GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.113

Public Capital, Public Pension, and Growth

Noritaka Maebayashi

January 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Public Capital, Public Pension, and Growth
Noritaka Maebayashi∗
January 2010

Abstract
This paper constructs an endogenous growth model with overlapping generations,whose engine of economic growth is productive public capital. The government facesa trade-off in public policy between public investment and social security provisionbecause of its budget constraint. Larger public investment accelerates economic
growth. On the other hand, larger public investment reduces the social security provision. This may reduce the consumption stream of agents. We first show that when
the government aims at growth maximization, it chooses no social security provision. However, we also show that the growth-maximizing policy does not maximize
welfare levels of each generation on the balanced growth path. Early generations
may demand social security provision because the benefits from economic growthcaused by an acceleration of public investment are relatively small. In contrast,
future generations may require no social security provision but a large amount ofpublic capital. Additionally, by setting the tax rate below the level that maximizes
the growth rate, the government can make the welfare levels of all generations fromthe initial state on the balanced growth path better off. Moreover, in an economy
facing an aging population, an increase in the social security provision to the oldrather than an increase in public investment can be preferable from the viewpointof social welfare.

JEL classification: E62, H54, H55
Keywords: public capital, social security, overlapping generations

1

Introduction

There are many theoretical studies on how publicly provided infrastructure affects
economic growth, for example, Barro (1990), Barro and Sala-i-Martin (1992), Futagami
et al. (1993), Turnovsky (1997), and Yakita (2008); however, they do not consider the
public pension. 1 Most developed countries have aging populations and some studies haveexamined social security policy in an aging economy, for example, Pecchenino and Pollard(1997), Pecchenino and Utendorf (1999), and Yakita (2001). If the rapid aging of the∗

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: gge010mn@mail2.econ.osaka-u.ac.jp
1
Yakita (2008) uses an overlapping generation model in contrast to the other studies that are representative agent models.

1

population in most developed countries is considered, then public policy that focuses only
on public investment does not fully capture the issues in developed countries. Therefore,
incorporating a social security system into such a growth model with public capital isworth investigating.
The present analysis is based on an overlapping generation model, in which the growthengine is public capital accumulation, such as Yakita (2008).2 We assume that the government imposes a distortionary tax on income and allocates its tax revenue between public
investment and social security provision. Therefore, the government controls both the tax
rate and the expenditure share between public investment and social security provision. In
this model, the social security provision has two effects on the economy.3 Firstly, because
the social security provision reduces public investment to satisfy the government’s budget,
it lowers the accumulation of public capital. Secondly, the social security provision decreases the amount of young agents’ saving because the social security provision increases
the agent’s income after retirement and thus reduces the incentive to save. Hence, thesocial security provision also lowers the accumulation of private capital.
This paper has two main objectives. One is investigating the policy that maximizes
the balanced growth rate. The other is examining welfare effects on the balanced growth
path when the government controls the tax rate and the expenditure share.
The findings of this paper on growth-maximizing policies are as follows. As describedabove, the social security provision lowers the accumulation of both public capital andprivate capital. Thus, when there is no social security provision, the growth rate is maximized by setting the tax rate at the elasticity of output with respect to public capital.4
We next show that, although the social security provision lowers economic growth, there
exists a tax rate that maximizes the growth rate of the economy in the presence of social
security provision. This tax rate becomes smaller as the social security provision becomeslarger. This finding implies that an economy with a large social security provision canadopt a small tax rate to maximize economic growth.
We next establish the following results regarding social welfare. The public policy thatmaximizes the growth rate, that is, the policy in which there is no expenditure on social
security and the tax rate is set at the elasticity of output with respect to public capital,
does not maximize the welfare level of every generation. Reducing the tax level from that
under the growth-maximizing policy improves the welfare levels of all generations fromthe initial state on the balanced growth path. In terms of the allocation of tax revenue
between public investment and social security, there exists intergenerational conflict. Early
generations may prefer the social security provision to be larger when they are old because
the benefits stemming from higher economic growth caused by an acceleration of public
investment are relatively small. In contrast, for future generations, benefits from economic
growth driven by the accumulation of public capital are relatively high. Therefore, they
demand no social security provision.
When the government conducts policies that take into account the weighted sum of2

