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Involuntary employment in contracts with risky job search.


The implications of partially insurable job search risk for incentive compatibility in a standard contracting framework are explored. When unemployment spells provoke job search, workers face risk in spell duration and reemployment wages. When search effort is not variable, contracts including unemployment insurance will yield involuntary employment. When search effort is diminished by UI benefits, firms shift some compensation back to wages, increasing the relative attractiveness of employment. The analysis begins with a theorem on incomplete insurance by Imai, Geanakoplos and Ito [1981] for which a simple proof and economic intuition are provided.


In seminal models of contingent labor contracts, Azariadis [1975] and Baily [1974] sought to explain the behavior of wages and employment over the business cycle. Much of the subsequent literature (for example, Grossman and Hart [1983] and Hart [1983]) has been devoted to exploring the impact of asymmetric information on such properties of contracting equilibria as over- or underemployment, wage flexibility, and whether or not utility falls when a worker is fired. This paper looks at the effects on contingent labor contracts of the asymmetry in risk faced on the one hand by workers who, being laid off when the contract is implemented, experience job search risk (which is partially uninsured due to asymmetric information) and on the other hand by workers who, retaining their jobs, return to a known status quo for the rest of the period.

In most contingent contracting models, all relevant information about the prevailing state of nature is revealed at once, and wage and employment levels are then determined. Once the contract is implemented, there is no residual uncertainty. This sequence of events--agreement upon a contingent contract, resolution of uncertainty, determination of wages and employment --is a reasonable approximation to the experience of workers who retain their jobs, since they know their wages and their working conditions. Workers who are laid off, however, often face considerable risk over the effectiveness of job search and the terms of future employment. This creates an asymmetry in the riskiness of the ex post outcomes of the contract: job retention means no more uncertainty within the period, while layoff entails further within-period risk associated with job search and reemployment. While severance pay or unemployment benefits can reduce some of the uncertainty associated with job loss, they will not generally provide complete insurance. This is because the intensity of job search and the arrival of job offers are private information; since the firm cannot verify, for example, the intensity of job search, it will insist that the worker bear some of the risk associated with the search's outcome. (This is a standard principal-agent problem; see, for example, Shavell [1979].) Equilibrium labor contracts will therefore offer workers a gamble between job retention with no residual risk and layoff with some uninsured job search risk.(1)

This asymmetry in residual risk affects the contingent contracts workers enter into. For example, if workers feel more comfortable facing a given risk when their incomes are higher, the residual risk associated with layoff may lead them to accept lower contingent wages in the employed state in exchange for greater severance pay when laid off. A problem with this kind of modification of the standard contract (i.e., one with no residual risk) is that, if enough compensation is shifted into the unemployed state, expected utility may actually be higher for laid-off workers than for those who are retained. This is a situation with voluntary layoffs and involuntary employment: at prevailing wages, retained workers would rather be laid off. If workers can increase their chances of being fired by behaving unproductively, involuntary-employment contracts may be unenforceable. Imposing an incentive compatibility constraint ruling out involuntary employment is likely to lead to an overemployment equilibrium as workers, now unable to reduce the disutility of risky job search by raising unemployment income, seek instead to reduce the likelihood of facing the risk by trading off expected compensation in exchange for reduced probabilities of layoff.

This paper introduces sequentially three elements of risky job search that affect the nature of contingent labor contracts. They are, first, uninsured reemployment wage risk; second, insurable unemployment spell duration risk; and third, moral hazard associated with the choice of search intensity. Section II begins with an economic interpretation for a theorem established by Imai, Geanakoplos and Ito [1981] and relates severance payments and risk aversion to voluntary layoffs (involuntary employment). Section III expands the analysis to the case of layoffs with uncertain unemployment spell duration, replacing severance payments with a duration-insuring flow of unemployment insurance (UI) benefit payments. These section deal with third-party insurers and establish the relationship between decreasing absolute risk aversion and involuntary employment.


Section IV turns to a contingent labor contract with unemployment insurance provided by the employer. Layoffs considered here are expected to be of long enough duration that it is in the firm's and workers' best interest for the workers to search for (perhaps temporary) employment elsewhere. Because firms cannot observe their workers' job search efforts or verify their job offers, the firms will not offer complete insurance against the risks involved in a stochastic search process. It is shown that uncertain duration increases the likelihood of observing involuntary employment. However, the effects of this on the contract's division of compensation between wages and UI may be offset by the effect on the optimal contract of moral hazard in the search process. This division, in turn, determines whether or not employment levels will be ex ante Pareto efficient. Section V relates the results under incomplete insurance to other issues addressed in the contracts literature.


