Discussion Papers In Economics
And Business

Aging, Pensions, and Growth
Tetsuo Ono
Discussion Paper 14-17-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAging, Pensions, and Growth
Tetsuo Ono
Discussion Paper 14-17-Rev.2

June 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANAging, Pensions, and Growth∗
Tetsuo Ono†
Osaka University

Abstract
This study presents an endogenous growth, overlapping-generations model featuring probabilistic voting over public pensions. The analysis shows that (i) the
pension–GDP ratio increases as life expectancy increases in the presence of an annuity market, while it may show a hump-shaped pattern in its absence; (ii) thegrowth rate is higher in the presence of the annuity market than its absence, but
the presence implies an intergenerational trade-oﬀ in terms of utility.
Keywords: Economic Growth; Population Aging; Probabilistic Voting; Public
Pensions; Annuity Market
JEL Classification: D70, E24, H55∗

This paper is a substantially revised version of an earlier paper entitled “Economic Growth and thePolitics of Intergenerational Redistribution.” This work was supported in part by a Grant-in-Aid for
Scientific Research (C) from the Ministry of Education, Science and Culture of Japan No. 15K03509.
The author would like to thank Yuki Uchida for his research assistance.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Tel.: +81-6-6850-6111; Fax: +81-6-6850-5256. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

Many Organisation for Economic Co-operation and Development (OECD) countries haveexperienced declining population growth rates and increasing life expectancy over the past
decades (OECD, 2011). This demographic change raises the share of the elderly in the
population, which is expected to strengthen their political power in voting. Therefore,
government spending for the elderly, such as on public pensions and long-term care, islikely to increase. One of the expected side eﬀects of this trend is an increase in the tax
burden on the young, which may result in a declining growth rate over time.
The prediction about the eﬀect of population decline on pensions is in line with observations in OECD countries. Panel (a) of Figure 1 suggests that the public pension
spending–GDP ratio is positively correlated to the declining population growth rate. On
the other hand, the aforementioned prediction on the eﬀect of increasing life expectancy is
not likely to fit the observation. In Panel (b) of Figure 1, the pension–GDP ratio shows aweak positive correlation with increasing life expectancy. However, France and Italy show
more than five times higher ratios than Iceland, Israel, and Mexico, although they all
share similar life expectancy. Therefore, the empirical evidence seems to be mixed. The
first aim of this study is to present a political economy theory that explains the diverging
evidence observed in Figure 1.
[Figure 1 here.]
The second aim is to provide a theory that fits the evidence on aging and economic
growth. The aforementioned argument suggests that lower population growth has a negative eﬀect on economic growth because of increasing pension burden. However, theevidence in Panel (a) of Figure 2 shows that lower population growth is associated withhigher per capita GDP growth. In addition, increasing life expectancy is associated withlower per capita GDP growth, as depicted in Panel (b) of Figure 2. The two aging factors
have opposite implications for economic growth. The present study demonstrates a modelthat explains these contradictory results.
[Figure 2 here.]
For the aims of this study, we develop an overlapping-generations model with individuals who live for a maximum of two periods, youth and old age, and competitive firms
endowed with AK technology, as in Romer (1986). Government spending financed by a
tax on the young includes public pensions that benefit the elderly. Within this framework, we consider and compare the following two cases. The first is a perfect annuity
case in which an agent’s wealth is annuitized and transferred to the other agents, who
1live throughout old age, if he or she dies young (Sheshinski and Weiss, 1981). The second
is a case of no annuity, in which agents’ unannuitized wealth is bequeathed to children as
an unintentional bequest (Abel, 1985).
Within the abovementioned framework, we demonstrate the conflict of interest between generations by assuming probabilistic voting a la Lindbeck and Weibull (1987),
where the government’s objective is to maximize the weighted sum of the utility of theyoung and elderly. We employ a Markov strategy in which the policy variable of pensions
is conditioned on a payoﬀ-relevant state variable, that is, the beginning-of-period capital in the present framework. This implies that the expected level of public pensions in
the next period depends on the next-period stock of capital, which is aﬀected by policy
decisions in the current period. Forward-looking individuals consider this inter-temporaleﬀect when they vote.
We characterize a political equilibrium of probabilistic voting, and obtain the following three results. First, diverging evidence on pensions in Figure 1 could be explained byfocusing on the presence or absence of an annuity market. The model analysis shows thatin the presence of the annuity market, the pension–GDP ratio increases as life expectancy
increases. However, in the absence of the annuity market, an additional eﬀect appears.
Agents increase savings as life expectancy increases. Given that savings are perfect substitutes with the present value of pensions, the politician is induced to oﬀer lower pension
benefits as life expectancy increases. This negative eﬀect, peculiar to the case of the
no-annuity market, may outweigh the positive eﬀect, depending on the political power ofthe elderly, and create a hump-shaped pattern of the ratio.
Second, a lower population growth rate leads to a higher per capita growth rate.
This result is consistent with the observed evidence in Panel (a) of Figure 2. However,higher life expectancy is shown to raise the growth rate, mainly because the young are
incentivized to save more for their consumption in old age. To explain this contradictory
finding, we focus on another aging factor, that is, the political power of the elderly. A
rise in their power incentivizes the government to increase public pension burden on theyoung and thereby lowers the growth rate. Because higher life expectancy implies a larger
share of the elderly in the population, the political power of the elderly is likely to bestrengthened as life expectancy increases (OECD, 2006; Smets, 2012). Therefore, thenegative correlation between life expectancy and per capita growth, which is observed forsome countries, as depicted in Panel (b) of Figure 2, could be explained when the eﬀects
of life expectancy and the political power of the elderly are examined together.
Third, there is an inter-generational trade-oﬀ in terms of utility. The growth analysis
shows that the growth rate is higher in the presence of the annuity market than its
absence. A higher growth rate generates more resources available for future generations,2which works to improve their utility. However, the presence of the annuity market raises
the return from savings and thereby lowers the present value of public pension benefits,
which, in turn, works to decrease lifetime consumption. This negative eﬀect is relevant
for all generations, while the positive growth eﬀect is relevant for all generations except
the initial one. Therefore, the presence of the annuity market makes the initial generation
worse oﬀ, and future generations better oﬀ.
The present study is related to the literature on the political economy of public pensions. Earlier studies consider the political sustainability of public pensions (see, e.g.,
Grossman and Helpman, 1998; Cooley and Soares, 1999; Boldrin and Rustichini, 2000;
Azariadis and Galasso, 2002; Forni, 2005; Mateos-Planas, 2008). Recently, the focus haschanged to the political eﬀect of population aging on pension provision. Examples are
Gonzalez-Eiras and Niepelt (2008), Bassetto (2008), Chen and Song (2014), and Lancia and Russo (2015). However, their focus is mainly on a decline in the population
growth rate. Increasing life expectancy, which is also a key factor for population aging,is abstracted away from their analyses.
Gonzalez-Eiras and Niepelt (2012) is, to the best of our knowledge, the first attempt toinvestigate the political eﬀect of rising life expectancy on pensions and economic growth.1
They conduct the analysis under an environment with perfect annuity markets. However,as reported in Rusconi (2008), the extent of annuitization varies across OECD countries,and some countries are still characterized by small annuity markets. Ono and Uchida(2016) overcome this issue by comparing the case of perfect annuity with the case of noannuity, but their analysis is confined to numerical simulation of the growth of humancapital. By contrast, the present study analytically solves the model in both the presence
and absence of annuity markets, shows aging eﬀects consistent with the observed data on
OECD countries, and provides welfare implications of annuities across generations.
The rest of the paper is organized as follows. Section 2 presents the model andcharacterizes an economic equilibrium. Section 3 characterizes a political equilibrium, investigates the eﬀects of population aging on pensions and economic growth, and evaluates
the welfare implications of annuities. Section 4 provides concluding remarks. Proofs are
given in the Appendix.

