# Firm specific training, layoffs, and unreliable workers.

FIRM SPECIFIC TRAINING, LAYOFFS, AND UNRELIABLE WORKERS

1. INTRODUCTION

Early work in implicit contract theory argues that a long-term relationship between a worker and a firm evolves due to the risk-sharing nature of the implicit contract between a risk-averse worker and a risk-neutral firm. Some authors have criticized this initial argument (see, for example, Haltiwanger [1983!), feeling that a long-term relationship between a worker and a firm (and the terms of an implicit contract between the two) in large parts is explained via hiring costs, training costs, and the like. This is the approach taken in the early work of Becker [1975!, Oi [1962! and the most recent work of Jovanovic [1979a!.

These latter explanations are also appealing because long-term relations between a worker and a firm are typically characterized by quits that decline with tenure and wages that increase with tenure. As discussed in Mortensen [1988!, explanations of these phenomena fall into two categories: job search theories or job training theories. Job search theories stress that there is a match specific component to the productivity of any worker-firm match. That a good match will result in a long-term relationship explains why wages are positively correlated with tenure. To explain a rising wage profile within a given match requires that the firm and the worker learn about the quality of their match over time. A good match will be a long-term one, and the wage will rise over time as the match is discovered to be a good one. (Jovanovic [1979b! and Barron and Loewenstein [1985! model such a learning process.) Job training theories maintain that long-term relationships with rising wages exist because of firm specific training that increases with tenure (again see Jovanovic [1979a! as an example). This paper integrates these two theories.

In this paper firm specific training evolves as an increasing function of a worker's tenure with a firm. This is so because there is a positive probability that a worker is unreliable in that he will be searching for another employer. The firm, being unable to identify in advance which workers are unreliable, will update its prior belief as to a worker's reliability as a function of the worker's tenure. The total level of training at any point in time will be an increasing function of this updated belief. Thus learning about the quality of a match, an aspect of the job search theories, plays an important role in explaining the level of firm-specific training.

This learning process can also play a role in determining optimal layoffs. If an unreliable worker is more likely to receive an acceptable outside offer when the worker is laid off than when he is employed, the firm can use layoffs to elicit information about the worker's reliabiliy. Based upon this fact, layoffs may occur even when the worker's marginal product exceeds his value of leisure.

Before discussing the model in detail, one crucial assumption deserves comment. Unlike previous work, such a Salop and Salop [1976!, enforceable self-revealing contracts are ruled out by assumption. Thus the firm here cannot commit itself to a wage contract promising low wages initially in exchange for high wages as job tenure increases, which contract would attract only reliable workers.

This assumption is of course a polar extreme, but as such is consistent with more recent work in contract theory which investigates the terms of self-enforcing contracts (see, for example, Holmstrom [1983!). The idea is to provide a comparison with contracts in an idealized economy with no enforcement problems. This assumption is justified if the cost of writing a legally enforceable contract is prohibitive or if, as suggested in charmichael [1984!, reputational forces are insufficient to deter a firm from cheating. Although workers here are risk neutral, the model could be amended to include risk averse agents. With this amendment, an alternative justification for restricting the contract in this fashion could be that the worker is unable to borrow (due to an incomplete loan market) against his future income. In this case the firm may attract a worker only by offering a contract with a constant per period income as is done here.

Other assumptions are discussed as they arise in section 2 where the model is described in more detail. A final section suggests possible extensions.

II. THE MODEL

The economy is composed of many separate competitive labor markets. Partial equilibrium analysis, concerned with the optimizing behavior of a single firm taking certain market determined parameters as fixed, is employed. The economy evolves through discrete time periods, and in each market at the beginning of a period a market clearing labor contract is determined. As hiring decisions are made at the beginning of a period before the price of the firm's output is determined, the contract offered to the firm's employee will specify a layoff probability s and a wage w for the upcoming period. Each worker is risk neutral and is indifferent between any contracts offering the same mean wage during any period. Thus this fixed wage contract is optimal. The income value of the market clearing wage contract specifies that the worker's expected per period income must be at least as large as y. Assuming that the income value of leisure for each worker is x, it must be the case that each firm's s and w satisfy

(1) y [is less than or equal to! sx + (1 - s)w

The economy is analyzed is steady state so y is constant over time thus the firm can hire and keep a worker as long as equation (1) is satisfied for each period's s and w.

The price available for the firm's output during a given time, labeled p, is drawn from a distribution with function G(p) and continuous density g(p). The price is drawn independently and identically across the discrete time periods. For a given time period, the firm's layoff rule will specify a price [p.sup.*' such that if p [p.sup.*! during the period the worker will be laid off. This implies s = G([p.sup.*!).

A worker's output at a firm is determined by the amount of firm specific training provided by the firm to the worker. Letting l be the amount of training, output will be a functio of l, call it

F(l), where F satisfies F' 0, F" 0 and

F'(0) = [infinity!. The marginal cost of training is assumed constant at c. At the beginning of each period, in addition to announcing its choice for s and w, the firm must decide how much, if any, additional training to provide its employee at the fixed marginal cost c. This decision is not determined by equating F'(l) with the marginal cost c for reasons to follow.

