Discussion Papers In Economics
And Business

Availability of Higher Education andLong-Term Economic Growth

Ryo Horii

Akiomi Kitagawa

Koichi Futagami

Discussion Paper 03-14-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAvailability of Higher Education andLong-Term Economic Growth

Ryo Horii

Akiomi Kitagawa

Koichi Futagami

Discussion Paper 03-14-Rev.

September 2005
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAvailability of Higher Education andLong-Term Economic Growth
Ryo Horii∗ Akiomi Kitagawa† Koichi Futagami‡
September 8, 2005

Abstract
This paper examines the economic growth eﬀects of limited availability ofhigher education in a simple endogenous growth model with overlapping generations. With limited availability, the scarcity of human capital keeps its price highand distributes a larger share of the aggregate output to young households. Undercertain conditions, it leads to greater aggregate savings in each period, thereby enabling the economy to grow faster than without any limitation. In such cases, anexcessive expansion in the availability causes a temporary boom followed by a serious deficiency in investible funds, resulting in a substantial slowdown in economic
growth.
JEL Classification Numbers: O41.
Keywords: Endogenous Growth; Human Capital; Slowdown; Intergenerational
Income Distribution.
∗correspondence: Graduate School of Economics, Osaka University,1-7 Machikaneyama, Toyonaka,

Osaka 560-0043, JAPAN; E-mail: horii@econ.osaka-u.ac.jp; Tel: +81-6-6850-5254; Fax: +81-6-68505274.
†

Graduate School of Economics and Management, Tohoku University.

‡

Graduate School of Economics, Osaka University.

11

Introduction
Does education promote economic growth? Although this may appear to have an

obvious answer, deeper examination has proved to be an uneasy task for economists,
primarily because there is no direct measure for the output of education. Despite thediﬃculty, recent empirical micro studies find strong evidence supporting the view that
individual private returns to education are quite high (see the survey by Card, 1999).
Given that human capital is individually productive, existing models of economic growth
predict that education should enhance growth (e.g., Barro and Sala-i-Martin, 2004). The
growth accounting literature devotes enormous eﬀort to confirming this prediction using
aggregate data, but so far, most studies only find weak and elusive connections between
education and economic growth.1
There are three candidates, at least, that may cause this discrepancy between the
micro-macro evidence. First, if education works as a signaling device, its positive private returns would not correspond to individual productivity of human capital (Spence,
1973; Weiss, 1995). Second, the amount and quality of human capital as inputs to the
production process have not yet been fully measured (see the survey by Caselli, 2005).
The third possibility, on which little work has been done so far but is the focus of thispaper, is that the individual benefits of education might be drowned by negative external
1

Using the Solow (1956) growth model, Mankiw, Romer and Weil (1992) argue that a large part ofcross-country diﬀerences in steady-state income is explained by a certain measure of human capital. Islam (1995) finds, however, that once diﬀerences in technologies (individual country eﬀects) are accounted
for by a dynamic panel data model, the role of human capital becomes insignificant. Romer (1989),
De Gregorio (1992), Barro and Lee (1994), Benhabib and Spiegel (1994), and Pritchett (2001) also
report that the direct eﬀect of human capital on growth is either insignificant or even negative. Topel(1999) and Temple (2001) argue that the growth eﬀect of education is found to be positive under moresophisticated specifications, but both admit that the literature connecting human capital investment toeconomic growth is still inconclusive.

2

eﬀects operating at the aggregate level, even when human capital is truly productive andappropriately measured at the micro level.
Specifically, this paper investigates how the long-term rate of economic growth isaﬀected by the pecuniary externality of an increased supply of human capital.2 A risein the aggregate supply in human capital would lower its price compared to other factors summarized as physical capital. Since the distribution of production factors is not
uniform across cohorts of diﬀerent age–that is, human capital belongs to working-age
generations whereas physical capital is largely owned by older generations–the impliedchange in factor prices shifts the distribution of income from the young to the old. If
the aggregate saving rate is adversely aﬀected by the reduced income of the young generation, the shift in income distribution would be critical for long-term growth. Usinga simple endogenous growth model with overlapping generations, this paper shows theoretically that, under a certain condition, the long-term rate of growth may be reduced
by an increased supply of human capital.
To examine changes in the supply of human capital, we focus on the availability ofhigher education. In many countries, aggregate investment in human capital is constrained by the number of higher education institutions and the enrollment capacityof each institution. These are not entirely determined by market forces but are also
determined by non-economic factors such as history, culture, and the social system of
the country, as well as the government’s education policies.3 The limited availability of2

A majority of endogenous growth models assumes the presence of technological externalities toknowledge or human capital, as also does the model we present in the next section (see the survey
by Klenow and Rodrı̀guez-Clare, 2005). A widespread perception, however, is that they are positive
externalities and therefore do not explain the micro-macro discrepancy.
3

For example, Duan (2003) reports that only 2.4 percent of Chinese university candidates could gaina place in 1981 due to the government’s strict control on the number of enrollments for higher education
institutions. In Japan, Kaneko (1997) documents that the Ministry of Education used to control the3higher education puts an upper bound on the rate at which the economy can accumulate
human capital, hence potentially restricts the rate of economic growth. At the sametime, however, the scarcity of human capital keeps its price high and thereby enables
young households to earn a larger share of the aggregate output than without such a limitation.4 Their increased savings contribute to maintaining a high rate of accumulationof both physical and human capital and therefore of growth.
The relative significance of the two opposing eﬀects is shown to depend on the stageof development to which the economy in question belongs. In agrarian countries, ormore precisely in economies where the nature of existing knowledge allows it to be transferred intergenerationally largely without higher education, the savings-enhancing eﬀect
is marginal and therefore expanding enrollment capacity promotes growth in the longrun.5 Conversely, in industrialized economies where the transfer of existing knowledge issubstantially dependent on higher education, there is a range of levels of the availability
within which the savings-enhancing eﬀect dominates the growth-restricting eﬀect. In thiscase, the economy has a balanced growth path on which human capital accumulation is
number of private universities through the rigorous interpretation of university establishment standards.
Private institutions did not argue against such a discretionary policy because it allowed them to operate
as a virtual cartel to maintain their prestige.
4This argument implicitly assumes that graduates, not higher education institutions, receive a largepart of the rent that is generated by the limited availability of higher education. Kaneko (1997) notesthat this was in fact the case in Japan since universities were often reluctant to raise their tuition forfear of losing students of higher academic ability.
5This prediction is consistent with the recent experience of rapid economic growth and drastic ex-

pansion of higher education in China. The proportion of university candidates that could be admitted
was increased dramatically from 2.4 per cent in 1981 to 52 per cent by 2003, mainly as a result of
the government’s policy of reducing the gap between strong demand for higher education and limitedaccess. The total enrollment in government-controlled higher education institutions almost doubled inonly three years, from 6.53 million in 1998 to 12.14 million in 2001 (Duan, 2003).

