Discussion Papers In Economics
And Business

Aging, Inequality and Social Security
Ryo Arawatari and Tetsuo Ono
Discussion Paper 08-19

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAging, Inequality and Social Security
Ryo Arawatari and Tetsuo Ono
Discussion Paper 08-19

April 2008

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Aging, Inequality and Social Security∗
Ryo Arawatari†and Tetsuo Ono‡
Graduate School of Economics, Osaka University

April 2008

Abstract
This paper develops an overlapping-generations model including wage inequality
within a generation and intra- and intergenerational resource reallocation via social
security. Based on the concept of a stationary Markov perfect equilibrium, the paper
focuses on the feedback mechanism between current individuals’ decisions on saving
and future voting on social security. The paper demonstrates the determination ofsocial security via probabilistic voting and its consequence for consumption inequality within a generation. It is shown that when the elderly are politically powerful,(i) the economy attains an oscillatory path of inequality and social security, and (ii)aging may reduce consumption inequality.
Keywords: Aging; Inequality; Social security; Political Economy; Stationary
Markov Perfect Equilibrium.
JEL Classification: D72, H55, J10.

∗

Arawatari acknowledges financial support from the Research Fellowships for Young Scientists of
the Japan Society for the Promotion of Science (JSPS). Ono acknowledges financial support from theJapan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Young Scientists (B) (No.
20730208).
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: ege001ar@mail2.econ.osaka-u.ac.jp‡
Corresponding Author. Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail address: tono@econ.osaka-u.ac.jp

1

Introduction

Pay-as-you-go (PAYG) social security systems are at the center of public sector expenditure in many OECD countries. Benefit levels and contribution rates are determined in
the political process and thus affected by political conflict between the beneficiaries (theelderly) and the contributors (the young). In order to assess the efficiency of existingsocial security and to obtain a prediction of the future of social security, there is a needto develop a politico-economic framework that incorporates this political conflict between
generations.
A key to the development of the framework is a feedback mechanism between social
security policy and individual decision making. Expectations about future policy affectcurrent individuals’ decision making on saving or education, which in turn has an effecton the future distribution of income and thus on the future voting behavior on the policy. Several researchers have provided politico-economic frameworks that incorporate thisfeedback mechanism based on the concept of a stationary Markov perfect equilibrium(see, for example, Grossman and Helpman, 1998; Azariadis and Galasso, 2002; Hassler etal., 2003; Forni, 2005; Hassler et al., 2005; Bassetto, 2007; Gonzalez-Eiras and Niepelt,
2007; Song, 2007).
Social security has a role of intragenerational redistribution via its Beveridgean partin addition to intergenerational redistribution. However, the above-mentioned studies abstracted the intragenerational redistribution aspect by assuming identical agents withina generation. Some researchers have considered the politics of intragenerational redistribution, but abstracted the feedback mechanism by assuming once-and-for-all voting(Casamatta, Cremer and Pestieau, 2000, 2001; Conde-Ruiz and Profeta, 2007; Koethenbuerger, Poutvaara and Profeta, 2008). They did not mention the political effect of social
security on inequality within a generation. Therefore, the question arises of whether or
not the politics of social security has an effect on intragenerational redistribution, andthus on inequality, within a generation in the presence of the feedback mechanism. The
answer is no, but it is only for the case of wealth inequality under a pure Bismarckian
scheme (Song, 2007).1
The aim of this paper is to characterize the dynamic political economy of social security
and consumption inequality under a social security system including the Beveridgean
and the Bismarckian parts. In addition, we consider population aging resulting from adecline in fertility rates, and investigate how aging affects social security and consumption
1

Hassler et al. (2003) and Hassler et al. (2005) also analyzed income inequality within a generation,
but there is no capital accumulation in their framework. In addition, their focus is different from Song(2007) and ours. They focus on the endogenous determination of income inequality affected by politics;
Song (2007) and our paper assume income inequality as given and focus on endogenous determination of
wealth inequality (Song, 2007) or consumption inequality (the current paper) affected by politics.

2

inequality through the political process. Therefore, the paper provides dynamic politicaleconomy implications of aging, inequality and social security.2
For the purpose of analysis, we develop a simple two-period overlapping-generations
model with a linear production function and a quasi-linear utility function. Agents differ
in their working ability and thus in their labor income in youth; they retire in old age andreceive social security benefits. The pension benefit consists of two parts: a contributory
part (Bismarckian) and a noncontributory part (Beveridgean). This pension benefit isdetermined in a probabilistic voting framework (Lindbeck and Weibull, 1987). We utilize
the concept of a stationary Markov perfect equilibrium where the social security tax rateis conditioned on capital as a state variable.
Under the framework presented above, the current paper shows the following two
results. First, the political power of the elderly related to aging plays a key role indetermining the dynamic patterns of inequality and social security. When the politicalpower of the elderly is weak, the economy displays a monotone path of inequality andsocial security toward the steady state. On the other hand, when their power is strong,
the economy displays an oscillatory convergence toward, or a period-two cycle around,the steady state.
Second, the model presents the opposite economic and political effects of aging on
the size of the social security similar to Galasso and Profeta (2004, 2007). Given thisresult, we examine the effect of aging on consumption inequality (evaluated in terms ofutility). It is shown that under a pure Beveridgean scheme, aging results in a lower levelof inequality when the political power of the elderly is strong. Under a pure Bismarckian
scheme, aging results in a lower level of inequality when the initial population growthrate is low. The result implies that aging may reduce consumption inequality through thepolitical process.
The organization of this paper is as follows. Section 2 develops the model and characterizes economic equilibrium. Section 3 characterizes a Ramsey allocation defined asa feasible plan chosen by a benevolent social planner. Section 4 characterizes a political equilibrium, shows its existence and stability, compares it to the Ramsey allocation
and analyzes the effect of aging on the amount of social security. Section 5 presents the
dynamics of inequality and investigates the effect of aging on consumption inequality.
Section 6 discusses the assumptions of this paper. Section 7 provides concluding remarks.
The proofs are provided in the appendix.

2

There are several studies analyzing the political economy of social security focusing on the role ofsocial security systems (see, for example, Casamatta et al., 2000, 2001; Cremer et al., 2007; Conde-Ruizand Profeta, 2007; Koethenbuerger, Poutvaara and Profeta, 2008). However, these studies focus on thesteady state by assuming once-and-for-all voting; the dynamic motion of social security and inequality isabstracted from their analysis.

3

2

The Model and Economic Equilibrium

Consider an overlapping-generations model where the economy is composed of individuals
and perfectly competitive firms. Time is discrete and denoted as t = 0, 1, 2, · · · . A newgeneration is born in each period t = 0, 1, 2, · · · . The size of a generation born in periodt (generation t) is given by N t . The population grows at a constant rate n > −1 :N t = (1 + n)N t−1 .
Individuals in each generation live for two periods: youth and old age. Each individualis endowed with one unit of labor in youth, supplies it inelastically to firms, and retires
in old age. An individual has a probability λ ∈ (0, 1) to be endowed with the highproductivity wh > 0 and a probability 1 −λ to be endowed with the low productivity wl ∈(0, wh ). Individuals with wi are referred to as type i(= h, l). The average productivity
is assumed to be fixed and given by w̄ = λwh + (1 − λ)wl . The aggregate effective labor
supply in period t is given by Lt = (λwh + (1 − λ)wl )N t .
To obtain analytical results, we assume a linear production function in which gross
output, Y, is produced using labor, L, and capital, K : Yt = Lt + RKt . The wage rate,
1, and the gross rental price of capital, R, are determined by the marginal productivity
conditions for factor prices and are already substituted into the production function. For
simplicity, the two types of labor are assumed to be perfect substitutes in production interms of efficiency units of labor input, and capital is assumed to fully depreciate at theend of the production process.
oi
A type-i individual in generation t evaluates consumption bundles (cyit , ct+1 ) by theyioiutility function U it = u(cyit ) + βct+1 where ct represents the consumption at time t of
type i young individual and coi
t+1 represents the consumption at time t + 1 of type i old
individual. β ∈ (0, 1) is the time discount factor. The function u is concave and twice
continuously differentiable with limc→0 u0 (c) = +∞. Quasi-linearity of U it implies that
all income effects are absorbed by old age consumption. The role of this assumption will
be discussed in Section 6.
A type-i individual is subject to a type-independent payroll tax τt ∈ [0, 1] and expects
the type-dependent old age pension bit+1 . He/she can allocate his/her disposable labori
income between consumption cyit and saving zt . When he/she retires, his/her consumptionii
coi
t+1 is equal to the gross return on saving, Rzt , plus the pension benefit, bt+1 . Formally,
type i individual solves the following problem:
oi
max
u(cyi
t ) + βct+1
i
zt =0

i
i
s.t. cyit + zt = (1 − τt )w ,iicoi
t+1 = Rzt + bt+1 .

