Discussion Papers In Economics
And BusinessOld age or dependence. Which social insurance?

Yukihiro Nishimura and Pierre Pestieau
Discussion Paper 19-03

April 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANOld age or dependence. Which social insurance?
Yukihiro Nishimuraa∗† and Pierre PestieaubaGraduate School of Economics, Osaka University, 1-7 Machikaneyama-cho, Toyonaka-shi
Osaka, 560-0043, Japan.
b

CREPP, Universite de Liège, CORE.

Voie du Roman Pays 34, L1.03.01, B-1348 Louvain-la-Neuve, Belgium.

April 3, 2019

Abstract
We consider a society where individuals diﬀer according to their productivity and their risk of mortality and dependency. We show that according to the most reasonable estimates of correlations among these threecharacteristics, if one had to choose between a public pension system anda long-term care social insurance, the latter should be chosen by a utilitarian social planner. With a Rawlsian planner, the balance between thetwo schemes does depend on the comparison between the probabilities of
the worst oﬀ individual and the probabilities of the rest of society.
JEL: H2, H5.
Keywords: long term care, pension, mortality risk, optimal taxation,liquidity constraints.

∗ Corresponding author. Tel: +81-6-6850-5230. E-mail: ynishimu@econ.osaka-u.ac.jp† This paper was developed while Pestieau visited Osaka University. The ﬁnancial support
from Osaka University (International Joint Research Promotion Program for Support of ShortTerm Personnel Costs) is gratefully acknowledged. Nishimura acknowledges the ﬁnancial support from the Grants-in-Aid for Scientiﬁc Research (C) (the Ministry of Education, Culture,
Sports, Science, and Technology, 24530348 and 18H00866) and Strategic Young Researcher
Overseas Visits Program for Accelerating Brain Circulation (Japan Society for the Promotionof Science, J2402).

1

Introduction

Assume that a government has to choose between a public pension
system and a long term care social (LTC) insurance. Admittedly
this is an odd dilemma that could be explained by limited publicfunds. We show in this short paper that priority should be given
to the LTC scheme. This ﬁnding is a bit paradoxical given that ina number of countries we observe generous pensions systems andquasi-inexistent public support to dependence in the old age.
To achieve this result, we consider a two-period economy withthree states of nature: the ﬁrst period (people work and save), thesecond period (people retire) with a healthy state, and the second
period with dependency (Cremer et al. (2010), Cremer and Pestieau(2014)). Society comprises a number of individuals who diﬀer intheir productivity and their probability of survival and dependence.
We have strong evidence concerning the correlation between those
probabilities. According to the recent waves of Survey of Health,Ageing and Retirement in Europe (SHARE), we indeed observe a
positive correlation between income and longevity, and a negative
correlation between income and dependence, conditional or not uponsurvival. Given these stylized facts, and assuming the existence ofsaving and LTC private insurance, it is intuitive that a redistributive
LTC scheme brings more utility to the poor than a redistributive
social security. One dollar devoted to a LTC public beneﬁts themmore than a dollar spent on public pensions. This is true as longas individuals, particularly those with low income, have a positivesaving and a positive LTC insurance, or alternatively they can havenegative saving and/or private LTC insurance (the absence of liquidity constraints). When these assumptions do not hold, we showthat the superiority of LTC social insurance over public pensions
still holds under some plausible conditions.
When the objective of the government is Rawlsian, then the desirability of public LTC scheme depends on the comparison between
the probabilities of the worst-oﬀ individual and the average probabilities. However, the superiority of LTC beneﬁts over social security
is more limited than in the utilitarian case.

