INTERNATIONAL COMPARISONS OF THE RESIDENTIAL DEMAND
FOR ENERGY: A PRELIMINARY ANALYSIS*
by
Robert S. Pindyck
Massachusetts Institute of Technology
September, 1976
Working Paper
#MIT EL 76-023WP
*
This work was supported by the National Science Foundation under
Grant #GSF SIA75-00739, and is part of a larger project to develop
analytical models of the world oil market. The author is indebted
to Jacqueline Carson, Ralph Chang, John Donnelly, Daniel DuBoff,
Ken Flamm, Ross Heide, Kevin Lloyd, and Eric Rosenfeld for their
excellent research assistance in many aspects of this work, to the
Computer Research Center of the National Bureau of Economic Research
for assistance in some of the computational work, and to Mel Fuss,
James Griffen, Edwin Kuh, and Leonard Waverman for comments and
suggestions.
INTERNATIONAL COMPARISONS OF THE RESIDENTIAL DEMAND FOR ENERGY:
A PRELIMINARY ANALYSIS
I.
Introduction
This paper reports on some initial results from an econometric study
of the world demand for energy.
The long-run objectives of this study
are to estimate the determinants of total energy demand and interfuel substitution in the residential and industrial sectors of about twelve industrialized
countries.
Here we concentrate on the residential sector, and examine some
very preliminary estimates of inter-country differences in the structure of
demand.
In modelling the residential demand for energy we assume that consumers
make two decisions in purchasing fuels.
First, they decide what fraction of
their total budgets will be spent on energy, as opposed to such other consump2
tion categories as food, clothing, etc.
Next, with the amount of money to be
spent on energy taken as given, consumers decide which fuels to purchase, i.e.
the fractions of energy expenditures allottedto oil, natural gas, coal, and
electricity.
Thus we assume (and we will empirically test this assumption) that
consumers' utility functions are separable between energy and other commodities,
i.e. that expenditure shares on fuels may depend on total energy expenditures,
but are independent of the expenditure shares for other consumption categories. 3
1
This study is itself part of a larger project to develop analytic models of the
world oil market.
2
For now we do not treat energy as a derived demand determined by the stock of
energy-using appliances
(in fact durable goods are a separate consumption
category). In the dynamic models to be discussed later the effects of changes
in appliance stocks will be included implicitly.
3
We thus have a "utility tree" along the lines described by Strotz [60], and the
marginal rate of substitution between any two variables in the class of energy
expenditures is independent of the expenditure on any other consumption category.
2
Estimating the demand for energy requires a model for the breakdown of
total consumption expenditures.
A number of such models have been constructed
by others, some of them additively consistent (in terms of shares adding to
one) and some inconsistent. 4
Typical model choices have included the additive
logrithmic model, the linear expenditure system, and the additive quadratic
model.
5
Usually these models have been estimated using time series data for
single countries, but in some cases cross-country comparisons have been made
using pooled time series-cross section data for a number of countries.6
We will extend this work by estimating both static and dynamic versions
of the indirect translog utility function with pooled data.
The advantage of
the translog function is that it is a general approximation to any utility
function, and therefore it does not a priori impose constraints of homotheticity
and additivity.
We will thus be able to obtain unrestricted estimates of own-
price, cross-price, and income elasticities; this is important as there is reason
to think, for example, that the income elasticity for energy demand differs from
unity.
In addition, we will be able to test whether homotheticity, additivity,
stationarity, and for that matter, utility maximization are reasonable assumptions.
We will also estimate alternative models for the breakdown of consumption
expenditures.
In particular, we will repeat Houthakker's 1965 study [31] using
more recent data and our own consumption categories.
This will permit us to
explore some basic issues in the pooling of heterogeneous data, and to look for
differences between short-run and long-run elasticities that might suggest
priori
specifications for dynamic models.
4
For an overview of models of consumer behavior see Brown and Deaton[ll] and
Phlips
[52].
5
See Houthakker [30], Pollak and Wales [55], Houthakker and Taylor [32], Phlips[52],
and Theil [63].
6
See Houthakker [31],
and Goldberger and Gamaletsos [23].
3
In estimating models of consumption expenditures we wish to explore
the extent to which higher energy prices might reduce the total consumption
of energy.
Although energy prices did not increase substantially during the
period covered by our data (pre-1974), there is enough cross-sectional variation in prices to allow us to obtain price elasticity estimates, as well as
estimates of income elasticities.
In addition, by estimating models using
country dummy variables and/or alternative groupings of countries we hope
to determine the extent to which elasticities vary across countries.
In estimating the demands for individual fuels we will also test a number
of alternative model structures.
We will again use our pooled data to estimate
both static and dynamic versions of the indirect translog utility function.
In
doing so we can obtain unrestricted estimates of own-price, cross-price, and
total expenditure elasticities, and test for homotheticity, additivity, and
stationarity.
Also, by estimating a translog model that includes both non-
energy consumption expenditures and fuel expenditures we can test for separability.
We will also use the multinomial logit model to break energy expenditures
down into fuel shares.
One advantage of the logit model is that it is relatively
easy to estimate; as long as the share data represents aggregated samples of
individual decisions (i.e. average shares for a large number of consumers) rather
than individual decisions, ordinary or generalized least squares can be used.
Another advantage of the logit model is that it allows considerable flexibility
for working a dynamic structure into the specification.
disadvantages, however.
share data.
The logit model also has
Estimates become inefficient when there are zeros in the
In addition, all cross-elasticities for a given own price are equal;
4
as Hausman and Wood have shown [26], they are the sum of the price elasticity
for total expenditure minus the own price elasticity weighted by the share.
This fact that cross elasticities are determined by total and own elasticities
is restrictive, but much less so than the restrictions inherent in the linear
expenditure system, additive quadratic model, and other "consistency" models.
In estimating demand models for individual fuels we will explore the extent
to which fuel shares shift in response to price changes and changes in total
energy expenditures over both the short- and long-run.
Again, there should be
enough variation in prices through the combined use of time-series and crosssection data to obtain reasonable elasticity estimates, and determine the extent
to which elasticities vary across countries.
As one might expect, in work like this we are continually bound by data
limitations.
For many countries there is no good data available for some or
all of the variables of interest to us.
For other countries data exists, but
obtaining that data can be an extremely time consuming and laborious task, so
that choices had to be made as to which data were to be collected.
These data
limitations were one of the factors that helped define and delimit the modelling
approaches used here.
In particular, it necessitated restricting our detailed
analysis of demand to a small set of countries.
7
Even for these countries,
however, the quality of the data varies, and compromises had to occasionally be
made.
8
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 MIT World Energy Demand Data Base." 9
7A much less detailed model of the demand for petroleum products is being constructed for a number of "secondary" countries for which only partial data is
available; the results of this work will be described in a forthcoming paper.
8
For example, one of our consumption expenditure categories is "food, alcohol,
and tobacco." For some countries a price index is available only for food, and
this index was used since food is by far the largest component of the category.
Working Paper No. MITEL76-01lWP, MIT World Oil Project, May 1976.
5
We stress that the results presented in this paper are preliminary and
in many ways incomplete.
Only some of our alternative model specifications
have been estimated at this point.
This report, however, does serve to provide
initial elasticity estimates, explores problems in the use of pooled international
data, and provides a framework for continuing modelling work.
Hopefully it
will also provide a source of feedback in the form of comments, criticisms,
and suggestions that can help correct deficiencies in the present approach.
In the next section we outline alternative specifications of alternative
models of residential energy demand, and discuss the characteristics of each
specification.
Section 3 discusses some methodological issues in the estimation
of energy demand models using pooled data.
These issues include the use of
purchasing power parities to make international comparisons, the question of
accounting for thermal efficiencies in the use of energy consumption data, the
formulation of an aggregate price index for energy, and the use of alternative
estimation methods.
Section 4 describes some of the characteristics and limita-
tions of our data, and Section 5 includes the statistical results.
6
2.
Alternative Specifications for Models of Residential Energy Demand
As explained above, all of our models of residential energy demand involve
a two-stage approach where first consumption expenditures are broken down into
energy and other consumption categories, and second energy expenditures are
broken down into expenditures on fuels.
We begin here by reviewing the proper-
ties of the indirect translog utility function with a time trend and discuss its
application to both stages of the residential model.
Next we describe some
alternative dynamic specifications of the indirect translog utility function.
As we will see, these specifications will permit us to explicitly include stock
adjustment or habit formation effects.
We then discuss the multinomial logit
model and its application to the estimation of fuel shares.
Finally we discuss
alternative model specifications, including simple models.
2.1
Use of the Indirect Translog Utility Function
The indirect translog utility function is a second-order approximation to
any indirect utility function.
The indirect translog function with time-varying
preferences, introduced by Jorgenson and Lau [37], has the form:10
log V =
+ ZEailog(Pi/M
) + att + Izi log(Pi/M)log(Pj/M
ij
+ EZi t log(Pi /M)' t +
2
Stt
)
~
~~~~~~~~(1)
2
When the indirect translog function is used to model expenditure shares for
energy and non-energy consumption categories, P i is the price index for consumption category i and M is total consumption expenditures.
10
When this function is
The indirect translog utility function without time was introduced by Christensen,
Jorgenson, and Lau [15]. The homothetic form of the indirect translog function
was used by Christensen and Manser [14] to study consumer preferences for food,
and the non-homothetic form was used by Jorgenson [35] to study a three-category
breakdown of consumer expenditures in the United States. Berndt, Darrough, and
Diewert [70] demonstrated empirically that the translog specification is more
robust than other generalized functional forms such as the generalized Leontief
or generalized Cobb-Douglass utility functions.
7
used to model fuel shares, Pi is the price of fuel i, and M is total
expenditures on energy.
The indirect translog function implies the budget
share equations:
si -
PSX
M
aj + iBjilog(Pi/M)+
++*I=l
j
5a
+
ilog-
t
.....
j~~~~t
i/M) +
n
(2)
m
·-
where X. is the quantity consumed of category i (or fuel
),
t is a time trend (equal to zero at the beginning of the estimation period),
and
i
Oi
'-
'Oki' 8Mtk
Note that the parameters ao,
kt
t, and Ott in equation (1) do not affect the
utility-maximizing quantities consumed, and therefore cannot be identified.
In addition note that the budget constraint implies that
S
=
1, so that only
(n-l) of the share equations need be estimated to determine all of the parameters.
The budget share equations are homogeneous of degree zero in the parameters,
and therefore a parameter normalization is required for estimation.
normalization
= Za
= -1.
We use the
A number of parameter restrictions are also
required if the share equations are indeed based on utility maximization.
particular, the parameters
equations.
and
t must be the same in each of the n
In
share
Since there are (n+l) parameters involved, and (n-l) equations are
estimated, this implies a total of
(n+l)(n-l) restrictions.
Also, we assume that
that log V is twice differentiable in its arguments, so the Hessian of log V
must be symmetric.
Bi =i
,
This implies the following
i
j, i,j =
, ..., n
(n-l)(n-2) symmetry restrictions:
(3)
8
There are an additional (n-l) restrictions resulting from the fact that the
parameters of the n
th
equation are determined from the parameters of the first
(n-l) equations and the definitions of
of parameter restrictions is
Mi and
Mt.
Thus, the total number
n(n-l).
There are other restrictions that might be imposed on the indirect translog function, and tests can be performed to determine whether such restrictions are supported by the data.
We will test some of these restrictions
in this work, so we list them here.
The indirect translog function is stationary if preferences do not
11
change with time.
Stationarity implies that the parameters
equal to zero, jail,. . ., n.
it are all
it
12
In estimating the consumption breakdown model, we might wish to test for
groupwise separability between energy and the other consumption categories.
Letting P1 and S1 be the price index and expenditure share for energy, and
P2' P3
.
' Pn and S2, S3 , . .. , S
be prices and shares for the other
categories, separability would imply that the underlying indirect utility
function can be written as
logV
= F(logVl(P2 /M,P3 /M, .
