Doshisha University Life Risk Research Center
Discussion Paper Series No. 2011-01

Healthy Life Expectancy, Dynamic Efficiency, anda Pareto-improving Subsidy for Long-Term Care
Kazutoshi Miyazawa
Faculty of Economics
Doshisha University

Discussion Paper SeriesHealthy life expectancy, dynamic eﬃciency, anda Pareto-improving subsidy for long-term careKazutoshi Miyazawa∗

Abstract
It has been recognized the quality of life is not related to life expectancy
at birth but to the Heathy Life Expectancy (HALE) at birth. Does thepolicy priority concerning long-term care for older people become lower
if they become healthier than before? In a theoretical model we show
the answer depends on the dynamic eﬃciency condition. Introducing a
subsidy for long-term care is eﬃcient on the Pareto principle if the economy is dynamically ineﬃcient. Our result suggests a proper public policy
concerning long-term care is necessary to improve the quality of life not
only for older generations but also for younger and future generations.

JEL Classification: E62, H21, J14
Keywords: Healthy life expectancy, Dynamic eﬃciency, Long-term care, Subsidy∗ Kazutoshi Miyazawa, Faculty of Economics, Doshisha University, Kamigyo, Kyoto 6028580 Japan. kazu@mail.doshisha.ac.jp

1

Introduction

Facing a rapid rate of population aging, many developed countries have tackledwith a tough task of reforming long-term care for fragile elderly people (OECD(2005)). Among most controversial issues, an intergenerational conflict concerning long-term care would be a central issue to solve. More generous benefits
for older people may impose a greater financial burden not only on working
generation but also on future generations by discouraging work incentives andeconomic growth.
[Figure 1 is here]
To deal with the issue in a theoretical model, we have to carefully identify
how population aging aﬀects the demand for long-term care. Figure 1 illustrates
the relation between life expectancy at birth (LE) and healthy life expectancy
at birth (HALE) for 193 countries (WHO (2010)). The horizontal axis measures
LE and the vertical axis measures the ratio of HALE to LE. In Japan, one of themost aging countries in the world, the life expectancy is about 83 in 2008 andthe healthy life expectancy is about 76 in 2007. On average, Japanese people
live for 92 per cent of their lifetime with good health . Figure 1 shows the ratio
tends to increase with LE. It suggests using LE as a proxy of population aging
would be misleading because old people with good health would not demand
for long-term care services so much.
In this paper, we analyze whether and to what extent a subsidy for long-termcare improves welfare of each generation in an overlapping generations modelwith two-sector production (a good production and a formal care production).
In the first period of life, individuals work, consume goods, and save for retirement. In the second period, they consume goods and care services. Bequest
motives are absent. The care service production is labor intensive relative tothe good production. Markets work well. In this setting, we introduce a subsidy for long-term care financed by a wage income tax. We have two analyticalresults. First, introducing a subsidy for long-term care is Pareto-improving if
the economy is dynamically ineﬃcient in a Neoclassical world or if the economy
exhibits sustainable growth. Second, the maximum rate of subsidy which makes
all the generations better-oﬀ decreases with HALE. This implies HALE plays acritical role in the reform of public policy concerning long-term care.
The mechanism is simple. Suppose that the government introduces the taxsubsidy scheme once-and-for-all and keeps it thereafter. The scheme increases
the demand for long-term care, which reallocates labor from the good sector toward the care service sector. In the short run, the capital-labor ratio in the good
sector increases, which increases the wage rate and decreases the interest rate.
In a Neoclassical world, the capital-labor ratio comes back to the initial steadystate. Comparing the long run steady states, both capital and labor employedin the good sector after the reform are smaller than before the reform. Thischange would improve welfare of future generations if the initial capital stockis overaccumulated, that is, the initial steady state is dynamically ineﬃcient.
Next, it can be easily proved that all the generations in the transition process
are better-oﬀ if the future generations are better-oﬀ. Finally, the old people1who live in the start period of reform could be better-oﬀ if the positive eﬀect ofsubsidy dominates the negative price eﬀect. The same mechanism works in an
endogenous growth setting.
Our paper is related to two field of research. First, we rely on a possibilityof dynamic ineﬃciency. It is well known that a transfer from young to old
is Pareto-improving if the population growth rate is larger than the interest
rate (Samuelson (1954) and Aaron (1966) among others). This argument hadceased since Abel et al. (1989) suggested most OECD countries are dynamicallyeﬃcient in its stochastic one sector growth model. In a two-sector growth model,
however, van Groezen et al. (2007) shows the economy is more likely to be
dynamically ineﬃcient if older people prefer services to good consumption, andthat a positive pay-as-you-go tax maximizes long-run welfare in the service
economy.1 The mechanism is similar to ours, but it relies on simulations anddoes not consider the Pareto optimality.
Second, some researchers examine a possibility of Pareto-improving reformin various circumstances. Much more attention has been paid on public pension
reforms (Breyer (1989), Breyer and Straub (1993), Wiedmer (1996), Belan et al.
(1998), Wigger (1999, 2001), Gyárfás and Marquardt (2001), and van Groezen etal. (2003)). The other are on unemployment insurance (Corneo and Marquardt(2000)), and on bequest taxation (Grossmann and Poutvaara (2009)). The
research on a Pareto-improving reform of long-term care seems fairly scarce.
The structure of the paper is as follows. In section 2 we introduce a basicmodel and analyze the characteristics of equilibrium. We know the subsidy
for long-term care reallocates labor from the good production sector to thecare service production sector. One of the merits of the scheme is that it isneutral to the capital-labor ratio in the good production sector, which implies
factor prices are not aﬀected in the long run. In Section 3 we examine whether
and to what extent introducing the scheme is Pareto-improving. We know
all the generations are better-oﬀ if the steady-state equilibrium is dynamicallyineﬃcient. In Section 4 we extend the basic model to an endogenous growthmodel. We know the suﬃcient condition for the Pareto optimality is weakened.
The final section concludes the paper.

