The Structure of Multi-Period Employment Contracts with Incomplete Insurance Markets

The Structure of Multi-Period Employment Contracts with Incomplete Insurance MarketsAuthor(s): Richard ArnottSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 15, No. 1(Feb., 1982), pp. 51-76Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/134669 .

Accessed: 12/06/2014 23:22

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Wiley and Canadian Economics Association are collaborating with JSTOR to digitize, preserve and extendaccess to The Canadian Journal of Economics / Revue canadienne d'Economique.

http://www.jstor.org

This content downloaded from 195.34.79.223 on Thu, 12 Jun 2014 23:22:27 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=black

http://www.jstor.org/action/showPublisher?publisherCode=cea

http://www.jstor.org/stable/134669?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


The structure of multi-period employment contracts with incomplete insurance markets R I C H A R D A R N O T T / Queen's University

Abstract. This paper examines the structure of multi-period employment contracts in an economy with identical workers and labour turnover when only incomplete insurance is provided against job-related contingencies. The incompleteness of insurance markets stems from an asymmetry of information between the worker and the insurer. In this situation the employment contract will provide implicit insurance by paying workers less than their marginal product during some periods and more during others. The paper explores how the characteristics of such contracts are altered by increases in uncertainty, worker risk aversion, hiring costs, and the amount of specific and general training provided. A sequel to this paper examines the efficiency properties of such contracts.

La struicture des contrats de travail sur plusieurs periodes dans l'hypoth'ese oiu l'employe ne peut s'assurer qu'incompletement contre les risques attenants 'a l'emploi. Ce memoire examine la structure des contrats de travail sur plusieurs periodes dans une economie oiu tant les travailleurs que le taux de roulement des travailleurs sont identiques et oii il n'y a possibilite de se procurer qu'une assurance incomplete contre les risques attenants a l'emploi. Le caractere incomplet des marches d'assurance emerge comme consequence d'une asymetrie de l'information entre le travailleur et l'assureur. Dans une telle situation, le contrat de travail contient une assurance implicite: on paiera les travailleurs moins que leur produit marginal durant certaines periodes et davantage a d'autres moments.

Le memoire examine comment les caracteristiques de tels contrats sont affectees par l'accroissement de l'incertitude, l'aversion au risque des travailleurs, les cofuts d'embauche et la quantite de formation professionnelle tant generale que specifique attachee a l'emploi. Dans un memoire subsequent, l'auteur entend s'attacher a examiner les proprietes de tels contrats au plan de l'efficacite.

A worker faces a host of uncertainties associated with his employment - will he be good at his job? will his worth be recognized? will he receive a favourable job offer from another firm? will he be fired? will he get along well

This paper is an outgrowth of joint research with, separately, Mark Gersovitz and Joseph Stiglitz. David Card, Andrew Oswald, Harry Paarsch, and two anonymous referees provided helpful comments but are in no way responsible for any remaining errors.

Canadian Journal of Economics/Revue canadienne d'Economique, XV, No. I February/fevrier 1982. Printed in Canada/Imprime au Canada.

0008-4085 I 82 I 0000-0051 $01.50 ? 1982 Canadian Economics Association



52 / Richard Arnott

with his co-workers? will he suffer from some job-related disability? and so on. He will be able to obtain full insurance' against some of these risks, but not against most. Thus, the worker faces considerable residual risk.

This paper addresses the question: in what ways does the incompleteness of markets for insurance against job-related risks affect the structure of multi-period employment contracts? The answer is of interest in a number of contexts. First, it may explain some puzzling features of employment contracts. Second, it may resolve some contended points concerning the sharing, between firms and workers, of the costs of and benefit from specific and general training. Third, since the standard theorems of welfare economics fail to hold in an economy with incomplete markets, competitive employment contracts may be (constrained) inefficient. If they are, what should the government do to correct the distortions? And lastly, an understanding of the structure of the employment contract may provide insight into various labour market phenomena associated with macro-economic fluctuations.

The gist of the argument in the paper is that when job-related insurance markets are incomplete, there is an incentive for the firm and the worker to share the residual risks associated with the worker's employment. This can be achieved by the firm's providing either explicit or implicit insurance. Insurance can be provided implicitly by the firm's paying workers more than their marginal products over some periods and less during others. This general line of argument is familiar from the principal-agent literature (e.g., Ross, 1973, and Stiglitz, 1974) but, to our knowledge has not previously been applied to explain characteristics of multi-period employment contracts. We treat an economy of identical individuals in which the incompleteness of insurance markets stems from an asymmetry of information between the worker and the insurer, which causes the provision of insurance to be characterized by moral hazard. As a result, only partial insurance is offered.

Most of the recent literature dealing with the employment contract examines those features of the contract, notably temporary lay-offs and downward-sticky wages, which arise from workers and firms sharing the risk associated with fluctuations in demand (e.g., Azariadis, 1975). This paper is only tangentially related to that literature, and instead draws on a body of research extending back to the early 1960s on the structure of multi-period employment contracts, the most notable contribution of which is Becker (1962). While Becker's analysis was discursive, he clearly recognized the importance of risk-sharing as a determinant of the features of such contracts. Subsequent papers on the topic (D6naldson and Eaton, 1976, Parsons, 1972, and Hashimoto, 1978, inter alia) have ignored this insight, treating all agents as being risk-neutral. Thus, this paper formalizes some aspects of Becker's classic paper which have been unjustifiably neglected.

1 By full (in contrast to partial) insurance we mean that amount of insurance which results in the insured's marginal utility of income being the same whether or not the insured-against event occurs.



The structuire of multi-period employment contracts / 53

This paper is the first in a series of three. Its emphasis is on describing the structure of competitive multi-period employment contracts when workers face residual employment-related risk. The second paper (Arnott and Stiglitz, 1981) focuses on the efficiency properties of these contracts, while the third (Arnott, Hosios, and Stiglitz, 1980) builds on the other two papers, incorpo- rating demand fluctuations, search costs, and unemployment.

The organization of the paper is as follows. In the first section we present the basic model and discuss some of the qualitative characteristics of multi-period employment contracts in economies with an incomplete set of insurance markets. The derivation of the comparative static properties of such contracts is presented in the appendix. Concluding comments are presented in the second section.