Yakita (2008) does not include social security but incorporates the maintenance of public capital.
It is well known that pay-as-you-go social security systems hinder economic growth in an economy
with no public capital, only private capital. See, for example, de la Croix and Michel (2002) Chapter 3
and Pecchenino and Pollard (1997).
4
Barro (1990) and Futagami et al. (1993) show that, when the government allocates the tax revenue
into only public investment, the growth rate is maximized by setting the tax rate at the elasticity of output
with respect to public capital.
32each generation’s welfares on the balanced growth path and if it puts heavier weight on
future generations, large public investment and small social security provision should be
carried out. If it puts heavier weight on early generations, small public investment and thelarge social security provision should be carried out. Furthermore, we find in numericalexamples that, in the process of population aging, the expenditure share of the social
security provision that maximizes the social welfare increases as population aging occurs.
Population aging reduces the social security benefit per one old agent. As a result, thisincreases the savings of agents and reduces the social welfare. A rise in the expenditureon social security will prevent agents from saving excessively.
This paper is organized as follows. Section 2 develops the model and characterizes
the equilibrium. Section 3 explores how the tax rate and the expenditure share affects
the growth rate on the balanced growth path. Section 4 discusses the welfare effects ofeach generation on the balanced growth path. Section 5 discusses the case in which abenevolent government controls the policy instruments and takes every generation on thebalanced growth path into consideration and derives implications of public policy in an
aging economy. Section 6 gives concluding remarks.

2

The model

We consider a two period overlapping-generations model in which lifetime uncertainty
is incorporated. The economy comprises identical agents, a large number of symmetricfirms, and a government. Agents live for at most two periods. A cohort born in period
t is called generation t. Therefore, there exist two generations in period t (t ≥ 0); thatis, generation t (as a young generation) and generation t − 1 (as an old generation). The
economy begins its operation in period 0 and those who are old in period 0 are called theinitial old.

2.1

Production

All outputs in this economy are homogeneous and produced by a large number ofsymmetric firms installing the private capital stock and labor for the production. The
production technology of firm j is represented as:α
Yjt = Kjt
(ht Ljt )1−α , (0 < α < 1)(1)

where Yjt , Kjt and Ljt are the output level, private capital stock, and labor employed byfirm j, and ht is the labor productivity in period t. Perfect competition prevails in thegood and the factor markets. Thus, denoting the rental price of private capital and thewage rate as qt and wt , the profit maximizing conditions of firm j are given by:
¶α−1
µ
Kjt
h1−α
− qt = 0,
(2)
α
t
Ljt
µ
¶1−α
Kjt
h1−α
− wt = 0.
(3)(1 − α)t
Ljt

3

We assume that the public capital stock provides a positive externality in production.
Specifically, we specify ht as follows:
ht =

Gt
.
Lt

(4)

From (2) and (3), the capital–labor ratio becomes the same for all firms, that is,
Kjt /Ljt = Kt /Lt and Yjt /Kjt = Yt /Kt hold in the equilibrium. Here, Kt is the
Paggregate
capitalPstock and Lt is the total labor supply of the economy as follows: Kt = j Kjt and
Lt = j Ljt . We assume that the private capital stock fully depreciates in each period.
Substituting (4) into (1), we obtain the aggregate production function as follows:.
Yt = Ktα G1−αtProfit maximizing conditions (2) and (3) can be rewritten as:µ ¶α−1Kt
qt = α,Gtµ ¶α µ ¶Kt
Gt
wt = (1 − α)
.
Gt
Lt

2.2

(5)

(6)
(7)

Government

In this economy, the government pursues both the accumulation of the public capitalstock, which increases labor productivity, and a social security policy based on a pay-asyou-go pension system. The government imposes a flat rate of income tax τ on wage andinterest income and allocates its tax revenue between public investment and public pension
payments. Let us denote public investment and public pension payments in period t as Et
and Pt , respectively. The government budget constraint in period t becomes:
Et + Pt = τ Yt .

(8)

The government allocates a constant ratio, φ ∈ (0, 1] of its tax revenue to public
investment and the rest of the tax revenue to public pension payments. Thus, we obtain:
Et = φτ Yt ,(9)

Pt = (1 − φ)τ Yt .

(10)

The evolution of the public capital stock is given by:
Gt+1 − (1 − δG )Gt = Et = φτ Yt ,where δG is the depreciation rate of public capital.

