GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.186

Growth or Welfare State?
Optimal Composition of Government Expenditure?

Yusuke Kinai

March 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Growth or Welfare State? Optimal Composition of Government
Expenditure∗
Yusuke KINAI†
Graduate School of Economics, Osaka University

March 29, 2011
<Preliminary>

Abstract

In developing countries, a government must often simultaneously promote growth-enhancing and generational
distribution policies. However, under most circumstances, it is difficult for any government to carry out both
policies. Therefore, it is unavoidable to determine which policy should be carried out and then perform allocationof tax revenue through political compromise. This paper incorporates both a pay-as-you-go type pension andpublic investment into an overlapping generations model and shows how the contents of economic policy vary
under a situation in which the policy determination is dependent on political issues.

Keywords : Commitment; Growth-enhancing vs. Redistribution Scheme; Overlapping Generations Model;
Structure-induced Equilibrium.

JEL Classification : E61, H54, H55.

∗ The author is thankful to Kazuo Mino and Masatsugu Tsuji for their variable suggestions. Comments from Akihiko Kaneko, Gregory
Gilpin, Hisakazu Kato, Nobuo Akai, Keigo Kameda, Hiroki Tanaka, Takashi Shimizu and the seminar participants at PET10 held at Boğazişi
University, Kansai Public Economics Seminar, The 16th Decentralization Conference, and JIPF meeting are also greatly acknowledged. The
usual disclaimer applies.
† Corresponding to: Graduate School of Economics, Osaka University, 1-7, Toyonaka, Osaka, 560-0043, Japan
E-mail : ykinai@js8.so-net.ne.jp

1 Introduction
Traditionally, for analyzing economic policies such as fiscal or monetary policy, economists have assumed that the
government is a monolithic organization that is intended mainly to maximize social welfare or raise the economic
growth rate. In this regard, the following points must be emphasized: First, various entities (voters, bureaucrats,
representatives, organizations, etc.) that are involved in policy determination foster conflicts. Therefore, it is
impossible for a government to organize a policy based on only one perspective or position. In other words, nogovernment can avoid determining a policy that incorporates implications of numerous opinions. Second, even if
a policy were derived that maximizes social welfare or the growth rate, carrying out such a policy with certainty
would be difficult: it is difficult to commit to such a policy. Some conflicts exist even within actual governments.
As one example, the Ministry of Finance and the Ministry of Health, Labour and Welfare in Japan engage infrequent conflicts. These two organizations have entirely different objectives: The former seeks to decrease thedeficit or debt, although the latter is responsible for promoting the nation’s social security system, even if it isvery costly. Consequently, a contraposition of duties pertains between the two organizations. For that reason,
we cannot regard actual governments as monolithic organizations, as many economists have done. Nevertheless,many studies have been made under the assumption that policy variables are one-dimensional. Given the existenceof such conflicts, “political compromise” is unavoidable. It is difficult to maintain the government commitment
entirely. As countermeasures for such a situation, it is apparent that it is necessary ex post to carry out somecoordination policy to attenuate inefficiencies that result from discretionary policy. Then, as a source of suchconflicts, we can point out that the policy determination is not necessarily one-dimensional but is often multidimensional. Therefore, it is necessary to expand the past analysis into multi-dimensional policy determination.
This paper specifically presents consideration of the situation in which the government has policy options of two
kinds: pension and public investment.
As described in this paper, we specifically consider a situation in which there exist two kinds of committee in thegovernment: one related to public investment and one related to the pension system. Under such circumstances,
how are policy contents determined? This paper describes an attempt to answer the question. This paper presents
analysis of how the contents of the policy vary over time, assuming that policy is determined based on the resultsof voting.
Then, we consider the case of Japan. Figure 1 depicts the flow of the proportion of pension and public investment
to gross domestic production1) . From this figure, it is apparent that the cost of pensions has been increasing, while
the cost of public investment remains unchanged. One purpose of these analyses is to give an explanation of thefact that the cost of pensions has been increasing rather than public investment, as in Japan. Why is the cost of
the one policy higher than the other? This paper describes that a pension is politically more favored than public

1) Data Source:Social Security: “the costs of social security” given by the System of National Accounts Statistics: SNA National Institute of Populationand Social Security Research.
Public Investment (Government gross fixed capital formation): SNA

1

Figure 1 Flow of the proportion of pension and public investment to gross domestic production in JAPAN.

