213
[ Journal of Labor Economics, 2005, vol. 23, no. 2]
᭧ 2005 by The University of Chicago. All rights reserved.
0734-306X/2005/2302-0001$10.00
Speciﬁc Training Sometimes Cuts
Wages and Always Cuts Turnover
Lalith Munasinghe, Barnard College, Columbia University
Brendan O’Flaherty, Columbia University
Turnover falls with tenure, but wages do not always rise (and sometimes
fall) with tenure. We reconcile these ﬁndings by revisiting an
old issue: how gains from ﬁrm-speciﬁc training are split between
workers and ﬁrms. The division is determined by a stationary distribution
of outside offers. The lower the wage a ﬁrm pays to a
speciﬁcally trained worker, the more proﬁt it makes but the more
likely the employee is to leave. The optimal time paths of wages and
turnover show that, if marginal product is increasing, wages need not
be increasing but it always implies a falling turnover rate.
I. Introduction
Most theories of the labor market predict that, if speciﬁc training occurs,
increasing tenure on a job will both raise wages monotonically and cut
turnover. Since current empirical evidence supports very strongly the
proposition that tenure cuts turnover but does not support the proposition
that tenure raises wages, we are left in a quandary: extant theory cannot
tell us whether speciﬁc training occurs or not. In this article, we resolve
this quandary with a new theory. If speciﬁc training occurs, increasing
tenure must cut turnover but it need not always raise wages. Wages that
do not rise monotonically do not indicate that speciﬁc training is absent.
The positive cross-sectional association between tenure and earningsorigWe
thank Emanuele Gerratana, Cynthia Howells, Tack-Seung Jun, and Nachum
Sicherman for many helpful comments. Contact the corresponding author, Lalith
Munasinghe, at lm25@columbia.edu.
214 Munasinghe/O’Flaherty
inally led researchers to conclude that wages rose with tenure. However,
more recent evidence based on longitudinal data shows that the positive
wage returns to tenure are small (Abraham and Farber 1987; Altonji and
Shakotko 1987; Abowd, Kramarz, and Margolis 1999), despite an ubiquitous
strong negative tenure effect on turnover.1
Within-job wage cuts are
not uncommon (Baker, Gibbs, and Holmstrom 1994), and some evidence
even points to negative returns to seniority (Ransom 1993). In summary,
this empirical literature shows that wage dynamics are more complicated
and variegated than the simple presumption that wages monotonically increase
with job tenure.
We reconcile these ﬁndings by revisiting an old issue: how the gains
from speciﬁc training are split between workers and ﬁrms. The division
is determined by a stationary distribution of outside offers. The model
is ex post monopsony: the lower the wage a ﬁrm pays to a speciﬁcally
trained worker, the more proﬁt it makes and the more eager it is to have
that employee stay, but also the more likely the employee is to leave. We
solve for the optimal time paths of wages and turnover probabilities. Even
if marginal product is increasing, wages need not be increasing; however,
rising marginal product always implies a falling turnover rate. While speciﬁc
training should always cut turnover, there is no theoretical reason
why it should always raise wages. We show why sometimes wage cuts
happen. Wages eventually rise if you stay employed forever, but nothing
in our model guarantees that your completed wage proﬁle will be upward
sloping. Whenever there is a wage cut, there is a probability that the
worker will leave in that period.
Our model is designed to address some leading empirical puzzles in
labor economics. The consensus among empirical labor economists is that
wage returns to tenure are small—not that they are zero. So wages on
average do increase with tenure, despite occasional within-job wage cuts.
The well-recognized puzzle in the literature is why an additional year of
tenure reduces turnover by a lot but increases wages by very little. Our
results are consistent with this asymmetric tenure effect on wages and
turnover. Moreover, turnover decreases are compatible even in the presence
of wage cuts. Thus, our model provides an explanation for monotonically
decreasing turnover, weak but positive tenure effects on wages,
and, especially, occasional wage cuts.2
The remainder of this article is organized as follows. In Section II, we
discuss some related theory and highlight various features of the model.
1
Topel (1991) shows that wages do rise substantially with seniority. However,
in a reassessment, Altonji and Williams (1997) argue that Topel overestimates
these returns.
2
Demand shocks and nominal wage rigidities are other obvious explanations
for real wage cuts. Our point is that within-job wage cuts could also occur for
less obvious reasons.
Speciﬁc Training and Turnover 215
Section III presents the model and shows that turnover always falls as
tenure increases. Section IV derives our main result: wage cuts can occur
even if marginal product is increasing. We show the conditions (on the
outside offer distribution) under which wage cuts will occur, and we argue
that there is nothing unusual or bizarre about such conditions. The article
concludes, in Section V, with a brief summary.