. .,
t), P1 /M, t)
(4)
If the underlying indirect utility function is groupwise separable, then the
following restrictions must hold:13
A dynamic translog function, in which long-run elasticities differ from
short-run elasticities, may still be stationary as long as the elasticities
themselves do not depend on the particular time in which prices or income
change. This is discussed further later.
12
13
Note that stationarity is equivalent to explicit neutrality. An indirect
utility function is explicitly neutral if it can be written as
log V = log V (P1/M, P 2 /M...,Pn/M) + F(t)
See Jorgenson and Lau [37] for a derivation of these restrictions.
9
B12
where
1
:
813
2
is a constant.
l
3,
in
.
Pl
(5)
Even if the underlying indirect utility function is
groupwise separable, the translog approximation need not be.
Explicit group-
wise separability ensures that the translog approximation is also groupwise
separable.
This requires the additional restriction that p1 = 0.
We will also estimate share equations based on homothetic indirect utility
functions.1 4
Under homotheticity the budget shares S. are independent of total
J
expenditures M.
This implies that the income elasticities of demand for
every commodity are the same and equal to unity.
The underlying indirect
utility function is homothetic if
=
where a is a constant.
oa.
j
=
1,
. .. , n
(6)
Explicit homotheticity will ensure that the translog
approximation is also homothetic, and this requires the additional assumption
that
and
a = 0.
If the indirect utility function is explicitly homothetic
t = 0, then it is also homogeneous.
Finally, it is straightforward to test for explicit additivity, since
a necessary and sufficient condition for explicit additivity in the commodities is that the indirect translog function is explicitly groupwise seperable
15
in any pair of commodities from the remaining commodity.5
constraints
1
are that
ij = 0, i
Thus the parameter
j.
is homothetic if it can be written as logV = F(logH(Pl/M,
where H is homogeneous of degree 1.
.
.
, Pn/Mt)t)
15An indirect translog utility function is explicitly additive if it can be
written in the form
log V = log Vl(P1 /Mt) + ... + log Vn(Pn/M,t).
10
We can test restrictions using a simple chi-square test.
The appropriate
test statistic is
A
-2
where '1
I and
J6I
log A
=
^~~
n (logQ r
- logl u)
(7)
are the determinants of the estimated error covariance
matrices for the restricted and unrestricted models respectively.
This
statistic is distributed as chi-square with degrees of freedom equal to the
number of parameter restrictions being tested.
It is important to remember that there are only certain ranges of inputs
over which the indirect translog utility function is a meaningful approximation to the underlying utility function.
Consider, for example, the marginal
utility of income (or of total expenditure),
V agV
- .
M alogM
=
X = 3V/M:
VP (ai+8Ealog4
--
M
i
Sij
4
+~
a.t)(8
ii
)
(8)M
The a. sum to -1 by the normalization, while the 3.. can be positive or nega1
1J
tive.
If some
ij are positive, then as M becomes zero X can become negative,
ij
and if some 8ij are negative, then as M becomes increasingly large X can become
negative.
Thus, there are ranges of input space for which the translog ap-
proximation may not be meaningful.
It is important therefore to check esti-
mated translog models by determining whether the marginal utility of income
is positive over the range of historical (and forecasted) input data.
After estimating models for the breakdown of consumption expenditures
it might be useful to compute
risch's welfare indicator for purposes of
cross-country comparisons, e.g. to determine the relative effects on different
countries of higher energy prices.
This indicator is simply the income
elasticity of the marginal utility of income, i.e.
nXM=
logX/alogM.
For
11
a utility function that is well-behaved over the entire input space (which
the translog is not)nkM
would range from a large negative number (when M
j
(i
.i-1
is zero) to zero (as M approaches infinity).
Taking the log of equation (8)
16
and differentiating, we have for the indirect translog function:
SE 8.
~~~~
- 1
- n>M =n~~~
=~~~
i
i
i i
Note that as M + 0, nqM +- l,
(9)
C
i -i
M + Z.t
log M
+ ZiS
ij ij
i itt
so that a very small M is clearly out of the
The same is true for M + o.
"meaningful" range.
We also need formulas for the calculation of income and price elasticities.
logX /logM,
The income elasticity of demand for good j, %M =
is found by
multiplying the share equation (2) by M/P. and differentiating:
(10)
1+/+
1+
- i
i ji/3
iMi
aM
ilO)+
+
a + jM
E..t3logPi +
Note that this is also the formula for the expenditure elasticity alog(PjX.)/alogM.
alogX/alogPjis
The own price elasticity njj
J//S i- «iM
a + 8Milo- +
j
= -1+
(ll)
t
i
and the cross price elasticities nji
=
logX/logPi
are
j/ji -6
;i
=
/S
-
Milog
aM +
S
(12)
+
m.t
i
16The same equation also applies to the direct translog utility function.
12
These elasticity formulas can be applied to each stage of our two-stage
demand model, but they cannot be used to determine the total effect of a change
in price or income on the demand for a particular fuel.
If the price of oil
changes, there will be a change in total expenditures on energy, and this will
also affect the demand for oil.
d log X/d
log P
is given by
r]
jj
and X
J
i
~J
J
where P
The total own price elasticity
=
P. fx
j ) ME PE
-j I
X. aP.
DhE
J
(13)
aP.J
JJ
PE
are the price and quantity of fuel j, ME is expenditures on
J
energy, and PE is the price index for energy.
Thus, to determine the total
elasticity we need an expression for the price of energy in terms of the
prices of individual fuels.
not determine P
E
Since fuels are not perfect substitutes we can-
as a simple weighted average of the fuel prices.
Instead
we view PE as the cost of producing heat from fuel inputs, and use a translog
cost function with constant returns to model this "production" process:
log PE
=
Yo + Z Yi log P
Pilog Pj
+ Z Z Yij log
i
(14)
ij
This is an energy price aggregator, and can be determined up to a scalar
by estimating the share equations1 7 Si
Y
+
Y
log P
p*
Given equation
(14) for the price of energy, we have
aPE
EaP'
ap .
17
PE
S
Sj
(15)
We will discuss the energy price aggregator in more detail later.
13
*
We can thus compute TjI from the fact that
ax.
X.
DP.=
P
i
where Tn
ji
(16)
3
is the partial own price elasticity for fuel j given by equation
(11),
ax.
X.
(17)
__TI
DME
where Tj
ME
i
}7
is the expenditure elasticity for fuel j given by equation(10),
and
3E
aPE
X (1 + nEE)
(18)
where XE is the total quantity of energy consumed and nEE is the partial
own price elasticity of energy consumption.
(17), and (18) into
Now substituting (15), (16),
(13), we have
*
nj
njj
+ njME(l+nEE
(19)
We can similarly compute the total cross price elasticity nji from
*i
'.Pax.
'"
i
LaP
ax. aE ap
aiI aPE aPi]
Ti
+ SinjME(1+nEE) (20)
and the total income elasticity njM from
*
_
'jM
Note that since
M
X
aXj
E
aM aM
ME/aM = (ME/M)nEM, where
(21)
EM is the income elasticity of
energy expenditures, we obtain
*
TI jM
=TI
J ME "71 EM
(22)
14
2.2
Dynamic Versions of the Indirect Translog Utility Function
A problem with the translog demand models described above is that they
do not explain differences between short-run and long-run elasticities.
Even when the time trend is included in the indirect utility function, the
model is not really dynamic - tastes can change slowly over time, but there
is no dynamic (lagged) response in demand to a sudden change in price.
Thus
adjustments (while possibly non-stationary) were assumed to occur instantaneously.
There are two basic approaches that can be used to introduce dynamic adjustments into the translog utility function.
The first approach involves specifying
the translog approximation to the utility function (direct or indirect) to
include lagged quantities, prices, or shares.
The advantage of this approach
is that "adding up" is always preserved in the resulting share equations without
the introduction of additional parameter constraints.
The disadvantage is that
the translog approximation makes the dynamic specification somewhat arbitrary.
As an example of this approach, we could write the indirect translog
utility function as:
log V =
0 + E'1log(Pi/M) +
Z
log(P i/M)lg(P J/M)
(23)
+ Edilog(Pi/M)Di,
t-l
i i g(i/
~
where Di t
1
is a lagged term in price, quantity, or share that is considered
18
an exogenous input to the determination of current share.
18
Logical choices for
We are assuming that consumers determine their budget shares via
utility maximization, i.e. they maximize utility at each instant
ignoring the future, rather than maximizing the sum over time of
utilities. The Di t 1 (together with current prices and income)
static
of time
discounted
simply
represent the current state of the world. As shown by Hoel [28], even in
a static model dynamic utility maximization can result in a different
marginal utility of income.
15
Diitt-1 would be the quantity Xi t 1 or the share Si't-19
i t 19
to equation (23) yields the share equations:
Si
ad + d D
=
j
+
M
Applying
Roy's
identity
p l. igRysiett2
t-l + Ea1 ilog(P /M)
EdiDit-1l +
Milog(Pi/M)
,j,...,
where aM and OMi are defined as before.
(24)
Note that unless all of the di are zero,
the homothetic form of equation (24) - for which the ai
nonlinear in the parameters.
are zero - is
As a result, estimation of (24) can be costly,
even under the assumption of homotheticity.
The shares in (24) will always
add to one, however, even if lagged shares are used as the D
tl'
and -
assuming that the errors are not serially correlated - the parameter estimates
will be invariant to the choice of share that is dropped.
A second approach that can be used is to introduce the dynamic adjustment
directly into the share equations.
This has the advantage of facilitating the
use of simple and intuitively pleasing adjustment mechanisms.
It has the
disadvantage that "adding up" will not be preserved unless additional (and
possibly highly restrictive) parameter restrictions are introduced.
We consider two examples of this approach.
is assumed
to adjust
to a desired
level:
Xi t = Xit 1 + 6i(X-Xi
t_
where X
i,t
In the first, each quantity
1)
(25)
is the desired quantity of commodity i
as determined from static
utility maximization, and 6i is an adjustment parameter.
equations
P
Sj
Jil
S
SJ
+ (1-
t/M(
t
)
[j
Jil=-1jPj/t
j,t-1
t-l/t
19
This yields the share
(26)
The form of equation (23) using lagged quantity Xi t1 was suggested by
Manser [44,45], who applied it to the estimation oftfood demand.
20See [57]. The identity is:
PiXi
M
=
i
alogV/a1ogPi
alogV/alogM
16
or, using the indirect translog function for St
S
i.ii + (1-6j)Sj
i
J=t a ++ Ea
Ilog(P
1 /M) +
iMi
t
ci
J t'l[Pj,t-l/Mt-
1
]
(27)
M
The parameters of the share equations (27) are estimated subject to the
constraints
*
Ea S
A
are the same in each equation, and
1.
k /6_
k-
Note that the shares St
need not add to one.
Adding up can be imposed
by estimating only n-I of the share equations, and determining the parameters
th
of the nth equation from ESt
It
= 1, but the estimated parameters will depend
on the particular equation that is not estimated.
Despite this deficiency,
however, the specification of equation (27) permits the introduction of dynamic
adjustments in a simple and appealing manner.
~~
Alternatively, we can assume that the shares adjust to the desired shares
as follows:2 1
Si,t
Si,t-
+
S 5 +
ij(t-sjt-l)
(28)
Adding up requires that the sum of all changes in shares be zero:
i
(Si
i
it-
,t)
=='
t-
1)
si,t-z)
(29)
so that
iij (Sjt
Since the
=
,t Cip-
t-630
0
t and Sj,t-l1 sum to one, this equation implies the necessary
condition that all of the columns of the matrix (ij
sum to the same arbitrary
constant, i.e.
~~~.·
21
This approach was suggested by Leonard Waverman in the context of dynamic
adjustments in the translog production function.
17
i' 6=
c'
(31)
is a vector of l's (ones), 6 is the matrix (6t),
where
22
constant.2 2
and c is an arbitrary
Note that if the number of shares is greater than two, there are
alternative constraints on the
ij
that can be imposed to satisfy (31).
