GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.189

Design of a Social Security System:
Pension System vs. Unemployment Insurance

Yusuke Kinai

April 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Design of a Social Security System:
Pension System vs. Unemployment Insurance∗
Yusuke KINAI†
Graduate School of Economics, Osaka University

March, 2011

Abstract

This paper presents consideration of how the social security system evolves as the attributes of voterschange. In our setting, policy determination is based on majority voting. The government has twocomponents of social security policy: a pension system and unemployment insurance. When workers
constitute most voters, the pension system is supported and when unemployed people are the majority,
unemployment insurance is adopted. Under this setting, employing the concept of structure-induced
equilibrium developed by Shepsle (1979), the present paper describes how the contents of the social
security system evolve depending on the dynamics of capital accumulation and the unemployment rate,and demonstrates the possibility that one or the other social security system ceases to exist in certain
instances.

Keywords : Social Security, Pension System vs. Unemployment Insurance, Majority Voting, Structureinduced equilibrium.

JEL Classification : E61, H53, H55.

∗ The author is grateful for helpful comments and suggestions from Masatsugu Tsuji and Kazuo Mino.

He also thanks
Hideki Konishi, Akihiko Kaneko, Ryuichi Tanaka, Akira Momota, Makoto Hanazono, Hikaru Ogawa, Kazutoshi Miyazawa,
Ichiro Daito, Akihiko Yanase, Yunfang Hu, and the seminar participants at Tokyo Institute of Technology, Nagoya University,
Tohoku University, PET 08 held at Hanyang University, FESAMES09 held at The University of Tokyo, the JEA meeting heldat Kinki University and the JIPF meeting held at Kyoto University. The usual disclaimer applies.
† Correspondence to: E-mail: ykinai@js8.so-net.ne.jp1

Figure 1 Expenditures on Pensions in Selected Countries: 1990–2005 (source: OECD
Social Expenditure Database)

Figure 2 Recipients of Unemployment Insurance in Selected Countries: 1990–2005(source: OECD Social Expenditure Database)

1 Introduction
The social security system includes medical care, public sanitation, social insurance (pension insurance, medical insurance, at-home care insurance, unemployment insurance, and workmen’s compensation insurance) and other services and schemes. Among them, pensions and unemployment insurance
are of particular importance. In Japan and some other countries, the increase in expenditures for social
security are increasing as the population ages and as fewer babies are born (See Fig. 1.). On the other
hand, from Fig. 2, it is apparent that the necessity for subsidies for unemployment (so-called “NEET”,
or “Working poor” in Japan) has also increased remarkably, and the expenditure in such a situation (i.e.
unemployment insurance) is overwhelmingly large. Considering that the contributions to both pension
and unemployment insurance are increasing, there might emerge a situation in which the government
must prioritize either the pension system or policies for employment, although the government should
carry out both policies.
The study examines the sustainability of social security systems. Although this is not an issue thatremains confined to social security policy, the sustainability of economic policy is mainly based onpolitical factors. In that regard, the determination of policy is based on the legislature (in Japan’s case,
the Diet) in developed countries, which are representative democracies. Given such a situation, what is
important is consideration of the relative amounts of power of all voters when we analyze an economic
policy in relation to a political issue. This paper is intended to model such a situation and to show how
the scheme of social security varies as time passes from the viewpoint of political economy.
A sketch of our model is the following: First, households of two kinds are included in the model, including workers and unemployed people. The former hopes for a pay-as-you-go (PAYG) type pension
system; the latter hopes for unemployment insurance. In that regard, decisions are based on majority
voting. As time passes, the relative numbers of workers and unemployed people vary. For that reason,
the contents of social security also vary. At this stage, its choice affects social welfare. In our setting,1policy determination is based on majority voting and the government has social security policies of
two kinds: a pension system and unemployment insurance. When workers constitute most voters, thepension system is supported. When unemployed people are the majority, unemployment insurance isadopted. Under such a situation, we show how the contents of the social security system evolve depending on the dynamics of capital accumulation and the unemployment rate, and show that the social
security system ceases to exist in certain instances.
■Relation with the Literature

Here, let us describe the relation of this paper with past studies in

the following two respects. Some studies have specifically examined unemployment using an overlapping generations (hereinafter, OLG) model. Roughly speaking, two directions exist. One is a searchmodel; the other incorporates trade unions1) . This paper takes the latter stance. Since Demmel and
Keuschnigg (2000) and Corneo and Marquardt (2000), which were the first works to model the behavior of a trade union, some studies have modeled trade unions. For instance, Imoto (2003) extends themodel of Corneo and Marquardt (2000)2) and shows that the existence of a trade union might cause
a business cycle depending on the value of substitution between the volume of employment and thewage rate3) . Kaas and Thadden (2004) propose the other type of wage setting by a trade union in a similar model, and Ono (2007) specifically examines the interaction between pension and unemployment
insurance and derives the unemployment dynamics dependent on social security policy. Bräuninger(2005) shows that the unemployment rate is constant under the assumption of an endogenous growthmodel and wage determination through Nash bargaining. What is common to those studies is thatthe kind of social security system is exogenous. To be more precise, how the policy is chosen is notconsidered. This paper extends these studies to endogenize the choice of social security system byintroducing voting behavior.
On the other hand, since the seminal papers of Meltzer and Richard (1981) or Hu (1982), many studies have specifically examined the social security system in an OLG model in the context of political
economy. These studies typically examine specifically how the ratio of voters varies as time passes
and show how the social security system is altered. Recently, Hassler, Mora, Storesletten and Zilibotti(2003) and Conde-Ruiz and Galasso (2005) investigate how the contents of the social security systemchange depending on changes of the wealth distribution. Unlike these studies, we consider two combinations of redistribution schemes, pensions, and other redistribution policies as intergenerational andintragenerational redistributive schemes, respectively. This paper specifically examines unemployment
insurance in the role of intragenerational policy.
To summarize, our paper specifically differs from the papers described above in the following two
1) See also Galor and Lach (1990) or Bean and Pissarides (1993) introduces a search model into an OLG model.
2) More precisely, Imoto extends the objective function of the trade union in the model of Corneo and Marquardt into the

