ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E21-8

Higher education, performance pay, andthe polarization of wages
by
Kazutoshi Miyazawa

September 2021

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Higher education, performance pay, and the polarization ofwages
Kazutoshi Miyazawa∗
Doshisha University

Abstract
Some recent studies assert that, in the US and some European countries, one-half of the growth
in income inequality can be attributed to education. Therefore, focusing on performance-related pay
for non-routine skilled workers and an opportunity for higher education, we present a mechanism
to generate the polarization of wages. If the opportunity for higher education is limited, then wage
distribution is polarized by the prevalence of performance pay. However, wage polarization may
vanish in the case where higher education provides high-school graduates with a second chance at
success. Our result highlights the important role of higher education in shaping the distribution ofwages.

JEL Classification: I24, I26, J24, J42, L13
Keywords: Wage inequality, Higher education, Performance pay, Skill-biased technical change∗ Facultyof Economics,
kazu@mail.doshisha.ac.jp

Doshisha

University,

Imadegawa-Karasuma,

Kamigyo,Kyoto602-8580

Japan.

1

Introduction

Polarization is a phenomenon in the labor market in the United States and some European countries sincethe 1990s (Autor, Katz, and Kearney 2006; Goldin and Katz 2007; Lemieux 2008; Goos, Manning, andSalomons 2009). The term ‘polarization’ has two meanings. Autor, Katz, and Kearney (2006) explainedthat wage growth has been polarized since 1988, with higher wage growth rate occurring in the bottom
quartile than in the middle two quartiles. However, the highest growth rate and sustained spreadingout of the wage distributions occur in the top quartile. Employment growth in the 1990s was polarizedwith the highest employment growth rate occurring in high-skill jobs, the lowest growth rate occurringin middle-skill jobs, and the modest growth rate occurring in low-skill jobs.
In addition to the polarization, the top-end wage inequality in the United States that began at the endof the 1970s has also attracted considerable attention (Katz and Murphy 1992; Bound and Johnson 1992;
Johnson 1997). Johnson (1997) points out two reasons for the rapid increase in the relative demand
for skilled workers in the US economy during the 1980s. One is the skill-biased technological change(SBTC), and the other is increased openness. In the context of SBTC, some researchers attribute thetop-end wage inequality to computerization or automation (Autor, Katz, and Krueger 1998; Autor, Levy,
Murnane 2003; Autor and Dorn 2013; Cortes, Jaimovich, and Siu 2017; Cavenaile 2021). In the context
of increased openness, some researchers attribute the top-end wage inequality to oﬀshoring or capital
mobility (Beladi, Marjit, and Broll 2011; Goel 2017; Cavenaile 2021).
Despite the emphasis on the SBTC and increased openness, the following two channels that could
polarize the labor market have not been fully examined, that is, (1) performance-related pay such as
bonuses and stock options (Piketty and Saez 2003; Lemieux, MacLeod, and Parent 2009; Alvaredo etal. 2013). Autor (2014) points out that, between 1980 and 2012, real hourly earnings of full-timecollege-educated US males rose anywhere from 20% to 56%, with the greatest gains going to thosewith a postbaccalaureate degree (p.849). The wage inequality within college-educated workers could be
attributed to performance pay for non-routine skilled workers with professional degrees.
(2) Individuals’ choice of education. Hoﬀmann, Lee, and Lemieux (2020) calculated the contribution
rates of education and other factors to the growth in the variance of total income. They show that
education accounted for 56% of the growth in income inequality over the late 1970s to the late 2010s in the
United States. In the United Kingdom and Germany, the contribution rate of education is approximately
50%, although the sample periods are shorter than in the United States. They also show that occupationand place significantly cause the evolution of income inequality for women, but modest for men.
The purpose of the paper is to explain the mechanism of wage polarization by focusing on performance
pay for non-routine skilled workers and postgraduate education. Our model is based on the following
three assumptions.
First, individuals are heterogenous in a two-dimensional cost of higher education, that is, the cost of
acquiring a college degree, and the cost of acquiring a postgraduate professional degree. This assumption
enables us to examine the wage inequality within college-educated workers explicitly. Further, the cost
approach has two merits from an empirical point of view. Because the education cost assumed corresponds
proportionally to the rate of return on higher education, our model can be assessed by using available
data on the rate of return on higher education. The other merit is related to the measurement problemof wages. Most literature assume that individuals are heterogeneous in a skill that would be eﬀective
only if they were employed in a specific task or occupation. Under this assumption, the wage rate in
the specific occupation is measured in eﬃciency units, which implies that an additional assumption on
the skill distribution is necessary to calculate hourly wage rates. The top 1% income distribution would
follow the Pareto distribution, but the distribution of the other 99% is diﬃcult to specify (Jones andKim 2018). Our cost approach is free from the measurement problem, assuming that working time is thesame for all workers.
Second, we assume that the three educational statuses correspond directly with three occupations, thatis, high-school graduates are employed in the services sector as unskilled workers, and college graduates
are employed in the final goods sector as routine skilled workers. Their wages are determined by themarginal product of labor in the corresponding competitive labor market. We assume that non-routine
skilled workers who get a postgraduate professional degree finance foreign capital to produce intermediate
goods. In a monopolistic competitive market, the earnings of non-routine skilled workers are tied to theirfirm’s profits.
We admit that occupations in the real world are not segmented by education status. Ph.D. graduates
may work in the service sector because they like to spend more time with their families or talk withcustomers. Our assumption is a natural extension of the literature that assumes that occupations aresegmented by skill (Autor and Dorn 2013; Cortes, Jaimovich, and Siu 2017; Cavenaile 2021).
Finally, we assume that the performance pay for non-routine skilled workers increases with the degree1

