Doshisha University Life Risk Research Center
Discussion Paper Series No. 2010-04

Live Longer, Work Longer: A Positive Role of
Public Pensions in a Service Economy
Kazutoshi Miyazawa
Faculty of Economics
Doshisha University

Discussion Paper Series

Live longer, work longer: a positive role ofpublic pensions in a service economy
Kazutoshi Miyazawa∗
Doshisha University

Abstract
Population aging may be an opportunity in a sense that it creates a
demand for less durable goods (services) and that it creates a job for skilled
older workers in the service sector. In this changing demographic andeconomic environment, we show that public pensions eﬀectively achieve asocial optimum by reallocating unskilled labor from a manufacturing to
the service sector.

JEL Classification: H55, J14, O41
Keywords: Population aging, Public pensions∗ Faculty of Economics,
kazu@mail.doshisha.ac.jp

Doshisha University,

Kamigyo,

Kyoto 602-8580 Japan.

1Introduction

The purpose of the paper is to give an answer to the question of whether aging
society is opportunity or challenge (Cutler et al. (1990)). Some people may
have a negative opinion in that an increase in the dependency ratio imposes
financial burdens on young workers, which discourages work incentives. Others
may believe that an increase in the political power of older people makes public
expenditures in favor of older rather than younger people. Such discussions
depend on whether older people are economically inactive or not. In fact many
OECD countries have experienced a rise in the proportion of older people aged
50-64 who are working since the mid-1990s (OECD (2006)).1 If older people
continue to work in the labor market, then they contribute to an expansion of
production possibility set, which not only improves their own welfare but also
may improve the welfare of younger people.
We interpret population aging as a continuous decline in an adult mortality
rate in a two-period overlapping generations model. It is assumed that one
period is 40 years, and that each individual in each generation enters the model
economy when he becomes 20 years old. Thus, supposed life expectancy at birthvaries from 60 years to 100 years.2
In our model population aging aﬀects economic outcomes through threechannels. First, it encourages capital accumulation because young people savemore than before for their prolonged second period of life. Since capital deepening attracts labor from a labor intensive sector to a capital intensive sector,
the capital deepening eﬀect changes the allocation of labor. Second, population
aging changes demand components. Older people would like to consume lessdurable goods (say, services) because their longevity is shorter than younger
people. The increased demand for services attracts the demand for labor inservice production sectors. Assuming that the service sectors are more labor
intensive than good production sectors, the demand eﬀect reallocates labor in
the opposite direction of the capital deepening eﬀect.
Finally, and most importantly, population aging increases an opportunity
for older people to work in the emerging service sectors. One of the reasons isthat older people have an advantage over younger counterparts in management
jobs because they have experienced longer time in the workplace and accumulated skills for management. Second, some kind of service sectors may well be
more flexible and less physically demanding than manufacturing sectors, which
improves employability for older workers. We assume that old-age labor andyoung-age labor are not substitutes but complements.3 In this setting, an increase in the employment of old-age labor attracts young-age labor from good
1 There are large diﬀerences between OECD countries in both the level of and the increasein the labor force participation rates among older people. In 2004, the level varies from over
80 per cent (Iceland) to 40 per cent (Turkey). In a decade from 1994 to 2004, Netherlands,
New Zealand, and Belgium have experienced over 10 percentage point increase, but the rateis almost constant in Japan, Korea, and Austria (ibid, p.29).
2 There are also large diﬀerences between OECD countries in life expectancy at birth. In
2000, it varies from 81.2 (Japan) to 68.1 (Turkey). The avearge among OECD countries is77.0. In 2050, it is expected that the average is 83.4 and that the diﬀerence shrinks (from 87.7
(France) to 78.5 (Turkey)) (ibid. p.18).
3 OECD (2006) suggests a countercharge against the lump of labor fallacy, which implies
there are a fixed number of jobs to go around and workers are perfectly substitutable for eachother. Across OECD countries, there is a positive rather than negative correlation betweenchanges in employment rates for younger and older people (Figure 7.1, p.140).