Note
ij
also that (31) implies that 6 cannot be diagonal unless the adjustment
coefficients for every share are the same, so that the adjustment of the
th
it h commodity share would generally not depend only on that share, but would
depend on other shares as well.
2.3
Multinominal Logit Models for Fuel Choice
Multinominal logit models have already been used to study the breakdown
of energy consumption into demands for fuels in the United States23 and
24
Canada.2 4
Although the logit model is not based on assumptions of utility
maximization, 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 a variety of alternative dynamic specifications.
22
2
For a discussion of adding up conditions for more general lag structures,
see Wall [65] and Berndt and Savin [9 ].
23See Baughman and Joskow [6 ]
24
See Fuss and Waverman [68].
18
We can write the logit model for the four fuel breakdown (oil, gas,
25
coal and electricity) as follows:2 5
Qi
=
efi(Xa)
QT
Qe4fi(Xo)
T'
ef x )
(32)
e
where Qi is the quantity (in
tcals) of fuel i, QT
EQi, 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(Qi/Q)
=
log(Si/S)
= fi(x)
- f((33)).
Note that only three equations are estimated, since the parameters of
the fourth equation are determined from the adding up constraint.
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
translog price aggregator described earlier.
Other attributes may include
per capita income, average temperature, and lagged quantity 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
25
In effect we are assuming that consumer preferences are represented by a choice
index which for the ith fuel, has the form fi(x)+ei(x), where i is an error
term. Then the probability that a consumer would choose fuel i is
Pi
=
Prob[fi(xo)+
i(x) > fj(x)+
cj(x)]
for i # J.
If the error terms
(x) are independently and identically distributed with
the Weibull distribution
Prob[ei(x)
<
]=
e
-e
then the probability that fuel 1I will be chosen is given by equation (32). For
f!lrthiert1srcusNLon see MFa(Iddeti (
1
.CtC And MCFilddn
1191, (COx 161.
Theil [61, and Chapter 8 of Pindyck and Rubinfeld [].
For an interesting
application to aggregate demand analysis, see Park [51].
19
= Pi/PE'
linear functions of the relative.fuel prices P
where PE is the
aggregate price of energy, as well as income Y and temperature T:
fi(xB) = ai + biP
(34)
+ CiY + diT
This yields the three estimating equations
(c-c)Y
i (ai-a
i4 44) + biPi
i i - bP
4 4
4 + (cgc4
log(S /S)
+ (i
(dd)T,
4)
i=1,2,3 (35)
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 habit
formation or stock adjustments) is to include the lagged share:
+ biPi + ciSit
fi(xB) = a
1.(36)
The three estimating equations are then
log(S /S)
= (ai-a4 )+
biPi
-bp bt,
+ ic
j
1 -
Note that two lagged shares appear in each equation.
i=1,2,3. (37)
The three equations
must again be estimated simultaneously, with both b 4 and c4 constrained to
be the same in each equation.
In Canada and the United States very little coal is used in the
residential sector, and coal shares are generally less than one percent.
In these countries, the residential fuel choice is effectively between
three fuels, and we therefore use a three-fuel breakdown for Canada and
the U.S., and a four-fuel breakdown elsewhere.
2.4
Other Models of Residential Energy Demand
When working with pooled cross-section time series data it is often
difficult to separately identify short-term and long-term effects, and
determine the relative contributions to each from the cross-section versus
time series variation in the data.
This can be particularly true in the case
20
of the translog function, where nonlinear estimation is
involved.
It may
therefore be useful to also estimate simple log-log demand equations for
expenditures on each consumption category and each fuel.
Houthakker in his
1965 study [3i] separated short-run and long-run elasticities by running
separate regressions across countries and across time.
It will be useful to
repeat Houthakker's study for our own consumption date, and also apply the
approach to the estimation of fuel demands.26 In addition, we will specify a
dynamic version of Houthakker's basic model that should enable us to isolate
the cross-section versus time series contributions to lag adjustments in demand.
The basic demand equation for commodity i
logqijt
where j
=
ai + logYjt + Yilog
Pi + t
is the country index and t
on commodity i
is
the time index, q
at constant prices, y
constant prices per capita, and Pi
+
iJt
(38)
is per capita expenditure
is total consumer expenditure at
is the relative price of commodity i.
These equations can initially be estimated using simple weighted least
squares, e.g. dividing each observation by the population of the country
in the year concerned.
Short-run and long-run effects can be identified by estimating "within
country" and "between country" regressions.
regression
26
The "within country" (short-run)
is
Houthakker's consumption breakdown had only five categories: food, clothing, rent, durables, and "other." We use our own consumption breakdown for
which energy is a separate category. Goldberger and Gamaletsos [3] also
reestimated Houthakker's log-log demand functions, but using pre- 1961 data
and the same categories that Houthakker used. What is more interesting is
their estimation of a linear expenditure system using the same data, and
the comparison of the two demand systems. The approach was also used by
Gregory and Griffen [21] to identify international differences and intertemporal change in industry structure.
21
log
j
log
-
(log
=
Yjt - log yj) +
i(1og PiJt
-
log Pij)
(39)
and the "between country" (long-run)regression is
log qij =
ilog Yj + Y
1 °g
o
Pi
+
i
Here the bar represents averaging over time.
(40)
Note that the "within country"
regression is pooled, while the "between country" is purely cross sectional,
and that deviations from means (over time) are used in the "within country"
regressions in order to eliminate long-run effects.
Equation (39) can also
be run for each country separately, in order to determine how elasticities vary
Alternatively, a pooled regression can be performed, with a
across countries.
multiplicative district dummy variable introduced to one coefficient at a time, e.g.:
= 10ijDj(log Yjt - log yj) + Yi(log Pijt - log
- log q
log qit
(41)
Pij
Equations (39) and (40) can also be estimated in first-differenced form.
This crudely reduces trend effects, and also eliminates problems associated
with the arbitrary choice of purchasing power parities.
version
is:
Alog qi
- Alog q
=
.~~~~~~~.
(
log Yjt - Alog yj) +
The "within country"
i(Alog Pijt
-
Alog
ij)
(42)
and the "between country" version is:
Alog qij = 8iAlog Y
(43)
PiJ
+ YiAlog
We can also specify a dynamic version of (39).
We assume that demand
qit depends not only on price and income in period t, but also on a "state
variable"
it:
log qit
If the demand is
be negative.
=
ai + blog
y
+ cl°g
Pit+ dil°g sit
(44)
for durables, then sit will represent a stock, and di should
If the demand is for a non-durable commodity, then sit will
22
represent a "habit" level, and di should be positive.
The dynamics of sit can
be expressed as
=
Alog sit
(45)
wil°g Si,t-
log qit
To obtain the demand equation
where wi is effectively a depreciation rate.
rewrite equation (44) as
1
sit = d-[log q
ai
clog
bilog y
(46)
it
which can be substituted into equation (45):
Alog sit
wi
og qit
di [log qi,t-1
ai -
bil°g Yt1-
]
c log
(47)
Now first-difference equation (46):
Alog it =d
[Alog qit
-
biAl°g yt
-
(48)
ciAlog Pi1
Substituting this into (47) and rewriting, we have the estimating equation
log q
=
it
0
+
+
11o g,
1_1
12
Alog y
'2
+ a31g
3 lo
Yt
~t-1
-
Al g
4 Alo
+
t
a
g
c 5 lo ~it-1'
(49)
This equation can also be estimated within countries and between countries.
If this results in differences in the estimated value of a,
it would indicate
that adjustment response is not constant over time.
3.
Methodological Issues in the Estimation of Residential Demand Models
There are a number of problems that make estimation of the models described
in the previous section less than straightforward.
This is due in part to the
fact that pooled international data are being used to obtain estimates, and in
part to the nature of the models themselves.
First, the comparison of expenditures
or prices in different countries requires valuing different currencies in terms
23
of a common unit.
Although using purchasing power parities for this purpose is
probably more desirable than using official exchange rates, the choice of a
particular index is not always clear.
Second, a choice must be made whether
to value energy quantities in gross or net terms, i.e. whether to adjust for
thermal efficiencies of different fuels.
Next, an energy price index must be
obtained that accounts for fuel choice differences across countries.
Finally,
there are a number of econometric issues associated with the estimation of our
models.
3.1
We examine these problems in this section.
Use of Purchasing Power Parities
Since all of the price and expenditure data for each country in our sample
aremeasured in terms of the local currency of that country, a method is needed
to convert these numbers into common units.
One method which has been used by
27
a number of researchers is to simply use official exchange rates.27
This can
be misleading, however, since official rates can differ considerably from
equilibrium exchange rates, and tariffs, quotas, subsidies, and other controls
can result in price structures that differ considerably from relative international
prices.
Alternatively, one could attempt to identify "free market" exchange
rates between individual countries over time periods thought to reflect
equilibrium conditions, e.g. during which trade balances were near zero.
Even
under free trade, however, equilibrium exchange rates only reflect the price
equalization of internationally traded goods, which for most countries repre-
27
This approach has been used by Adams and Griffen [1 ], and Goldberger and
Gamaletsos
[23].
24
28
sent a small subset of all market goods.28
A better approach is to use purchasing power parities (PPP's) to convert
national currencies to some base currency.
Purchasing power parities can be
obtained explicitly by making binary comparisons between a base country (e.g.
299
the U.S.) and various other countries, using a fixed set of quantity weights.
The problem, of course, is that two sets of price index numbers (Laspeyre and
Paasche) can be obtained depending on whether base country or other country
weights are used.
In this work we use a Fisher "ideal" index (a geometric mean
of these two index numbers) as a single index of relative purchasing power.
30
We use purchasing power parities by consumption category calculated by the
German Statistical Office (Statistiches Bundesamt [59]) by means of detailed price
comparisons.
31
These are binary index numbers with Germany the base country,
and we use Germany as a "bridge" to convert to the U.S. as base
country.3 2
28
As Chenery and Syrquin [12] point out, the relative prices of non-traded
goods can be expected to increase with real per capita income, so that the
use of official (or "free market") exchange rates leads to an underestimate
of the purchasing power of the currencies of lower income countries.
29
Purchasing power parities can also be obtained implicitly by dividing a
nominal national currency estimate of national product (or one of its components) by a base currency estimate of the same national product. This
procedure was used recently by Lluch and Powell [].
For a general discussion of explicit purchasing power parities, see Balassa [ ] and Allen [3 ].
30
The use of a Fisher "ideal" index is suggested on theoretical grounds by
Samuelson [68] and on empirical grounds by Kloek and Theil [36].
31
Binary purchasing power parities were more recently calculated by Kravis
et al. [39], but for only a subset of the countries in our sample.
32
Note that although binary PPP's permit us to make a transitive international
ordering of purchasing powers, this ordering is not invariant with respect to
the choice of "bridge" country. Kravis et. al. [39] also calculated multilateral PPP's by means of a regression model that estimates the purchasing
power parity for a single category of expenditure as a function of all other
international price ratios. Again, some of the countries in our sample are
not included in the Kravis study.
25
The resulting parities apply to a base year, but we must construct intertemporal
indices to deflate our time series.
We do this using implicit price indices for
each consumption category in each country, thus constructing an implicit ratio
of relative intertemporal purchasing power in terms of a base year numeraire
(normalized so that 1970 is our base year).
The resulting base year purchasing
33
power parities for each consumption category are shown in Table 1.
3.2
"Gross" versus "Net" Energy Consumption
As pointed out by Adams and Miovic
[2], alternative fuels are not equiva-
lent on a calorific basis as a result of the differing thermal efficiencies of
energy consuming equipment.
Since more efficient fuels are substituted for
less efficient fuels over time, the measurement of an overall "energy elasticity"
(i.e. the percent change in energy use associated with a 1 percent change in GNP)
34
will yield a larger number if thermal efficiencies are taken into account.3 4
This has led some individuals to suggest the use of "net" energy consumption
(adjusted for thermal efficiencies) rather than gross energy consumption in the
estimation of demand models.
