OSIPP Discussion Paper : DP-2006-E-002

Does the Diversity of Human Capital Increase GDP?
A Comparison of Education Systems
June 27, 2006
Katsuya Takii, Osaka University
Ryuichi Tanaka, Tokyo Institute of Technology【キーワード】Span of control, Complementarities, Human capital, Ability tracking【要約】This paper examines how different education systems affect GDP by influencing
the diversity of human capital. We construct an overlapping generation model in which
agents are heterogeneous in income and innate ability, and the final goods are producedwith differentiated intermediate goods. We analyze an economy in which an income
distribution converges to a stationary distribution. It is shown that the diversityof human capital induced by income inequality always lowers the GDP of the next period,
while the diversity of human capital induced by heterogeneous ability can increase
GDP, if the produced intermediate goods are sufficiently substitutable and firms havea large span of control. Hence, as public education equalizes education resources
across households, it mitigates the negative effect of income inequality on GDP, while
the effects of ability tracking crucially depend on the production structure of theeconomy.

We are grateful to Koichi Futagami, Ryo Horii, Kazuo Mino, Yasuyuki Sawada, Akihisa
Shibata and the seminar participants at Osaka University, Kyoto University, Doshisha
University, Nagoya City University, Kobe University, and GRIPS for their very helpfulcomments. Of course, all remaining errors are solely ours.

Financial support fromthe Japanese Ministry of Education, Culture, Sports, Science and Technology (theGrant-in-Aid for Scientific Research) is gratefully acknowledged. Katsuya Takii:

Osaka School of International Public Policy, Osaka University, 1-31, Machikaneyama,
Toyonaka, Osaka, 560-0043, Japan.(takii@osipp.osaka-u.ac.jp)

Ryuichi Tanaka:

Graduate School of Information Science and Engineering, Tokyo Institute of Technology,
2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan. (tanaka@mei.titech.ac.jp)

1

Introduction

Economists typically consider that education can improve the human capitalof workers and raise GDP. Several researchers estimate the level of humancapital from education attainment and examine the impact of human capitalon GDP or economic growth [e.g., Mankiw, Romer and Weil, 1992]. On theother hand, relatively little is known about the eﬀect of the diversity ofhuman capital on GDP.
Apparently, the diversity of human capital diﬀers across countries. Several recent international surveys reveal this variation. Although diﬀerent
surveys compare diﬀerent abilities at diﬀerent ages, some common tendencies can be found in the surveys.1 Brown, Micklewright, Schnepf and Waldmann (2005) find that among 18 OECD countries, results from three surveys(Trends in International Mathematics and Science Study, Programme for International Student Assessment, and International Adult Literacy Survey)
consistently indicate that Finland and the Netherlands have relatively small
inequalities of achievements; the United Kingdom, New Zealand and the
USA have relatively large inequalities of achievements.2
How does this diversity of human capital influence GDP? The impor1

Trends in International Mathematics and Science Study (TIMSS) covers achievement
in mathematics and science for early or middle teens, the Programme for International
Student Assessment (PISA) covers achievement in reading, mathematics and science forearly or middle teens, the International Adult Literacy Survey (IALS) examines ‘document’, ‘prose’ and ‘quantitative’ literacy for all people of working age, and the Progressin International Reading Literacy Study (PIRLS) examines the reading skills of young
children.
2
Brown, Micklewright, Schnepf and Waldmann (2005) compare the diﬀerence between
the 95th percentile and 5th percentile of achievement distributions for 18 OECD countries,
converted from the data in TIMSS, PISA, and the late teens and early 20s in IALS. The
18 countries include Finland, Denmark, Netherlands, Canada, Czech Republic, Sweden,
Australia, Portugal, Norway, Germany, Ireland, Italy, Belgium, Hungary, Switzerland,
UK, New Zealand and USA. The countries are ordered from the most equal achievement
to the least equal achievement using the average ranking of the three surveys.

1

tance of this question can be understood when we recognize that one of
the central aims of an education policy is to provide students with equal
education resources. For example, several reforms have been conducted toachieve equity in education outcomes in the United States. The 1971 landmark decision in Serrano v. Priest transformed the public education system
in California, and other states (e.g., Michigan in 1994 and Washington in1979) have also centralized their education systems in order to achieve equity in education resources. More recently, the “No Child Left Behind Act”
by the George W. Bush administration aims to achieve equity in and ahigh quality of education by raising the performance of the lowest achieving
students. Hence, the previous question leads us to ask a more importantquestion: can an egalitarian education policy raise GDP?
The answer is not obvious. On the one hand, if a government fails
to provide everybody with enough literacy skills, it would be diﬃcult forworkers to communicate and cooperate with each other. On the other hand,as top managers’ decisions are influential in a company, we want them tounderstand the varieties of opinions and to make sound decisions. Hence,some may insist that an education policy should target the bottom of ability
distribution; others may advocate the importance of education for the elite.
In order to evaluate the impact of an egalitarian education policy on
GDP, we need a model to link education reform, the diversity of humancapital and GDP in a unified framework. This paper aims to accomplish
this task. It constructs an overlapping generation model in which education systems influence GDP by changing the variance in human capital andcompares alternative education systems by their eﬀects on GDP.
This model is distinguished from the previous literature in two aspects.
First, we tractably parameterize the structure of industries and firms and2examine how an education system and the production structure of an economy have an interactive eﬀect on GDP. In particular, this paper pays special
attention to the span of control in a firm and the complementarity of goods
in an industry. A large span of control gives an individual the authority toreallocate large amounts of resources. Without authority, an able person
cannot fully utilize his/her unusual talents. Hence, a high level of control
favors an education system that produces a few highly educated workers.
If complementarity of goods exists, the value of a firm’s product depends
on other firms’ product, and a good produced by incompetent persons may
reduce the value of other firms’ product. Hence, high complementarity ofgoods demands an education system that produces many reasonably welltrained workers.3
Secondly, diﬀerent from the previous literature that analyzes education
policies in a dynamic general equilibrium model, education systems arecharacterized not only by their financing systems, but also by their abilitytracking programs. Hence, education systems change the way the heterogeneities of both income and ability influence the diversity of human capital.
A private education system yields more diverse human capital than a public
education system because the rich spend more on education than the poor.4
3For example, in a financial market, fund managers are allowed to allocate a large
amount of resources to buy diﬀerent stocks that are highly substitutable. It is likely todemand unusual talent. On the other hand, firms in the car industry need to combine a
number of complementary intermediate goods (e.g., the quality of tires is likely to influence
the value of brakes). This might demand many well-trained workers. The level of control
in the car industry would be influenced by the structure in the firm. If intermediate
good sectors are less vertically integrated, or bottom-up decision making is common, thelevel of control would be low. Again, it must demand reasonably well-trained workers.
It is interesting to note that the structure of industries and firms in the U.S., whichis regarded as having relatively heterogeneous human capital, seems to relatively favor
unusual talent, while that in Japan, which is seen as having a relatively homogeneoushuman capital, relatively favors reasonably well-trained workers.
4
A private education system is defined as an education system in which individuals
finance the costs of education, and a public system is defined as an education system in3

