University of Minnesota
Department of Economics
Econ 3102: Intermediate Macroeconomics
Handout 6
1
The representative household’s problem
The representative household enjoys consumption and leisure time in all periods t = 0,
1,...,
∞
. The household has no exogenous income, but it has fixed time endowments in each
period,
¯
h
, and an initial capital endowment,
K
0
.
Assume that the household’s utility function is timeseparable, and takes the form
U
(
C
0
, C
1
, ..., ‘
0
, ‘
1
, ...
) =
∞
X
t
=0
β
t
u
(
C
t
, ‘
t
)
where
C
t
and
‘
t
denote consumption and leisure in period t, u(
·
,
·
) is a period util
ity function which is strictly increasing, strictly concave and satifies the Inada Conditions
(lim
C
t
→
0
u
C
(
C
t
, ‘
t
) =
∞
,
lim
‘
t
→
0
u
‘
(
C
t
, ‘
t
) =
∞
; these help ensure that the solutions are in
terior).
β
∈
(0
,
1) is the household’s discount factor. Note that
N
t
=
¯
h

‘
t
.
1.1 The household’s investment decision and its budget constraints
The household makes investment decisions to increase its initial capital stock over time.
The law of motion for capital is given by:
K
t
+1
= (1

δ
)
K
t
+
I
t
,
t
= 0
,
1
, ...,
∞
where
δ
∈
(0
,
1) is the depreciation rate, and
I
t
denotes the household’s investment in period
t.
In all periods, the household has capital
K
t
and rents it to the firm at a rental rate
r
t
.
After the firm uses the capital, it returns the remaining stock (1

δ
)
K
t
to the household.
Once we include investment
I
t
, the household gets the next period’s capital stock
K
t
+1
.
The household’s budget constraint is given by
C
t
+
I
t
=
w
t
(
¯
h

‘
t
) +
r
t
K
t
t
= 0
,
1
, ...,
∞
We can then summarize the household’s utility maximization problem as
1
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{
C
t
,‘
t
,I
t
}
∞
t
=0
∞
X
t
=0
β
t
u
(
C
t
, ‘
t
)
(1.1)
s.t.
C
t
+
I
t
=
w
t
(
¯
h

‘
t
) +
r
t
K
t
t
= 0
,
1
, ...,
∞
(1.2)
K
t
+1
= (1

δ
)
K
t
+
I
t
,
t
= 0
,
1
, ...,
∞
(1.3)
K
0
given
C
t
≥
0
,
0
≤
‘
t
≤
¯
h
where
N
t
=
¯
h

‘
t
Under our assumptions over u(
·
,
·
), one can prove a solution exists.
(Technically, we
need a transversality condition. Instead we use an assumption that after some T,
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 Spring '08
 MINGYI
 Economics, Macroeconomics, Utility, Trigraph, UCI race classifications, Tour de Georgia

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