The Society for Economic Studies
The University of Kitakyushu
Working Paper Series No. 2011-7(accepted in March 2, 2012)

Public Investment, Life Expectancy and Income GrowthMinoru Watanabe and Masaya Yasuoka

Public Investment, Life Expectancy and Income Growth∗
Minoru Watanabe† Masaya Yasuoka‡
August 25, 2011

Abstract
Based on individual occupational choice in the model including a production function with publicinvestment, this paper presents an examination of how public investment affects the dynamics. Individuals work as skilled laborers or unskilled laborers. As in the model described by Caselli (1999),educational costs are necessary to work as a skilled laborer. Results show that life expectancy determines whether income growth occurs or not. Public investment can bring about income growth if life
expectancy is sufficiently high. However, with low probability, the government can not bring about
income growth with an increase in public investment.

JEL Classification: H54, E20
Key Words: Public investment, Life expectancy, Occupational choice, Economic growth∗ We would like to thank Shinji Miyake, Tamotsu Nakamura and seminar participants at the 2011 Spring Meeting ofJapan Association for Applied Economics for their helpful comments and suggestions. Research for this paper was supported
financially by a Grant-in-Aid for Scientific Research (No.21730159) and the University of Kitakyushu. Any remaining errors
are solely the authors’ responsibility.
† 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

1Introduction

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), Futagami and Morita and Shibata (1993) examine public capital accumulation and growth. Glomm and Ravikumar (1997) show public accumulation and growth with humancapital investment. Turnovsky (1997) also discusses public capital accumulation and growth and thedifference between a socially planned economy and a decentralized economy. Yakita (2008) discusses anendogenized fertility rate and an aging economy including public accumulation. Public investment inthese earlier studies serves a 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.
Based on Chen (2010), this paper presents the derivation of some results. Depending on public investment and life expectancy, it is decided whether income growth continues or the income level converges toa constant level. The economy can not escape from a poverty trap if life expectancy is low. Chen (2010)shows that increased life expectancy reduces fertility. Subsequently, capital per capita increases. Therefore the economy can avoid a poverty trap. Different from Chen (2010), this paper presents considerationof public investment, which raises productivity. However, with low life expectancy, public investment
can not bring about income growth. Therefore, both public investment and life expectancy should be
regarded as continued income growth.
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 our manuscript.

2The 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.

2.1Households

Households experience two periods: young and old. During the young period, each household supplies
labor to earn labor income. This economy accommodates labor of two types: skilled labor and unskilled2

labor. Education costs must be incurred to become a skilled laborer, as assumed by Caselli (1999), Meckland Zink (2004), Miyake, Muro, Nakamura and Yasuoka (2009), and Chen (2010). That cost is assumed
as σ. Herein, wts denotes the wage rate of skilled labor. The government imposes labor income taxationon the wage income of skilled labor to provide public investment. Each household allocates its labor
income between consumption in the young period and saving. Consequently, we obtain the following
budget constraint:
cs2t+1= (1 − τ )wts − σ,Rt+1cucu1t + 2t+1 = wtu .
Rt+1
cs1t +(1)(2)

Indexes s and u respectively denote skilled labor and unskilled labor. Also, 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 + p(1 − α) ln c2t+1 , 0 < α < 1, 0 < p < 1(3)Therein, p denotes the probability that the individual lives during the old period. These savings areallocated 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 + p(1 − α)p(1 − α)Rt+1c2t+1 =((1 − τ )wts − σ) .
α + p(1 − α)c1t =(4)(5)

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

2.2

(6)
(7)

Firm

This paper assumes the production function shown below1 .
Yt = AKtθ (Gt Lt )1−θ + B(1 − Lt ), 0 < θ < 1, 0 < A, 0 < B(8)Therein, Yt denotes the aggregate output. Gt and Kt respectively denote public investment and capitalstock. Lt denotes the skilled labor amount. Considering that the population size of each generation
1 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 unskilled sector. Neither Caselli (1999) nor Chen (2010) considered public investment.

3is unity, the unskilled labor amount is shown by 1 − Lt . With a perfectly competitive market, profit
maximization reduces the following equations, as
wts = A(1 − θ)G1−θt

Kt
Lt

θ,

(9)

wtu = B.

(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.
p(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. Then, the government budget constraint is presented as
Gt = τ wts Lt .

3(13)Equilibrium

This section presents derivation of the equilibrium of this model economy. If workers move between twosectors freely, then the indifference condition is described as
wts =B+σ
≡ ŵ.
1−τ

(14)

Considering (9), (13), and (14), Lt is given as
1−θ

Lt =

11(1 − τ )τ θ (1 − θ) θ A θ Kt
B+σθ! 2θ−1.

(15)Intuitively, an increase in Kt raises Lt because an increase in wts brings about an increase in the amountof skilled labor. Therefore, we assume θ > 12 . Additionally, we assume the total population size in eachgeneration as unity. Then, the dynamics of capital stock at Lt < 1 is derived as shown below:
Kt+1 =p(1 − α)p(1 − α)B(((1 − τ )wts − σ)Lt + B(1 − Lt )) =.
α + p(1 − α)α + p(1 − α)(16)

In contrast, an increase in Kt raises Lt . Therefore, the dynamics of capital stock at Lst = 1 is derived as
Kt+1 =

1−θ11p(1 − α) (1 − τ )τ θ (1 − θ) θ A θ Kt − σ .
α + p(1 − α)4(17)Then, wts and 1 + rt at Lt = 1 are
1−θ

11wts = τ θ (1 − θ) θ A θ Kt ,1−θ

1−θ

(18)

11 + rt = τ θ θ(1 − θ) θ A θ .
1−θ

(19)
1−θ

1

Therefore, calculating Yt = (1 + rt )Kt + wts Lt , we obtain Yt = τ θ (1 − θ) θ A θ Kt . The capital stockK̂ which brings about wts = ŵ and Lt = 1 isK̂ =B+σ(1 − τ )τ1−θθ11(1 − θ) θ A θ.