4

(11)

2.3

AgentsAgents live for at most two periods. Some agents die at the end of the first period of life.
We denote the probability that agents are alive in the second period of life by λ ∈ (0, 1).
In each period t, Nt persons are born, so that the size of generation t is expressed as Nt inperiod t (when young) and λNt in period t + 1 (when old). Therefore, the total populationin period t is λNt−1 + Nt . We assume that the size of the newly born cohort in each period
is constant over time, that is, Nt−1 = Nt = N .
Agents are endowed with one unit of labor and they supply it inelastically to firms intheir young period, and retire when old. Hence, total labor supply equals the population
size, that is, Lt = N . They derive utility from their own consumption when young andold. Therefore, the lifetime expected utility of a representative agent in generation t isrepresented as:ln ct + λρ ln dt+1 ,(12)where ct and dt+1 denote consumption levels in young and old ages, respectively, andρ ∈ (0, 1) is the discount factor.
The agents allocate their labor income wt between consumption ct and saving st , andthe payment of taxes is a proportion of his/her wage τ wt . Thus, the budget constraint fora young agent of generation t is given by:(1 − τ )wt = ct + st .

(13)

We assume there exists an actuarially fair insurance company in this paper, as usedin Yaari (1965) and Blanchard (1985). The company collects funds and invests them infirms. The company repays returns on the investment to the insured households still living
in old age. When the insurance company collects from each young agent (At = at Nt in
the aggregate) in period t, it obtains the total proceeds qt At in period t + 1. Because
only λNt old people are alive in period t + 1, each of them receives (qt+1 /λ)at from theinsurance company because of perfect competition between the companies. Because thegovernment imposes tax on this return, the net return for agents is [(1 − τ )qt+1 /λ]at . On
the other hand, if the young agents in period t invest at in firms directly, they will receive(1 − τ )qt+1 at whether they are alive or dead in period t + 1. As we assume that households
have no bequest motive, they accept the insurance contract, which yields a higher interest
rate than self-investment. In addition to the return on the annuity assets, agents receive
social security benefits from the government in old age. Let pt+1 be a social security
benefit for one agent of generation t when all agents survive in their retired period; then,total social security benefits pt+1 Nt must be equivalent to the government’s expenditureon public pension payment Pt+1 in period t + 1. However, because the public pension
payment Pt+1 is actually distributed among agents who are alive in old age λNt , an agent
can receive pt+1 /λ if he/she is alive in his/her second period. Thus, the budget constraintof each old agent is as follows:(1 − τ )qt+1 st pt+1+
= dt+1 .
(14)
λλEach agent of generation t maximizes (12) subject to (13) and (14), where wt , qt+1 ,pt+1 , and τ are given. The first-order condition results in the following:5

dt+1 = ρ(1 − τ )qt+1 ct .

(15)

From (13), (14), and (15) we obtain the following saving function:st =λρ(1 − τ )
pt+1
wt −
.
1 + λρ
(1 + λρ)(1 − τ )qt+1(16)

This saving function shows that a higher level of wage leads to higher savings, whereasa higher level of social security benefit reduces savings. A higher interest rate lowers thevalue of social security benefits evaluated in youth, and thus enhances savings.

2.4

Dynamics

The equilibrium condition in the capital market is given by:
Kt+1 = st N.

(17)

Substituting (16) into (17) yields:
Kt+1 =λρ(1 − τ )pt+1 N
wt N −.
1 + λρ
(1 + λρ)(1 − τ )qt+1(18)

From (5) and (10), the total social security benefit of generation t paid by the government in period t + 1 is given by:µpt+1 N = Pt+1 = (1 − φ)τ Yt+1 = (1 − φ)τ

Kt+1
Gt+1

¶α
Gt+1 .

(19)

Substituting (6), (7), and (19) into (18), we obtain:
Kt+1 =λρ(1 − τ )(1 − α)
1 + λρ

µ

Kt
Gt

³

¶α
Gt −

(1 − φ)τ

Kt+1
Gt+1

(1 + λρ)(1 − τ )α

´α
³

Gt+1
´α−1 .

