Doshisha University Center for the Study of the Creative Economy
Discussion Paper Series No. 2017-03

Beyond the scope of politicians:A growth effecton intergenerational redistribution policies in aprobabilistic voting modelKazutoshi Miyazawa

Discussion Paper SeriesBeyond the scope of politicians: A growth eﬀect
on intergenerational redistribution policies in aprobabilistic voting model
Kazutoshi Miyazawa∗

Abstract
We aim to understand why some developed countries adopt more generous social security systems than the Markov perfect equilibrium in aprobabilistic voting model predicts. Contrary to the literature, we assume that politicians are opportunistic in that they take market outcomes,
such as prices and externalities, as given. With this assumption, we provethat current politicians underestimate private savings, which increases theamount of income redistribution when capital accumulation has a positive
external eﬀect on labor productivity. Using data on trust in government
as an index of political opportunism, we show that more opportunistic
politicians adopt more generous public pensions.

JEL Classification: H11, H23, H55, O41
Keywords: Political opportunism, Social security, Capital externality, Economic
growth, Markov perfect equilibrium, Probabilistic voting∗ Faculty of Economics,
kazu@mail.doshisha.ac.jp

Doshisha University,

Kamigyo,

Kyoto 602-8580 Japan.

1Introduction

In order to keep pace with population aging, many developed countries arechanging their resource allocation in favor of the older population. Figure 1
illustrates the relationship between public spending on pensions and health andthe old-age dependency ratio1 (OECD, 2015b). The relationship is significantly
positive. A 1-percentage point increase in the old-age dependency ratio causes
a 0.79 percentage point increase in the public spending/GDP ratio.
[Figure 1 about here]
Although this relationship may be intuitive, it can be interpreted in several
ways. First, the governments in developed countries are benevolent in the sensethat they reallocate resources optimally in response to demographic change.
Second, the positive relationship could reflect that population aging makes theelderly population form a voting majority, which raises the policy priority ofintergenerational redistribution.
[Figures 2a and 2b about here]
The first interpretation seems doubtful. Figure 2a illustrates the relationshipbetween public pension/GDP ratio and an index of trust in the government(OECD, 2015a, 2015b, 2017b). This relationship seems negative, although notsignificant, which implies that less trusted governments are more generous atproviding public pensions. This provides a counterexample to the view thatgovernments are benevolent.
Since social, economic, and cultural factors have a large influence on diﬀerences in levels of trust in government across countries (OECD, 2017a), it would
be more reliable to examine the relationship between change in the public pension/GDP ratio and change in the trust in government across countries. Figure
2b clarifies that the relationship is negative. A 1-percentage point increase in
the trust in government is associated with a 0.048 percentage point decrease in
the pension/GDP ratio.
The purpose of this study is to reveal why less trusted governments are moregenerous at providing public pensions. This question is particularly important because trust plays a very tangible role in the eﬀectiveness of government(OECD, 2017b).
To this end, we propose an equilibrium concept that is diﬀerent from but
comparable to the Markov perfect equilibrium. In the growth model of political economy, a Markov perfect equilibrium consists of economic and politicalequilibria (see Figure 3a). First, economic equilibria in a period are given by aset of prices that clear the markets in which consumers and producers behave
as price-takers and maximize their objectives. In equilibrium, labor productivity is determined according to market outcomes. Overall, economic variables
1 The old-age dependency ratio is defined as the ratio of population aged 65 years and overto the population aged from 15 to 64 years. Figure 1 shows the number of dependents per
100 working-age population.