4

constrained by limited availability but nonetheless grows faster than in the case without
such a limitation. It implies, however, that a further expansion in enrollment capacity
will lower the long-term rate of growth since it causes a regime change beyond which
the limited availability is no longer binding.6 After the regime change, young households
no longer enjoy rent from the limited aggregate supply of human capital and therefore
their savings cannot maintain that high rate of growth in the long run. We examine
the dynamic properties of the slowdown process and show that it is characterized by atemporary boom followed by a serious deficiency in investible funds. This slowdown issubstantial not only in magnitude but also in that, due to its hysteresis, the recovery
from the slowdown involves a major restructuring of educational institutions.
In the literature on human capital accumulation and endogenous growth, the issue ofthe limited availability of education has been examined in the context of credit market
imperfections. De Gregorio (1996) constructs an endogenous growth model in which
the availability of education is limited by the borrowing constraints imposed on theyoung generation, and argues that relaxing these constraints has two opposing eﬀects
on growth. First, it makes possible a rapid accumulation of human capital throughincreased participation in education, which has a positive eﬀect on long-term growth.
Second, it enables young households to enjoy more consumption by borrowing more,which reduces the aggregate saving rate and therefore has a negative eﬀect on growth.7
6

For example, a recent report by the Japanese government predicts that a steady increase in thenumber of private universities will cause total enrollment capacity to exceed the number of all university
applicants by 2007.
7

Some authors have already pointed out the possibility that severe borrowing constraints rather

accelerate economic growth by encouraging aggregate savings. Modigliani (1986), for example, argues
that credit market imperfections prevent households from borrowing as much as would be required tocarry out an unconstrained optimum consumption plan, which has the general eﬀect of postponing consumption and increasing wealth as well as savings. Based on this argument, Jappelli and Pagano (1994)

5

If the second eﬀect dominates the first, relaxing the borrowing constraint reduces therate of economic growth. De Gregorio and Kim (2000), Christou (2001), and Azariadis
and de la Croix (2003) have extended this argument to the case in which each generation
contains heterogeneous households that are subject to diﬀerential borrowing constraints.
Distinct from these studies, this paper directly imposes a limitation on the level ofhuman capital investment, so that a relaxation of the limitation does not have a negativeeﬀect on aggregate savings through changing the consumption behavior of young households. In our model, however, it does have a negative eﬀect on aggregate savings throughencouraging human capital accumulation, because human capital aﬀects the distributionof income between generations. The cumulative eﬀect eventually causes a catastrophic
slowdown, pushing the rates of both physical and human capital accumulation, and hence
economic growth, to a considerably lower level. This is in sharp contrast to the aforementioned models that only emphasize the growth-enhancing eﬀects of human capital
accumulation.
The rest of the paper is organized as follows. After presenting the model in Section
2, temporary equilibrium is characterized in Section 3. Section 4 examines equilibrium
dynamics and steady states for possible situations. Section 5 evaluates the growth eﬀects
of a gradual increase in the availability of higher education. Section 6 gives some concluding remarks. Proofs of propositions are contained in an additional appendix, whichis available from the corresponding author upon request, or can be downloaded fromhttp://www2.econ.osaka-u.ac.jp/∼horii/.
constructed an endogenous growth model without human capital, in which the borrowing constraint
encourages physical capital accumulation and economic growth.

6

2

Model
The model is a variant of Diamond’s (1965) overlapping generations model. Time isdiscrete and extends from one to infinity. A continuum of households of measure one areborn at the beginning of each period, and every household lives for two periods. Thus,in any period, there are just two age groups of households of an equal size, which we callthe young and old households.
In each period, a single final good, the numeraire, is competitively produced fromphysical and human capital according to a Cobb-Douglas technology, Yt = AKtα Ht1−α ,where A > 0, α ∈ (0, 1), Yt , Kt and Ht , respectively, represent the total factor productivity, the share of physical capital, the aggregate output of the final good, the aggregate
input of physical capital, and that of human capital. Factor markets are perfectly competitive, so that both kinds of capital earn their marginal values, i.e.,
rt = Aα(Kt /Ht )α−1 ,wt = A(1 − α)(Kt /Ht )α ,(1)

where rt and wt , respectively, denote the market price of physical capital and humancapital. The good produced in one period can be either consumed in that period or
saved for the production in the next period. Once saved, the good cannot be consumed,
but can be used as either physical capital or an input to human capital investment,
which we call higher education. Moreover, the saved good perishes in one period, so thatphysical capital existing in the current period cannot be carried over to the next period.
The household maximizes a time-separable logarithmic utility function:
Ut = (1 − σ) log c1,t + σ log c2,t+1 ,(2)

where c1,t and c2,t+1 , respectively, represent the amounts of consumption in the young
and old ages, and σ ∈ (0, 1) is a parameter specifying the patience of households. This
formulation suggests that, given that households earn labor income only when young,they would save σ percentage of that income for their old age.
7

The main diﬀerence between our model and that of Diamond is that newly bornhouseholds are not only endowed with some amount of human capital through hometraining and free (costless) primary education, but also with the ability to augment their
human capital by participating in higher education. Higher education is described asa process that produces human capital using saved goods, relying upon the previous
generation’s aggregate stock of knowledge. Specifically, when the previous generation
possesses amount Ht−1 of human capital in aggregate (or equivalently, on average), a
young household can obtainht = G(et , Ht−1 )(3)units of human capital by investing et ≥ 0 units of the saved goods.8
We assume that the education production function G(·, ·) is homogeneous of degree
one with respect to et and Ht−1 , but that its output cannot increase unboundedly withthe value of et because investment opportunities are limited by the current availability ofhigher education. To specify this feature we assume that G(·, ·) has the following simple
functional form,G(et , Ht−1 ) ≡ δHt−1 + min[et , γHt−1 ],(4)where δ and γ are parameters satisfying 0 < δ < 1 and γ > 0. The first term in
the right-hand side of (4) represents the amount of human capital transferred from theparent’s generation through home training and primary education. Parameter δ measures
the percentage of the previous generation’s knowledge that can be transferred throughsuch channels. The value of δ tends to be smaller in industrialized economies than inagrarian economies, since agricultural knowledge is often passed down from parents tochildren whereas the transfer of industrial knowledge often requires higher education.
The second term is the amount of human capital obtained through higher education. By
8Similar specifications are found, for example, in Azariadis and Drazen (1990) and De la Croix andMichel (2002, section 5.2).

8

participating in higher education, young households can produce human capital linearlywith the input of saved goods,9 but the limited availability of higher education puts anupper bound on the individual investment in human capital at γHt−1 . The availability ismeasured by parameter γ, which is determined by such factors as the number of higher
education facilities that oﬀer graduate and undergraduate courses and the enrollment
capacity in each facility. Due to those limitations, there is an upper limit on the rate ofaggregate human capital growth;10 i.e., G(et , Ht−1 )/Ht−1 is bounded by δ + γ.
The life of a household born at the beginning of a generic period t proceeds as follows.
In period t, this household is endowed with δHt−1 units of human capital. In addition, shepurchases some amount of the saved goods from the physical capital market (the market
for the saved goods) to augment her own human capital through higher education. Thenshe supplies the sum of endowed and augmented human capital to the human capital
market, and in return receives wt ht units of the newly produced good at the end of that
period. Since the household has to pay rt et units of the produced good for the purchaseof the saved goods made at the beginning of that period, her net income becomes
wt ht − rt et = wt δHt−1 + (wt − rt )etfor et ≤ γHt−1 .

