A Theory of Wage and Turnover Dynamics∗
Lalith Munasinghe
Department of Economics
Barnard College, Columbia University
E-mail: [email protected]
March 2002
Abstract
This paper presents a theory — built on job search, firm specific
human capital, and disreputable contracting — to provide a unified
explanation for a host of widely documented findings on wage and
turnover dynamics. In particular, the theory reconciles two seemingly
inconsistent facts: evidence that past wage growth on a job reduces
turnover; and yet, using the same data, evidence that there is no
significant variation in wage growth rates among jobs because of lack
of serial correlation of wage growth. The main results of the paper are
as follows. First, average wage growth is higher in high learning jobs
because higher outside job offers are more likely to be matched in jobs
that generate more firm-specific rents. Second, past wage growth on
a job is negatively related to turnover since worker-firm separations
are less likely in high learning jobs. Third, although productivity
increases on the job are serially correlated, wage increases are not
necessarily serially correlated. Finally, the model implies asymmetric
tenure effects on wages and turnover, consistent with weak positive
wage effects and strong negative turnover effects that are both widely
documented in the labor literature.
∗
I thank Prajit Dutta, Rajiv Sethi, Nachum Sicherman, Paolo Siconolfi, and Karl Sigman for many helpful discussions. I also thank Stephanie Aaronson, Tavis Barr, Stephen
Cameron, Duncan Foley, Michael Gibbs, Erica Groshen, Cynthia Howells, Todd Idson,
Tack-Seung Jun, Brendan O’Flaherty, Sanjay Tikku, Joe Tracy, Michael Waldman, and
seminar participants at Columbia, Hunter, New York Federal Reserve Bank, Rutgers and
University of Delaware for comments and suggestions.
1
1
Introduction
How wages are determined, and what relationship wages have with other
labor market outcomes are central questions in economics. The literature
on wages is extensive, dating back to Malthus’ theory of subsistence wages,
and extending to modern compensation theories based on human capital,
search, and incentives. A key feature, especially of the modern theories, is
the dynamic aspect of wages. As a consequence wage growth ramifications
are all too obvious among these theories. For example: human capital theory
predicts wage growth due to on-the-job skill accumulation; job search theory
predicts wage growth due to imperfect information about the location of high
productivity jobs; mismatch theory predicts wage growth due to imperfect
information about the match quality of jobs; and, agency theory predicts
wage growth due to hiring and monitoring costs.1 But these theories do not
explicitly ask how differences in skill accumulation or search costs or monitoring costs may give rise to permanent differences in wage growth rates among
jobs, workers, or occupations. As a consequence, the role of wage growth as
a predictor of other labor market outcomes is not addressed. In this paper I
present a model of wage and turnover dynamics to directly address: How do
wages grow over the duration of an employment relationship? Why do wage
growth rates differ among jobs? Why is serial correlation of wage growth
an inconclusive test of wage growth heterogeneity? What is the relationship
between wage growth on a job and other labor market outcomes such as job
turnover?
The theory presented here is an elaboration of job search theory and firm
specific human capital theory. The key assumption in job search theory is
that each worker-firm pair is characterized by an idiosyncratic (implying of
course both a different and firm specific) productivity level. By contrast, my
model builds on the idea that each worker-firm pair is characterized by an
idiosyncratic productivity profile — that is, by both an initial productivity
level and a growth function that determines future productivity on the job.
Heterogeneity of productivity profiles across worker-firm pairs therefore underpins the nondegenerate distribution of jobs a worker faces in the labor
market. The theory retains the salient characteristic of search, namely, the
1
The empirical significance of wage growth is clearly not in dispute. The volumes of
studies especially in labor economics that document wage increases over the working life
and duration of jobs over the past several decades are ample evidence of the empirical
importance of wage growth.
2
optimal assignment of workers to jobs. But unlike in traditional job search
models where productivity of a worker-firm pair is static, in the model presented here the productivity of a worker-firm pair grows as the employment
relationship ages. More specifically, this increased productivity is assumed
to be firm specific and the rate of productivity growth to be different among
jobs. Since the implications of different rates of investment in firm-specific
human capital have not been fully explored in isolation before, it may explain
some otherwise anomalous findings on wage and turnover dynamics.
The motivation for this paper is not only to make a contribution to the
theory of wages, but also to provide an unified explanation for some recent,
and more importantly, puzzling findings on wage dynamics and turnover.
In the past two decades empirical studies using panel surveys of individual work histories and personnel records of large companies have repeatedly
documented within-job wage increases, persistence of wage growth, and correlations between wage growth and turnover. Bartel and Borjas (1981) find
evidence of positive correlation between completed tenure and within-job
wage growth. In a later and more conclusive study Topel and Ward (1992)
find that jobs offering higher wage growth are significantly less likely to end
in worker-firm separations than jobs offering lower wage growth (holding the
current wage constant). This finding not only implies heterogeneity of wage
growth rates among jobs, but also that the source of wage growth must have
a firm specific component.2
Although the negative correlation between wage growth and turnover implies wage growth heterogeneity, the direct evidence says that jobs do not
in fact differ in their prospects for wage growth. Two studies, based on the
time series properties of within-job wage changes, conclude that heterogeneity in permanent rates of wage growth among jobs is empirically unimportant
(Topel 1991, and Topel and Ward 1992). Note that the data of one of these
studies (Topel and Ward 1992) on which this conclusion rests, are the identical data that show past wage growth on a job reduces turnover.3 Hence
2
If wage growth is due exclusively to general human capital then these skills are equally
valued in other firms and wage growth should have no material impact on turnover. I
assume in this paper that the source of wage growth is exclusively firm specific. But none
of the main results would be altered if both general and specific human capital were the
source of observed wage growth.
3
In some sense this puzzle is quite dramatic. It is not unlike finding a significant and
positive schooling coefficient in a wage regression, and yet finding no evidence of any
significant variation in schooling levels in the same sample.
3
the puzzle laid out in the abstract of this paper: direct evidence says that
different jobs do not have different wage growth rates despite the fact that
past wage growth on a job reduces turnover.4
Taken together these two findings — (1) the negative relationship between
wage growth and turnover, and (2) the lack of evidence of serial correlation
of wage growth — pose a challenge for accepted theory. One widely accepted
theory is the “mismatch” theory of turnover. The mismatch theory says
that the current wage is a sufficient statistic for job value, and hence it is
consistent with studies that find no evidence of positive serial correlation of
wage growth. But the mismatch theory cannot explain the negative correlation between wage growth and turnover; it predicts that separations should
decline as a function of the wage level and not as a function of wage growth
(Jovanovic 1979a).5 On the other hand, theories that simply assume heterogeneity of wage growth rates can explain the negative correlation between
wage growth and turnover (Munasinghe 2000), but they are open to the objection that the evidence on wage growth persistence is inconclusive. One
main objective of this paper is to provide a unified explanation of why wage
growth is negatively related to turnover, and at the same time, why there is
no direct evidence of serial correlation of wage growth.
1.1
A theory of wage and turnover dynamics
The theory of wages in this paper builds on human capital, job search and
disreputable contracting. Its starting point is the idea that different employment relationships offer idiosyncratic learning opportunities.6 In particular,
4
In contrast, another study using personnel records of employees in a large firm finds
evidence of positive serial correlation of wage increases in adjacent time periods (Baker et
al. 1994). See Gibbons and Waldman (1999), and Prendergast (1996) for a more extensive
and detailed description of the empirical literature on wage dynamics. The point, however,
is that the evidence on serial correlation of wage increases is mixed. I return to this issue
in Section 4.2 below where I offer a conjecture, based on considerations presented in this
paper, to explain why the evidence is mixed across these particular studies.
5
Topel and Ward (1992) adopt the mismatch theory of turnover and acknowledge that
their turnover result is a puzzle for this theory.
6
The idea that different jobs or work activities offer different learning opportunities is
not new in the labor literature (for example see: Rosen 1972, and Sicherman and Galor
1991). The difference here is that learning is firm specific. Of course, the notion of
firm specific skills play a central role in modern labor economics since Becker formalized
the concept in the early 1960s. Nevertheless, the questions of why learning on a job
4
the theory assumes heterogeneity of firm specific learning rates among jobs.7
Hence productivity on a job increases because of pure firm specific learning.8
Although the accumulation of firm specific skills is often implicated as a
cause of wage increases with job tenure,9 wage determination is complicated
because outside firms do not value these skills. The question is: why would
firms increase wages in the absence of competition from other firms for specific skills?10 One answer based on reputation repercussions is: if firms renege
on such implicit contracts workers are unlikely to sample such firms in the
future.11 However, not all firms have the ability to make credible promises
of matching future productivity increases with wage increases. Hence the
question still remains: how do firms compensate workers in a time consistent
manner when the employment relationship generates firm specific rents over
time?