As noted, the firm operates in one of many labor markets. In the firm's market a constant fraction q of workers is searching for employment opportunities outside the market. (As one example, this search can be thought of as arising due to the different non-pecuniary characteristics of the various labor markets.) From the firm's point of view, this means that with probability q a hired worker will be unreliable in the sense that he will be searching for employment elsewhere and may quit the firm at some future time. For these unreliable workers an outside opportunity arrives as a Poisson process with a parameter varying with the employee's work status. More specifically, when working during a period the employee receives an outside offer with probability [alpha! and when not working due to a layoff during a period, he receives an outside offer with probability [beta!, with [beta!

[alpha! because a worker will have less time to seek an outside offer when working. (Jovanovic [1984! also assumes outside offers are more likely to arrive when the worker is unemployed.)

During each period the firm determines the level of firm specific training simultaneously with s and w prior to the realization of the firm's price. Since the firm will update its prior probability as to a worker's unreliabiliy as the worker's tenure with the firm increases, the firm may wish to make incremental changes at the beginning of a period in a worker's training level and may also make optimal changes in s and w.

Before discussing the firm's optimization problem, determining the optimal paths of l, s, and w as a worker's tenure increases, some additional comments are needed. First, as discussed in the introduction, enforceable self-revealing contracts are ruled out by assumption. Second, information about a worker's tenure with a firm cannot be transmitted to other firms in the market. Thus a worker's revealed reliability is information known only by the worker and his employer. (This assumption is made for analytical convenience. The implications of relaxing it are discussed in the conclusion.) Finally, in equilibrium y will take on that value yielding firms hiring new workers zero expected profits.

The Firm's Decision Problem

The analysis of the firm's problem begins by considering how q, the probability of a worker's being unreliable, is updated and then considers in a general context how the optimal layoff and training policies are determined.

Consider first how q is updated when the worker is employed during a period and does not leave at the beginning of the next period for another job. From Bayes' theorem i follows that after such a sequence of events the updated q, call it q - z, is given by

(2) q - z = prob (worker is unreliable numeric Y worker is employed during a period and remains for the next period) = (1 - [alpha!)q/(1 - q[alpha!).

Thus, under these circumstances q decreases by

(2') z = q(1 - q)[alpha!/(1 - q[alpha!).

Similarly, when the worker is unemployed due to a layoff and does not quit the firm, the updated q, call it q - z', is given by

(3) q - z' = (1 - [beta!)q/(1-q[beta!)

and q decreases by

(3) z' = q(1 - q)[beta!/(1 - q[beta!).

Note then that since [beta! [alpha!, z' [is greater than! z and therefore when the worker remains with the firm q will fall more quickly after a period of layoff unemployment than after a period of employment. In this sense a period of unemployment will yield more information about the probability of the worker's leaving than a period of employment. This fact will, of course, affect the optimal layoff policy.

Consider now the problem faced by a firm employing a worker who has received training at the level l and is deemed unreliable with probability q, where q and l are restricted to the intervals (0, q) and (0, [l.sub.max!) respectively and where [l.sub.max! is the optimal training level when q = 0. (Note that these intervals contain the q, l pairs of interest since by assumption * will always exceed any updated q and since training will never exceed [l.sub.max!.) This firm must determine what additional training the worker should receive at the beginning of the period, call it [DELTA!L, and must determine a layoff rule for the upcoming period so to maximize its expected future discounted profits. As noted previously, the layoff rule will be to determine a price, [p.sup.*!, such that the worker will be employed if p [is greater than or less than! [p.sup.*! and will be laid off if p [p.sup.*! during the period. The firm's optimal expected future discounted income is, of course, a function of the given values of q and l and is determined by the equation

[Mathematical expressions omitted!

where [(1 + r).sup.-1! is the discount factor, c the marginal cost of training, z and z' are given by equations (2') and (3'), and [p.sup.*! is defined to be the expected price when the worker is employed times the probability of the worker's being employed. That is, [p.sup.*! is

[Mathematical expressions omitted!

In equation (4) the term [p.sup.*! F(l+[DELTA!L) - [DELTA!LC is SELF-EXPLANATORY. Second, the term * - G([p.sup.*!)x is the firm's expected wage bill. This follows because the probability of a layoff is s = G([p.sup.*!), since layoffs occur when p [p.sup.*! and since G() is the appropriate distribution function. Therefore, the expected wage bill is (1-s)w = (1-G([p.sup.*!))w, which by equation (1) must be * - sx * * -G([p.sup.*!)x. Third, a worker will leave the firm with probability q[beta! when laid off and with probability q[alpha! when employed. In either case, when the worker quits the firm will be forced to employ a new untrained worker who is unreliable with probability *. Hence the firm's remaining expected income will be V(*, 0). In equilibrium * is at that level making V(*, 0) equal to zero, and thus V(*, 0) vanishes in determining V(q,l). Fourth, when laid off, a worker will return with probability (1 - q[beta!), and in this case the firm will update q to equal q - z' and thus will have an expected discounted future income of V(q-z', l+[DELTA!L). Fifth, the term (V(q-z, l+[DELTA!l) is explained as was V(q-z', l+[DELTA!l).