1

in this period are described by a set of functions of state and policy variables.
Second, political equilibria in this period consist of a set of policy functions.
Taking state variables as given, politicians compete over some policies in thisperiod, which generates a set of policies as functions of state variables. Solving
economic and political equilibria backward from the terminal period yields theMarkov perfect equilibria.
[Figures 3a and 3b about here]
Alternatively, we analyze an economy in which politicians are opportunistic,
which means they behave as price-takers just as consumers and producers do(see Figure 3b). We call this equilibrium an opportunistic political equilibrium.
The diﬀerence between the Markov perfect equilibrium and the opportunistic
political equilibrium is politicians’ expectation about market outcomes. Suppose that capital accumulation has an external eﬀect on labor productivity.
Then, opportunistic politicians would underestimate private savings, since they
do not recognize capital externality. Our modeling strategy is simple and intuitive. Suppose that governments that implement policies consistent with Markov
perfect equilibrium are more trusted than governments that implement policies
in the opportunistic political equilibrium. Then, we can conclude that a lesstrusted government tends to provide more generous public pensions by using
the fact that opportunistic politicians underestimate private savings, and thatpublic pensions are a substitute for private savings.
Furthermore, our model is applicable to normative analysis. For example, it
can be shown that the opportunistic political equilibrium could be superior to
the Markov perfect equilibrium in cases in which the social discount rate is high,and the political power of the elderly is low. On one hand, a high discount rate
implies the optimal growth rate would be low. On the other hand, the growthrate in a Markov perfect equilibrium is negatively related to political pressureof elderly people on income redistribution. Therefore, the equilibrium growthrate could be too high relative to the optimal rate if the political power of theelderly were low.
This study is related to the works of Song, Storesletten, and Zilibotti (2012),
Ono (2015, 2017), and Lancia and Russo (2016).
Song et al. (2012) construct a dynamic general equilibrium model of small
open economies in which voters choose domestic public goods provision and its
financing through debt and taxes. Debt and capital are traded in a worldwide
market, which implies that politicians in each country take the world interest
rate as given. Furthermore, countries are heterogeneous in that their preferences for public goods are diﬀerent. The equilibrium, which is called a stablestationary Markov-perfect political equilibrium, is characterized by an interesting pattern of economies. Countries that are most concerned with public goods
maintain fiscal discipline, while the other countries suﬀer from public poverty,
that is, an extremely high tax rate to finance public debt. Lancia and Russo(2016) derive a stationary Markov perfect equilibrium in a small open economy,
which is characterized by the coexistence of public education and pensions. The
structure of their model is similar to ours in that politicians take market prices
and private savings as given. However, our focus is on political opportunism
2

characterized by the external eﬀect of capital accumulation, which is ignored inSong et al. (2012) and Lancia and Russo (2016).
In a similar context to that of Lancia and Russo (2016), Ono (2015) develops a political economy model of intergenerational conflict over public educationand pensions. In the closed general equilibrium model, Ono (2015) shows thepossibility of multiple equilibria: one is characterized by a public education
regime, and the other by a private education regime. One of the critical differences between Ono (2015) and the present study is politicians’ expectation
about the outcome of current investment. In Ono (2015), politicians are assumed long-sighted in that they precisely predict to what extent public and/or
private education aﬀect children’s human capital whereas the politicians in ourstudy are short sighted and opportunistic.
Ono (2017) also constructs a theoretical model that shows that the public
pension/GDP ratio is positively related to life expectancy in the presence of anannuity market, and that the relationship might be hump-shaped if the market
is absent. The aim of this study is quite diﬀerent to that of Ono (2017) because
we do not consider how population aging and annuity markets aﬀect intergenerational redistribution policies. However, we can highlight the diﬀerence between
the opportunistic political equilibrium and the Markov perfect equilibrium byusing the results of Ono (2017).
In the following Section 2, we introduce the basic model. In Section 3,
we derive a series of policy response functions by backward solving politicians’
problem from the terminal period of the model economy. In Section 4, we derive
the amount of intergenerational income redistribution and the growth rate in
the opportunistic political equilibrium. In Section 5, we derive the Markov
perfect equilibrium, and compare it with the opportunistic political equilibrium.
In Section 6, we discuss the optimality of these equilibria. Finally, Section 7
concludes.

2

The model

2.1

Setup

In this section, we briefly outline the model structure. We use a two-period(young/old) overlapping-generations model with perfectly competitive markets
in a closed economy. The mass of individuals born at the beginning of period t
is called generation t. Denoting the population size of generation t by Nt , anda constant population growth rate by n > 0, population evolves according to
Nt+1 = Nt n.
The utility function of an individual in generation t is given by
ut = U (cyt , cot+1 ) = ln cyt + β ln cot+1(1)where cyt and cot+1 stand for young-age and old-age consumption, respectively.
0 < β < 1 is a private discount factor.
In the first period, the individual supplies one unit of labor to earn wage
income, and allocates his/her disposable income between consumption and savings. In the second period, he/she receives capital income and an income transfer, such as a public pension, and consumes all. The budget constraints in the3

first and second periods are given by= cyt + st
= cot+1(1 − τ t )wt
Rt+1 st + Pt+1

(2)
(3)

respectively. Here, st stands for savings, wt is the wage rate, and Rt+1 is thegross interest rate. 0 ≤ τ t < 1 is the tax rate, and Pt+1 is income transfers.
The production function is given by
Yt = F (Kt , Bt Lt ) = AKtα (Bt Lt )1−α(4)where Yt , Kt , and Lt represent output, capital, and labor, respectively. Bt
is labor productivity, A > 0 is time-invariant total factor productivity, and0 < α < 1 is the output elasticity of capital.
The government budget constraint is given byNt τ t wt = Nt−1 Pt(5)which implies that the transfer scheme is based on a pay-as-you-go method.
In a closed economy, the market-clearing conditions for labor, capital, andgoods are given by
Nt
Kt+1
Yt

