Doshisha University Life Risk Research Center
Discussion Paper Series No. 2010-05

Preferences for Redistribution in an Aging
EconomyKazutoshi Miyazawa
Faculty of Economics
Doshisha University

Discussion Paper SeriesPreferences for redistribution in an aging
economyKazutoshi Miyazawa∗

Abstract
It is one of the controversial issues whether and to what extent population aging aﬀects the size of redistribution. Introducing some kind of
altruism into a simple overlapping generations model with endogenous
labor supply, we examine the relation between population aging and aredistributive tax rate in a voting equilibrium. We can observe the relation tends to be negative in models of intergenerational altruism, but
the relation is not monotonic but complex in many cases. It suggests the
puzzle of Razin et-al. (2002) may be attributed to lack of information
about individual preferences for redistribution.

JEL Classification: D31, D64, D72, E62, H11
Keywords: Voting, Altruism, Redistribution∗ Kazutoshi Miyazawa, Faculty of Economics, Doshisha University, Kamigyo, Kyoto 6028580 Japan. kazu@mail.doshisha.ac.jp

1

Introduction

It has been one of the most controversial issues especially in the research fieldof public economics and population economics whether and to what extent population aging (an increase in the old-age dependency ratio) aﬀects the size of
the welfare state since Razin et-al. (2002) showed the increased size of the
nonworking population may well lead to lower taxes and transfers.1
The mechanism is simple. First, population aging makes the political powerof older people stronger, which encourages redistribution because they are net
beneficiaries of the related public policies. Second, higher taxes deteriorate
economic eﬃciency associated with labor supply (Meltzer and Richard (1981)),human capital investment (Razin et-al. (2002)), or fertility (Hirazawa et-al.
(2010)). This economic eﬀect discourages redistribution. Moreover, the eﬀect
becomes larger if young workers expect current distribution policies to be kept
in the future because they are not only net contributors now but also net beneficiaries in the future.2
In addition to the political and economic eﬀects on redistribution, some researchers examine another factors of individual preferences for redistribution.
In an empirical analysis, Corneo and Grüner (2002) shows not only ‘homo oeconomicus eﬀect’ but also ‘the public values eﬀect’ and ‘the social rivalry eﬀect’play a significant role in shaping individual preferences for redistribution insome countries. In a theoretical analysis, Galasso (2003) shows, by introducing
fair agents into the model of Meltzer and Richard (1981), a rise in the income
inequality between poor and middle class may not decrease redistribution. Further research would be necessary to solve the puzzle of Razin et-al. (2002).
To make our model as simple as possible, we introduce some kind of altruism
inferred from Corneo and Grüner (2002) into a two-period overlapping generations model with endogenous labor supply such as Meltzer and Richard (1981).
In each period individuals enter the economy with diﬀerent ability. They workin the first period and retire in the second period. In each period voters consistof the younger and older generation. They vote on the size of unfunded public
pensions.
We assume three types of altruism.3 First, individuals with ability w in one
generation care about the welfare of individuals with ability w in the other generation. This assumption reflects the conscious of family background or social
class, or the rivalry among heterogenous groups. We call it ‘the ability-biased
intergenerational altruism’. Second, individuals in one generation care about asocial welfare of the other generation. Since the altruism is not related to ability, we call it ‘the ability-neutral intergenerational altruism’. Finally, individuals
care about a social welfare of their own generation. It reflects the sympathy to1 See Disney (2007), Galasso and Profeta (2007), Razin and Sadka (2007), Simonovits(2007), Shelton (2008), and Hirazawa et-al. (2010) about the issue.
2 Alesina and La Ferrara (2005) shows support for redistributive policies is negatively affected by the likelihood of moving above an income threshold that is likely to separate thewinners and the losers from redistribution. Borck (2007) shows individuals with higher income have an incentive to vote on a larger size of redistribution because they live longer in theretirement than the lower income group. In a model of endogenous fertility, Hirazawa et-al.
(2010) shows population aging (a decrease in the adult mortality rate) may result in a higher
contribution rate and a higher fertility rate.
3 Assuming within-group altruism (fractionalization) and/or between-group altruism (antagonism), Lind (2007) shows that the former reduces the support for redistribution and thelatter has the opposite eﬀect.

