Why was there mandatory retirement?

Journal of Public Economics 39 (1989) 127-136. North-Holland

WHY WAS THERE MANDATORY RETIREMENT?

Kevin LANG*

Boston University. Boston MA 02215, USA, and National Bureau of Economic Research

Received March 1988, revised version received December 1988

Lazear argues that hours constraints and mandatory retirement form part of an efficient contract which inhibits worker shirking. He argues that initially workers earn less than their VMP and must work more than they want while later they earn more than their VMP and must work less. However, if bonding is costly, incentive compatibility requires that workers can never exceed the contracted hours. In addition, hours will be set so that VMP exceeds the disutility of employment. Consequently, being paid more than their VMP is only sufficient not necessary for workers to be constrained to work less than they wish.

1. Introduction

Lazear (1979, 1981) has argued that hours constraints, in general, and mandatory retirement, in particular, form part of an efficient labor market contract designed to increase output by inhibiting worker shirking. In effect, workers post bonds against malfeasance which are returned to them in the form of higher wages later in the work relation. Workers who are caught shirking are fired and forfeit the bond. Since wages deviate from the value of marginal product (VMP), the contract must specify hours as well as wages.

The Lazear agency model has received considerable attention in the literature for two reasons. First, there is some evidence that wages rise faster than productivity [Medoff and Abraham (1980)], and the model offers an explanation for this finding. Secondly, unlike the shirking-based efficiency wage model [Shapiro and Stiglitz (1984), Bulow and Summers (1986)], the contracts discussed by Lazear are first-best efficient. If Lazear is correct, legislative interference is welfare reducing. Yet, Congress has recently passed a law making mandatory retirement illegal in most cases.

*This research was supported by an Olin Fellowship at the National Bureau of Economic Research, a Sloan Faculty Research Fellowship and by NSF grant number SES-8707422. Many of the ideas in this paper were generated in the course of conversations with Shulamit Kahn concerning a related paper [Kahn and Lang (1987)]. I am grateful to Larry Katz for pointing out a significant oversight in the previous draft and to Bob Gibbons, Alan Krueger, Edward Lazear, Larry Summers and two anonymous referees for helpful comments.

004772727/89/$3.50 80 1989. Elsevier Science Publishers B.V. (North-Holland)

128 K. Lang, Whql was there mandatory retirement?

In his papers Lazear assumes that hours are set efficiently, at the point at which the shadow value of leisure equals the workers’ marginal product from an additional hour of work. Hours constraints arise because when wages exceed VMP, they therefore also exceed the shadow value of leisure, and workers want to work more, but additional hours are unprofitable for the firm. Similarly, when VMP is greater than the wage, the shadow value of leisure is greater than the wage, and workers want to work less, but reducing hours is unprofitable for the firm.

A principal point of this paper is that when bonding is costly (as I argue it must be for the Lazear model to produce sensible results), hours will not be set at the first-best efficient level. Moreover, the firm will not be willing to permit the worker to work more than the ‘contracted’ number of hours even if VMP exceeds the wage. The reason is that a constraint on maximum hours arises out of the incentive compatibility condition. Workers who worked additional hours would shirk. The maximum hours constraints arising from the incentive compatibility condition need not be binding on the worker if VMP exceeds the wage. Instead it is possible that workers who are paid less than their VMP may wish to work fewer hours. Consequently, being paid more than VMP is a sufficient but not necessary condition for workers to want to work more hours.

Before turning to the model, I note the reasons for assuming that bonding is costly. The first is theoretical: as Lazear himself notes, the wage profile is indeterminate unless bonding is costly. Thus, unless bonding is costly, the

bonding model has little or no predictive power. The second reason is empirical: Dickens, Katz, Lang and Summers (1989) point out that unless bonding is costly at the margin, firms will not engage in any monitoring, but instead will deter shirking solely by setting large bonds. Since we observe that firms do engage in monitoring, bonding must be costly at the margin.

2. The basic model

I assume that costly bonding arises from workers having higher discount rates than firms. Bonding will be costly because an increase in the bond that leaves the worker equally well off requires a repayment stream with a present value which exceeds the initial value of the bond to the firm. While the precise results may vary depending on the reasons for bonding being costly, the essential results do not appear to depend on the source of the cost.