= Lt
= Nt st= Nt cyt + Nt−1 cot + Kt+1

(6)
(7)
(8)

respectively. The goods market-clearing condition (8) can be derived from theother equations by Walras’ law.
Following Arrow (1962), Romer (1986), and Grossman and Yanagawa (1993),
we employ labor-augmenting technology, such that
Bt =

Kt
Lt

(9)

Finally, the political objective function in period t is given byΩt = ρVto + nVty

(10)

where Vto represents the welfare of an old individual in generation t − 1, and Vty
the welfare of a young individual in generation t. ρ > 0 is the political powerof the old generation relative to the young generation. The coeﬃcient of Vty
reflects the size of voters.

2.2

Households and firms

In this subsection, we derive demand and supply functions to obtain an economic
equilibrium. The household optimization problem is formulated as
max ut = ln cyt + β ln cot+1ocyt , ct+1

subject to the lifetime budget constant,
It = cyt +
4

cot+1
Rt+1

where It represents lifetime full income,
It = (1 − τ t )wt +

Pt+1
Rt+1

(11)

Solving this problem, we obtain the following consumption functions:
cyt

=

cot+1

=

1
It
1+β
β
Rt+1 It1+βWith them, the welfare of a young individual in generation t in period t isgiven by
Vty = (1 + β) ln It + β ln Rt+1 + β ln β − (1 + β) ln(1 + β)

(12)

The welfare of an old individual in generation t − 1 in period t is given by
Vto = ln(Rt st−1 + Pt )(13)

The saving function in period t is given byst =β
1 Pt+1(1 − τ t )wt −1+β
1 + β Rt+1

(14)

Firms maximize their profits in each period. In perfectly competitive factor markets, they employ capital and labor at a point at which the marginal
products are equal to the corresponding prices,

3

Yt
Kt

Rt

= α

wt

= (1 − α)Yt
Lt

Policy response functions

In this section, we outline the structure of policy response functions when politicians behave opportunistically. The formal derivation is outlined in the Appendix. From the government budget constraint, the tax payment in period t isgiven byPt
τ t wt =n
Therefore, political parties in period t compete for the amount of income redistribution, Pt , and the tax revenue in this period is determined endogenously.
Assume that the model economy terminates at the end of period T . Individuals in generation T would not save and consume all the income, cyT =(1 − τ T )wT = wT − PT /n. The old-age consumption in period T is given bycoT = RT sT −1 + PT . When politicians behave opportunistically, they compete
for PT , taking factor prices in period T , wT and RT , and private savings, sT −1 ,as given. The policy response function in period T is given byµ¶PT∗
PT = arg max ΩT = ρ ln(RT sT −1 + PT ) + n ln wT −
(15)
n
5

The policy response function, PT∗ , increases with wT , and decreases with
RT sT −1 . If the wage rate is high, then young individuals are rich, which implies
that the marginal welfare loss of income redistribution is small. If capital income
is high, then old individuals are rich, which implies that the marginal welfare
gain of income redistribution is small.
In period T −1, the welfare of an old individual is given by VTo−1 = ln(RT −1 sT −2 +PT −1 ). Omitting constant terms, the welfare of a young individual is given by
µ
¶
PT −1
P∗
VTy−1 = (1 + β) ln wT −1 −+ TnRTwhere PT∗ is a solution of equation (15).
Note that PT∗ contains sT −1 , and that sT −1 contains PT∗ ,
µ
¶
β
PT −1
1 PT∗
sT −1 =
wT −1 −−1+βn
1 + β RT
Political parties in period T − 1 expect that current policy, PT −1 , decreases
current private savings sT −1 , which, in turn, increases PT∗ chosen in the next
period. Furthermore, political parties expect that this policy reaction in the
future decreases current private savings. However, they do not recognize thatthe depressed capital accumulation has a negative impact on wT because theycannot control market conditions and the outcome of externality.
Specifically, the policy response function in period T − 1 is given by
µ
¶
PT −1
PT∗
∗
PT −1 = arg max ΩT −1 = ρ ln(RT −1 sT −2 +PT −1 )+n(1+β) ln wT −1 −+nRTIn the same way, we can calculate PT∗−2 as a function of PT∗ −1 and sT −2 .
The general result is summarized in the following proposition.
Proposition 1 Assume that the model economy terminates at the end of period
T . Then, the policy response functions when politicians behave opportunisticallyare given as follows.
(i) In period t ≤ T − 1,
Pt∗ =

T
X
ρ
1+β
nj−t
wj −
Rt st−1ρj−t
1 + β + n j=t Πi=1
1+
β + nρ
Rt+i(16)(ii) In period T ,
PT∗ =

ρ
1
RT sT −1
ρ wT −
1+ n
1 + nρ

Proof. See the Appendix.

