INTERFUEL SUBSTITUTION AND THE INDUSTRIAL DEMAND

FOR ENERGY: AN INTERNATIONAL COPARISON

by

Robert S. PindyckMassachusetts Institute of Technology

August 1977Working Paper

#MIT EL 77-026WP

This work was supported by the RANN Division of the National Science Foundationunder Grant #GSF SIA 75-00739, and is part of a larger project to developanalytical models of the world oil market. The author is indepted toJacqueline Carson, Vinod Dar, Ross Heide and Kevin Lloyd for their excellentresearch assi-sance in many aspects nof this work, to the Comr;ter ResenrchCenter of The 't'ic.mna i Breau of Eccnr c Rsearch for assistance in the comru-taEional work, and to Ernst 1lerndlt, '."Klvyn Ftess, James Griffen, and LeonardWavezman for co-=aents and suggestions.

INTERFUEL SUBSTITIrON AI.D T'E ID1;ST.AIL DEMAND FOR ENERGY:

AN INTEl'NATIONAL COPARISON

I. Introduction

Until recently most studies of production concentrated on the substitutability

of capital and labor, assuming that in the underlying production function these

factors were separable from energy and raw material inputs. Dramatic increases

in the price of energy, together with the appearance of studies such as that of

Berndt and Wood [9] which indicated that energy and capital may in fact be complement;

rather than substitutes, resulted in an increased interest in the substitutability

of energy and other factors of production. In addition, rapid changes in the prices

of individual fuels raised the issue of how rapidly and to what extent the in-

dustrial sector could substitute among different fuels. Finally, there has been

an increased concern with the growth in energy demand brought about by growth

in industrial activity.

In this paper we report on results of an econometric study of the industrial

demand for energy in a number of industrialized countries. Our objectives are to

determine the extent to which capital, labor, and energy can be substituted for

one another, and vary across time. In addition, we will attempt to measure the

extent to which alternative fuels can be substituted for one another both in the

short-run and the long-run, and estimate the total elasticities of demand for

individual fuels.

We model industrial energy demand using a two-stage approach. In the first

stage total energy demand in the industrial sector is modelled as a derived factor

demand base on a translog cost function. Factor inputs include capital, labor,

and energy; a lack of data makes it impossible to include materials as an explicit

Griffen and Gregory [34] have alreadv fitted translog cost functions to pooledinternational dota in c;d'r to ->assure easticits of sustitution for capial,labor, and energy, but that work oes not explain inter-country diffcrences inelasticities since only four observations were used for each country, and does notdeal with interfuel substitution.

input, so we must assume separability of materials from the other factors in the

underlying structure of production. In the second stage expenditures on energy

are broken down into expenditures on oil, natural gas, coal and electricity.

Here translog cost functions are again used, but we have also experimented with the

use of a multinominal logit model to explain fuel shares. The use of this two-

stage approach requires certain additional assumptions about the underlying struc-

ture of production. In particular, we must assume that the production function

is homothetically separable in the capital, labor, and energy aggregates, i.e.

first, that expenditure shares for fuels are independent of the expenditure shares

for capital and labor, and second, that expenditure shares for fuels are independent

of total energy expenditures.

The use of the translog cost function has the advantage that it permits us to

obtain relatively unrestricted estimates of elasticities of substitution and demand

elasticities. We need not a piori assume that the cost function is homothetic

(at least in modelling the demand for capital, labor, and energy inputs), and we

need not assume that the elasticities of substitution between different inputs are

all the same. Of course there are other "generalized" cost and production functions

that could be used that also impose no a priori restrictions on elasticities of sub-

stitution, and we might find that these provide tighter estimates than does the

translog function. For the time being, however, we use only the translog function.

As one might expect,' in work like this one is continually bound by data limita-

tions. For many countries there is no good data available for some or allcf the

variables of interest to us. For other countries data exists, but obtaining that

2The necessity for these assumptions is shown by Berndt and Christensen [8] and

Denny and Fuss [26]. Note that this second ass~uption of homotheticity of fuel

shares will not be violated n the use of a logit model if total energy expendi-

tures is not an explanatory variable in that model.

-For example,. the . ont'f c n and te generalized Cobb-. outzlas

function, both of wich were introued by Di:;ert [27,28]. Lau and T-.lura [44]

used the generalized Leontief production function in estimating input substitu-

tion in petrochemical refining. ZLaSnus [46] used the generalized Cobb-Douglas

cost function in estimating substitution between capital, labor, and energy inputsin Dutch manufacturing.

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data can be an extremely time consuming and laborious task. These data limitations

were one of the factors that helped define and delimit the modelling approaches

used here. In particular, it necessitated restricting this analysis of industrial

4demand to a set of only ten countries. Even for these countries, however, the

quality of the data varies, and compromises had .to occasionally be made. The

data used in this study is described briefly in this report; a much more detailed

description is provided in a separate report entitled "A User's Guide to the M.I.T.

World Energy Demand Data Base."5

In the next section we outline the specifications of alternative models of

industrial energy demand, and discuss the characteristics of each specification..

Section 3 discusses some methodological issues in the estimation of industrial

energy demand models using pooled data, and describes our estimation method.

Section 4 describes some of the characteristics and limitations of our data, and

Section 5 includes the statistical results.

2. Alternative Secifications for Models of Industrial Energy Demand… -=… ~-

- - --_…=…==-==-_-=…===---====---- 7---

As explained before, our industrial demand models are based on a two-stage

approach where energy demand is explained as a factor input share, and energy ex-

penditures are then broken down into expenditures on fuels. We begin here by sum-

marizing our assumptions about the structure of production. We then review the

properties of the static translog cost function, and discuss its application to

the industrial demand model. Next we describe some alternative dynamic specifi-

cations of the translog cost function; these specifications permit non-constant

returns to scale in the short-run, with an adjustment to constant returns in the

A much less detailed model of the demand for petroleum products is being con-structed for a number of countries for whnch only partial data is available;the results of this work will be described in a forthccring paper.

5Working Paper No. MIT EL 76-011WP, M.I.T. World Oil Project, May 1976. A re-vised version of this report will appear shortly.

long-run. Finally we discuss the use of the multinominal logit model as an al-

ternative means of describing fuel shares.

2.1' The Structure of Production.

Our approach involves certain assumptions about the structure of production.

First, we assume that capital, labor, and energy inputs are as a group weakly

separable from the fourth input, materials. 6 This assumption is made necessary

by the fact that we have no data from which to construct price indices of materials

inputs, and therefore we can only estimate unrestricted elasticities of substitu-

tion between capital, labor, and energy. Second, we assume that the production

function is weakly separable in the major categories of capital, labor, and energy.

This implies that the marginal rates of substitution between individual fuels is

independent of the quantities of capital and labor. The assumption permits us to

use aggregate price indices for capital, labor, and energy inputs; in particular

to construct an energy price index that aggregates the price of the four fuels,

and to construct a price index of capital services.that aggregates different types

of capital. Finally, we assume that the capital, labor, and energy aggregates

are homothetic in their components - in particular, that the energy aggregate is

homothetic in its oil, gas, coal, and electricity inputs.8 This last assumption

provides a necessary and sufficient condition for an underlying two-stage optimiza-

tion process, i.e. optimize the mix of fuels that make up the energy input, and

6Materials includes intermediate inputs as well as non-energy raw materials. Weak

separability here means that the marginal rate of substitution between any two of

the first three inputs is independent of the quantity of materials used as an input.

This is a necessary and sufficient condition for the production function to be of

the form Q = F[f(K,L,E);M]. For a proof and further discussion, see Berndt and

Christensen [8].

7Halvorsen and Ford [37] recently used translog cost functions to.test for separa-

bility of the energy agre-ate for each of eight individual two-digit industries

in the U.S. They found sparability to hold for four of the eight councries.

8The second and third assumptions together are referred to as homothetic separability

9then optimally choose quantities of capital, labor, and energy. Equivalently

we can express these three assumptions by writing the production function as:

Q = [(f(K,L,e(F1,F2,F3 F4));M]. (1)

where e is a homothetic function of the four fuels.

If the factor prices and output level are exogenously determined, the pro-

duction structure described by (1) can alternatively be described by a cost

function that is also weakly separable, i.e. a function of the form:l0

C =G[g(PK, PL,PE(PF'PF2'P3 ' P F4QQ );P'(2)C [g( K' L' E Fl' F2' F3' F4. (2)

Here PE is an aggregate price index of energy, i.e. a function that aggregates

the fuel prices PFi' This "aggregator function" is homothetic, and thus does not

include the total quantity of energy as one of its arguments.

2.2 Use of the Translog Cost Function

Our approach here is similar to that used recently by Fuss and Waverman [31]

in estimating the demand for energy in Canadian manufacturing. We first represent

the price of energy (which is the unit cost of energy to a producer choosing fuel

inputs) by a homothetic translog cost function with constant returns to scale.

Estimation of the share equations implied by this cost function gives us the own

and cross partial price elasticities for the four fuels. In addition, the cost

function itself provides an instrumental variable for the price of energy. The

second step is to represent the cost of industrial output by a non-homothetic

translog cost function. Estimation of the share equations implied by this cost

function gives us the elasticities of substitution and the own and cross price elas-

ticities for capital, labor, and energy.

See Denny and Fuss [26].

10This was shown by Shephard [54].

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It is important to point out that we could have chosen to use translog pro-

duction functions rather than cost functions in estimating elasticities. Since the

translog cost and production functions are not self-dual, different elasticity es-

timates would result, and as Burgess [11] has recently shown, the difference could

be significant. However, we choose to use the cost function since it is more

appropriate to take prices as exogenous than quantities.

11We begin by reviewing the properties of the translog cost function. The

translog cost function is a second-order approximation to an arbitrary cost func-

tion, and has the form

logC - a + aQloSQ + EacloEPi + yQQ(1ogQ)2 + gEYi+logP logPj + EyQilgQlogPi (3)

where C is total cost, Q is output, and Pi are the factor prices. From Shephard's

Lemma [54], the derived demand functions are found by differentiating the cost

function with respect to the prices, i.e. Xi = C/aP i. Thus the share equations

are given by S alogC/alogP i a (PiXi)/C, or

Si i +Yqil to Q + Yl o imi, ..., n. (4)

Since the shares must add to 1, only n-l of the share equations need be estimated.

Note, however, that the parameters a0 , Q, and yQQ are not-identified unless the

cost function itself is estimated.l

The translog production function and cost function were introduced by Christensen,

Jorgenson, and Lau [21]. Applications of the translog cost function can be found

in the work of Berndt and Christensen [7], Berndt and Wood [9], Christensen and

Greene [34], Fuss and Waverman [31], Fuss [30], Griffen and Gregory [34], Halvorsen

[36], and Hudson and Jorgenson [38]. The translog production function has been

used by Humphrey and oroney [39] and oroney and Toevs (47,48].

