Discussion Papers In Economics
And Business

Human Capital Investment, Credentialing, and Wage Differentials

Masashi Tanaka

Discussion Paper 13-31-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHuman Capital Investment, Credentialing, and Wage Differentials

Masashi Tanaka

Discussion Paper 13-31-Rev.

August 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHuman Capital Investment, Credentialing, and WageDiﬀerentials∗
Masashi Tanaka†

Abstract
This study considers how individuals determine the ratio of two kinds of educational investment. One kind contributes to labor skills and the other does not. We refer to the former
as human capital investment and the latter as unproductive investment which improves testscores, but has no beneﬁcial eﬀect on students’ human capital. We formulate an overlapping
generations economy in which the rich and poor invest in both types of education. We argue
that the ratio of human capital investment to unproductive investment is a U-shaped functionof the wage diﬀerentials between the rich and poor. Moreover, we identify three patterns ofstable steady states for these wage diﬀerentials, namely, no-inequality, high-inequality, andmultiple steady states. Using these results, we conclude that a rapid increase in the levelof skill-biased technology may switch the steady state from no inequality to high inequality.
Further, it causes a temporary or permanent increase in the ratio of unproductive investment
during the transition to the new steady state.
Keywords: human capital investment, unproductive education, wage diﬀerentials, overlapping generations, and multiple steady states
JEL classification: O11, D81, I24∗

I am grateful to Koichi Futagami, Yutaro Hatta, Tatsuro Iwaisako, Hideki Nakamura, Ryosuke Okazawa,

Tetsuo Ono, Thitima Puttitanun, Katsuya Takii, Kazuhiro Yuki, participants in the 2013 Japanese Economic
Association Spring Meeting at Toyama University, participants in the 2014 Asian Meeting of the EconometricSociety held at Academia Sinica, and participants in the Tenth Annual Conference of the Asia-Paciﬁc Economic
Association at Thammasat University for their helpful comments and suggestions.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
E-mail address: masashi.tanaka.0203@gmail.com

1

1

Introduction

For several decades, many countries have experienced expanded access to higher education.
One reason is an increase in demand for high skilled worker from skill-biased technologicalprogress, such as the IT revolution. In addition to the fact, it is known that higher educational
attainment serves as a signal and has a big impact on students’ postgraduation earnings.
Using US data, Jaeger and Page (1996) and Park (1999) ﬁnd evidence of diploma eﬀects:
earning gains of 9–10% associated with high school graduation, 11% with associate’s degrees,and 20–30% with bachelor’s degrees. Thus, higher education plays a dual role of “educating”
and “credentialing”.
The main purpose of this paper is to investigate whether higher education is socially
productive, or whether it is partly a social waste. From the perspective of credentialing,
students are concerned not with learning, but with being certiﬁed as having learned. In
other words, their main objective is to obtain higher test scores and better grades. We seethat these credentialing activities are individually valuable but may contain a lot of sociallywasteful expenditure. Ronald P. Dore, who is widely regarded as the leading authority onsociology, sates in his book that:
The eﬀect of schooling, the way it alters a man’s capacity and will to do things,
depends not only on what he learns, or the way he learns it, but also on why
he learns it. That is at the basis of the distinction between schooling which is
education, and schooling which is only qualiﬁcation, a mere process of certiﬁcating- or ‘credentialing’. (Dore (1976) [pp. 8])
According to the argument, we can see that whether schooling contributes to build one’s
capacity depends on what his/her purpose of schooling is.
In this study, we separate educational investment into schooling that is education, or
human capital investment, and schooling that is only qualiﬁcation, or unproductive investment. Unproductive investment in our model implies investment undertaken only to obtainan observable credential, which does not contribute to labor skills. Conversely, human capital
investment not only contributes to labor skills but is also helpful for obtaining the credential. Thus, the ratio of human capital investment to unproductive investment captures thedegree of credentialing: a lower ratio means that credentialing is households’ major purposeof receiving higher education.
We introduce an overlapping generations economy, in which parents are heterogeneousin their wages, being either rich or poor. Each parent invests in the above-mentioned two
2

types of education for their children. Moreover, uncertainty exists in the acquisition of thecredential, which is determined endogenously by the diﬀerences in each type of educational
investment by the rich and poor. That is, as the gap in the education level between rich andpoor increases, the probability that the child of rich parent obtains the credential increases,
whereas the probability that the child of poor parent obtains the credential decreases. In
other words, this model considers competition to obtain the credential between the two
young, and this competition becomes an important factor that aﬀects the ratio of humancapital to unproductive investment.
The results of our study are divided into two parts, static and dynamic analysis. In static
analysis, we examine how the ratio of human capital to unproductive investment changes withthe wage diﬀerentials. We ﬁnd that the ratio of human capital to unproductive investment
for both rich and poor becomes a U-shaped function of the wage diﬀerentials. That is to say,the ratio is high when the wage diﬀerential is small or large, while it is low with medium
levels of wage diﬀerentials. The ratio of the two types of investment well describes the degreeof credentialing in higher education; a lower ratio of human capital to unproductive investment implies that a larger amount of socially wasteful activities is involved in educational
investment.
Our model also contributes to explaining the dynamics of the wage diﬀerentials of middleaged generation. In dynamic analysis, we ﬁnd that three patterns of stable steady states ofwage diﬀerentials exist: a unique steady state with no inequality, a unique steady state withhigh inequality, and multiple steady states. These alternative steady states are distinguishedby the parameter representing the level of skill-biased technology, and we can show that anincrease in the level of skill-biased technology tends to lead to high inequality steady state.
Thus, this result well explains the expansion of wage diﬀerentials experienced in the periodof skill-biased technical change (SBTC) in many advanced countries. Further, and more
importantly, we show that rapid progress of skill-biased technology switches the dynamics ofthe economy from no-inequality to high-inequality system, and the ratio of human capitalto unproductive investment decreases at least temporarily during the transition to the highinequality steady state. Thus, we can say that SBTC, such as the IT revolution from the1970s to the 1990s, would enhance socially wasteful unproductive investment.
To understand credentialing activities, signaling theory developed by Spence (1973, 1974)
have often been used in economic models. Many empirical works, including those of Langand Kropp (1986), Hungerford and Solon (1987), Belman and Heywood (1991), and Bedard(2001), argue that higher education serves as a signal of higher productivity in addition to3increasing individual human capital, and that this signal is rewarded in the labor market.
Following these results, theoretical studies focusing on signaling as private educational investment have been developed. Futagami and Ishiguro (2004), Hendel et al. (2005), Yuki
(2009), D’Amato and Mookherjee (2012), and Balart (2016) study educational signaling andwage inequality in a dynamic model (using an overlapping generations model). Futagami and
Ishiguro (2004) address macroeconomic issues by considering how signaling behavior of individuals aﬀects physical capital accumulation of the economy. Hendel et al. (2005) and Balart(2016) examine the dynamics of education and wage inequality using a signaling frameworkthat incorporates walth heterogeneity and imperfect credit markets. Although our model is
not signaling model, our study have a certain similarity to the studies in that we consider
unproductive education under asymmetric information.1
Our model is also related to Blankenau and Camera (2006, 2009), in which investment
for obtaining qualiﬁcations is separated from skill-enhancing investment. Blankenau andCamera (2009) constructs a model in which homogeneous students choose their educational
investment as follows: skilled (with qualiﬁcation and skill), schooled (with qualiﬁcation but
no skill), and unschooled (no qualiﬁcation and no skill). They suggest that increased tuition
subsidies may not be beneﬁcial because they increase enrollment but they may lower the
incentives of education or making achievement.
In addition, our results in dynamic analysis are consistent with the arguments in theliterature on the relation between wage inequality and SBTC. They argue that SBTC canbe oﬀered as explanations for an increase in US wage inequality between college and highschool graduates from the 1980s (for seminal empirical and theoretical works, see Katz andMurphy, 1992; Acemoglu, 1998; Autor et al., 1998; and Galor and Moav, 2000). Some
subsequent theoretical studies analyze the relationship between the emergence of skill-biased
technologies and educational attainment of workers. They show that higher skill premiums
driven by SBTC increase educational attainment of individuals with higher ability or income(see Caselli, 1999; Miyake et al., 2009; and Restuccia and Vandenbroucke, 2013), while the1

