Working Paper, CP-83-6 International Institute forApplied Systems Analysis, 2361 Laxenburg, Austria,
February 1983, Chapter 17 G.M. Heal (ed) Critical
Writin,gs in the Economics of Exhuastible Resources ,Edward Elgar, 1993,p .339-90

Energy-Capital Substitution :A General Equilibrium
Analysis
Graciela Chichilnisky and Geofrey Heal

We consider an economy which imports energy from a monopolistic price-setter. The
domestic general equilibrium of this economy adjusts in response to the price of energy . We
define the total cross price elasticity of demand between energy and capital as the crossprice elasticity across general equilibria of the economy, as the equilibrium changes in
response to energy price changes . This corresponds to the price elasticity given by a total
demand curve, and incorporates adjustments on both supply and demand sides. It is shownthat whether this total elasticity implies energy-capital complementarity or substitutability
depends upon the parameters of the model and the price of energy : for a given model, there
may be a change from substitutability to complementariry as the price of energy rises. This
framework offers an additional way of reconciling apparently conflicting findings onenergy-capital complementarity and substitutability : an earlier suggestion was made by
Berndt and Wood (1979) . It is a natural extension of the general equilibrium approachinitiated by Hogan (1977) .
Introduction
The question of whether capital and energy are complements or substitutes is one that has
attracted a great deal of attention in the last decade. For a world with only a finite stock of
low-cost energy, its importance is obvious. The long-run growth potential of the economy
depends crucially on the magnitude of the capital-energy substitution elasticity (see Dasgupta
and Heal (1979), Chapter 6) ; consequently a variety of policy measures also hinge uponthis .
Unfortunately, the question, 'Are capital and energy complements or substitutes?' is notan easy one to answer. Indeed, the difficulty is compounded . as we shall show below, by
the fact that there are at least three different ways of posing it. There have been someattempts to see whether basic scientific and engineering principles can throw light on theissue: Berry, Heal and Salamon (1978) investigate the implications of thermodynamic

parameters of the production functions, but also on parameters of the demand side and on
the price of energy.
This leads to a framework within which one may obtain differing econometric results
about the complementarity of substitutability of energy and capital. even if all economies
have identical production technologies . These variations may arise from differences in theprice of energy (Europeans may face higher energy prices than Americans), or from differences in demand conditions (Europeans may be more willing to shift out of energyintensive consumption patterns). Of course, as in the earlier approaches . differences intechnologies could still cause international variations in the econometric findings . However, the fact that virtually identical technologies are available in most OECD countries
makes this explanation less appealing than the alternatives since demand conditions andprice regimes clearly do vary across countries.
Like the works cited above, we consider an economy which is a price-taker in theinternational energy market and investigate how its general equilibrium changes withvariations in the price of energy. If the price of energy changes from po toQthe generalequilibrium will alter and, with it. the amount of capital used . We define the total crossprice elasticity as the ratio of the proportional response of capital used to the proportionalchange in the energy price across the general equilibria.
Our method could in principle be used with any general equilibrium specification of anenergy-using economy, though in the following sections we shall adopt a simple two-sector
three-good general equilibrium model developed by Chichilnisky (1981) for studying theimpact of oil price changes on an oil-using economy. The model is complex enough toillustrate the concept of total elasticity and its dependence upon parameter and priceregimes, yet still simple enough to be tractable. This model is presented in the next section,
where we also use it to analyse the variation of capital demanded with energy prices acrossequilibria. establishing that substitution characteristics vary with the price of energy.
Intuitively, this is different from studying the gross elasticity as defined by Berndt andWood (1979) mainly because it takes account of the impact of changes in the relative pricesof goods on the composition of demand . It could be the case, for example, that capital andenergy are used in fixed proportions in all industries. so that at the micro level there is no
possibility of substituting capital for energy. However. when the price of energy rises, therelative prices of energy-intensive goods increase, and consumer demand shifts from theseto goods using energy and capital in a lower ratio. Consequently capital-intensive industries expand, energy-intensive industries contract, and overall more capital and less energy
are employed . Substitution has occurred through demand shifts as a result of relative pricechanges. This is at least reminiscent of the argument . originally due to Houthaker (1955),
that even though every firm in an economy has a fixed proportions production function, theeconomy as a whole may behave as if it had a Cobb-Douglas production function . It is alsoa generalization of the analysis of Akerlof and Burmeister (1970) of substitution in a
general equilibrium framework. This is a phenomenon which a one-sector model clearlycannot capture, yet which may be important at the aggregate level.

