Romer D. Advanced Macroeconomics

9.4 The Shapiro-Stiglitz Model 423

workers have been in their jobs. Processes like (9.24)are known as Poisson processes.

An equivalent way to describe the process of ,job breakup is lo say that it occurs with probability b per unit time, or to say that the hazard rare for Job breakup is b. That is, the probahhty thar an employed worker's job ends in the next dt units of time approaches bdt as dt approaches 0. To see thar our assumptions imply this, note that (9.24)implies P'(t) = -bP(r).

The second assumption concerning workers' transitions between states is that firms' detcction of workers'who are shirking is also a Poisson process. Specifically, detection occurs with probability q per unit time. q is exogenous, and detection is independent of job breakups. Workers who are caught shirking are fired. Thus if a worker is employed but s h i r h g , the probability that he or she is still employed time T later is e-qT (the probability that the workcr has not been caught and fired) times e (the probability that the job has not ended exogenously).

Third, unemployed workers find employmcnt at rate a per unit time. Each workcr takes a as given. In the economy as a whole, however, a is determined endogenously. When firms want to hire workers, they choose workers at random out of the pool of unemployed workers. Thus u is determined by the rate at which firms are hiring (which is determined by the number of employed workers and the rate at whlch jobs end) and the number of unemployed workers. Because workers are identical, the probability of finding a job does not depend on how workers become unemployed or on how long they are unemployed.

Firms' behavior is straightforward. A firm's profits at t are

where L is thc number of employees who are exerting effort and S is the number who are shirking. The problem facing the firm is to set w sufficiently high that its workers do not shirk, and to choose L. Because the firm's decisions at any date affectprofits only at that date, there is no need to analyze the present value of profits: the firm chooses w and L at each moment to maximize the instantaneous flow of profits.

The final assumption of the model is Z F ' ( Z ~ / N )> Z, or F'(i?r/N) > 1. This condition states that if each firm hires 1 / N of the labor force, the marginal product of labor exceeds the cost of exerting effort. Thus in the absence of imperfect monitoring, there is full employmcnt.

The Values of E , U , and S

Let V, denote the "value" of being in state i (for i = E, S,and U ) . That is. Viis the expected value of discounted lifetime utility from the present moment forward of a worker who is in state i . Because transitions among states are Poisson processes, the Vi's do not depend on how long the worker has been

424 C h a ~ t e 9r UNEMPLOYMENT

in his or her current state or on his or her prior history. And because WE are focusing on steady states, the V,'s are constant over rime.

To find VL,Vs, and Vr,, it is not necessary to analyze the various paths the worker may follow over the infinite future. Inslead we can use dynamic programming. The central idea of dynamic programming is to look at on11 z brief interval of time and use the LrZ'sthcmselvcs to summari~ewhat occurs after the cnd of the inter\,al.1° Consider first a worker who is employ-ed and exerting effort at time 0. Suppose temporarily that time is divided into inten7alsof length At, and that a worker who loses his or her job during one interval cannot begin to look for a new-job until the beglnning of the next interval. Let V , (At)and V,;(At) denote the values of employment and unemplo~menras of the beginning of an interval under this assumption. Ir. a moment we will let At approach 0. When we do this, the constraint that a worker who loses his or her job during an intcrval cannot find a new job during the remainder of that interval becomes irrelevant. Thus V,(At) will approach VE.

If a worker is employcd in a job paying a wage of w , CJL (At) is given b!-

The first term of (9.26)reflects utility during the interval (0,At). The prob~ ability that the worker is still employed at time t is e-h'. If the worker i% employed, flow utilit)~is w - Z.Discounting this back to time 0 yields an expected contribution to lifetime utility or e-'"bb" ( w - Z).ll

The second term of (9.26) reflects utility after At. At time At, the worker is employed with probability chAt,and is unemployed wjith probability 1 - e-har. Combining these probabilities with the V's and discounting yields the second lerm.

If we compute the integral in (9.261,wc can rewrite the equation as

+ e-parle-b~tvt ( a t ) + (1 - e ~ ~ ~ ~ ) v , ~ ( a t ) ] .