Kt+1
Gt+1

(20)

Dividing both sides of (20) by Kt and that of (11) by Gt , we obtain the dynamic systemof this economy as follows:
Kt+1
λρ(1 − τ )2 (1 − α)α=xα−1 ,
Kt(1 + λρ)(1 − τ )α + (1 − φ)τ t

(21)

Gt+1
= φτ xαt + 1 − δG ,
Gt

(22)

t
where xt ≡ K. We obtain following dynamic system of x by dividing both sides of (21)Gt
by (22).

xt+1 =λρ(1 − τ )2 (1 − α)αxαt.
(1 + λρ)(1 − τ )α + (1 − φ)τ φτ xαt + 1 − δG

In the steady state, x satisfies the following:
6(23)

λρ(1 − τ )2 (1 − α)α(x∗ )α−1 = φτ (x∗ )α + 1 − δG .
(1 + λρ)(1 − τ )α + (1 − φ)τ

(24)

Proposition 1 There is a unique x∗ that attains a balanced growth path along which
private capital, public capital, and aggregate output grow at the same rate: Kt+1 /Kt =Gt+1 /Gt = Yt+1 /Yt = γ. The steady state value x∗ is uniquely determined by (24) and isglobally stable.

Proof.

Equation (23) satisfies the following conditions.

dxt+1(1 − δG )αxα−1
t
=ψ
>0
α
dxt
(φτ xt + 1 − δG )2
d2 xt+1(1 − δG )αxα−2[φτ xαt (α + 1) + (1 − α)(1 − δG )]
t
=
−ψ
<0
(dxt )2(φτ xαt + 1 − δG )3
dxt+1=∞
xt →0 dxt
dxt+1
lim
=0
xt →∞ dxtlimψ≡λρ(1 − τ )2 (1 − α)α.
(1 + λρ)(1 − τ )α + (1 − φ)τ

Thus, (23) intersects the 45◦ line at x∗ (> 0) uniquely and the steady state is globallystable as depicted in Figure 1. When xt+1 = xt = x∗ , (24) holds and it implies Kt+1 /Kt =Gt+1 /Gt from (21) and (22).
Q.E.D.
[figure1]

3

The policy maximizing the long-run growth rate

In this section, we investigate the following questions. What should be the amount oftax to maximize the balanced growth rate? How should the government divide tax revenue
between investment in public capital and public pension payments so as to maximize thebalanced growth rate? To solve this policy set (τ, φ), we firstly analyze the expenditure
share of public investment φ for a given tax rate that maximizes the balanced growth rate.
7

Secondly, for a given expenditure share, we find the tax rate that maximizes the growth
rate.

3.1

Choice of the expenditure share φ

From (22), the balanced growth rate γ becomes:
γ = φτ (x∗ )α + 1 − δG .

(25)

By using (24) and (25), we obtain the effect of changes in the expenditure share of
public investment on the balanced growth rate. Appendix A provides the derivation of
the equation below.
dγ
τ (x∗ )α=dφφτ (x∗ )α + (1 − α)(1 − δG )

½
1−α+

αφτ
(1 + λρ)(1 − τ )α + (1 − φ)τ

¾
γ.

(26)

Because (26) takes a positive value, the long run growth rate is maximized when φ = 1.
Hence, when the government allocates all of the tax revenue to public investment, it
can maximize the balanced growth rate. The reason for this result is quite simple. The
social security provision reduces public investment to satisfy the government’s budget, andthus lowers the accumulation of public capital. In addition, the social security provision
decreases the amount of savings by young agents because social security decreases theinsentive of young agents to save. Therefore, the social security provision also lowers theaccumulation of private capital. We can examine the effect of changes in the expenditure
share of public investment on the balanced growth rate.

3.2

Choice of the tax rate with social security

In this subsection, we focus on the tax rate, taking φ ∈ (0, 1) as given, and investigate
the effects of changes in the tax rate on the balanced growth rate. From (24) and (25),
we obtain the effect of changes in the tax rate on the balanced growth rate. Appendix B
provides the derivation of the equation below.
dγ= Ω[((1 + λρ)α − (1 − φ))τ 2 − ((1 + λρ)(2 − α)α + 2α(1 − φ))τ + (1 + λρ)(1 − α)α]. (27)dτΩ≡φ(x∗ )2α λρ(1 − α)(1 − τ )α> 0.
[φτ (x∗ )α+1 + (1 − α)(1 − δG )x∗ ][(1 + λρ)(1 − τ )α + (1 − φ)τ ]2

To explore the relationship between τ and γ, we must analyze the following quadratic
function f (τ ):f (τ ) = [(1 + λρ)α − (1 − φ)]τ 2 − [(1 + λρ)(2 − α)α + 2α(1 − φ)]τ + (1 + λρ)(1 − α)α. (28)The growth rate γ increases with τ if τ is small enough so that f (τ ) > 0 and decreases
with τ if τ is large enough so that f (τ ) < 0, and the tax rate τ ∗ that maximizes the growthrate corresponds to the condition; f (τ ) = 0.
8

