Discussion Papers In Economics
And Business

Retirement and Social Security: A Political Economy
Perspective
Ryo Arawatari

Tetsuo Ono

Discussion Paper 10-04-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRetirement and Social Security: A Political Economy
Perspective
Ryo Arawatari

Tetsuo Ono

Discussion Paper 10-04-Rev.

September 2010

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRetirement and Social Security: A Political Economy
Perspective∗
Ryo Arawatari†

Tetsuo Ono‡

Shinshu University

Osaka University

September 4, 2010

Abstract
Countries with higher implicit taxes on continued work are associated with lower
labor force participation rates of the elderly. This paper constructs a politicoeconomic model that accounts for this feature based on multiple, self-fulﬁlling expectations of agents. In this model, agents are identical at birth and can become
skilled (or remain unskilled) through educational investment. When agents holdexpectations of larger social security beneﬁts, it provides a disincentive to engage
in educational investment, thereby resulting in an unskilled majority. In turn, thisunskilled majority supports larger social security beneﬁts, which induces the retirement of the elderly and thus results in a lower labor force participation rate. The
opposite applies when agents have expectations of smaller social security beneﬁts
in their old age.
Key words Political equilibrium; Retirement; Self-fulﬁlling expectations; Social
security
JEL classification H55, D72, J26∗

Arawatari acknowledges the ﬁnancial support of the Japan Society for the Promotion of Science(JSPS) through a Grant-in-Aid for Young Scientists (Startup) (No. 21830046). Ono also acknowledges
the ﬁnancial support of the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid
for Young Scientists (B) (No. 20730208).
†
Faculty of Economics, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano, 390-8621 Japan. Email: ryo a@shinshu-u.ac.jp‡
Corresponding author. Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp1

1

Introduction

The labor force participation rate of the elderly in the OECD is negatively correlated withtaxes on their continued work. For instance, Daval (2003) and Fenge and Pestieau (2005)
argue that countries in the OECD can be classiﬁed into two groups: one characterized
by a higher tax burden and lower labor force participation rate of the elderly, and theother characterized by a lower tax burden and higher labor force participation rate of
the elderly. The former includes Austria, France, Germany and Italy; the latter includes
Canada, Norway, Sweden, the United Kingdom, and the United States.
Several studies have attempted to reveal the mechanism underlying this interestingobservation (see, for example, Sheshinski, 1978; Crawford and Lilien, 1981; and the recent
survey by Feldstein and Liebman, 2002). In general, continued work in older age may besubject to the burden of payroll tax and forgone pension beneﬁts. This double burden,
regarded as an implicit tax on the elderly (Gruber and Wise, 1999), provides the elderlywith an incentive to retire earlier. That is, a higher tax burden results in earlier retirement
and thus a lower labor force participation rate of the elderly. Indeed, Gruber and Wise
(1999) argue that this implicit tax on the elderly explains more than 80% of the crosscountry variation in unused labor capacity for those 55 to 65 years of age.
All of the above-mentioned studies focus on the eﬀect of the tax burden on retirement
decisions. However, little attention has been given to the question why some countries
choose a higher tax burden on the elderly while others select a lower tax burden, eventhough they share a similar political and economic background. This paper develops apolitico-economic framework that responds to this important question, and demonstrates
how the retirement behavior of the elderly is aﬀected by voting on taxation. We thenshow the mechanism underlying the diﬀerences in social security and retirement behavior
across the OECD countries.
Our framework is based on that developed in Hassler, Storesletten and Zilibotti (2007).
In this approach, agents identical at birth can aﬀect their prospects in life with educational
investment. In particular, agents become either skilled or unskilled, and by undertaking
costly investment can increase the probability of becoming skilled in their youth. This
framework is further enriched by allowing for endogenous retirement decisions in oldage. Here, each type of old agent obtains the opportunity to continue to work withsome probability and makes his/her decision on retirement from the viewpoint of utility
maximization. The government is assumed to provide old-age social security ﬁnanced bytaxes levied on the young and the working old.
Under the above-mentioned framework, there are multiple, self-fulﬁlling expectationsof agents. In undertaking educational investment, young agents have expectations ofsocial security in their old age. When young agents hold expectations of higher social
security beneﬁts, it provides a disincentive to engage in educational investment, thereby
resulting in a lower proportion of the skilled. This implies a greater number of unskilled
young individuals, which in turn increases the demand for old-age social security andthus enhances retirement. The opposite applies when young agents have expectationsof lower social security in old age. The economy may then attain multiple equilibria,
including a skilled-majority equilibrium featuring a low tax burden and late retirement,
and an unskilled-majority equilibrium featuring a high tax burden and early retirement.
Ultimately, the particular outcome attained depends on the expectations of agents.
2

The ﬁnding of multiple equilibria well ﬁts the empirical evidence. In the skilledmajority equilibrium, the skilled old choose a lower level of social security and thus alower tax burden on the working old. This gives the old an incentive to continue to work,
thereby resulting in the higher labor force participation rate of the elderly. The opposite
applies in the unskilled-majority equilibrium. In terms of real-world observations, theskilled-majority equilibrium represents a group of countries including Canada, Norway,
Sweden, the United Kingdom, and the United States; and the unskilled-majority equilibrium comprises a group of countries including Austria, France, Germany and Italy (Daval,
2003; Fenge and Pestieau, 2005).
Interestingly, Norway and Sweden are much less unequal than Canada, the United
Kingdom and the United States, even though they present a similar level of taxes andcomparable labor force participation of the elderly. However, while the evidence of similarity at ﬁrst appears counterintuitive, this can be explained by self-fulﬁlling expectations.
That is, people in Norway and Sweden happen to have similar expectations, as do thosein Canada, the United Kingdom and the United States, regarding future social security.
This similarity of expectations leads to an identical pattern of taxation and labor force
participation in each of the two groups of countries.
We undertake a numerical analysis of the pension–GDP ratio to enable further investigation of these cross-country diﬀerences. We ﬁnd that the unskilled-majority equilibrium
attains a higher pension–GDP ratio than the skilled-majority equilibrium, and a moreequal society realizes a lower pension–GDP ratio. This numerical result appears to ﬁt theavailable empirical evidence. In fact, the ratio in Austria, France, Germany and Italy,
representing the unskilled-majority equilibrium, is about twice that of Canada, Norway,
Sweden, the United Kingdom and the United States, representing the skilled-majority
equilibrium. In addition, the evidence indicates a negative correlation between wageequality and the size of pensions across the OECD countries.
We also pursue a numerical investigation of the expected utility of the young in order
to consider the welfare implications of wage inequality and the multiple expectations ofagents. We ﬁnd that the young obtain a higher expected utility in the skilled-majority
equilibrium than in the unskilled-majority equilibrium. This is mainly because the former
equilibrium requires a lower tax burden on the old and thus realizes a higher probabilityof being successful in education. We also ﬁnd that more wage equality results in a higher
expected utility of the young in the skilled-majority equilibrium, and a lower expectedutility of the young in the unskilled-majority equilibrium. These dissimilar results appear
to arise from diﬀerences in voting behavior in the two equilibria.
The current paper relates to the following two strands of literature. The ﬁrst strand
is the literature on the political economy of social security and retirement, including
studies focusing on early retirement provisions (Conde-Ruiz and Galasso, 2003, 2004),
old-age social security provisions (Profeta, 2002; Casamatta, Cremer and Pestieau, 2005,
2006), and the legal retirement age (Lacomba and Lagos, 2006, 2007; Galasso, 2008). The
current paper is similar to that of Casamatta, Cremer and Pestieau (2005, 2006) in that
we consider the political determination of social security and its impact on retirement.
However, our work diﬀers in that we introduce a link between current economic decisions
and the expectations of future social security, and this creates the multiple, self-fulﬁlling
expectations of agents that help to explain the empirical evidence.
The second strand is the literature on the dynamic political economy of social security
3

based on the concept of Markov-perfect equilibrium. This body of work includes studies
demonstrating a unique equilibrium pinned down by the initial (Grossman and Helpman,
1998; Azariadis and Galasso, 2002) and multiple, self-fulﬁlling expectations of agents(Hassler et al., 2003; Hassler, Storesletten and Zilibotti, 2007). Conde-Ruiz, Galasso andProfeta (2005) extend the framework in the ﬁrst two studies by introducing the retirement
decisions of old agents. In contrast, the current paper extends the framework in the lattertwo studies to demonstrate multiple equilibria as regards social security and retirement.
The current paper also shows that the self-fulﬁlling expectations of agents explain why
two diﬀerent countries in terms of wage inequality display similar levels of taxes and labor
force participation of the elderly.
The organization of this paper is as follows. Section 2 develops the model. Section 3
demonstrates the multiple political equilibria. Section 4 undertakes the numerical analysis, and Section 5 discusses two extensions. Section 6 provides some concluding remarks.