II. Related Theory
The problem of wage determination when an employment relationship
generates ﬁrm-speciﬁc rents has been well known since Becker’s original
idea of sharing costs and returns of ﬁrm-speciﬁc investments as a means
of providing mutual insurance to each party’s investment (Becker 1962;
Parsons 1972; Hashimoto 1981). These models show the dual effects of
tenure, namely, wage increases and turnover decreases. Our model also
predicts turnover decreases, but, unlike these earlier models, it predicts
that wages do not increase monotonically with tenure even if marginal
product increases monotonically with tenure.
As is well known, Becker-type sharing rules imply inefﬁciently high
turnover because the party initiating turnover does not take into consideration
the resulting loss to the other party’s investment. Our assumption
of ex post monopsony also generates inefﬁciently high turnover. Mortensen
(1978) considers various employment agreements, including
“matching alternative offers,” that would induce both workers and employers
to pursue efﬁcient or joint wealth-maximizing search strategies.
Although offer matching has been widely adopted as a wage-setting mechanism
since Mortensen (e.g., Harris and Holmstrom 1982; Postel-Vinay
and Robin 2002; Munasinghe 2005), we exclude Mortensen-type ex post
bargaining in our model. So the question is whether more efﬁcient wage
policies such as offer matching can also generate wage cuts. We argue that
wage cuts are not an artifact of inefﬁciency. We show that a model of
efﬁcient turnover—offer matching a` la Mortensen—leads to both occasional
wage cuts and decreasing turnover. However, various considerations
make offer matching somewhat controversial.
Jovanovic’s article “Firm-Speciﬁc Capital and Turnover” (1979), like
our model, integrates speciﬁc human capital theory and job search theory.
Jovanovic shows how match quality jointly determines search effort and
investment in speciﬁc human capital. Jovanovic’s central result is that
turnover declines with tenure. He derives this result under the restrictive
assumption that wages always equal marginal product. We show the same
result under the weaker condition that ﬁrms have no commitment to pay
marginal product every period. Clearly, the wage sequence a ﬁrm can
offer a worker depends on the commitment ability of the ﬁrm—that is,
whether a ﬁrm can be trusted not to renege on a promised wage schedule,
216 Munasinghe/O’Flaherty
since productivity increases on the job are assumed to be ﬁrm speciﬁc.
Because we focus on the no-commitment-ability case—that is, on selfenforcing
contracts—our results, unlike Jovanovic’s, are time consistent.
However, we also show that, if ﬁrms have complete commitment ability,
then they will delay payments for as long as they can. This is an argument
for pensions quite independent of disciplinary considerations.
The model presented here is also closely related to another paper (Munasinghe
2005) that was designed to explain a variety of stylized ﬁndings
on wage and turnover dynamics. Both models build on a common set of
human capital and job search assumptions: speciﬁc training is strictly learning-by-doing,
and thus neither the ﬁrm nor the worker needs to make
either decisions or sacriﬁces in terms of investments;3
workers receive a
single job offer from a stationary distribution in each period;4
and, because
the market for prospective workers is competitive, ﬁrms offer zero-proﬁt
wage schedules.5
But, unlike the model here, Munasinghe assumes downward
wage rigidity and ex post bargaining. As a result, his model cannot
theoretically address the ﬁnding about within-job wage decreases.
Our assumption that speciﬁc training is strictly learning-by-doing contrasts
with other compensation models, such as up-or-out contracts (Kahn
and Huberman 1988) and promotion-based contracts (Prendergast 1993),
both of which are designed to create incentives for workers to invest in
(nonveriﬁable) ﬁrm-speciﬁc human capital. The common implication of
these models is that the worker is paid a higher posttraining wage, which
implies wage increases with job tenure.6
The objective of this literature,
however, is to understand labor practices as commitment devices to avoid
ﬁrm opportunism ex post in capturing rents from speciﬁc training. The
“no-commitment” wage policies, like the one presented here, are relevant
to employment relationships when ﬁrms cannot rely only on their reputation
to be credible.
III. Turnover
A. The Model
Time is discrete, and t designates the number of periods of completed
tenure that a worker has accumulated at the beginning of the current
3
As a consequence, both models are silent about layoffs.
4
The ﬁnite number of offers per period implies ﬁxed search costs, and hence
search effort is not endogenous.
5
Munasinghe (2005) presents a more complete discussion of how such competitive
market might arise when employment relationships generate speciﬁc rents.
6
Postinvestment wages may not increase by much for reasons given in Waldman
(1990). Waldman argues that up-or-out contracts are incentive compatible in an
environment where human capital is general but information is private. The return
for being retained is not the result of a precommitment to a high postinvestment
wage, but rather it is the result of the positive signal (to other ﬁrms) and bidding
by other ﬁrms.
Speciﬁc Training and Turnover 217
period. Let denote the marginal revenue product of a worker who hasyt
accumulated t periods on the job; thus, is the marginal product of ay0
new worker. We assume that ( ) is a monotonically increasing sequence;yt
speciﬁc training occurs.