1

production sectors to service sectors. Thus, the labor supply eﬀect enhances thedemand eﬀect mentioned above.
We introduce a pay-as-you-go public pension into the model economy, andanalyze whether the public pension is a useful device to achieve a social optimum. We show (1) when the average longevity exceeds a threshold (about 63
years old), the public pension eﬀectively achieves a socially optimal allocation
by reallocating young-age (unskilled) labor from manufacturing to service sectors, and that (2) the optimal contribution rate increases almost linearly withaverage longevity. For example, it is about 8 per cent when the longevity is77.0, and about 20 per cent when the longevity is 83.4.
Our model is relevant to some economic and policy debates. First, ourmodel explains the Petty-Clark’s law in a demographic point of view. In termsof employment as well as value-added, the share of service sector increases in aprocess of population aging. Acemoglu and Guerrieri (2008) presents an alternative model to explain the scenario. Assuming a complementarity between acapital intensive sector and a labor intensive sector and diﬀerent rates of technical progress between sectors, it shows both capital and labor move from theformer to the latter sector in a process of capital deepening. The reason is simple. Suppose that the rate of technical progress in the capital intensive sector
is higher than the labor intensive sector. Since two goods are complements, anaggregate steady growth requires more resources in the labor intensive sector.
We admit our model also depends on the complementarity, but the mechanism
is fairly diﬀerent. We assume a complementarity between old-age skilled labor
and young-age unskilled labor. Further our result fits more for the reallocationof labor.4
Second, our model gives some insights into the field of research on service
economy. In a growth model in which a good is produced by capital and labor,a service is produced by labor only, and the preference to service increases withage, van Groezen et al. (2005) shows that population aging enhances economic
growth in a closed economy if the elasticity of substitution between capital andlabor in the production sector is less than one. The fundamental mechanism issimilar to Acemoglu and Guerrieri (2008). The paper assumes the service sector
employs young-age labor only, so that the labor supply eﬀect in our model is
not examined. The paper incorporates a pay-as-you-go public pension scheme,
but the optimality is not discussed. In a similar model, van Groezen et al.
(2007) shows that the golden rule capital stock in a two-sector model is lower
than in a one-sector model. The paper gives a numerical example that, whenthe elasticity of substitution between goods and services is relatively small, themarket economy is dynamically ineﬃcient for a large set of parameter values(Fig.1, p.157). It suggests the rate of contribution to public pensions that
achieves the golden rule is positive, and that an increase in the dependency
ratio increases the contribution rate. However, the rates indicated in the paper
are fairly small (Table 2, p.162). Further the golden rule criterion is necessary
but not suﬃcient for optimality considerations. To discuss the optimality, weneed examine whether the allocation of labor and consumption are also optimal.
Our model shows that there exists a unique optimal contribution rate in somecircumstances.
4 Acemoglu and Guerrieri (2008) indicates the ratio of employment in the labor intensive
sector to that in the capital intensive sector is 1.03 in 1948 and 0.80 in 2005. The decline isabout 33 per cent. The basic model in the paper explains only 5 per cent (p.491).

2

Third, our model presents income components for older people on an optimality basis.5 It predicts that the share of labor income is 20 per cent regardlessof longevity, and that pension benefits crowd out capital income as the average
longevity increases. For example, the share of pension benefits is 20 per centwhen the longevity is 77.0, and about 37 per cent when the longevity is 83.4.
Finally, some argue that the government should reform the existing public
pension scheme because it discourages work incentives for old-age workers andinduces early retirement. In an extended model which incorporates a retirement
decision into the basic model, we show the labor participation rate of old-age
workers indeed decreases with longevity and the contribution rate. Our model
suggests that the rate of suspension related to work at an older age should belowered gradually in a process of population aging.
The structure of the paper is as follows. In section 2 we introduce a basicmodel. In section 3 we first derive the equilibrium and then examine whether
and to what extent pay-as-you-go public pensions are eﬀective in achieving asocially optimal allocation. In section 4 we extends the basic model to deal
with early retirement and employability for older workers. The final section
concludes the paper.

2

The basic model

2.1

Setup

We use a two-period overlapping generations model with an adult mortality risk.
In each period newly born individuals enter the economy. The population size isnormalized to unity. They live for two period at most. They face a mortality risk
when they enter the second period of life. The survival probability is denoted byθ ∈ (0, 1]. In the first period of life, each individual supplies one unit of unskilledlabor either in the good production sector or in the service production sector
to earn wage income. He consumes a (durable) good, purchases actuariallyfair private annuities, and contributes to a pay-as-you-go public pension. If hesurvives into the second period, he works as a manager in the service sector toearn management income. He also receives the return of private annuities andthe public pension benefit to consume the service provided. Our model contrasts
with usual two-period overlapping generations models in that individuals thatsurvive into the second period are economically active. They have not only achance to enjoy consuming services but also a chance to get money as managers
in their later stage of life.
The utility of an individual who is born at period t is specified by
ut = ln ct + γθ ln dt+1

(1)

where ct and dt+1 stand for the good consumption in the first period and theservice consumption in the second period, respectively. 0 < γ ≤ 1 is a private
discount factor.
5 OECD (2009) shows the share of public transfers, labor income, and capital income insome OECD countries. The share of public transfers is higher in France (85%), Belgium(81%), and Austria (80%). The share of labor income is higher in Korea (59%), Japan (44%),
and United States (34%).

