OSIPP Discussion Paper : DP-2009-E-005

On the Role of Job Assignment in a Comparison of Education Systems

July 06, 2009
Katsuya Takii, Osaka University
Ryuichi Tanaka, Tokyo Institute of Technology【キーワード】Job assignment, Human capital, Education system,
D31 D72 H42 I22 O11 O15【要約】This paper reexamines how differences in systems for financing educationinfluence GDP by highlighting a neglected function of education policy: it affects
the magnitude of gains from job assignment. When more productive jobs demand moreskill, privately financed education can increase productivity gains from matching
between jobs and skill by increasing the availability of highly educated people.

This

differs from the standard argument that publicly financed education increases thetotal amount of human capital by equalizing educational opportunities. It is shownthat if job opportunities have large variations in productivity, education policy
may face a serious efficiency--equity trade-off.

We are grateful to the seminar participants at the Osaka Workshop on Economics ofInstitutions and Organizations and Tohoku University for helpful comments. Financial
support from Seimeikai and the Japanese Ministry of Education, Culture, Sports,
Science and Technology (Grant-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

A publicly financed education system redistributes educational resources from richto poor households through taxes and aims to provide equal opportunity for children
to receive schooling regardless of their family background. One of the main concerns
is whether or not this egalitarian education policy causes a serious trade-oﬀ between
eﬃciency and equity. The literature on the economics of education points out several
mechanisms through which the public education system increases or decreases grossdomestic product (GDP), and provides useful insights for understanding when andhow the trade-oﬀ emerges.
On the one hand, it is common in the literature to assume declining marginal returns to educational investment, which favors a publicly financed system (e.g., Loury
1981, Glomm and Ravikumar 1992, Bénabou 1996, 2002, Fernández and Rogerson
1998, and Takii and Tanaka 2009).1

If the marginal returns from investment in

human capital decline as agents spend more resources on education, the public system can increase the total amount of human capital in an economy, and therefore
GDP, by equalizing educational opportunities because the poor’s marginal returns toinvestment are higher than those of the rich.2
On the other hand, Glomm and Ravikumar (1992) and Takii and Tanaka (2009)
point out that there are disadvantages of a publicly financed education system in
1

To give a public education system the role of equalizing educational opportunities, it is also

common to assume there is a liquidity constraint or that parents have warm-glow altruism and paternalistic altruism so that education can be treated as a consumption good as well as an investment
good.
2This mechanism is fairly strong. Takii and Tanaka (2009) show that when the intergenerational
income elasticity is less than one, a public system with equal provision of resources to students yields
higher GDP than a private system, even if the aggregate production function is convex in individual
skill.

1

comparison with a privately financed education system.

Glomm and Ravikumar(1992) show that the private system can generate higher GDP than the public system
if a child can choose to exert eﬀort in the accumulation of human capital. Takii andTanaka (2009) find that if private schools are better able to screen students based on
their ability than public schools, there is a production structure in which the private
system attains higher GDP than the public system.
The purpose of this paper is to contribute to these discussions by highlighting
an important function of education policy neglected in the literature: it aﬀects the
distribution of educated people and thus the magnitude of gains from job assignment
in the labor market. Without exception, the existing literature assumes that there isan aggregated measure of human capital that summarizes all the impacts of individual
human capital accumulation on GDP. We argue that this assumption masks how an
education policy influences the gains from assignment.
To highlight the novel role of an education policy, we incorporate a standard positive assortative assignment model based on Sattinger (1979) into an overlappinggenerations model with human capital accumulation through tax-financed publicschooling or tuition-financed private schooling, abstracting away the eﬀects of effort level and ability tracking on the accumulation of human capital.

Under thissetup, the public system always yields higher GDP than a private education system
if there is no assignment problem in the labor market. It is shown that a private system can attain higher GDP because it increases productivity gains from assignment
by increasing the availability of highly educated people.
We emphasize the importance of three distinctive assumptions behind the positive
assortative assignment model, which are essential to understanding this result. The
first assumption is that the quality and the quantity of workers are not perfect substitutes.3 When the quality of workers can be substituted by the quantity of workers,
3

Sattinger (1980) discusses convincingly the importance of imperfect substitution between the2the returns from employing doubly productive workers are equivalent to those fromdoubling the number of workers regardless of the structure of the industry and firm.
In this case, the total productivity of the economy is influenced only by the total
amount of human capital. However, when the quantity and quality of workers are
not perfect substitutes, a firm may seek a highly talented expert rather than a large
number of ordinary workers. If there is no expert in an economy, the firm will incura huge loss. An education policy changes the availability of qualified workers, so it
can influence not only the total productive capacity of workers but also the patternof assignment between skill and productive job opportunities.
The second assumption is that job opportunities are heterogeneous.4

Without

heterogeneity among jobs, there are no gains from assignment that may potentiallyprovide advantages to a private education system. Among the papers on the comparison between public and private education systems, Bénabou (1996) and Takii andTanaka (2009) analyze models in which the quality and quantity of human capitalare not perfect substitutes. However, as they do not have any variation in jobs, theirmeasures of aggregate human capital summarize all the impacts of individual humancapital accumulation on GDP.
The third assumption is that jobs that are more productive demand more skill.
That is, job-specific productivity is complementary to skill. With this complemenquality and quantity of workers for assignment.