Let R be the future surplus to the worker from keeping his job. If the worker is to be deterred from shirking, the expected value from not shirking must at least equal the expected value from shirking so that the no shirk condition is given by

w-e(h)+Rzw+(l -q(h))R

K. Lang, Why was there mandatory retirement? 129

or

R 2 e(Wq(4. (2)

where w is the wage, e is the disutility of effort on the job (value of shirking), h is the number of hours worked and q is the probability of shirking being detected.’ It seems reasonable to assume that e’ >O and that e” 20, i.e. that the marginal disutility of effort on the job is always positive and that it is nondecreasing in hours worked. In other words, the value of a coffee break does not fall as you work more hours. The probability of detection, q, is assumed to follow a Poisson process with constant arrival probability over any period of a given length. As a consequence, q” < 0.2

Under these assumptions, e/q is increasing in h and the no shirking constraint can be inverted to give a maximum hours constraint:

huh*, h*'>O, (3)

which says simply that for any future surplus there is a maximum number of hours the worker can be employed without shirking. The no shirking constraint and the maximum hours constraint are identical so that if the no shirking constraint is binding, so is the maximum hours constraint. The maximum hours form of the no shirking constraint is useful for the multiperiod model developed here and underscores one of the major points of this paper: if the incentive compatibility (no shirking) condition is binding, so is the maximum hours constraint. Workers will not be permitted to work more than the agreed upon hours.

R, the surplus from remaining with the firm, is the value of the pension, P,

plus the excess of future wage payments over the disutility of working on the job and the shadow value of time, s, all discounted at the worker’s discount rate, i, or

R,= f: (M~j-ej(hj)-sJ{hj))/(l +i)j-‘+P/(l +i)Tmr+l. j=t+ 1

(4)

It greatly simplifies presentation and does not alter any important results to assume that u, the value of a worker’s marginal product per hour, does not decline with hours worked. In this case, the firm maximizes profits at the time of hire which are given by

‘For simplicity it is assumed that the worker either shirks for the entire period or not at all. Formally, this assumption requires that e” be not too large.

‘Intuitively, while the probability of being caught in any particular hour remains constant, the probability of having already been caught rises as hours worked increases. Consequently, the marginal contribution of the last hours worked to the probability of being caught decreases with hours worked.

130 K. Lang, Why was there mandatory retirement?

(5)

where I is the firm’s discount rate and b is the upfront bond paid by the workers, subject to the no shirking/maximum hours constraints obtained by substituting (4) into (3) for each period:

(Wj-ej(hj)-sAhj))/(I +i)‘-‘+P/(l +i)T-‘+’ >3 . (6)

It should be noted that since the constraint requires that h, be less than or equal to h:, At is nonpositive.

Maximization of (5) is also subject to the constraint that over the lifetime the worker receives at least the competitive level of utility:

A,[-b+x(w,-e,(h,)-s,(h,))/(l+i)‘-’+P/(l+i)r].

The first-order conditions with respect to h, and w13 are given by

f-l

u/(l+r)‘-‘- c ~j(-e~-s~)hj*‘/(l+i)‘-j+A,(-e~-s~)/(l+i)’-’+A,=O 1

(8)

and

f-l

-(l+~)‘-t- C Ijhi*‘/(l+i)‘-‘+l,/(l+i)‘-‘=O. (9)

To prove that the hours constraint is always binding, combine the first-order conditions for w, and w,+ 1 and rearrange terms to obtain:

At = (r - i)/[h:‘( 1 + r)‘]. (10)

Provided that i is greater than I so that bonding is costly, the hours constraint will be binding. In contrast to costless bonding, which implies that workers will be allowed to work more if VMP exceeds the wage, with costly bonding, workers are never allowed to work additional hours.

3Given the set-up of the model, the firm is indifferent among combinations of b and w, which give the same value of w, - b. It simplifies presentation later to treat w1 as being determined by the same rule as wages in the other periods and allowing b to adjust to set w, - b equal to its equilibrium value.


3. Binding hours constraints

Although the costly bonding model implies that workers are not allowed to work additional hours, the view that workers want to work more hours because they want the opportunity to shirk does not seem to correspond to common notions of hours constraints and certainly does not correspond to the type of constraints found in Lazear’s work. In Lazear’s model workers are willing to work additional hours without shirking if and only if the wage exceeds the VMP. To examine this view of hours constraints, let us consider the conditions under which workers would want to work more hours even if they expected not to shirk. To clarify the thought experiment: the worker is unaware that he will shirk if he works additional hours, but the firm is aware of this and knows that it will have to increase the future surplus the worker receives in order to prevent him from shirking. I show that the Lazear condition (workers want to work more if and only if the value of marginal product is less than the wage) still does not apply.

The first step is to show that hours will be set below their optimal level, i.e. below the level where r equals the marginal disutility of employment. Combine (8) (9) and (10) to obtain:

v,-12--s;= -A,(1 +r)-i=(i-r)/((l +r)hjr’). (11)

Eq. (11) implies that hours are set at a level at which the marginal product of labor exceeds the marginal disutility of employment (the disutility of labor plus the marginal value of time) so that hours are suboptimal.