2

The Model and Economic Equilibrium

Consider an infinite-horizon economy composed of identical agents, perfectly competitive
firms, and annuity markets. A new generation, called generation t, is born in each period1

In addition, Zhang, Zhang, and Lee (2003), Gradstein and Kaganovich (2004), and Kunze (2014)
analyze the political eﬀect of increasing life expectancy on economic growth. However, pensions are
abstracted from their analyses. Their policy focus is on public education as an engine of economic
growth.

3t = 0, 1, 2, .... Generation t is composed of a continuum of Nt > 0 identical agents. We
assume that Nt = (1 + n)Nt−1 , that is, the net rate of population growth is n > −1.

2.1

Preferences and Utility Maximization

Agents live for a maximum of two periods, youth and old age. An agent dies at the endof youth with probability of p ∈ (0, 1). The probability p represents life expectancy or
longevity, both of which are used interchangeably in the following sections. If an agent diesyoung, his or her annuitized wealth is transferred to the other agents, who live throughout
old age, and his or her unannuitized wealth is bequeathed to children as unintentionalbequests.
In youth, each agent is endowed with one unit of labor, which is supplied inelasticallyto firms, and each agent obtains wages. An agent in generation t divides his/her wage
wt between his/her own current consumption cyt ; savings held as an annuity and invested
into physical capital for consumption in old age, st ; and tax payments as a proportion of
his or her wage, τt wt , where τt is the period-t pension contribution rate. Thus, the budget
constraint for a young agent in period t iscyt + st ≤ (1 − τt )wt + ut ,where ut is the per capita unintentional bequest from generation t − 1 to generation t.
If an agent is alive in old age, he/she consumes the returns from savings plus the
public pension benefit. The budget constraint for generation t in old age is
cot+1 ≤ (Rt+1 + αt+1 ) st + bt+1 ,where cot+1 is consumption in old age and bt+1 is the public pension benefit. The returnon savings is stated as the sum of the return of direct holdings of capital, Rt+1 , and thereturn from annuity, αt+1 .
Let γ ∈ {0, 1} denote the degree of annuitization in the economy. In particular, γ = 0(γ = 1) implies the absence (presence) of annuity markets. If γ = 0, the unannuitized
portion of agent’s wealth is distributed to his or her heirs as unintentional bequests.
However, if γ = 1, the annuitized wealth is transferred via annuity markets to the agents
who live throughout old age. Therefore, ut+1 satisfies Nt+1 ut+1 = Nt (1 − γ)(1 − p)Rt+1 st ,or{
(1 − p)Rt+1 st /(1 + n) if γ = 0,
ut+1 =(1)0
if γ = 1.
The return from annuity, αt+1 , satisfies pαt+1 st = γ(1 − p)Rt+1 st , where the left-handside denotes the aggregate payments to the agents who are alive in old age, while theright-hand side denotes return from annuity. Thus, αt+1 is given by{
0 if γ = 0,
αt+1 =(2)
Rt+1
if γ = 1.
p
4

We focus on the two extreme cases, γ = 0 and 1, to demonstrate the role of annuity
markets in a tractable way.
Agents consume private goods. We assume additively separable logarithmic preferences to obtain a closed-form solution. The utility of a young agent in period t is written
as ln cyt + pβ ln cot+1 , where β ∈ (0, 1) is a discount factor. Thus, the expected utilitymaximization problem for a period-t young agent can be written as:
max ln cyt + pβ · ln cot+1{cyt ,cot+1 }s.t. cyt + st ≤ (1 − τt )wt + ut ,
cot+1 ≤ (Rt+1 + αt+1 ) st + bt+1
given τt , wt , bt+1 αt+1 , and Rt+1 .
Solving the problem leads to the following consumption and saving functions:
[
]
1
bt+1
y
ct =
(1 − τt )wt + ut +,1 + pβ
Rt+1 + αt+1[]
pβ (Rt+1 + αt+1 )
bt+1o
ct+1 =(1 − τt )wt + ut +,1 + pβ
Rt+1 + αt+1
[
]
pβ
bt+1
st =
(1 − τt )wt + ut −.
1 + pβpβ (Rt+1 + αt+1 )In period 0, there are young agents in generation 0 and initial elderly agents in generation −1. Each agent in generation −1 is endowed with s−1 units of goods, earns returnR0 s−1 plus pension benefit b0 , and consumes them. The initial elderly agents’ measure is
pN−1 . The utility of an agent in generation −1 is (1 − θ) ln co0 + θ ln g0 .

2.2

Technology and Profit Maximization

There is a continuum of identical, perfectly competitive, profit-maximizing firms that
produce output with a constant-returns-to-scale Cobb–Douglas production function, Yt =At (Kt )α (Nt )1−α , where Yt is aggregate output, At is the productivity parameter, Kt isaggregate capital, Nt is aggregate labor, and α ∈ (0, 1) is a constant parameter representing capital share. The productivity parameter is assumed to be proportional to theaggregate capital per labor unit in the overall economy, that is, At = A(Kt /Nt )1−α . Capital investment, thus, involves a type of technological externality often used in theories ofendogenous growth (see, e.g., Romer, 1986). Capital is assumed to fully depreciate withina period.
In each period t, a firm chooses capital and labor to maximize its profits, Πt =At (Kt )α (Nt )1−α − Rt Kt − wt Nt , where Rt is the rental price of capital and wt is thewage rate. The firm takes these prices as given. The first-order conditions for profit
5

maximization are given by
Kt : Rt = αAt (Kt )α−1 (Nt )1−α ,Nt : wt = (1 − α)At (Kt )α (Nt )−α .

