Abstract
In this paper, we evaluate the capacity of the Translog cost share model to approximate the producer’s true demand system and introduces two non-linear functional forms, which have been achieved by altering and extending the standard quadratic logarithmic Translog model. The extensions have additional desirable approximation properties with respect to output and time variables, and thus allow more flexible treatments of non-homothetic technologies and non-neutral technical change than those provided by the standard Translog. The performances of the three models are assessed (1) on theoretical ground, by the size of the domain of regularity, (2) on their ability to provide plausible estimates of the economic and technological indicators being measured and finally (3) on their reliability in fitting input shares, input-output ratios and unit cost. The most important finding is that the standard model exhibits some weakness in fitting. We show via a series of experiments that those shortcomings are due to a lack of flexibility of the logarithmic model. The estimation results obtained with the new extended model are more satisfactory and promising.
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Notes
- 1.
The duality theory provides alternative equivalent representations of a producer’s technology (production, cost, profit, revenue, or distance function) or of a consumer’s preferences (direct utility, indirect utility, inverse indirect utility, or expenditure function). Under certain regularity conditions, given one of these functions the other can be determined and completely characterized. See Diewert (1974, 1982) for excellent surveys of the applications of duality theory in different areas of economics. See also Fuss and McFadden (1978) for a comprehensive treatment of the application of duality to production theory.
- 2.
The class of flexible functional forms was originally defined by Diewert (1971, 1974) to be the class of forms capable of attaining the level of the “true” function and of all its first and second order partial derivatives at some point. Barnett (1983a,b) later calls this local flexibility. Gallant distinguish between two concepts of flexibility: Diewert or Sobolev.
- 3.
Uzawa (1962) proved that it is impossible for any functional form that exhibits constant elasticities of substitution to provide simultaneously the capability to attain an arbitrary set of elasticities.
- 4.
According to Diewert’s (1971, 1974) definition, a function G(x) is a second order approximation to an arbitrary function F ∗(x) at a given point x ∗ if the level, the first and second order partial derivatives of these two functions are equal at these point. Lau (1974) distinguished between two definitions to the concept of ‘second order approximation’ that are usually used in economic literature, by referring to Diewert’s definition as ‘second order differential’ approximation and to Christensen, Jorgenson and Lau’s (1973) definition as ‘second order numerical’ approximation. Barnett (1983a,b) later proved that flexibility in the sens defined by Diewert is necessary and sufficient for a function to satisfy the mathematical definition of a local second order approximation. Furthermore, a second order Taylor series approximation is flexible in that sense.
- 5.
- 6.
As has been noted by Gallant and Golub (1984), a Sobolev-flexible form can endow a parametric methodology with a semi-nonparametric property.
- 7.
A functional form is parsimonious if it provides a second order approximation using a minimal number of parameters. Barnett (1985) refer to that property, first defined by Diewert (1971), as the minimality property. The term “parsimonious” is due to Fuss, McFadden, and Mundlak (1978). As these authors noted, an excessive number of parameters exacerbate problems of multicollinearity. Furthermore, when the sample is small, excess parameters mean a loss of freedom and hence a loss in the precision of estimation.
- 8.
The use of the translog for short-run studies in production and demand analysis entails a significant shortcoming in that the full equilibrium level(s) of fixed input(s) cannot be derived in analytical closed form(s), but instead must be calculated using iterative numerical techniques. However, some researchers have reported difficulties in obtaining numerical convergence with the translog variable cost function (assuming temporary equilibrium) and thus with computing estimates for long-run elasticities and capacity utilization (see, e.g., Brown and Christensen 1981; Berndt and Hesse 1986). Moreover, problems may rise with frameworks based on cost of adjustment or time to build dynamic models.
- 9.
The three functional forms considered belong to the generalized quadratic family of locally flexible functional forms and can be interpreted as second order Taylor series expansion about a point in powers of \(\ln x\), x 1∕2 and x respectively. They share the common characteristics of linearity in parameters and the “ability” of providing second-order approximations to an arbitrary twice continuously differentiable function at a point. All of these forms can be viewed as limiting or special cases of a more general quadratic form: the generalized Box-Cox due to Berndt and Khaled (1979).
- 10.
Notations: \(\boldsymbol{}x \geq 0_{N}\) means each element of the N-dimensional vector \(\boldsymbol{}x\) is nonnegative, and \(\boldsymbol{}x \gg 0_{N}\) means that each element of \(\boldsymbol{}x\) is positive.
