Energy Substitution, Technical Change and Rebound Effects

Transcription

1 nerges 2014, 7, ; do: /en Artcle OPN ACCSS energes ISSN nergy Substtuton, Techncal Change and Rebound ffects Steve Sorrell Centre for Innovaton and nergy Demand and Sussex nergy Group, SPRU Scence and Technology Polcy Research, Unversty of Sussex, Falmer, Brghton BN1 9Q, UK; -Mal: Tel.: ; Fax: Receved: 28 February 2014; n revsed form: 13 Aprl 2014 / Accepted: 16 Aprl 2014 / Publshed: 29 Aprl 2014 Abstract: Ths paper nvestgates the relatonshps between energy effcency mprovements by producers, the ease of substtuton between energy and other nputs and the sze of the resultng rebound effects. Fundamentally, easer substtuton leads to larger rebounds. Focusng upon conceptual and methodologcal ssues, the paper hghlghts the challenges of estmatng and modelng rebound effects wth the help of producton and cost functons and questons the robustness of the evdence base n ths area. It argues that the multple defntons of elastctes of substtuton are a source of confuson, the most commonly estmated elastcty s of lttle practcal value, the emprcal lterature s contradctory, prone to bas and dffcult to use and there are only tenuous lnks between ths lterature and the assumptons used wthn energy-economc models. Whle energy-augmentng techncal change provdes the natural choce of ndependent varable for an estmate of rebound effects, most emprcal studes do not estmate ths form of techncal change, many modelng studes do not smulate t and others smulate t n such a way as to underestmate rebound effects. As a result, the paper argues that current econometrc and modelng studes do not provde relable gudance on the magntude of rebound effects n dfferent ndustral sectors. Keywords: rebound effects; elastctes of substtuton; energy augmentng techncal change 1. Introducton The rebound effect s an umbrella term for a varety of economc mechansms that reduce the energy savngs from mproved energy effcency [1]. For producers, cost-effectve energy effcency mprovements encourage ncreased consumpton of energy servces, boost productvty, ncrease

2 nerges 2014, output and potentally mpact energy use throughout the economy. The rebound effect s commonly defned as the percentage of potental energy savngs that are offset by these dfferent mechansms. For producers, the potental energy savngs are typcally estmated by engneerng models that assume no economc responses to mproved energy effcency, whle actual savngs are estmated by energy-economc models that smulate those responses wth the help of econometrcally estmated producton or cost functons [1 4]. nergy effcency or energy productvty may generally be defned as the rato of useful outputs to energy nputs for a specfed system. Inputs and outputs may be defned n thermodynamc, physcal or economc terms [5] and dfferent defntons and measures, as well as dfferent choces of system boundary, may lead to dfferent conclusons regardng the nature, source, magntude and sgn of energy productvty mprovements, as well as of the energy savngs that result. But neoclasscal producton theory narrows the range of choces for these varables: defnng energy productvty solely n economc terms and focusng upon only two sources of energy productvty mprovements: namely the substtuton of energy by other nputs and techncal change. The latter provdes a common choce of ndependent varable for studes of rebound effects n ndustral producton [3,6]. In a wdely cted paper, Saunders [7] uses neoclasscal producton theory to argue that: the ease wth whch fuel can substtute for other factors of producton (such as captal and labour) has a strong nfluence on how much rebound wll be experenced the greater ths ease of substtuton, the greater wll be the rebound. Saunders [2,8 11] had extended ths analyss n subsequent works, but consstently concludes that rebound effects wll be larger when the greater s the flexblty of the economy to adapt to energy effcency gans va substtuton [10]. Saunders results therefore mply that emprcal studes of the ease of substtuton between energy and other nputs should allow the lkely magntude of rebound effects n dfferent sectors to be explored [7]. Ths paper nvestgates the relatonshps between energy productvty mprovements for producers, the ease of substtuton between energy and other nputs and the sze of the resultng rebound effects. It assesses the meanng and applcablty of the above statement by Saunders [7], the challenge of estmatng and modelng rebound effects wth producton and cost functons and the robustness of the evdence base n ths area. The paper focuses on conceptual and methodologcal ssues, partly wth the am of clarfyng these ssues for non-specalsts. The paper begns by outlnng how dfferent types of energy productvty mprovement are represented n neoclasscal producton theory, how these may lead to rebound effects and how energy substtuton contrbutes to those effects. The followng three sectons examne how the ease of substtuton between energy and other nputs s defned and measured and how these estmates are commonly used. The paper hghlghts the challenges n estmatng elastctes of substtuton, the dffcultes n nterpretng the avalable lterature, the contradctons n the emprcal results and the tenuous lnk between these results and the assumptons used n energy-economc models. Secton 8 compares the dfferent ways of estmatng and modelng techncal change and hghlghts the lmtatons of commonly used measures when appled to the estmaton of rebound effects. On the bass of these arguments, the paper argues that current econometrc and modelng studes do not provde relable gudance on the magntude of rebound effects n dfferent sectors. The paper concludes by ndcatng the requrements for provdng more realstc gudance.

3 nerges 2014, Key Concepts from Producton Theory Neoclasscal producton functons ndcate the maxmum possble economc output (Y) obtanable from captal (K), labor (L) and ntermedate nputs to a frm or sector gven the technology avalable at a partcular tme (t) [12]. Intermedates are commonly dsaggregated nto energy () and materals (M) (whle consstent wth the structure of nput-output tables, the treatment of captal and labor as prmary nputs and energy as a secondary or ntermedate nput s problematc from the perspectve of the natural scences): Y = f ( K, L,, M, t) (1) Producton functons are normally assumed to be postve, twce dfferentable and quas-concave wth constant returns to scale. Under standard assumptons a dual cost functon can be defned whch ndcates the mnmum possble cost (C) of producng output Y, gven the prces of each nput (p ) and the current state of technology (t): Dvdng through by output gves a unt cost functon (c = C/Y): C = g( p, p, p, p, Y, t) (2) K L M c = h( p, p, p, p, t) (3) K L Cost functons are preferred to producton functons n emprcal studes snce the ndependent varables (nput prces) are more lkely to be exogenous. Assumng perfect competton and proft maxmzaton, the margnal product of each nput should be equal to the nput prce (p ) and the elastcty of cost wth respect to ths prce should be equal to the value share (s ): where: Y = p X lnc = s ln p M (4) (5) p X s = (6) C The output produced from a gven quantty of nputs typcally ncreases over tme as technology mproves. The rate of change of total factor productvty (ε ft ) s then gven by: ε ft lny = (7) t The equvalent cost functon defnton s: ε gt lnc = (8) t Wth constant returns to scale: ε ft = ε gt.