investment as a result of capital accumulation.
■Relation with Past Studies

First, we explain the position of the present paper in the literature. Since theseminal work in this area such as that of Barro (1990), Futagami, Morita and Shibata (1993) or Alesina and Rodrik(1994), many studies have analyzed the relation between public investment and economic growth2) . However, few
studies have analyzed the relation in an overlapping generations (hereinafter, OLG) model. As starting studies,
we can point out Pestieau (1974) or Yoshida (1986). Since their studies, some others have been conducted:
Among others, for instance, Yakita (1994) and Burgess (2006) investigate the relation between the return of public
investment and the discount factor. Glomm and Ravikumar (1997) introduce public investment into an OLG modelin a framework of endogenous growth model. Recently, Kaas (2003) incorporates Majority Voting related to policy
determination and shows the existence of cyclical equilibria as for the tax rate. However, policy determination ofhis model is one-dimensional. Moreover, Hung (2005) specifically examines the monetary aspect by introducing
seigniorage. More recently, Yakita (2008) expands two-dimensional policy determination by incorporating not
only public investment but also maintenance activity into an OLG model, yet he does not consider political issues
related to the policy determination.
Regarding studies that analyze pension policy in an OLG model, there exist numerous efforts, but our survey
is limited to analysis of a context of political economy3) . For instance, we can point out Casamatta, Cremer
and Pestieau (2000), Wigger (1999), Razin, Sadka and Swagel (2002), Aı́bo, Mahieu and Patxot (2004), Boldrin
and Rustichini (2000), and so forth. What is common to these studies is that the policy determination is onedimensional, although the analyses in those earlier papers resemble those of this paper in the respect that policy
determination is conducted through voting.
As explained above, there exist many studies that analyze the effect of pension or public investment in an OLG
model. However, to the best of our knowledge, few reports describe incorporation of both pensions and public
2) For a survey, see Irmen and Kühnel (2008).
3) For a survey, see Galasso and Profeta (2002).

2

investment into an OLG model4) . As an exception, we can point out Maebayashi (2010) who introduces a pension
into the model of Yakita (2008), but his result is limited to the corner solution. In fact, Maebayashi shows that the
government allocates total tax revenue to public investment if they aim to maximize the growth rate. Regarding
other studies, although Creedy, Li and Moslehi (2008) expands past studies into two-dimensional policies (pension
and public goods provision), they limit the analysis to that of the balanced growth path.
In contrast to those studies, the features of this study are summarized as follows: First, we expand policy
determination to two-dimensional policy determination by introducing a pension into the model of Kaas (2003),
who incorporates public investment into an OLG model. Second, we regard public capital as a stock variable,
whereas Kaas treats it as a flow variable. That is because we specifically examine the transitional path as well asbalanced-growth-path. Third, this paper considers political issues related to policy determination, which differsfrom Yakita (2008). Additionally, we note that the model of this paper does not apply ordinal voting theory
because policy determination in our model is two-dimensional. We resolve such a difficulty using the concept ofstructure-induced equilibrium developed by Shepsle (1979).
In summary, we demonstrate the following:
1. Under the case without political processes, the growth-rate-maximizing tax rate does not coincide with thewelfare-maximizing one.
2. On the other hand, under the case with political processes, there are three cases, in which both pension andpublic investment survive or either of the two policies prevails.
The first result means that the result of Barro (1990) does not hold in an overlapping generations economy withpublic stock. More importantly, the second result means that we can show the possibility that only a pension or
public investment is politically supported.
The remainder of this paper is structured as follows. In section 2, we set up the model; then we analyze optimal
taxation in the case in which there is no political issue in the section 3. In section 4, we analyze the situation inwhich policy determination is dependent on the voting behavior. Final remarks are presented in section 5.

2 The ModelWe employ the Diamond (1965)-type two-period overlapping generations model without a bequest motive ina closed economy. There is population growth, i.e. Nt+1 = (1 + µ )Nt , by which µ can be both positive andnegative. Time is discrete and goes to infinity. We incorporate social security policy into the model of Kaas
(2003). Therefore, the government has policies of two kinds: pension and public investment. We introduce theheterogeneity of households: households vary depending on the labor productivity. We also consider politicalissue related to the policy determination as in Kaas (2003). The difference between Kaas (2003) and the analyses4) However, there exist some studies particularly addressing the combination of a pension and education. For instance, see Naito (2009),

Boldrin and Montes (2005), Lambrecht, Michel and Vidal (2005), and Kaganovich and Zilcha (1999). These studies specifically examine the combination of Backward Intergenerational Goods (BIG) and Forward Intergenerational Goods (FIG) using the terminology
presented in Rangel (2003).

3

presented in this paper is that we expand his analysis into two-dimensional policy determination. In what follows,
we consider the case in which there is no political issue (voting behavior) in section 3 as a benchmark, while insection 4, we consider the case in which there are political issues.

