Discussion Papers In Economics
And Business

Equilibrium Dynamics in an Overlapping
Generations Economy with Endogenous Labor
SupplyAtsue Mizushima
Discussion Paper 06-27

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEquilibrium Dynamics in an Overlapping
Generations Economy with Endogenous Labor
SupplyAtsue Mizushima
Discussion Paper 06-27
October 2006

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEquilibrium Dynamics in an Overlapping
Generations Economy with Endogenous Labor
Supply∗Atsue Mizushima†
October, 2006

Abstract
This note develops a one-sector, two-period, overlapping generations modelthat incorporates endogenous labor–care choice. Care choice is modeled byallowing young agents to participate in the production of household healthstatus. Using this model, we derive the steady-state equilibrium dynamics.

JEL Classiﬁcation Number: D13, J22
Keyword: care supply, equilibrium dynamics, household production∗

I would like to thank Koichi Futagami, Kazuo Mino, and Mitsu-yoshi Yanagihara for helpfulcomments and suggestions. All remaining errors are naturally my own.
†
Correspondence to: Graduate School of Economics, Osaka University, 1-7, Machikaneyama,
Toyonaka, Osaka, 560-0043, Japan; E-mail:cg089ma@srv.econ.osaka-u.ac.jp

1

Introduction

Along with rapidly aging populations, it has been argued that health care and longterm care are increasing in industrialized countries.1 Thus, this note demonstrates
an economy in which young agents supply their time for taking care of their parents
and veriﬁes the equilibrium dynamics of that economy.
Care supply is modeled by allowing young agents to participate in the productionof household health status. In this note, we analyze labor–care choice in an overlapping generations model that incorporates lifetime and health status uncertainty. Ourmodel has two key features. First, young agents are heterogeneous with respect totheir parents’ death–illness status (death, good health, and bad health), and exhibit
one-sided altruism (from children to parents) that is derived from a “joy of giving” totheir parents. Second, the health status of aged parents depends on the care supply
from their children.
Under this analytical framework, we show that the equilibrium dynamics can feature multiple equilibria. More speciﬁcally, there exist multiple equilibria, comprisinga unique saddle stable equilibrium and other equilibria, which have an inﬁnite numberof trajectory paths.
The remainder of this paper is organized as follows. Section 2 sets up the basicmodel. Section 3 derives the equilibrium dynamics. Section 4 concludes this paper.

2

The Model

In this paper, we use an overlapping generations model in which a continuum of
agents [0, 1] is born every period and each agent lives for a maximum of two periods.
Each agent is endowed with one unit of time in young age. Young agents have aprobability p of surviving throughout old age. In addition, an agent who is alive in
old age has a probability 1 − ψ of being in bad health. Thus a fraction pψ of agentsare of type g who have good health, a fraction p(1 − ψ) of agents are of type b who
have bad health. Type d agents, who constitute a fraction 1 − p of agents, die whenold. We express the death–illness status of each agent’s parents as index i.
1

In OECD countries as a whole, the average density of practicing nurses per 1,000 persons was
about 2.5 in 1960 and 8.1 in 2003 (OECD (2005)).

1

Each young agent of generation t allocates their unit of time between ﬁrms li ,tt andthe production of household health status qi ,tt . He or she earns wage income wt li ,tt ,and saves all wage income, where wt is the real wage rate. Old agents of generationt consume the proceeds of their savings, which we denote by ci ,tt+1 . We assume theexistence of actuarially fair insurance in this paper (see Yaari (1965) and Blanchard(1985)). Thus the rate of return on the annuities is Rt+1 /p if they are alive and 0
if they die at the end of period t, where Rt+1 is the real rental rate. Let us assume
that the health status of type g and type b old agents at time t is produced in thehousehold using the following household health production function:
hg ,t = dqg ,tt ,(1)

hb ,t = qb ,tt ,(2)where d > 1 is a productivity parameter. Thus the marginal productivity of care
supply of the agents with index i = g is higher than that of the agents with index
i = b. Each young agent of generation t solves the following optimization problem attime t:Ui ,t = β ln hi ,t +pci ,tt+1

i = g, b, ds.t.
ci ,tt+1 = Rt+1 wt li ,tt ,qi ,tt +li ,tt = 1,

(3)

0 ≤ qi ,tt ≤ 1, 0 ≤ li ,tt ≤ 1,(1), (2),where β ∈ (0, 1) measures the degree of altruism towards parents. The optimal care
supply to the production of household health status is derived as follows: β
if β ≤ Rt+1 wt ,tt
qg ,t = qb ,t = Rt+1 wt
1
if Rt+1 wt ≤ β.