I begin by restating a theorem due to Imai, Geanakoplos and Ito [1981] (hereafter IGI) on absolute risk aversion and the demand for actuarially fair insurance when some of the risk is uninsurable. Uninsurable risk arises, for example, when the insured party has an incentive to misrepresent private information, thereby limiting the set of contingencies that can be covered by the contract. Moral hazard, the adjustment of behavior to the presence of insurance that cannot be made contingent on that (unverifiable) behavior, is a common reason for some risk being uninsurable.

This theorem was first proved by IGI [1981]; a simpler proof is contained in Ito and Machina [1983]. The derivation presented here is simpler still and provides some intuition for the perhaps surprising result that if workers display decreasing absolute risk aversion they will demand such large lumpsum severance payments that expected utility actually rises if they are laid off.

Consider a representative worker with wage who faces a probability q of being retained. In a transaction that is completely separate from the employment contract, the worker can purchase actuarially fair contingent claims to provide a severance payment in the event of layoff. A worker who is not retained will find new employment at an uncertain wage w generated by a known density function. The size of the severance payment is not contingent on the realization of the reemployment wage; hence there is some uninsured risk associated with job loss. The worker agrees ex ante to relinquish claim to an amount of income x if retained, in return for an actuarially fair severance payment b = [q/(1-q)]x if laid off. The level of insurance is chosen to maximize a strictly concave von Neumann-Morgenstern utility function: max EU, where EU = qU(wo-x) + (1-q) EU [qx/(1-q) + w]. (1)

The first-order condition equates the expected marginal utilities:

U'(wo-x) = EU'(b + w) (2)

With this utility-maximizing level of insurance, a layoff is involuntary if expected utility is higher when retained than when laid off: U(wo-x) greater than or equal to EU(b + w). The theorem due to IGI states: given that U'(wo-x) = EU'(b+w), U(wo-x) greater than or equal to or lesser than or equal to EU(b+w)

(increasing) as U displays ( constant ) absolute risk aversion.

( decreasing )

To understand this result, let w be the certainty equivalent (see Pratt [1964]) of the risky reemployment wage, so that w is defined by EU(w + b) = U(w + b). The certainty equivalent will be a function of the level of certain income from severance pay: w = w(b). We can rewrite (1) as max qU(wo-x) + (1-q)U[b+w(b)]. The first order condition is U'(wo-x) = U'[b+w(b)][1+w'(b)]. Since U' > O, U" < O, this implies U(wo-x) greater than or equal to or lesser than or equal to U[b+w(b)]=EU(b+w)as w' greater than or equal to or lesser than or equal to O. By a theorem due to Pratt (theorem 2), w' greater than or equal to or lesser than or equal to O if and only if the utility

( increasing ) function U displays ( constant ) absolute risk aversion. Therefore

( decreasing )

U(wo-x) greater than or equal to or lesser than or equal to U[b+w(b)] = EU(b+w)

( increasing ) as absolute risk aversion is ( constant ).
 ( decreasing )

The risk from the uncertain future wage is uninsurable. If severance pay can make the risk matter less or more by adding a floor of certain income at which the risk is faced, then the marginal utility of any given level of severance payments is affected by the presence of the uninsured risk. In the case of decreasing absolute risk aversion, additions to severance pay reduce the worker's degree of risk aversion, thereby raising the value of (i.e., the certainty equivalent of) the risky reemployment wage. It is this greater investment in severance pay that makes expected utility higher under layoff than under retention when absolute risk aversion is decreasing. The opposite holds for increasing absolute risk aversion: the worker chooses less severance pay because each dollar reduces the certainty equivalent of the future wage.

Decreasing absolute risk aversion (DARA) is generally taken to be the empirically relevant case. DARA represents a fairly mild restriction on preferences: if a worker's portfolio includes a risky asset and a riskless one, for example, DARA is a necessary condition for the dollar amount invested in the risky asset to rise with income. The implication of decreasing absolute risk aversion for the demand for severance pay is that we expect to see involuntary employment.


At the time a worker is laid off, there is usually uncertainty not only over future wages but also over the length of time required to find a new job. If unemployment duration risk is viewed as uninsurable, its presence can be incorporated into the distribution of future income. If duration is insurable, however, risk-averse workers will prefer duration-dependent unemployment compensation to lump-sum severance pay.

