1. Art and Diseño

Journal of Economic Behavior & Organization
Vol. 44 (2001) 413–434
Does inequality of labor earnings
emerge in young days or later?
Labor earnings dynamics and learning about
individual ability in heterogeneous society
Futoshi Yamauchi K.∗
Department of Economics, Yokohama National University,
Tokiwadai 79-3, Hodogaya, Yokohama 240-8501, Japan
Received 11 January 1999; received in revised form 28 February 2000; accepted 9 March 2000
Abstract
This paper examines the evolution of labor earnings inequality in an environment, where individuals learn about their own ability (productivity) from wage realizations. It is shown that innate
ability heterogeneity and idiosyncratic income shock variance have distinct effects on the emergence of earnings inequality through changes in learning speed and effort decisions. Therefore,
given endogenous changes in individual perception, we are able to explain different patterns of
labor earnings inequality evolution in differently endowed societies, actually observed in UK, US,
Germany and Japan. Simulations illustrate inequality emergence patterns and earnings mobility in
different environments. © 2001 Elsevier Science B.V. All rights reserved.
JEL classification: J2; J3; D3; D8
Keywords: Labor earnings inequality; Learning; Ability heterogeneity; Income shocks
1. Introduction
The evolution of earnings inequality has long been in the center of research agenda for
economics profession as well as policy makers. 1 Labor earnings, one with the largest share
∗ Tel.: +81-45-339-3557; fax: +81-45-339-3518.
E-mail address: [email protected] (F. Yamauchi K.).
1 Levy and Murane (1992) surveys issues of earnings inequality in US, and Gottschalk and Smeeding (1997)
provide an extended survey on cross-country comparisons of earnings and income inequality. Gottschalk and
Joyce (1998) provide a comprehensive analysis of rises in earnings inequality in OECD countries and explain the
changes from market and institutional factors. For other international comparisons, see eg. Gittleman and Wolff
(1993), Green et al. (1992) and Wolff (1996). Murphy and Welch (1992), Katz and Murphy (1992), and Juhn et al.
0167-2681/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 2 6 8 1 ( 0 0 ) 0 0 1 4 4 - X
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
among different categories of income sources, show various patterns of the inequality emergence across countries (e.g., Deaton and Paxson, 1993). However, the way to interpret the
observed inequality of labor earnings, from which we may design an justifiable redistribution scheme, depends on what proportion of the variations is attributed to innate ability
heterogeneity and to stochastic nature of life, i.e. luck. Is the inequality of labor earnings
a revealed productivity heterogeneity predetermined prior to the entry to labor force, or
just an accumulated consequence of stochastic income process? The aim of this paper is to
identify the effects of ability heterogeneity and of income risks on the pattern in which labor
earnings inequality emerges as people age. In a theoretical framework, individual workers
learn about their abilities (productivities) and decide their effort levels sequentially. Through
learning behavior and sequential effort decisions, ability heterogeneity and income shock
variance are found to have distinct effects on the intertemporal pattern of labor earnings
inequality. From this point, we are able to explain different patterns of inequality emergence
observed in different societies and cohorts.
Previous studies attempt to decompose earnings differentials into those attributed to endowment heterogeneity and to income risks (Blundell and Preston, 1998; Burkhauser et al.,
1997; etc.). 2 Blundell and Preston (1998), using household-level data from UK, examine
the composition of permanent and transitory components in household income shocks, and
conclude that an increase in transitory income shock variance contributed to a rise in the consumption inequality in the 1980s. 3 Burkhauser et al. (1997) show that individual-specific
fixed components mainly contribute to the inequality of labor earnings in US, while persistence of income shocks contributes to labor earnings inequality in Germany. This finding
implies a possibility of cross-country difference in the proportions explained by permanent
and transitory components. For the US, Geweke and Keane (2000) show that about 60–70
percent of the variations of the log of earnings is accounted for by transitory income shocks
and that about 60 percent of the variation of lifetime earnings is attributed to unobserved
permanent individual characteristics uncorrelated with race, age and education. 4
However, the correlation between earnings and ability likely changes as workers age.
Farber and Gibbons (1996) show evidence that time-invariant variables, correlated with unobserved ability, become more strongly correlated to wages as workers experience, possibly
because employers learn about employees’ abilities over time and adjust wage rates. From
a different perspective, Behrman et al. (1980) using a sample of twin-individuals find that
(1993) examine the sources for the widely documented increase in wage inequality in the 1980s US. Most of the
empirical studies on earnings (income) distribution focus on the causes for rises in the inequality observed in US,
UK, and other developed countries.
2 In other studies, e.g., the relationship of idiosyncratic income risks and risk-taking behavior, and of its consequences for personal income distribution was analyzed in Kanbur (1979), and Kilmstrom and Laffont (1979).
3 They use consumption as a measure of inequality as it measures permanent component of income more precisely
than income does. Note also that in their benchmark framework, all the shocks (permanent and transitory) are
idiosyncratic. They show that the introduction of correlated shocks to households within a cohort does not change
the identification problem of permanent and transitory components. In our simple model below, since I focus on
the transitional dynamics (volatility) of investment without any permanent shift of returns except in the very initial
period, all the shocks (aggregate and idiosyncratic) are transitory.
4 In a related paper using the same data set, Moffit and Gottschalk (1993) found that, within age-education groups,
earnings variations due to differences in permanent component are much larger than that attributed to transitory
shock component.