of product diﬀerentiation of intermediate goods. Specifically, we examine the relationship between performance pay and the top-end and low-end wage inequality by parameterizing the elasticity of substitution
between intermediated goods. It would be better if the pay system for managers was endogenized byincorporating some forms of incentive contract (Lemieux, MacLeod, and Parent 2009), or that the degreeof product diﬀerentiation was endogenized by incorporating the firm’s investment strategy for researchand development (Jones and Kim 2018; Aghion et al. 2019; Akcigit, Pearce, and Prato 2020). These
important micro-foundations of our model are left for future research.
Based on the above assumptions, we obtained two results. First, we derive a simple formula thatsuggests that the top-end wage inequality is negatively related to the low-end wage inequality. Supposethat the profit rate in the intermediate goods sector increases. Then, the rich become richer, which
induces the poor to be richer. This is because the rich will increase demand for services that are a
substitute for household production. This will, in turn, increase the wage rate of workers employed in theservices sector (Manning 2004; Mazzolari and Ragusa 2013). Therefore, when the share of performance
pay increases, the top-end wage inequality increases and the low-end wage inequality decreases. This is apossible story in our model, which could explain the labor polarization observed in the US, the UK, andGermany.
Second, if the increased performance pay changes individuals’ education choices, both top-end andlow-end wage inequalities may decrease. In our model, the ultimate aim of working high-school graduates
who later acquire a college degree is to also acquire a postgraduate professional degree. Therefore, thepolarization of wages may vanish because the increased supply of non-routine skilled workers depresses
their earnings. This scenario could explain the relatively flat wage distribution observed in France andJapan. Our result suggests that diﬀerences in the accessibility to higher education reflect the diﬀerencesin the shape of the wage distribution.
The reminder of this paper is organized as follows: In Section 2, we briefly analyze the related
literature. In Section 3, we introduce the basic model. In Section 4, we conduct a comparative statics
exercise concerning the elasticity of substitution between intermediate goods. In Section 5, we provide
numerical examples. The final section concludes the paper.

2

Related literatureThis research is closely related to Beladi, Marjit, and Broll (2011), Autor and Dorn (2013), Cortes,
Jaimovich, and Siu (2017), and Cavenaile (2021).
Beladi, Marjit, and Broll (2011) assume a static economy in which three goods, that is, high-, middle-,
and unskilled goods, are produced by four inputs, high-, middle-, and unskilled labor and physical capitalin competitive markets. Production of domestic high-skilled and middle-skilled labor is subject to theendowment of educational capital. In this setting, the paper analyzes the response of sectoral employment
to a transition from a closed to an open economy. The paper shows that either the high-skilled or middleskilled sector vanishes when foreign capital flows into the country and that the middle-skilled sector may
vanish when foreign high-skilled labor flows into the country.
Our model is the same as Beladi, Marjit, and Broll (2011) in that education resources are allocatedto the production of high-skilled and middle-skilled labor. However, the theoretical method and the
purpose are quite diﬀerent. Our research focuses on the wage polarization in a closed economy, while
Beladi, Marjit, and Broll (2011) focuses on the employment polarization in a transitional economy.
Autor and Dorn (2013) is one of the most influential papers, which proves that computerization (afall of computer prices) generates labor market polarization. In a closed competitive economy, goods areproduced by abstract labor, routine labor, and computer capital. Computer capital is assumed to be a
relative complement to abstract labor and a relative substitute for routine labor. When technical progressdecreases the price of computer capital, the relative demand for routine labor to abstract labor decreases,
which results in an expansion of the top-end wage inequality, that is, the relative wage rate of routine
labor to abstract labor falls. Services are produced by manual labor. The number of high-skill workers
who supply abstract labor in the goods sector is assumed to be constant. Low-skill workers supply either
routine labor in the goods sector or manual labor in the services sector, depending on their intrinsic skill
that is eﬀective only if they are employed in the goods sector. Under those assumptions, employment
is polarized in the sense that low-skill workers move from the goods sector to the services sector when
computer prices fall.
Our model economy and the mechanism of generating wage polarization have characteristics similar
to Autor and Dorn (2013). However, our model is diﬀerent from Autor and Dorn (2013) in the following
two aspects. First, we focus on performance pay for non-routine skilled workers as a factor of labormarket polarization, while Autor and Dorn (2013) focus on computerization in the context of skill-biased2

