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This paper investigates the relationships between energy efficiency improvements by producers, the ease of substitution between energy and other inputs and the size of the resulting “rebound effects”. Fundamentally, easier substitution leads to larger rebounds. Focusing upon conceptual and methodological issues, the paper highlights the challenges of estimating and modeling rebound effects with the help of production and cost functions and questions the robustness of the evidence base in this area. It argues that the multiple definitions of “elasticities of substitution” are a source of confusion, the most commonly estimated elasticity is of little practical value, the empirical literature is contradictory, prone to bias and difficult to use and there are only tenuous links between this literature and the assumptions used within energyeconomic models. While “energyaugmenting technical change” provides the natural choice of independent variable for an estimate of rebound effects, most empirical studies do not estimate this form of technical change, many modeling studies do not simulate it and others simulate it in such a way as to underestimate rebound effects. As a result, the paper argues that current econometric and modeling studies do not provide reliable guidance on the magnitude of rebound effects in different industrial sectors.
The “rebound effect” is an umbrella term for a variety of economic mechanisms that reduce the “energy savings” from improved energy efficiency [
Energy efficiency or energy productivity may generally be defined as the ratio of useful outputs to energy inputs for a specified system. Inputs and outputs may be defined in thermodynamic, physical or economic terms [
In a widely cited paper, Saunders [
This paper investigates the relationships between energy productivity improvements for producers, the ease of substitution between energy and other inputs and the size of the resulting rebound effects. It assesses the meaning and applicability of the above statement by Saunders [
The paper begins by outlining how different types of energy productivity improvement are represented in neoclassical production theory, how these may lead to rebound effects and how energy substitution contributes to those effects. The following three sections examine how the “ease of substitution” between energy and other inputs is defined and measured and how these estimates are commonly used. The paper highlights the challenges in estimating “elasticities of substitution”, the difficulties in interpreting the available literature, the contradictions in the empirical results and the tenuous link between these results and the assumptions used in energyeconomic models. Section 8 compares the different ways of estimating and modeling technical change and highlights the limitations of commonly used measures when applied to the estimation of rebound effects. On the basis of these arguments, the paper argues that current econometric and modeling studies do not provide reliable guidance on the magnitude of rebound effects in different sectors. The paper concludes by indicating the requirements for providing more realistic guidance.
Neoclassical production functions indicate the maximum possible economic output (
Production functions are normally assumed to be positive, twice differentiable and quasiconcave with constant returns to scale. Under standard assumptions a dual
Dividing through by output gives a unit cost function (
Cost functions are preferred to production functions in empirical studies since the independent variables (input prices) are more likely to be exogenous.
Assuming perfect competition and profit maximization, the marginal product of each input should be equal to the input price (
The output produced from a given quantity of inputs typically increases over time as technology improves. The rate of change of
The equivalent cost function definition is:
With constant returns to scale: ε_{ft} = ε_{gt}.
Energy productivity (θ
Aggregate energy productivity therefore depends upon the level of each input, the current state of technology and the level of output, as well as upon how individual inputs are measured and aggregated (e.g., how different types of energy carrier are combined). The inverse of energy productivity (energy intensity) can be derived from the unit cost function using Shephard's Lemma:
Increases in energy prices encourage the
Substitution is conventionally represented as movement along an isoquant of a production or cost function and technical change as a shift of the isoquants (
In practice, technical change is frequently biased in that the productivity of some inputs increases more rapidly than others [
Technical change is said to be “input saving” if the value share of that input falls over time (ψ
In modeling studies, biases in technical change are commonly simulated by including “augmenting multipliers” (τ
Letting
Letting
The “effective production function” (
A key point relevant to the estimation of rebound effects is that “energyaugmenting” technical change is not the same as “energy saving” technical change. As described below, the former may not lead to the latter owing to substitution between inputs. Also, the direction of technical change is likely to be influenced by relative prices.
Saunders [
In general, energyaugmenting technical change will not lead to a proportionate improvement in aggregate energy productivity (θ
lower price effective energy will stimulate the
lower input costs will stimulate an increase in
In combination, these substitution and output effects will increase energy consumption above what it would have been in the absence of these responses. The sum of the two is the
The contribution of substitution to the direct rebound effect may be illustrated graphically. For simplicity, we assume that nonenergy inputs are “separable” from energy inputs and can therefore be grouped together, or “nested”. The meaning and implications of this assumption are described further below.