Let denote the wage of a worker who has completed t periods; wewt
refer to this as the wage of an age-t worker. At the beginning of each
period (except period 0), the worker receives an outside offer. The outside
offer has a present value of v, where v is a random variable drawn independently
each period from the same distribution, for which is theG(.)
cumulative distribution function (CDF). (If the worker would get a signing
bonus for taking the outside job, that signing bonus is included in v,
along with the expected present value of wages and transitions opportunities.)
We assume that is strictly increasing on its support. Sta-G(.)
tionarity reﬂects the assumption that it is speciﬁc training that we are
talking about; since the training affects only the worker’s productivity
within the ﬁrm, there is no reason for the distribution of outside offers
to change.
Let denote the worker’s value of optimal continuation with the ﬁrm.vt
This value is evaluated before the outside option becomes known. Assume
that the discount factor is and that work involves no utility ord ! 1
disutility. Then,
( )v p E max v, w ϩ dv , G t ≥ 1,[ ]t t tϩ1
and
v p w ϩ dv ,0 0 1
because the worker cannot receive an outside offer before going to work
(we take this as a deﬁnition of what an “outside” offer is). It will be
convenient to write
x p w ϩ dv .t t tϩ1
Then, the probability that a worker quits depends on ; speciﬁcally, axt
worker stays in period t if and only if (in the event of ties we assumev ≤ xt
that the worker stays). We call the worker’s “comparison value.”xt
Let , , denote the probability that an age-t worker stays on theh G t ≥ 1t
job. Then,
h p G(x ).t t
We refer to as the continuation hazard. Hence,ht
v p h x ϩ (1 Ϫ h )E(vFv 1 x ).t t t t t
218 Munasinghe/O’Flaherty
For future reference, note that
xt
v p x dG(v) ϩ vdG(v),t ͵ t ͵xt
and so
dv p G(x )dx 1 0. (1)t t t
On the ﬁrm side, let denote the optimal continuation value for aVt
ﬁrm with an age-t worker. Each period the ﬁrm chooses a wage offer
to maximize its expected present value. Since the ﬁrm has no precom-wt
mitment power, it can assume that its future wages, , , and sow wtϩ1 tϩ2
forth, will also be set optimally and dynamically consistently, and so we
can consider them as given when it chooses . Thus, we can think ofwt
the ﬁrm as choosing rather than . But since is strictly increasing,x w G(.)t t
it is an invertible function, and we can think of the ﬁrm as choosing ,ht
the continuation hazard. Let be the inverse of G:F(.)
h p G(F(h)).
Then, is given by the fundamental recursion equationVt
V p max [h(y Ϫ w ϩ dV ) ϩ (1 Ϫ h)V ], G t ≥ 1,t t t tϩ1 0
h
where
w p F(h ) Ϫ dv .t t tϩ1
We can further simplify by writing as the total rent from the rela-Rt
tionship,
R p V ϩ v .t t t
Then, the above recursion equation becomes
V p max [h(y Ϫ F(h) ϩ dR ) ϩ (1 Ϫ h)V ].t t tϩ1 0
h
Assume that . We make this assumption because we want to studyV p 00
ex post monopsony—that is, monopsony after the worker has been hired
and has acquired some speciﬁc skills—not ex ante monopsony as well.
So we must suppose that there are many ﬁrms identical to this one bidding
for the worker and that, if were positive, even more ﬁrms would enter.V0
The market for prospective workers is competitive. This assumption is
not crucial, but it simpliﬁes the mathematics and is consistent with much
other theory in labor economics.
Speciﬁc Training and Turnover 219
Finally, let denote expected surplus at time t contingent on theR*t
worker staying on the job that period. Hence,
R* p y ϩ dR .t t tϩ1
Note that is independent of the ﬁrm’s action at t.R*t
Then, the recursion equation for the ﬁrm is
V p max [h(R* Ϫ F(h))].t t
h
Let denote the optimal continuation hazard. Then,ht
R p V ϩ vt t t
p h R* ϩ (1 Ϫ h )E(vFv 1 F(h ))t t t t
p h R* ϩ vdG(v).t t ͵
Bargaining and wage setting are somewhat different in period 0. Since
we assume as a convention that no outside offers are possible in period
0 (since there is no “outside” until period 0 has been completed), we take
the “continuation hazard” to be undeﬁned, since there is no relationship
to be continued. A signing bonus (or penalty) has to be set to lure the
worker in, not to keep her from leaving, and our perfect competition
assumption implies that the bonus/penalty is high enough that , theV0
ﬁrm’s value of a new worker, is zero.7
Hence,
w p y ϩ dV .0 0 1
Since the signing bonus/penalty is set by a different process from wages
in other periods, we will not pay much attention to it. We do not consider
it a wage payment.
