A theoretical analysis of price elasticity of energy demand in multi-stage energy conversion systems

Energy Policy 31 (2003) 1699–1704

A theoretical analysis of price elasticity of energy demandin multi-stage energy conversion systems$

Robert Lowe*

Faculty of Health & Environment, Centre for the Built Environment, Leeds Metropolitan University, Leeds LS1 3HE, UK

Abstract

The objective of this paper is an analytical exploration of the problem of price elasticity of energy demand in multi-stage energy

conversion systems. The paper describes in some detail an analytical model of energy demand in such systems. Under a clearly stated

set of assumptions, the model makes it possible to explore both the impacts of the number of sub-systems, and of varying sub-system

elasticities on overall system elasticity. The analysis suggests that overall price elasticity of energy demand for such systems will tend

asymptotically to unity as the number of sub-systems increases.

r 2003 Published by Elsevier Science Ltd.

Keywords: Price elasticity; Analytical model; Multi-stage systems

1. Introduction

This paper has been written in an attempt tounderstand certain aspects of the impact of energy priceon demand in multi-stage energy conversion systems.For a variety of reasons ranging from problems of timelags and short time series, to the problem of non-stationarity, any analytical treatment of such systems isunlikely to give more than a rather incomplete picture oftheir behaviour. To make progress at all, the author hashad to assume that:

* time lags can be neglected,* the performance of each sub-system depends only on

the price of energy immediately upstream,* the additional costs imposed by each sub-system

relate only to the energy dissipated by that sub-system,

* the performance of each sub-system is reversible, andthat it can be represented analytically by a powerlaw.1

The nature of the energy conversion system issketched in Fig. 1.As noted above, the ith sub-system efficiency is

assumed to depend on the effective cost of energyfollowing the ði � 1Þth stage of energy conversion.2 Thus

Zi ¼ Zi;base ci�1=ci�1;base

� �ai ð1Þ

and

cnþ1 ¼cn

Znþ1¼

c0Qnþ1i¼1 Zi

: ð2Þ

2. Evaluation of energy costs

Since the conversion efficiency of each sub-system isassumed to depend on the upstream energy price, thefirst stage in the process of analysis is to calculatethese prices in terms of sub-system elasticities andprimary energy price, c0: We can then calculate the

ARTICLE IN PRESS

$The first version of this paper was written in August 1998.

*Tel.: +44-113-283-1724.

E-mail address: [email protected] (R. Lowe).1 In certain simple cases, engineering analysis would lead one to

expect such behaviour, and the expected exponents can be calculated.

Space heating in buildings is one such case. The predicted price

elasticity of space heating in buildings, based on re-optimisation of the

building thermal envelope alone, is approximately 0.5 (Lowe et al.,

1997).

2For small changes in energy cost, such that ci�1=ci�1;baseE1

1=Ei�1;base

� �dEi�1Eai 1=ci�1;base

� �dci�1:

Hence, the exponent, ai is the price elasticity of the ith sub-system

with respect to the effective cost of energy from the preceding stages of

conversion. It seems appropriate to refer to this exponent as a partial

or sub-system elasticity of demand.

0301-4215/03/$ - see front matter r 2003 Published by Elsevier Science Ltd.

doi:10.1016/S0301-4215(03)00064-8

corresponding energy fluxes. For a single stage system

c1 ¼c0

Z1¼

c0

Z1;base c0=c0;base

� �aið3Þ

and re-arranging

c1=c0;base ¼c0=c0;base

� � 1�a1ð Þ

Z1;base

: ð4Þ

For a two-stage system


Z2¼

c0=c0;base

� � 1�a1ð Þ

Z1;baseZ2;base c1=c1;base

� �a2 ; ð5Þ

but from Eq. (1)

c1;base ¼ c0;base=Z1;base ð6Þ

so


� � 1�a1ð Þ

Z1;baseZ2;base Z1;basec1=c0;base

� �a2 : ð7Þ

Substituting for Z1;basec1=c0;base from Eq. (4)


� � 1�a1ð Þ

Z1;baseZ2;base c0=c0;base

� � 1�a1ð Þa2ð8Þ

and re-arranging


� � 1�a1ð Þ 1�a2ð Þ

Z1;baseZ2;base

: ð9Þ

In general (a formal proof is presented in the Appendix)

cn=c0;base ¼c0=c0;base

� �Qn

i¼11�aið ÞQn

i¼1 Zi;base

: ð10Þ

3. Evaluation of energy fluxes

Conceptually, the evaluation of energy fluxes is donein the opposite direction from the evaluation of prices.