2

The Model

The model is a two-period-lived, overlapping-generations model based on that discussed in
Hassler, Storesletten and Zilibotti (2007). Time is discrete and denoted by t = 0, 1, 2, · · · .
The economy comprises a continuum of agents living for two periods, youth and old age.
Each generation has a unit mass. Agents are identical at birth.
Consider the young agents born in period t. They can aﬀect their prospects in life witheducational investment. In particular, they become either skilled or unskilled (denotedby s and u, respectively), and by undertaking costly investment, they can increase theprobability et of becoming skilled in their youth. The cost of investment, measured interms of disutility, is given by (et )2 /2. Skilled agents earn a high wage, normalized tounity, whereas unskilled agents earn a low wage, normalized to w ∈ [0, 1) over their life
cycle. A lower w implies higher wage inequality between the two types of agents.
At the beginning of period t + 1, there are two types of old agents: the skilled andthe unskilled. Our model departs from Hassler, Storesletten and Zilibotti (2007) in thefollowing respects. First, in old age, each agent faces job loss with a probability of
1 − µ ∈ [0, 1] because of job replacement or deterioration in health. However, they may
obtain opportunities to continue to work with a probability of µ. The situation wherethese opportunities diﬀer between skilled and unskilled old agents will be examined injSection 4. Second, the cost of continuing to work is given by (lt+1)2 /2 (j = s, u), wherejlt+1∈ [0, 1] is the amount of labor supplied by a type-j old agent. Figure 1 illustrates thetiming of events.
[Figure 1 about here.]
There is no storage technology in this economy: each individual uses his/her endowments within a given period. The government provides old-age social security, b, ﬁnancedby taxes levied on the young and the working old. The tax rates are age dependent: τ ofor the old and τ y for the young. In turn, the tax rates are determined before the young
agents decide on their investment and before the old agents decide on their retirement.

4

Therefore, the expected utility functions of the agents alive at time t are given as follows:{}(lts )2ossoVt = µ lt (1 − τt ) −+ bt ,(1)
2{}(ltu )2ouuoVt = µ wlt (1 − τt ) −+ bt ,(2)2(ey )2
Vty = et (1 − τty ) + (1 − et )w(1 − τty ) − t
2 )
[ { (
}
s
2
(l
t+1 )
s
o
+ β et µ lt+1 (1 − τt+1 ) −+ bt+1
2
}]
)
{ (
u
(lt+1
)2
y
o
u
+(1 − et ) µ wlt+1 (1 − τt+1 ) −+ bt+1 ,(3)
2
where Vtos , Vtou and Vty denote the utility of the skilled old, the utility of the unskilled
old, and the expected utility of the young, respectively. The utility levels of Vtos , Vtou
and Vty are computed prior to individual success or failure in education. The parameterβ ∈ (0, 1) is a discount factor.
Given these preferences, the skilled old agent chooses lts to maximize Vtos ; the unskilled
old agent chooses ltu to maximize Vtou ; and the young agent in period t chooses et tosmaximize Vty by taking account of the optimal labor supply in his/her old age, lt+1
andu
lt+1 . The optimal choices of the old and the young are respectively given by:ls∗ (τto ) = 1 − τto ; lu∗ (τto ) = w(1 − τto );[]ooey∗ (τty , τt+1
) = (1 − w) (1 − τty ) + βµ(1 + w)(1 − τt+1
)2 /2 .

(4)
(5)

Young agents are ex ante identical, and therefore agents in the same cohort choose
the same investment. This implies that at the beginning of period t + 1, the proportionof the unskilled old, ut+1 , is equal to the proportion of the unsuccessful period-t youngoagents, 1 − ey∗ (τty , τt+1
):
[
]
o
o
ut+1 ≡ 1 − ey∗ (τty , τt+1
) = 1 − (1 − w) (1 − τty ) + βµ(1 + w)(1 − τt+1
)2 /2 .
The proportion of the unskilled old at the time of voting, ut+1 , depends on the tax leviedon the young in period t, τty , and the tax levied on working old agents in period t + 1,oτt+1.
The tax revenues from the young and the working old are transferred to every old
agent in a lump-sum fashion. The government budget is balanced in each period so it canbe expressed as:o),bt = W (τto , ut ) + Z(τty , τt+1o) is the tax revenue from
where W (τto , ut ) is the tax revenue from the old and Z(τty , τt+1
the young, deﬁned by:

W (τto , ut ) ≡ {(1 − ut ) + ut w2 }µ(1 − τto )τto ;
}]
[
{
o
o
)2 /2 τty .
) ≡ w + (1 − w)2 (1 − τty ) + βµ(1 + w)(1 − τt+1
Z(τty , τt+1

5

3

Political Equilibria

This section characterizes the political equilibria where agents vote on taxation period
by period. Section 3.1 provides the deﬁnition of a political equilibrium based on theconcept of a stationary Markov-perfect equilibrium with majority voting. Sections 3.2
and 3.3 provide the characterization of political equilibria classiﬁed according to the typeof majority.

3.1

Deﬁnition of Political Equilibrium

Following Hassler, Storesletten and Zilibotti (2007), we assume that elections are held at
the end of each period and the elected politicians set the tax rates for the following period.
The old abstain from voting because they have no interest in the following period; only
the young participate in voting. This is observationally equivalent to assuming that only
old agents vote over current taxes at the beginning of each period. We adopt the latter interpretation in the following analysis. Our assumption about voting implies that we focus
on the intragenerational, rather than intergenerational, conﬂict over redistribution. Withooptimal choices ls∗ (τto ), lu∗ (τto ) and e∗ (τty , τt+1
) and the government budget constraint, theindirect utility functions of the skilled and the unskilled old are respectively given by:µo(1 − τto )2 + W (τto , ut ) + Z(τty , τt+1
),
2
µ
o
),
Vtou = (w)2 (1 − τto )2 + W (τto , ut ) + Z(τty , τt+1
2
Vtos =

where the term µ(1 − τto )2 /2 in the ﬁrst line is the expected after-tax income of the skilled
old, the term µ(w)2 (1 − τto )2 /2 in the second line is the expected after-tax income of theounskilled old, and the term W (τto , ut ) + Z(τty , τt+1
) is the old-age social security beneﬁt.
This paper focuses on stationary Markov-perfect equilibria with majority voting. The
proportion of the unskilled old (ut ) summarizes the state of the economy; the identity ofa decisive voter depends on this proportion. An oﬃce-seeking politician elected by voterssets policies to maximize the utility of the larger group. Given these features, we now
provide the deﬁnition of the political equilibrium as follows.
Deﬁnition: A (stationary Markov-perfect) political equilibrium is deﬁned as a tripletof functions {T o , T y , U }, where T o : [0, 1] → [0, 1] and T y are two public policy
rules, τto = T o (ut ) and τty = T y , and U : [0, 1] → [0, 1] is a private decision rule,
ut+1 = U (τty ), such that given u0 , the following functional equations hold.
1. T o (ut ) = arg maxτto ∈[0,1] W dec (τto , ut ) (dec = os, ou), where:{
if ut ≤ 1/2,
W os (τto , ut ) ≡ µ2 (1 − τto )2 + W (τto , ut )dec o
W (τt , ut ) =
µ
ou o
o
2
o 2
W (τt , ut ) ≡ 2 (w) (1 − τt ) + W (τt , ut ) if ut > 1/2oo= T o (U (τty ))), with τt+1
2. U (τty ) = 1 − e∗ (τty , τt+1oo
= T o (U (τty ))) subject to τt+1
3. T y = arg maxτto ∈[0,1] Z(τty , τt+1