3

The budget constraints are(1 − τ )wt
Rt+1 st + π t+1 + Pt+1

= ct + st= qt+1 dt+1

(2)
(3)

Equation (2) stands for the budget constraint in the first period. wt is a
wage rate and τ ∈ [0, 1) is a rate of contribution to the public pension. st
stands for the purchase of private annuities. Equation (3) stands for the budget
constraint in the second period. Rt+1 is a rate of return of private annuities
and Pt+1 is a public pension benefit. π t+1 stands for the management income,and qt+1 stands for a price of the service.6
Each individual chooses ct , dt+1 , and st to maximize equation (1) subject toequations (2) and (3). Solving the problem, we have
ct

=

dt+1

=

st

=

1
It
1 + γθ
γθ Rt+1 It
1 + γθ qt+1γθ
1 πt+1 + Pt+1
(1 − τ )wt −1 + γθ
1 + γθ
Rt+1

(4)
(5)
(6)

where It stands for lifetime full income,
It = (1 − τ )wt +π t+1 + Pt+1
Rt+1

From equation (6), the purchase of private annuities is positive whenγθ(1 − τ )wt >π t+1 + Pt+1
Rt+1

(7)

Hereafter, we assume equation (7) is satisfied7 .
In the good production sector, competitive firms employ capital and unskilled labor to produce a homogeneous good. The technology is specified by a
Cobb-Douglas production function,1−αy1t = f (kt , l1t ) = Aktα l1t(8)where y1t , kt , and l1t stand for the output, capital, and unskilled labor employedin the good sector, respectively. 0 < α < 1 is a share of capital income, and
A > 0 is a productivity parameter.
Competition in the factor market requireswt
1 + rt−α= (1 − α)Aktα l1t
1−α= αAktα−1 l1t

(9)
(10)

We assume capital is fully depreciated in one period.
6 We assume all individuals that survive into the second period of life continue to work as

managers. The problem of early retirement is discussed in Section 4.1.
7 We verify it is true in equilibrium below.

4In the service production sector, competitive firms employ managers andunskilled labor to produce a homogeneous service8 . The technology is specified
by a Cobb-Douglas production function,1−βy2t = g(ht , l2t ) = Bhβt l2t(11)

where y2t , ht , and l2t stand for the output, the number of managers, and unskilled labor employed in the service sector, respectively. 0 < β < 1 is a shareof management income, and B > 0 is a productivity parameter.
Competition in the factor market requires
wt
πt

−β
= qt (1 − β)Bhβt l2t=1−β
qt βBhβ−1
l2t
t

(12)
(13)

where qt is a price of the service, and π t is the management income.
In our model, there are 6 markets, that is, markets of unskilled labor, managers, capital, private annuities, goods and services. The following equations
stand for the corresponding market clearing conditions.
l1t + l2t
ht
kt+1
Rt
y1t
y2t

= 1
= θ
= st
1 + rt=θ= ct + kt+1
= θdt

(14)
(15)
(16)
(17)
(18)
(19)

Finally, the balanced budget condition for the public pension is given byτ wt = θPt(20)

The Walras law tells us one of the conditions is redundant. It can be shownthat both the good market and the service market clear if and only if9(1 + rt )kt = wt l2t − τ wt

2.2

(21)

Equilibrium

In this section we derive the equilibrium and examine the characteristics of key
variables.
Since the income share is constant, the capital income is proportional to theαlabor income: (1 + rt )kt = 1−αwt l1t . Substituting this into equation (21), we
have
α
l2t =
l1t + τ1−α8 It may be possible that managers are employed in the good production sector. We discuss
it in Section 4.2.
9 Substituting equations (2) and (16) into equation (18), we have y
1t = (1 − τ )wt . Fromequation (14), the capital income is given by (1 + rt )kt = (1 − τ )wt − wt l1t = wt l2t − τ wt .
Substituting equation (3) into equation (19), and using equations (12), (13), (17), and (20),
we have the same equation.

5

which implies the public pension moves unskilled labor from the good sector
to the service sector. From the labor market condition (14), the allocation of
unskilled labor is given byl1l2= (1 − α)(1 − τ )= α(1 − τ ) + τ

(22)
(23)

From equations (9), (10), and (22), we know
wt+1
kt+1
=
1 + rt+1α(1 − τ )(24)

From equations (17), (20), and (24), the present value of the pension benefit
is given by
τ
Pt+1
=
(25)
kt+1
Rt+1
α(1 − τ )From equations (12), (13), (17), and (24), the present value of the management income is given by
∙
¸
β
τ
π t+1
=
(26)
1+
kt+1
Rt+1
1−β
α(1 − τ )Substituting equations (25) and (26) into equation (6), and using equation(16), we haveαγθ(1 − τ )wt³´
kt+1 =(27)1ταγθ + 1−β
α + 1−τ

which regulates the law of motion of kt (with equation (9)).
Since y1t = (1 − τ )wt , equation (27) tells us the investment ratio is constantover time,
αγθ
kt+1
³
´
=
(28)
1
y1t
αγθ +
α+ τ
1−β

1−τ

The dynamics is simple. Given that τ and θ are constant over time, the allocation of unskilled labor is constant (equations (22) and (23)). The investment
ratio is also constant (equation (28)). Given an initial capital stock, the capitalstock converges monotonically to a unique and stable steady state (equations(9) and (27)). In a process of capital accumulation, the wage rate increases,
which induces both the price of the service, qt , and the management income,
π t , to increase (equations (12) and (13)).

