Econ 102 Winter 2012
Lecture Note 3
The Solow Growth Model (with no growth)
The Solow model is a simple model of economic growth, with one good that is produced by a CobbDouglas
production technology. The good is used for either consumption (
C
) or investment (
I
):
Y
=
C
+
I
; there is
no government and no trade with other countries. Investment is converted to capital (
K
), but not immedi
ately.
We will assume that the markets for consumption, capital and labor are all perfectly competitive (ﬁrms
are price takers). In addition, we assume that everyone works  there is no diﬀerence between
L
t
and the
population at time
t
. One of the key assumptions we make is that the savings rate of the country is given
fraction
s
and not chosen. This means a fraction
s
of output will be devoted to investment in each period.
In addition, capital
depreciates
at a constant rate
δ
. This means that capital, if left alone, disappears
at a rate
δ
. In order to sustain the same amount of capital, the economy needs to replenish its
capital
stock
(
K
t
) with new investment.
The Solow Model, Mathematically
The above assumptions are consistent with an economy described by these equations:
Y
t
=
F
(
K
t
,L
t
) =
AK
α
t
L
1

α
t
(1)
I
t
=
sY
t
(2)
K
t
+1
=
K
t
+
I
t

δK
t
(3)
(1) Output is produced by a CobbDouglas production function
(2) Investment is a constant fraction of output
(3) Capital depreciates at rate
δ
and accumulates with investment
The values for
A
,
α
,
δ
, and
s
are all given in this model. Variables and parameters that are given from
outside the model are called
exogenous
(Greek for “given from outside”), while variables like
Y
t
,
K
t
and
I
t
, which are determined within the model, are called
endogenous
.
1
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View Full DocumentPercapita Version
In order to simplify our analysis, we can set up a per capita version of the model. Since
F
(
K,L
) is constant
returns to scale,
Y
t
=
F
(
K
t
,L
t
) =
⇒
Y
t
L
t
=
F
(
K
t
/L
t
,
1)
Whenever we use lowercase variables, we will mean percapita. So
Y
t
is total output while
y
t
will stand for
percapita output. Mathematically,
y
t
≡
Y
t
L
t
,
k
t
≡
K
t
L
t
,
i
t
=
I
t
L
t
, etc. For our production function, this means:
y
t
=
F
(
k
t
,
1) =
Ak
α
t
(1)
1

α
Since this looks a bit weird, we’ll use a lowercase
f
(
k
t
) to mean
F
(
k
t
,
1)
y
t
=
f
(
k
t
) =
Ak
α
t
In order to discuss growth, ﬁrst we examine what happens in the model over time.
Our capital accumulation equation tells us that capital in the next period comes from capital in the
current period, plus investment, minus depreciation.
k
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 Spring '11
 d
 Steady State, Capital accumulation, KSS

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