General Equilibrium with Incomplete Markets
Jonathan Heathcote
Updated April 2006
1. Heterogeneity
Consider an economy with a continuum of agents of total mass equal to
1
.
Each agent’s
productivity process is independent of all other agents current and past endowments
and de
f
ned by a
f
rst order Markov process. How should we describe the equilibrium
distribution of households across asset holdings and endowments? We should use a
probability measure.
•
Let
a
max
denote the endogenous maxmum asset holdings in equilibrium. (In
the iid shock case described above
a
max
=
a
0
(
z
max
)
).
•
Let
ψ
be a probability measure de
f
ned on
(
S, β
S
)
where
S
=[
−
φ, a
max
]
×
E
and
β
S
is the Borel
σ
algebra (an appropriate set of subsets of
S
). Thus for any
set
B
∈
β
S
,ψ
(
B
)
is the mass of agents whose individual state vectors lie in
B.
•
Let
P
((
a, e
)
,B
)
be the probability than an agent with state
(
a, e
)
today has in
individual state in the set
B
∈
β
S
tomorrow. Formally the transition function
P
:
S
×
β
S
→
[0
,
1]
is given by
P
((
a, e
)
,B
)=
X
e
0
I
(
a
0
(
a,e
)
,e
0
)
∈
B
×
π
(
e
0

e
)
Since we are in an economy with constant prices we would hope that
ψ
will be
unchanged over time. A probability measure
ψ
is stationary provided that
ψ
0
(
B
)=
Z
P
(((
a, e
)
,B
)
dψ
=
ψ
(
B
)
for all
B
∈
β
S
Let
x
=(
a, e
)
be the individual state vector. Let
q
be the price of a noncontingent
bond that pays one unit of consumption in the next period. Given a constant
q,
the
individual’s problem, in recursive form is characterized by the following functional
equation:
v
(
x
;
q
)=
m
a
x
{
c,a
0
∈
Γ
(
x
;
q
)
}
½
u
(
c
)+
β
P
e
0
π
(
e
0

e
)
v
(
a
0
,e
0
;
q
)
¾
Γ
(
x
;
q
)=
{
(
c, a
0
):
c
+
qa
0
≤
a
+
e
;
c
≥
0;
a
0
≥−
φ
}
Thanks to the Principle of Optimality, a solution to this functional equation (as
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View Full DocumentGeneral Equilibrium with Incomplete Markets
2
1.1.
Equilibrium De
f
nition.
A stationary equilibrium is a
c
(
x
)
,a
0
(
x
)
,q
and
ψ
satisfying
1.
c
(
x
)
and
a
0
(
x
)
are optimal given
q
2. Markets clear. In Huggett’s (1993) economy this means
R
S
c
(
x
)
dψ
=
R
S
e
(
x
)
dψ
R
S
a
0
(
x
)
dψ
=0
3.
ψ
is a stationary probability measure
In Aiyagari’s 1994 and 1995 papers and Huggett’s 1997 paper the aggregate supply
of assets is endogenous and derived from a production technology. This is an impor
tant di
f
erence with respect to Huggett 1993 paper in which assets are assumed to be
in zero net supply. Aiyagari and McGrattan 1998 allow two sources of positive asset
supply: capital used in production and government debt. These alternative models
change the market clearing conditions, but do not a
f
ect the form of the individual
consumptionsavings problem.
1.2.
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