Proposition 1 The share of capital income, pension benefits, and labor incomeof old-age workers are respectively given by
Rt st−1
qt dt
Pt
qt dt
πt
qt dt

=
=

(1 − β)α(1 − τ )α(1 − τ ) + τ(1 − β)τ
α(1 − τ ) + τ= β

The income shares are independent of the survival rate, θ. The capital shareis decreasing in τ , and the labor share is independent of τ .

6

Proof. Straightforward from equations (3), (25), and (26).
At the end of this section, we briefly prove the interior condition (7) issatisfied in equilibrium. From equations (25), (26), and (27), we know⎡⎤α(1 + γθ)π t+1 + Pt+1 ⎣³´ ⎦ γθ(1 − τ )wt
= 1−
Rt+1
αγθ + 1 α + τ
1−β

1−τ

which is smaller than γθ(1 − τ )wt for ∀τ ∈ [0, 1) and ∀θ ∈ (0, 1].

3

An optimal pension scheme

In the previous section we treat the rate of contribution to the public pension,τ , as if it is independent of demographic factors. It seems more insightful torelate the pension scheme to a demographic factor in a meaning way.
Appendix A shows the optimal employment in the good sector is given byl1∗ =ρ(1 − α)ρ(1 − α) + (1 − ρα)(1 − β)γθand the optimal capital-output ratio is given by
¶∗
µ
kt+1
= ρα
y1t

(29)

(30)

where 0 < ρ < 1 is a social discount factor. An asterisk is used for the optimalvariables.
Comparing equations (22) and (29), we have a contribution rate that achieves
the optimal allocation of unskilled labor. Comparing equations (28) and (30),
we have a contribution rate that achieves the optimal resource allocation. If
the both rates coincide with each other, the public pension could achieve theoptimal allocation. The following proposition tells us it is the case.
Proposition 2 The decentralized economy is optimal if and only if the rate ofcontribution to the public pension isτ∗ =(1 − ρα)(1 − β)γθ − ραρ(1 − α) + (1 − ρα)(1 − β)γθ(31)

Proof. Straight calculation shows, under the contribution rate τ ∗ in equation(31), equation (22) is equal to equation (29), and equation (28) is equal toequation (30).
The reason is simple. Let us define the industry ratio by
IRt =

7

qt y2ty1t

In our model, the industry ratio in the competitive equilibrium is constantover time,
µ
¶
1
τ
IR =
α+
1−β
1−τ
Appendix A shows the optimal industry ratio is given byIR∗ =(1 − ρα)γθρ

It can be easily shown that IR = IR∗ is equivalent to l1 = l1∗ and that theEuler equation,
ct+1= ρ(1 + rt+1 )ct
is satisfied if IR = IR∗ . Therefore, the adjustment of the industry ratio makes
both intratemporal and intertemporal allocation optimal in our model.
From equation (31), we know τ ∗ ≥ 0 ⇔ θ ≥ θ̂ where
ρα
θ̂ =
(32)
γ(1 − ρα)(1 − β)A pay-as-you-go public pension eﬀectively achieves the optimal allocation if
the survival probability exceeds the threshold in equation (32). The reason is
simple. Equation (29) tells us, for a larger θ, the optimal employment in thegood sector is smaller because a larger θ makes the optimal size of the service
sector larger. Public pensions serve the purpose by reallocating unskilled laborfrom the good sector to the service sector. Roughly speaking, public pensions
assist industrial restructuring induced by population aging.
From equation (32), we know θ̂ is increasing in ρ, α, and β, and decreasing in
γ. In our model public pensions are more likely to be eﬀective when θ̂ is smaller.
It is the case when society attaches more importance to current generations
rather than future generations, the income share of unskilled labor is larger, orindividuals are more patient. The following proposition summarizes the results.
Proposition 3 In an aging economy (θ ≥ θ̂), pay-as-you-go public pensions
eﬀectively achieve an optimal allocation. The threshold survival rate θ̂ is lower
when the social discount factor is smaller, when the income share of unskilledlabor is larger, or when the private discount factor is larger.
[Figure 1 is here]
Let us provide a numerical example. Suppose that individuals start theireconomic activity at 20 years old, and that one period is 40 years. Assumingthat an annual rate of time preference is one per cent, we have γ = 1.01−40 =0.67. The income shares are α = 0.3 and β = 0.2, which implies the service
sector uses unskilled labor more intensively than the good production sector.
The social discount factor is ρ = 0.5. Equation (32) gives θ̂ = 0.33, which
implies the average longevity is 20 + 40 ∗ 1.33 = 73.2. Figure 1 illustrates the
optimal contribution rate which is given by equation (31). The horizontal axis
is measured by the average longevity instead of θ. We know the optimal rateincreases almost linearly and reaches to 38 per cent when the longevity is 100.
The eﬀect of public pension is fairly sizable.