We assume that the energy flux from the final stage ofconversion is fixed. Our objective is to calculate theinput of primary energy that is needed to obtain thisfixed quantity, under differing assumptions about theprice of primary energy. For a single stage system:

E0 ¼E1

Z1¼

E1

Z1;base c0=c0;base

� �a1 ð11Þ

and since E0;base ¼ E1=Z1;base

E0 ¼ E0;base c0=c0;base

� ��a1 ð12Þ

which, for reasons that will become apparent, we willwrite

E0 ¼ E0;base c0=c0;base

� �1� 1�a1ð Þ: ð13Þ

For a two-stage system

E0 ¼E2

Z1Z2: ð14Þ

Using Eq. (1) to expand Z1 and Z2:

E0 ¼E2

Z1;baseðc0=c0;baseÞa1Z2;baseðc1=c1;baseÞ

a2 ð15Þ

and since E0;base ¼ E2=Z1;baseZ1;base

E0 ¼E0;base

c0=c0;base

� �a1 c1=c1;base

� �a2 : ð16Þ

But from Eq. (2), c1;base ¼ c0;base=Z1;base

E0 ¼E0;base

c0=c0;base

� �a1 Z1;basec1=c0;base

� �a2 : ð17Þ

Substituting from Eq. (4) and simplifying

E0 ¼ E0;base c0;base=c0� �1� 1�a1ð Þ 1�a2ð Þ

: ð18Þ

ARTICLE IN PRESS

Nomenclature

Ei the energy flux following the ith stage ofenergy conversion

Zi the ith sub-system efficiencyZ0i the ith sub-system efficiency in the case that

up-stream sub-systems are inelasticai the ith sub-system exponent of demand, with

respect to the effective cost of energy followingthe ði � 1Þth stage of conversion

asystem the overall price elasticity of energy demandfor the whole energy conversion system

ai;effective the overall price elasticity of energy demandfor the whole energy conversion system

ci the effective cost of energy following the ithenergy conversion stage

c0 the cost of primary energy, up-stream of allenergy conversion stages

ci;base the effective value of ci in the base case, whenc0 ¼ c0;base

11,αη 22 ,αη 33 ,αη 44 ,αη ... nn

αη ,

00 ,cE 11,cE 22 ,cE 33 ,cE 44 ,cE ... nncE ,

Fig. 1. Components of multi-stage energy conversion system.

R. Lowe / Energy Policy 31 (2003) 1699–17041700

For an n-stage system (again, a formal proof ispresented in Appendix):

E0 ¼ E0;base c0;base=c0� �1�Qn

i¼1

1�aið Þ: ð19Þ

The overall system elasticity is given by

asystem ¼ 1�Yn

i¼1

ð1� aiÞ: ð20Þ

In the special case that all sub-system elasticities areequal, that is when ai ¼ a; the overall system elasticityreduces to

asystem ¼ 1� ð1� aÞn: ð21Þ

When sub-system elasticities are small, that isP

ai51;the overall system elasticity approximates to the sum ofthe sub-system elasticities:

asystemEXn

i¼1

ai �OðajakÞ: ð22Þ

4. Discussion and conclusions

For a system in which sub-systems are not priceelastic, ai ¼ 0; and Eq. (19) simplifies to