technological change. Second, the supply of non-routine skilled labor is endogenized in our model byincorporating education choice. The assumption of a constant supply of high-skill labor in Autor andDorn (2013) may overestimate the top-end wage inequality.
Cortes, Jaimovich, and Siu (2017) assume an aggregate production function with five inputs (nonroutine cognitive workers, non-routine manual workers, routine workers, automation capital, and capitalother than automation) when examining the eﬀect of introducing automation capital on occupational
choice. The number of non-routine cognitive workers is assumed to be constant. The other workers diﬀer
in work ability and disutility of labor, which divides them into routine workers, non-routine manual workers, and unemployed workers. Cortes, Jaimovich, and Siu (2017) show that introduction of automation
capital, a substitute for routine workers generates employment polarization. In other words, when automation capital is introduced, the number of routine workers decreases, while the number of non-routine
manual workers and non-employment increases.
Our model economy is the same as Cortes, Jaimovich, and Siu (2017) in that non-routine cognitive
workers earn the share of profits. However, we diﬀer from Cortes, Jaimovich, and Siu (2017) in that theobjective of our research is wage polarization while theirs is employment polarization. In addition, ourmodel endogenizes the supply of non-routine cognitive workers, which allows routine workers to become
non-routine cognitive workers.
Cavenaile (2021) assumes a small open economy in which domestic workers choose their occupationfrom four alternatives: two jobs (manager or worker) in two sectors (goods or services). In equilibrium,
occupations are segmented by intrinsic human capital. The highest pay occupation is a manager in the
goods sector, followed by a manager in the services sector. The third is a worker in the goods sector, andthe lowest pay occupation is a worker in the services sector because it is assumed that human capitalof workers is not eﬀective in the services sector. In this model economy, Cavenaile (2021) examines
the eﬀects of oﬀshoring and computerization on the distributions of employment and wages. When thewage rate of foreign skilled workers decreases, both employment and wages of domestic skilled workers
decrease. A reduction in the price of computer capital delivers the same result. With some empiricalevidence, the paper concludes that both oﬀshoring and computerization are the main factors of labormarket polarization.
Our model economy is the same as that of Cavenaile (2021) in that the managers in the goods and theservices sectors earn a share of profits. Another parallel is we both consider a segmented labor market. In
our model, three occupations are segmented by a two-dimensional cost of higher education. In Cavenaile(2014), four occupations are segmented by human capital.
A crucial diﬀerence between Cavenaile (2021) and our research is the mechanism of polarization. The
mechanism proposed in Cavenaile (2021) is rather conservative in the sense that both the skill-biased
technological change and increased openness can be traced back to Johnson (1997). Our research is
motivated by Hoﬀmann, Lee, and Lemieux (2020), who indicate that there is a sizable contribution of
education to income inequality in some developed countries.

3

The model

The model economy consists of three production sectors (services, final consumption goods, and intermediate goods) (See Table 1). Individuals are heterogeneous with respect to a two-dimensional cost of
higher education. Labor markets are education-segmented in the sense that high-school graduates work
as unskilled labor in the services sector, college graduates work as routine-skilled labor in the final goods
sector, and individuals with postgraduate professional degrees work as non-routine skilled labor in theintermediate goods sector. We assume that markets of services and final goods are perfectly competitive
and that the market of intermediate goods is monopolistic. Because our focus is on wage inequality, we
assume that firms in the intermediate goods sector can access the international capital markets, which
implies that capital income inequality is beyond the scope of the paper.
Table 1. Production sectors
OutputServices
Final goods
Intermediate goods

InputsUnskilled (L)Routine skilled (M )Composite goods
Non-routine skilled (H) Capital