QUALITATIVE PROPERTIES OF MULTI-PERIOD

EMPLOYMENT CONTRACTS

The basic model The economy is very simple. A single numeraire commodity is produced according to a single, constant returns to scale technology in which the sole factor is labour. The risk-averse workers are equally productive and attach the same relative valuations to (have the same utility function over) income and job satisfaction. They differ, however, in what they like and dislike about a particular job; some enjoy rotating shifts, others do not; some enjoy Muzak on the job, others do not; and so on. Firms are competitive and risk-neutral. They offer different work environments: some have rotating shifts, other do not; some have Muzak, others do not; and so on. At the time a worker joins a firm, he does not know what the firm's work environment is like, nor can the firm judge ex ante which workers will enjoy the particular work environment it offers. Thus, when joining a firm, a worker is uncertain about how much he will enjoy his work environment (or, synonymously, what his level of job satisfaction will be with the firm, or what the quality of the job match will be). This is the only kind of uncertainty in the economy. While workers' tastes vis-a-vis the work environment differ, they are symmetric in the sense that the probability distribution of a worker's job satisfaction when he starts employment with any firm is independent of both the firm and the worker.2 To

2 A worker's level of job satisfaction may be formalized as follows: the worker's tastes concerning his work environment can be characterized by his most preferred point in work environment space which is the shell-of an n-dimensional sphere. The work environment provided by a particular firm can be characterized by another point in this space. The quality of a job match may be measured by the distance between the two points. And the level of job satisfaction is some monotonic function of this distance, having the range (-oo, oo). If we assume that both workers' tastes and firms' work environments are uni- formly distributed over the shell of the sphere, then the probability distribution of the level of job satisfaction associated with a random match is independent of the firm and of the worker.



54 / Richard Arnott

simplify the analysis, we assume that a worker's job satisfaction in a particular job remains unchanged over time.

Workers live and are employed for two periods. If a worker finds out that he dislikes his job, he may quit the firm, but only after working there for one period. He will quit if doing so increases his expected utility. A worker therefore has one of two possible employment histories at the time he joins a firm: 'never worked before' or 'quit after working with a previous firm one period.' Competitive equilibrium requires that zero expected profits be made on each group of workers with a particular employment history at the time of hiring. If it were otherwise, then profits could be made by hiring only workers with particular histories. Thus, we may usefully imagine that there are two types of firms.3 Type I firms hire workers when they have just entered the labour market; their employment contracts are characterized by a first- and second-period wage. Type 2 firms hire workers at the beginning of the second period of their working lives, after they have quit a type 1 firm; their employment contracts are characterized by a single, second-period wage. Both firm types, so defined, must, from the argument above, make zero profits in competitive equilibrium. Finally, an insurer can observe whether or not a worker quits but not his job satisfaction.

Before presenting the model formally, we shall discuss some of its characteristics. The assumptions have been chosen so as to provide as simple as possible a general equilibrium model which contains a multi-period employment contract, labour turnover, and incomplete insurance markets. The qualitative results of the paper generalize to more complex and realistic models.

The assumptions imply the incompleteness of insurance markets, in contrast to most previous papers on incomplete markets which have instead assumed incompleteness.

In the context of the economy described above, complete insurance markets would involve the insurer's providing full insurance against job dissatisfaction; that is, specifying a schedule of net pay-outs (negative over some range) as a function of realized job satisfaction such that a worker's marginal utility of income is the same whatever his job satisfaction.

Recall that the insurer cannot observe a worker's job satisfaction but can ascertain whether or not the worker quits. Suppose, in these circumstances, that the insurer were to provide full insurance against a worker's first-period job satisfaction. And suppose he were to ask the worker what his level of job satisfaction was. In reply the worker would have an incentive to lie; specifically, he would state that his job satisfaction was such that he would receive the largest net pay-out consistent with his observed behaviour, whether or not he quits. Thus, if the insurer were to offer actuarially fair full

3 This is done solely for expositional ease. In the context of this paper it makes no difference whether, when a worker quits a job, he takes another job in the same firm (in the conven- tional sense of the word) or joins a different firm.



The structlure of mlulti-period employment contracts / 55

insurance against job dissatisfaction per se, he would lose money. In fact, there is no truth-revealing mechanism. Whatever menu of insurance contracts the insurer offers and whatever question he asks the worker, the worker will always choose that contract (from among the contracts offered conditional on his observed behaviour) which provides him with the largest net pay-out and state that his job satisfaction is consistent with this choice. Hence, all quitters receive one level of net pay-out and all stayers another level, which in turn implies that the insurer can do no better than to offer a single insurance contract which specifies one level of net pay-out if the worker quits and another if he stays; he can do no better than provide straight quit insurance.4 Thus, insurance against job satisfaction, when it is unobservable, is incomplete and takes the form of quit insurance.

The incompleteness of insurance markets in the model results from an asymmetry in information: the worker knows his job satisfaction, but the insurer cannot observe it. In an economy in which the incompleteness of insurance markets derives from an asymmetry of information, two generic problems may arise in the provision of insurance. First, if individuals differ in their riskiness and if each knows better than the insurer into what risk-class he falls, adverse selection phenomena arise. We have excluded this possibility by assuming that individuals are, in relevant respects, identical. In so doing we have eliminated a potentially important determinant of the structure of multi-period employment contracts.5 Second, moral hazard6 arises whenever the provision of insurance affects the probability of the insured-against event(s). In our model the provision of insurance against quitting increases the probability of quitting; such insurance is therefore characterized by moral hazard. It is well-known that when moral hazard is present, typically only partial insurance is provided.

We have argued thus far that insurance against uncertain job satisfaction will take the form of partial insurance against quitting. Two questions remain. Who will provide this insurance, and in what form? In the context of the model, the same results are obtained whether the firm or an independent insurance agent provides such insurance. In a more complex economy, however, the firm has an informational advantage over an independent insurance agent. For instance, suppose the worker is paid different amounts depending on whether his separation from a firm is due to his having quit or his having been fired. The firm can monitor this better than an external agent. Relatedly, an independent insurance agent might find it difficult to ascertain

4 The above argument is correct only when capital markets are perfect. When capital markets are imperfect, the argument needs to be modified, but the same result - that full insurance is not offered in equilibrium - applies.

5 Extending the analysis of this paper to treat adverse selection would be useful. Rothschild and Stiglitz (1976) provide an excellent discussion of some of the problems which arise from adverse selection.

6 Arrow (1965), Spence and Zeckhauser (1971), and Pauly (1974) together provide a good discussion of moral hazard.



56/ Richard Arnott

whether a worker's move to a subsidiary firm were just a transfer or a quit and rehire. Thus, we expect the firm to provide quit insurance.

The quit insurance could be provided by the firm either explicitly or implicitly. We assume, by appealing to what is observed, that it is provided implicitly. In the economy of the paper, the pair of equilibrium contracts is characterized by three parameters: first period remuneration, second period remuneration if the worker quit at the end of the previous period, and second period remuneration if the worker did not quit at the end of the previous period. We assume that insurance is provided implicitly by setting these levels of remuneration above or below the corresponding value of net marginal product. This can be achieved in a variety of ways: seniority increments which do not correspond exactly to productivity increases, pensions, severance pay, and so on. We treat the three parameters of the contract as three wage rates. It should be kept in mind, however, that they have these other interpretations.

To illustrate the implicit provision of insurance in the employment contract let us assume that capital markets are perfect, and that with implicit quit insurance a quitter's present value of remuneration exceeds his present value of net marginal product by x, while a stayer's present value of remuneration falls short of his present value of net marginal product by y. Now if the quit insurance had instead been provided explicitly, stayers would have paid the insurance premium but not received the pay-out, while quitters would have paid the premium and also received the pay-out. Hence, implicit insurance with the parameters indicated is equivalent to explicit insurance with premium y and gross pay-out x + y.