The maximization problem defined by equation (4) is a straightforward dynamic programming problem common in the literature (see, for example, Lucas and Prescott [1974! or Jovanovic [1979a, 1979b!). It can be shown that V as fixed point of the functional defined by the right side of equation (4) does exist and that V is a decreasing function of q and is an increasing function of l, as intuition would suggest. An increase in q means the worker is more likely to quit and thus means it is more likely the firm will be forced to hire a less well-trained worker in the future. An increase in l, ceteris paribus, means an increase in the worker's output and an increase in future expected profits. Assuming interior solutions for [DELTA!l and [p.sup.*!, the first order conditions are

[Mathematical expression omitted!

Equation (6) follows because a small increase in additional training, [DELTA!l, will increase expected revenue by [p.sup.*! F!(l+[DELTA!l) per unit of the increase and will cost c per unit of the increase. However, a small increase in additional training today will necessitate less additional training the next period if the worker is still at the firm. Thus the expected discounted training cost for the next period will be lowered by c[G([p.sup.*!)(1-q[beta!) + (1-G([p.sup.*!))(1-g[alpha!)'/(1+r). The optimal level of additional training will occur at that point where these effects on income just offset one another. This optimal level is determined by equation (6).

Equation (7) can be rewritten as

[Mathematical expression omitted!

The intuition behind this condition is now relatively straightforward. First, an increase in [p.sup.*! will tend to decrease expected revenues during the upcoming period by g([p.sup.*!)[p.sup.*! F(l+[DELTA!l) and decrease the expected wage bill by g([p.sup.*!)x. Second, an increase in [p.sup.*! will decrease expected future discounted profits since an increase in [p.sup.*! will make layoffs more likely and thus will increase the likelihood that an unreliable worker will find a better opportunity elswhere. This will force the firm to begin anew with an untrained worker, unreliable with probability q. The net expected loss from this source is g([P.sup.*!)q([beta!-[alpha!) V(q-z', l+[DELTA!l)(l+r), when [p.sup.*! is increased. Third, an increase in [p.sup.*!, and hence layoffs, has the benefit of increasing expected future discounted income because a worker's returning to the firm after a layoff provides more information about the worker's reliability than does a worker's remaining with the firm after a period of employment. This is so because, as noted earlier, the updated probability of unreliability after the former event, q - z', is less than that of the latter, q - z. The expected increase in profits from this source when [p.sup.*! is increased is g([p.sup.*!) (1-q[alpha!) [V(q-z', l+[DELTA!l)-V(q-z, l+[DELTA!l)!/(1+r). The optimal layoff price [p.sup.*! is that price where the effects of an increase in [p.sup.*! on profits offset one another. This optimal level is determined by equation (7) or equivalently by equation (7').

The implications of (6) and (7) are summarized in the following propositions.

Proposition 1: Firm specific training is a nondecreasing function of tenure, converging to [l.sub.max! as tenure converges to infinity.

Proposition 2: Layoffs may b higher or lower than those implied by the condition [p.sup.*!F(l+[DELTA!l)=x. As tenure increases to infinity, layoffs will converge to the levels implied by this condition.

Proposition 1 follows because as tenure increases, a worker's reputation for reliability increases. That is, q decreases and hence the firm's fear of a worker's quitting decreases. As tenure increases to infinity, q converges to zero and equation 6 collapses to the condition [*.sup.*!F' (l+[delta!l) = rc/(1+r), so that training is at the level l + [DELTA!l=[l.sup.max!. Proposition 2 follows from equation 7'. Layoffs will be below the level implied by the condition that [p.sup.*!F = x if the

expected loss to the firm from laying off a worker--caused by an increased quit probability--exceeds the gain--caused by an increased updated reliability probability if the worker remains with the firm after a layoff.

III. CONCLUSION

Rather than reiterate the preceding conclusions, in this section several possible extensions of the model are discussed. First, recall that a worker's reputation for reliability at a firm does not become public knowledge. This assumption was made for conveniece only and relaxing it will not materially affect the optimal layoff and training rules. However, if the assumption is voided then one would expect a worker who has established a record of reliability within the market to be viewed as having a low q and to command a higher per period expected income. In fact, in equilibrium one would expect wages for workers with good reputations to be pushed up to the point where firms entering the market will be indifferent between hiring workers with high or low q's. Thus in equilibrium the benefits of firm specific training, net of training costs, will accrue over time to workers. This would imply that a worker's per period expected income would rise with tenure. That some benefits of firm specific training accrue to the worker is consistent with the empirical work of Barron, Black, and Loewenstein [1989!.

This implication of allowing a worker's reliability to be public knowledge is similar to that derived by Salop and Salop [1976!, wven though these authors do not allow q to be updated a a worker's tenure increases. In their analysis a worker receives the full value of his firm specific training and pays for his own firm specific training costs. This follows because the firm writes a self-revealin contract that induces the worker to reveal his reliability level--the reliable worker chooses to pay training costs by accepting a low initial wage and receives a more than compensating higher subsequent wage. However, Salop and Salop assume the firm will abide by the terms of a contract even when in the absence of an explicit enforcement device it could retain the worker by lowering his subsequent wage. Without this assumption one would expect the firm to pay training costs with the worker accruing the net benefits of training as discussed here.

