The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No. 2011-6(accepted in March 2, 2012)

Public Investment, Health Infrastructure and Income Growth
Minoru Watanabe, Yusuke Miyake, and Masaya Yasuoka

Public Investment, Health Infrastructure and Income Growth∗
Minoru Watanabe† Yusuke Miyake ‡ Masaya Yasuoka§
December 19, 2011

Abstract
Based on individual occupational choice in a model including a production function with public
investment and public health infrastructure, this paper presents an examination of how allocation of
public investment and public health infrastructure affects the dynamics of income. Individuals workas skilled laborers or unskilled laborers, as in the model described by Caselli (1999), and educationalcosts are necessary to work as a skilled laborer. Results show that government should provide bothpublic investment and public health infrastructure to escape from the poverty trap with low income.
Moreover, based on an initial allocation between public investment and public health infrastructure,
it is decided how the government should form a policy to increase income growth.

JEL Classification: H54, E20
Key Words: Public investment, Health infrastructure, Life expectancy, Occupational choice, Economic growth∗ We would like to thank Shinji Miyake, Tamotsu Nakamura, and Toshiki Tamai for helpful comments. Research for thispaper was supported financially by the University of Kitakyushu. Any remaining errors are solely the authors’ responsibility.
† Graduate School of Economics, Kobe University‡ Graduate School of Economics, Kobe University§ Corresponding to: Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata
Kokuraminami, Kitakyushu, Fukuoka 802-8577 Japan, Tel.:+81-93-964-4318, E-mail: yasuoka@kitakyu-u.ac.jp1

1

Introduction

As explained by Foster and Briceño-Garmendia (2010) and UN-HABITAT (2011), African countries
must be provided infrastructure (Transport, Modern Energy, Telecoms, Water System, Sanitation, andso on) to foster economic growth and to escape from poverty. Providing infrastructure can achieve
the Millennium Development Goals. UN-HABITAT (2011) introduces to the macroeconomic empiricalliterature the idea that the development of infrastructure brings about economic growth and productivity
effects (Estache, Speciale and Veredas (2005), Ayogu (2007)). In developed countries, that infrastructure
is sufficiently provided, but not in developing countries, especially in Sub-Saharan African countries.1
Therefore, the government must carry out policies to increase the infrastructure. Governments in SubSaharan African countries for infrastructure spend, on average 6 − 12% of their gross domestic product(GDP) (UN-HABITAT (2011)).
Referring to these data, we consider the manner in which the government should provide infrastructure. Infrastructure of some kinds exists. This paper presents examination of the allocation of theinfrastructure of two types: one for public investment, which increases labor productivity (transport,
telecoms, and so on) and the other for health infrastructure, which raises life expectancy (water system,
sanitation, hospitals, and so on).
Based on simple OLG, we discuss occupational choice, either skilled or unskilled, in addition to
public investment with life expectancy. Especially, we address not only public investment but also life
expectancy against dynamics. Many papers describe studies of capital accumulation by government.
Barro (1990), Barro and Sala-i-Martin (1992), and Futagami, Morita and Shibata (1993) examine public
capital accumulation and growth. Glomm and Ravikumar (1997) show public accumulation and growthwith human capital investment. Turnovsky (1997) also discusses public capital accumulation and growth
and the difference between a socially planned economy and a decentralized economy. Yakita (2008)
discusses an endogenized fertility rate and an aging economy including public accumulation. Public
investment in these earlier studies serves an important role in increasing productivity.
Apart from public capital accumulation, many papers have described studies of educational choice.
Maoz and Moav (2000) examine skill acquisition and inequality including intergenerational mobility.
Caselli (1999) and Galor and Moav (2000) introduce an idea that new technology prohibits people fromlearning new skills. Chen (2010) uses an overlapping generations model with life expectancy and educational choice.
Some studies have been conducted on the assumption that life expectancy is set exogenously, such as
Chen (2010). However, some papers consider life expectancy as an endogenous variable. Chakraborty
1 As shown by UN-HABITAT (2011), in Sub-Saharan African countries, paved roads are 11.9% of all roads (2006), accessto electricity is available in only 18% of households (2004), water with improved water sources is accessible to only 58% of
population (2006), and only 31% of the population has access to improved sanitation facilities (2006).