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
415
the earnings correlation between twins is about 0.56 for white male veterans of about age
50, which has not been adjusted for differences in environments with which the twin individuals are provided. The estimated correlation can be regarded as the lower bound of the
contribution of ability to earnings variance. Therefore, the contribution of ability to earnings
increases as people age. The general consensus in the existing empirical studies would be
that both ability and income shocks contribute to the evolution of earnings inequality, and
that the roles and significance of the two factors vary over workers’ career.
Although the accumulated empirical findings on the evolution of earnings inequality are
still too limited to generalize our understanding, 5 casual observations and existing empirical studies suggest that different societies exhibit different patterns of inequality dynamics.
For example, it is perceived that the inequality emerges in relatively early career in US,
while it emerges more in late career in Japan. In Deaton and Paxson (1993), within-cohort
earnings and consumption inequality increase with age in Taiwan, US, and UK, but patterns of earnings inequality emergence are different across the three countries. For example,
earnings inequality emerges intensively around age of 50 in Taiwan, but it emerges much
earlier both in US and UK.
The model of this paper is capable of distinguishing different patterns of inequality evolution. An increase in the variance of idiosyncratic income shock relative to the variance
of individual ability intensifies the emergence of labor earnings inequality in late career,
while an increase in ability heterogeneity makes the inequality emergence in young days.
Even with an identical asymptotic variance of earnings, the inequality emerges early in
lifetime in a heterogeneous society that ability heterogeneity is relatively large, while the
inequality emerges later in lifetime in a homogeneous society that income risk is relatively
large. For creating different patterns of earnings inequality emergence, the time-varying
sensitivity of individual effort to stochastic realizations of wages plays an important role. If
ability variance is relatively larger than income shock variance, because wage observations
are informative, workers can learn about their ability from current wage realizations and
are willing to adjust their effort levels. As a consequence, this environment leads to an
early emergence of earnings inequality. On the contrary, if shock (noise) components are
relatively large in wage observations, since wages are less informative, workers become
cautious about learning and effort decisions are not responsive to current wage realizations.
This makes inequality emergence later. In similar spirit, Prendargast and Stole (1996) examine the role of individual-specific noise variance (defined as unknown ability in their
paper) in investment behavior. In their context, the sensitivity of investment decisions to
signals reveals the individual-specific noise variance, which contributes to the formation of
market-wide reputation agents concern. In this paper, the time-varying micro-level sensitivity of individual effort decisions to wage realizations plays a key role in determining the
timing and magnitude of earnings inequality emergence in aggregate level. 6
To motivate us on this issue more empirically, Section 2 shows different emerging patterns of within-cohort earnings inequality in US, UK, Germany, and Japan. Section 3 sets
5
For identifying earnings mobility over time, longitudinal data is necessary.
Our setting-up is similar to theirs in terms of objective function and Bayesian updating. However, we focus on
population variance of individual effort and income (labor earnings). We do not incorporate reputation, formed in
markets, into agents’ preference.
6
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
Fig. 1. Variance of household income by birth cohorts in UK.
up a model, and Section 4 characterizes intertemporal changes in effort inequality. Labor
earnings inequality is characterized in Section 5. To confirm our results, dynamics of earnings inequality and mobility are simulated for heterogeneous and homogeneous societies
in Section 6. Concluding remarks are made in Section 7.
2. Some evidence
In this section, we observe patterns of earnings or income inequality evolution in UK,
US, Japan and Germany. Different economies share a common phenomenon that earnings
inequality increases as people age, but the speed of its emergence differs across countries.
First, cohort-specific variances of household income in UK are computed by Blundell and
Preston (1998, Table 1) using the Family Expenditure Survey (FES) 1968–1992. Ten-year
bands for age of birth of household head are used for defining cohorts. Fig. 1 shows the time
(age) paths of income variances of four cohorts. The inequality paths take similar convex
shapes. 7
For US, Farber and Gibbons (1996, Table 1) computed standard deviations of wages for
each experience group for relatively young workers, using the National Longitudinal Survey
of Youth (NLSY) 1979–1988. Those who were of ages 14–21 on January 1, 1979 are in
the sample. Fig. 2 shows the experience path of wage variance. It is found that the wage
variance rises as years of experience increase up to at most 11 years, but the rate of increase
is the highest in the onset of their career and decreases as workers experience. Contrary to
the findings for the UK, the path of wage inequality exhibits a concave shape in US. Deaton
7 Deaton and Paxson (1993) use the same data and derive similar curvatures of age-earnings inequality relationship
for the country.
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
417
Fig. 2. The effect of years of experience on wage variance (NLSY).
and Paxson (1993, Fig. 6) using the Consumer Expenditure Survey 1980–1990 show that
age effect on the variance of log earnings exhibits a concave shape in the ages of 20–50,
consistent with Fig. 2. 8
I estimated the variances of log labor earnings 9 using the Panel Survey of Income
Dynamics, 1990–1997, and estimated the age effects following Deaton and Paxson (1993)
method. 10 Fig. 3 shows the estimated age effects for age 25–55. The earnings inequality
rises drastically in the 20–30s and it increases linearly thereafter, consistent with the findings
in Farber and Gibbons (1996).