2

Basic model

2.1

Setup

To make the analysis as simple as possible, we assume a gross rate of population
growth is constant n > 0. Denoting the population in generation t by Nt , wehave Nt+1 = nNt for all t.
The utility of an individual in generation t, who is born in period t, is givenbyut = ln c1t + β[ρ ln c2t+1 + (1 − ρ) ln mt+1 ](1)where c1t , c2t+1 , and mt+1 stand for good consumption when young, good
consumption when old, and consumption of care service, respectively. 0 <β < 1 is a subjective discount factor, and 0 ≤ ρ ≤ 1 stands for healthiness in
1 Cremers (2006) examines the dynamic eﬃciency condition in a more general two sector
OLG model.

2retirement. A higher ρ represents a higher healthy life expectancy at birth as
discussed in the previous section.2
The budget constraint in the first and the second period are given by(1 − τ t )wt
Rt+1 st= c1t + st
= c2t+1 + (1 − θ)pt+1 mt+1

(2)
(3)

where st stands for a private saving. wt , Rt+1 , pt+1 stand for a wage rate inperiod t, a gross interest rate in period (t + 1), and a price of the care service
in period (t + 1), respectively. 0 ≤ θ < 1 is a rate of subsidy for the purchase ofcare service , and 0 ≤ τ t < 1 is a tax rate.
From equations (2) and (3), the lifetime budget constraint is given by(1 − τ t )wt = c1t +c2t+1 + (1 − θ)pt+1 mt+1
Rt+1

(4)

The optimization problem for an individual in generation t is to maximize
equation (1) subject to equation (4). The first-order conditions require
1
c1t
βρ

= λt
λt
Rt+1
λt (1 − θ)pt+1
Rt+1

=

c2t+1
β(1 − ρ)mt+1=

where λt stands for a multiplier attached to equation (4).
Solving them, we have net demand functions such as
c1t