6

(17)

4

Opportunistic political equilibrium

In this section, we derive the amount of income redistribution and the growthrate by assuming that the economy exists forever, that is, T → ∞.
Factor prices in period t are given by
Rt
wt= R = αA= (1 − α)Akt

where kt = Kt /Nt represents the capital-labor ratio in period t. Per capita
income in period t is given by yt = Yt /Nt = Akt .
The opportunistic political equilibrium consists of {Pt∗ , kt , st }∞t=0 where⎧ ∗P∞ ¡ n ¢j−t
ρ1+β
(1 − α)Akj − 1+β+
ρ Rst−1⎪⎨ Pt = 1+β+ nρ j=t
h R
i
∗ n
Pt∗
β
1 Pt+1st = 1+β (1 − α)Akt − n − 1+β R⎪⎩
kt+1 = snt

and k0 is given.
Let us denote the gross growth rate of per capita income by g. Obviously,
this yields
yt+1
kt+1
wt+1
g=
=
=
yt
kt
wt
In the following, we assume that the economy is dynamically eﬃcient, thatis, the interest rate is greater than the growth rate of GDP,R > ng(18)

This condition is necessary for the first term in equation (16) to converge.
Regarding the amount of income redistribution and the growth rate of percapita income, we obtain the following proposition.
Proposition 2 (Income redistribution) The amount of intergenerational redistribution relative to GDP is given by¸∙
Nt−1 Pt∗1ρ(1 − α)
=
−
(1
+
β)nα
≡ Γ(g)(19)
Yt(1 + β)n + ρ 1 − ngRwhich is increasing in the growth rate of per capita income.
(Growth rate) The growth rate of per capita income is implicitly given by thefollowing equation,βA [1 − α − Γ(g)]
i
g= h
(20)
n 1 + β + Γ(g)αProof. See the Appendix.

One of the advantages of our model is that we can solve the equations in
Proposition 2 explicitly. The following proposition summarizes the result.

7

Proposition 3 The growth rate of per capita income and the amount of intergenerational redistribution relative to GDP are given byg∗β R
β + nρ n(ρ + βn)(1 − α) − (1 + β)nα(1 + β)n + ρ=

Γ(g ∗ ) =

(21)
(22)

Proof. See the Appendix.
The growth rate in equation (21) consists of economic and political eﬀects.
The term, R/n, represents an economic eﬀect in that the growth rate is positively related to the interest rate, as the Euler equation suggests. The term,β/(β + ρ/n), represents a political eﬀect because ρ/n is the political power ofthe old generation relative to the young generation. If this relative politicalpower of the old generation is large, then politicians increase intergenerational
redistribution, which decreases private savings as well as the growth rate.

5

Markov perfect equilibrium

In this section, we derive the Markov perfect equilibrium in the basic model,
and compare it with the opportunistic political equilibrium obtained in Section
4. It is shown that (1) the amount of income redistribution in the opportunistic
political equilibrium is larger than that in the Markov perfect equilibrium, and(2) the growth rate in the opportunistic political equilibrium is lower than that in
the Markov perfect equilibrium. Since opportunistic politicians do not recognize
capital externality, they underestimate private savings, which induces a large
amount of income redistribution and a low growth rate.
We should add two aspects to the basic model in order to obtain a stationary
Markov perfect equilibrium (Ono, 2017). The first consideration is related tocurrent politicians’ expectation about future policies. In our simple linearized
economy, politicians in period t expect that the amount of income redistributionin period t + 1 is proportional to per capita income in period t + 1,
Pt+1 = P (yt+1 ) = myt+1 ,

m≥0

(23)

The second consideration is related to politicians’ expectation about market
outcomes. From the capital market-clearing condition, the capital-labor ratio
in period t + 1 is given by∙¸
1
1 Pt+1βkt+1 =(1 − τ t )wt −
(24)
n 1+β
1+β R
Using yt+1 = Akt+1 and R = αA, equations (23) and (24) yield
kt+1 =β(1 − τ t )wt(1 + β)n + m
α(25)Using equation (25) and the government budget constraint, τ t wt = Pt /n,
the lifetime full income in period t is given by
!∙
Ã
¸
n+ m
Pt+1
Pt
α
−
(1
−
α)y
It = (1 − τ t )wt +
=
t
1 m
R
n
n + 1+βα8On the other hand, the capital income in period t is given by Rst−1 = Rnkt =nαyt . Therefore, omitting constant terms, the political objective function inperiod t is given by∙¸PtΩt = ρ ln(nαyt + Pt ) + n(1 + β) ln (1 − α)yt −n
The first-order condition requires
∂Ωt
ρ
1+β
=
−
=0
∂Pt
nαyt + Pt (1 − α)yt − Pnt
Solving this, we obtain a policy response function in period t,
PtM =ρ(1 − α) − (1 + β)nαyt
1 + β + nρ

