Contractual Models of the Labor Market By BENGT HOLMSTROM 1981

 https://translate.google.com/translate?sl=en&tl=ja&u=https%3A%2F%2Fnam-students.blogspot.com%2F2019%2F02%2Fcontractual-models-of-labor-market-by.html

American Economic Association

Contractual Models of the Labor Market Author(s): Bengt Holmstrom Source:  The American Economic Review,  Vol. 71, No. 2, Papers and Proceedings of the Ninety-Third Annual Meeting of the American Economic Association (May, 1981), pp. 308313 Published by: American Economic Association Stable URL: https://www.jstor.org/stable/1815736 Accessed: 13-02-2019 12:37 UTC REFERENCES Linked references are available on JSTOR for this article: https://www.jstor.org/stable/1815736?seq=1&cid=pdf-reference#references_tab_contents 

 Contractual Models of the Labor Market By BENGT HOLMSTROM* In this paper I discuss some approaches to modelling labor markets as contractually mediated. The central thesis of such models is that due to market imperfections- absence of contingent claims for labor and income-wage mediated auction markets will in general not be sustainable. There will be an opportunity for firms and workers to make a joint long-term contract which im- proves the welfare of both, and causes the auction market to collapse. This view, first suggested by implicit contract theory (see Costas Azariadis; Martin Baily), opens up quite new perspectives on the operation of labor markets. Most importantly, one finds that in the contractual paradigm wage and marginal product may differ so that what appears to be disequilibrium in the short run may be a consistent equilibrium in the long run. Whether or not this can account for involuntary unemployment I will comment upon later. It relates to my main concern, which is with the extent to which complex contingent contracts can be enforced, and the implications this has on the nature of equilibrium. I will start by surveying the main results of implicit contract theory in the light of a rather general model of contractual equi- librium (Section 1). Weaknesses of implicit contract theory will lead us to question the assumption of enforceable state-contingent contracts. Section I; looks at the alterna- tive of non-contingent fix-wage contracts whereas Section III presents a simple analy- sis of reputation as a means of enforcing more complex contingent contracts. I. Implicit Contracts Since it has been widely believed that implicit contracts can emerge only in markets where it is costly to move, let me start by showing that this is an artifact of the commonly used one-period model. As- sume initially that the labor market is cleared through sequential wage auctions and con- sider for simplicity two periods only. Cur- rent wage is wo and next period wage w1 is uncertain with known distribution G(wj). A worker's expected utility of participating in the market is assumed to be u(wo) + fu(wl)dG(wl), where u(.) is an atemporal risk averse utility function. Firms are as- sumed risk neutral. The expected wage bill for one worker is wo + fw1dG(wj). The claim is that firms can depart from the auction outcome to the benefit of both parties. To show that, consider a contract which pays the worker w' in the first period and guarantees w' in the second period as well. If market wage in the second period exceeds w', the firm has to follow suit or else the worker quits so the contract is called off. Choose w' so that (1) u(wo)+ fwu(wi)dG(wi) -u(w') I + )dG(w) that is, so that the contract offers the worker the same prospect as the auction market. Dividing (1) by (I+ fow'dG(wj)) and apply- ing Jensen's inequality, gives (2) Wo + f w,dG(w,) >w'(l + fwdG(w,) establishing the claim. Note that the role of two periods is to allow the firm to collect a premium in the first period (wo - w' >0) for the insurance it provides in the second period. The sug- gested contract looks like an option and (2) indicates that the selling price includes a risk premium which makes the contract *Associate professor, J. L. Kellogg School of Management, Northwestern University. I would like to thank Dale Mortensen and Edward Prescott for com- ments on an earlier manuscript. 308