Aside from studies of dynamics, several studies describe individual incentives of investment to acquire skills

by applying the basic model of signaling theory. Hopkins (2012) and Bidner (2014) show that by assuming
positive assortative matching of worker–ﬁrm or worker–worker pair production, a higher level of investment
not only serves as a signal and leads to matches with higher productive partners but also increases productivity
and leads to higher wages. Ordine and Rose (2011) introduce both innate and schooling ability in determining
the individuals’ educational choice, and analyze within wage inequality in terms of educational mismatch andineﬃcient self-selection into education. They shows that wage inequality rise along with ability-complementary
technological progress whenever education equality is low.

4

attainments of individuals with lower ability or income decrease because of an increased costof educational investment (see Crifo, 2008; Nakamura, 2013). These results, like our study,
explain the widening wage inequality between college and high school graduates and thecontemporary increase in college enrollment in many advanced countries in the decades after
SBTC.
The remainder of the paper is structured as follows. Section 2 describes the constructionof the model. Section 3 provides the results of static analysis. We illustrates the mechanism
underlying the U-shaped relationship between the ratio of human capital to unproductive
investment and the wage diﬀerentials. In Section 4, we analyze the dynamics of the wage
diﬀerentials of middle-aged and old workers. Section 5 provides micro-foundation theory forthe main assumption of this model which is about the description of the credential-gaining
process deﬁned in section 2.1. Section 6 concludes.

2

The Model

Consider a discrete-time overlapping generations economy in which each agent lives for three
periods: young, middle, and old. Each generation is composed of two agents, rich and poor.
We refer to a young agent born to a rich (poor) middle-aged parent as “rich (poor) young”.
We assume that there is no population growth and, for simplicity, that all agents have noaccess to an intertemporal storage technology.2
In the ﬁrst period of life, young agents receive education from their parents. We assume
both young agents are born with the same ability. At the end of the young period, either
the rich young agent or the poor young agent obtains a credential, and the probability isdetermined as a result of their educational investments. In the second period of life, middleaged agents work and obtain wage incomes. They give birth to one child and allocate income
between their own consumptions and investments in education for the child. Further, we
assume that ﬁrms can not observe the individual output of each middle-aged worker. In theﬁnal period of life, old agents work and consume, but they neither bear children nor invest in
education. We assume that, unlike the middle-aged worker, ﬁrms can observe the individual
output of each old worker.
Thus, our model divides individuals’ working period into two terms in order to introduce
2Thus, we assume that individuals are credit constrained due to ﬁnancial market imperfections. This

assumption is made in order to ensure that middle-aged agents remains in the two types regardless of theirincome levels in old age. This constraint can be relaxed without altering any of the qualitative features of theresults presented in this paper, if we assume a small open economy.

5both symmetric and asymmetric information cases. That is, having the credential works asa signaling device for middle-aged workers and positively aﬀects their wages. By contrast,
the wages of the second working period are paid according to the individuals’ actual working
skills, regardless of whether they have the credential.3

2.1

Educational investment and uncertainty of the credential

Parents invest in two types of education for their children: one is productive and the otheris not. Here, we refer to the former as human capital investment and and the latter as
unproductive investment. Assume that the human capital production function is given by
the following expression:
hi,t = ei,t µ ,(1)where hi,t is the human capital of individual i born in period t, ei,t is the amount of humancapital investment put in by his/her parent, and µ ∈ (0, 1) measures eﬃciency of humancapital investment. In addition, parents can invest in unproductive activities aimed only atcredentialing. We denote by si,t the amount of the unproductive investment. We assume
that the levels of e and s are unobservable to ﬁrms.
Now we deﬁne the probability that describes the uncertainty of the credential. We denoteby θi,t the probability that young agent i born in period t obtains the credential, and assume
that θi,t increases in hi,t and si,t . Since one of the two young agents can obtain the credential,
θi,t has to be equal to the probability that young agent j fails to obtain the credential, thatis, θi,t = 1 − θj,t . In particular, we deﬁne θi,t as the following function:()()
θi,t = Θ ehi,t , sei,t = 1 − Θ ehj,t , sej,t = 1 − θj,t ,(2)where ehi,t = hi,t /hj,t = 1/ehj,t and sei,t = si,t /sj,t = 1/esj,t . Further, we assume Θ : R+ × R+ →(0, 1), Θeh > 0, Θeheh < 0, Θse > 0, Θsese < 0, Θehes = 0, limeh→0,es→0 Θ = 0, Θ(1, 1) = 21 ,( )and limeh→∞,S→∞Θ = 1. In addition, we assume that θi,t = Θ ehi,t when si,t = sj,t = 0,e
( )θi,t = Θ sei,t when ei,t = ej,t = 0, and θi,t = 1/2 when ei,t = ej,t = si,t = sj,t = 0.
In this model, the two young agents compete for obtaining one credential. The assumptions Θeh > 0 and Θse > 0 mean that Θ is an increasing function of the educational gaps
3

For a justiﬁcation behind this assumption, see Altonji and Pierret (2001). They argue that ﬁrms in general

have only limited information about the quality of workers in the early stages of their careers, and distinguishamong workers on the basis of easily observable variables, such as years of education or degree. On the other
hand, with each successive period, ﬁrms can observe more of workers’ performance.