(5)At the end of period t, she consumes the 1 − σ percentage of the net income and saves
st = σ(wt ht − rt et ) units of the produced good for her old age. In period t + 1, the saved
9

In this respect, we follow the simplest setting employed in the growth literature. A more realistic

production function of human capital would take both goods and human capital (i.e., time) as inputs.
In that case, one can interpret (4) as the net production function. The one-to-one production function
is equivalent to the one that produces x units of human capital from one unit of saved good and x − 1units of (the learner’s and/or teachers’) human capital.
10

There are more ‘well-behaved’ production functions that satisfy this property as well as homogeneityof degree one, twice diﬀerentiability, and strict quasi-concavity; e.g., G(et , Ht−1 ) = δHt−1 + γet /[γ +(et /Ht−1 )]. The savings-enhancing eﬀect of limited availability can also be derived by assuming thosefunctions, but analytical results are harder to obtain than in the current linear setting.

9

Figure 1: Time structure of the model

goods are supplied to the physical capital market, being used for either the productionof the final good or that of human capital. At the end of that period, the household
consumes rt+1 st units of the produced good which is received in compensation for her
supply of the saved goods, and dies. Figure 1 summarizes the timing of transactions,where Ht , Et and St represent the aggregate levels of ht , et and st , respectively.
Taking the values of wt , rt and Ht−1 as given, every household born in period tchooses a level of et so as to maximize the net income at the end of period t, given by(5), because this choice leads to the maximization of her lifetime utility. Note that fromthe viewpoint of a newly born household, wt represents the rate of return from higher
education whereas rt is the interest rate that applies to the intra-period education loan.

10According to their relative magnitude, each household sets the level of et as
0
if wt /rt < 1,



et =
∀e ∈ [0, γHt−1 ] if wt /rt = 1,





γHt−1
if wt /rt > 1.

(6)

Once the level of et is determined, so are the levels of ht , st , c1,t and c2,t+1 .

To close the model, we assume that, in period 1, there are old households of measure
one, each of whom is endowed with S0 units of the saved goods. The young households
born in that period are assumed to have access to a human capital production function
h1 = G(e1 , H0 ), where H0 is a positive constant. The old households supply the endowed
good to the physical capital market, and in return receive r1 S0 units of the produced
good. They consume those goods and die at the end of that period. Their lifetime
utilities are given by U0 = σ log c2,1 = σ log r1 S0 .

3

Temporary Equilibrium
We begin the analysis by showing that the equilibrium of our model exhibits backward-

looking dynamics, or equivalently, that temporary equilibria of the markets for humanand physical capital in one period are uniquely determined by those in the previous
period.
Consider the human capital market equilibrium at a generic period t ≥ 1. From (4)and (6), we can derive the aggregate supply of human capital as a function of the ratioof factor prices, wt /rt ,

HtS =








δHt−1

if wt /rt < 1,∀h ∈ [δHt−1 , (δ + γ)Ht−1 ] if wt /rt = 1,





if wt /rt > 1.
(δ + γ)Ht−1(7)

Equation (1) implies that, given the aggregate level of physical capital used in the production of the final good, Kt , the aggregate demand for human capital can also be
11

Figure 2: Demand and supply of human capital

expressed as a function of wt /rt ,HtD =1 − α rt
Kt .
α wt

(8)

Figure 2 depicts the graphs of (7) and (8), the supply and demand curves of humancapital. As shown in figure 2, the two graphs have a unique intersection for any valueof Kt > 0. When Kt takes a value between αδHt−1 /(1 − α) and α(δ + γ)Ht−1 /(1 − α),the demand curve intersects with the flat segment of the supply curve, and the prices ofhuman and physical capital are equalized; i.e., wt /rt = 1. This is not a surprising result,since both kinds of capital are produced from the saved goods on a one-to-one basis.11 In
this case, the factor intensity is determined as Ht /Kt = (1 − α)/α, at which the marginal
products of human and physical capital equate with each other. When Kt takes a value
outside that interval, the demand curve intersects with one of the vertical sections of
the supply curve, and the prices of human and physical capital diﬀer in equilibrium. In
this case, the value of Ht has reached either the upper or lower limit, so that the factor
intensity cannot be adjusted to the level at which the marginal products are equalized.
11

The same result is also obtained by Barro and Sala-i-Martin (2004, section 4.2).

12

Figure 3: The HCME and PCME curves

To summarize, the human capital market determines the value of Ht as a function of Kt :


αδ

δHt−1
if Kt < 1−αHt−1 ,



1−α
αδ
(9)
Ht =
Kt
if Kt ∈ [ 1−αHt−1 , α(δ+γ)
Ht−1 ],
α
1−α




α(δ+γ)
 (δ + γ)H
t−1 if Kt > 1−α Ht−1 .
Next, consider the physical capital market equilibrium at period t. While the ag-

gregate supply of physical capital (the saved goods) equals the aggregate savings in theprevious period, St−1 , the aggregate demand for physical capital consists of Et and Kt ,the former of which represents the demand arising from the investment into human capital and the latter the demand arising from the production of the final good. Because
the level of Et is given by Et = Ht − δHt−1 , the equilibrium condition for the physicalcapital market can be expressed as
Ht − δHt−1 + Kt = St−1 .

(10)Given the values of Ht−1 and St−1 , the equilibrium values of Ht and Kt are uniquelydetermined by (9) and (10). To see this, we only need to draw the graphs of these conditions in the (Kt , Ht ) space. As depicted in figure 3, the graph of (9), which we call the13

HCME (human capital market equilibrium) curve, is a piece-wise linear curve consisting of an upward-sloping segment in the middle and two flat segments corresponding to
the upper and lower limits on Ht , whereas the graph of (10), which we call the PCME
(physical capital market equilibrium) curve, is a downward-sloping line with a gradientof −1. Note that locations of these curves are fully determined by the values of Ht−1
and St−1 , and that the two curves intersect each other only once. This means that theequilibrium values of Ht and Kt are uniquely determined, being expressed as
Ht = H(Ht−1 , St−1 ),(11)

Kt = K(Ht−1 , St−1 ).

(12)

We can also show that the aggregate level of savings in period t, St , is a function of
Ht−1 and St−1 . As seen in the previous section, young households save the σ percentageof their net income, and thus the individual level of savings in period t, st , is given by
st = σ(wt ht − rt et ) = σ{wt δHt−1 + (wt − rt )et } for et ≤ γHt−1 .

(13)Aggregating (13) over young households in period t and using the fact that Et = Ht −
δHt−1 , we obtain St = σ{wt Ht − rt (Ht − δHt−1 )}. As implied by (1), (11) and (12), theequilibrium values of rt , wt and Ht are uniquely determined, given the values of Ht−1
and St−1 . Thus the equilibrium value of St can be expressed as
St = S(Ht−1 , St−1 ).

(14)Note that (11) and (14) constitute a mapping from (Ht−1 , St−1 ) to (Ht , St ), whichgoverns the evolution of this economy. Given the values of H0 and S0 , it uniquely determines the equilibrium sequence, {(Ht , St )}∞t=1 , which, in turn, determines the sequenceof other aggregate variables, Kt , rt and wt through (1) and (12). Thus, to investigate thedynamical system of this economy, we need to derive an explicit form of that mapping.
This task necessitates dividing all possible situations into three cases by the value of
St−1 /Ht−1 .
14Case 1. When St−1 /Ht−1 < αδ/(1 − α), the PCME curve intersects with the bottom flat segment of the HCME curve, and so the equilibrium values of Ht and Kt are,respectively, determined as Kt = St−1 and
Ht = δHt−1 .