6

The ﬁrst equilibrium condition requires that the decisive voter chooses τto to maximize
the utility of the skilled old (if ut < 1/2) or the unskilled old (if ut > 1/2). In the case ofequal numbers of skilled and unskilled old agents (i.e., ut = 1/2), the skilled old are assumed to be decisive. The second equilibrium condition implies that all young individualsochoose their investment optimally, given τty and τt+1
, under rational expectations about
future taxes and the distribution of types. The third equilibrium condition requires thatthe decisive old voter chooses τty to maximize revenue from the young. Rational voters
understand that their choice over current redistribution aﬀects future redistribution via
the private decision rule and public policy.

3.2

The Determination of T o and U

We now solve the equilibrium conditions recursively. Condition 1 deﬁnes a one-to-one
mapping from the state variable to the equilibrium choice of taxation of the old: τto =T o (ut ). Suppose that the skilled old form the majority: ut ≤ 1/2. The objective functionof the majority is given by W os (τto , ut ) ≡ µ(1 − τto )2 /2 + W (τto , ut ), which is strictly
decreasing in τto : ∂W os (τto , ut )/∂τto < 0. The skilled old pay more than they receive
because the unskilled agents pay less than the skilled agents, but the revenue is distributed
equally between the skilled and unskilled old agents. Therefore, the skilled old preferτto = 0:
]
[
1
o
.
(6)
T (ut ) = 0 if ut ∈ 0,
2
Alternatively, suppose that the unskilled old form the majority: ut > 1/2. The
objective function of the majority is given by W ou (τto , ut ) ≡ µ(w)2 (1 − τto )2 /2 + W (τto , ut ),which has the following properties:∂W ou (τto , ut )= µ(1 − ut )(1 − (w)2 ) > 0,o∂τt
τ o =0tou∂W (τto , ut )1
= − µ(w)2 < 0,o∂τt2τto =1/2[]∂ 2 W ou (τto , ut )
2
=
µ
−2(1
−
u
)
+
(1
−
2u
)(w)
< 0.
t
t
∂τto2

Given these properties, there is a unique tax rate τto that maximizes W ou (τto , ut ) :()(1 − ut )(1 − (w)2 )
1
ou o
∈ 0,
.
arg max
W (τt , ut ) =τto2(1 − ut ) + (w)2 (2ut − 1)2
The optimal tax rate chosen by the unskilled old depends on the population of theunskilled old agents, ut . This makes it impossible to obtain an analytic solution. In fact,
the model without retirement decisions (Hassler, Storesletten and Zilibotti, 2007) could
derive an analytic solution because the unskilled old choose τto = 1 irrespective of thestate variable ut . The state-independent tax rates on the old enable us to attain analytically tractable solutions of multiple, self-fulﬁlling expectations of agents when employing
the Hassler, Storesletten and Zilibotti (2007) framework. In order to retain analyticaltractability, we impose the following assumption regarding the choice of the tax rate by
the unskilled old.
7

Assumption 1. The politically available tax rates for the old are limited within therange [0, τ o ], where τ o ∈ (0, 1) satisﬁes ∂W ou (τto , ut )/∂τto |τto =τ o > 0.
Figure 2 illustrates a situation where Assumption 1 holds. The unskilled old wish tochoose the tax rate that attains the top of the Laﬀer curve W ou (τto , ut ): τto = arg max W ou (τto , ut ).
However, Assumption 1 restricts their available choices to [0, τ o ] because of some institutional or political constraint. Given this restriction, the unskilled old choose τto = τ o fromthe viewpoint of utility maximization. Under Assumption 1, the mapping that satisﬁes
Equilibrium Condition 1 is summarized as follows:{
0 if ut ≤ 21 ,o
(7)
T (ut ) =
τ o if ut > 21 .
[Figure 2 about here.]
Next, we rewrite Equilibrium Condition 2 by substituting in the optimal investmentoey∗ (τty , τt+1). This yields the following functional equation:
{
}
βµ
2
y
y
y
o
U (τt ) = 1 − (1 − w) (1 − τt ) +(1 + w) (1 − T (U (τt ))) ,(8)2
where T o (·) is given by (7). We derive the solution to the functional equation (8) byassuming rational expectations. The solution to the functional equation (8) is given by:{}{
U (τty , 0) ≡ 1 − (1 −{w) (1 − τty ) + βµ
(1 + w)
if τty ≤ τ̄ yy
2}
U (τt ) =(9)2yy
U (τt , τ o ) ≡ 1 − (1 − w) (1 − τt ) + βµ
(1 + w) (1 − τ o )if τty > τ y2where τ̄ y and τ y are deﬁned as:−1 + βµ(1 − w)(1 + w),
2(1 − w)
−1 + βµ(1 − w)(1 + w) (1 − τ o )2
y
τ ≡1+
.
2(1 − w)τ̄ y ≡ 1 +The interpretation of (9) is as follows. Suppose that the agents in period t expectoτt+1= 0. Under this expectation, young agents choose their investment as ey∗ (τty , 0) =(1 − w) [(1 − τty ) + βµ(1 + w)/2]. By (7), this expectation is rational if 1 − ey∗ (τty , 0) ≤1/2; that is, if τty ≤ τ̄ y . Next, suppose that the young agents in period t expecto= τ o . Under this expectation, young agents choose their investment as ey∗ (τty , τ o ) =τt+1(1 − w) [(1 − τty ) + βµ(1 + w)(1 − τ o )2 /2]. By (7), their expectation is rational if 1 −ey∗ (τty , τ o ) > 1/2; that is, if τty > τ y . Figure 3 illustrates an example of the solutions to
the functional equation (9).
[Figure 3 about here.]
There are multiple, self-fulﬁlling expectations of U for the set of τty ∈ [τ y , τ̄ y ]. The
particular U that arises in equilibrium depends on the expectations of agents. To illustrate
U in equilibrium, we follow the method in Hassler, Storesletten and Zilibotti (2007) andintroduce the critical rate of τty : θ ∈ (max(0, τ y ), τ̄ y ]. The rate θ, which depends on the8expectations of agents, is the highest tax rate that can yield a majority of the unskilled
old. For τty > θ, the majority is the unskilled old; however, for τty ∈ (max(0, τ y ), θ], themajority can be either the skilled or the unskilled old depending on agents’ expectations.
Given the deﬁnition of θ, the solution is now given by:
U (τty , 0)ifτty ≤ max(0, τ y )yy{U (τt , 0), U (τt , τ o )} if max(0, τ y ) < τty ≤ θ
U (τty ) =
(10)

y
y
o
U (τt , τ )ifθ < τt .
Figure 4 illustrates three possible cases of the solution in (10).
[Figure 4 about here.]