4

The first-order condition associated with an interior solution of zti is:
u0 (cyit ) = βR for i = h, l,

(1)

y
0−1
cyi
(βR) for i = h, l.
t = c̄ ≡ u

(2)

or:

With (2) and the budget constraints, the saving and the old-age consumption of type iare given by, respectively:
zti = (1 − τt )wi − c̄y ,
i
y
i
coi
t+1 = R · {(1 − τt )w − c̄ } + bt+1 .

(3)
(4)

Because of the assumption of the quasi-linear utility function, the consumption in youth
is type-independent and constant over time, and the saving is independent of the pension
benefit bit+1 .3 We assume that pension benefit of type-i, bit+1 , consists of two parts: acontributory part that is directly related to individual earning, wi , and a noncontributory
part that depends on average earnings, w̄. With a pay-as-you-go (PAYG) scheme, theaverage return of the social security system is given by the population growth rate. These
properties yield the following expression for bit+1 :bit+1 = (1 + n) · τt+1 · (αwi + (1 − α)w̄).

(5)

The parameter α ∈ [0, 1] is the Bismarckian factor that is the fraction of the pension
benefits that is related to contributions. When α = 1, the pension scheme is pure Bismarckian or contributory; when α = 0, pension benefits are uniform and the scheme is
Beveridgean. The case of an endogenous choice of α by voting is discussed in Section 6.
At the start of the economy, time 0, there are young agents in generation 0 as well asi
the initial old agents. A type-i initial old agent is endowed with z−1
units of saving atithe beginning of period 0 and obtains the return Rz−1
and the pension benefit bi0 , which
are consumed.
iDefinition 1: For a given sequence of taxes, {τt }∞t=0 , the initial condition z−1 > 0(i =h, l) and a set of parameters, (n, α, wh , wl , λ, R), an economic equilibrium is a se3

The saving function indicates that for type-l individuals, there is a possibility of ztl < 0. This implies
that type-l individuals want to borrow from type-h individuals in the credit market in order to financetheir consumption in youth. This paper assumes that such borrowing is available for type-l individuals:
there is no limit of access to the credit market. Although a borrowing constraint may provide interesting
results and implications for the political economy (see, for example, Bellettini and Berti Ceroni, 2007),
this paper leaves this for future research and instead focuses on the interaction between aging, inequality
and social security under the condition of no credit constraint.

5

©
∞
oi
i i
quence of allocations, cyit , ct+1 , zt , bt , Kt }t=0 , such that (i) the utility maximization
problem is solved for each type i; (ii) the social security budget constraint is balanced in every period, N t (λbht+1 + (1 − λ)blt+1 ) = N t+1 (λwh + (1 − λ)wl )τt+1 , underthe benefit rule (5); and (iii) the capital market clears: Kt+1 = N t (λzth + (1 − λ)ztl ).
Let kt ≡ Kt /N t denote capital per young person in period t. Given a sequence of∞taxes, {τt }∞t=0 , and the initial condition k0 > 0, an equilibrium sequence of kt , {kt }t=0 , ischaracterized by:(1 + n)kt+1 = (1 − τt )w̄ − c̄y .
(6)3Ramsey Allocation

In this section, we characterize a Ramsey allocation defined as a feasible plan chosen bya benevolent social planner who can commit to a policy sequence at time zero. The
allocation maximizes the following objective function:
W =β

©

ol
λcoh
0 + (1 − λ)c0

ª

+

P∞

t=0 η

t+1

h n
o
n
oi
yh
yl
oh
ol
λ u(ct ) + βct+1 + (1 − λ) u(ct ) + βct+1

subject to
y oi
i
y
i
cyi
t = c̄ , ct+1 = R · {(1 − τt )w − c̄ } + bt+1 ,λzth + (1 − λ)ztl ≥ 0,oiwhere cyit and ct+1 are households’ consumption functions and the third inequality is thenonnegative constraint of aggregate saving. The planner discounts future generations’utilities with a discount factor of η ∈ (0, 1).
The Ramsey problem described above is dynamic in nature, but it admits a simple

static representation. To show this, we substitute households’ budget constraints into theobjective function to obtain:hl
W = βR{λz−1+ (1 − λ)z−1} + W̃ ,

where
W̃ ≡

P∞

t=0 η

t

[β w̄{(1 + n) − ηR}τ + η{u(c̄y ) + βR(w̄ − c̄y )}] .

The problem reduces to a sequence of identical static optimization problems over τ .
Let τ R denote the solution to the Ramsey problem described above. We establish thefollowing result.

6

Proposition 1: The Ramsey allocation has




y

= w̄−c̄
if 1 + n > ηR,w̄
£ w̄−c̄y¤Rτ∈ 0, w̄
if 1 + n = ηR,=0if 1 + n < ηR.
Figure 1 shows the tax rate in the Ramsey allocation as a function of the population
growth rate, n. For 1 + n < (>)ηR, the return from intergenerational transfer 1 + n
is lower (higher) than the return from saving, R, multiplied by the discount factor ηfrom the viewpoint of the social planner. Therefore, the planner chooses τ R = 0(= 1)
if 1 + n < (>)ηR. For 1 + n = ηR, the planner is indifferent between the two ways ofresource reallocation.
[Figure 1 about here.]

4

Political Equilibrium

This section characterizes a political equilibrium. Section 3.1 provides the definition ofpolitical equilibrium based on the concept of a stationary Markov perfect equilibrium withprobabilistic voting. Section 3.2 provides the characterization of the political equilibrium
path of capital and tax rate. We then focus on the steady-state political equilibrium andshow the existence and stability of the steady-state equilibrium. In Section 3.3, basedon the characterization of the steady-state equilibrium, we examine how the size of thewelfare state, represented by the tax rate, is affected by population aging resulting froma decline in the fertility rate.

4.1

The Political Environment

The tax rate τt is chosen by some repeated political process at the beginning of eachperiod. This paper assumes that τt is determined in a probabilistic voting framework(see, for example, Lindbeck and Weibull, 1987). In this framework, political decision
makers’ platforms in period t simply maximize a weighted average utility of all voters
alive in that period. Thus, the equilibrium policy maximizes a political objective function
given by:£¤ol
Gt = γ · λcoht + (1 − λ)ct
h n
o
n
oi
yl
oh
ol
+(1 + n) · λ u(cyh
)
+
βc
+
(1
−
λ)
u(c
)
+
βc
,
t
t
t+1
t+1

7

where the political weight on the old is given by γ ∈ [0, ∞) and the weight on the young
is given by the gross rate of population growth. When γ = 0, only the young are entitledto vote; when γ > 0, both the young and the old vote on the tax rate. A greater weighton the old implies that they have greater political power.
We focus on a stationary Markov perfect equilibrium, in which the state of the economy
is summarized by the stock of capital per young person in the economy. We restrict the
choice of political decision makers to the Markovian policy function:τt = θ(kt ),where θ(·) is a stationary, differentiable function. Let the political decision maker inperiod t, with expectations θe (kt+1 ) about the period t + 1 tax rate find it optimal toset the transfer θ(kt ). The decision maker’s expectations about the future are consistent
with the incentives that will actually face the future decision maker if and only if θ = θe .
Formally, the Markov perfect equilibrium is defined as follows.
oi
i i∞
Definition 2: A Markov perfect political equilibrium is a sequence of allocations, {cyit , ct+1 , zt , bt , kt }t=0
and a differentiable policy function θ(·) such that (i) conditions at Definition 1 (eco-nomic equilibrium) are satisfied; and (ii) θ(·) is a fixed point of the mapping fromθe (·) to θ(·), where θ(·) is the solution to the policy maker’s problem and θe (·) isthe expected policy function.