1

2

The Model

Consider a two-period model, where individuals work and save inthe ﬁrst period and retire in the second. In the second period people
face diﬀerent risks of mortality and dependence. There are I typesof individuals. The proportion of type i (i = 0, 1, ..., I) individuals
is denoted by ni , with the total numberPof individuals born in theIﬁrst period being normalized to unity:
i=0 ni = 1. Each individual of type i is characterized by three characteristics: (i) wi (laborproductivity in the ﬁrst period), (ii) πi (the probability to be alivein the second period), and (iii) pi (the probability of becoming dependent in the second period). From the Survey of Health, Ageing
and Retirement in Europe (SHARE), we know that at least for the
Eurpean countries covered the following relations hold:• longevity (πi ) increases with income.
• conditional upon survival, the probability of dependency (pi )decreases with income.
• the probability of dependency (πi pi ) decreases with income.
Consistent with these facts, we assume cov(wi , πi ) > 0, cov(wi , pi ) <0 and cov(wi , πi pi ) < 0.1¯
Let ci denote individual i’s ﬁrst period consumption, `i ∈ [0, `]denote labor supply, di represent the second period consumption if(s)he is healthy, and mi denote care expenditures in case of dependency. An individual’s expected lifetime utility is given by
Ui = u (ci − v(`i )) + πi (1 − pi )u(di ) + πi pi H(mi ).
In the following we denote xi = ci − v(`i ). We assume u0 > 0, u00 <¯ >0, v 0 > 0, v 00 > 0, H 0 > 0, H 00 < 0, v 0 (0) = 0 and v 0 (`)
00maxi=1,...,I wi . We also assume H (y) > u (y) for all y > 0, to reﬂect
costly needs for dependency.
Private saving is invested in a perfect annuity market with a zerointerest rate. From saving si , type i has a return si /πi . There is alsoa private insurance market against dependency. From the insurance
purchase Pi , type i receives Pi /(πi pi ), where the return is inversely
proportional to the individual’s risk of dependency.
1 Our stylized model in the following yields that life-time incomes are increasing in wage

wi .

2

The government imposes (i) linear income taxes (τ ≥ 0), (ii) ﬂatrate pension (r ≥ 0), and (iii) uniform long-term care beneﬁt (q ≥0). Individuals choose labor supply (`i ), savings (si ), and private
insurance (Pi ) while taking the government’s scheme as given:
Ui = u ((1 − τ )wi `i − si − Pi − v(`i ))(1)+πi (1 − pi )u(si /πi + r) + πi pi H(si /πi + Pi /(πi pi ) + q + r).
In the following, we assume:
Assumption 1 xi ≡ ci − v(`i ) > 0 holds for all i at the optimumof (1).
Namely, individuals would not transfer all his/her ﬁrst-period incomes to the second period through Pi and si . Here we assume thatwi ’s are suﬃciently high, and/or the expected cost for dependency
is suﬃciently low. The case where there is an individual with wi = 0
will be discussed in Section 4.
The FOCs with respect to `i , si and Pi are:u0 (xi ) ((1 − τ )wi − v 0 (`i )) = 0,−u0 (xi ) + (1 − pi )u0 (di ) + pi H 0 (mi ) ≤ 0,
−u0 (xi ) + H 0 (mi ) ≤ 0.

(2)
(3)
(4)

Let the solution values be `∗i , s∗i and Pi∗ respectively. The ﬁrst condition is written with an equal sign, implying for an interior solution
for labor. The two other solutions are not necessarily interior, implying that some individuals may be constrained to have a non-negative
level of saving or of LTC insurance. Formally: s∗i ≥ 0; Ii∗ ≥ 0. In
case of interior solutions, we haveu0 (ci ) = u0 (di ) = H 0 (mi ).
Concerning those solutions, we distinguish two cases in which
they are interior:• Given the parameters of the model, all the solutions are interior. This will be the case when both q and r are small, or
alternatively when the tax rate is low for some reason (politicaldecision or tax distortions).
• Liquidity constraints are assumed away, implying that individuals can have negative saving or insurance premium.
3

We now turn to the optimal level of public beneﬁts chosen by agovernment that is utilitarian or Rawlsian.

