LongRun Economic Growth
16 November 2010
1
Reading
•
pp.212231, Chapter 6 of Abel/Bernanke/Croushore
2
The Solow Growth Model
•
If an economy’s output is given by
Y
=
AF
(
K, N
)
,
the growth in output/income over time would be the result of the growth in
total factor productivity
A
, capital input
K
, and labor input
N
. The theory
of long—run economic growth is a theory of growth in productivity and factor
inputs over time.
2.1
Returns to scale
•
The marginal product of capital
∂
Y/
∂
K >
0
but that
∂
2
Y/
∂
N
2
<
0
; likewise
the marginal product of labor
∂
Y/
∂
N >
0
but that
∂
2
Y/
∂
N
2
<
0
.
Now what
if the two factor inputs increase at the same time? Of course, output should
increase, but should the increase be at an increasing or decreasing rate?
•
Suppose the two factor inputs each increases by
x
percent and that
AF
([1 +
x
]
K,
[1 +
x
]
N
)
>
[1 +
x
]
AF
(
K, N
)
.
This condition says that when both factor inputs increase by some
x
percent,
output increases by more than
x
percent. We say that production is subject to
increasing returns to scale (economies of scale). On the other hand, if
AF
([1 +
x
]
K,
[1 +
x
]
N
)
<
[1 +
x
]
AF
(
K, N
)
,
we say that production is subject to decreasing returns to scale (diseconomies
of scale).
1
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•
On an individual
fi
rm level, production can certainly be subject to scale economies
or scale diseconomies, for the economy as a whole, we do not think that either
assumption is plausible. If aggregate production is indeed subject to increasing
returns to scale, there should be higher labor productivities in more populous
countries. Think of two countries
i
and
j
, and that
j
is two times the size of
i
in that
K
j
= 2
K
i
and
N
j
= 2
N
i
. Labor productivity in
i
is
y
i
≡
Y
i
N
i
=
AF
(
K
i
, N
i
)
N
i
.
Labor productivity in
j
is
y
j
≡
Y
j
N
j
=
AF
(
K
j
, N
j
)
N
j
=
AF
(2
K
i
,
2
N
i
)
2
N
i
>
2
AF
(
K
i
, N
i
)
2
N
i
=
y
i
.
Conversely, if aggregate production should be subject to decreasing returns to
scale
y
j
≡
Y
j
N
j
=
AF
(
K
j
, N
j
)
N
j
=
AF
(2
K
i
,
2
N
i
)
2
N
i
<
2
AF
(
K
i
, N
i
)
2
N
i
=
y
i
.
But there does not seem to be any systematic relation between country size
and living standard.
The U.S. is a few times larger than Germany; yet the
two countries have just about the same living standard.
Such considerations
suggest that it is most reasonable to assume that aggregate production is subject
to constant returns to scale, meaning that
AF
([1 +
x
]
K,
[1 +
x
]
N
) = [1 +
x
]
AF
(
K, N
)
,
(1)
whereby an increase in the two factor inputs by
x
percent results in aggregate
output rising by just the same
x
percent.
•
An important consequence from assuming constant returns to scale is that a
country’s labor productivity depends only on the country’s capital—labor ratio
K/N
, but not on the levels of
K
and
N.
To see this, let
[1 +
x
] = 1
/N
in
(1) ;
then the condition becomes
AF
μ
K
×
1
N
, N
×
1
N
¶
=
1
N
×
AF
(
K, N
) ;
i.e.,
y
=
AF
μ
K
N
,
1
¶
≡
Af
(
k
)
,
where
k
=
K/N.
The function
Af
(
k
)
express the production function in per—
worker terms and is a function of the country’s capital—labor ratio. It is impor
tant to recognize that this representation is valid only if aggregate production is
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