1

ward the same generation or the rivalry between diﬀerent generations. We callit ‘intragenerational altruism’.
Numerical analysis shows that population aging is likely to decrease redistribution in (i) the ability-biased intergenerational altruism model and (ii) theability-neutral intergenerational altruism model. Interestingly, the majority in(ii) consists of the older generation. If the median voter belongs to the lower
income group in the younger generation, population aging increases redistribution. We also observe that the relation between population aging and the sizeof redistribution is not monotonic but complex in many cases. It suggests the
puzzle of Razin et-al. (2002) may be attributed to lack of information about
individual preferences for redistribution as Corneo and Grüner (2002) insists.
The structure of the paper is as follows. In section 2 we first setup thebasic model and then introduce three types of altruism into the basic model. In
section 3 we present numerical examples. The final section concludes the paper.

2

The model

2.1

Basic model

We use a simple two-period overlapping generations model. In each period,
individuals enter the economy with
diﬀerent ability w ∈ W . The probability
R
distribution function is F (w) = f (w)dw, where f (w) ≥ 0 is a density function.
Individuals are risk-neutral with respect to consumption. A gross interest
rate is normalized to one, and a gross rate of population growth, n > 0, isconstant over time. The old-age dependency ratio is 1/(1 + n), which increases
when n decreases. If n > 1, then unfunded public pensions are more eﬃcientthan funded pensions.
The optimization problem for an individual with ability w who is born inperiod t is
max Ut = ct − u(lt )subject to the lifetime budget constraint,(1 − τ t )wlt + Pt+1 = ct
where ct and lt stand for present-valued consumption and working time, respectively. The marginal disutility of labor is positive and increasing (u0 > 0and u00 > 0). τ t ∈ [0, 1) is a tax rate, and Pt+1 is the pension benefit in theretirement.
The first-order condition requires
u0 (lt ) = (1 − τ t )w
which gives a labor supply function, lt = l(wt , τ t ).
The balanced budget for the unfunded public pension requires
Pt = nτ t ȳ(τ t )(1)

where we define the labor income of an individual with ability w by y(w, τ t ) =wl(w, τ t ), and the average labor income by ȳ(τ t ),Zȳ(τ t ) =y(w, τ t )dF (w)W

2The indirect utility of an individual with ability w is given by
U (w, τ t , τ t+1 ) = (1 − τ t )y(w, τ t ) − u(l(w, τ t )) + nτ t+1 ȳ(τ t+1 )(2)

We assume each individual in the younger generation expects the tax rate inhis second period is the same as that in this period, τ t+1 = τ t . For a notational
simplicity, we omit time subscripts hereafter.
In each period, an individual with ability w in the younger generation prefersa tax rate which maximizes (2). Denote it by τ y (w). An individual with ability
w in the older generation prefers a tax rate which maximizes (1). Denote itby τ o (w). Without altruism, the preferred tax rate for the older generation is
independent of ability, τ o (w) = τ o . However, it does depend on ability if we
introduce ability-biased intergenerational altruism below.
With the envelope theorem, we have∂U
= −y(w, τ ) + nȳ(τ ) [1 − σ(τ )]∂τ(3)where σ(τ ) stands for the tax elasticity of the average income,σ(τ ) ≡ −τ ∂ ȳȳ ∂τAssumption 1. The tax elasticity of the average income is increasing, thatis,
σ 0 (τ ) > 0(4)

for any τ ∈ [0, 1).
Since ∂ 2 U/(∂τ ∂w) < 0, τ y (w) is decreasing in w. In general, ∂ 2 U/(∂τ ∂n) >0 because a higher n makes unfunded public pensions more eﬃcient. It suggests
the preferred tax rate τ y (w) shifts downward when n decreases.
The optimal tax rate for the older generation is given byσ(τ o ) = 1(5)Equation (5) implies all the individuals in the older generation prefer a tax
rate which maximizes the pension benefit. We know τ y (w) ≤ τ o for any w ∈ W
because (∂U/∂τ )τ =τ o = −y(w, τ o ) ≤ 0. The voting equilibrium in the basicmodel is given by
½ y y
n>1
τ (wm )
τ∗ =
if
n<1
τo
where the median voter in the younger generation is given by
µ
¶
1
1
y
F (wm
)=
1−
2
n

Population aging (an increase in the old-age dependency ratio) aﬀects theequilibrium in two ways. One is a political eﬀect. The median voter moves tolower income group, which increases the tax rate. The other is an economic
eﬀect. The preferred tax rate for the median voter decreases because unfundedpublic pensions become less eﬃcient.
In the following subsections, we introduce some kind of altruism into thebasic model to examine ‘the public value eﬀect’ and ‘the social rivalry eﬀect’ inCorneo and Grüner (2002).
3