A second extensionto the analysis would be endogenize the probability of an unreliable worker's quitting. For example, during each time period an unreliable worker could receive an outside offer whose value is drawn from a commonly known probability distribution. In such a setting the firm could use its wage policy to lower (but not necessarily eliminate) turnover (see, for example, Haltiwanger [1984!). This extension would also be of interest if workers were finitely lived rather than infinitely lived as here. In this case, an unreliable worker's outside alternative may vary with his life cycle and this fact would affect the firm's wage policy and worker turnover. Also, with finitely lived workers training would not necessarily continue with job tenure but might reach a peak at some finite tenure level. Nevertheless, with each of these extensions the firm could use layoffs as a means of determining the worker's reliability.

Another extension would be to embed the problem of determining firm specific training and the division of the proceeds from same in a sequential bargaining model like that of Sobel and takahashi [1983!, Rubinstein [1985!, or Rubinstein and Wolinsky [1985!. In this paper the expected rents from firm specific training accrue to the worker (recall that V(*, 0) = O as competing firms bid wages up to remove any expected net profits. If, however, there is no centralized labor market and workers and firms are matched only after costly search (see Mortensen [1982! and Rubinstein and Wolinsky [1985!) then in the resulting bargaining game almost any division of the proceeds is possible depending upon, say, the relative search technologies of the two parties (again, see Rubinstein and Wolinsky [1985!). Of course, the division of rents from firm specific training is a separate problem from determining the optimal level of rents, and one feels that the division of rents will have little or no impact on the training level itself. The papers of Sobel and Takahashi and Rubinstein do deal with sequential revelation of player type in a sequential bargaining game and this aspect of their models might appear of relevance to the present analysis. However, again if the firm is unable to commit itself to a contract with wages increasing with job tenure, there appears no way to a reliable worker to distinguish himself from an unreliable one in the bargaining game and thus training and layoffs will evolve as in equations (6) and (7').

Of course, the first order conditions of equiations (6) and (7') could be changed if firms and workers both are allowed to acquire reputations for reliability--the firms for abiding by the terms of a contract with a screening device and the workers for remaining in the market however, a reputational equilibrium with anything less than perfect knowledge about reliability will not yield first best outcomes (see, Charmichael [1984!).

Finally, the results in this paper hinge on the fact the firm cannot make intertemporal wage commitments and hence cannot use a rising wage profile to attract reliable workers. Without this constraint on contracts, other potential problems with respect to optimal training and layoffs can arise if the firm and worker are asymmetrically informed concerning the worker's productivity. In such a setting, Haltiwanger [1984! shows that labor turnover may still be inefficient and this fact will have important implications for job training.

(*1) Associate Professor, Virginia Polytechnic Institute and State University. I wish to thank Mark Loewenstein, Richard J. Sweeney, and anonymous referees for helpful comments.

REFERENCES

Barron, John M., Dan A. Black, and Mark A. Loewenstein. "Job Matching and On-the-Job Training." Journal of Labor Economics, January 1989, 1-19.

Barron, John M., and Mark A. Loewenstein. "On Employer-Specific Information and Internal Labor Markets." Southern Economic Journal October 1985, 368-76.

Becker, Gary S. Human Capital. New york: Columbia University Press, 1975.

Charmichael, H. Lorne. "Reputations in the Labor Market." American Economic Review, September 1984, 713-25.

Haltiwanger, John C. "Asymmetric Information, Multiperiod Labor Contracts, and Inefficient Job Separations." Southern Economic Journal, April 1984, 1005-24.

_____. "On the Relationship Between Risk Aversion and the Development of Long-Term Worker-Firm Attachments." Southern Economic Journal, October 1983, 572-77.

Holmstrom, Bengt. "Equilibrium Long-Term Contracts." Quarterly Journal of Economics, Supplement, 1983, 23-54.

Jovanovic, Boyan. "Firm Specific Capital and Turnover." Journal of Political Economy, December 1979a, 1246-60.

_____. "Job Matching and the Theory of Turnover." Journal of Political Economy, October 1979b, 972-90.

_____. "Matching, Turnover, and Unemployment." Journal of Political Economy, February 1984, 108-22.

Lucas, robert E. and Edward C. Prescott. "Equilibrium Search and Unemployment." Journal of Economic Theory 7(2), 1974, 188-209.

Mortensen, Dale T. "The Matching Process as a Noncooperative Bargaining Game," in The Economics of Information and Uncertainty, edited by J. J. McCall. Chicago and London: The University of Chicago Press, 1982, 233-54.

_____. "Wages, Separations, and Job Tenure: On the Job Training or Matching?" Journal of Labor Economics, October 1988, 445-71.

Oi, Walter Y. "Labor as a Quasi-Fixed Factor." Journal of Political Economy, December 1962, 538-55.

Rubinstein, Ariel. "A Bargaining Model with Incomplete Information About Time Preferences." Econometrica, September 1985, 1151-72.

Rubinstein, Ariel and Asher Wolinsky. "Equilibrium In A Market with Sequential Bargaining." Econometrica, September 1985, 1133-50.

Salop, Joanne and Steven Salop. "Self-Selection and Turnover in the Labor Market." Quarterly Journal of Economics 90(4), 1976, 619-27.

Sobel, Joel and Ichiro Takahashi. "A Multistage Model of bargaining." Review of Economic Studies 50(3), 1983, 411-26.

