INTERFUEL SUBSTITUTION AND THE INDUSTRIAL DEMAND
FOR ENERGY: AN INTERNATIONAL COPARISON
by
Robert S. Pindyck
Massachusetts Institute of Technology
August 1977
Working Paper
#MIT EL 77-026WP
This work was supported by the RANN Division of the National Science Foundation
under Grant #GSF SIA 75-00739, and is part of a larger project to develop
analytical models of the world oil market. The author is indepted to
Jacqueline Carson, Vinod Dar, Ross Heide and Kevin Lloyd for their excellent
research assi-sance in many aspects
nof this work, to the Comr;ter Resenrch
Center
of The 't'ic.mna
iBreau
of Eccnr
c Rsearch
for assistance in the comrutaEional work, and to Ernst 1lerndlt, '."Klvyn Ftess,James Griffen, and Leonard
Wavezman for co-=aentsand 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 industrial 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 pooled
international
dota in c;d'rto ->assure easticits
of sustitution for capial,
labor,
andenergy,
but that work oes not explain inter-country diffcrences in
elasticities since only four observations were used for each country, and does not
deal with interfuel substitution.
input, so we must assume separability of materials from the other factors in the
In the second stage expenditures on energy
underlying structure of production.
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 structure 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 substitution, 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
tions.
one is continually bound by data limita-
For many countries there is no good data available for
variables of interest to us.
2
this
some or allcf
the
For other countries data exists, but obtaining that
The 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 expenditures is not an explanatory variable in that model.
c n and te generalized Cobb-.outzlas
ont'f
.
by Di:;ert [27,28]. Lau and T-.lura [44]
function, both of wich were introued
used the generalized Leontief production function in estimating input substitution in petrochemical refining. ZLaSnus [46] used the generalized Cobb-Douglas
cost function in estimating substitution between capital, labor, and energy inputs
in Dutch manufacturing.
-For example,.
the
-3-
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
demand to a set of only ten countries.
4
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 expenditures 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 constructed for a number of countries for whnch only partial data is available;
the results of this work will be described in a forthccring paper.
5
Working Paper No. MIT EL 76-011WP, M.I.T. World Oil Project, May 1976.
vised version of this report will appear shortly.
A re-
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 substitution 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 optimization process, i.e. optimize the mix of fuels that make up the energy input, and
6
Materials 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
Q = F[f(K,L,E);M]. For a proof and further discussion, see Berndt and
the form
Christensen
[8].
7
Halvorsen and Ford [37] recently used translog cost functions to.test for separability of the energy agre-ate for each of eight individual two-digit industries
to hold for four of the eight councries.
in the U.S. They found sparability
8
The second and third assumptions together are referred to as homothetic separability
9
then optimally choose quantities of capital, labor, and energy.
Equivalently
we can express these three assumptions by writing the production function as:
Q
=
(1)
[(f(K,L,e(F 1 ,F2 ,F3 F4));M].
where e is a homothetic function of the four fuels.
If the factor prices and output level are exogenously determined, the production structure described by (1) can alternatively be described by a cost
function that is also weakly separable, i.e. a function of the form:l 0
=G[g(PK,
)
C
PL,PE(PF'PF2'P3
' QF2';P'(2)
P F3' F4Q
C
[g( K'
L' E Fl'
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
in estimating the demand for energy in Canadian manufacturing.
[31]
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 elasticities for capital, labor, and energy.
See Denny and Fuss [26].
10
This was shown by Shephard [54].
-6-
It is important to point out that we could have chosen to use translog proSince the
duction functions rather than cost functions in estimating elasticities.
translog cost and production functions are not self-dual, different elasticity estimates 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.
We begin by reviewing the properties of the translog cost function.
11
The
translog cost function is a second-order approximation to an arbitrary cost function, and has the form
logC - a
+ aQloSQ + EacloEP
i +
yQQ(1ogQ)2 + gEYi+logP
logPj
+
where C is total cost, Q is output, and Pi are the factor prices.
EyQilgQlogPi (3)
From Shephard's
Lemma [54], the derived demand functions are found by differentiating the cost
function with respect to the prices, i.e. X i =
imi,..., n.