1 2Corbo and Meller [23] recently estimated the ranslog production function directly

(instead of the derived share of equations) using capital and labor input data indi-

vidual firrms. This llc;:.d th;cn to cst '. hethar th ! lin .....rroduction functionis really translo,, and o test for cc:petitive bhavilor.

The cost function must be homogenous of degree 1 in prices, and mu§t satisfy

the conditions corresponding to a well-behaved production function. This implies

the following parameter restrictions that must be imposed:1 3

Ca -1 (5)i iy ~j 0 (6)

iYQi

yiJ Yi i j (7)

.EiJY i Ej 0. (8)

Note that the cost function as specified so far is non-homothetic, and may

have non-constant returns to-scale. The cost function would be homothetic if it

could be written as a separable function of output and factor prices. Thus the-

following parameter restrictions can be added to impose homotheticity:

YQi =0. (9)

The cost function is also homogenous if the elasticity of cost with respect to

output (logC/alogQ) is constant. This implies the additional restriction:

YQQ . (10)

Finally, we could impose the restriction that the elasticities of substitution be-

tween all factors are equal to 1 (so that the cost function corresponds to a

Cobb-Douglas production function). This implies the additional parameter restric-

tions: Y . (11)

1 3See Christensen, Jorgenson, and Lau [21].

14 Christensen and Greene [19] define the following index of scale economies:SCE - 1 - aloC/alogQ 1 - (lQ + yqrlOQ + ZYQioPi). Note that if

SCE is positiv> (n2c-ative), thoro is incrcasing (dacra3sin,) returns to scnle.

This is a useful . - an, d has a ntural it.rprctztion in prceltagea erms.However, it can only be computed if a and yg are known (which means estimating

the cost function) or are assumed to Qbe 1 and 0 respectively... . . _ . .. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,

Rather than impose these restrictions a priori, we can test them using a

simple chi-square test. The appropriate test statistic is

-2.logA = N(loglrI - loglIUl) (12)

where IRl and %1iuI are the determinants of the estimated error covariance

matrices for the restricted and unrestricted models respectively, and N is the-

number of observations. This statistic is distributed as chi-square with degrees

of freedom equal to the number of parameter restrictions being tested.

15Uzawa [56] showed that the Allen partial elasticities of substitution can

be computed from ij CCij/CiCj, so that for the translog cost function these

are given by

-ij =(Yij + SiSj)/SiSj, i j(13)

ii= [Yii + Si(Si-1)]/Si

It is easy to show that the own and cross price elasticities of demand are given

by ynii " algXal ogPi = (iisi

(14)

riJ - BlogXi/alogP Sj S

Note that these are partial price elasticities; when applied to fuels they account

only for substitution between fuels, under the constraint that the total quantity

of energy consumed remains constant. The total own price elasticity for each fuel

rii = d logXi/d logPi accounts for the effect of a change in the price of a fuel

on total energy consumption, and is given by

i ax1 + i +aE aE (15)iiXi PiE const E aPE aP

where E'is the total quantity of energy consumed, and PE is the price index for

energy. However, since the price of energy is given by the homothetic translog

cost function with constant returns to scale:

logPE a + logPi + CilogP + iyl1 g P '(16)

which implies the fuel share equations Si Yi + Zijl°gPj' we have

See Allen [3].

16Note that expenditures on energy will not remain constant.

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PE PE

-i - S (17)

aE E.

· , aPE-a E E E (18)

where'nEE is the own price elasticity of energy, and, since the energy cost

function is homothetic,

ax ax xiX i ME Xi X i

a E ME.E E (19)

where M = PE E is total expenditures on energy. Then, by substituting (17),

(18), and (19) into (15), we have

tlij T ii + EESi (20)

Similarly, we can compute the total cross price elasticity nrj from

P ax ax aE aP

'i-|const. + E aP ap li + EE (21)Xi -J E cons. E

17aud the total output elasticity from

d logXi 2_ ....T)* = oX -. (22)iQ d logQ x a? aQ

Since the energy cost function is homothetic, this reduces to

niQ - nEQ (23)

where EQ is the elasticity of energy with respect to output changes. This

17 thi s

This assumes that the value of output is equal to the value (cost).of inputs.

This would be the case under perfect competition, or under oligopoly pricingba~.d on a : ' ;:rl,.: ov---I'"33r c:,;. Or-~7.e this elasticity would

be better referre d :o a a t~'~i cz.t zi;..2. thz c.t cr.e inthe dmand for fuel i corrcspondin to a 1 percent change in the total cost

of production.

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elasticity can in turn be computed as follows:

d log E d lo E. a log CEQ d log Q d log C a log Q

logE _L__o_ +logQ aiogc aLogQ

gSE lo (24)logQ alogQ

Obtaining these derivatives from equations (3) and (4), we have

~QE+ I L E (25)nEQ = SE + aQ + QQlogQ + YQilogP i (25).E ~i= K,

Since the cost function is not estimated directly, we assume that aQ and yQQ

are 1 and 0 respectively.

We would like to calculate standard errors for our estimates of elasti-

cities, but since the elasticities are nonlinear functions of the estimated

parameters (since the shares are themselves functions of the parameters), there

is no straightforward way to do this without reverting to Monte Carlo simulation.

However, we can obtain approximate estimates of the standard errors. To do this

we follow Christensen and Greene [19] and calculate standard errors under the

assumption that the shares Si are constant and equal to the means (over the esti-

mation time bounds) of their estimated values. Under this assumption we have,

asympototically,

Var(8ij)- Var(Yij )/SiS

Var (ii) - Var(Yii/S (26)

Var(Tij) -Vcr('tj)/S2

Var (l) = ar (i) / S

Finally, it is useful to calculate the elasticity of the average cost

of production with respect to the price of energy, i.e. CE = log(AC)/3logPE

and the elasticities of the average cost of production with respect to the prices

of each fuel, i.e. Ci = alog(AC)/alogPi. This will enable us to calculate the

effect of a 1 percent change in the price of energy, or a 1 percent change in the

price of a single fuel, on the cost of industrial output. We follow Fuss(30] in

calculating point elasticities for nCE and nCi. From equation (3) we have

TCE + YEEl°gPE + YEKlogPK + YELlogPL + YQElogQ (Z)

We obtain qCi from

alog(AC) al EnCi alogPE !logP i CESi (28)

Let us now review the steps involved in estimating a translog model of energy

demand. First the fuel share equations

$ i ji + jijlgP J (29)

are estimated, and the estimated parameters are used to calculate partial price

elasticities. These equations are estimated subject to the parameter restrictions

Eac 1, Y , and Zy =Zy =0 We also test the additional restrictions

YiJ O. Next the estimated values of the ci and Yij are used in equation (16) to

obtain an aggregate price index for energy. To do this the parameter aO in

equation (16) is determined so that the price of energy is equal to 1 in the U.S.

in 1970. An energy price index is then calculated for each country.

Next we estimate the- factor share equations (4), with i and j equal to

capital, labor, and energy. In estimating these equations, we use our estimated

aggregate price index for energy as an instrumental variable; We estimate these

equations in stages, imposing additional parameter restrictions at each stage,

and testing each set of restrictions. In the first stage we impose only the

restrictions implied by neoclassical production theory, i.e. Ea = 1, YQi 0,

Yij - Yji' and 0ij s Yij . Next we add the homotheticity restrictions YQi= O.

Finally we test the restrictions that ij O, i.e. that the elasticities of

substitution between all three factors are equal to 1.ig

t

2.3 Dynamic Versions of the Translog Cost Function

A problem with the translog cost functions described above is that they

do not describe differences between short-run and long-run elasticities, or

how the adjustment to the long-run takes place. It is reasonable, for example,

to expect that in the long-run the aggregate production function exhibits

constant returns to scale, but that it exhibits non-constant returns in the

short-run. Our objective here is to specify a dynamic translog cost function

that permits adjustment to constant returns over time.

Constant returns requires that aQ P 1, YQQ 0, and YQi 0 ° in equation (3).

One way, then, to build in an adjustment to constant returns in the long-run is

to make these parameters functions of changes in output or prices. For example,

the parameters aQ and yQQ could be specified as:

K

a = exp[O l(AQt-k) (30)

Kand yQQ 02klX (AQtk) 03 (31)

Here the parameters 81, 2 and 03 are estimated. If 83 is not equal to zero,

then non-constant returns can exist even in the long-run. Thus if long-run

constant returns is taken as given a priori, the estimate of 83 (i.e. whether

it is significantly different from 0) provides a test of the correctness of the

kspecification of the lag distribution. The lag distribution parameters Xk and

Xk could be estimated if the data permitted or they could be specified a priori

We will also test restrictiocn: pertaiinc. to tea ch:ractcristics of inter-cot..try

diffrecnces in the cost functicn. This !..:o1vxs tha ue o du'.y vari'lcs th:'permit some of the parameters of the translog cost function to vary across countries.

This is discussed later.

(perhaps declining linearly). In either case K might be 3to 5 years. Of course

estimation of the parameters in (30) and (31) requires that the cost function (3)

be estimated simultaneously with the share equations (4).

The parameters of YQi can also be made to adjust to zero in long-run

equilibrium by making them functions of a distributed lag in changes in prices:

K

YQi = O kQi (AlogP )-k2 (32)

Thus if the data do not permit the simultaneous estimation of the cost function

with the share equations, aQ and' yQQ could be assumed equal to 1 and 0 respectively,

and adjustments could occur through the YQi. In this case the production structure

would be homothetic in the long-run.

An alternative approach is to introduce the dynamic adjustment directly into

the share equations. This can be done by assuming that the shares adjust to a set

of desired shares as follows:1 9

,t ' s + ES (s -s it Sit-l + i(Jtjt-1 (33)

where S is given by equation (4). Adding up requires that the sum of allj,t

changes in shares be zero:

s(S - 0 (34)i t Si, t -

so that

Z6 i(S* t Sj = *t (S* 't E - S X6t-s) = o.ij J ' j

Since the St and Sj sum to one, this equation implies the necessary condi-

tion that all of the columns of the matrix (6ij) sum to the same arbitrary

constant, i.e.' c 2 ' (36)

This approach was suggested by L. averman and M. Fuss.

where is a vector of l's (ones), 6 is a matrix (6ij), and is an arbitrary con-

20stant. Note that if the number of shares is greater than two, there are alterna-

tive constraints on the ij that can be imposed to satisfy (36).

2.4 Multinomial Logit Models for Fuel Choice

Multinomial logit models have already been used to study the breakdown

of energy consumption into demands for fuels in the residential sectors of

the United States and other countries2 1 and in the industrial sector of the

22United States. Although the logit model is not based on assumptions of cost

minimization, it has properties that make it appealing for this work. The

model is consistent in terms of shares adding to one, and shares respond to

price changes in a way that is intuitively appealing; as the share of, say,

natural gas becomes small, it requires increasingly large price changes to

make it still smaller. Finally the logit model is easy to estimate and permits

us to easily introduce alternative dynamic specifications.

