Discussion Papers In Economics
And Business
Education policy and R&D-based growth in an
overlapping-generations model
Kohei Okada

Discussion Paper 20-07

May

2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANEducation policy and R&D-based growth in an
overlapping-generations model∗
Kohei Okada†

Abstract
Employing an overlapping-generations model of R&D-based growth with endogenous education decision-making and government’s education policy, we examine how government’s education policy and human capital accumulation influence R&D activity. We show that an increasein government’s public education expenditure has an inverted U-shaped eﬀect on the growthrate at the steady state. We examine how increased public education expenditure aﬀects welfare
and show that an increase in the public education expenditure has an inverted U-shaped eﬀect
on the steady state level of welfare.
Keywords: Education expenditure, Human capital accumulation, R&D
JEL Classification: H52, I25, J24, O30∗

I especially thank Koichi Futagami for his helpful comments and encouragement.
Graduate School of Economics, Osaka University. Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: koheiokada257@gmail.com†

1Per capita GDP growth rate (%)

9
7
5
3
1
-1 0

2

4

6

8

10

12

-3
-5

Government expenditure on education (% of GDP)

Figure 1: The hump-shaped relationship between per capita GDP growth rate and government
expenditure on education. Source: World Bank. Cross-country data for the period from 1975 to
2005.

1

Introduction

Many countries have achieved economic development and a concomitant increase in the numberof people who receive education. Throughout this process, governments have adopted various
educational policies. At the same time, R&D activity has played a significant role in economic
growth. Our analysis focusses on how government’s education policy aﬀects R&D activity andeconomic growth. From Figure 1, we find that there is a hump-shaped relationship between percapita GDP growth rate and government expenditure on education1 . This indicates that an increasein government expenditure on education tends to increase the economic growth rate when the levelof government expenditure on education is low. However, as government expenditure on education
increases, it tends to reduce the economic growth rate.
To explain this phenomenon, we construct a simple overlapping-generations model of R&D-based
growth, with endogenous education decision-making and education expenditure. We assume that
1
The World Development Indicators (World Bank, 2019) are used to calculate government expenditure on education and per capita output growth rates. We use data on 180 countries for the period from 1975 to 2005 andestimate simple regressions, in which the average per capita output growth rate (Growth) is the dependent variable
and is a function of government expenditure on education (Education1) and the value of its square (Education2). The
following equation generates simple estimation results using ordinary least squares:

Growth = 0.162(0.249) + 0.714(2.783)Education1 − 0.065(−2.846)Education2,
where the figures in parentheses are the values of the t-statistics. The equation above suggests that there is a humpshaped correlation between government expenditure on education and the per capita GDP growth rate.

2each individual life comprises three periods. In the first period, individuals receive public education
provided by government. In the second period, individuals invest in their own education, supply
eﬃcient units of labor, pay income tax, consume diﬀerentiated goods, and save any remaining
income. In the final period, individuals retire and consume diﬀerentiated goods. In this model,
there is a production sector and an R&D sector. In the production sector, a single firm produces
a diﬀerentiated good by using eﬀective labor. Similarly, in the R&D sector, R&D firms use theeﬀective labor to invent new diﬀerentiated goods. The equilibrium dynamics of this economy arecharacterized by the level of human capital. We show that increasing the government’s public
education expenditure has an inverted U-shaped eﬀect on the economic growth rate at the steadystate. We also examine how increased public education expenditure aﬀects welfare and show that
increasing the government’s public education expenditure has an inverted U-shaped eﬀect on thewelfare level at the steady state.
This study is related to studies on human capital accumulation and R&D-based growth, forinstance, Grossmann (2007), Chu et al. (2016), Hashimoto and Tabata (2016), and Okada (2020).
Grossmann (2007) examines how education expenditure aﬀects R&D-based growth. However, hedoes not find a hump-shaped relationship between growth rate and education expenditure. Chu etal. (2016) find a hump-shaped relationship between growth rate and cultural preference for education. Hashimoto and Tabata (2016) find a hump-shaped relationship between growth rate and life
expectancy. Okada (2020) find a hump-shaped relationship between growth rate and government’s
public health expenditure. However, Chu et al. (2016) and Okada (2020) do not consider education
expenditure. Hashimoto and Tabata (2016) examine how education expenditure aﬀects the growthrate. However, they do not find a hump-shaped relationship between growth rate and education
expenditure.
The rest of this paper is organized as follows. Section 2 shows the basic structure of the model,
Section 3 analyzes equilibrium and dynamics. Section 4 analyzes education policy. Finally, theconclusion is given in Section 5.