2.1 Behaviors■Households Hereinafter, we call households created in t period as generation t. Generation t whose laborproductivity is lti solves the following problem:oi),
max U(ctyi , ct+1

(1)

sti

o denote cosumption in young and old period, respectivley. Here, we impose some assumptions
where cty and ct+1

on the utility function, following Kaas (2003) or Kaas and von Thadden (2003). First, U(·) is twice differentiablewith respect to each variable, homothetic, strictly increasing, and quasi-concave. Second, with respect to eachvariable, the utility function is homogeneous to the degree of 1 − γ , in which γ ∈ [0, 1] denotes the degrees ofcu′′relative risk aversion (γ ≡ − ′ ). Third, consumption in the young period and old period are substitutes.
uTaxes of two kinds exist: a pension tax (τt ) and public investment tax (θt ), which are imposed on both labor
income and saving. Households formed at period t allocate after-tax labor income (w̃t ) to consumption (cty ) ando ) as after-tax savings and pension (d ) when they are old. Consequently, the youngsavings (st ), and consume (ct+1tperiod and old-period budget constraints are represented respectively asoi
ctyi + sti = w̃t lti , ct+1
= R̃t+1 sti + dt+1 ,(2)

where w̃t ≡ (1 − τt − θt )wt and R̃t+1 ≡ (1 − τt+1 − θt+1 )Rt+1 . Each household determines its own saving bysolving the following problem.
max U(−sti + w̃t lti , R̃t+1 sti + dt+1 )sti

By solving the equation presented above, the saving rate (st (·)i ) is dependent solely on R̃t+1 , and the saving
function is denoted as the multiplicity that is separable as follows:5) .
sti = si (R̃t+1 )w̃t lti(3)

By substituting this into the utility function, the indirect function is denoted as
1−γ

V (·) = U(1 − sti (R̃t ), sti (R̃t )R̃t+1 )w̃t.

(4)

The following equation holds because consumption in the young and old period is assumed to be substituted as∂ s(·)≥ 0.
∂ R̃t+1(5)

We then assume that the economy is dynamically efficient, i.e. the following equation holds:
Assumption 1

Rt > 1 + µ .

(6)5) Regarding the reason why the saving rate is dependent only on R
t+1 , not wt , see De La Croix and Michel (2002, pp.53–54). In the casein which the utility function is homothetic, the saving function can be denoted as multiplicity separable.

4

This equation means that the return of saving is larger than that of pension and the condition, which eliminates thetrivial situation in which all households support a pension system.
■The Government Next, we examine government behavior. They have schemes of two kinds: a public in-vestment scheme (growth-enhancing scheme) and the PAYG-type pension system (intergenerational redistribution
scheme). They determine which policy should be carried out, or the allocation of tax revenue based on voting.
The budget constraint of each scheme is balanced in each period. Tax revenue of public investment and pension
respectively as Et and Mt . The policy variables are θt (contribution to public investment) and τt (contribution topension). Both schemes are covered with labor-income and capital-income taxes. In what follows, we normalize
the total labor input as 1, i.e. ∑i lti ≡ Lt = 1.
Growth-enhancing scheme:
The government covers the cost of public investment with capital-income and labor-income taxes as in Barro(1990), although our case is a stock variable, not a flow variable. We assume that public investment has nocongestion effect. Therefore, the budget constraint is written as presented below.
Et = θt Yt = ∑ θt wt lti + θt Rt st−1 = θt (wt Lt + Rt st−1 ) = θt (wt + Rt st−1 )(7)iTherein, Gt and Et respectively denote public capital stock and public investment. The evolution of public investment is written as

Gt+1 − (1 − δG )Gt = Et .

(8)

Arranging the above equation into per-capita terms, we have
(1 + µ )gt+1 = et + (1 − δG )gt ,(9)where δg denotes the depression rate of public capital.
Intergenerational redistribution scheme:
Next, we move to the explanation of pension as a role of intergenerational redistribution. Here, we assume thatthe pension is pay-as-you-go type and Beveridgean, which means that the amount of pension received in old age isindependent of the labor productivity6) . In the aggregate, noting that the pension system is a pay-as-you-go type,the budget constraint of this scheme is written as shown below.
Nt−1 dt = τt Yt = τt Rt st−1 + ∑ τt wt lti = τt (Rt st−1 + wt Lt ) = τt (Rt st−1 + wt ) = Mt

(10)

i

Dividing both sides of the above equation with Nt , we obtaindt|{z}= (1 + µ ) τt (wt + Rt st−1 ) ≡ Mt .
|{z}pension received by households(11)

contribution

Merging the two schemes, the budget constraint of the government is expressed as
Et + Mt = (1 + µ )(θt + τt )(wt Lt + Rt st−1 ) = (1 + µ )(θt + τt )(wt + Rt st−1 ).
6) There are pensions of two types: Bismarckian and Beveridgean. Regarding the difference between Bismarckian and Beveridgean types,

see also Casamatta et al. (2000), or Conde-Ruiz and Profeta (2007).