7
This signing bonus plays a similar role to the front-loaded wage contracts in
Black and Loewenstein (1991). Their model is based on heterogeneity of mobility
costs, and employers have monopsony power because it is costly for workers to
switch employers. Although the source of monopsony power in our model is
speciﬁc training, in both cases, workers, in anticipation of future monopsony
power of the employer, will demand higher wages up front. Another interesting
point is Black and Loewenstein’s claim that, in the absence of speciﬁc training,
wages are a decreasing function of tenure. However, in their model, speciﬁc training
tends to increase wages as tenure lengthens. By contrast, as we will see in the
next section, our model can generate falling wages even when speciﬁc training is
ongoing, depending on the distribution of outside offers. The requirement that
the value of an incipient relationship to an employer be zero rules out the sort
of equilibrium where workers always go hopping quickly from employer to
employer, picking up huge signing bonuses with every hop. Employers would
not pay such bonuses.
220 Munasinghe/O’Flaherty
B. General Results
Now we prove that ( ) is a weakly increasing sequence: turnover alwaysht
falls as tenure increases.
Proposition 1. The sequence is (weakly) increasing.(h )t
Proof of proposition 1. See the appendix.
Thus, in our model, no matter what the distribution of outside offers
is, turnover decreases as tenure lengthens. The value of the job to the
worker, , also monotonically increases as tenure grows.F(h )t
This does not imply, however, that wages grow monotonically. Current
wages are only part of the worker’s inducement to reject outside offers.
The other part is the continuation value. The continuation value may or
may not be growing faster than the total inducement that the ﬁrm wants
to provide. Only if the total inducement is growing faster than the continuation
value will wages rise. This may or may not happen. Section IV
analyzes this issue.
We can also prove that turnover will be inefﬁciently high in this equilibrium.
This result, however, is well known, intuitively obvious (since it
is a monopsony), and the proof would introduce more notation. We
therefore omit it.
Turnover in this model depends on , the rent the relationship wouldR*t
generate if the worker turned down this period’s outside offer. Speciﬁc
training decreases turnover because it increases rent. More speciﬁc training
implies lower turnover in the sense that a job where marginal productivity
is higher every period than it is in another job will have (weakly) lower
turnover every period. Rent drives turnover in Mortensen’s (1978) matching-offers
model, too.
The relationship between rent and turnover is ﬁrst order, however, not
second order: bigger increases in do not necessarily correspond withR*t
bigger increases in . This is because the function that links andh h R*t t t
is, in general, nonlinear. Mortensen’s model also has this property.
In one case, however, we can derive a result that resembles “more
quickly rising marginal product implies more quickly falling turnover
rate.” Let ( ) and ( ) be two sequences of marginal products, and lety yt t
( ) and ( ), respectively, be the associated sequences of retention rates,h ht t
all other parameters being the same. The following corollary follows.
Corollary 1. Let for all , and let for ally p y t ≥ T 1 1 y 1 yt t t t
. Then, for all and for all , with strictt ! T h p h t ≥ T h ≥ h t ! Tt t t t
inequality if .h ! 1t
Proof of corollary 1. See the appendix.
C. Other Closely Related Models
The sequence of wages that the ﬁrm offers depends on its commitment
ability. We have focused on the case where the ﬁrm has no commitment
Speciﬁc Training and Turnover 221
ability. However, if the ﬁrm has complete commitment ability, then the
solution to the ﬁrm’s problem is trivial: it postpones payment as long as
it can. In the appendix, we show this for the uniform case.
Another distinctive feature of our model is the inefﬁciency of ex post
monopsony. One alternative model with efﬁcient turnover is Mortensen’s
(1978) model of offer matching. It is easy to see that this model will also
have declining turnover with tenure. The greater the value of rent, the
higher the outside offer the employer will match and so the smaller the
probability that the worker will leave.
IV. Wages
A. Motivating Example: Two-Point Distribution
The intuition for why wage cuts sometimes occur is best imparted by
considering a particular example from one of the simplest distributions
of outside offers. The next section shows how this example generalizes.
Suppose, then, that the outside offer can take one of only two values:
with probability p and with probability . The sequence of mar-v v 1 Ϫ pl h
ginal products is strictly increasing, but the differences between suc-(y )t
cessive values are small in a sense that we will make more precise shortly.
Assuming that marginal products are high enough that the employer
would hire the worker in the ﬁrst place but not so high that the employer
would never take a chance on the worker’s departure, it would be easy
to see what the sequence of wages would have to accomplish. When the
worker is relatively junior and so not especially productive, the employer
will be prepared to let her go when she receives the high outside offer
. Thus, with junior workers, the employer will set the worker’s com-vh
parison value at —so that the junior worker will stay when she receivesx vt l
the low offer and leave when she receives the high offer .v vl h
However, when the worker is senior and sufﬁciently productive, the
employer will want her to stay no matter what. So the employer will set
, and the worker will stay forever.x p vt h
At some point, the worker will transition from being junior (and dispensable)
to senior (and indispensable). We call the transition time T.
Speciﬁcally, T is the ﬁrst period for which and we consider thex p vt h
worker to be senior. It is easy to calculate T, but we will not do so here.
Our implicit assumption about slowly rising marginal product can now
be translated into the explicit assumption that . This informal ar-T 1 2
gument can be summarized in ﬁgure 1, which graphs as a function ofxt
the worker’s tenure on the job.
