Download Solow (1957) “Technical Change and the Aggregate

Solow (1957) “Technical Change and the
Aggregate Production Function”
Robert Solow won a Nobel prize for his work on economic growth that
identified the importance of technological progress
This discovery proceeded from an attempt to decompose GDP growth
into growth attributable to inputs into the aggregate production
function (aggregate capital and labor)
Today, virtually all economists believe that social welfare improvements
in the long run depend more on economic growth and improvements
in labor productivity than on other macroeconomic factors
Technological progress, managerial improvements, and innovation in
general are regarded as key contributors to economic growth
Empirical Approach
For simplicity, assume the aggregate production function can be expressed as:
Q  A(t ) f ( K , L)
Q aggregate output (GDP)
A(t ) a function of time that allows for neutral technological change
f ( K , L) a function of capital and labor
Treat the aggregate production function as an identity, and differentiate both
sides with respect to time
Formal Analysis
Q A
f K
f L

f ( K , L)  A
A
t
t
K t
L t
which can be expressed as
f .
f .
Q  A f ( K , L)  A
K A L
K
L
.
.
Now divide both sides by Q to obtain
.
.
.
Q A
A f . A f . A A f . A f .
 f ( K , L) 
K
L 
K
L
Q Q
Q K
Q L
A Q K
Q L
Some Microeconomic Theory
In a competitive equilibrium, factors are paid their marginal products
f
f
Thus, the wage w  A
and the rental rate r  A
L
K
Substitute these into the derived equation to obtain
.
.
Q A r . w .
  K L
Q A Q
Q
Further Analysis
The derived equation can be expressed as
.
.
.
.
Q A rK K wL L
 

Q A Q K Q L
Note that wL is the aggregate income of labor; rK is the aggregate income
of capital
wL
rK
(this is labor's share of GDP);  
; these can be computed
Q
Q
from data
Let  
Technical Progress as a Residual
.
.
.
.
Q A
K
L
   
Q A
K
L
.
.
.
.
A Q
K
L
Rearrange:     
A Q
K
L
.
All of the terms on the right-hand side can be measured directly (
Q
is the
Q
change in GDP divided by GDP, and so on)
.
A
Thus,
can be determined as a residual based on the growth in GDP that
A
cannot be explained by increasing capital and labor; this series can be used to
construct the technical change index
One Additional Consideration: Returns to Scale
If we assume the aggregate production function exhibits constant returns to
scale, then   1   , and
.
.
.
.
A Q
K
L
    (1   )
A Q
K
L
which can be expressed as
.
.
 .

 .

A Q L
K
L


   
K L
A Q L




.
.
.
.
Q
q Q L
Let q  . Working through the calculus establishes that  
L
q Q L
.
Similarly, if k 
.
.
K
k K L
, then  
L
k K L
Results
.
.
.
A q
k
 
A q
k
Now the technical change index can be determined using series for output
per man hour (labor productivity), capital per man hour, and the share of
capital
Solow examines the series for the period 1909-1949; labor productivity
doubles over this period
Capital's share is constant over the period, and increases in capital per man
hour account for only about 1/8 of the increase in labor productivity; the
rest is due to technical change
Conclusions
Solow’s analysis has been followed by many studies of economic
growth and many attempts to decompose growth into contributing
factors using more complex formulations that allow for such factors
as human capital, technological improvements embodied in plants
and equipment, multiple sectors, and so on
Many growth economists disagree about the fraction of economic
growth that can be explained by technological progress, but virtually
all agree it is important
Solow’s analysis also gives us one simple way to conceptualize
innovation: all improvements in output that cannot be attributed to
growth in quantities of inputs such as labor and capital