- 11.
This cost function is derived under the assumption that the observed production technology is instantaneously in full static equilibrium. Thus, all inputs are assumed to adjust fully to their long-run equilibrium levels within one sample period. A variable (or restricted) cost function can also be specified to allow for short-run fixity of some inputs in studies based either on partial-static or dynamic equilibrium models.
- 12.
- 13.
- 14.
From Shephard’s lemma, ∂ C∕∂ p i = x i . Consider a logarithmic transformation of the cost function, we find that the logarithmic derivatives of the cost function are input cost shares.
- 15.
In both measures, \(\sigma _{ij}\) and \(\varepsilon _{ij}\) the quantities of all inputs are allowed to adjust in response to a change in p j , when output is held constant.
- 16.
Hicks defined technical change as being neutral if the marginal rate of technological substitution between each pair of inputs is independent of technical change. Thus, when technical change is Hicks-neutral all factor demands are affected equi-proportionally. Hicksian bias is defined as the change in factor-ratio (x i ∕x j ) or in factor share-ratio (S i ∕S j ) that is not attributable to price changes.
- 17.
Under constant returns to scale, the unit cost function is independent of y.
- 18.
Flexibility here is meant in the sense of Diewert. It is measured in terms of the number of free parameters in the model. See Diewert and Wales (1987).
- 19.
The SYSNLYN procedure (option ITSUR) contained in SAS was used for the estimation of the different models.
- 20.
Imposing this kind of restrictions on the parameters of the input share equations, including those for the technical progress, is unavoidable for assuring linear homogeneity of the TL cost function.
- 21.
The motivation for imposing the restrictions implied by symmetry and linear homogeneity in input prices are three. The first is the more pressing and it is that reported results and policy recommendations at least appear reasonable. The second motivation is to gain statistical efficiency in the estimation of parameters. Finally, imposition of these properties globally is very easy in practice since it involves simple linear restrictions on the parameters.
- 22.
For instance, Diewert and Wales (1987) showed that the use of Jorgenson and Fraumeni (1981) procedure for imposing concavity on the translog cost function at all positive input price space (globally), not just in the historical sample on which the parameter estimates are based (locally), will seriously restrict the substitution possibilities allowed for by the technology and will generate a Cobb-Douglas representation of technology over certain areas.
- 23.
In another experience, I found that concavity is satisfied by GL and MF throughout the historic period for equipment and consumer goods, but is often violated for intermediate goods.
- 24.
- 25.
From an economic viewpoint the own price elasticity of demand is of vital importance. It gives a measure of input’s conservation proportionate to an increase in that input’s price, when output remains constant. The greater the substitutability of other factors for the input in question, the greater the opportunity for resource conservation \((\varepsilon _{ii} = -\sum _{i\neq j}\varepsilon _{ij})\).
- 26.
For brevity, I only report results for consumer goods. Note that the curvature properties required by theory are less violated in this branch.
- 27.
However, at the sample mean point, the elasticities are evaluated at means of exogenous variables which correspond to the average of the logarithm of input prices in the TL case. Thus the analysis of their derivatives with respect to input prices at this point does not allow a useful interpretation. We have arbitrarily chosen the 1980 point which corresponds to the reference year of the sample: All input prices are equal to one at this point.
- 28.
On the other hand, for the intermediate goods branch, variations in levels are accompanied with variations in signs.
- 29.
In fact, with TL, since the cost function is to be linear homogeneity in input prices, Euler theorem implies that: ∑ i = 1 N α it = 0. Then the parameter α it is not input i specific.
- 30.
- 31.
The unit cost function is a linear combination of the endogenous variables of the systems of demand GL and MF expressed in terms of input-output ratios. By contrast, it is impossible to recalculate directly the unit cost or the input-output ratios from the system of input cost shares equations derived from TL.
- 32.
Though the second order parameter of the autonomous technical change estimated for the three branches is always statistically significant, it has no effect on the precision of fit of translog when the first order autonomous parameter is estimated. A second experience relaxing the restriction on the returns to scale has been attempted. The gain in precision is modest.
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Baccar, S. (2016). Limitations of the Approximation Capabilities of the Translog Model: Implications for Energy Demand and Technical Change Analysis. In: Greene, W., Khalaf, L., Sickles, R., Veall, M., Voia, MC. (eds) Productivity and Efficiency Analysis. Springer Proceedings in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-23228-7_15
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