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NOTES ON SOLOW GROWTH MODEL
L
t
= Labor Force at beginning of year t
K
t
= Capital Stock at beginning of year t
C
t
= Consumption over year t
I
t
= Investment over year t
Y
t
= Output over year t
k
t
= K
t
/L
t
= capital per worker
c
t
= C
t
/L
t
= consumption per worker (“standard of living”)
i
t
= I
t
/L
t
= investment per worker
y
t
= Y
t
/L
t
= output per worker
Distinguish between LR economic growth versus fluctuations around the trend
Assumptions
:
1.
NX = 0 and G = 0
b
Y = C + I
2. Labor force grows at the same rate as the population (let n = population growth rate)
3. CobbDouglas production function: Y
t
= A*K
t
α
L
t
1
α
.
PERWORKER PRODUCTION FUNCTION
To convert this production function into the “perworker production function”, divide both sides
by L
t
:
(Y
t
/L
t
)= (A*K
t
α
L
t
1
α
)/L
t
= A*K
t
α
L
t

α
= (A) (K
t
/L
t
)
α
b
y
= Ak
α
. This is the perworker
production function.
For convenience, I
will be dropping the time subscript for the perworker
variables (with a few exceptions) and use lowercase letters.
y
(
output per worker)
y
= Ak
α
Δ
y
Δ
k
k
(capital per worker)
Slope = (
Δ
y/
Δ
k) = MPK
Notice that the slope gets flatter and flatter. This is due to diminishing marginal returns to capital
(i.e.,
MPK falls as k rises, other things constant
)
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View Full DocumentThe conclusion of the Solow Growth Model is that in the absence of technology changes, the
economy ends up in a steadystate. A
steadystate
is, in this context, a situation in which k,i,c,
and y are constant. NOTE: It is the
ratio
that is constant NOT the
level
.
To see this,
we need to first study investment in a steadystate.
1. I
t
=
δ
K
t
+
(K
t+1
– K
t
)
where
δ
is the depreciation rate of capital
Investment to
Net investment
replace worn
out capital
Gross investment (the lefthand side—I
t
) is used to replace worn out capital (the term
δ
K
t
) and to add
to the capital stock. The addition to the capital stock is called net investment
and is given by the
term (K
t+1
– K
t
) .
Question
: What is
net
investment (the last term on the righthand side of the above equation) in a
steady state
?
The answer is nK
t
.
To understand this, consider the following three points:
a. k = K
t
/L
t
.
b. k is constant in a steady state.
c. L
t
grows at annual rate of n.
As you can see, if the denominator grows at rate n, and the ratio k is constant, then the
numerator must also grow at rate n.

NOTE
—Perhaps a better way to understand this last point is as follows.
1.
L
t+t
= (1+n)L
t
by the assumption that the labor force grows at rate n.
2.
K
t+1
= (1+g)K
t
with g = growth rate of capital.
If we divide (2) by (1) we get:
(K
t+1
/ L
t+1
) =
[(1+g)/(1+n)] (K
t
/ L
t
)
k
t+1
= [(1+g)/(1+n)] (k
t
)
In a steady state,
k
t+1
= k
t
which can only be true if g = n.
And so
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 Spring '08
 Serra
 Macroeconomics

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