2.3

Government Budget Constraints

The government budget for pensions is assumed balanced in each period. Fiscal policy
is determined through elections. A period-t budget constraint on pensions is Nt τt wt =pNt−1 bt . Dividing both sides of the constraint by Nt , we obtain the per capita form of
the government budget constraint:
τt wt =

2.4

p
bt .
1+n

Economic Equilibrium

The market-clearing condition for capital is Kt+1 = Nt st , which expresses the equalityof total savings by young agents in generation t, Nt st , to the stock of aggregate physicalcapital. Dividing both sides by Nt leads to(1 + n)kt+1 = st ,where kt ≡ Kt /Nt is per capita capital.
Definition 1. An economic equilibrium is a sequence of prices, {wt , Rt , αt }∞t=0 , allo-∞
cations, {cyt , cot , st , ut }∞t=0 , capital stock {kt }t=0 with the initial condition k0 (> 0),
and policies {τt , bt }∞t=0 , such that: (i) utility is maximized with the budget constraints in youth and old age; (ii) profit is maximized; (iii) the government budget

is constrained; and (iv) the annuity and capital markets clear.
Assuming productive externality, At = A(Kt /Nt )1−α , the first-order conditions forprofit maximization are
Rt = R ≡ αA and wt = (1 − α)Akt .
Using the saving function and the first-order conditions for profit maximization, we rewrite
the capital market-clearing condition as:


p
b
pβ
( t+1 )  ,· (1 − α)Akt −bt + (1 − γ)(1 − p)Rkt −(1 + n)kt+1 =1 + pβ1+npβ 1 + γ(1−p) Rp(3)
where γ = 1(= 0) if the annuity market is present (absent).
6

In an economic equilibrium, the indirect utility of a young agent in period t, Vty , andthat of an elderly agent alive in period t, Vto , can be expressed as functions of government
policy and capital stock:


bt+1
) 
Vty = (1 + pβ) ln (1 − τt )(1 − α)Akt + (1 − γ)(1 − p)Rkt + (γ(1−p)
R1+ p)(pβ 1 + γ(1−p)
R
p
1
+ ln
+ pβ ln,
1 + pβ1 + pβ(())γ(1 − p)o1+R(1 + n)kt + bt .
Vt = lnp
The first term of the young agent’s indirect utility function corresponds to the utility of
consumption in youth and old age; and the term of the elderly agent’s indirect utility
corresponds to the utility of consumption.

3

Political Equilibrium

This study assumes probabilistic voting to demonstrate the political mechanism. In eachperiod, the government in power maximizes a political objective function. Formally, thepolitical objective function in each period t is given byΩt = ωpVto + (1 + n)Vty ,where ωp and (1 + n) are the relative weights of elderly and young agents, respectively.
In particular, the parameter ω(> 0) represents the political power of the elderly, whichreflects the recent age gap in voter turnout in developed countries (OECD, 2006; Smets,2012). The government’s problem in period t is to maximize Ωt subject to its budget
constraints, given the state variable, kt .2
The scope of this study is restricted to a stationary Markov-perfect equilibrium.
Markov perfectness implies that outcomes depend only on the payoﬀ-relevant state variable, that is, capital k. The stationary property implies that our focus is on equilibrium
policy rules that do not depend on time. Therefore, the expected level of public pensions for the next period, bt+1 , is given by a function of the next period’s stock of capital,
bt+1 = B(kt+1 ). Using recursive notation with x′ denoting the next period x, we can definea stationary Markov-perfect political equilibrium in the present framework as follows.
2

An explicit micro-foundation for this modeling is explained in Persson and Tabellini (2000, Chapter
3) and Acemoglu and Robinson (2005, Appendix). Song, Storesletten, and Zilibotti (2012) outline theprocess to derive the political objective function under probabilistic voting in the framework of overlapping
generations.

7Definition 2. A stationary Markov-perfect political equilibrium is a set of functions,⟨T, B⟩, where T : ℜ++ → [0, 1] is a pension contribution rule, τ = T (k), andB : ℜ++ → ℜ+ is a public pension rule, b = B(k), such that the following conditions
hold:(i) the capital market clears


′
pβ 
B(k )
(
) ,
(1+n)k ′ =· (1 − T (k)) · (1 − α)Ak + (1 − γ)(1 − p)Rk −γ(1−p)
1 + pβpβ 1 +
Rp(4)(ii) ⟨T (k), B(k)⟩ = arg max Ω(k, b, τ, b′ ) is subject to the expectation of future pensions,b = B(k ′ ), the capital market-clearing condition in (4), and the government budget
constraint,p
T (k)(1 − α)Ak =· B(k),
1+nwith a non-negativity constraint, b ≥ 0, where Ω(k, b, τ, b′ ) is defined by{()}γ(1 − p)Ω(·) ≡ ωp ln
1+R(1 + n)k + b
p




′
b
)
+ (1 + n)(1 + pβ) ln (1 − τ )(1 − α)Ak + (1 − γ)(1 − p)Rk + (,γ(1−p)
1+Rpwhere irrelevant terms are omitted from the expression of Ω(·).
The first condition describes agents’ response to the government’s choice of a tax, T (k),under the expectation that future pensions will be set according to the rule, b′ = B(k ′ ).
The second condition states that the government chooses its fiscal policy to maximize its
objective, subject to the government budget constraint, the capital market-clearing condition representing the agents’ response to the fiscal policy choice, and their expectationof the future pensions, b′ = B(k ′ ). The solution to the government problem constitutes astationary Markov-perfect equilibrium if b = B(k). For example, suppose the expectation
is given by b′ = B̃ · k ′ , and the solution is b = B̂ · k, where B̃ and B̂ are constant. Then,the solution is said to constitute a stationary Markov-perfect equilibrium if B̃ = B̂.