On the other hand, ability tracking students into separate groups according
to their ability restricts those with whom they can interact as schoolmates orclassmates. Since advantaged students interact with advantaged students,
ability-tracking benefits advantaged students more than disadvantaged students through the peer eﬀects. Hence, a streamed program yields more diverse human capital than an untracked program by amplifying the benefits
from high innate ability.
This paper analyzes an aggregate economy in which an income distribution converges to a stationary distribution, and shows that a public systemyields higher GDP than a private system regardless of industry and firm
structure, while the eﬀect of an ability-tracking program on GDP depends
on the production structure.

As far as the variance of log income con-verging to a finite value, this paper shows that, given a current GDP andan ability distribution, a larger income diﬀerence reduces GDP at the next
period. Since the public system always lowers income inequality more thanthe private system through the redistribution of income, it always attains ahigher GDP than the private system.
A similar mechanism is emphasized in the previous literature when thehuman capital accumulation function is concave in expenditure on educationand the production function is linear in human capital (e.g., Loury, 1981,
and Glomm and Ravikumar, 1992). Our result shows that this still holds
even if the diversity of human capital increases GDP on the production side.
However, when the diversity of human capital is enhanced by ability
tracking, the structure of the production side becomes important. Diﬀerent from the dynamics of income distribution, the distribution of ability is
exogenously given and the variance of ability is always finite. It is shown
which revenues from labor income tax are assumed to finance the costs.

4

that a rise in inequality in ability can increase GDP at the next period.
It is also shown that an ability-tracking program can attain higher GDP if
goods in an industry are fairly substitutable and if the span of control in a
firm is suﬃciently large. Hence, the private education system may increase
GDP if a private school has more incentives or advantages to screen students
through its entrance examination.5 This result highlights a distinctive roleof ability tracking in macroeconomics.
This paper is based on the literature that compares the performance of
diﬀerent education systems in a dynamic general equilibrium model (e.g.,
Glomm and Ravikumar, 1992, Bénabou, 1996 and Fernández and Rogerson,1998). These papers compare diﬀerent financing methods for education. In
particular, Bénabou (1996) examines the eﬀect of diversity of human capital on economic growth when the human capitals of individual agents haveinteractive eﬀects on GDP. His main focus is to examine the role of complementarity of human capital at the community level and at the productionlevel. By contrast, we do not consider a local interaction at the community
level and pay more attention to the interaction at a production level. In
particular, we explicitly examine the role of complementarity of products
and a manager’s span of control to compare education systems. We also
explicitly analyze the eﬀect of ability sorting on GDP.
Ability tracking has been examined by Epple, Newlon and Romano(2002) and Brunello, Giannini and Ariga (2004). Although these papers
examine the benefits and costs of ability tracking, they do not examine therole of the production structure. The interaction between an education system and the structure of an industry and firm is the main focus of our5

A model in Epple and Romano (1998) predicts that a private school attracts moreable students than a public school.

5paper.
The rest of this paper is organized as follows. Section 2 describes themodel, and Section 3 shows some results about the relationship between
income inequality and GDP. Section 4 compares the two education systems,
the public and the private. Section 5 considers the case with ability tracking
in public schools. Section 6 discusses some extensions and concludes. All
proofs are in the Appendix.

2

The Model

In this section, we construct an overlapping generation model with humancapital accumulation. The model features complementarities among intermediate goods in the final good production and the span of control in the
production of intermediate goods. Individuals are required to participate inboth tax-financed (public) schools and self-financed (private) schools in theirlife. In our benchmark model, private schools are allowed to sort students
based on their ability, while public schools are not. This assumption is relaxed later. Using this model, we derive the dynamics of individual income
and examine its aggregate behavior in the next section.

2.1

Technology

Consider an economy in which there is only one final (numeraire) good. The
final good is produced by combining intermediate goods {xi } where i is theindex for ith intermediate good. The production function of the final goodis given by:
Yt =

∙Z

¸1/ρ

(xit )ρ di6, 0 < ρ < 1,

where xit is the amount of the intermediate good i at date t.6
The final good producers sell their products in a competitive market.
Taking the prices of the intermediate goods {pit } as given, a (representative)
final good producer chooses {xit } to maximize profits:
max

∙Z

¸1/ρ

(xit )ρ di−Z

pit xit di.