(20)

Then, we can depict the dynamics of Kt as shown below.
[Insert Fig. 1 around here.]
There exist dynamics of three 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). If K0 is greater than K̃,income growth occurs. The last is that no income growth occurs (Fig. 1-3). Life expectancy p determines
which case is adopted.
The condition that the slope of Kt+1 shown by (17) is greater than unity is shown asp>
1−θ1α ≡ p1 ,1−θ
11(1 − α) (1 − τ )τ θ (1 − θ) θ A θ − 1(21)1where (1 − τ )τ θ (1 − θ) θ A θ > 1 is assumed. Otherwise, no p exists that holds this inequality. The
condition that K̂ has no point of intersection with the 45-degree line (Kt+1 = Kt ) is shown asσα B+1 ≡ p2
p>1−θ11σ−1(1 − α) (1 − τ )τ θ (1 − θ) θ A θ − B1−θ1(22)1σwhere (1 − τ )τ θ (1 − θ) θ A θ > 1 + B
is assumed. Otherwise, no p exists that is true for this inequality.

We obtain 0 < p1 < p2 . Therefore, with p2 < p, the dynamics of Kt+1 is depicted as Fig. 1-1. With
p1 < p < p2 , the dynamics of Kt+1 is depicted as Fig. 1-2. However, with p < p1 , the dynamics of Kt+1
is depicted as Fig. 1-3. Then, the following proposition is established.
Proposition 1

With p2 < p, income growth continues for any initial capital stock. With p1 < p < p2 ,income growth continues for initial capital stock that is greater than K̃. With p < p1 , no income growth
exists and the economy remains in a poverty trap.

This proposition shows that life expectancy, i.e. aging, determines whether income growth occurs ornot. An increase in p, which signifies an aging society, raises the saving and capital stock. An increasein capital stock raises the wage rate in skilled labor, which increases capital stock. This paper presents
the steady state with K ∗ as a poverty trap because no increase in income occurs.

5

Next, we examine whether an increase in τ can decrease K̂. If the economy stays in the case given inFig. 1-3, then a decrease in K̂ brings about the economy given by Fig. 1-1, which continues increasing1−θincome. Defining F (τ ) = (1 − τ )τ θ , F (τ ) is maximized at τ = 1 − θ. Therefore, this substitutes into
p2 , and we obtainασ
B +1 ≡ p̃2 .
2−θ1σ(1 − α) θ(1 − θ) θ A θ − B−1(23)Then, the following proposition is established.
Proposition 2With p < p̃2 , an increase in τ can not bring about income growth.

An increase in τ increases public investment, which raises productivity. However, this increase reduces
capital accumulation, which thereby decreases productivity. Capital accumulation is not large if life
expectancy p is low. Therefore, a decrease in capital accumulation greatly reduces productivity.
In developed countries, we regard p as small. Even if the government increases public investment topull up the income level, an income level converges to a poverty trap as long as the government levies anincome tax. In fact, p must become large to bring about income growth, such as a government providing
medical service appropriately.

4Concluding and Remarks

This paper described a model with public investment and illustrated how an increase in public investment
affects capital stock, the amount of skilled labor, and the wage rate. First, when the government provides
public investment, the government can achieve income growth if life expectancy is high. However, withlow life expectancy, public investment does not engender income growth, meaning that the income level
converges to a constant level: a poverty trap. Therefore, the government should provide not only publicinvestment but also medical services to increase life expectancy and escape from the poverty trap.

6References[1] Barro R.J. (1990) “Government Spending in a Simple Model of Endogenous Growth,” Journal of
Political Economy, vol.98, pp.S103-S125.
[2] Barro R.J. and Sala-i-Martin X. (1992) “Public Finance in Models of Economic Growth,” Review ofEconomic Studies, vol.59, pp.645-661.
[3] Caselli F. (1999). “Technological Revolutions,” American Economic Review, vol.89-1, pp.78-102.
[4] Chen H. (2010). “Life Expectancy, Fertility, and Educational Investment,” Journal of Population
Economics, vol.23, pp.37-56.
[5] 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.
[6] Galor O. and Moav O. (2000) “Ability Based Technological Transition, Wage Inequality and Economic Growth,” Quarterly Journal of Economics, vol.115, pp.469-497.
[7] Glomm G. and Ravikumar B. (1997) “Productive Government Expenditure and Long-Run Growth,”
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[8] Maoz Y.D. and Moav O. (2000) “Intergenerational Mobility and the Process of Development,”
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[9] Meckl J. and Zink S. (2004) “Solow and Heterogeneous Labor: A Neoclassical Explanation of Wage
Inequality,” Economic Journal, vol.114, pp.825-843.
[10] 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.
[11] Turnovsky S.J. (1997) “Fiscal Policy in a Growing Economy with Public Capital,” MacroeconomicDynamics, vol.1, pp.615-639.
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7

Kt+1

Kt+1 = Kt

Kt

K̂

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

Kt+1 = Kt

K ∗ K̂

K̃

Kt

Fig. 1-2: Dynamics of Kt (Income Growth or No Income Growth).
Kt+1

Kt+1 = Kt

Kt

K ∗ K̂

Fig. 1-3: Dynamics of Kt (No Income Growth).

8