5

■Firms

We then describe the firms’ behavior. We assume that factor markets are perfectly competitive and thatfirms maximize their profits. We herein specify the production function as follows:
Yt = F(Kt , At Lt ) = Ktα (At Lt )1−α .
Here, we specifically define At as(
At ≡ a

β

1−β

Kt Gt
Lt

)
, β ∈ [0, 1](12)where Gt denotes public capital. Presented as a narrative in Yakita (2008) and Kaas (2003), we can state theassumption that public investment has a positive externality in the sense that it takes up the marginal labor productivity. The model of Kaas (2003) corresponds to the case of β = 0 in eq. (12). Using this definition, the production
function is rewritten as

α +β (1−α )Yt = Kt(1−β )(1−α )Gt= Ktω Gt1−ω ,(13)where ω ≡ α + β (1 − α ) ∈ (0, 1). Firms are assumed to solve the following problem.
max Π = F(Kt , At Lt ) − Rt Kt − wt Lt
Therein, Kt , Lt , and Yt respectively denote capital, labor, and gross output. We define kt ≡ AKt Lt t , and F( AKt Lt t , 1) =f (kt ). Then, by solving the profit maximization problem presented above, we can obtain the following.
( )ω −1
Kt
Rt = At f ′ (kt ) = α≡ R(θt )Gt( )−ω ( )Gt
Kt′
wt = At { f (kt ) − kt f (kt )} = (1 − α )≡ w(θt ),
{z
}
|
Gt
Lt

(14a)
(14b)

w(kt )

These two factor prices are functions of public investment tax because Gt is financed with θt . Taking eqs. (14b),(7), (12), and (8) into consideration, we have1w(kt ; θ ) =
aθt

(

Gt
Kt

)β
.

(15)

f (kt )
Kt .
kt

(16)

Here, taking τt and θt as given, F(·) can be rewritten as
F(Kt , At Lt ) = At Lt f (kt ) =Therefore, we can rewrite the production function of F(·) in the form of an AK-type production function, takingt)
the policy variables as given. Here, f (kkt is decreasing in kt .

Moreover, net labor income can be written in the form of a linear function of capital:
wt (1 − τt − θt )Lt = w̃t Lt
where Ψ(θt ; τt ) ≡w(kt )(1 − τt − θt )Kt = Ψ(θt ; τt )(1 − τt − θt )Kt ,
kt

( )β
1
Gt
, and τt is given. Moreover, we assume that Ψ(·) satisfies the following:aτt k(θt ) Kt

Assumption 2 There exists only one solution, θ ∗ that maximizes Ψ(θt ; τt ).
This assumption gives the condition that ensures the existence of the solution.
6(17)

2.2 Market Equilibrium
Finally, we describe the equilibrium condition of each market.
(1) Capital Market
In aggregate terms,∑ sti (·)w̃t lti = Kt+1

(18)

i

From eq. (17), considering the saving rate st (R̃t ), we have the following.
Kt+1 = ∑ si (R(τt+1 ))Ψ(θt ; τt )(1 − τt − θt )Kt

(19)

i

Transforming the above equation into per-capita terms, we obtainkt+1 = s̄(R(θt+1 ))Ψ(θt ; τt )(1 − τt − θt )kt .

(20)

Here, s̄(R̃(θt+1 )) denotes the average saving rate.
To summarize, the capital market clearing condition is written as
kt+1 =1s(R̃(θt+1 ) w̃(kt )lti .
1 + µ | {z }(21)saving rate(2) Goods Market
In the aggregate, we can state this condition as follows.
cty Nt + cto Nt−1 + (Kt+1 − Kt ) + (Gt+1 − (1 − δG )Gt ) + Mt = Yt
Dividing both sides of the above equation with Nt yields the expression shown below.
cty +

cto
+ mt + (1 + µ )(kt+1 + gt+1 ) = Rt kt + wt lt + (1 − δG )gt
1+µ

(22)

(3) Labor Market
Denoting labor demand as Lt , the condition is
Nt = Lt .

(23)

Here, we describe the definition of the competitive equilibrium.
Defnition 1 Taking K0 and G0 as given, we define {cty , cto , st , kt , lt , Rt , wt , gt , θt , τt } as a competitive equilibriumsuch that
1. For all t, taking {Rt , wt , θt , τt } as given, the condition of utility maximization for generation t holds.
2. For all t, taking {τt , θt } as given, the condition of profit maximization holds.
3. Taking {cty , cto , st , kt , lt , Rt , wt } as given, {Gt , Et , Mt , θt , τt } meet the budget constraint of the government.
4. All markets clear.

