This note develops a one-sector, two-period, overlapping generations model that incorporates endogenous labor-care choice. Care choice is modeled by allowing young agents to participate in the production of household health status. Using this model, we derive the steady-state equilibrium dynamics.care supply, equilibrium dynamics, household production

Atsue Mizushima

English

Research Papers in Economics

 
 
 
Discussion Papers In Economics 
And Business 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Graduate School of Economics and 
Osaka 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 
 
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 and 
Osaka 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 
 
Discussion Paper 06-27 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 model
that incorporates endogenous labor–care choice. Care choice is modeled by
allowing young agents to participate in the production of household health
status. 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 helpful
comments 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.jp1 Introduction
Along with rapidly aging populations, it has been argued that health care and long-
term 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 production
of household health status. In this note, we analyze labor–care choice in an overlap-
ping generations model that incorporates lifetime and health status uncertainty. Our
model has two key features. First, young agents are heterogeneous with respect to
their 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” to
their 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 fea-
ture multiple equilibria. More speciﬁcally, there exist multiple equilibria, comprising
a unique saddle stable equilibrium and other equilibria, which have an inﬁnite number
of trajectory paths.
The remainder of this paper is organized as follows. Section 2 sets up the basic
model. 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 a
probability 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 agents
are 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 when
old. We express the death–illness status of each agent’s parents as index i.
1In 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)).
1Each young agent of generation t allocates their unit of time between ﬁrms li,t
t and
the production of household health status qi,t
t. He or she earns wage income wtli,t
t,
and saves all wage income, where wt is the real wage rate. Old agents of generation
t consume the proceeds of their savings, which we denote by ci,t
t+1. We assume the
existence 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 the
household using the following household health production function:
hg,t = dqg,
t
t , (1)
hb,t = qb,
t
t , (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 at
time t:
Ui,t = β lnhi,t +pci,
t
t+1 i = g,b,d
s.t.
ci,
t
t+1 = Rt+1wtli,
t
t ,
qi,
t
t +li,
t
t = 1, (3)
0 ≤ qi,
t
t ≤ 1, 0 ≤ li,
t
t ≤ 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:
qg,
t
t = qb,
t
t =



β
Rt+1wt
if β ≤ Rt+1wt,
1 if Rt+1wt ≤ β.
(4)
Due to the quasi-linear utility function, when Rt+1wt is suﬃciently large, the oppor-
tunity cost of care supply to the production of household health status is high, and
2each 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,
t
t = 0. (5)
Using (3), (4), and (5), we have the aggregate labor supply as follows:
Lt ≡ ltNt =