The importance of the imperfect substitution

between skilled workers and unskilled workers can also be understood from the arguments regarding
skill-biased technological change (e.g., Katz and Murphy 1992).
4
Considerable evidence suggests that job opportunities are heterogeneous.

For example, theliterature finds that there are substantial and persistent diﬀerences in productivity between plants
and between firms (e.g., Baily, Hulten and Campbell 1992). Researchers also find that the observed
and unobserved characteristics of a firm influence wage payments after controlling for the individual
eﬀect using matched employer—employee data (e.g., Abowd, Kramarz and Margolis 1999).

3

tarity, it is well known that talented workers are assigned to productive jobs.5 Hence,the existence of a few highly skilled workers causes disproportional impacts on aggregate productivity compared with that of ordinary workers because they are not
only productive, but can also utilize their skill to succeed in highly productive jobs.
Hence, the aggregate productivity might be larger by educating a few highly skilledworkers rather than a large number of reasonably trained workers. This mechanism
provides a novel channel, through which the private system increases GDP.
This paper shows that, in the short run, an increase in the variations in productivity across jobs (in the sense of mean preserving spread) increases the productivity
gains from a private education system more than those from a public education system. Because of the complementarity between the productivity of jobs and skill, theexistence of highly productive job opportunities significantly increases the marginal
5

There is plenty of suggestive evidence showing that more productive jobs demand more skill andthat there is positive assortative matching in the labor market. For example, evidence shows thatnew technologies are skill biased, which means that advanced technologies demand more skill (e.g.,
Krusell et al. 2000). Other factors also demand high levels of skill. Chandler (1977) demonstrates
that the modern organizational structure demands skilled workers to process information. Bresnahan, Brynjolfsson and Hitt (2002) also show that reorganization accommodated by IT investment
demands more skilled workers.

More directly, using a model similar to this paper, Takii (2008)shows that the assumption of complementarity between firm-specific productivity and skill is consistent with observed persistent productivity diﬀerences across firms.

Recently, researchers have

tried to find more direct evidence using matched employer—employee data. Although controversial
evidence is found by some researchers (e.g., Abowd, Kramarz and Margolis 1999), other researchers
criticize the identification strategy used by these authors (e.g., Eeckhout and Kircher (2009)) anddevelop more sophisticated methods to examine empirically the positive assortative matching hypothesis in a labor market. Among them, Mendes, Van den Berg and Lindeboom (2007) and Lopes
de Melo (2008) provide evidence based on their methods that supports a positive assortative matching hypothesis. At least, there is no evidence against a positive assortative matching hypothesis if
observable characteristics are not controlled for, which is more relevant in this paper.

4

product of skill. Hence, an increase in the availability of highly skilled workers improves aggregate productivity. As the private education system can produce more
highly qualified workers than the public system, an increase in the variations in productivity increases the relative advantage of private education.
This result may not be true in the long run. As the large productivity diﬀerences
bring better opportunities for advantaged people, it increases income inequality inthe long run, which may make income transfer by a public education system more
attractive. Nonetheless, it is shown that the private system generates higher long-run
GDP than the public system does if jobs have suﬃcient variations in productivity.
Hence, if jobs have large variations in their productivity, a serious eﬃciency—equity
trade-oﬀ may occur even in the long run.
As well as contributing to the literature on the comparison of education systems,
this paper also contributes to the literature of positive assortative assignment models. Positive assortative assignment models have a long history (e.g., Becker 1973,
Sattinger 1979, and Kremer 1993 ).

However, these papers typically assume theexistence of exogenous distributions of assigned variables. More recently, researchers
have started to endogenize the distribution (e.g., Acemoglu 1997, Jovanovic 1998,
2009 and Takii 2008). In particular, similarly to this paper, Acemoglu (1997) and
Jovanovic (1998, 2009) endogenize the distribution of skill. However, no paper analyzes how an education policy influences the gains from assignment.
The rest of this paper is organized as follows. Section 2 describes the model.
Sections 3 and 4 compare the two education systems in the short run and in the longrun, respectively. Section 5 discusses some extensions and concludes.

5

2

The Model

In this section, we incorporate positive assortative job assignment between jobspecific productivity and the human capital of workers into a standard overlappinggenerations model with an educational choice. We first describe the production sideand solve the static equilibrium, and then model the investment in human capital.

Static Equilibrium: Our static assignment model is a standard one, which was
originally analyzed by Sattinger (1979). The labor market of the economy is represented by a continuum of workers and jobs in each period. Although this paper callsa unit of production “a job”, a job can alternatively be interpreted as a firm with one
unit mass of workers.
The populations of both jobs and workers are normalized to one. Each job consistsof job-specific productivity6 , z, the logarithm of which is normally distributed withmean µz and variance σ 2z : ln z ∼ N(µz , σ 2z ). The distribution of z is assumed to beconstant over time. We interpret a large z as a result of large fixed specific resourcesthat workers can use to make the job productive. Hence, the variance of ln z, σ 2z ,can be interpreted as the variability of job opportunities.
Each worker is characterized by his/her human capital, ht , which is also lognormally distributed: ln ht ∼ N(µh,t , σ 2h,t ). Note that the distribution of ln ht is afunction of time, t. We endogenize this distribution later.
Assume that a job with productivity z employs one worker with human capital
ht , and produces y(z) by a production function, y(z) = zhυt ,where υ > 0. This
production function exhibits the complementarity between z and ht . For a simple
6

Sattinger (1979, 1980) interprets his model as positive assortative matching between the skill ofworkers and a firm’s real physical capital.