Q+ Ylo
i +Yqil to
shares
Since
the
Note,
however,
Thus the share equations
or
alogC/alogP i a (PiXi)/C,
are given by S
Si
C/aP i.
must
that
(4)
add to 1, only n-l of the share equations need be estimated.
the parameters
a0 ,
Q, and yQQ are not-identifiedunless 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].
12
Corbo and Meller [23] recently estimated the ranslog production function directly
(instead of the derived share of equations) using capital and labor input data indi-
This llc;:.d
vidualfirrms.
is really translo,,
th;cn to
.....
! lin
rroduction
bhavilor.
hethar th
cst '.
and o test for cc:petitive
function
The cost function must be homogenous of degree 1 in prices, and mu§t satisfy
This implies
the conditions corresponding to a well-behaved production function.
the following parameter restrictions that must be imposed:1
3
iCai -1
(5)
y ~j
(6)
0
iYQi
yiJ
Yi i
EiJY
.
iEj
(7)
j
0.
(8)
Notethatthe cost function as specified
have non-constant returns to-scale.
so far is non-homothetic, and may
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
is constant.
output (logC/alogQ)
This implies the additional restriction:
(10)
.YQQ
Finally, we could impose the restriction that the elasticities of substitution between all factors are equal to 1 (so that the cost function corresponds to a
Cobb-Douglas production function).
Y
tions:
13
This implies the additional parameter restric(11)
.
See Christensen, Jorgenson, and Lau [21].
14 Christensen and Greene [19] define the following index of scale economies:
Note that if
+ ZYQioPi).
1 - (lQ + yqrlOQ
SCE - 1 - aloC/alogQ
SCE is positiv>
(n2c-ative), thoro
is incrcasing
an,d has a ntural
This is a useful . -
(dacra3sin,)
it.rprctztion
returns
in prceltagea
to scnle.
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
The appropriate test statistic is
simple chi-square test.
-2.logA = N(loglrI
(12)
%1iuI
are the determinants of the estimated error covariance
IRl and
where
- loglIUl)
matrices for the restricted and unrestricted models respectively, and N
is the-
This statistic is distributed as chi-square with degrees
number of observations.
of freedom equal to the number of parameter restrictions being tested.
Uzawa
15
can
[56] showed that the Allen partial elasticities of substitution
CCij/CiC j,
ij
be computed from
so that for the translog cost function these
are given by
=(Yij
-ij
ii=
i
+ SiSj)/SiSj,
j
(13)
[Yii + Si(Si-1)]/Si
It is easy to show that the own and cross price elasticities of demand are given
ynii
" algXal
by
ogPi =
riJ - BlogXi/alogP
Note
(iisi
(14)
SjS
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.
rii
The total own price elasticity for each fuel
= d logXi/d logP i accounts for the effect of a change in the price of a fuel
on total energy consumption, and is given by
iiXi
i ax1
PiE const
+
+aE
aE
i
(15)
E aPE aP
where E'is the total quantity of energy consumed, and PE is
energy.
cost
the
price
index
for
However, since the price of energy is given by the homothetic translog
function with constant returns to scale:
logPi+
logPE a + CilogP
which implies the fuel share equations Si
iyl1
gP
'(16)
Yi + Zijl°gPj'
we have
See Allen [3].
16Note that expenditures on energy will not remain constant.
-9-
PE
PE
-
-i
(17)
E E
(18)
E.
aE
aPE-aE
· ,
S
where'nEE is the own price elasticity of energy, and, since the energy cost
function is homothetic,
ax iX
axi
ME
a
E
xX i
Xi
ME.E
E
(19)
Then, by substituting
= PE E is total expenditures on energy.
where M
(17),
(18), and (19) into (15), we have
tlij
T
(20)
ii + EESi
Similarly, we can compute the total cross price elasticity nrj from
ax
P
'i-|const.
Xi
+
E cons.
ax
aE
aP
E
aP
E
ap
li +
(21)
EE
-J
17
aud the total output elasticity from
d logX i
oX
T)* =
iQ d logQ
2_ ....
-.
x
(22)
a? aQ
Since the energy cost function is
homothetic, this reduces to
(23)
niQ - nEQ
where
EQ is the elasticity of energy with respect to output changes.
This
s
thi
17
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 pricing
ba~.d
on a
be better
'
:
c:,;.
ov---I'"33r
;:rl,.:
d :o a a t~'~icz.t
referre
for fuel
the dmand
production.
of
i
corrcspondin
zi;..2.