(
1 −
pβ
Rt+1wt
)
Nt if β ≤ Rt+1wt,
(1 − p)Nt if Rt+1wt ≤ β.
(6)
Firms are perfectly competitive proﬁt maximizers that produce output using a
production function of the Cobb–Douglas form Yt = AKα
t L
1−α
t , where Yt is aggregate
output, 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)Ak
α
t , Rt = αAk
α−1
t , (7)
where kt ≡ Kt/Lt.
Capital market equilibrium in period t requires total savings in the previous period
equal to:
Kt+1 = stNt = wtltNt. (8)
3 Equilibrium Dynamics and Indeterminacy
Using (6) through (8), we obtain the following complete dynamic system:
<Regime I: β ≤ Rt+1wt >
l
1−α
t+1 =
pβl
1−α
t
αA(1 − lt)((1 − α)Akα
t )α, (9)
k
1−α
t+1 =
(1 − α)Akα
t αA(1 − lt)
pβ
. (10)
2If 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 order
to take care of their parents, the data show a negative correlation between the rate of resident and
income. (See U.N. (2005)).
3<Regime II: Rt+1wt ≤ β >
lt = 1 − p, (11)
kt+1 = (1 − α)Ak
α
t . (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 and
lt ≤ 1 − p.
The equations (9) through (12) characterize the economic equilibria that are rep-
resented 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” loci
from (9) through (12):
<Regime I: 1 − p ≤ lt >
LL1 : lt = 1 −
pβ
αA((1 − α)Akα
t )α, (14)
KK1 : kt =
((1 − α)AαA(1 − lt)
pβ
) 1
1−2α
(15)
LL1 and KK1 loci respectively intersect the borderline lt = 1 − p at the points
All and Akk, where: All ≡ (kt,lt) = (((
β
αA)
1
α 1
(1−α)A)
1
α, 1 − p) and Akk ≡ (kt,lt) =
((
(1−α)AαA
β )
1
1−2α, 1 − p).
<Regime II : lt ≤ 1 − p >
LL2 : lt = 1 − p, (16)
KK2 : kt = ((1 − α)A)
1
1−α. (17)
For simplicity of analysis, we assume the following:
Assumption 1 (i) α <
1
2
, and (ii)
(β
α
)1−α( 1
1 − α
)α
< A.
4Under Assumption 1(i), the KK locus is decreasing in lt, and the KK and LL loci
intersect in Regime I under Assumption 1(ii).3
The initial point at which the economy starts can be derived from:
l1 =
K1
k1N1
. (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 initially
on 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 not
satisfy 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/k1N1 in Figure 1. If the economy initially happens to be on the
line 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 of
trajectory paths that leads to the equilibria E1 (E3). Therefore, for a given level of
the 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 the
equilibrium E1 in Figure 1. On the equilibrium path towards E1, any trajectory that
starts above the line “SS” at G in Figure 1, initially has a higher labor supply for
ﬁrms and savings than other equilibrium paths. Higher savings are related to the
larger 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 the
3Assumption 1(i):∂k
∂l = − 1
1−2α(
(1−α)AαA(1−lt)
pβ )
2α
1−2α < 0, if α < 1
2.
Assumption 1(ii): We ﬁnd the point k becomes All < ((1−α)A)
1
1−α < Akk, if (
β
α)1−α( 1
1−α)α < A.
4See Appendix A for the conditions for the stability of the steady state.
5capital–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, the
economy is doomed on this equilibrium.
Next, let us consider the equilibrium E3 in Figure 1. On the equilibrium path
of E3, any trajectory that starts below the line “SS” at I in Figure 1, initially
has 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 decreased
labor 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.
4 Conclusion
We have investigated the steady-state equilibrium dynamics in a model that incorpo-
rates lifetime and health status uncertainty; and endogenous labor–care choice. Care
choice is modeled by allowing young agents to participate in the household health
production. 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 the
local dynamics of equilibrium E2, we take a ﬁrst-order Taylor expansion of the system
around the steady state (k∗,l∗). With ˇ lt ≡ lt − l∗ and ˇ kt ≡ kt − k∗, the linearization
6is expressed as:
(ˇ kt+1
ˇ lt+1
)
=


 

α
(1 − α)
−
αA
pβ
·
((1 − α)A)
1+α
1−α
1 − α
−
l∗α2
(1 − α)((1 − α)A)
1
1−α
1 +
((1 − α)A)
α
1−αl∗αA
(1 − α)pβ


 

(ˇ kt
ˇ lt
)
,
where l∗ ≡ 1 −
pβ
αA((1−α)A)
α
1−α . The characteristic polynomial becomes:
P(κ) = κ
2 − Tκ + D,
T =
α
1 − α
+ 1 +
((1 − α)A)
α
1−αl∗αA
(1 − α)pβ
,
D =
α
1 − α
+
α
(1 − α)
·
((1 − α)A)
α
1−αl∗αA
pβ
.
Azariadis (1993) checks that the steady state is a saddle point, when 1 − T + D < 0
holds. It is clear that:
1 − T + D = −
((1 − α)A)
α
1−αl∗αA
pβ
< 0.
Therefore, 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 Polit-
ical 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 Na-
tions Publications.
Yaari, M. E. (1965): “Uncertain lifetime, life insurance, and the theory of the
consumer,” Review of Economic Studies, 32(2), 137–150.
7lt
kt
1
((1 − α)A)
1
1−α
1 − p
G
G′
E1
E2
E3
SS
I
I′
I”
LL
KK
l1 =
K1
k1N1
Regime I
Regime II
J
All Akk
Figure 1: The phase diagram analysis
8

Intertemporal Macroeconomics. Basil

Living Arrangements of Older Persons Around the World.

OECD Health Data 2005: Statistics and Indicators for 30 Countries. Organization for Economic.

Uncertain lifetime, life insurance, and the theory of the consumer,”

Equilibrium Dynamics in an Overlapping Generations Economy with Endogenous Labor Supply

http://www2.econ.osaka-u.ac.jp/library/global/dp/0627.pdf

Equilibrium Dynamics in an Overlapping Generations Economy with Endogenous Labor Supply

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