CES-type function.
3) Coimbra, Lloyd-Braga and Modesto (2005) also implicitly derives the unemployment dynamics under the different modelfrom Corneo and Marquardt (2000). The difference from those studies is the objective function of the trade union.

2

Trade Union
wage setting

Households
workers the unemployed

Capital Market

Firms

The employee’s contribution
to unemployment insurance

Government

The firms’ contribution
to unemployment insurance

unemployment insurance
Figure 3 Structure of the model
Note: This figure depicts the intratemporal flow of goods, not intertemporal.

respects. First, unlike past studies under the first part, in our model, the policy determination is endogenized by introduction of the voting model. In that regard, we specifically examine the notion ofissue-by-issue voting as a mode of policy determination. Second, we specifically examine unemployment insurance as an intragenerational redistribution scheme, which differs from the second part.
The remainder of this paper is organized as follows: Section 2 sets up the model. We investigate thedynamics of this economy in section 3. In section 4 section, we show how the contents of the social
security system changes and show the possibility of annihilation of social security. Section 5 presents
the conclusion.

2 The Model
We consider an infinitely-lived economy comprising households, firms, trade unions, and a government. Although our model is fundamentally similar to that of Kaas and Thadden (2004), our model
differs from theirs in the following two respects. The first point is wage determination through bargaining other than Nash bargaining. The second point is that the determinant of the social security systemis specifically examined in our model. Households exist for two periods: young and old periods. The
population growth rate is µ , i.e., Nt+1 = (1 + µ )Nt . To avoid complication, the model structure is
depicted in Fig. 3, which summarizes the intratemporal, not intertemporal, flow of goods.

2.1 Behavior of Each Agent2.1.1 Households
Households exist for two periods in a closed economy without a bequest motive. Dynasties derive
utility from consumption in young and old periods. For simplicity, the preferences of the dynasty’s

3cohort that survives at t period is described by the following additively separable function:max
U i (·) = Et [u(ctyi ) +yo
ct ,ct+1= lt [u(ctye ) +1oiu(ct+1
)]
1+ρ
1
1
oe
ou
u(ct+1)] + (1 − lt )[u(ctyu ) +u(ct+1
)].
1+ρ
1+ρ

(1)

Therein, ρ denotes the discount factor, and i = {e, u}. Furthermore, ctyi and ctoi respectively denoteconsumption during young and old periods. Two subscripts “e” and “u” respectively denote employedpeople and unemployed people. We then specify the utility function as u(·) = ln c.
In Ono (2007), he assumes that whether households include workers or retired people is determinedat birth, whereas we assume that households can be either workers or retired people at birth. Therefore,
lt denotes the employment rate defined by lt ≡ NLtt , which can be interpreted as the probability ofobtaining jobs during the youth period.
The budget constraints of workers and unemployed people are shown, respectively, as follows.
• Case of being Employed:
When people are young, they work and divide after-tax labor-income into savings, contributions
to pensions, and consumption. When they are old, they consume savings and pension payments.
oeectye + ste = (1 − τt − θwt )wt , ct+1
= Rt+1 ste + dt+1,

(2)

In those equations, τ and θw respectively signify the contributions to the pension and unemployment insurance. In addition, st and dt+1 respectively stand for savings and pension paymentsthat old people receive. Furthermore, wt and Rt+1 denote the wage rate and the rental rate ofcapital stock. Maximization of the utility function of employed people under the constraint, eq.
(2), yields
{
e }
dt+1
(1 − θwt − τt )wt +
,
2+ρ
Rt+1
{
e }
dt+1
1
oe
ct+1 =(1 − θwt − τt )wt +
,
2+ρ
Rt+1
{
e }
dt+1
1
e
st =
(1 − θwt − τt )wt − (1 + ρ )
.
2+ρ
Rt+1
1+ρ
ctye =

(3a)
(3b)
(3c)

• Case of being unemployed:
When they are young, people receive unemployment insurance (bt ) and divide it into savings,
contributions to pensions (dtu ), and consumption. When they are old, they consume savings andpension payments asouuctyu + stu = (1 − τt )bt , ct+1
= Rt+1 stu + dt+1
.

4

(4)

Maximization of the utility function of unemployed people under the constraint, eq. (4), yields
{
u }
dt+1
1+ρ
yu
ct =
(1 − τt )bt +
,
(5a)
2+ρ
Rt+1
{
u }
dt+1
1
ou
ct+1 =(1 − τt )bt +
(5b)
,
2+ρ
Rt+1
{
}
du
1
stu =
(1 − τt )bt − (1 + ρ ) t+1 .
(5c)
2+ρ
Rt+1
Here, we assume the following.
A1. dte > dtu eliminates the trivial case in which all households choose unemployment.
A2. Rt > 1 + µ means that the economy is dynamically efficient.