1. INTRODUCTION

Early work in implicit contract theory argues that a long-term relationship between a worker and a firm evolves due to the risk-sharing nature of the implicit contract between a risk-averse worker and a risk-neutral firm. Some authors have criticized this initial argument (see, for example, Haltiwanger [1983!), feeling that a long-term relationship between a worker and a firm (and the terms of an implicit contract between the two) in large parts is explained via hiring costs, training costs, and the like. This is the approach taken in the early work of Becker [1975!, Oi [1962! and the most recent work of Jovanovic [1979a!.

These latter explanations are also appealing because long-term relations between a worker and a firm are typically characterized by quits that decline with tenure and wages that increase with tenure. As discussed in Mortensen [1988!, explanations of these phenomena fall into two categories: job search theories or job training theories. Job search theories stress that there is a match specific component to the productivity of any worker-firm match. That a good match will result in a long-term relationship explains why wages are positively correlated with tenure. To explain a rising wage profile within a given match requires that the firm and the worker learn about the quality of their match over time. A good match will be a long-term one, and the wage will rise over time as the match is discovered to be a good one. (Jovanovic [1979b! and Barron and Loewenstein [1985! model such a learning process.) Job training theories maintain that long-term relationships with rising wages exist because of firm specific training that increases with tenure (again see Jovanovic [1979a! as an example). This paper integrates these two theories.

In this paper firm specific training evolves as an increasing function of a worker's tenure with a firm. This is so because there is a positive probability that a worker is unreliable in that he will be searching for another employer. The firm, being unable to identify in advance which workers are unreliable, will update its prior belief as to a worker's reliability as a function of the worker's tenure. The total level of training at any point in time will be an increasing function of this updated belief. Thus learning about the quality of a match, an aspect of the job search theories, plays an important role in explaining the level of firm-specific training.

This learning process can also play a role in determining optimal layoffs. If an unreliable worker is more likely to receive an acceptable outside offer when the worker is laid off than when he is employed, the firm can use layoffs to elicit information about the worker's reliabiliy. Based upon this fact, layoffs may occur even when the worker's marginal product exceeds his value of leisure.

Before discussing the model in detail, one crucial assumption deserves comment. Unlike previous work, such a Salop and Salop [1976!, enforceable self-revealing contracts are ruled out by assumption. Thus the firm here cannot commit itself to a wage contract promising low wages initially in exchange for high wages as job tenure increases, which contract would attract only reliable workers.

This assumption is of course a polar extreme, but as such is consistent with more recent work in contract theory which investigates the terms of self-enforcing contracts (see, for example, Holmstrom [1983!). The idea is to provide a comparison with contracts in an idealized economy with no enforcement problems. This assumption is justified if the cost of writing a legally enforceable contract is prohibitive or if, as suggested in charmichael [1984!, reputational forces are insufficient to deter a firm from cheating. Although workers here are risk neutral, the model could be amended to include risk averse agents. With this amendment, an alternative justification for restricting the contract in this fashion could be that the worker is unable to borrow (due to an incomplete loan market) against his future income. In this case the firm may attract a worker only by offering a contract with a constant per period income as is done here.

Other assumptions are discussed as they arise in section 2 where the model is described in more detail. A final section suggests possible extensions.

II. THE MODEL

The economy is composed of many separate competitive labor markets. Partial equilibrium analysis, concerned with the optimizing behavior of a single firm taking certain market determined parameters as fixed, is employed. The economy evolves through discrete time periods, and in each market at the beginning of a period a market clearing labor contract is determined. As hiring decisions are made at the beginning of a period before the price of the firm's output is determined, the contract offered to the firm's employee will specify a layoff probability s and a wage w for the upcoming period. Each worker is risk neutral and is indifferent between any contracts offering the same mean wage during any period. Thus this fixed wage contract is optimal. The income value of the market clearing wage contract specifies that the worker's expected per period income must be at least as large as y. Assuming that the income value of leisure for each worker is x, it must be the case that each firm's s and w satisfy

(1) y [is less than or equal to! sx + (1 - s)w

The economy is analyzed is steady state so y is constant over time thus the firm can hire and keep a worker as long as equation (1) is satisfied for each period's s and w.

The price available for the firm's output during a given time, labeled p, is drawn from a distribution with function G(p) and continuous density g(p). The price is drawn independently and identically across the discrete time periods. For a given time period, the firm's layoff rule will specify a price [p.sup.*' such that if p [p.sup.*! during the period the worker will be laid off. This implies s = G([p.sup.*!).

A worker's output at a firm is determined by the amount of firm specific training provided by the firm to the worker. Letting l be the amount of training, output will be a functio of l, call it

F(l), where F satisfies F' 0, F" 0 and

F'(0) = [infinity!. The marginal cost of training is assumed constant at c. At the beginning of each period, in addition to announcing its choice for s and w, the firm must decide how much, if any, additional training to provide its employee at the fixed marginal cost c. This decision is not determined by equating F'(l) with the marginal cost c for reasons to follow.