Solving this expression for Vf(At) gives

As described above, VE equals the limit of VL(At)as At approaches 0. (Sim-

Because of the steady-state assumption, if it is optimal fur ihr wnrkrr to exert effort initially, it continues to he optimal. Thus rve du nu1 habe lo allow for r h r posslbilif).of the worker beglnning to shirk.

9.4 The Shapiro-Stiglitz Model 425

wc apply I'HApnal's rulr to i9.78). This yields

Equation (9.29) can also be derived intuitively. Think of an asset that pays dividends at rate w Z per unit time when the worker is employcd and no dividends whcn the worker is unemployed; in addition, assume that the asset is being priced by risk-neutral investors with required rate of return p. Since the expected present valueuf lifetime dividends of this asset is the same as rhe worker's expccted present value of lifetime utility, the asset's price must be V Fwhcn the worker is employed and Vr, when the worker is unemployed. For the asset to be held, it must provide an expccted rate of return of p. That is, its dividends per unit time, plus any expected capital gains or losses per unit time, must equal ~ V EWhcn. the worker is employed, dividends per unit time are MI Z , and there is a probability b per unit time of a capital loss of VE- VLIThus,

Rearranging this expression yiclds (9.29).

If the worker is shirking, the "dividcnd is w per unit time, and thc expected capital loss is (h + q ) ( V s - VU) per unit time. Thus reasoning parallel to that used to derive (9.30)implies

Finally, if the worker is unemployed, the dividend is 0 and the expccted capital gain (assuming that firms pay sufficientlyhigh wages that employed workers excrt effort) is a(V E- V1,)per unit time.12 Thus,

T h e No-Shirking Condition

The firm must pay enough that VE t VS; otherwise its workers exert no effort and produce nothing. At the same time, since effort cannot exceed P , there is no need lo pay any excess OT-erthc minimum uecdcd to induce effort. Thus the firm chooses w so that \'E just equals Vs:'"

Since V Fand VS must be equal, (9.30)and (9.31)imply

''Equations (9.31)and (9.32) can also be d i r i t c d by d e h g Vc(At l and V5(A11 and ~roceedingalong the lines uscd to derive (Y . ?o s

l 3 Since all firms arc the same, lhcy choose :?c same wage. Thus Vi and Vs do not d e p ~ n don what firm a worker is employcd b!

426 Chapter 9 UNEMPLOYMENT

Equation (9.35)implies that firms set wages high enough that workers stricrly prefer employment to unemployment. Thus workers obtain rents. ?'hi size of the premium is increasing in the cost of exerting effort, Z,and d e creasing in firms' efficacy in detecting shirkers, q .

The next step is to findwhat the wage must be for the rent to employmento equal Z l q . Equations (9.30) and (9.32) imply

This expression implics that for VL- Vn to equal Z l q , the wage must be

This condition states that the wage needed to induce effort is increasing ir the cost of effort (f),the ease of finding jobs (a),the rate of job breakup ( b and the discount rate (p),and is decreasing in the probability that shirkcri are detected (q).

It turns out to be more convenient to express the wage needed to preven: shirking in terms of employment per rirm. L, rather than the rate at which the unemployed find jobs, a. To substitute for a , we use the fact that, since the economy is in steady state, movements into and out of unemploymen: must balance. The number of workers becoming unemployed per unit time is N (the number of rirms) times L (the number of workers per firm) timer b (the rate of job breakup).14 The number of unemployed workers finding ,jobs is 1- NL times a. Equating these two quantities yields

a = - NLb

L - N L '

Equation (9.38) implies a ib = ib/(i- NL). Substituting this into (9.3; yields

Equation (9.39) is the no-shirking condition. It shows, as a function of the level of employment, the wage that firms must pay to induce workcrs to exert effort. When more workers are employed, there are fewer unemployed workcrs and more workers leaving their Jobs; thus it is easier for unemployed workers to find employment. The wage necded to deter shirking is therefore an increasing function of employment. At full employment. unemployed workers [ind work instantly, and so there is no cost to being fircd and thus no wage that can deter shirking. The set of points in (NL,w) space satisfying the no-shirking condition (NSC) is shobn in Figure 9.2.

l4 W r are assuming that the economy is large enough that dthouph the breakup of any individual job is random, aggregate brrakups are not.