The stage is now set to relate the structure of the employment contract to workers' tastes, the amount of risk, and the technology of production.

The following notation is employed: wi: the ith period (i = 1, 2) wage of a (type 1) firm which hires workers when

they enter the labour force W: the wage of a (type 2) firm which hires workers only for the second

period of employment (i.e., those who quit another firm at the end of the first period)

m: marginal product of labour ji: a worker's job satisfaction during his ith (i = 1, 2) period of employment

fLj]: the probability distribution of job satisfaction at the time a worker enters either a type 1 or type 2 firm (square brackets are used to enclose the arguments of a function),

F[j] = f [j']dj' -00

UH: utility (EU, expected utility) q: the proportion of workers who quit at the end of the first period



The struicture of multi-period employment contracts / 57

Since firms act competitively, the employment contracts offered in long-run equilibrium will result in each firm making zero expected profit. Meanwhile, knowing the employment contracts offered by all firms, a worker will choose to work for that type 1 firm whose contract provides him with the highest level of expected utility. Consequently, the equilibrium type 1 contracts will be those that maximize a worker's expected utility subject to yielding type 1 firms zero profit. Type 2 firms have no choice but to pay workers their (net) marginal product.

We first employ the model to investigate a particularly simple issue: if insurance markets are incomplete and if hiring a new worker entails no cost, will type 1 firms pay workers their marginal products or will some implicit insurance be provided in this case? The answer is that a positive amount, zero, or even a negative amount of implicit quit insurance may be provided, depending on the substitutability between consumption and job satisfaction. With perfect capital markets and separability between consumption and job satisfaction, workers' marginal utilities of income are equalized if the present values of their incomes are equalized. Since the present value of a worker's net marginal product is the same whether or not he quits the type 1 firm, then workers' marginal utilities of income are equalized by paying each the present value of his net marginal product, that is, by providing no implicit insurance. However, if consumption and job satisfaction are complementary and each worker is paid the present value of his net marginal product, the average marginal utility of income of stayers exceeds that of quitters, since stayers have higher average job satisfaction. Negative quit insurance should be provided, since this reduces the difference between the average marginal utility of income of stayers and that of quitters.

To illustrate the application of the model we examine this issue formally. We may view the determination of w1 and w2 as the solution to a two-stage maximization problem. In the first stage, workers decide on the subset of levels of job satisfaction corresponding to which they would quit and the complementary subset corresponding to which they would not,7 taking wl, w2, and ivw as given. This gives expected utility and the quit rate as functions of w1, w2, and w. In the second stage a type 1 firm chooses w1 and w2 to maximize workers' expected utility, subject to its making zero profits. In so doing it takes into account the dependence of the quit rate on w1 and w2.

A worker's utility if he quits may be expressed as U[w1, w, il, i2]; he receives wages of w 1 and iw, and has different levels of job satisfaction during the two periods. Meanwhile, the utility of a worker who stays with the same firm for both periods is U[w1, w2,j1, jI]; his wages are w1 and w2 in the two

7 Ordinarily, workers with lower job satisfaction will quit and others will stay. However, if a worker quits his future job satisfaction is uncertain, while if he does not quit it is certain. It is possible that a reduction in period-one job satisfaction makes a worker so much more risk averse that he is more reluctant to quit. Thus, it may occur that the subset of levels of job satisfaction corresponding to which workers quit is not connected.



58 / Richard Arnott

periods, and his job satisfaction is the same during both periods with the firm. It will normally (see fn. 7) be the case that there is a critical level of job satisfaction 0; a worker whose job satisfaction exceeds 0 stays, while other workers quit. For the worker with job satisfaction 0, the (expected, since second-period job satisfaction is uncertain) utility from quitting equals the (certain) utility from staying:

00

{U1w1, W 0,j2]dF[j2] = U[w1, w2, 0, 0]. (1) -00

(1) characterizes 0 as a function of w1, w2, and w:

0 = 01W1, W2, W]. (1')

Since everyone withj1 < 0 quits, then

q = F[0]. (2)

And (1') and (2) together imply

q = q[wl, W2, W]. (3)

Expected utility may be written as

EU-| U[w1, iiJ1,J2]dF[J1]dFli2] + U[w1, w2,j1,j1]dF[jl]. (4)8

The first integral on the right-hand side is the expected utility of quitters times the probability of quitting; the second is the corresponding term for stayers.

The budget constraint of a type 2 firm is

w -isn, (5)

which determines wj. From the substitution of (5) and (1) into (4), and (5) into (3), we may write

EU = EU[w1, w2j and q = q[w1, w2], (6a, b)

where iw is suppressed to simplify notation. This characterizes the solution to the first stage of the maximization problem.

In the second stage of the maximization problem, a type 1 firm chooses w1 and w2 to maximize (6a) subject to (6b) and its (expected) budget constraint. A type 1 firm faces the budget constraint

m +m(l -q) (1- q) (7) +r l +r

where r is the discount rate. To make zero expected profit it must, on 8 This expression and the results of the paper can be modified to take account of the point

made in fn. 7.



The strulcture of multi-period employment contracts / 59

average, make zero profit on each worker it hires. m + m(1 - q)I(1 + r) is the average discounted output per worker hired, while w1 + (1 - q)w21(1 + r) is the expected discounted payment. To simplify the analysis it is assumed here and throughout the rest of the paper that capital markers are perfect,9 and that the discount rate is zero. Thus,

U[w1, iw,jl,j2] U[w1 + W ,j1,J2],

U[w1, w2,j1,j1] = U[w1 + w2,j1j,1, and

m(2- q) = w1 + (1 - q)W2. (7')

The type 1 firm's maximization problem is therefore

max EU[W1, w2] s.t. m(2 - q[wl, w2]) =- wI + (1 -- q[wl, w2])w2. (8) WI, W2

Where X is the Lagrange multiplier on the budget constraint, the first-order conditions are:

w : a - A(1- -- (w2-m) =O,and (9a)

O9EU /- I q q(b aW2 aW2

which together imply

1 + q (m - w2) OEU/,W1 OW1 (9c) OEU/0W2 q

(1 -q) + --(m -W2) OW2

Since stayers pay the implicit insurance premium but do not receive the implicit pay-out, the implicit premium is calculated as stayers' discounted vmP minus their discounted remuneration, 2m - w- w2. The implicit net (of premium) pay-out, meanwhile, is calculated as quitters' discounted remun-

9 When capital markets are imperfect, the employment contract will usually provide not only implicit insurance but also implicit banking services. Implicit banking services will take the form of paying a worker above his VMP over those periods of time when his marginal utility of income is high, and below his VMP when it is low. Treating this aspect would complicate the analysis considerably.