2.2

Ability-biased intergenerational altruism

In this subsection, we assume individuals with ability w in one generation careabout individuals with w in the other generation. The intergenerational altruism
is ability-biased. The positive correlation of ability between the generations
reflects the conscious of family background or social class, or the rivalry among
heterogenous groups.
The objective function of an individual w in the younger generation is
W1y (w, τ ) = U (w, τ ) + ρy P (τ )(6)where ρy ≥ 0 stands for an altruism parameter of the younger generation.
The objective function of an individual w in the older generation is
W1o (w, τ ) = P (τ ) + ρo U (w, τ )(7)where ρo ≥ 0 stands for an altruism parameter of the older generation.
We have
∂W1y
∂τ
∂W1o
∂τ

= −y(w, τ ) + (1 + ρy )nȳ(τ )[1 − σ(τ )](8)= −ρo y(w, τ ) + (1 + ρo )nȳ(τ )[1 − σ(τ )](9)which give the preferred tax rates, τ y (w) and τ o (w).
The voting equilibrium is given byτ∗yo= τ y (wm) = τ o (wm)1yo) + F (wm) =nF (wm(1 + n)2[Figure 1 is here]
Figure 1 shows the voting equilibrium in the model of ability-biased intergenerational altruism. The vertical axis measures the tax rate, and the horizontal
axis measures the ability of the younger generation rightward from the origin,and the ability of the older generation leftward from the origin. The dashedcurve stands for the preferred tax rate in the basic model, and the solid curve
stands for the preferred tax rate in the model of ability-biased intergenerational
altruism. Comparing to the basic model, the tax curve for the younger generation shifts upward, and the tax curve for the older generation shifts downward
because both generations care about each other. Individuals with higher ability
in the older generation prefer lower tax rates because they care about individuals
with higher ability in the younger generation who prefer lower tax rates. Thereyexist a marginal voter in the younger generation, wm, and a marginal voteroin the older generation, wm , such that the lower ability group in the youngeryogeneration, wy ≤ wm
, and in the older generation, wo ≤ wm
, form the majority.
In section 3, we examine the eﬀect of population aging on the the equilibriumtax rate numerically.

4

2.3

Ability-neutral intergenerational altruism

In this subsection, we assume individuals in one generation care about a social
welfare of the other generation. The intergenerational altruism is ability-neutral.
The objective function of an individual w in the younger generation is thesame as (6), W2y (w, τ ) = U (w, τ ) + ρy P (τ ).
The objective function of an individual w in the older generation isZooW2 (τ ) = P (τ ) + ρ
U (w, τ )dF (w)(10)
W

Diﬀerentiating equation (10) with respect to τ , we have∂W2o= ȳ(τ ) [(1 + ρo )n (1 − σ(τ )) − ρo ]∂τwhich gives the optimal rate for the older generation such asσ(τ o ) = 1 −ρon(1 + ρo )(11)

Without altruism, τ o is the same as the basic model. τ o is decreasing inρ , and increasing in n given that ρo > 0. If the altruism is stronger, the older
generation agree on less redistribution. If the population growth rate decreases,
they also agree on less distribution because the decreased eﬃciency of unfundedpensions makes the younger generation worse-oﬀ.
o[Figure 2 is here]
Figure 2 shows the voting equilibrium in the model of ability-neutral intergenerational altruism. The tax curve for the younger generation is the same as
Figure 1. Contrary to the ability-biased case, the tax curve for the older generation shifts downward in parallel. Thus, one of the possible equilibria in which
the older generation form the majority, τ o , is politically stable because the otherequilibria could be realized only if some group in the younger generation form
the majority.
Specifically, denote by ŵy the ability of individuals in the younger generation
who would prefer the same tax rate as the older generation. First, assume thatnF (ŵy ) < (1 + n)/2 and n[1 − F (ŵy )] < (1 + n)/2. The voting equilibrium isτ o because the younger generation cannot form a majority group. Second, if
nF (ŵy ) > (1 + n)/2 or if n[1 − F (ŵy )] > (1 + n)/2, then the younger generation
can form the majority. In the former case, the tax rate, τ H , is high because thelower ability group becomes the majority. In the latter case, the tax rate, τ L ,is low because the higher ability group does.
The voting equilibrium in the ability-neutral intergenerational altruism model
is given by⎧
F (ŵy ) < 12 (1 − n1 )
⎨ τL
∗
1
1
1
1
o
y
τ
τ =
if
2 (1 − n ) < F (ŵ ) < 2 (1 + n )
⎩
1
1
y
τH
2 (1 + n ) < F (ŵ )5where τ y (ŵy ) = τ o , and
τH
τL