The first order necessary conditions for profit maximization imply that thedemand function for ith good at date t is:pit = Yt1−ρ (xit )ρ−1 .

(1)Intermediate goods are produced by monopolistically competitive producers. Specifically, the production function of the intermediate good i isgiven by:xit = hit (kti )λ , 0 < λ < 1,(2)

where hit is the human capital of the producer i at date t and kti is thephysical capital employed by the producer i at date t. This production
function captures the idea of the span of control a la Lucas (1978) in thesense that the marginal product of capital declines as capital accumulates.
Hence, it is not productive for one manager to manage all capital.

Notethat the larger is λ, the larger is the marginal product of capital. That is, alarger λ allows the manager to productively operate more capital. Hence,
λ measures the degree of the span of control.
For a given (gross) interest rate rt and his/her own human capital hit ,the intermediate good producer i chooses {kti } to maximize his/her profits:

6

max pit xit − rt kti ,The elasticity of substitution of intermediate goods is given by 1/(1 − ρ) > 1.

7

subject to the demand function (1) and the production function (2). From
the first order necessary conditions, we have the demand for physical capitalby the producer i:

∙

kti =

¸

1

λρ 1−ρ i ρ 1−λρ
Y
(ht )
,
rt t

and thus the profit function of the producer i is given by:
π it = (1 − λρ)

µ

λρ
rt

¶ λρ

1−λρ

1−ρ

ρ

Yt 1−λρ (hit ) 1−λρ .

(3)Note that for a given interest rate and total output, the producer i
produces the intermediate good i by the following amount:
xit =

∙

¸

λ

1
λρ 1−ρ 1−λρ i 1−λρ
Yt
(ht )
,
rt

and thus the total output of the economy at date t is given by:
Yt =where Ht =hRρi 1−λρ

(hit ) 1−λρ di

ρ

µ

λρ
rt

¶ λ

1−λ

1

Ht1−λ ,

(4)is the aggregate level of human capital.

Note that if the elasticity of substitution in intermediate goods in the final good production and/or the degree of the span of control are so large thatρ11−λρ > 1 (i.e., ρ > 1+λ ), Ht is increasing in the variance of human capital.

One of the immediate implications of this observation is that the heterogeneity in human capital across intermediate good producers is a source ofgains from society’s point of view if the intermediate goods are less complementary in the production of the final good, and/or the degree of the spanof control is large enough.
Let Πt denote the total profit of the intermediate producers, that is,Πt ≡R

π it di. The relationship between total output and total profit is Πt =(1 − λρ)Yt . Hence, the share of physical capital is 1 − λρ, and the share ofintermediate good producers is λρ.
82.2

Preferences and Human Capital Formation

There is a continuum of overlapping-generation families i of unit measure.
In each period, each family consists of one child, one young adult and one
aged person. Each individual lives for three periods. In the first period,
children obtain an education while they are still supported by their parents.
In the second period, young individuals work, spend part of their income forcurrent consumption and the education of their children, and save the rest.
In the third period, old individuals consume their savings.
Human capital is formed with individual ability ξ it+1 (i.i.d. with mean1), and public and private schooling. It is assumed that a child must spend
1 − θ fraction of his/her time at a public school and θ fraction of his/hertime at a private school. The production function of human capital is givenby:hhit+1 = (ξ it+1 )φ (Rti )θ (Ut )1−θiα

, φ > 0, 0 ≤ θ ≤ 1, 0 < α < 1,

where Ut and Rti are the quality of public and private education, respectively.
The quality of schooling consists of education expenditure and the averageability (that is, a measure of the quality of a peer group) in each type ofschool:P ub

P ri,iUt = (ξ̄ t+1 )ν gt , Rti = (ξ̄ t+1 )ν eit , ν ≥ 0,
P ub

where ξ̄ t+1 is the average ability of students in public school, gt is the public
P ri,ieducation expenditure per pupil, ξ̄ t+1 is the average ability of students inthe private school the child of household i attends, and eit is the tuition costof this private school. A similar specification of human capital formation
can be found in, for example, Bénabou (1996), Epple and Romano (1998),
Nechyba (2000, 2003), Epple, Newlon and Romano (2002), and Brunello,9Giannini and Ariga (2004).
The quality of a peer group in each type of school crucially depends
on the feasibility of sorting based on student ability. As a benchmark, we
consider the case in which private schools are allowed to choose students
based on their ability (and thus students are perfectly sorted in school byability), but public schools are not. This is a standard case studied by, forexample, Epple and Romano (1998), and we think that this is a plausible
case because private schools have more incentives and advantages to screen
students through their entrance exams than public schools. However, we will
discuss how our results change if public schools are allowed to sort students
based on ability in a later section.
The quality of a peer group in each type of school is determined asfollows. On the one hand, since students are randomly assigned to a publicschool, the average ability in each public school is the same as the population
P ubmean ( that is, ξ̄ t+1 = 1). On the other hand, since students are assumed tobe sorted in the private school by ability, the average ability of students in a
P ri,iprivate school for students with ability ξ it+1 is given by ξ̄ t+1 = ξ it+1 . Hence,
the reduced form of the production function of human capital is given by:³

hit+1 = (ξ it+1 )φ+αθν (eit )θ (gt )1−θ

´α

.