2.1.2 Firms
We assume that factor markets are perfectly competitive and that firms maximize their profits. Laborand capital stock are used for production; production technology is assumed to be neoclassical productfunction with constant returns to scale, Yt = F(Kt , Lt ), where Yt , Kt and Lt respectively represent
output in aggregate terms, capital stock and the number of the workers at t period4) . The firms’ profit
maximization problem is written as
Πt = F(Kt , Lt ) − Rt Kt − (1 + θ f t )wt Lt ,where θ f t represents the contributions to unemployment insurance and pensions which the firms bear.
We then specify the production function as Cobb–Douglas type, F(Kt , Lt ) = Ktα Lt1−α . Then, FOCs arederived as∂ Πt= 0 ⇐⇒ Rt = α Ktα −1 Lt1−α ,
∂ Kt
∂ Πt
= 0 ⇐⇒ (1 + θwt )wt = (1 − α )Ktα Lt−α .
∂ Lt

(6a)
(6b)

Defining k̂t ≡ KNtt = KLtt NLtt = kt lt , we obtain the following.
Rt = α k̂tα −1 lt1−α ,

(7a)

wt =

(7b)

(1 − α )k̂tα lt−α
1 + θwt

If Lt = Nt (perfect employment), then lt = 1; the wage rate is derived as(1 + θwt )w̄t = (1 − α )k̂tα ,(8)where w̄t denotes the wage rate at perfect employment.
4) Although Ono (2007) adopts the Grossman and Yanagawa (1993) type production function, we adopt the neoclassicalgrowth model because our concern is not directed to the effect on growth rate.

52.1.3 Trade Union
Following Corneo and Marquardt (2000) or Ono (2007), the wage is determined by the monopolistic
trade union. The trade union strives to maintain both high wages and low unemployment rates simultaneously. Following Imoto (2003), who extends the model of Corneo and Marquardt, the problem of
the trade union is:[]− 1
wt ≡ arg max W (·) ≡ γ (wt − w̄t )−σ + (1 − γ )(lt )−σ σ , σ ∈ (−1, ∞), and γ ∈ (0, 1),wt(9)

under the constraint of eq. (7b). Two parameters α and γ are exogenous parameters. The type of
bargaining is “Right-to-Management”: firms accept the wage requested by the trade union and thendecide the amount of employment to maximize their profit. On the other hand, the trade union requests
the wage based on that condition to maximize its objective function, eq. (9). The trade union decides
the wage rate to maximize their objective function for given w̄t treated as the reference wage rate. The
first order condition for this problem is written as∂ W (·)σ (1 − γ )= 0 ⇐⇒ −γσ (wt − w̄t )−σ −1 +
∂ wt
α
σ

((

1−α
1+θf

−(1+σ ) and R ≡ w α −1 , where V = (1− γ )Denoting L ≡ αγ1t
V1 (wt − w̄t )

)−σ

)1

((

α

1−α
1+θ f

σ −1

wtα

k̂t

(10)

)−σ

)1

α

= 0.

k̂t

. Determinationof the wage rate wt is shown in Fig. 4.
In what follows, we assume that the following equation (second order conditions) holds.

∂ 2W (·)<0∂ wt2
2.1.4 The Government
Finally, let us describe government behavior. It has redistribution schemes of two kinds: a PAYGtype pension system (intergenerational redistribution scheme) and unemployment insurance (intragenerational redistribution scheme). The budget constraint under each scheme is balanced and written as
follows.
• PAYG-type pension system (intergenerational redistribution scheme)
In the aggregate, the budget constraint under this scheme is written aseu
Lt × dt+1+ (Nt − Lt ) × dt+1
= τt+1 Lt+1 wt+1 + τt+1 (Nt+1 − Lt+1 )bt+1 ,where bt denotes the benefit from unemployment insurance. The left side term of the above
equation denotes the pension entitlement. The first and second terms of the right side respectively denote contributions to pensions of workers, and unemployed people. Dividing both sidesof the above equation with Nt yieldseu
lt dt+1+ (1 − lt )dt+1= (1 + µ ){τt+1 wt+1 lt+1 + τt+1 bt+1 (1 − lt+1 )}.

6

(11)

R(σ ≥ 2α )LHS, RHS

LHS, RHS

R(σ < 2α )

R
L
wt

L
wt

(a) −1 < σ < 0(b) 0 < σ < ∞Figure 4 Wage Determination through Trade Union• Unemployment insurance (intragenerational redistribution scheme)
Similarly, in the aggregate, the budget constraint is written as(Nt − Lt )bt = θwt Lt wt + θ f t Lt wt.
The left side of the equation presented above signifies the entitlement of unemployment insurance; the first and second terms of the right side respectively denote contributions to unemployment insurance of workers and firms. Dividing both sides of the above equation with Nt
yieldsbt =lt(θwt + θ f t )wt .
1 − lt(12)

Which policy is chosen is dependent on the voting constituency: if young unemployed people arein the majority, then unemployment insurance is supported. Old and young workers, if constituting
the majority, will support the PAYG-type pension system. Therefore, we investigate the transitionalchange of voters in the next section.

2.2 Timing of Decision Making
Next, we summarize the sequence of decision-making (or political process). The decision-making
sequence is also depicted in Fig. 5.
Stage 1. At the tth period, a new generation is born. Household members can be both employed andunemployed at this stage.

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 5 Decision-Making Sequence in the t-th period.