2.4

µ
µ
¶¶
n+1
= τ y F −1
2n
µ
µ
¶¶
n
−1
= τ y F −12nIntragenerational altruism

In this subsection, we assume individuals care about a social welfare of theirown generation. It reflects the sympathy toward the same generation or therivalry between diﬀerent generations.
The objective function of an individual w in the younger generation is givenbyZ
W3y = U (w, τ ) + ρy
U (w, τ )dF (w)(12)
W

The objective function of an individual w in the older generation is
W3o = (1 + ρo )P (τ )(13)which gives τ o in the basic model.
From (12), we have∂W3y
= −y(w, τ ) + ȳ(τ ) [n(1 + ρy )(1 − σ(τ )) − ρy ]
∂τ

(14)

which gives τ y (w).
[Figure 3 is here]
Figure 3 shows the voting equilibrium in the intragenerational altruism
model. Comparing to the basic model, individuals with higher ability in theyounger generation prefer higher tax rates because they care about the welfareof the lower income group. The voting equilibrium is given by½ y y
τ (wm )
n>1
τ∗ =
if
n<1
τo
¡
¢
y
where F (wm) = 12 1 − n1 .
When n > 1, population aging could decrease the tax rate because themedian voter who belongs to the lower income group in the younger generation
may well prefer a lower tax rate by the intragenerational altruism.

3

Numerical analysis

In this section we present numerical examples because it is diﬃcult to get general
results. Assumptions may be unrealistic, but we hope the results would be
insightful.
Assumption 2. The ability w is uniformly distributed in [0, 1].4
4 More realistic cases are analyzed in section 3.5.

6

Assumption 3. The disutility of labor is a quadratic function such asl22βu(l) =

where β > 0 stands for a parameter associated with the willingness to work.
The labor supply function is given byl = β (1 − τ ) w
which gives the labor income,y = wl = β (1 − τ ) w2
and the average labor income,
ȳ =

Z 1

ydw =

0

1
β (1 − τ )3

The tax elasticity of the average income isσ(τ ) =

τ1−τwhich satisfies Assumption 1.

3.1

Basic model

From equation (5), the optimal tax rate for the older generation is
τo =

1
2

The optimal tax rate for an individual w in the younger generation is givenby5yτ (w) =½

n−3w2
2n−3w2if0p0 ≤ w ≤ n3pn
3 ≤w ≤1

(15)

∗
o
∗
y
y
The voting equilibrium¡¢ is τ = τ for n < 1, and τ = τ (wm (n)) for n > 1,11ywhere wm (n) = 2 1 − n .
py(n) < n3 for any n > 0. Therefore, we have
We can verify wm¡¢2
n − 34 1 − n1
τ (wm (n)) =¡¢22n − 34 1 − n1y(16)[Figure 4 is here]

Figure 4 illustrates the relation between τ ∗ and n in the basic model. The
relation is not monotonic. For a larger n, population aging decreases the tax
rate because the economic eﬀect dominates the political eﬀect. For a smaller n,
the tax rate increases because the political eﬀect becomes dominant.
5 See Appendix for the derivation.

7

3.2

Ability-biased intergenerational altruism

The optimal tax rate for an individual w in the younger generation is given by6(pn(1+ρy )−3w2
0 ≤ w ≤ n3 (1 + ρy )yy )−3w22n(1+ρ
pτ (w) =
if
(17)
n
y
0
3 (1 + ρ ) ≤ w ≤ 1
The optimal tax rate for an individual w in the older generation is⎧q⎨ n(1+ρo )−3ρo w20 ≤ w ≤ n3 (1 + ρ1o )
2n(1+ρo )−3ρo w2
if qτ o (w) =n
1⎩
0
3 (1 + ρo ) ≤ w ≤ 1(18)

The voting equilibrium is summarized in the following proposition.

