Disclaimer: These notes were prepared based on lectures of Prof Salai
Martin’s 2008 Fall
Course of Intermediate MacroW3213. Contents of these notes might not match completely
with the
current
teachings
in
class. An updated version would be available later
in
the semester.
2.13 Measuring Productivity: The method of the Solow Residual.
Imagine that the production function is Cobb Douglas
𝑌
=
??
𝛼
?
1
−α
2. A2.1
.
We have learnt that technology is the source of growth in the long run. Now we want to
MEASURE the growth rate of technology (or the rate of technological progress). Unfortunately,
we cannot do it directly since “technology” cannot be measured. Solow came up with a trick to
measure the growth rate of A:
?
?
=
𝛥?
?
indirectly.
Take logarithms of the production function
𝑌
=
??
𝛼
?
1
−α
.
:
??? 𝑌
=
??? ?
+
???
(
?
𝛼
) +
???
(
?
1
−𝛼
)
Using the laws of logarithms we can bring the exponents down and write the equation as:
???
𝑌
=
???
?
+
𝛼???
?
+ (1
− 𝛼
)
???
?
.
Now remember that the derivative of log x is :
𝛥
log
?
=
𝛥?
?
.
Use this rule to take derivatives of both sides of our equation and get:
𝛥𝑌
𝑌
=
𝛥?
?
+
𝛼
𝛥?
?
+
1
− 𝛼
𝛥?
?
2.10
The key now is to realize that WE CAN MEASURE the following components of the above
equation:
𝛥𝑌
𝑌
is the growth rate of aggregate GDP (this is measured in the national accounts),
𝛥?
?
is the growth rate of the capital stock (this is also measured in the N.A.),
𝛥?
?
is the growth rate of labor (again, this is measured),
(1
α)
is the fraction of total national income that accrues to workers (=wage bill divided by
national income i.e.
??
𝑌
).
α is the fraction of national income that accrues to capital.
Hence, we can measure everything in the above equation except for
𝛥?
?
.
As a result, we can
measure
𝛥?
?
indirectly by rearranging the equation:
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𝛥?
?
=
𝛥𝑌
𝑌
−
𝛼
𝛥?
?
−
1
− 𝛼 𝛥?
?
2.11,
i.e., the growth rate of productivity can be measured as the difference (or the residual) between the
growth rate of GDP
𝛥𝑌
𝑌
and the growth rate of the two inputs
(
𝛥?
?
and
𝛥?
?
) each of which is
weighted by its share or contribution to the production of GDP.
Some Results
In class we presented some results for a bunch of countries (taken from Chapter 10 of the book
“ECONOMIC GROWTH” by Barro and Sala
iMartin.) The main message of the table is that
productivity growth in MIRACLE COUNTRIES (Hong Kong, Singapore, Taiwan, and South
Korea) is NOT really spectacular in the sense that it is comparable to the growth rates for Mexico
or Brazil. Hence, the miracle is NOT why they have so much productivity growth but HOW they
achieved enormous savings and investment rates (which led to large increases in the capital stock
and the stock of labor).
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 Spring '11
 SALAIMARTIN
 Poverty, Productivity model

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