6

between the two young agents. That is, as ei,t and si,t increase, young agent i can obtainthe credential more easily compared to young agent j. Likewise, the increase in ej,t and sj,tmakes young agent j easier to obtain the credential compared to young agent i. See Section
5 for a more detailed discussion of these assumptions.
The reason for calling sit unproductive investment is apparent now. Equation (2) indicates
that, a young agent with higher productive skill is likely to get the credential on one hand,but on the other hand, a young agent with larger amount of the unproductive investment isalso likely to obtain the credential. As will become apparent shortly, the rich young agent
receives larger amount of investments on both education in a unique equilibrium in this model.
Further, the equilibrium wage of the worker with the credential is higher than that of theworker without the credential in middle-age. Therefore, not only human capital investment
but also unproductive investment are privately worthwhile for expected middle-aged wages,
but the latter is socially wasteful. Thus we interpret et /st as the degree of credentialing: alower ratio indicates households invest more in socially wasteful education.

2.2

Production

Firms produce ﬁnal goods using only labor. We assume that, for agent i and j in eachgeneration, if hi,t > hj,t , the individual outputs of i and j are ϕhi,t and hj,t , where ϕ > 1.
That is, only the higher skilled worker can work with the superior technology, and thus ϕcan be interpreted as the level of skill-biased technology.
We assume that more than two risk-neutral ﬁrms simultaneously make wage oﬀers toworkers. If all ﬁrms make the same oﬀers, one randomly selected ﬁrm acquires all workers
and produce.4 For each middle-aged worker, ﬁrms can observe whether the worker has
the credential, but cannot observe the individual output and the education level. Hence,
according to the expected outputs of the worker with the credential and the worker withoutm ∈ {w c , w n } denote thethe credential, ﬁrms make wage oﬀers to the workers. Let wt+1
t+1
t+1

wages of the middle-aged worker with the credential (m = c) and the worker with no credential(m = n) in period t + 1. On the other hand, ﬁrms can observe the individual output of eachold-age worker. Therefore, old age workers are paid their individual outputs, that is, thewages of old workers are paid regardless of the credential.

4Thus, in equilibrium, the competition between ﬁrms drives their proﬁts down to zero.

7

2.3

Household

The preferences of middle-aged agents in period t (born in period t − 1) are deﬁned overconsumption and the wage of their oﬀspring. They are represented by a log-linear utility
function:
[
)]
(
o
m
o
m
o
m
o
+
(1
−
α)E
ln
w
+ β 2 (1 − α) ln wt+2
, (3)
+
β
α
ln
c
,
w
,
w
)
=
α
ln
c
,
c
U (cm
t+1
t+1
t+1
t+2
t
t+1
t
o
where cmt is own middle-age consumption in period t, ct+1 is own old-age consumption inm ∈ {w c , w n } is the wages that their oﬀspring earn in middle age, w operiod t + 1, wt+1
t+1
t+1
t+2

is the wage that their oﬀspring get in old age, α ∈ (0, 1) is an inverse measure of parental
altruism, β ∈ (0, 1) is the time discount rate, and E is the expectation operator. The budget
constraint of middle age and old age are expressed as follows:mcm
t + e t + s t ≤ wt ,(4)ocot+1 ≤ wt+1
.

(5)

and

2.4

Optimization and equilibrium

We now deﬁne a competitive equilibrium. Let us denote by hH,t and hL,t the human capitallevels of higher skilled and lower skilled worker in generation t. That is, hi,t = hH,t andhj,t = hL,t if hi,t > hj,t . Further, let ϑt denotes the value of ﬁrms’ expectation about theprobability that the worker with the credential has higher skill hH,t . A Competitive Equilibc , w n , w o }, individuals’ allocation {cm , co , e , s },rium consists of sequence of wages {wt+1
t
t+1
t+2
t+1 t tand the probabilities {θi,t , θj,t , ϑt }, such thatc , wn , e• taking as given wt+1
j,t and sj,t , middle-aged parent i maximizes her utility
t+1

subject to the budget constraints and the wages that her oﬀspring earns in old age (8):

[ (
)
(
(
))
]

m
c
n
e
e

max
α
ln
c
+
β(1
−
α)
Θ
h
,
s
e
ln
w
+
1
−
Θ
h
,
s
e
ln
w

i,t
i,t
i,t
i,t
i,t
t+1
t+1

m

c ,e,s
[
]
o
o
(6)
+β
α
ln
c
+
β(1
−
α)
ln
w
,
t+2
i,t+1





 s.t. cm + e + s = wm , co
o
i,t
i,t
t
i,t+1 = wt+1 , and (8);i,t• taking as given ei,t , si,t , ej,t and sj,t , and formulating the expectation about the probability ϑt , ﬁrms make wage oﬀers to the middle-aged workers as:


c
wt+1
= ϑt ϕhH,t + (1 − ϑt )hL,t ,w n

t+1 = ϑt hL,t + (1 − ϑt )ϕhH,t ;8(7)• observing the individual outputs of each old-age worker, ﬁrms oﬀer wages that equal tothe actual outputs of each old-age worker:

o
wt+2
=



 ϕhH,t if the worker has higher skill,

h

L,t

(8)

otherwise;• ﬁrms’ expectations about ϑt coincide with the actual value:

ϑt =



 θi,tif hi,t > hj,t ,θotherwise.

j,t

(9)

We now solve the equilibrium. Without loss of generality, we assume the middle-aged
agent i has the credential, and thus, her wage in period t is wtc . Note that wc ≤ wn iscn .
never realized for any period in equilibrium. Suppose, by contradiction, that wt+1
≤ wt+1

From (7), it implies ﬁrms expect that the worker with the credential is more likely of being
low skilled, that is, ϑ ≤ 1/2. Then, households’ maximization decision with respect to the( )unproductive investment becomes si,t = sj,t = 0. Given that θi,t = Θ ehi,t when si,t =sj,t = 0, then the worker with the credential is more likely of being high-skilled. However, it
contradicts the equilibrium condition (9), we thus have wc > wn in equilibrium.
The lifetime events of young agents born in period t are illustrated in Figure 1 below.
The solid lines represent the case where rich young (i) obtains the credential, and the dashedlines represent the case where poor young (j) obtains the credential.

Figure 1 : Lifetime events of young agents born in period t

9

First, we solve the household’s problem (6). We have the ﬁrst order conditions for ei,tand si,t as follows:
[
c ]
wt+1
∂Θ(e
hi,t , sei,t ) eei,tβ(1 − α)µei,t : c=β+hi,t ln n ,wt − ei,t − si,t
α
wt+1
∂e
hi,t

(10)

wc
si,t
hi,t , sei,t )β(1 − α) ∂Θ(e=sei,t ln t+1cn ,wt − ei,t − si,t
α
∂e
si,t
wt+1

(11)

si,t :()()
Likewise, by using Θ ehj,t , sej,t = 1 − Θ ehi,t , sei,t , the optimal condition of the middle-aged
agent j can be expressed as follows:
[
c ]
wt+1
ej,t
∂Θ(e
hi,t , sei,t ) eβ(1 − α)µej,t : n=β+hi,t ln n ,wt − ej,t − sj,t
α
wt+1
∂e
hi,t

