Discussion Papers In Economics
And Business

Who Benefits from a Better Education Environment?

Akiomi KITAGAWA

Ryo HORII

Koichi FUTAGAMI

Discussion Paper 04-15

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWho Benefits from a Better Education Environment?

Akiomi KITAGAWA

Ryo HORII

Koichi FUTAGAMI

Discussion Paper 04-15

July 2004
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Who Benefits from a Better Education Environment?
Akiomi KITAGAWA∗

Ryo HORII†

Koichi FUTAGAMI‡

Abstract
Using an overlapping generations model, this note shows that an improvementin the eﬃciency of human capital production decreases the net income of the young
household while increasing that of the old. Without compensating redistribution, it
deteriorates lifetime utilities of all generations except for the initial old households.
JEL Classification Number: J24, O15, I22
Keywords: human capital; intergenerational income distribution; overlapping generations.

∗Faculty of Economics and Business Administration, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, JAPAN; E-mail: akiomi@yokohama-cu.ac.jp†

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,

JAPAN; E-mail: horii@econ.osaka-u.ac.jp‡

Graduate School of Economics, Osaka University; E-mail: futagami@econ.osaka-u.ac.jp

1Introduction

The importance of the education environment has a prominent place in the literature ofendogenous growth, where the education environment determines the eﬃciency of humancapital production and thereby critically aﬀects the long-term path of aggregate output.1
This note demonstrates that the education environment also has a nontrivial eﬀect on theintergenerational distribution of income and that a better education environment does notnecessarily improve the welfare of households in the long run. We present an overlapping
generations model in which young households invest in the process of human capital production (i.e., education). A better education environment increases the supply of humancapital. The increased supply of human capital, on one hand, raises the productivity ofphysical capital and the interest rate, thereby raising the income of old households. On theother hand, the net income of young households is reduced because the increased supply
lowers the price of their labor endowments (i.e., wages) while the increased revenue fromselling human capital is oﬀset by education expenses.2 Although the initial old households unilaterally benefit from increased income, the overall welfare eﬀect on subsequent
generations is shown to be negative under plausible parameter values.

2Model

Production Technology. A version of Diamond’s (1965) overlapping generations model
is considered, in which time is divided into periods t = 1, 2, · · · , ∞. In each period, a single
final good, denoted by Yt , is competitively produced from physical capital Kt and humancapital Ht by a Cobb-Douglas technology. The production function is Yt = AKtα Ht1−α ,1In a model with a representative agent, Lucas (1988) and many others have shown that a better

education environment accelerates economic growth. In our separate paper (Kitagawa, Horii and Futagami,2003), we showed in an overlapping generations model that an expansion in the availability of education(i.e., the maximum amount of human capital that can be accumulated through education) has nonmonotonic eﬀects on the long-term rate of growth.
2In a static model of child labor, Basu and Van (1998) also showed that an exogenous expansion oflabor-supply capacity does not necessarily benefit its suppliers.

1where A > 0 and α ∈ (0, 1), respectively, represent total factor productivity and the shareof physical capital. Factor markets are perfectly competitive, so that the market price ofphysical capital rt and that of human capital wt , in terms of the final good, are determinedby their marginal productivities:
rt = Aα(Kt /Ht )α−1 ,wt = A(1 − α)(Kt /Ht )α .

(1)

The final goods produced in a certain period can either be consumed in that period or besaved for production in the next period. Once saved, the good can either be used as an
input to human capital production or be used directly as physical capital, which implies
that the price of the saved good (i.e. the interest rate) is rt . Human and physical capitaldepreciates within one period and therefore cannot be carried over to subsequent periods.
Households. At each period, there are two generations of households, which we callthe young and the old. Each generation contains a unit mass of households and lives fortwo periods. The objective of the generation t households (those born at period t) is tomaximize their lifetime utility
ut = (1 − β) ln c1t + β ln c2 t+1 ,(2)where β ∈ (0, 1) is a parameter specifying the patience of agents and c1t and c2 t+1 represent
their consumption in youth and old age. The young households are endowed with one unitof human capital. In addition, they can augment their human capital through education,
which must be financed by borrowing from the old. Let et ≥ 0 be the amount of saved
goods borrowed from the old generation to invest in this process. Then the amount of
their human capital isHt = 1 + γet ,(3)where parameter γ ≥ 0 represents the quality of the education environment, which determines the eﬃciency of human capital production. They sell oﬀ their human capital atmarket price wt and in return receive wt (1+γet ) units of the final good. After repaying rt etunits of the final good, they consume part of their net income at the end of that period andsave the remainder for consumption in their old age. The intertemporal budget constraint2