6explanation, we assume that jobs are perfect substitutes in a firm. In this case,the firm’s profit maximization problem is equivalent to the job’s profit maximization
problem. As discussed later, this assumption can be relaxed without any changes inour result. The job chooses the human capital of a worker to maximize the profit:χt (z) = arg max {zhυt − wt (ht )} ,

(1)

ht

where wt (h) is the wage payment for the workers with h at date t. Note that thejob determines the quality of workers but not the quantity. This means that thequantity of workers cannot be substituted for the quality.

This is the assumptionthat deviates from standard human capital theory.
As the production function exhibits complementarity between job opportunities
and skill, it is well known that job matching is positively assortative (e.g., Becker1973). We characterize a positive assortative matching equilibrium. Let Φ(•) denotea standard normal distribution. Because ln z ∼ N(µz , σ 2z ) and ln ht ∼ N(µh,t , σ 2h,t ), a
positive assortative equilibrium implies that:µ¶µ¶ln χt (z) − µh,t
ln z − µz
1−Φ=1−Φ, for all z ≥ 0.
σz
σ h,t

(2)

This equation means that a job in the top x percentile of z must select a worker inthe top x percentile of h. For the sake of simplicity, we assume that jobs and workers
have a reservation wage and profit of zero, respectively.

As the number of jobs isequal to the number of workers, nobody chooses the outside option and every agent
can find a partner. Hence, a static equilibrium is fully characterized by χt (·) andwt (·) satisfying equations (1) and (2).
Because the function χt (·) must satisfy the equation (2),¡¢σ z ln ht − µh,t−1
+ µz .
ln χt (ht ) =
σ h,t
³ ´
ĥ . This job requires a worker with ĥ by definition
Consider a job with ẑ = χ−1tof χt (·).

This means that if the maximization is characterized by the first- order7

condition, the following condition must be satisfied in the equilibrium:³ ´³ ´³ ´ ³ ´υ−1 υ y̌t ĥ
wt0 ĥ = υχ−1
ĥ ĥ
=
,
t
ĥ
³ ´
³ ´
n
´³
´o
³
σz +υσh,twhere y̌t ĥ = y(χ−1
ĥ
)
=
exp
µ
ln
ĥ
−
µ
. Because
+
υµ
+
t
z
h,t
h,t
σh,t

the reservation wages of all workers are equal to zero and the choice of ĥ is arbitrary,
we can solve the wage function by integrating wt0 (h) with respect to h:³ ´ Z ĥwt0 (s) ds = LSt y̌t (ĥ),
wt ĥ =

(3)

0

υσ

h,t
is the labor share in production. As this wage function guarwhere LSt = σz +υσh,t³ ´antees that the problem (1) is a convex problem for the job with ẑ = χ−1ĥ , thetfirst-order condition fully characterizes the job’s profit maximization problem. Hencethe derived χt (·) and wt (·) satisfy the static equilibrium conditions (1) and (2), which

proves the existence of the equilibrium.
We can derive the GDP of the economy, Yt , as follows:
Yt = E [zhυt ] = E (z) E (hυt ) ASt ,

(4)

³
´
´
³
2
υ2 σ2
where E (z) = exp µz + σ2z , E (hυt ) = exp υµh,t + 2h,t and ASt = exp [υσ z σ h,t ].

Note that GDP can be decomposed into three components: average productivity,
E (z), aggregate human capital E (hυt ) and the productivity gains from assignment,

ASt . Note that an increase in σ h,t increases ASt . Because job-specific productivity
and the skill of workers are complementary to each other, the matching of highly
productive jobs and exceptionally high skill creates disproportional impacts on productivity.

As an increase in σ h,t means that more extremely talented workers areavailable, it increases the productivity gains from assignment. This is the mechanism
by which a private education system can increase aggregate productivity.
Note that if σ z = 0, Y = E (z) E (Ht ) and w (ht ) = E (z) Ht , where Ht = hυt .
Hence, the result is equivalent to that of a linear production function with perfectly
8substitutable human capital Ht . This is the standard production function consideredin previous studies (e.g., Glomm and Ravikumar 1992).

It means that, without

the heterogeneity of jobs, there are no gains from assignment and aggregate humancapital can summarize all the impacts of human capital accumulation on GDP. In
the following section, we take σ z = 0 as the benchmark case and investigate how theresults change when σ z > 0.