8[Figure 2, 3, and 4 are here]
Figure 2 illustrates the optimal allocation of unskilled labor which is given
by equations (29) and (31). At a starting point where the average longevity is73.2, the labor share in the good sector is 70 per cent. In a process of population
aging, it decreases gradually to the bottom of 45 per cent when the longevity
is 100. Our model generates a fairly sizable reallocation of labor compared to
Acemoglu and Guerrieri (2008).
Figure 3 illustrates the industry share in terms of value-added. When the
average longevity is 73.2, we have qt y2t /y1t = α/(1 − β) = 0.375. Thus, theshare of good production sector is y1t /(y1t + qt y2t ) = 0.73. The share decreases
gradually to the bottom of 48 per cent when the longevity is 100. Our model
suggests demographic changes and the related public policy account for the
Petty-Clark’s law in terms of both employment and value-added.
Figure 4 illustrates the optimal income components of old-age workers which
is given by equations in Proposition 1 and 2. Without public pensions, theincome share of labor is β = 0.2 and that of capital is 1 − β = 0.8. The labor
share is independent of population aging. In a process of population aging,
however, public pension benefits crowd-out capital income. The pension sharereaches to 50 per cent when the average longevity is 95.

4

Discussions

4.1

A decision on retirement

In the basic model, we assumed that every survivor works in the service sector.
It seems restrictive because someone may well choose early retirement. In thissubsection, we introduce an opportunity to retire into the basic model in a
simplest way.
Suppose that a fraction σt ∈ (0, 1] of individuals born at period t would liketo work in the service sector if they survive into the second period of life andthat a fraction (1 − σ t ) of individuals would not.
The budget constraints for a candidate to be a old-age worker are(1 − τ )wt
Rt+1 st + π t+1 + (1 − η)Pt+1= ct + st= qt+1 dt+1

where 0 ≤ η ≤ 1 stands for a suspension rate for old-age workers.
Utility maximization gives
cet

=

det+1

=

set

=

Ite
1 + γθ
γθ Rt+1 Ite
1 + γθ qt+1γθ
1 π t+1 + (1 − η)Pt+1(1 − τ )wt −1 + γθ
1 + γθ
Rt+1

where Ite stands for lifetime full income,Ite = (1 − τ )wt +πt+1 + (1 − η)Pt+1
Rt+1
9

A superscript e denotes the variables of old-age workers.
The budget constrains for a candidate to be a retiree are(1 − τ )wt
Rt+1 st + Pt+1

= ct + st= qt+1 dt+1

Utility maximization gives
cut

=

dut+1

=

sut

=

Itu
1 + γθ
γθ Rt+1 Itu
1 + γθ qt+1γθ
1 Pt+1(1 − τ )wt −1 + γθ
1 + γθ Rt+1

where

Pt+1
Rt+1
A superscript u denotes the variables of retirees.
If Ite > Itu , everyone would like to work. Otherwise everyone would liketo retire. In equilibrium, the fraction σ t is endogenously determined where
Ite = Itu , i.e.,
π t+1 = ηPt+1(33)
Itu = (1 − τ )wt +Aggregation givesct= σ t cet + (1 − σ t )cut¸∙
1
σ t πt+1 + (1 − ησ t )Pt+1=(1 − τ )wt +1 + γθ
Rt+1

dt+1= σ t det+1 + (1 − σ t )dut+1∙¸
γθ Rt+1
σ t π t+1 + (1 − ησ t )Pt+1=(1 − τ )wt +1 + γθ qt+1
Rt+1

st

= σ t set + (1 − σ t )sutγθ1 σ t πt+1 + (1 − ησt )Pt+1=(1 − τ )wt −1 + γθ
1 + γθ
Rt+1(34)

Conditions for the market of managers and the budget of public pension are
modified as follows,ht
τ wt= θσt−1= (1 − ησ t−1 )θPt

(35)
(36)

The allocation of unskilled labor is given by equations (22) and (23).
In the same way as the basic model, the present value of pension benefits
and the present value of management income are respectively given by
Pt+1
Rt+1
πt+1
Rt+1

=
=

kt+1
τ
α(1 − τ ) 1 − ησ t
∙
¸
β
τ
kt+1
1+
1−β
α(1 − τ ) σt
10

(37)
(38)

Substituting equations (37) and (38) into equation (34), and using equation(16), we have
kt+1
αγθ
³
´
=
1
yt
αγθ +
α+ τ
1−β

1−τ

which is the same as equation (28).
Finally, substituting equations (37) and (38) into equation (33), we haveσt =1 βα + β(1 − α)τ
η βα + (1 − βα)τ(39)

for ∀t. Equation (39) shows the fraction of old-age workers is decreasing in τ andη, as expected. The pension scheme (τ , η) is restricted from the lower boundin order to ensure the interior condition (0 < σ t ≤ 1), that is, the contribution
rate and the suspension rate have to satisfyητβ(1 − α)1 − βα(1 − η)βα≥ τ≡η(1 − βα) − β(1 − α)> η≡When α = 0.3 and β = 0.2, we have η = 0.15. Assuming that the suspension
rate is 50 per cent, we have τ = 0.09. The opportunity of early retirement rises
the threshold longevity that a public pension becomes eﬀective.
[Figure 5 is here]
Figure 5 illustrates the rate of labor force participation of old-age workers
when the contribution rate is set to be optimal (equation (31)). The suspension
rate is set to be η = 0.5. The model works well when the average longevity
exceeds 77.5 years old. Our model suggests the suspension rate should be lower
when longevity is higher because a higher contribution rate induces more old-age
workers to leave labor market.