We follow our recent work in applying the logit model to the estimation

of residential fuel demands [53 . The logit model for four fuels can be written

as

Qi efi(xS)

Qe 4 f(37)J-1

where Qi is the quantity (in tcals) of fuel i, QT ZQi, and the fi are functions

of a vector of attributes x and vector of parameters . Given this model, the

relative shares of any two fuels can be represented as

log(Q1/Qj) = og(Si/Sj) = f(x) - f(x4). (38)

Only three equations are estimated, since the parameters of the fourth equation

are determined from the adding up constraint.

For a disc ussio n z :d .- Ir: -;i " c.s 1foe .. ore ui.era lag structures, seeWall [ ] and iernLit nd Savln [ j.

21See Baughman and Joskow t ], Fuss and Waverman [31], and Pindyck [53].22See Joskow and Baug.man [421.

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In estimating fuel shares we include as attributes the relative price of

each fuel. The relative oil price, for example, is the ratio of the real price

of oil to the real price of energy, the latter being measured by the energy

price index described earlier. We do not include total energy expenditures of

industrial output as attributes since we a priori impose homotheticity on the

fuel share model. Other attributes, however, can include lagged quantity or

share variables that allow shares to adjust dynamically to changes in price.

Functional forms for the fi are somewhat arbitrary, but in the simplest model

they might be linear functions of the relative fuel prices Pi = Pi/PE, where

PE is the aggregate price of energy, and output Q:

fi(x ) = ai + biPi + ciQ .(39)

This yields the three estimating equations

log(Si/S 4) = (a i-a4) + bi b4 + (i-c 4)Q, i = 1,2,3. (40)

Note that these three equations must be estimated simultaneously, with b4

constrained to be the same in each equation.

The simplest means by which the preference functions can be made dynamic

(e.g. to account for stock adjustments) is to include the lagged share:

fi(xB) = ai + biP i + iSi,t-l' (41)

The three estimating equations are then

og(Si/S4) = .(ai-a4) + ib4P 4+ CiSi t-l 4 4t-1 2

(42)

Note that two lagged shares appear in each equation. The three equations must

again be estimated simultaneously, with both b4 and c4 constrained to be the

same in each equation.

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3. Some Methodological Issues in Model Estimation.

There are a number of issues that must be resolved before the model pre-

sented in the last section can be estimated. Some of these issues have already

been dealt with in some detail in an earlier paper by this author [53]. For

example, purchasing power parity indices, rather than official exchange rates,

are used to convert data measured in local currency units into U.S. dollars.2 3

Secondly, we followed the approach used in our residential energy demand study

in measuring energy consumption in "gross" rather than "net" terms.2 4

There are two other issues, however, that must be treated. The first has

to do with the identification of inter-country differences in the structure of

production. The second important issue is the choice of estimation method. We

deal with these in turn.

3.1 Identifying Inter-country Differences in Production Structure

One of our objectives in estimating energy demand models is to determine the

extent to which elasticities vary across countries, and the possible reasons for

such variations. To identify regional variations in elasticities, we must specify

alternative ways of allowing for regional parameter variation when our models are

estimated with pooled data.

At the one extreme, we could assume that the parameters of our models are the

same for all countries (the resulting elasticities could still vary across countries

since relative prices and total output levels are different in different countries).

At the other extreme we could estimate our models for each country separately;

23As before we use a Fisher "ideal" index (a geometric mean of a Laspeyre andPaasche index numbers) as a single index of relative purchasing power. Ourpurchasing power parities are binary index numbers with the U.S. as base country.

2 4Thpt i, we do not adjust fua! qu.ntities or prices by thermal efficiencies ofutilization. Thrn, a.~re t-o r-o-n for this; first, thr e no gocod cinmatesof thezal afficigr.zcs a;vl iae., and second, there ara ochier economic' effi-ciency measures that could be equally important in affecting consumer demand. Fora further discussion of this issue, see Pindyck [ ], pages 25 to 28.

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this would be infeasible, however, due to insufficient data. Instead we use a

compromise approach that we followed in our earlier work in residential energy

demand [53]. The countries are pooled, but regional dummy variables are intro-

duced that allow a subset of a model's parameters to vary across countries. In

the translog models this could be done by assuming that the coefficients a i of

the first-order terms in the Taylor series approximation can vary across countries,

while the coefficients ?Qi and Yij are the same for each country. This would mean

estimating the following share equations:

Si a D. + QilgQ + Yi :(43)k ikk Qi k lo P

where Dk are country dummy variables (Dk = 1 for country k and 0 otherwise).

Note that the usual restrictions on the yij and the YQi apply, but Eaik = 1

for each country k. Note that an advantage of this method is that it partially deals

with heteroscedasticity of the error terms within each equation; it is essentially

the covariance method for estimation with pooled data.

Alternatively, we could assume that the coefficients Yij or YQi of the second-

order terms can vary across countries, while the ai's are the same for each country.

For example, we could estimate share equations of the form

Si i + YilogQ + .DlogP j. 44)kj I ksk

Note that the restrictions on the yijk's are now that Yijk Yjik for each country

k, and that Yik Yi = 0 for every country k.i ijk - ijk

There is no a riori reason for preferring either specification (other than

the econometric convenience of the first specification). However, for either

specification, the null hypothesis that the corresponding coefficients are the

same across countries can be tested using the straightforward chi-square test.

3.2 Estimation Methods.

The choice of estimation methods involves a trade-off between the richness

of the stochastic specification (and hopefully a resulting gain in efficiency) and

computational expense. This trade-off is particularly severe given that all of

our models involve systems with equations. Ideally one would like to estimate a

stochastic specification for which the error terms are heteroscedastic and auto-

correlated both across time and across countries within each equation, and are

correlated across equations in the system. Estimating such a specification (which

amounts to full generalized least squares) however, would be unreasonably costly

given the computer software available to us. We must therefore settle for a more

restrictive stochastic specification that would still capture the more important

characteristics of the error terms.

When estimating our models we ignore error term autocorrelation within equa-

tions, but account for error correlations across equations. In particular, we

use iterative Zellner estimation which (under the assumption of no heteroscedasti-

city or autocorrelation within equations) is equivalent to full-information maximum-

likelihood estimation. However, we limit the number of iterations on the error

covariance matrix to five; this reduces computational expense while still capturing

at least 90% of the added efficiency that results from accounting for cross-equation

error correlations.

We will attempt to account for within-equation heteroscedasticity (at

least as far as we can given the ccnstraints on our computer budget). This

-19-

is done using the following procedure. First each equation in the system is

estimated using ordinary least squares. The resulting regression residuals,

which we can label Ukt, are then used to obtain consistent estimates of the

regional (country) error variances oF 21c , .

Ok T-m- Ukt) (45)

where T is the number of annual observations for country k and m is the number

of independent variables in the equation. Different estimates of these error

variances will of course be obtained for each equation in the system. We then

transform the data by dividing each observation by the appropriate estimatedA

error term standard deviation aok , and then re-estimate the entire system of

equations using iterative Zellner estimation. At this point, new estimates of

the regional error variances can be computed, again using equation 43. Iterative

.,Zellner estimation can then be repeated. Ideally this process should be iterated

until convergence occurs; because of computational expense, however, we limit

.A:the process to one iteration.

Our estimation work has been carried out at the computer research center of

the National Bureau of Economic Research, using the GRELIN experimental non-

linear estimation package on the TROLL Econometric software system. Some work

was also done using the new version of TSP at M.I.T.'s Information Processing

Center.

4. Characteristics.of the Data

Estimation of the models described in Section 2 requires data for capital,

labor, and energy price indices and expenditure shares of manufacturing output,

and for the prices and quantities of petrolemi, natural gas, coal and electricity

used in the industrial sector. In so-me cases data was available from standard

sources such as the UN Statistical Office or OECD publications, in other cases

-20-

we relied on the data collection efforts of the International Studies Division

of the Federal Energy Administration. Finally, in some cases it was necessary

to turn to the national statistical yearbook of individual countries.

Ten countries are included in our sample: Canada, France, Italy, Japan,

The Netherlands, Norway, Sweden, U.K., U.S.A. and West Germany. The data col-

lected for these countries are described briefly below.2 5

Expenditures on Labor: Expenditures on labor include wages and salaries

plus supplements paid to the manufacturing sector. For some countries,

Canada, Italy, The Netherlands, Norway and West Germany, this was avail-

able from the United Nations' Growth of World Industry. For other countries,

where the UN publication lacked data on supplements for all years, it was

necessary to extend supplements by using the national percentages of supple-

ments indicated by GWI, UN National Accounts or the International Labor Or-

ganization's Statistical Yearbook. For Sweden and the United Kingdom it

was necessary to determine what percentage of total national compensation

went to manufacturing, using data from UN National Accounts and ILO Statis-

tics. Finally, for the U.S., Japan and France, national statistical year-

books were used. Data is in local currency units and is converted to U.S.

dollars using the purchasing power parity numbers for GDP.

Price of Labor: The price of labor was determined implicitly by dividing

labor expenditures by total manhours of employees. Manhours of employees

was calculated for 1967 by multiplying manhours of operatives by the ratio

2 5The data used here are part of a larger international energy data base assembledfor use in this and several related studies. For a more detailed description ofthat data base, see "A User's Guide to the M.I.T. World Energy Demand Data Base'.'(M.I.T. Energy Laboratory Workin, Paper). Other researchers wishing to replicateor extend this stu-:dy or .ror; st:-es of -hir can cn ccess te data dircct lythrough the TROLL computer svste-i of the 'N"B.

-21-

of numbers of employees to number of operatives, for Canada, Italy, Japan,

Norway, Sweden, U.S.A. and W. Germany. Data is from the UN Growth of World

Industry. Where GWI did not have information, manhours were calculated

from UN data on number of employees and ILO data on average working hours.

Then, for every country except Norway, a wage index (1967 = 100) from the

U.S. Bureau of Labor Statistics, which includes wages and supplements,

was used to convert our price/hour for 1967 to a time series, 1955 to 1974.

The time series for Norway was available directly from Growth of World In-

dustry. (Note that the resulting index is not quality adjusted.)

Price of Capital Services: We compute a capital service price index sepa-

rately for non-residential structures (PNR) and producers durables (PD),

and aggregate these two series into a final price of capital services

using a Divisia index, where the investment shares of non-residential struc-

tures and durables serve as the Divisia weights. The computation of the

price of capital services of each component is based on Christensen and

Jorgenson [20], i.e. we assume that the investment price of an asset q is

equal to the present value of its future services evaluated at the ser-

vice price P (which is the price we wish to ascertain). We aiso assume

that the service from an asset declines geometrically over time. Then,

disregarding taxes, the asset price is related to the service price by

j+l 1

qt = jJt [(l-d) tpj+l n +rs (46)

where d is the depreciation rate and r is the appropriate interest rate.

From this we can obtain the equations that relate the price index for each

type of capital service to the corresponding asset price index:

2 6See also Hall and Jorgenson [35] and Coen' [22].