(12)

c
wt+1
sj,t
hi,t , sei,t )β(1 − α) ∂Θ(e
=
s
e
ln
i,t
n .
wtn − ej,t − sj,t
α
∂e
si,t
wt+1

(13)

sj,t :A derivation of these equations can be found in Appendix. We can see from (10) and (12)that ei,t /ci,t = ej,t /cj,t , and from (11) and (13) that si,t /ci,t = sj,t /cj,t . Using them, we have
ei,t
si,t
wc
=
= nt .
ej,tsj,twt
The above expression shows an important equilibrium property in this model. First, giventhat wtc /wtn > 1, the rich parent who has the credential invests more in both education. Let
eH,t (eL,t ) and sH,t (sL,t ) denote the equilibrium investment levels made by the rich (poor)and w
et ≡ wtc /wtn denote the equilibrium wage diﬀerentials of the middle-aged generation.
Rewriting the above expression in terms of the equilibrium values, we have
sH,t
eH,t
=
=w
et ,
eL,t
sL,t

(14)

which indicates that the the levels of the educational gaps between rich and poor young are
consistent with the levels of parental wage diﬀerentials. (14) also indicates that the rich andpoor always choose the same level of e/s, that is,
eH,t
eL,t
=
.
sH,t
sL,t

(15)

We deﬁne e
ht = hH,t /hL,t and set = sH,t /sL,t as the gap of education levels between rich andpoor in equilibrium, then we have)(eH,t µsH,te=w
etµ and set =
=w
et .
ht =
eL,t
sL,t

(16)

Moreover, from (9) and (16), the equilibrium value of θ is expressed as a function of w
et :
(
)
( µ
)
ϑt = Θ eht , set = Θ w
et , w
et .
10

(17)

( µ
)
[
)
Let us deﬁne Θ w
et , wet ≡ Γ(wet ) where Γ satisﬁes Γ(wet ) : [1, ∞) → 21 , 1 , Γ′ (wet ) > 0,
Γ′′ (wet ) < 0,and limwet →∞Γ(wet ) = 1.5 Finally, we can express the equilibrium

Γ(1) = 21

dynamics of w
et , by using (7), (16) and (17), as follows:w
et+1 =cwt+1
Γ(wet )ϕw
etµ + (1 − Γ(w
et ))
=
.
n
wt+1
Γ(w
et ) + (1 − Γ(wet ))ϕw
etµ(18)

Thus, we can express the right-hand sides of (10)–(13) as a function of w
et , so we can derive
the equilibrium value of each endogenous variable, eH,t , sH,t , eL,t and sL,t , when we specify
the functional form of Θ. However, we omit this work because it is not necessary for our
subsequent analysis.

3

Ratio of Human Capital to Credentialing Investment

This section proposes a static analysis to examine how the ratio of human capital to unproductive investment changes with w
et . That is, here we explore the relationship between
et /st and w
et by regarding et /st as a function of w
et . As shown in (15), rich and poor parents
divide educational expenditure into human capital and unproductive investment in the sameratio. Let us deﬁne ηt as the ratio of human capital to unproductive investment, that is,ηt ≡ eH,t /sH,t = eL,t /sL,t . Then, by using (10) and (11), we can express ηt in terms of theequilibrium values, e
ht , set and w
et+1 , as follows:
∂Θ(e
ht ,e
st ) eht ln w
et+1 + β∂ehtηt = µ
∂Θ(e
ht ,e
st )
set ln w
et+1
∂e
st

[

εh
β
=µ
+
∂Θ(e
ht ,e
st )
εs
se ln w
e
∂e
st

where

e

εh = ∂Θ(het ,est )
∂ ht

/

Θ(e
ht ,e
st )
e
ht

,

t

t+1

e

/

ht ,e
st )
εs = ∂Θ(∂e
st

and

]
(19)

Θ(e
ht ,e
st )
.
set

In the following analysis, we focus on the denominator of the second term in (19) andshow that, regardless of the value of εh /εs , ηt is a U-shaped function of w
et . First, we examine
how ∂Θ(eht , set )/∂est varies with w
et . From (16) and the assumptions about the Θ function, we
have
d
dw
et

(

∂Θ(e
ht , set )
∂e
st

)
=

∂ 2 Θ(eht , set ) deht ∂ 2 Θ(eht , set ) de
st
+
< 0,
2
e
d
w
e
d
w
et
∂e
s
∂e
st ∂ ht |{z}t |t{z} |{z}
|
{z
}
>0

=0
5

<0

(20)

>0

Thus, we obtain a negative relationship between income inequality and intergenerational social mobility

since an increase in θ decreases the mobility. This relationship is supported by both empirical and theoreticalstudies (see Andrews and Leigh, 2009; Ferrer, 2005; Solon, 2004).

11

e

e

ht ,e
st )
ht ,e
st )
limwet →∞ ∂Θ(∂e= 0, and limwet →1 ∂Θ(∂e> 0. Next, we examine set ln w
et+1 . We obtainststfrom (16) and (18) that this term is increasing in w
et and that limwet →∞ (est ln w
et+1 ) = ∞.
Further, as it is clear from (18) that w
et+1 = 1 when w
et = 1, we have limwet →1 (est ln w
et+1 ) = 0.
From the above analysis, we see that the denominator of the second term in (19) converges
to zero when w
et → 1 and that it takes positive and ﬁnite values for ﬁnite w
et > 1. Further,
we have the following lemma for the case where w
et → ∞.

Lemma 1.

∂Θ(e
ht ,e
st )
set ln w
et+1 → 0 when w
et → ∞ (see Appendix for details).
∂est

It is clear from the above discussion and Lemma 1 that ηt in (19) approaches inﬁnity
when w
et → 1 and w
et → ∞. Conversely, for ﬁnite w
et > 1, since εh /εs is positive and ﬁnite, ηt
is also positive and ﬁnite. Therefore, we have the following proposition.

Proposition 1. Finite value we′ > 1 exists such that ηt decreases in wet for all wet < we′ , andﬁnite value w
e′′ (> we′ ) exists such that ηt increases in w
et for all w
et > we′′ .
From Proposition 1, we can say that ηt is a U-shaped function of w
et , because regardless ofthe value of εh /εs , we have ηt = ∞ at the boundaries. Strictly speaking, single-peakedness ofthe objective does not hold for every possible Θ function. However, if we restrict Θ to someclasses of functions, we obtain single-peaked function as shown in the next ﬁgure. Figure 2
shows a numerical example of ηt . A numerical example in Figure 2 shows a case of w
e′ = we′′ .
That is, in this example, ηt is a single-peaked.

Figure 212N otes : Figure 2 visualizes equation (19) by using Θ(eht , set ) = 1 −

1
4

(

1
e
ht

)a1

−

1
4

(

1
set

)a2
, a1 = 3, a2 = 3,

µ = 0.85, ϕ = 4, and β = 0.2. In this case we have w
e′ = we′′ = 1.36, that is, ηt is a single-peaked.