7

Stage.2
Stage.3
Stage.4
Stage.1
a new generation Households vote Firms decide volume The government
over the policy. of employment and go determines which
is born
into production.

policy is adopted.

Stage.5{τt , θwt } isdetermined.

Figure 2 Sequence of Decision Making in the t-th period.

2.3 Timing of Decision Making
The timing of decision-making in t period is summarized as follows, which is also depicted in Fig. 2.
Stage 1. Households are born in t period.
Stage 2. Households vote over the two policy variables: contributions to the pension τt and that to public investment θt during the young period.
Stage 3. Firms produce.
Stage 4. Based on the voting results, the government determines the allocation of tax revenue (or which scheme
the government admits).
Stage 5. A new generation t + 1 is born in the next period.

3 Policy Determination without Political Process
In this section, we assume the case in which there is no political issue as a benchmark. In other words, we
treat the policy determination as a solution of maximization problem. We consider the case in which the objectivefunction of each scheme is the growth rate and social welfare.

3.1 Growth-Rate Maximizing Tax Rate
In what follows, we limit our analysis to the Balanced Growth Path. Here we seek the growth rate at a balanced
growth path. The growth rate of each variable is the following:
Substituting the saving function, eq. (3) into the capital market-clearing condition, eq. (??), and using eqs.
(14b), (8),(7), we have the following.
Kt+1s(R̃(τt+1 ))Ψ(τ , θ )(1 − τ − θ )
=
Kt
Kt
Gt+1
τt wt Lt
= (1 − δG ) +Gt
Gt
Defining γ =

Kt+1
Nt+1
Kt
Nt

=

Gt+1
Nt+1
Gt
Nt

=

Yt+1
Nt+1
Yt
Nt

(24a)
(24b)

, then the intersection of the above two equations (24a) and (24b) is BGP.

8

xt+1

x∗

xt

Figure 3 Determination of the Balanced Growth Path.

Letting xt be

Ktdx, we then must investigate the sign of | dxt+1| and its slope. The two equations above yieldtGt
xt+1 =s(R̃(τt+1 ))Ψ(τ , θ )(1 − τ − θ ).
(1 − δG ) + τt wxtt Lt(25)

By differentiating with respect to xt , we then have
dxt+1
> 0,
dxt
and
lim

xt →∞

dxt+1
= ∞.
dxt

Therefore, we then can state this result in the form of a lemma.
Lemma 1 There exists at least one BGP.
Then, by differentiating BGP with respect to τ and theta, we can derive the growth-rate-maximizing tax. We
define τ and θ as τ ∗ and θ ∗ such that
dγ
=0
dτ
dγ
=0
dθ

(26)
(27)

3.2 Welfare-Maximizing Tax Rate
Define the objective function as follows:1−γ

V i (Rt ) = si (R̃t+1 )w̃t

∂ V (·)i=0∂θ∂ V (·)i=0∂τ
By solving the above equations, we can derive the welfare-maximizing tax τ ∗∗ , and θ ∗∗ .
9

(28)

(29)
(30)

Comparison of Tax Rates Then, let us investigate the relation between the growth-rate maximizing andwelfare-maximizing tax rate. The following proposition answers such a question:
Proposition 1 The relation between the growth-rate-maximizing and social-welfare maximizing tax rate is given
as shown below.

τ ∗ ̸= τ ∗∗ , θ ∗ ̸= θ ∗∗Proof See Appendix.
Intuitively, the reason is explained as follows: They try to allocate the tax revenue to the greatest extent possible
to public investment, which has a role of promoting economic growth if the government aims to maximize thegrowth rate. Therefore, the growth-rate maximizing tax is higher than the welfare-maximizing one.
This result differs from that of Barro (1990). In Barro (1990), he claimed that the tax rate that maximizes thegrowth rate equals that which maximizes social welfare. However, this result shows that his claim is not robust.

4 Policy Determination with Political Process
4.1 Equilibrium Concept and Some Assumptions
Given the discussion in the previous section, we advance the analysis by endogenizing policy determination.
In this section, we introduce the political issue (i.e. voting behavior). Two policy variables exist in our model.
Generally, no Condorcet winner exists in voting over multiple issues such as a combination of policy of two kinds,
without imposing additional conditions on voter preference7) . To avoid such a problem, following Conde-Ruizand Galasso (2005), we adopt the concept of a structure-induced equilibrium developed by Shepsle (1979)8) .
We consider the following situation. There exist committees of two kinds: a committee that determines the contribution to pension and one that determines the contribution to public investment. The preferences to each policyof committee member are the same as those of voters. The policy determination itself is achieved independently,
and that policy determination is based on the other policy determination. This situation can be regarded as thestate in which there exist the following two reaction functions.
{τ = τ (θ ) : taking θ as given.
θ = θ (τ ) : taking τ as given.
We regard the intersection of the above two response functions as a (politico-economic) equilibrium.
We then consider the voting behavior related to determination of contributions to a pension and public investment. We assume here that
A 1. Voting is conducted in each period, which means issue-by-issue voting under direct democracy.
7) Regarding this issue, see Persson and Tabellini (2000), for instance.
8) Regarding studies of those who employ the structure-induced equilibrium, see Table 1.

