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:

( ) ( , )

aggregate output (GDP)

( ) a function of time that allows for neutral technological change

( , ) a function o

Q A t f K L

Q

A t

f K L

f capital and labor

Treat the aggregate production function as an identity, and differentiate both

sides with respect to time

Formal Analysis

. . . .

. . .. . . .

( , )

which can be expressed as

( , )

Now divide both sides by to obtain

( , )

Q A f K f Lf K L A A

t t K t L t

f fQ A f K L A K A L

K L

Q

Q A A f A f A A f A ff 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

Thus, the wage and the rental rate

Substitute these into the derived equation to obtain

f fw A r A

L K

Q A r wK L

Q A Q Q

Further Analysis

. . . .

The derived equation can be expressed as

Note that is the aggregate income of labor; is the aggregate income

of capital

Let (this is labor's share of GDP); ; these c

Q A rK K wL L

Q A Q K Q L

wL rK

wL rK

Q Q

an be computed

from data

Technical Progress as a Residual

. . . .

. . . .

.

.

Rearrange:

All of the terms on the right-hand side can be measured directly ( is the

change in GDP divided by GDP, and so on)

Thus, can be determined as a residu

Q A K L

Q A K L

A Q K L

A Q K L

Q

Q

A

A

al based on the growth in GDP that

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

(1 )

which can be expressed as

Let . Working th

A Q K L

A Q K L

A Q L K L

A Q L K L

Qq

L

. . .

. . .

rough the calculus establishes that

Similarly, if , then

q Q L

q Q L

K k K Lk

L k K L

Results

. . .

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 pe

A q k

A q k

riod 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