3

3.1

Individuals

We assume that the total number of individuals is constant, and is denoted by N . Individuals are classified
into three groups: high-school graduates (denoted by L), college graduates (M ), and college graduateswith a postgraduate professional degree (H).
Individuals choose their education and the allocation of consumption. The allocation problem ofindividual i = L, M , and H is formulated as
max ui = (1 − α) ln ci + α ln disubject to yi = ci + pd dici , d i(1)where ci and di represent goods consumption and services consumption, respectively. yi represents thedisposable income of an individual i, which is specified below. Assuming that goods are numeraire,
pd represents the relative price of services. 0 < α < 1 is a preference parameter attached to services
consumption.
Solving the problem, the demand functions are given bycidi= (1 − α)yi
yi
= α
pd

(2)
(3)

Individuals are heterogenous with respect to a pair of education costs (e1 , e2 ) ∈ R2+ . e1 represents thecost incurred in acquiring a college degree, and e2 represents the cost incurred in acquiring a postgraduate professional degree. The disposable income of high-school graduates (say, unskilled labor), college
graduates (routine-skilled labor), and college graduates with a professional degree (non-routine skilledlabor) are respectively given by⎧⎨ yL = wL
yM = wM − e1 (wM − wL )(4)⎩
yH = wH − e1 (wM − wL ) − e2 (wH − wM )where wL , wM , and wH represent the wage rates for unskilled labor, routine skilled labor, and non-routine
skilled labor, respectively. We assume, in equilibrium,
wH > wM > wL
Our setup is not so specific. Eq.(4) yields
µ
¶
wM
yM
− 1 = (1 − e1 )
−1
yL
wL
µ
¶
wH
yH
− 1 = (1 − e2 )−1yM
wM
Note that (wM /wL −1) represents the rate of return on college education, and (wH /wM −1) represents
the rate of return on postgraduate education. The greater e1 is, the smaller the net rate of return oncollege education. The greater e2 is, the smaller the net rate of return on postgraduate education.
Therefore, the pair of (e1 , e2 ) represents the heterogeneity among individuals with respect to the rates ofreturn on college and postgraduate education.
Formally, the optimal choice of education is specified as follows, an individual (e1 , e2 ) becomes:(i) Unskilled labor L, if½½
yL > yM
e1 > 1⇒yL > yHe2 > φ(e1 ; w)

whereφ(e1 ; w) =(ii) Routine skilled labor M , ifor (iii) non-routine skilled labor H if½½

wH − wL
wM − wL−e1
wH − wM
wH − wM

yM > yL⇒
yM > yH

yH > yL
yH > yM

⇒

½

4

½

e1 < 1e2 > 1e2 < φ(e1 ; w)
e2 < 1(5)

[Figure 1 here]
Figure 1 illustrates the regions of L, M , and H on (e1 , e2 ). The region above the horizontal line e2 = 1
corresponds to a traditional two-class economy: Individuals e1 < 1 go to college, and individuals e1 > 1
do not. The region below line e2 = 1 is the core of our model. Low-cost individuals (e1 < 1) go to college
to obtain a postgraduate degree. High-cost individuals on the right side of the border e2 = φ(e1 ; w)
become high-school graduates. In the middle triangle region (e1 > 1 and e2 < φ(e1 ; w)), individuals go
to college even though the net rate of return on college education is negative. The reason is that they
have a chance to become non-routine skilled workers by getting a postgraduate degree. Notice that thetriangle is widened when wH and wL increase, or when wM decreases. In this sense, wage polarizationencourages a second chance at success provided by higher education.

3.2

Firms

3.2.1

Services sector

For simplicity, we assume a linear technology of the services sector,
Yd = f (L) = L(6)

where L is unskilled labor, and Yd is the output of the services.
Perfect competition makes the wage rate of unskilled labor equal to the price of services,
wL = pd
3.2.2(7)

Final goods sector

Final consumption goods are produced by routine-skilled labor and composite goods which include intermediate goods. Following Dixit and Stiglitz (1977), the technology is specified byYZ= M 1−β Z β
σ
# σ−1
"Z
H
σ−1
x(j) σ dj
=

(8)
(9)

0

where Y is the output of the final goods, and M and Z represent the inputs of routine-skilled labor andthe composite goods, respectively. x(j) represents the input of an intermediate good j ∈ [0, H], where H
is the variety of intermediate goods. 0 < β < 1 is a constant production share of the composite goods,
and σ > 1 is a constant elasticity of substitution between the diﬀerent intermediate goods.
The optimization problem is broken up into two parts. First, firms minimize expenditure for intermediated goods, taking the level of composite goods as given,e(Z) = min
x(j)

