Dynamic translog and linear logit models: a factor demand analysis of interfuel substitution in US industrial energy demand
Introduction
In examining interfuel substitution possibilities in industrial energy demand, several authors have found themselves comparing two popular flexible functional forms for their factor demand equations, the translog and the linear logit (for example Considine, 1989a, Jones, 1995 amongst others). These studies usually conclude that the linear logit model is more likely to satisfy neo-classical regularity conditions, especially in a dynamic formulation. Industrial energy demand is often thought to have the greatest potential for interfuel substitution so obtaining sensible elasticity estimates is of considerable practical importance (especially since the industrial sector accounted for some 37% of total net energy consumption in the US in 1992).
In general, these global curvature conditions in static models are verified ex post without imposing them from the outset, because this would exclude complementarities amongst inputs.1 In the case of dynamic models, where long-run and short-run coefficients are jointly estimated, the validity of concavity conditions depends both upon the functional form adopted and especially on the dynamic specification of the adjustment of producer behaviour. The main aim of this paper is to show the crucial role played by these dynamic formulations in the analysis of factor demands.
In this sense, our work is closest to Jones (1995) who compares the dynamic linear logit model with a representative dynamic translog formulation—the partial adjustment model (representative in the sense that it has been widely used, see, for example Taheri, 1994 and references). Using Considine's (1989a) data updated through 1992, Jones estimates both static and dynamic translog and linear logit models of cost shares for coal, oil, electricity and natural gas. He concludes both translog models violate concavity conditions over nearly two thirds of the sample, whereas the linear logit models do not.
A natural extension of this work is therefore to compare the partial adjustment linear logit model with a richer dynamic specification of the translog. In fact, we show that this representative dynamic translog formulation is rejected by the data, which is presumably why it is inferior to the dynamic linear logit specification. In fact, a more general model, the partially generalised error-correction model (ECM) dominates (nests) alternative dynamic formulations including this representative partial adjustment mechanism.
This ECM is possible because we utilise a new approach to estimating singular systems of demand equations, an approach which is able to solve the parameter identification problem within the singular system—a problem originally noted by Anderson and Blundell (1982) and re-examined by, amongst others, Urga (1996) and Allen and Urga (1999). We use restrictions implied by the short run effective cost function to obtain identification, restrictions which are intuitively more appealing than ad hoc restrictions (such as those used in Friesen, 1992) as they may be derived from such a cost function. The weakness of our approach is that it cannot be derived directly from an explicit dynamic optimisation problem, unlike the dynamic linear logit model (Treadway, 1971, Considine and Mount, 1984). We therefore directly compare the results of the richer dynamics of our approach with those of the more theoretically pleasing partial adjustment linear logit model.
Although this theoretical framework is applicable to any singular system, in Section 2 we present a general long-run translog cost function and alternative dynamic specifications. The dynamic linear logit model is presented in Section 3. In order to embed our work in a familiar context, we conduct our empirical evaluation of these two models using the same data as Considine (1989a) and Jones (1995)—our results on interfuel substitution in US industrial energy demand are reported in Section 4. Empirical support for the ECM translog is weak, concavity violations do not appear to be an artefact of either dynamic mis-specification or inclusion of price-unresponsive non-energy fuels in the aggregate consumption data. And the dynamic logit model produces much more sensible estimates. Section 5 concludes.
Section snippets
Longterm specification
A general long-run translog cost function with non-neutral technical change may assume the following form:where γij=γji for all i, j=1,…, N, ln indicates the natural logarithm, Pjt (Pit) denotes the price of input j (i) at time t, yt is the level of output in period t and t denotes a time trend reflecting biased technical change. Necessary and sufficient (adding up)
The dynamic linear logit model
Although logit models are frequently associated with discrete choice problems, they can be used whenever ‘outcomes’ must be non-negative and sum to one. The logistic distribution provides a close approximation to the cumulative normal distribution and computations are far simpler since a closed form solution of the multivariate normal integral does not exist (Berkson, 1944).
The logistic approximation of a set of n non-homothetic cost shares with non-neutral technical change takes the following
Empirical application
The data used in this study are those in Jones (1995) and were very kindly provided to us by Clifton T. Jones. They are described in detail there but represent annual data on aggregate US industrial fuel consumption from 1960 to 1992 for four major fuels: coal (both steam and coking); oil (all petroleum products, regardless of use); electricity; and natural gas. The data are taken from standard government and industry publications. Use of aggregate fuel consumption permits comparability of our
Conclusions
In this paper, two alternative dynamic cost frameworks are compared and evaluated. A more theoretically pleasing dynamic logit model is compared with an optimal dynamic version of the translog specification. The dynamic logit model emerges with a remarkably clean bill of health, whereas the dynamic translog does not-but not for reasons of dynamic mis-specification in previous studies or inclusion of price-unresponsive non-energy use fuels in our aggregate fuel use data.
The rate of adjustment of
Acknowledgements
We are indebted to Chris Allen, Steven Hall and Clifton T. Jones for discussions and suggestions during the preparation of this paper, which was part of an ESRC research project at the Centre for Economic Forecasting entitled ‘Macroeconomic Modelling and Policy Analysis in a Changing World’ (grant no. L116251013). We would also like to thank Clifton T. Jones for kindly providing us with his data. The usual disclaimer applies.
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