Theoretical framework
The costbased estimate for capacity output and capacity utilization was revolutionized in scholars aiming to unravel the productivity shocks in the aftermath of the 1970s oil price shock (Berndt and Hess 1986; Morrison 1985, 1988, 1999). Conceptually, capacity utilization is measured by the ratio of actual to optimal steady state output or the desired level of output (Y/Y*). This measurement is a shortrun concept. Accordingly, the desired level of output is the one that corresponds with the tangency of the shortrun average cost (SRATC) curve with the longrun average cost (LRATC). This tangency occurs at the minimum of the SRATC if we assume a constant returns to scale technology. Accordingly, the capacity utilization rate estimated in this manner can be both above and below one.
Figure 4, adapted from Morrison (1988), illustrates the costbased theory that underpins conceptualization and calculation of capacity utilization based on cost functions. Figure 4 summarizes this framework. If the output level is at A, it is the desired level since there is no costreduction potential by changing capacity utilization, given the production technology and fixed capacity (scale), which implies that the capacity utilization rate is equal to one. On the other hand, if realized output is equal to C, there is capacity overutilization so that the costbased capacity utilization is above one. The opposite occurs if output is at B. Since the position and shape of the shortrun average cost curve depend on the level of factor prices and the amount of capital, it is straightforward to imply that the capacity utilization rate depends on factor prices. The information that indicates how a factor price affects capacity utilization is the elasticity of capacity utilization rate with respect to factor prices (Berndt and Hess 1986).
Figure 4 also illustrates the relationship between capacity utilization and the costbased measure of productivity, which measures the decline in the average cost of producing a given level of output. If, for example, we consider the case of capacity underutilization (output level B), producers could either change their production technology or adjust their capacity, which changes capacity output. Both of these could improve the capacity utilization rate. This is the theoretical linkage between input prices and capacity utilization (Morrison 1985, 1988), a relationship that can be measured using the elasticity of capacity utilization with respect to variable input prices. In addition, the Hicksian theory of induced innovation implies that an increase in the price of one production factor relative to other factor prices induces a sequence of technical changes that can reduce the use of the factor whose cost has risen (Kennedy 1964). This is known as “biased technological progress” in the sense that it saves the use of the relatively expensive inputs. Accordingly, estimation of the factorbias of technological progress allows us to establish the longrun relationship between input prices and technological progress. That is, if a disembodied technical progress is factoriusing (saving), then a rise in the price of factor i, all else remaining the same, reduces (increases) the rate of multifactor productivity growth (Jorgenson and Wilcoxen 1993).^{Footnote 4}
The restricted translog cost function
Specification of the restricted (shortrun) cost function allows us to analytically derive a measurement for costbased capacity utilization and its elasticity with respect to input prices. It also allows us to derive an expression for the growth rate in total factor productivity. Following Berndt and Hess (1986), the translog restricted cost specification is given as follows:
$$\begin{aligned} \ln VC & = \alpha_{0} + \alpha_{Y} \ln Y + \frac{1}{2}\gamma_{YY} \left( {\ln Y} \right)^{2} + \beta_{K} \ln K + \frac{1}{2}\gamma_{KK} \left( {\ln K} \right)^{2} + \alpha_{t} t \\ & \quad + \frac{1}{2}\alpha_{tt} t^{2} + \mathop \sum \limits_{i} \alpha_{i} \ln P_{i} + \frac{1}{2}\mathop \sum \limits_{i} \mathop \sum \limits_{j} \gamma_{ij} \ln P_{i} \ln P_{j} + \mathop \sum \limits_{i} \rho_{Yi} \ln P_{i} \ln Y \\ & \quad + \mathop \sum \limits_{i} \rho_{Ki} \ln P_{i} \ln K + \mathop \sum \limits_{i} \rho_{ti} t\ln P_{i} + \rho_{tY} t\ln Y + \rho_{tK} t\ln K, \\ \end{aligned}$$