(5)

The utility maximization problem of the young individual in household
i is characterized as follows. The young individual of household i chooses
consumption when young cy,it , his/her child’s private education expenditure
eit and consumption when old co,i
t+1 to maximize his/her utility:o,ii
max ln cy,it + β ln ht+1 + γ ln ct+1 ,10

subject to the budget constraint:
o,i
i
i
i
i
cy,i
t + et + st = (1 − τ t )π t , ct+1 = rt+1 st ,and the production function of human capital (5), where τ t ∈ [0, 1] is a tax
Rrate for public education and gt = τ t π it di is per capita public education
expenditure.

From this utility maximization problem, we have the optimal private
education and saving for the young individual i at date t:
eit =

αβθ
γ
(1 − τ t )π it , sit =(1 − τ t )π it .
1 + γ + αβθ
1 + γ + αβθHence, each household allocates constant fractions of his/her own disposable
income to each current and future consumption and the private educationof his/her child.

2.3

Physical Capital Market Clearing

The total supply of physical capital at date t + 1 (call it Kt+1 ) is given by
the total saving at date t:
Kt+1 ≡

Z

sit di =Zγ(1 − τ t ) iγ(1 − τ t )π di =
Πt .
1 + γ + αβθ t
1 + γ + αβθ(6)On the other hand, the total demand for physical capital at date t + 1 isgiven by:
Z

i
kt+1
di =

Z ∙

¸

1

λρ 1−ρ i ρ 1−λρλρYt+1 (ht+1 )di =Yt+1 .
rt+1
rt+1

Hence, the market-clearing interest rate at date t + 1 is given by:
rt+1 = λρ

1 + γ + αβθ −1
Yt+1
Πt .
= λρYt+1Kt+1γ(1 − τ t )

11

(7)

2.4

Choice of Tax Rate

Each young individual votes for a tax rate that maximizes his/her lifetime
utility. We assume that when voters choose their favorite tax rate, theytake the future interest rate as given.7 Hence, they vote for a tax rate that
resolves the trade-oﬀ between the benefit from and the tax burden for public
education.
The indirect lifetime utility of the young individual in the family i isgiven by:v(τ t ; π it , ξ it+1 , Πt , rt+1 ) = A(π it , ξ it+1 , Πt , rt+1 )+(1+αβθ+γ) ln(1−τ t )+(1−θ)αβ ln τ t ,where A(π it , ξ it+1 , Πt , rt+1 ) = (1+αβθ+γ) ln π it +βφ ln ξ it+1 +αβ(1−θ) ln Πt +hiγ ln rt+1 + ln (αβθ)αβθ γ γ /(1 + αβθ + γ)1+αβθ+γ .
The favorite tax rate of the household with income yi is given by:τ ∗t =αβ(1 − θ).
1 + αβ + γ(8)Note that the favorite tax rate is independent of household income (and anyother household characteristics). Although we assume that the tax rate ischosen by majority voting, equation (8) implies that τ t = τ ∗t is chosen byunanimity no matter what the income distribution. That is, equation (8)shows that the tax rate chosen in this paper is constant over time.8
7

In principle, the choice of tax rate aﬀects the next period interest rate through theaggregate human capital in the next period. Here we assume that voters ignore this eﬀectof tax choice.
8
If only the young individuals have the voting right, then τ ∗t is chosen by unanimity.
Even if both the young and the old individuals are franchised, τ t = τ ∗t is chosen by amajority because τ ∗t is always supported by at least half of the population (that is, all the
young individuals always support τ t = τ ∗t ).

12

3

Evolution of Income Distribution

In this section, we study the evolution of income distribution using the model
described in the previous section. In particular, we show that the heterogeneity in household income reduces the GDP of the next period, but theeﬀects of the heterogeneity in ability can be positive or negative dependingon the production structure. These results are crucial to understanding thecomparison of education systems in the next section.

3.1

Distribution Dynamics

To study the evolution of an income distribution, we track the dynamics ofthe distribution of (pretax) profit income of young individuals (π it ), because
this corresponds to the present value of lifetime (pretax) income in our model(hereafter, we call it simply “income”).
For the given total output in period t + 1, the tax rate and the total(profit) income in period t, and the income of the young individual i inperiod t, the income of his/her child (and thus the income of the young
individual i in the next period) π it+1 is given by:
π it+1 = (ξ it+1 )where B̄ ≡ (1 − λρ)³ λ(φ+αθν)ρ
1−λραθρ

1−ρ(1+λ)ρ[λ+α(1−θ)]
1−λρ(π it ) 1−λρ Yt+11−λρ Πtγ (αβθθ (1−θ)1−θ )α(1+γ+αβ)α+λ

´

ρ
1−λρ

B̄,

(9)

.

In order to characterize the dynamics of income distribution, we assume
that income and ability are distributed across individuals by the following
log-normal distributions: ln π it ∼ N (mt , ∆2t ) and ln ξ it+1 ∼ N (−σ 2 /2, σ 2 ).
Under these assumptions, income of the next period is also distributed by
the log-normal distribution ln π it+1 ∼ N (mt+1 , ∆2t+1 ). The mean and the13variance of log income in the next period are given by:
h

i

∆2t
1−ρ(1+λ) (αθρ)22 + λ + α(1 − θ)ρ2(1−λρ)hi 2
1−ρ(1+λ) ρ2 (φ+αθν)σ+(φ + αθν)ρ(1−λρ)2 − 1 2 + B̃,

mt+1 = (α + λ)mt +∆2t+1 =(10)

ρ2 (φ + αθν)2 2(αθρ)2σ+∆2 ,(1 − λρ)2(1 − λρ)2 t(11)

where B̃ ≡ (1 − λρ)/ρ ln B̄ + [ρ(1 + λ) − 1]/ρ ln(1 − λρ).
The evolution of income distribution is fully described by equations (10)and (11). However, in tracking aggregate growth, we shall focus on GDP, Yt .
For this purpose, we would like to transform equation (10) to the dynamics
of ln Yt . As ln Yt = mt + ∆2t /2 − ln(1 − λρ), we can rewrite the equation (10)as follows:
∙

¸

∙

¸

αθρ
∆2t
(φ + αθν)ρ
σ2−1+(φ+αθν)−1+B,
ln Yt+1 = R ln Yt +αθ
1 − λρ
2
1 − λρ
2(12)
where B ≡ B̃ + (R − 1) ln(1 − λρ) and R ≡ α + λ. In the rest of this paper,
we mostly analyze the property of equation (11) and (12).