The Model
The model we shall use to illustrate our concept of total elasticity and its dependence on
price and parameter regimes is that developed by Chichilnisky (1981) . It is a two-sector
model with three productive factors: capital, labour and oil. Within each sector, production
functions display fixed input proportions, so that no substitution is possible . However, thetwo sectors differ in their factor intensities, so that changes in relative factor prices lead tochanges in relative goods prices and hence to substitution on the demand side . This in turnleads to changes in the proportions in which the various factors are employed at theaggregate level. We shall characterize the equilibria of the model and then study how
equilibrium factor usage changes in response to changes in factor prices .
The economy has two sectors and produces a consumption good and an industrial good
denoted B and 1 respectively. There are three inputs : labour (L), capital (K) and oil (t9) . Oilis not produced domestically, so the economy takes the price of oil po as given by themonopolistic oil exporter. In order to simplify the analysis, the production functions of thiscountry are assumed to be of the fixed proportion type
B= min(LB /a,,t9 B lb,,KB lc i )(1)where LB, 04 and KB denote inputs of labour, oil and capital into the production of theconsumption good, and a,, b, and c, are the technical factor-output coefficients . Similarly,
the production function for the industrial good is
QSSoU etl. L° ^_C/= min(L!/a,,t91/b~,K'/c2).

(2)

The associao*Ycompetitive price equations are then,_,
ps =a,w+b,pa +c,rp,
Pi = a2w + bZPO + CZrP1

and

(3)
(4)

where w denotes wages, po denotes the price of oil, r the quasi rent of capital, p, the price of
the industrial good, and pB the price of the consumption good. p,r is then the user's cost ofcapital2 which enters as a cost . Although the two production functions show fixed proportions in the use of factors, they are assumed below to have very different oil-output
coefficients . We assume that factor supplies are sensitive to prices . If the price of theconsumption good is the unit of measurement, then labour supply is responsive to real
wages :
r ,1
L=a I
IPBJ
and available capital is a function of the rate of profit) r, i .e.

Next we formulate the demand behaviour, postulating that at equilibrium the value ofconsumption B equals wage income :PBBD = wL .
The market equilibrium conditions are
K=c,Bs +c2 13(i .e . Ks =KD)(ga)

L= at BS +a2 Is (i .e. LS = 2)

(gb)

BD =Bs(gc)

ID +X= 1/(Sd)

PIX =Pot9(Se)

where X denotes exports of I and the superscripts D and S indicate domestic demand andsupply respectively. The last equation is a balance of payments condition.
At equilibrium, the national income identity (national demand equals national income)for this model is
PBBD + Pt ID = wL + rp,K .
To summarize, the model's exogenous variables are the technical coefficients (a,, a, b, .
b:, c,, c~j the parameters or and Q denoting the responses of domestic factor supplies toprices . and the price of oil, po. The model can be formalized as a general equilibrium
system given by eleven equations in twelve endogenous variables. The equations are : (1),(2), (5), (6), (7), (8a-e) and (9). The endogenous variables are: supply of 1, IS: demand for I.
10; exports of 1, X: supply of B, Bs : demand for B, BD ; rate of profit, r, price of B, pB ; priceof 1, p,; wages, w ; labour employed . L; oil used, t~ and capital used, K. The accounting
identity (9) is always satisfied when all markets are in equilibrium .
As there are eleven equations and twelve unknowns, the system can be solved in theusual general equilibrium fashion by considering one good as a numeraire. The prices thatemerge for the other goods are therefore relative prices . We choose B to be the numeraire,so that pB = 1 .
For any given price of oil and set of technological and behavioural parameters, equations(1)-(9) determine a locally unique general equilibrium of the economy. Our next step is to
study how this equilibrium changes as the price of oil changes, and in particular how theamount of capital used varies across equilibria in response to changes in the price of oil .
The following is very much an exercise in computing the general equilibrium of themodel . As the price of oil varies, the equilibria of this model will generally describe a oneparameter family, i .e . a curve in the space of endogenous variables. Along this curve theprice of goods, wages and interest rates, total output of each good, the relative price ofimports and exports (i .e . terms of trade) and the amount of exports are all endogenously
related. Our next goal will be to study their relation across equilibria .
First note that, from the production functions (1) and (2), one can obtain demand
equations for factors L, K and fl at each level of output, assuming that factors are usedefficiently:
L° = Bs a, + ISa2

(10)

K ° = BSc, + I'c,(11)

O D =B Sb,+ISb, .