2

MODELS OF LABOR MARKET EQUILIBRIUM 309 favorable to the firm. However, the argu- ment can be amended for risk averse firms. Also, identifying the worker's utility func- tion is inessential. What is essential though is that beliefs are not too dispersed and, most importantly, that the firm will not lay off or exchange the worker when w1 <w'. This last point I will return to. Once all firms recognize the value of long- term contracts, wage auctions get replaced by contractual auctions in which the market will be equilibrated through the expected utility contracts offer. Since the market still is incomplete there will be a need to reopen it each period. What is envi- sioned to happen in future markets will in- fluence current contract design and vice versa so the natural notion of equilibrium is one based on rational expectations. One can show such an equilibrium to exist. It is closely related to the pioneering models of Roy Radner and Oliver Hart, with the dis- tinction that here securities are created en- dogenously by firms, and by assumption can be bought only through labor attach- ment at the respective firm. Let me now turn to some properties of an equilibrium in implicit contracts, basing my discussion on an explicit treatment of a two- period version of the model sketched above (see my earlier paper). With risk-neutral firms and homogeneous labor, there will be a unique optimal state-contingent contract that firms offer to their workers. If firms differ in their risks, contracts will differ and different firms may therefore pay different wages. In general contracts will involve lay- offs in bad states and if the market turns favorable some workers may quit. Wage- wise, an optimal contract will look like the earlier described option. As long as market forces do not push up equilibrium expected utility levels, retained workers will enjoy a constant wage, but with increased labor de- mand, wages will rise (due to legal con- straints on involuntary servitude). It should be stressed that the implied downward rigid- ity of wages relate to individual contracts rather than aggregate wage levels. There is no presumption that new generations will receive the same wage as old ones-in gen- eral they will not-and a laid-off worker who has to find a job elsewhere is normally forced to take a wage cut. Thus, optimal contracting creates endogenously seniority classes within an otherwise homogenous labor force. Therefore, in aggregate, the wage level will be flexible both up and down, but at a more sluggish pace than wage auc- tions would imply. This sluggishness is fur- ther increased by rights to be recalled (at previous wage) before any new workers are employed; a provision which will be part of an optimal contract. Regarding the employment part of con- tracts I note that the model accomodates both quits and layoffs. Layoffs will occur for the same reason as in Azariadis' original treatment, namely the outside opportunity a worker has. But somewhat more acceptably this outside opportunity could be another firm rather than an exogenous benefit (for example, household income). Essential for understanding the determination of employ- ment is the fact that since the implicit con- tract is ex ante efficient and state-contingent it is also ex post efficient (as a necessary condition for ex ante efficiency). It follows immediately that a worker who can produce more within the firm than outside will not be laid off. A more specific condition is obtained by equating the marginal rates of substitution between wage and employment probability (defined as the percentage of workers retained), which gives (3) pf'(nr) = w' u(w') u(w ) where p = output price (random), f(.) =pro- duction function, n= labor pool, r = proportion retained, and the other variables have been defined earlier. The right-hand side is decreasing in w' and achieves its maximum for w'=w1 (since w'>w1 is re- quired or else the worker quits). Thus, we find that actually labor will be retained not only beyond the point where marginal prod- uct equals contract wage but market wage! Two important conclusions follow: wage and marginal product generally differ and a divergence of the two does not signal dis- equilibrium (which casts some doubt on re- cent fix-price modelling); and, secondly,