Proposition 2 (i)There exists a tax rate τ ∗ ∈ (0, 1) for any φ ∈ (0, 1) that maximizes thebalanced growth rate in an economy with a social security system. This tax rate is given
by:
h
1
τ =
(1 + λρ)(2 − α)α + 2α(1 − φ)2[(1 + λρ)α − (1 − φ)]ip2− [(1 + λρ)(2 − α)α + 2α(1 − φ)] − 4[α(1 + λρ) − (1 − φ)](1 + λρ)(1 − α)α . (29)∗(ii)The tax rate τ ∗ is monotonically increasing in the expenditure share φ and, when φ = 1,
τ ∗ takes the largest value 1 − α.

Proof.
(i) We will show that the tax rate τ ∗ given by (29) exists within (0,1) for ∀φ ∈ (0, 1) andmaximizes the balanced growth rate.
The quadratic function of (28) satisfies the following condition:f (0) = (1 + λρ)(1 − α)α > 0,f (1) = −(1 − φ)(1 + α) < 0.
Hence, the tax rate τ ∗ exists within (0,1) for ∀φ ∈ (0, 1)(ii)Differentiating f (τ ) with respect to φ, we obtain:∂f (τ )= τ (1 + 2ατ ) > 0.
∂φ
Therefore, when φ increases, f (τ ) shifts upward so that τ ∗ increases.

Q.E.D.

In Subsection 3.1, we have shown that an increase in the ratio of public investment
raises the balanced growth rate. Therefore, the tax rate τ ∗ |φ=1 = 1 − α leads to the highest
growth rate. Thus, the growth-maximizing policy is given by (τ, φ) = (1 − α, 1). This
corresponds to the result of Barro (1990) and Futagami, et al. (1993). They show that,when the government sets the tax rate on income at the elasticity of output with respect
to public capital, the balanced growth rate takes its maximum value. In the economy witha social security transfer to old generations, when the government allocates a larger shareof its tax revenue to public investment, the tax rate that maximizes the balanced growthrate becomes higher.55This result is similar to that in Section 5 in Barro (1990). Barro (1990) shows that, when thegovernment expenditure finances some service that increases household’s utility, the growth-maximizing
share of productive government spending is smaller. In our analysis, social security provision is a kind ofservice that enters into the household’s utility in the second period.

9

4

Welfare level of each generation

In this section, our task is to investigate the welfare effects of each generation on thebalanced growth path when the government controls its policy (τ, φ). We show that thegrowth-maximizing policy does not maximize welfare levels of generations on the balanced
growth path. In particular, we find that, when the government aims at growth maximization, it chooses no social security provision. Although social security reduces the benefitsof economic growth, it increases consumption after retirement. We show that some expenditure on social security provision improves the welfare of the early generations. However,later generations lose some of the benefits of economic growth if the government chooses
a large expenditure on social security provision.
Because on the balanced growth path, the wage rate w∗ and the rental price of private
capital q ∗ are constant over time, the consumption by the young agents in period t is
written as follows:
1−τ ∗
pt+1
w +
1 + λρ
(1 + λρ)(1 − τ )q ∗½¾
(1 − τ )(1 − α) ∗ αλρ(1 − φ)τ
=(x ) 1 +[φτ (x∗ )α + 1 − δG ]t G0 ,(1 + λρ)N(1 + λρ)(1 − τ )α + (1 − φ)τ
(30)

ct =

where G0 denotes the initial level of public capital. From (15), the consumption by old
agents of generation t becomes as follows:
dt+1 = ρ(1 − τ )q ∗ ct .

(31)

Substituting (30) and (31) into (12) yields the utility of generation t on the balanced
growth path.
G0N+(1 + λρ)α + λρ(α − 1) ln x∗ + (1 + 2λρ) ln(1 − τ )+(1 + λρ) {ln[(1 − τ )α + (1 − φ)τ ] + ln[(1 + λρ)(1 − τ )α + (1 − φ)τ ]}+(1 + λρ)t ln γ.

ut =λρ(ln ρ + ln α) + (1 + λρ) ln(1 − α)(32)