As noted, the firm operates in one of many labor markets. In the firm's market a constant fraction q of workers is searching for employment opportunities outside the market. (As one example, this search can be thought of as arising due to the different non-pecuniary characteristics of the various labor markets.) From the firm's point of view, this means that with probability q a hired worker will be unreliable in the sense that he will be searching for employment elsewhere and may quit the firm at some future time. For these unreliable workers an outside opportunity arrives as a Poisson process with a parameter varying with the employee's work status. More specifically, when working during a period the employee receives an outside offer with probability [alpha! and when not working due to a layoff during a period, he receives an outside offer with probability [beta!, with [beta!

[alpha! because a worker will have less time to seek an outside offer when working. (Jovanovic [1984! also assumes outside offers are more likely to arrive when the worker is unemployed.)

During each period the firm determines the level of firm specific training simultaneously with s and w prior to the realization of the firm's price. Since the firm will update its prior probability as to a worker's unreliabiliy as the worker's tenure with the firm increases, the firm may wish to make incremental changes at the beginning of a period in a worker's training level and may also make optimal changes in s and w.

Before discussing the firm's optimization problem, determining the optimal paths of l, s, and w as a worker's tenure increases, some additional comments are needed. First, as discussed in the introduction, enforceable self-revealing contracts are ruled out by assumption. Second, information about a worker's tenure with a firm cannot be transmitted to other firms in the market. Thus a worker's revealed reliability is information known only by the worker and his employer. (This assumption is made for analytical convenience. The implications of relaxing it are discussed in the conclusion.) Finally, in equilibrium y will take on that value yielding firms hiring new workers zero expected profits.

The Firm's Decision Problem

The analysis of the firm's problem begins by considering how q, the probability of a worker's being unreliable, is updated and then considers in a general context how the optimal layoff and training policies are determined.

Consider first how q is updated when the worker is employed during a period and does not leave at the beginning of the next period for another job. From Bayes' theorem i follows that after such a sequence of events the updated q, call it q - z, is given by

(2) q - z = prob (worker is unreliable numeric Y worker is employed during a period and remains for the next period) = (1 - [alpha!)q/(1 - q[alpha!).

Thus, under these circumstances q decreases by

(2') z = q(1 - q)[alpha!/(1 - q[alpha!).

Similarly, when the worker is unemployed due to a layoff and does not quit the firm, the updated q, call it q - z', is given by

(3) q - z' = (1 - [beta!)q/(1-q[beta!)

and q decreases by

(3) z' = q(1 - q)[beta!/(1 - q[beta!).

Note then that since [beta! [alpha!, z' [is greater than! z and therefore when the worker remains with the firm q will fall more quickly after a period of layoff unemployment than after a period of employment. In this sense a period of unemployment will yield more information about the probability of the worker's leaving than a period of employment. This fact will, of course, affect the optimal layoff policy.

Consider now the problem faced by a firm employing a worker who has received training at the level l and is deemed unreliable with probability q, where q and l are restricted to the intervals (0, q) and (0, [l.sub.max!) respectively and where [l.sub.max! is the optimal training level when q = 0. (Note that these intervals contain the q, l pairs of interest since by assumption * will always exceed any updated q and since training will never exceed [l.sub.max!.) This firm must determine what additional training the worker should receive at the beginning of the period, call it [DELTA!L, and must determine a layoff rule for the upcoming period so to maximize its expected future discounted profits. As noted previously, the layoff rule will be to determine a price, [p.sup.*!, such that the worker will be employed if p [is greater than or less than! [p.sup.*! and will be laid off if p [p.sup.*! during the period. The firm's optimal expected future discounted income is, of course, a function of the given values of q and l and is determined by the equation

[Mathematical expressions omitted!

where [(1 + r).sup.-1! is the discount factor, c the marginal cost of training, z and z' are given by equations (2') and (3'), and [p.sup.*! is defined to be the expected price when the worker is employed times the probability of the worker's being employed. That is, [p.sup.*! is

[Mathematical expressions omitted!

In equation (4) the term [p.sup.*! F(l+[DELTA!L) - [DELTA!LC is SELF-EXPLANATORY. Second, the term * - G([p.sup.*!)x is the firm's expected wage bill. This follows because the probability of a layoff is s = G([p.sup.*!), since layoffs occur when p [p.sup.*! and since G() is the appropriate distribution function. Therefore, the expected wage bill is (1-s)w = (1-G([p.sup.*!))w, which by equation (1) must be * - sx * * -G([p.sup.*!)x. Third, a worker will leave the firm with probability q[beta! when laid off and with probability q[alpha! when employed. In either case, when the worker quits the firm will be forced to employ a new untrained worker who is unreliable with probability *. Hence the firm's remaining expected income will be V(*, 0). In equilibrium * is at that level making V(*, 0) equal to zero, and thus V(*, 0) vanishes in determining V(q,l). Fourth, when laid off, a worker will return with probability (1 - q[beta!), and in this case the firm will update q to equal q - z' and thus will have an expected discounted future income of V(q-z', l+[DELTA!L). Fifth, the term (V(q-z, l+[DELTA!l) is explained as was V(q-z', l+[DELTA!l).