A mathematical interpretation of Proposition 1 is that the U-shapedness of ηt is derived( e)∂Θ(ht ,e
st )
w
et+1
d
mainly from dw< 0 and d lndw
> 0. Hence, to see the intuition of Proposition
et
∂e
st
et
1, here we focus on the two opposite eﬀects of increasing the wage diﬀerentials. First, we( e)∂Θ(ht ,est )dconsider the intuition of dw< 0. Given that Θ is increasing in set but with anet∂estupper bound of 1, the marginal change in Θ becomes progressively smaller as w
et increases.
That is, when w
et is large, marginal increases in e and s lead to very little change in theprobability of obtaining the credential for both rich and poor households. Therefore, asw
et increases, households reduce their unproductive investment ratio because it does notcontribute to promoting their productivities. Thus, from the perspective of the probability ofobtaining the credential, an increase in w
et decreases households’ incentives to make additional
investment in unproductive education.
w
et+1
On the other hand, from d lndw
> 0, we can see that it becomes more valuable forethouseholds to obtain the credential as w
et increases. Clearly, larger wage diﬀerentials of theparental generation lead to larger wage diﬀerentials of the oﬀspring generation. Further, notethat a higher value of w
et+1 implies a larger beneﬁt from obtaining the credential becausew
et+1 represents the wage gap between the worker with the credential and the worker withno credential in the oﬀspring generation. Therefore, from the perspective of the beneﬁt fromobtaining the credential, a larger w
et leads households to have more incentive to invest in
unproductive education. Thus, these two opposite eﬀects of wage diﬀerentials creates the
U-shape of ηt .
Proposition 1 implies that private investment in higher education may contain muchwasteful investment in economies with medium-sized wage diﬀerentials. That is, at medium
wage diﬀerentials, because obtaining the credential is of great concern to households, they
may spend a lot on unproductive training that contributes only to test scores. The same logic
can be applied to some publicly-funded educational investments. If households have strong
incentive to obtain the credential, part of public expenditure (such as education vouchers)
may be used wastefully as unproductive investment by households. As is shown theoreticallyby Blankenau and Camera (2009), our result in Proposition 1 also shows the mechanism of
weak connection between education expenditure and accumulation of human capital.
Furthermore, note that an increase in Θeh or a decrease in Θse increases η. Θeh and Θsehere are interpreted as the sensitivity of the credential-gaining process to human capital and13unproductive investment respectively. Therefore, we can say that as test scores that lead
students to better earnings become more sensitive to human capital, households’ private
educational investments in human capital increase relatively.

4

The Dynamics of Wage Diﬀerentials

4.1

The dynamics of w
et

In this section, we investigate the dynamics of middle-aged wage diﬀerentials w
et by using the
dynamic equation (18). In order to clarify the analysis, here we specify the functional form[)of Γ(wet ). Note that Γ(wet ) derived in (17) has to satisfy Γ(wet ) : [1, ∞) → 21 , 1 , Γ′ (wet ) > 0,
Γ′′ (wet ) < 0,Γ(1) = 12 , and lim Γ(wet ) = 1. Further, we obtain from (17) that Γ(wet ) isw
et →∞increasing in µ. For simplicity, we specify Γ(wet ) asΓ(wet ) = 1 −

1
,
2w
etµ

(21)

which satisﬁes the above conditions. Using (21), we can rewrite (18) as follows:w
et+1 =2ϕw
et2µ − ϕw
etµ + 1.
(2 + ϕ)w
etµ − 1(22)

Our purpose here is to examine how the dynamics of middle-aged wage diﬀerentials varies
with the parameters in (22), namely ϕ and µ. We show in Appendix that the dynamics of w
et
has three patterns of steady states depending on the values of ϕ and µ: a unique steady statewith no inequality, a unique steady state with high inequality, and multiple steady states.
For the following proposition, we deﬁne ϕ̄ and ϕb as follows:ϕ̄ =1 + 2µ1+µand ϕb =.
2µ − 13µ − 2(23)

Proposition 2. Assume that µ > 2/3. Further, we specify Γ(wet ) as in (21), and deﬁne ϕ̄ andϕb as in (23). Then,(i) When ϕ > ϕ̄, there is a unique stable steady state in which w
e > 1.
b there is a unique stable steady state in which w(ii) When ϕ ≤ min{ϕ̄, ϕ},e = 1.
√(iii) When µ > (1 + 5)/4 and ϕb < ϕ ≤ ϕ̄, multiple steady states may exist.
Proof: See Appendix for the technical proof of the statements (i) and (ii). Figure 3 illustrates
the existence of multiple steady states mentioned in (iii).

14b = 3.07.
For Γ(wet ) = 1 − 1/(2wetµ ), µ = 0.87 and ϕ = 3.80. In this case, we have ϕ̄ = 3.70 and ϕ

b = 3.07.
For Γ(wet ) = 1 − 1/(2wetµ ), µ = 0.87 and ϕ = 1.50. In this case, we have ϕ̄ = 3.70 and ϕb = 2.71.
For Γ(wet ) = 1 − 1/(2wetµ ), µ = 0.90 and ϕ = 3.28. In this case, we have ϕ̄ = 3.50 and ϕ
N otes : In Figure 3, we provides numerical examples of Proposition 2. It plots w
et+1 − w
et . As shown in (iii) in
Figure 3, we verify the existence of multiple steady states. In the following, we denote by we∗ the steady statevalue of (i) High-inequality steady state.

Figure 315

The most notable parameter here is ϕ, which represents the level of skill-biased technology. From Proposition 2, we can say that a high-inequality steady state tends to be realized
for higher values of ϕ. Therefore, it can be said that if an economy with an initial steadystate of no inequality experiences rapid increases of skill-biased technologies, the dynamics
of the economy may switch from (ii) to (i) in Figure 3. As a result, its steady state changes
from no-inequality to high-inequality, and the economy suﬀers a large increase in wage differentials during the transition to the high-inequality steady state. This is consistent with acommonly asserted notion that SBTC from the 1970s to the 1990s caused large increases ofwage inequality between highly educated and poorly educated workers. In turn, we see from(23) that ϕ̄ and ϕb are decreasing in µ. That is, it implies that a high-inequality steady state
tends to be realized for higher values of µ. The eﬀects of µ are also intuitive. This parameter
represents the productivity of human capital investment. Therefore, in addition, Proposition
2 states that more productive education leads to a high-inequality steady state.