Stage 2. Households vote over the policy variables of social security system; the contribution to pension (τt ) and unemployment insurance (θwt ).
Stage 3. Then firms decide the volume of employment and go into production. At this stage, households are divided into two types, those with employed people and those with unemployedpeople.
Stage 4. The government determines which policy is adopted, that is, the contents of social security
system is determined, based on the result of voting. In other words, the contribution topension or unemployment insurance is also determined as {τt , θwt }. At the same time, θ f t isalso determined.
Stage 5. The t + 1th generation is newly born.

2.3 Market Equilibrium
We finally formulate equilibrium conditions for each market.
• Commodity market
In the aggregate, we can state this condition as Ctye + Ctoe + Ctyu + Ctou + Kt+1 = Yt , where Cti
denotes the aggregate consumption of type i in t period. Dividing both sides of this equation
with Nt yields
lt ctye + (1 − lt )ctyu +lt−1 oe 1 − lt−1 ouc +c + (1 + µ )kt+1 = yt ,
1+µ t
1+µ t

(13)

where yt ≡ NYtt .
• Capital market
In the aggregate, we can state that Kt+1 = ε Yt = Lt ste + (Nt − Lt )stu , where ε is national savings

8

wt

wt

Labor Supply

Unemployment

w̄t

Labor Demand

lt∗

l¯t

lt

Figure 6

rate5) We can write the following(1 + µ )k̂t+1 = lt ste + (1 − lt )stu .

(14)This equation determines the dynamics of capital accumulation in this economy.
• Labor market
In this market, the demand for labor is expected to equal to the supply of labor. Therefore,
combining the solution of eq. (6b) and that of eq. (10) yields the labor market equilibrium
condition.
Lt = Nt lt(15)

The left and right side of the above equation respectively signify the labor demand and labor
supply. Figure 6 depicts the situation of labor market equilibrium. In that figure, the heavy
and middle lines respectively show the labor demand curve and the indifferent curve. The
combination (w̄t , l¯t ) denotes the wage and volume of employment at full employment. The
direction between lt∗ (labor supply curve) and l¯t denotes the amount of unemployment.
Finally, let us define the competitive equilibrium.
i , si }∞ , i ∈Defnition 1 (Competitive Equilibrium) An economic equilibrium is a sequence {cti , ct+1
t t=1

{e, u} that accords with the following condition.
∞ , each agent (employed or unemployed) determines policy(i) Given the sequence {τt , θwt , θ f t }t=1

variables that maximize their individual utility: the optimal policy variables meet the following5) K is a constant that does not depend on policy variables of the social security system at t periods or t + 1:tue
(τt−1 , θw,t−1 ).
Kt = Lt st−1
(τt−1 , θw,t−1 ) + (Nt − Lt )st−1

Therefore, we can treat the policy variables as given and treat the state variable as fixed. This point is related to theremark on an earlier page 16.

9maximization problem:
max ln(ctyi ) +1oiln(ct+1), i ∈ (e, u).
1+ρ(ii) Budget constraints of pension and unemployment insurance are balanced in each period.
(iii) Firms maximize their profit.
(iv) The condition for the trade union’s wage setting holds.
(v) Finally, the commodity market clears, i.e. eq. (13) holds.

3 Analysis
In this section, we consider the equilibrium dynamics treating the policy variable τt and θwt as
constant. In other words, we assume that a government can commit to the policy once it is determined.
The justification of this assumption is explained in the next section.

3.1 Dynamics of Capital Accumulation
From eqs. (3c), (5c), and (14), considering the case in which τ = τt = τt+1 and θw = θwt = θw,t+1 ,let us derive the dynamics of kt as follows6) :1{se × lt + stu × (1 − lt )}1+µ t
α (1 − (1 + θwt )τt + (1 − τt )θ f t )(1 + θ f t )k̂α l 1−α .
=(1 + µ )(1 + θ f t )}(2 + ρ ) + (1 + ρ )(τt (1 + θwt + θ f t ))} t tk̂t+1 =(16)1α (1−(1+θwt )τt +(1−τt )θ f t )(1+θ f t )ε yt , and ε ≡ (1+µ )(1+θ )}(2+ρ )+(1+ρ )(τt (1+θwt +θ f t ))}ft
1+µdenotes the national saving rate. Here, subscript t is omitted. From this equation, we find that theWe can rewrite the equation above as k̂t+1 =increase in τ and θw causes decrease in the capital stock per capita at t + 1 period because of thedecrease in the saving rate.

3.2 Dynamics of the Employment Rate
Next, we must investigate the dynamics of the employment rate. From eqs. (10) and (16), we canderive the following equation:1−γ
σ (1+α )(1 − ltα )−(1+σ ) =ltαγThis equation describes the relation k̂t and lt .

6) For calculation details, see Appendix A.1.

10

(

1−α α
k̂
1+θf t)σ.

(17)Considering the labor and capital market equilibrium conditions in eqs. (15) and (14), the above
equation can be rewritten as
1+α

α
lt+1

α
(1 − lt+1

ε
1+σ =
1+µ
) ασ

This equation can be rewritten as

(

1−γ
αγ

(

1−α
1+θf

1 (1−α ))σ ) ασlt2

1+σ(1 − ltα ) σ

.

