Fuel conserving (and using) production functions☆
Introduction
Energy analysts wishing to explore the relationship between engineering efficiency gains and energy use at a sectoral or macro level almost always rely on some specification of a production function (or its dual cost function).
In this article we examine and compare eight such functions for their capability to flexibly depict this complex relationship. For pedagogical purposes, we first analyze the traditional Leontief and Cobb–Douglas production functions. Then we examine the Generalized Leontief, CES (Solow), and Translog functions, popular among energy economists. Finally, we investigate three cost functions used in the broader economics community: the Symmetric Generalized Barnett, Symmetric Generalized McFadden, and Gallant (Fourier) functions. The version of the Symmetric Generalized Barnett we use is identical to the cost version of the Generalized Leontief (given constant returns to scale) so we refer to this form as the GLc/SGB version, giving precedence of timing to the GLc form, although both forms in any case are due to Diewert (1971) and Diewert and Wales (1987).
The examination focuses on the propensity of each function to exhibit energy consumption rebound and decomposes this propensity into intensity and output/income effects. We conclude that one of the traditionally-designated “flexible” forms, namely the Gallant (Fourier) form, is the only one among those examined that is wholly “rebound flexible.” The GLc/SGB, being fairly rebound flexible, may also be suitable for analyses in this field.
Unfortunately, the Generalized Leontief production function and Symmetric Generalized McFadden cost function do not appear well suited, and the Translog cost function is suitable only under certain restrictive conditions. A particular form of the “inflexible” CES (Solow) function is fairly rebound flexible, but requires strong a priori assumptions as to the structure of real-world production possibilities.
“Rebound flexible” here means the function can exhibit a full numeric range of rebound when rebound is quantified and that it honours the standard regularity restrictions including, notably, concavity. This quantification, and the meaning of “full numeric range,” are later described. If a function meets these regularity restrictions and exhibits most of the rebound range, we call it “fairly rebound flexible.”
The article also provides a general methodology for exploring these questions for an arbitrarily defined constant return to scale production function/cost function. We further provide analytic expressions and spreadsheets for rapidly calculating these effects for those practitioners who have econometrically measured any of the production/cost functions examined here and provide numerical examples.
It is hoped this article will add to the theoretical underpinning of this burgeoning field and will allow practitioners both to make better choices in structuring their analyses and to understand their own results more deeply.
It scarcely needs mention that current attention on global warming remedies, in particular those on the energy consumption side, increases the stakes for analyses of this kind.
Section snippets
Theoretical framework
A (nearly) ideal practitioner's model of energy consumption and technical change would consider a multi-sector, multi-country economy in a general equilibrium framework that characterizes production possibilities using multi-product production functions incorporating capital vintaging, and consumption using multi-product utility functions, with energy supplied by non-competitive producers. Then one would introduce generalized engineering efficiency gains and inquire as to the effect these would
Cautions and limitations
The assumptions above are insufficient for our analysis. To the more standard assumptions for such analyses – including perfectly competitive behaviour, continuity, monotonicity, positivity, and concavity of production/cost functions – we add certain others: the functions examined are all constant returns to scale (CRS). We do not consider nested functions.1
Background of the article
The article is an updated version of a 1996 working paper that has been in informal circulation in the community entitled “Fuel Conserving Production Functions.” The results reported here supersede the results reported there, except for the inclusion in that paper of a more detailed energy services sector. The primary difference is that in the original paper, energy services were depicted by means of nesting an energy services function within the overall function. Largely because we used a
Structure of the article
In the following section, we give a brief background on previous work in the field. Then we delineate and describe the eight production/cost functions that are the subject of our examination. In doing so, we categorize production functions separately from cost functions to accommodate later methodological considerations. Then we introduce the concept of a “fuel conserving condition,” which allows us to discriminate among production functions for their power to analyze the rebound question, and
Underpinnings
This field owes a great debt to Brookes, 1979, Brookes, 1990a, Brookes, 1990b, Brookes, 1990c, Brookes, 1992, Brookes, 2000 and Khazzoom, 1980, Khazzoom, 1987, Khazzoom, 1989, who first put forward the proposition that engineering energy efficiency gains may not conform to the naïve presumption they will reduce energy use in a one-for-one fashion, as had been the popular notion to that point.9 Early students of this
Production functions for energy consumption analysis
We review eight production/cost functions that are candidates for analysis of energy use given engineering energy efficiency gains. We make the distinction between fuel (F) and energy services (E) to distinguish the two conceptually. From here forward, we use F to refer to physical fuel use, which is, after all, the parameter of interest in analyzing questions of global warming.