=

c2t+1

=

mt+1

=

st

=

(1 − τ t )wt1+ββρ(1 − τ t )wt Rt+1
1+β
β(1 − ρ) (1 − τ t )wt Rt+1
1+β
(1 − θ)pt+1
β(1 − τ t )wt
1+β

(5)
(6)
(7)
(8)

Note that
c2t+1
mt+1

= ρRt+1 st(1 − ρ)Rt+1 st=(1 − θ)pt+1

(9.1)
(9.2)

2 More generally, the utility in the second period may be

vt+1 = ρ(ln cg2t+1 + ln mgt+1 ) + (1 − ρ)(ln cb2t+1 + ln mbt+1 )where a superscript g stands for good health and b stands for bad health. Let us definecomposite consumption of good and service by

 1−ρ
ρ
c2t+1 = cg2t+1 cb2t+1
mt+1

=

 ρ

1−ρ
mgt+1
mbt+1

Then, the utility above is reduced to equation (1).

3The retirement income Rt+1 st is optimally allocated between the expenditures of good consumption c2t+1 and care service (1 − θ)pt+1 mt+1 . Equations(9.1) and (9.2) imply the healthiness in retirement, ρ, plays a critical role in
the allocation of labor between the good production sector and the care service
sector.
The technology in the good production sector is specified by a Cobb-Douglasform such as
Ytg = F (Kt , Lgt ) = AKtα (Lgt )1−αAssuming factor markets are competitive, the factor prices are given bywt
Rt= (1 − α)AKtα (Lgt )−α= αAKtα−1 (Lgt )1−α

(10)
(11)

It seems reasonable to assume the technology in the care service sector islabor-intensive relative to the good production sector. To simplify the analysis,
we assume one unit of labor produces one unit of care service in the care service
sector. The production function ism
Ytm = f (Lmt ) = Lt

Competition in the labor market makes the price of care service and thewage rate equal,
pt = wt(12)
The balanced budget of tax-subsidy scheme requires
Nt τ t wt = Nt−1 θpt mt(13)

The left-hand side is the revenue collected from young generation in periodt, and the right-hand side is the expenditure paid to old generation in period t.
There are four markets in this economy, i.e., the market of labor, capital,
care service, and good. The market clearing conditions are respectively given
by
Nt
Kt+1
Ytm
Ytg

=
=
=
=

Lgt + Lmt
Nt st
Nt−1 mt
Nt c1t + Nt−1 c2t + Kt+1

(14)
(15)
(16)
(17)

where one of them is redundant to solve the equilibrium by Walras law.

2.2

Equilibrium

In this subsection, we derive (i) the allocation of labor between the two sectors,(ii) the required contribution rate to balance the budget, (iii) the law of motionof capital, and (iv) a condition for the dynamic eﬃciency at a steady state.
First, let us define the labor share in the good sector by
γt =

Lgt
Nt

(18)

The labor share in the care service sector is Lm
t /Nt = 1 − γ t . First, we havethe following lemma.
4Lemma 1 The labor share in the good sector is constant over time and givenby(1 − α)(1 − θ)
γt =
(19)
α(1 − ρ) + (1 − α)(1 − θ)which is increasing in ρ, and decreasing in θ and α.

Proof. Substituting equation (9.2) into equation (16) with Ytm = Lm
t ,
Lm
t

(1 − ρ)Rt Nt−1 st−1
(1 − θ)pt(1 − ρ)Rt Kt(1 − θ)pt==

where the second equality comes from equation (15).
Note that the ratio of capital income to labor income in the good sector isconstant over time,
α
Rt Kt
g =
wt Lt
1−α
Substituting this and equation (18) into the equation above, we have
1 − γt =1−ρ αγ1−θ1−α twhich gives equation (19).
Note that per capita income in period t is given byyt

Ytg + pt Ytm
N
µ t¶α
£
¤
Kt
= A
α(γ t )1−α + (1 − α)(γ t )−αNt=

Given Kt and Nt , the per capita income decreases with γ t because a smaller
labor share in the good sector pushes up wages, which dominates a loss of capitalincome.
Second, we derive the relation between the subsidy rate and the tax rate.
Substituting equations (12) and (16) with Ytm = Lmt into equation (13), we have
τ t = θ(1 − γ t ) =α(1 − ρ)θα(1 − ρ) + (1 − α)(1 − θ)which is obviously increasing in the subsidy rate θ.
Third, let us define the capital-labor ratio in the good sector by
kt =