(1)
where lnVC is log of variable costs, lnP_{i}s are the variable input prices, t is the time trend, i and j are labor (L), energy (E), raw materials (M), and Services (S); and lnK and lnY are logs of capital input and output, respectively. Restrictions regarding symmetry (γ_{ij} = γ_{ji}) and linear homogeneity of degree one in prices are still applicable but, restrictions guaranteeing homogeneity of a constant degree in K and Y are also imposed in addition. These are given as:
$$\begin{aligned} \mathop \sum \limits_{i} \alpha_{i} & = 1; \mathop \sum \limits_{i} \gamma_{ij} = 0 \forall i \& j; \mathop \sum \limits_{i} \rho_{ti} = \mathop \sum \limits_{i} \rho_{Ki} = \mathop \sum \limits_{i} \rho_{Yi} = 0 \\ & \quad \gamma_{YY} + \gamma_{KK} = \rho_{Yi} + \rho_{Ki} = \rho_{tY} + \rho_{tK} = 0 \\ \end{aligned}$$
(2)
so that α_{Y} + β_{K} = η, a measure of longrun returns to scale. After imposing the restriction of homogeneity of a constant degree in K and Y, we can write Eq. 1 as
$$\begin{aligned} \ln VC & = \alpha_{0} + \alpha_{Y} \ln Y + \beta_{K} \ln K + \frac{1}{2}\gamma_{YY} \left( {\ln \left( \frac{Y}{K} \right)} \right)^{2} \\ & \quad + \alpha_{t} t + \frac{1}{2}\alpha_{tt} t^{2} + \mathop \sum \limits_{i} \alpha_{i} \ln P_{i} + \frac{1}{2}\mathop \sum \limits_{i} \mathop \sum \limits_{j} \gamma_{ij} \ln P_{i} \ln P_{j} \\ & \quad + \mathop \sum \limits_{i} \rho_{Yi} \ln P_{i} \ln \left( \frac{Y}{K} \right) + \mathop \sum \limits_{i} \rho_{ti} t\ln P_{i} + \rho_{tY} t\ln \left( \frac{Y}{K} \right) \\ \end{aligned}$$
(3)
Optimal variable cost share equations are obtained by logarithmically differentiating Eq. 3 with respect to the input prices and is given as
$$s_{i} = \alpha_{i} + \mathop \sum \limits_{i} \gamma_{ij} \ln P_{j} + \rho_{Yi} \ln \left( {Y/K} \right) + \rho_{ti} t$$
(4)
Equations 3 and 4 are estimated by preimposing the restrictions pertaining to the addingup, symmetry, and homogeneity of degree one in prices. Estimating the cost function along with the share equations permits efficiency gains without loss of a degree of freedom since doing so does not imply estimating more parameters than those of the cost function. What this requires is estimation of the system of equations by preimposing the relevant restrictions. The adding up restriction \((\sum {\alpha }_{i}=1)\) requires the exclusion of one of the share equations. This can be done by excluding any of the share equations since the results are not sensitive to the share equation chosen to be excluded. We estimate by excluding the service input share equation. Thus, the elasticities for this variable will be estimated using the identity that the service share equation is equal to \(1{s}_{s}=1\sum_{i}{s}_{i}\) where i in \({s}_{i}\) stands for labor, energy, and raw materials. Homogeneity of degree one in prices requires restrictions \(\sum_{i}{\gamma }_{ij}=0 \forall i \&j; \sum_{i}{\rho }_{ti}=\sum_{i}{\rho }_{Yi}.\) In addition to these, estimating the cost function along with the share equations also requires crossequation restrictions that guarantee the estimated coefficients are identical in cost and share equations. For example, the coefficient of the unit price of labor in the labor share equation must be equal to the coefficient of the square of the unit labor price in the cost equation. All together, the estimation using Eq. 3 and its corresponding share equation involves 22 restrictions. Then, the elasticities with respect to Y and K are computed as follows:
$$\frac{\partial \ln VC}{{\ln Y}} = \alpha_{Y} + \gamma_{YY} \ln \left( \frac{Y}{K} \right) + \mathop \sum \limits_{i} \rho_{Yi} \ln P_{i} + \rho_{tY} t$$
(5)
$$\frac{\partial \ln VC}{{\ln K}} = \alpha_{K}  \left( {\gamma_{YY} \ln \left( \frac{Y}{K} \right) + \mathop \sum \limits_{i} \rho_{Yi} \ln P_{i} + \rho_{tY} t} \right)$$
(6)
After estimating the parameter values, we follow the following definitions to obtain the summary measurements needed for our interpretations and generalizations:

i.
The bias of technological progress is determined by the coefficient \({\rho }_{it}\). If \({\rho }_{it}<0 (>0)\), technological progress is inputisaving (using), and therefore, an increase in the price of that input contributes to improvements in (worsening of) total factor productivity growth.