The employer keeps the worker’s comparison value at up until timevl
T and then increases it to . (Since the worker’s termination hazard is pvh
before T and 0 at T and after, this example clearly shows declining
turnover.)
222 Munasinghe/O’Flaherty
Fig. 1.—Comparison value ( ) for two-point distributionxt
What sequence of wages ( ) implements the sequence of comparisonwt
values ( ) in ﬁgure 1? A moment’s thought shows that holding wages atxt
a low value before T and then raising them will not create the sequence
of comparison values that the employer wants. Such a sequence of wages
would make continuation values start rising well before T, and thisvt
would mean that comparison values would rise gradually for a juniorxt
worker, instead of remaining constant. With constant wages before a jump
up, the job gets more valuable as the worker approaches the jump—
contrary to the optimal pattern shown in ﬁgure 1.
Implementing the sequence of continuation values in ﬁgure 1 therefore
requires that wages go down from period to . The wage cutT Ϫ 2 T Ϫ 1
in must be just big enough that the prospect of losing wages atT Ϫ 1
and then getting a raise at T and beyond is just the same in presentT Ϫ 1
value as getting forever the regular junior wages (the wages at andT Ϫ 2
earlier). Then as a junior worker approaches T, she becomes no better
off—the prospect of the wage rise at is exactly offset by the prospectT Ϫ 1
of the wage cut at . This creates ﬁgure 1’s ﬂat proﬁle of comparisonT Ϫ 1
values that we know is optimal.
Algebra conﬁrms this verbal argument. For a senior worker, with t ≥
, , since continued employment is certain. For a junior workerT v p vtϩ1 h
not approaching the raise, that is, for ,t ≤ T Ϫ 2
C
v p pv ϩ (1 Ϫ p)v p v,tϩ1 l h
Speciﬁc Training and Turnover 223
Fig. 2.—Iso-value curves for given R*tϪ1
since she will continue if she gets the low outside offer (with probability
) or leave if she gets the high outside offer (with probability ). Thus,p 1 Ϫ p
w p x Ϫ dv p v Ϫ dv.t t tϩ1 l
But for a junior worker about to get the raise, ,t p T Ϫ 1 v p v ptϩ1 T
, and sovh
w p x Ϫ dv p v Ϫ dv ,TϪ1 TϪ1 T l h
which is smaller than all preceding wages. To maintain the same termination
hazard, the employer has to offer less in current wages to a junior worker
about to be promoted.
B. How Special Is This Example?
What was it about the two-point distribution that made wages decrease?
In terms of fundamentals, we will argue in this section that a steep CDF
of outside offers is the driving force behind this result. The argument in
this section will again be graphical and intuitive.
To make this argument, we must take a step back from ﬁgure 1 and
examine the employer’s decision problem in a bit more detail. At any
period t, we can consider the problem in space. First, since value is(v, h)
, once we know , we can draw iso-value curves, as in ﬁgureh(R* Ϫ x) R*tϪ1 tϪ1
2. These curves slope upward and asymptote at . Of course, thex p R*tϪ1
employer cannot choose h and x independently: he is bound by the constraint
, which we illustrate for the two-point case in ﬁgure 3.h p G(x)
Thus, when is fairly small (but larger than ), the employer’s optimumR* vtϪ1 l
224 Munasinghe/O’Flaherty
is at the lower kink, as we show in ﬁgure 4 (and we consider the worker
junior). Small changes in do not change the employer’s optimum,R*tϪ1
and so long as the worker stays junior, does not change. That is thext
nature of a kink in a constrained optimization program (and the fact that
the kink in this example is a right angle is not necessary). If , forG(.)
instance, looked as in ﬁgure 5, the employer would still stay on the lower
kink for many periods if marginal product increased slowly.
Now consider how the ﬁrst difference in wages relates to the ﬁrst
difference in comparison values. From the deﬁnition of comparison value,
w Ϫ w p (x Ϫ x ) Ϫ d(v Ϫ v ).tϩ1 t tϩ1 t tϩ2 tϩ1
Notice that the second term in (1) is nonpositive, since worker’s value is
weakly monotonically increasing. This term, moreover, depends solely on
events occurring after . Thus, wages can rise only if the differencet ϩ 1
is sufﬁciently great. But when the kink captures comparison(x Ϫ x )tϩ1 t
value at the same place for a long period of time, no force is present to
raise wages.
Of course, as long as , no force is present in the junior(v Ϫ v ) p 0tϩ2 tϩ1
phase to reduce wages either, and so they stay constant. But when t p
,T Ϫ 2
v Ϫ v p v Ϫ v 1 0,tϩ2 tϩ1 T TϪ1
and so wages go down.