P R(t) = R(t)qNR(t-1) + dRq(t) - (qNR(t) - qNR(t-1) )NR N RN RN (47)

PD(t) = R(t)qD(t-l) + dDqD(t) - ( qD(t) - qD(t-) ) (48)

Here R is a long-term government bond interest rate (source: International

Finance Statistics of the IMF), and qNR and qD are the asset price indices

for non-residential structures and durables.27

:For some countries (Canada, France, Italy, The Netherlands, U.K.

and U.S.) asset price indices and depreciation rates were obtained from

Christensen et al. [12,13,14,15,16,17,18]. For the remaining countries it

was necessary to compute implicit asset price indices from gross fixed capi-

tal formation in current and constant units using national statisical year-

books, the U.N. or OECD National Accounts. Remaining depreciation rates

were obtained from life of capital figures in Denison [25], or implicit

rates from OECD National Accounts were used. Asset price indices were de-

flated and then converted into indices relative to the U.S. using the ap-

propriate purchasing parity indices. Data on the investment shares (gross

fixed capital formation for producer durables and non-residential structures)

used to compute the Divisia index were obtained from national statistical

yearbooks, or UN or OECD National Accounts. Note that this method of compu-

ting the price of capital does not take into account differences in corporate

tax structures across countries; we simply did not have access to the data

needed to take taxes into account. This means, however, that our price

index for capital services must be viewed as approximate.

Expenditures on Capital Services: Expenditures on capital services were

determined by subtracting labor expenditures from value added. Data on

value added at factor cost was obtained from the United Nations' Growth of

World Inr.:strv or Annral Ycar:boo"k Value added for France and Germany was

2 7For West Germany the discount rate was used as the interest rate, since thegoveriument bond yield was unavailable.

-23-

available only at producer costs, and value added tax data obtained from the

EEC Tax Yearbook was used to arrive at a factor cost figure. All of this

data is measured in local currency, was deflated using the local GDP price

deflator, -and converted to U.S. dollars using the purchasing power parity

for GDP. Note that this does not include depreciation. Since the concept of

depreciation varies between countries and comparable data is not available,

the gross figures are used.

Fuel Quantities: Quantities of fuels used in the industrial sector (ex-

eluding energy conversion) are all obtained from OECD energy publications.

Two different publications were used, Energy Balances of OECD Countries:

1960-1974, Paris, 1976, and Energy Statitics of OECD Countries. The 1976

publication is used for 1960-1974 since it contains the most recent and re-

vised data and clearly excludes chemical feedstocks. These data series are

related to those in the earlier OECD publications via simple linear regres-

sions, together with the earlier data, are then used to extrapolate our

1960-1974 series back to 1955. The U.S. was treated differently from other

countries in that the 1976 publication showed a large amount of "crude and

NGL" consumed by industry. Investigations into other publications and con-

sultations with the Paris office of the OECD and the International Studies

Division of the FEA have led us to conclude that this category probably er-

roneously contains some petroleum products used for petrochemical feedstocks,

non-petroleum hydrocarbons and other refinery.gas. To keep our accounting

consistent with other countries, this category was not included in our petro-

leum total.

Fuel Prices: Industrial price of heavy fuel oil, natural gas, coal and elec-

tricity were obtained from EEC publications and the OECD statistical office.

These data are measured in local currency units, and converted to U.S. dollars

using the appropriate purchasing power parities. Final units are U.S.

dollars/tcal.

Purchasing Power Parities: Purchasing power parities for gross domestic

product, producers durables, and non-residential structures were obtained

from Gilbert and Kravis 58], Gilbert et al. [59], and Kravis et al. [60],

and are all bilateral indices with the U.S. base country.

Our basic models require cost shares and price indices for capital, labor.

and energy, expenditure shares and prices for the four fuels, and the value of

output. The available range of our data is shown in Table 1. Note that for

France and the U.K., factor share data is not available for some of the early

years. It is useful to examine some of the share and price data before turning

to the estimation results. Data for 1962 and 1970 are shown in Table 2. (Note

that the energy price index is computed from a "preferred" translog fuel share

model; this model is presented and discussed in the next section.) We see from

this table that there is considerable variation in fuel expenditure shares across

countries, and through time in any one country. Fuel prices also vary considerably

across countries, and have generally decreased over time. Factor shares and prices

show much more variation across countries than across time, so that our capital,

labor and energy elasticity estimates should probably be viewed as long-term.

5. Statistical Resultsa==_=.__=- -====------

In this section we present the results of the estimation of the models set

forth in Section 2. We have estimated static translog models of fuel shares and

factor shares, and static and dynamic logit models of fuel shares. At this point

we have not yet estimated dynamic versions of the translog model. We begin with

the two stages of the transicg rlodel - first the fuel share model, which in turn

is used to generate a price index for energy, and then the factor share model.

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From these we determine partial and total elasticities for fuel demand, and elas-

ticities of substitution and demand for capital, labor and energy. Finally, we

describe the results (which were less successful) of estimating the logit models.

5.1 Fuel Share Model

In estimating the fuel share equations (29) a number of choices must be made

regarding the pooling of data and the choice of time bounds. First, regional

dummy variables can be used to allow the intercept terms of the share equations

to vary across countries as in equation (43), or to allow the ij parameters to

vary across countries as in equation (44). We found the latter alternative to

involve a considerable reduction in degrees of freedom, and the resulting elas-

ticities had large standard errors, so in our "preferred" model we use regional

dummy variables for the intercept terms. We also consider using no dummy vari-

ables at all, and we use the standard chi-square test to determine the need for

intercept dummies. Second, since fuel prices in Canada and the U.S. have been

significantly lower than in the other eight countries in our sample, this might

have resulted in a production structure different enough to suggest pooling

these countries separately. We therefore estimate models in which all ten coun-

tries are pooled together, and in which Canada and the U.S. are pooled separately.

Third, although our data span the period 1959-1974, there is a question as to

whether the 1974 data should be considered to have come from the same popula-

tion as the 1959-1973 data, i.e. whether the 1974 data point lies on the same

long-run cost function. We therefore estimate models both including and excluding

the 1974 data. Finally, it is useful to test whether we are indeed estimating

a long-run cost function. To do this we estimate one of the models using data at

three-year intervals, and compare it to the same model estimated with annual

data. If the resulting estimates are nearly the same, we can conclude that we have

estimated a long-run cost function.

-29-

Estimated parameters for the various versions of the model are shown in

Table 3. (Standard errors are in parentheses.) In the first version all ten

countries are pooled and the 1959-1973 data are used, but the model is restric-

ted in that no intercept dummy variables are included. We can test this re-

striction by comparing the model to the equivalent unrestricted model of column

6. The value of the chi-square statistic is 628, and given that there are nine

parameter restrictions, this is well above the critical 1% level of 27.8. We

therefore include intercept dummy variables in all other versions of the model.

(A model is also estimated using regional dummy variables for the second-order

terms, but we do not report the estimated parameters here. The resulting own

price elasticities, however, are shown in Table 6, and as can be seen from that

table, many of the elasticities are statistically insignificant.)

In columns 2, 3, 4 and 5 Canada and the U.S. are pooled separately, and

in columns 6 and 7 they are pooled together with the European countries and

Japan. Also, the effects of including the 1974 data can be seen by comparing

columns 2 and 3, columns 4 and 5, and columns 6 and 7. Note that in all cases

the ij estimates change considerably when the additional data is added, leading

us to believe that it should not be included.with the 1959-1973 sample. Also

note by comparing columns 2, 4 and 6 that the Bij parameter estimates for Canada

and the U.S. are quite different from those for Europe when these countries are

pooled separately. In addition, the parameter estimates for Canada and the U.S.

are still statistically significant when the pooling is separate. This is an

indication that it is probably preferable to pool Canada and the U.S. separately.

By comparing columns 8 to column 4 we see that using data at three-year inter-

vals results in little change in the estimated parameters. The resulting price

elasticities also do not change much (compare Tables 4 and 7), so that we conclude

that we are indeed estimating long-run elasticities.

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0 I 3S j .i. . 0 0 4. 6 4 0 4

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H H H H H v-4 H H4 H H H Ns N> N q C Nc ('4 js

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to

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O C1l TJ C1 M CI C) *1q m~ 1- --T * r- 1- lz r1 co 'A cn _4- ITJ-l rs 1- T ^; * r- - T _1q %D -4 xD 04 tM " 0 rr O -0 ) O . u O'1,-i ,C) 0 0o O Oq O c O r9 O,.o O4 c O C)9 C) o O C> C O cO u1 C D O '

O O c O O O O , O- O -O O r-40 O O O O O O O O '0 -40 4 o c o -'

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co at rs a- C- 0 C x ) %O 1 r- M m r r r 1 r r-I 00 m, T Lf) 40¢, C,4 C7, r-q 0 cs ~l(- 4 C-4

M~~~~ ~ ~~~~~~~~~~~ (3J r- 0 H 0 r- 0 N C) 0C r-4 C 4sH qo4 ) r- o C oa .z 0 Cn u

en O- -1 O- %D O) csO O Cf O- a, cc C) O OO H O LA O cn O %D O H 'd- O 0 % HO °. . . % . * . .. . . . p . . .p. .. p . .% . . .% . . . .% . . . .V C14 ON , ,-T D , H -4 co ,-4 or1 ,-i 0 N , -4 4 ,- 0 Cl0 '40 CI zt L C0cn H ' ')0 C)OH O H O H O C 0 0 00 H 0 CC) )0--40 H O C0 C 0 C-4 O-0 C' O O OO-OO.O,-O,-OOOOOOOOOOOOOOOO O O O O,-t O(%10~ -IO,-40 O~ OO O O O G

,' 0 0 . 0 0 0 0 0 00' I'~ ~D t?'l 0 0 0 t' ~. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0D HoLof oc.J, e0l C r HO In. rO. C .,0* 0 .4 0 C14 10 en M co

u~eq ~,-!-.T ,-4 ~O ,- ,- 00 0 0 0 0 0 0~~,~. -~ 0 0 0 0 0 0 0 -~ 4 O ,-4 0 0 0 0 00C~ 0 , - tO,-0O OC'q 0 O O OC' O- O C,-0OC" 0CN C" O' 0 C)O H-

C0 .i i

C; C; C; C;~~~~~~~~~n H '6' e ' ml0i -01

N I,0 000O c 0 c 1 00_ 0 0 1

p., p., p.. p L..oO n f cr. co O N , N H xD e *H n ;r D 00 , bb s~t co v i,H < ^ < vD N n m H ('11~ oo~ t[ ('~ ( cq O¢,") D Hr

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gg_

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,- ¢'qnC Ha) 'v0

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>~~~~~~~~~c C'4 0) tol O , rQl m< t r ) C-4 4 CqN rt4 0 M r - 1-4 M CN N fo 'T 0 Cf .nC14 MI CfU C C) Cl N C) t- .' C) C-W , ) r "- - (cc CIA r- N c V) CN CO Cr r- .- q cn YM - (P

It C) 1 t'- VI -) C') r' -4 CI #r' - 1 ()M l r-4 r- O_ -4 V1 fq rH t- *-t r - 1^ C:)C) r - C ) r-4 rsV 0 C0 C 0 0- C) O C) C) C) rO 0 I r r O O O O O C) O C) O r4 O C) O ~ C: C

I **C) . . . . . . . . .. . . . I .~10 *,I C:O C10 O C O O-- O O O O O C) O O O O OC C) O C; O O O O O C; O O O O C C; qO~ OOI C | -4000OIOI OOOOOI IQ~r .40I I)OCNIC

-40~~~~~~~~~~~~~~~~~~~~~~~~~~~-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r r.