If reemployment opportunities arise exogenously and are verifiable, duration risk is insurable. Suppose that instead of offering severance pay b, insurance markets allow workers to choose a level z of the flow of unemployment compensation that lasts for the fraction sigma of the period for which the worker remains unemployed.(2) Fair insurance sets contingent premiums x so that qx = (1-q)(Esigma)z. The worker's problem is max EU, where U = qU(wo-x) + (1-q)EU[sigmaz + (1-sigma)w] and qx = (1-q)(Esigma)z. The first order condition gives U'(wo-x) = EU'(y) + (Esigma)-1 cov [sigma, U'(y)], where y denotes income in a period begun with a layoff [y = sigmaz+(1-sigma) w]. Assuming that within-period income is higher the more quickly a new job is found (i.e., one only accepts jobs that are income-increasing), the covariance of spell duration and marginal utility is positive. The effect of a positive covariance is to depress the expected marginal utility in periods beginning with layoff below the marginal utility of the wage. Insurance, now more attractive because it helps to hedge some of the duration uncertainty that follows a layoff, is demanded to an extent such that with DARA and even with constant absolute risk aversion, layoff will be preferred to retention.


Up to this point, we have been dealing with insurance provided by a third party who takes as given the layoff probabilities of the industry. Offering UI alternatively as part of the employment contract allows the layoff and insurance decisions to be made simultaneously, increasing the achievable level of expected profits associated with any given level of expected utility for workers. In this section, we consider labor contracts that offer UI to workers who are laid off. For now, assume that firms can costlessly monitor work effort and that shirkers fired for just cause along with workers who quit do not collect UI. Otherwise, we rule out contracts that yield involuntary employment, since these would be unenforceable. The case where monitoring is not feasible is considered below.

We also drop the assumption that unemployment spell duration is unaffected by laid-off labor's behavior. If UI benefit levels are set by firms and cannot be made contingent on the level of search effort, UI creates a moral hazard problem and induces inefficient levels of search activity. With risk neutral firms and risk-averse workers, the presence of moral hazard in search behavior will induce contracts in which the firm bears some but not all of the risk of uncertain unemployment spell duration. When UI payments are offered by employers and high levels of UI payments increase the average spell length by reducing search effort, firms attempt to avoid part of these extra costs by offering contracts that deviate from the no-moral-hazard contract by paying a little less in UI and a little more in wages. This shift in compensation toward wages will in turn make retention a more attractive prospect.

This section looks at such contracts when the firm offers a constant flow of UI benefits. Under the constant benefit flow offered by the public UI program in the U.S., there has been accumulated strong empirical evidence that an increase in the level of UI payments will induce longer expected unemployment spells. In interpreting the model below, this positive relationship is taken to be an outcome of the worker's expected utility maximization process and the level of search effort is taken to be the intervening variable.(3) Because there is little empirical support for the possibility that longer search leads to a higher reemployment wage, we further assume that the reemployment wage is independent of unemployment spell duration.

Knowing this relationship between spell duration and UI, the firm seeks to maximize expected profit. Each possible state i occurs with probability qi (Summation q=1) and is revealed through the price pi at which the firm can sell its output. Output prices are observable by all; the job search process remains the only source of asymmetric information. The contract is assumed to be ex ante Pareto efficient for the workers and the employer; we model it as an expected profit maximization problem, subject to the constraint that workers receive an expected level of utility Uo. Output is related to the level of employment by a strictly concave function f(.). The firm contracts with a workforce of size L and sets state-contingent values for the wage wi, retention rate ri, and the level zi of weekly UI benefit payments.(4) A worker's utility is assumed not to be affected by the timing of income receipts within the period. Denoting income in a period begun by layoff as y and the proportion of the period spent unemployed as sigma, the firm's problem can be represented by the constrained maximization Z = Summation qi[pif(riL) - riwiL - (1-ri )LziEsigma i] + Lambda{Summation qi{riU(wi) + (1-ri)EU(yi)] - Uo},

(5) where yi=sigma(zi-search intensity) + (1-sigma) (reemployment wage). The first order conditions are
 aZ/ari = qiL(pif'(riL) - wi + ziEsigma) + Lambdaqi[U(wi) - EU(yi)] = 0 (6)
 i = 1,2...,N
 aZ/awi = qiri [-L+LambdaU'(wi)]=O
 i=1,2...., N

aZ/azi=qi(1-ri) {-L[Esigma + zi(dEsigma/dz) + LambdaE(U'(yi)dyi/dz)]}=O

i = 1,2.,N

Equation (7) reveals that the wage is equalized across states. By the envelope theorem, we know that E[U'(y), dyi/dz] = E[U'(yi)ayi/az] = E(U'(yi)sigma).