Thus, the feature of the two-point distribution that makes wage cuts
possible is the steepness of the CDF in the vicinity of that makesvl
successive values of the same for at least two periods. But makingxt
successive values of the same was not necessary for the argument wext
used—all that was needed was for them to be sufﬁciently small. Thus,
kinks in the CDF are not required either—all that is needed is that the
CDF be sufﬁciently steep in some region and sufﬁciently ﬂat nearby that
successive values of be sufﬁciently close.xt
Wage cuts, therefore, do not require that the distribution of outside
offers be atomic. It is also easy to construct examples with multiple wage
cuts (if marginal products rise slowly enough, then if the distribution of
outside offers is an n-point distribution, a worker will experience n Ϫ 1
wage cuts if she stays with the same ﬁrm throughout her career). We can
even construct examples with wage cuts in arbitrarily many successive
periods. (Even if wages are cut in arbitrarily many periods, both the
workers and the employer must anticipate that, at some point, wages will
rise if the employment relationship continues long enough. In this sense,
wage cuts are “temporary.” But there is never a guarantee that the relationship
will last long enough for wages to rise again.) In the next section,
we will generalize this intuition and show more rigorously which sorts
of outside offer CDFs lead to wage cuts.
Fig. 3.—Likelihood of staying for two-point distribution
Fig. 4.—Employer’s optimum
Fig. 5.—Alternative distribution of outside offers
226 Munasinghe/O’Flaherty
C. General Results on Wage Cuts
Wages will fall when the CDF of the outside offer distribution in the
neighborhood of is rising steeply enough, provided that the employ-xt
ment relationship is productive enough and the value of the relationship
to the worker will grow quickly enough in the future. In other words,
constructing examples with wage cuts is relatively easy and does not
require unusual distributions of outside wage offers or bizarre marginal
product sequences. Of course, since little or nothing is known empirically
about the kinds of outside offer distributions workers actually experience,
we cannot make any statements about the probability of wage cuts.
In this section, we will show some sufﬁcient conditions for wages at
period to be lower than wages at period t, for . These conditionst ϩ 1 t ≥ 1
will not be expressed directly in terms of primitives, but we will show
that, for some sets of primitives, they will be satisﬁed.
In this section, we will assume that the CDF of outside offers isG(.)
twice differentiable. We have already shown, with the example of the twopoint
distribution, that wage cuts are possible when the CDF of outside
offers is not differentiable.
Let denote the probability distribution function (PDF) correspond-g(.)
ing to the CDF G. Our main result about wage cuts is given below.
Proposition 2. Let be twice differentiable for and satisfyG(.) x ! xtϩ1
12
[g(x)] 1 g (x)G(x), (G x). (2)
2
If for all ,x ≤ xtϩ1
G(x)
! R* Ϫ x ϩ d(v Ϫ v ), (3)t tϩ1 tϩ2 tϩ1
g(x)
then .w 1 wt tϩ1
Proof of proposition 2. See the appendix.
The intuition behind this proposition, especially the condition shown in
(3), is that, if the slope of the CDF G is sufﬁciently steep in the neighborhood
of , as in ﬁgure 6, then will be very close to . (Steepnessx x xt t tϩ1
is reﬂected in being large relative to and so being small.)g(x) G(x) G(x)/g(x)
“Sufﬁcient” steepness in this case depends on how productive the relationship
is, , and how quickly the value of the relationship to the workerR*t
will grow in the future, .d(v Ϫ v )tϩ2 tϩ1
D. Alternative Models
It is trivial to show that, in Mortensen’s offer-matching model, wages
will sometime fall. When the worker gets a bad draw from the outsideoffer
distribution after a good draw, then the employer will cut the value
of the job to the worker, not just the wage, even with marginal product
Speciﬁc Training and Turnover 227
Fig. 6.—Employer’s optimum with differentiable cumulative distribution function
increasing. Hence, offer matching, like our model, can generate both wage
cuts and always-decreasing turnover.
Of course, wage cuts with offer matching will only occur if outside
offers must be accepted or rejected within a ﬁnite time period—that is,
if offers have time limits.8
(If workers report only “good” offers to their
current employers, wages will still fall when bad offers are received, since
employers can infer from a worker’s silence that her offer this period was
bad.) If offers last only one period, wages will go down whenever a bad
offer follows a good offer. The frequency of wage cuts will, of course,
depend inversely on the length of time it takes for an offer to expire.
Mortensen implicitly assumes downward wage rigidity despite his claim
that outside offers must be accepted or rejected immediately—that is, “no
recall” of outside offers.9
However, various considerations make offer matching somewhat controversial.
First, offer matching (with ﬁnite recall) leads to implausible
life-cycle wage dynamics. Individual wages over time will follow a meanreverting
process. This runs counter to considerable empirical evidence
8
With inﬁnite recall of outside wage offers, offer matching would never generate
within-job wage cuts (Munasinghe 2005). But permitting inﬁnite recall does not
seem like an always reasonable strategy to follow for employers making outside
offers.
9
In Mortensen’s continuous time framework, current wage remains the same
as the previous period wage unless a better outside offer is matched by the ﬁrm
(or the worker separates).