4.2

The dynamics of ηt

Now that we have solved for the dynamics of w
et , we turn to the dynamics of ηt . We showed
in section 3 that the ratio of human capital to unproductive investment, ηt , is a U-shaped
function of w
et . Our purpose of this subsection is to examine how the educational investment
ratio varies with SBTC.
b We also assume that the economy
We assume here that ϕ initially satisﬁes ϕ ≤ min{ϕ̄, ϕ}.
is initially in the neighborhood of the no-inequality steady state.6 Further, let us consider
that, in period t0 (> 0), the economy faces a rapid increase in the level of skill-biased technology and ϕ rises to a level greater than ϕ̄. It means that the steady state of the economy
switches from no-inequality to high-inequality in period t0 , and then, the wage diﬀerentials
increase toward the high-inequality steady state w
e∗ . What is important here is that η decreases for at least a couple of periods from period t0 because ηt0 = ∞. Then we have thefollowing proposition.
b holds and w
Proposition 3. Suppose that, at t = 0, ϕ ≤ min{ϕ̄, ϕ}
e0 is in the neighborhoodof w
e = 1. Further, suppose that ϕ increases to a level higher than ϕ̄ in t = t0 . Then, there
exists t1 ≥ t0 such that ηt+1 < ηt holds for all t ∈ [t0 , t1 ].

6This assumption can be relaxed by considering a suﬃciently long but ﬁnite time period. This is because,b for any initial condition w
when ϕ ≤ min{ϕ̄, ϕ},e0 > 1, w
et converges to no-inequality steady state w
e = 1.

16

We can interpret Proposition 3 to mean that the rapid technological progress from the
1980s may have enhanced educational investment for the purpose of credentialing. Thatis, the recent increase in higher educational attainment experienced in many countries may
stem from an increase in socially wasteful educational expenditure caused by SBTC. If this is
the case, the increase in educational attainment would make a slight contribution to humancapital accumulation.
Further, Proposition 3 indicates that the decrease in η after period t0 may be temporary
or permanent. That is, t1 in Proposition 3 may be ﬁnite or inﬁnite. When the high-inequality
steady state value we∗ is less than w
e′ (which was deﬁned as the smallest value of w
et satisfying
dηt /dw
et = 0 in Proposition 1), η decreases for all t ∈ [t0 , ∞). On the other hand, if w
e∗ > we′ ,the decrease in η is temporary and t1 is ﬁnite. Instead of giving an analytical proof, we
provide numerical examples of the existence of the above two cases in the following Figure
4. Note that, in both cases, the ratio of unproductive investment increases for some period
after the switch of steady state, as shown in Proposition 3.

N otes : We draw η function and transition path of w
et which converges to the high-inequality steady statewe∗ . Here, we assume that the paths start from around the w
e = 1. The ﬁgure on the left is drawn by using
Γ(wet ) = 1 − 1/(2wetµ ), µ = 0.83, ϕ = 4.03 and β = 0.1. In this case, we have we∗ = 1.40 and w
b′ (= wb′′ ) = 1.50.
On the other hand, the ﬁgure on the right is drawn by using Γ(wet ) = 1 − 1/(2wetµ ), µ = 0.85, ϕ = 4.00 and
β = 0.2. In this case, we have we∗ = 2.71 and w
b′ (= wb′′ ) = 1.36.

Figure 417

4.3

The dynamics of the wage diﬀerentials of old workers

In this paper, we have focused on the wage diﬀerentials of middle-aged workers because it
determines the educational investment ratio η. However, it is also worthwhile to examine thef the wage diﬀerentials of old workers.
wage diﬀerentials of old workers. Let us denote by W
ft = ϕeThen, from (8), we obtain W
ht−2 . Further, from (16), its equilibrium value is as follows:µft = ϕw
et−2
W
.

(24)

ft , is expressed as an(24) indicates that the wage diﬀerentials of old workers in period t, W
increasing function of the wage diﬀerentials of middle-aged workers in period t − 2, w
et−2 .
ft , we can see that W
ft varies in the sameThus, without yielding a dynamic equation of W
direction as w
et but with a two-period lag.75Discussion

In this section, we discuss the two main features of this study: ﬁrst, only one of two students
obtains the credential, the probability of which is determined by the education gaps between
the two students; second, we treat productive and unproductive educational investment separately.

5.1

Competition for the credential

The model describes the competition between the students for the credential. It involves
a broad concept that includes competition in high school or college admissions, in highlevel academic transcripts, and in certiﬁcation exams. In this subsection, we ﬁrst oﬀer a
simple interpretation to the competition for the credential and provide a micro-foundation
for the credential-gaining process between two students, which is expressed in equation (2).
Then, we argue that the credential-gaining process in equation (2) can be supported by the
aforementioned broad concept.
The most characteristic aspect of this model is that only one of two students can obtainthe credential. Simply stated, we can interpret the credential in this model as a diploma of7

ft as follows:
Using (22) and (24), we have the dynamics of W[]µft2 − ϕWft + ϕ2
W
ft+1 = ϕW.
ft − ϕ(2 + ϕ)W

18

higher education. In the case where the enrollment limit of the educational institution is ﬁxed
at one, only one student can obtain the diploma at the institution. Then, we can say that theprobability that the highly (poorly) educated student gets the diploma increases (decreases)
as the gap of education levels between two students increases, as we supposed in equation (2).
However, this situation might be seen as ad hoc because the enrollment limit is always ﬁxed
regardless of the education levels of the two students. Generally, governments control thepupil strength in higher education in accordance with the education level of the population.
In order to tackle this problem, we next show a micro-foundation model for the credentialgaining process in equation (2) and then discuss a broader concept of the competition for thecredential by using the micro-foundation model.
Here, we suppose there is an educational institution that can provide comprehensive
education from primary to post-secondary curriculum. The households invest in two typesof education by paying e and s to the educational institution. After receiving education, thetwo students, i and j, with the education levels, (hi , si ) and (hj , sj ), take a test, the score ofwhich is given by
Score = εi Φ(hi , si )where Φ is an increasing function of both hi and si , and εi is an i.i.d. individual level shockwith E(εi ) = 1. Then, the winner and the loser of the test receive credentials H and L,
respectively, from the educational institution. Then, the probability that a young agent bornto parent i obtains the credential H can be written as
θi = prob [εi Φ(hi , si ) > εj Φ(hj , sj )][]Φ(hi , si )= prob>ξ ,Φ(hj , sj )where ξ ≡ εj /εi and E(ξ) = 1 because εi and εj are independent. Now, we assume that
Φ(hi , si )/Φ(hj , sj ) can be expressed as a function of the educational gaps, φ(ehi , sei ), whereehi = hi /hj and sei = si /sj . Then we have prob[φ(ehi , sei ) > ξ] : R2 → (0, 1), which is anincreasing function of ehi and sei , and satisﬁes prob[φ(1, 1) > ξ] = 1/2. Further, note thatsince θi = 1 − θj , that isprob [εi Φ(hi , si ) > εj Φ(hj , sj )] = 1 − prob [εj Φ(hj , sj ) > εi Φ(hi , si )] ,then we haveprob[φ(ehi , sei ) > ξ] = 1 − prob[φ(ehj , sej ) > ξ].
Therefore, we can derive the deﬁnition of Θ in equation (2) by applying the second-orderderivatives, φeh,eh < 0, φse,es < 0, and φeh,es = 0.
19