3

PAPERS AND PROCEEDINGS MAY1981 there will be less rather than more unem- ployment in the contractual model com- pared to wage auctions. The last conclusion has been viewed as a failure of implicit contract theory to explain involuntary unemployment, and if one de- fines involuntary unemployment as unex- ploited opportunities to trade, indeed it is. However, it is not clear to what extent op- portunities are left unexploited in the real economy when one looks at the labor market in isolation; what we observe taking place could be consistent with (3), with inef- ficiencies being due to faults in the coordi- nation between product and labor markets instead. In order to alter the conclusion about the level of unemployment it is necessary to bring in some elements of incomplete or asymmetric information (otherwise con- tracts will be ex post optimal). More im- portantly, such modifications will help to patch a logical inconsistency of implicit contract theory, namely the lack of sever- ance payments with fully insured income. One simple change in informational as- sumptions is that w1 cannot be observed by the firm and it will instead have to act on the conditional expectation E(w1 p). Then it is clear that, even if it wanted to, the firm could not guarantee the worker a constant income, and in some states (recessions) the level of unemployment would fall below that of an auction market (the rule would be as (3) with Eu(w1 I p) replacing u(w,)). Another change, explaining incomplete severance payments, would take note of the fact that laid off workers would have no incentive to search if their income was fully insured. With search, layoffs would result in some unemployment rather than pure transfers as in the simple model. The third change, which I will pay more attention to, calls into question the firms assumed honesty. Generally, it is hard (for a single worker at least) to observe the margi- nal product of the firm and hence see if the required rule (3) is being followed. It ap- pears therefore tempting for the firm to de- viate from this behavior and insofar that this can be expected to happen, implicit contracts are rendered infeasible. Though I will indicate in the last section how a con- cern for reputation in the labor market may induce the firm to behave as if an implicit contract was written and honored, the doubt about enforceability of contracts prompts us to look at alternative contracts, which are not contingent on states that cannot be ob- served. II. Fix-Price Contracts The simplest non-state-contingent con- tract is one in which the firm guarantees the worker a nominal wage, but adjusts employ- ment at its own discretion. Let me first analyze whether such contracts can be ex- pected to arise endogenously as a mutually beneficial arrangement compared to the wage auction outcome. If so, we would again have an explanation for the breakdown of wage auctions and have a legitimate reason to study fix-wage contracts further. I look at an example only. Consider an industry in which there is a large number of identical risk neutral firms characterized by the production function f(n), f'(l) = 1. The output price fluctuates randomly between 1 and pe(O, 1), which both occur with equal frequency. Workers supply their unit of labor inelastically and have an atemporal utility function u(w). The num- ber of workers per firm (appropriately scaled) is n = 1. In a wage auction therefore, wage will equal output price (1 or p). Ex- pected utility from such a market is Eu= 1/2(u(1)+u(p)), and expected profit E1= (1 +p)(f(l) - 1). Suppose now a firm would offer its workers a fixed wage w and lay off the unprofitable ones when output price is p. Letting r be the number (and proportion) retained, we have (4) pf'(r) = w In order for this contract to be favorable (compared to the auction outcome) both to the worker and the firm we must have (5) Eu-_--2(1 r)u(w)+ '(l-r)u(p) >EuThis content downloaded from 126.235.28.162 on Wed, 13 Feb 2019 12:37:24 UTC