The theory presented in this paper explains why within-job wages inis specific and why learning rates might differ among jobs warrant some discussion. One
neglected aspect of increased division of labor (specialization) characteristic of the modern
workplace is that it typically entails greater and more complex human transactions. Given
the particularities (and peculiarities) of individual personality, it is not surprising that it
takes time and effort to work, communicate, and transact effectively with new colleagues
and bosses in an evolving hierarchy of production. Such learning curves are not only
specific to members of teams and work groups, but they are likely to be more or less steep
depending on various idiosyncratic factors, including production technologies and culture
of the work environment. Hence it is perhaps not altogether absurd to suppose that the
idea of idiosyncratic productivity profiles rests on fundamental principles of individual
psychology and science of hermeneutics.
7
A question that arises is whether learning rates are attached to jobs or workers. This
paper assumes that it is tied to the job. The reason for making this assumption is because
the negative correlation between wage growth and turnover holds despite corrections for
unobserved individual heterogeneity (Topel and Ward, 1992). If wage growth is a purely
individual effect then corrections for individual fixed effects should nullify the wage growth
effect on turnover.
8
The accumulation of specific knowledge is also a key feature in Weiss’ (1971) article,
“Learning by Doing and Occupational Specialization.” He argues that inter-occupational
mobility is discouraged in the presence of occupation-specific elements. A similar conclusion is reached in this paper about inter-firm mobility.
9
See Topel (1991) for an excellent summary of this literature.
10
This point is forcefully made in Felli and Harris (1996). They raise this issue in the
context of interpreting positive returns to tenure over and above the returns attributable
to general labor market experience. They argue in their model that competition for wage
increases due to firm specific human capital comes from the fact that workers also learn
about their productivities in other firms while working in the current firm.
11
For example, see Jovanovic, 1979a and 1979b.
5
crease even when both productivity increases are purely firm specific and
firms do not have the reputation to make credible promises of future wage
increases: firms increase wages because workers search for and receive better
outside job offers. This wage policy of matching counter offers or of disreputable contracting draws on a fundamental insight of search theory: workers
do not immediately find the employer willing to pay the most for their services because finding this employer is costly (Stigler 1962). Since Stigler,
search theory has been recast as a sequential decision problem of rational
workers with imperfect information. Modern renditions of search behavior have introduced more realism to the theory by assuming that workers
face non-degenerate offer distributions because of idiosyncratic match components across worker-firm pairs (Jovanovic 1979a), and that workers search
for better jobs even while they are employed (Burdett 1978). Hence the theory presented in this paper not only assumes heterogeneity of learning rates
among jobs, but also that workers search for better jobs while employed from
a stationary, non-degenerate distribution of outside job offers.
To summarize the main results of this paper: (1) average wage growth is
higher in high learning jobs than in low learning jobs;12 (2) past wage growth
on a job reduces turnover since worker-firm separations are less likely in high
learning jobs than in low learning jobs; and (3) serial correlation of wage
increases is an inconclusive test of heterogeneity of learning rates, and hence
of permanent differences in wage growth rates among jobs.
The intuition for these results are relatively straightforward. First, although firms cannot pre-commit to future wage growth, they will match
higher outside wage offers in high learning jobs because they generate more
firm specific rents as time on the job progresses. Hence observed wage growth
is higher in high learning jobs than in low learning jobs. Second, optimal
wage renegotiation implies that turnover is determined by total match value
of a job and not by the actual wage the worker receives. As a consequence,
the model predicts lower turnover in high learning jobs than in low learning jobs.13 Third, the stochastic nature of wage changes implies that wage
growth in adjacent time periods is not necessarily positively correlated, even
though productivity increases are positively correlated by construction. In
12
The terms high and low learning jobs are used interchangeably with high and low
productivity growth jobs throughout the paper.
13
Even if match values are the same at the time of job start, in subsequent time periods
the value of a high learning job is greater than the value of a low learning job because
match values grow faster in high learning jobs.
6
particular, for a given sequence of productivity increases, a large wage increase leads to a smaller subsequent expected wage increase because the wage
in any given period is the lower bound for the next period wage.
The model presented here also relates to other widely documented facts
about wages and turnover. Modern theories of compensation and turnover,
ranging from firm specific human capital to Lazear type bonding models
(Lazear 1981), imply dual tenure effects on wages and turnover — commonly
described as “wage increases with job seniority” and “negatively sloped tenure
turnover profile,” respectively. Since this duality is at the core of these models the vast empirical effort over the past several decades to establish the size
and significance of these tenure effects is certainly not surprising. What is
surprising, however, is the current consensus of small (if not absent) positive
wage returns to job seniority,14 despite the ubiquitous fact of a strong negative tenure effect on turnover. The model presented here implies not only
the dual effects of tenure but also this asymmetry of tenure effects on wages
and turnover, respectively.
On a related note, the effect of wage growth on turnover suggests that
high wage growth jobs are likely to be over sampled in empirical analyses
because they are more likely to survive. Such over sampling of course would
lead to biased estimates of wage returns to tenure. This selection problem is
rarely addressed in the empirical literature; and even when it is recognized
as a potential source of bias, the lack of evidence of serial correlation of
wage growth it put forth as a reason for its empirical irrelevance.15 This
paper shows that serial correlation tests are inconclusive about permanent
differences in wage growth rates among jobs.
14
For example, see Abraham and Farber (1987). Abraham and Farber argue that earlier
findings of significant positive wage returns to tenure, based on cross sectional data, could
be an artifact of the higher survival rate of high productivity or high wage jobs. Using
panel data they confirm that much of the cross sectional wage return to tenure is due
to omitted variable bias, and that earnings do not rise very much with job seniority. I
argue in this paper that a low wage return to job seniority is consistent with substantial
accumulation of firm-specific human capital on the job. The reason is because dynamic
consistency of wage determination over the duration of an employment relationship implies
a relative high initial wage as an up-front payment to future investments in firm specific
skills since within-job wage increases occur only in response to better outside offers.
For more recent evidence on wage returns to job seniority, and an excellent summary of
this literature, see Altonji and Williams (1997).
15
See Topel (1991) and Altonji and Williams (1997).
7
1.2
Related theory
The model presented in this paper is related to a variety of other compensation and turnover models. In a previous study, Munasinghe (2000) argues that past wage growth on a job reduces turnover by also showing that
turnover is lower in high learning jobs than in low learning jobs. In that paper wages are assumed to equal productivity, and thus questions about wage
determination and the mixed evidence on serial correlation of wage increases
are not addressed. This paper extends the turnover result by showing that
the negative relationship between past wage growth on a job and turnover
holds with dynamic wage renegotiation where the direct and contemporaneous link between wages and productivity is de-coupled.
Since Becker’s (1962) original idea of sharing costs and returns of firm
specific investments as a means of providing mutual insurance to each party’s
investment, the problem of wage determination has been well known.16 As a
solution to the inefficiency of Becker’s sharing hypothesis, Mortensen (1978)
considered the notion of wage renegotiation when an employment relationship
generates specific rents. However, Mortensen did not encompass dynamic
wage renegotiation due to on-going learning throughout the life of a job. As a
consequence the idea of matching counter offers has never been developed into
a theory of wage dynamics with ramifications for wage growth heterogeneity
and impact of wage growth on other labor market outcomes such as turnover.
The job search aspect of the model is based on imperfect information
about the location of the best match a la Burdett (1978). In Burdett the
productivity level is a sufficient statistic for job value. In this paper, however, each worker-firm pair is characterized by an idiosyncratic firm-specific
productivity profile, and the match value of a job increases with time on
the job due to productivity growth. The distribution of such initial match
values across all worker-firm pairs supports the assumption of a nondegenerate distribution of outside job offers. Hence, the match value derived in this
paper applies not only to the incumbent job but also to all prospective jobs
represented by the entire job offer distribution.17 Embedding firm-specific
productivity growth within the standard job search framework generates implications for within-job wage growth and within-job turnover dynamics, un16
The standard references here include, Parsons (1972), Kuratani (1973) and Hashimoto
(1981). Also see Malcomson (1997) for an excellent summary of this firm specific rent
literature, and of the general issue of contracts and hold-up.
17
See section 4.3.
8
like Burdett’s model that only generates implications for wage changes due
to job switches and turnover dynamics over the life cycle.