The first and second terms are common to all generations for t = 0, · · · , ∞ but thelast term is different between generations. The last term represents the benefit fromaccumulated private capital and public capital from an initial state along the balanced
growth path and it is increasing in t. Therefore, later generations obtain more benefits
from economic growth.
Differentiating (32) with respect to τ and evaluating it at (τ, φ) = (1 − α, 1), we obtain:
1 + λρ
dut|(τ,φ)=(1−α,1) = −< 0.
dτ(1 − α)α(33)

Hence, lowering τ from 1 − α increases the welfare levels of all generations from the initialstate along the balanced growth path.
10

Next, we investigate how an increase in the social security provision affects the utilityof each generation with the tax rate set at 1 − α. To get a clear result we evaluate thiseffect at φ = 1, that is, no provision of public pension. Taking the derivative of (32) withrespect to φ, and evaluating it at (τ, φ) = (1 − α, 1), we obtain:
dut
1 dx∗
λρ(1 − α)1 dγ|(τ,φ)=(1−α,1) = [λρ(α−1)+α(1+λρ)] ∗|(τ,φ)=(1−α,1) −|(τ,φ)=(1−α,1) .
+(1+λρ)t
dφ
x dφ
α2
γ dφ
(34)
When the government decreases φ from 1, there are the following effects on the welfare
of generation t. The first term depends on the sign of dx/dφ. This term indicates thata change in the ratio of private capital to public capital because of a fall in φ affects theagent’s consumption in both his/her young and old periods. The second term reflects
the following two opposite effects because of a fall in φ . First, it has positive effects on
consumption because of an increase of the social security benefit. On the other hand, italso has negative effects on consumption caused by a fall in the growth rate of both private
capital and public capital because of a decrease in public investment. Because the second
term takes a negative value, this indicates that the positive effects dominate the positiveeffects. The first term and the second term are common to all generations. However, thelast term differs between generations. The last term causes the differences in the policy
effects on each generation’s welfare. The last term indicates that economic growth exerts
positive effects on the wage income and the social security benefit of the agents alive
today. Because the last term is increasing in t, an increase in the social security provision(a decrease in φ) negatively affects the welfare level of the later generations.
However, it is difficult to evaluate (34) analytically because the sign of dx/dφ remains
ambiguous. Therefore, we present numerical examples to show differences in the welfare effects between generations. We set the parameters as follows: (α, ρ, λ, δG , τ )=(0.3,0.6,0.8,0.01,0.7).
Figure 2 indicates the relationship between the welfare levels of generations from zero to
five when the tax rate is fixed at 1 − α = 0.7. An increase in the social security provision(smaller public investment) raises the welfare levels of generations close to the initial generation; however, it reduces those of the later generations. Therefore, there is a conflict
between generations as to the level of the social security provision.

[figure 2]

5Social welfare

In this section, we make the assumption that the government is benevolent and its
objective is to improve the social welfare by taking all generations on the balanced growth
path into consideration. This objective function consists of the weighted sum of eachgeneration’s welfare on the balanced growth path. Let θ denote the time discount factorof the benevolent government, which evaluates all future generations’ utilities at the initialperiod 0 on the balanced growth path. A higher value of θ means that government is moreconcerned about future generations.

11

U=

∞
X
t=0

µ
θ

t

¶
λρ
ln ct +
ln dt .
θ(35)

On the balanced growth path, the initial old generation’s consumption level d0 is written as follows:µ ¶(1 − τ )q ∗ s−1 p0(1 − τ )α + (1 − φ)τ ∗ α G0
d0 =
+
=
(x )
,
(36)
λ
λ
λ
N
Substituting (30), (31), and (36) into (35), we obtain:
1 + 2λρλρ(α − 1) + α(1 + λρ)ln(1 − τ ) +
ln x∗
1−θ
1−θ
1 + λρ
+{ln[(1 − τ )α + (1 − φ)τ ] − ln[(1 + λρ)(1 − τ )α + (1 − φ)τ ]}1−θ
θ(1 + λρ)+ln γ
(1 − θ)2λραλρ+ D2 +
ln x∗ +ln[(1 − τ )α + (1 − φ)τ ] .
θ
θ {z
|
}

U =D1 +

(37)

the welfare of the initial old

D1 ≡1
1 + λρ (1 − α)G0λρ G0(λρ ln ρ + λρ ln α) +ln
, D2 ≡
ln
.
1−θ
1−θ
N
θ
λN

Let us calculate the welfare effect of φ by using a numerical example. We set theparameters as follows: (α, ρ, λ, δG , τ, θ) = (0.3, 0.6, 0.8, 0.01, 0.7, 0.6). The relationshipbetween the social welfare U and the expenditure share: φ can be calculated in Figure 3.