9.4 The Shapiro-Stiglitz Model 42 7

LL'

L NL

FIGURE 9.2 The Shapiro-Stiglitz model

Closing the Model

Firms hire workers up to the point where the marginal product of labor equals the wagc. Equation (9.25)implies that when its workers are exerting effort, a firm's flow profits arc F ( e L ) - wL. Thus the condition for the marginal product of labor to equal the wage is

The set of points satisfying (9.40) (which is simply a conventional labor demand curve) is also shown in Figure 9.2.

Labor supply is horizontal at e up to the number of workers, f;, and then vertical. In the absence of imperfect monitoring, equilibrium occurs at the intersection of labor demand and supply. Our assumption that the marginal product of labor at full emp1o)ment exceeds thc disutility of effort (F'(Zz/N) > 1)implies that this intersection occurs in the vertical part of the labor supply curve. The Walrasian equilibrium is shown as Point EWin the diagram.

With imperfect monitoring, equilibrium occurs at the intersection of the labor demand curve (equation L9.401) and the no-shirking condition (equalion [9.391). This is shown as Point E in the diagram. At the equilibrium. there is unemployment. Unemployed workers strictly prefer to be employed

428 Chapter 9 UNEMPLOYMENT

at the prevailing wage and to exert effort,rather than to remainunemploycd. Nonetheless, they cannot bid the wage down: firms know that if they hire additional workers at slightly less than the prevailing wage, the workers will prefer shirking to exerting effort. Thus the wage does not fall, and the unemployment remains.

Two examples may help to clarify the workings of the model. First, a rise in q-an increase in the probability per unit rime that a shirker is detectedshifts the no-shirking locus down and does not affect the labor demand curve. This is shown in F i y r e 9.3. Thus the wage falls and employmen! rises. As q approaches infinity, the probability that a shirker is detecled in any finite length of time approaches 1. As a result, the no-shirking wage approaches Z for any level of employment less than full employment. Thus the economy approaches the Walrasian equilibrium.

Second, if there is no turnover ( b = O), unemployed workers are ncvcr hired. A s a result, the no-shrking wage is indcpendcnt of the level of employment. From (9.3'31,the no-shirking wage in this case is Z + pZ/q. Intuitively, the gain from shirking relative to exerting effort is ?i per unit time. The cost is that thcrc is probability q per unit time of becoming permanentl) unemployed and thereby losing the discounted surplus from the job, which is ( w - Z ) / p . Equating the cost and benefit givcs w = e t pZ/q. This case is shown in Figurc 9.4.

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I. NI.

FIGURE 9.3 The effects of a rise in q in the Shapiro-Stiglitz model

9.4 The Shapiro-Stiglitz Model 429

," I

FIGURE 9.4 The Shapiro-Stiglitz model without turnover

Implications

The model implies the existence of equilibrium unemployment and suggests various factors that are likely to influence it. Thus the model has some promise as a candidate explanation of unemployment. Unfortunately, thc model is so stylized that it is difficult to determine what level of unemployment it predicts or to use it to dcrivc spccific predictions concerning the behavior of unemployment over time.

With regard to short-run fluctuations, consider the impact of a fall in labor demand, shown in Figure 9.5. w and L move down along thc no-shirking locus. Since labor supply is perfectly inelastic, employment necessarily responds more than it would nlthout imperfect monitoring. Thus the model suggests onc possible reason that wages may respond less to dcmand-drivcn output fluctuations than they would if workers were always on their iabor supply curves. Again, howevcr, the nlodcl is sufficiently stylized that it is difficult to gauge its quantitative implication^.'^

.-

'j This disruss~onneglects hvo important isiuc5. First. wc have brrn r o m ~ a r i n gsrrady states with different levels of labor demand rather than analyzing the dynamir rftects of

cun-e up, unemployment trends dobrn. One p r o r ~ c l ;route to eliminating this counterfac- tunl prediction is tu makr lhc cosl or tvrrlinq r ' -'T endogenous, and to structure the