Workers who did not intend to shirk would want to work additional hours if and only if their hourly wage exceeded the marginal disutility of employment (e’+s’). It follows immediately from (11) that workers will be constrained to work fewer hours than they wish if the wage is greater than or not too much below the marginal product of an additional hour of work. Thus, as discussed by Lazear, workers who earn more than their VMP will want more work while those who earn sufficiently less will want less work if we ignore workers wanting to work more in the expectation that they will shirk.

The similarity to Lazear should not be exaggerated. We can let fatigue cause the VMP to decrease with the number of hours worked per period so that

r/;= lqh,), V’>O, V”<o, (12)

where v is the marginal product of an additional worker. I”, the value of marginal product for an additional hour’s work, is identical to v in the model above. In this case, it is quite possible for V, to be greater than total per


period wage or ‘salary’ but for v’ to be less than the hourly wage. This ensures that workers want to work more. Thus, the extent to which I/ must exceed wages in order for workers to want to work less is understated by eq. (11). The use of continuous time in Lazear’s model obscures the importance of distinguishing between the VMP of an additional worker and the VMP of an additional hour of work by that worker.

It is plausible that workers will want to work more in every period with the exception of an arbitrarily short first period in which the bond is posted. To see this, we need to solve for the earnings profile. The first step is to solve for hours of work as a function of exogenous variables. This is straightforward since (11) is a function only of h and exogenous variables. We can then eliminate h from the maximum hours or no shirking constraint. Since this constraint is binding in every period, the wage profile can be obtained by backwards induction, as in Becker and Stigler (1974) and Akerlof and Katz (1989) who assume hours are fixed exogenously. The last period constraint is a function of only the pension and exogenous variables and thus can be used to find the pension. The next to last period constraint depends only on the pension, the last period wage and exogenous variables, and so on.

In order to stop the worker from shirking in the last period, the pension must be sufficiently large to satisfy the no shirking condition:

P/(1 +i)-ee,=(l -q,)P/(l +i) (13)

or

P=(l +i)e,/q,. (14)

In every period before the last, the no shirking condition can be solved to give:

w,=e,+s,+(l +i)e,/q,-etm,/qtpl. (15)

In the first period, the wage is indeterminate, but it is possible to solve for the wage minus the bond. If the first period wage is treated as being determined by (15) and the bond is allowed to adjust to set w1 - b equal to its equilibrium value, the bond can be derived from constraint (7). In the special case where the u, h and e functions are all constant with respect to t, the bond will be given by

b=(l +i)e/q,

and the wage will be given by

(16)

w=e+s+ie/q, (17)


so that, in effect, workers receive interest on their bond as long as they are working and have the principal returned when they retire. It should be noted that in this model, whether actual upfront bonds are required or whether a lower first period wage is sufficient to clear the market depends on the value of wi -b and not just on the value of b. This point is discussed in greater detail in Akerlof and Katz (1989).

If we divide (17) by h, we observe that the hourly wage will be less than the marginal disutility of employment if and only if

w/h = e/h + s/h + ie/(qh) < e’ + s’ = u + (i - r) (e/q)'. (18)

There is no presumption that this inequality will hold. It is straightforward to construct examples with the direction of inequality in either direction, even though the VMP exceeds the wage.

It is worth noting in passing that market clearing is not essential to the results of this section. If the competitive utility constraint (7) were replaced with a limit on the size of the bond, as in a standard efficiency wage model, eq. (11) would be unchanged. This conclusion contrasts sharply with Bulow and Summers (1986) who argue that in an efficiency wage model, firms will prevent workers from being employed part time. In essence, Bulow and Summers assume that both e” and q” equal zero. If this assumption were correct, the firm would set hours until either u’ equalled zero or the workers’ time endowment was exhausted. It seems likely that in either event, the hours chosen would be quite high and that therefore s(h) would be sufftciently high to make the labor market constraint binding. In effect, it is probable that under the Bulow/Summers assumption, firms would clear the market by setting very high work hours. However, as noted above, q” =0 is not a natural assumption.

4. Mandatory retirement

Just as hours are set suboptimally and workers may be constrained to work fewer hours than they wish even though the VMP exceeds the wage, mandatory retirement occurs before the optimal age and may occur even though workers are paid less than their VMP. To determine the mandatory retirement age, we need to consider the effect of increasing the contract by an extra year. The change in profits consists of two parts: firms gain or lose the difference between the value of marginal product and the wage and gain or lose from delaying repayment of the bond:

d prot3s=(u,-w,)/(l+r)T+P,/(1+r)T-PP,+,/(1+r)T+1. (19)

Mandatory retirement occurs when the change in profits equals zero.


Substituting from (14) for the pension and from (15) for the wage and rearranging terms gives the condition for mandatory retirement to be

uT-eT-ST=(i-r)eT+ll(qT+l(l +4),

or, in other words, at a point when the value of marginal product exceeds the disutility of employment. Thus, just as hours are chosen suboptimally in the bonding model, the mandatory retirement age is chosen suboptimally. It is worth noting that if the pension needed to deter shirking rises sufficiently rapidly with age or tenure, mandatory retirement may occur when the wage is less than the VMP, as can be seen in eq. (19). Thus, this special form of hours constraint may also arise because of the cost of ensuring incentive compatibility.