3

Utilitarian Case

The problem of the utilitarian government is to maximize the following Lagrangian:X£ =ni {u ((1 − τ )wi `∗i − s∗i − Ii∗ − v(`∗i ))+πi (1 − pi )u(s∗i /πi + r) + πi pi H(s∗i /πi + Ii∗ /(πi pi ) + q + r)}X+µni (τ wi `∗i − πi r − πi pi q),(5)
For simplicity, the stars with respect to xi , di , mi and `i are dropped
in the remainder of the paper. The FOCs on q and r are as follows:∂£ X=
ni πi pi H 0 (mi ) − µπp,
(6)
∂q
∂£ X
=
ni πi {(1 − pi )u0 (di ) + pi H 0 (mi )} − µπ,(7)∂r
where the bar denote the population average of the respective parameter.
We adopt the view point of tax reform wherein we consider thatthe tax is given, not necessarily optimal, and we look at the welfare
incidence of increasing q at the expense of r while keeping a balanced
budget. This is given by:∂£c∂£ ∂£ ∂r ¯¯
≡
+
¯
∂q
∂q
∂r ∂q dτ =0
µ
¶
πp X
πp X
= 1−
ni πpi H 0 (mi ) −ni πi (1 − pi )u0 (di ),
π̄
π̄
or
Ã
!
!
µ
¶ Ã
µ
¶
π
p
π
(1
−
p
)
∂£c
πp
i
i
i
i
= 1−
πp cov H 0 (mi ),− cov u0 (di ),+ ∆ , (8)∂qπ̄πpπ(1 − p)P
where ∆ ≡ ni {H 0 (mi ) − u0 (di )}.
Regarding the second-period consumption (mi and di ), the following property holds (regardless of the liquidity constraints). The
proof is given in the Appendix:
4Lemma 1 ∂mi /∂wi ≥ 0 and ∂di /∂wi ≥ 0 with strict inequality
when s∗i > 0.
We assume in the following that wage diﬀerentials represent thedominant source of heterogeneity of mi and di . Namely, we assume
that w varies suﬃciently across individuals.
If π is constant or has a low variance, then, from our stylized
facts concerning the correlations between πp and π(1−p) on the one
hand and w on the other hand, we have cov(H 0 (mi ), πi pi /πp) > 0
and cov(u0 (di ), πi (1 − pi )/π(1 − p)) < 0. The LTC social insurance
realizes targeted expenditures but the public pension favors the productive individuals who also live longer. Notice that we assume
that the tax distortions are independent of the type of insurances.
Therefore, as long as ∆ = 0 (H 0 (mi ) − u0 (di ) = 0 for all i), it isalways desirable to increase q at the expense of r; in other words,∂£c /∂q > 0 in the case of interior solutions. ∆ = 0 holds if both s∗iand Ii∗ are positive for all individuals. For low values of both q andr (and τ ), this condition is fullﬁlled.
∂£c
To get≤ 0, we have to assume that ∆ is negative and large∂qenough. For an illustrative purpose, suppose that H(y) = u(y − L)
where L > 0 stands for the resources needed to compensate for the
dependency.
Lemma 2 Suppose that H(y) = u(y − L). If q ≤ L, then ∆ ≥ 0.
Only if q > L (the government fully compensates resources the needfor the dependency), we have H 0 (mi ) = u0 (s∗i /πi + q + r − L) <u0 (s∗i /πi + r),2 so that ∆ < 0. However, note that q > L is a∂£cnecessary but not suﬃcient condition for≤ 0. This leads us to∂q
our ﬁrst theorem.
Theorem 1 It is always desirable to have a balanced budget increasein LTC beneﬁts at the expense of social security beneﬁts, as long as
the liquidity constraint is not binding for all individuals. In case partof the population is subject to a liquidity constraint, this dominanceof LTC over social security still holds as long as LTC beneﬁts are
not too high relative to pension beneﬁts.
2 From (3) and (4), −u0 (x ) + H 0 (m ) < −u0 (x ) + (1 − p )u0 (d ) + p H 0 (m ) ≤ 0, so that
i
i
i
i
i
i
i
Ii∗ = 0 for all i.