8 Geweke and Keane (2000) in their study using PSID provide some interpretable evidence on age-varying
magnitudes of income risks, and of education effect. First, disturbance variance in income determination is large
when young and it is decreasing as people age. In other words, stochastic mobility is large when young, not when
old. Disturbance variances for young men in one version of model are 0.614 (age 25), 0.455 (age 30), 0.442 (age
45), and 0.442 (age 60), and those in another version are 0.599 (age 25), 0.473 (age 30), 0.445 (age 45), and
0.445 (age 60). Second, education effects also vary by ages. The marginal effects on earnings of 16-year education
relative to 12-year education for young men in the first version are 0.195 (age 25), 0.341 (age 35), 0.374 (age 45),
and 0.284 (age 55), and those in the second are 0.173 (age 25), 0.450 (age 35), 0.389 (age 45), and 0.400 (age 55).
The first effect works for widening earnings inequality in relatively early career, but the second effect contributes
to widening the inequality in late career.
9 For the self-employed, labor earnings include income from their assets. Therefore, it is possible to take a negative
value. The observations of negative value are dropped from the sample. The age groups of age less than 24 and
more than 56 were not used in the estimation because the sample sizes are too small and possibly cause biases in
the variance estimates. Compared with Deaton and Paxson (1993) estimates for US, my variance estimates for the
ages more than 56 certainly are excessively large.
10 Estimated variances of log earnings are regressed on age, birth-year and year dummies. Birth-year (cohort)
effects capture the effects of distribution of observable time-invariant factors such as those of schooling years
under the assumption that those returns are constant over time. Age effects plotted in Figs. 3–5 are estimated
marginal effects of age dummies (age 25–55).
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
Fig. 3. Age effects on log earnings variances in the United States 1990–1997.
For Japan, though data source is limited in the country, Wage Structure Survey 1961 and
1976 can be used for a comparison of within-cohort inequality of earnings between the years.
Atoda and Tachibanaki (1991), using this data, compute variances of log earnings in different
birth cohorts sorted by educational attainment. Strikingly, the inequality had decreased as
workers age in that period for all the cohorts they investigate. 11 However, more recently,
Ohtake and Saito (1998, Figs. 3-2 and 4-1) use the National Survey on Family Income and
Expenditure 1979, 1984, and 1989 and show a more comprehensive picture of within-cohort
log income variance dynamics, in which age effect on income variance is found to be positive
and convex. Iwamoto (2000), on the other hand, also decomposed the variance of log income
into age and cohort effects, using merged large-sample cross-sectional household data from
1989 to 1995 (Comprehensive Survey of Living Condition of the People on Health and
Welfare). Fig. 4 shows the estimated age effects from the Iwamoto estimates of log income
variances (Iwamoto, 2000). An increasing and convex age-curve is depicted for the ages
25–55. From the last two studies, the income (earnings) inequality of Japanese households
is smaller than those for US and UK, and it emerges slower.
For the case of Taiwan, Deaton and Paxson (1993, Fig. 6), use the Personal Income
Distribution Surveys 1976–1990 and find that earnings variance is convex in age. The
pattern is similar to the case of Japan, but the inequality emerges more intensively around
the age of 50.
For Germany, I estimated the variance of log earnings from the German Socio-Economic
Panel (GSOEP)1984–1989 (before the German unification). Fig. 5 shows estimated age
effects. 12 The interesting finding for Germany is that earnings inequality rises intensively
in the age 35–45. The inequality evolves in an S-shape. In this sense, it is a hybrid type of
Japan and US.
11 This observation is the only one which shows a negative age effect on earnings inequality. Compared to other
studies about Japan and other OECD countries, I conclude that the generality of this finding is questionable.
12 The cohort effects of 10-year band are controlled in this specification.
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
419
Fig. 4. Age effects on variance of log income in Japan.
Different from the countries previously surveyed, the case of Germany alarms that, to
begin with our investigation, it is important to recognize heterogeneities across countries
in labor-market institutions which generate earnings inequality.
The observations from these countries motivate me to formulate a framework for understanding the mechanism which generates different patterns of earnings inequality. In the
Fig. 5. Age effects on variance of log earnings in Germany 1984–1989.
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
next section, a simple model is formulated to enable us to interpret a variety of patterns in
which earnings inequality emerges as workers age.
3. Model
Individual i in a cohort (or simply agent i), uniformly distributed over [0, 1], decides
his/her effort level eti before observing wage rate wti in each time. Production shocks affect
the marginal productivity of labor. Wage rate wti is a sum of individual ability (endowment)
and a stochastic shock: 13
wti = θi + εti ,
where θi is time-invariant individual ability, and εti is idiosyncratic shock. True productivity
θ is not known to both employers and individual workers. εti is a real productivity shock
to worker’s output. Assume that εti ∼ i.i.d. N(0, σε2 ), where σε2 > 0. The extent to which
ability can be inferred from wage observations, measured by σε2 , is apparently different
across labor market institutions. Utility function is assumed to be separable over time and
additive for consumption and leisure. 14
Uti = u(cti ) − v(eti ) = wti eti − 21 (eti )2 .
Some reservations on the form of our wage equation follow. First, for simplicity, years of
schooling, experience, on-the-job training, and other determinants of individual productivity
are normalized to be zero. 15 Second, returns on ability is normalized to be one (constant); it
is assumed that α = 1 in wti = αθi + εti . In general, the value of α (market price of ability or
skill) depends on demands for abilities and varies over time. The production in this section
has a simplified structure in that individual marginal productivity depends only on his or
her own ability plus an idiosyncratic shock. It is also important to recognize that different
labor-market institutions have different functioning for determining α. The framework also
does not exclude a possibility that α differs across countries, i.e. magnitudes of earnings
inequality attributed to ability heterogeneity are different. If we attempt to decompose the
earnings inequality into αθ and ε variations, it is important to identify the metric of ability
in wage terms (the value of α). However, since the aim of this paper is to disentangle the
patterns, not the magnitude, of cohort-specific inequality evolution, it is thought to be a
minor issue. 16
13 It is assumed here that wage rate is a linear function of individual ability, not of his/her own and the others’.
A rather simple production technology is assumed so that the others’ abilities do not affect his/her marginal
productivity in production.