Proposition 1 In the ability-biased intergenerational altruism model, the voting equilibrium is given byτ∗ =n(1 + ρy ) − 34
2n(1 + ρy ) − 34

where
ρ≡

s

h

ρ(1+n)
1+ρnhi2

ρ(1+n)
1+ρni2

ρo (1 + ρy )1 + ρo(19)(20)

stands for a strength of altruism.
y, and the median voter in
The median voter in the younger generation, wmothe older generation, wm , are respectively given by
y
wm

=

o
wm

=

ρ(1 + n)2(1 + ρn)1+n2(1 + ρn)yo, and decreases wm. Given that ρ < 1, a larger n
A larger ρ increases wm
yoincreases both wm
and wm.

Proof. See Appendix.
[Figure 5a and 5b are here]
Figure 5a and 5b illustrate the relation between τ ∗ and n in the ability-biased
intergenerational altruism model. Figure 2a is a case in which intergenerational
altruism is weak (we assume ρy = 0 and ρo = 0.01, which gives ρ = 0.0995.),
and Figure 2b is a strong case ρy = ρo = 0.1, and ρ = 0.316).
When the intergenerational altruism is weak, the relation is not monotonic.
The reason is the same as the basic model. On the one hand, a decrease in n
makes the public pension less eﬃcient, which decreases the optimal tax rate.
6 See Appendix.

8On the other hand, a decrease in n shifts the median voter towards the lower
income group. This eﬀect increases the optimal tax rate.
When the intergenerational altruism is strong, the relation between τ ∗ andn is positive in a broader region. This result support for Razin et-al. (2002).
The strength of altruism, ρ in equation (20), is a key factor. Specifically, we
have the following proposition.
Proposition 2 Under Assumption 2 and 3,(i) If 0 < ρ < 1/9, the tax rate is increasing in n ∈ (0, n), (n, ∞), anddecreasing in n ∈ (n, n), wherep1 − 3ρ − (1 − ρ)(1 − 9ρ)n =2ρ
p1 − 3ρ + (1 − ρ)(1 − 9ρ)n =2ρ(ii) If 1/9 ≤ ρ < 1, the tax rate is increasing in n.
Proof. See Appendix.

3.3

Ability-neutral intergenerational altruism

The optimal tax rate for an individual with ability w in the younger generation
is given by equation (17).
From equation (11), the optimal tax rate for the older generation is given byτo =n(1 + ρo ) − ρo2n(1 + ρo ) − ρo(21)Given that ρo > 0, the optimal tax rate for the older generation is strictly
smaller than the optimal tax rate for the lowest ability group in the younger
generation, τ o < τ y (0). Since τ y (w) is decreasing in w, and τ o > 0, there
exists a unique ability level which satisfies τ y (w) = τ o . Denote it by ŵy . Fromequations (17) and (21), we haveρŵy = √
(22)
3
where ρ is given by equation (20).
Specifically, we have the following proposition.
Proposition 3 The voting equilibrium in the ability-neutral intergenerational
altruism model is⎧ŵy < 12 (1 − n1 )
⎨ τL
∗
1
1
1
1
o
y
τ
if
τ =
2 (1 − n ) < ŵ < 2 (1 + n )
⎩
1
1
y
τH
2 (1 + n ) < ŵ

where

τH
τL

µ µ
¶¶
1
1
= τ
1+
2
n
µ µ
¶¶
1
1
y
= τ
1−
2
n
y

τ y (w), τ o , and ŵy are respectively given by equations (17), (21), and (22).
9[Figure 6 is here]
Figure 6 illustrates the relation between τ ∗ and n in the model of abilityneutral intergenerational altruism. We assume ρy = ρo = 0.1. The criticalability is ŵy = 0.183. It implies τ ∗ = τ o if n < 1.577, and τ ∗ = τ L if n > 1.577.
In a region where the critical mass consists of a higher income group in theyounger generation, the tax rate moves as the basic model. In the process of
population aging, however, the political power of the older generation becomesstrong, and finally becomes the majority. It is noteworthy that this politicalchange does not increase the tax rate but decrease the tax rate because theolder generation care about a social welfare of the younger generation.