Z H

p(j)x(j)dj

subject to Z =

0

"Z
H

x(j)

0

σ−1
σ

dj

σ
# σ−1

where p(j) is the price of the intermediate goods j, and e(Z) is the expenditure function.
Solving the problem, the demand for the intermediate good j is given by¸σ∙Px(j) = Zp(j)where P represents a price index
P =

"Z
H

p(j)

0

1−σ

dj

1
# 1−σ

(10)

(11)

Eqs.(10) and (11) yield e(Z) = P Z, which implies that P represents the price of the composite goods.
Second, firms minimize total cost of production, taking the output as given,min wM M + P Z

M, Z

subject to Y = M 1−β Z βwhere wM is the wage rate of routine-skilled labor.
5The optimality condition is given by
1−β Z
wM
=
P
β M

(12)

which implies that the factor price ratio is equal to the marginal rate of transformation.
Solving the problem, the factor demand is given by
M

µ

=

Z

µ

=

β wM
1−β P
β wM
1−β P

¶−β

Y

¶1−β

Y

Finally, the zero-profit condition yields the factor price frontier,
1=
3.2.3

µ

wM
1−β

¶1−β µ

P
β

¶β

(13)

Intermediate goods sector

Intermediate goods are produced by managers (non-routine skilled labor) and capital. Focusing on wage
income, we assume that capital is owned by foreigners or domestic owners outside of the model.1 We
assume a linear technology in sector j ∈ [0, H] as
x(j) = k(j)(14)

where x(j) and k(j) represent output and capital, respectively.
Denoting a constant world interest rate by r > 0, the production cost is rk(j) = rx(j). The manager
in sector j earns wH (j) as performance pay. Specifically, the optimization problem is formulated bywH (j) = max

p(j), x(j)π(j) = [p(j) − r]x(j)subject to Eq.(10).
Solving the problem, the monopoly price is given by the markup pricing
1p(j) − r
=
p(j)
σ

(15)

for all j ∈ [0, H].
From Eq.(15), the profit share, that is, the ratio of earnings of the manager to total revenue is σ −1 ,which implies that the performance pay is negatively related to the elasticity of factor substitution.
Therefore, we interpret a reduction of σ as meaning that performance pay for managers increases.
Omitting the index of the intermediate good j, price index P , output of intermediate goods x, andthe wage rate of non-routine skilled labor wH are given by
σ
r
σ−1
σ
x = H 1−σ × Z
σrZ
wH = H 1−σ ×
σ−1
P

1

= H 1−σ ×

(16)
(17)
(18)

Because σ > 1, the price index and wage rate of the non-routine skilled labor decrease with thenumber of non-routine skilled labor, H.

3.3

Market equilibrium

Let us denote the numbers of non-routine skilled labor, routine skilled labor, and unskilled labor by nH ,
nM , and nL , respectively. The population constraint is given by
N = nH + nM + nL
1 We can extend the basic model to incorporate capital endowments among individuals. Because capital income ispositively associated with labor income (Hoﬀmann, Lee, and Lemieux 2020), the extension would strengthen our result.

6where N represents a constant total number of individuals.
Because labor markets are education-segmented, the market clearing conditions are given by
L = nL
M = nM
H = nHThe services market clearing condition is given by
X
Yd =

di

(19)

i∈L,M,H

Since the model is closed, the goods market clearing condition,
Y =X

i∈L,M,Hci + (wM − wL )

X

i∈M,H

e1 + (wH − wM )X

e2 + ri∈H

Z H

k(j)dj0can be derived by Walas’ law. Notice that the distribution of education cost (e1 , e2 ) ∈ R2+ is given
exogenously.

3.4

Equilibrium

In our model, individuals’ choice of education depends on the distribution of wages, while the equilibrium
wages depend on the outcome of individuals’ education choice. Therefore, we define the equilibrium
formally in the following way.
Denote the distribution of wages by w = (wH , wM , wL ) ∈ R3+ and the outcome of education choiceby n = (nH , nM , nL ) ∈ R3+ .
1. A mapping of the wage space R3+ onto the population space R3+ , denoted by n = ϕ(w), is given by
nM
nH
nL

= # {(e1 , e2 )| e1 < 1 and e2 > 1}
= # {(e1 , e2 )| e2 < φ(e1 ; w) and e2 < 1}
= N − nH − nM

where the symbol # represents the number of points (e1 , e2 ) in the corresponding region.
2. A mapping of the population space onto the wage space, denoted by w = ψ(n), is given by
wH
wM
wL

= ψ H (n)= ψ M (n)= ψ L (n)3. Equilibrium distribution of wages, denoted by w∗ is defined as a fixed point in the composite mapping:w∗ = ψ ◦ ϕ(w∗ ).
In the next section, we derive the mapping w = ψ(n) explicitly from equilibrium conditions.