4 nerges 2014, nergy productvty (θ ) s defned as the rato of economc output to energy nputs: (,,,, ) θ = Y f K L M t = (9) Aggregate energy productvty therefore depends upon the level of each nput, the current state of technology and the level of output, as well as upon how ndvdual nputs are measured and aggregated (e.g., how dfferent types of energy carrer are combned). The nverse of energy productvty (energy ntensty) can be derved from the unt cost functon usng Shephard s Lemma: c 1 = = p Y θ (10) Increases n energy prces encourage the substtuton of other nputs for energy, thereby mprovng aggregate energy productvty but snce costs have ncreased output may fall. In contrast, techncal change s assumed to mprove energy productvty ndependently of changes n relatve prces and wthout reducng output. Total factor productvty measures the net mpact of techncal change on all nputs. Techncal change has frequently been assumed to be tme-dependent, exogenous and neutral (N), wth the productvty of all nputs ncreasng at the same constant rate (ε ft = λ N 0). In ths form, techncal change may be represented as a tme-dependent multpler on the producton functon: Y = τ N f( K, L,, M) (11) where: τ N () t e = λn t (12) Substtuton s conventonally represented as movement along an soquant of a producton or cost functon and techncal change as a shft of the soquants (Fgure 1). However, the dstncton between the two s less clear from an engneerng perspectve and both may reflect a complex mx of nvestment, operatonal changes and shfts n the composton of output [13]. Fgure 1. Substtuton and neutral techncal change. Other nputs - X Y Y Substtuton Techncal change nergy - In practce, techncal change s frequently based n that the productvty of some nputs ncreases more rapdly than others [12]. In emprcal studes, ths s commonly measured by the rate of change n the value share of each nput, ndependent of changes n relatve prces:

5 nerges 2014, ψ s t = (13) Techncal change s sad to be nput savng f the value share of that nput falls over tme (ψ < 0) and nput usng f t ncreases (ψ > 0) [14,15]. Note that ψ = 1. In modelng studes, bases n techncal change are commonly smulated by ncludng augmentng λ multplers ( τ () t t = e where λ > 0) on one or more nputs to the producton functon: Y = f(τ KK, τ LL, τ, τ MM) (14) Lettng X = τ X, ths can be expressed as: ~ ~ ~ ~ ~ Y = f ( K, L, M) (15) Lettng p = p / τ, the equvalent effectve cost functon s: ~ K L M ~ C = g( p ~, p~, p ~, p ) (16) The effectve producton functon ( ~ f ) ndcates the maxmum output obtanable from the effectve ( X ~ ) rather than real nputs. Input-augmentng techncal change s defned as Y / τ > 0 ; mplyng that the margnal productvty of nput X ncreases over tme and ts effectve prce ( falls. The energy-augmentng multpler (τ ) converts energy nputs () to effectve energy ( ~ ). Although the latter s sometmes referred to as energy servces, t s dfferent from the engneerng nterpretaton of that term. A key pont relevant to the estmaton of rebound effects s that energy-augmentng techncal change s not the same as energy savng techncal change. As descrbed below, the former may not lead to the latter owng to substtuton between nputs. Also, the drecton of techncal change s lkely to be nfluenced by relatve prces. 3. Techncal Change, nergy Substtuton and Rebound ffects Saunders [7] defned rebound effects n relaton to energy-augmentng techncal change (τ ). Ths represents a pure energy productvty mprovement that does not affect the productvty of other nputs or negatvely affect economc output. But as descrbed below, emprcal studes may not drectly estmate ths form of techncal change, energy-economc models may not smulate t (or may smulate t n dfferent ways) and studes of rebound effects may make a dfferent choce for the ndependent varable. Also, solatng energy-augmentng techncal change n ths way s somewhat artfcal as new technologes frequently mprove the productvty of multple nputs smultaneously. In general, energy-augmentng techncal change wll not lead to a proportonate mprovement n aggregate energy productvty (θ ) because: lower prce effectve energy wll stmulate the substtuton of (effectve) energy for other (effectve) nputs; and lower nput costs wll stmulate an ncrease n output whch n turn wll drag up energy consumpton. p~ )