Or-~7.e
this
thz
would
elasticity
c.t
cr.e
in
to a 1 percent change in the total cost
-10-
elasticity can in turn be computed as follows:
d lo E. a log C
d log C a log Q
d log E
d log Q
EQ
logE
logQ
_L__o_
+
aLogQ
aiogc
gSE
lo
(24)
alogQ
logQ
Obtaining these derivatives from equations (3) and (4), we have
~QE+
nEQ =
I
L E
(25)
.E ~i= K,
+ aQ +
SE
QQlogQ +
YQilogP
i
(25)
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 elasticities, 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.
we follow Christensen and Greene
To do this
[19] and calculate standard errors under the
assumption that the shares Si are constant and equal to the means (over the estimation time bounds) of their estimated values.
Under this assumption we have,
asympototically,
Var(8ij)-
Var (ii)
Var(Yij)/SiS
-
Var(Yii/S
Var(Tij)
-Vcr('tj)/S2
Var (l)
=
ar (i)
/
S
(26)
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.
TCE
From equation (3) we have
(Z)
+ YEEl°gPE + YEKlogPK + YELlogPL+ YQElogQ
We obtain qCi from
nCi
alog(AC) al
alogPE !logP
E
i
(28)
CESi
Let us now review the steps involved in estimating a translog model of energy
demand.
First the fuel share equations
$
i
+
ji
jijlgP
(29)
J
are estimated, and the estimated parameters are used to calculate partial price
elasticities.
Eac
1,
YiJ
O.
Y
These equations are estimated subject to the parameter restrictions
, and Zy
=Zy
=0
We also test the additional restrictions
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
Yij - Yji'
and
s
Yij
=
1,
YQi
0,
Next we add the homotheticity restrictions YQi =
0ij
.
Finally we test the restrictions that
O.
O, i.e. that the elasticities of
ij
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
Our objective here is to specify a dynamic translog cost function
short-run.
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
(30)
l(AQt-k)
K
and
yQQ
02klX (AQtk)
Here the parameters 81,
(31)
03
2 and 03 are estimated.
If 83 is not equal to zero,
Thus if long-run
then non-constant returns can exist even in the 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
specification of the lag distribution.
k
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.
This
cost
functicn.
in
the
diffrecnces
to
tea ch:ractcristics
!..:o1vxstha ue
o
of inter-cot..try
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
2
)-k
kQi
(AlogP
Thus if the data do not permit
(32)
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
1
of desired shares as follows:
,t
it
'
+
sSit-l
+
9
(s -s
ESi(Jtjt-1
where S
is given by equation (4).
j,t
(33)
Adding up requires that the sum of all
changes in shares be zero:
-
s(S
i
t tSi, -
(34)
0
so that
Z6 i(S* t
ij
Since the St
j
Sj
and Sj
=
(S**t'tE
J
'
- S
X6t-s) = o.
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.
'
This
c 2'
approach was suggested by L.
(36)
averman and M. Fuss.
where
is a vector
20
stant.
(ones),
6 is a matrix
(6ij), and
is an arbitrary
con-
Note that if the number of shares is greater than two, there are alterna-
tive constraints
2.4
of l's
on the
ij that can be imposed
to satisfy
(36).
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
22
United 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
Finally the logit model is easy to estimate and permits
make it still smaller.
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
ef i(xS)
Qi
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/S
j) = 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
Wall
2 1See
22
[
z :d .- Ir: -;i
] and iernLit
nd Savln
Baughman and Joskow t
[
"c.s 1foe
..
ore
], Fuss and Waverman [31],
See Joskow and Baug.man [421.
ui.eralag structures,see
j.
and Pindyck
[53].
-15In 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) = (ai-a4 ) + bi
b4
+ (i-c
4 )Q,
i = 1,2,3.
(40)
Note that these three equations must be estimated simultaneously, with b 4
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) = a i + biP i +
(41)
iSi,t-l'
The three estimating equations are then
og(Si/S4)
= .(ai-a
4) +
b4P
i
4+
CiSi t-l
Note that two lagged shares appear in each equation.
4 4t-1
2
(42)
The three equations must
again be estimated simultaneously, with both b4 and c4 constrained to be the
same in each equation.