3.1

Characterization of Political Equilibrium

Given that preferences are specified by the logarithmic utility function, we assume a linear
policy function of public pensions for the next period, B(k ′ ) = Bγ · Ak ′ , where Bγ (> 0),γ ∈ {0, 1}, is a constant parameter. In this scenario, we solve the problem and determine
the political equilibrium outcome as follows.
8

Proposition 1. There is a stationary Markov-perfect political equilibrium with b > 0 if
γ = 1 and α <1,
1 + (1 + pβ)(1 + n)/ωp

γ = 0 and α <1.
p + (1 + pβ)(1 + n)/ωor

For the case of b > 0 the policy function of public pensions is{
B1 Ak if γ = 1,
Bγ Ak =
B0 Ak if γ = 0,
where
B1 ≡

ω(1 − α) − (1 + pβ) αp (1 + n),ωp
(1 + pβ) + 1+nω {(1 − α) + (1 − p)α} − (1 + pβ)α(1 + n)
B0 ≡(> B1 ) .
ωp
(1 + pβ) + 1+nProof. See Appendix A.1.
The result in Proposition 1 implies that for both cases of γ = 1 and 0, public pensions
are more likely to be provided in the political equilibrium if the political power of theelderly (ω) is larger and/or the population growth rate (n) is lower. Greater politicalpower of the elderly attaches a larger weight to the utility of the elderly in the politicalobjective function. This incentivizes the politician to provide higher pension benefits tothe elderly. A smaller population growth rate implies lower savings per head for given
capital stock, and thus, a lower consumption level of the elderly. To maintain theirconsumption level, the politician oﬀers a larger pension benefit to the elderly.
The eﬀects of longevity (p) on the provision of public pensions diﬀer between the twocases. In the presence of annuity markets (γ = 1), greater longevity has two opposing effects on pension provision. First, greater longevity leads to a lower rate of return from annuity and thus, a lower consumption level in old age. To compensate for this loss of old-age
consumption, the politician oﬀers a larger pension benefit to the elderly. This is a positiveeﬀect on the public pension represented by the term α/p in the numerator of B1 . Second,
greater longevity attaches a larger weight to the lifetime utility of the young. To improvetheir utility, the politician cuts the tax burden on the young by reducing public pension
provisions for the elderly. This is a negative eﬀect on the public pension represented
by the term pβ on the numerator of B1 . The condition α < [1 + (1 + pβ)(1 + n)/ωp]−1
implies that the former positive eﬀect outweighs the latter negative eﬀect when γ = 1.
In the absence of annuity markets (γ = 0), the negative eﬀect remains, but the positiveeﬀect through the annuity returns disappears. Instead, there is an additional negativeeﬀect through the accidental bequest represented by the term (1 − p)α in the numerator
9

of B0 . That is, an increase in longevity decreases the accidental bequests and thereby
creates a negative income eﬀect on the young. This negative eﬀect, accompanied with thenegative eﬀect that remains in the absence of annuity markets, incentivizes the politician
to cut the tax burden on the young and thereby to reduce the pension benefit to theelderly. Therefore, in the absence of annuity markets, public pensions are less likely to beprovided if longevity is higher.

3.2

Aging and Pensions

Based on the characterization of the political equilibrium, we now examine how aging
factors, n, ω, and p, aﬀect public pension provision.
Proposition 2. Consider an equilibrium with b > 0 demonstrated in Proposition 1.
(i) The pension–GDP ratio increases with a lower population growth rate and greater
political power of the elderly.
(ii) Assume γ = 1. The pension–GDP ratio increases with greater longevity.
(iii) Assume γ = 0. With greater longevity, the pension–GDP ratio increases if α ≤{ω/(1 + n)}/ {(1 + β) + ω/(1 + n)}2 ; and shows a hump-shaped pattern otherwise.
Proof. See Appendix A.2.
To confirm the statement in Proposition 2, we first compute the pension–GDP ratio,
pNt−1 bt /Yt = (p/(1 + n)) · (b/Ak), as follows:

1− α if γ = 1,1+pβ 1+npb· p +1ω·=
1 + n Ak  1+pβ ·11+n +1 − αp if γ = 0.
ωpA lower population growth rate implies a smaller weight to the utility of the young,whereas greater political power of the elderly implies a larger weight to the utility of theelderly. This puts less value on the cost of public pensions for the young and more value
on its benefit for the elderly. Therefore, the politician is incentivized to increase public
pension provision in response to a decrease in n and an increase in ω.
When γ = 1, the ratio is aﬀected by greater longevity in two ways. First, greater
longevity implies that the current young attach a larger weight to their utility of consumption in old age. They prefer saving for their own future consumption to public
pension spending on the currently elderly. This preference of the young incentivizes thegovernment to cut current public pension spending. Therefore, greater longevity has anegative eﬀect on the pension–GDP ratio, as observed by the term pβ.
10Second, greater longevity leads to a higher dependency ratio, and thereby a higher
pension–GDP ratio. This positive eﬀect is observed by the term (1 + n)/p. Therefore,
greater longevity has two opposing eﬀects on the ratio, but the result in Proposition
1(ii) shows that the latter always outweighs the former in the political equilibrium if
γ = 1. This result is consistent with the prediction of Gonzalez-Eiras and Niepelt’s(2008) model using a neoclassical growth framework. The present analysis indicates that
their prediction also holds in an endogenous growth model with AK technology. The solid
curve in Figure 3 illustrates a numerical example of the ratio when γ = 1.
[Figure 3 here.]
When γ = 0, there is an additional negative eﬀect represented by the term αp. Greater
longevity implies that more weight is attached to the return from savings; agents increasesavings as longevity increases. The politician takes account of this economic behavior inpolicy decision making, and given that savings are perfectly substitutable with pensions,
he or she oﬀers lower pension benefits as longevity increases. When γ = 1, this negativeeﬀect on pension is oﬀset by the decrease in the return from annuity, R̃ = R/p. However,there is no such cancellation eﬀect when γ = 0.
The term αp, representing the aforementioned negative eﬀect, becomes larger as theinterest rate, R = αA, increases. In particular, if α is small, such that α ≤ {ω/(1 +n)}/ {(1 + β) + ω/(1 + n)}2 , the sum of the negative eﬀects is outweighed by the positiveeﬀect; greater longevity leads to a higher pension–GDP ratio. However, if α is above thecritical value, an increase in longevity produces an initial increase followed by a decreasein the ratio. The dotted curve in Figure 3 illustrates a numerical example of the humpshaped pattern of the ratio when γ = 0.
To recap, the pension–GDP ratio increases as longevity rises in the presence of anannuity market. In its absence, the ratio increases or exhibits a hump-shaped pattern as
longevity rises. These diﬀerent eﬀects of longevity could be ascribed to the presence (orabsence) of annuities. Therefore, the extent of annuitization may be an important featurein providing an explanation for the mixed evidence on the pension–GDP ratio observedfor high-longevity countries in Figure 1.