6 nerges 2014, In combnaton, these substtuton and output effects wll ncrease energy consumpton above what t would have been n the absence of these responses. The sum of the two s the drect rebound effect for producers. In addton, there wll also be varous ndrect and economy-wde rebound effects. For example, f the relevant product forms an ntermedate nput to other sectors (e.g., steel n car producton), reductons n product prces may stmulate ncreased output from those sectors and hence further ncrease economy-wde energy consumpton. But ths paper s concerned solely wth drect rebound effects. The contrbuton of substtuton to the drect rebound effect may be llustrated graphcally. For smplcty, we assume that non-energy nputs are separable from energy nputs and can therefore be grouped together, or nested. The meanng and mplcatons of ths assumpton are descrbed further below. Fgure 2 shows a two-nput conventonal producton functon, where the optmal mx of energy () and a nest of other nputs (N) to produce output Y for an expendture of C s gven by the ntersecton of the soquants wth the so-cost lne (N 0, 0 ). nergy-savng techncal change shfts the soquants to the left and changes ther slope. If other nputs reman unchanged, the potental energy savngs are 0 1. However, over tme producers wll shft to a new, lower cost nput mx (N 1, 2 ), leadng to actual energy savngs of 0 2 whch are less than the potental savngs. The sze of the rebound effect (( 2 1 )/( 0 1 )) depends upon the ease of substtuton between energy and other nputs ndcated by the curvature of the soquant. Fgure 2. nergy-augmentng techncal change encourages the substtuton of energy for other nputs. Techncal change may also ncrease output (not shown) snce: frst, a hgher level of output can be produced for a gven expendture on nputs; and second, reductons n product prces may ncrease aggregate supply and hence output. Ths ncrease n output wll further ncrease energy use. The total drect rebound effect s the sum of these substtuton and output effects. In what follows, we focus on the substtuton effect whch Saunders [11] found to be more mportant. 4. Rebound ffects wth a Constant lastcty of Substtuton (CS) Producton Functon These examples llustrate the mportance of substtuton for rebound effects and suggest that these effects wll be larger when the scope of substtuton between energy and other nputs s easer. Saunders [10] demonstrates ths formally, usng the followng defnton of the drect rebound effect (R):

7 nerges 2014, R = 1+ ητ ( ) (17) where ητ ( ) s the elastcty of energy consumpton wth respect to energy-augmentng techncal change: We expect 0 ητ ( ) 1 η τ ln ( ) = ln τ (18) and hence 0 R 1. But the drect rebound effect could theoretcally exceed unty ( backfre ) or be negatve ( super conservaton ). Snce = Y/θ, quaton (18) can be decomposed: η τ ( ) = ητ ( Y ) ητ ( θ ) (19) If η τ (θ ) 1, there s a substtuton effect; and f η τ ( Y ) 0, there s an output effect. Saunders [9] derved these elastctes for a varety of producton and cost functons, but hs orgnal analyss [7] employed a CS producton functon smlar to that used n the maorty of energy-economc models. The partcular formulaton rests upon the assumpton that captal and labor are separable from energy and may therefore be grouped together: { } 1 γ 1 γ ρ ρ ρ ( ) ( ) ) (τ ) Y = υn a K L + b (20) As shown n [9], the substtuton, output and rebound effects are (Appendx 1 derves the substtuton effect): η τ (θ) = 1 σ (21) σ η τ ( Y ) = 1 (22) s σ R = 1 s (23) where σ = 1/(1 ρ) s the Hcks elastcty of substtuton (HS) between the captal-labour composte and effectve energy a standard measure of the ease of substtuton. If σ = 0, there s no rebound effect; whle f σ > 1, energy-augmentng techncal change reduces aggregate energy productvty (Δθ < 1) and ncreases energy consumpton (Δ > 1). Ths appears unlkely, snce t mples that energy s a non-essental nput. But Saunders [7] demonstrates that alternatve nestng structures of the same functon (.e., (K)L and (L)K) always lead to rebound effects greater than 100%, regardless of the HS between the nests. Hence, the estmated magntude of rebound effects appears senstve to the partcular choce of nestng structure and functonal form. Ths dscusson suggests that sectors where substtuton towards energy s easer (.e., σ s larger) wll be more vulnerable to rebound effects. Hogan and Manne [16] used analogous arguments to suggest that sectors where substtuton away from energy s easer wll be less vulnerable to rsng energy prces. Both conclusons have mportant polcy mplcatons and hghlght the mportance of accurately estmatng both the ease of substtuton and the rate of energy-augmentng techncal change n dfferent sectors.

8 nerges 2014, So does the exstng evdence base allow the ease of substtuton and rate of energy-augmentng techncal change wthn dfferent sectors to be accurately dentfed? How well do energy-economc models reflect ths evdence base? And can the avalable emprcal and modelng studes provde relable gudance on the magntude of rebound effects n dfferent sectors? The remander of the paper addresses these questons by examnng: frst, the defnton and estmaton of energy substtuton (Sectons 5 and 6); second, the use of those estmates wthn energy-economc models (Secton 7); and thrd, the estmaton and representaton of energy-augmentng techncal change (Secton 8). 5. Defnng Substtuton the Choce of Measure We frst look more closely at how the ease of substtuton s defned. The prevous dscusson used the Hcks elastcty of substtuton (σ or HS) whch measures the ease wth whch a decrease n one nput () can be compensated by an ncrease n another () whle holdng output fxed [17]. It s defned as: HS ln( X / X ) = ln(π / π ) (24) where π s the margnal productvty of nput X ( Y / X ). Ths defnton refers to movement along an soquant of a producton functon and s a scale-free measure of the curvature of ths soquant: the less the curvature the easer t s to substtute between two nputs and the larger the HS (HS 0). The extremes are: a lnear producton functon ( HS = ) and a Leontef (fxed proportons) producton functon (HS = 0). For a Cobb Douglas producton functon HS = 1, whle for a CS producton functon, HS s constant (as the name suggests) between 0 and nfnty. If HS > 1, each nput can fully substtute for the other (.e., the soquants cross the axes). Assumng proft maxmzaton and perfect competton, the HS can also be defned n relaton to a change n relatve prces: HS ln( X / X ) = ln( p / p ) (25) The HS was orgnally defned for a two-nput producton functon (captal and labor). When appled to mult-nput producton functons t s sometmes termed the Hcks Allen elastcty of substtuton (HAS), but ths extenson creates dffcultes snce the ease of substtuton between and may depend upon the level or prce of other nputs. Ths may not be the case f and are separable from other nputs but, as dscussed below, ths assumpton often does not hold. Also, the value of the HAS depends upon the partcular prce changes beng consdered. For ths reason, most emprcal studes defne substtuton n relaton to changes n a sngle nput prce, wth the most common measures beng the Cross prce elastcty (CP ), the Allen-Uzawa elastcty of substtuton (AS ) and the Morshma elastcty of substtuton (MS ). The relevant defntons are gven n Table 1 and Box 1. Substtuton between two nputs s easer when these measures are larger and ther sgn s commonly used to defne nputs as ether substtutes (+ve) or complements ( ve). However, whether two nputs may be descrbed as substtutes or complements depends upon the partcular elastcty that s beng used. Ths, together wth nconsstent termnology, complcates the nterpretaton of the emprcal lterature.

