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 = AKδL1−δ , where 0 < δ < 1 (6.14)
In equation (6.14) the exponent on the capital stock δ measures the elasticity of
output with respect to capital and the exponent on the labour input (1 – δ)
measures the elasticity of output with respect to labour. The weights δ and 1 – δ
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
KδL1 δ (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 δ and hence 1 – δ 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 Kδ plus the growth rate of L1–δ, 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 (ΔY/Y)
depends on the contribution of changes in total factor productivity (ΔA/A),
changes in the weighted contribution of capital (δΔK/K) and changes in the
weighted contribution of labour (1 – δ)(ΔL/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.
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.
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 = AKδL1−δ , where 0 < δ < 1 (6.14)
In equation (6.14) the exponent on the capital stock δ measures the elasticity of
output with respect to capital and the exponent on the labour input (1 – δ)
measures the elasticity of output with respect to labour. The weights δ and 1 – δ
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
KδL1 δ (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 δ and hence 1 – δ 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 Kδ plus the growth rate of L1–δ, 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 (ΔY/Y)
depends on the contribution of changes in total factor productivity (ΔA/A),
changes in the weighted contribution of capital (δΔK/K) and changes in the
weighted contribution of labour (1 – δ)(ΔL/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.
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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