-16-
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;
23
As before we use a Fisher "ideal" index (a geometric mean of a Laspeyre and
Paasche index numbers) as a single index of relative purchasing power. Our
purchasing power parities are binary index numbers with the U.S. as base country.
24
we do not adjust fua! qu.ntities
or prices by thermal efficiencies of
first, thr
e
no gocod cinmates
of thezal afficigr.zcs a;vl
iae.,
and second, there ara ochier economic' efficiency measures that could be equally important in affecting consumer demand. For
a further discussion of this issue, see Pindyck [ ], pages 25 to 28.
Thpt i,
utilization.
Thrn,
a.~re t-o r-o-n
for this;
-17-
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 lo P
Qi
k
k ikk
:(43)
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 secondorder 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
.
kj I ksk
Note that the restrictions on the yijk's are now that Yijk
k,
and that
Yik
i ijk -
Yi
ijk
= 0
for every country k.
44)
Yjik for each country
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
computational expense.
(and hopefully a resulting gain in efficiency) and
This trade-off is particularly severe given that all of
Ideally one would like to estimate a
our models involve systems with equations.
stochastic specification for which the error terms are heteroscedastic and autocorrelated 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 equations, but account for error correlations across equations.
In particular, we
use iterative Zellner estimation which (under the assumption of no heteroscedasticity or autocorrelation within equations) is equivalent to full-information maximumlikelihood 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
least as far as we can given the ccnstraints on our computer budget).
(at
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 2
1c
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 estimated
A
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
.,Zellnerestimation 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 available 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 supplements indicated by GWI, UN National Accounts or the International Labor Organization'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 Statistics.
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
25
The data used here are part of a larger international energy data base assembled
for use in this and several related studies. For a more detailed description of
that 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 replicate
or extend this stu-:dyor .ror;
through
the TROLL computer
st:-esof
svste-i of
the 'N"B.
cancn ccesste data dircct ly
-hir
-21-
of numbers of employees to number of operatives, for Canada, Italy, Japan,
Norway, Sweden, U.S.A. and W. Germany.
Industry.
Data is from the UN Growth of World
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 Industry.
(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 structures 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 service 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
qt
=
jJt [(l-d) tpj+l
j+l
1
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:
26
See also Hall and Jorgenson [35] and Coen' [22].
- (qNR(t)
RN - qNR(t-1) )
+ dRq(t)
PNRR(t) = R(t)qNR(t-1)
N
RN
PD(t) = R(t)qD(t-l) + dDqD(t) -
qD(t)
- qD(t-)
(
)
(47)
(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.
For the remaining countries it
[12,13,14,15,16,17,18].
was necessary to compute implicit asset price indices from gross fixed capital formation in current and constant units using national statisical yearRemaining depreciation rates
books, the U.N. or OECD National Accounts.
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 appropriate 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.:strvor Annral Ycar:boo"k Value added for France and Germany was
27
For West Germany the discount rate was used as the interest rate, since the
goveriument 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 revised data and clearly excludes chemical feedstocks.
These data series are
related to those in the earlier OECD publications via simple linear regressions, 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 erroneously 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 petroleum 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.
Results
Statistical
5.
a==_=.__=-====-----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.
we have not yet estimated dynamic versions of the translog model.
At this point
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.
P2
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From these we determine partial and total elasticities for fuel demand, and elasticities 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
vary across countries as in equation (44).
ij parameters to
We found the latter alternative to
involve a considerable reduction in degrees of freedom, and the resulting elasticities 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 population as the 1959-1973 data, i.e. whether the 1974 data point lies on the same
long-run cost function.
the 1974 data.
We therefore estimate models both including and excluding
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 restricted 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
We
parameter restrictions, this is well above the critical 1% level of 27.8.
therefore include
intercept dummy variables in all other versions of
(A model is also estimated using
the model.
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 intervals results in little change in the estimated parameters.
elasticities also do not change much
The resulting price
(compare Tables 4 and 7), so that we conclude
that we are indeed estimating long-run elasticities.
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%
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'C,
Estimated elasticities are shown in Tables 4, 5, 6 and 7, (again, with standard 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
The same elasticities are shown in Table 5 for all ten countries
time bounds.
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, elasticities 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 (
little across countries.
33)
that vary
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
This is not surprising.
smaller and in many cases insignificant.