Solving (7) for Lambda and substituting into (8) gives 6

(Esigma + zi dEsigma/dz)U'(w) = E[U'(yi)sigma].

Dividing by Esigma, this can be written

(1+esigmaz)U'(w)=EU'(yi)+ (Esigma)-1 cov[U'(yi),sigma], where esigmaz is the elasticity of unemployment spell duration with respect to the benefit level. Workers have an incentive to search only if speedy reemployment will increase income; imposing this condition gives a positive covariance between spell duration and the marginal utility of income.

The following observations can be made:

OBSERVATION ONE. With no moral hazard (esigmaz) = O, equation (10) reduces to equation (4). As long as the level of expected utility guaranteed workers by the ex ante labor contract is the same as that observed with third-party insurance, these alternative arrangements for insurance provision lead, in the absence of moral hazard, to the same level of insurance. Insurance provision through labor contracts neither improves nor hinders the insuring process.

OBSERVATION TWO. When search effort varies with the UI benefit level (esigmaz > O), its effect on the contract is to induce more compensation through wages and less through UI. Equation (10) shows that positive values of esigmaz depress U'(w) relative EU'(yi) as firms reduce their reliance on distortion-inducing UI benefits. Empirical estimates of esigmaz vary but suggest that 0.2 is not an unreasonable value (see Welch (1977), Hamermesh (1982)). As before, the bigger the covariance of the marginal utility of income and unemployment spell duration, the smaller is the expected marginal utility of income in periods of initial unemployment. Dividing equation (10)by EU'(y) shows that if any utility function exhibiting DARA is to be consistent with voluntary employment, the unemployment spell duration elasticity must be greater that the normalized covariance: in other words, unless esigmaz greater than or equal to cov (U'(y),sigma)/EU'(y)Esigma, the labor contract described here results in involuntary employment for any utility function exhibiting DARA.

The covariance between marginal utility and unemployment spell duration will be small when either consumption is not too dependent on spell duration or marginal utility is not too dependent on unemployment-induced shortfalls in consumption. Dissaving or borrowing could reduce the dependence of consumption on unemployment duration. The covariance will also be small when workers are not too risk averse, i.e., when marginal utility does not change much with duration-related changes in the level of consumption.

When workers can increase their chances of being fired by behaving unproductively, the labor contract must not offer involuntary employment; i.e., it must satisfy the constraint U(w) greater than or equal to EU(y). The previous section shows that, unless the elasticity of unemployment spell duration is large relative to the normalized covariance of marginal utility and spell duration, this constraint is likely to be binding. In this case, it can be shown that the contract yields overemployment. Labor is not being allocated efficiently in the sense that, even after netting out search costs, the expected value of the marginal worker's output is higher outside the firm than within it.(5)

OBSERVATION THREE. When the marginal conditions describe contracts that offer involuntary employment and such contracts are unenforceable, the incentive compatibility constraint (U(w) greater than or equal to EU(y) is binding with equality. In this case, the contract implies a retention rate that is too high relative to the level that is ex ante (pre-search) efficient.

Proof. When U(w) = EU(y), equation (6) reduces to dividing by qi, dropping subscripts and substituting Lambda from equation (7)

pf'(rL) = w - zEsigma. (11)

By the strict concavity of U(.), U(w) = EU(y) also implies w < Ey. This gives

pf'(rL) < Ey - zEsigma. (12)

The left-hand side of (12) is the value of labor's marginal product in the firm. The right-hand side is the expected value of the worker's alternative productivity net of search costs, since expected net income Ey exceeds expected net product by zEsigma, the income transfer through UI. Since f(.) is concave, a lower marginal product within the firm that without implies an inefficiently high retention rate, i.e., overemployment.QED.

It should be clear from the structure of this argument that it is not the costliness of search that leads to overemployment. Rather, it is the residual income risk associated with unemployment that matters.

In full-information contracts and in many contracts where all parties are risk-neutral, the value of labor's marginal product within the firm equals the alternative wage; i.e., there is neither over- nor underemployment. Since a spot market model predicts that the value of labor's marginal product equals the firm's wage rather than the alternative wage, these predictions provide an empirically testable hypothesis to discriminate between contracts and spot markets as allocators of labor. Little empirical support is found for the simple contracting model (see Brown and Ashenfelter (1986), Card (1986). It follows from observation three that uninsured search risk gives rise to contracts in which the value of labor's marginal product is not equated to the expected alternative wage. The model here, with risk-averse workers and some uninsured search risk, does not fall into the set of contract models called into question by these empirical results.