9 nerges 2014, HS HS Defnton HAS ln( X / X ) = ln( π / π ) ln( X = ln( p / X ) / p ) Table 1. Comparng common defntons of substtuton elastctes. Source: Broadstock et al. [18]. Output Other nputs X, k X Other nput prces p, k p Type Substtutes (complements) Input shares 1 ( s / s ) Fxed Fxed Fxed Two nput, two prce HS > 0 0 ( X / X ) ln( X / X ) ( s / s ) = ln( p / p ) Fxed Varable Fxed Two nput, two prce HAS > 0 (<0) 0 ( X / X ) ln X lns CP = ln p Fxed Varable Fxed One nput, one prce CP > 0 (<0) 0 ln p 1 ln X AS = s ln p Fxed Varable Fxed One nput, one prce AS > 0 (<0) MS ln( X / X ) ln( s / s ) = ln( p ) Fxed Varable Fxed Two nput, one prce MS > 0 (<0) 0 ln p Note: X = level of nput ; p = unt prce of nput ; π = margnal productvty of nput ( f / x ); and s = share of n nput costs (s = X p /C). Under perfect competton, s s equal to the share of n the value of output. lns ln p 0 Symmetrc Yes No No Yes No

10 nerges 2014, Box 1. Relatonshps among common substtuton elastctes. CP AS = s MS = CP CP HAS = MS ln p p ln p ln s ln p = s MS ( AS 1) ln p p ln p ln( s / s ln p ) = ( MS 1) Source: Broadstock et al. [18], Frondel [19,20], Sato and Kozum [21]. If only p changes and p s fxed, then HAS = MS, whle f only p changes and p s fxed, then HAS = MS. The great maorty of emprcal studes estmates the AS and uses ts sgn to classfy nput pars as ether substtutes or complements. Ths reflects the most common understandng of these terms, whch s the effect of a change n the prce of one nput on the demand for another. However, exactly the same nformaton s provded by the CP the AS ust dvdes the CP by the value share of one of the nputs (Box 1). Ths s not very helpful snce t mples that quanttatve estmates of the AS lack meanng and are dffcult to compare snce they depends upon value shares [19]. The AS s also symmetrc (AS = AS ) whch s equally unhelpful snce the mpacts of nterest usually depend upon whch prce s changng (p or p ). Hence, n most crcumstances, t seems preferable to estmate the CP, snce ths s asymmetrc ( CP CP ) and measures the change n demand for a sngle nput followng a prce change, rather than the change n a rato of nputs [19]. The MS s closer to the orgnal Hcks defnton snce t measures the percentage change n a rato of nputs and ndcates the curvature of an soquant. Lke the CP, t s also asymmetrc. However, the sgn of the MS s of lttle value for defnng substtutes or complements snce the MS should generally be postve. A negatve estmate of MS s lkely to ndcate problems wth the specfcaton, snce t mples substtuton away (towards) from an nput despte a fall (ncrease) n ts relatve prce. Ths also follows from quaton (2) n Box 1, snce we expect CP < 0 and CP > CP. 6. Measurng Substtuton The nergy-captal Debate Wth the CS producton functon of quaton (20), the magntude of rebound effects from energy-augmentng techncal change depends upon the magntude of the HS between effectve energy and the captal-labor composte. Many energy-economc models use a smlar functonal form, mplyng that ther estmates of rebound effects wll be senstve to the assumed value of the HS between energy and other nputs. But most emprcal studes do not use the CS functonal form owng

11 nerges 2014, to the constrants t mposes on the potental for substtuton. Instead, they use more flexble functonal forms such as the translog and derve the CP, the AS or MS from the estmated parameter values. Translog producton and cost functons were orgnally ntroduced by Chrstensen et al. [22], and are wdely used for emprcal work because they do not mpose any restrctons on nput substtutablty. stmaton usually nvolves applyng Shepard s Lemma to derve lnear cost share equatons and mposng varous restrctons on the parameter values. These dfferences create consderable dffcultes n usng the emprcal lterature to nfer approprate values for the HS to use wthn energy-economc models. Moreover, these dffcultes are greatly exacerbated by the fact that the emprcal lterature s tself confusng, contradctory and dffcult to nterpret. To llustrate these dffcultes, we take the long-standng debate over energy-captal substtuton as an example. ngneerng studes have long ndcated sgnfcant potental for cost-effectvely mprovng energy effcency through varous forms of captal nvestment whch could be nterpreted as substtutng physcal captal for energy. Ths potental s reproduced n the assumptons used for many energy-economc models. But begnnng wth Berndt and Wood [23], a large number of econometrc studes have suggested that energy and captal are AS and CP complements, mplyng that an ncrease n energy prces wll reduce the rate of captal nvestment [18]. Such a result mples that energy and captal are closely lnked n economc producton and that ncreases n energy prces could reduce output growth. Berndt and Wood [23] estmated a four-nput (KLM) translog cost functon for US manufacturng over the perod of , and found all nput pars to be substtutes apart from captal and energy (CP K = 0.16; CP K = 0.17). Snce then, more than one hundred studes have estmated dfferent types of substtuton elastcty between captal and energy n dfferent countres and sectors and tme perods, but have faled to reach a consensus on whether these nputs may generally be regarded as substtutes or complements. In a revew of over 50 studes provdng elastcty estmates, Broadstock et al. [19] found that ~40% of estmates suggested that energy and captal were complements (CP K < 0) and ~55% suggested they were substtutes (CP K > 0) wth wdely dfferng values. In a meta-analyss of ~40 studes, Koetse et al. [24] found energy and captal to be substtutes, wth a base case. CP K estmate of for tme seres data and for cross-sectonal data. Followng earler authors, e.g., [25], Koetse et al. [24] nterpret the former as representng short-run adustments and the latter long-run, hence suggestng greater scope for substtuton as the captal stock rotates. However, Frondel and Schmdt [26] demonstrate that alternatve explanatons for ths fndng are equally plausble. Specfcally, when a statc translog cost functon s estmated, the cost share of energy strongly nfluences the magntude of the estmated CP K [26]. When materal nputs are ncluded n the specfcaton, the cost shares of captal and energy becomes smaller, together wth the estmated CP K. Snce studes usng tme-seres data are more lkely to nclude materal nputs, they are more lkely to fnd energy-captal complementarty. Broadstock et al. [18] showed how dfferent studes have analyzed dfferent countres, sectors and tme perods usng dfferent specfcatons, data sets and methods of estmaton and come to qute dfferent conclusons on energy-captal substtutablty. Whle ths may be expected f the degree of substtutablty depends upon the sector, level of aggregaton and tme perod, t s notable that many studes reach dfferent conclusons for the same sector and tme perod, or for the same sector n