4.2

A chance to work in the good sector

In the basic model we assumed that old-age workers are employed only in theservice sector. It may be restrictive because some kind of management abilities would be useful in the good production sector. It can be shown that thisgeneralization does not change the main result qualitatively.
Let us assume that the production functions in the good and the service
sector are given byy1t

H 1−αK −αH= f (kt , h1t , l1t ) = AktαK hα1t l1ty2t1−β= g(h2t , l2t ) = Bhβ2t l2twhere h1t and h2t stand for the employment of skilled labor in the good andthe service sector, respectively. αK and αH are the income share of capital andskilled labor in the good sector.
11

The factor prices are given bywt= (1 − αK − αH )

πt

= αH

1 + rty1t
y2t= qt (1 − β)
l1t
l2t

y2t
y1t
= qt β
h1t
h2t
y1t
= αK
kt

The industry ratio is given byIRt =(1 − αK − αH )l2t
αH h2t=(1 − β)l1t
βh1t

(40)

Equation (40) implies the allocation of skilled labor is determined by theallocation of unskilled labor in order to adjust the marginal productivity between
the good sector and the service sector.
The market of skilled labor is modified as
h1t + h2t = θ(41)

In the same way as the basic model, the allocation of unskilled labor is givenby
l1t = (1 − αK − αH )(1 − τ )(42)and l2t = 1 − l1t . Substituting equations (41) and (42) into equation (40), we
have
α θ
³ H
´
h1t =
(43)
β
τ
αH + 1−β αK + αH + 1−τwhich is decreasing in τ . Public pensions reallocate skilled labor from the good
sector to the service sector.
Finally, we derive the investment ratio as follows. The present value ofpension benefits and that of management income are given by
Pt+1
Rt+1
π t+1
Rt+1

1 − αK − αH kt+1
αK
l1t+1
αH θ
kt+1
αK h1t+1= τ=

Substituting them into the capital market clearing condition, and using equations (42) and (43), we have
α γθ
kt+1
³K
´
=
1
τ
y1t
αK γθ + 1−βαK + αH + 1−τ

Appendix B shows the optimal solutions are
µ
¶∗
kt+1
= ραK
y1t
1
l1∗ =
1 + 1−α1−β
IR∗
K −αH
h∗1

=

θ

1 + αβH IR∗
12

(44)

(45)
(46)
(47)

where the industry ratio is given byIR∗ =(1 − ραK )γθρ(48)Comparing equations (44), (42), and (43) with equations (45), (46), and (47),respectively, we know there exists a unique contribution rate which achieves the
optimal allocation. Specifically, we have the following proposition.
Proposition 4 The optimal contribution rate is given byτ∗ =(1 − ραK )(1 − β)γθ − ρ(αK + αH )ρ(1 − αH − αK ) + (1 − αK )(1 − β)γθ(49)

which is increasing in θ. Public pensions eﬀectively achieve the social optimum
when θ ≥ θ̃ whereρ(αK + αH )
(50)
θ̃ =
(1 − ραK )(1 − β)γθ̃ is increasing in ρ, αK , αH , and β, and decreasing in γ.

5Conclusions

In this paper we construct a theoretical model to examine whether and to what
extent pay-as-you-go public pensions achieve a socially optimal allocation. In
contrast to the existing literature we assume older people are economically active. They demand services more than younger people, and supply their labor
resource in the emerging service sector. In a process of population aging, it issocially desirable to reallocate young-age unskilled labor from manufacturing toservice sectors. We show that public pensions serve the purpose and that thepositive role of public pensions increases with population aging.
The title of OECD (2006) cited in Introduction is “Live longer, work longer.”
The foreword reads as below. We hope our research gives an answer to at least
one of the questions.
‘Employment and social policies and practices that discourage work at anolder age eﬀectively deny older workers choice in when and how they retire.
Moreover, in an era of rapid population ageing, they result in a waste of valuable
resources that business, the economy and society can ill-aﬀord. This has to stop.
Policy reforms are needed to reverse the trend towards ever-earlier retirement.
But what can governments do to bring about changes that are often unpopularwith many voters? How can workers, employers and governments work together
to guide our ageing society to a prosperous future?’13

Acknowledgments
The author would like to thank Akira Yakita, Tatsuya Omori, Hikaru Ogawa,
Ken Tabata and seminar participants at Mie Chukyo University, Nagoya University, and Kwansei-gakuin University for useful comments. Financial support
from the Grant-in-Aid for Scientific Research of JSPS (No.19530163) and theJapan Economic Research Foundation is greatly acknowledged. The remaining
errors are of course mine.