14 As long as the cost function is increasing and convex, the qualitative results coming below hold. E(θ |Ω i ) =
t
c0 (eti∗ ) ≡ g(eti∗ ). Then, eti∗ = g −1 (E(θ |Ωti )).
15 In other words, ability (productivity) is assumed to be constant. This assumption is necessary for exclusively
focusing on ability learning and resulting effort decisions. However, when we assess earnings data empirically, it
is necessary to incorporate some frameworks for distinguishing ability learning and productivity increase due to
human capital investments in and out firms.
16 Of course, if the variances of ability are compared across societies, we need to take into account the contribution
of α variations.
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
421
Assume that there is no publicly observable correlates of individual ability
by which employers (or market) can infer individual ability, but individual output is measurable at each time with inclusion of transitory shocks. Since relevant information for
ability learning is the history of individual wage realizations exogenously given in
market, the evolution of Ωti does not depend on effort decisions. Thus at each time,
each worker optimize his/her effort level soP
as to maximize contemporaneous utility subT
t−1 E(U i |Ω i ) ⇔ max E(U i |Ω i ) for
ject to the information set (i.e. max{ei }T
t
t
t
t
t=1 δ
eti
t t=1
each t).
The true value of individual ability θi is unknown, but Bayesian agents update their
perception on it given the observations of wage rates. If agents have full information on θi ,
it is optimal to set the effort equal to ability: eti∗ = θi . Thus, θi can be thought of a true
potential, which is a target level for effort decision for i. 17 Note that, in this model, workers
learn nothing but their own true potential (ability). In this sense, the learning of this model is
different from workers’ on-the-job skill acquisition (human capital formation). Although it
is possible to incorporate into the framework time-varying heterogeneities in productivity,
I assume that workers learn only about the best effort level for their time-invariant ability.
The model is intended to create the evolution of earnings inequality in a time-invariant
environment. 18
Heterogeneity of ability is measured by σθ2 . Before the entry to working age, agents
only know the population distribution of θi ∼ N (µθ , σθ2 ). I assume that the initial prior
is also N (µθ , σθ2 ) for all the agents. 19 Under this assumption, e0i∗ = µθ . Assume that
2 < σε2 /σθ2 < +∞, i.e. noise variance in wage signals is large relative to the prior variance.
Bayesian learning (updating) provides the law of motion for µiθ |t ≡ E(θi |Ωti ):
i
− µiθ|t−1 ],
µiθ|t = µiθ|t−1 + ωt [wt−1
2 ), which measures the sensitivity of mean prior to wage signals.
where ωt = σθ2|t /(σε2 + σθ|t
Since the variance of noise in wage observations is identical for all workers, ωt is the same
2 = σ 2σ 2
2
2
for all. Given the updation of variance prior σθ|t
ε θ|t−1 /(σε + σθ|t−1 ), the identity
of noise variance results in a property that the subjective variance is updated identically
by all. Therefore, wherever a worker is located in the ability distribution (µiθ |t ), his or her
2 ). I choose this setup because, on a
subjective uncertainty is the same as the others (σθ|t
prior ground, it is hard to suppose which agents experience large noise and therefore weak
effort sensitivity to wage signals (i.e. small ωt ). Secondly, heterogeneities in noise variance
in addition to the one in ability unnecessarily complicates the characterization of effort and
labor earnings variance dynamics.
17 An alternative interpretation for the (θ , ei ) relationship is that θ measures a type of occupation and individuals
i t
i
search for a best match of her occupation choice eti and her most suitable occupation θi . In this case, an individual
receives some signal about her best occupation, θi + εti .
18 If in the current framework workers approach (learn) θ from below, the workers’ learning can be interpreted
i
as on-the-job learning (skill acquisition) with different capacities for knowledge θi . In this model, however, the
initial prior for θi is assumed to be identical for everyone, µθ for normalization.
19 In Appendix A, where ability heterogeneity does not exist, I drop this assumption temporarily and characterize
effort inequality evolution given that true identical ability is just unknown.
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
We know, at this stage, that eti∗ = µiθ|t−1 . Thus, effort decision follows:
!−1
σε2
i∗
i∗
i
i∗
+t
[θi + εt−1
− et−1
].
et = et−1 +
σθ2
i∗ is
The conditional variance of eti∗ given et−1
"
i∗
i∗
i
i∗
, θi ) = (ωt )2 Var(θi − et−1
+ εt−1
|et−1
, θi ) = t +
Var(eti∗ |et−1
σε2
σθ2
(1)
!#−2
σε2 , (2)
2 and σ 2 . Note that the variance above is objective in the
which is decreasing in t given σθ|1
ε
sense that the variance is conditional on θi (constant), i.e. deterministic although agents
do not know. What happens if their guess was actually correct but ambiguous (i.e. with
subjective uncertainty).
i∗ = θ , Var(ei∗ |ei∗ , θ ) > 0 since 0 < σ 2 /σ 2 < +∞.
Remark 1. Even if et−1
i
t
ε
θ
t−1 i
4. Individual effort variations over time
i∗ , θ ), and then
We first examine the effects of income shock volatility on Var(eti∗ |et−1
i
proceed to characterizing Var(eti∗ ). The derivative of the conditional variance of eti∗ given
i∗ and θ with respect to σ 2 is
et−1
i
ε

!2 
i∗ , θ )
2
∂Var(eti∗ |et−1
σ
i
ε
.