Kt
Lgt

The following lemma states the law of motion of capital.
Lemma 2 The capital-labor ratio in the good sector follows
kt+1 = Γktα
5(20)

whereΓ=β(1 − ρα)An(1 + β)(21)

Given k0 = K0 /(γN0 ), kt converges monotonically to a unique steady state1
k∞ = Γ 1−α . The public scheme aﬀects the initial value, k0 , but does not aﬀect
the steady state value.
Proof. Substituting equation (8) into equation (15), we havekt+1 =β(1 − τ t )wtn(1 + β)γ t+1

Substituting equations (10), (19), and (20) into this, we have equation (21).
The neutrality of the subsidy policy relies on our specification of preference
and production technologies. On the one hand, the subsidy for care service
decreases employment in the good sector, which increases the capital-labor ratio.
On the other hand, the related wage income tax reduces private savings, which
decreases the capital-labor ratio. The opposite eﬀects are exactly cancelled out
in our specification.
Finally, we derive a condition for the dynamic eﬃciency at a steady state.
From equations (11) and (21), the gross interest rate is given by
α−1
R∞ = αAk∞=

α(1 + β)nβ(1 − ρα)The steady state equilibrium is dynamically (in)eﬃcient if R∞ > (<)n. If
R∞ = n, the steady state capital stock satisfies the Golden Rule. Specifically,
we have the following lemma.
Lemma 3 The steady state equilibrium is dynamically (in)eﬃcient if
ρ > (<)1−α1− ≡ ρ̂αβ(22)ρ̂ is increasing in β and decreasing in α. The investment ratio is higher when
individuals are more patient, and/or when the income share of labor (1 − α)is larger.3 Therefore a larger ρ is necessary to satisfy the dynamic eﬃciency
condition for a larger β and/or for a smaller α.4
3 It can be shown the investment ratio is constant over time and given byβKt+1(1 − ρα)=Ytg
1+β4 Assuming that an annual discount rate is 1.5 percent, and that one period is 35 years,
we have β = 1.015−35 = 0.59. Under a plausible assumption that the income share of laboris 70 per cent (α = 0.3), we have ρ̂ = 0.65. Assuming that individuals enter the economy atage 20, the life expectancy is 20 + 35 ∗ 2 = 90, and the healthy life expectancy when ρ = ρ̂ is
20 + 35 ∗ 1.65 = 77.7.

6

3

Pareto-improving reformIn this section we examine whether introducing a subsidy for long-term care
improves welfare of each generation. Suppose that the economy without thesubsidy stays at the steady state in period 0. Then the government introduces
the scheme once-and-for-all in period 0, and keeps it thereafter. We examine
the welfare of (i) future generations, (ii) generations in the transition process,and (iii) the old generation in period 0. If they become better-oﬀ, introducing
the scheme is welfare-improving.
[Figure 2 is here]
Figure 2 illustrates the eﬀect of introducing the public scheme on the accumulation of capital. Before the reform, the economy stays at point A. Afterthe reform, the capital-labor ratio in the good sector jumps to k0 because theemployment in the good sector decreases. Since the capital-labor ratio is higher
than the steady state value, capital decreases over time according to equation(21). In the long run, the capital-labor ratio comes back to the initial steadystate, but capital itself is smaller than that before the reform. In this model,
public subsidies for long-term care suppress the accumulation of capital both in
the short run and in the long run.