3.3

The Determination of T y and Characterization of the Political Equilibria

Given the characterization of T o and U satisfying Equilibrium Conditions 1 and 2, respectively, we now consider the political determination of τty that satisﬁes Equilibrium
Condition 3. Because there are two possible cases of a majority, we introduce corresponding deﬁnitions of the political equilibria: a skilled-majority equilibrium and an unskilledmajority equilibrium. In particular, the ﬁrst equilibrium condition given by (7) implies
that when the majority are skilled, there is a skilled-majority equilibrium where agentsoexpect no taxation on the old in the future (τt+1= 0) and choose τty to induce a majorityof skilled at time t + 1 (ut+1 ≤ 1/2). In contrast, when the majority are unskilled, there isan unskilled-majority equilibrium where agents expect taxation on the old in the futureo(τt+1= τ o ) and choose τty to induce a majority of unskilled at time t + 1 (ut+1 > 1/2).
In order to ﬁnd a τty that satisﬁes Equilibrium Condition 3, we note the following
properties: (i) Z(τty , 0) > Z(τty , τ o )∀τty ∈ [0, 1]; (ii) Z(τty , 0) and Z(τty , τ o ) attain the topsof the Laﬀer curves at τty = τ ys and τ yu , respectively, where:
[
{
}]
βµ
1
y
2
ys
w + (1 − w) 1 +(1 + w) ,τ ≡ arg max Z(τt , 0) =2(1 − w)2
2
[
{
}]
1
βµ
y
yu
o
2
o 2
τ ≡ arg max Z(τt , τ ) =w + (1 − w) 1 +(1 + w)(1 − τ );2(1 − w)22oand (iii) the expectations of τt+1
= 0 and τ o are rational if and only if τty ≤ θ(≤ τ̄ y ) andyτt > τ y , respectively. Given these properties, the revenue from the young is illustrated
in Figure 5.

[Figure 5 about here.]
Panels (a) and (b) in Figure 5 depict the skilled-majority equilibrium; that is, theo= 0. In particular,
revenue from the young is maximized under the expectation of τt+1yys
Panel (a) demonstrates the case where τ ≤ τ holds. Under this condition, agents canchoose τty = τ ys irrespective of the expectation parameter θ. Revenue is maximized bychoosing τty = τ ys . In contrast, Panel (b) illustrates the case where τ ys > τ y holds. Agentso= 0, but the
may have an opportunity of choosing τty = τ ys under the expectation of τt+1
choice depends on the expectation of θ. The choice of τty = τ ys is feasible if θ is high such9that θ ≥ τ ys ; however, it is unfeasible if θ is low such that θ < τ ys . Under the expectationoof τt+1= 0, revenue is maximized by setting τty = min(θ, τ ys ), where τty = τ ys is included
as a special case.
Panels (c) and (d) illustrate the unskilled-majority equilibrium; that is, revenue fromothe young is maximized under the expectation of τt+1
= τ o . In particular, Panel (c)yudemonstrates the case where τ ≤ 1, that is, it is feasible to attain the top of theLaﬀer curve Z(τty , τ o ), while Panel (d) demonstrates the case where τ yu > 1, that is,
it is unfeasible to attain the top of the Laﬀer curve. In both cases, the expectation θ
is assumed to be low, implying that the concerned choice that produces the unskilledo
majority equilibrium dominates the option of (τty , τt+1
) = (θ, 0) that produces the skilledmajority equilibrium in terms of revenue from the young.
Before proceeding to the analysis, we deﬁne the following four equations that help us
characterize the political equilibria:w(2 − w),
(1 + w)(1 − w)2 {(1 − τ o )2 − 1/2}
2w2 − 5w + 1
βµ = f 2 (w) ≡,(1 − w)(1 + w)[−(1 + w) + 2(1 − w)2 (1 − τ o )2 ]√(1 + w) − {{w + (1 − w)2 }(1 − τ o )2 } − y3βµ = f (w) ≡,(1 − τ o )4 (1 − w)2 (1 + w)/22(w2 − 3w + 1)
βµ = f 4 (w) ≡,(1 + w)(1 − w)2 (1 − τ o )2βµ = f 1 (w) ≡where y in f 3 (w) is given by:[]2y ≡ {w + (1 − w)2 }(1 − τ o )2 − (1 + w) − (1 − τ o )4 (w(2 − w))2 .
The implications of these equations are described below. The graphs of these equations
are illustrated in Figure 6. As depicted in the ﬁgure, there is a unique level of w, denotedby ŵ, where f 2 , f 3 and f 4 cross: f 2 (w) = f 3 (w) = f 4 (w) at w = ŵ.
[Figure 6 about here.]
Using these equations, we provide a characterization of each type of equilibrium inturn. We ﬁrst provide the characterization of the skilled-majority equilibria.
Proposition 1(i) Suppose the following conditions hold:
(
√ )
2
−
2
τ o ∈ 0,and βµ ≥ f 1 (w).
2
There exists a set of skilled-majority equilibria such that ∀t, T o is given by (7), U
is given by (10), and T y = τ ys . The equilibrium outcome is unique and such that∀t, τty = τ ys , τto = 0, and ut = 1 − (1 − w){(1 − τ ys ) + βµ(1 + w)/2}.
10

(ii) Suppose the following three conditions hold:

[ √ )
(
√ )
2− 2 1
2− 2
o
o

τ
∈
,
,
or
τ
∈
0,
and βµ < f 1 (w);222βµ ≥ f 2 (w); andβµ ≥ f 3 (w).
There exists a set of skilled-majority equilibria such that ∀t, T o is given by (7), U
is given by (10), and T y = min(θ, τ ys ). The equilibrium outcome is indeterminate
such that ∀t, τty = min(θ, τ ys ), τto = 0, and ut = 1 − (1 − w){(1 − min(θ, τ ys )) +βµ(1 + w)/2}.
Proof. See Appendix 7.1.
Areas P.1(i) and P.1(ii) in Figure 6 indicate the set of parameters (w, βµ) satisfying
the equilibrium conditions in Statements (i) and (ii) in Proposition 1, respectively. The
assumption in Statement (i) ensures that it is feasible to choose τty = τ ys under theoexpectation of τt+1
= 0 irrespective of the expectation parameter θ. Because this choiceattains the top of the Laﬀer curve Z(τty , 0), the choice dominates any other pair of tax rates
from the viewpoint of tax revenue maximization (see Panel (a) in Figure 5). A uniqueoequilibrium with (τty , τt+1
) = (τ ys , 0) then exists when the assumption in Statement (i)holds.
The ﬁrst assumption in Statement (ii), which is the exact opposite of the ﬁrst assumption in Statement (i), implies that the choice of τty depends on the expectations parameteroθ when agents expect τt+1
= 0. In other words, it is unfeasible to set τty = τ ys irrespective of θ. For example, the decisive voter cannot choose τty = τ ys if θ is low such that
θ < τ ys ; however, he/she can choose τty = τ ys if θ is high such that θ ≥ τ ys . The choiceof the decisive voter depends on the expectations of agents represented by the parameter
θ. The revenue from the young is maximized by setting τty = min(θ, τ ys ), and thus theequilibrium outcome becomes indeterminate, as illustrated by Panel (b) in Figure 5.
Under the current situation, there is either of the following alternative choices for theodecisive voter: setting τty = 1 under the expectation of τt+1
= τ o if τ yu > 1, and setting
y
yu
o
o
yu
τt = τ under the expectation of τt+1 = τ if τ ≤ 1. The concerned choice is sustained
against these two alternatives if Z(θ, 0) ≥ Z(1, τ o ) and Z(θ, 0) ≥ Z(τ yu , τ o ); that is, ifθ ≥ θ̃ and θ ≥ θ̂ where:√θ̃ ≡ τ ys − (τ ys − 1)2 + 2(τ ys − τ yu );√θ̂ ≡ τ ys − (τ ys )2 − (τ yu )2 .
Figure 7 illustrates the determination of θ̃ and θ̂. Details of the derivation of θ̃ and θ̂are given in the Appendix. Because θ is bounded from above τ̄ y , Z(θ, 0) ≥ Z(1, τ o ) andZ(θ, 0) ≥ Z(τ yu , τ o ) hold if θ ∈ [θ̃, τ̄ y ] and θ ∈ [θ̂, τ̄ y ]. The second and third assumptionsin Statement (ii) ensure that [θ̃, τ̄ y ] and [θ̂, τ̄ y ] are nonempty.
[Figure 7 about here.]
The next proposition establishes the conditions for the existence of the unskilledmajority equilibria.
Proposition 2
11