5Combining (6) and the revenue-side optimization, we obtain:∂£ ∂£ ∂q ¯¯
∂£c
=
+
¯
∂τ
∂τ
∂q ∂τ dr=0
XX
y + τ ∂y/∂τ=
ni u0i (xi )(−wi `i ) +ni πi pi H 0 (mi )(9)πp
π i pi∂y= −cov(u0 (xi ), yi ) + cov(H 0 (mi ),)y − Γ y + µτ ,πp∂τPwhere yi ≡ wi `i and Γ ≡ ni {u0 (xi ) − H 0 (mi )} ≥ 0. This implies:¶µ¶µπi p iyi
00−cov u (xi ),+ cov H (mi ),
−Γ
y
πp
∗
τ =
> 0.
(10)
−µ∂y/∂τ · 1/y
The denominator is the conventional eﬃciency term. It is positive. The ﬁrst term of the numerator is the traditional equity
term −cov(u0 (xi ), yi ) > 0. These two terms correspond to the conventional optimal tax formula (e.g., Sheshinski (1972) and Hellwig(1986)). This redistributive impact of the conventional ﬁrst termof the numerator is reinforced by the second term, which is positive and reﬂects the redistributive impact of the LTC beneﬁt. Notethat if instead of using the tax proceeds for LTC they were used
for pensions, the second term of the numerator would be negative,
reﬂecting the fact that pensions tend to beneﬁt the high-income individuals. The last term of the numerator represents the cost of thebinding liquidity constraints (4).
Below we have a further discussion for the case of H(y) = u(y −L). Lemma 2 implies the following:
Theorem 2 Suppose that H(y) = u(y − L). Let (τ ∗ , q ∗ , r∗ ) be theutilirarian social optimum. If q ∗ ≤ L, then r∗ = 0.
Evaluated at q ≤ L and r = 0, taking account of the government
budget balance, the total eﬀect of the tax increase for the increaseof q is given by (9). Whether there exists the optimum at q ∗ ≤ L
depends on the sign of (9) at q ≤ L. The qualitative features are
as follows. Other things being equal, (9) is lower (and q ∗ is lower)
when the distribution of income and risk of dependency are moreequal, or when the tax distortions are high.3
3 Whenq ≤ L and r = 0, we can show that Γ = 0.

6

Dividing (9) by yu0 (xi ), it isBetween q and r, the priority is given to q until q = L. r∗ > 0
might happen only when q ∗ > L (individuals are overly insured
under dependency), which we do not observe in reality.

4

Rawlsian Case

Suppose that there is an individual 0 whose w0 = 0. For this individual, we have s∗0 = 0 and P0∗ = 0. Suppose that the government’ssocial objective is to maximize the second-period utility of individual0:X£ = π0 (1 − p0 )u(r) + π0 p0 H(q + r) + µni (τ wi `∗i − πi r − πi pi q),(11)Since individual 0 does not pay the payroll tax, the optimal tax rate
under this social objective is the peak of the Laﬀer curve: τ ∗ =y. The issue here is how to allocate the tax revenue between−∂y/∂τ
r and q.
∂£= π0 p0 H 0 (r + q) − µπp,∂q∂£
= π0 {(1 − p0 )u0 (r) + p0 H 0 (r + q)} − µπ.
∂r

(12)
(13)

From these FOCs, we have:
µ
¶
πp
πp
∂£c
= 1−
π0 p0 H 0 (r + q) −π0 (1 − p0 )u0 (r) (14)∂qπ̄π̄In other words, a compensated increase of LTC beneﬁts, q, is desirable if and only if:
H 0 (r + q) > u0 (r)Φ,(15)

0 πp/π̄where Φ = 1−p< 1.
p0 1−πp/π̄The parameter Φ depends on the avarage probability of dependency (πp/π̄) and the probability of dependency of the poorest individuals (p0 ). Note that in case of πi = π for all i, Φ < 1 since
p0 > p.

increasing in −cov(u0 (xi )/u0 (xi ), yi /y) and decreasing in ∂y/∂τ · 1/y. Noting that µ =cov(u0 (xi ), πi pi /πp) + u0 (xi ), evaluated at the left side of the Laﬀer curve (y + τ ∂y/∂τ > 0),
the value is increasing in cov(u0 (xi )/u0 (xi ), πi pi /πp).

7Clearly in the Ralwsian case, the superiority of LTC beneﬁts oversocial security is more limited than in the utilitarian case.
Theorem 3 In the Rawlsian case, a balanced budget increase in
LTC beneﬁts is desirable as long as these are not too high relative
to pension beneﬁts and if the dependence probability of the poorest
is higher than that of the average population.

5

Conclusion

This paper has studied the design of both a social LTC insurance anda public pension system. Both beneﬁts were uniform as well as the
payroll tax rate. Under the realistic assumption of a positive correlation between income and the survival probability and of a negative
correlation of the dependency probability and income for the skilled
and a lower probability of turning dependent, we show that a utilitarian government should give priority to the LTC scheme relative
to the pension program. When the government adopts a Rawlsian criterion, both programs are needed and the relative advantage
of one over the other will depend on the comparison between the
average probabilities of survival and dependency and those of theworst-oﬀ individuals. In this paper, we use linear instruments. In a
companion paper (Nishimura and Pesitieau (2016)) we instead use
non-linear instruments but with just two types of individuals. In the
optimal non-linear scheme, our stylized facts determine the featuresof the optimal tax policies on the saving and the LTC schemes.