4

Wage inequality

In this section, we examine the characteristics of the top-end and low-end wage inequality, focusing on
the performance pay of non-routine skilled workers. Because the performance pay is associated with thedegree of product diﬀerentiation of intermediate goods, we measure the eﬀect of performance pay on thewage inequality by parameterizing the elasticity of factor substitution, σ.

4.1Top-end wage inequality

The following proposition summarizes the top-end wage inequality.
Proposition 1 The top-end wage inequality is given by
nM
wH
β
×
=
wM
σ(1 − β)
nH

7

(20)

Proof. Combining Eqs.(12), (16), (18), M = nM , and H = nH , we obtain Eq.(20).
Eq.(20) implies that the relative demand for (non-) routine skilled labor is negatively related to therelative wage rate of (non-) routine skilled labor. Eq.(20) corresponds to Johnson’s (1992) relative demand
curve.
From Proposition 1, we obtain the following corollary.
Corollary 2 Assume that the elasticity of factor substitution σ falls and that this technical change keeps
both nH and nM constant. Then, the top-end inequality increases.
From Eqs.(13), (16), and (20), the wage rates of routine labor and non-routine labor are explicitly
given by
β
¸ 1−β
∙
1
β(σ − 1)(nH ) σ−1(21)wM = (1 − β)
σr
β
∙
¸ 1−β
1
β nM β(σ − 1)(nH ) σ−1
(22)
wH =
σ nH
σr
which represent ψ M (n) and ψ H (n) in Section 3.4, respectively.
The wage rate of routine skilled labor wM increases with nH and σ. When either nH or σ increases,
the price index P falls (See Eq.(16)), which increases wM along the factor price frontier in Eq.(13). wM
is neutral to the number of routine skilled labor, nM .
As for the wage rate of non-routine skilled labor wH , we obtain following lemma.
Lemma 3 The wage rate of non-routine skilled labor wH is (i) hump-shaped with respect to nH and σ,
∂wH
∂wH
R0
R 0,
∂nH
∂σ

σQ

if

1
1−β

(23)

and (ii) increasing in the number of routine-skilled labor, nM .
Proof. The power nH in Eq.(22) is β/[(1 − β)(σ − 1)] − 1, which is positive (negative) if and only if
σ < (>)1/(1 − β).
For σ, wH increases (decreases) with σ if and only if (σ−1)β /σ increases (decreases). Log-diﬀerentiation
yields the condition in Eq.(23).
In a special case, the number of non-routine skilled labor nH is equal to the number of routine skilledlabor nM , the equilibrium condition wH > wM requires σ < β/(1 − β). In this case, Lemma 3 shows∂wH /∂nH > 0 and ∂wH /∂σ > 0, which implies that wH and wM move in the same direction when eithernH or σ changes.
If the non-routine skilled labor is scarce enough to satisfy nH < βnM , then σ > 1/(1 − β) is possible.
In this case, we obtain that ∂wH /∂σ < 0, which implies that a reduction of σ increases wH and decreases
wM . The higher the degree of product diﬀerentiation is, the larger the wage inequality among college
graduates.

4.2

Low-end wage inequality

In this section, we derive the wage rate of unskilled labor, wL = ψ L (n), in Section 2.4 explicitly.
Substituting Eqs.(3), (7), and (8) into Eq.(19), the wage rate of unskilled labor is given by
X
α
yi
wL =
nL

(24)

i∈L,M,H

where the aggregate disposable income by type i = L, M , and H are given byX
yi = nL wL
i∈L

X

yi

i∈M

X

i∈H

yi

= nM wM − (wM − wL )= nH wH − (wM − wL )

X

e1

i∈M

X

i∈H

e1 − (wH − wM )

X

e2

i∈H

respectively. Substituting them into Eq.(24), and rearranging terms, we obtain the following proposition.
8

Proposition 4 Assume that
(1 − α)nL − αXe1 > 0i∈M,HThen, the low-end wage inequality is given byh¢ ³ wH ´i
¡
P
P
P
α
n
−
e
+
e
+
n
−
e
M
H
i∈M,H 1
i∈H 2
i∈H 2
wM
wL
P
=
wM
(1 − α)nL − α i∈M,H e1(25)