This paper differs from those studies in thesense that they specifically examine the combination of income redistribution schemes, whereas this paper specifically examines the
combination of social security policy and other kinds of economic policy.

10

Table1 Past studies that use the concept of structure-induced equilibrium.

Conde-Ruiz and Galasso (2005)

Class of Social Security
PAYG-type pension vs. Redistribution Policy
Class of Social Security

Poutvaara (2006)

Pension vs. Public Education
Type of Pension

Conde-Ruiz and Profeta (2007)

Bismarckian vs. Beveridgean
Financial Resource of Social Security

Konishi (2008)

Consumption Tax vs. Labor-income Tax
Class of Social Security

Bethencourt and Galasso (2008)

Pension vs. Health Investment
Class of Social Security

Kinai (2008)

PAYG-type Pension vs. Unemployment Insurance
Class of Economic Policy

This Paper

Public Investment vs. PAYG-type Pension

A 2. Voting on contributions to pension and public investment takes place simultaneously.
A 3. Voters are young and old people who co-exist in the same period.
A 4. Policy determination is based on the median voter theorem.
A 5. Voting is repeated among successive generations of voters.
Before entering into the analyses, we must present the following lemma:
Lemma 2 Defining V (·) as the indirect utility function, the following equations hold:∂ 2V (·)
< 0,
∂θ2

∂ 2γ
<0
∂ τ2

.
Proof As presented in (4), we can obtain the indirect utility function as follows.
γ −1

V (·) = U(1 − st (Rt ), st (Rt )R̃t+1 )It= si (Rt+1 )w̃t

(31)

From this equation, by differentiating twice, we obtain∂ 2V (·)< 0.
∂ τ2This lemma shows that the indirect utility function is concave. Therefore, we can employee the median voter
theorem relative to policy determination.
11

■Determination of the contribution of public investment• We employ the Median Voter Theorem9) .
• As in the previous analysis, the indirect utility function is derived as follows.
γ −1

V (·) = U(1 − st (Rt ), st (Rt )R̃t+1 )It= si (Rt+1 )w̃t

(32)

Applying the Median Voter Theorem to the policy determination of contribution to public investment, we obtain∂ V (·)
= 0.
∂τ

(33)

From this equation, this relation can be written as shown below.

τ med = τ med (θ )■Determination of Contribution to Pension System We also employ the Median Voter Theorem. The indi-

rect utility function is written asγ −1

V (·) = U(1 − st (Rt ), st (Rt )R̃t+1 )It= si (Rt+1 )w̃t .
Differentiating with respect to θ , we obtain

The median voter preference is(34)∂ V (·)=0∂θ∂ V (Rt )med
= 0.
∂θ

(35)

Solving this equation can be stated in the following form:θ med = θ (τ ).

(36)Here, we can show that the two reaction functions are downward-sloping by total differentiation.

4.2 The Case with Commitment
First, we consider the case with commitment. We assume the tax rate is constant over time: θ = θt = θt+1 ,τ = τt = τt+1 . Here the shapes of the two response functions τ = τ (θ ) and θ = θ (τ ) are down-sloping. Then,
there are three plausible cases as depicted in Figs. 4–6: To summarize the discussion presented above,
Proposition 2 depending on the parameters, three plausible cases exist:
1. Both policies (pension and public investment) survive.
2. Only the pension survives.
3. Only public investment survives.
9) In this respect, our approach resembles that of Alesina and Rodrik (1994).

12

θw

θw

θ = θ (τ )
τ = τ (θ )
θ = θ (τ )
τ = τ (θ )ττFigure 4 Case 1..

Figure 5 Case 2..

θw

θ = θ (τ )
τ = τ (θ )τFigure 6 Case 3..

4.3 Case without Commitment
Next we examine analysis under no commitment. Before entering into analysis, let us define the game structure of our model. We then formally define the voting game. The public history of the game at t period,
ht = {(τ0 , θw0 ), (τ1 , θ1 ), ..., (τt−1 , θt−1 )} ∈ Ht is the sequence of social security system (pension and public investment). Actually, Ht is the set of all possible history at time t. An action profile for those who support the pensionis, {τt , bt } ∈ [0, 1] × [0, 1]. Analogously, an action for unemployed individuals at time t is {τt , bt } ∈ [0, 1] × [0, 1].
A strategy for those who support the pension at t period is a mapping from the history of the game into the action
space, i.e. σ pension : ht → {τt , θt }. Analogously, a strategy for those who support public investment is at t period
is σ pu : ht → {τt , θwt }. The strategy profile played by both individuals at t period is denoted by σt ≡ σte ∪ σtu .
At t periods, the objective function for young each player isiVti (σ0i , σ1i , ....σti , σt+1...) = Vti (τt , θwt , τt+1 , θw,t+1 ).