[Figure 3]

The expenditure share that maximizes the social welfare shown in Figure 3 when the
tax rate is fixed at 1 − α = 0.7 is contingent on how highly the government values thewelfare levels of future generations. If θ takes a relatively large value, the government
cares about later generations and, hence, they accelerate public investment to enhance
economic growth, that is, φ takes a larger value. If θ takes a relatively small value,
the government chooses higher expenditure on social security provision to increase early
generations’ welfare levels.
We next study how population aging influences the welfare effects of φ. In this model,
population aging can be expressed as an increase in λ. Here, we use the same parameteras before. We calculate social welfare when λ = 0.6 and 0.8 respectively, and obtain theeffects of population aging on social welfare.
12[figure 4]

Figures 4-(i) and 4-(ii) respectively show the relationship between social welfare U andexpenditure share φ when λ = 0.6 and 0.8. From these figures, as aging progresses from
λ = 0.6 to 0.8, the expenditure share φ that maximizes the social welfare for a given tax
rate decreases. This fact indicates that the government should expand the expenditure onsocial security in accordance with population aging.
The reason for this is as follows. Population aging reduces the social security benefit
per old agent, so that each agent is likely to save more. Consequently, this increase in thesaving of agents reduces social welfare. Hence, enhancing social security provision prevents
agents from saving excessively.

6

Concluding remarks

This paper investigates public policy when the government allocates tax revenue between public investment and social security payments in an endogenous growth model witha growth engine of productive public capital. We obtain the following results. Firstly, there
is a unique transition path converging to the steady-state equilibrium. Secondly, in termsof growth-maximizing policies, when there is no social security provision, setting the tax
rate at the elasticity of output with respect to public capital as in Barro (1990) and Futagami et al. (1993) leads to the highest growth. With social security provision, there
is a tax rate to maximize the balanced growth rate for each expenditure share of social
security provision and, if the economy adopts a larger amount of social security, higher
tax revenue is necessary to maximize the growth rate. Thirdly, as for welfare analysis,
the policy that maximizes the growth rate does not maximize social welfare. One way to
improve social welfare is to reduce the tax rate below the elasticity of output with respect
to public capital. This increases the welfare levels of all generations from the initial statealong the balanced growth path. The other way is introducing social security expenditure.
However, this causes intergenerational conflict. A larger social security system increases
the welfare level of the generations close to the initial because the loss of consumption
stemming from higher economic growth is smaller; however, for future generations, it becomes larger. Therefore, the future generations prefer a smaller the social security system.
Finally, as population aging occurs, agents increase saving and reduce consumption because of a fall in social security provision per an agent when old; hence, the government
may increase social security expenditure in accordance with population aging.

A

Appendix

Appendix A

Differentiating γ with respect to φ yields

13

µ
¶
dγ
φα ∂x∗∗ α= τ (x ) 1 + ∗
.
dφ
x ∂φ

(A.1)

By totally differentiating (24), we obtain the following derivative of x∗ with respect to φ :noλρ(1−τ )2 (1−α)α
∗
∗ α
−τ
(x
)
x
−
∗
[(1+λρ)(1−τ )α+(1−φ)τ ]2
∂x
=
.
(A.2)
∂φ
φτ (x∗ )α + (1 − α)(1 − δG )There are the following two effects. A rise in the expenditure share of public investment φleads to an acceleration of the accumulation of public capital. This effect reduces x∗ . On
the other hand, the rise in φ means a reduction in social security benefits for agents, which
enhances saving and private capital accumulation. This effect increases x∗ . Therefore, arise in φ increases not only the public capital stock but also the private capital stock. If
the increase in the private capital stock is larger (smaller) than that of the public capitalstock, (A.2) takes a positive (negative) value.
Substituting (A.2) into (A.1), we obtain (26).

Appendix Bµ¶dγ
τ α ∂x∗∗ α= τ (x ) 1 + ∗
.
dτ
x ∂τ

Totally differentiating (24) yields:noλρ(1−τ )(1−α)α[(1+λρ)(1−τ )α+(1−φ)(1+τ )]∗ α∗−(x ) φx +[(1+λρ)(1−τ )α+(1−φ)τ ]2
∂x∗
=
< 0.
∂τ
φτ (x∗ )α + (1 − α)(1 − δG )

(A.3)

(A.4)

Substituting (A.4) into (A.3), we obtain (27).