3

2

The model

2.1

Individuals

Time is discrete and denoted by t = 0, 1, 2, · · · . Each individual lives for three periods (childhood,
adulthood, and old age). In the first period (childhood), individuals do not make any decisions
and receive public education. In the second period (adulthood), individuals invest in their own
education, supply eﬃcient units of labor, pay income tax, consume diﬀerentiated goods, and save
any remaining income. In the final period (old age), individuals retire and consume diﬀerentiatedgoods. Members of the cohort born in period t − 1 become active workers in period t. Thus, we callthis cohort generation t. The population size is constant and normalized to 1. Individuals derive
their utility from their own consumption during adulthood C1,t , and their own consumption during
old age C2,t+1 . The lifetime utility of individuals in generation t is expressed as

ut = log C1,t + β log C2,t+1 ,(1)where the positive γ denotes the weights of the children’s level of human capital. β ∈ (0, 1] denotes
the discount factor. We specify the subutility function Ck,t for k ∈ {1, 2} as

Ck,t ≡[∫ A t] α1ck,t (i) di ,
α

(2)

0

where ck,t (i) represents the consumption of diﬀerentiated good i ∈ [0, At ]. At denotes the variety ofdiﬀerentiated goods or the level of technological knowledge in this economy, which grows throughR&D. In individuals’ second period of life (adulthood), they are endowed with one unit of time,
which is devoted to working lt in the labor market and investing in their own education et . Individuals divide their disposable income (1 − τ )wt ht lt between consumption and saving st for their oldage. Here, wt and τ ∈ [0, 1) are the wage rate for eﬃcient units of labor, and the income tax rate,
respectively. Thus, the budget and time constraints for individuals in generation t are expressed as4

follows:
E1,t = (1 − τ )wt ht lt − st ,(3)

E2,t+1 = Rt+1 st ,∫ At
E1,t =
pt (i)c1,t (i)di,0∫ At+1E2,t+1 =pt+1 (i)c2,t+1 (i)di,

(4)
(5)
(6)

0

lt + et = 1,

(7)where E1,t , E2,t+1 , and p(i) denote the expenditure of an individual in adulthood, the expenditureof an individual in old age, and the price of good i, respectively.
We assume that the human capital production function is given by the following expression:1−σ,ht+1 = ϕeσt+1 lE,tσ ∈ (0.1].

ϕ > 0,(8)

where ϕ and σ are parameters, and. lE,t reflects the eﬀective labor of teachers who teach in publicschool.
We next consider the individual’s utility maximization. By maximizing the subutility function(2) subject to the budget constraint (5), we obtain the demand for diﬀerentiated good i as follows:

ck,t (i) =pt (i)−ϵEk,t ,
Pt1−ϵ1where ϵ ≡ 1−αand Pt is the price index defined by Pt ≡[∫At
1−ϵ dj0 pt (j)

(9)
] 1

1−ϵ

. This demand function

implies that indirect utility becomes a linear function of expenditure as follows:

Ck,t =

5

Ek,t
.
Pt

(10)

Let us denote the total demand for good i as xt (i). xt (i) is given by

xt (i) = c1,t (i)Nt + c2,t (i)λt−1 Nt−1 ,=pt (i)−ϵ(E1,t Nt + E2,t λt−1 Nt−1 ).
Pt1−ϵ

(11)

By maximizing (1) subject to (3), (4), (7), (8), and (10), we obtain the following solution:
σ
,
1+σ
1
,
lt =
1+σ
β(1 − τ )wt ht
st =.
(1 + β)(1 + σ)
et =

2.2

(12)
(13)
(14)

Production

There are diﬀerentiated goods indicated by i ∈ [0, At ]. A single firm produces each good. Eachfirm supplies a diﬀerentiated good monopolistically and sets its price. The monopoly is protectedby perfect patent protection. Each monopolistic firm produces one unit of good by using one unitof eﬀective labor. The producer of good i maximizes the following profit:
πt (i) = pt (i)xt (i) − wt xt (i),(15)

subject to the total demand function for good i (11). From the profit maximization condition, theprice of good i is

pt (i) =1
wt ≡ p t .
α(16)Hence, all goods have the same price. Thus, the firm-specific index i in the diﬀerentiated goods
sector can be dropped. By substituting (16) into the demand function (11), we obtain the output
level of the diﬀerentiated good:E1,t Nt + E2,t λt−1 Nt−1p−ϵ.
xt = ∫ At t1−ϵ (E1,t Nt + E2,t λt−1 Nt−1 ) =At pt
0 pt dj