We can interpret from the above micro-foundation that the credential in this model canbe seen as a certiﬁcate for distinguishing the winner of the test from the loser. Generally,educational institutions make relative evaluations on students’ achievements across various
subjects or classes, and thus, certiﬁcates oﬀer substantial information about each student.
However, for simplicity, this model describes an economy with only two students, a onedimensional productive skill, and one educational institution that certiﬁes the winner and theloser on the test. Therefore, one credential is enough to distinguish between the two students.
This is why we assume, in section 2, that only one of two students can obtain the credential.
More importantly, now, we do not have to rely on the limited interpretation mentioned above.
The two students freely progress toward a higher stage of educational curriculum without
being bound to any enrollment limits, but their success or failure are ﬁnally evaluated bynoisy test scores. In reality, workers could have many certiﬁcates such as high school or college
admission acceptances, academic transcripts, and diplomas, and the ﬁnal judgment of successor failure is determined by comprehensive assessments of these. Therefore, the competitive
educational investments in this model capture broad private expenditures on education.
In economics, education has frequently not been interpreted as competition. However,
especially in East Asia including South Korea, Japan and China, students traditionally havebeen in tough competition for high-stake tests.8 Becker (2013) argues that such competitionin East Asian countries has spread to other nations owing the worldwide boom in higher
education. For example, he indicates that, in the US, competition to get into top colleges has
become much tougher, and families are spending a lot more on private tutoring partly fordirect preparation for the STA exams.9 Further, Becker (2013) states that “These results do
not answer the basic question of whether the extra eﬀort by high school students is socially
productive, or whether it is partly a social waste (an “arms race”)”. Our study can provide
one answer to the question.

5.2

Productive and unproductive educational investment

The model assumes that there exists an unproductive education that can be invested separately from a productive education. To our knowledge, few studies address these two kindsof education, while some models of multi-task principal agent theory have a concept close
8

College Scholastic Ability Test in South Korea, National Center Test for University Admissions in Japan

and National Higher Education Entrance Examination in China can be seen as representative examples ofhigh-stake tests.
9
Scholastic Assessment Test

20

to our study. For example, in Acemoglu et al. (2007), teachers with career concerns chose
two types of eﬀort separately, one that inﬂuences the human capital of their students andone that improves the test scores but has no beneﬁcial eﬀect on students’ human capital.10
They interpret that the latter involves rote learning, where a teacher forces students to cram
certain essential facts or methods, without explaining the concepts or phenomena behindthem.
Further, the existence of productive and unproductive education at schools is acceptedamong experts in education. Popham (2001) identiﬁes two kinds of instruction methodsof teachers: “curriculum teaching” and “item teaching”. Curriculum teaching involves the
body of knowledge and skills regarding a subject in accordance with the curricular content,
while item teaching narrows the instruction and uses clone items of the particular questions
most likely to appear in the test. Critics argue that students who receive item teachinglack a comprehensive and lasting understanding of the subject matter, because item teachers
exclude the strengthening of creative skills and abstract-thinking ability.
Our study diﬀers from the perspective of Popham (2001) in that recipients of education
decide the educational choice between productive and unproductive investments. However,when we focus on private educational investments, there is no substantial diﬀerence between
our study and Popham (2001). When we consider that private education is competitive, we
can see that private teachers provide exactly what households want, and thus, the households
can buy the two types of educational services by paying tuition fees. Examples of these
private educations include after-class tutoring and coaching schools. In another respect,
especially in higher education, students themselves, not teachers, may be able to decidelearning methods among productive and unproductive eﬀorts. For example, college students
can either exert eﬀort to achieve a comprehensive understanding of a particular topic or forachieving higher test scores. Thus, the unproductive investment can exist both in secondaryand post-secondary private education.

6

Conclusions and Remarks

In this study, we presented a simple dynamic model that contains both human capital investment and unproductive investment. By introducing two opposing types of education, human10

Acemoglu et al. (2007) show that high-powered incentives can induce more unproductive eﬀort of teachers.

Therefore, when principals have a commitment problem because of their intention of providing high-powered
incentives to teachers, government provision may be useful as a credible commitment to low-powered incentives.

21

capital investment and unproductive investment, into the model, we described the degree ofcredentialing from the investment ratio. Furthermore, we focused on the wage diﬀerentials
between the rich and poor. Because parents care not only about the productivity of theiroﬀspring but also about the credential, the rich and poor compete with one another to obtainthe credential. Therefore, the investment ratio of both the rich and poor depends on the wage
diﬀerentials.
Our static analysis proved that the ratio of human capital investment to unproductive
investment becomes a U-shaped function of the wage diﬀerentials. On the other hand, fromthe dynamic analysis, we showed that three patterns of stable steady states exist with respect
to the wage diﬀerentials: no-inequality, high-inequality, and multiple steady states. Undera higher level of skill-biased technology, an economy is more likely to have a high-inequality
steady state. Furthermore, we showed that if the economy were initially around the noinequality steady state, rapid progress of skill-biased technology would switch the steadystate from no inequality to high inequality, and the ratio of unproductive investment would
increase, at least for some periods. From this result, we see that the substantial observed
increase in educational attainment experienced after SBTC in many countries may haveresulted not only in an increase in human capital, but also in a large increase in wastefulinvestment aimed at obtaining credentials.
Clearly, it may be hard to distinguish between productive and unproductive investment
in existing educational investment. In reality, no educational investment would contributenothing to human capital accumulation. However, the ratio of human capital investment tounproductive investment in our model well describes whether households’ primary objectives
for educational attainment are learning or credentialing. Because private decisions between
learning and credentialing are less observable, our theoretical analysis makes a signiﬁcant
contribution to explaining households’ decisions.
Moreover, our model is related to the analysis of the negative relationship between income
inequality and intergenerational social mobility, known as “The Great Gatsby Curve”. In ourmodel, social mobility is driven by the probability of obtaining the credential, then parents
invest in dissipative expenses in order to improve these chance to their oﬀspring. Introducing
heterogeneity in innate ability into the model may lead to a richer analysis of social mobility
or occupational mobility. Although these extensions are beyond the scope of the present
paper, they are worth examining.