12 nerges 2014, dfferent countres. For example, Ra and Veall [27] found that studes usng the orgnal Berndt and Wood dataset have produced 38 estmates of AS K, rangng from 3.94 to Studes cte a range of possble causes for the varaton n results, ncludng dfferences n data type, level of sectoral aggregaton, measurement and aggregaton of nputs, functonal form, relatve value share of each nput and assumptons about homogenety and seperablty (see below) [18]. However, dfferent studes cte dfferent causes, and there appears to be no consensus on ether the relatve mportance of each cause or ther lkely drecton of nfluence. Ths ongong debate llustrates how the estmaton of substtuton elastctes rases a varety of theoretcal and methodologcal ssues that collectvely make t very dffcult to nterpret the results and draw useful conclusons [18]. Of partcular mportance to the present dscusson are the explct or mplct assumptons about the separablty of nputs. Separablty mples that the HS between two nputs s unaffected by the level or prce of the other nputs. These two condtons are only equvalent when relatve nput shares are ndependent of the level of output. In addton, f two nputs (,) are separable from a thrd (k), then the ease of substtuton between and k (as measured by the CP, AS or MS) s equal to that between and k (e.g., CP k = CP k ) [28]. Separablty assumptons are commonly used to ustfy ether the omsson of nputs for whch data s unavalable (notably materals) or the groupng, or nestng, of dfferent nputs. Nestng mples that producers engage n a two-stage decson process: frst optmzng the combnaton of nputs wthn each nest, and then optmzng the combnaton of nests requred to produce the fnal output. Two nputs may only be legtmately grouped wthn a nest f they are separable from nputs outsde of the nest. For example, Saunders (KL) nestng structure requres that captal and labor are separable from energy [7]. One of the contrbutons of Berndt and Wood [23] was to show that captal and labor was not separable from ether energy or materals wthn ther dataset. But even when two nputs (e.g., and ) wthn a nest are separable from a thrd (e.g., k), ths does not mean that measures of the CP, AS or MS between and are unaffected by the prce of k. As Frondel and Schmdt [29] have shown, even f captal and labor were separable from energy under the standard defnton (as n quaton (20)), the ease of substtuton between captal and labor (as measured by the CP, AS or MS ) may stll be affected by the prce of energy. Frondel and Schmdt [29] defne a strcter condton of emprcal dual separablty, n whch the value of CP s unaffected by the prce of k. Stablty of AS requres the addtonal condton that the value shares are unaffected, whch seems unlkely. Hence, the emprcal measures of substtuton between and are lkely to depend upon the prce of other nputs, even when and are separable from those nputs. Ths suggests that estmates of substtuton elastctes are lkely to be based f separablty s assumed where not supported by the data, or f measures of any nput are omtted. The latter stuaton s common, partcularly wth regard to the omsson of materals. In practce, studes that exclude materals more often ndcate captal-energy substtutablty, whle those that nclude materals ndcate complementarty [18,23,26]. In sum, the multple defntons of substtuton elastctes, the range of factors nfluencng emprcal estmates and the senstvty of results to those factors make the evdence base n ths area confusng, contradctory, prone to bas and dffcult to use. At a mnmum, statements about substtutablty need to be qualfed by the countres, sectors and tme perods to whch they apply the manner n whch nputs are dsaggregated and measured and the specfc assumptons that are made wth the latter