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 between the short- and long-run.
we are estimating a static
This, however, is of little value given that
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
PARTTAI, FUEL PRICE
4 -
Count:rv DurramaVariables)*
(US & Canada Estimated Separately,
,,,,,,,--cllc--.----r---------·l
*(a)
_
0
0
-4
a:
I~ %.
I
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lqr o
C)
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---
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4
r %
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1959 - 1973
------
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rl
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1959 -. 1974
(b)
----
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9.4
94
C7
es
Z
TABLE 4 - PARTIAL FUEL PI:ICE EL,.ASTICITIES
(US
& CanandaEstimated
Separately,
_
I
(a)
Country__ Dummy
____Variables)*
__
1959 - 1973
(b)
1959 - 1974
_
m
oo%,
C-,
-
v v
.
.
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cc-v cc'
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2TASTICITIES
TABLE 5 - PARTIAT FUEL ,'T1C
O10 Countries Pooled, Country D:..y' Vriablce)*
0
co c oo
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---
m
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eq
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9,
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C7
'I
I
TABLE 5 -
PARTIAL
(CONT.)
FUEL PRP.ICE ELASTICITIES
(10 Countries Pooled, Country DunmyVariables)*
(a)
1959 - 1973
-
Cd
---
1t o
Co
aD It n
D
H-- H O
w
0
1959 - 1974
(b)
v
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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 consumption).
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 relatively 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.
tion
This is done by applying the estimated parameters to equa-
16) (note that the first-order parameters
and choosing
i will vary across countries),
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
(U.S.
00r-
C O 1-)(I
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Fuel Cice
Dr tw d Frr:m"Preferred"
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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
data.
28
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.
the cost function is homothetic (Qi
homothetic.
0, i
=
In column 1
K,L,E), and in column 2 it is non-
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.
28
Inclusion of 1974 data is not a question here; our derived capital price data
extends only through 1973.
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-44-
A second test for homotheticity is perfomned by estimating homothetic versions of the cost functions in columns 4 and 5 (the resulting parameter estimates
are not reported here).
non-homothetic model.
These models can then be compared to the corresponding
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 dunmmyvariables 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.
29
We also tested the hl-uethsee.sthat the y,. are all zero, i.e. that the rouzticn
I'm ., t st at_s cs are significant at the 1% level,
function is Cobb--Dv.I. ' .
so that this hypotr.:ses
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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)
tive, although small (0.61 to 0..86).
is posi-
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 substitution 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
elasticity.
31
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
30
Berndt and Wood [9], for example, found strong complementarity between energy
and capital. Griffen and Gregory [34], using pooled international data at
four-year intervals,
31
obtain results
ver silar to ours.
Our estimate is close to that found by Griff`tn and Gregory [34], who also used
international 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.
however, is significantly less than unity.
The output elasticity of energy demand,
Thus, even if energy prices remain con-
stant relative to other prices, as output increases there will be substitution
32
away 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
We attribute this to
to -2.34), while those for oil are small (-0.6 to -.56).
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.
the natural gas elasticities upwards.
This might have tended to bias
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 reversed - 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.
32
This result is not surprising given the data. Note from Table 2 that for
large-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)
.
q
1EQ
SCE
Country
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)
-.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)
Netherlands
Standard errors are computed based on constancy of shares and
prices at their mean values. The standard error of SCE is thus
computed from:
L,M
Var(SCE)
=
i(igP i) Var(YQi) + I
i=K,
and the standard error of
logPilogPjCovar
(QiyQ
Q~ j
)
Q is computed from:
VarOlEQ) = Var(SCE) + [2 logPE/SE + 1/SE]Var(YE
Q)
(QEyQL
+ (2 /SE)logPKCovar(QEYQK) + (2/SE)logPLCovar
)
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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
since certain coefficients are constrained to be the same across equations.
34
We
therefore use iterative Zellner estimation to estimate all of the models.
Our static logit models are of the general form
10
1
3In
(i/s4)
i
4
klai4kDk
+ biPi
i - b44 P 4 + ci4Q, i=-1,2,3
comparing these elasticities across countries, remember that they are dependent
on the shares
the largest
of enerv,y. and the fuel
sh.ares.
Thus Italy,
apan and Sed;len hv-:
valu.s oz 1C in
_ part bucause they have the largest
in the cost of cutout.