is
c1t + c2 t+1 /rt+1 = wt (1 + γet ) − rt et .

(4)

For every t ≥ 0, the generation t households choose et , c1t and c2 t+1 so as to maximize(2) under constraint (4). Net income in the right hand side of (4) is maximized by choosing=0if wt − γrt < 0;
(5)et ∈ [0, +∞) if wt − γrt = 0;


= +∞
if wt − γrt > 0.

Since human capital investment must be finite in equilibrium, condition (5) implies thatwt − rt ≤ 0 with equality whenever et > 0.

(6)

From (2), (4) and (6), the consumption and savings of generation t ≥ 0 households inequilibrium are written in terms of factor prices:c1t = (1 − β)wt , c2 t+1 = βrt+1 wt , St = βwt .

(7)

At period 0, the initial old (generation −1) households are endowed with S0 > 0 units ofsaved goods and consume r0 S0 units of final goods in exchange for their endowment.
Equilibrium. Substituting (1) and (3) into condition (6) gives a relation between two
kinds of capital,½¾
1−αKt , 1 .
Ht = max γα(8)Aggregate demand for the saved good consists of demand for physical capital Kt anddemand for input to human capital investment. From (3), the latter is γ −1 (Ht − 1). Thus,
the market-clearing condition for saved goods is
Kt + γ −1 (Ht − 1) = St−1 ,(9)where St−1 is the savings of generation t − 1 households. Given St−1 , (8) and (9) determine
the equilibrium (Kt , Ht ) pair,¢¡
if St−1 ≤ α/(γ(1 − α));¢  St−1 , 1¡
Kt , H t = ¡ α(St−1 + 1/γ), (1 − α)(γSt−1 + 1)¢ if St−1 > α/(γ(1 − α)),3(10)Panel 1: γ ≤ γbPanel 2: γ > bγ

Figure 1: Saving Dynamics

which shows that young households invest in human capital if and only if St−1 > α/(γ(1 −α)). Substituting (1) and (10) into (7) yields the saving dynamics:αβA(1 − α)St−1if St−1 < α/(γ(1 − α));St =βAαα (1 − α)1−α γ −α ≡ S(γ)
eif St−1 ≥ α/(γ(1 − α)).

(11)saving. As shown by Figure 1, the sequence monotonically converges to(βA(1 − α))1/(1−α) ≡ S if γ ≤ bγ;∗S (γ) =S(γ)
eif γ > γb,(12)Given initial S0 > 0, equation (11) generates the equilibrium sequence of aggregate

where γb ≡ α (βA(1 − α)2−α )−1/(1−α). Observe from Panel 2 of Figure 1 that the economy

settles to the steady state in one period whenever γ > b
γ and et > 0. The pair of factor
prices in the steady state is obtained by substituting (12) for (1):¢¡
 S/β, α/((1 − α)β) if γ ≤ γb;¢¡ ∗w (γ), r∗ (γ) = ¡¢ S(γ)/β,ee
γ S(γ)/βif γ > b
γ.
4(13)

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Implications of a Better Education Environment

Welfare eﬀects. Equation (13) implies that in the long run households receive education
only if γ > γb. When γ is in this range, a better education environment (i.e., highereﬃciency of human capital production) increases the interest rate but reduces the wage
rate in the steady state. Then, from (7), consumption of young households decreases,

whereas that of old households tends to increase.3 Substituting (7) and (13) into theutility function (2) gives the lifetime utility of consumers in the steady state:u∗ (γ) = constant + ln w∗ (γ) + β ln r ∗ (γ)(14)= constant + (β(1 − α) − α) ln γ.
Lifetime utility is decreasing in γ if (and only if)
β < α/(1 − α).