Whether or not capital markets are perfect should be derived rather than assumed. Capital markets will be perfect if there is no risk of default. If the penalty for default on a loan made during the first period is confiscation of a worker's second-period wage plus his savings, his second period consumption is zero. If we assume that

&olt[C1, C2,1,2 =2] 0 C'C2 1?2=0

where ci, i = 1, 2 is consumption during period i, and Il [] is the worker's direct utility function, the worker will choose not to default. Thus, the assumption of perfect capital markets is consistent with our model for at least a certain subset of utility functions.



60 / Richard Arnott

eration minus their discounted vmP, w1 + w - 2m. Thus, m - w2 is the implicit gross pay-out. We may imagine that in choosing w1 and w2 the firm is choosing the implicit insurance premium and gross pay-out. The cost per entering worker to the firm of raising w 1 by one unit may therefore be viewed as the reduction in the implicit insurance premium paid by each worker (unity) plus the average increase per entering worker in the implicit gross pay-out which equals the implicit gross payout (m - w2) times the increase in the quit rate induced by the unit rise in wI (OqlOw1). The term (1 - q) + (9qkOw2) (m - w2) has an analogous interpretation. Thus, (9c) states that w 1 and w2 should be set so that the ratio of the average marginal benefit per entering worker from raising w1 to that from raising w2 equal the corresponding ratio of marginal costs.

To examine whether firms will pay workers their marginal product, we set W2 = m, which implies that all workers are paid their marginal product, and determine under what circumstances (9a) and (9b) are satisfied. (9a), (9b), and W2= m together imply that

OEU 1 OEU .wi

. q Q

= ...,(10)

IW 1-q OW2'

Now, OEU/aW1 is the average marginal utility of income of all workers, since all workers receive w1. And (CEU/OW2) *. (1 - q) is the average marginal utility of income of stayers, since only they receive w2. Thus, (10) states that the average marginal utility of income of stayers equals that of quitters. Workers should therefore be paid their marginal products if and only if doing so results in the average marginal utility of income of quitters equalling that of stayers. 10 If income and job satisfaction (or equivalently consumption and job satisfaction) are strongly separable in demand, that is,

U[Y,j1Y j2] = U[y] + VU[jlj2],

where y is income, this condition holds, since a worker's income and hence his marginal utility of income are the same whether he quits or stays. However, if job satisfaction is a substitute for income, that is,

U[y, Z[U,1j2]], Uy, < 0,

then with income the same for quitters and for stayers, the average marginal utility of income of quitters exceeds that of stayers. In this case expected utility can be increased by transferring income from stayers to quitters, which can be achieved by setting w1 > m and w2 < m. After a certain point, as more income is transferred, thus increasing the quit rate, average job satisfaction falls. In the equilibrium the gain in utility from transferring extra income (the reduction in the difference between the average marginal utilities of income of 10 Becker implicitly assumed in his argument that when there is no cost to hiring a worker

and no training, workers will be paid the value of their marginal product at each point in time.



The striucture of multi-period employment contracts / 61

quitters and stayers) just offsets the loss (the fall in average job satisfaction). Thus, as (9c) indicates, the implicit insurance provided will stop short of equalizing the average marginal utility of income of the two groups. The analogous argument for the case where income and job satisfaction are complements is left to the reader.

The results of the above argument are summarized in proposition 1.

Proposition 1 Even where there is no cost to hiring a worker and no training, when workers are risk averse, when insurance against job-related risks is incomplete, and when there is labour turnover, multi-period employment contracts will not in general pay workers the value of their marginal product at each point in time.

However, in the special case where capital markets are perfect and consumption is strongly separable in demand from the outcome of the random variable which determines whether or not a worker quits, multi-period employment contracts will pay workers the value of their marginal product at each point in time.

It should be evident that proposition 1 applies very generally. There can be many time periods, many factors of production, and many sources of uncertainty which influence a worker's decision whether or not to quit, and the proposition still holds.

The next two subsections extend the argument to treat hiring and training costs.

Hiring costs There are typically fixed costs to hiring a worker. They include the costs of advertising and interviewing, as well as the costs of training the worker in the basic skill he is to perform. We term these fixed costs 'turnover costs' and distinguish them from variable training costs which we associate with the upgrading of skills. To simplify the analysis we assume that the turnover cost per worker is T for both type 1 and type 2 firms. If workers were paid their marginal product less the turnover costs they cause (net marginal product), stayers would receive an income over the two periods of 2m - T, while quitters' income would be 2m - 2T. Since quitters would then receive a lower income, then unless job satisfaction and income are strongly complementary, they would have a higher marginal utility of income, on average, than stayers. In this case, therefore, one expects quitters to be paid more and stayers to be paid less than their net marginal product.

A type 2 firm has no choice but to pay workers their net marginal product, w = m - T. A type 1 firm faces the maximization problem

max EU[W1, W2] s.t. m(2 - q[w1, w2) - T WI, W2

-W1 + (1 - q[w1, W2)W2, (11)



62 / Richard Arnott

which is the same as (8) except for training costs. The first-order conditions are the same as (9a) and (9b) and have an analogous interpretation. Intuitively, one expects the amount of implicit insurance provided in the employment contract (which may be measured by the size of the premium (2m - T - w - w2) or the size of the (gross) pay-out (m - w2)) to be larger, ceteris paribus, the more risk-averse are workers, the greater the uncertainty associated with job satisfaction, and the less sensitive is average job satisfaction to the amount of insurance provided. Now, if an increase in the amount of insurance provided has no effect on the quit rate, it will also have no effect on average job satisfaction; more generally, the more sensitive average job satisfaction is to the amount of insurance, the more sensitive is the quit rate. One therefore expects more implicit insurance to be provided the less sensitive the quit rate is to the size of the insurance pay-out. The above intuition is checked in the appendix, which examines the comparative static properties of the type 1 employment contract.

This is a convenient point at which to note that the first-order conditions may characterize multiple local optima. This may be seen by rewriting (9c). Subtracting one from both sides and multiplying the resulting expression by (1 - q)/q gives

( 1 + )L~? ( (m w2)! q aW1 aW2 aWi aW2 q 1 &EU / 1 (12)

1 + -(m - W2) I--q aW2 aw2 1-q

Now (aEU/aW2) . (1 - q) is the average marginal utility of income of a stayer and ((aEU/Iw1) - (aEU/aw2))/q the corresponding term for a quitter. As the amount of insurance provided rises, the former term increases while the latter decreases. Thus, the expression on the LHS of (12), which we shall refer to (inexactly) as the marginal benefit from insurance (MB), falls as the amount of insurance provided increases. The expression on the RHS of (12), which we call the marginal cost of insurance (Mc), may, however, rise or fall as the amount of implicit insurance provided increases. There may therefore be several levels of insurance at which (12) is satisfied, with those levels at which MB cuts MC from above being local maxima. These results are shown in figure 1.

The results of this subsection are summarized in proposition 2.

Proposition 2 When workers are risk-averse, when insurance against job-related risks is incomplete, when there are costs to hiring a worker, and when there is labour turnover, multi-period employment contracts will typically provide implicit insurance. This implicit insurance will stop short of equalizing the average marginal utility of income of quitters and stayers.