228 Munasinghe/O’Flaherty
about how wages evolve within and between jobs over time.10
By contrast,
although our model generates (occasional) wage cuts, job value always
increases (because productivity increases), and sooner or later wages must
also increase. Second, offer matching may involve high transactions costs
(Hashimoto 1981). Third, if information about outside offers is nonveriﬁable,
then offer matching is not feasible, and thus, ﬁrms would have to
resort to time-consistent wage policies like the one proposed in this article.
If employers have full commitment power, then wage cuts are impossible
but the wage-tenure proﬁle is different from empirically observed
proﬁles. The worker is paid only in the last period and never before the
last period.
V. Conclusion
We present a model to reconcile recent ﬁndings on wage and turnover
dynamics. Although the evidence of a positive cross-sectional association
between earnings and seniority is widespread, the fairly recent use of
longitudinal data has allowed researchers to ask whether individual wages
do rise with seniority. Somewhat surprisingly, the evidence on wage
changes with tenure appears to be far more variegated than the simple
presupposition that wages rise with seniority. Average wage increases with
tenure appear to be small, if not negligible, and within-job wage cuts are
not uncommon either. The challenge to theory is to reconcile this complex
picture of wage dynamics with the fact that the negative relation between
turnover and tenure remains as ubiquitous as ever. Our model, built on
the workhorse theories of speciﬁc training, job search, and self-enforcing
contracts, shows that, even if marginal product is increasing due to ongoing
speciﬁc training, wages need not be increasing; however, rising
marginal product always implies a falling turnover rate.
Appendix
Proof of Proposition 1
We proceed by a series of lemmas.
Lemma 1. is an increasing function of .V R*t t
Proof. Let maximize for and assume thath h(R* Ϫ F(h)) i p 1, 2i i
. Since is a maximizer,R* 1 R* h1 2 1
h (R* Ϫ F(h )) ≥ h (R* Ϫ F(h )). (A1)1 1 1 2 1 2
10
Of course, average wage increases over the life cycle and within jobs could
be due to general skill improvements. However, if speciﬁc training is important,
then the wage dynamic predictions of offer matching with ﬁnite recall does not
square with the empirical evidence.
Speciﬁc Training and Turnover 229
Since and ,h 1 0 R* 1 R*2 1 2
h (R* Ϫ F(h )) 1 h (R* Ϫ F(h )). (A2)2 1 2 2 2 2
And so combining (A1) and (A2),
h (R* Ϫ F(h )) 1 h (R* Ϫ F(h )).1 1 1 2 2 2
Lemma 2. is a weakly increasing function of .h R*t t
Proof. Retain the same notation as in the proof of lemma 1. Proof
is by contradiction. Suppose that . From (A1),h 1 h2 1
h F(h ) Ϫ h F(h ) ≥ R*(h Ϫ h ).2 2 1 1 1 2 1
Then
h F(h ) Ϫ h F(h )2 2 1 1
R* ≤ . (A3)1
h Ϫ h2 1
Similarly, since is a maximizer,h2
h (R* Ϫ F(h )) ≥ h (R* Ϫ F(h )),2 2 2 1 2 1
and so
(h Ϫ h )R* ≥ h F(h ) Ϫ h F(h ).2 1 2 2 2 1 1
Thus,
h F(h ) Ϫ h F(h )2 2 1 1
R* ≤ . (A4)2
h Ϫ h2 1
Combining (A3) and (A4), we obtain
R* ≥ R*,2 1
which is a contradiction.
Lemma 3. is a weakly increasing function of .R R*t t
Proof. Retain the same notation as lemmas 1 and 2, and let denotevi
the corresponding continuation value, and .i p 1, 2 x p F(h ), i p 1, 2i i
Since and by lemma 1 we have shown that is a weaklyR p V ϩ v Vt t t t
increasing function of , we need only know that is a weakly increasingR* vt t
function of . Note that lemma 2 implies that . Then,R* x ≥ xt 1 2
x x1 2
v Ϫ v p x dG(v) ϩ vdG(v) Ϫ x dG(v) Ϫ vdG(v)1 2 ͵ 1 ͵ ͵ 2 ͵x x1 2
x x2 1
p (x Ϫ x )dG(v) ϩ (x Ϫ v)dG(v) ϩ (v Ϫ v)dG(v)͵ 1 2 ͵ 1 ͵x x2 1
≥ 0,
230 Munasinghe/O’Flaherty
since the ﬁrst two integrals are nonnegative and the third is zero.
Lemma 4. An increase in for any increases .y t ≥ 0 Rtϩt t
Proof. Let . If increases, increases, and by lemma 2,t p 0 y R* Rt t t
increases.
Let . If increases, increases. By lemma 3, the increase int 1 0 y R*tϩt tϩt
increases . That implies thatR* Rtϩt tϩt
R* p y ϩ dRtϩtϪ1 tϩtϪ1 tϩt
increases. The increase in , in turn, increases , which increasesR* RtϩtϪ1 tϩtϪ1
, and so on, until increases.R* RtϩtϪ2 t
Thus we can write
R p Q(y , y , y , … y , …),t t tϩ1 tϩ2 tϩt
given G and d, and note that is a weakly increasing function of eachQ(.)
of its arguments.