Households: There is a continuum of overlapping-generation families i, the total
mass of which is also normalized to unity. 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 theirparents. In the second period, young individuals work, spend part of their income
on current consumption and the education of their children, and save the rest. In
the third period, old individuals consume their savings.
Human capital is formed according to individual ability ξ it+1 (i.i.d. with mean1), public education expenditure per pupil gt and private education expenditure per
pupil eit . Public education is financed by the proportional wage income tax: gt =Zτ t w(hit )t di where τ t ∈ [0, 1] is the tax rate. The production function of humancapital is given by:£¤α
hit+1 = ξ it+1 (eit )θ (gt )1−θ , θ ∈ {0, 1} , α ∈ (0, 1),(5)where θ = 0 means a purely public education system and θ = 1 means a purely
private education system.

This is a standard assumption in the literature (e.g.,

Glomm and Ravikumar 1992, Bénabou 1996, Fernandez and Rogerson 1998, andTakii and Tanaka 2009).

When θ ∈ {0, 1}, public education diﬀers from private

education only in its financial system: the costs of education are financed by tuitionin the private education system and by taxes in the public education system. Hence,9

we can ignore technical diﬀerences that have nothing to do with the definition of
public education and private education.
The utility maximization problem of the young individual in household i is characterized as follows. The young individual of household i chooses consumption wheniyoung cy,it , his/her child’s private education expenditure et and consumption whenold co,i
t+1 to maximize his/her utility:o,ii
ln cy,it + βE(ln ht+1 ) + γ ln ct+1

subject to the budget constraint:
o,i
i
i
i
i
cy,i
t + et + bt = (1 − τ t )wt (ht ), ct+1 = Rt+1 bt

and the production function of human capital (5), where Rt+1 is the gross market
interest rate and E(·) denotes the expectation operator.
From this utility maximization problem, we have the optimal private educationand savings for the young individual i at date t:eit = λ(θ)(1 − τ t )wt (hit ),γ(1 − τ t )wt (hit ),bit =1 + γ + αβθ

(6)

αβθ
. Hence, each individual allocates constant fractions of his/her
where λ(θ) = 1+γ+αβθ

own disposable income to each current and future period’s consumption, and theprivate education of his/her child only under the private education system.

Stock Market: Let Bt denote total savings at date t, that is:
Bt =γ
LSt Yt .
1 + γ + αβθThe young individual invests all his/her savings in an index fund. The index consistsof all stocks issued by all the firms. The total return from the index fund at date t
10

is equal to the total dividend plus the value of the index fund at date t + 1. As thetotal savings must equal the value of the index fund at each period, the stock market
clearing condition implies that:
Rt+1 Bt = Bt+1 + Πt+1 ,where Πt+1 = (1 − LSt+1 )Yt+1 is the total profits earned by all jobs at date t + 1, and
Rt+1 is the gross rate of return from the index fund. This market clearing condition
determines endogenously the gross rate of return Rt+1 .

Choice of Tax Rate: Each young individual votes for a tax rate that maximizes
his/her lifetime utility.7 Following Takii and Tanaka (2009), we assume that when
voters choose their favorite tax rate, they take the future interest rate as given.8
7

If the young individuals have the right to vote, then τ ∗ is chosen by unanimity. Even if bothyoung and old individuals are enfranchised, τ t = τ ∗ is chosen by a majority because τ ∗ is always
supported by at least half of the population (that is, all the young individuals always support
τ t = τ ∗ ). Moreover, our model and the results are fully robust to the presence of positive population
growth if job opportunities grow at the same rate and keep the same distribution. With positive
population growth, the young are a strict majority, and thus τ t = τ ∗ is chosen in equilibrium even
if the old are also enfranchised.
8
In principle, the choice of tax rate aﬀects the next period’s interest rate through the aggregate
variables. Here, we assume that voters ignore this eﬀect of tax choice. Some may wonder whether
this ignorance contradicts the assumption of rationality. However, to estimate the impact of tax on
the interest rate, a household must calculate how other households in the current generation respondto the changes in the tax rate, how their responses change the distribution of human capital in the
future generation, how the changes in the distribution of human capital influence the future tax ratethat is voted on by future generations, and so on. These calculations allow a household to computethe changes not only in current aggregate savings, but also in future aggregate savings and future
aggregate profits that determine the interest rate. We find that this process requires unreasonablerationality from agents and it is realistic to assume that agents take the future interest rate asgiven. Alternatively, we could assume a small open economy to derive our results without relying

11Hence, they vote for a tax rate that resolves the trade-oﬀ between the benefit fromand the tax burden for public education.
The objective function (that is, indirect lifetime utility) in the tax choice problemof the young individual in family i is given by:
∙
¸
γ
i
i
i
ln (1 − τ t )w(ht )t − λ(θ)(1 − τ t )wt (ht ) −(1 − τ t )wt (ht )1 + γ + αβθ
µ ∙
µ
¶α ¸¶
Z
i
i θ
i
1−θ
+βE ln ξ t+1 (λ(θ)(1 − τ t )wt (ht )) (τ t wt (ht )di)¸∙γi
(1 − τ t )wt (ht ) .
+γ ln Rt+1
1 + γ + αβθThe favorite tax rate of the household with labor income wt (hit ) is given by:τ ∗t =(1 − θ)αβ≡ τ ∗ (θ).
1 + γ + αβ(7)Hence, τ ∗ (θ) is chosen by unanimity.