(15)Recall that α is the share of physical capital while β is the young agents’ propensity tosave. Using a conventional value of 0.3 for α, condition (15) becomes β < 0.428, whichis met under plausible values for β. In addition, (15) coincides with the condition for theeconomy to be dynamically eﬃcient for all γ in the steady state because applying (15) for(13) gives r∗ (γ) ≥ α/((1 − α)β) > 1.4 Therefore, a better education environment reduces
the lifetime utility of agents in the long run given that parameters are within an empiricallyplausible range or in a range that guarantees the economy’s dynamic eﬃciency.
Who benefits? One may wonder why relaxing one of the resource constraints in theeconomy results in an adverse consequence. To be precise, the economy with a high γ
is not Pareto inferior to the economy with a low γ, because consumption of the initial
old households is higher in the economy with a high γ.5 This implies that existing old
3

When γ > γb, consumption of old households is (βAαα (1 − α)1−α )2 γ 1−2α , which is increasing in γ

given that α < 1/2. We assume, reasonably, that α < 1/2 because α is the share of physical (non human)capital.
4

Since population is constant over time, dynamic eﬃciency requires the gross interest rate to be higher

than 1.
5

Recall that initial old households consume r0 S0 units of goods, where r0 is (weakly) increasing in γ5households in each period have an incentive to implement various policies that aim toimprove the educational environment for young households whenever possible. Contrary to
the usual perception, these seemingly altruistic policies actually benefit the old generation
themselves while future generations are thrust into an ‘educational rat race’.
Combination of policies. When combined with appropriate redistributional policies, however, a better education environment can improve the welfare of all generations.
Substituting (10) into the production function and then diﬀerentiating it with respect toγ yields

0∂Yt=³´α ³´∂γαα
A−S>0t−1γ(1−α)γ(1−α)

αif St−1 ≤ γ(1−α)
.

(16)

α.
if St−1 > γ(1−α)

Given St−1 , (16) shows that aggregate output is increasing in γ whenever young households
invest in human capital. Thus, old households benefit from a larger γ even when theauthority implements a lump-sum redistribution policy that transfers income from the oldto the young so that the income of young households (and therefore their savings) are
unaﬀected by the increase in γ. When continued forever, this combination of a larger
γ and the intergenerational transfer benefits all generations because they can enjoy moreconsumption when old while consumption in their youth is unchanged. In an economy witha highly developed education system, this argument legitimizes income transfers from old
to young in the forms of grants and scholarships funded by taxes on the elder generation.6

and S0 is historically given.
6

In the U.S., the percentage of students receiving grants and the average amounts received by students

with grant aid have increased between 1990 and 2000, which seems to be mitigating the problem. (Choy,2004). By contrast, the Japanese government has recently abolished a scholarship loan forgiveness program
so that all scholarships received by Japanese residents must be repaid after graduation. Although thesupply of scholarship loans has been increased in compensation, our model predicts that this change will
aggravate intergenerational income inequality in the long run.

6References
Basu, K. and P. H. Van, 1998, The economics of child labor, American Economic Review
88, 412-427.
Choy, S. P., 2004, Paying for college: changes between 1990 and 2000 for full-time dependent undergraduates, NCES 2004-075, U.S. Department of Education, National Center
for Education Statistics.
Diamond, P. A., 1965, National debt in a neoclassical growth model, American Economic
Review 55, 1126-1150.
Kitagawa, A.; R. Horii; and K. Futagami, 2003, Availability of higher education andlong-term economic growth, Discussion Papers in Economics and Business 03-14, OsakaUniversity.
Lucas, R. E., 1988, On the mechanics of economic development, Journal of Monetary
Economics 22, 3-42.

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