430 Chapter 9 UNEMPLOYMENT

FIGURE 9.5 The effects of a fall in labor demand in the

Shapiro-Stiglitz model

Finally, thc model implies that the decentralized equilibrium is inefficient. '1'0 see this, note that since the marginal product of labor at fuL employment. Z F ' ( Z / N ) , exceeds the cost to workers of supplying effon Z, the first-best allocation is for everyone to be employed and exert effon Of course, the government cannot bring this about simply by dictating thav firms move down the labor demand curve until full employment is reached this policy causes workers to shirk, and thus results in zero output. Bu' Shapiro and Stiglitz note that rr7agesubsidies financed by lump-sum taxes or profits taxes improve welfare. Such a policy shifts the labor demand c u n t up, and thus increases the wage and employment along the no-shirking locus. Since the value of the additional output excecds the opportunity cost of producing it, overall welfare rises. How thc gain is divided between workers and firms depends on how the wage subsidies are financed.

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rnodrl so that V and uulpur per worker grow af the same rate i n the long run. This cause? the NSC c w e tu shift up at the same ratc as the labor demand curve in the long run, and thus eliminatt's rhc doanward trend in unemplu)mml. Rut ~ 7 t he endugcnous, one lhrc has tu reexamme the shurt-run rffects of a ski1 in labor demand accounting fur any cffects through e.

9.4 The Shapiro-Stiglitz Model 43 1

Extensions

'The basic model can be extended in many ways. Here we discuss three. First, an important question about the 1ahor.market is why, given that

unemp1o)ment appears so harmful to workers, employers use layoffs rather than M-ark-sharingarrangements when they reduce the amount of labor they use. One might cxpcct workers to place sufficient valuc on reducing the risk of unemployment that they would accept a lower wage to work at a firm that used work-sharing rather than layoffs. Shapiro and Stiglitz's model (modified so that the number of hours cmployees work can vary) suggests a possible explanation for the puzzling idrequency of work-sharing. A reduction in hours or work lowers the surplus that employees are getting from their jobs. As a result, the wage that the firm has to pay to prevcnt shirking rises. If the firm lays off some workers, on the other hand, the remaining workers' surplus is unchanged, and so no increase in the wage is needed. Thus the firm may find layoffs preferable to work-sharing even though it subjects its workers to greater risk.

Second, Bulow and Summers (1986) extend the model to include a second type of job where effort can be monitored perfectly. 'These jobs could be piece-rate jobs where output is observable, for example. Since there is no asymmetric information in this sector, the jobs p r o ~ ~ dnoe surplus and are not rationed. Under plausible assumptions, the absence of surplus results in high turnover. The jobs with imperfect monitoring continue to pay more than the market-clearing wage. Thus marginal products in these jobs are higher, and workers, once they obtain such jobs, are reluctant to leave them. If the model is extended further to include groups of workers with different job attachments (different b's), a higher wage is needed to induce effort from workers with less job attachment. As a result, firms with jobs that require monitoring arc reluctant to hire workers with low job attachment, and so these worlters are disproportionately employed in the low-wage, high-turnover sector. 'Thcsc predictions concerning wage levels, turnover, and occupational segregation fit the stylized facts about primary and secondary jobs identiried by Doeringer and Piore (1971)in their theory of dual labor markets.

The third extension is more problematic for the rheory. So far, we have assumed that compensation takes the form of conventional wage payments. Rut, as suggested in the general discuss~onof potential sources of efficiency wages, more complicated compensation policies can dramatically change the effects of imperfect monitoring. I'KOexamples of such compensation policies are bonding and job selling. Bonding occurs when firms require each new worker to post a bond that must be forfcited if he or she is caught shirhng. By requiring sufficientlylarge bonds, the firm can induce workcrs not to shirk even at the market-clearing \\-age;that is, it can shift the no-shirking locus down until it cnincidts hit11 the labor supply curve. If firms are able to require bonds, the! 1\11: do so, and unemployment will be