where the superscript M indicates the Markov perfect equilibrium.
The amount of income redistribution relative to GDP is given byρ(1 − α) − (1 + β)nαNt−1 PtM=≡ ΓM
Yt(1 + β)n + ρFrom equation (25), we obtain the growth rate of per capita income,¢¡βA 1 − α − ΓM
M
´
g = ³
M
n 1 + β + Γα

The following proposition summarizes the result2 .

Proposition 4 The amount of income redistribution in a stationary Markov
perfect equilibrium is given byΓM =ρ(1 − α) − (1 + β)nα(1 + β)n + ρ

The growth rate of per capita income is given by¢¡
βA 1 − α − ΓM
M
´
g = ³
M
n 1 + β + Γα

(26)

(27)

Comparing Propositions 2 and 4, we obtain the following proposition.
Proposition 5 (Income redistribution) The amount of income redistribution
relative to GDP in Proposition 2 is strictly greater than that in Proposition 4.
Specifically, Γ(g ∗ ) > Γ(0) = ΓM .
(Growth rate) The growth rate in Proposition 2 is strictly smaller than that in
Proposition 4.
2 Proposition 4 corresponds to Propositions 1 and 3 in Ono (2017). ΓM in equation (26) is

the same as B1 with p = 1 (p.174), and g M in equation (27) is the same as k0 /k with γ = 1(p.177).

9

[Figures 4 and 5 about here][Tables I and II about here]
Figure 4 illustrates the growth rate in Proposition 2. The equilibrium is
unique because the left-hand side of equation (20) increases with g, while theright-hand side decreases with g. Under our specifications (see Table I), theequilibrium growth rate is g ∗ = 1.134. Assuming that one period is 25 years,
the annual growth rate is 0.5 per cent (Table II).
Figure 5 illustrates Γ(g) in equation (19). In the opportunistic politicalequilibrium, the amount of income redistribution relative to GDP is Γ(g ∗ ) =0.245. In the Markov perfect equilibrium, the size of income redistribution isΓM = 0.0527, and the growth rate is g M = 2.16 (the corresponding annual rate
is 3.13 per cent). Regarding the size of income redistribution, the opportunistic
political equilibrium is more realistic than the Markov perfect equilibrium inmany developed countries.

6

Optimality

In the previous section, we show that the growth rate in the opportunistic political equilibrium is strictly smaller than that in the Markov perfect equilibrium.
However, this result does not imply that the Markov perfect equilibrium is superior to the opportunistic political equilibrium, because a decentralized economy
might induce over-investment from a social point of view. To this end, we derivea socially optimal allocation, and examine the optimality of the two equilibria.
Let us assume that the social welfare function is utilitarian,∞Xδ t Nt U (cyt , cot+1 )

(28)

t=0

where 0 < δ < 1/n represents a social discount factor.
A social planner maximizes equation (28) subject to the resource constraints
in period t = 0, 1, ..., taking the initial capital as given. The Appendix showsthat the optimal growth rate of per capita income is given by
g S = δA

(29)

where the superscript S indicates the social optimum.
Comparing equations (21), (27), and (29), we obtain
µ
¶
1 n
∗
S
g = g
⇔ ρ = αβ−≡ ρ1 (δ)δ αµ¶αβ(1 + β) 1
M
S
g
= g
⇔ ρ=
− n ≡ ρ2 (δ)
1 + αβδ
Using the fact that ρ2 (δ) > ρ1 (δ) for 0 < δ < 1/n, and that ρ1 (δ) and ρ2 (δ)
decrease with δ, we can summarize the order of the three growth rates in thefollowing proposition.

10

Proposition 6 The order of the three growth rates is given by⎧ S
0 < ρ < ρ1 (δ)⎨ g < g∗ < gM
ρ1 (δ) < ρ < ρ2 (δ)
g∗ < gS < gMif⎩ ∗
ρ2 (δ) < ρ
g < gM < gS
[Figure 6 about here]
Figure 6 illustrates the result of Proposition 6. The horizontal axis measures
the social discount factor, δ, and the vertical axis measures the political powerof the elderly, ρ. The left curve is ρ = ρ1 (δ), and the right curve is ρ = ρ2 (δ).
This figure shows g S < g ∗ in the left region, g ∗ < g S < g M in the middle, andg M < g S in the right. In the right region, the Markov perfect equilibrium issuperior to the opportunistic political equilibrium in that g M is closer to g S thanto g ∗ . However, the opposite could be true in the left region. The opportunistic
political equilibrium is superior to the Markov perfect equilibrium if both thesocial discount factor and the political power of the elderly are low.