Nordhaus
In recent studies of energy demand, for example,
['8, Adams and Griffen [£ ], and Fuss and Waverman [68], made the
assumption that within each sector fuels are perfect substitutes, so that
(given equal levels of non-fuel cost) interfuel competition is determined by
relative net prices of fuels.
Net consumption and net price are given by
33
For those countries also covered by Kravis et. al., our numbers are at all
times within 10 percent of the 1970 Kravis numbers.
34
Adams and Miovic estimate an overall energy elasticity for the U.S. and
several European countries of about 0.8 when gross energy quantities are
used, and about 1.0 when energy quantities are adjusted for thermal
efficiencies.
26
Table 1 - Base Year (1970) Purchasing Power Parities
Consumption
Category
Transportation
ConsumptionComnctn
Country
Consumption
~~-
.
Belgium
-
Apparel
Food
Durables
Energy
,
36.37
46.73
~ ~~~~~
~~~~~
.
Canad
.866
.828
France
4.82
4.26
Italy
467.2
364.5
,
32.40
~
1.022
.
2.21
5.90
,
.576
5.78
4.00
5.16
5.76
3.13
606.5
400.0
523.1
343.7
2.94
4.77
7.77
.
..
.818
473.3
.,
2.59
42.48
27.91
~
~~~~~~~~~~~~~~~~~~~~~~~~~
~
,
.689
.
2.14
i iim
.
45.30
~
.792
m m,.i
Norway
Other
.
42.05
Netherlands
T
and
Communication
.
1.47
,
i
I
.
.
3.01
2.17
5.65
4.06
L~~.
.
4.06
i
,
2.60
i
.
,.
.K.
.282
.208
.314
.235
.280
.281
.220
U.S.A.
1.0
1.0
1.0
1.0
1.0
1.0
1.0
3.13
2.26
3.76
2.84
2.84
3.40
2.44
est Germany
For Belgium, Canada, Netherlands, and Norway, no "durables" PPP
exists. A PPP for "other household" was used in going from these
countries to Germany, and the "durables" PPP was used in bridging
from Germany to the U.S.
**
For Belgium and the Netherlands, the PPP relative to Germany refers
to "electricity, gas, and water."
27
QNii
qij
and
PN
=
flJQi
j iJij
(50)
P
(51)
Pij = ij/ij
where nij is the efficiency of fuel i in sector J.
We see two problems with this approach.
First, it is difficult to
obtain reliable estimates of thermal efficiencies.
Identification problems
make econometric estimates infeasible unless unduly restrictive structural
assumptions are imposed, and engineering estimates differ considerably from
35
source to source.35
As an example of this problem, we show in Table 2
engineering estimates of thermal efficiencies cited or used in four different
studies.
Note that these estimates differ considerably from study to study.
A second and more fundamental problem is that fuels are not perfect
substitutes (particularly in the short run), and there are non-thermal
efficiencies (which we could label "economic") that also affect consumer
demand.
Fuel choice is also based on convenience, controllability, cleanliness,
capital costs, etc., and the effects of these "efficiencies" (as well as thermal
ones) will hopefully be manifested in the estimated parameters of our demand
35
Adams and Miovic [ ] attempted to measure thermal efficiences by assuming
that fuel inputs are a constant proportion of aggregate economic output,
and that there is no substitution between fuel inputs and labor and capital.
Their production function was thus
Y = min(aF, f(L,K))
where fuel input F is given by F =
of fuel i (Fi ).
EnihiFi,
where
h
is the calorific content
Since the hi's are known, they can estimate the n
s up to a
scalar multiple. The assumptions are extremely restrictive, however, and
their results differ considerably from engineering estimates that they
cite. For an engineering discussion of thermal efficiencies, as well as
a set of estimates, see Hottel and Howard [29].
28
Table 2 - Alternative Engineering Estimates of Thermal Efficiencies
Citation or
Use of Estimate Adams and Miovic
Use of Estimatel
Fuel
~ue~
Residential
Industrial
Adams and Griffen
Residential
Industrial
Nordhaus
Residential
Industrial
Fuss and Waverman
Residential
Industrial
Gas
.65-.72
.39
.60
.65
.70
.85
.75
.85
Solid
.05-.60
.33
.50
.45
.20
.70
.50
.87
.65
.59
.60
.80
.65
.87
.80
.95
.99
1.00
1.00
Liquid
.65
.40
Electricity
.80
.80
models.
--
It thus does not seem particularly relevant to measure fuel consump-
tion in efficiency-adjusted thermal units, any more than it would be to
measure food consumption in net calorific terms.3 6
We therefore choose to measure all of our energy quantities in "gross"
rather than "et"
terms.
We assume that both thermal and non-thermal efficiencies
have effects on interfuel competition, and that these effects will be picked
up in the way that estimated fuel expenditure shares change as relative prices
and income change.
3.3
A Price Index for Energy
Estimation of our consumption breakdown models requires a price index
for energy, and since price series for individual fuels are available, it
would be preferable to use this data rather than an implicit index constructed
from nominal and real energy expenditure series.
36
Since fuels are not perfect
This is discussed further by Turvey and Nobay [6q].
29
substitutes, however, a price index that truly reflects the unit cost of
energy will not equal a simple weighted average of fuel prices.
A typical
37
approach is to construct an approximate Divisia index as a means of aggregation.
An alternative approach is to specify (and estimate) an aggregator function that
relates the aggregate price index to the component prices.
Any unit cost function
could be used to represent the aggregate price of energy, but a logical choice is
38
the translog cost function.
As an incidental advantage, the translog cost function
(or "aggregator") provides us with an instrumental variable for estimation purposes.
The translog cost function (which is equivalent to a homothetic and stationary
indirect utility function with unit total expenditure) is given by
log E =
+
YilogP
+
logP logP.
Y
(5
Assuming cost-minimizing behavior, the fuel share equations are then
Si
=
Yi+
YijlogPj
, i=l,...,n
(5
The first (n - 1) share equations are estimated subject to the restrictions
Zyi= 1
Yi
= Yji
and
Yij = 0.
The estimated parameters Yi and Yij are then
A
substituted in equation (52) to yield the estimated price index PE
A
Note that the energy price index PE is determined only up to an unknown
E
scalar multiple YO.
The procedure is to pick one country (say the U.S.) as a
base country, and then solve equation (52) for yo so that
37
38
See Jorgenson and Griliches [36]
the price of energy
and Hulten [].
This is appealing as an unrestrictive representation of unit cost. Also, as
Diewert [8 ] has shown, the Divisia index is "exact" for the translog aggregator function, i.e. it retrieves the actual values of the function.
30
in the base country is equal to 1 in some base year (say 1970).
Relative price
indices are thereby determined for all of the other countries.
A problem remains regarding the number of fuels to be included in equations
(52) and (53).
Although four fuels are included in our demand model, very little
coal is consumed in the residential sectors of the U.S. and Canada.
This suggests
that equations (52) and (53) should apply to a three-fuel aggregation (oil, gas,
and electricity) for the U.S. and Canada, and a four-fuel aggregation for the
remaining countries.
Should this approach be used--as opposed to a four-fuel
aggregation for all countries--a method is needed to "bridge" the U.S.-Canadian
aggregator with the aggregator for the remaining countries.
We use the follow-
ing bridging method:
(1)
Equation (53) is estimated for four fuels for all countries except
the U. S. and Canada.
The unidentified parameter y0 in equation (52)
is chosen so that the price of energy is 1 in Belgium in 1970.
This
permits the calculation of the price of energy for all countries
except the U. S. and Canada relative to Belgium in 1970.
(2)
Equation (53) is estimated for three fuels (leaving out coal) for
all countries.
The parameter
in equation (52) is chosen so that
the price of energy is equal to 1 in the U. S. in 1970.
The relative
price of energy in Belgium in 1970 to that in the U. S. in 1970 is
then computed.
(3)
Equation (53) is estimated for three fuels for the U. S. and Canada
only.
yO is chosen so that the price of energy is
1970.
Now using the Belgium-to-U.S. conversion ratio determined in
1 in the U.S. in
step (2), the price indices calculated in step (1) are converted to
a U. S. 1970 base.
31
We said before that it is not clear whether a bridging approach such as
that described above or a simple four-fuel aggregation should be used to
construct the energy price index.
using both methods.
We therefore estimate energy price indices
If the resulting indices are nearly the same, this would
indicate that the relative size of the coal shares in the U. S. and Canada do
not distort the fit of a fuel choice model that includes four fuels for all
countries; this would indicate that coal should be included for all countries
in the fuel demand models.
If the results are significantly different, then
coal should not be included in the U. S. and Canada demand models, and the
bridging method should be used to calculate the energy price index.
3.4
Identifying Inter-Country Differences in Elasticities
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 variation.
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 might assume that the parameters of our models
are the same for all countries.
Estimating the translog share equations (2)
by simply pooling all of the data would restrict the parameters aj, 8ji, and
8t,
to be the same in each country.
it,
The resulting elasticities could still
vary across countries since relative prices and total expenditures are
different in different countries, but such variation would be slight.
At
the other extreme we could estimate our models for each country separately;
in the translog case the a's,
in~
~~~~~
country.
gcsetea',i
trn
th
and
sI
jt's could be different for every
While this specification is least restrictive, it is likely to be
infeasible due to insufficient data.
32
There are two compromise approaches that could be followed.
One is to
estimate models by pooling subsets of countries, so that parameters can
differ across subsets but are the same within each subset.
This might involve,
for example, pooling the U. S. and Canada, pooling France, U.K., Italy and
Germany, and pooling the Netherlands, Belgium and Norway.
Re-estimation using
alternative groupings could then be used to determine the validity of constraining parameters to be the same across countries in a subset.
A second approach is to pool all of the countries, but to introduce
regional dummy variables that would allow a subset of a model's parameters to
vary across countries.
coefficients a
In the translog case, we might assume that the
of the first-order terms in the Taylor series approximation
can vary across countries, while the coefficients
for each country.
Ea
Si
D
=
ji and
jt are the same
This would involve estimating the following share equations:
+
ilog(Pi/M) +Sj
+-
.t
, j
o
=
1,..., (n
-
1)
(54)
k MkD + Zi
&log(Pi/M)+ · Mtt
where D
are country dummy variables (Dk =1
kk
Note that the usual restrictions on the
k =
for country k, and 0 otherwise).
ji and
jt apply, but
ajk = -1 for each country k.
jJk
Alternatively we might assume that the coefficients Bji of the secondorder terms can vary across countries, while the a s
for each country.
and Bjt's are the same
The share equations are then
aj +U.ikDklog(Pi/M)+ jt.t
j
=
ki
J
n
a + Sik Dlog(Pi/M)+ Mtt
ki Mk k
m)t+
1
, j = 1,...,(n - 1)
(55)
33
Note that the restrictions on the jik 's are now that
jik
ijk for each
country k, and aMik is the same in each share equation for every country k.
Finally, note that variables whose variation is largely regional (as
opposed to time-wise) can be introduced in additional to the regional dummy
variables.
In the translog case, for example, we might assume that a
is a
function of temperature T (which has both regional and time-wise variation),
aj
with a
= a
(56)
+ bT
varying across countries, so that the share equations are
Ea=kDk
+ bT
+ Eajilog(Pi/M)+
jtt
kk
i
i(57)
k
S.
ZaMkDk
+ EbT + Ealog(P/M) + 8Mt
i
k
with aMk =
aik
=
i
-1 for each country k.
We will estimate demand models by pooling data for alternative subsets
of country, and by using regional dummy variables as described above.
Hope-
fully this would enable us to identify sources of inter-country elasticity
variation.
3.5
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 of equations (even though for some models, e.g.
the log-log models, the systems are not consistent, i.e. "adding up" does not
hold).