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N C C C f- C4 C, CA* - C4 :D r -4 ,-- C) 4 C'-) CD C,. " r ) ) C 0 D 0 ) W- 01 t) 0 %Ov C O _°;

~~~~~~~~~~~~~~~~~~~~~~~~~ '0H 0 0 H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00

O o - OH ' C" 0 0 O 0 i '0 0 O ON O'O C\5 O O 0 ONON O0 iC 1O - 1-O cO Czt

v I '_ I .v _- I '_ I v _ I v i v v i ''I v I' - I v i ~ - _-'H O 'r 0 "_ 4 0 -. 0 ,, N C 0 N C)0 00 C -- .¢ N C C .0 C H N C0 ' 0" N C 'CNI ~! " O C O C;. O C; O 1 C C; O C; C; C; l C; O C 8) C; O C) OOI r' C; G

°o O14 co°C-4 o T- O O O e O O O O O -O O O C) O O C M) O OC O O O OOO O O v , I # vE v vEs_ I _ v_ vJ I ' I v ~-~ _ v I I - I i I v INCClrJH(NC'lHC4..~~H1r4HNHCNC) ell.0C f0%HH0lu- _ .

co %D cn 04 cDOe t ) 0 T a<r 0 fl .r m rl C 0 0 It n . o rvo m h N o- co u- ) 4 m rq'O esq -N C) r- C-3 - GT 0 ffi a) 0 o (,'I <N T r-- c- 0 k o U'

°:' ~1 0 C 0I " H - - r- O -- C C H C", r'q e O 'IO CO N O C o %'~ ~ o O r~-HoC; o o, o o Co O o O O o O O o o o O O o o o o o C oI v v ~' I v ~ v I v I v v I ~ .~ v I ~ I v~~~~ I I i - ~

i ,i i i i at~~~~~~~~ C14 in - c el 0D N) i n 1-4 r- %( O 19 t C) 0 Le C 0 1 - 04 0%\o ~ ~ ~~ ~~D e a 4 i LI) - INS 00 C4 _ o V- o, -t - C-- ko -( C:T r- ¢ M c: Lr v-o3 M r- O T- MC toC4 r-" .N N - O cs o cs H H 8 c4 6 ( oC; C; C; cs 8 8 8 ; 8 C; cs 8 coG8 sH ;Vx

N~~~~~~~r O t O O rH) O- C3- O1 OD O en O O , O- O' T c O~ O c O -O .O i O .O 11 O- O c OHO **

O1 O O O O O O O O O O O O O O- O O ON O O O O O O C) O O O COv O~~I I \/ 'I v v I v _s I I v I I v _- I ,.)I v ~._ ~,~~~~~~ ~ im ~

V- O c On r- co C 0i 0 Un , o o o o o o 0 ro c o 0 ar 04 O C-C C; C; C; .8 C; c; C; C; ~o o c C, o o rsr nc <nca o ;

C1 C\ 0% Cbl V) C r L4 N C) -zq N CI I V , LI) CO N Lj N %O U- \ - Cq %D %O VI %O C? HO O O NO O O G O O O O O O O O O OCO O O O-O O O O OO O O OO O*- a- 4 -8 L% -- : 0 0 * S S L N C 0 r C C ) _ N r \* > O F-4 0 q

ggC C; C4' O', O C C; C O O O OOO O O O O C O'

' C,e <) uEo cst < < uE c uN 0> ̂ t N < C vt M 04 < ,t t Nc clb <s

.' .X ,. ,. ~ ,.q %'t N \4 V o U'q e' {,I ev > ) r

C'~ (N LI H Cl Cl'0 N 1'~ tH H '< ') 00 u;) C Cl '00 N H L~ NO '0 N 0 - f N (. _t . 2

C c n C C = ca ca ca ea ea Cc e in C°COl* (N H *N C( N'* .- . .. : . , @ H. . , . , , . . HC. , , O . , . .OO H O O OOOHQOOO: > ^^>;O 0 vO' O O OO CO clC> O c.I I _OOOOOOOOOOOIOOOOOOOOOOOOOOOCOOs_

* _1 co} r~~~~~~*

n~~~~~ a~ a' % eJ -X _% S f _ F_ eqcCJ1

, ~ ~ ~ ~ v .. . i N.~' . f)C )~.~ - ) . .1- N C . lH , .-4. --% -l H 7

X

I%O

.-.--

jv

C4

A

0

0

0

IH

.

.-44.1

4

4-I

1

P

41

00

0

4,0

f-4

:-: I I �- 4'. .

i

Estimated elasticities are shown in Tables 4, 5, 6 and 7, (again, with stan-

dard errors in parentheses). Table 4 shows partial own and cross price elasticities

for Canada and the U.S. pooled separately, for both the 1959-1973 and 1959-1974

time bounds. The same elasticities are shown in Table 5 for all ten countries

pooled together. Note that while elasticities for solid fuel and electricity

are more or less the same for the four different versions of the model, elas-

ticities for liquid fuel (largely residual fuel oil) and natural gas vary across

the four versions. In Table 5 we see that pooling all ten countries results in

own price elasticities for liquid fuel (Nl2) and natural gas ( 3 3) that vary

little across countries. We see in Table 4 that pooling Canada and the U.S.

separately results in the oil elasticity becoming much larger for these two

countries and smaller for the other countries, while the opposite is true for the

natural gas elasticity. In addition the own and cross price elasticities for oil

are statistically significant for Canada and the U.S. in Table 4, whereas many

of them are insignificant in Table 5. This is a further indication that Canada

and the U.S. should be pooled separately.

Note in Table 4 that including the 1974 data results in a large change

in the price elasticities for oil, and in particular these elasticities become

smaller and in many cases insignificant. This is not surprising. Oil prices

rose considerably in that year, but demand can adjust to these higher prices

only slowly. The 1974 data thus lies on the short-run cost function, and in-

cluding this data gives us demand elasticities for oil that are somewhere be-

tween the short- and long-run. This, however, is of little value given that

we are estimating a static cost function. By leaving out the 1974 data we

can safely assume that our estimates represent long-run elasticities. (As

mentioned before, this assumption is supported by the elasticity estimates of

Table 7.) We therefore choose as our "preferred" model version (a) in Table 4 -

i.e. Canada and the U.S. pooled separately, and estimation over the 1959-1973

time bounds.

TABLE 4 - PARTTAI, FUEL PRICE FLA:TICITISS ·

(US & Canada Estimated Separately,

*(a) 1959 - 1973

Count:rv Durrama Variables)*

(b) 1959 -. 1974

0 0 -4 4

HOO Cd O 0000 r %

I~ %. I C) cn rl ~ rlL o lqr 7 3c

-, 0 oc; ` ;,~ o o o

O Ln \ C\0 ( %o cq

* *

O. Ltn ~o -1~t0~00000

%.T %v

r a) a\ tC 00 C4 r-.n-4 H O H-4,0000 , 0-o ,

`3cJ`3` c;`3 ~Nc

-% ~-% -'% --

co C)y4 0.. C V- -1 n

I I

o co tn Ln-4 NO0 C1

0c`3`3`31 -..O I--

C M oN N - CIN Cr 0 ,HC.J- I H e4 O -4

_- %- I - I

Do 0D 0 r,- VIO O O O O O H` O

I - I - I %.

6 e n CN 45 O \ Ln 1n M LtN H- rI -I C H , O UL O 0bl O

I %- I . _ %. %,

O r-.c,I r- 00 -1-0 N , , H

It C S ` cC c3 C" `8 8 ( 86 8

-1% 1% ,-% -%r- .* . C4 -It cn O 04cn q C"4 4 m r(y-4my 4

0000 HOHO "9- '9

r( 1 45 yC r rq r-

-99'8 000000C Ln coa\ o o' oC; C; 8

r- %O co

v r -, -%

co r qHO00 0 0'9- 9'- r-99 d9Cvt4H

9 co mI

-(o 0 000008 C;

'99, 9999

c 0 0 0 000 I . * * . I v

* .'l . % ..-- .-

r---4 r H r-I 0 P-

-, - %-

0 r-l -'i d JI

.99- -%9 -9% -9'

e-(V-h r r 04U') \0 ON84H '3%00000 000

`N r4C c;-400 0 0

%, o o o

`_. c;`0000 -4\.,*-i C) T C

I-_. H3 a oC .' coc c co C Co c o

_ ` ̀

0 u c"O - N o4 co C, o D c v- )

(' c1 _ -d i000 0000V

00 l cn c - 0 0 00C* r-i 0 . .r ., 0 6668 86861 - - %-

-9% 0-% -9% -9%0 C9 r .4 '%0 .r- Oq o o 0 0 o

*dd * *H ** Ivoo Ivlv

,'9 -%

co -so I - i --X o <

* . *

Or-0 0000

-l -%- i*% -9% -9% ,-% -9 '- -99% ,- -% 0- -9% 0-0 .0 0 .0 co .0 0 .0 0 .0 co .0 co ,0%.O %-,- %. I %..I - %.. , , %..

C'9.4

Cc494

C7

0a:

e r' 1-i0I".-4 9-I14 814

cc o\ cc o

coO -o o

I' I a C) O c

I % I

H c c3

0000'9.0 99-d

N O _ co

1 _( I -. O0000

00 0 DO 9 cO '-*O* * oooo3`

*1 -. I . .LA O )O

_ _

0 ' tLO L'Cf) e O0c0 c;

v '9

CN -- r- r-

0000i, I 'v

z

:E

.- % i0'900 cn -I r-O Cn 00 c

"940 -I O-

r C r4\ C- 99 %C3

1r 4 (l

b Ds t Nfi- 40w- I1%il--

-9l% -%

r-i V) _4 U)H-f'iHU ,-IOHO0000

o0

11) Ln rn000000

0 '99

JJ

)

0wa

c.9

0r4.

CI

-,

c.

r i

1I I

0C,

r-4

0c;

c ̀ 3 c

- D 000"d3 - l9 - %

N rn r-4 nCn H N rt40H0 "'%-I'--

-% ,-%

v-4 r-4 c0 14 614 1 %, I%-H

%O VC 0 e

00o00s 8 8 (d

v .

000O'

* * I

C14 - 0 -4i .. i I1%- ,

O 0 o 0"N 9-I' r-.-- .

` o `3`I v !

%c0

-99% -%

Co 0 C O 000000000I -I

000000 O om Ln CI _

.- I 0 . O* * * .

H o o Il l

-9. -C) O .;, "co c cn1 C 8r- Id

I--

` o `* *' * .-

0000'9-i %9

,_4 c :T 0\-99%QN 'C-H

* * * 80000o

,- u r - tL

,-I d . .

- 09`O `3 -I c;I C) ; C)

% o 9o

*%9c r.

i d --rv C

09.0

-rq14

4-,

rHl

C) t- 0O

0000* . .-

.- % o o

% 0 0OH i

O

-1

0 clC'9 4t

t ·.-.

-4

I. CO L)

r

Z

r.rN 0*99 '

es

,,�,,,,,�--�cllc-�-.�--�--r�---------�·l

_ I ------ � --- -------- -- ---- c�--C --- II�-C---LI---L·-C C I

m

M

TABLE 4 - PARTIAL FUEL PI:ICE EL,.ASTICITIES

(US & Cananda Estimated Separately,

(a) 1959 - 1973

Country Dummy Variables)*

(b) 1959 - 1974

oo% , -

v v.11 . . -cO o co oq1 I -JHOd

cc-v cc'

0 00 0 0 00 c c-I r H H C H Ov . In *~ . * * *

v % O O I v O vvv- l

F ; 4 C;4

N o 01 _ I -r f~%_ccc a10c'J0L(' o t'o

0000* o o o -

N Ln c 4cnc cr4 e n owO 0 00 a 0

el% - e%- '-I

Iv %- ( v gdggI gdgc;

co LH - " m o' cOc r n H c 0 C O

4I C O O OI I v ? v

0000I ~ Iv

-4C " C

0 0 0 dv v

0e% .c ,-% .1

c'q o , I c -o IoC40CN 0 0 0 0

0o N % O oK - * C Kc K * ' . O 4 H O0

-x * * C ) * K O

C K * C *K K 4 K * K 00 00* *r ·0 00 0

%.O -.

% \0 0ulCD r-I 0 r410' 10

t o HHOHO

C C 4 0

0 ri I

HOY0 -4 0

r -I r O r DI. .V .

4 O 'HH O

L r %C r(

cM LO Cl O

U r \0 0 C .C4 c* )

HOHO4 0 10

rH O T

ddI Iv vC, c O,4 CS* * i vI0 % I

CV G r CO(4 O C' OC0 00

Ivi r- v % CY It u) H O o ac r CNH 1 * ;t- 1-T r-i _ rt H J O

I I - I I - I I

0dd0r'S-I C r-

.* *1%N 'JH 04,.

I'SO I C)I . I %-,

o O O 0 tn 0 0

-IH rA n 4 n 0r ) < C) H 0 0 0 '

IvIv I vIv

0 0 0 0 0 0 0 * *

-I C4N W-400000 000

v0% 0

' C 0C

c N co c4 0o eN 0000 0000c;d c;d c;c;d cvv v v %O N 4 Itr( ( N N

H-O H O 0000

0 00 0 0000 Oc cr)O C c c cnHO 0 c 0000

0 ) . M co .M ,

0 (' Q O 0 0 0

0000 o 00 0 0 0 ; c ; c; 0

co r( o ro* a a 0

I 1 I03000

enN cn C4000000 0v _

tn W-f L r-000000 001'-I'--I %-, I %.

OH

00000

C) O O CJ0dd

I O -i 0000

I I v

l HO 10 N co C_IO O _I O O C0 -I0 I0 0 0

0000H N H00 000000

00 00C-4 C r-q0000. . .

0000

1 - I

O0 CO4 O C0000O 00'd

, Cs ' C

I-'.

-CI 0 C

CV ,- . , ~ ~ Od rVN C -4 en 4 's C '40000 H 000Iv- I v I v- I -

0000i * Iv

0000c-I c; Sdd d * * *

,- 0 VA C)

I - I,%0 10-; ; c;

t M rl V)

Oo r-iO. 9 .* * * C)I %-. I %-

V ,0 0 .0 V V V 0 - 0, 0 co ' 0%..O - %.., %.." %.e %., %.., %-, %-ol %.O el %.., %-O %-., %..O %-, ..,

onC

C-*

.)e4

O9:

.ttO

O*.0

(CON'T. )

C-,

R

