One-Sector Growth Model Notes
1 The representative household's problem
The representative household enjoys consumption and leisure time in all periods t = 0, 1,.
..,
f
.
The household has no exogenous income, but it has fixed time endowments in each period,
h
,
and an initial capital endowment,
0
K
.
Assume that the household's utility function is time-separable, and takes the form
01 01
0
(,,
.
,,,
)
(,
)
t
tt
t
UC C L L
uC h L
E
f
±
¦
where
t
C
and
t
hL
±
denote
consumption
leisure
in
period
t
,
u(·,·) is a period utility
function which is strictly increasing, strictly concave and satifies the Inada Conditions
(
0
lim
(
,
)
, lim
(
,
)
Ct
t
Lt
t
CL
h
uCh L
oo
±
f
±
f
; these help ensure that the solutions are
interior).
(0,1)
²
is
the
household
's
discount
factor
.
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:
1
(1
)
,
0,1, .,
t
KK
I
t
G
³
±³
f
where
(0,1)
²
depreciation rate
,
t
I
denotes
investment in
period t.
In all periods, the household has capital
t
K
rents
it
to the
firm
at a
rental rate
t
r
. After
the firm uses the capital, it returns the remaining stock (1
)
t
K
±
to
household. Once
we
include investment
t
I
, the household gets the next period's capital stock
1
t
K
³
.