of Work and Corresponding Earnings." Monthly Labor Review, November 1986, 40-44.

Shapiro, Carl, and Joseph E. Stiglitz. "Equilibrium Unemployment as a Worker Discipline Device." American Economic Review, June 1984, 433-44.

Stiglitz, Joseph E. "The Causes and Consequences of the Dependence of Quality on Price." Journal of Economic Literature, March 1987, 1-48.

Summers, Lawrence. "Some Simple Economics of Mandated Benefits." American Economic Review Papers and Proceedings, May 1989, 177-83.

Weiss, Andrew. Efficiency Wages: Models of Unemployment, Layoffs, and Wage Dispersion. Princeton, N.J.: Princeton University Press, 1990.

Lowell J. Taylor is Associate Professor, Sloan School of Management, Massachusetts Institute of Technology, and Assistant Professor, H. John Heinz III School of Public Policy and Management, Carnegie Mellon University. Helpful comments from two referees are gratefully acknowledged.