Using Eqs.(25), (21), and (22), we obtain wL = ψ L (n). Eq.(25) implies that wL /wM is positively
related to the top-end wage inequality wH /wM because the coeﬃcient of wH /wM in Eq.(25) is positive(note that e2 < 1 for ∀i ∈ H). The reasons for this are as follows: assume that the wage rate of the nonroutine skilled labor wH increases by one dollar. For an individual of type H, his/her disposable income
increases with the net rate of return on graduate education, 1 − e2 . Then, he/she increases expenditure
for services by α(1 − e2 ). Given the number¡ skilled
P labor¢is constant, the aggregate
P of the non-routine
expenditure for services is increased by α i∈H (1 − e2 ) = α nH − i∈H e2 . This demand-side eﬀect
increases the price of services, which increases the wage rate of unskilled labor. When the rich becomesricher, the poor also becomes richer.
Combining Corollary 2 and Proposition 4, we obtain the following corollary.
Corollary 5 Assume that the elasticity of factor substitution σ falls and that this technical change keeps
the allocation of labor constant, then, we can observe that the top-end inequality increases, while thelow-end inequality decreases.

5

Numerical example

In this section, we present numerical examples. In Section 4.1, we examine the case where n = ϕ(w) is
a constant mapping, that is, individuals’ education choice does not depend on the distribution of wages.
As shown in Corollary 5, we can observe a negative relationship between the top-end and low-end wage
inequality. In Section 4.2, we examine a more general case. When wH /wM and wL /wM increase, thetriangle in Figure 1 is widened. This means that some individuals change their education status fromhigh-school graduates to college graduates with a postgraduate degree. Therefore, the supply of nonroutine skilled labor increases and the supply of unskilled labor decreases during the transition process.
We show that both top-end and low-end wage inequalities decrease in transition. On the one hand, thereduction of unskilled labor shifts the supply curve to the left, which increases the wage rate of unskilledlabor. On the other hand, an increased variety of intermediate goods depresses profits, which decreases
the wage rate of non-routine skilled labor.

5.1

1-2-3 economy: An immobile case
[Figure 2 here]
Table 2. Immobile case
e1
e2
ni
1
H
1
1
2
1
M
2 ¡ 2 ¢ (2)
3
L
3
(1)
2

Figure 2 illustrates the distribution of population on the plane (e1 , e2 ) (See also Table 2). There is
one person whose education cost is (e1 , e2 ) = (1, 1/2). He/she optimally chooses to get a postgraduate
degree. Two persons with education cost (e1 , e2 ) = (1/2, 2) become college graduates, and three persons
with (e1 , e2 ) = (3/2, 1) are high-school graduates.
The aggregate costs of college education and postgraduate education are respectively given byX1
e1 = 1 × 1 + 2 × = 22i∈M, H

X

i∈H

e2

= 1×
9

1
1
=
2
2

From Eqs.(20) and (25), the top-end and low-end wage inequalities are as follows:
wH
wM
wL
wM

=
=

2β
σ(1 − β)
∙
¸
1
β
α
+
3 − 5α 2 σ(1 − β)Table 3. Factor substitution and wage inequality
wH
wL
σ
wM
wM
1.1
7.27
0.414
1.2
6.67
0.383
1.3
6.15
0.358
1.4
5.71
0.336
1.5
5.33
0.317
2.0
4.00
0.250
2.5
3.20
0.210
3.0
2.67
0.183
3.5
2.29
0.164
4.0
2.00
0.150
Note. β = 0.8, α = 0.2
Table 3 shows the relationship between the elasticity of factor substitution σ and both top-end andlow-end wage inequalities. When σ = 4, the top-end wage inequality is wH /wM = 2 and the low-end
wage inequality is wL /wM = 0.15. When σ drops to 2, the top-end wage inequality is doubled, and thelow-end wage inequality decreases to wL /wM = 0.25. Notice that education choice is not aﬀected bychanges in the wage distribution in this example.

5.2

1-2-2-1 economy: A mobile case
[Figure 3 here]
Table 4. Mobile case
ni
e1
e2
1
H
1
1
2
1
M
2
(2)
¡ 23 ¢
L
2
(1)
2
1
Q
1 1+ε
2

The previous section assumes that individuals do not respond to changes in wage distribution. Thissection allows for a case in which some individuals optimally change their education choice.
Figure 3 illustrates the distribution of the population (See also Table 4). The types i = H, M , andL are the same as in the previous section. We added a group Q, in which individuals do not want to beroutine skilled workers because e1 = 1 + ε > 1, but may want to be non-routine skilled workers if ε issmall and the slope of the border-line is flat.
Assume that θ persons in group Q become H and (1 − θ) persons become L. Then, the aggregate
education costs are given byX
e1 = 2 + (1 + ε)θ
i∈M, H