3

Public vs. Private Education System in the ShortRun

In this section, we derive the dynamics of GDP and examine how a change in the education system influences GDP dynamics. It is shown that an increase in σ 2z increases
the benefits from private education.
Substituting equations (3), (6) and (7) into equation (5), we can derive the following dynamics of human capital of the dynasty i.

ln hit+1¶¢
σ z + υσ h,t ¡ln hit − µh,t + ln ξ it+1
= αθ
σ h,t
£
¤
+α ln LSt + αθ µz + υµh,t + α(1 − θ) ln Yt + C (θ) ,µon this controversial assumption. This alternative assumption makes the interest rate constant andtherefore removes the eﬀect of tax on the interest rate without changing our results.

12

i
h
where C (θ) = ln (λ(θ)(1 − τ ∗ (θ)))αθ (τ ∗ (θ))α(1−θ) .

Assume that ability is i.i.d.

drawn from the log-normal distribution ln ξ it+1 ∼ N(−σ 2a /2, σ 2a ).

Because ln hit ∼N(µh,t , σ 2h,t ), the human capital of the next period is also log-normally distributed:
ln hit+1 ∼ N(µh,t+1 , σ 2h,t+1 ) where¸α(1 − θ) (σ z + υσ h,t )2 σ 2a
υσ h,t+− +αµz +ln C(θ), (8)µh,t+1 = αυµh,t +α ln
σ z + υσ h,t
2
2
∙

and
σ 2h,t+1 = σ 2a + α2 θ2 (σ z + υσ h,t )2 .

(9)These two equations fully describe the dynamics of the distribution of ln ht . Substituting equations (8) and (9) into equation (4), we can derive the dynamics ofthe logarithm of GDP and the standard deviation of log income as functions of theparameter of the education system θ.
Proposition 1 The dynamics of ln Yt+1 and σ w,t+1 can be given as functions of θ:£¤ln Yt+1 = ln E (z) + ln E hυt+1 + ln ASt+1 ,¤£ln E hυt+1 = αυ ln wte + HCt + C (θ) ,ln wte = ln LSt + ln Yt

and
σ w,t+1 = σ z + υσ h,t+1 = σ z + υ
where
ln ASt+1q
σ 2a + α2 θ2 σ 2w,t ,q= υσ z σ h,t+1 = υσ z σ 2a + α2 θ2 σ 2w,t ,σ 2w,tσ2,
ln HCt = −υ (1 − υ) a − αθυ(1 − αθυ)
2¶
2
µ
σz
.
ln LSt = ln 1 −
σ w,t

13

(10)
(11)
(12)

(13)

(14)
(15)
(16)

Equation (10) shows that Yt+1 can be decomposed into average productivity E (z),£¤aggregate human capital E hvt+1 , and the productivity gains from assignment ASt+1 .

Equation (11) shows that aggregate human capital is decomposed into the contribution of average labor income wte , the reduction in aggregate human capital becauseof the heterogeneity of income and ability HCt , and other constants C (θ). Finally,equation (12) shows that average labor income increases if either current GDP Yt or
labor share LSt increases. These three equations describe the dynamics of Yt given
the dynamics of σ w,t . The dynamics of σ w,t are determined solely by equation (13),which shows that the variation of labor income depends not only on the variation ofhuman capital, but also on that of job-specific productivity.
The component HCt summarizes a reduction in aggregate human capital becauseof the heterogeneity of income and ability. To guarantee the existence of a globallystable stationary distribution, it is shown later that we must assume αυθ < 1. Given
this assumption, equation (15) shows that a rise in σ w,t lowers the aggregate measure£¤of human capital E hυt+1 . This is the standard eﬀect in previous studies: an increase

in inequality lowers aggregate human capital.

The novel part of this paper is summarized by the component ASt+1 . It captures
the productivity gains from assignment. As argued before, a rise in σ h,t+1 increases
this gain. Because a rise in σ w,t increases σ h,t+1 , it might be possible that a rise ininequality raises GDP in the next period.
Next, we identify the benefits and costs of an egalitarian education policy using
the dynamics derived above. To do so, we compare a purely public education system(θ = 0) with a purely private education system (θ = 1), following the previous
literature such as Glomm and Ravikumar (1992), Fernandez and Rogerson (1998),
and Takii and Tanaka (2009).
Imagine an economy with the distribution of household income ln wt (hit ) ∼ N(µw,t , σ 2w,t ).

14

Suppose that the government of the economy is about to choose either the purely private (θ = 1) or the purely public (θ = 0) education system.

Starting from thisincome distribution, the diﬀerence in the next period’s GDP under these two systems
is given by:
£
¤ £
¤
Pr i
P ub
ln ASt+1− ln ASt+1+ ln HCtPr i − ln HCtP ubhqi αυ (1 − αυ) σ 2w,t,
σ 2a + α2 σ 2w,t − σ a −= υσ z2

P ri
P ub
ln Yt+1− ln Yt+1=(17)

where XtP ub and XtP ri are Xt under the public education system (θ = 0) and theprivate education system (θ = 1), respectively. Note that as C (0) = C (1), C (θ) is
canceled out in equation (17).
The first term of this equation captures the productivity gains from assignment,
and the second term captures human capital loss because of heterogeneous income.
On the one hand, the productivity gains from assignment are always positive because
σ h,t+1 is larger under the private education system. On the other hand, the private
education system always suﬀers from human capital loss, because income inequality
reduces aggregate human capital in the next period.