E0 ¼ E0;base: ð23Þ

For a completely elastic system, ai ¼ 1; and Eq. (2)simplifies to

E0 ¼ E0;base c0;base=c0� �

: ð24Þ

In such a system, energy use is inversely proportional toenergy price. More importantly, Eqs. (20) and (21)suggest a tendency for overall system elasticity ofcomplex, multi-stage systems to tend to unity, evenwhere all aio1: This key result can be seen more clearlywhen all partial elasticities are equal: ai ¼ a for all i: Inthis case the overall elasticity, given by Eq. (21), is anexponential function of the number of stages, n: Thisfunction is plotted for a ¼ 0:25 and a ¼ 0:5 in Fig. 2.In practice, all stages in a multi-stage system may not

be able to respond immediately to a change in upstreamenergy price. Possible reasons for this include longresponse times associated with long physical lifetimes ofparticular pieces of infrastructure such as powerstations, and hysteresis induced by network effects.Where upstream stages do not respond quickly or at all,the effect is to induce a larger response from down-stream stages, coupled with reduced overall response.The cost of the larger short-term response from down-stream stages is likely to be short-term over-investmentin these stages, coupled with a tendency for the responseof the whole system to overshoot in the long term.This can be illustrated as follows. If all sub-systems

respond to an energy price change, then the efficiency of

sub-system i is given by Eq. (1):

Zi ¼ Zi;base ci�1=ci�1;base

� �ai

¼ Zi;base ci�1=c0� �Yi�1

j¼1

Zj

!ai

ð25Þ

and substituting for ci�1=c0� �Qi�1

j¼1 Zj from Eq. (10):

Zi ¼ Zi;base c0=c0;base

� �ai

Qi�1j¼1

1�ajð Þ: ð26Þ

In this case, the effective elasticity of the ith sub-system,with respect to changes in c0 rather than ci�1; is not ai

but

ai;effective ¼ ai

Yi�1j¼1

ð1� ajÞ: ð27Þ

The later any particular sub-system appears inthe chain of conversion, the smaller will be its effectiveelasticity. Elastic upstream sub-systems attenuate theprice signal experienced by downstream sub-systems andthus the response of those upstream sub-systems to anoverall change in energy price. In the simple case that allsub-system elasticities are equal, this upstream shieldingfactor simplifies to an exponential function:

ð1� aÞi�1: ð28Þ

The shielding effect of upstream sub-systems is illu-strated in Fig. 3 for a ¼ 0:5:In a two-stage system, with a1 ¼ a2 ¼ 0:5; in which

both stages are allowed to respond to a change in rawenergy price, the effective elasticity of the second stage is14rather than 1

2and the overall system elasticity is 3

4:

Though this is just one of several possible theoreticalexplanations for such phenomena, it is easy to see howshielding, combined with time lags in upstream sub-systems and asymmetric price responses (Gately, 1992;

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7

number of stages

sys

tem

ela

sti

cit

y

alpha=0.5

alpha=0.25

Fig. 2. System elasticity versus number of sub-systems.

R. Lowe / Energy Policy 31 (2003) 1699–1704 1701

Walker & Wirl, 1993) can give rise to empirical long-runsystem elasticities greater than unity.If on the other hand we assume that only sub-system i

of an n-stage system is elastic, then the change inefficiency of sub-system i is given by

Z0i ¼ Zi;base c0=c0;base

� �ai : ð29Þ

In this case the effective elasticity of the ith sub-systemis, as we would expect, simply ai: Thus, non-response ofupstream stages increases the effective elasticity of theith sub-system with respect to changes in the price ofraw energy, c0; by a factor of

Yi�1j¼1

ð1� ajÞ ð30Þ

and reduces the overall system elasticity by a factor of

1�Qn

j¼1ð1� ajÞ

ai

ð31Þ

compared with the case where all systems respond tochanges in price. In the simple case that all sub-systemelasticities are equal, the reduction in overall elasticitysimplifies to

1� 1� að Þnð Þ=a: ð32Þ

The results of this paper may be of significance in thatmany practical energy conversion systems do in factconsist of chains of linked processes. Four simpleexamples within the built environment are:

* mechanical ventilation systems—inlet and exhaustresistance, motor efficiency, fan efficiency, distribu-tion system, supply and extract terminals (N^rg(ardet al., 1983);

* lighting systems—electricity supply system, lamp,luminaire, lighting control system, building (Verder-ber & Rubinstein, 1984; Ne’eman, 1984);

* space heating in buildings—energy supply system,space heating system, thermal envelope;

* space cooling in buildings—energy supply system,space cooling system, thermal envelope.