(4)

Due to the quasi-linear utility function, when Rt+1 wt is suﬃciently large, the opportunity cost of care supply to the production of household health status is high, and2each agent decreases his or her transfers of care supply.2 If parents die, young agents
do not derive any utility from household health status; thus we obtain:
qd ,tt = 0.
Using (3), (4), and (5), we have the aggregate labor supply as follows:()
 1 − pβNt
if β ≤ Rt+1 wt ,Rt+1 wt
Lt ≡ lt Nt =
(1 − p)Nt
if Rt+1 wt ≤ β.

(5)

(6)

Firms are perfectly competitive proﬁt maximizers that produce output using a
production function of the Cobb–Douglas form Yt = AKtα L1−α
, where Yt is aggregatetoutput, A > 0 is a productivity parameter and Kt is the aggregate capital stock.
We assume that capital depreciates fully in the process of production. Thus proﬁt
maximization in the competitive market equates the marginal products of private
labor and capital to the real wage and the real rental rate, respectively:
wt = (1 − α)Aktα ,Rt = αAktα−1 ,(7)where kt ≡ Kt /Lt .
Capital market equilibrium in period t requires total savings in the previous period
equal to:Kt+1 = st Nt = wt lt Nt .

3

(8)

Equilibrium Dynamics and Indeterminacy

Using (6) through (8), we obtain the following complete dynamic system:<Regime I: β ≤ Rt+1 wt >pβlt1−α,αA(1 − lt )((1 − α)Aktα )α(1 − α)Aktα αA(1 − lt )
=
.
pβ

1−α
lt+1
=
1−α
kt+1
2

(9)
(10)

If a log-linear utility function is assumed, the care supply to aged parents becomes constant.
This is not consistent with the actual data. When we consider that the care supplied is in orderto take care of their parents, the data show a negative correlation between the rate of resident andincome. (See U.N. (2005)).

3

<Regime II: Rt+1 wt ≤ β >lt = 1 − p,(11)

kt+1 = (1 − α)Aktα .

(12)

Before stating the equilibrium, we consider the borderline between the regimes.
Substituting (7), (8), and (9) into the borderline, we express the borderline as follows:
lt = 1 − p.

(13)

Thus, Regime I and Regime II is respectively feasible on the area 1 − p ≤ lt ≤ 1 andlt ≤ 1 − p.
The equations (9) through (12) characterize the economic equilibria that are represented sequences of {kt , lt }∞t=1 with an initial condition (k1 , l1 ) ≥ 0. Now let us draw
the phase diagram on the (kt , lt ) plane. We refer to the loci representing kt+1 = kt as“KK” and that representing lt+1 = lt as “LL”. We have the “KK” and “LL” locifrom (9) through (12):<Regime I: 1 − p ≤ lt >pβ,αA((1 − α)Aktα )α1( (1 − α)AαA(1 − l ) ) 1−2αt: kt =pβ

LL1 : lt = 1 −

(14)

KK1

(15)

LL1 and KK1 loci respectively intersect the borderline lt = 1 − p at the points
1

1

β α
1
All and Akk , where: All ≡ (kt , lt ) = ((( αA) (1−α)A) α , 1 − p) and Akk ≡ (kt , lt ) =1(( (1−α)AαA) 1−2α , 1 − p).
β<Regime II : lt ≤ 1 − p >LL2 : lt = 1 − p,

(16)
1

KK2 : kt = ((1 − α)A) 1−α .
For simplicity of analysis, we assume the following:( β )1−α ( 1 )α1
< A.
Assumption 1 (i) α < , and (ii)
2
α
1−α

4

(17)

Under Assumption 1(i), the KK locus is decreasing in lt , and the KK and LL lociintersect in Regime I under Assumption 1(ii).3
The initial point at which the economy starts can be derived from:
l1 =

K1
.
k1 N 1(18)