34
(49)
shares of er erv
Even without cross-equation coefficient constraints, simultaneous equation estimation is desirable in that insofar as errors are correlated across equations,
it yields more efficient par;eter
stim.ateS.
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-54-
where ai4 k = 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
tive prices and output.
's
rela-
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
+ dS
iSi,t-
bii
dS
d 4 ,t1'
b4 4
i4
112,3
=1,2,3
Note that this is not a Koyck adjustment model.
(50)
The coefficients d i 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 b 2 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 estimates might differ from these.
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1 5 (ont.)
Tab]
Elas.
Country
Price Elas.
(cont.)
Canada
Source
Estimate
K
nEE
0'.76, LL
-0.49,
nLE
KE = -0.05,g
0.55
= 0.05, nL
NetherlandsKK
nEE
-0.49,
-026,
-0.90
9 industrialized
KK = -0.18 to -0.38,
r LL =-0.12 to -0.27,
countries
LEE = -0.79
KE
six-country
composite
composite
to -0.80
0.13, XE
0.11
EE = -0.30
XleE
(h)
U.S.
elec: -0.66, oil: -2.75
gas: -1.30, coal: -1.46
(i)
U.S.
elec., S.R.: -0.14
elec., L.R.: -1.20
U.S.
elec., S.R.: -0.06
elec., L.R.: -0.52
U.S.
elec: -0.92, oil: -2.82,
-
Fuels own price
elasticities, partial
(k)
own price
elastici-
gas: -1.47, coal: -1.52
ties, total
elec: -0.74, oil:. -1.30,
gas: -1.30, coal: -0.48
Canada
Sources:
(a)
(b)
(c)
(d)
(e)
Berndt and Wood [9]
Halvorsen and Ford [37]
Fuss and Waverman [31], translog
Fuss and W-averman [31], generalized Leontief
Magnus
[46]
(f)
Gr.i,cn a-c Grcory
(g)
Fruss
(h)
Nordhuo [50]
(i)
(j)
(k)
[34]
3)]
Halvorsen
[36]
Mount, Chapman, and Tyrrell [49]
Griffcn [33]
(i)
(g
Table 15 - Al.ternative E'stimatesof Idustrfal
tion
1.01, a
a
Factor Inputs elasticities
of substitu-
KL
U.S.
CLE
tion
Source
Estimate
Country
Elas.
Demand Elasticities
=
-3.25,
(a)
KE
0.64
U.S. (2-digit
a
industries)
aKE = .48 to 2.88 (prod. workers),
ilE = -2.02 to 5.59 (non-prod.
= -1.03 to 2.02,
workers)
LE
KL = 0.72,
Canada
(b)
aKE = 0.42,
(c)
OLE = 1.70
aKL = 5.46, aKE = -11.91,
Canada
(d)
aLE =4.89
OKL = 0.30,
Netherlands
KE
(e)
-4.50,
aLE = 3.80
.KL= 0206 to 107.52,
9 industrialized
countriLes
Factor Inputs
price elastici-
= 0.72 to 0.87
K
U.S.
ties
0.44,
T1EE = -0.49,
L
-0.45,a)
KE = -0.15,
)LE = 0.03
U.S. (2-digit
-0.67 to -1.16,.
nKK -0.67 to -1.16,
industries)
(b)
nLL -0.28 to -1.55,
nEE = -0.66 to -2.56
nKK = -0.79,
Canada
(c)
LL = -0.45,
.]EE -0.36
nKK
Canada
-0.31, nLL =
0.77,
TEE = -0.59
!
,.,
,,,
,
,
!
i
,
.
,
,
.
,E
4.
Note that there is mixed evidence on the substitutability of energy and capital.
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 substitutability, 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.
of Magnus
Our estimate (about -0.8), together with those
(-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,
tained an estimate of -0.92 and Fuss [30)]-0.74.
here !Halvorsen [36] ob-
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.
There is less disagreement
this discrepancy will probably require further work.
over the elasticities for coal and natural gas.
An explanation for
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 provides means of estimating demand elasticities for aggregate energy use and for individual fuels.
In addition, by pooling international time series-cross-section
data we can obtain a sample large enough to provide low-variance estimates of essentially 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 apWe attribute these results to
pear to be substitutes, rather than complements.
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 outout 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 impact on the total cost of production.
This is due in part to cnergy's small s
production, and in part to substitution possibilities.
ihare
i:
-62-
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