3.3

Aging and Economic Growth

The result in Proposition 1 enables us to derive the growth rate of per capita capital, k ′ /k,and to investigate how the growth rate is aﬀected by population aging. The following
proposition summarizes the result.
Proposition 3. Consider an equilibrium with b > 0.
11

(i) The growth rate of per capita capital is ′(1+pβ)A
 kk γ=1 ≡′ωpωk(1+n){(1+pβ)+ 1+n}+ βp·(α/p)=′(1+pβ)A
 kk γ=0 ≡kωpω(1+n){(1+pβ)+ 1+n}+ βp·α

if γ = 1,
if γ = 0.

The growth rate is higher in the presence of an annuity market than its absence.
(ii) The growth rate increases with a lower population growth rate, less political power ofthe elderly, and greater longevity: ∂(k ′ /k)/∂n < 0, ∂(k ′ /k)/∂ω < 0, and ∂(k ′ /k)/∂p >0.
Proof. See Appendix A.3.
The growth rate of per capita capital is constant over time because the model exhibitsa constant interest rate inherited from AK technology. In addition, the presence of theannuity market increases the growth rate. In order to understand the role of the annuity
market, recall the capital market-clearing condition in (3). With the use of b = Bγ Ak, wereformulate it as


k′
=
k

pβ
1+pβ
pβ
(1 + n) + 1+pβ
·

Bγ

)
α
pβ 1 + γ(1−p)
p
|
{z
}
(


·
(1 − α) −pBγ + (1 − γ)(1 − p)αA. (5)
|
{z
}
1
+
n
| {z }
(#2)

(#3)

(#1)

The presence of the annuity market has eﬀects on the growth rate in the following
three ways. First, the per capita pension–GDP ratio, Bγ , is lower and the return fromsavings is higher in the presence of the annuity market than its absence. This implies alower present value of pension benefits, which incentivizes individuals to save more for
future consumption. This is a positive eﬀect of the annuity market on the growth rate,
represented by the term (#1) in Equation (5).
Second, the presence of the annuity market lowers the tax burden for pension provision.
This creates a positive income eﬀect on savings and the growth rate, represented by theterm by (#2). Finally, the unintentional bequest disappears owing to the presence of
the annuity market. This produces a negative income eﬀect presented by the term (#3).
Overall, there are two positive eﬀects by (#1) and (#2), which dominate the negativeeﬀect by (#3). Therefore, the growth rate is higher in the presence of the annuity market
than its absence.
To observe the eﬀect of population aging factors on economic growth, let us recall thegrowth rate of per capita capital when γ = 1,
[
(
)]−1
k′
1
ω
= A · (1 + n) +
.
ωp +
k γ=1
1 + pββp · (α/p)
12Thus, the growth rate is aﬀected by population aging via the terms, (1 + n), 1 + pβ, ωp,and ω/ (βp · (α/p)). When γ = 0, the term ω/ (βp · (α/p)) is replaced by ω/ (βpα).
Population growth, the political power of the old, and longevity have the following
implications for economic growth via these terms. First, a lower population growth rateincreases per capita capital equipment in the economy. This eﬀect, which is observedby the term (1 + n), accelerates capital accumulation and economic growth. Second,
greater political power of the elderly attaches a larger weight to the utility of the current
elderly. This incentivizes the government to increase the public pension expenditure,
thereby resulting in a greater tax burden on the young. This eﬀect, which is observedby the terms ωp and ω/ (βp · (α/p)), discourages saving of the young and thus, impedes
economic growth.
Greater longevity has the following eﬀects on the growth rate via the terms 1 + pβ,
ωp, and ω/ (βp · (α/p)) when γ = 1. First, greater longevity attaches a larger weight tothe lifetime utility of the young as represented by the term pβ. The young save more
for their consumption in old age in response to an increase in longevity. This creates a
positive eﬀect on economic growth. Second, greater longevity attaches a larger weight tothe utility of the elderly, too. As described earlier in this subsection, this increases thetax burden on the young and thus, has a negative eﬀect on economic growth.
Third, greater longevity lowers the rate of return from annuity, R̃ = R/p, and thus,increases the present value of public pension benefits. In response to this increase, individuals save less because pension benefits are perfect substitutes for savings. This negativeeﬀect on economic growth is observed in the term ω/ (βp · (α/p)) and is peculiar to thecase of γ = 1. Thus far, the three eﬀects imply that longevity has mixed growth eﬀects,
but the net eﬀect is positive when γ = 1. The result is qualitatively unchanged whenγ = 0 because the last negative eﬀect disappears.
The model prediction regarding population growth is consistent with the observationin Panel (a) of Figure 2, but the prediction regarding longevity is inconsistent with theobservation in Panel (b); the evidence shows that higher life expectancy results in alower growth rate. This inconsistency is resolved when we focus on another aging factor,
that is, the political power of the elderly. Greater longevity implies a larger share of
the elderly in the population, which might, in turn, lead to larger political power of theelderly (OECD, 2006; Smets, 2012). Given the negative eﬀect of the power on economic
growth, we might argue that the negative correlation between life expectancy and per
capita economic growth observed in Panel (b) of Figure 2 could be explained when theeﬀects of longevity and the political power of the elderly are examined together.

13

3.4

Welfare Analysis

The previous subsection shows that the growth rate is higher in the presence of the annuity
market than in its absence. A higher growth rate generates more resources available
for future generations. This implies that future generations are made better oﬀ by thepresence of the annuity market. However, the presence of the annuity market increases
the rate of return from saving and thus, lowers the present value of public pensions. This
works to decrease the present value of lifetime income and consumption. Hence, it may
be possible that the present generation is made worse oﬀ by the presence of the annuity
market compared to its absence.
In order to investigate such a possibility, we compare the indirect utility functions
when γ = 1 and γ = 0 for a given k as follows:
V y |γ=1 ≷ V y |γ=0