Jovanovic (1979b) is one of the first theoretical articles explicitly to integrate human capital theory and job search theory. In that sense, this paper
is similar to his. In Jovanovic’s model, match quality determines expected
job duration which in turn jointly determines optimal search effort and investment in firm specific human capital. Jovanovic’s central result is that
turnover declines with tenure. Although wages are endogenously determined
in the Jovanovic model, as in the model presented in this paper, his model
is not designed to study wage dynamics and the effects of wage growth on
turnover. The model here is explicitly designed to do so. Also, in Jovanovic’s
model the employer makes a wage offer to the worker that is equal to marginal product. The justification for such a wage policy is based on reputation
repercussions. As Jovanovic says, “employers offering wages below marginal
product will acquire bad reputations and will consequently not be sampled
by workers” (p. 1249, Jovanovic 1979b). As argued earlier, firms that do not
have the requisite reputation will need to offer time consistent wage policies.
The policy of wage renegotiation considered here is immune to charges of
time inconsistency.
Two recent articles (Gibbons and Waldman 1999, and Chiappori et al
1999) have proposed models to explain a host of findings on wage and promotion dynamics. The Gibbons-Waldman model is specifically designed to
explain various aspects of internal labor markets documented by Baker et al
(1994). Chiappori et al (1999) derive a late-beginner property, namely, the
negative correlation between past and future wages (holding current wages
constant), in a model of wage formation characterized by learning and downward wage rigidity.18 But neither of these models focus on turnover, unlike
the model presented here. An interesting issue is whether promotion dynamics can be derived in the context of the model presented in this paper by
incorporating different job levels.19 If firm specific skills are required to move
from one job level to the next higher job level, then indeed wage increases
are likely to predict future promotions, an implication explicitly derived in
Gibbons and Waldman. However, if promotions also signal higher general
skills such as managerial skills to competitor firms then turnover implications
18
They confirm this prediction using personnel data on executives of a French stateowned firm.
19
I thank Michael Gibbs and Michael Waldman for raising this issue.
9
following a promotion are likely to be amended. More empirical work on the
relationship between promotions and turnover will suggest how precisely to
model job levels within the current modeling framework.
The remainder of the paper is organized as follows. The next section
presents the basic model of wage and turnover dynamics. Section 3 presents
comparative results across high and low learning jobs. The following section
discusses some of the key assumptions of the model. Section 5 concludes
with a short summary and discussion of further implications. The Appendix
at the end contains some of the details of the proofs omitted in the text.
2
Model of wage and turnover dynamics
The model of wage and turnover dynamics is presented in the two sections
that follow. The focus in section 2 is on formalizing key assumptions and determining the optimal wage policy for a job characterized by a given productivity profile. In the last subsection of section 2, wage and turnover dynamics
are derived. The following section extends this basic model and compares
wage and turnover dynamics across jobs with different growth paths, and
derives the relationship between wage growth and turnover. Section 3 concludes with an analysis of serial correlation of wage increases in adjacent time
periods.
2.1
Assumptions
Following Jovanovic (1979a), assume that the firm’s production function exhibits constant returns to scale and that labor is the only factor of production. Then under competitive conditions firm size is indeterminate. Further
assume that a worker’s (marginal) productivity profile is know to both the
worker and the firm, and that within-job productivity increases deterministically as follows: pt+1 = g(pt ), where pt+1 > pt , ∀t ≥ 0, and the productivity
growth function g is bounded. Initial productivity (p0 ) and the growth function (g) completely determine the productivity profile for a given worker-firm
pair. This productivity profile is firm specific.
In each period the worker samples a single cluster of outside job offers20
20
This assumption of a constant outside offer arrival rate is largely for analytical simplicity. Section 4.2 below discusses the implications of relaxing this assumption.
10
from a stable offer distribution.21 All job offers in this single cluster have
the same value σ.22 But even though each productivity profile within this
cluster has equivalent value, they are still idiosyncratic — i.e., skills acquired
in any one job is not portable to any other job in the cluster. Assume that
the density of outside offers is non-atomic, and denote φ(σ) as the cumulative
distribution function of σ. Also assume that φ(σ) is strictly increasing. This
outside distribution function is supported by idiosyncratic pairs of (p0 , g)
across all worker-firm pairs.
Firms maximize profits. And finally, assume that mobility costs are zero,
that workers and firms have a constant discount factor β, and that they are
infinitely lived.
2.2
2.2.1
Model
Job value: present discounted value of wage payments
Since firms cannot make credible promises of future wage increases they make
an initial wage offer, and subsequently increase wages if and only if the worker
receives a better outside offer. In particular, suppose the firm renegotiates
next period wages to match outside offers, and denote zt as the highest
outside offer the firm would match at the beginning of period t + 1. If the
worker receives an offer σ that is greater than zt , the worker will quit, and
receive the full value of the outside offer σ because firms within the cluster
of offers provide the requisite competition. Denote wt as the wage at time t,
and define job value V as the present value of a worker’s expected lifetime
21
The stationarity of the offer distribution makes it clear that learning on the job is firm
specific. In addition, jobs are treated as inspection goods, and hence the value of a job —
given by the pair (p0 , g) since they are sufficient statistics for job value — is known to both
the worker and the firm at the time of the offer.
22
I assume that the worker receives a cluster of equivalent job offers instead of a single
job offer because the former is consistent with a competitive labor market for prospective
workers. If a worker receives a single outside job offer then the outside firm would only
need to pay an infinitesimally higher wage than the highest wage the incumbent firm is
willing to match to lure the worker away. This assumption of sampling from a cluster of
equivalent job offers is not critical for the main results of the paper. See section 4.1 for a
more detailed discussion of this assumption.
11
wage payments:
V (wt ) = wt + β
"Z
V (wt )
V (wt )dφ +
Z
zt
V (wt )
0
σdφ +
·
¸
Z ∞
σdφ
= wt + β V (wt )φ(V (wt )) +
V (wt )
Z ∞
max(V (wt ), σ)dφ
= wt + β
Z
∞
zt
#
σdφ
0
The Bellman representation of job value at time t is the current wage wt
plus the discounted expected job value in the next period. The terms in the
bracket of the first line details the various components of job value in the next
period. payment. The first term states that if the outside offer is less than
the current job value then the next period wage will remain unchanged from
the wage in the current period.23 The second term represents the conditions
of wage renegotiation. If an outside offer falls between the current job value
V (wt ) and zt , then next period wages are revised to match the value σ of the
outside offer. Thus observed wage increases on the job are entirely due to
search.24 The third and final term says that if the outside offer exceeds zt ,
the worker quits and takes the outside offer σ. Since we assume a competitive
labor market, the worker receives a compensation equivalent to the full value
of the offer in the new firm. Note however that the worker is indifferent
about the reservation value zt , and hence job value V is not a function of z
as highlighted in the second and third line; it is only a function of the wage
the worker receives.25 But of course firms will determine zt so as to maximize
profits.
23
Hence, the theory assumes non-decreasing wages. Needless to say, this assumption
makes the analysis simpler. But note that for firms to lower wages due to a lower outside
offer the firm must be privy to this information. Since workers elicit outside offers they
are likely to treat it as private information. Also there is evidence that workers rarely
experience nominal wage decreases on a job. See Baker et al. 1994 for empirical evidence.
Also see Chiappori et al. 1999 for further discussion regarding downward wage rigidity.
This downward rigidity of wages is also a feature of their model of wage formation.
24
Note, however, that from a worker’s perspective it does not matter whether wages
increase within the firm or by switching to another firm since the source of this wage
increase is driven by receiving a superior outside offer in either case.
25
The growth function will of course impact turnover since the highest outside offer zt
the firm is willing to match is a function of the productivity growth rate. See the derivation
of optimal zt below.
12
2.2.2
Match value: present discounted value of productivity
Next define the match value W at time t as the present value of a worker’s
expected lifetime productivity:26
·Z zt
¸
Z ∞
W (pt , zt ) = pt + β
W (pt+1 )dφ +
σdφ
0
zt
·
¸
Z ∞
σdφ
= pt + β W (pt+1 )φ(zt ) +
zt
The Bellman formulation says match value at time t is current productivity
plus the discounted expected match value in the next period. The match
value in the next period is equal the next period match value in the current
job W (pt+1 ) if the worker does not quit in the next period. And it will be
equal the match value of the outside job offer σ if the worker does quit in the
next period.27 Clearly match value at time t is a function of zt , the highest
outside job offer the firm is willing to match in he next period. The firm
will of course determine zt so as to maximize the presented discounted value
of profits at time t (see below). Note that the formulation of match value
incorporates productivity increases due to learning on the job.28
2.2.3
Firm’s problem: to maximize profits
Define a firm’s profits (Π) as the present discounted value of profits, and
suppose that profit at time t is given by the difference between match value
26
The term match value is somewhat of a misnomer here because it typically refers to the
productivity of a specific employment relationship. Here the term refers to the expected
lifetime productivity of a worker which includes not only the expected productivity in the
incumbent firm but also the expected productivity in future firms. Hence two jobs that
have the same match value may not necessarily have the same expected productivity on
their current jobs.