4.2

Existence and Stability of Political Equilibrium

In order to find the solution to the problem of maximization Gt , we take the derivativeof the objective function Gt with respect to θ(kt ), which yields:
¡
¢
¡
¢
∂Gt
= γλ(1 + n) αwh + (1 − α)w̄ + γ(1 − λ)(1 + n) αwl + (1 − α)w̄∂θ(kt )
|
{z
} |
{z
}
(∗1)

(∗2)

½
¾
¢
∂θ(kt+1 ) ∂kt+1 ¡ hh+(1 + n)λ −βRw + β(1 + n)
αw + (1 − α)w̄
∂kt+1 ∂θ(kt )
{z
}
|
(∗3)

½
¾
¢
∂θ(kt+1 ) ∂kt+1 ¡ ll+(1 + n)(1 − λ) −βRw + β(1 + n)
αw + (1 − α)w̄ ,∂kt+1 ∂θ(kt )
|
{z
}
(∗4)

where the term (*1) ((*2)) represents the marginal benefit for high-productivity (lowproductivity) old people, and comes from the increase in θ(kt ). The term (*3)((*4)) represents the sum of the marginal cost and benefit for high-productivity (low-productivity)
young people, and comes from the increase in θ(kt ) and the increase in θ(kt+1 ) through8

a change in θ(kt ), respectively. The derivation of the above equation is provided in the
appendix.
Because we have (1 + n)kt+1 = (1 − θ(kt ))w̄ − c̄y in equilibrium, it follows that
∂kt+1 /∂θ(kt ) = −w̄/(1+n). Substituting this derivative into the condition of ∂Gt /∂θ(kt ) =0 and rearranging the terms, we obtain the differential equation ∂θ(kt+1 )/∂kt+1 = −µwhere:
βR − γµ≡.
β w̄
The parameter µ takes a positive or negative value depending on the sizes of β, γ and R.4
Integrating the differential equation, ∂θ(kt+1 )/∂kt+1 = −µ, we obtain:
θ(kt+1 ) = A − µkt+1 .

(7)

The constant of integration, A, is a free parameter that is determined by the first decision
maker’s expectation of future policies. Any period-t decision maker, who expects the next
decision maker to choose the policy according to (7), is indifferent between all feasible
taxes. That is, given the expectation that θe (kt+1 ) = A − µkt+1 , ∂Gt /∂θ(kt ) = 0 holds
for all θ(kt ) ∈ [0, 1]. This is because a change in θ(kt ) would be exactly compensatedby a corresponding change in θe (kt+1 ), driven by a variation in kt+1 . Therefore, every
decision maker is willing to act according to (7), lagged one period, in order to validate
the previous decision maker’s expectations.
The equilibrium policy function θ(kt ) = A − µkt can be combined with the accumulation equation (6) to derive the following law of motion for the stock of capital:
kt+1 =µw̄
w̄(1 − A) − c̄y
+
kt .
1+n
1+n

(8)

Using this law of motion and the policy function, we can write the law of motion of theequilibrium tax as:µ(w̄ − c̄y )µw̄
θt+1 = A −
+
θt .
(9)
1+n
1+n
We impose the following assumption (Assumption 1) to ensure that given kt > 0, kt+1 > 0
and θt ∈ [0, 1) hold.5
Assumption 1: w̄ > c̄y = u0−1 (βR).
4For example, assume that a generation is 30 years in length and set β = (0.98)30 and γ = 1. The
value of β is based on the single-period discount rate found by Auerbach and Kotlikoff (1987) of 0.98.
The value of γ implies that the old have equal political power with the young. If the single-period grossinterest rate is given by 1.02, then βR = (0.98 × 1.02)30 < γ = 1 : µ takes a negative value. Alternatively,
if it is given by 1.03, then βR = (0.98 × 1.03)30 > γ = 1 : µ takes a positive value.
5
In the Appendix (Section 8.4), it is shown that under Assumption 1, there exists a nonempty set of
kt that leads to kt+1 > 0 and θt ∈ [0, 1).

9

The political equilibrium attains a steady state if a sequence of capital and tax isstationary over time: kt = kt+1 = k̄ and θt = θt+1 = θ̄. Suppose that there is a steadystate with k̄ > 0 and θ̄ ∈ [0, 1). Equations (8) and (9) imply that the equilibrium sequenceof {kt , θt } displays monotone convergence toward the steady state if µw̄/(1 + n) ∈ (0, 1);constant levels over time if µw̄/(1+n) = 0; oscillatory convergence toward the steady state
if µw̄/(1+n) ∈ (−1, 0); and a period-two cycle around the steady state if µw̄/(1+n) = −1.
Therefore, we impose the following assumption (Assumption 2) on the set of parameters toensure the stability of the interior steady state. Figure 2 illustrates the set of parametersthat satisfies Assumption 2 in a R − γ space.
Assumption 2: 0 < (1 + n) − µw̄ and 0 ≤ (1 + n) + µw̄, i.e., γ ∈ (β(R − (1 + n)), β(R +(1 + n))].
[Figure 2 about here.]
Define two critical values of the free parameter A as:
Alow ≡µ(w̄ − c̄y )w̄ − c̄y, Ahigh ≡
.
1+n
w̄

(10)

Using these definitions, we establish the existence of a steady-state equilibrium.
Proposition 2: If A ∈ [Alow , Ahigh ), there exists a steady-state equilibrium withk̄ =w̄(1 − A) − c̄y(1 + n)A − µ(w̄ − c̄y )> 0 and θ̄ =∈ [0, 1).
(1 + n) − µw̄(1 + n) − µw̄

If A = Alow , then θ̄ = 0; if A ∈ (Alow , Ahigh ), then θ̄ ∈ (0, 1).
To understand the result shown in Proposition 2, consider the case of µ > 0(that is,
βR > γ) as an example. First, k̄ > 0 is rewritten as A < Ahigh . This implies that if the
expectation parameter A is high such that A ≥ Ahigh , the tax rate is too high to save:there is no resource remaining for saving and thus no accumulation of capital. Second,θ̄ < 1 always holds under the condition of A < Ahigh that requires k̄ > 0. Finally, θ̄ ≥ 0
is rewritten as A ≥ Alow . If A < Alow , the tax rate becomes negative. Therefore, there
exists a steady-state political equilibrium with k̄ > 0 and θ̄ ∈ [0, 1) if A is set within therange [Alow , Ahigh ).
The result established in Proposition 2 indicates that the steady-state levels of capitaland tax are independent of the parameter α representing the degree of redistribution via
the pension system. This result depends on the following two assumptions: (i) the quasilinear utility function of individuals; and (ii) the political objective function defined as theweighted sum of utilities of individuals. Because of these assumptions, the terms related
10

to the parameter α are displayed in a linear form and thus redistribution effects on types
l and h offset each other. Therefore, the parameter α has no effect on the determinationof capital and tax. However, as we will show in Section 5, the redistribution effect on
consumption still remains under these assumptions, implying that the pension system has
an effect on consumption inequality.
Having shown the existence of a steady-state equilibrium with k̄ > 0, we next examine
stability of the steady state. Equations (8) and (9) indicate that if µ = 0(βR = γ), thelevels of capital and tax are constant over time. On the other hand, if µ 6= 0(βR 6= γ),
the time paths of capital and tax display dynamics tending toward or around the steadystate. The stability property for the latter case is formally summarized in the following
proposition.
Proposition 3: Assume that µ 6= 0(βR 6= γ). Given k0 ∈ ((c̄y − w̄)/µw̄ + A/µ, A/µ] ifµ > 0 (i.e., γ < βR) or k0 ∈ [A/µ, (c̄y − w̄)/µw̄ +A/µ) if µ < 0 (i.e., γ > βR), there
exists an equilibrium sequence of (kt , θt )∞t=0 displaying (i) monotone convergence toward the interior steady-state equilibrium if γ ∈ (β(R − (1 + n)), βR); (ii) oscillatory
convergence toward the interior steady-state equilibrium if γ ∈ (βR, β(R + (1 + n)));and (iii) a period-two cycle around the interior steady-state equilibrium if γ =β(R + (1 + n)).
The conditions of the parameters β, R, γ and n for the stability property are immediately found by equations (8) and (9). For the result established in Proposition 3, tworemarks are in order. First, the relative magnitude between the political weight on theold, γ, and the gross interest rate multiplied by the discount factor, βR, plays a key rolein determining the stability property of the equilibrium path. A larger γ leads to a higher
tax rate because the old have greater political power in policy decision making, while alarger βR leads to a lower tax rate because a higher return from saving ensures a higher
level of consumption in the second period of the life cycle, thereby giving the policy maker
an incentive to set a lower current tax rate. Given these competing effects, if γ < (>)βR,
i.e., if µ > (<)0, the former effect is smaller (larger) than the latter effect, implying thatan increase in the current capital stock, kt , results in a lower (higher) tax burden and thus
an increase (a decrease) in the capital stock in the next period. Therefore, the political
economy displays a monotone (an oscillatory) sequence of capital if γ < (>)βR, i.e., ifµ > (<)0.
Second, the property of the equilibrium path depends on the expectation parameter
A. To see this, suppose first that γ < βR: the economy displays a monotone convergence
toward the steady state. If A is set within the range (Alow , Ahigh ), the economy displays an
increasing or decreasing sequence of the tax depending on the initial condition. However, if
A = Alow , there is no increasing sequence of the tax because the steady-state tax is given11