Appendix
Proof of Lemma 1: Let f (τ, wi ) ≡ (1 − τ )wi `∗i − v(`∗i ). From
the Envelope Theorem, ∂f /∂wi = (1 − τ )`∗i > 0. When s∗i > 0
and Pi∗ > 0, (3) and (4) imply u0 (f (τ, wi ) − s∗i − Pi∗ ) = H 0 (mi ) =f (τ, wi ) − Pi∗ − rs∗+ r. Diﬀerentiatingu0 (s∗i /πi + r), so xi = i + r =πi
1 + πi¶µ¶µf (τ, wi ) − Pi∗ − rPi∗
f (τ, wi ) − Pi∗ − r
00+ r +H++q+r =−u
1 + πi
1 + πiπ i pi0, we obtain:∂Pi∗(u00 (xi ) − H 00 (mi ))/(1 + πi ) · ∂f /∂wi
= 00
.
∂wi
(u (xi ) − H 00 (mi ))/(1 + πi ) + H 00 (mi )/(πi pi )8(16)The denominator of (16) is negative due to the µsecond-order ¶
con∗
1
∂f
∂Pi
∂xi
dition with respect to Pi∗ . Then
=
−
=
∂wi
1 + πi ∂wi∂wi
H 00 (mi )/(πi pi )∂f /∂wi∂mi
> 0. Since H 00 (mi )
=
00
00
00
u (xi ) − H (mi ) + H (mi )(1 + πi )/(πi pi )
∂wi
∂xi
∂mi
∂di
∂di
= u00 (xi ), we have> 0 and> 0.
u00 (di )
∂wi
∂wi
∂wi
∂wi
When the individual optimum faces the liquidity constraint forPi∗ , then (3) and (4) are characterized by −u0 (f (τ, wi ) − s∗i ) + (1 −pi )u0 (s∗i /πi + r) + pi H 0 (s∗i /πi + q + r) ≤ 0 and Pi∗ = 0. When s∗i > 0,∂s∗i
diﬀerentiating the former equation, we obtain=∂wi
u00 (xi )∂f> 0. When s∗i = 000
0000
u (xi ) + (1 − pi )u (di )/πi + pi H (mi )/πi ∂wi
and Pi∗ = 0, ∂mi /∂wi = 0.
∗∗When−u0 (f (τ, wi ) − Pi∗ )+¶ Pi > 0, then diﬀerentiatingµ ∗ si = 0 and
∗
∂f
∂xi
∂f
∂Pi
H 00 (mi )/(πi pi )Pi
+ q + r = 0,
=
−
= 00
>
H0
00
π i pi∂wi∂wi∂wi
u (xi ) + H (mi )/(πi pi ) ∂wi
∂mi
∂xi
∂mi
0. Since H 00 (mi )= u00 (xi ), we have> 0. Q.E.D.
∂wi
∂wi
∂wi
Proof of Lemma 2: Suppose that q ≤ L. If (4) holds inequality, then (3) implies that −u0 (xi ) + (1 − pi )u0 (di ) + pi H 0 (mi ) =(1 − pi )(u0 (di ) − H 0 (mi )) ≤ 0 for all i. When q ≤ L and Pi∗ = 0,
u0 (di ) = u0 (s∗i /πi + r) ≤ H 0 (s∗i /πi + r + q) = H 0 (mi ) for all i. Therefore, u0 (di ) ≤ H 0 (mi ) for all i when q ≤ L. We therefore have ∆ ≥ 0.
Q.E.D.

References[1] Cremer, Helmuth, Jean Lozachmeur, and Pierre Pestieau. (2010)“Collective annuities and redistribution.” Journal of Public Economic Theory 12.1, 23-41.
[2] Cremer, Helmuth and Pierre Pestieau. (2014) “Social long-termcare insurance and redistribution.” International Tax and Public
Finance 21.6, 955-974.
[3] Hellwig, Martin F. “The optimal linear income tax revisited.”
Journal of Public Economics 31.2, 163-179.
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[4] Nishimura, Yukihiro and Pierre Pestieau. (2016) “Eﬃcient taxation with diﬀerential risks of dependence and mortality.” Economics Bulletin, Vol. 36.1, 52-57.
[5] Sheshinski, Eytan. (1972), “The optimal linear income-tax.” Review of Economic Studies 39.3, 297-302.

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