3.2Transition

We first analyze the eﬀect of diversity of human capital on GDP during
the transition process, and later discuss the same eﬀect when the aggregate
economy reaches a steady state.

Transition dynamics of GDP are char-acterized by equation (12). Results from direct investigations on (12) aresummarized by Proposition 1.
Proposition 1: For given Yt ,1. Yt+1 is strictly increasing in σ 2 if and only if ρ > 1/(φ + αθν + λ).
14

αθρ
2. Yt+1 is strictly decreasing in ∆2t if and only if 1−λρ
< 1.

The first part of Proposition 1 says that heterogeneity in ability may
have positive eﬀects on the next period’s GDP. The variance of ability canaﬀect the next period’s GDP through the following two channels. The firstchannel is ability tracking.

If ability tracking has substantial peer groupeﬀects, the overall variance of ability in the economy aﬀects the average levelof the next period’s human capital. In particular, if the elasticity of peergroup eﬀects in the private school ν is so high that the total elasticity ofhuman capital to ability (φ + αθν) is larger than unity, the human capital
accumulation function is convex in ability and therefore the next period’s
average human capital is increasing in the variance of ability.
The second channel is the span of control. Remember that a large spanof control gives an individual the authority to reallocate a large amount of
resources, and thus a large span of control favors workers with very highability. If the degree of the span of control is large (and complementarity
among produced goods is relatively small), the diversity of ability increasesρ> 1, aGDP. This eﬀect is reflected in equation (4). It shows that if 1−λρlarge variance in human capital increases GDP.
Consequently, if the “total” elasticity of income to ability ((φ+αθν)ρ/(1−λρ)) is larger than unity, the variance of ability has positive eﬀects of thenext period’s GDP.9
The second part of Proposition 1 shows that the eﬀect of inequality inαθρ.
current income on GDP at the next period depends on the parameters 1−λρ

The mechanical reason can be understood by equation (9). Note that the9Note that the variance in ability raises the next period’s GDP even if GDP is decreasing in the variance of human capital when ρ ∈ (1/(φ + αθν + λ), 1/(1 + λ)). In this case,the first (positive) eﬀect dominates the second (negative) eﬀect.

15

left-hand side of the equation (9) is strictly concave in π it if and only ifαθρ/(1 − λρ) < 1.

Hence, the high variance of π it reduces the expectedvalue of π it+1 , and, therefore, GDP, if and only if αθρ/(1 − λρ) < 1.
On the other hand, equation (11) shows that the variance of log income
converges to a finite value if and only if αθρ/(1 − λρ) < 1.

Hence, thefollowing corollary is the immediate result.
Corollary 1: Suppose that the variance of log income does not diverge.
An increase in current income inequality lowers GDP at the next period.
This negative eﬀect of the variance of income on GDP is interpreted as
the result of the concavity of the production function of human capital withrespect to household income, as in the previous literature.

For example,

Loury (1981) and Fernandez and Rogerson (1998) derive the same results
when the production function of human capital is concave in current income
and the production function of consumption goods is linear in human capital.
In their setting, the mapping from current income to next-period income
is concave and, therefore, their income dynamics converge to a stationary
process.
Diﬀerent from the previous literature, our production function in the
consumption goods sector can be strictly convex in human capital if the
span of control in a firm is large enough. It provides a possibility that theincome in the next period can be a strictly convex function of current income. Nonetheless, Corollary 1 implies that the concavity of the production
function for human capital must outweigh the convexity of the production
function for consumption goods when the variance of log income converges
to a finite value.
An intuitive reason can be understood as follows. Whenever an increase16

in inequality in current income always raises the average income at thenext period, the mapping from current income to next income must exhibit
convexity. However, the convexity of the mapping implies that the current
rich are likely to receive disproportionately high income at the next period.
This means that inequality at the next period is larger than that in thecurrent period. Hence, the variance of log income is infinite in the long run.
In other words, insofar as we restrict our attention to the income dynamics
that have a finite variance in the long run, the mapping from the current
income to the future income must exhibit concavity for some income level.10

3.3

Steady State

Next, we examine the relationship between the steady-state variance in income and steady-state GDP. For this purpose, we must know when two
aggregate state variables, ln Yt and ∆2t , converge to steady states.

Equa-tion (11) shows that the variance of log income converges to a stationaryαθρ< 1.
point if and only if 1−λρ

Provided that the variance of log income

converges to a stationary point, equation (12) shows that the logarithm of
GDP converges to a stationary point if and only if R < 1. Note that R < 1αθρ< 1. This reasoning proves the following lemma.
guarantees 1−λρLemma 1: An aggregate economy converges to a stationary distribution
if and only if R < 1.
Since we would like to analyze the relationships among income inequality,
10
Technically, we obtain Corollary 1 because local concavity (around a steady state)
implies global concavity in equation (9). This is a standard assumption in this literature
and we can still apply Corollary 1 when a deviation from the assumption is small. However,
if a mapping from current income to future income exhibits a more general form, we
may need additional assumptions on the variance of ability to guarantee the negative
relationship between current inequality and future GDP.