Technical change may also increase output (not shown) since: first, a higher level of output can be produced for a given expenditure on inputs; and second, reductions in product prices may increase aggregate supply and hence output. This increase in output will further increase energy use. The total direct rebound effect is the sum of these substitution and output effects. In what follows, we focus on the substitution effect—which Saunders [
These examples illustrate the importance of substitution for rebound effects and suggest that these effects will be larger when the scope of substitution between energy and other inputs is easier. Saunders [
We expect 0 ≥ η_{τ}
If, η_{τ}
As shown in [
This discussion suggests that sectors where substitution
So does the existing evidence base allow the ease of substitution and rate of energyaugmenting technical change within different sectors to be accurately identified? How well do energyeconomic models reflect this evidence base? And can the available empirical and modeling studies provide reliable guidance on the magnitude of rebound effects in different sectors? The remainder of the paper addresses these questions by examining: first, the definition and estimation of energy substitution (Sections 5 and 6); second, the use of those estimates within energyeconomic models (Section 7); and third, the estimation and representation of energyaugmenting technical change (Section 8).
We first look more closely at how the “ease of substitution” is defined. The previous discussion used the
The
Source: Broadstock
The great majority of empirical studies estimates the
The
With the CES production function of
These differences create considerable difficulties in using the empirical literature to infer appropriate values for the
Engineering studies have long indicated significant potential for costeffectively improving energy efficiency through various forms of capital investment—which could be interpreted as substituting physical capital for energy. This potential is reproduced in the assumptions used for many energyeconomic models. But beginning with Berndt and Wood [
Berndt and Wood [
Broadstock
This ongoing debate illustrates how the estimation of substitution elasticities raises a variety of theoretical and methodological issues that collectively make it very difficult to interpret the results and draw useful conclusions [
Separability assumptions are commonly used to justify either the omission of inputs for which data is unavailable (notably materials) or the grouping, or
But even when two inputs (e.g.,
This suggests that estimates of substitution elasticities are likely to be biased if separability is assumed where not supported by the data, or if measures of any input are omitted. The latter situation is common, particularly with regard to the omission of materials. In practice, studies that exclude materials more often indicate capitalenergy substitutability, while those that include materials indicate complementarity [
In sum, the multiple definitions of substitution elasticities, the range of factors influencing empirical estimates and the sensitivity of results to those factors make the evidence base in this area confusing, contradictory, prone to bias and difficult to use. At a minimum, statements about substitutability need to be qualified by the countries, sectors and time periods to which they apply the manner in which inputs are disaggregated and measured and the specific assumptions that are made—with the latter being supported, where possible, by statistical tests. But the resulting estimates may still not be useful for particular applications, including the parameterization of energyeconomic models. To illustrate this, the next section examines how substitution elasticities are used in energyeconomic models and highlights the limited basis for the assumptions made.
Energyeconomic models based upon computable general equilibrium (CGE) techniques are widely used for exploring policyrelevant questions including the estimation of rebound effects. Such models almost invariably use CES production functions and assume that some inputs are separable from others. Parameterisation requires assumptions about the
For such results to be robust, the assumed parameter values should be firmly based upon empirical research. Unfortunately, it is common practice to assume these values with only limited reference to the empirical literature. Moreover, even when such references are made, there are considerable difficulties in using empirical studies to infer values of the
differ from the cited empirical studies in the manner in which individual inputs are aggregated and in the level of sectoral aggregation;
assume values for
impose separability between groups (nests) of inputs while most empirical studies do not; and
require estimates of the
These points are briefly elaborated below.
Blackorby and Russell [
The structure of this “twolevel” nested CES, in which capital
With a nested CES, the
Hence, it is possible for
Hence, estimates of the
In sum, the assumptions made for production structures and substitution elasticities within most CGE models appear to be only weakly linked to an empirical literature that is itself contradictory and inconclusive. This suggests that the results of such models, including their estimates of rebound effects, should be treated with considerable caution. Unless more flexible functional forms can be adopted, e.g., [
Sections 2 and 3 argued that the magnitude of rebound effects from energyaugmenting technical change should be proportional to the flexibility of producers to adapt to those efficiency gains via substitution [