(i) Suppose the following conditions hold:
[ √ )
(
{
√ )
τ o ∈ 2−2 2 , 21 , or τ o ∈ 0, 2−2 2 and βµ < f 1 (w); andβµ ≤ f 4 (w).
There exists a set of unskilled-majority equilibria such that ∀t, T o is given by (7), U
is given by (10), and T y = τ yu . The equilibrium outcome is unique such that ∀t,τty = τ yu , τto = τ o , and ut = 1 − (1 − w){(1 − τ yu ) + βµ(1 + w)(1 − τ o )2 /2}.
(ii) Suppose the following conditions hold:
[ √ )
(
{
√ )
τ o ∈ 2−2 2 , 12 , or τ o ∈ 0, 2−2 2 and βµ < f 1 (w); andβµ > f 4 (w).
There exists a set of unskilled-majority equilibria such that ∀t, T o is given by (7),
U is given by (10), and T y = 1. The equilibrium outcome is unique such that ∀t,τty = 1, τto = τ o , and ut = 1 − (1 − w)βµ(1 + w)(1 − τ o )2 /2.
Proof. See Appendix 7.2.
Areas P.2(i) and P.2(ii) in Figure 6 indicate the set of parameters (w, βµ) satisfying theequilibrium conditions in Statements (i) and (ii) in Proposition 2, respectively. The ﬁrst
assumptions in Statements (i) and (ii) are identical to the ﬁrst assumption in Statement(ii) of Proposition 1. These mean that it is not always feasible to choose τty = τ ys underothe expectation of τt+1
= 0. When this choice is always feasible, the choice dominates
any pair of tax rates that induces the unskilled-majority equilibrium.
The second assumption in each statement determines the tax rate on the young τtyothat maximizes the revenue from the young under the expectation of τt+1
= τ o . The
second assumption in Statement (i) is equivalent to τ yu ≤ 1, implying that it is feasible
to choose a τty = τ yu that attains the top of the Laﬀer curve under the expectation ofoτt+1= τ o . In contrast, the second assumption in Statement (ii) is equivalent to τ yu > 1,
implying that it is unfeasible to choose τty = τ yu .
Given the argument thus far, we can conclude that the revenue from the young isomaximized by choosing τty = min(θ, τ ys ) under the expectation of τt+1
= 0, or by choosingyyuooτt = min(τ , 1) under the expectation of τt+1 = τ . The latter dominates the former
if Z(min(1, τ yu ), τ o ) > Z(θ, 0); that is, if θ < min(θ̃, θ̂). There then exists an unskilledmajority equilibrium under the conditions presented in Proposition 2 as long as agents
attach a low value to the expectation parameter θ.
The following proposition summarizes the results established thus far. The proof ofthe proposition is immediately apparent from Figure 6.
Proposition 3(i) Suppose the following conditions hold:
(
√ )
2
2
−
and βµ ≥ f 1 (w).
τ o ∈ 0,2
There exists a unique skilled-majority equilibrium as in Proposition 1(i).
12

(ii) Suppose the following conditions hold:

[ √ )
(
√ )
2− 2 12− 2oo, 2 , or τ ∈ 0, 2
and βµ < f 1 (w); τ ∈2βµ ≥ f 3 (w) for w ∈ [0, ŵ];βµ ≥ f 2 (w) for w ∈ (ŵ, 1).
Then the equilibrium is indeterminate. There exists both a skilled-majority equilibrium, as in Proposition 1(ii), and an unskilled-majority equilibrium, as in Proposition 2(i), if βµ ≤ f 4 (w); there exist both a skilled-majority equilibrium as in
Proposition 1(ii) and an unskilled-majority equilibrium, as in Proposition 2(ii), if
βµ > f 4 (w).
(iii) Suppose the following conditions hold:{βµ < f 3 (w) for w ∈ [0, ŵ];βµ < f 2 (w) for w ∈ (ŵ, 1).
There exists a unique unskilled-majority equilibrium, as in Proposition 2(i), if βµ ≤f 4 (w); there exists a unique unskilled-majority equilibrium as in Proposition 2(ii) if
βµ > f 4 (w).
√)(
Statement (i) in Proposition 3 imposes two assumptions, τ o ∈ 0, (2 − 2)/2 andβµ ≥ f 1 (w), for the existence of a unique skilled-majority equilibria with τ y = τ ys andτ o = 0. The ﬁrst assumption implies a low tax burden in old age; the second assumption
implies a low wage for unskilled workers. These assumptions jointly create an incentive
for young agents to invest in education, thereby resulting in a high probability of being
successful in education and thus a high ratio of skilled to unskilled workers within ageneration. Unskilled-majority equilibrium cannot arise under these assumptions because
they strongly induce a skilled-majority equilibrium.
If one of the above assumptions is relaxed, there arises an unskilled-majority equilibrium, as in Statements (ii) and (iii). The incentive for educational investment is thenreduced because of a higher tax burden on the old and/or a higher wage for unskilled workers. In addition, the skilled workers may fail to attain the top of the Laﬀer curve. Thatois, under the expectation of τt+1
= 0, the revenue from the young is maximized by settingyyysτt = θ rather than τt = τ . This equilibrium outcome depends on the expectations ofagents and thus becomes indeterminate in the skilled-majority equilibrium.
A skilled-majority equilibrium and an unskilled-majority equilibrium coexist if thewage for unskilled workers is low such that the assumptions in Statement (ii) hold. Whichparticular equilibrium arises depends on the expectation of agents: the skilled-majority(the unskilled-majority) equilibrium arises when θ is high (low) such that Z(θ, 0) ≥ (<)Z(min(τ yu , 1), τ o ) holds. In contrast, when the wage for the unskilled is high such thatthe assumptions in Statement (iii) hold, young agents have little incentive for educational
investment. There no longer remains a skilled-majority equilibrium; that is, there is aunique unskilled-majority equilibrium, as in Statement (iii).
The model prediction of multiple equilibria in Statement (ii) is consistent with theempirical evidence for OECD countries. As demonstrated in Daval (2003), we can classify
countries in the OECD into two groups in terms of their implicit taxes on continued workand the labor force participation of the elderly. The ﬁrst, featuring high tax rates and13low labor force participation rates, includes Austria, France, Germany and Italy. The
second, featuring low tax rates and high labor force participation rates, includes Canada,
Norway, Sweden, the United Kingdom and the United States. The former group ofcountries corresponds to the unskilled-majority equilibrium with τto = τ o , ls∗ (τ o ) = 1 − τ oand lu∗ (τ o ) = w(1 − τ o ), whereas the latter group of countries corresponds to the skilledmajority equilibrium with τto = 0, ls∗ (0) = 1 and lu∗ (0) = w. The current model thenprovides at least one possible explanation for the diﬀerences in implicit tax on continued
work and labor force participation rates of the elderly across the OECD. The key to this
explanation is the multiple, self-fulﬁlling expectations of agents.
We should note that Norway and Sweden are much less unequal than Canada, theUnited Kingdom and the United States, even though all present similar levels of taxes
and labor force participation of the elderly. This evidence of similarity at ﬁrst appears
counterintuitive. However, the similarity can be explained, using our framework, by the
expectation-based tax rate on the young in the skilled-majority equilibrium. That is,people in Norway and Sweden have similar expectations θ as in those in Canada, theUnited Kingdom, and the United States. This similarity of expectations leads to thesame pattern of taxation and comparable labor force participation of the elderly for thetwo diﬀerent groups of countries.