Whether or not the private

education system increases GDP depends on the strength of these two opposite eﬀects.
The following proposition is immediate from equation (17).
Proposition 2 The relative advantages of the private education system depend on
σ z in the following sense.
1. When σ z = 0, the public education system always generates larger GDP thanthe private system does.
2. A rise in σ z increases the relative advantages of the private education system:
¡
¢
i
hq
P ri
P ub
d ln Yt+1− ln Yt+12
2
2=υ
σ a + α σ w,t − σ a > 0.
dσ z
15

3. The private education system attains higher GDP than the public system does
if:
σz ≥

α (1 − αυ) σ 2w,t
i.
hq
2
2
2
2
σ a + α σ w,t − σ aProposition 2 shows how the relative advantages of the private education systemchange as σ z becomes large. When σ z = 0, as there is no variation in jobs, there
is no assignment problem. Hence, there are no productivity gains from assignment.
In this case, the public education system always dominates the private education
system. This is the standard result emphasized in previous studies.
When σ z > 0, diﬀerences in education systems can influence the pattern of assignment. This proposition shows that the advantage of the private education systemincreases as σ z increases. Note that a rise in σ z increases the marginal productivity of highly talented people.

As the private education system can provide moretalented people, a rise in σ z increases productivity gains from the private education
system. This proposition also tells us that the productivity gains from assignment
can outweigh the human capital loss if σ z is suﬃciently large. Hence, if job opportunities have large variations, an education policy may face a serious eﬃciency—equity
trade-oﬀ.
Compared with the impacts of σ z , it is not clear how a rise in σ 2w,t influences therelative advantages of the private education system. On the one hand, it increases
human capital loss because of a rise in inequality. It makes equalizing income andthus the public education system more beneficial. On the other hand, as a rise in σ 2w,t
makes it likely to have more highly skilled workers, it also increases the productivity
gains from assignment. Taking the derivative with respect to σ 2w,t in equation (17),
it is shown that:
¡
¢
P ri
P ub
d ln Yt+1− ln Yt+1
σ 2z
σ 2a
2
>
0,
if
f
σ
<
−
.
w,t
dσ 2w,t(1 − αυ)2 α216

If σ 2w,t is small enough, a rise in inequality increases the relative advantages of a
private education system; if it is large enough, the opposite is true. The cutoﬀ point
depends on σ 2z .

Note that if σ 2z = 0, a rise in inequality always moves in favor of

the public education system.
assignment.

When σ 2z = 0, there are no productivity gains from

Hence, equalizing income is always beneficial.

However, when σ 2z islarge, the demand for highly skilled labor is large, which makes private education
more beneficial.
This analysis suggests that certain results in this section may not be relevant for
the long run. As a rise in σ z also increases σ 2w,t+1 , there are indirect eﬀects that may
outweigh the direct eﬀect. This is the theme in the next section.

4

Public vs. Private Education System in the LongRun

In this section, we analyze how a rise in σ z influences the relative advantages of
education systems when an aggregate economy reaches a steady state. The steadystate level of σ 2w,t is endogenously determined as a function of σ z . Hence, it is notclear how a rise in σ z changes the impacts of changes in education systems on GDP
in the long run. First, we show that an aggregate economy converges to a uniquesteady state. Later, we analyze how a rise in σ z changes the relative advantages of a
private education system when the economy is in its steady state.
First, we prove that there is a globally stable stationary income distribution. Notethat both ln Yt and σ w,t converge to stationary points if and only if µh,t and σ h,t inequations (8) and (9) converge to stationary points. Assuming that αυ < 1, it is easy
to see that µh,t converges to a stationary point if σ h,t converges to a stationary point.
Hence, by examining the convergence of σ h,t , we can prove the following proposition.

17

Its formal proof is in the appendix.
Proposition 3 Assume that αυθ < 1. The economy converges monotonically to aunique and globally stable stationary income distribution.
When the economy reaches the stationary distribution, σ w,t , which aﬀects GDP,is endogenously determined. The next proposition summarizes the dependent relationship of GDP on the exogenous parameters.
Proposition 4 The logarithm of the steady-state GDP is expressed by:ln Y ∗ =1[ln E (z) + αυ ln LS ∗ + ln HC ∗ + ln AS ∗ + C (θ)] ,1 − αυ

where
ln AS

∗

q
= υσ z σ 2a + α2 θ2 (σ ∗w )2 ,(σ ∗ )2
σ2
ln HC ∗ = −υ (1 − υ) a − αθυ(1 − αθυ) w ,
2
¶2
µ
σz
ln LS ∗ = ln 1 − ∗ ,σw
andσ ∗w =qσ z + α2 θ2 υ 2 σ 2z + (1 − α2 υ 2 θ2 )υ 2 σ 2a1 − α2 υ 2 θ2.