by θ̄ = 0; the economy displays only a decreasing sequence of the tax. Alternatively,
suppose that γ > βR: the economy displays an oscillatory convergence toward, or aperiod-two cycle around, the steady state. In this case, the policy parameter A must be
set within the range (Alow , Ahigh ). If A = Alow , the sequence of tax is unfeasible because
θ1 < 0 holds for a given θ0 > 0.
Figure 1 compares the politically determined tax rate (the political tax rate) to the tax
rate in the Ramsey allocation (the Ramsey tax rate). Comparison is made for the following three cases. First, if the gross rate of population growth is low such that 1 + n < ηR,
the political tax rate is generally higher than the Ramsey tax rate. The political equilibrium resembles the Ramsey allocation if and only if the expectation parameter A is
unexpectedly set at A = Alow . Second, if the gross rate of population growth is highsuch that 1 + n > ηR, the political tax rate is lower than the Ramsey tax rate. The
Ramsey allocation requires 100% taxation because resource reallocation via intergenerational transfer yields a higher return than that via private saving. However, politicalequilibrium attains a lower tax rate than that required by the Ramsey allocation becauseof the political power of the young, who want to avoid a tax burden. Finally, if the grossrate of population growth is by chance equal to ηR, any political tax rate is efficientbecause the Ramsey allocation attains indeterminate tax rates given by the set [0, 1].
Therefore, the comparison provides the following implication for political equilibrium interms of efficiency: the political economy attains an antiwelfare state when it has a highrate of population growth whereas it attains a prowelfare state when it has a low rate of
population growth.

4.3

Population Aging and the Size of the Welfare State

Based on the characterization of the political equilibrium, we now consider how the size of
the welfare state, represented by the tax rate θ, is affected by population aging represented
by a decrease in population growth (n).
Proposition 4: Suppose that γ ∈ (β(R − (1 + n)), β(R + (1 + n))): the equilibrium
path displays a stable convergence toward the steady state with θ̄ ∈ (0, 1). A lower
population growth rate leads to a smaller (constant, larger) size of the welfare state
if and only if µ > (=, <)0: ∂ θ̄/∂n R 0 if and only if µ R 0, i.e., βR R γ.
The result for the case of µ = 0 is immediately apparent from the fact of θ̄ = A. To
understand the results for the case of µ 6= 0, we first focus our attention on the capital
market-clearing condition in the steady state derived from (8): (1 + n)k̄ = (1 − (A −µk̄))w̄ − c̄y and consider the effects of the parameters n and γ on the steady-state level12

of capital. We then turn our focus to the policy function θ = A − µk and consider theeffects of n and γ on the tax rate via the steady-state level of capital.
Figure 3 illustrates the steady-state level of capital given by the point at the intersection of (1 + n)k with (1 − (A − µk))w̄ − c̄y . As illustrated in Figure 3, a lower population
growth leads to a higher level of per capita capital for both cases of µ > 0 and µ < 0.
This is because, given the aggregate level of capital, a smaller size of population leads toa higher level of per capita capital. Given the policy function θ̄ = A − µk̄, a higher level ofper capita capital results in a lower (higher) level of tax rate if and only if µ > (<)0, i.e.,βR > (<)γ. The relative magnitude between βR and γ is important for the determinationof the effect of aging on the size of the welfare system.
[Figure 3 about here.]
The terms βR and γ indicate that population aging has two opposite economic andpolitical effects on the size of the welfare state (see, for example, Razin, Sadka and Swagel,
2002; Galasso and Profeta, 2004, 2007). The economic effect is captured by the increasein the share of retirees to workers, which reduces the profitability of a pay-as-you-go social
security system compared to saving. This effect is represented by the term βR. A larger
R implies that the pay-as-you-go social security system is less profitable than saving,which gives individuals an incentive to downsize the social security system. The politicaleffect is captured by a decrease in the political weight for the young, that is, an increasein the political weight for the old. This effect is represented by the term γ. A larger
γ implies that the political power of the old becomes stronger and thus a larger social
security system is supported by voting. The overall effect of aging depends on the relative
magnitude between the two terms βR and γ.

5Inequality

This section introduces an index that measures an inequality between the two types of
agents in terms of consumption, i.e., lifetime utility. We focus on inequality of economic
welfare by consumption rather than income because consumption reflects not only current
income but also lifetime resources. For the purpose of analysis, we define the index of the
lifetime utility inequality as:£¤u(c̄y ) + β R{(1 − θ(kt ))wh − c̄y } + (1 + n)θ(kt+1 )(αwh + (1 − α)w̄)It =,u(c̄y ) + β [R{(1 − θ(kt ))wl − c̄y } + (1 + n)θ(kt+1 )(αwl + (1 − α)w̄)]
where the numerator and the denominator represent the lifetime utilities of types h andl, respectively. Based on this measure, we investigate how inequality changes over time13

(Section 5.1). We also consider how inequality is affected by population aging (Section
5.2). For the tractability of analysis, we make the following assumption.
Assumption 3: u(c̄y ) ≥ βRc̄y .
Assumption 3 guarantees that the level of the lifetime utility for each type of agent isgreater than zero. Since the utility level of type h is greater than the utility level of typel, the index takes a value between one and infinity under Assumption 3: It ∈ (1, ∞). A
higher It implies greater inequality in terms of lifetime utility. If u(c̄y ) = βRc̄y , the index
is identical to that measuring the lifetime income inequality.

5.1

Dynamics of Inequality

This section investigates the time path of inequality. The inequality index It is rewritten
as a function of kt by using the law of motion for capital (8). Thus, we can display thetime path of the lifetime utility (or income) inequality based on the dynamics of capital
accumulation presented in Proposition 3.
Proposition 5(i) Suppose that µ > 0, i.e., γ < βR. The inequality is increased by capital accumulation:∂It /∂kt > 0.
(ii) Suppose that µ < 0, i.e., γ > βR. The inequality is nonincreased by capital accumulation, i.e., ∂It /∂kt ≤ 0, if c̄y < (1+n)w̄/{(1+n)−µw̄} and A ∈ (−µc̄y /(1+n), Ahigh ).
In order to interpret the result established in Proposition 5, we differentiate It withrespect to kt and rearrange the terms to obtain:1
2 ∂It[u(c̄y ) + βcol
t+1 ]
β
∂kt
∂θ(kt )(−wh + wl ) [(u(c̄y ) − βRc̄y ) + (1 + n)θ(kt+1 )(1 − α)w̄]
= R
∂kt
|
{z
}
Labor income effect Bismarckian effectBeveridgean effect
z
}|
{
z
}|
{
∂θ(kt+1 ) ∂kt+1 h+(1 + n)(w − wl ) α(u(c̄y ) − βRc̄y )−(1 − α)w̄βR(1 − θ(kt )),∂kt+1 ∂kt{z}|
Pension effect(11)
where the first term shows the effect of capital on inequality through labor income andthe second term shows the effect of capital on inequality through the pension benefit. The
14

