# 6.13 Measuring Technology Shocks: The Solow Residual

If technology shocks are the primary cause of business cycles, then it is important

to identify and measure the rate of technological progress. Given the

structure of real business cycle models, the key parameter is the variance of the

technology shock. Prescott (1986) suggests that Solow’s method of measuring

this variance is an acceptable and reasonable approach. Solow’s (1957) technique

was to define technological change as changes in aggregate output minus

the sum of the weighted contributions of the labour and capital inputs. In short,

the Solow residual measures that part of a change in aggregate output which

cannot be explained by changes in the measurable quantities of capital and

labour inputs. The derivation of the Solow residual can be shown as follows.

The aggregate production function in equation (6.13) shows that output (Y) is

dependent on the inputs of capital (K), labour (L) and the currently available

technology (A) which acts as an index of total factor productivity:

Y AF(K, L) (6.13)

Output will change if A, K or L change. One specific type of production

function frequently used in empirical studies relating to growth accounting is

the Cobb–Douglas production function, which is written as follows:

Y AKL1−, where 0 1 (6.14)

In equation (6.14) the exponent on the capital stock measures the elasticity of

output with respect to capital and the exponent on the labour input (1 – )

measures the elasticity of output with respect to labour. The weights and 1 –

measure the income shares of capital and labour, respectively (see Dornbusch

et al., 2004, pp. 54–8 for a simple derivation). Since these weights sum to unity

this indicates that this is a constant returns to scale production function. Hence

an equal percentage increase in both factor inputs (K and L) will increase Y by

the same percentage. By rearranging equation (6.14) we can represent the

productivity index which we need to measure as equation (6.15):

Solow residual−A

Y

KL1 (6.15)

Because there is no direct way of measuring A, it has to be estimated as a

residual. Data relating to output and the capital and labour inputs are available.

Estimates of and hence 1 – can be acquired from historical data.

Since the growth rate of the product of the inputs will be the growth rate of A

plus the growth rate of Kplus the growth rate of L1–, equation (6.15) can be

rewritten as (6.16), which is the basic growth accounting equation that has

been used in numerous empirical studies of the sources of economic growth

(see Denison, 1985; Maddison, 1987).

Equation (6.16) is simply the Cobb–Douglas production function written in a

form representing rates of change. It shows that the growth of output (Y/Y)

depends on the contribution of changes in total factor productivity (A/A),

changes in the weighted contribution of capital (K/K) and changes in the

weighted contribution of labour (1 – )(L/L). By writing down equation

(6.15) in terms of rates of change or by rearranging equation (6.16), which

amounts to the same thing, we can obtain an equation from which the growth

of total factor productivity (technology change) can be estimated as a residual.

This is shown in equation (6.17).

A

A

Y

Y

K

K

L

L

−−

(1 ) (6.17)

In equation (6.17) the Solow residual equals A/A. Real business cycle theorists

have used estimates of the Solow residual as a measure of technological

progress. Prescott’s (1986) analysis suggests that ‘the process on the percentage

change in the technology process is a random walk with drift plus some

serially uncorrelated measurement error’. Plosser (1989) also argues that ‘it

seems acceptable to view the level of productivity as a random walk’. Figure

6.9 reproduces Plosser’s estimates for the annual growth rates of technology

and output for the period 1955–85 in the USA. These findings appear to

support the real business cycle view that aggregate fluctuations are induced,

in the main, by technological disturbances. In a later study, Kydland and

Prescott (1991) found that about 70 per cent of the variance in US output in

the post-war period can be accounted for by variations in the Solow residual.

We will consider criticisms of the work in this area in section 6.16 below. In

particular, Keynesians offer an alternative explanation of the observed

procyclical behaviour of productivity.

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