Ideally one would like to estimate a stochastic specification for which
the error terms are heteroscedastic and autocorrelated both across time and
34
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 even if the individual
equations were linear in the parameters.
If individual equations are nonlinear in
the parameters (as is the case with our non-homothetic translog model), the
estimation might be computationally impossible.
We must therefore settle for a
more restrictive specification that would hopefully capture the more important
characteristics of the error terms.
When estimating translog models (which can be nonlinear in the parameters
and/or have cross-equation parameter constraints that are nonlinear), we ignore
error term heteroscedasticity and autocorrelation within equations, and account
only for error correlations across equations.399
In particular, we use iterative
nonlinear Zellner estimation, which (under the assumption of no heteroscedasticity
or autocorrelation within equations) is equivalent to full-information maximumlikelihood estimation.4 0
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 crossequation error correlations.
Our logit models and logarithmic models are all linear in the parameters,
and therefore our stochastic specification can be somewhat richer here.
When
39
Accounting for within-equation heteroscedasticity and autocorrelation is
certainly possible even if the equation is nonlinear in the parameters.
One might use an algorithm that repeatedly linearized each equation and
iteratively computed an error covariance matrix and estimated the linear
equation for each linearization (see, for example, Eisner and Pindyck []).
There is no guarantee, however, that final convergence would ever occur,
and if it did the process would be extremely expensive.
40
See Zellner [67] and Gallant [22]. Oberhofer and Kmenta [49] prove that
iterative Zellner estimation (iterating to convergence on the crossequation error covariances) is equivalent to full-information maximum
likelihood.
35
estimating the equations of these models, we also account for within-equation
heteroscedasticity.
This is done using the following procedure.
equation in the system is estimated using ordinary least squares.
First each
The resulting
regression residuals, which we can label ukt, are then used to obtain consistent
2
estimates of the regional (country) error variances aok:
.2
=
IT
(58)
(kt)
T-m-t
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 the
ak s will of course be obtained for each equation in the system.
We then
transform the data by dividing each observation by the appropriate estimated
error term standard deviation e, and then re-estimate the entire system of
k'
41
equations using iterative Zellner estimation.4 1
All of our estimation work has been carried out at the Computer Research
Center of the National Bureau of Economic Research, using the GREMLIN experi42
42
mental nonlinear estimation package on the TROLL econometric software system.
This package permits one to perform iterative nonlinear Zellner estimation
conveniently and with reasonable computational expense.
4
MacAvoy and Pindyck [j~] used a similar approach to single-equation
estimation that also accounted for time-wise autocorrelation.
42
For details on the estimation algorithm and its use, see Belsley [ ].
For a discussion of alternative nonlinear estimation algorithms, see
Berndt,-Hall, Hall, and Hausman [8 ], Chow [3], and Gallant [22].
36
4.
Characteristics
of the Data
Unfortunately much of the data for this study could not be obtained from
In some
standard sources such as the OECD or the U.N. Statistical Office.
cases the needed data (such as retail fuel prices) are not collected by
these sources, and in other cases the data have been collected, but have
been aggregated or categorized in ways limiting their usefulness for this
study.
As a result it was often necessary or desirable to go to the national
statistical yearbooks of individual countries to obtain data.
Nine countries are included in our sample:
Belgium, Canada, France,
Italy, the Netherlands, Norway, U.K., U.S., and West Germany.
The data
collected for these countries are described briefly below. 4 3
Consumption Expenditures.
categories:
These are broken down into six
food (including alcohol and tobacco), clothing,
durable goods, transportation and communication, energy, and
"other".
This last category includes housing expenditures
(actual and imputed rental payments), expenditures on health
services, and any other consumption expenditures.
Data were
obtained from the OECD's National Accounts, the national accounts
publications of the EEC Statistical Office, the U.N. Yearbook of
National Accounts, and national statistical yearbooks.
The data
are measured in current local currency units.
,,,
43
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 [66 ]. Other researchers wishing to replicate or extend
this study, or perform studies of their own, can access the data directly through
the TROLL computer system of the NBER.
37
Price Indices for Consumption Expenditures.
A retail price
index (1970=100) was collected for each of the categories of
consumption expenditures listed above.
Although for some countries
retail price indices were available directly, we constructed
implicit price indices for all countries from consumption expenditure series in current and constant monetary units.
Although
price indices for energy are available, we use the energy price
aggregator function described earlier in estimating our consumption
breakdown model. 4 4
Data were obtained from the OECD's National
Accounts, and the national statistical yearbooks of individual
countries.
Fuel Expenditures.
Data were collected for total residential
consumption expenditures on petroleum products (largely light
fuel oil), natural gas, coal, and electricity.
These data were
generally obtained from the Statistical Office of the EEC's
Energy Statistics and national accounts publications, or from
national statistical yearbooks.
In a few cases figures were
obtained by multiplying the retail price of the fuel by the
physical quantity of the fuel consumed;
physical quantity data
were obtained from the OECD's Energy Statistics tape.
The data
are measured in current local currency units.
Fuel Prices.
For each fuel, the data are countrywide averages of
the average retail price.
4
In the cases of natural gas and
The energy price aggregator requires data on fuel prices. For some countries
our data on fuel prices does not go back as far as our consumption expenditure
data. In these cases the estimated energy price aggregator was regressed
against the implicit energy price index, so that data for the index, could be used
to extend the aggregator backwards.
38
electricity, for countries with tariffs the price level chosen
was the average price facing an average size household.4 5
When
the price of natural gas differs from that of manufactured gas,
an average of the prices weighted by the relative amounts
consumed was calculated.
Data were obtained from the Statistical
Office of the EEC's Energy Statistics and Studien und Erhabungen,
and from national statistical yearbooks of individual countries.
All prices are measured in local currency units per tcal, but
have been converted to 1970 U.S. dollars per tcal for estimation
purposes.
Fuel Quantities.
for fuels.
Some of our models use physical quantity data
All quantities are implicitly derived from data on
fuel expenditures and fuel prices.
Other Variables.
Units are tcals.
Data were also collected for net disposable income,
population, and temperature.
The income data represent total
net disposable income of all households, although for some countries
only total private income data (personal income plus income going
to non-profit institutions) were available.
Income data were
obtained from the OECD's National Accounts and the U.N.'s Yearbook
of National Accounts, and have been converted to 1970 U.S. dollars.
Data for the total population of each country came from the U.N.
Demographic Yearbook, and are measured in millions of people.
45
Some researchers, e.g. Halvorsen [2], have used the marginal price of
electricity as a measure of price. The marginal price alone is inappropriate,
however, as has been demonstrated by Taylor [61]. The correct procedure is
to use the average price at a normalized and constant rate of consumption,
or to incorporate both average and marginal prices. We use only the average
price, since that is the only data that is available.
39
Finally, our temperature data represent the average temperature
over the five winter months (November - March) averaged over
the principle city or cities of each country.
The source is the
U.S. Weather Bureau's Monthly Climatic Data for the World, and
the units of measurement are degrees Fahrenheit.
In some cases data for one or two variables was not available over as long
a time period as was the case for the other variables used in the models.
For
example, for Italy the price of electricity was available only for 1963 to 1973,
while other fuel prices and all fuel expenditures were available beginning
in 1960.
In constructing our fuel choice model, more efficient parameter estimates
can be obtained by constructing an instrumental variable for the Italian price
of electricity and using that to extend the price series back to 1960, rather
than dropping the first three years of data for all of the other variables. 4
Instrumetnal variables were used in this way to fill in missing data points
in a number of instances.
In Table 3 we show the time bounds for all of the
data described above; years in parentheses represent data points constructed
via instrumental variables.
Note that these time bounds do not represent
those used in model estimation.
Because of the need for overlapping bounds,
only a subset of this data can be used for estimation work.
46
Letting PE1 represent the price of electricity in Italy, this variable
I
was regressed against a set of variables Z1 ... Zn that would include the
prices of electricity in other countries and the prices of other fuels in
Italy. This regression is used to generate values of PE , for 1960-1962.
In using this data to perform the final regressions for uel shares, efficient estimates are obtained if weighted least squares is used. If 1 is
the coefficient of PEI,
is the error term for the fuel share equation,
and
is the error term for the instrumental variable regression used to
construct kI, the error term in the final regression will be (v + £)
for 1960-62 and
after 1962. The resulting gain in efficiency is probably small, however, so we do not use weighted least squares. We thus
obtain consistent, but slightly inefficient, parameter estimates. For
a general discussion of approaches to the problem of missing data, see
Pindyck and Rubinfeld [3], pps. 194-202.
40
0
I,
CY)
N-
0
Uj
0
0
N-
U)
U,
0
rU)
U
0
N-
0
N-
U)
Ln
Ln
U,
CV,
ce)
cn
r-
o
r-
I
I
L()
LO)
LO)
Lf)
r-
r-
I
NU)
o
r-
Lv,
Lv,
0
U,
U)
CY)
I
0
O
U,)
U,
Lv,
0
0
r
U,)
Lf)
_
N-
L-1
Ur)
0
U,
o
Nq
I
I
r-
U,
U,)
r-
U,)
U,)
U,
U,
NI
NI
0
cl)
U,)
O
N
r-.
r-
I
L1
U)
I
U,
U)
0
rI
U)
U,
O
r-
CY)
I
I
Lf)
U)
00
CY)
CY)
N-
0
Lf,
0
U,)
Cv)
CY)
0
Lr)
0
11)
-
I 0
I
o' OrLr)1 -
N-
r-
I
0
I
C
L)
U)
N-I
'IO
V)
I_ I
I I %ID
I
%Dun)
ur) r- Cv ,
rO
U
CY)
(le)
-
r-
I,
IU)
Un)
CY)
CY
tr)
U,
Lr
Lr
l-
'-I
r-It
Lv,
N-
0U,)
Cf,
N-
NIt
Le)
U,
CY)
ce~)
Cv,)
I
I
I
r-
rl_
r-
r-
r-
r-
Lv,
0I,
U,
Cr)
0-
0U,)
CY)
CY)
0-
r-
Lv,
CY)
M
rI
r-
rI
r-
L)
U)
CY)
CY)
0lU)
0-
I,
r-
I
NI
Lv)
U,
0
r-
CY) M
V)
0I 0I
0
%OI-C-C
-
Cv,
0-
N-
I
I
0
U,)
U,)
LO
Lv
m
clv
r- .
U,
0
00-
N
N-
l
0
U,)
U)
Uv)
0-
N-
0
0
0-
I,
U,)
0
I
r-
H-
0U,
U,)
0
VCo
Ul
V)
0
0I,
0-co
U,)
cv,
I,
0
r-
N-
U,
CY)
0-
0
N-
U,
U)
Mv
U,)
0r-
r-.I
co
Lr
U,)
HII
r-
-i
rI
U,)
Lt]
U)
N-
0-
0
0
I I
Lr2II
U
U,)
0
0U,
Lf,
0
Loi,
V,
0
Co
U,I:
OD
0
NI
U,
0
0I
Co
?,
°iX
O
rI
U,)
0)
r-
I
Co
U,
0
0H-
U,
0
V,
Lv,
0-
r-
Co
Lv)
Lv)
CY)
U,)
Ur
U,
0
r-
r-
0
U,)
I,
U)
Lr)
H-
Lf,
U,)
i 0I,
Lr
0
0-
o
NI
U)
Lf)
Lv,
U,)
0)
0
H-
U,
I,
I,
O
0
0
CO
Co
L
OD
U,
Ui
U,
Ln
0
-j
0
Lv
0
0Lv,
0
o0
I'D
ID
o
or-
N-
I
U,)
I
0
o
U,)
%-C
IC)
?
o
0
o
lZ
oI
u)
U,I
I
O
Lrl
Co
I
U)
r-
0
uLf
0o
%D
I
0
r-
0)
r-
IC
~o
0
0-
fl
o
0
10
o1.1J'1C
uNC1
:N %O
D I-
Co
H
I-C
r--
I
I
0
U,
Ln
I-0
0
r-
o 1
I'll4
_
I
I0
0
N1
I
I
o0
I-C
-
U,)
U,)
Cv,
r-
N-
IC4
0f
Lr)
r0
iv,
01
O
CY)
I
Lir)
I
C)
r-
IOC
Co
I-O
I'
--
r-
Lr)
0
0
Ln
I
L?