Acknowledgments
I would like to thank Koich Futagami,Kazuo Mino, and Shinsuke Ikeda for their helpfulcomments. I am responsible for all remaining errors.

References[1] Barro, R.J., 1990. Government spending in a simple model of endogenous growth.
Journal of Political Economy 98:S103–S125[2] Barro, R.J., Sala-i-Martin, X., 1992. Public finance in models of economic growth.
Review of Economic Studies 59:645–661
14[3] Blanchard, O.J., 1985. Debt, deficits, and finite horizons. Journal of Political Economy
93:223–247[4] de la Croix, D., Michel, P., 2002. A Theory of Economic Growth. Dynamics and Policyin Overlapping Generations. Cambridge University Press[5] Diamond, P.A., 1965. National debt in a neoclassical growth model. American Economic Review 55:1126–1150[6] Futagami, K., Morita, Y., Shibata, A., 1993. Dynamic analysis of an endogenous
growth model with public capital. Scandinavian Journal of Economics 95:607–625[7] Maxime, F., Marcel, M., 1999. Population ageing and economic growth in sevenOECD countries. Economic Modelling 16:411–427[8] Pecchenino, P., Pollard, R., 1997. The effects of annuities, bequests, and aging in an
overlapping generations model of endogenous growth. Economic Journal 107:26–46[9] Pecchenino, P., Utendorf, K., 1999. Social security, social welfare and the aging population. Journal of Population Economics 12:607–623[10] Turnovsky, S.J., 1997. Fiscal policy in a growing economy with public capital. Macroeconomic Dynamics 1:615–639.
[11] Yaari, M.E., 1965. Uncertain lifetime, life insurance, andthe theory of the consumer.
Review of Economic Studies 32:137–150[12] Yakita, A., 2001. Uncertain lifetime, fertility and social security. Journal of Population
Economics 14:635–640[13] Yakita, A., 2008. Aging and public capital accumulation. International Tax and Public
Finance DOI 10.1007/s10797-007-9041-0

15

Figure1

Dynamics of x

xt +1
XX

45°

0

x0 = GK00x*xt

xt +1 =λρ (1 − τ ) 2 (1 − α )αxtα⋅
α(1 + λρ )(1 − τ )α + (1 − φ )τ φτxt + 1 − δ GThe effects of φ on distinctive generations’ utility

Figure 2

Generation 0uGeneration 1
u

Welfare-4.45

Welfare

-4.32
-4.34

-4.5

-4.36
-4.38

WelfareWelfare

-4.55

-4.6

-4.65

-4.4
-4.42
-4.44
-4.46

-4.7

-4.48
-4.75

0

0.1

0.2

0.3

0.4

0.5
phi

0.6

0.7

0.8

0.9

1

-4.5
0

φ

0.1

0.2

0.3

Generation 3
u

0.5
phi

0.6

0.7

0.8

0.9

1

φ

0.7

0.8

0.9

1

φ

Generation 5
u

Welfare-3.8

0.4

Welfare

-3.2

-3.9

-3.4

-4

-3.6

-4.1

WelfareWelfare

-3.8
-4.2

-4

-4.3

-4.2

-4.4

-4.4

-4.5

-4.6

0

0.1

0.2

0.3

0.4

0.5
phi

0.6

0.7

0.8

0.9

1

φ

-4.6

0

0.1

0.2

0.3

0.4

0.5
phi

0.6

The relationship between φ and U

Figure 3

Social welfare: U
Welfare

-12.6

-12.7

Welfare

-12.8

-12.9

-13

-13.1

-13.2

0

0.1

0.2

0.3

0.4

0.5
phi

0.6

0.7

0.8

0.9

1

φ

The welfare effects of φ in an aging economy

Figure 4(i)

λ＝0.6Social welfare: U
Welfare

-11.2

-11.3

-11.4

Welfare

-11.5

-11.6

-11.7

-11.8

-11.9

(ii)

0

0.1

0.2

0.3

0.4

0.5
phi

0.6

0.7

0.8

0.9

1

φ

λ=0.8

Social welfare: U

Welfare

-12.6

-12.7

Welfare

-12.8

-12.9

-13

-13.1

-13.2

0

0.1

0.2

0.3

0.4

0.5
phi

0.6

0.7

0.8

0.9

1

φ