7

Concluding remarks

We presented an alternative equilibrium concept in the probabilistic voting
model. Politicians behave opportunistically in that they take market outcomesas given, in the same way that households and firms do. If capital accumulation has a positive externality on labor productivity, politicians underestimate
private savings, which induces them to adopt generous intergenerational redistribution policies. In fact, the opportunistic political equilibria proposed in thisstudy could explain the negative relationship between trust in government andgenerosity of public pensions in OECD countries.
Our analytical method is applicable to the related fields of research in public
economics. In a similar model of ours, Lancia and Russo (2016) explain thecoexistence of public education and pensions in modern economies. Our study
could complement their argument by showing a variety of combinations of public
education and pensions, which arises from diﬀerences in the degree of political
opportunism. Using a multicountry model with incomplete markets, Azzimonti
et al. (2014) show that governments choose higher levels of debt when financial
markets become internationally integrated. Our study could complement theirargument about public debt by incorporating into our model a debt policy forwhich opportunistic politicians compete. These extensions of our study are left
for future research.

11

Appendix[Proof of Proposition 1]
In the terminal period T , the policy response function is given byµ¶PT∗
PT = arg max ΩT = ρ ln ln(RT sT −1 + PT ) + n ln wT −n
The first-order condition requires
∂ΩT
ρ
1
=
−
=0
∂PT
RT sT −1 + PT
wT − PnTThe second-order condition is satisfied. Solving this yields
PT∗ =

ρ
1
RT sT −1
ρ wT −
1+ n
1 + nρ

which is equation (17) in Proposition 1.
In period T − 1, the policy response function is given byPT∗−1 = arg max ΩT −1 = ρ ln(RT −1 sT −2 +PT −1 )+n(1+β) ln

(A1)

µ
¶
PT −1
PT∗
wT −1 −+
nRT

where PT∗ is given by equation (A1), and sT −1 is given by
µ
¶
β
PT −1
1 PT∗
sT −1 =
wT −1 −−1+βn
1 + β RT

(A2)

From equations (A1) and (A2), we obtain the present value of the amountof income redistribution in period T ,
³
´
PT −1
β
wT
∗
−
−
w
ρ
T
−1
RT
1+β
n
PT
=
β
ρ
RT
1+β + nwhich determines the lifetime full income in period T − 1,
µ
¶
ρ
n
PT −1
wT −
IT −1 = β n ρ wT −1 +RTn1+β + nUsing this, the first-order condition for PT −1 requiresρ1+β∂ΩT −1=−=0∂PT −1
RT −1 sT −2 + PT −1 wT −1 + n wT − PT −1RTnwhich gives the policy response function in period T − 1,
µ
¶
ρ
n
1+β
PT∗−1 =+w
RT −1 sT −2
w
−
T −1
T
1 + β + nρRT
1 + β + nρ

12Let us assume that the policy response function in period t ≤ T − 1 isexpressed as
ρ
1+β
Pt∗ =
Wt −
Rt st−1(A3)
1 + β + nρ
1 + β + nρ
where Wt stands for population-adjusted human wealth in period t,
Wt = wt +nnT −t
wt+1 + · · · +wT
Rt+1
Rt+1 Rt+2 · · · RT

(A4)

In period t − 1, the policy response function is given byµ¶Pt−1 Pt∗∗= arg max Ωt−1 = ρ ln(Rt−1 st−2 +Pt−1 )+n(1+β) ln wt−1 −
+
Pt−1
n
Rt
where Pt∗ is given by equation (A3), and st−1 is given by
µ
¶
β
Pt−1
1 Pt∗
st−1 =
wt−1 −−1+βn
1 + β Rt

(A5)

From (A3) and (A5), we obtain
³
´
Pt−1
t
ρW
Rt − β wt−1 − n
Pt∗=Rt
β + nρ
which gives the lifetime full income in period t − 1,
µ
¶
ρ
n
Pt−1
n
Wt −
wt−1 +
It−1 =
β + nρ
RtnUsing this, the first-order condition for Pt−1 requires
ρ
1+β
∂Ωt−1
=
−
=0
∂Pt−1
Rt−1 st−2 + Pt−1
wt−1 + n Wt − Pt−1
Rtnwhich gives the policy response function in period t − 1,
µ
¶
1+β
ρ
n
∗
Pt−1
w
=
+
W
Rt−1 st−2t−1t −1 + β + nρRt
1 + β + nρNote that population-adjusted human wealth in period t − 1 is given by
Wt−1 = wt−1 + (n/Rt )Wt . Therefore, the mathematical induction proves equation (16) in Proposition 1.
[Q.E.D.]13