second term includes the two effects via the Bismarckian and Beveridgean parts of thepension benefit.
Suppose that µ > 0: a higher level of capital results in a lower tax rate and a higher
level of the following period’s capital: ∂θ(kt )/∂kt = ∂θ(kt )/∂kt = −µ < 0 and ∂kt+1 /∂kt =µw̄/(1 + n) > 0. Given these results, there are three effects of capital on inequality. First,
both types of agents benefit from a decrease in the tax burden, but the marginal benefit
for type h, wh , is greater than the marginal benefit for type l, wl . The effect via labor
income is negative in terms of equality. Second, because of a decrease in the tax rate,
both types of agents are harmed by a decrease in the Bismarckian part of the pension
benefit, but the marginal loss for type h, wh , is greater than that for type l, wl . The effectvia the Bismarckian part is thus positive in terms of equality. Third, both types of agents
suffer the same degree of loss in the Beveridgean part of the pension benefit. This implies
that the effect via the Beveridgean part is negative in terms of equality. In equilibrium,
the sum of the first and the third negative effects overcomes the second positive one.
Therefore, if µ > 0, inequality is increased by capital accumulation.
Alternatively, suppose that µ < 0: a higher level of capital results in a higher tax rate
and a lower level of the next period’s capital: ∂θ(kt )/∂kt = ∂θ(kt )/∂kt = −µ > 0 and∂kt+1 /∂kt = µw̄/(1 + n) < 0. The three effects observed for the case of µ > 0 are reversed:
the effect via the labor income effect is positive; the effect via the Bismarckian part isnegative; and the effect via the Beveridgean part is positive. The sum of the positiveeffects is equal to or greater than the negative one if A ∈ [−µc̄y /(1 + n), Ahigh ). This set
is nonempty if c̄y < (1 + n)/{(1 + n) − µw̄}.
The result established in Proposition 5 suggests that the dynamic pattern of inequality
depends on the political power of the elderly. When the political power of the elderly is
weak such that µ > 0 (i.e., γ < βR), the economy experiences an expanding (shrinking)inequality along the transition toward the steady state if the initial level of capital is
below (above) the steady-state level. However, when the political power is strong suchthat µ < 0 (i.e., γ > βR), it may experience fluctuation of inequality along the transition
toward the steady state. The result indicates that the property of inequality dynamics
depends heavily on the political power of the elderly.

5.2

Population Aging and Inequality

This section focuses on the steady-state level of inequality and investigates how inequality
is affected by population aging via the political determination of social security. For
the tractability of analysis, we focus on two extreme social security systems, the pure
Beveridgean (α = 0) and the pure Bismarckian (α = 1), and consider how population
growth rate (n) affects the steady-state level of inequality in each case.
15

The steady-state level of inequality is given by:
£
¤
y
h
y
h
u(c̄
)
+
β
R{(1
−
θ(
k̄))w
−
c̄
}
+
(1
+
n)θ(
k̄)(αw
+
(1
−
α)
w̄)
£
¤.
I¯ =
u(c̄y ) + β R{(1 − θ(k̄))wl − c̄y } + (1 + n)θ(k̄)(αwl + (1 − α)w̄)By using this, we first consider the effect of a decrease in population growth on inequality.6
Proposition 6(i) Suppose that the pension system is purely Beveridgean: α = 0. An inequality is (a)¯increased by a lower population growth, i.e., ∂ I/∂n
< 0 if µ ≥ 0; and (b) decreased¯
by a lower population growth, i.e., ∂ I/∂n
> 0 if µ < 0 and A ∈ (Alow , Â] where:−µ(w̄ − c̄y )µw̄Â ≡.
(1 + n){(1 + n) − 2µw̄}(ii) Suppose that the pension system is purely Bismarckian: α = 1. An inequality is not¯increased by lower population growth, i.e., ∂ I/∂n
≥ 0 if (a) µ > 0 and 1 + n ≥ R,(b) µ = 0, or (c) µ < 0 and 1 + n ≤ R.
We differentiate I¯ with respect to n to obtain:1∂ I¯[u(c̄y ) + βc̄ol ]2
β
∂n
µ
¶
£
¤
∂θ(k̄)=−R(wh − wl ) (u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)(1 − α)w̄
∂n
|
{z
}
Effect via labor income½¾£¤∂θ(k̄)+ (1 + n)+ θ(k̄) α(wh − wl ) (u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)(1 − α)w̄
∂n
|
{z
}
Effect via Bismarckian part of the pension benefit½¾£¤∂θ(k̄)− (1 + n)+ θ(k̄) (1 − α)w̄β(wh − wl ) R(1 − θ(k̄)) + (1 + n)θ(k̄)α .
∂n
|
{z
}
Effect via Beveridgean part of the pension benefit(12)This shows that population growth has effects on inequality through the three factors oflabor income and the Bismarckian and Beveridgean parts of the pension benefit. In what
follows, we consider how these three factors may interact to affect the steady-state levelof inequality.
6

The effect of α on inequality is straightforward. Given that the tax rate θ̄ is independent of α
(Proposition 2), the numerator of the index is increasing in α and the denominator of the index is¯decreasing in α. Therefore, we obtain ∂ I/∂α> 0: a shift toward the Bismarckian part leads to an
expansion of the steady-state inequality.

16

Let us consider an economy where the pension system is purely Beveridgean (α = 0):
the effect via the Bismarckian part disappears in this case. First, suppose that µ > 0. A
lower population growth rate leads to a lower tax rate (Proposition 4(i)) and thus produces
benefits for both types of young agents via labor income and losses for both types of old
agents via the Beveridgean part of the pension benefits. However, the marginal benefit
for type l young agents, wl , is lower than the marginal benefit for type h young agents,
wh , implying that the effect of a lower population growth via labor income is negative interms of equality. In addition, both types of old agents suffer the same degree of loss in
the Beveridgean part of the pension benefits, implying that the effect via the Beveridgeanpart is negative in terms of equality. Given these two negative effects on equality, a lower
population growth rate leads to a higher level of inequality if α = 0 and µ > 0.
Second, suppose that µ = 0. In this case, population growth has no effect on thepolitically determined tax rate (Proposition 4(i)), implying that the effect via labor income
disappears. There remains only the effect via the Beveridgean part of the pension benefit.
Therefore, a lower population growth rate leads to a higher level of inequality if α = 0 andµ = 0. Third, suppose that µ < 0. A lower population growth rate leads to a higher tax
rate (Proposition 4(i)) and thus produces losses for both types of young agents via labor
income. However, the marginal loss for a type l agent, wl , is smaller than the marginalloss for a type h agent, wh , implying that the effect of a lower population growth rate via
labor income is positive in terms of equality. The effect via the Beveridgean part of thepension benefit is ambiguous because it contains two competing effects: a positive effectcaused by a decrease in the pension benefit because of a lower population growth rate,
and a negative effect caused by a decrease in the tax rate. The condition A ∈ (Alow , Â]
ensures that the former overcomes the latter: inequality is decreased by a lower population
growth rate.
Next, consider the economy where the pension system is Bismarckian (α = 1). The
effect via the Beveridgean part disappears in this case. First, suppose that µ > 0. As in
the case when α = 0, a lower population growth rate yields a negative effect via labor
income in terms of equality. However, it yields a positive effect via the Bismarckian partof the pension benefit in terms of equality. The condition 1 + n ≥ R ensures that thepositive effect overcomes the negative one. Second, suppose that µ = 0. In this case,
population growth has no effect on the politically determined tax rate (Proposition 4(i)),
implying that the effect via labor income disappears in this case. There remains only
the positive effect via the Bismarckian part of the pension benefit. Finally, suppose thatµ < 0. As in the case when α = 0, a lower population growth rate yields a positive effectvia labor income in terms of equality. However, the effect via the Bismarckian part isambiguous because it contains the two competing effects as explained in the case when17

α = 0. If 1 + n ≤ R, the sum of the positive effects overcomes the negative one: theinequality level is decreased by a lower population growth rate.
The result established in Proposition 6 has the following policy implication. For countries with an aging population, the condition µ < 0 (i.e. βR < γ) is more likely to holdbecause they feature greater political power of the elderly. Under this condition, the sizeof the social security is increased by aging (Proposition 4). Given this result and thecondition that the population growth rate is low such that 1 + n ≤ R, aging leads toa reduction of inequality via the increasing size of redistribution (Proposition 6). This
result is in sharp contrast to the effect of a decrease in mortality rates on consumption
inequality. When the decrease in mortality rates is responsible for population aging, consumption inequality is expanded by aging because uninsured idiosyncratic events havepersistent impacts on individual income (Deaton and Paxson, 1994; Ohtake and Saito,
1998). Although our result is based on theoretical analysis with some restrictive assumptions, it indicates that the source of population aging is important in determining theeffect of aging on inequality.