4

Numerical Analysis

Until now, we have characterized the political equilibrium and qualitatively assessed theimpact of wage inequality and the expectations of agents on the determination of thetaxation and labor force participation rate of the elderly. To facilitate understanding,this section numerically investigates how wage inequality and the expectations of agentsaﬀect the pension–GDP ratio as well as the expected utility of the young.
For the purpose of the analysis, we assume a generation to be 20 years in duration.
The ﬁrst and second periods correspond to, for example, ages 26–45 and 46–65 years,
respectively. The parameters µ and τ o are assumed to be 0.8 and 0.25, respectively. Our
selection of β is based on the single-period discount rate of 0.95. Because the agents under
the current assumption plan over generations that span 20 years, we discount the futureby (0.95)20 .
In the current environment, there are three threshold levels of w, as illustrated in
Figure 6: w = 0.0090, 0.2227, and 0.3642. For w ∈ (0, 0.0090), there exists a unique
skilled-majority equilibrium as in Proposition 1(i); for w ∈ (0.0090, 0.2227), there exist
multiple political equilibria: one is the skilled-majority equilibrium as in Proposition
1(ii), and the other is the unskilled-majority equilibrium as in Proposition 2(i); for w ∈(0.2227, 0.3624), there exists a unique unskilled majority equilibrium as in Proposition
2(i); and for w ∈ (0.3642, 1), there exists a unique unskilled-majority equilibrium as in
Proposition 2(ii). We hereafter focus on the range (0.0090, 0.2227) that creates multiple
political equilibria because these provide an explanation for the cross-country diﬀerencesin the tax and labor force participation rate of the elderly as presented in the previous
section.

14

4.1

Cross-country Diﬀerences in the Pension–GDP Ratio

Panel (a) in Figure 8 demonstrates how the pension–GDP ratio is aﬀected by wage
inequality in the current sample economy. The upper curve shows the pension–GDP
ratio in the unskilled-majority equilibrium, whereas the lower set of downward-sloping
curves indicates the pension–GDP ratio in the skilled-majority equilibria. As illustrated,
the unskilled-majority equilibrium attains a higher pension–GDP ratio than the skilledmajority equilibrium. This is mainly because the tax on the old in the former equilibrium
is higher than that in the latter equilibrium.
[Figure 8 about here.]
The numerical result in Panel (a) ﬁts the empirical evidence of Panel (b) in the following ways. First, the evidence indicates that the ratios in Austria, France, Germany and
Italy are about twice those in Canada, Norway, Sweden, the United Kingdom and theUnited States. Multiple equilibria in the current framework can provide a possible explanation for this diﬀerence in the pension–GDP ratio between the two groups of countries
sharing similar levels of wage inequality.1
Second, the evidence indicates a negative correlation between wage equality and thepension–GDP ratio in most countries, except Austria, France, Germany and Italy. A
more equal society generally realizes a lower pension–GDP ratio. This correlation can be
explained in the current framework by focusing on the set of downward-sloping curves inthe skilled-majority equilibria, where more equality in wages is associated with a lower
tax on the young2 , which results in a lower per capita tax revenue and thus a smaller
pension.

4.2

Expected Utility of the Young

Panel (a) in Figure 9 illustrates the expected utility of the young. As shown, the young
obtain a higher expected utility in the skilled-majority equilibrium than in the unskilledmajority equilibrium. This is mainly because the skilled-majority equilibrium requires alower tax burden on the old and thus realizes a higher probability of being successful ineducation than the unskilled-majority equilibrium.
[Figure 9 about here.]
One noteworthy feature is that eﬀects of wage equality on expected utility are diﬀerent
between the two equilibria. In the skilled-majority equilibrium, more equality (i.e., ahigher w) leads to the higher expected utility of the young. However, in the unskilledmajority equilibrium, more equality results in lower expected utility. These diﬀerencesin the consequences of wage equality arise from the tax on the young (see Panel (b) in
Figure 9).
1

We should note that the range of w in Panel (a) is qualitatively diﬀerent from that in Panel (b). Thisdiﬀerence comes from the assumption of the ﬁxed tax rate on the old, τ o , and the model speciﬁcation of
the disutility functions of education and the labor supply. Changes in τ o and the utility functions would
derive the range of w that ﬁts the empirical evidence.
2
The mechanism underlying this eﬀect is explained in Subsection 4.2.

15

In the unskilled-majority equilibrium, more equality results in a higher tax rate on
the young. A higher w gives agents a disincentive to invest in education because they canobtain a higher wage, even when they fail in education and become poor. This results in
fewer skilled agents and thus a smaller tax base. Given the shrinkage of the tax base, thedecisive voter chooses to impose a higher tax on the young to maintain tax revenue andthus the level of pension beneﬁt. Therefore, more equality results in a higher tax rateon the young and thus the lower expected utility of the young in the unskilled-majority
equilibrium.
In the skilled-majority equilibrium, more equality results in a lower tax rate on theyoung and thus the higher expected utility of the young. This opposing result to theunskilled-majority equilibrium comes about in the following way. As indicated above, ahigher wage for the unskilled gives agents a disincentive to invest in education, thereby
resulting in fewer skilled agents. A lower tax on the young oﬀsets this negative size eﬀect
and thus keeps the majority skilled. Therefore, more equality results in a lower tax rateon the young and thus a higher expected utility of the young in the skilled-majority
equilibrium.

5

Extensions and Further Analysis

The analysis thus far has assumed that: (i) opportunities to continue to work are common
to both the skilled and the unskilled; and (ii) the same pension beneﬁts are paid to theskilled and the unskilled. This section brieﬂy considers how the analysis and the results
are changed when each of these assumptions is relaxed.

5.1

Diﬀerent Opportunities for Working in Old Age

The analysis in the previous sections assumed that opportunities to continue to work arecommon to both the skilled and the unskilled. However, the skilled (i.e., the rich) may
have more opportunities to work than the unskilled (i.e., the poor) because the rich arehealthier than the poor, as observed in some empirical studies (see Smith, 1999, and thereferences therein). This subsection introduces diﬀerences in opportunities into the modeland brieﬂy considers the consequences.
Assume that the probability of obtaining opportunities to work in old age is µ for theunskilled (as in the original model), whereas it is γµ(> µ) for the skilled where γ > 1
holds. Under this assumption, the utility function of the unskilled old is still given by (2),whereas the utility function of the skilled old is given by:{}(lts )2osos
Vt = γµ lt (1 − τt ) −+ bt .
2
Educational investment by the young is now given as follows:}[{]βµ γ − 1
y
o
2
y∗ y
o
+ (1 + w) (1 − τt+1 ) .
e (τt , τt+1 ) = (1 − w) (1 − τt ) +2 1−w
This equation indicates that a larger γ results in a higher probability of becoming skilled inyouth. The economy would then be more likely to attain the skilled-majority equilibrium
if the skilled could obtain a higher probability of opportunities to work in old age.
16