X

e2

=

i∈H

1
(1 + θ)2

Substituting them into Eqs.(20) and (25), the top- and low-end wages inequality are respectively given
by
wH
wM
wL
wM

=
=

2
β
×
σ(1 − β) 1 + θ
∙
µ
¶
¸
1
1
1
wH
α
−
+ ε θ + (1 + θ)
3 − 5α − (1 + αε) θ 2
2
2
wM
10

The disposable income of an individual in group Q, who becomes H, is given by
10= wH − (1 + ε)(wM − wL ) − (wH − wM )yH
2which yields
0yH
R yL ⇔

wH
wL+ 2ε
R 1 + 2ε
wM
wM

(26)

0
If an individual expects yH> yL , then he/she becomes H. Otherwise, he/she becomes L. The
0equilibrium is given by yH = yL , which determines θ, the share of H in group Q.
Figure 4 illustrates both sides of Eq.(26) in the plane with the θ axis. A downward sloped curve
represents wH /wM + 2εwL /wM , which shifts upward when σ decreases. The horizontal line is 1 + 2ε.
When σ = 4, the curve lies below the line, which implies that no one becomes H (θ = 0). When σ = 2,
the curve intersects with the line at θ = 0.934. The equilibrium is stable because the excess demand forhigher education is positive (negative) when θ < (>)0.934.
Table 5 shows the relationship among the factor substitutions, the share of H in group Q, and both
top-end and low-end wage inequalities. In cases of the corner solution (θ = 0 or θ = 1), the reductionin σ increases the top-end inequality and decreases the low-end inequality in the same manner as theprevious section. However, in cases of the interior solution, a reduction of σ decreases both top-end andlow-end wage inequalities. This is because some individuals change from unskilled labor to non-routine
skilled labor, which shifts the supply curve of unskilled labor to the left, and that of non-routine skilledlabor to the right. This example sheds light on the important role of a chance of higher education in
shaping the distribution of wages.

Figure 4. Education choice

4

3

2

1

0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

theta

Note. Each curve represents wH /wM + 2εwL /wM as a function of θ when σ = 4 (bottom), σ = 3.546(middle), or σ = 2 (top). The horizontal line is 1 + 2ε. As σ decreases, the curve shifts upward, which
implies that θ increases. β = 0.8, α = 0.2, ε = 0.75.

11

Table 5. Factor substitution, education choice, and wage inequality
wH
wL
σ
θ
wM
wM
1.1
1
3.636
0.679
1.2
1
3.333
0.608
1.3
1
3.077
0.548
1.4
1
2.857
0.496
1.5
1
2.667
0.451
2.0
0.934
2.068
0.288
2.5
0.462
2.189
0.207
3.0
0.195
2.231
0.179
3.5
0.014
2.254
0.164
3.546
0
2.256
0.163
4.0
0
2.000
0.150
Note. β = 0.8, α = 0.2, ε = 0.75

6Conclusions

In a simple static model of higher education, we examine the relationship between performance pay for
non-routine skilled workers and the distribution of wages. In an economy where the chance of higher
education is limited, we can observe a negative relationship between the top-end and low-end wage
inequalities. This result is consistent with the stylized facts presented in Lemieux (2008). However, topend wage inequality could be positively related to low-end inequality if individuals can access a second
chance at success provided by higher education. This scenario could explain the relatively flat wage
distribution observed in France and Japan.
Our research can be extended in several directions. For example, we use the elasticity of substitution
between intermediate goods as a proxy of the prevalence of performance pay. Research is necessary tobuild micro-foundations of both the performance pay (Lemieux, MacLeod, and Parent 2019) and thedeterminant of product diﬀerentiation (Jones and Kim 2018; Aghion et al. 2019; Akcigit, Pearce, andPrato 2020). Focusing on wage inequality, this research passes over the inequality of capital income.
Incorporating capital ownership is a natural extension to capture the overall income inequality. We also
assumed that occupations are segmented by education status, however, it would be plausible to alleviate
the assumption by determining the relationship between education and occupation in a probabilistic
setting. Our result highlights the important role of higher education in shaping the distribution of wages.
Further, analyses on the public policy for higher education would complement this paper because thepublic policy for higher education will aﬀect not only the top-end wage inequality but also the low-end
wage inequality. These extensions are left for future research.
Acknowledgments
I would like to thank Toshiki Tamai, Akira Kamiguchi, and the seminar participants at Nagoya
University for their useful comments. This research is financially supported by the Japan Society for the
Promotion of Science (JSPS KAKENHI Grant No. JP21K01472). Any remaining errors were my own.

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