27
The value of the outside offer is the present discounted value of a new job that is
characterized by its own idiosyncratic firm-specific productivity profile. Thus this value is
directly comparable to the current match value. Note that incumbent and outside firms
are treated symmetrically.
28
Productivity increases are strictly firm specific, and the formulation here ignores general human capital accumulation. Although general human capital can be incorporated,
it would add little of interest, since the model is specifically designed to generate turnover
predictions across high and low productivity growth jobs.
13
(W ) and job value (V ) at t.29 Hence assume:
Π(pt , zt , wt ) ≡ W (pt , zt ) − V (wt )
 R V (wt )
= pt − wt + β 

= pt − wt + β 
0
+
(W (pt+1 , zt+1 ) − V (wt ))dφ
R zt
V (wt )
(W (pt+1 , zt+1 ) − σ)dφ
W (pt+1 , zt+1 )φ(zt )
−V (wt )φ(V (wt )) −
= pt − wt + βEΠ(pt+1 , zt+1 , wt )
R zt
V (wt )
σ)dφ




The fourth line highlights the fact that the present value of profits at time
t is equal to current profits plus the discounted present value of expected
profits in the next period. The present value of next period profits is defined
in terms of the difference between match value (W (pt+1 )) and job value in
the next period (as shown in lines two and three above). But notice that the
wage in the next period is a random variable due to the wage renegotiation
process. Hence the present value of expected profits in the next period is
the expected difference between match value and job value over all possible
values of wages in the next period.
The firm’s objective is to find zt that maximizes profits:
Max Π(pt , wt , zt )
zt
Since only the last term in the profitR equation is a function of zt , we need to
z
only consider the zt that maximizes V t(wt ) (W (pt+1 )−σ)dφ. From integrationby-parts, let u = W (pt+1 ) − σ and dv = dφ (σ). Then du = −dσ and
v = φ (σ). Rewriting:
29
The proof is given below.
14
Z
zt
V (wt )
(W (pt+1 , zt+1 ) − σ)dφ
= (W (pt+1 , zt+1 ) −
σ) φ (σ) |zVt(wt )
+
Z
zt
φ (σ) dσ
V (wt )
= (W (pt+1 , zt+1 ) − zt ) φ (z) − (W (pt+1 , zt+1 ) − V (wt )) φ (V (wt ))
Z zt
+
φ (σ) dσ
V (wt )
Taking the derivative of Π with respect to zt and setting the first order
condition implies:
d
d
Π = −φ (zt ) + (W (pt+1 ) − zt ) φ (zt ) + φ (zt )
dzt
dz
= (W (pt+1 ) − zt )
d
φ (zt ) = 0.
dz
d
φ (zt ) 6= 0; and since this condition holds for any period t, profit
Note that dz
maximization implies:
zt = W (pt+1 ), ∀t ≥ 0.
This first order condition says that firms will match outside job offers up
to the match value of the current job. And since we assume Π = W − V ,
the profit maximizing condition implies that firms match outside offers up
to a zero profit condition. The intuition is simple: if the highest outside
offer firms match is less than W (pt+1 ), then some workers will be allowed to
quit even though a further renegotiation of wages leads to positive expected
profits; and if firms match outside offers that are higher than W (pt+1 ), then
they will make negative expected profits on these workers who stay. Hence
profit maximization implies the above zero-profit condition, and further the
implied turnover rate at time t is given by:
1 − φ(W (pt+1 ))
Since profit maximization implies zt = W (pt+1 ), the profit function is
15
defined by:

Π(pt , wt ) = pt − wt + β 
W (pt+1 )φ(W (pt+1 ))
−V (wt )φ(V (wt )) −
= pt − wt + βEΠ(pt+1 , wt ).
R W (pt+1 )
V (wt )
σdφ


(1)
Job value function V and match value function W , can also be rewritten to
incorporate the profit maximizing value of z. Thus:
·
¸
Z ∞
σdφ
V (wt ) = wt + β V (wt )φ(V (wt )) +
V (wt )
"
= wt + β V (wt )φ(V (wt )) +
and
Z
W (pt+1 )
σdφ +
V (wt )
Z
∞
W (pt+1 )
·
Z
W (pt ) = pt + β W (pt+1 )φ(W (pt+1 )) +
∞
W (pt+1 )
¸
σdφ
#
σdφ
(2)
(3)
These functions are well defined: they exist and they are unique. Further V
and W are increasing in w and p, respectively, and Π is increasing in p and
decreasing in w.30
The next step is to show that W − V is indeed the profit function.
Proposition 1 Π(pt , wt ) = W (pt ) − V (wt ), ∀t ≥ 0
Proof. See section 6.1 of the appendix.
2.2.4
Determination of first period wages
The final issue is to determine first period wages. Earlier we assumed that
workers receive the full match value of a prospective job offer because they
receive a cluster of equivalent job offers. Hence first period wages will satisfy
the zero-profit condition.
Lemma 2 w0 s.t. V (w0 ) = W (p0 , g)
30
The details are available upon request from the author.
16
Proof. Each firm f in the cluster sets initial wages w0f , given the initial
wages set by the other firms in the cluster, to maximize the present value of
profits. The worker is indifferent among the firms in the cluster since they
all have the same match value. It follows, by Bertrand competition, that w0f
is set by each firm in the cluster so that the present value of profits is zero:
w0f such that V (w0f ) = W (pf0 , g f ). Since W (pf0 , g f ) = k, ∀f , where k is some
constant, it implies that w0f = w0 , ∀f .
In the next subsection, implications for wage and turnover dynamics are
derived.
2.3
Wage growth and turnover implications
Wage growth and turnover implications follow straightforwardly from the result that W (match value) is a strictly increasing function of p: W (pt+1 ) >
W (pt ), ∀t ≥ 0. First, define the initial wage w0 consistent with a competitive
labor market such that V (w0 ) − W (p0 ) = 0. Notice that this initial zeroprofit wage exists for every productivity profile, and that it is a sufficient
statistic for job value (at the time of job start) with future renegotiation.31
Hence the outside offer distribution of match values can be reduced to an initial wage offer distribution with no loss of generality. Denote the cumulative
distribution function of this wage offer distribution as F (x). Further, define
a sequence of wages w
bt such that V (w
bt ) = W (pt ), ∀t > 0. This sequence of
wages is the highest outside wage the firm matches in each time period. (It
is also the wage that implies zero expected profits from that time period onwards.) Since W (pt+1 ) > W (pt ) and V is a monotonically increasing function
of w it follows that w
bt+1 > w
bt . Hence the highest wage a firm would match
increases with job tenure. Denote the random variable Wt as the observed
wage and W t as the expected wage at time t. The first result of the model
is that expected wages increase with job tenure:
Proposition 3 W t+1 > W t , ∀t > 0.
R wb
Proof. Rewrite W t+1 = αwt + (1 − α) wtt+1 xdF (x) > wt , where α =
bt+1 ). Since W t+1 is greater for every possible previous period
F (wt )/F (w
wage wt , it follows that W t+1 > W t .
Turnover is determined by the highest wage the firm is willing to match.
The firm matches outside wage offers up to the maximum wage (w
bt+1 ) that
31
Clearly w0 is a function of both p0 and g.
17
implies zero profits. This turnover rule is efficient, and the second result is
that the turnover rate falls with job tenure:
Proposition 4 1 − F (w
bt+1 ) < 1 − F (w
bt ), ∀t ≥ 0.
Proof. Since w
bt+1 > w
bt , ∀t ≥ 0, and F is a strictly increasing function.
Note that the initial zero-profit wage (w0 ) a firm pays is higher than
initial productivity, i.e., w0 > p0 . But as the employment relationship ages
productivity will increase more rapidly than average within-job wages. Hence
average wages increase with job tenure, but less rapidly than the increase in
productivity. But the decline in turnover with tenure is a direct function of
the productivity growth rate because the highest outside wage offer a firm
would match (w)
b is a direct function of productivity. The model therefore
implies the standard dual tenure effects, but the positive tenure effect on
wages, unlike the negative tenure effect on turnover, is attenuated on account of the wage renegotiation policy. These asymmetric tenure effects are
consistent with the broad pattern of empirical findings, namely, the small
and not highly significant positive effect on wages and the large and highly
significant negative effect on turnover.
The model presented here is an infinite-horizon, constant-discount-rate
problem where information about the location of the best employment matches
is costly. In the model as time goes to infinity workers do get optimally assigned to the best jobs. But, of course, workers do not live forever. The
point, however, is that the rich wage and turnover dynamics generated in the
early time periods in the model, might indeed mimic the actual labor market
outcomes of workers.