3.4

Intragenerational altruism

The optimal tax rate for the older generation is given σ(τ ) = 1, that is,
τo =

1
2

The optimal tax rate for an individual w is given by7⎧
q(1+ρy )n−ρy⎨ n(1+ρy )−ρy −3w2
0
≤
w
≤
3
y
y )−ρy −3w2
2n(1+ρ
if q
τ (w) =
(1+ρy )n−ρy
⎩
0
≤w≤1

(23)

3

τ y shifts downward when n decreases as the basic model. Introducing theintragenerational altruism makes the tax rate flat. Specifically, we have∂τ y1
R 0 ⇔ w2 Ry∂ρ
3The voting equilibrium is given by½ y y
τ (wm (n))
∗
τ =
1
2

if

n>1
n<1

¡
¢
y
where wm(n) = 12 1 − n1 . Substituting this into equation (23), we have¡¢2n(1 + ρy ) − ρy − 34 1 − n1
τ =¡¢22n(1 + ρy ) − ρy − 34 1 − n1∗(24)

[Figure 7a and 7b are here]

Figure 7a and 7b illustrate the relation between τ ∗ and n in the intragenerational altruism model. Figure 7a is a case in which the altruism is weak(ρy = 0.01), and Figure 4b is a strong case (ρy = 0.1). We can observe that a
median voter who belongs to the lowest income group in the younger generation
could prefer a lower tax rate if the intragenerational altruism is fairly strong.
7 See Appendix.

10

3.5

Distribution

In this subsection we briefly discuss about the distribution of ability. We havequalitatively the same result in more realistic distribution as the uniform one.
First, let us assume a Pareto distribution function such as³
w ´−aF (w) = 1 − 1 +w ∈ [0, ∞)(25)bwhere a > 2 and b > 0. The average w̄ = b/(a − 1) is greater than the median1
wm = b(2 a − 1).
With assumption 3, the average income is given byȳ(τ ) =2b2 β(1 − τ )(a − 1)(a − 2)which gives σ(τ ) = τ /(1 − τ ).
Without altruism, the optimal tax rates are τ o = 0.5 and⎧q2n⎨ 2nb2 −(a−1)(a−2)w2w ≤ b (a−1)(a−2)
4nb2 −(a−1)(a−2)w2q
τ y (w) =if
2n⎩0
w > b (a−1)(a−2)
Solving 1 + nF (w) = 1+n
2 , the median voter is given by
y
=b
wm

"µ

2n
n+1

¶ a1

#

−1

y
) if n ≥ 1.
The equilibrium tax rate is τ ∗ = τ o if n < 1, and τ ∗ = τ y (wm[Figure 8 is here]
Figure 8 illustrates the relation between τ ∗ and n when ability follows thePareto distribution (We assume a = 20).8 The shape looks like Figure 4, although the quantitative eﬀect of population aging is small because the distribution is skewed to the lower income group.
Next, assume an exponential distribution function such as
F (w) = 1 − e−dw

w ∈ [0, ∞)(26)

where d > 0. The average w̄ = d−1 is greater than the median wm = d−1 ln 2.
With assumption 3, the average income is given byȳ(τ ) =2β(1 − τ )d2which gives σ(τ ) = τ /(1 − τ ).
8 The equilibrium tax rate is independent of b. On the one hand, a larger b increases
preferred tax rates because the average income increases, which also increases pension benefits.
On the other hand, median voters move to the right, which decreases the preferred tax rate.
The opposite eﬀects are fully cancelled out in our model.

11Without altruism, the optimal tax rate for individuals with ability w in theyounger generation is given by(√
2n−(dw)2w ≤ √d2ny24n−(dw)
τ (w) =
ifw > d2n0
The median voter is given byywm= d−1 ln2nn+1which gives the equilibrium tax rate when n ≥ 1 as

´2³
2n2n − ln n+1y)=τ ∗ = τ y (wm³´22n4n − ln n+1[Figure 9 is here]

Figure 9 illustrates the relation between τ ∗ and n when ability follows the
exponential distribution.9 We observe the shape looks like Figure 4 again. Theysuggests the analysis in a case of uniform distribution would be, at least qualitatively, robust.

4

Concluding remarks

We can observe population aging decreases the size of redistribution especiallyin a case of the ability-biased intergenerational altruism, and the ability-neutralintergenerational altruism with the median voter belonging to the older generation. This result supports for Razin et-al. (2002). However, we can also observethat the relation is not monotonic but complex (See for example Figure 6 and7b). Our results suggest ‘the public values eﬀect’ and ‘the social rivalry eﬀect’in Corneo and Grüner (2002) play an important role to answer the question ofwhether population aging aﬀect the size of the welfare state.