5.2

Contribution-related Pension

Pension schemes in many OECD countries feature a mixture of Beveridgean and Bismarckian characteristics. The purely Beveridgean system provides a lump-sum beneﬁt to
every agent, irrespective of his/her contribution (as in the model in previous sections).
In contrast, the purely Bismarckian system gives the beneﬁt to each agent depending on
his/her contribution in their youth.
Under a mixture of the Beveridgean and Bismarckian systems, the government budget
constraint is given by:o[(1 − ut ) + ut {αw + (1 − α)}] bt = W̃ (τto , ut ) + Z̃(τty , τt+1),

where the parameter α ∈ [0, 1] represents the Bismarckian factor, and W̃ (τto , ut ) andoZ̃(τty , τt+1
) are the tax revenues from the old and young, respectively:
W (τto , ut );(1 − ut ) + ut {αw + (1 − α)}oZ(τty , τt+1)yoZ̃(τt , τt+1 ) ≡.
(1 − ut ) + ut {αw + (1 − α)}W̃ (τto , ut ) ≡The pension system becomes purely Beveridgean if α = 0; the system is purely Bismarckian if α = 1.
The objective functions of the old then change to:µo(1 − τto )2 + W̃ (τto , ut ) + Z̃(τty , τt+1),2µ
o
Vtou = (w)2 (1 − τto )2 + W̃ (τto , ut ) + Z̃(τty , τt+1).
2
Vtos =The rich old prefer τto = 0, whereas the poor old prefer τto = τ o under Assumption 1. The
qualitative properties of the analysis and the results of previous sections would then notchange if a contribution-related pension system were introduced into the model.

6Conclusion

This paper focuses on the negative correlation between the labor force participation rateof the elderly and the implicit tax on continued work in older ages. Some previous studies
demonstrate the mechanism of how a higher tax gives the elderly an incentive to retire
early and thus results in their lower participation rate. However, few studies show whysome countries choose a lower tax rate while others choose a higher tax rate. We developa politico-economic model with endogenous retirement decisions to answer this question.
A key feature of our analysis that is of importance to the result is the multiple,
self-fulﬁlling expectations of agents. When young agents hold expectations of a higher
social security beneﬁt, it provides a disincentive to engage in educational investment,
thereby resulting in a lower proportion of the skilled. This implies a majority of unskilledindividuals choosing a higher social security beneﬁt. This induces retirement and resultsin the lower labor force participation rate of the elderly. The opposite applies when young
agents have expectations of lower social security in their old age. This ﬁnding of multiple
political equilibria well ﬁts the empirical evidence for the OECD.
17

We undertake numerical analysis to consider how wage inequality between the skilled
and the unskilled aﬀects the size of the pension as well as the expected utility of theyoung. The results show that the predictions of our model well ﬁt observations of thepension–GDP ratio in the OECDṪhe results also show that a reduction in wage inequality
has opposite eﬀects on the expected utility of the young depending on the equilibria.
To obtain these results, we imposed a restriction on the set of tax rates available for theold. Because of this restriction, we were able to obtain an analytically tractable solution
for the model. While we would expect that expanding the set of tax rates available would
provide additional insights, we defer this to future work.

18

7

Appendix

7.1

Proof of Proposition 1o(i) Suppose that at time t, agents know that τty = τ ys and expect τt+1
= 0. Then:{}βµ(1 + w)
1
ys
ut+1 = 1 − (1 − w) (1 − τ ) +
≤ .
2
2
o
By (7), this implies that τt+1
= 0, fulﬁlling initial expectations. Therefore, there existsoa skilled-majority equilibrium with τt+1
= 0 if the decisive voter ﬁnds it optimal to setyysτt = τ .
o
To establish that setting τty = τ ys is optimal under the expectation of τt+1
= 0 for theydecisive voter, we note the following properties of the function Z : Z(τt , 0) attain the topof the Laﬀer curve at τty = τ ys ; and Z(τty , 0) > Z(τty , τ o )∀τty ∈ [0, 1]. These properties
imply that setting τty = τ ys is optimal for any θ if τ ys ≤ τ y (≤ θ) (see Panel (a) in Figure5). The inequality τ ys ≤ τ y is rewritten as:
{
}
1
o 2
w(2 − w) ≤ (1 − τ ) −(1 + w)(1 − w)2 βµ,2where (the left-handright-hand side is positiveif √and only√()side is positive;ysand the
) if
y
o
o
τ ∈ 0, (2 − 2)/2 . Therefore, τ ≤ τ is rewritten as τ ∈ 0, (2 − 2)/2 andβµ ≥ f 1 (w) ≡ w(2 − w)/[(1 + w)(1 − w)2 {(1 − τ o )2 − 1/2}].
o(ii) Suppose that at time t, agents know that τty = θ and expect τt+1
= 0. Then:}{1βµ(1 + w)< .
ut+1 = 1 − (1 − w) (1 − θ) +22oBy (7), this implies that τt+1
= 0, fulﬁlling initial expectations. Therefore, there existsoa skilled-majority equilibrium with τt+1
= 0 if the decisive voter ﬁnds it optimal to setyτt = θ.
To establish that setting τty = θ is optimal for the decisive voter, we ﬁrst note that theﬁrst assumption in Statement (ii) lies opposite to the assumption in Statement (i); thatis, τ ys > τ y holds under the ﬁrst assumption. Under this condition, the decisive voterﬁnds it optimal to choose τty = max(θ, τ ys ) from the viewpoint of revenue maximizationowhen he/she expects τt+1
= 0.
Given the result above, either of the following options can oppose the choice ofyoo= τ o if τ yu > 1; or) = (θ, 0): setting τty = 1 under the expectation of τt+1
(τt , τt+1
y
o
yu
yu
o
setting τt = τ under the expectation of τt+1 = τ if τ ≤ 1. The concerned choice issustained against the ﬁrst option if Z(θ, 0) ≥ Z(1, τ o ); that is:(θ)2 − 2τ ys θ − (1 − 2τ yu ) ≤ 0 ⇔ θ ≥ θ̃,where:θ̃ ≡ τ ys −√(τ ys − 1)2 + 2(τ ys − τ yu ).

As θ is bounded above τ̄ y , Z(θ, 0) ≥ Z(1, τ o ) holds if and only if θ ∈ [θ̃, τ̄ y ]. The set[θ̃, τ̄ y ] is nonempty if and only if θ̃ ≤ τ̄ y , which is rewritten as the second assumption inStatement (ii).
19oSetting τty = θ under the expectation of τt+1
= 0 is sustained against the second optionyuoif Z(θ, 0) ≥ Z(τ , τ ); that is:(θ)2 − 2τ ys θ + (τ yu )2 ≤ 0 ⇔ θ ≥ θ̂,where:θ̂ ≡ τ ys −√(τ ys )2 − (τ yu )2 .

As θ is bounded above τ̄ y , Z(θ, 0) ≥ Z(τ yu , τ o ) holds if and only if θ ∈ [θ̂, τ̄ y ]. The set[θ̂, τ̄ y ] is nonempty if and only if θ̂ ≤ τ̄ y , which is rewritten as the third assumption inStatement (ii).