17

the heterogeneity in innate ability, and GDP in the long run, we assume
R < 1 in the rest of this paper.11 As the variance of log income has a finite
value on the stationary distribution, an increase in inequality in current
income lowers GDP at the next period during the transition process. We
want to show how the result changes in the long run.
Using equations (11) and (12), the logarithm of GDP in the steady stateis given by:
ln Y∞ =∙¸

1
1
D(ρ; α, λ, θ, ν, φ)∆2∞ + ln B̃ ,1 − R 2ρ2 (φ + αθν)where:
D(ρ; α, λ, θ, ν, φ) = ρ2 (αθ(αθ−φ−αθν)−λ(φ+αθν+λ))+ρ(φ+αθν+2λ)−1,
(13)
and
∆2∞ =

ρ2 (φ + αθν)2σ2 .
(1 − λρ)2 − (αθρ)2

If D > 0, then the greater is the variance of income, the higher the long-run
GDP (and vice versa). The eﬀects of income inequality on steady-state GDP
is summarized as Proposition 2.
Proposition 2: (1) If λ < 1−φ+(1−ν)αθ, the higher is the steady-statevariance of income, the lower is the steady-state GDP. (2) If λ > 1 − φ +(1 − ν)αθ, the higher is the steady-state variance of income, the higher is thesteady-state GDP if and only if ρ ∈ (ρ, 1) where ρ = 1/(φ + αθ(ν − 1) + λ).
This proposition tells us that when the production of the final goodis suﬃciently substitutable and the degree of the span of control is large
11
Even if we allow endogenous growth (that is, R = 1), the results regarding transition
dynamics in the rest of the paper remain unchanged. Almost all results from the steadystate analysis also can be applied to the analysis of the long run growth if we relabel
steady-state GDP to “the long-run growth rate”.

18enough, the relationship between income inequality and GDP is positive inthe steady state. Note that, similar to the case of transition, the varianceof income aﬀects the next period’s GDP negatively.

However, since thesteady-state variance of income is determined by the variance of ability,
which can have a positive impact on GDP, the total association between
income variance (and hence the variance of ability) and GDP in the steadystate depends on the combination of these two eﬀects. When the final good
production is suﬃciently substitutable and the degree of the span of control
is large enough, the positive eﬀect of the high variance in ability dominates
the negative eﬀect of the variance of income.
In summary, the (causal) eﬀect of income inequality on GDP is negativein our stationary economy, but heterogeneity in ability can have positiveeﬀects on GDP due to ability tracking and the span of control. Moreover,the long-run association between income inequality and GDP can be positiveor negative depending on the production structure of the economy.

4

Public vs Private Education System

In the benchmark model, we allow both public and private education investment in human capital accumulation (when θ ∈ [0, 1]). However, comparison
between public and private education systems may be of interest becausesuch comparison is useful to identify the advantages and disadvantages ofeach education system. In the literature of education macroeconomics, many
researchers study the costs and benefits of each education system throughsuch comparison (e.g., Glomm and Ravikumar 1992, Bénabou 1996, Fernandez and Rogerson 1998, and so on). In this section, we also study thecomparison between the public and the private education systems in the19short run and in the long run.

4.1

Short Run

Imagine that the economy with the distribution of household income ln π it ∼N (mt , ∆2t ) is about to choose between the private (θ = 1) or the public(θ = 0) education system. Starting from this income distribution, the next
period’s GDP is given by the equation (12). The diﬀerence of the next
period’s GDP under the two systems is given by:
P ub
P ri
− ln Yt+1= −α
ln Yt+1

µ

¶

∙

¸

αρ
∆2t
ρ(2φ + αν)
σ2
−1
− αν
−1
, (14)
1 − λρ
2
1 − λρ
2

P ub and Y P ri are the next period’s GDP under the public (withwhere Yt+1
t+1

θ = 0) and the private system (with θ = 1), respectively.
Under the assumption that the aggregate economy converges to a steadystate in the long run, the first term of this equation is always positive. Thisis because income inequality reduces average human capital in the next
period in the private system, but this eﬀect is absent in the public system
as expenditure on education does not depend on households’ income in the
public system.
The total comparison between the public and the private systems crucially depends on the sign of the second term and the relative size of thevariance in household income and ability. This comparison is summarized
as Proposition 3.
Proposition 3: (1) If ρ ≤ 1/(2φ + αν + λ), the public system generates
higher GDP in the next period than the private system. (2) If ρ > 1/(2φ +αν + λ), the private system generates higher GDP in the next period thanσ2 .
the public system if and only if ∆2t < ν[ρ(2φ+αν+λ)−1]1−ρ(α+λ)20

Observe that the second term of the equation (14) is positive if ρ <1/(2φ + αν + λ). In this case, the public system is always better than theprivate system. Intuitively, if the production of the final good is complementary enough in terms of human capital, public education is better because
it generates low variance of human capital.
However, the second term of the equation (14) is negative if ρ > 1/(2φ +αν + λ). In this case, the private system has an advantage because the variance of human capital is a source of gains. As the private system generates
a higher variation in human capital than the public system, it generates ahigher output.
The total comparison of these two systems crucially depends on thevariance of household income when the production of the final good hasless complementarity. If the variance of household income is small enoughrelative to the variance of ability, the private system is better than the public
system because the positive eﬀect in the second term dominates the negativeeﬀect in the first term.