6(17)

The total expenditure is treated as a numeraire (E1,t Nt + E2,t λt−1 Nt−1 = 1). Therefore, we canrewrite (17) by using (16) as follows:

xt =

1
α
=
.
At p twt At(18)

The profit of each diﬀerentiated good firm is given by
(
πt =

2.3

)
1
− 1 wt x t .
α

(19)

R&D

R&D firms use the eﬀective labor to invent new diﬀerentiated goods. After invention, the firms sella blueprint of a new good to an entrepreneur who produces the diﬀerentiated good. Developmentof At+1 − At new blueprints requires lR,t units of eﬀective labor input. Let us define ∆At as∆At ≡ At+1 − At . Thus, given research productivity of ψt , output is expressed as follows:∆At = ψt At lR,t ,(20)

where At implies the spillover from general knowledge accumulated by past innovations. We specify
ν−1productivity as ψt ≡ ψlR,twhere ψ > 0 and the parameter ν ∈ (0, 1) inversely measures the negative

duplication externality discussed in Jones (1995) and Jones and Williams (2000). In accordancetwith Jones (1995), the research productivity is given for each firm. gt ≡ ∆A
At is the growth rate ofproduct variety. Given lR,t , the growth rate of product variety features decreasing returns to scale
in lR,t . The R&D firms’ profits πtR are given by
πtR = vt ∆At − wt lR,t ,()
wt= vt −
∆At ,
ψt At

(21)

where vt is the price of a blueprint of a newly invented good. Because an entrepreneur pays the priceof the blueprint of the new good to the R&D firm, this price corresponds to the price of the equity

7

that is sold to the household. Free entry into R&D leads to the following zero-profit condition:
vt ≤

wt
ψt At

with equality if∆At > 0.

(22)

As shown in Appendix A, R&D is always undertaken and (22) holds with equality. The value of vt
equals the present value of future profits as follows:

vt =

∞
∑

πT
.
T
Π
R
T =t+1 ν=t+1 νAfter some manipulations, we obtain the following no-arbitrage condition:

Rt+1 =

vt+1 + πt+1
.
vt

(23)

Each individual saves at the gross rate of interest Rt+1 determined by (23).

2.4

Government

The government collects income tax from individuals and invests it in public education. We assume
that the government employs teachers and they supply public education services. Let us define lG,t
as the eﬀective labor that is employed as teachers. Public education expenditure is financed by thegovernment’s balanced budget. Therefore, the government’s budget constraint is

wt lE,t = τ wt ht lt .

3

Equilibrium

3.1

Market equilibrium(24)

We first describe the equilibrium condition of the asset market. Individuals’ savings must be directed
to purchase either the newly issued stocks for R&D or the existing stocks of the firms operating thathave been owned by the preceding generation. Therefore, the asset market equilibrium condition is

st = wt lR,t + vt At .

8

(25)

The left-hand side of (25) is the savings of the adult individuals. The first term on the right-handside is the total investment in newly issued stocks, which is equal to the total cost of R&D activities.
The second term is the purchase of existing stocks. Note that the second term implicitly assumes
that ex-dividend stocks are traded between adult and old generations in each period. In otherwords, the old generation always receives all dividends before selling the stocks.
We next describe the labor market equilibrium condition. Eﬀective labor is used for productionof diﬀerentiated goods, R&D, and public education services. The labor market-clearing conditionisht lt = lR,t + At xt + lE,t ,(26)

where lR,t , At xt , and lE,t denote the quantities of eﬀective labor engaging in R&D, production
activities, and the supply of public education services, respectively.
From (14), (20), (22), and (25), the growth rate of product variety gt is determined by thefollowing equation:1βψ ν (1 − τ )ht − 1−ν
gt + 1 =g ν .
(1 + β)(1 + σ) t

3.2

(27)

Dynamics

The dynamics of this economy is characterized by human capital ht , and is given by

ht+1 =σσ ϕ(τ ht )1−σ ≡ Φ(ht ; τ ).
1+σ

(28)

Let us define the right-hand side of (28) as Φ(ht ; τ ). Figure 2 shows the phase diagram of ht . we
obtain the following proposition.
Proposition 1 The economy has a unique steady state.
Proof: see Appendix B