(15)

Substitution of these values into (1) yields
µ
µ
¶α
¶α−1
St−1
St−1
wt = A(1 − α)< Aα= rt .
δHt−1
δHt−1
Recall that wt can be viewed as the rate of return from higher education. When thisrate of return is smaller than the interest rate rt , young households in period t are notwilling to participate in higher education and merely supply the endowed human capitalto the market, receiving in return wt δHt−1 units of the produced good. They save the σpercentage of that income, and thus the aggregate level of savings in period t is given byα.
St = σwt δHt−1 = σA(1 − α)(δHt−1 )1−α St−1

(16)

Case 2. When αδ/(1−α) ≤ St−1 /Ht−1 ≤ (αδ+γ)/(1−α), the PCME curve intersects
with the upward-sloping segment of the HCME curve. In this case, factor intensity andprices are determined as Kt /Ht = α/(1 − α) and wt = rt = Aαα (1 − α)1−α . From thisresult and (10), we obtain
Ht = (1 − α)(St−1 + δHt−1 ).

(17)

Since wt = rt , the cost and return of higher education are equalized with each other,
implying that the level of human capital investment through education does not aﬀect
the net income of young households in period t. In this case, the net income is determinedsolely by the earnings of endowed human capital, wt δHt−1 . They save the σ percentageof that income, and so the aggregate level of savings in period t is given by
St = σwt δHt−1 = σAαα (1 − α)1−α δHt−1 .
15

(18)

Case 3. When St−1 /Ht−1 > (αδ + γ)/(1 − α), the PCME curve intersects with the
top flat segment of the HCME curve, and so the equilibrium values of Ht and Kt are,respectively, determined as Kt = St−1 − γHt−1 and
Ht = (δ + γ)Ht−1 .

(19)

Substitution of these values into (1) yields
wt = A(1 − α)µSt−1 − γHt−1(δ + γ)Ht−1

¶α

> Aα

µ

St−1 − γHt−1(δ + γ)Ht−1¶α−1= rt .

This inequality shows that the rate of return from higher education exceeds the interest
rate. Thus, young households in period t are willing to participate in higher education
as much as availability permits, implying that their net income equals wt (δ + γ)Ht−1 −rt γHt−1 . Since they save the σ percentage of that income, the aggregate level of savingsin period t is given by
St = σ(wt (δ + γ)Ht−1 − rt γHt−1 )µ¶1−α(δ + γ)Ht−1.
= σA ((1 − α)St−1 − γHt−1 )St−1 − γHt−1

4

(20)

Dynamics and Steady States
Now we are in a position to analyze the dynamic behavior of this economy. As

shown in the previous section, the equilibrium dynamics is completely described by the
movement of two variables Ht and St , which evolves according to (15)-(20) given theinitial values of H0 and S0 . In this section, we further simplify the dynamic system byredefining variable and parameters as follows:
Xt ≡

St
,
σAHt

∆≡

δ
,
σA

Γ≡

γ
.
σA

(21)

By definition, the values of Xt , ∆ and Γ are all positive. The new variable Xt summarizes
how households born in period t contribute to the increased production of goods in16

period t + 1. Recall that their activities can increase the production in period t + 1 boththrough enhancing the supply of the saved goods in that period (St ) and through theexternal eﬀect of aggregate human capital on the next generation’s achievement in higher
education (Ht ). A large value of Xt suggests that a greater part of their contributions
has been made through the first channel, rather than the second channel. The newparameters, ∆ and Γ, respectively, are normalized measures of δ and γ.
Using (21), we can rewrite (15)-(20) as a dynamic system with one variable and two
parameters,
Xt = Ψ(Xt−1 ; ∆, Γ)

for all t ≥ 1.

(22)

In (22), the shape of function Ψ is given by a combination of three curves:¡ α∆ ¢
Ψ1 (Xt−1 ; ∆) if Xt−1 ∈ 0, 1−α
,




£
¤
Ψ(Xt−1 ; ∆, Γ) ≡ Ψ2 (Xt−1 ; ∆) if Xt−1 ∈ α∆ , α∆+Γ ,1−α 1−α Ψ3 (Xt−1 ; ∆, Γ) if Xt−1 ∈ ¡ α∆+Γ , ∞¢,1−αwhere those three curves are defined byα(1 − α)Xt−1
Ψ1 (Xt−1 ; ∆) ≡,αµ ∆ ¶αα∆Ψ2 (Xt−1 ; ∆) ≡,1−αXt−1 + ∆(1 − α)Xt−1 − Γ.
Ψ3 (Xt−1 ; ∆, Γ) ≡(∆ + Γ)α (Xt−1 − Γ)1−α

(23)
(24)

Equilibrium sequences of Xt , Ht and St are derived as follows. Given the initial value
X0 ≡ S0 /(σAH0 ), equation (22) uniquely determines the equilibrium sequence {Xt }∞t=1 ina recursive manner. Then, given the obtained sequence of Xt , the equilibrium sequencesof Ht and St are determined by the relations,¡ α∆ ¢


∆σA
if
X
∈
0, 1−α ,

t−1



£
¤
Ht /Ht−1 = (1 − α)(∆ + Xt−1 )σA if Xt−1 ∈ α∆ , α∆+Γ ,
1−α
1−α





 (∆ + Γ)σA if Xt−1 ∈ ¡ α∆+Γ , ∞¢.
1−α
17

(25)

Figure 4: The Ψ curveand St = σAHt Xt for all t ≥ 1, both of which are derived from (15)-(21). Since thesteps of deriving the sequences of Ht and St are straightforward, we concentrate on the
behavior of Xt in the rest of this section.
Figure 4 depicts the graph of function Ψ, which we call the Ψ curve. It is comprisedα∆ α∆+Γ, 1−α ], the arbitrageof three segments called Ψ1 , Ψ2 and Ψ3 curves. When Xt−1 ∈ [ 1−αbetween two sorts of capital equalizes their marginal productivities with each other,
keeping the factor intensity constant.12 In this case, the aggregate production is linear
in the amount of overall capital stock (Kt + Ht ), which is equal to the sum of saved
goods and transferred human capital: (Kt + Ht ) = St−1 + δHt−1 (see equation 10). This
representation suggests that, as long as Xt stays within this interval, the workings ofour model are basically similar to the AK-type endogenous growth models.13 In thisAK Regime the Ψ curve is downward sloping, which can be explained as follows. Note
12

Recall Case 2 in the previous section.

13

To see this point, define Kt ≡ Kt + Ht and A ≡ αα (1 − α)1−α A. Then, the amount of production

can be expressed as Yt = AKt .