We define a production function as one that takes factor inputs K, L, and E, and converts them to marketable output Y:
Cost functions for energy consumption analysis
Many practitioners find it more convenient to measure and use cost functions than production functions. This is primarily because cost functions readily accommodate an assumption all producers face the same prices, thus facilitating aggregation.21 This creates challenges for our rebound analysis methodology, but the remaining functions we analyze are all of this kind.
The Cobb–Douglas and CES (Solow) production functions are “self-dual,”
The fuel conserving condition
Four of the functions analyzed are production functions; four are cost functions. We begin with the easier task of analyzing the production functions. We wish to test these functions to see whether an increase in τ will decrease fuel use in the short term. We define the Fuel Conserving Condition as:
This condition says if increasing the fuel efficiency decreases the marginal productivity of fuel when fuel F is fixed, fuel use will go down when fuel is not fixed. We for the moment
Decomposing rebound
Much deeper insight is available if we decompose the elements of rebound. Analysts have come to recognize34 there are two effects at work here: one, a fuel efficiency gain reduces the effective price of fuel and makes it more attractive; and two, it increases the space of production possibilities and the resulting increased output “drags up” fuel use. We here formally decompose these two effects, which
The long-term fuel conserving condition
The Fuel Conserving Condition previously developed is a necessary, but not sufficient condition to prevent backfire. The assumption of short-term fixed capital and labour (with only fuel allowed to freely adjust) causes an understatement of rebound. This is because the efficiency gain will increase the marginal productivity of these other two factors,41 thus drawing in more of them and increasing output if more supply is
Fuel conserving conditions for cost functions, short- and long-term
The rebound analysis of cost functions is what ultimately leads us to our recommendations. The methodology is very different from the production-side methodology and is developed in Appendix D Components of short-term rebound — cost functions, Appendix E Long-term rebound — cost functions. We invoke a slight change of terminology by referring here to the second component of rebound as the “income effect” in place of “output effect.” As with the production functions, the intuition is τ can both
CES (Solow)
The CES (Solow) function appears to remain a viable candidate. However, it is critical to note only the Hogan–Manne–Richels nesting of the CES (Solow) form, [(KL), F] nesting, shows the needed flexibility in depicting rebound. It has been shown elsewhere other nesting schemes can depict only backfire.45 This leaves one with an uncomfortable feeling of arbitrariness in choosing this function to depict reality. Accordingly, we recommend practitioners use this function only if
A note on super-conservation
Super-conservation is a highly counter-intuitive phenomenon. How can, say, a 1% increase in fuel efficiency result in a 2% decline in fuel use? The answer lies in the fact that profit maximization in response to a change in τ invokes a host of complex interactions: cross-substitutions among factors, changes in their marginal productivities (and therefore prices), changes in the way prices affect the cost function, and changes in the magnitude of the cost function itself. In their impact on
Quantitative examples
In this section, we compare the cost functions using quantitative examples. Quantifying the CES (Solow) cost function is straightforward. For the other cost functions (Translog, Generalized Leontief cost/Symmetric Generalized Barnett, Symmetric Generalized McFadden and Gallant (Fourier) functions), the parameter sets used are shown in the spreadsheets posted online with this article. The parameter sets were chosen somewhat arbitrarily — they are simply intended to illustrate the range of
Conclusions
We are led to the following conclusions:
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Practitioners' choice of production/cost function can pre-determine the results related to rebound. Practitioners accordingly should be judicious in their selection of functional forms.
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We recommend using the Gallant (Fourier) form, which is wholly rebound flexible, that is, can depict the full range of rebound conditions from super-conservation to backfire while honouring global concavity.
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The Symmetric Generalized Barnett/Generalized Leontief cost form is
Acknowledgements
This research has benefited greatly by comments from many researchers. However, I would like to specifically acknowledge extensive and substantive comments from Steve Sorrell and John Dimitropoulos. Taoyuan Wei also made helpful comments. This article was vastly improved by the quality critiques of four anonymous referees. My very deep gratitude goes to the reviewer who suggested looking at the Symmetric Generalized Barnett, Symmetric Generalized McFadden, and Gallant (Fourier) functions, which
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This article is dedicated to the memory of Alan Manne, who died September 27, 2005.