= (ωt )4 t 2 −
(3)
∂σε2
σθ2
i∗ , θ ) in early periods, but increases
Therefore, an increase in σε2 decreases Var(eti∗ |et−1
i
i∗
i∗
Var(et |et−1 , θi ) in later periods. This is different from a monotonically increasing relationship, stated in most of literature. A rise in wage uncertainty may lessen the conditional
fluctuations of effort. Formally, the condition is
i∗ , θ )
∂Var(eti∗ |et−1
i
∂σε2
R 0⇔t R
σε2
.
σθ2
There is a non-monotonic relationship between income shock variance and the conditional
effort volatility.
We are interested in the lifetime path of effort variations and the dynamics of its crossindividual variations, which is captured by the unconditional variance of effort. Note that
since the population is uniformly distributed over [0, 1], the distinction of sample and population does not exist. Given the above preliminary results, we characterize the unconditional
variance of eti∗ .
Theorem 1.
Var(et∗ ) = tωt2 Var(ε) + (tωt )2 Var(θ).
(4)
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
423
Proof. See Appendix B.
By Theorem 1 and the assumption that prior variance is identical to population variance,
the expression for effort variance is simplified as
Var(et∗ ) =
tσθ2
(σε2 /σθ2 ) + t
= tωt σθ2 .
(5)
Since tωt → 1 monotonically, Var(et∗ ) → σθ2 monotonically. The effects of idiosyncratic income shock variance and ability variance on effort variations are characterized as
follows.
Proposition 1. Comparative statics of effort inequality
1. A rise in idiosyncratic income shock variance decreases effort inequality.
2. Large ability variance raises effort inequality.
3. Effort inequality is increasing over time.
Proof. See Appendix B.
The roles of income shock variance and ability variance are distinct: while income shock
variance decreases effort variations, ability variance increases its variations. First, there are
two ways in which an increase in ability variance influences effort variations: (i) an increase
in asymptotic effort variance, and (ii) an increase in learning speed (sensitivity to wage
observations). The first point is a natural consequence of heterogeneities in ability and wage.
The second point results in early emergence of cross-agent effort variations. Therefore, both
contribute to raising effort inequality. Appendix A proves that ability heterogeneity is not
necessary for effort variations, in a special case that agents are identical with ability, but it
is unknown (rational expectations do not hold).
However, a seemingly counter-intuitive point is (i). Since income shock variance increases
wage uncertainty and effort decisions are responsive to wage realizations, it seems that a rise
in income shock variance raises effort inequality. However, this reasoning does not incorporate the role of learning. A rise in income shock variance reduces the sensitivity of prior mean
to wage signals because the observations from which agents try to learn about their productivities contain larger noise (less informative), and it, therefore, makes effort decisions
less responsive to wage realizations. This results in smaller cross-agent variations of effort
levels.
5. Labor earnings inequality in lifetime
In this section, we characterize the variance of labor earnings, based on the results in
the previous sections. Recall that income is generated as the product of wage and effort,
i.e. yti = wti eti∗ = (θi + εti )eti∗ . The next result shows the dynamics of labor earnings
variance.
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
Theorem 2. Labor earnings variance
Var(yt ) = (tωt )2 α + ωt2 [β + tγ ],
(6)
where
α = Var(θ 2 ) + σε2 E(θ 2 ),
β = µ2θ [Var(θ ) + σε2 ],
γ = σε2 [E(θ 2 ) + 2µ2θ + σε2 ].
Proof. See Appendix B.
Proposition 2. Var(yt ) → α as t → +∞.
Proof. The result follows from that (tωt )2 → 1, ωt2 → 0, and tωt2 → 0 as t → +∞. From (6), we understand that there are three components in the labor earnings variance. First, the variance of earnings converges to α = Var(θ 2 ) + σε2 E(θ 2 ) since tωt is
monotonically increasing to one. Note that θ 2 follows a chi-squared distribution. Second,
β = µ2θ [Var(θ) + σε2 ] determines the decreasing portion of earnings variance since it is associated with ωt2 . Third, γ = σε2 [E(θ 2 )+2µ2θ +σε2 ] determines a temporally increasing and
decreasing portion since tωt2 is increasing initially and decreasing to zero asymptotically.
The three components jointly determine the dynamics of labor earnings inequality.
The next proposition shows its comparative statics. Essentially, the same intuitions are
confirmed as in the results on effort inequality.
Proposition 3. Comparative statics of within-cohort labor earnings inequality
1. An increase in ability variance raises labor earnings inequality.
2. An increase in idiosyncratic income shock variance raises labor earnings inequality for
large t, and decreases the inequality for small t given that σε2 /σθ 2 is large and σθ2 /µ2θ
is small.
Proof. See Appendix B.
Even if asymptotic variance of earnings α is identical, various compositions of σθ2 and
create different levels of inequality in the society as a whole. Now assume that σθ2 and
take values such that α is constant, i.e. asymptotic variance of labor earnings is constant.
If the same size of new cohort is born each
P year, the average earnings inequality over T
cohorts is expressed as Var(y) = (1/T ) Tt=1 Var(yt ), where T is the age at which people
exit labor force. This measure incorporates both between- and within-cohort inequality.