Lighting, heating, cooling and air movement accountfor most of the energy used in the built environment.Two further examples illustrate the potentially wide-spread applicability of this analysis:

* IT systems—electricity supply system, power supply,energy management systems, CPU, screen (Norfordet al., 1989); and

* vehicles—oil refinery, engine, gearbox, transmission,vehicle mass and air resistance (von Weizs.acker et al.,1997)

A more detailed examination of these systems revealsmany additional sub-systems, but also structures thatare significantly more complex than the simple chain ofconversion that forms the conceptual basis for theanalysis presented in this paper. Moreover, in practice,many aspects of these systems are not determined bymicro-economic optimisation. For example, the thermalproperties of building envelopes are substantiallydetermined by regulation. Nevertheless, the demandsof regulation are themselves influenced by economicanalysis, and perceptions of future energy price (DETR,2000).Despite these caveats, it would appear likely that

technological advance and economic development gen-erally lead to an increasing proportion of complex multi-stage energy conversion systems. The analysis presentedhere would lead one to expect total price elasticities ofsuch systems to be larger than sub-system analyseswould suggest, and to approach unity. Moreover, theanalysis suggests that in attempting to predict orunderstand overall empirical elasticities of energyconversion systems, one should place at least as muchweight on the structure of such systems, as on the detailsof any particular sub-system. It is not the author’s intentto present a complete review of empirical work on priceelasticity of energy demand in support of this conten-tion, but some work suggests that this might indeed bethe case. Von Weizs.acker and Jesinghaus (1992) suggesta price elasticity for energy use in cars in the region ofunity based on comparison of energy use and price datain 14 countries, Berkhout et al. (2000) state that thelong-term elasticity for passenger transport is in therange 0.8–1.0, while Goodwin (1992) gives a value of 1.2for the long-term price elasticity of energy demand fortransport in the UK. In other areas such as lighting andIT, while a combination of short runs of data and non-stationarity make it difficult to confirm this empirically,

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7

sub-system

up

-str

ea

m s

hie

ldin

g f

ac

tor

Fig. 3. Shielding effect of up-stream sub-systems on effective elasticity

of final sub-system in an n-stage energy conversion system.

R. Lowe / Energy Policy 31 (2003) 1699–17041702

engineering analyses suggest that elasticities should behigh.The analysis presented here does not apply directly to

systems other than simple energy conversion chains. Athorough treatment of the problem of price elasticity oftransport demand cannot, for example, avoid consider-ing sub-systems such as social attitudes to cycling andwalking, logistics strategies for freight distribution,urban density and form, and living and workingpatterns, none of which satisfies this condition. Whileit would be interesting to attempt to extend theargument presented here to a wider range of systemsand to include cost categories other than energy, it maynot be easy to do this analytically. Nevertheless thereappears to be no obvious reason why the basic results ofthis paper should not apply, at least qualitatively, tosuch systems.To conclude, the main policy implications of this

paper are that:

* the overall structure of energy conversion systemsmay be at least as important in determining systembehaviour as the details of any particular sub-system;

* long run price elasticities for many energy conversionsystems may approach unity;3

* pricing policies may therefore have a significantimpact on energy demand and carbon emissions formany energy using systems; and

* the impact of such policies is likely to be maximised ifenergy or carbon taxation is levied at the earliestpossible point in energy conversion chains.

Acknowledgements

The author wishes to thank Harry D. Saunders for hisadvice and encouragement during the writing of thispaper.