VOL. 71 NO. 2 MODELS OF LABOR MARKET EQUILIBRIUM 311 In (5) I use the assumption that the firm is small so a laid-off worker can take a job in the spot market at wage p. Rewriting (5) and (6), (7) r(u(w) -u(p)) > u(I) -u(w) (8) (I +p)/p-f'(r)(l +r) >f(l) -f(r) It is easy to see that (4), (7), and (8) can be simultaneously satisfied; for instance, if w < (1 +p)/2, then (4) implies that (6) (hence (8)) is satisfied for all f, and we can take f and u so that (7) holds. To understand what factors determine when (4)-(6) will hold, we can look at comparative statics. Assume that the equations above hold. Then they will remain intact either when the absolute risk aversion of u is increased, or if f becomes more kinked in the sense that f'(r) and f(l) -f(r) decrease (f becomes flatter be- tween r and 1 and steeper between 0 and r). These conditions accord with intuition, since it is easy to understand (and check from (4)-(6)) that when the worker is risk neutral or when the production function exhibits constant returns to scale, there can be no gains to a fix-wage contract. Regarding changes in p, the effect is ambiguous. Note, however, that if there is a firm for which productivity does not fluctuate at all, it will always pay this firm to offer a fix-wage contract. The assumption that the market is fric- tionless is rather unfavorable for a fix-wage scheme. Robert Gordon was first to suggest that a fix-wage scheme may be a response to the temptation a firm would have to lower the wage (claiming low marginal product) at will. Such behavior is, of course, limited by the worker's outside opportunity, but with costly labor mobility the firm's opportunity to exploit workers would quickly become rather substantial. My intention with the model above was to show that even without such transaction costs one can make a case for a fix-wage arrangement, and to indicate what factors will influence the benefits thereof. In general, fix-wage arrangements will im- ply unemployment in excess of voluntary levels. Yet it may be an outcome of equi- librium and moreover an efficient arrange- ment subject to informational constraints, since as recent work on efficiency under asymmetric information emphasizes, it is false to apply standard ex post efficiency measures in the type of situation described. The incomplete information paradigm has some apparent advantages over implicit contract theory. Wages are rigid without the unrealistic assumption of firm risk neutral- ity. With risk neutrality one is left wonder- ing why a firm does not pay severance to laid off workers and thereby insure income (the model above begs the same question; a justification would require a state in which severance is infeasible). Indeed, the role of unemployment benefits is quite unclear- why do firms not provide those privately? When wages are fixed due to incentive con- siderations, firms can be assumed risk averse, explaining incomplete severance and the incentive to pool risks through jointly paid unemployment benefits. One may also expect that optimal levels of such benefits, the degree to which they should be experience rated, and any additional sever- ance payments could be determined. These questions appear fruitful for future research. However, one should keep in mind that, if there would be large gains to state contin- gent contracts, we would expect to see prox- ies for those states enter into contracts (a host of public data would be available for that) or observe direct monitoring of re- quisite states. But we do not, and before we understand why, enthusiasm over the in- complete information paradigm, in particu- lar its implication for involuntary unem- ployment, should be controlled. III. Reputation I turn to the final point: can a concern for reputation lead the firm to act as if it honored an implicit contract? As we know from the growing literature on incentives in agencies (and more generally from the the- ory of repeated games), multiperiod consid- erations will allow a rather richer set of opportunities to combat averse effects of informational asymmetries. To illustrate how, I will again use a simple example.

 312 AEA PAPERS AND PROCEEDINGS MAY1981 Consider a firm operating in an economy that can be in one of two states s =0 or s = 1. The firm's revenue function is (1 -as) xf(n), where n is labor input. A suggested interpretation is that s = 1 corresponds to a recession and the parameter a indicates how sensitive the firm is to the recession. The sensitivity parameter stays fixed over time. It is initially unknown to workers, and this creates the opportunity for the firm to build a reputation by signalling the value of a through layoff behavior. Workers work only in one period for which they sign a fix-wage contract as in the previous section. The wage demand will de- pend on their assessment of the probability of layoff, labelled as the firms reputation pc(O, 1). In order to attract workers the firm has to pay w(p), which is defined from the expected utility expression: (9) u(w(p))(p +(1-p)p) +u(w)( -p)(l -p) =v where w- is income as unemployed, p = Pr(s = 0), and v is the expected utility offered by the market (assumed constant over time). Reputation p is a function of how the firm behaved in the previous recession. If a per- centage r was laid off, this will be inter- preted as indicating that the firm's type is a(r) (a signalling function to be determined in equilibrium), which in turn will give a prediction for how the firm will behave in the next recession p(a). Since a does not change over time in the simple version pre- sented here, equilibrium will have p(a(r)) = r, that is, the firm will lay off the same amount in each recession and the predict- ions obtained by assuming that the firm will repeat its behavior will become self-fulfilling in equilibrium. Therefore, I take reputation formation to progress as: p= p,-, if s, =0, p, = r, if s, = 1, where t is a time index. Let the firm's discount factor be 8, and V(p) be the optimal discounted expected profit function given current reputation p. Then (10) V(p)=maxt(f(n)-w(p)n)p n,r + ((1 -a)f(nr)-w(p)nr) x (1 -p)+ SV(p)p + dV(r)(I -p)) Here the firm is viewed as deciding the proportion retained r in case of a reces- sion at the time it hires labor. Note that TT(n, w, r)=_ (f(n)-wn)p + ((1 -a)f(nr) - wnr)(I -p), is the one-period expected profit and that an implicit contract that is optimal solves maxn, rv(n, w(r), r). In a stationary state we want r = p* to solve (10), were p* is the stationary reten- tion value. Thus, ( 11) V(p*) = (n*, w(p*), p*) where n*=argnmaxv(n,w(p*),p*). For p* we get a condition by noting that it must not benefit the firm to move to any other stationary layoff policy. Thus, (12) p*= argmax {(1-oa)f(n*r) r -w(p*)n*r+ l _ (n*, w(r), r) This is the main equation of interest. Myopic behavior (as assumed in the previ- ous section) results if 8 = 0. Then layoffs will equate current marginal product to wage. But when 8 >0, so that there is a concern for the future, reputation will force the firm to set layoffs below the myopically optimal value; ((d/dr)v(n*, w(r), r) > O for myopic r). Thus, reputation will provide for in- creased employment insurance. On the other hand, unless 8= 1, p* will not be as high as in an implicit contract (which just maxi- mizes TT(n*, w(r), r) over r), since there is a cost of signalling through the excess em- ployment. The optimal level of retained workers p* will therefore lie between the myopic solution and the implicit contract solution, and be closer to the latter the closer 8 is to 1 (the case of no discounting). The paradigm suggested above is, of course, extremely simple and stylized, but I take it to indicate that reputation may have