(12)

Equations (10) and (11) imply that, when factors are used efficiently,
BS =(c2 L-a2 K)/ D(13)

IS =(a2K - c2L) / D(14)

where D is the determinant of the matrix

( at

I\ c,

a2 ~ .
c2

The price equations (3) and (4) can be regarded as a system of two equations in two
variables, w and r, when p o is a constant . From these equations one obtains
w= (po - apn)c2 - (Pr - b7po)c,'(15)_ a, (pt - b2pn) a, (PB - b~ po)

(16)

D

DPI

Substituting K and L from (5) and (6) into (13), and then w and r from (15) and (16), one
obtains the equilibrium values of the supply of basic goods B as a function only of theirprice pt
3 - (c,aw-a,fr) =

(l7)

D

Ctc,

(c2+PON-CIPI)+P~(

M

p

al

+P~-ailwhere M = a l b, - a,6, and N = c,b 2 - b,c2 . Similarly, from (14) one obtains

4-t,40J'- e-Az e

Is = Qal (ai- PoM D

'\,J

PI

PI

)+

0

l (PIci - cz - PoM(18)

Now, from the demand relation (7), B° = wL . and from (9).

at equilibrium . Therefore, from Bs = B° and Is' 1° + X, one obtains the following expressions from (17) when ps = 1 :
(20)

=VI

- blP0)cz - (Pt - b2Po)CI7 z.

The implicit function theorem implies that from (20) one can obtain, at least locally, afunction p, = p,(po.) . Therefore, since po is given, an equilibrium value of p, can beobtained . This, from (17) and (18), yields the supply of B and 1, Bs and 1s, at equilibrium .
From (15) and (16) one obtained wages and profits w and r, and from (5) and (6) theequilibrium use of inputs K and L. This determines 1° (see (19)), so that the volume ofexports X is also known; therefore imports of oil can be computed from (8e) . Thus themodel is 'closed' . i.e . its equilibria are determined (and locally unique) when po is given .
When po changes. the equilibrium values of all endogenous variables will change, inparticular the use of capital. Our next goal is to compute this relationship across equilibria .
We now make an assumption that simplifies the computations : we assume that c, = 0, i.e.
that B requires no capital inputs . This is not strictly necessary to obtain the results: all thatis required is that B be significantly less capitai-intensive than 1 in order to obtain substitution in the aggregate use of factors. One can think of B as a non-traded . relatively labourintensive commodity, such as services .
From (20), and using this assumption, one obtains an explicit expression for p,= pXp o)P~ _a,+p0 M
)iblPo(bipo - 1)+a,(21)

where y = ac;'- / Pa,Consider now the po ible range of variation of po. From the price equation (3), since c,w -10 implies l/b,=0,
o z 0. Now, from (3)---jw= I - bipn
W

(22)

Therefore (4) implies
1'-J~ + POM
a,c2 p,(23)

r= r- (po - ~ po ).
~aia2

(24)

c2

Substituting forp, from (21) we obtain

Therefore r = 0 both when po is zero and when po assumes its maximum value 1/b, . Thechange in the rate of profit as the price of oil varies is_ar = ac2b,

Qa+a2

aPo

(1-2PA).

(25)

Since
_ar
dPo

=0= po=

l

2A

(26)

and r is quadratic in po, it follows that the rate of profit as an increasing function of po for po< 1/26,, and a decreasing function for po > 1126, . Since r takes its maximum value when po= 1126,, the maximum value of r is
r. =

I ac,
- .
4fa,a,(27)

Figure I shows the relationship between r and the price of oil. po. The intiutive explanation of this relationship is straightforward . An increase in the price of oil has two opposing
effects on the demand for capital - a substitution effect and an income effect. The substitution effect occurs because the relative price of the oil-intensive produced good rises,
shifting demand to the capital-intensive good and thus raising the demand for and the levelof use of capital. This bids up the return to capital. The income effect occurs because anincrease in the price of oil reduces aggregate demand and thus the demand for factors,
tending to lower the price of capital. The income effect can be shown to dominate at higher
price levels (see Chichilnisky (1981)).

Rate of prcfit,rPrice of oil, G a

Figure l

The rate of profit and the price of oilWith this information, we are in a position to analyse the response of capital used tochanges in the price of oil. From (6), the supply of capital is just Br and from (8a) capitaldemanded and supplied are equal at equilibrium . Hence if K is equilibrium capital use,
_aK >

apo

< I

<o as Po >26,.

Thus . within this model, capital and energy are total substitutes at low energy prices andtotal complements at high energy prices. Whether capital and energy are total complements
or substitutes is a characteristic of the equilibrium position of the model, which depends on
the price of energy and the parameters of the model . For the particular model presented
here, the total cross elasticity of demand actually has a very simple analytical form . Whendefined as_ dKpodp,Kwhere all values of variables are equilibrium values, derivatives are evaluated acrossequilibria . Using (6), (8a), (24) and (25), it can be shown that_ 1-2ApoI - t)POThe total cross price elasticity is thus positive and is between one and zero for 0 <- po < 1/

26,, is zero for pa = 1/2b,, and varies from zero to minus infinity as po varies between 1/2b,
and 1/b, . This is summarized in Figure 2.
Total cross elasticity

0.

