Homework 6. Solution
Graduate Macroeconomics. Viktor Tsyrennikov
Question 1.
Your task is to solve a model in which consumption growth
process resembles that in the data and compare the model’s predictions about
the equity premium with the data.
Model environment is as follows. Uncertainly is modeled by a stochastic
shock
λ
t
that follows a twostate firstorder Markov process with a transition
matrix Π. That is
π
(
λ
t

λ
0
) =
π
(
λ
t

λ
t

1
)
...π
(
λ
2

λ
1
)
π
(
λ
1

λ
0
)
.
An agent’s preferences are given by
U
(
c
) =
E
[
∞
summationdisplay
t
=0
β
t
u
(
c
t
)

λ
0
] =
∞
summationdisplay
t
=0
summationdisplay
λ
t
β
t
π
(
λ
t

λ
0
)
u
(
c
t
(
λ
t
))
.
Assume that
u
(
c
) =
c
1

γ
/
(1

γ
). The agent faces the (sequential) budget
constraint:
c
t
(
λ
t
) +
summationdisplay
λ
t
+1
Q
t
(
λ
t
+1

λ
t
)
a
t
+1
(
λ
t
+1
)
lessorequalslant
y
t
(
λ
t
) +
a
t
(
λ
t
)
,
where income of the agent is given by
y
t
(
λ
t
) =
λ
t
...λ
2
λ
1
,
∀
λ
t
, t
greaterorequalslant
1 and
y
0
(
λ
0
) = 1.
Assume annual frequency.
a) Construct the transition matrix and the two states to match the fol
lowing data facts:
1
M1. Expected growth rate is 0.018%.
M2. Std. deviation of the growth rate is 0.035%.
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 Spring '11
 ViktorTsyrennikov
 Macroeconomics, Riskfree interest rate, Viktor Tsyrennikov, Solution Graduate Macroeconomics

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