[Proof of Proposition 2]
Using the capital market-clearing condition,st
kt+1 =n
and wt+1 /wt = g, the policy response function in equation (16) is given by∞ ³Xρ(1 + β)nng ´j−t
wt −
Rkt
Pt∗ =
ρ
1 + β + n j=t R
1 + β + nρ
Assume that R > ng. Using wt = (1 − α)yt and Rkt = αyt , this equation istransformed into¸∙
1ρ(1 − α)
Pt∗ =
−
(1
+
β)nα
yt
1 + β + nρ 1 − ngRwhich is equation (19) in Proposition 2.
On the other hand, substituting the saving function into the capital marketclearing condition, we obtain
nkt+1 =βnΓ(g)[wt − Γ(g)yt ] −yt+11+β(1 + β)R

Using wt = (1 − α)yt , R = αA, and yt = Akt , we obtaing=

kt+1
βA [1 − α − Γ(g)]i= hkt
n 1 + β + Γ(g)αwhich is equation (20) in Proposition 2.
[Q.E.D.][Proof of Proposition 3]
Let us denote the ratio of the growth rate to the interest rate by x = ng/R <1.
Equation (19) becomes∙¸
1ρ(1 − α)Γ(g) =− (1 + β)nα(1 + β)n + ρ
1−x
Substituting this into equation (20), we obtaing=(1 + β)n − ρ(1−α)xβA
1−xn (1 + β)(ρ + βn) + ρ(1−α)α(1−x)

which is transformed intox=

(1 + β)n − ρ(1−α)xβ
1−x
α (1 + β)(ρ + βn) + ρ(1−α)α(1−x)

Solving this equation, we obtain two solutions, x = 1/α andx=β
β + nρ

Because x < 1, the relevant solution is x = β/(β + ρ/n), which yields
equations (21) and (22) in Proposition 3.
[Q.E.D.]14

[Socially optimal allocation]
The planner’s problem is formulated as
max

∞
X
t=0

¡
¢
δ t Nt ln cyt + β ln cot+1

subject to the resource constraints,
AKt = Nt cyt + Nt−1 cot + Kt+1
Let us setup the Lagrangian,
L0 =

∞
X
t=0

¤
£ ¡
¢
δ t Nt ln cyt + β ln cot+1 + μt (AKt − Nt cyt − Nt−1 cot − Kt+1 )where μt represents a multiplier attached to the resource constraint in period t.
The first-order conditions for cyt , cot+1 , and Kt+1 require
1
− μt
cyt

= 0

− δμt+1
cot+1

= 0

β

−μt + δμt+1 A = 0
These equations imply that per capita consumption grows at a rate of δA.
The resource constraint in per capita terms gives a first-order diﬀerence
equation of μt kt ,µ¶1
βμt kt =
1++ nδμt+1 kt+1
Aδn
Because nδ < 1, the unique solution is1
μt kt =
A(1 − nδ)

µ
¶
β
1+
δn

for all t ≥ 0. The transversality condition, limt→∞ δ t μt Kt+1 = 0, is satisfied.
Using this, the growth rate of the capital-labor ratio is given bykt+1
μ= t = δAkt
μt+1
and the optimal resource allocation is given by
Nt cytYt
Nt−1 cot
Yt
Kt+1
Yt

=
=

nδ
(1 − nδ)β + nδ
β(1 − nδ)β + nδ

= nδ

15

Acknowledgments
The author thanks Taiki Susa, Akira Yakita, Ken Tabata, Akira Kamiguchi,
Koji Kitaura, Makoto Hirazawa, and Yasuo Niwa for helpful comments on an
earlier version of the paper, as well as the participants in JAAE 2017 at Tokai
University, and seminar participants at Hokusei Gakuin University and Chukyo
University. This research is financially supported by the Japan Society for the
Promotion of Science (Grant Nos. 16H03625, 16H03635, and 17K03784). Anyerrors are of course mine.

