Homework 5, Econ 606
Jonathan Heathcote
Due in class, Tuesday April 4th
Consider the following consumptionsavings problem:
An individual faces two possible realizations for their wage in each period,
w
∈
{
w
l
,w
h
}
where
0
<w
l
<w
h
.
At time zero,
w
0
may be high or low with equal probability.
In subsequent periods, wages evolve stochastically according to a Markov
process de
f
ned by the transition probability matrix
π.
The individual must choose consumption
c
t
, savings in a noncontingent
bond
a
t
+1
and hours worked,
n
t
at each date
t
to maximize expected lifetime
utility, where utility associated with an allocation
{
c
t
,n
t
}
∞
t
=0
is given by
∞
P
t
=0
β
t
u
(
c
t
,n
t
)
u
(
c
t
,n
t
)=
1
1
−
γ
"
c
t
−
ψn
1+
1
ε
t
1+
1
ε
#
1
−
γ
where
ψ,ε,γ >
0
(these are known as Greenwood, Hercowitz and Hu
f
mann
preferences)
Suppose that initial wealth at time zero is given by
a
0
and that borrowing
is not permitted:
φ
=0
.
Hours and consumption must be nonnegative (there
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 '05
 would
 Economics, Utility, Probability theory, Markov chain, Andrey Markov

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