Since the within-cohort evolution of inequality varies as the ratio of σε2 to σθ2 changes,
Var(y) will also vary. Early emergence of earnings inequality in a relatively heterogeneous
society results in a larger Var(y), therefore more inequality in the society.
Finally, intertemporal earnings mobility is given as:
σε2
σε2
Proposition 4. Intertemporal earnings change is decomposed as follows:
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
425
i
i
1y(t+1,t)
= ηt+1 [wti − eti∗ ] + 1ε(t+1,t)
eti∗ ,
where
ηt+1 =
i
θ i + εt+1
(σε2 /σθ2 ) + (t + 1)
,
i
i
1ε(t+1,t)
= εt+1
− εti .
Proof. See Appendix B.
The intertemporal change in labor earnings is decomposed into two components, both
of which are interpretable. The first term is what is generated from the prediction error
in effort decision at time t, and the second term comes from transitory shocks at times t
and t + 1. Since wage realization at time t signals ability, if agents observe that wage is
higher than predetermined effort level (a positive prediction error), they have an incentive
to raise their effort levels in the next period, vice versa. However, the sensitivity of earnings
change to the prediction error decreases as workers age, and the speed depends on the
composition of noise and ability variances. If σε2 increases (i.e. ηt+1 decreases), the role of
the prediction error in effort decision shrinks and the transitory shock component becomes
dominant in earnings mobility. On the contrary, if σθ2 increases, the relative importance of
Fig. 6. Homogeneous society: Var ⇒ σθ2 , σ ⇒ σε2 , µ ⇒ µθ .
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
the prediction error becomes larger (i.e. ηt+1 increases). However, whatever the ratio of
prior-noise variance ratio is, transitory shock component becomes dominant as t → +∞,
i.e. earnings process converges to a stationary one.
6. Simulations
6.1. Earnings inequality
In this section, we simulate labor earnings inequality and earnings mobility. Different
situations are assumed: (i) ability is heterogeneous and it is unknown to agents in a situation, where ability variance is relatively small, and (ii) ability is heterogeneous and it is
unknown to agents in a situation, where ability variance is relatively large. For both cases,
the initial variance prior on ability is assumed to be the variance in population. Appendix
A characterizes the case that ability is identical among agents, but it is unknown to agents.
This case is also simulated for comparison.
We illustrate the lifetime dynamics of labor earnings variance in two polar cases: homogeneous and heterogeneous societies. The computation is based on (6) in Theorem 2. For sake
Fig. 7. Heterogeneous society: Var = 45, σ = 100, µ = 100.
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
427
Fig. 8. No ability heterogeneity, but imperfect information, Var = 10, σ = 100.
of comparability, we fix the variance of idiosyncratic income shock and population mean
of ability, σε2 = 100 and µθ = 100. Two values of ability variance are chosen here: σθ2 = 1
for homogeneous society and σθ2 = 45 for heterogeneous society. 20 Given the discussions
in the previous section, it is predicted that a homogeneous society exhibits a relatively late
emergence of labor earnings inequality, while a heterogeneous society exhibits an early
emergence of the inequality.
Fig. 6 shows labor earnings variance up to 50 years in a homogeneous society, given
the parameter configurations specified above. As predicted, relatively late emergence of
earnings inequality is illustrated. This is because learning about individual ability is slow due
to a relatively large magnitude of noise (idiosyncratic income shocks) in wage observations.
On the other hand, Fig. 7 shows a similar experiment for a heterogeneous society. A
clear concave shape is illustrated in the figure. Note that since the variance of income shock
has been fixed, the asymptotic variance of income is larger than the previous case. In this
case, since learning is fast in early career due to a relatively large prior variance (subjective
uncertainty) and innate productivity differentials are large in population, the inequality of
labor earnings emerge in young days.
20 Since our focus is on the evolutionary patterns of labor earnings variance, different asymptotic variances in the
two cases are allowed.
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
Fig. 9. No ability heterogeneity, but imperfect information, Var = 10, σ = 400.
The next exercise focuses on the case that ability is identical among agents, but it is
unknown to them. Figs. 8 and 9 illustrate two polar cases with relatively small idiosyncratic
income shock (σε2 = 100) and large shock (σε2 = 400), respectively. It is assumed for
the prior variance that σθ2 = 10. While, as expected, the emergence of transitional earnings inequality is concentrated in early lifetime if income noise is small (Fig. 8), earnings
inequality persists if income noise is relatively large (Fig. 9).
Interestingly, the earnings dynamics appeared in Figs. 7 and 9 look similar. The two cases
are, however, opposite situations in the sense that ability variance is large (and income shock
is relatively small) in Fig. 7, while ability is identical but the magnitude of income shock
is relatively large in Fig. 9. It seems to be difficult to identify the two cases in empirical
studies. 21
6.2. Earnings mobility
Earnings mobility is quantified in this section. Proposition 4 decomposed the intertemporal earnings change into those attributed to effort level adjustment, and to stochastic income
21
In this paper, any empirical examination is not attempted however.
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
429
Fig. 10. Mean of absolute change in labor earnings.
shocks. Although, in both homogeneous and heterogeneous societies, the effort adjustment
is more responsive to wage signals in young career than in late career, the tendency is much
stronger in a heterogeneous society than in a homogeneous society. This difference likely
generates a larger variability of earnings in young career in a heterogeneous society. In the
following experiments, assume that µθ = σε2 = 100, and that the initial mean prior on ability is the same as the true one: µθ|0 = 100. The prior variance for ability takes two values,
σθ2 = 1 and 45, corresponding to homogeneous and heterogeneous cases, respectively.
