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1N
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O
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1.1
Introduction
The overlapping generations (OLG) model, introduced by Sameulson (1958),
is a dynamic economic model with many interesting properties. It contains
agents who are born at di®erent dates and have ¯nite lifetimes, even though
the economy goes on forever. This induces a natural heterogeneity across
individuals at a point in time, as well as nontrivial lifecycle considerations
for a given individual across time. These features of the model can also
generate di®erences from models where there is a ¯nite set of time periods
and agents, or from models where there is an in¯nite number of time periods
but agents live forever. In particular, competitive equilibria in the OLG
model may not to be Pareto optimal. A closely related feature of the model
is that it has a role for ¯at money. This means we can use OLG models to
address a variety of substantive issues in monetary economics.
1.2
The Basic Model
Suppose that
t
=1
;
2
;:::
, and that at every date
t
there is born a new
generation
G
t
of individuals who live for two periods. More realistic (longer)
lifetimes can be studied, but two periods is the simplest case where the
generations overlap. There is also a generation
G
0
around at
t
=1who
only live for one period, called the \initial old." For now, every generation
consists of a [0
;
1] continuum of homogeneous agents. Let
c
t
1
and
c
t
2
denote
consumption of an individual from
G
t
,
t
¸
1, in the 1st and 2nd periods of
life, and let
e
1
and
e
2
denote his (timeinvariant) endowments in the 1st and
2nd periods of life. His utility function
u
(
c
t
1
;c
t
2
) is strictly increasing and
quasiconcave. Members of generation
G
0
consume only
c
02
and are endowed
1
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e
2
.
One can de¯ne a
Walrasian Competitive Equilibrium
(WCE) for this econ
omy as follows. Let
p
t
be the price of a unit of the consumption good at date
t
. Now clearly every member of generation
G
0
simply consumes his endow
ment,
c
02
=
e
2
. For all
t
¸
1, every member of
G
t
maximizes
u
(
c
t
1
;c
t
2
)
subject to
p
t
c
t
1
+
p
t
+1
c
t
2
=
p
t
e
1
+
p
t
+1
e
2
(1)
and
c
tj
¸
0.
1
We always write budget constraints with strict equality because
u
is strictly increasing. Then a WCE is a sequence of prices and allocations
f
p
t
t
1
t
2
g
such that:
c
02
=
e
2
;g
iven
f
p
t
g
,(
c
t
1
t
2
)so
lvesthemax
im
izat
ion
problem of
G
t
for all
t
¸
1; and markets clear in the sense that for all
t
c
t
1
+
c
t
¡
1
;
2
=
e
1
+
e
2
:
(2)
One can also de¯ne a
Recursive Competitive Equilibrium
(RCE) as follows.
Let
s
t
denote savings or loans by a member of
G
t
at
t
,and
R
t
the gross
(principal plus interest) return on savings between
t
and
t
+ 1. Then for all
t
¸
1, every member of
G
t
maximizes
u
(
c
t
1
t
2
)subjectto
c
t
1
=
e
1
¡
s
t
(3)
c
t
2
=
e
2
+
R
t
s
t
(4)
and
c
tj
¸
0.
2
ARCEisasequence
f
R
t
t
1
t
2
g
such that:
c
02
=
e
2
1
We could write (1) more generally as
P
j
p
j
c
tj
=
P
j
p
j
e
tj
,where
c
tj
is the consump
tion and
e
tj
is the endowment in period
j
of an agent born at
t
, but it is obvious from the
speci¯cation of preferences and endowments that
c
tj
and
e
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 Spring '02
 Krueger

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