(
)t
(
)t
⇔ pβ ln(1/p) + (1 + pβ) ln p + (1 + pβ) ln k ′ /k|γ=1 ≷ (1 + pβ) ln k ′ /k|γ=0 .
(6)From this condition, we obtain the following result.
Proposition 4. Consider an equilibrium with b > 0.
(i) For generation 0, the expected utility of the young is higher in the absence of theannuity market than its presence.
(ii) Along the equilibrium path with k ′ /k > 1, there is some T (> 1) such that the expectedutility of generation t( ≥ T ) is higher in the presence of the annuity market than its
absence.
Proof. See Appendix A.4.
To understand the argument in Proposition 4, let us look at the condition in (6). The
first term on the left-hand side, pβ ln(1/p), shows a positive eﬀect of the annuity market.
The presence of the annuity market increases the return from saving, which works toincrease the old-age consumption. However, an increase in the interest rate lowers thepresent value of public pension benefits. This works to decrease lifetime consumption
and thereby creates a negative eﬀect on utility. This negative eﬀect is represented by thesecond term on the left-hand side, denoted by (1 + pβ) ln p. The net eﬀect of these two
opposing eﬀects through the interest rate is negative.
()t
The third term on the left-hand side, (1+pβ) ln k ′ /k|γ=1 , and the term on the right()thand side, (1 + pβ) ln k ′ /k|γ=0 , show the lifetime income (i.e., lifetime consumption)that is aﬀected by the growth rate. Given the result in Proposition 3, we find that the14

lifetime income when γ = 1 is larger than that when γ = 0. The growth eﬀect works
from generation 1 onward, and the diﬀerence in the lifetime income becomes larger astime passes.
Overall, the presence of the annuity market creates a negative eﬀect through theinterest rate and a positive eﬀect through the growth rate. For generation 0, the growtheﬀect is irrelevant, and thus, it is made better oﬀ in the absence of the annuity market
compared to its absence. However, from generation 1 onward, the positive growth eﬀect
is relevant, and this eﬀect becomes larger as time passes. Therefore, there is some T (> 0)such that the expected utility of generation t(≥ T ) is higher in the presence of the annuity
market than its absence. A numerical example is demonstrated in Figure 4.
[Figure 4 here.]

3.5

Alternative Production Function

In previous subsections, we conduct the analysis by assuming AK technology that exhibitsa constant interest rate. Within this assumption, we show that the pension–GDP ratio
increases with longevity in the presence of an annuity market, while it may show a humpshaped pattern in the absence of the market. In other words, greater longevity is lesslikely to increase the pension–GDP ratio in the absence of the annuity market compared
to in the presence of the annuity market.
In order to investigate whether this result still holds when the interest rate is endogenous, we here undertake a brief analysis by alternatively assuming a neoclassical production function, yt = A (kt )α . The interest rate and wage are given by Rt = αA (kt )α−1 and
wt = (1 − α)A (kt )α , respectively. The interest rate is now decreasing in capital.
The indirect utility function of the young isV y = (1 + pβ) ln (1 − α)A (k)α + (1 − γ)(1 − p)αA (k)α + (()
1−α+ pβ ln 1/ (k ′ ),1 + γ(1−p)
p

b′
)


αA (k ′ )α−1where the second term on the right-hand side, representing the utility of the return fromsaving, indicates that the choice of public pension aﬀects the lifetime utility of the young
through the interest rate. This eﬀect is not included in the model with AK technology.
Following the procedure in Subsection 3.1, we guess that the future public pension is
given by b′ = B0 · A (k ′ )α . Given this guess and the capital market-clearing condition, the()
term pβ ln 1/ (k ′ )1−α is reformulated as
[
]
(
)
1
1−α
pβ ln 1/ (k ′ )= pβ ln.
pb(1 − α) + (1 − γ)(1 − p)α − 1+n· A(k)
α15The expression suggests that a higher pension–GDP ratio, (p/(1 + n)) · (b/A (k)α ), is
associated with lower capital in the next period and thereby a higher interest rate.
The expression shows that longevity has two additional eﬀects on the pension–GDP
ratio, (p/(1 + n)) · (b/A (k)α ), through the terms pβ and (1 − γ)(1 − p)α. First, the termpβ indicates that greater longevity attaches a larger weight to the utility of the interest
rate and thereby works to increase the pension–GDP ratio. This positive eﬀect appears
for both cases. Second, the term (1 − γ)(1 − p)α shows that greater longevity reduces
the unintentional bequests. To compensate for this loss of bequests, the government may
attempt to lower the pension–GDP ratio. This negative eﬀect, peculiar to the case of noannuity market, implies that greater longevity is less likely to increase the pension–GDP
ratio in the absence of the annuity market compared to its presence. This property isqualitatively equivalent to that in the model with AK technology.

4

Concluding Remarks

This study attempted to examine how an aging population aﬀects voting on pension
expenditure, and how this expenditure in turn aﬀects economic growth. To address theseissues, we used an endogenous growth, overlapping-generation model in which pension
expenditure is financed by tax on the working young. The expenditure is determined via
probabilistic voting that captures the intergenerational conflict caused by the three factorsof population aging—a decline in the population growth rate, a rise in life expectancy,
and an increase in the political power of the elderly.
We considered two alternative cases, in which an annuity market is either present
or absent, and showed the following. First, the pension–GDP ratio increases as life expectancy increases in the presence of the annuity market. However, the ratio may show
a hump-shaped pattern in the absence of the annuity market. Second, the growth rate isincreased by a lower population growth rate, less political power of the elderly, and greater
longevity. However, when the longevity and political power of the elderly are examined
together, greater longevity could be associated with a lower growth rate. These results
are consistent with the observed evidence in OECD countries.
To evaluate the role of the annuity market, we compared the two cases, the presence
and absence of the annuity market, in terms of growth and welfare. We showed that (i)the growth rate is higher in the presence of the annuity market than its absence; and (ii)due to this growth eﬀect, future generations are made better oﬀ by the presence of theannuity market, but the current generation cannot benefit from future growth. These
results suggest that the development of annuity markets is beneficial from the viewpointof economic growth, but implies an intergenerational trade-oﬀ in terms of utility.

16

A

ProofsA.1

Proof of Proposition 1

Suppose that public pensions are provided in equilibrium in the next period, b′ > 0.
Assume a linear policy function of public pensions for the next period, B(k ′ ) = Bγ Ak ′ ,where Bγ (> 0) is a constant parameter. Given this assumption and the government
budget constraint, the capital market-clearing condition in Definition 2 becomes


′
pβ
p
B Ak
( γ
) ,
(1 + n)k ′ =· (1 − α)Ak −B(k) + (1 − γ)(1 − p)Rk −γ(1−p)
1 + pβ1+npβ 1 +
Rp

which is rewritten as
k

′

[
]
pβ
p
1+pβ
=
· (1 − α)Ak −B(k) + (1 − γ)(1 − p)Rk .
Bγ A
pβ
1+n
(1 + n) + 1+pβ·γ(1−p)pβ (1+)R(7)

p

Using (7) and the assumption B(k ′ ) = BAk ′ , we write the present value of lifetime
income as
Bγ A
p
) k′
B(k) + (1 − γ)(1 − p)Rk + (γ(1−p)
1+n
1+ p
R


pβ
Bγ A
1+pβ


) ·
= 1+ (Bγ Apβγ(1−p)
(1
+
n)
+
·
1+ p
R
1+pβ pβ (1+ γ(1−p) )R[]pp
× (1 − α)Ak −B(k) + (1 − γ)(1 − p)Rk .
1+n(1 − α)Ak −(8)