3
High and low learning jobs
For simplicity consider a high learning job (H) and a low learning job (L), and
let pH and pL denote the productivities of the two jobs, respectively. Further
H
L
let: (i) pH satisfy the difference equation pH
t+1 = h(pt ), (ii) p satisfy the
difference equation pLt+1 = l(pLt ), and (iii) h(z) > l(z) > z, ∀z. The first
inequality in (iii) says that the learning rate is higher in the first job than in
the second job, and the second inequality says that learning occurs in both
jobs. Denote W H and W L as the match values, and V H and V L as the
job values, of the high and low learning jobs, respectively. Results from the
18
previous section imply that these functions exist, are unique, and increasing
in productivity and wages, respectively.
To compare wage and turnover dynamics across high and low learning
L L
jobs, consider two jobs with equivalent match value: W H (pH
0 ) = W (p0 ),
H
L
32
where p0 and p0 are the initial productivities. Since labor markets are
competitive, firms will offer matching wages such that:
L L
L
L
V H (w0H ) = W H (pH
0 ) = W (p0 ) = V (w0 ),
where w0H and w0L are the initial wages in the high and low learning jobs,
respectively. Note, however, that with a policy of wage renegotiation if
V H (w0H ) = V L (w0L ) then w0H = w0L since the job value to the worker is
only a function of the wage level.33 Hence let w0 denote the initial wage in
both the high learning job and the low learning job.34
In the following subsections wage growth across high and low learning
jobs are compared, the relationship between wage growth and turnover is
derived, and serial correlation of wage increases in adjacent time periods is
shown to be indeterminate.
3.1
Wage growth in high and low learning jobs
The key result is that although the initial wages are the same, average wage
increases are higher in high learning jobs than in low learning jobs. ProposiH
L
tion 5 below states this formally. Denote W t and W t as the expected wages
at time t in high and low learning jobs, respectively. Then:
32
Becker (1975, p. 55) and Mincer (1993, p. 65) discuss the equalization of present values
among jobs with different learning opportunities due to labor mobility. As a consequence,
they interpret learning-by-doing as a human capital investment decision because labor
mobility implies an opportunity cost of learning. A policy of wage renegotiation can be
viewed as an extreme case where the worker bears neither the costs nor the benefits of
increased productivity due to accumulation of firm specific skills.
33
As a result the superscripts H and L on the V functions are superfluous. Recall, from
the workers perspective the maximum wage the firm matches does not affect job value.
But the superscripts are kept in the text to stress the fact that the highest outside offer
the firm is willing to match is different across the high and low learning jobs.
34
L
H
L
Note that pH
0 < p0 < w0 . The reason p0 < p0 is because a high growth function
implies a higher match value, and hence initial productivity in the high learning job will
be lower than the initial productivity in the low learning job if they both have equivalent
match values at the time of job start. Further, the initial wage with a policy of wage
L
renegotiation will be higher than either pH
0 or p0 because subsequent wage increases occur
only if the worker receives a better outside offer.
19
H
L
Proposition 5 (W t − w0 ) > (W t − w0 ), ∀t > 0
Proof. See lemma 6 below and section 6.1 of the Appendix.
Parenthetically, it is important to highlight the fact that equality of match
values across high and low learning jobs does not imply equality of expected
productivity across the two jobs. Recall that match value refers to lifetime
productivity which includes not only expected productivity in the current job
but also expected productivity in outside firms because of search. In fact,
expected productivity in the current job is higher in the high learning job
than in the low learning job if their respective match values are equivalent.
This also explains why expected wage payments are higher and turnover rates
are lower (shown in below in section 3.2) in the high learning job.
The proof of Proposition 5 follows by first showing that the highest outside
wage offer a firm would match is higher in the high learning job than in the
low learning job in all subsequent time periods. Lemma 6 below states it
formally. First define w
btH and w
btL as the “maximum” wage a firm would
match at time t, in the high and low learning jobs, respectively.
Lemma 6 If w0H = w0L then w
btH > w
btL , ∀t > 0.
Proof. The proof begins with the following claim:
L L
H H
L L
If W H (pH
0 ) = W (p0 ) then W (pt ) > W (pt ), ∀t > 0.
The match value of the high learning job increases faster than the match value
btH and w
btL such that V H (w
btH ) = W H (pH
of the low learning job.35 Define w
t )
L
L
L L
H H
L L
bt ) = W (pt ). Since W (pt ) > W (pt ), and V is a monotonically
and V (w
increasing function of wages, w
btH > w
btL , ∀t > 0.
Note that w
btH and w
btL are the highest wage levels to which the firm would
raise previous period (t − 1) wages at the beginning of time t. Recall from
earlier that these “maximum” wages satisfy the zero profit condition.
Proof of Proposition 5 follows from the above lemma because both jobs
sample from the same offer distribution in each period, and hence the expected wage in the high learning job is greater than the expected wage in the
H
L
H
L
low learning job, i.e., W t > W t , ∀t > 0. Hence (W t −w0 ) > (W t −w0 ), ∀t >
0, since the start wages in both jobs are the same, which completes the proof.
35
This result is explicitly derived in Munasinghe (2000).
20
The result follows trivially for the first period because the distribution
of wages in the first period is the same in the high and low learning jobs
except that the right truncation point is higher in the high learning job
(w
b1H > w
b1L ). For subsequent time periods the argument is more intricate
because the observed wage is conditional on surviving the previous period,
the probability of which is different across high and low learning jobs. The
details of the proof are given in section 6.1 of the Appendix.
The above result implies that average wage growth on a job is a proxy for
firm specific learning rates. It also implies heterogeneity of permanent rates
of wage growth among jobs.
3.2
Wage growth and turnover
Turnover in this model is determined by match value, and not by the actual
wage the worker receives. If the worker receives a better offer that does
not exceed the match value then the firm increases the wage so that the
job value matches the value of the outside offer, and the worker stays. The
worker quits the firm only when the outside offer exceeds the match value.
Since this is the same criterion for separation if wages equal productivity, the
model generates efficient turnover. Since the match value is higher in high
learning jobs than in low learning jobs, and because all jobs sample from the
same offer distribution, turnover is lower in high learning jobs. The following
proposition states this formally.
Proposition 7 1 − φ(V (w
btH )) < 1 − φ(V (w
btL )), ∀t > 0.36
Proof. Since w
btH > w
btL , ∀t > 0.
An obvious corollary is that past wage growth on a job is negatively
correlated with quit rates since average wage growth is higher in high learning
jobs than in low learning jobs. This result addresses a key finding in the
empirical literature that jobs offering higher wage growth are significantly
less likely to end in worker-firm separations than jobs offering lower wage
growth (Topel and Ward, 1992).
36
Alternatively Proposition 7 can be stated in terms of the wage offer distribution:
1 − F (w
btH ) < 1 − F (w
btL ), ∀t > 0.
21
3.3
Serial correlation of wage increases
Since within-job wage increases occur because firms make counter offers,
the correlation of wage increases in adjacent time periods is indeterminate
even though productivity increases are positively correlated. For expositional
c1 is a
convenience, consider wage determination in two periods. Suppose W
c
c
c1 and
random variable and define W2 = (1 + α)W1 , where α > 0. Interpret W
c2 as the highest wages the firm would match in periods 1 and 2, respectively,
W
c1 < W
c2 .
and denote the initial wage as w0 (a constant), such that w0 < W
c1 , W
c2 } mimics various productivity profiles
Thus sequences given by {w0 , W
c1 simply refers to a steeper
of equivalent job value: a higher draw from W
productivity profile, and the constant w0 implies that we are considering
only jobs of equivalent value. Then productivity increases in adjacent time
periods are positively correlated by construction:
c2 − W
c1 ) = αV ar(W
c1 ) > 0
c1 − w0 , W
Cov(W
Next suppose workers sample outside job offers from the same outside
wage offer distribution in the two subsequent periods, and denote X1 and X2
as the outside wage offers in periods 1 and 2, respectively. Now consider two
further random variables, W1 and W2 , and interpret them as the observed
wages in periods 1 and 2, respectively. Due to wage renegotiation, W1 and
W2 , are defined as follows. In the first period: if X1 ≤ w0 then W1 = w0
c1 then W1 = X1 . In the second period: if X2 ≤ W1
and if w0 < X1 ≤ W
c1 then W2 = X2 . (Otherwise the
then W2 = W1 and if W1 < X2 ≤ (1 + α)W
offered wage is higher than what renegotiation can handle, and the worker
moves to the other job.) For any given productivity sequence the covariance
of observed wage increases in adjacent time periods is always negative:
b1 , w
b2 }
Proposition 8 Cov(W1 − w0 , W2 − W1 ) < 0, for a given {w0 , w
Proof. See section 6.3 of the Appendix.