3.1

Future generations

In the long run, the individual welfare is given byu = ln c1 + β[ρ ln c2 + (1 − ρ) ln m](23)

The wage rate and the interest rate after the reform are the same as those
before the reform in the long run. Thus, we havec1c2

= (1 − τ )ĉ1
= (1 − τ )ĉ2
1−τ
m =
m̂
1−θ

where ĉ1 , ĉ2 , and m̂ stand for the steady state values before the reform.
The long run welfare eﬀect of the public long-term care is (i) positive because
the demand for formal care increases, and (ii) negative because both consumption when young and old decrease.
Let us define the welfare diﬀerence asϕ(θ) = u − û
where û stands for a steady state welfare before the reform.
From equation (23), we haveϕ(θ) = (1 + β) ln(1 − τ ) − β(1 − ρ) ln(1 − θ)
= (1 + βρ) ln(1 − θ) + (1 + β) ln
71 − ραα(1 − ρ) + (1 − α)(1 − θ)(24)The second equality comes from equation (20). We know ϕ(0) = 0. If
ϕ0 (0) > 0, then introducing the scheme is welfare-improving in the long run.
Otherwise, it worsens welfare. Specifically, we have the following proposition.
Proposition 4 Introducing the tax-subsidy scheme improves welfare of future
generations if the steady state equilibrium is dynamically ineﬃcient. The optimal rate of subsidy for future generations is given byθ∗ = 1 −α(1 + βρ)β(1 − α)(25)which is increasing in β and decreasing in ρ and α.
If the economy is dynamically eﬃcient, then the reform worsens the welfareof future generations.
Proof. Diﬀerentiating equation (24) with respect to θ, we have
ϕ0 (θ) =(1 − ρ)[β(1 − α)(1 − θ) − α(1 + βρ)](1 − θ)[α(1 − ρ) + (1 − α)(1 − θ)]

ϕ0 (0) > 0 is equivalent to equation (22). Solving ϕ0 (θ) = 0, we have equation
(25).

3.2

Generations in transition

First, note that the capital-labor ratio in the good sector decreases monotonically to a unique steady state and that the steady state is the same as that
before the reform,k0 > k1 > · · · > k∞ = k̂(26)where k̂ stands for the initial capital-labor ratio.
The welfare of generation t ≥ 0 is given by
ut = ln c1t + β[ρ ln c2t+1 + (1 − ρ) ln mt+1 ]where c1t , c2t+1 , and mt+1 are given by the following lemma.
Lemma 5 The consumption of good and care service are given as
c1t
c2t+1
mt+1

=

1−α
(1 − τ )Aktα
1+β2= ραnγAΓα ktα(1 − ρ)αnγ=(1 − α)(1 − θ)(27.1)(27.2)(27.3)

where Γ and γ are respectively given by equations (19) and (21).
Proof. Substituting equations (10) and (18) into equation (5), we haveequation (27.1).
Substituting equation (15) into equation (9.1), and using Rt Kt /(wt Lgt ) =α/(1 − α), we have
ραnγ
c2t+1 =
wt+1
1−α
8

α
From equations (10) and (21), we know wt+1 = (1 − α)Akt+1= (1 −α)A(Γktα )α . Substituting this into the equation above, we have equation (27.2).
Substituting equations (12) and (15) into equation (9.2), and using Rt Kt /(wt Lgt ) =α/(1 − α), we have equation (27.3).

Equations (27.1)-(27.3) give the welfare of generation t such as
ut = ln(1 − τ ) + β ln γ − β(1 − ρ) ln(1 − θ) + α(1 + βρα) ln kt
where we omitted the constant terms.
Substituting γ in equation (19) and τ in equation (20) into this, we have (byomitting the constant terms)
ut = (1+βρ) ln(1−θ)−(1+β) ln[α(1−ρ)+(1−α)(1−θ)]+α(1+βρα) ln kt (28)Comparing equations (24) and (28), we are able to prove the optimal subsidy
rate θ∗ in equation (25) is also optimal for generation t ≥ 0 except for the capitaleﬀect. We know that kt is larger than the value before the reform (see equation(26)), and that the impact of capital stock on welfare is positive (see equation(28)). Therefore, we have the following proposition.
Proposition 6 If the future generations are better-oﬀ by introducing the taxsubsidy scheme, then all the generations in the transition process are also betteroﬀ.

