+
S
K
_
K
K
GDP per capita (or worker in our case) is
GDP
L
and
rate
growth in GDP per capita is
d
dt
Ln
´
GDP
L
µ
=
±
GDP
GDP
±
_
L
L
Thus, subtract
_
L=L
from both sides to obtain
G
_
DP
GDP
±
_
L
L
=
s
1
_
p
1
p
1
+
s
2
_
p
2
p
2
+
S
L
_
A
A
+
_
L
L
!
+
S
K
_
K
K
±
_
L
L
)
G
_
DP
GDP
±
_
L
L
=
s
1
_
p
1
p
1
+
s
2
_
p
2
p
2
+ (
S
L
±
1)
_
L
L
+
S
L
_
A
A
+
S
K
_
K
K
Why does labor productivity,
G
_
DP
GDP
±
_
L
L
>
0
;
tend to increase in a recession?
2.5
Contributions to growth
The literature often discusses "contributions to growth." What is this? Con-
sider
G
_
DP
GDP
=
s
1
_
p
1
p
1
+
s
2
_
p
2
p
2
+
S
L
_
A
A
+
_
L
L
!
+
S
K
_
K
K
16

What is the contribution to
G
_
DP
GDP
of each of the right-hand side variables?
These are
C
p
1
=
0
@
s
1
_
p
1
=p
1
±
GDP=GDP
1
A
¶
100
C
p
2
=
0
@
s
2
_
p
2
=p
2
±
GDP=GDP
1
A
¶
100
C
L
=
0
@
S
L
_
L=L
±
GDP=GDP
1
A
¶
100
C
K
=
0
@
s
K
_
K=K
±
GDP=GDP
1
A
¶
100
C
A
=
0
@
S
L
_
A=A
±
GDP=GDP
1
A
¶
100
Suppose we have an economic growth model that predicts in the
long-run
equilibrium, (i.e., the steady state)
_
p
j
=p
j
=
0
_
L=L
=
n
_
K=K
=
x
+
n
_
A=A
=
x
17

Then
C
ss
p
1
=
0
@
s
1
0
±
GDP=GDP
1
A
¶
100
C
ss
p
2
=
0
@
s
2
0
±
GDP=GDP
1
A
¶
100
C
ss
L
=
0
@
S
L
n
±
GDP=GDP
1
A
¶
100
C
ss
K
=
0
@
s
K
x
+
n
±
GDP=GDP
1
A
¶
100
C
ss
A
=
0
@
S
L
x
±
GDP=GDP
1
A
¶
100
What does
C
L
±
C
ss
L
C
K
±
C
ss
K
C
A
±
C
ss
A
tell us?
How far the economy is from long-run equilibrium?
Yes
And,
it provides insight into contributions to growth during "transition" growth.
Finally, if we had identi³ed speci³c sectors, (manf., ag., services) we could
test the "balanced growth" hypothesis.
3
Measurement of capital stock
(see Hall and Jones, and/or Fanjzylber and Lederman Appendix for formula.
See also Hulten, p41-51)
3.1
Method I: often used but not desirable
Here, we skim the surface of this topic.
18

A common method is to estimate the stock of capital for some initial
period, which is referred to as
t
= 0
:
The formula is
K
(0) =
I
g
+
³
(12)
where
I
=
Gross ³xed capital formation in constant LCU
g
=
Rate of growth of GDP in constant LCU
³
=
Rate of depreciation
Once we have calculated
K
(0)
;
we use the law of motion equation
K
(
t
) =
K
(0) (1
±
³
) +
I
(0)
; t
= 0
;
1
;
² ² ²
; T
Does this seem to simple? Yes, it is.
1. Why the formula (12)?
In the steady state ..with balanced growth..
we expect the rate of growth in GDP to equal the rate of growth in
capital stock, i.e.,
_
G
G
=
_
K
K
G
(
t
)
±
G
(
t
±
1)
G
(
t
±
1)
=
K
(
t
)
±
K
(
t
±
1)
K
(
t
±
1)
(13)
K
(
t
) =
K
(
t
±
1) (1
±
³
) +
I
(
t
±
1)
)
K
(
t
)
±
K
(
t
±
1) =
±
K
(
t
±
1)
³
+
I
(
t
±
1)
;
Now
·
by
K
(
t
±
1)
)
K
(
t
)
±
K
(
t
±
1)
K
(
t
±
1)
=
±
³
+
I
(
t
±
1)
K
(
t
±
1)
)
given assumption (
??
)
G
(
t
)
±
G
(
t
±
1)
G
(
t
±
1)
=
±
³
+
I
(
t
±
1)
K
(
t
±
1)
Now solve for
K
(
t
±
1)
G
(
t
)
±
G
(
t
±
1)
G
(
t
±
1)
K
(
t
±
1)
=
±
³K
(
t
±
1) +
I
(
t
±
1)
G
(
t
)
±
G
(
t
±
1)
G
(
t
±
1)
K
(
t
±
1) +
³K
(
t
±
1)
=
I
(
t
±
1)
¶
G
(
t
)
±
G
(
t
±
1)
G
(
t
±
1)
+
³
·
K
(
t
±
1)
=
I
(
t
±
1)
19

to obtain (12)
K
(
t
±
1) =
I
(
t
±
1)
²
G
(
t
)
³
G
(
t
³
1)
G
(
t
³
1)
+
³
³
(14)
2. Problems with
K
(selected)
(a) If we are studying economies in the process of economic growth,
they are unlikely to be in a steady state, so (13) is unlikely to hold
which implies
_
G=G <
_
K=K
in which case (14)
over
-estimates
K
(0)
:

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