ii.
The AllenUzawa partial elasticity of substitution is computed as
$$\delta_{ij} = \frac{{\gamma_{ij} + \widehat{{s_{i} }}\widehat{{s_{j} }}}}{{\widehat{{s_{i} }}\widehat{{s_{j} }}}}, i \ne j\;{\text{and}}\;\delta_{ii} = \frac{{\gamma_{ii} + \widehat{{s_{i} }}^{2}  \widehat{{s_{j} }}}}{{\widehat{{s_{i} }}^{2} }},$$
(7)
and the price elasticities are computed from the AllenUzawa partial elasticity as
$$\varepsilon_{ij} = \widehat{{s_{j} }}\delta_{ij} = \frac{{\gamma_{ij} }}{{\widehat{{s_{i} }}}} + \widehat{{s_{j} }}\;{\text{and}}\;\varepsilon_{ii} = \widehat{{s_{i} }}\delta_{ii} = \frac{{\gamma_{ii} }}{{\widehat{{s_{i} }}}} + \widehat{{s_{1} }}  1$$
(8)
where \(\widehat{{s}_{i} } \mathrm{and} \widehat{{s}_{j} }\) denote the fitted shares, that must be positive according to the monotonicity restrictions. Since the shares for service inputs is not estimated, their values are generated using \({\widehat{s}}_{s}=1{\widehat{s}}_{L}{\widehat{s}}_{E}{\widehat{s}}_{M}\).
Positive values of crosselasticity suggest that the inputs are substitutes, while negative values suggest that they are complements. Note that these elasticities are not symmetric; that is, \({\varepsilon }_{ij}\ne {\varepsilon }_{ji}\) since \({\varepsilon }_{ij}=\widehat{{s}_{j}}{\delta }_{ij}\) while \({\varepsilon }_{ji}=\widehat{{s}_{i}}{\delta }_{ij}.\) While they are different in magnitude, the signs must be the same since they depend on the AllenUzawa partial elasticities (\({\delta }_{ij}).\)
It is also important to estimate Morishima elasticity of substitution since the AllenUzawa elasticity are conceptually relevant to the cases involving only two inputs (Blackorby and Russell 1989). The Morishima elasticity of substitution are computed as
$$M_{ij} = \varepsilon_{ij}  \varepsilon_{jj}$$
(9)

iii.
Technological progress is calculated using \(\varepsilon_{ct} =  \partial \ln cv/\partial t\), which is equal to
$$\varepsilon_{ct} =  \left( {\alpha_{t} + \alpha_{tt} t + \mathop \sum \limits_{i} \rho_{ti} \ln P_{i} + \rho_{tY} \ln \left( \frac{Y}{K} \right) } \right)$$
(10)
and can be decomposed into neutral \(({\alpha }_{t}{\alpha }_{tt}t)\), biased \((\sum {\rho }_{ti}\mathrm{ln}{P}_{i})\) and scaleaugmenting \(\left({\rho }_{ty}\mathrm{ln}(Y/ K)\right)\). Technological progress entails the downward shift in longrun average cost and measures growth in total factor productivity.

iv.
Following Morrison (1988) and Paul (1999), capacity utilization (CU) is computed as
$${\text{CU}} = \frac{{\varepsilon_{cY} }}{{\alpha_{Y} + \beta_{K} }},$$
(11)
where \({\alpha }_{y}+{\beta }_{k}\) is the longrun returnstoscale, and the elasticity of shortrun cost with respect to output (\({\varepsilon }_{cy}\)) is given by Eq. 5. Then, the elasticity of capacity utilization with respect to a variable input price depends on \({\rho }_{yi}\), calculated as
$$\varepsilon^{CU} = \frac{\partial CU}{{\partial P_{i} }} \cdot \frac{{P_{i} }}{CU} = \frac{{\rho_{yi} }}{CU},$$
(12)
where \({{\varepsilon }_{i}}^{cu}\) is the elasticity of capacity utilization with respect to the price of a variable input i. Since \({\rho }_{yi}={\rho }_{ki}\), a positive \({\rho }_{yi}\) implies that capital and input i are substitutes. In other words, the elasticity of capacity utilization with respect to the price of a variable input depends on the relationship between the variable input and capital.