Similarly to the previous section, θ influences GDP through the productivity gains
from assignment AS ∗ , a reduction in aggregate human capital because of the heterogeneity of income and ability HC ∗ , and other constants C (θ).
However, a change in θ has an additional indirect eﬀect. As it changes σ ∗w , it alsochanges the labor share LS ∗ . A rise in the variation of labor income σ ∗w increases thevariation of skill at the stationary point σ ∗h , which increases the rent from skill. To
understand the logic behind this argument, imagine what would happen if σ ∗h = 0.
In this case, everybody has the same skill, and thus a firm can always find alternative18

workers. Therefore, the wage payments for all workers are at the reservation wage,
which is 0 by assumption, and a firm takes all the surplus.

This mechanism isequivalent to that of a Ricardian rent model. When workers have more variation inskill, skilled workers receive more than the reservation wage because it is diﬃcult tofind alternative workers. As the reservation wage does not change, unskilled workers
are still guaranteed to receive the reservation wages. This mechanism increases thelabor share. As an increase in the labor share increases the average labor income, it
raises the average amount of investment in human capital and, therefore, GDP.
Similarly to the previous section, we compare the productive impact of the purely
private education system (θ = 1) and the purely public education system (θ = 0):ln (Y ∗ )Pr i − ln (Y ∗ )P ub
(18)
⎧
i h
i ⎫
h
⎪
⎬
⎨ αυ ln (LS ∗ )Pr i − ln (LS ∗ )P ub + ln (AS ∗ )Pr i − ln (AS ∗ )P ub ⎪
i
h
⎪
⎪
⎭
⎩
+ ln (HC ∗ )Pr i − ln (HC ∗ )P ub,=
1 − αυ
where
σz
υσ a
) − ln(),
ln (LS ∗ )Pr i − ln (LS ∗ )P ub = ln(1 −Priσ z + υσ a
(σ ∗w )"r#hi2
ln (AS ∗ )Pr i − ln (AS ∗ )P ub = σ z(υσ a )2 + (αυ)2 (σ ∗w )Pr i − υσ a ,ln (HC ∗ )Pr i − ln (HC ∗ )P ub = −(σ ∗w )Pr i =hi2αυ (1 − αυ) (σ ∗w )P ri

,
2
q
£
¤
σ z + (αυ)2 σ 2z + (1 − (αυ)2 (υσ a )2
1 − (αυ)2.

Note that C (θ) is canceled out in equation (18) for the same reason as in the previous
section.
Because a rise in σ z increases (σ ∗w )Pr i , not only the productivity gains from assignment ln (AS ∗ )Pr i − ln (AS ∗ )P ub but also human capital loss because of the variation of
19

¯
¯
¯
¯
income ¯ln (HC ∗ )Pr i − ln (HC ∗ )P ub ¯ increases. In addition, ln (LS ∗ )Pr i − ln (LS ∗ )P ubd(ln Y Pr i −ln Y P ub )is also influenced by σ z and (σ ∗w )Pr i . These observations imply thatdσ z
can be positive or negative, and thus there is no monotonic relationship. However,
we can provide the following weaker statement about the relationship between therelative advantage of the public education system and the variations in jobs as in
Proposition 5 (the proof is in the appendix).
Proposition 5 There exist σ z and σ z (0 < σ z ≤ σ z < ∞) such that: (i) if σ z < σ z ,ln Y P ub > ln Y Pr i ; (ii) if σ z < σ z , ln Y P ub < ln Y Pr i .
This proposition says that if σ z is large enough (small enough), productivity gains
from the private education system (the public education system) are larger. Therefore, this proposition confirms the main messages of this paper: if job opportunities
have large variations, an education policy may face a serious eﬃciency—equity tradeoﬀ.

5

Concluding Remarks

This paper compared private and public education systems when the labor market
is characterized by positive assortative matching between the productivity of jobs
and skill. It pointed out that there is a neglected role for an education policy thatcannot be captured by a measure of aggregate human capital. More specifically, it
showed that private education can increase the productivity gains from assignment
by increasing the availability of highly educated people, and that if job opportunities
have large variations in productivity, an education policy may face a serious eﬃciency—
equity trade-oﬀ.
The results of the model must be interpreted with caution. We have no intentionof claiming that a private education system always increases gains from assignment.
20

Of course, it depends on the structure of the production side. We selected Sattinger’s(1979) assignment model as our production structure because it clarifies how an
education policy influences gains from assignment by changing the distribution ofhuman capital and how the novel mechanism changes the results in previous studies.
It is possible to develop a production structure for which a public education system
can improve gains from assignment.

In the rest of paper, we speculated aboutalternative specifications of the production structure to examine the robustness of
our results.
First, we presumed that productivity diﬀers across jobs because there exist jobspecific resources. This creates Ricardian rent, which can be shared with assigned
workers. However, the advantage of a job might be the result of other factors suchas unnecessary regulation. If this is the case, skilled workers may be misallocated toan unsuitable position, which may lower productivity. This reasoning suggests thatthe relative advantages of a private education system may be lower when a market isheavily distorted.
Second, we assumed that jobs are perfect substitutes in a firm . In fact, our assumption of perfectly substitutable jobs is not as restrictive as it looks. Alternatively,
we could assume that a firm has the following production function and maximizes its
profits:y (z) = z

∙Z 1
0

¸ ψυ
¡ i ¢ψ
ht di , ψ < 1, υ > 0.