The structure of multi-period employment contracts I 63

MB MC

MB LOCAL MAXIMA

FULL AMOUNT OF INSURANCE PROVIDED

FIGURE 1 Possibility of multiple local optima

Training The treatment of specific and general training is a straightforward extension of the analysis of hiring costs. We measure the amount of training provided by the firm in terms of the increase it induces in a worker's marginal product. 1I The cost to the firm of providing a worker with g units of general training and s units of specific training is c[s, g]. We assume that there are positive but diminishing returns to training. 12 To simplify the analysis the inconsequential assumption is made that the productivity gains from training occur in the period subsequent to that in which the training occurs. An implication of this in the model is that type 2 firms do not train workers, while type 1 firms train them only in the first period.

With general and specific training the implicit insurance premium is 2m + g + s - T - c[s, g] -w - W2, the implicit net (of premium) pay-out is w, + w- 2m - g + 2T + c[s, g], and the implicit gross pay-out is iw + T + s -W2 = m + g + s - w2.

11 This involves no ambiguity, since we have assumed that the marginal product of labour is constant. If, however, there were diminishing marginal productivity of labour, the way in which the amount of training is measured would have to be modified.

12 c8 cg0, C881, c , CssCg - (c sg)2 > 0.



64 / Richard Arnott

To simplify the discussion, we make three additional assumptions:

Ai) workers have zero time preference; Aii) a unit of consumption is perfectly substitutable for a unit of job

satisfaction; Aiii) workers' job satisfaction in type 2 firms is certain and equals zero.

Thus,

U[w1, w,J1,J2] = U[W1 + w + i]

U[W1, W2,jl,jl] = UjW1 + W2 + 2ilj]

Under these assumptions a worker will quit if w1 + w +1l > w1 + w2 + 2j1 or - i > W2 - Wi; that is, the gain in job satisfaction from quitting exceeds the financial penalty. Note that with (Ai)-(Aiii), a worker's quit decision does not depend on wl; thus, aqlaw1 = 0.

We first treat general training. We investigate how the equilibrium contract for firms of type 1 changes as the amount of general training provided increases.

Consider a type 1 firm which is offering an equilibrium contract. It now increases general training by one unit, which reduces the worker's net marginal product by C9 in the first period, and increases his second-period marginal product by one unit. In what way will the firm alter the wage structure in response to this increase in general training? To answer this question we suppose that the firm lowers the first-period wage by c9 and raises the second-period wage by unity - and trace through the consequences.

The reader can check that the hypothesized change in training cum wage structure:

1 does not alter the amount of implicit insurance provided; 2 does not alter the quit rate (which depends on w - w2) and keeps the

firm on its budget constraint; and 3 changes all workers' incomes by 1 - c9.

Thus, the hypothesized change in wage structure results in too much / the right amount of / too little implicit insurance being provided if the change in all workers' incomes by 1 - cg decreases / does not affect / increases the demand for insurance. A rise in income decreases / does not affect / increases the demand for insurance as tastes exhibit decreasing / constant / increasing absolute risk aversion (ARA). Putting these two results together implies that the hypothesized change in wage structure provides too much implicit insurance if either 1 - cg > 0 and ARA is decreasing, or 1 - c9 < 0 and ARA iS

increasing, etc. It may be shown that, at least under the stated assumptions,



The structure of multi-period employment contracts / 65

the profit-maximizing amount of general training to provide is given by C= 1.13 We present these findings in proposition 3.

Proposition 3 In the model of the paper, under (Ai)-(Aiii), an increase in the amount of general training increases the amount of implicit insurance provided in the employment contract if either the amount of general training is less than the profit-maximizing amount and ARA is increasing or the amount of general training is greater than the profit-maximizing amount and ARA is decreasing.

A cataloguing of the other cases is left to the reader. We note, however, that if the amount of general training is profit-maximizing, then a small change in the amount of general training will not affect the amount of implicit insurance provided in the employment contract. Furthermore, if ARA is constant, the amount of implicit insurance is independent of the level of general training. 14,15

We now turn to specific training. As Becker (1962, 19-21) rightly pointed out, the effects of specific training on the wage strucIture are considerably more complex than those of general training. To simplify the discussion, we retain assumptions (Ai)-(Aiii). We shall perform a conceptual experiment

13 The object of the firm is to

max, ,W2ssEu = J U[wI + w +j]dF[J] + U[w1 + W2 + 2j]dF[j]

s.t. i) O = m + g-T-W2

ii) m(2 - q) + (1 q)(g + s) - w - c[s, g] - T -(1 - q)w2 = 0

iii) q = F[O]

iv) w = m + g - T

The first-order conditions are

s: A((1 -q) - c.) = O

g: (EU1 -EU2) + X((1 -q) -cg-(aqlag)(m + g + s -w2)) = 0

W1: EU,-X=0

w2: EU2 - A((1 - q) + (aq/aW2)(m + g + s -- W2)) = 0

Substituting the F.O.C.S with respect to w, and w2 into that for g gives X(1 - cg) = 0. Thus, at least with assumptions (Ai)-(Aiii), the formulae characterizing the optimal amounts of specific and general training are the same as in the absence of uncertainty.

14 Results simplify with an additive random variable (r.v.) when there is constant absolute risk aversion, and with a multiplicative r.v. when there is constant relative risk aversion. Here the assumption of perfect substitutability between consumption and job satisfaction (the random variable) has the effect of making the r.v. additive. This is why the results of the paper are so neat when there is constant absolute risk aversion.

15 Becker incorrectly argued that in response to a unit increase in general training, workers' wages will always fall by the cost of the general training in the training period, and rise by the increase in marginal product in subsequent periods. Becker's arguments are correct, however, for the special case of perfect capital markets, perfect substitutability between consumption and job satisfaction, and constant absolute risk aversion.



66 / Richard Arnott

analogous to that for general training - we suppose that the firm increases the amount of specific training provided by one unit, and simultaneously increases w2 by one unit and lowers w1 by cs units - and trace through the consequences.

First, note that the posited change does not alter the amount of implicit insurance provided. The way we shall proceed is to investigate how, at this level of insurance, the change alters the marginal benefit from and the marginal cost of insurance, as they were defined after equation (12). From (12) (making an appropriate modification for the presence of training),

1 (aEU aEU

q Jwlm -W2 MBlS ' 1 aEU

1-q aW2

which, recall, gives the ratio of the marginal utility of income of quitters to that of stayers, while

+1 +(-- (m +g +S-W2)- awl aW2 q

MC = - __

1+ (m +g+s-W2)1q

Consider first how the change affects marginal benefit. As a result of the change a quitter's income falls by cs, while a stayer's income rises by 1 - c,. Thus, if 1 - cs > 0 (the profit-maximizing amount of specific training is given by (1 - q) = cs (see fn. 13)) marginal benefit unambiguously rises, while if 1 - cs < 0 marginal benefit rises unless absolute risk aversion is strongly increasing.