2

(2004) and Hashimoto and Tabata (2005) set the model that life expectancy depends on public expenditure for health infrastructure, such as hospitals, clean water supply, and so on. By virtue of public
expenditure, income per capita increases because capital accumulation is stimulated.2 Chakraborty andDas (2005) and Bhattacharya and Qiao (2007) use an economy model in which private health investment
can raise life expectancy and income per capita because of an increase in the saving rate.
In fact, we can consider two reasons that an economy might become enmired in a poverty trap with
low income: low productivity and a low saving rate. If a government provides public infrastructure,then the productivity of labor and capital increases. Thereby, the model economy escapes from thepoverty trap. However, an increase in public health infrastructure raises the saving rate and then laborproductivity rises thanks to an increase in capital accumulation. Finally, the model economy escapes fromthe poverty trap. Our paper presents examination as the following process. First, based on Chen (2010),
the paper presents public expenditure of two types (public investment and public health infrastructure)and examines what the government should provide to escape from the poverty trap. Second, after escaping
from the poverty trap, the paper presents derivation of how the government allocates tax revenue between
public investment and public health infrastructure to increase income growth.
As derived in this paper, the allocation for public health infrastructure to escape from the poverty trap
should be within a certain range. If this allocation is large because of a decrease in public infrastructure
and low productivity of labor, then the economy can not escape from the poverty trap. However, if theallocation for public investment is large, then capital accumulation is prevented and productivity of laboris low and the economy can not escape from the poverty trap. Moreover, based on the initial allocation,
the government is expected to provide public infrastructure or public health infrastructure to raise theincome growth rate. The results obtained in this study show how the government provides a policy tobring about income growth with a given tax revenue.
The remainder of this paper is organized as follows. Section 2 introduces the model. Section 3 presents
a description of equilibrium and macroeconomic dynamics and derives the conditions under which income
growth continues. Lastly, we summarize the salient points of the paper.

2

The Model

The model economy is based on a two-period (young and old) overlapping generations model. Thiseconomy has agents of three types: households, firms, and a government.

2.1

Households

Households experience two periods: young and old. During the young period, each household supplies
labor inelasticity to earn labor income. This economy accommodates labor of two types: skilled labor
2 Hashimoto and Tabata (2005) derived the relation between health infrastructure and fertility.

3and unskilled labor. Education costs must be incurred in order to become a skilled laborer, as assumed
by Caselli (1999), Meckl and Zink (2004), Miyake, Muro, Nakamura, and Yasuoka (2009), and by Chen
(2010). That cost is assumed as σ. Herein, wts denotes the wage rate of skilled labor. The government
imposes labor income taxation on the wage income of skilled labor to provide public investment andpublic health infrastructure.3 Each household allocates its labor income between consumption in theyoung period and saving. Consequently, we obtain the following budget constraint:
cs2t+1= (1 − τ )wts − σ,Rt+1cu
cu1t + 2t+1 = wtu .
Rt+1

cs1t +(1)(2)

Indexes s and u respectively denote skilled labor and unskilled labor. In addition, c1t and c2t+1 respectively denote consumption in the young period and old period. wtu denotes the wage rate of unskilled
sector. Rt+1 signifies an interest rate for annuitized savings. τ signifies labor income tax rate (0 < τ < 1).
Finally, t denotes the period. A household’s utility function ut is given as shown below.
ut = α ln c1t + pt (1 − α) ln c2t+1 , 0 < α < 1, 0 < pt < 1(3)

Therein, pt denotes the probability that the individual lives during the old period. These savings are allocated among older living people if the individual dies: this is annuitized wealth. The optimal allocations
at skilled labor are determined asα((1 − τ )wts − σ) ,α + pt (1 − α)pt (1 − α)Rt+1
c2t+1 =((1 − τ )wts − σ) .
α + pt (1 − α)
c1t =

(4)
(5)