0-I
UI
U,
Lv
H
00
O
V,
CY
rU)
CY
N-
U)
'IC
CY)
r-
Lr
r-
NI
0
U,
CY
Lrv
NI
0
O
V
U)
Cl)
Lv,
rI
r,
U)
Lr
U)
C'-,
NLv,
0
Lr)
r-
Lv,
CY)
r-
Ch
V)
m
0-
0-
0'
U,)
cvn
0-
0l-
Ln
N
0CY)
00U,)
0
0U,)
H-
0I,
0rU)
Ul
cv
r,
Cf)
UI
0
-.
U,)
V)
0
0-
0
Lv,
U,)
U,)
Hr-
HN-
I
I
H-
0)
'%C
U,
o
o
I
I
U,
U,)
0-4
I,
0
0-
U,)
0
NOIQldlSNOD
Lv,
0
U)
0-
0
Lv,
0
0I,
0
U,
U,)
HI
H
r-l
II
u)
U,
-
I
0
U,
0
10rr-I
I
U,
Lv,
I
I
10
0-
l-
o
U,
-q
o
0
00
NIC
%.
o-
%O
l
0I
0)
U,
H
rI
00
U,)
N-
IH
Co
00
-
0
-
N1
Cv,
r-
I
I
0
0'-
Lv,
I,
0
I-C
NU,
U,
U,
U,
4-
I
U')
Ln
01
cv,
0U,
I
Ln,
rI
cv
Lv,
a)
.1i
0
of
-H10
0
p
0
4A
0
C-.)
Ca
Er-l
0)
rxz
I-,
IC
4-~
-H
-e
C-,
0-~
z
H
xn
SfDINI aoIldc
I'TrJT
"'fn.T.T
rp
fl
J7
0
a)
H-
cv,
0-_
UlI
Un
Cr
-Ii
CD
co
Lv,
C
-4
0
U,
cJ
4-i
-C)
.- 4
.H
0
C,)
SHDIUC
c
sd
rl
UI
I-i
1.4
.C
SNOIf.JJSNODX
Cm
14
4-i
HC-
Lv,
i
i
cd
I-4
E
Lv,
0I
4-.J
0
c-
r-
r-I
Lv,
H
014
CY)
H
N-f
C
CE
N-1
I
Ln
U,)
41
Our basic translog models requre consumption expenditure shares and
prices.
It is useful to examine some of the share and price data before
turning to the estimation results.
Table 4.
Data for 1962 and 1970 are shown in
Note that the energy price index is not shown, since we compute a
price aggregator from the fuel prices.
5.
Statistical Results
In this section we present the results of the estimation of the models set
forth in Section 2.
We begin with the estimation of the translog energy price
aggregator, and present relative energy price indices for our nine countries.
Next we present estimation results - and the implied demand elasticities - for
Finally, we describe the results of
the static and dynamic translog models.
estimating the logit and log-log models.
5.1
The Price of Energy
A price index for energy was obtained using two methods.
First, a translog
aggregator was estimated assuming a choice among four fuels in all nine countries.
The estimated parameters for this aggregator are shown in column 1 of Table 5
(standard errors are in parentheses).
tricity, (2) oil, (3) gas, (4) coal.
Fuels are indexed in the order (1) elecNote that the parameter Yo is determined
so that the price of energy is 1.0 in the U.S. in 1970.
Second, the "bridging" method described in Section 3.3 was used.
This involves
first estimating a four-fuel aggregator for all countries exeept the U.S. and
Canada; here Yo is chosen so that the price of energy is 1.0 in Belgium in 1970.
The estimated parameters are shown in column 2 of Table 5.
Next a three-fuel
aggregator (leaving out coal) is estimated for all nine countries in order to
find the price of eergy
in Belgium in 1970 relative to that in the U.S. in 1970.
42
I
-
_~
N
:n
H.
0O
-It
.0
)
-
N
N
O
%D
rD
IF
N
Lr,
NS
'.0
.)
'.
U'
cc
H
H
O
H
H
H
*
V)
i
)l
,I
U)
c,~
ND
cc
00
AD
t-Q
'0
U.)
-a
ao
0%
-I0
CO
,-4
Ha
O
f.cn
4m
O
'.0
u0n
oM
(V.
Ln
-t
(',
N
H
lV,
0
N
0
0
-t
N
a
N
H
.*
C--A
0%
Ln.
Sr
4t
00
C'4
--T
'0
Ce
cO
H-
-t
oc -It
Ln.
0H-
0%
Ln
00
o0
u~
0
p°
Cv)
O
o0o
-*
r-
v.D
Cc
C)
cc
0
M
o
o
O
N
C
O
U.)
rC)
o')
'o
H
OD
o
.)*
10
CYN N-
o
o
0
N
H)
cc
0
O
PH
on
o
U)
x
a
om
00
c^l
0
I.-O o0H
Cc
cc
D
0
A
0
H*
oN
0H
H
o
r-
cq
H
rH
o00
C)
co
O
o
H
H
H
o
a)
A
U)
0
C)
cs
C
0
C
o
-r
-t
0
oO
r-
-Iu
co
o'Q
Ci~
H
H
H
H
N
N
a
OD
ON
0o
0
-t IC0%
O0
0C)
*1.c0
CY.)
Ln.
H
U
O
4(I
N
CD
F.-
C)
01
O
C1
o~
O
r-
cc
o~
ooou,-
H
00
H
co
eo
0s H
O
H
N
CY)
H
'.0%4
0oDo 0%
N
00
H
(V.)
H
H
C
C)
H
ccD
oo
C)
0
u%
0o
C)
H
0LV,
4t
If)
0.,'0
0
oo
co
tn
cc
co
H
'.0
N
oo
00
UX
H
0%
-
OC)
0
C
-0
U)
N
.)
Q
00
0
oV,
o
C4
cc
00
N
roo
CO
rn
O
-~
r-
rD
Ol
'o
O
00
0;
H--,I
0
'.0
00
r-
H
10
-o
0
-t'
-
N
N
N
N
O>
0
N
0a
H
C)
r-
r-
N 0N
C'.
CY)
--T
cc
CV,
*
rC)
as
I-.
-s
H
H
C)
H
ONO
H
H
r-N
O
O
CO
CO
O
U.)
0
C)
U,
0
00
CV,
o0
C)
O0
H
-
0)
r-
H-
N
H-
N
0%*% C%
0
9
4
HH
co
PI
H
0O
N
cc
uL)
N
0%A
H
C)
N
H
H
00
-
C)4
CY)
0
C
(V.)
*~
0%
(V
O
H
Ir)
0
C)
C)
C)
C)
H
NCO
CY
O
H
H
(V.
O)
0o
C)
Hs
N
H
Ce)
H
N
%D
C)
H
N
C1
o
CV
*~
*
H
H
CO
H
N
H
CY~)
N4
CU
-IC
w
.iJ
..q
oo
cc
CY0
?--I
I
H
N
0
N
0ae
o o
*~
c
H
C)
H
00
0%
HI
ONO
O
H
O
O
-t
C)
O
0% C)
H
0
0
U)
Ot.
O
(V00
01%
o
CO r-.
N
U.)
C
Ln
u,
CO
-It
Cl
If)
co
0o
00
0 1.
N
cc
0De~
O
oo
C)
oC
Cf)
04.
H
0S
C)
cO
o o'
U)
00
cc
C)
H
H
0%r
co
-
o
N
OD
C)
-
0%
L.)
H
,H
00
C)
0%
I_H
OH
-I
CO
cc
CO
O
-n*~ H
H
*
p
-N
u~
c1o
r<
0
*Y~ N
'
4
,C)
H
'
..
o
H
H
CY)
ccl
C)H
co
H
H
Ns
'0
14
N
N
It
Lr
00
· 1 .~.
N
oh
00
-
-
-
On
r- a
H
H
-
H<
o£.
%o
-4r
(cV.
o
r-la
00
Cn.
-,1
-t
'.0
H
Ct
%~t~
N
cn
H
-It
-It
0
0o
H
-
'0
U,-
0}
-
N
0%
U)
o
0
I
H
N
r0
|
-0
r_
to
u~
C)
m
%D
_O
-t
o~
o4
I
I
0%
H1
co
H
cn
%O
'0
%D
J I II
NE
N
*~C
N0
.-I
zU)
*I
l
0D
H
CD
.
C-
l
N -0
a%
%D
.H
. a)
U
CN
1-i
co
N .
'.
0
H
C)
C0%
H-
N..0
%D
H
H
O
H
m
H
C.0
0%
H
r
F.-
H
N
'.0
0
HO
,.-
'.o
0%
H
C)
f.
N1
C)
0%
ao
r-.
H4
OH
r.
CD
-
5-I
U)
~4J
s-i
4J
a)
U
u
o
0
o
43
En
,-4
'0
P-.-I
N
i
o
o
o
o
o0
0
o
Un
-It
-I
oo
,l_
-I
cn
H
N
N
0
r-_
N
00
cn*
*4
o
0
o
o
w C
4 10
un
Y
CJ
0
CN
I
cf)
c'
L)
en
a) 4-I)
H
oo
0o
*
ai U
I'D
E-1 *ri
N
z
.r-
.-
o-It*
X4)
0
ki
r-
0Cq
L()
-o
I
c0
o4
N
r_
-;T.
0
o
C_
0
H0
0
0
0
#-
0.'
HC',
M'
m.
H
M
0
0
: 4
P4 0
:j
(
o
p4
$i0
i CU
o00
(ii
. a)
N C/
H
o
o
c-
O
.
D
4
:
rX
*-
r4
z
U
to t
N-
H
o
o
o
-
Lf)
U)
O
*
TI
C;
-
r-
0
10
r-l
o
0
O
CS
1
0
o
O
O
c;
0
*-
N-
ON'
'.
NT
N-
0
o%
Nt
cn
N-_
00
I
4)
-
;>
H
4
H-
rH
,r-
',T
en
-n
H
0-
C.
Ih
I
'.0
0
C',
H
'
Lr)
-T
Cn
N
r-
;>
U-)
C,
r
-<
r-
L(
00
H
0
00
o0.
CI
I
I
I
>- ;,.
N
0
.
00
0
0
H4
U-
0
c;
1
0
N-
oo00
Hq
,-O
rH
CN
ON
00O
-H
H
0
N4
N
0.'
-It
0.'
0
L
0
0
Ln
'D
00
VL
0
o- 0
0
(S
'.O
0
LO)
H
0
.
0
0
*
Ln
0
0
*
I
O
Un
n
I
Ln.
O
,
N
0
C;
-
CN
0
0
0
*
c;
0
0
N'.0
0r
0c
cn
,H4
'.0
O
0
C;
H
e'
0
'
V)
U)
O
0I
N
cn
O
4N
0
0
*
I
0
CN
HN
00
0
0
Ul)
NT
C
'.0
0cI
CY)
0
"v
%-.
0.
0
0
0
H 0 O
o o
o
o
C,
-I
'D
cI ;
I
;>
0
0
0
00
00
0
-%e
0
Nt
H
,-
r)
Nt
o
-
uL')
.0
o
Vu
IT
0
H-
%
%0
H-q
'.0
N-
c; ;o 1
I
;..
*
0-
o0
C
-H
N
*
o
-
0
H-
0
0
O'
0
*- 1-
rZ
4i
o
O
O
SI*
H
I
o
0o
0
04
H
C-*
':r-
0
%.
o
e\
CN
~o
o
o
L)
H
eci
-I
0S
fLO
*
t
-
I
00
0
o~~~~~~~
c;
0
Nl
U-)
N
0
0
H
fN-
o
o
H
0
o
0
r4 d
0O
o*
o
-cn
C)
N
0
0
0
<4
,-%
0o
i/1
C0
0;
Lt)
o
N
OI
N
<
'.
