Final Exam, Macro II, Spring 2008
Prof. Dean Corbae.
This version: May 13, 2008
Question I. (25 points)
Consider the following in
fi
nite horizon, exchange economy with a single nonstorable consumption
good in each period. There are a large number of each of two types of agents: type
S
and type
B
.
Type
S
are more patient than
B
; that is,
β
S
> β
B
.
The preferences of an agent of type
i
∈
{
B, S
}
are given by
P
β
i
u
(
c
i
t
)
where
u
(
c
i
t
) =
£
(
c
i
t
)
1
−
σ
¤
/
(1
−
σ
)
.
Both consumers have the same constant
endowment stream
y
in each period. Each period
t
they can trade a risk free, one period bond
b
t
+1
at price
q
t
and both start with no bonds (i.e.
b
0
= 0
). Suppose further that each period there is a
borrowing constraint
b
t
+1
≥
b
.
a. (2.5 points)Write down the maximization problem of an agent.
Answer.
max
{
c
i
t
≥
0
,b
i
t
+1
≥
b
}
X
β
i
u
(
c
i
t
)
s.t.c
i
t
+
q
t
b
i
t
+1
=
y
+
b
i
t
b. (2.5 points) De
fi
ne a symmetric equilibrium (where symmetry implies that all agents of type
i
take the same actions).
Answer.
A symmetric eqm is where given prices
q
∗
,
©
c
i
∗
t
ª
∞
t
=0
and
©
b
i
∗
t
+1
ª
∞
t
=0
solve the above
problem for each
i
∈
{
B, S
}
and markets clear:
c
S
∗
t
+
c
B
∗
t
= 2
y
for each
t
. (Note that this also implies
asset market clearing:
b
S
∗
t
+1
+
b
B
∗
t
+1
= 0
)
c. (12.5 points) Is there a steady state equilibrium where the borrowing constraint is not binding?
If so, characterize it (i.e. what are prices and consumption allocations in the steady state?). If not,
characterize the dynamic path of consumption and bond holdings of each type of agent.
Answer.
Suppose the constraint is not binding. Then the euler equations of each type of agent
are given by
q
t
=
β
S
Ã
c
S
t
+1
c
S
t
!
−
σ
=
β
B
Ã
c
B
t
+1
c
B
t
!
−
σ
which implies
c
S
t
+1
c
S
t
>
c
B
t
+1
c
B
t
so that there can be no steady state. Market clearing implies
c
S
t
+1
c
S
t
>
1
>
c
B
t
+1
c
B
t
so that
c
S
t
→
2
y
and
c
B
t
→
0
as
t
→ ∞
.
Further,
q
t
< β
S
and
q
t
→
β
S
as
t
→ ∞
.
Finally, the
bondholdings of the type
B
agent converges to
−
y/
(1
−
β
S
)
as
t
→ ∞
.
1
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d.
(7.5 points) Is there a steady state equilibrium where a borrowing constraint which says
b
t
+1
≥
b
=
−
(
y
−
ε
)
/
(1
−
β
S
)
is binding? If so, for which type of agent is it binding and what are
steady state equilibrium prices and allocations?
Answer.
Given the answer above, it is the type
B
agent who has the binding constraint. In the
steady state his budget set is given by
c
B
=
y
+
b
B
(1
−
q
)
where
q
=
β
S
Since the constraint binds,
c
B
=
y
+
b
B
(1
−
q
) =
y
−
(
y
−
ε
)(1
−
β
S
)
/
(1
−
β
S
) =
ε.
Market clearing implies
c
S
= 2
y
−
ε
. Finally,
b
S
= (
y
−
ε
)
/
(1
−
β
S
)
.
Question II. (55 points)
Consider the following search model with indivisible commodities and indivisible money. There is
a unit measure of agents and a unit measure of nonstoreable indivisible commodity (i.e. consumption
good) types. The proportion of commodity types that can be consumed by any given agent is denoted
x
∈
(0
,
1)
and
x
also equals the proportion of agents that can consume any given commodity. If an
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