22Appendix
Derivation of the first order conditions (10)–(13)First, we solve the utility maximization problem of middle-aged parent i represented in (6).
o , we exclude it from
Given that the optimal allocations in old age are simply coi,t+1 = wt+1o
the optimization. Further, because ϕ in wt+2
does not aﬀect the optimization, the objectivefunction in (6) can be rewritten as follows:
[ (
]
) wc
t+1
c
n
e
α ln(wt − ei,t − si,t ) + β(1 − α) Θ hi,t , sei,t ln n + ln wt+1 + β 2 (1 − α) ln hi,t .
wt+1
The ﬁrst order condition with respect to eit is
(
)
e
c
∂Θ
h
,
s
e
i,t
i,t
µeµ−1
µeµ−1
wt+1
α
i,t
i,t
2
ln n + β (1 − α)=0− c
+ β(1 − α)wt − ei,t − si,t
hj,t
wt+1
hi,t
∂e
hi,t
⇔

[
c ]
wt+1
ei,t
∂Θ(e
hi,t , sei,t ) eβ(1 − α)µhi,t ln n .
=β+
wtc − ei,t − si,t
α
wt+1
∂e
hi,t

Thus we have (10). In the same manner, we have the ﬁrst order condition with respect tosit represented in (11). Next, we solve the problem of j. Using (2), the objective function of
middle-aged parent j can be written as follow:
[ (
]
) wc
t+1
n
n
e
α ln(wt − ej,t − sj,t ) + β(1 − α) Θ hj,t , sej,t ln n + ln wt+1 + β 2 (1 − α) ln hj,twt+1
=α ln(wtn − ej,t − sj,t ) + β(1 − α)

]
[ (
) wn
t+1
c
e
Θ hi,t , sei,t ln c + ln wt+1 + β 2 (1 − α) ln hj,t .
wt+1

The ﬁrst order condition with respect to ejt is
(
)(
)
e
∂Θ
h
,
s
e
i,t
i,t
µeµ−1
wn
hi,t µ−1
α
j,t
2
− n
+ β(1 − α)
− 2 µej,t
ln t+1
+
β
(1
−
α)
=0
c
e
wt − ej,t − sj,t
w
h
h
j,t
∂ hi,t
t+1
j,t
⇔

[
c ]
wt+1
ej,t
∂Θ(e
hi,t , sei,t ) eβ(1 − α)µ=β+hi,t ln n .
wtn − ej,t − sj,t
α
wt+1
∂e
hi,t

Thus we have (12). In the same manner, we have the ﬁrst order condition with respect to sjt
represented in (13).
23

Proof of Lemma 1
To simplify the exposition, we use Θ instead of Θ(eht , set ). Since we have ln w
et+1 < ln ϕeht <ϕ ln eht when w
et → ∞, using the denominator of the second term in (19) we have:∂Θ∂Θ
set ln w
et+1 < ϕset ln e
ht .
∂e
st
∂e
st
( )µ
In addition, we have eht = set ν , and thus, from the right-hand side of the above equation,
we have
ϕ

∂Θ
µϕ
∂Θ
(
) → 0 (when w
set ln eht =et → ∞).
∂estν ∂ ln ln set

The last term of limit to zero is obtained by properties that Θ → 1 when w
et → ∞, whileln(ln set ) → ∞ when w
et → ∞. Then, we obtain:∂Θ(eht , set )set ln w
et+1 = 0.
∂e
st
w
et →∞
lim



Proof of Proposition 2
Rewriting (22), we havew
et+1 − w
et =

(
)
1
Φ(w
et ) − Ψ(wet )µ(2 + ϕ)w
et − 1

(25)

where
Φ(w
et ) = 2ϕw
et2µ − ϕwetµ ,Ψ(wet ) = (2 + ϕ)w
et1+µ − w
et − 1.
We can see from (25) that the sign of w
et+1 − w
et depends on Φ(wet ) − Ψ(wet ), and thus we focus
on the relative sizes of Φ(wet ) and Ψ(wet ) in the following analysis.
It is immediately seen that Φ(1) = Ψ(1) = ϕ. It means that w
e = 1 is a steady state of the
dynamics of w
et . Note that the stability of the steady state at w
e = 1 is unclear at present.
Further, we have limwet →∞ (wet+1 − wet ) < 0, and thus we see that w
et does not diverge toinﬁnity.
Next, let us examine the shapes of functions Φ(wet ) and Ψ(wet ). The ﬁrst and second
derivatives of Φ(wet ) areΦ′ (wet ) = 4ϕµw
et2µ−1 − ϕµw
etµ−1 ,
Φ′′ (wet ) = 4ϕµ(2µ − 1)w
et2µ−2 − ϕµ(µ − 1)w
etµ−2 ,
24

which satisfy Φ′ (wet ) > 0 and Φ′′ (wet ) > 0 for all w
et ≥ 1 when µ > 1/2. Further we getΨ′ (wet ) = (2 + ϕ)(1 + µ)w
etµ − 1,Ψ′′ (wet ) = (2 + ϕ)(1 + µ)µw
etµ−1 ,
which satisfy Ψ′ (wet ) > 0 and Ψ′′ (wet ) > 0 for all w
et > 1. Therefore, when µ > 1/2, Φ(wet ) andΨ(wet ) are both convex and increasing in w
et . In the following analysis, we show that three
types of steady states may arise depending on the intersections of Φ(wet ) and Ψ(wet ).

To begin with, we can see that Φ′′ (wet ) − Ψ′′ (wet ) is decreasing with w
et when µ > 1/2.
Therefore, if Φ′ (1) > Ψ′ (1) holds, Φ(wet ) intersects with Ψ(wet ) at w
et = 1 and w
et > 1,
as depicted in Case 1 in the above ﬁgure. Let us denote by we∗ the steady state value atw
et > 1. From (25), we see that w
et+1 > w
et holds for w
et ∈ (1, we∗ ), and w
et+1 < w
et holds forw
et ∈ (we∗ , ∞). Thus, we see that a steady state at w
et = we∗ is stable, but that at w
et = 1 isunstable. Using the above ﬁrst-order derivatives and rewriting Φ′ (1) > Ψ′ (1), we haveϕ>

1 + 2µ≡ ϕ̄.
2µ − 1

Then we can say that, when µ > 1/2 and ϕ > ϕ̄, there exists a unique stable steady statewith positive wage diﬀerentials.
Next, we consider ϕ ≤ ϕ̄. That is, Φ(wet ) is lower than Ψ(wet ) at least in the neighborhoodof w
et = 1. Since Φ′′ (wet ) − Ψ′′ (wet ) is decreasing with w
et , if Φ′′ (1) ≤ Ψ′′ (1) holds, we haveΦ(wet ) < Ψ(wet ) for all w
et ∈ (1, ∞). That is, Φ(wet ) intersects with Ψ(wet ) only at w
et = 1, as
depicted in Case 2. Assuming that µ > 2/3 and rewriting Φ′′ (1) ≤ Ψ′′ (1), we have
ϕ≤

1+µ
b
≡ ϕ.
3µ − 2b w
Hence, when µ > 2/3 and ϕ ≤ min{ϕ̄, ϕ},e = 1 is a unique stable steady state.
√
Finally, let us consider ϕb < ϕ ≤ ϕ̄. Here, we have to suppose that µ > (1 + 5)/4 in orderto satisfy ϕb < ϕ̄. In this case, Φ(wet ) may have three intersections with Ψ(wet ), as depicted in25

Case 3, that is, multiple steady states may exist (but not always). Otherwise, Φ(wet ) intersects
with Ψ(wet ) only at w
et = 1 (that is, we have Case 2). Instead of giving suﬃcient conditions
for the existence of multiple steady states, we provide a numerical example of this case. See
Figure 3 for numerical examples of the three types of steady states.



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