If a worker is an unskilled laborer, thenαwu ,α + pt (1 − α) tpt (1 − α)Rt+1 u
c2t+1 =w .
α + pt (1 − α) t
c1t =

2.2

(6)
(7)

Firm

This paper assumes the production function shown below.4
Yt = AKtθ (Gt Lt )1−θ + B(1 − Lt ), 0 < θ < 1, 0 < A, 0 < B(8)

3 This paper assumes that the government imposes income tax for skilled labor wage income in terms of redistribution.
Assuming that the government can not capture an unskilled labor wage, the government can collect tax revenue from only
skilled labor wage income.
4 Some papers consider a production function with public investment. For example, Barro (1990) assumed Y =K θ (GL)1−θ . In addition, Caselli (1999) assumed that not only labor but also capital stock is inputted as a productive
factor in the unskilled sector. Neither Caselli (1999) nor Chen (2010) considered public investment.

4

Therein, Yt denotes the aggregate output. Gt and Kt respectively denote public investment and capitalstock. Lt denotes the skilled labor amount. Assuming that the population size of each generation isunity, then the unskilled labor amount is shown as 1 − Lt . With a perfectly competitive market, profit
maximization reduces the following equations, as
wts = A(1 − θ)G1−θ
t

Kt
Lt

θ
,

wtu = B.

(9)(10)

The interest rate is shown as
1 + rt = Aθ

Kt
Gt Lt

θ−1
.

(11)

It is noteworthy that
Rt+1 =

1 + rt+1
.
pt

(12)

Capital stock is assumed to be fully depreciated in one period.

2.3

Government

The government imposes labor income taxation at a tax rate τ on skilled labor to provide public investment Gt and public health infrastructure Ht . Public health infrastructure is regarded as hospitals, cleanwater systems, and so on to raise life expectancy. Then, the government budget constraint is presentedas
Gt + Ht = τ wts Lt .

(13)

Our paper assumes the following allocation rule.
Gt = βτ wts Lt(14)

Ht = (1 − β)τ wts Lt(15)

In those equations, β (0 < β < 1) denotes the ratio of public investment to tax revenue and 1 − β denotesthat of public health infrastructure. Moreover, this paper includes the assumption of life expectancy as
pt = min [CHt , 1] (C > 0 and 0 < pt < 1).

3

Equilibrium

This section presents derivation of the equilibrium of this model economy. If workers move freely between
two sectors, then the indifference condition is described as
wts =

B+σ
≡ ŵ.
1−τ

5

(16)

Considering (9), (14), and (16), Lt is given asθ! 2θ−11−θ
1−θ
11θ (1 − θ) θ A θ β θ K
(1
−
τ
)τ
t
Lt = min , 1 .
B+σ(17)

Intuitively, an increase in Kt raises Lt because an increase in wts brings about an increase in the amountof skilled labor. Therefore, we assume that θ > 21 .5 Additionally, we assume the total population size ineach generation as unity. Then, the dynamics of capital stock at Lt < 1 is derived as shown below:
pt (1 − α)pt (1 − α)B(((1 − τ )wts − σ)Lt + B(1 − Lt )) =,(18)
α + pt (1 − α)α + pt (1 − α)




τ (B+σ)t6where pt = C τ (1−β)(B+σ)L
.
This
paper
assumesC< 1. In contrast, an increase in Kt
1−τ
1−τ
Kt+1 =

raises Lt . Therefore, the dynamics of capital stock at Lst = 1 is derived aspt (1 − α)
(XKt − σ) ,(19)
α + pt (1 − α)
h 
 i
1−θ
1−θ
1−θ
1
1
1
1
1
where pt = min C (1 − β)τ θ A θ (1 − θ) θ β θ Kt , 1 and X = (1 − τ )A θ (1 − θ) θ β θ τ θ .7 Then,
Kt+1 =

wts and 1 + rt at Lt = 1 are
1−θ

1−θ

1−θ

1−θ

1

1

wts = β θ τ θ (1 − θ) θ A θ Kt ,
1−θ

(20)

1

1 + rt = β θ τ θ θ(1 − θ) θ A θ .
1−θ

(21)
1−θ

1−θ

1

Therefore, calculating Yt = (1 + rt )Kt + wts Lt , we obtain Yt = β θ τ θ (1 − θ) θ A θ Kt . The level ofcapital stock K̂ that brings about wts = ŵ and Lt = 1 is
K̂ =

B+σ
(1 − τ )τ

1−θ
θ

1−θ

1

1

β θ (1 − θ) θ A θ

.