References[1] Acemoglu, Daron, and Veronica Guerrieri (2008) Capital deepening andnonbalanced economic growth, Journal of Political Economy 116, 467-498.
[2] Cutler, David M., Poterba, James M., Sheiner, Louise M., and Lawrence
H. Summers (1990) An aging society: opportunity or challenge?, Brookings
Papers on Economic Activity 1990-1, 1-73.
[3] van Groezen, Bas, Meijdam, Lex, and Harrie A. A. Verbon (2005) Serving
the old: ageing and economic growth, Oxford Economic Papers 57, 647-663.
[4] van Groezen, Bas, Meijdam, Lex, and Harrie A. A. Verbon (2007b) The casefor pay-as-you-go pensions in a service economy, Scottish Journal of Political
Economy 54, 151-165.
[5] OECD (2006) Live longer, work longer, OECD, Paris.
[6] OECD (2009) Pensions at a Glance 2009, OECD, Paris.

14

Appendix A [Social optimum]
The social optimum is characterized by solving the following problem
max

∞
X

ρt [uy (cyt , dyt ) + γθuo (cot+1 , dot+1 )]t=0

subject to
y1t
y2t
l1t + l2t
ht

=
=
=
=

cyt + θcot + kt+1
dyt + θdot
1
θ

(A1)
(A2)
(A3)

where
uy (cyt , dyt ) = (1 − μ) ln cyt + μ ln dyt
uo (cot+1 , dot+1 ) = (1 − λ) ln cot+1 + λ ln dot+1y1t

1−α= Aktα l1t

y2t

1−β
= Bhβt l2t

and 0 < ρ < 1 stands for a social discount factor. μ and λ stand for the relative
preference to service consumption when young and when old, respectively. We
assume λ ≥ μ, i.e., the preference to service consumption increases with age. In
a case where λ = 1 and μ = 0, the structure is reduced to the basic model.
Let us set up the Lagrangian,
L0

=

∞
X
t=0

ρt [uy (cyt , dyt ) + γθuo (cot+1 , dot+1 ) + π1t (y1t − cyt − θcot − kt+1 )+π 2t (y2t − dyt − θdot ) + π3t (1 − l1t − l2t )]where π 1t , π 2t , and π 3t stand for multipliers attached to equations (A1), (A2),and (A3), respectively.
The first-order conditions require∂uy− π 1t∂cyt∂uy− π 2t∂dyt∂uo− ρπ 1t+1 θ∂cot+1∂uoγθ o − ρπ 2t+1 θ∂dt+1∂y1t+1−π 1t + ρπ 1t+1
∂kt+1
∂y1t
−π 3t + π1t
∂l1t
∂y2t
−π 3t + π2t
∂l2t
γθ

15

= 0

(A4)

= 0

(A5)

= 0

(A6)

= 0

(A7)

= 0

(A8)

= 0

(A9)

= 0

(A10)

and the transversality condition is limt→∞ ρt π1t kt+1 = 0.
From equations (A9) and (A10),π 3t l1t
π 3t l2t= (1 − α)π 1t y1t= (1 − β)π2t y2t

(A11)
(A12)

Substituting equations (A11) and (A12) into equation (A3), we have theoptimal allocation of unskilled labor,
∗
l1t
=

1
1−β
1 + 1−αIRt(A13)where IRt stands for the industry ratio of the service sector to the good sector,
IRt =π2t y2tπ1t y1t

Substituting equations (A5) and (A7) into equation (A2), we know, for ∀t,
π 2t y2t = μ + λ

γθ
ρ

(A14)

Substituting equations (A4) and (A6) into equations (A1), we have
π 1t y1t = 1 − μ + (1 − λ)γθ+ π 1t kt+1
ρ

(A15)

Equation (A8) gives
π 1t kt+1 = ραπ1t+1 y1t+1(A16)

From equations (A15) and (A16), we have a first-order diﬀerence equationof λ1t y1t ,γθ
π 1t y1t = 1 − μ + (1 − λ)+ ραπ 1t+1 y1t+1
ρFrom the transversality condition, we know, for ∀t,∙¸
1γθπ 1t y1t =1 − μ + (1 − λ)1 − ραρ(A17)Equations (A14) and (A17) tell us the industry ratio is constant over time,´³(1 − ρα) μ + λ γθ
ρ
(A18)
IR∗ =
1 − μ + (1 − λ) γθρSubstituting equation (A18) into equation (A13), we haveih(1 − α) 1 − μ + (1 − λ) γθ
ρ
h
i
³
´
l1∗ =
γθ
(1 − α) 1 − μ + (1 − λ) γθ
+
(1
−
β)(1
−
ρα)
μ
+
λ
ρ
ρ

(A19)