6

Discussions

This section discusses the roles of two assumptions in the current analysis: the quasi-linear
utility function and an exogenously given pension scheme.

6.1

The Role of a Quasi-linear Utility Function

So far, we have assumed that the marginal utility of consumption in old age is constantoiover time: U it = u(cyit ) + βct+1 . To consider the role of this assumption, let us make thealternative assumption that the marginal utility of consumption in youth is constant overoitime: U it = cyit + βv(ct+1 ). Under this assumption, it is shown below that the politicalequilibrium generally fails to attain an interior solution.
oi
Under the assumption of U it = cyit + βv(ct+1 ), the first-order condition of utilityo0−1
oi(1/βR). This implies thatmaximization is given by βRv 0 (coit+1 ) = 1, or ct+1 = c̄ ≡ v
all income effects are absorbed by the consumption in youth. With this condition, thesaving of type i agent in period t is:¡¡¢¢zti = (c̄o − bit+1 )/R = c̄o − (1 + n)θ(kt+1 ) αwi + (1 − α)w̄ /R.
With this saving function, the capital market clearing condition is written as:(1 + n)kt+1 = c̄o /R − (1 + n)θ(kt+1 )w̄/R,
18

which leads to ∂kt+1 /∂θ(kt ) = 0.
Given the βRv 0 (coit+1 ) = 1, the derivative of Gt with respect to θ(kt ) is:
·
¸
∂Gt
γ
1 + n ∂θ(kt+1 ) ∂kt+1= (1 + n)w̄
−1+
·
·
.
∂θ(kt )
βRR
∂kt+1
∂θ(kt )
We substitute ∂kt+1 /∂θ(kt ) = 0 into this equation to obtain:∂Gt= (1 + n)w̄∂θ(kt )

µ

¶
γ
−1 .
βR

This implies that the solution to the problem is generally given by θ(kt ) = 0 or 1. Therefore, the result implies that as long as we adopt a quasi-linear utility function for thetractability of analysis, we must assume that the utility of consumption in old age islinear in order to attain an interior solution.

6.2

An Endogenous Choice of a Pension System

We have considered the case in which the pension system, represented by the parameterα, is exogenously given. We now consider an alternative case where α is chosen by voting,and show that voters are indifferent between any levels of α under the current economic
environment.
Because we focus on Markov perfect equilibrium, we restrict the choice of a pension
system to the following Markovian policy function:
αt = ψ(kt ),where ψ is a stationary, differentiable function. The derivative of Gt with respect to ψ(kt )leads to:£¤∂Gt
= γ · λ(1 + n)θ(kt )(wh − w̄) + (1 − λ)(1 + n)θ(kt )(wl − w̄)∂ψ(kt )
|
{z
}
£

(#1)

¤
+(1 + n) · λβ(1 + n)θ(kt+1 )(wh − w̄) + (1 − λ)β(1 + n)θ(kt+1 )(wl − w̄)
|
{z
}
(#2)

∂ψ(kt+1 ) ∂kt+1·∂kt+1∂ψ(kt )
= 0 ∀ψ(kt ) ∈ [0, 1],×

where the last equality holds because the terms (#1) and (#2) are zero. Therefore, there
is a continuum of solutions that maximize the objective function.
19

The reason for this result is that in the current environment, there is no distortion
produced by α. In order to obtain a unique, interior solution of α, there is a need to
introduce a distortionary effect associated with the social security system. An example isa tax base net of distortion given by φ(α) = [1−η(1−α)] where η identifies a distortionary
effect associated with the noncontributory part of the social security system (CondeRuiz and Profeta, 2007). Given this distortion, the pension benefit for type i is bit+1 =(1 + n) · τt+1 · (αt+1 wi + (1 − αt+1 )w̄) · φ(αt+1 ). This extension may provide a richer political
implication, but definitely makes it impossible to solve the model analytically. Becauseof this limitation, we have assumed α to be constant over time, and focused our attention
on the determination of the tax rate and its impact on inequality.

7

Conclusion

This paper presents a simple political-economy model that includes intergenerational aswell as intragenerational distribution parts of social security. In particular, based on theconcept of a stationary Markov perfect equilibrium, the paper focuses on the feedback
mechanism between current individuals’ decision on saving and future voting over social
security. In the presence of this mechanism, it considers how politics affects dynamic pathsof inequality and social security. In addition, it considers population aging resulting from adecline in fertility rates, and analyzes the effects of aging on inequality and social security.
The main contribution of this paper is twofold. First, the feedback mechanism createsvarious dynamic patterns of inequality and social security. In particular, these are affectedby the political power of the elderly. When their power is weak, the economy displays
a monotone increase or decrease in the size of social security and the level of inequality.
However, when the elderly’s political power is strong, the economy displays an oscillatory
path of inequality and social security. This result suggests that an aging economy associated with great political power of the elderly will tend to fluctuate in inequality andsocial security over time.
Second, the effects of aging on inequality differ between the two pension systems.
Under a pure Beveridgean scheme, aging increases (decreases) inequality if the politicalpower of the elderly is weak (strong). Under a pure Bismarckian scheme, aging decreases
inequality when the political power of the elderly is strong and the population growth rate
is low. These two results indicate that regardless of the social security system, an aging
economy may experience a decrease in inequality when a fertility decline is responsible
for population aging.
Our analysis is subject to a number of caveats. First, we ignore population aging
resulting from a decrease in mortality rates. Joint analysis of decreases in mortality and20fertility rates would produce a richer set of policy implications. Second, we abstract
the political determination of the social security scheme. An interesting extension would
be to consider the joint determination of the size of social security and the degree ofredistributiveness as discussed in Section 6.2. Third, for analytical convenience, we assume
a quasi-linear utility function and a linear production function. A natural extension of
the model is to relax this assumption. In a related work, Song (2007) presents a method
to derive an analytical solution under the assumption of a log-linear utility function anda Cobb–Douglas production function. Utilizing his method would produce a wider set ofresults and policy implications for aging, inequality and social security.

21

8

Appendix

8.1

The derivative of Gt with respect to θ(kt )The derivative of Gt with respect to θ(kt ) yields:
£
¡
¢
¡
¢¤
∂Gt
= γ λ(1 + n) αwh + (1 − α)w̄ + (1 − λ)(1 + n) αwl + (1 − α)w̄∂θ(kt )
´
h n
³
∂zth
h
+(1 + n) λ u0 (cyh)−w−t∂θ(kt )³¢´oh∂zt
∂θ(kt+1 ) ∂kt+1 ¡h+ β R ∂θ(kt ) + (1 + n) ∂kt+1 ∂θ(kt ) αw + (1 − α)w̄
n
´
³
∂ztl
l
+(1 − λ) u0 (cyl)−w−t∂θ(kt )
³
¡ l
¢´oi
l
∂z
t+1 ) ∂kt+1+ β R ∂θ(ktt ) + (1 + n) ∂θ(k
αw
+
(1
−
α)
w̄
∂kt+1 ∂θ(kt )· ½¾¢∂θ(kt+1 ) ∂kt+1 ¡ hh= γ(1 + n)w̄ + (1 + n) λ −βRw + β(1 + n)
αw + (1 − α)w̄
∂kt+1 ∂θ(kt )½¾¸¢∂θ(kt+1 ) ∂kt+1 ¡ ll+ (1 − λ) −βRw + β(1 + n)
αw + (1 − α)w̄,
∂kt+1 ∂θ(kt )¡ h¢where the second equality comes from u0 (cyht ) = βR and λ(1 + n) αw + (1 − α)w̄ + (1 −¡¢λ)(1 + n) αwl + (1 − α)w̄ = (1 + n)w̄. Because we have (1 + n)kt+1 = (1 − θ(kt ))w̄ − c̄yin equilibrium, it follows that ∂kt+1 /∂θ(kt ) = −w̄/(1 + n). We substitute this into theabove equation to obtain:· ½¾¢∂θ(kt+1 ) ¡ h∂Gth= γ(1 + n)w̄ + (1 + n) λ −βRw − β w̄
αw + (1 − α)w̄∂θ(kt )
∂kt+1
½
¾¸
¢
∂θ(kt+1 ) ¡ ll+ (1 − λ) −βRw − β w̄
αw + (1 − α)w̄.
∂kt+1
Under the assumption of an interior solution, ∂Gt /∂θ(kt ) = 0 leads to:∂θ(kt+1 )βR − γ= −µ ≡ −
.
∂kt+1
β w̄

8.2

Proof of Proposition 1

We choose τ to maximize W̃ under the constraint of λz h + (1 − λ)z l ≥ 0. The solution
must satisfy the following first-order condition:η t β w̄{(1 + n) − ηR} − ξt = 0,(13)where ξt is the Kuhn–Tucker multiplier associated with the constraint of λz h + (1 − λ)z l ≥0. If 1 + n > ηR, (13) holds if and only if ξt > 0. This implies that λz h + (1 − λ)z l = 0,22that is, τ R = (w̄ − c̄y )/w̄. If 1 + n = ηR, (13) holds if and only if ξt = 0. This implies
that λz h + (1 − λ)z l ≥ 0, that is, τ R ∈ [0, (w̄ − c̄y )/w̄]. Finally, if 1 + n < ηR, (13) implies
that τ R = 0.