9 The equlibrium tax rate is independent of d. The reason is the same as Footnote 8.

12

Acknowledgments
The author would like to thank Akira Yakita, Makoto Hirazawa, Hikaru
Ogawa, Akihiko Kaneko, Taiki Susa and seminar participants at Nagoya University and Waseda University for useful comments on an earlier version of thepaper. Financial support from the Grant-in-Aid for Scientific Research of JSPS
(No.22530190) and the Japan Economic Research Foundation is gratefully acknowledged. The remaining errors are of course mine.

References[1] Alesina, A., and La Ferrara, E. (2005) Preferences for redistribution in theland of opportunities, Journal of Public Economics 89, 897-931.
[2] Borck, R. (2007) On the choice of public pensions when income and life
expectancy are correlated, Journal of Public Economic Theory 9, 711-725.
[3] Corneo, G., and Grüner, H.P. (2002) Individual preferences for political
redistribution, Journal of Public Economics 83, 83-107.
[4] Disney, R. (2007) Population ageing and the size of the welfare state: Is
there a puzzle to explain?, European Journal of Political Economy 23, 542553.
[5] Galasso, V. (2003) Redistribution and fairness: a note, European Journalof Political Economy 19, 885-892.
[6] Galasso, V., and Profeta, P. (2007) How does ageing aﬀect the welfare
state?, European Journal of Political Economy 23, 554-563.
[7] Hirazawa, M., Kitaura, K., and Yakita, A. (2010) Aging, fertility, social
security and political equilibrium, Journal of Population Economics 23,559-569.
[8] Lind, J.T. (2007) Fractionalization and the size of government, Journal of
Public Economics 91, 51-76.
[9] Meltzer, A.H., and Richard, S.F. (1981) A rational theory of the size ofgovernment, Journal of Political Economy 89, 914-927.
[10] Razin, A., Sadka, E., and Swagel, P. (2002) The aging population and thesize of the welfare state, Journal of Political Economy 110, 900-918.
[11] Razin, A., and Sadka, E. (2007) Aging population: The complex eﬀect offiscal leakages on the politico-economic equilibrium, European Journal of
Political Economy 23, 564-575.
[12] Shelton, C.A. (2008) The aging population and the size of the welfare state:
Is there a puzzle?, Journal of Public Economics 92, 647-651.
[13] Simonovits, A. (2007) Can population ageing imply a smaller welfare state?,
European Journal of Political Economy 23, 534-541.

13

Appendix[Derivation of equation (15)]
From equation (3), we have
¡
¢ ¤
β£
∂U
=
n − 3w2 − 2n − 3w2 τ∂τ
3First, we know (∂U/∂τ )τ =1 < 0. Second, assume that
w2 ≤

n
3

Then, the optimal tax rate is given byτ y (w) =n − 3w2
2n − 3w2

which is decreasing in w, and increasing in n.
Finally, assume thatnw2 >3
It implies ∂U/∂τ < 0 for ∀τ ∈ [0, 1]. Therefore we have τ y (w) = 0.
[Derivation of equations (17) and (18)]
From (8), we have
ª
β©
∂W1y
=
n(1 + ρy ) − 3w2 − [2n(1 + ρy ) − 3w2 ]τ∂τ
3

We know (∂W1y /∂τ )τ =1 < 0. Assume that w2 ≤ n(1 + ρy )/3. Then we have
τ y (w) =n(1 + ρy ) − 3w2
2n(1 + ρy ) − 3w2

which is decreasing in w, and increasing in n and ρy . Next, assume that w2 >n(1+ρy )/3. Then, we know ∂W1y /∂τ < 0 for any τ ∈ [0, 1]. It implies τ y (w) = 0.
From equation (9), we have
£
¤ ª
∂W1o
β©
=
n(1 + ρo ) − 3ρo w2 − 2n(1 + ρo ) − 3ρo w2 τ∂τ3

We know (∂W1o /∂τ )τ =1 < 0. Assume that w2 ≤ n(1 + ρo )/(3ρo ). Then wehaven(1 + ρo ) − 3ρo w2
τ o (w) =2n(1 + ρo ) − 3ρo w2
which is decreasing in w and ρo , and increasing in n. Next, assume that w2 >n(1 + ρo )/(3ρo ). Then, we know ∂W1o /∂τ < 0 for any τ ∈ [0, 1]. It impliesτ o (w) = 0.