The maximization problem defined by equation (4) is a straightforward dynamic programming problem common in the literature (see, for example, Lucas and Prescott [1974! or Jovanovic [1979a, 1979b!). It can be shown that V as fixed point of the functional defined by the right side of equation (4) does exist and that V is a decreasing function of q and is an increasing function of l, as intuition would suggest. An increase in q means the worker is more likely to quit and thus means it is more likely the firm will be forced to hire a less well-trained worker in the future. An increase in l, ceteris paribus, means an increase in the worker's output and an increase in future expected profits. Assuming interior solutions for [DELTA!l and [p.sup.*!, the first order conditions are

[Mathematical expression omitted!

Equation (6) follows because a small increase in additional training, [DELTA!l, will increase expected revenue by [p.sup.*! F!(l+[DELTA!l) per unit of the increase and will cost c per unit of the increase. However, a small increase in additional training today will necessitate less additional training the next period if the worker is still at the firm. Thus the expected discounted training cost for the next period will be lowered by c[G([p.sup.*!)(1-q[beta!) + (1-G([p.sup.*!))(1-g[alpha!)'/(1+r). The optimal level of additional training will occur at that point where these effects on income just offset one another. This optimal level is determined by equation (6).

Equation (7) can be rewritten as

[Mathematical expression omitted!

The intuition behind this condition is now relatively straightforward. First, an increase in [p.sup.*! will tend to decrease expected revenues during the upcoming period by g([p.sup.*!)[p.sup.*! F(l+[DELTA!l) and decrease the expected wage bill by g([p.sup.*!)x. Second, an increase in [p.sup.*! will decrease expected future discounted profits since an increase in [p.sup.*! will make layoffs more likely and thus will increase the likelihood that an unreliable worker will find a better opportunity elswhere. This will force the firm to begin anew with an untrained worker, unreliable with probability q. The net expected loss from this source is g([P.sup.*!)q([beta!-[alpha!) V(q-z', l+[DELTA!l)(l+r), when [p.sup.*! is increased. Third, an increase in [p.sup.*!, and hence layoffs, has the benefit of increasing expected future discounted income because a worker's returning to the firm after a layoff provides more information about the worker's reliability than does a worker's remaining with the firm after a period of employment. This is so because, as noted earlier, the updated probability of unreliability after the former event, q - z', is less than that of the latter, q - z. The expected increase in profits from this source when [p.sup.*! is increased is g([p.sup.*!) (1-q[alpha!) [V(q-z', l+[DELTA!l)-V(q-z, l+[DELTA!l)!/(1+r). The optimal layoff price [p.sup.*! is that price where the effects of an increase in [p.sup.*! on profits offset one another. This optimal level is determined by equation (7) or equivalently by equation (7').

The implications of (6) and (7) are summarized in the following propositions.

Proposition 1: Firm specific training is a nondecreasing function of tenure, converging to [l.sub.max! as tenure converges to infinity.

Proposition 2: Layoffs may b higher or lower than those implied by the condition [p.sup.*!F(l+[DELTA!l)=x. As tenure increases to infinity, layoffs will converge to the levels implied by this condition.

Proposition 1 follows because as tenure increases, a worker's reputation for reliability increases. That is, q decreases and hence the firm's fear of a worker's quitting decreases. As tenure increases to infinity, q converges to zero and equation 6 collapses to the condition [*.sup.*!F' (l+[delta!l) = rc/(1+r), so that training is at the level l + [DELTA!l=[l.sup.max!. Proposition 2 follows from equation 7'. Layoffs will be below the level implied by the condition that [p.sup.*!F = x if the

expected loss to the firm from laying off a worker--caused by an increased quit probability--exceeds the gain--caused by an increased updated reliability probability if the worker remains with the firm after a layoff.

III. CONCLUSION

Rather than reiterate the preceding conclusions, in this section several possible extensions of the model are discussed. First, recall that a worker's reputation for reliability at a firm does not become public knowledge. This assumption was made for conveniece only and relaxing it will not materially affect the optimal layoff and training rules. However, if the assumption is voided then one would expect a worker who has established a record of reliability within the market to be viewed as having a low q and to command a higher per period expected income. In fact, in equilibrium one would expect wages for workers with good reputations to be pushed up to the point where firms entering the market will be indifferent between hiring workers with high or low q's. Thus in equilibrium the benefits of firm specific training, net of training costs, will accrue over time to workers. This would imply that a worker's per period expected income would rise with tenure. That some benefits of firm specific training accrue to the worker is consistent with the empirical work of Barron, Black, and Loewenstein [1989!.

This implication of allowing a worker's reliability to be public knowledge is similar to that derived by Salop and Salop [1976!, wven though these authors do not allow q to be updated a a worker's tenure increases. In their analysis a worker receives the full value of his firm specific training and pays for his own firm specific training costs. This follows because the firm writes a self-revealin contract that induces the worker to reveal his reliability level--the reliable worker chooses to pay training costs by accepting a low initial wage and receives a more than compensating higher subsequent wage. However, Salop and Salop assume the firm will abide by the terms of a contract even when in the absence of an explicit enforcement device it could retain the worker by lowering his subsequent wage. Without this assumption one would expect the firm to pay training costs with the worker accruing the net benefits of training as discussed here.