1+α
lt+1
1−α
= A σ lt2αα
1 − lt+1

(18)

(19)

(
(
) )
ε
1−γ 1−α σwhere A ≡
.
1+µ
αγ
1+θf
Equation (19) implicitly shows the relation between lt+1 and lt . From this equation, we can show
the relation as

lt+1 = φ (lt ),(20)

which determines the dynamics of the employment rate. Depending on the value of σ , we have threecases. Figures 7(a)–7(c) depict the dynamic patterns of the employment rate. Unemployment characterizes this economy if the employment rate is less than half of the population.
Incidentally, we can derive the equilibrium at the steady state, which holds k̂t = k̂t+1 = k̂∗ or lt =lt+1 = l ∗ . k̂∗ and l ∗ meets the following equations:k̂∗ =α (1 − (1 + θw )τ + (1 − τ )θ f )(1 + θ f )(k̂∗ )α (l ∗ )1−α ,(1 + µ )(1 + θ f )((2 + ρ ) + (1 + ρ )(τ (1 + θw + θ f )))1−α(l ∗ )1+α= A σ (l ∗ )2α .
∗α1 − (l )

(21a)
(21b)

We find that increases in τ and θw decrease the capital stock per capita at the steady state k∗ . The
increase in τ and θw decreases the employment rate at a steady state l ∗ if σ ≥ 0. On the other hand,
if σ < 0, then the increase in τ and θw increase l ∗ . In addition, the increase in µ engenders a similar
result. Then, we establish the following proposition.
Proposition 1
The capital per capita at the steady state decreases if the contribution rate of pension and unemployment insurance which the worker bears and the population growth rate increase. Therefore, if the
trade union prefers complementarity between wage rate and employment rate, then the employment
rate decreases. On the other hand, under the opposite preferences, the employment rate increases.
Proof We can derive the result presented above by differentiating eqs. (21) with respect to τ and θw .

11

lt+1

lt+1
l∗

l∗

(a) 0 < σ < ∞

l∗

lt

l∗
(b) −1 < σ < 0, & σ < 2α

lt

lt+1

l∗

l∗

(c) −1 < σ < 0, & σ > 2αlt

Figure 7 Dynamics of Employment Rate lt under Commitment

4 Politico-Economic Equilibria
Given the discussion in the previous section, we advance the analysis by endogenizing policy choice.
We then consider the voting behavior related to pension and unemployment insurance. We first assume
that
A 1. Voting is held in each period, which means issue-by-issue voting under direct democracy.
A 2. Voting takes place simultaneously on contributions to pension and unemployment insurance.
A 3. Voters consist of young people (employed people and unemployed) and old people who are alivein the same period.
A 4. Policy determination is based on majority voting.
A 5. Voting is repeated among successive generations of voters.

12

4.1 Case with Commitment
To characterize the equilibria of this voting game, we first assume the government has commitment
technology; that is, θwt = θw,t+1 = θw and τt = τt+1 = τ , following Poutvaara (2006), Conde-Ruiz andGalasso (2005) and so forth. In other words, we assume that each generation alive in the same period
considers τt and θwt chosen through election will be in place over its entire lifetime. From anotherangle, this assumption can be regarded as once-and-for-all voting or a static expectation. Under thiscase, voters determine the constant sequence of parameters of the welfare state. Under the existenceof commitment, we can advance the analysis similarly to the case of static analysis7)
It is known that, 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’s preference8) . To
avoid such a problem, following Conde-Ruiz and Galasso (2005), we adopt the concept of a structureinduced equilibrium developed by Shepsle (1979)9) . We then investigate the preferences to each policy
variable. The indirect utility functions of the worker and unemployed people are derived, respectively,
as follows.
• Employed people:(

1+ρ
V (·) = ln2+ρe

• Unemployed people:

(

)

[
]
2+ρ
dt+1
+
ln(1 − τt − θwt )wt +
1+ρ
Rt+1

1+ρ
V (·) = ln
2+ρ
u

)

[
]
2+ρ
dt+1
+
ln(1 − τt )bt +
1+ρ
Rt+1

(22)

(23)

Therefore, the indirect utility function is written as shown below.
E[U(·)] = lt V e (·) + (1 − lt )V u (·)
)
[
{ (
]}
dt+1
1+ρ
2+ρ
= lt ln+ln(1 − τt − θwt )wt +
2+ρ
1+ρ
Rt+1
{ (
)
[
]}
1+ρ
2+ρ
dt+1
+
ln(1 − τt )bt ++ (1 − lt ) ln
2+ρ
1+ρ
Rt+1

(24)

We then have the following reaction functions of each agent by solving the following first orderconditions.

∂ E[U(·)]=0∂ θw∂ E[U(·)]
=0
∂τ

(25a)
(25b)

7) Some readers might wonder whether this setting is considered to be commitment. Regarding this issue, see the remarkof page 16.
8) Regarding this issue, see Persson and Tabellini (2000), for instance.
9) This approach is also adopted in Konishi (2008), Bethencourt and Galasso (2008), Conde-Ruiz and Profeta (2007),

Poutvaara (2006) and so forth.