[Proof of Proposition 1]
The voting equilibrium is given by
τ∗
y
o
+ wm
nwm

y
o
= τ y (wm) = τ o (wm)
1=(1 + n)2

14

Assume that
y
wm
o
wm

r

n
(1 + ρy )3
r
1n<(1 + o )3ρ<yoFirst, τ y (wm) = τ o (wm) is equivalent to
y 2
)
(wm
ρo
=
(wo )2y1+ρ
1 + ρo mwhich gives
y
o
wm
= ρwm
y
o
Substituting this into nwm+ wm= 12 (1 + n), we have
o
wm

=

y
wm

=

1+n
2(1 + ρn)ρ(1 + n)2(1 + ρn)yo) or τ o (wm), we have
Substituting one of them into τ y (wmτ∗ =n(1 + ρy ) − 34
2n(1 + ρy ) − 34hρ(1+n)
1+ρnhi2

ρ(1+n)
1+ρni2[Proof of Proposition 2]oo 2
From equation (18), we know τ ∗ = τ o (wm) is decreasing in (wm) /n. Observingo
wm in Proposition 1, defineφ(n) =(1 + n)2n(1 + ρn)2

We know the sign of φ0 is the same as−ρn2 + (1 − 3ρ)n − 1
If 0 < ρ < 1/9, then φ0 < 0 when n < n or n < n, and φ0 > 0 whenn < n < n, wherep1 − 3ρ − (1 − ρ)(1 − 9ρ)n =2ρ
p1 − 3ρ + (1 − ρ)(1 − 9ρ)n =2ρTherefore τ ∗ is increasing in n ∈ (0, n), (n, ∞), and decreasing in n ∈ (n, n).
If 1/9 ≤ ρ < 1, then φ0 < 0 for any n > 0. In this case, τ ∗ increases with n.
[Derivation of equation (23)]15

From equation (14), we have
ª
β©
∂W3y
=
n(1 + ρy ) − ρy − 3w2 − [2n(1 + ρy ) − ρy − 3w2 ]τ∂τ
3

We know (∂W3y /∂τ )τ =1 < 0. Assume thatw2 ≤n(1 + ρy ) − ρy3Then, the optimal tax rate is given byτ y (w) =n(1 + ρy ) − ρy − 3w2
2n(1 + ρy ) − ρy − 3w2

Next, assume thatn(1 + ρy ) − ρy3which implies ∂W3y /∂τ < 0 for ∀τ ∈ [0, 1]. Therefore we have τ y (w) = 0.
w2 >

16

Figure 1. Ability-biased intergenerational altruismτ1τ (w )

τ (w )
τ∗

w

w

O

w

w

Figure 2. Ability-neutral intergenerational altruismτ1τ (w )
τ

w

O

w

w

Figure 3. Intragenerational altruism

τ
1

τ∗

τ (w )

w

O

w

w

Figure 4. Basic model

t

0.51

0.50

0.49

0.48

0.47

0.46
0

1

2

3

4

5

6

n

Figure 5a. Ability-biased intergenerational altruism (ρy = 0, ρo = 0.01)t0.49500.49480.49460.49440.49420.49400.49380.4936
0.4934
0.4932
0.4930
1

2

3

4

5

6

n

Figure 5b. Ability-biased intergenerational altruism (ρy = ρo = 0.1)

t

0.50

0.49

0.48

0.47

0.46

0.45

0.44

0.43

0.42
1

2

3

4

5

6

n

Figure 6. Ability-neutral intergenerational altruism (ρy = ρo = 0.1)

t

0.50

0.49

0.48

0.47

0.46
1

2

3

4

5

6

n

Figure 7a. Intragenerational altruism (ρy = 0.01, ρo = 0)

t

0.51

0.50

0.49

0.48

0.47

0.46
0

1

2

3

4

5

6

n

Figure 7b. Intragenerational altruism (ρy = 0.1, ρo = 0)

t

0.51

0.50

0.49

0.48

0.47

0.46
0

1

2

3

4

5

6

n

Figure 8. Pareto distribution

t

0.510
0.508
0.506
0.504
0.502
0.500
0.498
0.496
0.494
0.492
0.490
0

1

2

3

4

5

6

n

Figure 9. Exponential distribution

t

0.510
0.508
0.506
0.504
0.502
0.500
0.498
0.496
0.494
0.492
0.490
0

1

2

3

4

5

6

n