An interesting consequence is that if in fact heterogeneity of permanent
wage growth rates among jobs is unimportant, as Topel and Ward claim, then
the model here predicts that the observed correlation between wage increases
in adjacent periods should be negative. The key result, however, is that in
the presence of high and low growth jobs serial correlation of observed wage
growth is indeterminate.
22
Proposition 9 Cov(W1 − w0 , W2 − W1 ) T 0.
This prosition seems counter intuitive since for any given growth rate the
serial correlation is negative (see proposition 8 above). However, consider
two random variables where the covariance between them is negative in two
distinct populations. If the means of the two random variables are different
across the two populations then the covariance between the two random variables is indeterminate when the two populations are aggregated because the
covariance measures the linear association of the deviations of each random
variable from their new means. Since average wage growth in both the first
period and second period are higher in high learning jobs than in low learning
jobs, the covariance of wage increases in adjacent time periods when there
are high and low learning jobs is indeterminate.37
Hence serial correlation of wage growth is not necessarily a test of heterogeneity of productivity growth rates among jobs. More importantly, studies
that fail to find positive serial correlation of wage increases such as Topel
(1991) and Topel and Ward (1992) do not present evidence against differences in permanent rates of wage growth among jobs. A possible objection
is to say that if wages are observed with some measurement error then wage
increases in adjacent time periods will, of course, be negatively correlated.
The problem with this explanation, however, is that although it might explain the lack of serial correlation of wage growth, it certainly cannot explain
why past wage growth reduces turnover. The advantage of the model here is
that it can jointly explain both these findings.
4
Key assumptions and related issues
This section discusses the implications of relaxing some of the key assumptions of the model. Since the search assumptions play an important role in
generating various predictions of the model, the focus here will be on the
salient aspects of search.
4.1
Job clusters and labor market competition
One of the assumptions of the model is that workers receive a single cluster of
equivalent job offers in each period. The reason for making this assumption
37
Although a formal proof is not offered here, examples of indeterminacy are available
upon request from the author.
23
is because it is consistent with a competitive labor market for prospective
workers. The cluster of equivalent job offers creates the necessary competition to assure that the worker will be paid an initial wage commensurate
with the match value. The issues are whether this assumption is reasonable,
whether competitive markets can arise under alternative assumptions, and
finally whether it is crucial for the modeling results of the paper.
One implication of this assumption is that when a worker locates and
identifies a match with a prospective employer, she can then without cost
identify other similar employers — i.e., employers with similar technologies,
work environments and growth potential. Perhaps it is not entirely unreasonable to suppose that when a worker locates a job and identifies its salient
characteristics that she would then be able to more easily locate other employers with similar job characteristics and hence jobs with comparable match
values. But of course whether this is a reasonable is an empirical issue.
The next question is whether firms will offer a compensation package that
implies zero profits in the absence of competition from a cluster of equivalent
job offers. A single job offer in conjunction with reputation considerations
may also imply a zero profit wage because workers are unlikely to sample
firms that offer compensation packages less than match value in the future.
If this reputation factor can be costlessly learned then firms may in fact end
up paying workers the full value of the match even though they are likely to
have some monopsony power because workers only receive a single job offer.38
The final question is whether the key results of the paper are robust to
different specifications of this search assumption. If workers only receive a
single outside offer and if prospective firms never pay an initial wage higher
than the “maximum” wage incumbent firms are willing to match, none of the
qualitative results of the model change except for some of the technical details
of the value functions, including, of course, the determination of initial wages
of workers entering the labor market for the first time. In essence, withinjob wage and turnover dynamics are unaffected by the exact compensation
the worker receives at the new firm. The renegotiation region is determined
38
Note that the standard usage of the term “firm reputation” typically refers to a “commitment ability” to pay higher wages in the future when there is no competition for a
worker’s increased productivity. Hence reputability in the standard literature is a retort
to the inherent time inconsistency of setting wages equal to productivity. The policy of
wage renegotiation considered in this paper, however, is immune to charges of time inconsistency. The idea expressed here is that if firms try to extract rents up-front workers are
unlikely to sample such firms in the future.
24
by productivity growth and competition between the incumbent and outside
firm that compels the incumbent firm to match the outside offer. Of course,
if firms pay an initial wage just sufficient to lure the worker away, rents will
shift from workers to firms.
4.2
Endogenous search effort and the mixed evidence
on serial correlation of wage increases
The other key assumption about the search process is the fact that workers receive a constant cluster of outside job offers in each period. More specifically,
the worker receives a single cluster of equivalent job offers. But note that the
restrictive aspect of this assumption is that search effort is not endogenously
determined in the model. Workers are, of course, likely to influence the arrival rate of outside offers by searching more or less intensely depending on
their current job value. Workers will determined their search effort based on
a comparison of marginal gains and marginal costs of an extra unit of search
effort. Under standard assumptions — an increasing marginal cost function
— optimal search effort will be a function of the wage level since the current
wage is a sufficient statistic for job value. Lower wages mean higher marginal
gains to search effort implying that optimal search effort will be a negative
function of current wages. The various ramifications of endogenous search
are easily discerned.
The explicit incorporation of search costs shows that at lower wage levels
workers are likely to search more intensely. This addition to the theory does
not affect the qualitative results of the paper. The quit results across high
and low learning jobs are likely to be strengthened because workers in low
growth jobs will search more intensely than their high growth counterparts
since their average wages are lower. Also note that wage growth is likely to
be more rapid at lower wage levels and less rapid at higher wage levels than
what the theory predicts under the assumption of a constant offer arrival
rate.
Although endogenous search effort does not change the qualitative results
of the paper, the current formulation of the model does hide the inefficiency
inherent in the wage renegotiation policy. The policy of wage renegotiation
implies inefficiently high levels of search intensity. The intensity of a worker’s
search effort is a function only of the wage the worker receives. Hence, optimal search effort is inefficiently high because the worker does not take into
25
consideration the capital loss incurred by the firm.39 This issue of inefficiently
high intensity of search effort raises other interesting implications, including
the fact that it provides an economic rationale for why firms that have “commitment ability” are likely to increase wages in line with productivity. Such
a policy would lead to a lower and efficient level of search intensity. A plausible conjecture is that with endogenous search firms that have the ability
to commit to future wage increases will earn a profit premium over their
counterparts that do not have this commitment ability.40
This in turn also resolves an apparent paradox in the current model.
Firms that might have the reputation to make credible promises of future
wage increases have no incentives to set wages equal to productivity in the
current model setup. And the reason is because there is no profit premium
to be earned by setting wages equal productivity instead of implementing
the time consistent policy of wage renegotiation. As mentioned above, with
endogenous search reputable firms are likely to have a profit motive to set
wages equal to productivity.
The difference in wage policies across firms with and without this commitment ability also provides sharper insight into why the evidence on serial
correlation is mixed across the various studies. If, as theory seems to suggest,
reputable firms set wages equal to productivity then it follows trivially that
wage growth will be serially correlated because of the assumed serial correlation of productivity increases. Hence it is perhaps not a mere coincidence
that the evidence of positive serial correlation (Baker et al. 1994) is based
on personnel records of a large, and presumably, reputable firm. In contrast,
the two studies that fail to find evidence of serial correlation (Topel 1991,
and Topel and Ward 1992) are based on panels of individual surveys representing a much wider cross section of employers, many of which are small
size firms, and presumably, lacking the requisite commitment ability. And
theory suggests that such firms are more likely to implement a policy of wage
renegotiation.41
39
This point is also made in Mortensen (1978).
This is currently on going research.
41
Gibbons and Waldman (1998) look at this welter of findings and suggest that serial
correlation (of wage growth) maybe exhibited only by a small fraction of workers restricted
to professional and managerial careers. The discussion here suggests that reputability of
firms may be key to understanding these divergent findings. In particular, reputation
considerations may be more important in managerial tracks due to internal labor markets.
Even in firms that lack reputations vis-a-vis the larger, outside labor market, it is much
40
26
Endogenous search also suggests an alternative solution to the determination of first period wages. If workers elicit more than a single outside offer
in each period, the firm with the highest match value would need to only pay
the worker an initial wage that would make the job value to the worker equal
the second highest match value. The new firm would of course make positive
profits. This also solves the problem of determining first period wages of new
entrants in to the labor market.