3.3

The old generation at the reform

The welfare of generation −1 in period 0 is given by
v0 = ρ ln c20 + (1 − ρ) ln m0

(29)

where
R0 s−1 = c20 + (1 − θ)p0 m0

Note that s−1 is predetermined. The welfare eﬀect is (i) positive because thepurchase of formal care is subsidized, and (ii) negative because the retirement
income decreases and the price of formal care (the wage rate) increases by thegeneral equilibrium eﬀect.
After the reform, old individuals in generation −1 reallocate retirement income between good consumption and care service consumption. Solving the
problem, we havec20
m0

= ρR0 s−1
(1 − ρ)R0 s−1=(1 − θ)p0

Note thatp0
R0= w0 = (1 − α)AK0α (γN0 )−α= αAK0α−1 (γN0 )1−α

Substituting them into equation (29), we have (by omitting the constantterms)
v0 = (1 − ρα) ln γ − (1 − ρ) ln(1 − θ) + (1 − ρα)(ln N0 − ln K0 ) + ln s−1 (30)9

A second term in equation (30) stands for a direct positive eﬀect related tothe subsidy for formal care, and a first term stands for the indirect negative eﬀect
related to changes in factor prices (note that ∂γ/∂θ < 0). If the positive eﬀect
is dominant, then introducing the public scheme improves welfare. Otherwise,
it worsens welfare. Specifically, we have the following proposition.
Proposition 7 Introducing the tax-subsidy scheme strictly improves welfare of
the old generation at the reform. The optimal subsidy rate is given asθ∗∗ = 1 − ρα(31)

Proof. Substituting γ in equation (19) into equation (30), and diﬀerentiating it with respect to θ, we have∂v0
(1 − α)(1 − ρ)(1 − ρα − θ)=∂θ
(1 − θ)[α(1 − ρ) + (1 − α)(1 − θ)]which gives equation (31).

3.4

ResultComparing equations (25) and (31), we have:
Lemma 8 The optimal subsidy rate for future generations is less than that forthe old generation at the reform, θ∗ < θ∗∗ .
With this lemma, and Proposition 4, 6, 7, we conclude:
Proposition 9 Introducing the tax-subsidy scheme is Pareto-improving if thesteady state is dynamically ineﬃcient. Increasing the subsidy rate until θ∗ inequation (25) is also Pareto-improving.
[Figure 3 is here]
Figure 3 illustrates the optimal rate of subsidy for future generations, θ∗ ,and that for the old generation at the reform, θ∗∗ . The horizontal axis measures
the healthiness in retirement, ρ. The steady state is dynamically ineﬃcient if
ρ < ρ̂, and dynamically eﬃcient if ρ > ρ̂. For a ρ < ρ̂, there exists a subsidy
rate θ ≤ θ∗ such that it achieves Pareto improvement. The more health status
in old-age improves, the smaller the optimal subsidy is.

4Discussions

In this section we extend the basic Neoclassical growth model to an endogenous
growth model with capital externality (Arrow (1962) and Romer (1988)). We
show that the result obtained in the previous section is robust if the economy
exhibits sustainable growth.
10The production function in the good sector is modified as
Ytg = F (Kt , Bt Lgt ) = AKtα (Bt Lgt )1−α
where Bt is a labor-augmenting technology and specified by
Bt =

Kt
Lgt

(32)

With equation (32), the good output and factor prices are respectively given
by
Ytg

= AKt

wt

= (1 − α)

Rt

= αA

AKt
Lgt

The per capita income in period t, yt = (Ytg + pt Ytm )/Nt , is given byµ¶
1−αyt = Aκt α +γtwhere γ t is the labor share in the good sector, and κt is the over-all capital-laborratio,Ktκt =Nt
In the same way as the basic model, the per capita income decreases with
γ t for a given κt . It can be easily proved that the labor share γ t is given byequation (19), and that the tax rate τ t is given by equation (20).
The growth rate of per capita income is
yt+1
κt+1
β(1 − ρα)A
=
=
=Γ
yt
κt
n(1 + β)(33)

which implies the tax-subsidy scheme is neutral to the growth rate.
Let us consider a public reform in the same way as the basic model.
The welfare of generation t ≥ 0 is given by
ut = ln c1t + β[ρ ln c2t+1 + (1 − ρ) ln mt+1 ]
where
c1t
c2t+1
mt+1