Further challenges are created by the use of energyaugmenting technical change as the independent variable for an estimate of rebound effects. This is because most empirical studies do not directly estimate this form of technical change and many energyeconomic models do not simulate it. To illustrate, we summarize the most common approaches to estimating and modeling technical change and highlight the implications of using two different approaches within a CES production function.
As noted, empirical studies typically employ flexible cost functions such as the translog and use Shepards Lemma to derive equations for the value share of each input. With a suitable functional form, this allows the “energy price bias” to be estimated:
In a widely cited study, Hogan and Jorgenson [
Modeling studies typically employ CES production functions. While some model energyaugmenting technical change in a similar manner to Saunders [
But again, the
In other words, the energy price bias is the “share weighted” deviation of the autonomous energy efficiency trend from the trend in total factor productivity (ε_{gt}). Positive values for ε_{gt} imply improvements in total factor productivity (declining costs per unit of output), while positive values for
Different models use different CES formulations and nesting structures and implement either the
In contrast, Saunders' version of this function simulates energyaugmenting technical change by a positive growth rate for parameter τ
These two approaches to representing technical change are not equivalent. Combining both approaches,
The Manne Richels approach has λ
In contrast, the Saunders' approach has δ = 0 [
In sum, the estimation of direct rebound effects for producers requires specification of the magnitude and direction of energyaugmenting technical change. Most empirical studies do not estimate this form of technical change, many energyeconomic models do not simulate it and others simulate it in a manner that precludes the accurate modeling of rebound effects. As a result, the available evidence provides insufficient guidance on the magnitude of rebound effects within different industrial sectors. When combined with the difficulties in specifying substitution elasticities discussed earlier, the result is considerable uncertainty over the magnitude of rebound effects in industrial production.
This paper has explored the relationships between energy productivity improvements for producers, the “ease of substitution” between energy and other inputs and the size of the resulting rebound effects. It has shown how easier substitution drives larger rebounds, but the relevant mechanisms are not straightforward and are difficult to capture empirically. There are three main findings.
First, the multiple definitions of substitution elasticities are a source of confusion, the most commonly estimated elasticity is of little practical value and the empirical literature is contradictory, prone to bias and difficult to use. For example, engineering studies suggest a large potential for improving energy efficiency by substituting capital for energy, but three decades of econometric research has achieved no consensus on whether these inputs are best described as substitutes or complements, and no consensus on how different factors influence empirical results.
Second, there are only tenuous links between the empirical literature on substitution elasticities and the assumptions used within energyeconomic models. Most models employ nested CES production functions and make assumptions about the
Third, while energyaugmenting technical change provides the natural choice of independent variable for an estimate of rebound effects, most empirical studies do not estimate this form of technical change, many modeling studies do not simulate it and others simulate in a manner that precludes the accurate modeling of rebound effects. As a result, the available evidence base provides only limited guidance on the magnitude of direct rebound effects for producers and widely used modeling tools may overestimate the future energy savings from improved energy efficiency.
These conclusions provide pointers to how future studies may more adequately capture direct rebound effects. For empirical studies, the most important requirement is the explicit estimation of inputaugmenting technical change, such as achieved recently by Saunders [
For modeling studies, the requirements include (where possible) greater use of flexible functional forms, abandonment of the
The research described in this paper formed part of a larger study on rebound effects by the UK Energy Research Centre [
The author declares no conflict of interest.
The functional form used by Saunders [
Using the chain rule, the marginal product of energy is given by:
The term in brackets is output (
Assuming perfect competition and cost minimization, this equals the unit price of energy:
Solving for energy:
Therefore, aggregate energy productivity (θ =
Aggregate energy productivity therefore depends upon energy prices, energyaugmenting technical change, the
Expressing this in elasticity terms gives:
In other words, for this production function, the elasticity of energy productivity with respect to energyaugmenting technical change is equal to one minus the
Beginning again with Saunders' functional form [
Manne and Richels [
As shown in
Using σ = 1/(1 − ρ), this becomes:
Differentiate with respect to time to derive an expression for the
Substituting for growth rates:
Substitution and neutral technical change.
Energyaugmenting technical change encourages the substitution of energy for other inputs.
Comparing common definitions of substitution elasticities. Source: Broadstock

Fixed  Fixed  Fixed  Two input, two price 

Yes  

Fixed  Variable  Fixed  Two input, two price 

No  

Fixed  Variable  Fixed  One input, one price 

No  

Fixed  Variable  Fixed  One input, one price 

Yes  

Fixed  Variable  Fixed  Two input, one price 

No 
Note:
Nesting structures and assumed values of the
Bosetti 
( 

Burniaux 

Edenhofer 

Gerlagh and van der Zwaan [ 
( 

Goulder and Schneider [ 

Kemfert [ 
( 

Manne 
( 

Popp [ 

Sue Wing [ 
( 