Now consider how the change affects marginal cost. We obtain the following intermediate results. The hypothesized change:

1 does not alterm +g +s -w2; 2 decreases the quit rate, since this is an increasing function of w -W2 = m +

q - T - w2;and 3 has an ambiguous effect on aqlaw2, while aqlaw1 = 0.

Putting these results together, and noting that aqlaw2 < 0 and m + g + s - w2> 0, we obtain that the hypothesized change has an ambiguous effect on MC, depending in a complicated way on the quit rate, the elasticity of the quit rate with respect to wT - w2, and the size of m + g + s - w2, all of which depend in turn on the amount of specific training and the characteristics of f [j] and of worker risk aversion. Generally there are no simple results to be



The struictuire of multi-period employment contracts / 67

had concerning the effect of an increase in specific training on the equilibrium wage structure when the employment contract provides implicit insurance. 16

The above arguments concerning how changes in the levels of general and specific training provided by a firm affect its wage structure can be modified to ascertain the effect on the amount of implicit insurance provided by an increase in hiring or turnover costs. An increase in the hiring costs of a type 1 firm decreases all workers' net marginal products by the same amount and therefore has the same qualitative effects on the wage structure as a decrease in general training for the case 1 - c, > 0. Meanwhile, an increase in the hiring costs of a type 2 firm by one unit is equivalent to a unit increase in specific training combined with a unit decrease in general training, when these changes in levels of training do not affect training costs.

The formal comparative static results of the effects of changes in s, g, and T on w1 and w2 are given in the appendix. The only result there which we have not touched on is that the share of marginal specific training costs borne by the worker may be less than zero. We have been unable to establish whether an analogous result applies for total specific training costs.

Becker's intuition is consistent with most of our results, although he erred on a few points. He took it for granted, for instance, that in the absence of hiring and training costs workers are paid their marginal products, and we have seen that this is generally true only with perfect capital markets and strict separability between income and the outcome of the random variables which affect the quit decision. Furthermore, he argued that general training costs are borne fully by workers while the full increase in VMP which results from such training accrues to them. We have seen that this is generally true only for

16 Becker (1962, 19-21) discussed the sharing of the costs and benefits of specific training in the context of an economy in which a job separation could be due either to a worker's being fired or to his quitting: 'If a firm had paid for the specific training of a worker who quit to take another job, its capital expenditure would be partly wasted, for no further return could be collected. Likewise, a worker fired after he had paid for specific training would be unable to collect any further return and would suffer a capital loss. The willingness of workers or firms to pay for specific training should, therefore, closely depend on the likelihood of labor turnover ... The shares [of specific training costs paid] depend on the relation between quit rates and wages, layoffs and profits, and on factors not discussed here, such as the cost of funds, attitudes toward risk, and desires for liquidity.' If our model were elaborated to include firing, it would substantiate Becker's argument.

Hashimoto (1978) solved for the optimal sharing rule in an economy with risk-neutral firms and workers. The quality of a job match has two dimensions: worker productivity and job satisfaction. The proportion of costs and benefits from specific training assumed by workers is determined so as to maximize the sum of output and output-equivalent job satisfaction. This will occur when the average gain in job satisfaction from marginal workers' leaving the firm just offsets the average loss in output from these workers quitting.

Donaldson and Eaton (1976) argue that the firm will impose a very high turnover penalty to protect its investment in specific human capital. Hashimoto pointed out that their argument is erroneous, since it ignores the fact that the higher is the turnover penalty, the lower is (the analogue to) average worker job satisfaction (in his model), and the higher will wages have to be to attract workers to the firm.



68 / Richard Arnott

constant absolute worker risk aversion or for incremental training in the neighbourhood of the profit maximum. Becker did perceive the most important point, however: that the way in which the costs of and the benefits from training are shared between firm and worker depends on the risk associated with such training and the risk aversion characteristics of the two groups of agents.

We now digress briefly to treat a related issue. To maximize profits, training should take place until, at the margin, the cost to the firm of an extra unit of training equals the increase in revenue generated thereby. The cost of providing infra-marginal units of training, however, will fall short of the benefits. Thus, the provision of training results in infra-marginal surplus. There has been a debate in the literature, which to our knowledge has never been resolved, concerning the disposition of the surplus from training. In our model the entire surplus accrues to labour. 17 However, if there were capital in production which were not completely elastically supplied to the industry, if labour were elastically supplied to the industry at a given level of expected utility, and if the good produced were perfectly elastically demanded, then the entire surplus would accrue to the owners of capital. Alternatively, if both capital and labour were perfectly elastically supplied to the industry but demand were not elastic, the entire surplus would accrue to consumers. It should be evident from these three cases that the determination of the disposition of the surplus from training is a standard exercise in incidence analysis 'a la Harberger. The disposition between factors and consumers will depend on the elasticities of supply of the factors to the industry in question, the elasticity of demand facing the industry for its product, the partial elasticities of substitution between the various factors, and the relative shares of the factors.

Proposition 4 The determination of the disposition of the surplus from specific and general training is a standard exercise in incidence analysis.

Some other comparative static reslults The comparative static properties of the equilibrium type 1 employment contract are derived in the appendix. Here we present and comment on the principal results concerning the effects of increased risk and risk aversion on the amount of implicit insurance provided in the employment contract. These results are summarized in proposition 5.

Proposition 5: In the economy treated in the paper, with (Ai)-(Aiii): a) An increase in workers' degree of risk aversion causes the amount of

implicit insurance provided in the employment contract to increase; and

17 Becker assumed that the entire surplus accrues to labour.



The structutre of multi-period employment contracts / 69

b) An expected-utility-preserving increase in risk, holding the quit rate and its sensitivity to the amount of implicit insurance fixed, causes the amount of implicit insurance provided to increase.

Both of these results are what one would have expected. Less obvious is that an expected-utility-preserving increase in risk generally has an ambiguous effect on the amount of implicit insurance provided in the contract, since it has an ambiguous effect on the quit rate and its sensitivity to the amount of implicit insurance provided, both of which influence the equilibrium quantity of implicit insurance.

CONCLUDING COMMENTS

The object of this paper was to investigate the characteristics of multi-period employment contracts in economies in which, as a result of asymmetric information, markets for insurance against those job-related contingencies which affect a worker' s quit decision are incomplete. We constructed a simple model in which to examine the structure of such contracts: The only factor of production was labour; labourers were ex ante identical and worked two periods; and the only source of uncertainty was the quality of a worker-firm match or, equivalently, job satisfaction. Despite the simplicity of the model, the qualitative results obtained clearly generalize:

When insurance markets against job-related contingencies are incomplete, multi-period employment contracts will contain implicit insurance. Since moral hazard is present, only partial insurance is provided.

In our model insurers were able to observe only whether or not a worker quit, not his job satisfaction. As a result, (partial) insurance was provided against quitting, which resulted in moral hazard. Quitters were the unlucky workers. The insurance typically compensated them by paying them in excess of their net marginal product. In other circumstances in which quitters are the unlucky ones and are unable to purchase full insurance against the outcome of the random variable(s) which determines the quit decision, one expects the same result. If quitters are the lucky ones, however, which would be the case when workers quit exclusively to take higher-paying jobs, then insurance would normally result in the firm' s providing the implicit insurance pay-out to stayers rather than to quitters.