13 nerges 2014, beng supported, where possble, by statstcal tests. But the resultng estmates may stll not be useful for partcular applcatons, ncludng the parameterzaton of energy-economc models. To llustrate ths, the next secton examnes how substtuton elastctes are used n energy-economc models and hghlghts the lmted bass for the assumptons made. 7. Assumng Substtuton nergy-conomc Modelng nergy-economc models based upon computable general equlbrum (CG) technques are wdely used for explorng polcy-relevant questons ncludng the estmaton of rebound effects. Such models almost nvarably use CS producton functons and assume that some nputs are separable from others. Parametersaton requres assumptons about the HS between nput groups [30,31] and these can have a maor nfluence on results [4,32,33]. For example, Grepperud and Rasmussen [4] estmate the rebound effects from energy-augmentng techncal change to be substantally hgher n the Norwegan prmary metals sector than n the fsheres sector, owng to the (assumed) greater opportuntes for energy substtuton n the former [4]. For such results to be robust, the assumed parameter values should be frmly based upon emprcal research. Unfortunately, t s common practce to assume these values wth only lmted reference to the emprcal lterature. Moreover, even when such references are made, there are consderable dffcultes n usng emprcal studes to nfer values of the HS for CG models. Ths s because most of these models: dffer from the cted emprcal studes n the manner n whch ndvdual nputs are aggregated and n the level of sectoral aggregaton; assume values for HS parameters wthn CS producton functons, whle most emprcal studes use flexble cost functons to estmate the AS, CP and/or MS; mpose separablty between groups (nests) of nputs whle most emprcal studes do not; and requre estmates of the HS between those groups, whle most emprcal studes provde estmates of the AS, CP and/or MS between ndvdual pars of nputs. These ponts are brefly elaborated below. Blackorby and Russell [34] showed that the AS, MS and HS are dentcal f (and only f) there are only two nputs to the producton functon, or the producton functon has a Cobb ρ ρ ρ ρ 1/ ρ Douglas or non-nested CS structure (e.g., Y = ( ak K + all + a + am M ) ). But the two-nput case s of lmted nterest, the Cobb-Douglas structure s excessvely restrctve and the non-nested CS requres the HS between all nputs to be dentcal [35] whch appears unlkely. In order to provde greater flexblty n substtuton possbltes, most CG models mpose separablty assumptons to create a nested CS functonal form [36] n whch nputs are grouped n pars such as the (KL) structure of quaton (20). A more general nested CS s: ρ Y = [ a( K* ) + (1 a)( *) where both the captal-labor composte (K*) and the energy-materals composte (*) are CS functons as well: ρ ] 1 ρ (26)

14 nerges 2014, * α α K = ( bk + (1 b) L ) a (27) 1 * β β β = ( c + (1 c) M ) (28) The structure of ths two-level nested CS, n whch captal K and labour L are nested as well as energy and materals M, rests on the assumpton that K* s separable from *. Saunder s functonal form (quaton (20)) follows the same structure, but omts materals and uses a smpler (Cobb Douglas) form for the KL nest [7]. Alternatve nestng schemes (such as (K)(LM) or (L)(KM)) are wdely used, but the approprate choce s rarely tested [37,38]. If a dstncton s made (as t should) between dfferent types of captal, labour or energy nputs (e.g., electrcty and non-electrcty), a multlevel CS can be formed, wth more than one functon nested wthn the orgnal one. Lecca et al. [39] demonstrate the senstvty of model results to these assumptons and crtcze the arbtrary choce of both nestng structure and parameter values n the maorty of energy-economc models. But such a structure s dffcult to ether estmate drectly or to parameterze from the results of exstng research. Sanstad et al. [40] observe that: There appears to be no publshed econometrc estmaton of a nested CS model wth general factor-augmentng techncal change, even n a degree of complexty less than s common n ntegrated assessment and energy smulaton models. To llustrate, we take a closer look at the mpled values of the HS, AS and CP wthn such a structure. Wth a nested CS, the AS between a par of nputs belongng to dfferent nests s equal to the HS between the nests [28]. For example, the (KL)(M) nestng structure mples that: AS K = AS KM = AS L = AS LM. = HS K**. But the AS between a par of nputs belongng to the same nest s not equal to HS between those nputs. Indeed, whle two nputs wthn an ndvdual nest are necessarly HS substtutes, they may at the same tme be AS (and CP) complements (.e., AS < 0). The AS between these two nputs s only equal to the HS f the output of the nest s held constant [36]. Takng labor and captal as an example n quaton (26): 1 ASLK = HS * * + ( HSLK HS * * ) K K (29) a Hence, t s possble for AS LK to be negatve, provded HS * * > HS. In other words, the scope for substtuton between the captal-labor composte (K * ) and the energy-materals composte ( * ) s greater than the scope for substtuton between captal and labour n the producton of K *. Hence, estmates of the AS, CP or MS between two nputs provde lttle gudance n choosng the approprate values of the HS between those nputs that are requred for the nested CS functons used n CG models. If the separablty assumptons were vald, a partcular nested CS could be parametersed f the functon was estmated drectly. But the maorty of emprcal studes estmate flexble functonal forms such as the translog and do not mpose separablty restrctons. Moreover even f separablty restrctons were to be mposed, ths would not ensure that estmates of the AS or CP between two nputs were nvarant to the prce of other (possbly omtted) nputs snce ths would requre the strcter condtons descrbed by Frondel and Schmdt [26,29]. Furthermore, even f the strcter condton were to hold, the mpled nestng structure may not correspond to that used wthn a partcular energy-economc model. K LK

15 nerges 2014, Table 2 compares the nestng structures and assumed values of HS n a number of contemporary CG models. Over half of these models exclude materal nputs, so therefore mplctly assume that these are separable from other nputs. These models further vary n how they dsaggregate and nest ndvdual nputs (e.g., fuel and electrcty) and how they model techncal change. The bass for the assumed values for the HS between dfferent nputs and nests of nputs s rarely made clear, senstvty tests are uncommon and the values chosen vary wdely between dfferent models. Table 2. Nestng structures and assumed values of the Hcks elastcty of substtuton (HS) n a selecton of contemporary CG models. Source: based on van der Werf [38]. Authors Nestng structure Assumed HS Bosett et al. [41] (KL) HS K,L = 1.0; HS KL, = 0.5 Burnaux et al. [42] (K)L HS K, = 0 or 0.8; HS K,L = 0 or 1.0 denhofer et al. [43] KL HS K,L, = 0.4 Gerlagh and van der Zwaan [44] (KL) HS K,L = 1.0; HS KL, = 0.4 Goulder and Schneder [45] KLM HS K,L,,M = 1.0 Kemfert [46] (KLM) HS KLM, = 0.5 Manne et al. [47] (KL) HS KL = 1.0; HS KL, = 0.4 Popp [48] KL HS K,L, = 1.0 Sue Wng [49] (KL)(M) HS K,L = ; HS.M = 0.7; HS KL,M = 0.7 In sum, the assumptons made for producton structures and substtuton elastctes wthn most CG models appear to be only weakly lnked to an emprcal lterature that s tself contradctory and nconclusve. Ths suggests that the results of such models, ncludng ther estmates of rebound effects, should be treated wth consderable cauton. Unless more flexble functonal forms can be adopted, e.g., [50], senstvty analyss of nestng structures and parameter values should be employed. 8. Representng Techncal Change Competng Approaches Sectons 2 and 3 argued that the magntude of rebound effects from energy-augmentng techncal change should be proportonal to the flexblty of producers to adapt to those effcency gans va substtuton [7,10]. But Sectons 4 7 hghlghted the dffcultes n both estmatng ths flexblty and n usng these results to parameterze energy-economc models. The multple defntons of substtuton elastctes are a source of confuson, the most commonly estmated elastcty s of lttle practcal value, the emprcal lterature s contradctory, prone to bas and dffcult to use, and there are only tenuous lnks between ths lterature and the assumptons used wthn energy-economc models. Whle smply assumng values for substtuton elastctes s tantamount assumng the answer [10], t appears to be very common. Further challenges are created by the use of energy-augmentng techncal change as the ndependent varable for an estmate of rebound effects. Ths s because most emprcal studes do not drectly estmate ths form of techncal change and many energy-economc models do not smulate t. To llustrate, we summarze the most common approaches to estmatng and modelng techncal change and hghlght the mplcatons of usng two dfferent approaches wthn a CS producton functon.