Equations (7) and (8) enable us to write the political objective function as follows:(())γ(1 − p)R(1 + n)k + B(k)Ω = ωp ln
1+
p
[
]
p
+ (1 + n) (1 + pβ) ln (1 − α)Ak −B(k) + (1 − γ)(1 − p)Rk ,1+nwhere the terms unrelated to political decisions are omitted from the expression.
The first-order condition with respect to B(k) is given by
B(k) : (1 + γ(1−p)p)

p(1 + n)(1 + pβ) 1+n.
=p(1 − α)Ak − 1+nB(k) + (1 − γ)(1 − p)Rk
R(1 + n)k + B(k)ωpRecalling that R = αA, this expression is reduced toB(k) =()
ω {(1 − α) + (1 − γ)(1 − p)α} − (1 + pβ) 1 + γ(1−p)α(1 + n)p
ωp
(1 + pβ) + 1+n

17

· Ak,

or,

{
B(k) =

Bγ=1 Ak if γ = 1,
Bγ=0 Ak if γ = 0,

where Bγ=1 and Bγ=0 are defined by
Bγ=1 ≡

ω(1 − α) − (1 + pβ) αp (1 + n),ωp
(1 + pβ) + 1+nω {(1 − α) + (1 − p)α} − (1 + pβ)α(1 + n)Bγ=0 ≡.
ωp(1 + pβ) + 1+n
Therefore, we obtain b > 0 if
α(1 + pβ)(1 + n)
when γ = 1,
(1 − α)pα(1 + pβ)(1 + n)Bγ=0 > 0 ⇔ ω >when γ = 0.
(1 − α) + (1 − p)αBγ=1 > 0 ⇔ ω >■A.2

Proof of Proposition 2

Using the policy function B(·) in Proposition 1, the pension–GDP ratio, pNt−1 bt /Yt =(p/(1 + n)) · (b/Ak), is computed as follows:{p· ω−α{ω+(1+pβ)(1+n)/p}
if γ = 1,pb1+n(1+pβ)+ωp/(1+n)·=ω−α{pω+(1+pβ)(1+n)}p
1 + n Ak
· (1+pβ)+ωp/(1+n)
if γ = 0.
1+nThat is,1− α if γ = 1,(1+pβ)(1+n)
p
b
+1
ωp
·
=
1
− αp if γ = 0.
1 + n Ak  (1+pβ)(1+n)+1ωpThis expression shows that the ratio is decreasing in n and increasing in ω. In addition,
the ratio is increasing in p if γ = 1.
To determine the eﬀect of p when γ = 0, we take the first- and second-order diﬀerentiations with respect to p and obtain[]ω(1 + n)
p
b
/∂p =
∂
·
− α,
1 + n Ak γ=0[(1 + pβ)(1 + n) + ωp]2
]
[
b
p
∂2
·
/∂p2 < 0.
1 + n Ak γ=0
The ratio is strictly concave in p with
[
]
ω
p
b
=
∂
·
/∂p
− α,
1 + n Ak γ=0
1+n
p=0
[
]
ω/(1 + n)
p
b
=
∂
·
/∂p
2 − α.
1 + n Ak γ=0
{(1
+
β)
+
ω/(1
+
n)}
p=1
18

The result suggests that there are two threshold values of α, ω/(1 + n) and ω/(1 +n) {(1 + β) + ω/(1 + n)}2 , such that
[
]
p
b
ω
∂
·
/∂p < 0∀p ∈ (0, 1) if≤ α,
1 + n Ak γ=0
1+n
[
]
p
b
ω/(1 + n)∂·/∂p > 0∀p ∈ (0, 1) if≥ α.
1 + n Ak γ=0{(1 + β) + ω/(1 + n)}2
Recall that when γ = 0, b > 0 holds if α < [p + (1 + pβ)(1 + n)/ω]−1 (Proposition 1).
After some manipulation, we find that the following relation holds:1ωω/(1 + n)<.
2 <p + (1 + pβ)(1 + n)/ω
1+n{(1 + β) + ω/(1 + n)}This relation indicates that the threshold value ω/(1+n) is irrelevant for the current case.
Therefore, we can conclude that[]pbω/(1 + n)∂·/∂p > 0∀p ∈ (0, 1) if α ≤,1 + n Ak γ=0{(1 + β) + ω/(1 + n)}2
[
]
]
[
p
b
b
p
∂
·
/∂p
·
/∂p
> 0 and ∂<0
1 + n Ak γ=0
1 + n Ak γ=0
p=0

p=1

if

1
ω/(1 + n).
2 < α <
p + (1 + pβ)(1 + n)/ω{(1 + β) + ω/(1 + n)}■

A.3

Proof of Proposition 3(i) Recall the capital market-clearing condition in Definition 1. With the use of thegovernment budget constraint, τ (1 − α)Ak = pb/(1 + n), the condition is reformulated as
k

′

[
]
pβ
p
1+pβ
· (1 − α)Ak −=B(k) + (1 − γ)(1 − p)αAk .
Bγ
pβ
1+n
(1 + n) + 1+pβ·γ(1−p)pβ (1+)αp

Substituting the policy function derived in Proposition 1, we reformulate the above expression as presented in Proposition 3(i). Direct comparison leads to k ′ /k|γ=1 > k ′ /k|γ=0 .
(ii) The expressions in Proposition 3(i) indicate that∂ k ′ /k|γ=1 /∂ω < 0, ∂ k ′ /k|γ=1 /∂n < 0,
∂ k ′ /k|γ=0 /∂ω < 0, ∂ k ′ /k|γ=0 /∂n < 0.