0U

zE-4:x:

EHH

.00

44

cnU)

rq0

0O)

4.

0

o4-i

0)

r4U4a)

4jc :

o

0

0 O04 a

-4

Jl,.J

,4 t4qC )

! H

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v-I- *OII·~'K4

_ I __ � ____ __

m _

_ ____ ___ __ __

I

TABLE 5 - PARTIAT FUEL , 'T1C 2TASTICITIES

O10 Countries Pooled, Country D:..y' Vriablce)*,. __ 4. _ _ou

(a) 1959 - 1973 (b) 1959 - 1974

'.., O (13 \D Ln c) VI enC 0 --0 --T 0 -. 00000 0000

%-.. %*1 %_1 %V-p

%-t0 i0NT 0 -

i r- r

· %*cN -H O r q 0 N 00 I I I Iv l o oo Oo

.- 1% 1-% -% -41

L~tVH~-~t CH0CIr-i 1 0 14 4 4V 4 c r4

0000 00006 ; ;C;C;8C;a

0O r,- O o Ln V-4000 o c - --oC4 -

00 ,-4 VI 0 0 r n

000 '0 0000

co u)co c'oCcl0*f 'O -o O -I on

0000 000066.c ; 6. 4-

1- ffi< o-4 m 0 m 0

% . .00004- 44

0 00e 000

6 o6 I66 I ,,~ i ,-i- I 'I J

In r u c 0 o c CI - I - %V n -V r

I vI

,H C U H r(N I

.-. ' I 4- '-' 4--

.I CO 1-,00044 HO ...' O -I

c; H 6 C;4 v

O N O -4c; r- c; r

%4.- %4.I

- 4 uM , r4 .n )I-HH -0 Hv0 1 I I I I % O

LA C! ' rC, r Cq rH

I v I

in r-t rlAH-~2HOHO

, O 0 O c CO Ln CO ui000 0 00 00c;-p 8 c; 4c-

45-M r m f-8 8 8 C;r

%..' %-.

O O O \0

O O6 00%.., 4-1

-4£D \D <- \O

000 04 ,-4

0 0 n 0

co co " -N O O

* .('0 NI C 0000_ 4

M-4 -

-4 -

ai C3 WI co" OCq 00000

4- 44-p

CO r- o H- Cn L)o'l,-4 o4,-- 0,- 0

I v I I v

m P5

4 d-4C.- 0 ct, C

Cj 0 O* * * 00004-I O -C; C CI - I

-v 0 - 00 0 0 06

0f l C t, c? -1 0 :en o 0co ot O of0 0 0 0 4- -.-

. o ,- co r, ,-4 co oN o N0 r- H H

*~r~ * ** **Ho Vo Voo

P d -. 4 -4t P-4 .1

t -. 0 H O C% v

c 0 -I

i . I I

N f- Co %O It r- Ln VON OH O i m C0

0 0 0 I -0 00 000I % I ,-' I·dd · ·'vl cv~~ o

.- % e-, .-. ,-4 e-% -% J% -% -% -%

U 40 c , - X ,0 0 o 3 45- 44-p 4- p %41f %-1 % _v 4-p %- I % -I 4- I

.- 4-4cr dr~9 m.4

* eq('I

0 0

C* '%.I -

N

,.41 41- -l% - 40 .0 0 40

%4.- 4- -I 45-p %4.o

9,

Ca.1'$

C7

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r-4 0 04

5·

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o -

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rl N C-I r.- 4 '-4'

4- r 441 v v

(7 m 00 f--00 0. . .; 8 l - I ,

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c( o 1 I O

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cvHe4I00000

04

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l . 4o 0 c000

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044-

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o 4-8

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.-. I-.

H- 0 r-lO0%-"I --

%O O M L V

-. 4 %-.

0%1 -. 40'%. H-TLOHi-d

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I

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C

44-

I-

I - - -.---

TABLE 5 - PARTIAL FUEL PRP.ICE ELASTICITIES (CONT.)

(10 Countries Pooled, Country Dunmy Variables)*

(a) 1959 - 1973 (b) 1959 - 1974

1 o t D Co aD It n CO CJ \0 0 H -- H O C% Cn -- e H--I O O O e O q

v v Iv I v I I I v I I

* . *

A A·- O yL . lc~ c5 PI c

r~ u asm-t . -I O 5O OO O 0 o ooI 0 I v I v I

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0000d %.'9.v

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I - I v

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I %

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C; C C; c;

-% dei

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4~ ~ '54 5

D M (y NC4 H C') H

15 I %ANi C') Hddd"5-I"-

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-

HO cIS,-l

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c HO o

I

eq% %(4 cr' 00 N

-5% d %

cs o c oI I

O CY oo n b N rqI c0000 0 0 0 o0

I I v v

O 0 oO (n 4 r C"L( rHt r rH 0 H 0o0000 0000

cN r O r CO N O C4(0 O(0O 00000000 0000

% .-

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O03

c'q N r H cOH r tc - C O A000 O O O O H0H000 0000 0000v Iv I I Iv I v

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I 0 000 I -" I -, - %Il

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C' r-7 C'

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i -- _

Note that, except for electricity, the elasticities of our "preferred"

model are large in magnitude. (Remember that these are partial price elastici-

ties, i.e. based on constant energy consumption, so that the total own price

elasticities will be even larger in magnitude.) Own price elasticities for

coal range from -1 in France and West Germany to about -2 in Norway, Canada,

and the U.S. (where coal has had a smaller share of industrial energy con-

sumption). Own price elasticities for natural gas are less than -1 in the

European countries and Japan, but -.33 in Canada and -.52 in the U.S. This is

reasonable given that natural gas prices have been much lower in Canada and the

U.S. than in the rest of the world. One the other hand, Canada and the U.S.

have the largest own price elasticities for oil, even though they had relative-

ly low prices. We can only explain this on the basis of a greater availability

:of alternative fuels at low prices (notably natural gas), so that producers

chose technologies allowing for greater interful substitution possibility.

Finally, note that the own price elasticities for electricity are all quite

small in magnitude. This is not surprising; since electricity is a much more

expensive fuel on a tcal basis, we expect it to be used only where there is no

possibility of using an alternative fuel.

Our "preferred" model can now be used to generate the aggregate price

index for energy. This is done by applying the estimated parameters to equa-

tion 16) (note that the first-order parameters i will vary across countries),

and choosing go, the unobservable parameter of the equation, so that the price

of energy PE is equal to 1.0 in the U.S. in 1970. Then using the data on fuel

prices, we can generate the relative energy price index over time for each country.

The resulting energy price indices are shown in Table 8. These indices will

in turn serve as the instrument variable for the price of energy in the estimation

of our factor share model..

Table 8 - Enery Price Indico, Dr tw d Frr:m "Preferred" Fuel Cice Model(U.S. n 1 7 i 0)

00 r - C O 1-) (I ) 0 Cr a) C1 CC % -J 4 O r - £ ; U O N r C c 1 O: n CM Cf- r( a % 3 ° to ¢<) -o4 o0 ro o C) r O r- ON - O C -t.T 9 O C) NN

Z CY ° D r- \ <r 00 (I r C C I' r - C,% ¢) M C X N s o q N Ci c O n P C% ur N. O In 00 ) .¢ r" ' U c o "3 r. C, c. C3u 3 .. oC) 'n .N CN.,I ¢O O. Co c .-'c

V; c4 c4 c4 c4 C-4' c4 c4 c4 ' H ,, c _I c _I c r.. -;-z ,. O -.:- ,- .C) Ci C, C,

N O rW O rU I n < % n t m U N m U n O O . N N l OC M M N 1 0 C r N O '3 MOZ· · · '. · · ·1C"'Ci 0C 0 CO r ° " " O " c~ C" N ¢-4 % 0 0 C- O r- 0 vT % ) V N N t 7 r! ¢ C-4 _0 * U N .l 0 .-t ...............................

0 c O H 4D M t- - -Ln coMC HN H H H H H H H O I o O H C -')

(0D u ) m CT 0 0 oO I ~ s m M C co n .- r-1 c4 N H C H r H o C C.r C cW W 0 -0

HI C) ) 7 43e t~ C CcV CI~ (~ C CS N tl N r( 1i _( _ - N ~cMl N tn N -,- N C N N C , C"oa cl en H C"

C!. Z C r "s " --I c. o N .' N Cor N NO O ..-I ar r HN..

n i cq u O - , 04, o u O O. t r C L - ,-4 V1 00 r- r4 O o on , - ,- o T H 0O Go 3.. r-s ,- n T,,-4 O O' o o o r-. % o %o o r- r r - O %O . . . . ..

c . . . .. en * C m N C C b N C CO > yJ 4 L" H - ,< , co co -- O q o o c/D c . co co o t- o cO O - . .< rs uN r_ tn - C C s £ u n C,- ' I t VD 1 4 O C C1 C H Y) - 0 0` )O C4 4 - -i H ,- i C O CO O C O O ON 0 ; CO ° O 4 '.-I . C L -T C, CO O-T c. N O c4 ,') 0 -T C.'

./I U'~ u'N u"N , _ l.O r.-'3 O O ) OC O- O' C, CO O C'' ,- O * .- c

Inl D r t0 0 O en 'T If) vA t.9 t_ 00 O1 "l T i n "" tP " O O .O ". I, ', .n co ON co .:o = N m tt) n n Ln L 1- \3 vO \3 3 \ D .-_ %:! · , 1 ) . r- ( -, LeN U L L t Ics ) ',0 ',-') I k O .I Z t- - - . -

0 O 0N i7 O 0 C C . C7N N CTN t C! J C OO Ch ON C,% CJ C'% C C cC O CA QN -1% Ch Oh N IN N N H H ,, -r H H H q H H , 4 H H H H H H H H H H H

-41-

5.2 Factor Share Model

We turn now to the model of capital, labor, and energy shares. Once again,

choices must be made regarding the use of dummy variables and the pooling of

28data. In addition, we cannot assume homotheticity, but must test this as a

possible restriction on the model.

Parameter estimates for several forms of the model are given in Table 9,

although other forms were estimated as well. In columns 1 and 2 the share

equations have been estimated without regional dummy variables. In column 1

the cost function is homothetic (Qi 0, i = K,L,E), and in column 2 it is non-

homothetic. These two models can be used as a first test for homotheticity;

the value of the test statistic is 68.4, and this is significant at the 1% level,

indicating that homotheticity cannot be accepted. In column 3 the cost function

is non-homothetic, and intercept dummy variables are added to the share equations.

A comparison of this model with that in column 2 allows us to test for constan-

-cy of the first-order parameters across countries, The test statistic is

.359.8, which is significant at the 1% level (9 degrees of freedom), so that

intercept dummy variables are retained in the share equations.

In columns 4 and 5 a non-homothetic cost function is again estimated

with intercept dummy variables in the share equations, but now Canada and the

U.S. are pooled separately. Note that the yij parameter values change con-

siderably, with the values in column 3 generally midway between those in columns