8.3

Proof of Proposition 2

From the capital market clearing condition (8), the steady-state level of capital is givenby:w̄(1 − A) − c̄yk̄ =.
(1 + n) − µw̄k̄ > 0 is rewritten as:w̄ − c̄y.
w̄
From the law of motion of the equilibrium tax (9), the steady-state level of tax is given
A < Ahigh ≡

by:
θ̄ =

(1 + n)A − µ(w̄ − c̄y ).
(1 + n) − µw̄

Since the denominator is positive under Assumption 2, the inequality θ̄ ≥ 0 is rewritten
as (1 + n)A − µ(w̄ − c̄y ) ≥ 0, that is,
A ≥ Alow ≡µ(w̄ − c̄y ).
1+nUnder Assumption 2, θ̄ ≤ 1 is rewritten as:
A≤(1 + n) − µc̄y.
1+n

By direct calculation, we can find that Ahigh < {(1 + n) − µc̄y }/(1 + n) holds under
Assumption 2. Therefore, we have k̄ > 0 and θ̄ ∈ [0, 1) if A ∈ [Alow , Ahigh ).

8.4

Proof of Proposition 3

The conditions of the parameters β, R, γ, and n for the stability property are immediately
found by equations (8) and (9). The remaining task is to provide the range of k0 that
ensures kt+1 > 0 and θt ∈ [0, 1) along the equilibrium path.
(i) The case of µ > 0
The law of motion for the stock of capital (8) implies that kt+1 > 0 is equivalent to:kt >c̄y − w̄ A
+ .
µw̄
µ

23

(14)

The policy function θt = A − µkt implies that:½ y¾c̄ − w̄ A
w̄ − c̄y+=< 1,
θt < A − µµw̄
µw̄
where the first inequality comes from (14) and the second inequality comes from Assumption 1. Given that µ > 0, θt ≥ 0 is rewritten as:
kt ≤

A
.
µ

(15)

(14) and (15) lead to the set of kt that ensures kt+1 > 0 and θt ∈ [0, 1) along the equilibriumpath:c̄y − w̄ A
A
+ < kt ≤ .
(16)
µw̄
µ
µ
Therefore, setting kt = k0 in (16) leads to the set of the initial condition that ensures
kt+1 > 0 and θt ∈ [0, 1) for t ≥ 0.
(ii) The case of µ < 0Taking the same procedure as in the case of µ > 0, we can show that kt+1 > 0 holds
if:
kt <

c̄y − w̄ A
+ ,
µw̄
µ

θt < 1 always holds under Assumption 1; and θt ≥ 0 holds if A/µ ≤ kt . We have kt+1 > 0
and θt ∈ [0, 1) along the equilibrium path if:Ac̄y − w̄ A
≤ kt <
+ .
µ
µw̄
µ

(17)

Therefore, setting kt = k0 in (17) leads to the set of the initial condition that ensures
kt+1 > 0 and θt ∈ [0, 1) for t ≥ 0.
¥

8.5

Proof of Proposition 4

When µ = 0, the steady-state tax rate is θ̄ = A and is thereby independent of n and γ.
In what follows, we calculate ∂ θ̄/∂n and ∂ θ̄/∂γ for the case of µ 6= 0.
(i) The differentiation of θ̄ with respect to n leads to:
·
¸
∂ θ̄
µw̄
w̄ − c̄yµw̄
=−A +(−A + Ahigh ).
=2∂n{(1 + n) − µw̄}w̄{(1 + n) − µw̄}2Given that A < Ahigh , ∂ θ̄/∂n ≷ 0 if and only if µ ≷ 0.

24

(ii) The differentiation of θ̄ with respect to γ leads to:∂ θ̄(1 + n)w̄
∂µ=(A − Ahigh ) .
2∂γ{(1 + n) − µw̄} ∂γ
Because A − Ahigh < 0 and ∂µ/∂γ < 0 hold, we obtain ∂ θ̄/∂γ > 0.

8.6¥

Proof of Proposition 5

We differentiate It with respect to kt and rearrange the terms to obtain:
1
2 ∂It
(18)
[u(c̄y ) + βcol
t+1 ]
β
∂kt
∂θ(kt )(−wh + wl ) [(u(c̄y ) − βRc̄y ) + (1 + n)θ(kt+1 )(1 − α)w̄]= R∂kt∂θ(kt+1 ) ∂kt+1 h+(1 + n)(w − wl ) [α(u(c̄y ) − βRc̄y ) − (1 − α)w̄βR(1 − θ(kt ))] .
∂kt+1 ∂kt
From the policy function and the capital market clearing condition, ∂θ(kt )/∂kt and∂kt+1 /∂kt are given by∂θ(kt )/∂kt = ∂θ(kt )/∂kt = −µ,∂kt+1 /∂kt = µw̄/(1 + n).
(i) Suppose that µ > 0. We can rewrite (18) as1∂θ(kt )2 ∂It[u(c̄y ) + βcol
= R(−wh + wl )(u(c̄y ) − βRc̄y )
t+1 ]
β
∂kt
∂kt
∂θ(kt )
+R(−wh + wl )(1 + n)θ(kt+1 )(1 − α)w̄∂kt
∂θ(kt+1 ) ∂kt+1 h+(1 + n)(w − wl )α(u(c̄y ) − βRc̄y )∂kt+1 ∂kt
∂θ(kt+1 ) ∂kt+1 h+(1 + n)(w − wl )(−1)(1 − α)w̄βR(1 − θ(kt )),∂kt+1 ∂kt
where the first and second terms are nonnegative, the third term is nonpositive, and thefourth term is positive. Therefore, ∂It /∂kt > 0 holds if the sum of the first and the third25

terms are nonnegative. The sum is written as:∂θ(kt )
∂θ(kt+1 ) ∂kt+1 h(−wh + wl )(u(c̄y ) − βRc̄y ) + (1 + n)
(w − wl )α(u(c̄y ) − βRc̄y )∂kt∂kt+1 ∂kt
·
¸
µw̄
h
l
y
y
= (−µ)(w − w )(u(c̄ ) − βRc̄ ) R − (1 + n)
α
1+n
·
¸
γ
h
l
y
y
= (−µ)(w − w )(u(c̄ ) − βRc̄ ) (1 − α)R + α
β
≥ 0,
R

where the inequality comes from −µ < 0, wh − wl > 0 and u(c̄y ) − βRc̄y ≥ 0. Thus,∂It /∂kt > 0 if µ > 0.
(ii) Suppose that µ < 0. Dividing both sides of (18) by (−µ)(wh − wl ) > 0, we have:
1
1
2 ∂It
[u(c̄y ) + βcol
t+1 ]
h
l
(−µ)(w − w ) β∂ktyy= (−R) [(u(c̄ ) − βRc̄ ) + (1 + n)θ(kt+1 )(1 − α)w̄]µw̄
+(1 + n)[α(u(c̄y ) − βRc̄y ) − (1 − α)w̄βR(1 − θ(kt )]1+n= (−1)(R − µw̄α)(u(c̄y ) − βRc̄y ) − R(1 − α)w̄ [(1 + n)θ(kt+1 ) + µw̄β(1 − θ(kt ))] .
Under the assumption of µ < 0, we have:0 > µw̄β(1 − θ(kt )) > µw̄(1 − θ(kt )),implying that
−R(1 − α)w̄ [(1 + n)θ(kt+1 ) + µw̄β(1 − θ(kt ))]< −R(1 − α)w̄ [(1 + n)θ(kt+1 ) + µw̄(1 − θ(kt ))] .
Therefore, we obtain:
1
1
1
1
2 ∂It
2 ∂It
[u(c̄y ) + βcol[u(c̄y ) + βcol
t+1 ]
t+1 ]
h
l
h
l
(−µ)(w − w ) β∂kt (−µ)(w − w ) β∂ktyy< (−1)(R − µw̄α)(u(c̄ ) − βRc̄ ) − R(1 − α)w̄ [(1 + n)θ(kt+1 ) + µw̄β(1 − θ(kt ))]= (−1)(R − µw̄α)(u(c̄y ) − βRc̄y )·½¾¸µ(w̄ − c̄y )µw̄
−R(1 − α)w̄ (1 + n) A −+θ(kt ) + µw̄β(1 − θ(kt )) ,1+n1+n
where the equality in the last line comes from the law of motion of the equilibrium tax.