The surface representing (18) can be drawn in (kt , lt ) space as the initial per capita
capital stock K1 /N1 is given exogenously. Therefore, the economy must be initiallyon the line (18). As can be veriﬁed immediately from (18), the contour of (18) drawn
in the (kt , lt ) plane is downward. Thus, if an initial condition is given, the dynamics
of this economy are determinable. The phase diagram of this economy is depicted in
Figure 1.
Let us ﬁrst consider the case in which the initial level of capital is suﬃciently
large. In this case, the initial contour is drawn in the upper-right corner in Figure
1, and the trajectory is drawn like J. Since any trajectory that is above J does notsatisfy the physical condition, we can exclude these trajectories from the equilibrium.
Thus contour J shows the boundary trajectory in this economy. Next, let us consider
the case l1 = K1 /k1 N1 in Figure 1. If the economy initially happens to be on theline SS, it converges to E2 .4 If the economy initially happens to be above (below)
the line SS, it converges to E1 (E3 ). In this case, there exists an inﬁnite number oftrajectory paths that leads to the equilibria E1 (E3 ). Therefore, for a given level ofthe initial capital stock, the economy converges to one of the equilibria E1 , E2 , or(E3 ) in the long run.
In order to obtain intuitive implications for this result, let us ﬁrstly consider theequilibrium E1 in Figure 1. On the equilibrium path towards E1 , any trajectory thatstarts above the line “SS” at G in Figure 1, initially has a higher labor supply forﬁrms and savings than other equilibrium paths. Higher savings are related to thelarger capital stock and an increased labor supply in next period. Thus, the economy
initially increases both labor supply and the capital–labor ratio, however, when it
crosses the KK locus at G′ in Figure 1, the increased labor supply decreases the32α(1−α)AαA(1−lt ) 1−2α1)Assumption 1(i): ∂k< 0, if α < 12 .
∂l = − 1−2α (
pβ
1

β 1−α
1
Assumption 1(ii): We ﬁnd the point k becomes All < ((1 − α)A) 1−α < Akk , if ( α)( 1−α)α < A.
4
See Appendix A for the conditions for the stability of the steady state.

5capital–labor ratio. A lower capital–labor ratio increases labor supply (see (9)), thus,
any trajectory of this economy converges to E1 . For the equilibrium E1 , the capital–
labor ratio converges to zero; that is, no production can be carried out. Thus, theeconomy is doomed on this equilibrium.
Next, let us consider the equilibrium E3 in Figure 1. On the equilibrium pathof E3 , any trajectory that starts below the line “SS” at I in Figure 1, initiallyhas a lower labor supply for ﬁrms and savings than other equilibrium paths. The
economy initially increases both labor supply and the capital–labor ratio, however,when it crosses the LL locus at I ′ in Figure 1, labor supply is restrained. A decreasedlabor supply increases the capital–labor ratio, making labor supply lower–bounded(at I ′′ in Figure 1). On the economy where labor supply becomes lower–bounded,
the capital–labor ratio decreases and converges to E3 .

4Conclusion

We have investigated the steady-state equilibrium dynamics in a model that incorporates lifetime and health status uncertainty; and endogenous labor–care choice. Carechoice is modeled by allowing young agents to participate in the household healthproduction. Using this model, we have shown that the equilibrium dynamics obtained
from the model has multiple equilibria. More speciﬁcally, there exists a unique saddle
stable equilibrium and other equilibria, which have an inﬁnite number of trajectory
paths.

Appendix
Appendix A
We examine the stability of the equilibrium E2 in this Appendix. To examine thelocal dynamics of equilibrium E2 , we take a ﬁrst-order Taylor expansion of the system
around the steady state (k ∗ , l∗ ). With ˇlt ≡ lt − l∗ and ǩt ≡ kt − k ∗ , the linearization6

is expressed as:

(

ǩt+1
ˇlt+1

)

where l∗ ≡ 1 −



=

−

α
(1 − α)l ∗ α21(1 − α)((1 − α)A) 1−αpβααA((1−α)A) 1−α
1+ααA ((1 − α)A) 1−α
−
·
( )
pβ
1 − αα ǩt,

((1 − α)A) 1−α l∗ αA  ˇlt
1+(1 − α)pβ. The characteristic polynomial becomes:
P (κ) = κ2 − T κ + D,αα((1 − α)A) 1−α l∗ αA
T =
+1+
,
1−α
(1 − α)pβαα((1 − α)A) 1−α l∗ αA
α
D =
+
·
.
1 − α (1 − α)pβAzariadis (1993) checks that the steady state is a saddle point, when 1 − T + D < 0holds. It is clear that:α((1 − α)A) 1−α l∗ αA
1−T +D =−< 0.
pβTherefore, E2 is a saddle point.

References
Azariadis, C. (1993): Intertemporal Macroeconomics. Basil Blackwell, Oxford.
Blanchard, O. J. (1985): “Debt, Deﬁcits, and Finite Horizons,” Journal of Political Economy, 93(2), 223–47.
OECD (2005): OECD Health Data 2005: Statistics and Indicators for 30 Countries.
Organization for Economic.
U.N. (2005): Living Arrangements of Older Persons Around the World. United Nations Publications.
Yaari, M. E. (1965): “Uncertain lifetime, life insurance, and the theory of theconsumer,” Review of Economic Studies, 32(2), 137–150.

7

lt
1

E1
LL
J

1
l1 = kK
1 N1

G′

SS

G
E2

Regime I

I′
KK
I

1−p

All

E3

Akk

I”
Regime IIkt

1((1 − α)A) 1−αFigure 1: The phase diagram analysis

8