Noah Williams
Economics 714
Department of Economics
Macroeconomic Theory
University of Wisconsin
Spring 2009
Midterm Examination Solutions
Instructions:
This is a 75 minute exam with two questions worth a total of 100 points.
Points are indicated at the start of each question.
Allocate your time wisely.
In order
to get full credit, you must give a clear, concise, and correct answer, including all necessary
explanations and calculations. Notes, books, and calculators are not permitted.
1. (
50 points
) Consider the following version of a Lucas asset pricing model, in which an
agent wants to minimize ﬂuctuations in consumption around a mean. For simplicity
we normalize the mean to zero. In other words, a representative agent has preferences:

1
2
E
0
∞
X
t
=0
β
t
c
2
t
over the single nonstorable consumption good
c
t
(“fruit”). The endowment process
x
t
is also mean zero and follows a Gaussian
AR
process:
x
t
+1
=
ρx
t
+
ε
t
+1
where 0
< ρ <
1 and
ε
t
+1
is an i.i.d.
N
(0
,σ
2
) random variable.
(a) (
15 points
) Deﬁne a recursive competitive equilibrium with a market in claims
to the endowment process (“trees”), with pricing function
p
(
x
).
Solution.
A recursive competitive equilibrium is a value function
V
(
a,x
)
, a pair
of policy functions
c
(
a,x
)
and
a
0
(
a,x
)
, and a pricing function
p
(
x
)
such that
(
i
)
V
(
a,x
) = max
c,a
0
{
u
(
c
) +
βE
[
V
(
a
0
,x
0
)

x
]
}
st
c
+
p
(
x
)
a
0
=
(
p
(
x
) +
x
)
a
x
0
=
ρx
+
ε
0
(
ii
)
c
(
a,x
)
, a
0
(
a,x
)
attain the maximum above,
(
iii
)
markets clear:
a
0
= 1
, c
(1
,x
) =
x.
Here we can substitute in
u
(
c
) =

1
2
c
2
and the expectation can be written as:
E
[
V
(
a
0
,x
0
)

x
] =
Z
V
(
a
0
,ρx
+
ε
0
)
d
Φ(
ε
0
)
where
Φ
is the standard normal distribution function.
1