VOL. 71 NO. 2 MODELS OF LABOR MARKET EQUILIBRIUM 313 the power to enforce implicit contracts (or nearly so) even when the worker cannot verify the state of nature. This deserves fur- ther study. One would hope to learn how sophisticated the contingencies can be in reputation contracts, what influence the frequency of events has on reputation for- mation and could wage variation (when de- sirable) be supported by reputation as well. Intuition suggests that only rather simple, regularly occuring contingencies can be in- cluded in reputation contracts and that wages cannot be varied in such contracts because the wage part of a contract is a zero-sum game whereas the employment part is not, but the art of modelling reputa- tion is still too primitive to confirm this intuition. IV. Concluding Remarks Implicit contract theory has been influen- tial in suggesting a contractual view of labor markets, which carries the promise of a much improved understanding of how labor markets operate. The theory itself is incom- plete at some crucial points. The main question taken up here concerns the en- forceability of state-contingent contracts. On one hand, if we abandon the assumption that finely state-contingent contracts can be enforced, it leads to a fix-wage model which at least casually looking displays more rea- sonable features than the implicit contracts. On the other hand, the previous section indicates that a concern for reputation leads to behavior which appears as if implicit contracts were enforced. A consolidate view is that wages will be downward rigid largely due to enforcement problems (rather than risk sharing), whereas employment rules will, at least to some degree, reflect a concern for reputation and therefore come closer to what would be implied by implicit contracts, per- haps close enough to explain why more complex contingent contracts that would be feasible to write are not written. REFERENCES C. Azariadis, "Implicit Contracts and Under- employment Equilibria," J. Polit. Econ., Dec. 1975, 83, 1183-1202. M. N. Baily, "Wages and Employment under Uncertain Demand," Rev. Econ. Stud., Jan. 1974, 41, 37-50. R. J. Gordon, "Recent Developments in the Theory of Inflation and Unemployment," J. Monet. Econ., Apr. 1976, 2, 185-219. 0. Hart, "On the Optimality of Equilibrium when the Market Structure is Incomplete," J. Econ. Theory, Dec. 1975, 11, 418-43. B. Holmstrom, "Equilibrium Long - Term Labor Contracts," DP No. 414, Center for Mathematical Studies in Economics and Management Science, Northwestern Univ. 1980. R. Radner, "Existence of Equilibrium of Plans, Price, and Price Expectations in a Sequence of Markets," Econometrica, Mar. 1972, 40, 283-303.