The extension of the analysis of the paper to n periods or to continuous time is conceptually if not notationally straightforward. Two interesting complications arise. First, the quit rate from a particular firm for workers of a given seniority depends not only on that firm's wage structure, but on those of other firms as well. With this interdependence employment contracts still provide implicit insurance, but generally not the optimal amount. We return to this point later. Second, unless job satisfaction and consumption are



70 / Richard Arnott

strongly separable, a worker's attitude towards risk will generally depend on his job satisfaction history. Thus, different employment contracts will typically be chosen by workers with the same employment history, some more risky, others less.

When there is more than one source of uncertainty, one expects the implicit insurance pay-out provided to workers of a given seniority to differ depending on the reason for the worker's leaving the firm, to the extent that this is observable. Early retirement provisions in the firm's pension, disability payments, and the conditions under which severance pay is granted all reflect this. Relatedly, the complexity of actual employment contracts is explainable to a large degree as resulting from their providing many forms of implicit insurance.

The employment contract is but one of many social institutions which involve an (implicit) multi-period contract. Friendship and family are the two most obvious other such social institutions. Just as with the employment contract, one expects these institutions to provide partial insurance against contingencies. The theory we have developed can therefore be applied to a wide variety of problems. In its current form, however, it is unrealistic in an important respect; we assumed that all individuals were ex ante identical. This was a useful assumption, since it allowed us to abstract from adverse selection problems. However, one expects these problems to influence significantly the form of multi-period contracts, and in particular to result in less insurance being provided than would be the case if the insurer could ascertain to which risk class each insured belonged.18 Thus, extending the model of the paper to treat adverse selection would be most useful.

The goal of this paper was to describe competitive multi-period employment contracts. The next paper in the series (Arnott and Stiglitz, 1981) examines the efficiency properties of such contracts. It is argued there that these contracts are generally inefficient. In deciding how much insurance to provide, a firm's manager neglects to consider that the contract he offers affects the profitability of the contracts offered by firms of other types. 19 The third paper in the series (Arnott, Hosios, and Stiglitz, 1980) modifies the structure of the model so that some workers are unable to find another job immediately and may become unemployed, and it investigates whether the amount of unemployment is efficient.

This paper also touched on a couple of issues in the traditional labour economics literature. First, Becker (1962) argued that the share of the costs of specific training borne by workers and of the benefits accruing to them

18 Adverse selection also leads to the possibility of non-existence of equilibrium in the Rothschild-Stiglitz (1976) sense.

19 This argument applies to an economy in which workers are employed for more than two periods. However, in a two-period economy, since the type 2 firm wage, w, is fully determined by its budget constraint, firms of neither type can influence the profitability of firms of the other type.



The structlure of multi-period employment contracts / 71

depends on workers' risk aversion, among other things. The subsequent literature on the sharing of specific training has ignored this point by assuming risk-neutral workers. We formalized Becker's insight. Second, we argued that the disposition of the surplus from training, which appears to be an unresolved issue, can be determined using standard incidence analysis.

APPENDIX: DERIVATION OF THE COMPARATIVE STATIC

PROPERTIES OF THE EQUILIBRIUM CONTRACTS

In this appendix we provide a fairly thorough cataloguing of the comparative statics properties of the wage structure of competitive employment contracts in our hypothetical economy. We formally derive those comparative static results which were obtained in the body of the paper by verbal argument. We also derive a number of other results not discussed there; in particular, we examine the effects of increases in uncertainty and worker risk aversion.

To keep the analysis manageable we retain assumptions (Ai)-(Aiii). We then obtain

[m+g-T-W2

EU J U[w1 + m - T + g +j]f[]dj J -oo~~~~~~~~~~~~~c

+ f U[w1 + W2 + 2j]f Li]dj (13) m +g-T-W2

The budget constraint of a type 1 firm is

m( 2 - q) + (1-q) (g + s)-w1-c[s, g]-T -(1 - q)w2 = 0, (14)

while that for a type 2 firm is given by

m - T + g - 0, (15)

which determines w. Also,

q = F[O] = F[w-w 2] = F[m + g - T -w2]. (16)

To simplify notation, we define EUi aEU/awi and EUij = O2EU/0wiwj, where EU is given in (13). Then the first-order conditions of a type 1 firm's profit maximization problem are:

wl: EU1 - X = 0, (17a)

and

w2: EU2- ((1-q)+ q p O

or EU2 - X 1 - F[O]) -f [fO]p) = 0, (17b)

where p m + g + s -- w2 is the implicit (gross) insurance pay-out. Total



72 / Richard Arnott

differentiation of (17a), (17b), and (14), using 0 = m + g - T - w2 gives

EU11 EU12 -1 dwl EU11 - EU12

EU12 EU22- Xh -H dw2 = -U'[flf [0] + X(h-f[O]) dT

-1 -H 0 dX H+ F[0] V (18)

EU12 -EU22 h EU12 -dEU(

+ U' [Qlf[O] - Xh dg + -xf [0] ds + U ' [Qlf [0] - Xh dm,

L-H + c, I cs(l - F[O] - 1- H

where h = 2f[0] + f'[0]p, fQ = w1 + w2 + 20, and H = 1- F[0] - f [0]p. Hereafter we suppress the argument of F, f, andf'.

It will prove instructive to examine two special cases. First, for risk-neutrality,

dw_ dw2 dw1 dw2

dT -1, dT = ?

dg -cg, dg 1

(19) dw- -Cs dw2 dw= 1, dw2 1. ds ds dm dm

Workers are paid the value of their net marginal products, and assume the full costs of and benefits from training. Second, for constant absolute risk aversion of degree A = - (U"/U').

dw l Xf dw2 Xf dw1 dw2 dT = - q - ( 1 - q) ' dg cl,yd 1 g (20) dw1 Xfc s dw2 Xf dw1 dw2 ds A' ds A dm 1 dm

where A = AHX(1 - H) - U'[jlf + Xh (> 0 for a maximum). We ex- plained the effect of a change in the amount of general training provided for the constant absolute risk aversion case in the text. dw2lds may be re- written as:

dw2 Xf ds2 AtI [fil +-LP AH(I - H . (21)

(f (+ A ) f A

From this equation it can be seen that the share of the costs of marginal specific training borne by the firm and of the benefits derived from it is directly pro- portional to the degree of worker risk aversion and to the sensitivity of the quit rate to w - w2. Note also that dw2IdT = (dw2lds) - (dw2ldg) and dw1ldm = dw2ldm = 1, which accord with arguments made in the text.



The structuire of multi-period employment contracts / 73

Since

lim A = Xf, A --*

the results for the constant absolute risk aversion case, as the degree of risk aversion approaches zero, collapse to the risk neutrality case, as one would expect. One may rewrite (21) as dw2lds = 1 - D, where

U'[Q] f'p AH(1-H) 1-AI + +-A

D= (22) ( I'Q Ifi ',p +AH(I1-H))

is interpreted as a dampening coefficient, which equals zero when the worker is risk-neutral and one when he is completely risk-averse. Note, however, that for intermediate values of A, D is not necessarily positive.