16 nerges 2014, As noted, emprcal studes typcally employ flexble cost functons such as the translog and use Shepards Lemma to derve equatons for the value share of each nput. Wth a sutable functonal form, ths allows the energy prce bas to be estmated: s ψ = (30) t where ψ 0 (ψ 0) ndcates energy savng (usng) techncal change. But energy-savng techncal change s not equvalent to energy-augmentng techncal change because captal, labor and materals augmentng techncal change, as well as substtuton between nputs, also affect the energy value share (s ). Whle τ represents techncal change for a sngle nput before nput shares adust, ψ represents the net effect of techncal change on all nputs after nput shares adust. Substtuton towards energy may lmt the reducton n the energy value share brought about by energy-augmentng techncal change and n some crcumstances may even ncrease the energy value share. Hence, n prncple, t s possble for energy-augmentng techncal change (τ 0) to coexst wth energy-usng techncal change (ψ 0). In a wdely cted study, Hogan and Jorgenson [51] fnd energy-usng techncal change n US ndustry over the perod Saunders [10] argues that ths result suggests rebound effects greater than unty n US ndustry mplyng that effcency mprovements ncreased aggregate energy consumpton. However, snce Hogan and Jorgenson measure energy-usng rather than energy-augmentng techncal change, ther results do not allow the magntude of the rebound effect (as defned by quaton (17)) to be drectly estmated. Whle they fnd an ncreasng value share of energy, ths may have derved from a number of sources and s not necessarly ndcatve of backfre followng energy augmentng techncal change. Modelng studes typcally employ CS producton functons. Whle some model energy-augmentng techncal change n a smlar manner to Saunders [7] (quaton (20)), others ncorporate an autonomous energy effcency ndex (AI) to ndcate the rate of growth of aggregate energy productvty [40,52,53]: ln θ AI = (31) t But agan, the AI s not equvalent to energy-augmentng techncal change (τ ), snce labor, captal and materals augmentng techncal change, as well as nput substtuton, wll also affect aggregate energy productvty (θ ). Hogan and Jorgensen [51] derve the followng relatonshp between the AI and the energy prce bas (ψ ): ψ (ε ) = s gt AI (32) In other words, the energy prce bas s the share weghted devaton of the autonomous energy effcency trend from the trend n total factor productvty (ε gt ). Postve values for ε gt mply mprovements n total factor productvty (declnng costs per unt of output), whle postve values for AI mply mprovements n energy productvty (declnng energy ntensty) over tme. If aggregate energy productvty s mprovng at the same rate as total factor productvty (ε gt = AI, then the energy prce bas s zero. If energy productvty s mprovng faster (slower) than total factor productvty, then the energy prce bas s negatve (postve) and techncal change s energy-savng (energy-usng). Normally, we would expect the AI and the energy prce bas to be opposte n sgn

17 nerges 2014, (e.g., f AI > 0, we expect ψ < 0). But energy-savng techncal change may also result from fallng total factor productvty (ε gt < 0), even f energy productvty s mprovng (AI > 0) provded > AI [40]. Sanstad et al. [40] fnd evdence of ths wthn developng countres. ε gt Dfferent models use dfferent CS formulatons and nestng structures and mplement ether the AI or energy-augmentng techncal change (τ ) n dfferent ways [38]. For example, the verson used by Manne and Rchels [52] nests a Cobb-Douglas functon for value added (captal and labor) wthn a CS and smulates autonomous energy effcency mprovements by a negatve growth rate of the dstrbuton parameter b (b = e δt where δ 0): 1 γ 1 γ ρ ρ ρ Y = [ a( K L ) + b( ) ] (33) In contrast, Saunders verson of ths functon smulates energy-augmentng techncal change by a λ postve growth rate for parameter τ ( τ () t = e t where λ 0) [7]: 1 γ 1 γ ρ ρ ρ τ Y = [ a( K L ) + b( ) ] (34) These two approaches to representng techncal change are not equvalent. Combnng both approaches, Appendx 2, shows that wth ths functonal form: AI = σδ + (1 σ)λ (35) The Manne Rchels approach has λ = 0, therefore, AI = σδ. Snce σ 0 and δ 0, a negatve growth rate for parameter b always leads to a postve AI. Hence, wth ths approach techncal change for energy nputs always leads to a proportonate reducton n aggregate energy productvty. As a result, ths approach s ncapable of smulatng any substtuton response to ths techncal change and hence of smulatng the substtuton component of the drect rebound effect. The model may stll allow the output component of the drect rebound effect to be smulated. A lkely outcome s that the energy savngs from mproved energy effcency wll be overestmated. In contrast, the Saunders approach has δ = 0 [7], therefore, AI = (1 σ)δλ. Snce λ 0, the mpact of energy-augmentng techncal change on aggregate energy productvty depends upon the elastcty of substtuton between energy and value-added. As shown n Secton 3, ths form of techncal change only leads to a postve AI when σ 1. In other words, substtuton contrbutes to a rebound effect that reduces energy savngs. Consstent wth ths, we fnd that models that depct energy-savng techncal progress nvarably assume a value for the HS between energy and other nputs that s less than unty [54 59]. In sum, the estmaton of drect rebound effects for producers requres specfcaton of the magntude and drecton of energy-augmentng techncal change. Most emprcal studes do not estmate ths form of techncal change, many energy-economc models do not smulate t and others smulate t n a manner that precludes the accurate modelng of rebound effects. As a result, the avalable evdence provdes nsuffcent gudance on the magntude of rebound effects wthn dfferent ndustral sectors. When combned wth the dffcultes n specfyng substtuton elastctes dscussed earler, the result s consderable uncertanty over the magntude of rebound effects n ndustral producton.