19

To find the eﬀect of p, we first reformulate the expressions as
[
(
)]−1
1
k′
ω
p+
,
= A · (1 + n) +k γ=1
1 + pβ
βα
(
)]−1
[
k′
ω
1
p+
.
= A · (1 + n) +k γ=1
1 + pβpβα()ω
1The diﬀerentiation of the term 1+pβ p + βα in (9) with respect to p is
{
(
)}
(
)
ω
1
1
1
∂
p+
/∂p =
1 − p(1 − β) −< 0.
1 + pββα
βα(1 + pβ)2()
1
ωThe diﬀerentiation of the term 1+pβ p + pβα in (10) with respect to p is
{
(
)}
(
)
ω
1
1
1
2
∂
p+
/∂p =
1−
−
< 0.
1 + pβpβα
βαp2 αp(1 + pβ)2

(9)
(10)

Therefore, ∂ k ′ /k|γ /∂p > 0 holds for both cases.
■

A.4

Proof of Proposition 4

Let V y |γ=1 and V y |γ=0 denote the lifetime utility functions of the young when γ = 1 and
γ = 0, respectively. For a given k, the functions are
[
]
pβ
p
Bγ=1
1+pβ
y
V |γ=1 = (1 + pβ) ln (1 − α)Ak −Bγ=1 Ak + (1 + pβ) ln·
1+nα/p (1 + n) + pβ · Bγ=1
1+pβ pβα/p1
pβα+ pβ ln· A,
1 + pβ1 + pβ p[]py
V |γ=0 = (1 + pβ) ln (1 − α)Ak −Bγ=0 Ak + (1 − p)αAk +
1+n
+ ln

pβ

(1 + pβ) ln

1
Bγ=0
pβ
1+pβ
·
+
ln
+
pβ
ln
· αA.
Bγ=0
pβ
α
1 + pβ1 + pβ(1 + n) + 1+pβ
· pβα

We first reformulate the first term in each expression as follows:][()t
1 + pβp′Bγ=1 Ak = (1 + pβ) ln
k0 ,(1 + pβ) ln (1 − α)Ak −ωp A k /k|γ=11+n(1 + pβ) + 1+n][()t
1 + pβp′Bγ=0 Ak + (1 − p)αAk = (1 + pβ) ln
A
k
/k|
k0 .
(1 + pβ) ln (1 − α)Ak −
ωp
γ=0
1+n
(1 + pβ) + 1+nSecond, we compare the second term in each expression and obtain
pβ

pβ

Bγ=1
Bγ=0
1+pβ
1+pβ
(1 + pβ) ln
·
≷
(1
+
pβ)
ln
·
Bγ=1
pβ
pβ
γ=0
α/p (1 + n) +α·
(1 + n) + 1+pβ
· Bpβα1+pβ pβα/p⇔ (1 + pβ) ln p ≷ (1 + pβ) ln 1 = 0.
20Given the result thus far, we obtain
)t
)t
(
(
y
y
′
′
V |γ=1 ≷ V |γ=0 ⇔ (1 + pβ) ln k /k|γ=1 + pβ ln(1/p) + (1 + pβ) ln p ≷ (1 + pβ) ln k /k|γ=0
)t
)t
(
(
′
′
⇔ (1 + pβ) ln k /k|γ=1 + ln p ≷ (1 + pβ) ln k /k|γ=0 .
If t = 0, then we obtain V y |γ=1 < V y |γ=0 because ln p < 0. For t ≥ 1, k ′ /k|γ=1 > k ′ /k|γ=0holds, as demonstrated in Proposition 3. Therefore, there is some T (> 1) such thatV y |γ=1 > V y |γ=0 for t ≥ T .
■21

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23pension-GDP ra!o average 2001-2010

popula!on growth and pension-GDP ra!o
12
10

ITA

AUT
PRT
GRC
JPN
FIN
POLBEL
SWE
HUN
SVN
CZE
CHE
ESP
SVKDNKNLD
GBRNOR

8
6
EST

4

FRA
DEU

USA
NZL
CAN LUXAUS TUR
CHL

IRL
ISL
KOR

2

ISR
MEX

0
-2

-1

0

1

2

3

4

popula!on growth rate average 1991-2000(a)pension-GDP ra!o average 2001-2010

life expectancy and pension-GDP ra!o
12
FRA
ITA

10
AUT
PRT
DEU

8
HUN

6
EST

4

POL
CZE
SVK

FIN BEL

SVN

TUR
2
KOR

GRC
JPN
SWE
ESPCHE

USA
DNK
NLD
NZL
GBR NORCAN LUX
AUS
CHL
IRL
ISR
MEX

ISL

0
14

15

16

17

18

19

20

21

males' life expectancy at 60 averagea 1991-2000(b)

Figure 1: Panel (a): The relationship between average pension–GDP ratio, 2001–
2010, and average population growth rate, 1991–2000, for OECD countries. Panel(b): The relationship between average pension–GDP ratio, 2001–2010, and average
male life expectancy at 60 years, 1991–2000, for OECD countries. Source: OECD.Stat(http://stats.oecd.org/).

24

per capita GDP growth average 19912000

popula!on growth and per capita GDP growth
6
5

SVK
EST

4

KOR

POL

3

CHL
TUR

CZE
SVN
HUN

2

SWEGRCISLAUS
FIN
NZL
IRL
LUX
AUT
CHE
BEL
GBR
NLD
CAN
ESP
USA MEX
JPN
NOR
FRA
PRT
DNK
ITA

1
0

ISR
DEU

-1
-2

-1

0

1

2

3

4

popula!on growth average 1991-2000(a)per capita growth average 2001-2010

life expectancy and per capita growth
6
5

SVK
EST

4

POL

3

CZE
HUN

2

KOR
CHL
TUR

SVN
IRL

1
0

AUS
SWE ISL
GRC
FIN
ISR
LUX
AUT NZL
DEU
BEL
GBR
NLD
CAN
ESPCHE JPN
USA MEX
NOR
FRA
PRT
DNK
ITA

-1
14

15

16

17

18

19

20

21

males life expectancy at 60 average 1991-2000(b)

Figure 2: Panel (a): The relationship between average per capita growth, 2001–2010, andaverage population growth rate, 1991–2000, for OECD countries. Panel (b): The relationship between average per capita growth, 2001–2010, and average male life expectancy at
60 years, 1991–2000, for OECD countries. Source: OECD.Stat (http://stats.oecd.org/).

25

Pension/GDP ratios

0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2

gamma=1
gamma=0

-0.25
-0.3
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: Eﬀects of longevity on pension–GDP ratio. Note: we assume each period
lasts 30 years. Parameters are set at α = 0.3, ω = 1.5, A = 8.8, β = (0.99)30 , and1 + n = (1.006)30 .

26

expected utility of the young, from generation 0 to generation 5

2

1.5

1

0.5

0

-0.5

-1
gamma=1
gamma=0

-1.5

-2
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 4: Expected lifetime utility of the young from generation 0 to generation 5. Note:parameters are set at α = 0.3, ω = 1.5, A = 8.8, p = 0.7, β = (0.99)30 , and 1 + n =(1.006)30 . The initial condition is k0 = 1.

27