(22)

Then, we can depict the dynamics of Kt as presented below.
[Insert Fig. 1 around here.]
1

1

1−θ

1−θ

Assuming (1 − α)(1 − τ )A θ (1 − θ) θ β θ τ θ > 1, there exist dynamics of two types. The first is that
income growth occurs for any K0 (Fig. 1-1). The second is that income growth occurs or does not occur
for given K0 (Fig. 1-2). In Fig. 1-1, given an initial K0 , income growth continues. Then, only skilledlabor exists. However, in Fig. 1-2, given K0 less than K̃, the capital stock converges to K ∗ , which exists
for both unskilled labor and skilled labor. It is the poverty trap bringing about low income B. The
pt (1−α)B
condition not to have the poverty trap is α+p>t (1−α)1>B+σ(1−τ )τ

1−θ
1−θ
1
1
θ β θ (1−θ) θ A θB+σ(1 − α)B(1 − τ )τ1−θ
θ

α + pt (1 − α)1θ(1 − θ) A

1
θ

1−θ

pt β θ

; that is,

,

(23)

5 1 − θ denotes the elasticity of output Y for public investment G, i.e., dY G . Our paper assumes small elasticity suchdG Y
as Barro (1990). Barro (1990) assumed 1 − θ = 0.25 and simulated the model economy.
6 Considering (17) and p = Ctτ (1−β)(B+σ)Lt1−τ, we obtain

∂Kt+1
∂2 K
> 0 and ∂Kt+1
< 0.
2
∂Kt
t

7 See Appendix for the form of this dynamics equation.

6
where pt = C

τ (1−β)(B+σ)
1−τ



.

Defining the right-hand-side of this inequality as F (β), we obtainαθ βlimβ→0 F (β) = ∞ and limβ→1 F (β) = ∞. In addition, β ∗ exists to hold 1−θ1−β = (1 − α)pt + α.

With β < β ∗ , the sign of dFdβ(β) is negative and the sign is positive if β > β ∗ . Then the following figureis shown.
[Insert Fig. 2 around here.]
The solid line is given by the following condition as1>

where pt = C

τ (1−β ∗ )(B+σ)
1−τ

B+σ(1 − α)B(1 − τ )τ



1−θ
θ

α + pt (1 − α)1θ(1 − θ) A

1
θ

1−θ

pt β ∗ θ,(24). Otherwise, the dashed line is given. At the solid line, if β exists between

β0 and β1 , then the dynamics is shown in Fig. 1-1 and the poverty trap does not exist. Therefore, even
if the economy stays in the poverty trap shown by K ∗ , the economy escapes from the poverty trap andincome growth continues as the government sets β within β0 < β < β1 . However, with the parametriccondition to hold the dashed line, the economy can not escape from the poverty trap. Then, the following
proposition is established.
Proposition 1

If Eq. (24) holds, then the economy can escape from the poverty trap and income

growth continues to set β within β0 < β < β1 .

This proposition is intuitive. If β is small, then public health infrastructure is large but public
investment is small. Small public investment decreases labor productivity and the income level. Then,the saving is small and capital accumulation is not large. Therefore, the economy can not escape fromthe poverty trap. However, if β is large, then public investment is large but public health infrastructureis small. Small public health infrastructure brings about short life expectancy and the saving rate is low.
However, if the condition of (24) does not hold, then the government can not induce the economy toescape from the poverty trap with the allocation of β. The government must collect more tax revenue
and allocate public investment and public health infrastructure. However, an increase in tax burden
decreases capital accumulation because of a decrease of the saving. If the following condition


1−θ
α
1
−
(α + (1 − α)pt ) +
<0
1−τ
θ
1−β

(25)

is held, then an increase in τ shifts down the curve of F (β).8 Defining τ ∗ to equalize (25), F (β ∗ ) at
τ = τ ∗ is larger than one, then the economy can not escape from the poverty trap even if the governmentchanges τ and β. The following proposition is established.
8 We obtain this condition as dF (β)
dβ< 0. This condition shows that the effect to decrease capital accumulation by anincrease in τ is small.