26

Rearranging the terms of this equation, we obtain:
1
1
1
1
2 ∂It
2 ∂It
[u(c̄y ) + βcol[u(c̄y ) + βcol
t+1 ]
t+1 ]
h
l
h
l
(−µ)(w − w ) β∂kt (−µ)(w − w ) β
∂kt
y
y
y
< (−1)(R − µw̄α)(u(c̄ ) − βRc̄ ) − R(1 − α)w̄{(1 + n)A + µc̄ },where the first term on the right-hand side is nonpositive. This implies that ∂It /∂kt ≤ 0
holds if (1 + n)A + µc̄y ≥ 0, i.e.,
−µc̄y
A≥
.
(19)
1+n
Given the definitions of Alow and Ahigh , we always have
Alow ≡−µc̄yµ(w̄ − c̄y )
<
,
1+n
1+n

and we have −µc̄y /(1 + n) < Ahigh ≡ (w̄ − c̄y )/w̄ if and only if:
c̄y <

1+n
w̄.
(1 + n) − µw̄

Therefore, from (19) and (20), we can conclude that:
·
¶
−µc̄y
1+n
∂It /∂kt ≤ 0 if c̄ <w̄ and A ∈, Ahigh .
(1 + n) − µw̄
1+n
y

¥

8.7

Proof of Proposition 6

We differentiate I¯ with respect to n and rearrange the terms to obtain:∂ I¯
1[u(c̄y ) + βc̄ol ]2
β
∂n
·
½
¾
¸
∂θ(k̄) h∂θ(k̄)h= −R
w + (1 + n)
+ θ(k̄) αw
∂n
∂n
£
¤
× (u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)(1 − α)w̄¾½¤£
∂θ(k̄)+ θ(k̄) (1 − α)w̄β R(1 − θ(k̄))wl + (1 + n)θ(k̄)αwl+ (1 + n)
∂n
·
½
¾
¸
∂θ(k̄) l∂θ(k̄)l− −R
w + (1 + n)
+ θ(k̄) αw
∂n
∂n
£
¤
× (u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)(1 − α)w̄¾½£¤
∂θ(k̄)+ θ(k̄) (1 − α)w̄β R(1 − θ(k̄))wh + (1 + n)θ(k̄)αwh ,− (1 + n)
∂n
27

(20)

which is reduced to:1∂ I¯[u(c̄y ) + βc̄ol ]2
β
∂n
¶
µ
£
¤
∂θ(k̄)=−R(wh − wl ) (u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)(1 − α)w̄
∂n
½
¾
£
¤
∂θ(k̄)+ (1 + n)+ θ(k̄) α(wh − wl ) (u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)(1 − α)w̄
∂n
¾
½
£
¤
∂θ(k̄)+ θ(k̄) (1 − α)w̄β(wh − wl ) R(1 − θ(k̄)) + (1 + n)θ(k̄)α .(21)− (1 + n)∂n(i) Suppose that α = 0. (21) is reduced to:

¶
µ
¯
ª
∂θ(k̄) ©
1
yol 2 ∂ Ihl [u(c̄ ) + βc̄ ]= (w − w )  −R(u(c̄y ) − βRc̄y ) + β(1 + n)θ(k̄)w̄
β
∂n
∂n

|
{z
}
(∗1)



½
¾

∂θ(k̄)
− (1 + n)+ θ(k̄) w̄βR(1 − θ(k̄)) .
∂n

{z
}
|

(22)

(∗2)

If µ > 0, ∂θ(k̄)/∂n > 0 (Proposition 4(i)). This implies that the sign of each term in¯
the square brackets is negative and thus ∂ I/∂n
< 0. If µ = 0, ∂θ(k̄)/∂n = 0 (Proposition
4(i)). Given this, (22) is rewritten as:1∂ I¯[u(c̄y ) + βc̄ol ]2= (wh − wl )(−1)θ(k̄)w̄βR(1 − θ(k̄)) < 0.
β∂nIf µ < 0, the sign of the term (*1) in (22) is positive; the sign of the term (*2) in (22) is¯
nonnegative if (1 + n)(∂θ(k̄)/∂n) + θ(k̄) ≤ 0. Therefore, ∂ I/∂n
> 0 if (1 + n)(∂θ(k̄)/∂n) +θ(k̄) ≤ 0. The direct calculation leads to:(1 + n)∂θ(k̄)(1 + n)A − µ(w̄ − c̄y )−Aµw̄ + µ(w̄ − c̄y )+ θ(k̄) =+ (1 + n)∂n(1 + n) − µw̄{(1 + n) − µw̄}2(1 + n)A{(1 + n) − 2µw̄} + µ(w̄ − c̄y )µw̄.
={(1 + n) − µw̄}2This implies that:(1 + n)−µ(w̄ − c̄y )µw̄∂θ(k̄)+ θ(k̄) ≷ 0 ⇔ A ≷ Â ≡.
∂n
(1 + n){(1 + n) − 2µw̄}

28

(23)

The critical level of A, Â, is larger than Alow under the assumption of (1 + n) − µw̄ > 0(Assumption 2), and is smaller than Ahigh since Â < 0 < Ahigh under the assumption ofµ < 0. Therefore, (1 + n)(∂θ(k̄)/∂n) + θ(k̄) ≤ 0 holds if µ < 0 and A ∈ (Alow , Â].
(ii) Suppose that α = 1. (21) is reduced to:·µ¶¯
1∂θ(k̄)yol 2 ∂ Ihl[u(c̄ ) + βc̄ ]= (w − w ) −R(u(c̄y ) − βRc̄y )
β
∂n
∂n
¾
¸
½
∂θ(k̄)yy+ θ(k̄) (u(c̄ ) − βRc̄ )+ (1 + n)∂n·¸∂θ(k̄)hlyy= (w − w )(u(c̄ ) − βRc̄ ) (−R + (1 + n))+ θ(k̄) .
∂n
Because (wh − wl )(u(c̄y ) − βRc̄y ) ≥ 0 holds (Assumption 3), the sign of the term (−R +¯(1 + n))(∂θ(k̄)/∂n) + θ(k̄) is a key to determine the sign of ∂ I/∂n.
Recall that ∂θ(k̄)/∂n ≷ 0 if and only if µ ≷ 0 (Proposition 4 (i)). Given this result,
we have (−R + (1 + n))(∂θ(k̄)/∂n) + θ(k̄) > 0 if µ > 0 and 1 + n ≥ R; or if µ < 0 and
1 + n ≤ R. If µ = 0, (21) is further reduced to:1∂ I¯[u(c̄y ) + βc̄ol ]2= (wh − wl )(u(c̄y ) − βRc̄y )θ(k̄) ≥ 0,β∂nwhere the inequality comes from wh − wl > 0 and u(c̄y ) − βRc̄y ≥ 0 (Assumption 3).

29¥

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30

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31

Figure 1: The figure illustrates the Ramsey tax rate and compares it to the politicallydetermined tax rates.

32

Figure 2: Stability property of political equilibrium paths.

33

Figure 3: The figure displays the effect of a change in a population growth rate on thesteady-state level of capital. Panel (a) illustrates the case of µ > 0. Panel (b) illustrates
the case of µ < 0.

34