4.3
Match values and the distribution of job offers
The search assumption of the paper not only generates turnover but also
within-job wage growth. Although the idea of a distribution of wage offers
dates back to Stigler (1962), Jovanovic (1979a) is the first to provide an
equilibrium interpretation of it by arguing that heterogeneity of productivity
levels across worker-firm pairs supports a non-degenerate distribution of wage
offers. By contrast, heterogeneity of productivity profiles across worker-firm
pairs underpins the search assumption in this paper. As a consequence the
characterization of match value and the implied within-job wage and turnover
dynamics apply to both the incumbent and outside firms. The model is
consistent across all the potential jobs a worker faces in the labor market.
If a worker receives an outside offer which has a higher match value than
the current match, the worker moves to this outside firm. But the match
value in the new firm is again, just like in the incumbent firm, determined
by its own idiosyncratic productivity profile. This match value evolves over
time in exactly the same manner as it did in the previous incumbent firm,
although how quickly the match value increases with tenure will depend on
the growth function specific to the new match. The worker continues to
sample outside job offers while working in the new firm, which is now the
incumbent firm. If the worker receives a better outside job offer either the
wage is renegotiated up to the value of the current match (if the outside offer
is less than the value of the current match but higher than the value of the
current job to the worker), or, if the outside offer exceeds the current match
value, the worker quits and moves yet again to this other new firm, where
the same job search and wage renegotiation process recommences.
A final issue to consider is the distributions of match quality among difeasier to establish a reputation within the firm, especially where managerial positions
represent the top levels of a hierarchical internal labor market.
27
ferent growth sectors. Note, workers can receive future job offers from high
or low growth jobs and are free to move across high and low growth jobs.
Labor mobility implies comparable distributions of match values across the
different growth sectors. For example, if the match values in low growth jobs
were systematically higher than the match values in high growth jobs, there
will be a net flow of workers from high to low growth jobs till the distributions
of match values become comparable across the growth sectors.42
5
Conclusion
This paper presents a model of within-job wage increases with firm specific
learning. Wage increases occur because firms match outside job offers. Job
search while employed provides the impetus for within-job wage increases.
Average wage increases are higher in high learning jobs because they generate
more firm specific rents. Past wage growth on a job negatively predicts quits
because turnover is lower in high learning jobs than in low learning jobs. A
further implication of the wage renegotiation process is that the correlation
of wage increases is indeterminable.
The model in this paper shows that tests of serial correlation of wage
increases are inconclusive about heterogeneity of permanent rates of wage
growth among jobs. The model also shows that average wage growth is higher
in high learning jobs than in low learning jobs. A joint implication of these
two results is that wage increases over a short duration of tenure are likely
to be noisy, and hence less informative about permanent wage growth rates.
It should also be noted that although longer panels provide more accurate
information, they are likely to over sample high growth jobs because they are
more likely to survive. Hence, variance estimates of wage growth measures
from longer panels are likely to underestimate true heterogeneity of wage
growth rates among jobs.
The results of this paper also have implications for Topel’s (1991) two
step procedure for estimating returns to experience and tenure. In the first
step Topel computes the within-job wage growth rate and argues that it is an
unbiased estimate of the joint returns to tenure and experience. Of course, if
high growth jobs are more likely to survive, as argued in this paper, then this
first step would yield an upwardly biased estimate. But Topel claims there
is no evidence of differences in permanent rates of wage growth among jobs.
42
See footnote 31.
28
What the theory here shows is that serial correlation of wage growth is an
inconclusive test of wage growth heterogeneity. Hence there is indeed doubt
whether Topel’s first step yields an unbiased estimate of the joint returns to
tenure and experience. The more important issue this raises is how to account
for this selection bias in estimating the returns to tenure and experience.
A related issue is the further implication of the model that observed wage
growth on the job is an underestimate of firm specific productivity growth.
Note that the wage-tenure profile will be flatter on average with a policy of
wage renegotiation compared to the productivity profile. Thus, estimates
of returns to tenure are likely to always underestimate the true returns to
firm specific skills.43 Of course, the tenure effect on turnover is driven by
the increase in productivity. Hence these aspect of the model may help
to address why empirical estimates of returns on tenure are typically tiny
if not absent,44 despite the fact that strong tenure effects on turnover are
universally documented, even after controlling for individual fixed effects.45
The point is that the empirical estimates of wage returns to tenure should not
be interpreted as the true returns to the acquisition of firm specific skills — the
initial wage may well represent an up-front payment to the future acquisition
of specific skills.
An implicit assumption of the model presented here is that competitive
labor markets provide a means for workers to extract a share of employment
rents. Thus an interesting empirical implication is whether within job wage
growth rates are lower in less competitive labor markets like in small cities
and in rural areas. A corollary is that workers in less competitive labor
markets will need to resort to alternative schemes to extract a share of rents.
For example, are unions more likely to be formed in less competitive labor
markets? Is the union wage premium a rent or front-loaded compensation
for accumulation of firm specific skills in the future?
The search assumption plays a critical role in wage determination. The
fact that workers search for better matches provides the competition for
within-job wage increases with firm specific learning. The novel contribution
of this paper is that it adds growth rates of worker-firm productivities as
another idiosyncratic feature of an employment relationship. In this paper,
43
This would also be true for Becker type rent-sharing policies, but less so.
See Altonji and Williams (1997) and Topel (1991).
45
Note that Lazear type bonding models generate negative tenure turnover profiles precisely due to the back loading of compensation. Thus the evidence of small or nonexistent
returns to tenure is more troubling for such models.
44
29
this growth rate is treated as an inspection good. It may be interesting to
consider a model where the learning rate on a job is treated as an experience
good, especially if uncertainty about growth prospects is difficult to resolve
by inspection alone.
A final point to consider is the consequence of a reduction in search costs.
Perhaps with the recent advent of internet companies that provide employment matching services at substantially reduced costs compared to traditional head-hunter methods, it is reasonable to suppose that search costs are
likely to reduce in this internet age. Besides the obvious implication of higher
turnover, the model here also predicts that the renegotiated wage will track
the productivity profile more closely, which in turn implies a steeper wage
profile or higher returns to tenure. Also note that reputation effects are likely
to be less important.
6
Appendix
6.1
Proof of Proposition 1
Proposition 1: Π(pt , wt ) = W (pt ) − V (wt ), ∀t ≥ 0
Consider a sequence of finite horizon problems. Define W T , V T , and ΠT as
the match value, job value, and profit function, respectively, with T periods
remaining. The proof strategy is to show that ΠT = W T − V T holds for any
T (proof by induction), and then to let T −→ ∞ and show that Π = W − V
as defined in the above proposition. The proof proceeds in three steps.
Step 1: Show Π1 = W 1 − V 1
Denote p and p0 (> p) as current and next period productivity on the job.
Then define W 1 , V 1 and the difference between them as follows:
"Z 0
#
Z
p
W 1 (p) = p + β
p0 dφ(σ) +
V (w) = w + β
"Z
w
wdφ(σ) +
0
W 1 (p) − V 1 (w) = (p − w) + β
σdφ(σ)
p0
0
1
∞
Z
p0
σdφ(σ) +
w
"Z
0
w
(p0 − w)dφ(σ) +
30
Z
∞
p0
Z
p0
w
#
σdφ(σ)
#
(p0 − σ)dφ(σ) ,
since with zero periods remaining W 0 (p0 ) = p0 and V 0 (w) = w. Note that
(p −h w) is the current period profits, and
i terms inside the discount factor,
R p0 0
Rw 0
i.e. 0 (p − w)dφ(σ) + w (p − σ)dφ(σ) , is the expected profits in the last
period. Hence:
Π1 (p, w) = W 1 (p) − V 1 (w)
Step 2: Show If ΠT −1 (p, w) = W T −1 (p) − V T −1 (w) then ΠT (p, w) =
W T (p) − V T (w).
First define ΠT −1 :
ΠT −1 (p, w) = W T −1 (p) − V T −1 (w)
" R T −2
= (p − w) + β
Second define W T − V T :
V
0
(w)
(W T −2 (p0 ) − V T −2 (w))dφ(σ)
R W T −2 (p0 ) T −2 0
+ V T −2
(W
(p ) − σ)dφ(σ)
#
#
" R T −1
V
(w)
T −1 0
T −1
(W
(p
)
−
V
(w))dφ(σ)
0
W T (p) − V T (w) = (p − w) + β
R W T −1 (p0 ) T −1 0
(W
(p ) − σ)dφ(σ)
+ V T −1
The terms inside the discount factor represent the next period profits for all
possible values of w. Hence it is the expected value of profits with T − 1
periods remaining. Hence:
" R T −1
#
V
(w)
T −1 0
T −1
(W
(p
)
−
V
(w))dφ(σ)
0
= EΠT −1 (p0 , w)
R W T −1 (p0 ) T −1 0
(W
(p ) − σ)dφ(σ)
+ V T −1
Therefore:
W T (p) − V T (w) = (p − w) + βEΠT −1 (p0 , w)
Since (p − w) is current period profits and βEΠT −1 (p0 , w) is the discounted
value of expected profits with T − 1 periods remaining (by assumption),
profits with T periods remaining is indeed equal to the difference between
match value and job value, i.e. ΠT (p, w) = W T (p) − V T (w). Hence in
conjunction with Step 1 we have shown:
ΠT (p, w) = W T (p) − V T (w), ∀T ≥ 0
Step 3: Show that W T (p) −→ W (p) and V T (w) −→ V (w) as T −→ ∞.