13

Regarding agents, those who support public investment,Vt (σ0 , σ1 , ....σt , σt+1 ...) = Vti (τt , θwt ).
These solutions describe the relation between the policy at t period and that at t + 1 period.
We describe the definition of equilibrium.
Defnition 2 (The Definition of Markovian Structure-Induced Equilibrium)
1. σ meets the property of Markov perfect equilibrium.
2. For all t, at t period, the equilibrium outcome associated to σt is a structure-induced equilibrium of thestatic game with commitment.
As contrasted with the analysis in the previous subsection, we assume in this subsection that the government has
no commitment to technology. Then, let us define the history of the game Ht as
Ht0 ≡ {ht ∈ Ht |θt = θw ∗, t ∈ {0, 1, ...}},
andHtσ ≡ {ht ∈ Ht |θk = 0, k = 0, 1, ...t0 , and θwt = 0, t ≥ t0 .}.

Moreover, the strategy profiles of people those who support pension and public investment are denoted respectively
as σte and σtu . We then investigate whether each player has an incentive to deviate from the solution under fullcommitment, as discussed in the previous subsection. Under this setting, we first verify that unemployed people
have no incentive to deviate from the strategy. We assume that unemployed people adopt the following strategy:∗ < τ deviate . However, employed people do not obtain an additional payoff by deviation because
θt0deviate > θw∗ and τt0tthey punish others by reducing the payment of contributions to the pension system, τ , which exerts negative effects
on the welfare of both agents. Therefore, it is apparent that unemployed people have no incentive to deviate fromthe commitment solution.
Regarding people who support public investment, presuming that those who support public investment deviate
from equilibrium, i.e. they avoid paying contributions to pensions, then the workers will punish others by not
paying contributions to public investment. They would pay contributions to pensions to avoid being punished.
Therefore, it is apparent that they have no incentive to deviate. To summarize, neither those who support pensions
nor those who support public investment have an incentive to deviate.
From the discussion, we have:
Proposition 3
Policies discussed in the previous subsection (with commitment case) coincide with those without commitment. In
other words, the strategies with commitment are time-consistent.

14

5 Concluding Remarks
As described in this paper, we consider the case in which the government has policies of two kinds: a pension
system and public investment. In our setting, policy determination is based on majority voting. The government
has social security policy mechanisms of two kinds: pension and public investment. Under this setting, we show
how the contents of economic policy vary.
The extension of this research is to increase policy variables. For instance, taking pensions as an example, there
are at least three variables: pension benefits, pension contributions, and retirement age. It is necessary to consider
these variables to obtain policy implications.

References
Aı́bo, G., G. Mahieu, and C. Patxot (2004) “On the optimality of PAYG pension systems in an endogenous fertility setting,”
Journal of Pension Economics and Finance, Vol. 3, No. 01, pp. 35-62.
Alesina, Alberto and Dani Rodrik (1994) “Distributive Politics and Economic Growth,” Quarterly Journal of Economics, Vol.
109, No. 2, pp. 465-490.
Barro, Robert J (1990) “Government Spending in a Simple Model of Endogenous Growth,” Journal of Political Economy, Vol.
98, No. 5, pp. S103-26.
Bethencourt, Carlos and Vincenzo Galasso (2008) “Political Complements in the welfare state: Health Care and Social Security,” Journal of Public Economics, Vol. 92, No. 3-4, pp. 609-632.
Boldrin, Michele and Ana Montes (2005) “The Intergenerational State: Public Education and Pensions,” Review of Economic
Studies, Vol. 72, No. 3, pp. 651-664.
Boldrin, Michele and Aldo Rustichini (2000) “Political Equilibria with Social Security,” Review of Economic Dynamics, Vol.
3, pp. 41-78.
Burgess, David (2006) “Public Investment Criteria in Overlapping Generations Models of Open Economies,” International
Tax and Public Finance, Vol. 13, No. 1, pp. 59-78.
Casamatta, Georges, Helmuth Cremer, and Pierre Pestieau (2000) “The Political Economy of Social Security,” Scandinavian
Journal of Economics, Vol. 102, No. 3, pp. 503-522.
Conde-Ruiz, J. Ignacio and Vincenzo Galasso (2005) “Positive Arithmetic of the Welfare State,” Journal of Public Economics,
Vol. 89, No. 5-6, pp. 933-955.
Conde-Ruiz, J. Ignacio and Paola Profeta (2007) “The Redistributive Design of Social Security Systems,” Economic Journal,
Vol. 117, No. 520, pp. 686-712.
Creedy, John, Shuyun May Li, and Solmaz Moslehi (2008) “The Composition of Government Expenditure in an Overlapping
Generations Model,” Department of Economics - Working Papers Series 1043, The University of Melbourne.
De La Croix, David and Philippe Michel (2002) A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations: Cambridge University Press.
Diamond, Peter A. (1965) “National Debt in a Neoclassical Growth Model,” American Economic Review, Vol. 55, No. 5, pp.
1126-1150.
Futagami, Koichi, Yuichi Morita, and Akihisa Shibata (1993) “Dynamic Analysis of an Endogenous Growth Model with Public
Capital,” Scandinavian Journal of Economics, Vol. 95, No. 4, pp. 607-25.
Galasso, Vincenzo and Paola Profeta (2002) “The political economy of social security: a survey,” European Journal of Political
Economy, Vol. 18, No. 1, pp. 1-29, March.
Glomm, Gerhard and B. Ravikumar (1997) “Productive government expenditures and long-run growth,” Journal of Economic
Dynamics and Control, Vol. 21, No. 1, pp. 183-204.
Hung, Fu-Sheng (2005) “Optimal Composition of Government Public Capital Financing,” Journal of Macroeconomics, Vol.