18

that a large value of Xt−1 (≡ St−1 /δAHt−1 ) means that the supply of saved goods, St−1 ,constitutes a large portion of overall capital in period t. As shown by equations (17) and(18), however, a large St−1 raises Ht but not St , reducing the percentage of saved goodsin the overall capital supplied for the next period. Therefore, a large Xt−1 is followed bya small Xt .
α∆or Xt−1 > α∆+Γ, the growth rate of Ht becomes constant.14 In
When Xt−1 < 1−α1−αthis case, we can interpret Ht as the eﬃciency unit of labor per worker growing at aconstant rate, and Yt = (AHt1−α )Ktα as a production function of the neoclassical growthα∆model with exogenous labor-augmenting technological progress. When Xt−1 < 1−α,in particular, our model becomes essentially the same as that of Solow (1956). In thisSolow Regime, a fixed percentage of the aggregate output in period t−1 is saved and used
entirely as physical capital in period t, i.e., Kt = St−1 = σ(1 − α)Yt−1 . This structure,
which is also observed in the Solow model, defines the shape of the Ψ1 curve in such a
way that it is increasing, concave, and starting from the origin. In the Modified Solow, a certain amount of the saved goods are used for higher
Regime, where Xt−1 > α∆+Γ
1−αeducation and the remainder becomes physical capital. This makes Ψ3 curve locatedbelow the extension of the Ψ1 curve, but the increasing and concave properties of thecurve are preserved.15
The global dynamics of Xt is critically dependent on intersecting patterns of the Ψ
curve and the 45-degree line, which are summarized in Figure 5. Intersection points ofthe Ψ curve and the 45-degree line represent steady states of this economy, in which
14

Recall Cases 1 and 3 in the previous section.

15

Our approach of dividing the patterns of the dynamics into the neoclassical and endogenous growth

regimes is similar to that of Matsuyama (1999), who classified the dynamics into the Solow and Romer
regimes. A novel property of our model is that the economy behaves as described in the neoclassicalgrowth models not only when Xt is small but also when it is large, giving rise to the possible existenceof multiple steady states.

19

Figure 5: Four patterns of dynamics

the aggregate amounts of output, consumption, physical capital, and human capital are
all growing at a common rate. Our model has at least one steady state, since Ψ(·) is acontinuous function satisfying limX→+0 Ψ0 (X; ∆, Γ) = ∞ and limX→∞ Ψ0 (X; ∆, Γ) = 0
for all values of ∆ and Γ. Parameter values determine the number of steady states. As
depicted in figure 5, the Ψ curve and the 45-degree line can intersect with each otheras many as three times, dependent on the values of ∆ and Γ. Thus, to examine thelong-run behavior of the economy, we need to clarify how the parameter values restrict
patterns of dynamics.
A straightforward calculation shows that the economy converges to a steady state in
the Solow regime as shown by panel (a) whenever16∆ ≥ αα−1 (1 − α)2−α ≡ ∆.
16

Condition (26) is derived from αα (1 − α)1−α < α∆/(1 − α). See Figure 4.

20

(26)

We ignore this possibility by assuming that ∆ < ∆, since the convergence to the Solow
regime implies that no human capital is accumulated in the long run and therefore theavailability of higher education is irrelevant for those economies.
4.1

Economies with Insuﬃcient Availability
Let us consider the dynamics for economies where the availability of higher education

is low. If Γ is below a threshold level defined by17
s
Ã
µ
¶α !
∆
4
α
−(1 + α) + (1 − α) 1 +,Γ(∆) ≡
2
∆ 1−α

(27)

the economy converges to a steady state in the modified Solow regime from any initial state, as depicted by panel (b) of figure 5. The unique steady state XL∗∗∗ (∆, Γ) ∈, +∞) is given by the larger root of equation Ψ3 (X; ∆, Γ) = X, where Ψ3 (X; ∆, Γ)[ α∆+Γ
1−αis defined by (24).
In the steady state, the aggregate amounts of output, consumption, human, andphysical capital are growing at a constant rate of δ + γ = (∆ + Γ)σA. In these economies,
the availability of higher education is too limited relative to the economy’s ability tosupply the saved goods. The limited availability keeps the rate of return from higher
education greater than the interest rate in the long run, which gives an incentive for young
households to participate in higher education as much as the availability permits. Thus
the long-term rate of economic growth as well as the rate of human capital accumulation
is determined by the availability of higher education.
4.2

Possibility of Multiple Steady States
When Γ ≥ Γ(∆), the dynamical system (22) has a steady state in the AK regime(See panels (c) and (d) of Figure 5). Solving equation Ψ2 (X; ∆) = X, where Ψ2 (X; ∆)17Γ(∆) is the unique positive root of equation (α∆ + Γ)/(1 − α) = αα (1 − α)1−α ∆/(∆ + Γ). See

Figure 4.

21

is defined by (23), we see that the steady state is located at
X ∗∗ (∆) ≡α∆ + Γ(∆)
.
1−α

(28)

In this steady state, the aggregate amounts of output, consumption, human, and physicalcapital are growing at the rate of δ + σAΓ(∆) = (∆ + Γ(∆))σA, which is between δ andδ+γ. In this steady state, marginal products of human and physical capital are equalizedwith each other, and the amount of education is reaching neither zero nor the maximum
limit implied by the availability. Thus, a small change in γ has no influence on thelong-term growth rate.
As panel (d) suggests, however, X ∗∗ (∆) is not necessarily the unique steady stateof the economy. The 45-degree line may intersect not only with the Ψ2 curve but alsowith Ψ3 , giving rise to further two steady states in the modified Solow regime. For the
existence of multiple steady states, both ∆ and Γ must satisfy certain conditions. First,
note that the smaller parameter ∆ is, the steeper the Ψ3 curve becomes (see the definitionof function Ψ3 in equation 24). Since the Ψ3 curve is concave, for it to intersect withthe 45-degree curve twice, ∆ must be small enough so that the slope of the Ψ3 curve at) must be greater than one. In particular, ∆ must be
the left end (i.e. at Xt−1 = α∆+Γ
1−αsmaller than18

´−1
³
p
−1
b
∆
∆ ≡ 3/2 − α + α − 3/4(29)

for the case in which the left end of the Ψ3 curve is exactly on the 45-degree curve, as
illustrated by point A in Figure 6 (that is, when Γ = Γ(∆)). As will be clear from Figureb is necessary for the
6 and the discussion below, for every Γ ≥ Γ(∆), condition ∆ < ∆existence of multiple steady states.19

b is smaller
This condition is obtained from 1 < Ψ03 ((α∆ + Γ(∆))/(1 − α); ∆, Γ(∆)). Note that ∆ppb is available
than ∆, since 3/2 − α + α−1 − 3/4 > 3/2 − 1 + 1 − 3/4 = 1. A detailed derivation of ∆18

22

Figure 6: Existence of multiple steady states. It depicts the Ψ curve for two values of Γ, keeping
b If ∆ > ∆,b the slope of Ψ3 curve at point A becomes less steeper thanthe value of ∆ fixed at ∆ < ∆.

the 45-degree line, and the stady state is always unique.

Second, observe from (24) that a larger Γ pushes the whole Ψ3 curve downward.20
Therefore, for the existence of multiple steady states, Γ should not be too large so thatthe position of the Ψ3 curve is not too low. From the concavity of the Ψ3 curve, there
must be a value of Γ ∈ (Γ(∆), ∞) such that the Ψ3 curve touches the 45-degree line atb
one point, as illustrated by point B in Figure 6. Letting Γ(∆)
denote this value, it isb
The following
apparent that multiple steady states exist only when Γ ∈ [Γ(∆), Γ(∆)].

proposition shows that this upper threshold is decreasing in ∆.

b Γ(∆)bb0 (∆) ∈ (−1, 0). In addition,
Proposition 1. For all ∆ < ∆,is diﬀerentiable and Γbblim∆→∆b Γ(∆)= Γ(∆).

