Econ 387L: Macro II
Spring 2008, University of Texas
Instructor: Dean Corbae
Midterm Exam
I. Consider the following OG model of production. Each period a new generation of
individuals are born and live for 2 periods. There is population growth of size
n
≥
0
so that
the measure of young alive at time
N
t
= (1+
n
)
t
N
0
with
N
0
= 1
(i.e. the measure of agents
of born in period
t
= 0
is normalized to 1) and the initial old are taken to be of measure
N
−
1
= 1
/
(1 +
n
)
.
Each agent from a generation born at time
t
has preferences given by
U
t
= ln(
c
t
t
) +
β
ln(
c
t
t
+1
)
where
c
g
t
denotes individual consumption (i.e. per capita) of generation
g
at time
t
(so that
c
t
t
represents consumption in youth and
c
t
t
+1
represents consumption in old age of a generation
t
individual) and
β
∈
(0
,
1)
such that
β
(1 +
n
)
<
1
.
Each individual can only work at most a
unit amount of time in youth. There is a constant returns to scale production function where
aggregate output
Y
t
is given by
Y
t
=
K
α
t
L
1
−
α
t
where
K
t
is aggregate capital,
L
t
is aggregate labor input, and
α
∈
(0
,
1)
.
There is complete
depreciation of capital (i.e.
δ
= 1)
so that the law of motion for aggregate capital is given by
K
t
+1
=
I
t
with
K
0
given. In what follows let per capita and aggregate variables be related
as
x
t
=
X
t
/N
t
so that
k
t
=
K
t
/N
t
.
1. Suppose that a social planner chooses
{
c
t
t
, c
t
−
1
t
, k
t
+1
}
∞
t
=0
to maximize the following
welfare function
1
1 +
n
ln(
c
−
1
0
) +
∞
X
t
=0
β
t
N
t
U
t
.
a. (2.5 points) What is the per capita feasibility constraint facing the planner in any given
period
t
?
b. (5 points) What are the
fi
rst order conditions for this problem?
c. (2.5 points) Manipulate the
fi
rst order conditions to arrive at a condition like an Euler
equation.
d. (5 points) What do the
fi
rst order conditions imply about consumption of the young
and the old in any given period (i.e.
c
t
t
and
c
t
−
1
t
)?
e. (2.5 points) Assume that the sequence
{
k
t
+1
}
∞
t
=0
implied by the
fi
rst order conditions
converges. What is the steady state capital stock in the planner’s solution? What is the
implied steady state interest rate?
f. (15 points) Linearize the necessary conditions around the steady state. What conditions
need to be satis
fi
ed for the system to be locally stable? If you are running out of time, at
least discuss how you would do this step.
g. (5 points) Is there balanced growth of aggregate output? At what rate?
h. (2.5 points) How do these results relate to what you would
fi
nd in a steady state of the
RBC model?
1

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