15

27, No. 4, pp. 704-723.
Irmen, Andreas and Johanna Kühnel (2008) “Productive Government Expenditure and Economic Growth,” Working Papers
0464, University of Heidelberg, Department of Economics.
Kaas, Leo (2003) “Productive Government Spending, Growth, and Sequential Voting,” European Journal of Political Economy,
Vol. 19, No. 2, pp. 227-246.
Kaas, Leo and Leopold von Thadden (2003) “Unemployment, Factor Substitution and Capital Formation,” German Economic
Review, Vol. 4, pp. 475-495.
Kaganovich, Michael and Itzhak Zilcha (1999) “Education, social security, and growth,” Journal of Public Economics, Vol.
71, No. 2, pp. 289-309.
Konishi, Hideki (2008) “Financing Social Security by Consumption Tax: a Political Economy Perspective.” mimeo, Tokyo
Institute of Technology.
Lambrecht, Stephane, Philippe Michel, and Jean-Pierre Vidal (2005) “Public pensions and growth,” European Economic Review, Vol. 49, No. 5, pp. 1261-1281.
Maebayashi, Noritaka (2010) “Public Capital, Public Pension, and Growth,” Discussion Papers in Economics and Business
10-03, Osaka University, Graduate School of Economics and Osaka School of International Public Policy (OSIPP).
Naito, Katsuyuki (2009) “From the Cradle to the Grave? A Politico-economic Approach on Social Security and Public Pension.” Kyoto University, mimeo.
Persson, Torsten and Guido Tabellini (2000) Political Economics: Explaining Economic Policy: MIT Press.
Pestieau, Pierre (1974) “Optimal taxation and discount rate for public investment in a growth setting,” Journal of Public
Economics, Vol. 3, No. 3, pp. 217-235.
Poutvaara, Panu (2006) “On the Political Economy of Social Security and Public Education,” Journal of Population Economics,
Vol. 19, No. 2, pp. 345-365.
Rangel, Antonio (2003) “Forward and Backward Intergenerational Goods: Why is Social Security Good for the Environment?”
American Economic Review, Vol. 93, No. 3, pp. 813-834.
Razin, Assaf, Efraim Sadka, and Phillip Swagel (2002) “The Ageing Population and the Size of the Welfare State,” Journal of
Political Economy, Vol. 110, No. 4, pp. 900-918.
Shepsle, Kenneth (1979) “Institutional Arrangements and Equilibrium in Multidimensional Voting Models,” American Journalof Political Science, Vol. 23, No. 1, pp. 27-59.
Wigger, Berthold U. (1999) “Pay-as-you-go financed public pensions in a model of endogenous growth and fertility,” Journalof Population Economics, Vol. 12, No. 4, pp. 625-640.
Yakita, Akira (1994) “Public investment criterion with distorted capital markets in an overlapping generations economy,”
Journal of Macroeconomics, Vol. 16, No. 4, pp. 715-728.
(2008) “Ageing and public capital accumulation,” International Tax and Public Finance, Vol. 15, No. 5, pp. 582-598.
Yoshida, Masatoshi (1986) “Public Investment Criterion in an Overlapping Generations Economy,” Economica, Vol. 53, No.
210, pp. 247-63.

16