N-
'IO
'.0
o
0
O0
o I
CN
0
0U
-It
un
*~o o*
r-
N
Hn
N
c
L
U
04
O
c;
c;
I
0
U)
L1
C
r-.
0
0
-
CN 0
*
*
0
N
U-)
N-
Lu
Lf)
N
Hd 0
- 0
*
I
T
,'
Ln
CN
'.0
Nt
-I
T
,-4
-;
On
0
CY)
T
cn
cn
I
U%O
-a
O
0
-I
O
H
cV)
0
t
N
'.0
0
rH
Nt
N
-
NT
Ln
-i
-'
cV)
C,
O
0I
-C
-IT
c
Hd
M'
H00
- 00
I
I
0.'
O'
LO
O
co
HCe
0oII ;; .a
I
-,
T
--
0
0
,9
1:34
-
44
This relative price was found to be 1.5239 (parameter estimates are shown in
column 3).
Finally a three-fuel aggregator was estimated for the U.S. and
Canada (see column 4).
Now using the Belgium-to-U.S. conversion price, the
price indices can be converted to a U.S. 1970 base.
The two price indices are shown for all nine countries in Table 6.
Note
that for all countries these two indices are quite close to each other, indicating
that the relative size of coal shares in the U.S. and Canada do not distort
the fit of a four-fuel aggregator (and thus a four-way choice model) that
is estimated over all nine countries.
We therefore use the four-fuel price
index in estimating our consumption breakdown models.
5.2
Static Translog Models
We begin with the breakdown of consumption expenditures.
were estimated, and the results are shown in Tables 7a and 7b.
Eight models
All of these
models are homothetic; the computational expense of estimating a non-homothetic
translog system with six categories of expenditures proved to be inordinate.
In the first four models all of the parameters are assumed to be the same
for every country.
Model (1) allows for time-varying preferences, so that the
only restriction imposed is that of homotheticity.
additional restriction of stationarity.
Model (2) imposes the
A likelihood ratio test was performed
for this restriction, and the test statistic is 112.3.
This is well above the
critical 5% level of 11.07 (5 degrees of freedom), so that the hypothesis of
stationarity cannot be accepted, conditional on model (1).
Model (3) imposes
the restriction of explicit groupwise seperability between energy (category 5)
and the other categories of consumption.
The test statistic for this model
45
mo
-I
cq m oo Oa -- C1 oo aW un
0
C,
-
, - , ,-,-,
oN
1
%D
cr
fl
c" 0.c ,, -Itr-i
w0" ,'4
O0 r'.
- ,-
,
,- -,T,
I
r---
r- o
I I
'4
Ca
J.JH cs
.,(
U
n
o'o o
0 o m
o 0
o
H
"0
H
H
H
00
-,
0
N
I'D
4
I
In
c
-
O
D
-
- H
I I
(ON
0Nv
un %D"
n
HHH 0 I
V-i
to
O
0
0 oo
W_
I I
I I I r- r-q r-I r-I r-I r-
,H
-t u'n ..t ,.
HI
I
'-4
un
n
m
w
O
w
unCOC
r- un
H
I
I
I
CH C,
I
I
I
m -4
H M'MHH DIn
H
I
III
T
m m
oun W a r-
0 i*.r- %.I U)I'l*n. -, . -,*' -,
m* m
4 *
, , ,~ , ~~,H
,~
-T
n -I -T m m --
n xD -. ,-i
, -,-I o
o
I 0
o.,.,1
r. u' , o,4u'l oo
o,, oN
ao
0o
flr
C rsN
<@oCnO r-4
I HHHH<H<H H HHH H-<H
C1l
4-J
O)
N
o
N r-OCOOS
e a c)N
<
v
CN
0
U)
:j
L -
r
-1 r- -I 0 0
t)
o o u) koo ul
r
/,-I 0 Co " H H r- "
N H c-o , ---"OX
Z
oo
oC7- oo ,_ .
ur'lL
C.. .I- c-4 -
OI
H
Ln r-q
m~ rl-I
mn0
oCLr
O
O %D O
b
% n
H
0
-H <1~ H
0
-
-e-
4.C
I
"0
0
I:
r"
a)
,C
bo
Or
e
_.o
"0
rCq
-I
0
ri
C"
00
C"
II
rO
OaN HO
1. O
Ln O
0
%D a
H
%O L
rH
10
D
o
C- %'
o Oo L-I
)
Nt -I
HH
H
n
o
o o-,,
Trn
CY)
-z
o0
ci
I
-)
HH
HH
HH
I
.-
, o ,m
oo .- ,o , It o o
,-
' U,-1 o-H,
,,D
0
oI
,,
,-
1
o
o
oo
r
-
-
O
-I
,-
I-.,o
c',
0o
o
o -r- ,-
,1
-
oO
CY) ,-,
CY)
OD
- ON o -I
,
.
O
I I I
HHH
0a
O
0o
~
~
~oo n<Ho
~
N--0crcOaHOO'0
, 0 0|
,a crLOcrN-N-OOC)H
££ ocuiC<
-r- < n
co-
I-,--
M C4
4%
HH
-IoC.
-i C
,.I cV3CY)"
N
,
M
* *CO*C*.-
. * .
.t-*
I I IH<H H HHH H HH HHH
I (_ un
H H
"
"s
r-i
r-
Ha
r-
r
" o CN
oo %o-T
, r- O o H-HO
0C 9 0 0
I I I
H
4 H4
H
(,4
oC
cmN OO
mOmc
cOH-i
r
r
r, Lm
o
LI
oo
01-
o cun
00
Ln
-1
(N
oq4
V-
I
vo
IHHHHHHHoo
X
cH
H-o
HH
W
I
I
V)
.
m
14
oci)
I II I
(a/
I~
:3
0
u.
H~oooo~oooo
O
ON --
M
0000
t 00 000 4 N
Ln
-I
V-
I I , * H* *HH *oH H* H* H rHH
IHHH ~I0o HHHH * *
00
Nm Mn
u
Ln
00
O
0
O
O
uLn un
N -O
- MD Ds
COo oM n COn - c aMn0
0 H
0 H0 OHHH
0H
00
CO r C Lf
COcNIO T r.
'4O-ArC4
%Dw
IHHH<HHHHHHH
co
CON)-OCOOOm
C
- m -I n M ' M n %I'D
%O%OD 'D %C)% %DLn %D
urt
Ln
(-N
I
0
I
l~(3HO~
~ %1
, ~i~,~~~~~~,- ,-~H ,-~ lH~ HA
~i ir-
lllll~~~l~
I
00
44
r
HU,
,YmmC
44
0
o0 o
0
o Ln r-
I ONil Ln Ln on
o r-
o . C-0
Ln csT
%D
o
II
IHH
HH
r, r-%O~~L ~
(N
L
UnH H HH
-I
HHm
C-
m
w
I
.oTO
oN
· ,-oo o o
u}~o
0,-h
LO
cN H 0'n
0a
Lm
o0 CO0 0
r r--r-
-o <
V-'
OU
:3
Fx.4
O
%D r
I
I
I
I I
I I
o ooooooo
I
I
I I
I
I
-H
aI
a af M
cr
r o ooj W %,
pq-V
I I I
H
HHHHH
4H
HH H
- -O0 '
:2
un
t U
I I I
"'
r
0 0 -I C
.. L,
U)
r- Owa0 0
,-iCN
, n n n n ,D ,-,- - %I0
,- ,- , -,- -1-,- -, aw rO O
4 oI r4
-0 r o- a o- r Ca a- oJ - a- r-
I
I
I
I
an %O
o- c
un
Ln
I
I
,
oo oH0 O-r-
C;
c
ON0 HI "c O
rn
o oC;
o
NCmOO O
CN CO
n n n I' %OD xQD
1
1. 1.0 ID %D
£ . rDN- P r,
t0NmC
HH
I
C -.ln
I' I o
r-1 HH 4
-
<r
-
O
r
-q
o
a c
_{ -4 H H H
N
H
46
Table 7a - Parameter Estimates: Consumption
,
3. Homothetic,
I
2. Homothetic,
Parameter
Ol
1
a2
3
a(4
a,
5
a1.6
t11
12
13
'14
15
B16
21
~22
~'23
P'24
P'25
~'26
'31
~'32
33
34
335
36
41
42
43
44
,45
'46
51
s52
53
54
5 55
5s6
61
62
63
f64
65
66
B1T
~2T
B3T
B4T
135T
II6T
1.
stationary
omothetic
-0.1132
-0.1274
-0.4449
-0.0541
-0.0202
-0.2402
-0.0767
-0.0112
-0.0314
-0.0568
-0.0076
(.0043)
-0.0908
-0.1240
-0.3182
-0.0340
-0.0319
-0.4011
0.0722
(.0095)
(.0178)
(.0083)
(.0028)
(.0053)
(.0014)
(.0080)
(.0073)
(.0029)
0. 0139
-0.0140
-0.0332
0.0119
-0.0408
0.0139
-0.0032
0.0334
-0.0032
0.0012
-0.0420
-0.0132
0.0334
0.1377
-0.1650
0.0079
-0.0073
-0.0112
-0.0024
+0.0257
-0.0348
0.0021
-0.0018
-0.0314
0.0257
0.0798
-0.0740
0.0129
-0.0130
-0.0568
-0.0348
-0.0740
0.1657
0.0225
-0.0226
0.0076
0.0021
0.0129
0.0225
-0.0452
+0.0001
-0.0073
-0.0018
-0.0130
-0.0226
-0.0001
0.0446
+0.0017
+0.0004
+0.0096
+0.0016
-0.0008
-0.0125
(.0030)
(.0051)
(.0026)
(.0086)
(.0258)
(.0172)
(.0052)
0.0000
-0.0332
-0.0032
-0.1650
0.0325
0.0238
0.1451
(.0180)
(.0048)
(.0023)
(.0033)
(.0028)
(.0002)
(.0006)
0
0
0
(.0011)
(.0005)
(.0002)
, ,,,,
.-
-0.1083
-0.1253
(.0111)
-0.4311
(.0041) -0.0381
(.0014) -0.0453
-0.2519
(.0056) 0. 0760
(.0015) 0.0126
(.0085) -0.0233
(.0073) -0.0392
(.0029)
(0.0043)
(0.0096)
(0.0173)
(0.0082)
(0.0029)
-0.1018
-0.1109
-0.3022
-0.0752
-0.0401
-0.3697
(.0018)
(.0029)
(.0057)
(.0034)
(.0008)
(0.0049)
(0.0049)
(0.0076)
(0.0074)
0
-0.0261
0.0126
(.0028) -0.0013
(.0059) 0.0235
(.0026) -0.0020
(.0009)
(0.0029)
(0.0049)
(0.0026)
0
-0.0328
-0.0233
0.0235
(.0288)
0.0745
(.0172) -0.1626
(.0051)
(0.0243)
(0.0171)
0
0.0879
-0.0392
-0.0020
-0.1624
(.0176)
0.0423
(.0045)
(0.0186)
0
0.1613
0. 0119
0.0012
0.0079
0.0238
-0.0283
-0.0067
-0.0408
-0.0420
0.0000
0.1451
-0.0067
-0.0556
non-stationary
separability
of energy
Homothetic,
stationary,
separability
of energy
0
0
0
0
(.0024)
0
0
-0.0261
-0.0328
0.0879
0.1613
0
-0.1903
0.0015
0.0004
0
0. 0091
0. 0010
0
0
0.0003
-0.0123
(0.0002)
(0.0006)
(0.0011)
(0.0005)
(0.0002)
~~~~~~~~~~~l
Consumption
are: 1
1 - Apparel, 2 - Durables, 3 - Food, 4 Transport *
categories are:
Consumption categories
Communication, 5 - Energy, 6 - Other.