Proof: available upon request.
Intuitively, if ∆ is larger, the Ψ3 curve is flatter, and therefore there will be a narrower

upon request.
19

If the gradient at the left end is less than unity when Γ = Γ(∆), then the concavity of Ψ3 curve

implies that there is no section of the Ψ3 curve which is above the 45-degree line. Since the Ψ3 curve
shifts down when Γ increases, there would be no multiple steady states for any Γ ≥ Γ(∆).
20

Equation (24) can be written as Ψ3 (X; ∆, Γ) = (X − Γ)α (∆ + Γ)−α (1 − αX/(X − Γ)). It is straight-

forward to confirm that this expression is decreasing in Γ for all X ≥ α∆+Γ
1−α .

23

range of Γ for which the Ψ3 curve intersects with the 45-degree line twice. The following
two subsections in turn examine the dynamics of the economy without and with multiple
steady states.
4.3

Economies with Excessive Availability
Now consider the dynamics of the economy where the availability of higher education

is high. The analysis in the previous subsection implies that the pattern of the dynamicsbb and when Γ ≥ Γ(∆)
looks like panel (c) of figure 5 when Γ > Γ(∆)
in the case of ∆ < ∆,b ∆). The following proposition demonstrates that the economy
in the case ∆ ∈ [∆,converges to the unique steady state X ∗∗ (∆) from any initial state.

Proposition 2. Suppose that the values of ∆ and Γ satisfy either of the conditions thatb and Γ > Γ(∆),
bb ∆) and Γ ≥ Γ(∆). Then, the sequence of Xt∆<∆
or that ∆ ∈ [∆,converges to X ∗∗ (∆) from any value of X0 .
Proof: available upon request.

Starting from any initial state, the sequence of Xt enters a neighborhood of X ∗∗ (∆)
in finite time. Then, it begins to oscillate around X ∗∗ (∆), but eventually converges tothat steady state. This oscillation implies that, in converging to the steady state, theeconomy alternately experiences accelerated and decelerated growth. These economies
have excessive availability of higher education not only in that the education facilities
are not fully utilized, but also in that a higher rate of economic growth could havebeen sustained if the availability were not so high. The latter point will be confirmed
immediately.
4.4

Economies with Intermediate Availability and Low ∆bb the Ψ curvein an economy with ∆ < ∆,When Γ is within the interval [Γ(∆), Γ(∆)]

and the 45-degree line have three intersection points, as depicted in panel (d) of figure24

5.21 In addition to X ∗∗ (∆), there are two steady states in the modified Solow regime.
These are XS∗∗∗ (∆, Γ) and XL∗∗∗ (∆, Γ), respectively defined as the smaller and the larger
roots of equation X = Ψ3 (X; ∆, Γ).
Panel (d) suggests that, in this case, an economy’s destination depends critically on
its initial state. If X0 > XS∗∗∗ (∆, Γ), then the sequence of Xt monotonically converges
to XL∗∗∗ (∆, Γ). This is similar to the dynamics observed in the case of insuﬃcient availability. If X0 is given in a neighborhood of X ∗∗ (∆), the sequence of Xt follows dynamics
essentially the same as the case of excessive availability, converging to that steady state.
Although the dynamics become less obvious when the initial state is far from these twostable steady states, the following proposition shows that, from almost all initial states,
the sequence of Xt eventually converges either to X ∗∗ (∆) or to XL∗∗∗ (∆, Γ), thus allowing
us to regard these steady states as plausible long-run states.
b and Γ ∈ [Γ(∆), Γ(∆)].
b
Proposition 3. Suppose that the values of ∆ and Γ satisfy ∆ < ∆Then,a. Xt converges to one of X ∗∗ (∆), XL∗∗∗ (∆, Γ), and XS∗∗∗ (∆, Γ).
b. From almost all initial states, Xt converges either to X ∗∗ (∆) or XL∗∗∗ (∆, Γ).
c. If X0 > XS∗∗∗ (∆, Γ), Xt monotonically converges to XL∗∗∗ (∆, Γ).
d. X ∗∗ (∆) is a locally stable steady state except for the case of X ∗∗ (∆) = XS∗∗∗ (∆, Γ).
Proof: available upon request.
A comparison of the two stable steady states reveals that the economy is growing
faster in steady state XL∗∗∗ (∆, Γ) than in X ∗∗ (∆).22 In the former steady state, the21bStrictly speaking, panel (d) depicts the Ψ curve when Γ ∈ (Γ(∆), Γ(∆)).

When Γ = Γ(∆),

b
XS∗∗∗ (∆, Γ) coincides with XL∗∗∗ (∆, Γ).
XS∗∗∗ (∆, Γ) coincides with X ∗∗ (∆). When Γ = Γ(∆),
22

This may seem tricky because XL∗∗∗ (∆, Γ) corresponds to the unique steady state in an economy

where higher education is insuﬃciently available, whereas X ∗∗ (∆) corresponds to one in an economy
with excessive availability. The point is that the economy is growing fully utilizing existing facilities for25

supply of the saved goods in one period is so large relative to the availability of higher
education in the economy that the marginal products of human and physical capitalcannot be equalized even if the availability is fully utilized. This keeps the rate of returnfrom higher education above the interest rate, distributing a considerable percentage ofthe current output to the young generation, and thereby realizing a large supply of the
saved goods in the next period. Due to this virtuous circle, this steady state maintains
large amounts of savings and hence fast growth, the rate of which is as high as (∆+Γ)σA.
In the latter steady state X ∗∗ (∆), by contrast, the supply of the saved goods in one
period is not so large relative to the availability of higher education. In this case, themarginal products of human and physical capital are equalized through higher education
available in the economy. This equalizes the rate of return from higher education with
the interest rate, distributing only a small percentage of the current output to the young
generation, and thereby realizing a small supply of the saved goods in the next period.
Due to this vicious circle, this steady state maintains only small amounts of savings andhence slower growth, the rate of which is as low as (∆ + Γ(∆))σA.
Despite its restricting eﬀects on the accumulation of human capital, the limited availability of higher education enables the economy to exhibit a fast growth through ensuring
a suﬃcient supply of investible funds in each period. If there is no limit in the availabilityof higher education, the marginal products of human and physical capital would always
be equalized to each other, and, as a result, X ∗∗ (∆) would be the unique steady state in
our model.
4.5

Summary
So far, we have seen how an economy’s long-term growth rate is related to valuesof Γ and ∆. The observed relations are summarized by figure 7 and Table 1. Figure 7
higher education in steady state XL∗∗∗ (∆, Γ), while the availability is not fully exploited in X ∗∗ (∆).

26

Figure 7: Division of the parameter space. The figure is numerically drawn by fixing the valueof α at 0.3. Given the value of α, the locations of the three graphs are uniquely determined, since thebb depend solely on α. In addition,functional forms of Γ(·) and Γ(·),
and hence the values of ∆ and ∆,Proposition 1 and the definitions of ∆ and Γ(∆) jointly guarantee that the shape and relative locationof three graphs are maintained for other values of α.

regionstable steady states

growth rate

regime(a)X ∗ (∆)

∆σA

Solow

(b)

XL∗∗∗ (∆, Γ)(∆ + Γ)σAmodified Solow(c)

X ∗∗ (∆)
X ∗∗∗ (∆, Γ)(∆ + Γ(∆))σAAK(∆ + Γ)σAmodified Solow(∆ + Γ(∆))σA

AK

(d)

L


 X ∗∗ (∆)

Table 1: Steady states and long-term growth rates in each region27

depicts how the whole (∆, Γ) space is divided into four regions by graphs of Γ = Γ(∆),
bΓ = Γ(∆)
and ∆ = ∆. Each region corresponds to a panel of the same label in figure

5, from which we can observe the pattern of the dynamics of the economy. Table 1 lists
stable steady states and their long-term growth rates for each region.