31
First define a transformation R such that:
" R T −1
W T (p) ≡ RW T −1 (p) = p + β
W
0
+
(p0 )
R∞ W
T −1
0
(p )dφ
σdφ
W T −1 (p0 )
#
If the mapping R is a contraction operator then the existence and uniqueness
of W are assured. Similarly define a transformation Q such that:
#
"
R V T −1 (w) T −1
V
(w)dφ
0
R∞
R W T −1
V T (w) ≡ QV T −1 (w) = w + β
(p0 )
+ V T −1 (w) σdF + W T −1 (p0 ) σdφ
If the mapping Q is a contraction operator then the existence and uniqueness
of V are also assured. Since both these mappings satisfy Blackwell’s two
sufficient conditions for a contraction, W T (p) −→ W (p) and V T (w) −→
V (w) as T −→ ∞. Hence:
ΠT (p, w) ≡ W T (p) − V T (w) −→ Π(p, w) ≡ W (p) − V (w) as T −→ ∞,
which completes the proof.
6.2
Proof of Proposition 5
H
L
Proposition 5 : (W t − w0 ) > (W t − w0 ), ∀t > 0
The above proposition claims that the expected wage in the high growth
job is higher than the expected wage in the low growth job. Since the start
wage is the same in both jobs, it implies that the expected wage growth
is higher in the high growth job. This proposition is proved by induction.
Note that if the distribution of wages in the high growth job stochastically
dominates the wage distribution in the low growth job (for time periods
greater than 0), Proposition 2 follows trivially. First a lemma, to be used
in the induction proof, is established. Second, stochastic dominance of the
wage distribution in the high growth is proved for t = 1. Finally, stochastic
dominance is assumed for t = n, and it is shown that stochastic dominance
holds for t = n + 1, which then concludes the proof.
Define a r.v. X from the distribution as F : P (X ≤ x) = F (x), x ≥ 0.
Next define two further r.v.s, RL and RH , such that:
RL = X if a1 < X ≤ b1
= a1 if X ≤ a1
= ∞ if X > b1 , a1 < b1
32
and similarly for RH :
RH = X if a2 < X ≤ b2
= a2 if X ≤ a2
= ∞ if X > b2 , a2 < b2
Note further that a1 < a2 < b1 < b2 . The following Lemma claims that RH
stochastically dominates RL , conditional on both been finite:
Lemma 10 P (W L > x|W L < ∞) ≤ P (W H > x|W H < ∞)∀x ≥ 0
Proof. Define
P (RL > x, RL < ∞)
P (RL < ∞)
P (RL > x, X ≤ b1 )
=
F (b1 )
L
P (R > x, X ≤ a1 ) + P (RL > x, a1 < X ≤ b1 )
=
F (b1 )
P (a1 > x, X ≤ a1 ) + P (X > x, a1 < X ≤ b1 )
=
(≡ GL (x))
F (b1 )
P (RL > x|RL < ∞) =
Similarly define the distribution of wages in the high wage growth job,
P (RH > x|RH < ∞), as:
P (a2 > x, X ≤ a2 ) + P (X > x, a2 < X ≤ b2 )
(≡ GH (x))
F (b2 )
We now show that for all x ≥ 0 the inequality in the above Lemma holds.
If x ≤ a1 then GL (x) = GH (x) = 1. If a1 < x ≤ a2 then GH (x) = 1. If
a2 ≤ x < b1 then
GL (x) =
F (b2 ) − F (x)
F (b1 ) − F (x)
<
= GH (x),
F (b1 )
F (b2 )
since b2 > b1 . Finally if b1 ≤ x < b2 then GL (x) = 0. Hence the inequality
holds for x ≥ 0, which completes the proof of the lemma.
The remainder of the proof proceeds by considering wages in the first
time period, and using the above Lemma to show that the wage distribution
33
in the high growth job stochastically dominates the wages in the low growth
job. Then the Lemma is used repeatedly to show that it holds for all time
periods.
Consider wages in the high and low growth jobs in the first period: W1L ,
and W1H , respectively. In period 0 the wages are the same in both the high
and low growth jobs. Hence let w
b0H = w
b0L = a, and let the upper barriers:
b1 = w
b1L < b2 = w
b1H . W1L is thus distributed as (RL |RL < ∞) and W1H as
H
H
(R |R < ∞). Thus from the lemma, W1L is stochastically dominated by
W1H . We can then take copies such that W1L = a1 < a2 = W1H . Re-define
b1 = w
b1L < b2 = w
b1H and proceed again with the Lemma to show that W2L is
stochastically dominated by W2H . By continuing in this manner, we conclude
that the result holds for all t.
6.3
Proof of Proposition 8
Proposition 8: Cov(W1 − w0 , W2 − W1 ) < 0, for a given p1
This section shows that the covariance of wage increases is always negative for any given sequence of productivity increases. First a more formal
statement of the correlation between wage increases in adjacent time periods
is presented.
Let f (x)R denote a probability density function on (0, ∞); f (x) ≥ 0, x ∈
∞
(0, ∞) and 0 f (x)dx = 1. Let
Z x
F (x) =
f (y)dy.
0
and let F̄ (x) = 1 − F (x),where F is the underlying wage offer distribution.
Let {Xn : n ≥ 1} denote an independent and identically distributed
sequence of r.v.s. distributed as F : P (X ≤ x) = F (x), x ≥ 0.
Let 0 < p0 < p1 < · · · denote an increasing sequence of numbers tending
to ∞.
Let V0 = p0 , V1 = (X1 | X1 < p1 ) and in general
Vn = (Xn | Xn < pn ), n ≥ 1,
which means that Vn is an independent copy of a wage X conditional on it
falling in the interval (0, pn ).
34
Now define W0 = p0 , W1 = W0 I{V1 ≤ W0 } + V1 I{V1 > W0 } and in
general
Wn = Wn−1 I{Vn ≤ Wn−1 } + Vn I{Vn > Wn−1 }, n ≥ 1.
Here, I{A} denotes the r.v. which is 1 if the event A occurs, and 0 if it does
not. So, for example, W1 = W0 if V1 ≤ W0 and W1 = V1 if V1 > W0 .
Wn thus denotes the nth renegotiated wage.
The objective is to show that successive increments ∆n = Wn − Wn−1 is
negatively correlated for any given F and {pn }.46 To be precise, consider the
sign of
def
Cov(∆n , ∆n+1 ) = E(∆n ∆n+1 ) − E(∆n )E(∆n+1 ).
In particular, consider
Cov(∆1 , ∆2 ) = Cov(W1 − p0 , W2 − W1 ),
and note that by conditioning on W1 = w one can equivalently consider
Cov(W1 − p0 , E(W2 − W1 |W1 )).
For negative correlation, it thus suffices to show that E(W2 − W1 |W1 ) is
decreasing in W1 .
But W2 − W1 is conditionally independent of W1 given W1 , and only
depends on a random draw X of F , so it is necessary to only consider showing
for V = (X|X ≤ b) that the overshoot
E(V − a; V > a) = E(V − a | V > a)P (V > a),
is a decreasing function of a for b = p2 . Since P (V ≤ x) = P (X ≤ x)/P (X ≤
b), x ≤ b, and b is constant throughout our analysis here, we can equivalently
use X and consider the non-normalized version of the overshoot
M(a) = E(X − a; a < X < b) = E(X − a | a < X < b)P (a < X < b).
Proposition 11 M 0 (a) < 0
46
Note, however, if wage increases are generated by different sequences of {pn } such
that the successive increments, pn − pn−1 , are positively correlated, then it is possible that
wage increments could also be positively correlated.
35
Proof. Compute the overshoot as:
M(a) =
Z
∞
0
=
Z
∞
P (X − a > x, a < X < b)dx
P (X > x, a < X < b)dx
a
=
Z
∞
P (x < X < b)dx
a
=
Z
b
P (x < X < b)dx
a
Hence the derivative of M(a) is given by:
M 0 (a) = −P (a < X < b) < 0,
as was to be shown.
36
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