Intuitively, we expect that the more rapidly the degree of absolute risk aversion is decreasing in income (or more precisely in income plus money- equivalent job satisfaction) the greater the decrease in implicit insurance provided in the employment contract, in response to a change which increases workers' expected utility. Unfortunately, we have been unable to formalize this intuition.

We next examine the effects of an increase in workers' risk aversion, and then of changes in the characteristics of F[].

We may write a type 1 firm's maximization problem as

max EU[w1[w2], w2], W2

where w I1w2] is solved for from a type 1 firm's budget constraint. Theorem 4 of Diamond and Stiglitz (1974, 349) states:

Let a* be the level of control variable which maximizes f V(U(4, a), p)dF(). If increases in p represent increases in risk-aversion ..., then a* increases (decreases) with p if there exists a 1* such that Ur > (0 0) for 0 S * and Ua, - 0 (? 0) for

In our problem, j corresponds to +, and w2 to a. Thus, to obtain an unambiguous comparative static result, it is sufficient to show that there exist a j* such that Uw2 0 for j - j* and Uw2 : 0 for j : j*. We show below that with j* = 0, these conditions are satisfied.

From (13) and (14),

JU[m(3 - q) + g(2 - q) + s(1 - q) - c[s, g] - 2T

U[j, w2] = - (1 - q)w2 +j] forj < 0 (23)

-U[m(2-q)+(I -q)(g +s)-c[s,g]-- T+qw2+2j] forj > 0.



74 / Richard Arnott

Thus - (I1q)+ Oq )fo j 0

UW2,U, W2]= -aW2 /

(24)

q- Oq forj j 0. OW2

"

Since EU2 > 0, then from (17b), Uw12 < 0 forj < 0. Also since EUI > EU2, then UW2> O for ] > 0. Thus, an increase in worker risk aversion results in a decrease in w2 and an increase in the implicit insurance pay-out, the expected result.

The most natural form of increasing risk in the context of this problem preserves expected utility (Diamond and Stiglitz, 1974). Let e denote a shift parameter, with larger values of e indicating more expected-utility-preserving risk. Since changes in e do not affect expected utility, EUe = 0. We obtain

EUll EU12 -1I dwi -EUie

EU12 EU22- Xh -H dw2= -EU2e-(Fe + fep) de. (25) 1 -H 0 d LFep

Again, we focus on the case of constant absolute risk aversion. Now,

EUe Ufedj + {Ufedj, (26)

where the arguments of U (see (13)) are suppressed to simplify notation. Integrating (26) by parts gives

6 ( X 60 EUe UFe - JU'Fedi + UFe 2 U'Fedj

- - | U'Fedj - 2 U'Fedj. (27)

Proceeding similarly, one obtains

ro r0 EUie - J U"Fedj - 2 U"Fedj, (28a)

and rX

EU2e = - U' []Fe[] - 2J U"Fedj. (28b)

With constant absolute risk aversion, EUe - 0, (27), and (28a) together imply that EUie = 0. Thus,

de2 - - U'[Q])Fe[O] + Xfe[O|p) - 2 U"Fedj A (29)

(1) (2)



The structure of multi-period employment contracts /75

It follows from (29) that an expected-utility-preserving increase in risk has two effects. The quit rate effect, which results from changes in F[O] andf [0], is captured by term (1) and is of ambiguous sign. The pure uncertainty effect, reflected in term (2), is negative, since A > 0 and

J U"FTdj = A U'Fedj < 0,

which follows from the definition of an expected-utility-preserving increase in risk. With more uncertainty, c.p., the employment contract contains more implicit insurance.

(25) can also be used to determine the effect of small changes in F andf in the neighbourhood of 0, by interpreting e as the size of a perturbation in F[O], holding f [0] constant, and of a perturbation in f [0], holding F[0] constant, respectively. For such changes EUie : EU2e 0 O. For an increase in F[0], dw2lde = (X - U'[0])Fe[0] . A, the sign of which is ambiguous. For an increase in f[0], dw2lde = Xfe[O]p + A > 0. Such a change increases the responsiveness of the quit rate to the amount of insurance provided, thereby increasing the marginal cost of insurance and lowering the equilibrium amount provided.

REFERENCES

Arnott, R.J. (1978) 'The control of labor turnover with incomplete insurance markets: the one-group case.' Discussion paper 292, Institute for Economic Re- search, Queen's University

Arnott, R.J., A. Hosios, and J.E. Stiglitz (1980) 'Implicit contracts, labor mobility, and unemployment.' Mimeo, December

Arnott, R.J. and J.E. Stiglitz (1981) 'Labor turnover, wage structures and moral hazard: the inefficiency of competitive markets.' Mimeo

Arrow, K. (1965) Aspects of the Theory of Risk-Bearing (Helsinki: Yrjo Jahnssonin Saatio)

Azariadis, C. (1975) 'Implicit contracts and underemployment equilibria.' Journal of Political Economy 83, 1183-202

Becker, G. (1962) 'Investment in human capital: a theoretical analysis.' Journal of Political Economy 70, Suppl., 9-49

Diamond, P. and J. Stiglitz (1974) 'Increases in risk and risk-aversion.' Journal of Economic Theory 8, 337-60

Donaldson, D. and B.C. Eaton (1976) 'Firm-specific human capital: a shared investment or optimal entrapment?' This JOURNAL 9, 462-72

Hashimoto, M. (1978) 'Firm-specific human capital as a shared investment.' Discus- sion paper 78-13, Institute for Economic Research, University of Washington

Oi, W. (1962) 'Labor as a quasi-fixed factor.' Journal of Political Economy 70, 538-55

Parsons, D.O. (1972) 'Specific human capital: an application to quit rates and lay-off rates.' Journal of Political Economy 80, 1120-43

Pauly, M. (1974) 'Overinsurance and the public provision of insurance: the roles of moral hazard and adverse selection.' Quarterly Journal of Economics 88, 44-62

Ross, S. (1973) 'The economic theory of agency: the principal's problem.' Ameri- can Economic Review 63, 134-9



76 / Richard Arnott

Rothschild, M. and J. Stiglitz (1976) 'Equilibrium in competitive insurance markets: an essay on the economics of imperfect information.' Quarterly Journal of Economics 90, 629-49.

Spence, M. and R. Zeckhauser (1971) 'Insurance, information, and individual action.' American Economic Review, Papers and Proceedings 61, 380-7

Stiglitz, J. (1974) 'Incentives and risk sharing in sharecropping.' Review of Economic Studies XLI, 219-55

-(1975) 'An economic analysis of labor turnover.' Working paper no. 53, The Economics Series, Institute for Mathematical Studies in the Social Sciences, Stanford University



Documents

The Structure of Multi-Period Employment Contracts with Incomplete Insurance Markets