13

Here, the situation is identical in the sense that the desired contribution pension isτ ∗,old = 1
for both employed and unemployed when they are young because both kinds of elder generations
desire to receive as much pension money as possible. Therefore, in the case in which old people
are in the majority, unemployment insurance ceases to exist and only the pension system survives.
Consequently, capital accumulation does not proceed. To eliminate such a trivial case, we advance theanalysis assuming that old people desires the same amount of pension as young workers does. Then, it
is necessary to consider the ratio of voters. The relative frequencies of young unemployed households,
young people, employed households, and old people is 1 − lt : lt : 1+1 µ . To support the pension system,µ +2
the following equation is expected to hold: lt + 1+1 µ ≥ 12 (1 + 1+1 µ ) ⇐⇒ lt ≥ 2+2µ . Therefore, it isnecessary to classify the analyses into three cases depending on the value of lt .
Then, the preferences for unemployment insurance and pensions are given respectively as follows:• contribution to unemployment insurance:
∗
θwt
= arg

max Et [V (·)] =θw ∈[0,1]{
θw that satisfies (25a)0µ +2
if lt ≤ 2+2µ

otherwise(26)• contribution to pension:τt∗ = arg max Et [V (·)] =τ ∈[0,1]{
τ that satisfies (25b)0µ +2
if lt ≥ 2+2µ

otherwise(27)

The optimal solution can be derived as an intersection of reaction functions of the following two
kinds:

θwt = θwt (τ )τt = τt (θw )

(28a)
(28b)

Depending on the patterns of intersections, three plausible cases exist10) . As in the case of Fig. 8,
both the pension and the unemployment insurance are adopted; either the pension or the unemployment
insurance is supported as in the case of Fig. 9 or Fig. 10. Case 1 is that corresponding to the situationin which both pension and unemployment insurance survives. Cases 2 and 3 show the situation inwhich either the pension system or unemployment insurance survives. Case 2 is the situation in which
the pension system does and case 3 is the case in which only unemployment insurance survives.
To summarize, depending on the dynamics of capital accumulation and the unemployment rate, thecontents of social security system vary as in the following cases.

10) In the Appendix A.4, we show the two response functions are downward-sloping.

14

θw

θw

θw = θw (τ )
τ = τ (θw )θw = θw (τ )
τ = τ (θw )ττFigure 8 Case 1.

Figure 9 Case 2.

θw

θw = θw (τ )
τ = τ (θw )τFigure 10 Case 3.

Case 1. Both pension and unemployment insurance policies survive.
Case 2. Only the pension policy survives.
Case 3. Only the unemployment insurance survives.
Here, let us explain the change of the social security system under each case, i.e., the case of σ ≥ 0and σ ∈ (−1, 0) (the case of σ < 2α or σ > 2α ). First, regarding the former case, until the time
when unemployed people are in the majority, the social security is the same as in case 2 (Only pension
policy survives). Subsequently, the situation is the same as in case 3 (Only unemployment insurance
survives.). On the other hand, we can explain how the social security system varies under the case of
Fig. 7(b). Under this regime, both pension and unemployment insurance policies survive as in case
1 when the level of capital accumulation is not so high. After that, as the employment rate rises, it is15

likely that only unemployment insurance is adopted.
µ +2Stage. 1 lt < 2+2µ : Only unemployment insurance.

µ +2Stage. 2 lt = 2+2µ : Both pension and unemployment insurance.
µ +2Stage. 3 lt > 2+2µ : Only the pension system.

Stage. 4 After that, only the pension system survives.
Turning to the second case, depending on the value of l0 , both stages 1 and 2 emerge alternately andeventually; finally, either of the two policies is adopted.
µ +2Stage. 1 lt < 2+2µ : Only unemployment insurance.

µ +2Stage. 2 lt = 2+2µ : Both pension and unemployment insurance.

µ +2Stage. 3 lt > 2+2µ : Finally, either a pension or unemployment insurance is adopted.

Finally, regarding the third case, the contents vary as follows: both stages 1 and 2 emerge alternately.
Eventually, either of the two policies is adopted. The second and third cases are similar, but the patternsof fluctuation mutually differ.
µ +2Stage. 1 lt > 2+2µ : Only the pension system is adopted.

µ +2Stage. 2 lt < 2+2µ : Only unemployment insurance is adopted.

µ +2Stage. 3 lt = 2+2µ : Both pension and unemployment insurance.

Stage. 4 Finally, either the pension or unemployment insurance is adopted.
To summarize the discussion presented above,
Proposition 2 The patterns of policy change are summarized as follows.
1. Case of −1 < σ < 0:
For the last time, only the pension system survives.
2. Case of 0 < σ < 1, & σ < 2α :The case in which only unemployment insurance survives and the case in which only the pension
system survives emerge alternately.
3. Case of 0 < σ < 1, & σ > 2α :The case in which only unemployment insurance survives and the case in which only the pension
system survives emerge alternately. Eventually, either pension or unemployment insurance isadopted, depending on the value of the employment rate at the steady state.
■Remark

Regarding the assumption of commitment, we have two remarks. First, once thegovernment determines a policy based on voting, we assume that this policy lasts for periods whensome generation is alive. It is possible to consider this situation as a steady state. However, strictly
speaking, this differs from the steady state. The difference between the steady state is explained16

as follows: In the steady state case, all variables are constant through time t. In marked contrast,
assuming a commitment by the government in this paper means that we assume that policy variables
are constant while some generation is alive.
Secondly, some readers might wonder why the tax rate can be treated as constant despite the existence of the state variable. They consider that the tax rate is dependent on the state variable, asτt = τ (kt ), and that the tax rate cannot be treated as constant as long as it is dependent on the statevariable. We avoid such a question by assuming that the voting is held only once. Our answer to sucha question is the following. From the capital market condition, we have
Ks = Lt ses−1 (τs−1 , θw,s−1 ) + (Nt − Lt )sus−1 (τs−1 , θw,s−1 ).
This equation shows that capital at s period is dependent only on the past policy variables. Therefore, we can treat Ks as constant because Ks is not dependent on policy variables at s period andthereafter. Conversely, it is apparent that the policy variables at s period do not depend on the statevariable. Therefore, we can avoid the effects of a change in the state variables. Most studies which apply the structure-induced equilibrium (Conde-Ruiz and Galasso (2005), Konishi (2008), Bethencourt
and Galasso (2008), for instance) avoid this criticism by dropping state variables (i.e. capital) fromtheir model. However, it is the case with commitment that the situation in which policy variables areconstant while some generation is alive in Conde-Ruiz and Profeta (2007), Poutvaara (2006), and thepresent paper, as long as capital is not taxed.