Absolute values of one-period earnings change is used to measure the intertemporal
variability of earnings change. Since the sensitivity of effort to wage realizations is a key
to earnings change, it is appropriate to take absolute value of earnings change. Fig. 10
shows the mean of absolute values of simulated one-period income changes over 10,000
individuals for a homogeneous society (lower line) and a heterogeneous society (upper line).
In the former case, the average magnitude of one-period income change does not change
noticeably over the 50 years. In the case of heterogeneous society, however, the magnitude of
one-period earnings change is decreasing as workers age. Labor earnings fluctuate largely
when workers are young, but it looks like a stationary process after 20 years of work
experience. The simulation result for a heterogeneous society is consistent with empirical
findings that earnings mobility is high when workers are young, e.g., in US.
7. Conclusions
Differently endowed societies exhibit different patterns of within-cohort labor earnings
inequality over time. Even with an identical asymptotic earnings variance for a cohort,
the earnings inequality emerges early in lifetime in a heterogeneous society, where ability
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F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
variance is relatively large, and the inequality likely emerges more intensively late in lifetime
in a homogeneous society, where income shock variance is relatively large. In this sense,
the model of this paper is capable of empirically explaining different emerging patterns of
labor earnings inequality observed in US and Japan.
In the analysis, however, I ignored other factors which generate time-varying earnings
inequality. These include changes in the returns to schooling and skills, and differences
in productivity increase on the job training. Of course, the question of whether or not our
approach is relevant to explaining the observed inequality even after controlling for the
effects of observable individual characteristics, such as differences in years of schooling,
is worth seriously asking and testing in empirical nature. It is interesting as a continuing
future research to identify the parameters and make cross-country comparisons from the
inequality paths observed in the OECD countries.
Acknowledgements
I thank Yasushi Iwamoto for providing his data, and Ian Preston and Henry Farber for
their permission to use a part of their results on UK and US, respectively. I also thank
Akira Kawaguchi, Soichi Ohta and especially an anonymous referee of the journal for valuable comments. Partial financial support was provided for the initial stage of this project
from Institute of Economic Research, Kyoto University. I am responsible for any remaining
shortcomings.
Appendix A
In this appendix, we show that ability heterogeneity is not necessary for effort (earnings)
inequality, rather subjective uncertainty on ability generates effort inequality. Drop the assumption of rational expectations for the above purpose. Agents are endowed with identical
ability θ , but it is unknown ex ante. Now, by Theorem 1
Var(et∗ ) = tωt2 Var(ε).
Proposition 5.
1.
2.
3.
4.
There is t ∗ such that ∂Var(et∗ )/∂σε2 < 0 if t < t ∗ and ∂Var(et∗ )/∂σε2 ≥ 0 if t ≥ t ∗ .
There is t ∗∗ such that ∂Var(et∗ )/∂t > 0 if t < t ∗∗ and ∂Var(et∗ )/∂t ≤ 0 if t ≥ t ∗∗ .
t ∗ = t ∗∗ = σε2 /σθ2 .
Var(eti∗ ) → 0 as t → +∞.
Proof.
(1)
"
#
2
σ
2
σε2
∂Var(et∗ )
ε
2
1
−
R
0
⇔
t
R
=
tω
.
t
∂σε2
(σε2 /σθ2 ) + t σθ2
σθ2
(2)
σ2
∂Var(et∗ )
= ωt2 σε2 [1 − 2tωt ] R 0 ⇔ ε2 R t.
∂t
σθ
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
431
σε2
.
σθ2
(3)
From (1) and (2), t ∗ = t ∗∗ =
(4)
Since tωt2 → 0, Var(eti∗ ) → 0 as t → +∞.
Even in the case of no ability heterogeneity, if agents have imperfect information on their
abilities, a rise in idiosyncratic income shock variance intensifies the emergence of effort
inequality in late career. Therefore, although effort levels converge to an unique value and
wages have the same distribution for all, ability learning (subjective uncertainty) generates
effort and earnings inequality temporarily.
Appendix B. Proofs
Proof of Theorem 1.
i∗
i∗
, θi )|θi ) + Var(E(eti∗ |et−1
, θi )|θi )
Var(eti∗ |θi ) = E(Var(eti∗ |et−1
i∗
i∗
, θi ) + (1 − ωt )2 Var(et−1
|θi ).
= Var(eti∗ |et−1
By recursively substituting
Var(eti∗ |θi )
t
X
i∗
=
φs Var(esi∗ |es−1
, θi ),
s=1
where φt = 1 and φs =
show
φs =
t−1
Y
q=s
Qt -1
q=s (1
(
i
(1 − ωq+1
)2 =
i
− ωq+1
)2 if s < t. By the definition of wsi , it is easy to
σε2 + sσθ2
σε2 + tσθ2
)2
.
Then,
i
)+
Var(eti∗ |θi ) = ωt2 Var(εt−1
=
i
)+
ωt2 Var(εt−1
i
)+
= ωt2 Var(εt−1
(
)2 (
t−2
X
σε2 + sσθ2
σε2 + tσθ2
s=0
t−2
X
s=0
(
σθ2
σε2 + tσθ2
σθ2
σε2 + sσθ2
)2
Var(εsi )
)2
Var(εsi )
t−1
t−2
X
X
(ωti )2 Var(εsi ) = ωt2 Var(εsi ) = tωt2 Var(εti ).
s=0
s=0
432
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
Var(et∗ ) = Eθ (Var(eti∗ |θi )) + Var θ (E(eti∗ |θi ))

t
t
t
X
Y
Y
ωs
(1 − ωj ) +
(1 − ωs )E(e0i∗ |θi )
= tωt2 Var(ε) + Var θ θi

s=1
j =s+1
s=1
= tωt2 Var(ε) + (tωt )2 Var(θ),
where e0i∗ = µθ for all i by assumption.