9

ℎ&'(
45°

ℎ&'( = Φ ℎ& ; 𝜏

ℎ∗

0

ℎ&

Figure 2: Dynamics of ht

4

Education policy

In this section, we consider how government policy aﬀects this economy. The government can set
the income tax rate in this model. Therefore, we focus on how changes of the tax rate aﬀect thiseconomy. First, we consider the eﬀects on economic growth. Let us denote g ∗ as the growth rate ofproduct variety of the steady state. We obtain the following proposition.
Proposition 2 An increase of tax rate τ has an inverted U-shaped eﬀect on the steady state growthrate g ∗ and it is maximized at τ = 1 − σ.
Proof: see Appendix C
Let us define h∗ as the level of human capital in the steady state. If τ increases, Φ(ht , τ ) shifts upward
in Figure 2 and h∗ increases. Therefore, individuals save more and R&D investment increases. Then,the growth rate increases (the positive eﬀect). However, if the tax rate τ increases, disposal income
decreases. Individuals’ savings consequently decrease, R&D investment decreases, and thus the2 ∗growth rate decreases (the negative eﬀect). In addition, we can find that ∂∂τh2 < 0 holds. Therefore,
the positive eﬀect on the growth rate decreases when τ increases. Hence, an increase of tax rateτ has an inverted U-shaped eﬀect on the steady-state growth rate g ∗ . Figure 3 shows a numericalexample of the inverted U-shaped eﬀect of the tax rate on the steady-state growth rate. We obtainthe following proposition.
10

Figure 3: The inverted U-shaped eﬀect of the tax rate on the steady-state growth rate. We assume
that β = (0.98)25 , ν = 0.5, σ = 0.635, ϕ = 1, and ψ = 4.9347.
Proposition 3 The steady state level of welfare is maximized at τ = 1 − σ.
Proof: see Appendix D
The government can maximize not only the steady state growth rate but also the steady state levelof welfare by setting τ = 1 − σ.2

5Conclusion

We constructed an overlapping-generations model of R&D-based growth with endogenous education
choice. We showed that increasing the government’s public education expenditure has an inverted
U-shaped eﬀect on the growth rate at the steady state. We examined how increased public education
spending aﬀects welfare, and showed that increasing the government’s public education expenditure
has an inverted U-shaped eﬀect on the steady state level of welfare.

2

Barro (1990) shows that there is an inverted-U shaped relationship between growth rate and government expenditure for production. Our results are consistent with his. In contrast to Barro (1990), we focus on education policy
and R&D-based growth.

11Appendix
Appendix A: Proof that R&D is always undertaken
We show that R&D is always undertaken in this model. Suppose that R&D is not undertaken and∆At = 0. Then, (22) holds with strict inequality as follows:

vt <

wt
.
ψ t At(A1)

From (20), we have
∆At
gt =
=ψ
At

(

lR,t
ht Nt

)ν
.

(A2)

From (A2), if ∆At = 0, lR,t = 0. By using (13), (18), (22), (24), and (26), we obtain the wage rate
wt as follows:
α

wt =

1−τ
1+σ ht −( )1 ,
gt
ψ

(A3)

ν

α(1+σ)
From (A3), when ∆At = 0 (i.e., gt = 0), wt = (1−τ)ht . In addition, when lR,t = 0, ψt approaches

infinity. Therefore, the right-hand side of (A1) converges to 0 when ∆At = 0. Meanwhile, when
∆At = 0 (i.e., lR,t = 0), we obtain the following equation from (14) and (28),vt =β(1 − τ )wt ht
st
=
> 0.
At
(1 + β)(1 + σ)At

(A4)These results contradict (A1). Therefore, ∆At > 0 always holds in this economy.

Appendix B: Proof of Proposition 1σt)= σ (1−σ)ϕτ
From (28), we obtain ∂Φ(h
∂ht
1+σ

1−σ
σ+1
∂ 2 Φ(ht )= − σ (1−σ)ϕ
h−σ−1
< 0.
t
τ
∂h2t

1−σ

t)
t)
= ∞, limht →∞ ∂Φ(h= 0,h−σ> 0, limht →0 ∂Φ(h
t
∂ht
∂ht

Therefore, the economy has a unique steady state as shownin Figure 2.

12Appendix C: Proof of Proposition 2
From (28), we obtain
∗

(

h =

σσ ϕ
1+σ

)1

σ

1−σ

τ σ .

(A5)

From (27), we obtaing ∗ + 1 = Ω(1 − τ )τ σ (g ∗ )− ν ,
1−σ

1
ν

βψ
where Ω ≡ (1+β)(1+σ)( σ )1
σ ϕ
1+σ

σ

1−ν

(A6)

. Totally diﬀerentiating (A6) with respect to g ∗ and τ , we obtaindg ∗σΩτ σ (g ∗ )− ν (1 − σ − τ )=1dτ
1 + (1−ν)(1−τ )Ω (g ∗ )− ν
1−2σ

1−ν

(A7)

ν

∗

From (A7), if τ < 1 − σ (τ > 1 − σ), dg
dτ > 0(dg ∗dτ < 0).