4.2 Case without Commitment
In this subsection, the assumption of commitment over future social security policies is relaxed.
Before reviewing the analyses, let us define the equilibrium concept. Then, we investigate whether
each agent has an incentive to deviate or not.
First, in the spirit of Krusell, Quadrini and Rios-Rull (1997), let us define the equilibrium concept(politico-economic equilibrium11) ).
Defnition 2 (Politico-Economic Equilibrium) A (Markov perfect) politico-economic equilibrium is∞ that accords with the following.
defined as a pair of functions {ctiy , ctio , τt , θwt }t=0
∞ , each agent (employed people and unemployed people) determines(i) Given the sequence {θ f t }t=1

policy variables that maximize their individual utility. The optimal policy variables meet thefollowing maximization problem.
max Et [U(·)], subject to Kt+1 = Ψ(Kt , τt , θwt )τ ,θw11) This concept corresponds to so-called Markov-perfect equilibrium. These conditions are dependent on the relation be-

tween the t and t + 1 period. Therefore, this concept meets the Markov property. See also Forni (2005) who specificallyexamines the Markov-perfect equilibrium in an OLG model.

17

(ii) Budget constraints of pension and unemployment insurance are balanced in each period.
(iii) Firms maximize their profits.
(iv) The condition of the trade union’s wage setting holds.
(v) Finally, the following markets clear.
Commodity Market, eq. (13); Capital Market, eq. (14); Labor Market, eq. (15)We then formally define the voting game.

The public history of the game at t period,

ht = {(τ0 , θw0 ), (τ1 , θw1 ), ..., (τt−1 , θw,t−1 )} ∈ Ht is the sequence of social security system (pension and unemployment insurance). In fact, Ht is the set of all possible history at time t. An action
profile for the employee is, {τt , bt } ∈ [0, 1] × [0, 1]. Analogously, an action for unemployed individual
at time t is {τt , bt } ∈ [0, 1] × [0, 1].
Then, a strategy for the employee is at t period is a mapping from the history of the game into
the action space, i.e., σ e : ht → {τt , θwt }. Analogously, a strategy for unemployed people is at t
period is σ u : ht → {τt , θwt }. The strategy profile played by both individuals at t period is denoted asσt ≡ σte ∪ σtu .
At t periods, the objective function for each young player (i ∈ {e, u}) isi...) = Vti (τt , θwt , τt+1 , θw,t+1 ).
Vti (σ0i , σ1i , ....σti , σt+1

Regarding old agents,

Vt (σ0 , σ1 , ....σt , σt+1 ...) = Vti (τt , θwt ).

These solutions describe the relation between the policy at t period and the one at t + 1 period.
Moreover, we describe the definition of equilibrium.
Defnition 3 (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 the static game with commitment.
As contrasted with the analysis in the previous subsection, we assume that the government has nocommitment technology in this subsection. Then, let us define the history of the game Ht as
Ht0 ≡ {ht ∈ Ht |θw,t = θw ∗, t ∈ {0, 1, ...}},
andHtσ ≡ {ht ∈ Ht |θwk = 0, k = 0, 1, ...t0 , and θwt = 0, t ≥ t0 .}.

Moreover, the strategy profiles of employed people and unemployed people are respectively denoted
as σte and σtu . We then investigate whether each player has an incentive to deviate from the solution
under full commitment, as discussed in the previous subsection. Under this setting, we first verify that18

unemployed people have no an incentive to deviate from the strategy. We assume that unemployeddeviate > θ ∗ and τ ∗ < τ deviate . However, employed people dopeople adopt the following strategy: θt0,w
w
tt0

not obtain an additional payoff by deviation because employed people punish employed people byreducing the payment of contributions to the pension system, τ , which exerts negative effects on thewelfare of both agents. Therefore, it is apparent that unemployed people do not have an incentive todeviate from the commitment solution.
Regarding employed people, presuming that unemployed people deviate from equilibrium, i.e. they
avoid paying contributions to pensions, then the workers will punish unemployed people by not paying
contributions to unemployment insurance. Unemployed people would pay contributions to pensions to
avoid being punished. Therefore, it is apparent that they have no incentive to deviate. To summarize,
neither workers nor unemployed people 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.

5 Conclusion
This paper describes how the social security system evolves as voter attributes change. In our setting, policy determination is based on majority voting. The government has social security policy
mechanisms of two kinds: a pension and unemployment insurance. When younger workers and oldpeople constitute most voters, the pension system is supported. When young unemployed people arein the majority, unemployment insurance is adopted. Under such a situation, we show how the contents of the social security system evolve depending on the dynamics of capital accumulation and theunemployment rate, and show that the social security system ceases to exist in certain instances. This
result might explain the future of social security policy in developed countries, including Japan.
Finally, we conclude this paper by mentioning problems to be solved in the future. First, when unemployment insurance (intragenerational redistribution scheme) is supported, the following conditions
are needed: extremely highly population growth and/or a high unemployment rate. These parametersmust be more concrete. Second, it is necessary to demonstrate the possibility that neither the pension
nor unemployment insurance is adopted. Although the condition for the existence of such a situation
is not derived, the present paper might be absorbing by deriving the possibility that the social security
system vanishes completely. Finally, we presented the pattern of fluctuation of the social security system, but it might be necessary to fortify persuasion of our obtained result using a numerical simulation.