It is straightforward to show that in the special case when the e, s and u profiles, and thus h and w, are all independent of age, firms need never establish a mandatory retirement age. Firms are indifferent with respect to the length of the contract. To see this note that firms will employ workers up to the point that over the length of the contract the value of marginal product equals compensation, or

C(v-w)/(l+r)‘_’ +b-P/(1 +r)T=O. (21)

Substituting for w, b and P gives the result that the profit maximization condition (21) and the mandatory retirement condition (20) are identical in this model. In other words, extending the contract by one year has no effect on profits in this case, and the firm is indifferent with respect to when the worker retires. In general, however, the u, e and s schedules will not be constant over the worker’s lifetime. Provided that the disutility of employment rises faster than the VMP, the firm will set a mandatory retirement age.

5. Concluding remarks

The hours constraints discussed above cast light on the bonding/efficiency wage debate. A major issue between proponents of the two models has been whether the agency problem imposed by shirking results in involuntary unemployment [Carmichael (1985), Shapiro and Stiglitz (1984, 1985)]. Carmichael maintains that the involuntary unemployment which arises in shirking versions of the efficiency wage model depends critically on the assumption that workers cannot post bonds. He argues that even with capital market imperfections, the level of bonds will adjust so that the marginal worker is just indifferent between working and not working.

In the model presented in this paper, it seems reasonable to argue that workers who have passed the mandatory retirement age are involuntarily unemployed although the issue may be more one of semantics or ideology


than of substance. They are voluntarily unemployed since they are unwilling to work at the best offer any employer is willing to make them. On the other hand, their reservation wage is below their value of marginal product. The reason that they cannot reach an agreement with a firm is, of course, that the wage they require plus the cost of ensuring incentive compatibility is less than their VMP.

In addition, the results suggest that whether or not the agency model leads to involuntary unemployment, it definitely leads to involuntary underemployment of which involuntary unemployment is an extreme case. Involuntary underemployment may be as significant an economic phenomenon as involuntary unemployment. Kahn and Lang (1987) estimate that 40-45 percent of hourly workers are involuntarily underemployed (i.e. want to work more hours) while only about 3 percent are involuntarily overem- ployed. While any model in which wages and the VMP diverge will lead workers to desire a different number of hours from those they actually work, most such models do not lead naturally to underemployment being much more prevalent than overemployment. This paper suggests that the agency problem may provide an explanation for the prevalence of such underemployment.

Finally, I note that the market equilibrium with costly bonding is not even second best. The reason is that by serving as potential employers of fired workers, firms impose an externality on each other which results in larger bonds and increased bonding costs. Government could improve on the market equilibrium by making it more costly to shirk and be fired (or more absurdly by allowing firms to execute shirkers). Thus, the agency model with costly bonding retains the same general flavor as efficiency wage models in which government can improve on the market equilibrium by making unemployment more costly.

References

Akerlof, George and Lawrence F. Katz, 1989, Workers’ trust funds and the logic of wage profiles, Quarterly Journal of Economics, forthcoming.

Becker, Gary S. and George Stigler, 1974, Law enforcement, malfeasance, and the compensation of enforcers, Journal of Legal Studies 3, l-18.

Bulow, Jeremy I. and Lawrence H. Summers, 1986, A theory of dual labor markets with application to industrial policy, discrimination, and Keynesian unemployment, Journal of Labor Economics 4, 37G414.

Carmichael, H. Lorne, 1985, Can efficiency wage unemployment be involuntarily?, American Economic Review 75, 1213-1214.

Dickens, William T., Lawrence F. Katz, Kevin Lana and Lawrence H. Summers. 1989. Employee crime and the monitoring puzzle, Journal o? Labor Economics 7, forthcoming.

Kahn, Shulamit and Kevin Lang, 1987, Constraints on the choice of work hours, NBER Working paper, no. 2238.

Lazear, Edward, 1979, Why is there mandatory retirement?, Journal of Political Economy 87, 1261-1284.


Lazear, Edward, 1981, Agency, earnings profiles, productivity and hours restrictions, American Economic Review 71, 606-620.

Medoff, James L. and Katherine G. Abraham, 1980, Experience, promotion and earnings, Quarterly Journal of Economics 95, 703-736.

Shapiro, Carl and Joseph E. Stiglitz, 1984, Equilibrium unemployment as a worker discipline device, American Economic Review 74, 433-444.

Shapiro, Carl and Joseph E. Stiglitz. 1985, Can unemployment be involuntary? Reply, American Economic Review 75, 1215-1217.

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