Proof of Proposition 1.
(1)
−t
∂Var(et∗ )
=
< 0.
2
2
∂σε
[(σε /σθ2 ) + t]2
(2)
t[2(σε2 /σθ2 ) + t]
∂Var(et∗ )
=
> 0.
∂σθ2
[(σε2 /σθ2 ) + t]2
(3)
σε2
∂Var(et∗ )
=
> 0.
∂t
[(σε2 /σθ2 ) + t]2
Proof of Theorem 2.
Var(yt ) = Var θ [E(yt |θi )] + Eθ [Var(yt |θi )] = Var θ [θi E(eti∗ |θi ) + E(εti eti∗ |θi )]
+Eθ [θi2 Var(eti∗ |θi ) + Var(εti |θi )[E(eti∗ |θi )2 + Var(eti∗ |θi )]],
where Var(yt |θi ) was further conditioned on εti and the last term was derived. By the results
of Theorem 1, it is equivalent to
Var θ [θi (tωt θi + ωt µθ ) + 0] + Eθ [θi2 tωt2 σε2 + σε2 {(tωt θi + ωt µθ )2 + tωt2 σε2 }]
= (tωt )2 Var θ (θi2 ) + (ωt µθ )2 Var θ (θi ) + tωt2 σε2 Eθ (θi2 ) + σε2 (tωt )2 Eθ (θi2 )
+2tωt2 µ2θ σε2 + ωt2 µ2θ σε2 + tωt2 σε4 = (tωt )2 [Var(θ 2 ) + σε2 E(θ 2 )]
+ωt2 [µ2θ (Var(θ) + σε2 ) + tσε2 [E(θ 2 ) + 2µ2θ + σε2 ]].
Proof of Proposition 3.
(1)
"
#
∂ωt2 2
∂β
∂γ
∂Var(yt )
2
2 ∂α
=
[t α + β + tγ ] + ωt t
+
+ t 2 > 0.
∂σθ2
∂σθ2
∂σθ2
∂σθ2
∂σθ
(2)
"
#
ωt 2
∂Var(yt )
2
2
2
2
2
= ωt −2 2 {t α + β + tγ } + (t + t)E(θ ) + (2 + t)µθ + 2tσε
∂σε2
σθ
R 0 ⇔ [σε2 + tσθ2 ][(t 2 + t)E(θ 2 ) + (2 + t)µ2θ + 2tσε2 ]
R 2[t 2 α + β + tγ ].
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
433
Divide both sides by t 3
2
1
2
1
1 2
1
σε
1
1
2
2
2
+ σθ
E(θ ) + 2 +
µθ + 2 σε R 2 α + 3 β + 2 γ .
1+
t
t
t
t
t
t
t
t
As t → +∞, LHS→ σθ2 E(θ 2 ) > 0 and RHS→ 0. Therefore, ∂Var(yt )/∂σε2 > 0 for
sufficiently large t.
For t = 1, the condition for ∂Var(yt )/∂σε2 < 0 is
[σε2 + σθ2 ][2E(θ 2 ) + 3µ2θ + 2σε2 ] < 2[α + β + γ ],
[σε2 + σθ2 ][E(θ 2 ) + 23 µ2θ + σε2 ] < Var(θ 2 )
+σε2 E(θ 2 ) + µ2θ [Var(θ) + σε2 ] + σε2 [E(θ 2 ) + 2µ2θ + σε2 ].
By rearranging
σθ4 < Var(θ 2 ) + µ2θ σε2 + 23 (σε2 − σθ2 )µ2θ .
It is equivalent to
#
"
2
2
2
µ
σ
σ
5
3
Var(θ 2 )
ε
.
+ µ2θ
+ 2 θ2 − θ2 −
0<
2 σθ2
2
σθ2
σθ
µθ
Since Var(θ 2 )/σθ2 > 0, sufficiency comes from
0<
2µ2θ
σθ2
3
5 σε2
+
−
− .
2
2
2
2 σθ
2
σθ
µθ
The statement (sufficient condition) follows from this condition.
Proof of Proposition 4. By definition of wage equation, the first difference of income is
i
i∗
i
i∗
i∗
− yti = θi (et+1
− eti∗ ) + (εt+1
− εti )et+1
+ (et+1
− eti∗ )εti
yt+1
i∗
i
i∗
i∗
i
i∗
= (θi +εti )(et+1
−eti∗ ) + (εt+1
− εti )et+1
= wti 1e(t+1,t)
+ 1ε(t+1,t)
et+1
.
By the law of motion of effort, it is written as
#
"
wti − eti∗
wti − eti∗
i
i
i∗
+ 1ε(t+1,t) et +
wt
(σε2 /σθ2 ) + t + 1
(σε2 /σθ2 ) + t + 1
i
]
= [wti + 1ε(t+1,t)
=
i
wt+1
wti − eti∗
i
+ 1ε(t+1,t)
eti∗
(σε2 /σθ2 ) + t + 1
(σε2 /σθ2 ) + t + 1
i
(wti − eti∗ ) + 1ε(t+1,t)
eti∗ .
434
F. Yamauchi K. / J. of Economic Behavior & Org. 44 (2001) 413–434
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