A second extensionto the analysis would be endogenize the probability of an unreliable worker's quitting. For example, during each time period an unreliable worker could receive an outside offer whose value is drawn from a commonly known probability distribution. In such a setting the firm could use its wage policy to lower (but not necessarily eliminate) turnover (see, for example, Haltiwanger [1984!). This extension would also be of interest if workers were finitely lived rather than infinitely lived as here. In this case, an unreliable worker's outside alternative may vary with his life cycle and this fact would affect the firm's wage policy and worker turnover. Also, with finitely lived workers training would not necessarily continue with job tenure but might reach a peak at some finite tenure level. Nevertheless, with each of these extensions the firm could use layoffs as a means of determining the worker's reliability.

Another extension would be to embed the problem of determining firm specific training and the division of the proceeds from same in a sequential bargaining model like that of Sobel and takahashi [1983!, Rubinstein [1985!, or Rubinstein and Wolinsky [1985!. In this paper the expected rents from firm specific training accrue to the worker (recall that V(*, 0) = O as competing firms bid wages up to remove any expected net profits. If, however, there is no centralized labor market and workers and firms are matched only after costly search (see Mortensen [1982! and Rubinstein and Wolinsky [1985!) then in the resulting bargaining game almost any division of the proceeds is possible depending upon, say, the relative search technologies of the two parties (again, see Rubinstein and Wolinsky [1985!). Of course, the division of rents from firm specific training is a separate problem from determining the optimal level of rents, and one feels that the division of rents will have little or no impact on the training level itself. The papers of Sobel and Takahashi and Rubinstein do deal with sequential revelation of player type in a sequential bargaining game and this aspect of their models might appear of relevance to the present analysis. However, again if the firm is unable to commit itself to a contract with wages increasing with job tenure, there appears no way to a reliable worker to distinguish himself from an unreliable one in the bargaining game and thus training and layoffs will evolve as in equations (6) and (7').

Of course, the first order conditions of equiations (6) and (7') could be changed if firms and workers both are allowed to acquire reputations for reliability--the firms for abiding by the terms of a contract with a screening device and the workers for remaining in the market however, a reputational equilibrium with anything less than perfect knowledge about reliability will not yield first best outcomes (see, Charmichael [1984!).

Finally, the results in this paper hinge on the fact the firm cannot make intertemporal wage commitments and hence cannot use a rising wage profile to attract reliable workers. Without this constraint on contracts, other potential problems with respect to optimal training and layoffs can arise if the firm and worker are asymmetrically informed concerning the worker's productivity. In such a setting, Haltiwanger [1984! shows that labor turnover may still be inefficient and this fact will have important implications for job training.

(*1) Associate Professor, Virginia Polytechnic Institute and State University. I wish to thank Mark Loewenstein, Richard J. Sweeney, and anonymous referees for helpful comments.

REFERENCES

Barron, John M., Dan A. Black, and Mark A. Loewenstein. "Job Matching and On-the-Job Training." Journal of Labor Economics, January 1989, 1-19.

Barron, John M., and Mark A. Loewenstein. "On Employer-Specific Information and Internal Labor Markets." Southern Economic Journal October 1985, 368-76.

Becker, Gary S. Human Capital. New york: Columbia University Press, 1975.

Charmichael, H. Lorne. "Reputations in the Labor Market." American Economic Review, September 1984, 713-25.

Haltiwanger, John C. "Asymmetric Information, Multiperiod Labor Contracts, and Inefficient Job Separations." Southern Economic Journal, April 1984, 1005-24.

_____. "On the Relationship Between Risk Aversion and the Development of Long-Term Worker-Firm Attachments." Southern Economic Journal, October 1983, 572-77.

Holmstrom, Bengt. "Equilibrium Long-Term Contracts." Quarterly Journal of Economics, Supplement, 1983, 23-54.

Jovanovic, Boyan. "Firm Specific Capital and Turnover." Journal of Political Economy, December 1979a, 1246-60.

_____. "Job Matching and the Theory of Turnover." Journal of Political Economy, October 1979b, 972-90.

_____. "Matching, Turnover, and Unemployment." Journal of Political Economy, February 1984, 108-22.

Lucas, robert E. and Edward C. Prescott. "Equilibrium Search and Unemployment." Journal of Economic Theory 7(2), 1974, 188-209.

Mortensen, Dale T. "The Matching Process as a Noncooperative Bargaining Game," in The Economics of Information and Uncertainty, edited by J. J. McCall. Chicago and London: The University of Chicago Press, 1982, 233-54.

_____. "Wages, Separations, and Job Tenure: On the Job Training or Matching?" Journal of Labor Economics, October 1988, 445-71.

Oi, Walter Y. "Labor as a Quasi-Fixed Factor." Journal of Political Economy, December 1962, 538-55.

Rubinstein, Ariel. "A Bargaining Model with Incomplete Information About Time Preferences." Econometrica, September 1985, 1151-72.

Rubinstein, Ariel and Asher Wolinsky. "Equilibrium In A Market with Sequential Bargaining." Econometrica, September 1985, 1133-50.

Salop, Joanne and Steven Salop. "Self-Selection and Turnover in the Labor Market." Quarterly Journal of Economics 90(4), 1976, 619-27.

Sobel, Joel and Ichiro Takahashi. "A Multistage Model of bargaining." Review of Economic Studies 50(3), 1983, 411-26.

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Author: | Cothren, Richard |
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Publication: | Economic Inquiry |

Date: | Jan 1, 1991 |

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