Assuming that λ = 1 and μ = 0, equation (A19) is equal to equation (29).
16

Equation (A14) gives the allocation of services,
dyt
y2t

=

θdot
y2t

=

μ
μ + λ γθρλ γθρμ + λ γθρEquation (A17) gives the allocation of goods,
cyt
y1t

=

(1 − ρα)(1 − μ)
1 − μ + (1 − λ) γθρ(1 − ρα)(1 − λ) γθ
ρ

θcot
y1t

=

kt+1
y1t

= ρα

1 − μ + (1 − λ) γθ
ρ

(A20)

Assuming λ = 1 and μ = 0, we have dyt = cot = 0, which corresponds to thebasic model. Equation (A20) is equation (30) in the main body.
Let us denote the preference distance byε=λ−μ≥0
Eliminating μ in equation (A18) by equation (A20), we have
´
i
h ³
−ε
(1 − ρα) λ 1 + γθ
ρ
³
´
IR∗ =
(1 − λ) 1 + γθ
+ε
ρ

(A21)

When ε = 0, i.e., the preference of individuals does not change, the industry
ratio is independent of θ,(1 − ρα)λIR∗ =1−λWhen ε > 0, i.e., the preference to service consumption increases with age,the industry ratio is increasing in θ.

17Appendix B [Social optimum in Section 4.2]
The social optimum is characterized by solving the following problem
max

∞
X

ρt (ln ct + γθ ln dt+1 )

t=0

subject to
y1t
y2t
l1t + l2t
h1t + h2t====

ct + kt+1
θdt
1
θ

(B1)
(B2)
(B3)
(B4)

where
y1t

H 1−αK −αH= AktαK hα1t l1ty2t1−β= Bhβ2t l2t

Let us set up the Lagrangian,
L0

=

∞
X
t=0

ρt [ln ct + γθ ln dt+1 + λ1t (y1t − ct − kt+1 )+λ2t (y2t − θdot ) + λLt (1 − l1t − l2t ) + λHt (θ − h1t − h2t )]where λ1t , λ2t , λLt , and λHt stand for multipliers attached to equations (B1),(B2), (B3), and (B4), respectively.
The first-order conditions require
1
− λ1t
ct
γθ
− ρλ2t+1 θ
dt+1∂y1t+1−λ1t + ρλ1t+1
∂kt+1
∂y1t
−λLt + λ1t
∂l1t
∂y2t
−λLt + λ2t
∂l2t
∂y1t
−λHt + λ1t
∂h1t
∂y2t
−λHt + λ2t
∂h2t

= 0

(B5)

= 0

(B6)

= 0

(B7)

= 0

(B8)

= 0

(B9)

= 0

(B10)

= 0

(B11)

and the transversality condition is limt→∞ ρt λ1t kt+1 = 0.
From equations (B8) and (B9),
λLt l1t
λLt l2t= (1 − αK − αH )λ1t y1t= (1 − β)λ2t y2t18

Substituting them into equation (B3), we have the optimal allocation of
unskilled labor,
1
∗
l1t
=
(B12)
1−β
1 + 1−αK −αH IRt
where IRt stands for the industry ratio of the service sector to the good sector,IRt =λ2t y2t
λ1t y1t(B13)

Equation (B12) is equation (46) in the main body.
From equations (B10) and (B11),
λHt h1t
λHt h2t= αH λ1t y1t= βλ2t y2t

Substituting them into equation (B4), we have the optimal allocation ofskilled labor,θh∗1t =
1 + αβH IRt
which is equation (47) in the main body.
From equations (B2) and (B6), we have
λ2t y2t =

γθ
ρ

(B14)

Substituting equations (B5) and (B7) into (B1), we have
λ1t y1t = 1 + ραK λ1t+1 y1t+1
Since 0 < ραK < 1, the transversality condition requires, for ∀t,
λ1t y1t =1
1 − ραK(B15)

From equation (B15), we have
kt+1
= ραK
yt
which is equation (45) in the main body.
Finally, substituting equations (B14) and (B15) into equation (B13), we haveIRt =(1 − ραK )γθρwhich is equation (48) in the main body.

19

Figure 1. Optimal contribution rate

tau

0.4

0.3

0.2

0.1

0.0
60

65

70

75

80

85

90

95

100

longevity

Figure 2. Industry share, Unskilled labor

labor

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
74

76

78

80

82

84

86

88

90

92

94

96

98

100

longevity

The employment of young-age workers moves from the good to the service
sector.

Figure 3. Industry share, Value-added

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
74

76

78

80

82

84

86

88

90

92

94

96

98

100

longevity

The share of value-added in the good production sector decreases withpopulation aging.

Figure 4. Income components of old-age workers

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
74

76

78

80

82

84

86

88

90

92

94

96

98

100

longevity

The share of labor income, capital income, and pension benefits
from top to bottom.

Figure 5. Participation rate of old-age workers

sigma

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
74

76

78

80

82

84

86

88

90

92

94

96

98 100longevity