Lemma 5. is a (weakly) increasing sequence.(R )t
Proof. Consider and .R Rt tϩ1
R p Q(y , y , … y , …),t t tϩ1 tϩt
R p Q(y , y , … y , …).tϩ1 tϩ1 tϩ2 tϩtϩ1
Since , , , . . . , by repeated applicationy 1 y y 1 y y 1 y R ≥ Rtϩ1 t tϩ2 tϩ1 tϩtϩ1 tϩt tϩ1 t
of lemma 4.
Proof of proposition. By assumption, is a weakly increasing se-(y )t
quence. By lemma 5, is a weakly increasing sequence. Hence, sinceRt
R* p y ϩ dRt t tϩ1
is a weakly increasing sequence, by lemma 1, is a weakly increasinght
sequence also.
Proof of corollary 1. For any t, , and t, , we have0 ! t ! T t ≥ T
y Ϫ y 1 y Ϫ y ,t t t t
the double-prime marginal product is increasing faster than the singleprime
marginal product, and
(1 Ϫ h ) Ϫ (1 Ϫ h ) ≥ (1 Ϫ h ) Ϫ (1 Ϫ h ),t t t t
the double-prime turnover rate is falling faster than the single-prime turnover
rate.
Complete Commitment Case
With complete commitment ability, the solution to the ﬁrm’s problem
is trivial: it postpones payment as long as it can. It is easiest to understand
this conclusion if we impose the restriction that no worker can work for
more than periods—for biological reasons, say. Then, the ﬁrm’s prob-T*
lem is to choose a sequence of wages ( ), , that maximizesw t p 0, … , T*t
the expected present value of its proﬁt. Let denote this objective func-V0
Speciﬁc Training and Turnover 231
tion, evaluated, as it must be, just as a worker is hired. We assume that
outside offers are uniformly distributed between 0 and 1.
It is easiest and most intuitive to consider the case ﬁrst. Then,T* p 1
max ( ) ( ) ( )V p y Ϫ w ϩd w y Ϫ w ϩ 1 Ϫ w V , G t ≥ 1.{ }[ ]0 0 0 1 1 1 1 0
w , w0 1
Clearly, proﬁt is maximized by setting (optimal is more dif-w p 0 w0 1
ﬁcult to calculate). Increases in merely reduce ﬁrst-period proﬁts; theyw0
do not increase period 1 retention because they are just irrelevant history
when the worker has to make his period 1 decision.
Now consider . Clearly . The probability of not quittingT* p 2 w p 00
at the beginning of period is
x p w ϩ dv ,1 1 2
where is an increasing function of and can be made as large asv w2 2
desired up to a maximum of one by making sufﬁciently large for anyw2
value of . Conditional on not quitting at the beginning of period 1, thew1
value of additional proﬁt is
( ) ( ) ( )p w , w py Ϫ w ϩ d[w y Ϫ w ϩ 1 Ϫ w V ].1 1 2 1 1 2 2 2 2 0
But is clearly maximized by setting , since any desired valuep w p 01 1
of can be obtained by setting and making sufﬁciently large.x w p 0 w1 1 2
Optimal wage scheme has , since proﬁts after period zero dependw p 01
only on and .x p1 1
By repeated arguments like this, we can establish that optimal wage
schemes when the ﬁrm has unlimited commitment power have positive
payments only in the last period; these schemes have maximum retentive
power for any expected outlay. Clearly, this result holds for any distribution
of outside offers, not just the uniform.
Proof of Proposition 2
Consider the employer’s problem at period t:
max G(x)[R* Ϫ x].t
x
The ﬁrst-order condition for a maximum is
g(x)(R* Ϫ x) Ϫ G(x) p 0,t
or
G(x)
p R* Ϫ x. (A5)t
g(x)
The second-order condition is
g (x)(R* Ϫ x) Ϫ 2g(x) ! 0.t
232 Munasinghe/O’Flaherty
Substituting from (A5), this becomes
G(x)
g (x) Ϫ 2g(x) ! 0,
g(x)
or
12
[g(x)] 1 g (x)G(x),
2
which is condition (2) in proposition 2. Hence, the second-order condition
holds, and (A5) is necessary and sufﬁcient for a maximum of the employer’s
problem. Hence,
G(x)
p R* Ϫ x ,t t
g(x)
or
G(x)
x p R* Ϫ . (A6)t t
g(x)
Substituting in (1),
G(x)
w Ϫ w p x Ϫ R* ϩ Ϫ d(v Ϫ v )tϩ1 t tϩ1 t tϩ2 tϩ1( )g(x)
G(x)
p Ϫ {R* Ϫ x ϩ d(v Ϫ v )}.t tϩ1 tϩ2 tϩ1
g(x)
If condition (3) in proposition 2 holds, this expression is negative. Hence,
w 1 w .t tϩ1
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