7.2

Proof of Proposition 2(i) The second assumption βµ ≤ f 4 (w) is equivalent to τ yu ≤ 1. Given the expectationoof τt+1= τ o , agents can choose τty to attain the top of the Laﬀer curve Z(τty , τ o ).
oSuppose that at time t, agents know that τty = τ yu and expect τt+1
= τ o . Then,o
ut+1 = U (τ yu , τ o ) > 1/2. By (7), this implies that τt+1
= τ o , fulﬁlling initial expectations.
oTherefore, there exists an unskilled-majority equilibrium with τt+1
= τ o if the decisiveyyuvoter ﬁnds it optimal to set τt = τ .
Given the properties of the payoﬀ functions, we have the following two options: settingoτty = τ ys under the expectation of τt+1
= 0; or setting τty = θ under the expectation ofoτt+1= 0. When the ﬁrst option is available, setting τty = τ yu under the expectation ofoτt+1= τ o is not sustained against the ﬁrst option because Z(τ ys , 0) > Z(τ yu , τ o ) holds.
Therefore, we impose the condition that the ﬁrst option is unfeasible, τ ys ≥ τ y , which isrewritten as:
{
}
1
2
o 2
1 − (1 − w) ≥ (1 − τ ) −(1 + w)(1 − w)2 βµ,2where the left-hand side is (always positive√) as w ∈ [0, 1); and the right-hand side isopositive if and only if τ ∈ 0, (2 − 2)/2 . Therefore, the above inequality condition√[)
holds if the right-hand side is either nonpositive, i.e., τ o ∈ (2 − 2)/2, 1/2 , or positive,√()
i.e., τ o ∈ 0, (2 − 2)/2 and βµ < f 1 (w). These conditions are summarized as the ﬁrst
assumption in Statement (i).
oSetting τty = τ yu under the expectation of τt+1
= τ o is sustained against the second
option if Z(τ yu , τ o ) > Z(θ, 0); that is, if
}]
[
{
βµ
yu
o 2
2
τ yu
(1 + w)(1 − τ )w + (1 − w) (1 − τ ) +
2
[
{
}]
βµ
2
> w + (1 − w) (1 − θ) +(1 + w) θ.
2This is rewritten as:(θ)2 − 2τ ys θ + (τ yu )2 > 0 ⇔ θ < θ̂,
where:θ̂ ≡ τ ys −√(τ ys )2 − (τ yu )2 .

Therefore, Z(τ yu , τ o ) > Z(θ, 0) holds if and only if θ < θ̂ as illustrated by Panel (b) in
Figure 7.
20

(ii) The second assumption βµ > f 4 (w) is equivalent to τ yu > 1. Given the expectationoof τt+1= τ o , agents cannot choose τty = τ yu to attain the top of the Laﬀer curve. The taxorevenue from the young is maximized at τty = 1 as long as the expectation is τt+1
= τ o.
yooSuppose that at time t, agents know that τt = 1 and expect that τt+1 = τ . Then,o
ut+1 = U (1, τ o ) > 1/2. By (7), this implies that τt+1
= τ o , fulﬁlling initial expectations.
oTherefore, there exists an unskilled-majority equilibrium with τt+1
= τ o if the decisiveyvoter ﬁnds it optimal to set τt = 1.
Given the properties of the payoﬀ functions, we have the following two options: settingoτty = τ ys under the expectation of τt+1
= 0; or setting τty = θ under the expectation ofoτt+1= 0. When the ﬁrst option is available, setting τty = 1 under the expectation ofoτt+1= τ o is not sustained against the ﬁrst option, because Z(τ ys , 0) > Z(1, τ o ) holds.
Therefore, we impose the condition that the ﬁrst option is unfeasible, τ ys ≥ τ y , which isrewritten as the ﬁrst assumption in Statement (ii).
oSetting τty = 1 under the expectation of τt+1
= τ o is sustained against the second
option if Z(1, τ o ) > Z(θ, 0); that is, if
[
]
2 βµ
o 2
w + (1 − w)
(1 + w)(1 − τ ) · 1
2
[
{
}]
βµ
2
> w + (1 − w) (1 − θ) +(1 + w)· θ.
2This is rewritten as:(θ)2 − 2τ ys θ − (1 − 2τ yu ) > 0 ⇔ θ < θ̃,
where:θ̃ ≡ τ ys −√(τ ys − 1)2 + 2(τ ys − τ yu ).

Therefore, Z(1, τ o ) > Z(θ, 0) holds if and only if θ < θ̃, as illustrated by Panel (a) in
Figure 7.
21References[1] Azariadis, C., and Galasso, V., 2002. Fiscal constitutions. Journal of Economic Theory 103, 255–281.
[2] Casamatta, G., Cremer, H., and Pestieau, P., 2005. Voting on pensions with endogenous retirement age. International Tax and Public Finance 12, 7–28.
[3] Casamatta, G., Cremer, H., and Pestieau, P., 2006. Is there a political support forthe double burden on prolonged activity? Economics of Governance 7, 143–154.
[4] Conde-Ruiz, J.I., and Galasso, V., 2003. Early retirement. Review of Economic Dynamics 6, 12–36.
[5] Conde-Ruiz, J.I., and Galasso, V., 2004. The macroeconomics of early retirement.
Journal of Public Economics 88, 1849–1869.
[6] Conde-Ruiz, J.I., Galasso, V., and Profeta, P., 2005. The evolution of early retirement. CEPR w.p. no. 4863.
[7] Crawford, V.P., and Lilien, D.M., 1981. Social security and the retirement decisions.
Quarterly Journal of Economics 95, 505–529.
[8] Daval, R., 2003. Retirement behaviour in OECD countries: Impact of old-age pension
schemes and other social transfer programmes. OECD Economic Studies 37, 7–50.
[9] Feldstein, M., and Liebman, J.B., 2002. Social security. In: Auerbach, A.J., andFeldstein, M. (Eds), Handbook of Public Economics Volume 4, Elsevier Science,
Amsterdam.
[10] Fenge, R., and Pestieau, P., 2005. Social Security and Early Retirement. CESifo Book
Series, MIT Press.
[11] Galasso, V., 2008. Postponing retirement: the political eﬀect of aging. Journal of
Public Economics 92, 2157–2169.
[12] Grossman, G. and Helpman, E., 1998. Intergenerational redistribution with shortlived governments. Economic Journal 108, 1299–1329.
[13] Gruber, J., and Wise, D. (Eds.), 1999. Social Security and Retirement around theWorld. University of Chicago Press, Chicago.
[14] Hassler, J., Rodriguez Mora, J.V., Storesletten, K., and Zilibotti, F., 2003. The
survival of the welfare state. American Economic Review 93, 87–112.
[15] Hassler, J., Storesletten, K., and Zilibotti, F., 2007. Democratic public good provision. Journal of Economic Theory 133, 127–151.
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22[17] Lacomba, J.A., and Lagos, F.M., 2007. Political election on legal retirement. Social
Choice and Welfare 20, 1–17.
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[22] Smith, J.P., 1999. Healthy bodies and thick wallets: The dual relation between healthand economic status. Journal of Economic Perspectives 13, 145–166.

23

Figure 1: The timing of events and the distribution of the skilled and unskilled.

24

Figure 2: The range of politically available tax rates for the old under Assumption 1.

25

Figure 3: An example of solutions to the functional equation (9).

26

Figure 4: Three possible cases of the solution (10).

27

Figure 5: Revenue functions of the young. The skilled-majority equilibrium is illustrated
in Panels (a) and (b). Panel (a) is the case of τ ys ≤ τ y ; Panel (b) is the case of τ ys >τ y .
The unskilled-majority equilibrium is illustrated in Panels (c) and (d). Panel (c) is thecase of τ yu ≤ 1; Panel (d) is the case of τ yu > 1.

28

Figure 6: Illustration of function f i (i = 1, 2, 3, 4) under the assumption of τ o = 0.25.

29

Figure 7: Determination of θ̃ (Panel (a)) and θ̂ (Panel (b)).

30

ITA
FRA AUT
POL

DEU
PRT
BEL

HUN

FIN

ESP
SWE

TUR
LUX

CHE

CHE
USA

GBR
NLD
CAN
IRE

DEN
NOR
NZL
AUS

KOR

Figure 8: Panel (a) demonstrates the numerical result of the pension–GDP ratio in
the current example; Panel (b) plots the cross-country data on wage equality and thepension–GDP ratio. Source: OECD (2009a) for wage equality; OECD (2009b) for thepension–GDP ratio.

31

Figure 9: Panel (a) depicts the expected utility of the young; Panel (b) depicts the tax
rate on the young.

32