Problem Set 9
Econ 702, Spring 2004
April 8, 2004
Keep in mind that so far we looked at the following models of heteroge
nous agents economies and in the first four questions, we will look at the
stationary equilibria of these economies.
1. Heterogenous agents economy with no trade: Archipelago with a con
tinuum of islands where fishermen cannot swim.
2. Economy with heterogenous agents and trade: Huggett Economy.
3. Economy with heterogenous agents and production: Aiyagari Economy.
Problem 1 (Fishermen Economy)
Solution:
Given
¯
B
∈ F
,
B
a
and
B
e
are fixed,
The agent decision rule,
g
(
e, a
) must be in the asset space so that
g
(
e, a
)
∈
A
∀
(
e, a
)
∈
(
E
×
A
). Since
¯
B
a
⊂
A
, we know that g(e,a) is either in
¯
B
a
or
not in
B
a
. Then 1
g
(
e,a
)
∈
¯
B
a
= 1 or 0.
Γ is a transition matrix for Markov chain, so that Γ
ee
is measurable. Then
∑
e
∈
B
e
Γ
ee
is measurable as well.
∑
e
∈
B
e
Γ
ee
multiplied by 1 or 0 is measurable
as well. Then
Q
(
.,
¯
B
is measurable.
Problem 2 (Heterogenous Agents Economy with Trade)
Solution:
A stationary recursive competitive equilibrium in the loan economy is
{
q
*
, g
(
s, a
;
q
*
)
, V
(
s, a
;
q
*
)
, x
*
(
q
*
)
, Q
(
s, a, B
;
q
*
)
}
such that,
1
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1. (Agents optimize) Given
q
*
,
{
g
(
s, a
;
q
*
)
, V
(
s, a
;
q
*
)
}
solve the agent’s
problem.
2. (Consistency)
Q
(
s, a, B
;
q
*
) is a transition function associated with Γ
ss
and
g
(
s, a
;
q
*
),
Q
(
s, a, B
;
q
*
) = 1
[
g
(
s,a
;
q
*
)
∈
B
a
]
s
∈
B
s
Γ
ss
(1)
3. (Stationarity)
x
*
is the unique stationary distribution associated with
Q
(
s, a, B
;
q
*
),
x
*
(
B
) =
(
S
×
A
)
Q
(
s, a, B
;
q
*
)
dx
*
(
B
)
∀
B
(2)
4. (Markets Clear)
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 Spring '08
 Staff
 Economics, Equilibrium, Probability theory, Markov chain, Economic equilibrium, Aiyagari Economy

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