References[1] Arrow, K.J. (1962). The economic implications of learning by doing. Review
of Economic Studies, 29(1), 155—173.
[2] Azzimonti, M., De Francisco, E., Quadrini, V. (2014). Financial globalization, inequality, and the rising public debt. American Economic Review,
104(8), 2267—2302.
[3] Grossman, G.M., Yanagawa, N. (1993). Asset bubbles and endogenous
growth. Journal of Monetary Economics, 31(1), 3—19.
[4] Lancia, F., Russo, A. (2016). Public education and pensions in democracy:
A political economy theory. Journal of the European Economic Association,
14(5), 1038—1073.
[5] OECD (2015a). How’s life? 2015: Measuring well-being. OECD Publishing,Paris.
[6] OECD (2015b). Pensions at a glance 2015: OECD and G20 indicators.
OECD Publishing, Paris.
[7] OECD (2017a). Government at a glance 2017. OECD Publishing, Paris.
[8] OECD (2017b). Trust and public policy: How better governance can help
rebuild public trust. OECD Public Governance Reviews, OECD Publishing,Paris.
[9] Ono, T. (2015). Public education and social security: A political economy
approach. Economics of Governance, 16 (1), 1—25.
[10] Ono, T. (2017). Aging, pensions, and growth. FinanzArchiv, 73(2), 163—
189.
[11] Romer, P.M. (1986). Increasing returns and long-run growth. Journal of
Political Economy, 94(5), 1002—1037.
[12] Song, Z., Storesletten, K., Zilibotti, F. (2012). Rotten parents and disciplined children: A politico-economic theory of public expenditure and debt.
Econometrica, 80(6), 2785—2803.

16

Figure 1. Old-age dependency ratio and public spending in pension and healthOld-age dependency ratio (%, 2013)
45

JPN

40

35
DEU

ITA

SWE

30

FIN

PRT

DNK

EST

GBR
HUN
CHE
NLD
CZE
SVK NOR

25

FRA
BEL

AUT

ESP

CAN
AUS

NZL

ISL

20
KOR

15

USA

LUX

ISR
CHL

Public spending in pension and health (% of GDP, 2013)

TUR

10
5

10

15

20

25

Figure 2a. Public pensions and the trust in government
Trust in government (%, 2016)
90

80

CHE

70

LUX

NOR
CAN

60
NZL

TUR

NLD

DEU

IRL
50
AUS

FIN

SWE
DNK
ISR

CZE

GBR

40
ISL

SVK

POL
JPN

LVA
USA

HUN

EST

30

PRT
ESP

FRA

SVN

KOR
20

AUT

BEL

ITA

CHL

Public Pension/GDP (%, 2013)
10
0

3

6

9

12

15

18

Figure 2b. Public pensions and the trust in government (% change)
Trust in government (% change, 2007-2016)
30

ISR

DEU

SVK

20
POL

CHE
CZE
ISL

JPN

10
HUN

GBR

-0.5

0.5
NZL
CAN
SWE
-10

AUS

NLD

Public Pension/GDP (% change, 2005-2013)

TUR

KOR

0
-1.5

LVA

NOR 1.5
EST

2.5
IRL
FRA

AUT
USA

3.5

4.5

ITA
PRT

DNK
LUX
BEL

ESP

-20
CHL

SVN
FIN
-30

Figure 3. Timing of decisions
a. Markov perfect equilibrium
Period t
Consumers

Period t+1
Market equilibrium Productivity

Politicians

Producers
Economic equilibrium
Political equilibrium

b. Opportunistic political equilibrium
Period t
Consumers
Producers

Period t+1
PoliticiansMarket equilibrium Productivity

Table I. Parameters
Capital share
Discount factor
Population growth
Political power
TFP

α = 0.3
β = 0.98825 = 0.74
n = 1.00625 = 1.16
ρ = 1.1
A = 10Note. The gross interest rate is R = αA = 3, which implies theannual interest rate is 4.5 per cent when one period is 25 years.

Table II. Results
Annual growth rate (%)
Pension/GDP ratio (%)

OPE
0.50
24.5

MPE
3.13
5.3

Note. OPE and MPE are abbreviations for opportunistic
political equilibrium and Markov perfect equilibrium.

Figure 4. Equilibrium growth rate

2.0

1.5

1.0

0.5

0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2g

Note. The horizontal axis measures the growth rate.
The upward-sloping line is a 45 degree line, and the downwardsloping curve is the right-hand side of equation (20).

Figure 5. Income redistribution and growth rate

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0

0.5

1.0

1.5

2.0

g

Note. The horizontal axis measures the growth rate, and the verticalaxis measures the amount of income redistribution relative to GDP.
The upward-sloping curve is Γ(g) in equation (19).

Figure 6. Optimality of OPE and MPE

rho

3

2

1

0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

delta

Note. The horizontal axis measures the social discount factor, δ,and the vertical axis measures the political power of the old generation, ρ.
This figure shows g S < g ∗ < g M in the left region, g ∗ < g S < g M in the middle,
and g ∗ < g M < g S in the right.