Figure 2

Price of oii,pa

Behaviour of the elasticity 4 with the price of oil

Conclusions
We have introduced the concept of the total cross price elasticity of demand between
energy and capital: it is an elasticity that takes into account the full range of adjustmentsthat occur in a multisector economy after the price of energy changes. It is a comparative
static measure, recording the change in the general equilibrium configuration of factor use
consequent upon a factor price change . It incorporates the effects of potentially importantsources of substitutability, such as those on the demand side that are not adequately
captured by analysis of a production function, and so measures the total response from all
sources to a price change . It is obviously a long-run concept, applicable on a time-scale
over which all adjustments may be assumed to have been completed. It is clear that there
are a number of contexts in which this total elasticity will be more relevant than any partial
concept - for example, when predicting the effect on total energy consumption of a pricechange .
The concept of total cross rice elasticity has been illustrated by reference to a model
which is simple enough to be tractable, yet highlights the main feature. It shows, forexample. that capital and energy may be substituted at the aggregate level, even though inevery production process they are used in fixed proportions . This illustrates clearly theimportance of considering the full range of responses to a price change . For the particularmodel considered, capital and energy are substitutes at low energy prices and complements
at high energy prices . This very simple relationship is heavily dependent on our simplifying
assumption that c,, the capital-output coefficient in the consumption goods sector, is zero .
Without this simplification, the relationship between po and r, and hence K, would be of thefourth order. This would lead to three regime switches between complementarity andsubstitutability, as opposed to the one transition of the present model. Such a possibility has
to be borne in mind when comparing the predictions of the model with the stylized facts ofcapital-energy compiementarity in the US and substitutability in Canada and Europe. It
also has to be borne in mind that the price of oil. po, is in fact the price of oil relative to theprice of consumption goods: it is not immediately clear whether this is higher or lower inEurope than in the US . Finally, it is of course, the case that existing studies have been
designed to measure a different elasticity : a rather different model specification would be
required to estimate the total cost price elasticity. This will be the subject of a forthcoming
paper.
References
Akerlof. G.A . and E. Burmeister (1970), 'Substitution in a General Equilibrium Framework' . Journal of Economic Theory, 2 (4), 411-22 .
Berndt . E.R . and D.O . Wood (1975), 'Technology, Prices and the Derived Demand for Energy',
Review of Economics and Statistics . 56. 259-68
Berndt. E.R . and D.O. Wood (1979), 'Engineering and Econometric Interpretations of EnergyCapital Complementarity', American Economic Review, 69, 342-54 .
Berry, R.S ., P Salamon and G.M. Heal (1978), 'On a Relation between Economic and Thermodynamic Optima', Resources and Energy, 1. 1125-37.
Chichilnisky, G. (1981), 'Oil Prices, Industrial Prices and Output : A General Equilibrium Macro
Analysis'. Essex Economic Paper. 1980 : presented at the AEA meetings, December 1981 .
Dasgupta, PS . and G.M. Heal (1979), Economic Theory and Ezhaustible Resources, Cambridge
University Press.
Denny, M., M.A . Fuss and L. Waverman (1980), 'The Substitution Possibilities for Energy : Evidence from US and Canadian Manufacturing Industries' . Working Paper No. 8013, Institute for
Policy Analysis . Toronto.
Griffin, J .M . and PR . Gregory (1976) . 'An Intercountry Translog Model of Energy Substitution
Responses' . American Economic Review, 66, 845-57 .
Hahn . F.H . (1977), 'Exercises in Conjectural Equilibria', Scandinavian Journal of Economics, 79 .
(2), 210-26 .
Hogan. W.W. (1977), 'Capital-Energy Complementarity in Aggregate Economic Analysis' . Energy
Modelling Forum Report EMF 1.10. Stanford University .
Hogan. W.W. and A.S . Marine (1979), 'Energy-Economy Interactions : The Fable of the Elephantand the Rabbit?' in R.S . Pindyck (ed.), Advances in the Economics of Energy and Resources. JAI
Press. Vol. 1, 7-26.
Houthaker, H. (1955) . 'The Pareto Distribution and the Cobb-Douglas Production Function in
Activity Analysis', Review of Economic Studies. 23, 27-31 .
Pearce. I.F. (1953), 'Total Demand Curves : A General Equilibrium Approach' . Review of Economic
Studies, XX (3), No . 53, 216-27.
Solow, J. (1979), 'A General Equilibrium Approach to Capital-Energy Complementarity', Economics Letters . 2, 91-4 .

Notes1.
2.
3.

Columbia University and the University of Essex. Valuable comments from Helen Gasking, AI
Klevorick and Kerry Smith are gratefully acknowledged. Chichilnisky's research was supported
by grants from the Rockefeller Foundation and UNITAR .
We have in mind an interpretation of this model as a temporary general equilibrium model .
Industrial goods produced in this period may be used inter alia to augment the capital stock in
the next period.
One could think of a situation where the capital stock consists of a number of machines ofdifferent ages and productivities . As the rate of profit rises, an increasing number of these will
be brought into operation.