7

Proposition 2

If F (β ∗ ) at τ = τ ∗ is larger than one, then the economy always stays in the poverty

trap even if the government changes τ and β.

Next, we consider that the economy escapes from the poverty trap and that income growth continues
and examine how the allocations of β affect income growth. Calculating dKdβt+1 at (19) in pt < 1, we
obtain the following equation.
(1 − α)pt
dKt+1
=
dβ
βθ(α + (1 − α)pt )(1 − θ)XKt +α(1 − β − θ)
(XKt − σ) .
(α + (1 − α)pt )(1 − β)(26)

If 1 − β − θ > 0, then the sign of dKdβt+1 is positive. In 1 − β − θ < 0, ifα(β + θ − 1)α−,(1 − α)(1 − β)(1 − θ) 1 − αα(1 − θ − β)σ,Kt <((1 − β)(1 − θ)(α + (1 − α)pt ) + α(1 − θ − β)) X
pt <

(27)
(28)

the sign of dKdβt+1 is positive. Then the following proposition is established.
Proposition 3

If income growth continues and 1 − β − θ > 0, then an increase in β can always raise

the income growth rate. If 1 − β − θ < 0 and the condition given by Eqs. (27) and (28) holds, then an
increase in β can raise the income growth rate.

An increase in β increases wage rate wts and the saving increases, too. However, an increase in β
reduces public health infrastructure. This effect decreases capital accumulation. Moreover, an increasein β raises tax revenue because of an increase in public investment and reduces tax revenue because of adecrease in capital accumulation because of a decrease in life expectancy. If 1 − β − θ > 0, then the effectof an increase in β raises tax revenue dominates; then life expectancy rises. However, if 1 − β − θ < 0,then the effect of an increase in β decreases life expectancy predominantly. If this effect is large, then an
increase in β reduces income growth.
= X. We note that limKt →∞ Kσt = 0
The income growth rate in this model economy converges to KKt+1t
and limKt →∞ pt = 1. Then, an increase in β can always raise income growth because an increase in β
does not affect life expectancy, which is a sufficiently large level.

4

Concluding and Remarks

This paper described a model with public investment and public health infrastructure and illustrated
how an increase in public investment and public health infrastructure affect capital stock, the amountof skilled labor, and the wage rate. First, the government must adequately allocate public investment
and public health infrastructure to escape from the poverty trap. Public investment and an increase8

in life expectancy with public health infrastructure can stimulate capital accumulation. Therefore, thegovernment sets these allocations not to be disturbed these both effects. Second, if the economy canescape from the poverty trap and income growth continues, then an increase in public investment can
not always raise income growth. However, if the allocation of tax revenue for public health infrastructureis large or capital accumulation is small, an increase in public investment can raise income growth.