19

AppendixA
A.1 Derivation of eq. (16)1
{se × lt + stu × (1 − lt )}
1+µ t
{
{
e }
u }
]
dt+1
dt+1
1 [ 1
1
=
(1 − θwt − τt )wt − (1 + ρ )× lt +(1 − τt )bt − (1 + ρ )× (1 − lt )1+µ 2+ρ
Rt+1
2+ρ
Rt+1
e
u
d
d
1
1
=
{(1 − θwt − τt )wt − (1 + ρ ) t+1 } × lt + (1 − τt )bt − (1 + ρ ) t+1 × (1 − lt )}1+µ 2+ρ
Rt+1
Rt+1
1
1
1+ρ
u
=
{(1 − θwt − τt )wt lt + (1 − τt )bt (1 − lt ) −{(lt d e ) + ((1 − lt )dt+1)}{z} Rt+1 | t+1
1+µ 2+ρ |{z}k̂t+1 =(∗)(∗∗)

Here, using eqs.(11) and (12), the term (∗) and the second term (∗∗) are rewritten respectively as}{(1 − θwt − τt )wt lt + (1 − τt )bt }(1 − lt ) and (1 + µ )lt+1 wt+1 τt+1 (1 + θwt+1 + θ f ) . Moreover, usingeq. 7a and 7b, we obtain the following.
k̂t+1 =α (1 − (1 + θwt )τ + (1 − τt )θ f t )(1 + θ f t )k̂α l 1−α
(1 + µ )(1 + θ f t )((2 + ρ ) + (1 + ρ )(τt (1 + θwt + θ f t ))) t t= ε k̂t

A.2 Derivation of eq. (20); the dynamics of lt
At the steady state, we have

σ −α

1−σ

α

(l) σ (1 − l α ) σ = A 1−α .

(29)Letting the right side of the equation above be ψ (l), we obtain the expression shown below.

ψ ′ (l) = [(1 −■Case of 0 < σ < ∞

α
1−σ
α
1−σ α
)−α
l (1 − l α )−1 ](l)− σ (1 − l α ) σσσ

In this case, ψ ′ (l) is positive. By totally differentiating eq. (19), we have:1

dlt+1
ψ α + ψ (1 − l)ψ ′= 1<1
dlt lt+1 =lt =l ψ α + ψ (1 − l)ψ ′(30)

Therefore, the steady state equilibrium is locally stable.
■Case of −1 < σ < 0, & σ < 2α

We investigate the relation between |φ ′ (l)| and 1. In this case,ψ ′ (l) < 0 and ψ (0) = ∞ and ψ (1) = 0. We then obtain1

dlt+1
ψ α + ψ (1 − l)ψ ′= 1
dlt lt+1 =lt =l ψ α + ψ (1 − l)ψ ′
20

(31)

At l = 1, we have
dlt+1
=0
dlt lt+1 =lt =l=1
and at l = 0, we have, by L’Hôpital’s rule,1

dlt+1{ψ α + ψ (1 − l)ψ ′ }′=dlt lt+1 =lt =l=0 {ψ α1 + ψ (1 − l)ψ ′ }′Then, we haveαψ α + α1 ψ α −1 ψ ′ (1 − l)ψ ′ + ψ α (1 − l)ψ ′′1

11αψ α + α12 ψ α −1 ψ ′ (1 − l)ψ ′ + α1 ψ α (1 − l)ψ ′′
1

1

1

>0

. Therefore, the dynamics of lt is depicted as Fig. 7(b).
■Case of −1 < σ < 0, & σ > 2α

We then have1

dlt+1
ψ α + ψ (1 − l)ψ ′= 1.
dlt lt+1 =lt =l ψ α + ψ (1 − l)ψ ′For the interval 0 < l < 1, there exists l¯ such that

dlt+1
dlt l

t+1 =lt =l(32)= 0m because φ ′ (·) < 0. Therefore,¯ 1],¯ dlt+1 < 0 and for [l,for the interval 0 < l < 1, there exists at least one solution that satisfies [0, l],
dlt
dlt+1
dlt > 0. The dynamics under this case is depicted as Fig. 7(c).

A.3 Shapes of the Response functions
Here, we show the two response functions are downward-sloping. Then, we investigate the sign of∂ θw∂τ∂ θw and ∂ τ . Each type of individual determines the contributions to unemployment insurance, θw andpension, τ to maximize personal utility. Preferences to θw and τ are derived by solving the following
first order conditions:()∂ Et [Vt ]2+ρ−wt= 0 ⇐⇒ lt
{
}=0
∂ θw
1+ρ
1 − τ − θw()
∂ Et [Vt ]2 + ρ −bt
2+ρ−wt
} + (1 − lt )(= 0 ⇐⇒ lt
{
){
}=0
∂τ
1+ρ
1 − τ − θw
1+ρ 1−τFirst, we define the LHS of (33a) as(

G(·) ≡ lt

2+ρ
1+ρ

)
{

−wt
}.
1 − τ − θw

From the implicit function theorem, we find
∂ G(·)∂τ∂ θw= − ∂ G(·)
<0
∂ θw
∂τ

Similarly, we have

∂ θw
< 0,
∂τ
which shows that the two response functions are downward-sloping.
21

(33a)
(33b)

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