4 and 5. In addition, the parameter estimates generally become more significant

for Europe and Japan, but less significant for. Canada and the U.S. This pro-

vides us with no firm indication of whether Canada and the U.S. should be pooled

separately.

28Inclusion of 1974 data is not a question here; our derived capital price dataextends only through 1973.

, -4 t~~~~~~~~~~~

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o o o o H o o o o o o · H n n O O H N N H < < O < O O O O NC 0 L

HI N '. 1_ I {_ 0 0 0 0 . 0 0 '0 0 0 ·o c; ~ ~~~~~~~~~~~~~~~ ~~; ; c; c; c; c;

.. . . . . . . . . . . . . . . .I I .I I I I

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O I I ~ · I I I I I I I I I I -

4 0) t g-. w a .- HH

* .4 Ncr '-4 N ) .-4 N en c(

C" 0 0n I" m m en m 4 C I 04 N lN 0 )4 n c co Y C (D6e * 6o 6s 64 6 6 r6 6o 6

• o 'I:IC i

·N.

I

0-% 1-1 -% 1-1 -% 0-1 -% .11% 1-1 e-, 0-1 0- 1-11 -% I-N -%

A second test for homotheticity is perfomned by estimating homothetic ver-

sions of the cost functions in columns 4 and 5 (the resulting parameter estimates

are not reported here). These models can then be compared to the corresponding

non-homothetic model. For Europe and Japan the test statistic is 8.58, and

this is significant at the 2.5% level, so that homotheticity cannot be accepted

here. For Canada and the U.S. the test statistic is 1.25, which is not signifi-

cant at the 10% level, so that homotheticity could be accepted. However, if the

regional dunmmy variables are eliminated, the same test for homotheticity gives

a test statistic of 32.4, which is significant at the 1% level. In addition,

testing constancy of the first-order terms in the non-homothetic model gives a

test statistic of 2.35 which is not significant at the 10% level. We therefore

consider the test for homotheticity to be inconclusive, and retain a non-homothetic

29 model.

In column 6 the non-homothetic cost function is estimated, with all ten

countries pooled, using data at 3-year intervals. Comparing columns 3 and 6 we

see that many of the estimated parameter values do change considerably. We are

thus less confident that we have estimated a long-run cost function than we are

for the fuel share model.

The resulting elasticities of substitution and price elasticities of demand

are shown in Table 10a and 10b for the model in which all ten countries are pooled

together, and the model in which Canada and the U.S. are pooled separately. Note

that the choice of pooling method has little effect on the elasticities of the

European countries and Japan, but almost all of the elasticities for Canada and

the U.S. are larger in magnitude when these countries are pooled separately.

29We also tested the hl-uethsee.s that the y,. are all zero, i.e. that the rouzticn

function is Cobb--Dv.I. ' . I'm ., t st at _s cs are significant at the 1% level,so that this hypotr.:ses c be rejected.

-44-

f--"- r 0 HH -4N ; NC' IO v 1 IVI O, C T C) C-tO (OH 00 V-i t-A -#H

00.00 00 00 00 0 IO

co tO -4 C-% "Oo4o' r~ ~ c-~ ~0C -4 , r-4 C CV4 0O

HO HO HO rL('%. v v wIY

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-48-

Pooling Canada and the U.S. separately, however, also results in elasticities with

larger standard errors, so we are inclined to choose as a "preferred" model

that for which all of the countries are pooled.

Let us now consider the implications of these elasticities for energy

demand and the substitutability of energy with other factors of production. First

we see that the elasticity of substitution for energy and capital ( KE) is posi-

tive, although small (0.61 to 0..86). We thus find that energy and capital are

substitutes, and not complements as earlier studies for the U.S. had indicated.

We find that labor and energy are also substitutes (with elasticities of substi-

tution close to 1); this is not surprising, and is supported by most other work.

Similarly, we find that capital and labor are substitutes, as expected. The own

price elasticities of demand for capital and labor are around -0.3 to -0.5,

which is in agreement with most earlier work. The own price elasticities for

energy, however, are larger in magnitude than most of those obtained earlier by

others. We find this elasticity to be about -0.8, whereas most earlier estimates

were around -0.4 to -0.5. Most earlier work, however, was based on data for a

single country, so that it is more likely that we have estimated the long-term

31elasticity. Finally, note from the cross-price elasticities of energy and capi-

tal that over the long-run a doubling in the price of energy should result in a

5% increase in the demand for capital and a 6% or 7% increase in the demand for

labor as substitution away from energy takes place.

Since our model is non-homothetic, it is also interesting to note that what

our results imply about economies of scale in aggregate production, and about

3 0Berndt and Wood [9], for example, found strong complementarity between energyand capital. Griffen and Gregory [34], using pooled international data atfour-year intervals, obtain results ver silar to ours.

31Our estimate is close to that found by Griff`tn and Gregory [34], who also usedinternational data.

the elasticity of energy demand with respect to output changes (EQ). In Table

11 we show the index of scale (SCE) introduced by Christensen and Greene [19] (see

footnote 14), and the elasticitiy of energy demand with respect to output changes,

as given by equation (25). (The indices and elasticities for each country are calcu-

lated at the point of means.) Note that the index of scale economies is insignifi-

cantly different from zero for each country, so that the aggregate cost functions

exhibit nearly constant returns to scale. The output elasticity of energy demand,

however, is significantly less than unity. Thus, even if energy prices remain con-

stant relative to other prices, as output increases there will be substitution

32away from energy.

We can now examine the total price elasticities for the individual fuel

demands. These elasticities are computed using equations (20) and (21), and are

shown in Table 12. The own price elasticities of energy are obtained from Table

10b based on all ten countries pooled together. Note that these total elas-

ticities are larger than the partial elasticities of Tables 4 and 5, since they

account for decreased use of energy as well as interfuel substitution. We find

that coal has the largest own price elasticities, ranging from -1.29 to -2.24.

For Europe and Japan, own price elasticities for natural gas are large (-1.37

to -2.34), while those for oil are small (-0.6 to -.56). We attribute this to

the fact that for two countries (Netherlands and W. Germany), as oil and gas prices

fell, there was a large increase in the share of natural gas (from almost zero)

as supplies became available for the first time. This might have tended to bias

the natural gas elasticities upwards. It is more difficult to explain the low

oil price elasticities; oil prices on a tcal basis were generally the lowest of

any fuel, but oil did not gain a dominant share in Europe. For Canada and the

U.S. (which was pooled separately in the fuel choice model) the situation is re-

versed - the price elasticities for oil are larger than those for natural gas.

Here natural gas prices were lower than oil prices, and the share of oil was small

(in the U.S.) and roughly constant over time.

3 2This result is not surprising given the data. Note from Table 2 that forlarge-output countries like to U.S., the share of energy is smaller than for

low-output countries.

Table 11 - Index of Scale Economies and

Output Elasticity of Energy Demand

(all ten countries pooled)

Standard errors are computed based on constancy of shares and

prices at their mean values. The standard error of SCE is thus

computed from:

Var(SCE) =L,M

i(igP i) Var(YQi) + I logPilogPjCovar (QiyQ )i=K, Q~ j

and the standard error of Q is computed from:

VarOlEQ) = Var(SCE) + [2logPE/SE + 1/SE]Var(YE Q )

+ (2/SE)logPKCovar(QEYQK) + (2/SE)logPLCovar (QEyQL)

Country SCE 1EQ

Canada .0015 (.0080) 0.785 (0.108)

France .0086 (.0383) 0.783 (0.113)

Italy .0032 (.0335) 0.855 (0.078)

Japan .0105 (.0487) 0.849 (0.087)

Netherlands -.0124 (.0170) 0.818 (0.093)

Norway .0056 (.0172) 0.807 (0.097)

Sweden . 0019 (.0205) 0.864 (0,'070)

UK .0041 (.0327) 0.778 (0.113)

U.S.A. .0003 (.0037) 0.624 (0.188)

W. Germany .0012 (.0316) 0.761 (0.122)

. q

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In Table 13 we show the elasticities of average cost of output with respect

to the price of energy and the prices of the individual fuels. These elasticities

are based on equations (27) and (28), and are shown for two versions of the factor

share model - Canada and the U.S. pooled separately, and. all ten countries pooled

together. These elasticities give us the effects of increases in energy prices

on the cost of output, assuning the level of output stays: fixed. They thus pro-

vide information about the inflationary impact of.an energy price rise, but they

do not provide information about the effect o the level of GNP. Note that in the

United States a 10% increase in the cost of energy.would result in about a 0.3%

increase in the cost of output, whereas for Italy, Japan, and Sweden the cost of