5

Impacts of Expanded Higher Education Opportunities
We now evaluate how economic growth is aﬀected by an increased enrollment capacityin each facility and the establishment of new higher education institutions.23 In themodel, those changes are described by gradual increases in parameter γ (and thereforeΓ ≡ γ/(σA)). Results are dependent on another parameter ∆.
Economies with high ∆24b ∆), the pair of parameters (∆, Γ) is moving upward from region (b) to
When ∆ ∈ [∆,region (c) in figure 7 as the availability expands. The economy always has a globallystable steady state and, according to Table 1, the long-term growth rate is (∆ + Γ)σAwhen Γ ≤ Γ(∆) and (∆ + Γ(∆))σA when Γ > Γ(∆). Therefore, the expansion ofhigher education opportunities raises the long-term growth rate up to (∆ + Γ(∆))σA,and thereafter has no eﬀect.
Recall that ∆ ≡ δ/σA is a product of the low dependence on higher education intransferring existing knowledge over generations, a low saving rate, and also a low TFP.
These are typical characteristics of developing agrarian countries. The conventional
wisdom that the expansion of education opportunity is beneficial for growth, at least
when the availability is binding, applies to these countries.
23

Throughout this section, those changes in the availability of higher education are treated as exoge-

nous. We assume, however, that the availability changes only gradually over generations.
24

We mean ‘high ∆’ within the assumption that ∆ ≤ ∆.

28

Figure 8: Evolution of steady states. Parameters: ∆ = 0.1, α = 0.3

Economies with low ∆We obtain diﬀerent results when ∆ is low, which is likely to be the case for more developedb the pair of parameters (∆, Γ) is moving
and industrialized economies. When ∆ < ∆,upward from region (b) through region (d) to region (c) in figure 7 as Γ takes a larger
value. According to Table 1, the long-term growth rate is (∆ + Γ)σA in region (b),whereas it is (∆ + Γ(∆))σA in region (c). Our remaining task is to determine thelong-term growth rate when the pair belongs to region (d).
To this end, we must determine to which steady state the economy is converging
because there are two stable steady states in region (d) and the long-term growth rate is
diﬀerent among them. Figure 8 is a bifurcation diagram that plots all steady states valuesb
Xt stays in the neighborhoodof Xt against Γ. It suggests that until Γ reaches Γ(∆),

of XL∗∗∗ (∆, Γ) as long as the increase in Γ is not too fast, because this steady statebis (locally) stable. However, at the moment when Γ exceeds Γ(∆),
the steady state
XL∗∗∗ (∆, Γ) disappears and Xt is forced to converge to X ∗∗ (∆), which is the unique andglobally stable steady state in region (c). The implied variation in growth rate is striking.
b
As long as Γ is below the threshold value Γ(∆),
a gradual expansion of higher education
29

Figure 9: Responses of aggregate variables to a permanent expansion of higher education
opportunities At date 0, γ is increased from 1 to 1.05 while other parameters are fixed at A = 10,
α = 0.3, σ = 0.5 and δ = 0.5.

opportunities enhances the long-term growth rate, which is given by (∆ + Γ)σA (seeTable 1). However, once the availability exceeds the threshold value, the long-termbgrowth rate drops sharply from (∆ + Γ(∆))σAto (∆ + Γ(∆))σA. The magnitude ofslowdown is largest in highly industrialized economies with the smallest ∆.25 It implies

that the most serious slowdown occurs in economies that rely most heavily on higher
education to maintain their growth. In addition, these economies are the ones in which
the expansion of higher education opportunities is most eﬀective, unless it exceeds thethreshold.26
The movements of aggregate variables in transition help us understand the causeof this slowdown. Panel (i) of figure 9 depicts the evolution of the one-period growth
rates of Ht , Kt , Yt and St , while panel (ii) shows the movement of wt and rt against the25b
The magnitude of slowdown is measured by (Γ(∆)− Γ(∆))σA, which is proportional to the verticalbdistance between Γ(∆)
curve and Γ(∆) curve in figure 7.
26

By expanding higher education opportunities, the long-term growth rate can be increased frombb
∆σA up to (∆ + Γ(∆))σA.
Since Γ(∆)
is a decreasing function of ∆ (See proposition 1), the maximumbcontribution of higher education to long-term growth, Γ(∆)σA,is larger in economies with smaller ∆.

30

number of periods elapsed after a sudden and permanent 5% increase in the value of γ,which suﬃces to push the pair (∆, Γ) from region (d) to region (c). At the period when
the availability is expanded, households fully utilize the increased availability of higher
education, increasing the growth rate of human capital while decreasing that of physicalcapital. Observe that this reallocation of resources increases the output of the economy,
creating a temporary boom. This is a natural response: since the constraint on humancapital formation is relaxed, the resources are allocated more eﬃciently, at least in astatic sense, and therefore a larger amount of the consumption good is produced from agiven amount of the saved goods.
To see how this boom turns into a slowdown, we need to notice that the growth
rates of savings and physical capital keep decreasing during that boom. This process iscaused by the accelerated growth of human capital. Specifically, it initially lowers therate of return from higher education, wt ,and raises the interest rate (see Panel ii). Thesemovements shift income from the young households who participate in higher education
to the old who receive interest earnings, thereby causing a slight slowdown in the growthof aggregate savings. Combined with the accelerated growth of human capital, thedecelerated growth of savings lowers the growth rate of physical capital, which leads
to further decline in the return from higher education and another slight slowdown in
the growth of savings, and so forth. As this process runs on, the gap between the rateof return from higher education and the interest rate is narrowed gradually, but at anaccelerated pace. Eventually the gap vanishes, and the young households lose incentives
to utilize the availability of higher education up to its limit. Thus, given that the supplyof saved goods has been significantly reduced by that time, the growth rate of humancapital drops sharply. From that period on, the factor intensity as well as the aggregate
savings rate is constant over time, and the amount of output, human, and physical capital
exhibits a common oscillation, ending up at their new, lower steady state levels.

31

We close this section by noting the seriousness of the slowdown. While a marginal
expansion of higher education opportunities causes a significant slowdown, the high rateof growth cannot be recovered by simply reducing the availability of higher education to
the previous level. Figure 8 explains why the slowdown is irreversible. Once Xt is driven
into the neighborhood of X ∗∗ (∆) by an excessive expansion, it will be trapped near thisstable steady state as long as this steady state exists. To restore a high rate of economicb
growth, it is not suﬃcient to reduce Γ to below Γ(∆),
the level at which the slowdown istriggered.27 Figure 8 shows that Γ must temporarily fall below Γ(∆) so that the economy

can return to XL∗∗∗ (∆, Γ). Then Γ must be increased to the former level, perhaps to nearb
Γ(∆),
in order to raise the long-term growth rate to the pre-slowdown level. In reality,

such a drastic change in the availability of higher education would necessitate a major
restructuring of educational institutions. This is especially true when ∆ is small, sincebin that case the distance between Γ(∆)
and Γ(∆) is considerable.