Appendix D: Proof of Proposition 3
From (3) and (14), we obtain

E1,t =(1 − τ )wt ht.
(1 + β)(1 + σ)

(A8)From (4) and (14), we obtain

E2,t = Rt+1

By using (16), we rewrite Pt ≡[∫At
1−ϵ dj0 pt (j)β(1 − τ )wt ht.
(1 + β)(1 + σ)

(A9)

] 1

1−ϵ

as follows:1

A 1−ϵ wt
Pt = t
.
α

13

(A10)

Combining (18) and (19), we obtain
1−α
.
At

πt =

(A11)

ν−1
From (20) and ψt ≡ ψlR,t, we obtain1

ν−1ψt = ψ ν gt ν .

(A12)Note that ht+1 = ht = h∗ and gt+1 = gt = g ∗ hold in the steady state. Then, from (22), (A3),(A11) and (A12), we can rewrite (23) as follows:
wt+1 ψt At(1 − α)ψt At+,
wt ψt+1 At+1
At+1 wt[](1 − α)ψ ∗R∗ = (1 + g ∗ ) 1 +,w∗

Rt+1 =→(A13)

where R∗ , ψ ∗ and w∗ denote steady state values of Rt , ψt and wt , respectively. Combining (1), (10),(A5), (A8), (9), and (A13), we obtain the welfare level of t th generation u∗t as follows:1−σβu∗t = Λ + (1 + β) log(1 − τ )τ σ + log(1 + g ∗ ) + β log

α[](1 − α)ψ ∗α(1 + β)1++log At ,
∗
w
1−α
(A14)

α
where Λ ≡ (1 + β) log (1+β)(1+σ)( σ )1σ ϕ
1+σσ+ β log β. We can easily confirm that the second term on

the right-hand side of (A14) is maximized at τ = 1 − σ. From Proposition 2, the third term on the∗right-hand side of (A14) is also maximized at τ = 1 − σ. Let us define Z(τ ) as Z(τ ) ≡ (1−α)ψ
.
w∗
Noting (A3), (A5), and (A12), we rewrite Z(τ ) as follows:Z(τ ) =]1−σ1
1 − α ∗ ν−1 [(g ) ν (1 + β)(1 − τ )τ σ Ω − (g ∗ ) ν .
αβ

(A15)

Diﬀerentiating Z(τ ) with respect to τ , We obtain() 1−2σ
1−σ
ν−2
σ
(1
−
α)(1
+
β)(1
−
ν)(1
−
τ
)
1
−
τ
στ σ Ω2 (g ∗ ) ν (1 − σ − τ )∂Z(τ )[]=.
)Ω ∗ − 1∂τν(g)
αβ 2 ν 1 + (1−ν)(1−τ
ν

14

(A16)

From (A16), we can find that Z(τ ) is maximized at τ = 1 − σ. Therefore, the fourth term on theright-hand side of (A14) is maximized at τ = 1 − σ. Given At , the welfare level of t th generationu∗t is maximized at τ = 1 − σ. From Proposition 2, for any time t, At is maximized at τ = 1 − σ inthe steady state. Therefore, the steady state level of welfare is maximized at τ = 1 − σ.

References[1] Barro, R. J. (1990). Government spending in a simple model of endogeneous growth. Journalof political economy, 98(5, Part 2), S103-S125.
[2] Chu, A. C., Furukawa, Y., and Zhu, D. (2016). Growth and parental preference for educationin China. Journal of Macroeconomics, 49, 192-202.
[3] Grossmann, V. (2007). How to promote R&D-based growth? Public education expenditure on
scientists and engineers versus R&D subsidies. Journal of Macroeconomics, 29(4), 891-911.
[4] Hashimoto, K. I., and Tabata, K. (2016). Demographic change, human capital accumulation
and R&D-based growth. Canadian Journal of Economics, 49(2), 707-737.
[5] Jones, C. I. (1995). R&D-based models of economic growth. Journal of Political Economy,
103(4), 759–784.
[6] Jones, C. I., and Williams, J. C. (2000). Too much of a good thing? The economics of investment
in R&D. Journal of Economic Growth, 5(1), 65–85.
[7] Okada, K. (2020). Dynamic analysis of demographic change and human capital accumulationin an R&D-based growth model. Journal of Economics, 1-24.

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