18 nerges 2014, Summary Ths paper has explored the relatonshps between energy productvty mprovements for producers, the ease of substtuton between energy and other nputs and the sze of the resultng rebound effects. It has shown how easer substtuton drves larger rebounds, but the relevant mechansms are not straghtforward and are dffcult to capture emprcally. There are three man fndngs. Frst, the multple defntons of substtuton elastctes are a source of confuson, the most commonly estmated elastcty s of lttle practcal value and the emprcal lterature s contradctory, prone to bas and dffcult to use. For example, engneerng studes suggest a large potental for mprovng energy effcency by substtutng captal for energy, but three decades of econometrc research has acheved no consensus on whether these nputs are best descrbed as substtutes or complements, and no consensus on how dfferent factors nfluence emprcal results. Second, there are only tenuous lnks between the emprcal lterature on substtuton elastctes and the assumptons used wthn energy-economc models. Most models employ nested CS producton functons and make assumptons about the HS between these nests, whle most emprcal studes use translog cost functons and estmate the AS, CP or MS between nput pars. Usng the latter to parameterze the former s problematc. In addton: the process of complng parameter values s rarely transparent; senstvty tests are uncommon; the emprcal studes (when cted) frequently apply to dfferent sectors, tme perods and levels of aggregaton to those represented by the models; and dfferent models use wdely dfferent assumptons. All these observatons suggest that the results of CG models, ncludng the estmates of rebound effects, should be treated wth cauton and that senstvty tests should be more extensvely employed. Thrd, whle energy-augmentng techncal change provdes the natural choce of ndependent varable for an estmate of rebound effects, most emprcal studes do not estmate ths form of techncal change, many modelng studes do not smulate t and others smulate n a manner that precludes the accurate modelng of rebound effects. As a result, the avalable evdence base provdes only lmted gudance on the magntude of drect rebound effects for producers and wdely used modelng tools may overestmate the future energy savngs from mproved energy effcency. These conclusons provde ponters to how future studes may more adequately capture drect rebound effects. For emprcal studes, the most mportant requrement s the explct estmaton of nput-augmentng techncal change, such as acheved recently by Saunders [11] for the US and Stern and Kander [60] for Sweden. Interestngly, both studes suggest substantal drect rebound effects (e.g., 60% or more) from energy augmentng techncal change [10]. In practce, smultaneous mprovements n the productvty of other nputs may amplfy these effects. For modelng studes, the requrements nclude (where possble) greater use of flexble functonal forms, abandonment of the AI n favor of nput-augmentng techncal change, the ncluson of materals nputs, much more careful attenton to the emprcal bass for elastcty assumptons and extensve senstvty tests of parameter values and nestng structures. These recommendatons could be challengng to mplement. But n ther absence, our knowledge of rebound effects n ndustral producton s lkely to reman lmted and our confdence n future energy savngs may be msplaced.

19 nerges 2014, Acknowledgments The research descrbed n ths paper formed part of a larger study on rebound effects by the UK nergy Research Centre [1]. The fnancal support of the UK Research Councls s gratefully acknowledged. The author s grateful for comments on earler versons from Davd Broadstock, Lester Hunt, Manuel Frondel and Harry Saunders. The usual dsclamers apply. Author Contrbutons Steve Sorrell s the sole author of ths work, but the research draws upon prevous collaboratons wth John Dmtropoulos [1]. Appendx 1 The functonal form used by Saunders [7] nests a Cobb Douglas functon for captal and labor ( value added ) wthn a CS functon ((KL)) and ncorporates energy-augmentng techncal change (τ ): 1 β 1 β ρ ρ ρ τ Usng the chan rule, the margnal product of energy s gven by: Y = [ a( K L ) + b( ) ] (A1-1) 1 1 ρ ρ 1 β 1 β ρ ρ ρ Y = bτ [ a( K L ) + b(τ )] (A1-2) The term n brackets s output (Y) n the (1 ρ) power: Y ρ ρ 1 1 ρ = bτ Y (A1-3) Y = bτ ρ Y ( ) Assumng perfect competton and cost mnmzaton, ths equals the unt prce of energy: 1 ρ (A1-4) Y 1 ρ ρ Y = p = bτ (A1-5) Solvng for energy: Therefore, aggregate energy productvty (θ = Y/) becomes: 1 p ρ ρ 1 = τ 1 ρ Y (A1-6) b p θ= b 1 ρ 1 τ ρ 1 ρ (A1-7) Aggregate energy productvty therefore depends upon energy prces, energy-augmentng techncal change, the HS (σ = 1/(1 ρ)) and the parameter b. By takng the partal dervatve of ths expresson