output would rise about 0.7%.

5.3 Logit Models of Fuel Shares

We estimate a number of static and dynamic logit models to describe the de-

pendence of fuel shares on prices and the level of output. The "decision functions"

in these models are linear or logarithmic functions of relative fuel prices Pi

(the price of fuel i divided by the price of energy), the level of output, and in

the case of the dynamic models, lagged shares. Recall from equations (40) and (42)

that this leads to a set of three equations that must be estimated simultaneously

34since certain coefficients are constrained to be the same across equations. We

therefore use iterative Zellner estimation to estimate all of the models.

Our static logit models are of the general form

10

1 (i/s4) klai4kDk + biPi - b4P4 + ci4Q, i=-1,2,3 (49)i 4 i 4

3In comparing these elasticities across countries, remember that they are dependenton the shares of enerv,y. and the fuel sh.ares. Thus Italy, apan and Sed;len hv-:the largest valu.s oz 1C _ in part bucause they have the largest shares of er ervin the cost of cutout.

3 4Even without cross-equation coefficient constraints, simultaneous equation esti-mation is desirable in that insofar as errors are correlated across equations,it yields more efficient par;eter stim.ateS.

It

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-54-

where ai4k = aika 4 k, and ci4 = ci-c4, as in equation (40). The Dk are country

dummy variables (countries are ordered alphabetically), and the fuels are ordered

(1) solid, (2) liquid, (3) gas, and (4) electricity.

The dynamic models are based on the assumption that .the choice of fuels this

period depends on the relative shares last period, as well as this period 's rela-

tive prices and output. The dependence on past shares is intended to incorporate

partial adjustment of the captJal stock. It leads- to equations of the form

10

log(Si/S4) klai4kDk i b i b4 4 i4

+ dS dS 112,3 (50)iSi,t- d 4 ,t1' =1,2,3

Note that this is not a Koyck adjustment model. The coefficients di can be greater

than 1 (although we would expect them to be positive), and in general a change in

price will not lead to geometrically declining changes in shares over time.

The results of estimating various versions of these models are shown in

Table 14. Unfortunately, the results are disappointing. In all of the models that

we have estimated, the own price elasticities are positive for both solid and

liquid fuel (note that b and b2 are positive for all of the models). In addition

the coefficients b3 and b4 are insignificant or positive for some of the models.

The logit models provides essentially an ad hot description of fuel choice by

industrial consumers of energy, and that description is not consistent with the

data and our a priori expectations regarding the characteristics of fuel demands.

6. Comparison of Results ith Other Studies.

In Table 15 we present a survey of recent estimates by others of industrial

energy demand elasticities. ],e can compare these estimates with our results to

get an idea of what "concensus" elasticities would be, and how and why our esti-

mates might differ from these.

C) 1.4 U) m (,) 1

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Tab] 1 5 (ont.)

Elas. Country Estimate Source

Price Elas.

(cont.) Canada K 0'.76, LL -0.49,

nEE -0.49, KE = -0.05,g

nLE 0.55

NetherlandsKK = 0.05, nL -026,

nEE -0.90

9 industrialized

countries KK = -0.18 to -0.38,r LL =-0.12 to -0.27,LEE = -0.79 to -0.80

KE 0.13, XE 0.11

six-countrycomposite XleE = -0.30 (h)composite - EE

Fuels -

own price U.S. elec: -0.66, oil: -2.75 (i)

elastici- gas: -1.30, coal: -1.46

ties, partialU.S. elec., S.R.: -0.14

elec., L.R.: -1.20

U.S. elec., S.R.: -0.06(k)

elec., L.R.: -0.52

own priceelastici- U.S. elec: -0.92, oil: -2.82,

(i)ties, total gas: -1.47, coal: -1.52

Canada elec: -0.74, oil:. -1.30, (ggas: -1.30, coal: -0.48

Sources:

(a)(b)(c)(d)(e)(f)(g)

(h)(i)(j)(k)

Berndt and Wood [9]Halvorsen and Ford [37]Fuss and Waverman [31], translog

Fuss and W-averman [31], generalized LeontiefMagnus [46]

Gr.i,cn a-c Grcory [34]

Fruss 3)]

No rdhuo [50]Halvorsen [36]

Mount, Chapman, and Tyrrell [49]

Griffcn [33]

Table 15 - Al.ternative E'stimates of Idustrfal Demand Elasticities

Elas. Country Estimate Source

Factor In- a 1.01, a = -3.25,puts - U.S. KL KE (a)elasticities CLE 0.64of substitu-tion tion U.S. (2-digit a = -1.03 to 2.02,

industries) aKE = .48 to 2.88 (prod. workers), (b)

ilE = -2.02 to 5.59 (non-prod.LE workers)

Canada KL = 0.72, aKE = 0.42, (c)

OLE = 1.70

Canada aKL = 5.46, aKE = -11.91, (d)

aLE =4.89

Netherlands OKL = 0.30, KE -4.50, (e)

aLE = 3.80

9 industrialized .KL = 0206 to 1 07.52,countriLes = 0.72 to 0.87

Factor Inputs price elastici- U.S. K 0.44, L -0.45,a)ties T1EE = -0.49, KE = -0.15,

)LE = 0.03

U.S. (2-digit -0.67 to -1.16,.industries) nKK -0.67 to -1.16,

nLL -0.28 to -1.55, (b)nEE = -0.66 to -2.56

Canada nKK = -0.79, LL = -0.45, (c)

.]EE -0.36

Canada nKK -0.31, nLL = 0.77,

TEE = -0.59! ,., ,,, , , ! i , . , , . ,E

4.

Note that there is mixed evidence on the substitutability of energy and capi-

tal. Berndt and Wood [9], Fuss [30], and agnus [46] find energy and capital to

be strong complements, but they worked with time series data for a single country,

and might have estimated a short-run cost or production function. Halvorsen and

Ford [37] and Fuss and Waverman [31] obtain mixed results on energy-capital sub-

stitutability, depending on the particular disaggregated industry or the particular

form of the cost function. Only Griffen and Gregory [34] find strong evidence of

capital-energy substitutability, and their estimate of the Allen elasticity of

substitution is close to ours (1.01 compared to about 0.8). This is reassuring

since both their study and ours use international data and presume to estimate

long-run elasticities. As for elasticities of substitution between other factors,

our results are close to Griffen and Gregory for labor and energy, but we find

greater substitution of capital and labor.

The own price elasticity of aggregate energy use is an important parameter

to any energy policy debate. Our estimate (about -0.8), together with those

of Magnus (-0.9) and Griffen and Gregory (also -0.8) are larger than most other

estimates, which fall in the.range of -0.3 to -0.6. Again, most other estimates

are based on time series data for a single country, and may be short-run.

It is more difficult to find a concensus on partial and total fuel price

elasticities. Although most would agree that electricity demand is less elastic

than the demands for other fuels, partial long-run elasticities for the U.S. range

from -0.5 ,to -1.2. Our study finds electricity demand to be even less elastic;

we found partial own price elasticities to range from -.08 to -.16. Our total

own price elasticity estimates, however, are closer to the estimates of others

(largely because of our higher estimate of the own price elasticity of energy).

We find this elasticity to rane from -0.54 to -0.63, here !Halvorsen [36] ob-

tained an estimate of -0.92 and Fuss [30)] -0.74. Our own price elasticity estim.,te

(total) for oil is also well below the estimates of others; -.22 to -1.17 as com-

-61-

pared to Halvorsen's estimate of -2.82 and Fuss's of -1.30. An explanation for

this discrepancy will probably require further work. There is less disagreement

over the elasticities for coal and natural gas. Our estimates of the total own

price elasticities for coal (-1.29 to -2.24) and natural gas (-0.41 to -2.34) are

generally in line with other estimates.

7. Summary

We have seen that the use of a "two-stage" weakly separable cost function pro-

vides means of estimating demand elasticities for aggregate energy use and for in-

dividual fuels. In addition, by pooling international time series-cross-section

data we can obtain a sample large enough to provide low-variance estimates of es-

sentially long-run elasticities.

We have found that the'own-price elasticity of aggregate industrial energy

demand is larger than had been thought previously, and that energy and capital ap-

pear to be substitutes, rather than complements. We attribute these results to

the long-run nature of our estimates. We found the:total own-price elasticities

of coal and natural gas to be large, as expected, but we found the total own-price

elasticities of oil and electricity to be below 1 in magnitude. While we expect

the small elasticity for electricity (there is little flexibility in its use), it

is harder to justify the-elasticities for oil.

We also found that the aggregate cost functions are mildly, but significantly.

non-homothetic, so that the elasticity of aggregate energy use with respect to out-

out changes is below l (generally around .7 or .8). This is not due to economies

of scale in the long-run (we found cost functions to exhiboit constant returns to

scale), but rather to substitution away from eergy as output increases. Finally

we found that further increases in the price of energy would have only a small im-

pact on the total cost of production. This is due in part to cnergy's small s ihare i:

production, and in part to substitution possibilities.

-62-

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