(19)

Workers are complementary to each other, but this production function derives exactly the same results as those in this paper. In this case, a firm employs the same
quality of workers and segregation occurs across firms. The assumption of perfectly
substitutable jobs is made only for simplicity.
Of course, this does not mean that complementarity is unimportant.

As em-

phasized by Bénabou (1996) and Takii and Tanaka (2009), if the firm faces a mo-

21

nopolistically competitive market, the complementarity among goods favors a morehomogeneous skill distribution. Hence, it is likely that the complementarity mattersin this context.
These arguments suggest that if we wish to identify from data how an education
policy influences the gains from assignment, we need to be careful not only about
the heterogeneity across jobs, but also about the source of the heterogeneities andhow heterogeneous factors interact. Incorporating these aspects into the model andquantitatively assessing an education policy would be interesting future research.

A

Appendix

Proof of Proposition 3: As αυ < 1 is guaranteed by the assumption, it is obviousthat equation (8) converges to the steady state if equation (9) converges to the steadystate. Hence, we only prove the convergence of equation (9). Define:£¤1/2
σ h,t+1 = σ 2a + α2 θ2 (σ z + υσ h,t )2≡ f (σ h,t ).

Because f (σ h,t ) is continuous in σ h,t and f (0) > 0, it is suﬃcient for the existence ofa steady state to show that limσh,t →∞ f 0 (σ h,t ) < 1. In fact:s
α2 θ2 (σ z + υσ h,t )2
lim f 0 (σ h,t ) = lim αυθ= αυθ < 1.
σh,t →∞
σ h,t →∞
σ 2a + α2 θ2 (σ z + υσ h,t )2Hence there exists at least one steady state σ h,t (and thus a steady-state distributionof income). For the uniqueness and stability of the steady state, note that:sα2 θ2 (σ z + υσ h,t )20
f (σ h,t ) = αυθ∈ [0, 1].
σ 2a + α2 θ2 (σ z + υσ h,t )2

Proof of Proposition 5: Note that when σ z = 0, ln Y P ri − ln Y P ub < 0. Becauseln Y P ri − ln Y P ub is continuous in σ z , there must exist σ z > 0 such that ln Y P ri −22

ln Y P ub < 0 for any σ z ∈ [0, σ z ). Hence, it is enough to show that there exists σ z < ∞such that ln Y P ri − ln Y P ub > 0 for any σ z ∈ (σ z , ∞).

hiFor the first term in equation (18), note that αυ ln (LS ∗ )P ri − ln (LS ∗ )P ub =hiaαυ ln(1 − (σ∗σ)zP ri ) − ln( σzυσ)> 0. This is because the steady-state standard de+υσaw

viation of income under the private system is always larger than that under the
public system, and because the labor share is increasing in the standard deviation ofincome. This observation implies that ln Y P ri − ln Y P ub > 0 if the sum of the secondhiand the third terms in equation (18) is nonnegative: ln (AS ∗ )P ri − ln (AS ∗ )P ub +hiln (HC ∗ )P ri − ln (HC ∗ )P ub ≥ 0.
i hih
∗ P ri∗ P ub∗ P ri∗ P ub+ ln (HC ) − ln (HC )≥ 0 ifNext, note that ln (AS ) − ln (AS )and only ifhi2(σ ∗w )P ri ≤ σ̄ w ≡

∙
¸
4σ z
σz
σa
−
.
1 − αυ 1 − αυα(20)This observation comes from direct calculations. It implies that for a given σ z , ifhi2
(σ ∗w )P ri ≤ σ w , then ln Y P ri − ln Y P ub > 0. Hence, defining σez ≡ σσaz , we must showi2h
∗ P riez ∈ ( 1−αυ
,
∞)
such
that
(σ
)
≤ σ̄ w for any σez ≥ σez .
that there exists a unique σwαThen, for a given σ a > 0, the threshold level of σ z (σ z in the proposition) is uniquely³ ´
ez σ a .
given by σ z = σConsider the following function of σez ∈ ( 1−αυ
, ∞):
α
g (e
σz ) =
2

Observe that g (eσz ) =[(σ∗w )P ri ]
σ 2w

µ

1+υ

t
α2 +(1−α2 υ 2 )1(hσ z )21−α2 υ2

4
1−αυ

h

1
− αh1σz
1−αυ

i

¶2

.

for σ
ez ∈ ( 1−αυ, ∞). Hence, it is enough to show there
α23

exists a unique σez ∈ ( 1−αυ, ∞) such that for any σ
ez > σez , g (eσ z ) < 1. Note thatαlim g (eσz ) =

σ
hz →∞

=

√ ´2
1+υ α21−α2 υ 2³(1+αυ)2(1−αυ)2 (1+αυ)24(1−αυ)2

=

³

4
1−αυ

£

1
1−αυ

¤=

1+αυ
(1−αυ)(1+αυ)
4(1−αυ)2

´2

1
< 1,
4

and limσhz → 1−αυ +0 g (eσ z ) = ∞. Because g (eσ z ) is continuous and strictly decreasing inα, ∞),there exists a unique threshold level of σez , σez ∈ ( 1−αυ, ∞) such thatσez ∈ ( 1−αυ
α
α

g (e
σ z ) < 1 for any σ
ez > σez . This completes the proof.

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26