4.2

Long Run

Next, we compare steady-state GDP under the two systems. In the shortrun, in the comparison of education systems the initial variance of householdincome is taken as given and common to the two systems. However, inthe long run, the degree of income inequality depends on which education
system the economy adopts. Hence, we need to compare steady-state GDP
and income inequality under these two systems.
The diﬀerence in the steady-state GDP under the two systems is given

21

by:
P ub
P ri
−ln Y∞
=
ln Y∞

"

µ

¶

ri
αρ
∆2,P
1
∞
−1
− αν
−α
1−R
1 − λρ
2µ¶ri = {ρ2 (φ + αν)2 σ 2 }/{(1 − λρ)2 − (αρ)2 }. Similar to the case ofwhere ∆2,P∞transition, the first term is positive, and the sign of the second depends on
the production structure of the economy. We state the results of the longrun comparison between the public and the private system in Proposition
4.
Proposition 4: (1) When ρ ≤ 1/(2φ + αν + λ), the public systemgenerates a higher steady-state GDP than the private system. (2) Whenρ > 1/(2φ + αν + λ), the public system generates a higher steady-stateri > ν[ρ(2φ+αν+λ)−1] σ 2 .
GDP than the private system if and only if ∆2,P∞1−ρ(α+λ)Moreover, for each (ρ, α, λ, φ) satisfying the assumptions R < 1 and ρ >1/(2φ + αν + λ), there exists a unique ν ∗ (ρ, α, λ, φ) such that the private
system generates higher steady-state GDP than the public system if and only
if ν > ν ∗ (ρ, α, λ, φ).
The first part of this proposition states that the public system generates
higher GDP when the production of the final good is complementary enougheven in the steady state.
More interesting, the second part of this proposition tells us that when
the final good production is less of a complement the long-run comparison
of GDP under the two systems depends on the long-run variance under theprivate system. In this case, the private system generates higher steadystate GDP than the public if the elasticity of human capital production to
the quality of a peer group is large enough. Intuitively, when the elasticity of22#

ρ(2φ + αν)σ2−1,
1 − λρ
2(15)

human capital production to the quality of a peer group is large, the production function of human capital becomes more convex in individual ability,
and thereby sorting by ability in private schools contributes to generating alarge amount of human capital.

5

Ability Tracking in Public Schools

So far, we assume that the ability sorting occurs only in private schools.
Recently, however, ability tracking in public schools is one of the main issues
in the debate on education reforms (e.g., Lefgren 2004). In this section, we
first examine how our results on the system comparison change if we allow
for public schools choosing students based on their ability. Next, we compare
the public system with ability tracking to that without tracking in order toevaluate the eﬀect of ability tracking on GDP.

5.1

Public vs Private System

When the public school is allowed to select students based on ability, theaverage ability of students in a public school for students with ability ξ it+1
P ub,iis given by ξ̄ t+1 = ξ it+1 . The reduced form of the production function of³

human capital is given by: hit+1 = (ξ it+1 )φ+αν (eit )θ (gt )1−θ
the equations (14) and (15) become:
P ub
P ri
− ln Yt+1= −α
ln Yt+1
P ub
P ri
ln Y∞
− ln Y∞
=

"

µ

¶

´α

αρ
∆2t
−1
> 0,
1 − λρ
2µ¶. In this case,

#

ri
αρ
∆2,P
1
∞
−1
−α
> 0,
1−R
1 − λρ
2

ri = {ρ2 (φ + αν)2 σ 2 }/{(1 − λρ)2 − (αρ)2 }. Therefore, the public
where ∆2,P∞

system generates higher GDP than the private system both in transition
and in the steady state. In this case, the only diﬀerence between the public
23and the private system is their source of finance.

As a reduction in thecurrent variance of income increases GDP at the next period, the public
system generates a higher GDP than does the private.12

5.2

Tracking vs Mixing in Public Schools

Can ability tracking in public schools increase GDP? To answer this question, we compare GDP under the public system with ability tracking to that
under the public system without tracking.
The diﬀerence of the logarithm of GDP under these two education systems during transition is given by:
P ub,SP ub,M− ln Yt+1= αν
ln Yt+1
P ub,SP ub,Mln Y∞− ln Y∞="µ¶ρ(2φ + αν)σ2−1,
1 − λρ
2

1
αν
1−R

µ

¶

#

ρ(2φ + αν)σ2−1,
1 − λρ
2where YtP ub,S and YtP ub,M are GDP under the public system with and without ability tracking, respectively.

From these equations, we have the fol-lowing result:
Proposition 5: Ability tracking in public schools generates higher GDP
in both transition and steady state if and only if ρ > 1/(2φ + αν + λ).
From Proposition 5, we may conclude that ability tracking in publicschools increases GDP when the production of the final good is a suﬃciently
substitutable (large ρ), the degree of the span of control (λ) is large enough,the production function of human capital is less concave (large α), and/or
the elasticity of human capital to the peer group eﬀect (ν) is large enough.
12
In the case where neither public nor private schools are allowed to select students
based on ability, the public system generates higher GDP than the private system bothin transition and in the steady state.

24

In summary, when public schools are allowed to select students based on
their ability, the public system generates a higher GDP than the private bothin transition and in the steady state. However, whether the ability tracking
in public schools increases GDP crucially depends on the interaction between
the production structure of the final and the intermediate goods and thetechnology of human capital accumulation.