9References[1] Ayogu M. (2007) “Infrastructure and Economic Development in Africa: A Review,” Journal of
African Economics, Supplement 1, pp.75-126.
[2] Barro R.J. (1990) “Government Spending in a Simple Model of Endogenous Growth,” Journal of
Political Economy, vol.98, pp.S103-S125.
[3] Barro R.J. and Sala-i-Martin X. (1992) “Public Finance in Models of Economic Growth,” Review of
Economic Studies, vol.59, pp.645-661.
[4] Bhattacharya J. and Qiao X. (2007) “Public and Private Expenditure on Health in a Growth Model,”Journal of Economics Dynamics & Control, vol.31, pp.2519-2535.
[5] Caselli F. (1999) “Technological Revolutions,” American Economic Review, vol.89-1, pp.78-102.
[6] Chakraborty S. (2004) “Endogenous Lifetime and Economic Growth,” Journal of Economic Theory,
vol.116, pp.119-137.
[7] Chakraborty S. and Das M. (2005) “Mortality, Fertility, and Child Labor,” Economics Letters, vol.86,
pp.273-278.
[8] Chen H. (2010) “Life Expectancy, Fertility, and Educational Investment,” Journal of Population
Economics, vol.23, pp.37-56.
[9] Estache A., Speciale B. and Veredas D. “How Much Does Infrastructure Matter to Growth in SubSaharan Africa?” The World Bank.
[10] Foster V. and Briceño-Garmendia C. (2010). “Africa’s Infrastructure – a Time for Transformation,”
the World Bank.
[11] Futagami K., Morita Y. and Shibata A. (1993) “Dynamic Analysis of an Endogenous Growth Modelwith Public Capital,” Scandinavian Journal of Economics, vol.95, pp.607-625.
[12] Galor O. and Moav O. (2000) “Ability Based Technological Transition, Wage Inequality and Economic Growth,” Quarterly Journal of Economics, vol.115, pp.469-497.
[13] Glomm G. and Ravikumar B. (1997) “Productive Government Expenditure and Long-Run Growth,”
Journal of Economic Dynamics and Control, vol.21, pp.183-204.
[14] Hashimoto K. and Tabata K. (2005) “Health Infrastructure, Demographic Transition and Growth,”
Review of Development Economics, vol.9-4, pp.549-562.
[15] Maoz Y.D. and Moav O. (2000) “Intergenerational Mobility and the Process of Development,”
Economic Journal, vol.109, pp.677-697.

10[16] Meckl J. and Zink S. (2004) “Solow and Heterogeneous Labor: A Neoclassical Explanation of Wage
Inequality,” Economic Journal, vol.114, pp.825-843.
[17] Miyake A., Muro K., Nakamura T. and Yasuoka M.(2009) “Between- and Within-Group Wage
Inequalities, and the Advent of New Technology,” Journal of Economic Inequality, vol.7-4, pp.387394.
[18] Turnovsky S.J. (1997) “Fiscal Policy in a Growing Economy with Public Capital,” Macroeconomic
Dynamics, vol.1, pp.615-639.
[19] UN-HABITAT (2011) “Infrastructure for Economic Development and Poverty Reduction,” United
Nations Human Settlements Programme Nairobi 2011.
[20] Yakita A. (2008) “Aging and Public Capital Accumulation,” International Tax and Public Finance,
vol.15-5, pp.582-598.

11

Appendix
Form of the dynamics equation2∂ Kt+1
t+1
The sign of dK
is calculated as follows.
dKt of (19) is positive. The sign of ∂K 2t∂Ht
X
2α(1 − α)p0t ∂K
∂Kt+1
t
=
2
∂Kt
(α + (1 − α)pt )2



2
2
∂Ht
∂H
p00t ∂K+ p0t ∂∂KH2t (α + (1 − α)pt ) − 2(1 − α)p02 ∂Kttα(1 − α)(α + (1 − α)pt )(XKt − σ),+(α + (1 − α)pt )4


1−θ
1
1
1
∂p
where p0 = ∂H
, Ht = (1 − β)τ θ β θ (1 − θ) θ A θ Kt . The first term of the right hand in this equationis positive. The sign is ambiguous because the second term is negative. However, an increase in Kt(1−α)pt
the slope α+(1−α)pfrom zero to (1 − α). We find that (XKt − σ) increases with Kt . Therefore,t(1−α)ptα+(1−α)pt × (XKt − σ) increases with Kt and the amount of increase raises with Kt .
dKt+1
∂ 2 Kt+1
dKt > 0 and ∂Kt2 > 0.

12

Consequently,Kt+1

Kt+1 = Kt

Kt

K̂

Fig. 1-1: Dynamics of Kt (Income growth).
Kt+1

Kt+1 = Kt

K ∗ K̂

Kt

K̃

Fig. 1-2: Dynamics of Kt (Income growth or no income growth).
F (β)

1

β0

β∗

β1

β

Fig. 2: Range of β not to stay in the poverty trap.

13