6

Concluding Remarks
Empirical studies have shown that higher education raises individual earnings, butthat its contribution to economic growth in the long run has been less clear. This paper demonstrates that even when education raises everyone’s productivity, its aggregate
consequence on growth in the long run is not always positive. An excessive availabilityof higher education may trigger a serious economic slowdown, by shifting the intergenerational income distribution toward the old and reducing aggregate savings. Thispossibility is more relevant in countries where the transfer of existing knowledge is highly
dependent on higher education.
Throughout this paper, we treated the availability of higher education as exogenous.
27

Recall that in steady state X ∗∗ (∆), the availability of higher education is not fully utilized. Thus,a marginal change in the availability does not really aﬀect the working of the economy.

32

If the government could restrict the expansion of higher education, is there any possibility
that it would, willingly or not, cause the slowdown? There are two possibilities. First,
the government might not have enough knowledge to detect the onset of the slowdown
process during the period of temporary boom. When the economy experiences a decline
in the rate of economic growth, the supply of investible funds is already too small for the
economy to return to the fast-growing steady state. Second, even when the government
correctly anticipates the forthcoming slowdown, it might willingly leave the slowdown
process to take its own course because it usually places greater weight on the utility ofcurrent generations than on that of future generations. During the temporary boom, oldhouseholds enjoy a rapid growth of consumption owing to the higher interest rate, while
the growth rate of the young households’ consumption is gradually declining due to thedecreasing return from higher education. If the government heavily discounts the utility
loss of future generations, it may be justified to pursue this temporary gain.28
We have shown that expansion of higher education opportunities may result in adverse
consequences even without considering the costs of implementation. If these costs are
explicitly introduced, such expansion can lead to a more serious slowdown in economic
growth. However, when the associated costs are substantial, the authorities would be
more reluctant, or simply unable, to establish suﬃcient opportunities to trigger the
slowdown process. Similarly, a limited ability of households to learn may eﬀectively
prevent the speed of human capital accumulation from becoming too fast. There is no
denying that those limitations have been preventing overinvestment in human capital,
thereby contributing to sustained growth.
28

We found that, in the settings given in Figure 9, social welfare is improved by the excessive expansionof higher education opportunities whenever the intergenerational discount factor is smaller than 0.72.

33

Acknowledgements
We are grateful to Fumio Hayashi, Yoshiyasu Ono, Makoto Saito, Yoshihiko Seoka,
Akihisa Shibata, Yoshimasa Shirai, Katsuya Takii, Makoto Yano and seminar participants at Hitotsubashi University, ISER, Keio University, Kansai Macroeconomics Workshop, Kyoto Conference on Education and Training, the University of Tokyo, and theWorkshop on Spatial and Public Economics for their helpful comments and suggestions.
All remaining errors are naturally ours.

References
Azariadis, C. and D. De la Croix (2003) “Growth or Equality?” mimeo, Université
catholique de Louvain.
Azariadis, C. and A. Drazen (1990) “Threshold Externalities in Economic Development,” Quarterly Journal of Economics, 105, 501-526.
Barro, R. (1991) “Economic Growth in a Cross Section of Countries,” Quarterly Journalof Economics, 106, 407-444.
Barro, R. and J. H. Lee (1994) “Sources of Economic Growth,” Carnegie Rochester
Conference Series on Public Policy, 40, 1-46.
Barro, R. and X. Sala-i-Martin (2004) Economic Growth, 2nd Ed. MIT Press, Cambridge, Mass.
Benhabib, J. and M. Spiegel (1994) “The Role of Human Capital in Economic Development: Evidence from Aggregate Cross-Country Data,” Journal of Monetary
Economics, 34, 143-174.

34

Card, D. (1999) “The Causal Eﬀect of Education on Earnings,” Handbook of Labor
Economics, Volume 3, Amsterdam, Elsevier Science.
Caselli, F. (2005) “Accounting for Cross-Country Income Diﬀerences,” in P. Aghion and
S. Durlauf (Eds.), Handbook of Economic Growth, Amsterdam, Elsevier Science.
Christou, C. (2001) “Diﬀerential Borrowing Constraints and Investment in Human Capital,” Journal of Macroeconomics, 23(1), 277-295.
Diamond, P. A. (1965) “National Debt in a Neoclassical Growth Model,” American
Economic Review, 55, 1126-1150.
De Gregorio, J. (1992) “Economic Growth in Latin America,” Journal of Development
Economics, 39(1), 59-84.
De Gregorio, J. (1996) “Borrowing Constraints, Human Capital Accumulation, and
Growth,” Journal of Monetary Economics, 37, 49-71.
De Gregorio, J. and S. J. Kim (2000) “Credit Market with Diﬀerences in Abilities:
Education, Distribution and Growth,” International Economic Review, 41(3), 579607.
De la Croix, D. and P. Michel (2002) A Theory of Economic Growth. Cambridge
University Press, Cambridge, U.K.
Duan, X. (2003) “Chinese Higher Education Enters a New Era,” Academe, 89(6), 22-27.
Islam, N. (1995), “Growth Empirics: A Panel Data Approach,” Quarterly Journal of
Economics, 110(4), 1127-70.
Jappelli, T. and M. Pagano (1994) “Saving, Growth, and Liquidity Constraint,” Quarterly Journal of Economics, 109(1), 83-109.

35

Kaneko, M. (1997) “Eﬃciency and Equity in Japanese Higher Education,” Higher Education, 34, 165-181.
Klenow P.J. and A. Rodrı̀guez-Clare (2005) “Externalities and Growth,” in P. Aghion
and S. Durlauf (Eds.), Handbook of Economic Growth, Amsterdam, Elsevier Science.
Mankiw, N. G., D. Romer and D. N. Weil (1992) “A Contribution to the Empirics ofEconomic Growth,” Quarterly Journal of Economics, 107, 407-437.
Matsuyama, K. (1999) “Growing through Cycles,” Econometrica, 67, 335-347.
Modigliani, F. (1986) “Life Cycle, Individual Thrift, and the Wealth of Nations,” American Economic Review, 76(3), 297-313.
Pritchett, L. (2001) “Where Has All the Education Gone?” World Bank Economic
Review, 15, 367-391.
Romer, P. M. (1986) “Increasing Returns and Long-Run Growth,” Journal of Political
Economy, 94, 1002-37.
Romer, P. M. (1989) “Human Capital and Growth: Theory and Evidence,” NBER
Working Papers, No. 3173.
Solow, R. M. (1956) “A Contribution to the Theory of Economic Growth,” Quarterly
Journal of Economics, 70, 65-94.
Spence, A. M. (1973) “Job Market Signalling,” Quarterly Journal of Economics, 87,355-74.
Temple, J. (2001) “Generalizations that Aren’t? Evidence on Education and Growth,”
European Economic Review, 45, 905-918.

36

Topel, R. (1999) “Labor Markets and Economic Growth,” in O. Ashenfelter and D. Card
(eds.), Handbook of Labor Economics, Volume 3, Amsterdam, Elsevier Science.
Weiss, A. (1995) “Human Capital vs. Signalling Explanation of Wages,” Journal of
Economic Perspectives, 9(4), 133-154.

37