6

Concluding Remarks

In this paper, we examined how diﬀerent education systems change GDP
through their influence on the diversity of human capital. We analyze an
economy in which an income distribution converges to a stationary distribution. We show that the diversity of human capital due to income inequality
always lowers GDP, while diverse human capital induced by heterogeneous
ability can increase GDP if produced goods are suﬃciently substitutable andfirms have a large span of control. Hence, we may conclude that a public
education system always yields higher GDP than a private education system, though the eﬀect of ability tracking on GDP depends on the structureof industries and firms.
It is worth mentioning some cautions to the interpretation of our results.
We focus on the eﬀect of diﬀerent education systems on GDP. But obviously,
GDP is not the only variable that governments should be concerned. In ourmodel, both a private system and ability tracking increase the variance of log
income and lower social mobility. Hence, our model indicates that equalizing
school resources can also promote eﬃciency, but that ability tracking may
induce a serious trade-oﬀ between eﬃciency and equity in some societies.
One of messages from this paper is that if a government wants to know

25how serious the trade-oﬀ is, it must carefully investigate the structure ofindustries and firms.
Although we examine the eﬀect of ability tracking on GDP in this paper,
we can extend our analysis to other policies that can change a mapping fromability to human capital. More specifically, we can extend our analysis topolicies that influence the human capital accumulation function through³

the following path, hit+1 = (ξ it+1 )φ(θ,p) (eit )θ (gt )1−θ´α, where p is a policy

parameter. For example, when policy allows individuals more flexibility inchoosing how many credits they can take in a year, it is likely that talented
students take more courses than ordinary students do.13 In this case, moreable students accumulate a larger human capital not only because they are
talented, but also because they study harder. Hence, the eﬀect of ability
on human capital is more than proportional and φ (θ, p) becomes larger by
this policy. According to our analysis, the impact of this policy on GDP
depends on the structure of production. In this way, our analysis suggeststhat we cannot evaluate the economic consequences of a liberal education
policy without the knowledge of how industries and firms are organized.
As indicated by this example, there will be more unexplored issues about
the relationship between education policy and the structure of the production sector. We hope that this paper helps researchers in unraveling part ofthe complicated relationships.

13

The Japanese government substantially reduced the number of subjects that students
must study in elementary school, junior high school and high school during 1990s. According to Kariya (2001), as a result of this policy change, students who previously received
poorer grades reallocate their time more to leisure than those who previously received abetter grade.

26

A

ProofsA.1

Proposition 1

Proof. Obvious from the equation (12). Q.E.D.

A.2

Proposition 2

Proof. We first show (2) of the proposition. The second part of the proposition is true if D > 0 when ρ ∈ (ρ, 1) and D < 0 when ρ ∈ (0, ρ). Solving
equation D(ρ; α, λ, θ, ν, φ) = 0 for ρ, we have ρ = 1/(φ + αθ(ν − 1) + λ), ρ̄ =1/(λ + αθ). Since αθ(αθ − φ − αθν) − λ(φ + αθν + λ) < 0, φ + αθν + 2λ > 0
and D(0; α, λ, θ, ν, φ) = −1 < 0, D > 0 if and only if ρ < ρ < ρ̄. Under
the assumption λ > 1 − φ + (1 − ν)αθ, ρ < 1 < ρ̄. Therefore, D > 0 whenρ ∈ (ρ, 1) and D < 0 when ρ ∈ (0, ρ).
Next we show (1) in the proposition. For the first part, note that ρ > 1
if λ < 1 − φ + (1 − ν)αθ. This implies that D(ρ; α, λ, θ, ν, φ) < 0 for anyρ ∈ (0, 1). Therefore, the association between GDP and income variance inthe steady state is negative when λ < 1 − φ + (1 − ν)αθ. Q.E.D.

A.3

Proposition 3

P ub − ln Y P ri > 0.
Proof. (1) When ρ ≤ 1/(2φ + αν + λ), ln Yt+1
t+1
P ub − ln Y P ri < 0 if and only if(2) When ρ > 1/(2φ + αν + λ), ln Yt+1
t+1

σ 2 . Q.E.D.
∆2t < ν[ρ(2φ+αν+λ)−1]1−ρ(α+λ)A.4

Proposition 4

Proof. For the first part, observe thatP ub − ln Y P ri > 0.
αν + λ). Hence ln Y∞
∞

27

³

´

ρ(2+αν)1−λρ − 1< 0 when ρ ≤ 1/(2 +P ub − ln Y P ri > 0 if
For the first half of the second part, note that ln Y∞∞

ri > ν[ρ(2+αν+λ)−1] σ 2 .
and only if ∆2,P∞1−ρ(α+λ)Finally, we prove the last half of the second part. The long-run varianceof household income under the private system is given by:
ri
=
∆2,P
∞

ρ2 (1 + αν)2σ2.
(1 − λρ)2 − (αρ)2P ub − ln Y P ri > 0 if and only if:
Hence, ln Y∞∞ν[ρ(2 + αν + λ) − 1] 2
ρ2 (1 + αν)2σ ,σ2 >(1 − λρ)2 − (αρ)21 − ρ(α + λ)and this inequality is equivalent to:αρ(1 − λρ)ν 2 + [(1 − λρ)(ρ(2 + λ) − 1) + αρ(ρλ − α)]ν − ρ2 < 0.
P ub −
When ν = 0, the left-hand side (LHS) is negative and thus ln Y∞
P ri > 0. When ν → ∞, LHS is infinity and thus ln Y P ub − ln Y P ri < 0.
ln Y∞
∞
∞

Since αρ(1 − λρ) > 0 and LHS is negative when ν = 0, there exists
a unique ν ∗ (ρ, α, λ) > 0 such that LHS = 0 when ν = ν ∗ (ρ, α, λ) andP ub < ln Y P ri if and only if ν > ν ∗ (ρ, α, λ). Q.E.D.
ln Y∞
∞

A.5

Proposition 5

Proof.

³´

ρ(2φ+αν)1−λρ − 1

P ub,S > Y P ub,M ) if and only if> 0 (and thus Y∞∞

ρ > 1/(2φ + αν + λ). Q.E.D.

28

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30