Under standard assumptions about risk aversion, implicit labor contracts are likely to yield equilibria in which employment is involuntary if there is uninsurable risk associated with the future income of laid-off-worker. When absolute risk aversion is decreasing, severance pay increases the certainty equivalent of the risky future wage; this effect on the certainty equivalent raises the marginal utility of a dollar devoted to insurance, and in equilibrium enough insurance is demanded to make layoff an enviable prospect.

Severance pay has become a standard feature of labor contract models (for example, see Grossman and Hart (1981), Green and Kahn (1983). When there is uncertainty over the (observable) length of unemployment spells, such lump-sum compensation represents a suboptimal allocation of risk. The option of paying unemployment insurance, where total unemployment compensation depends on (and helps insure against risk in) unemployment duration, provides an even greater incentive to demand insurance when longer spell duration is positively correlated with lower within-period income.

The inclusion of job search by risk-averse agents in a contracts model makes the distinction between severance pay and UI benefits a substantive one. If the provision of UI creates a moral hazard in workers' choice of job search effort, employers have an incentive to shy away from large UI payments in order to induce more intense job search by laid-off workers, thereby reducing the length of unemployment spells that must be compensated. Without duration-dependent benefits that adversely affect job search effort, decreasing absolute risk aversion and voluntary employment are mutually exclusive. When contracts offering involuntary employment are not enforceable, the incentive constraint yields contracts will overemployment in adverse states. Switching from severance pay to duration-dependent benefits will change this if the potential moral hazard in search behavior outweighs the added value to workers of the partial insurance against the stochastic nature of unemployment spell duration. If the moral hazard effect is not big enough to bring about voluntary employment and if these contracts are not enforceable, the incentive compatibility constraint again leads to contracts with overemployment in adverse states of nature.

* Pomona College, Claremont, CA. I am especially indebted to Takatoshi Ito and Richard Romano for helpful discussions. I would also like to thank Howard Kaufold, Doug Waldo, and an anonymous referee for their suggestions, and this journal for its generous editorial assistance. Any shortcomings of this work remain my responsibility alone.

(1.) For a contracting model with asymmetric information arising from on-the-job search, see Ito [1988].

(2.) While this case is interesting in that it mirrors the structure of the federal UI benefit program (up to a maximum number of weeks), it will not in general be privately optimal: there will in general be some other pattern of duration dependence of total UI compensation that will provide a given level of expected utility to the worker at a lower expected cost to the firm. Shavell and Weiss [1979] have characterized such benefit streams for one utility function under various assumptions on access to savings, loans, etc.

(3.) An explicit risk-averse search problem is not modeled since we have empirical evidence on the magnitudes that concern us here. Search models have been integrated into contracts models, but search is almost always modeled as a risk-neutral strategy in which it is expected income that is maximized. Burdett and Mortensen [1980], who were the first to integrate layoff-induced job search into the implicit contracts framework, model the search problem as one of expected income maximization. Cothren [1983] retains the assumption of risk-neutral job search in his contracting model that generates both temporary and permanent layoffs. Pissarides [1982] has a model of layoff-induced job search in which workers choose time paths for the reservation wage and search intensity, and firms choose the timing of recall of workers on layoff; here, too, workers are risk neutral. The predictions derived from expected-utility-maximizing search models are substantially different from those of expected-income-maximizing search: see for example Lippman and McCall [1979], Danforth [1979].

(4.) If one wished to constrain the benefit level to be state-invariant, equation (9) below would be summed over all possible states. A state-invariant benefit level arises as an outcome of the more general specification if the duration and wage distribution s and w are state invariant: see equation (9). The specification here does not treat the distribution of s as state dependent, but this extension is straightforward.

The relationship between the state of nature affecting the firm's output market and conditions in the labor market is not modeled. Variation in the zi across states arises if the labor market's conditions are related to the firm's output market. If good times for the firm are positively related to good times for job search, then zi will be negatively related to Pi.

(5.) Overemployment often refers to excessive levels of market employment vis-a-vis nonmarket activities. Here the relevant allocation is between initial jobs and the alternative of job search followed by other market work. Because of the unemployment involved in switching from one job to another, the excessively high retention rate by initial firms increases the level of measured employment in the economy. Overemployment in this model can be thought of as an allocation of too little labor to job search, leading to the usual characterizations of overemployment, i.e., marginal product that is too low relative to other opportunities and a measure d employment rate that is correspondingly too high.

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