(1 − α)A 1 − τ t
κt
1+β
γt
= ραnAΓκt(1 − ρ)αnγ t+1=(1 − α)(1 − θ)=By omitting the constant terms, we haveut

= ln(1 − τ ) − [1 − β(1 − ρ)] ln γ − β(1 − ρ) ln(1 − θ) + (1 + βρ) ln κt= −β(1 − ρ) ln[α(1 − ρ) + (1 − α)(1 − θ)] + (1 + βρ) ln κt11which implies
∂ut
>0
∂θ
for ∀t ≥ 0. The reason is that mt+1 increases with θ while c1t and c2t+1 are notaﬀected by θ for a given κt . Since the per capita capital growth is neutral to θ,
the increased formal care caused by the scheme improves the welfare of all thegenerations after the reform.
The welfare of generation −1 in period 0 is
v0 = ρ ln c20 + (1 − ρ) ln m0
where
c20
m0

= ρR0 s−1
(1 − ρ)R0 s−1=(1 − θ)p0

Note thatp0R0
γ(1 − α)Aκ0γ= αA(1 − α)(1 − θ)=α(1 − ρ) + (1 − α)(1 − θ)=Substituting them into v0 , we have
v0 = −(1 − ρ) ln[α(1 − ρ) + (1 − α)(1 − θ)] − (1 − ρ) ln κ0 + ln s−1
which implies
∂v0
>0
∂θ
Two eﬀects are recognized. First, the subsidy increases care service consumption. Second, the price of care service increases. The former eﬀect dominates thelatter eﬀect, so that introducing the tax-subsidy scheme improves the welfare ofgeneration −1.
We conclude introducing the tax-subsidy scheme is Pareto-improving because all the generations are better-oﬀ. It is not the dynamic eﬃciency condition but a condition for positive growth, Γ > 1, that is required for sustainablegrowth.55Concluding remarks

In this paper we show introducing a subsidy for long-term care is Paretoimproving if the economy is dynamically ineﬃcient in a Neoclassical world or if
the economy exhibits sustainable growth. The maximum rate of subsidy which
5 Since the per capita income grows at a rate Γ, the dynamic eﬃciency condition is modifiedas

R T Γn ⇔ ρ T ρ̂12

makes all the generations better-oﬀ decreases with the healthy life expectancy
at birth (HALE). Our results implies HALE plays a critical role in the reform of
public policy concerning long-term care. Since our model is highly stylized andbased on strong assumptions, the policy implication should be treated carefully.
However, our message would be useful for a debate on reforming long-term care.
A proper public policy concerning long-term care is necessary to improve thequality of life not only for older generations but also for younger and future
generations.

13

Acknowledgments
I would like to thank Akira Yakita, Kouki Sugawara, Masako Kimura, Tatsuya Omori, participants at the annual congress of the Japan Association forApplied Economics, Chukyo University, 2011, and Applied Regional Science
Conference, Toyama University, 2011, and seminar participants at Nagoya University and NIRA for useful comments. This research is financially supportedby the Japan Society for the Promotion of Science (No.22530190), the Japan
Economic Research Foundation, and the Zengin Foundation. The remaining
errors are of course mine.

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15

Figure 1 Life expectancy (LE) and Healthy life expectancy (HALE)
HALE/LE
1.00
1 00

0 95
0.95

0.90

0.85

0.80

0.75

LE

0.70
40

50

60

70

80

90

Figure 2 Dynamics

k

k

Γk

A

45°
O

k

k

k

k

Figure 3 Optimal subsidy

θ

1
θ

θ

1

ρα

1

ρ

O
Dynamic Inefficiency

1
Dynamic Efficiency

ρ