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Review Questions:
TwoSector Models
Econ720. Fall 2009. Prof. Lutz Hendricks
Question 1.
Habit Formation
1
Consider an economy composed of a continuum of infinitely lived, identical households who
maximize discounted utility. At date 0 the household’s preferences are:
∑
∞
=
σ
−
ρ
σ
−
−
−
β
0
1
1
]
)
1
(
)
[(
t
t
t
t
t
v
x
c
,
where
v
t
denotes time allocated to the production of consumption or investment goods and
x
is the
habit stock:
1
−
=
t
t
c
b
x
. Its interpretation is: If the household consumed a lot in the past, it dislikes
the idea of reducing consumption today.
There are two production sectors, one produces consumption goods according to
α
−
α
ψ
ϕ
=
1
)
(
)
(
t
t
t
t
t
v
K
c
. The other sector produces new capital according to
α
−
α
+
ψ
−
ϕ
−
+
δ
−
=
1
1
)
]
1
([
)
]
1
([
)
1
(
t
t
t
t
t
t
v
K
K
K
.
(a) Formulate and solve the planning problem using a Lagrangean. Interpret the Euler equations.
(b) Formulate and solve the planning problem using Dynamic Programming.
(c) Formulate the competitive equilibrium. Show that the Euler equations coincide with those for
the planning problem.
Answer: Habit Formation
Note that the two technologies are the same, so that a onesector reduced form exists.
Also note that in a Dynamic Programming setup the state vector is (
K, x
), not just
K
.
(a) The first trick is to note that this is really a onesector economy because the technologies in both
sectors are identical. We can therefore combine the two resource constraints into
t
t
t
t
t
c
v
K
K
K
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 '09
 LUTZHENDRICKS
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