Appendix

To avoid overburdening an already lengthy argumentwith detail, equations describing costs and energy use inan n-stage system were stated in the body of this paperwithout proof (Eqs. (10) and (19)). The purpose of thisappendix is to provide inductive proofs for these twostatements.From Eqs. (2) and (10) we have

cnþ1=c0;base ¼c0=c0;base

� �Qn

i¼11�aið Þ

Znþ1Qn

i¼1 Zi;base

: ðA:1Þ

Using Eq. (1) to expand Znþ1; we get


� �Qn

i¼11�aið Þ

Znþ1;base cn=cn;base

� �anþ1Qni¼1 Zi;base

: ðA:2Þ

From Eq. (2) we note that

cn;base ¼c0;baseQni¼1 Zi;base

; ðA:3Þ


� �Qn

i¼11�aið Þ

cn=c0;base

Qni¼1 Zi;base

� �anþ1Qnþ1i¼1 Zi;base

: ðA:4Þ

Substituting from Eq. (10), we get


� �Qn

i¼11�aið Þ

c0=c0;base

� �anþ1

Qn

i¼11�aið ÞQnþ1

i¼1 Zi;base

ðA:5Þ

and simplifying


� �Qnþ1

i¼11�aið ÞQnþ1

i¼1 Zi;base

: ðA:6Þ

Since we proved earlier that Eq. (10) holds for n ¼1 and2 , it follows that it is valid for all n:Eq. (19) for the primary energy flux of an n-stage

system can be derived as follows. By definition:

E0 ¼EnQni¼1 Zi

ðA:7Þ

Using Eq. (1) to expand Zi gives

E0 ¼En

Z1;base c0=c0;base

� �a1Z2;base c0=c0;base

� �a2yZn;base c0=c0;base

� �an

ðA:8Þ

and substituting for En; we have

E0 ¼E0;baseQn

i¼1 ci�1=ci�1;base

� �ai: ðA:9Þ

From Eq. (2):

ci�1;base ¼c0;baseQi�1j¼1 Zj;base

: ðA:10Þ

Hence

E0 ¼E0;baseQn

i¼1 ci�1=c0;base

� �Qi�1j¼1 Zj;base

ai: ðA:11Þ

Substituting from Eq. (10):

E0 ¼ E0;base

Yn

i¼1

c0;base=c0� �ai

Qi�1j¼1

1�ajð Þ0B@

1CA ðA:12Þ

ARTICLE IN PRESS

3While this conclusion is not new, its derivation from the structures

of energy consuming systems is original.

R. Lowe / Energy Policy 31 (2003) 1699–1704 1703

The product in the body of this expression can beexpanded as follows:

Yn

i¼1

yð Þ ¼ c0;base=c0� �a1 ðÞ 1�a1ð Þa2 ðÞ 1�a1ð Þ 1�a2ð Þa3

yðÞ 1�a1ð Þy 1�an�1ð Þan : ðA:13Þ

The first two terms on the left can be combined to give

Yn

i¼1

yð Þ ¼ c0;base=c0� �1� 1�a1ð Þ 1�a2ð ÞðÞ 1�a1ð Þ 1�a2ð Þa3


The first pair of terms in (A.14) can be similarlycombined. After j such operations, we have

Yn

i¼1

yð Þ ¼ ðÞ1� 1�a1ð Þ 1�a2ð Þy 1�ajð ÞðÞ1� 1�a1ð Þ 1�a2ð Þy 1�ajþ1ð Þ

y <ð Þ 1�a1ð Þy 1�an�1ð Þan : ðA:15Þ

Combining the first two terms in Eq. (15), we get

Yn

i¼1

yð Þ ¼ ðÞ1� 1�a1ð Þ 1�a2ð Þy 1�ajð Þ 1�ajþ1ð Þ


Without more elaboration, it is obvious that allfurther terms can be combined to give the resultpresented earlier as Eq. (19):

E0 ¼ E0;base c0;base=c0� �1�Qn

i¼1

1�aið Þ: ðA:17Þ

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A theoretical analysis of price elasticity of energy demand in multi-stage energy conversion systems