47
Table 7b - Parameter Estimates: Consumption
7
5
$4
6
Homothetic,
stationary,
country dummy
variable
p4
Homothetic,
stationary,
country dummy variables,
seperability
of energy
8
Homothetic,
stationary,
country dummy
variables,
additivity.
Homothetic,
non-stationary,
country dummy
variables
i i~~~~~~~~~~~
n
acxD2
DI
D32
alc1 D
2
1D
aD3
aD 4
aD 5
1D
a2D
1D7
a21
1D
22
24
:tD7
r
D
9
;vD
a3D
aa 3 D2
D3
3
c3 D
rvD2
a4D 5
a 4D
4 D8
a3D
7
a D9
a4D2
a5 D
D4
4
OL4
avD
aD5
a
D7
D
8
6
CLS
-0.0986
-0.1113
-0.1146
-0.1027
-0.1141
-0.1775
(.0047)
(.0039)
(.0048)
(.0051)
(.0028)
(.0134)
-0.0882
(.0046)
-0.0852
-0.1266
-0.0899
-0.0592
-0.0587
-0.0286
-0.1358
0.0807
-0.0610
-0.1473
-0.1041
-0.0329
-0.3081
-0.4380
-0.4995
-0.3471
-0.6594
-0.2721
-0.2488
-0.3854
-0.1189
-0.1619
-0.1093
-0.1064
-0.0558
-0.0882
-0.0564
-0.0362
-0.1150
-0.0507
-0.0282
-0.0381
-0.0325
-0.0546
-0.0291
-0.0504
-0.0357
-0.0405
(.0011)
(.0051)
(.0049)
(.0052)
(.0046)
(.0051)
(.0027)
(.0193)
(.0041)
(.0016)
(.0046)
(.0141)
(.0152)
(.0130)
(.0147)
(.0064)
(.0380)
(.0116)
(.0041)
(.0133)
(.0068)
(.0068)
(.0063)
(.0069)
(.0034)
(.0210)
(.0059)
(.0020)
(.0065)
(.0038)
(.0029)
(.0036)
(.0037)
(.0019)
(.0100)
(.0034)
(.0008)!
(.0036)
-0.1094
-0.1153
-0.1262
-0.1123
-0.1198
-0.1806
-0.0992
-0.0864
-0.1366
-0.0944
-0.0635
-0.0632
-0.0337
-0.1376
0.0627
-0.0647
-0.1476
-0.1089
-0.3993
-0.3240
-0.4474
-0.5129
-03519
-0.6362
-0.2870
-0.2541
-0.3989
-0.0858
-0.1388
-0.0807
-0.0761
-0.0416
-0.0697
-0.0255
-0.0303
-0.0847
-0.0510
-0.0316
-0.0335
-0.0300
-0.0522
-0.0309
-0.0477
-0.0371
-0.0371
(0.0032)
(0.0037)
(0.0036)
(0.0042)
(0.0023)
(0.0117)
(0.0035)
(0.0011)
(0.0043)
(0.0037)
(0.0048)
(0.0036)
(0.0041)
(0.0021)
(0.0183)
(0.0029)
(0.0015)
(0.0035)
(0.0118)
(0.0150)
(0.0109)
(0.0129)
(0.0054)
(0.0361)
(0.0092)
(0.0043)
(0.0111)
(0.0056)
(0.0073)
(0.0054)
(0.0062)
(0.0029)
(0.0221)
(0.0045)
(0.0022)
(0.0053)
(0.0006)
(0.0006)
(0.0007)
(0.0005)
(0.0008)
(0.0008)
(0.0006)
(0.0005)
(0.0005)
-0.0868
-0.0839
-0.1050
-0.0926
-0.1233
-0.1346
-0.0904
-0.0850
-0.1242
-0.1192
-0.0850
-0.0879
-0.0616
-0.1484
-0.0782
-0.0822
-0.1478
-0.1336
-0.3138
-0.2351
-0.3641
-0.4206
-0.3279
-0.2818
-0.2299
-0.2504
-0.3206
-0.0941
-0.1455
-0.0902
-0.0868
-0.0463
-0.1089
-0.0331
-0.0303
-0.0949
-0.0510
-0.0316
-0.0335
-0.0300
-0.0522
-0.0309
-0.0477
-0.0371
-0.0371
(.0019)
(.0020)
(.0019)
(.0021)
(.0017)
(.0025)
(.0018)
(.0016)
(.0017)
(.0018)
(.0019)
(.0018)
(.0020)
(.0016)
(.0024)
(.0016)
(.0015)
(.0016)
(.0066)
(.0072)
(.0066)
(.0075)
(.0059)
(.0090)
(.0063)
(.0056)
(.0061)
(.0023)
(.0025)
(.0023)
(.0026)
(.0021)
(.0031)
(.0021)
(.0020)
(.0021)
(.0006)
(.0006)
(.0006)
(.0007)
(.0005)
(.0008)
(.0006)
(.0005)
(.0005)
-0.1013
-0.1135
-0.1180
-0.1046
-0.1199
-0.1567
-0.0953
-0.0936
-0.1292
-0.1000
-0.0697
-0.0685
-0.0421
-0.1329
-0.0593
-0.0611
-0.1356
-0.1147
-0.4226
-0.3509
-0.4672
-0.5273
-0.4085
-0.4007
-0.3444
-0.3389
-0.4165
-0.0806
-0.1244
-0.0791
-0.0714
-0.0325
-0.0835
-0.0163
-0.0114
-0.0812
-0.0548
-0.0345
-0.0399
-0.0372
-0.0550
-0.0412
-0.0518
-0.0355
-0.0442
(.0059)
(.0047)
(.0057)
(.0060)
(.0043)
(.0052)
(.0067)
(.0034)
(.0062)
(.0049)
(.0054)
(.0046)
(.0049)
(.0033)
(.0052)
(.0049)
(.0034)
(.0046)
(.0096)
(.0093)
(.0088)
(.0096)
(.0057)
(.0109)
(.0091)
(.0053)
(.0091)
(.0065)
(.0068)
(.0059)
(.0064)
(.0041)
(.0071)
(.0063)
(.0043)
(.0060)
(.0061)
(.0042)
(.0055)
(.0053)
(.0040)
(.0036)
(.0065)
(.0033)
(.0054)
48
Table 7b, cont.
Parameter
5
6
7
8
mI
-0.2590
(x6 D
-0.2414
-0.2304
-0.2926
-0.1265
-0.4719
-0.4468
-0.2284
D3
(63
(I
D4
(6 D
4
(X D
(.6
D
(x D
(~6
6
6D7
!hD8
(I6
lS1z
I'
Q
DI 2
p1
0.0407
I
r
^
-u. UU4/
0.0381
-0.0067
-0.0117
-0.0558
-0.0047
0.0331
-0.0769
0.0091
-0.0004
f314
t32l
13 5
16
1321
322
B3 3
324
325
-0.0769
0.1769
0.0107
-0.0120
-0.1368
-0.0067
0.0091
332
133
t34
1335
336
1341
342
-0.2338
.
(.0072)
",
""
k. uuoo)
(.0081)
(.0057)
(.0033)
(.0051)
(.0088)
(.0052)
(.0024)
(.0278)
(.0138)
(.0054)
0.0107
(.0113)
134 5
0.0245
(.0037)
1354
0.0245
1355
0.0004
-0.0009
-0.0558
1356s
is6
136
0.0399
362
-0.1368
-0.0232
-0.0009
0.1769
33
364
0
(0.0029)
0.0324
O.0056
(0.0083)
(0.0060)
0
0
0
0
0
-0.0681
-0.0052
0.0298
-0.0659
(0.0049)
(0.0086)
0
0
0
0
0.0077
(0.0056)
0
0
-0.0659
0.2037
-0.0206
(0.0283)
(0.0155)
0
0.0056
0.0077
-0.0143
352
1353
(0.0071)
-0.1496
1343
-0.0232
-0.0117
-0.0004
-0.0120
0.0353
-0.0052
0.0336
0.0324
134 4
1346
351
-0.3352
-0.4189
-0.3193
-0.3084
-0.3018
-0.3656
-0.5166
-0.4495
-0.2897
-0.4446
0.0399
0.0382
f336
f3 1
-0.0206
-0.0025
0
0.0099
(.0028)
0
0
0
0
0
(0.0135)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0336
0
0
0
0
-0.1496
0
0. 0099
0
-0.0681
§T
f2 T
3T
0
0
0
0
0
0
0
0
0
0
0
0
0
4T
~i5T
0
0
0
365
364
6T
-
-0.2601
-0.3268
-0.2490
-0.2349
-0.2968
-0.1453
-0.4759
-0.3313
-
0
0
0
0.1743
0
0
0-
-0.2408
-0.3071
-0.2273
-0.2174
-0.2513
-0.2585
-0.4310
-0.3851
-0.2142
0.0351
0.0050
0.0133
(.0079)
(.0040)
(.0076)
0.0017
(.0055)
-0.0081
-0.0470
0.0050
(.0043)
0.0057
(.0022)
-0.0063
-0.0026
-0.0065
0.0048
0.0133
-0.0063
0.0335
-0.0172
(.0071)
(.0057)
(.0026)
0.0158
(.0053)
(.0173)
(.0105)
-0.0391
0.0017
-0.0026
-0.0172
0.0204
(.0118)
0.0074
-0.0097
-0.0081
-0.0065
0.0158
0.0074
-0.0027
-0.0060
-0.0470
(.0035)
(.0051)
0.0048
-0.0391
-0.0097
-0.0060
0.0970
0.0005
0.0007
0.0054
(.0002)
(.0002)
(.0003)
-0.0012
-0.0001
-0.0053
(.0002)
(.0002)
-
49
(against model (1)) is 112.8, which would indicate that the assumption
that energy is separable from other commodities is not supported.
Model
(4) imposes the restriction of explicit additivity (together with stationarity), and since the model is homothetic, this means that all of the secondorder coefficients are zero.
The test statistic for this model (against model
(2)) is 442.3, and this is also well above the critical 5% level (25.0 with
15 degrees of freedom), so that additivity cannot be accepted, conditional on
model
(2).
In the last four models country dummy variables are introduced so that
the first-order coefficients (ai) can differ across countries.
Model (5) is
stationary, so that the hypothesis that the a i's are the same across countries
is tested by computing the likelihood ratio statistic for model (2) against
model (5).
The value of the test statistic is 1345.5, which is well above
the critical 5% level of 64 (48 degrees of freedom), so that we must assume
that the ai's indeed vary across countries.
differ considerably across countries.)
(Note that the a i's in model (5)
In model (6) we allow the a.i's to vary
across countries, but we add the restriction that energy is explicitly groupwise
separable from the other categories.
The test statistic for this model (against
model (5)) is 2.92, which is below the critical 5% level of 11.07 (5
freedom), allowing us to accept the assumption of energy separability.
egrees of
We view
this test of energy separability as more definitive than the last one, since it
is based on a model for which the ai's vary across countries, which we have seen
is essential.
Model (7) adds the restriction of explicit additivity.
The
test statistic for this model (against model (5)) is 180.65, which is above the
47
5% level, so that we do not accept additivity.
47
Finally, model (8) includes no
We also test the hypothesis that the ai's are the same across countries conditional
on the restriction of additivity (model (7) against model (4)). The test statistic
is 1607.6, which is well above the 5% level, again supporting retention of the
country dummy variables.
50
restrictions other than homotheticity.
It differs from model (5) in that pre-
ferences are allowed to vary over time.
The results of these tests indicate that a non-stationary model based on a
non-additive indirect utility function with first-order coefficients that very
across countries is needed to estimate price elasticities of consumption expenditures.48 We now retain models (5) and (8), and present their implied price
elasticities in Tables 8a and 8b.
Note that the own price elasticity for energy
is close to -1 in both models, which is what would be implied by energy separability.
Own price elasticities vary between -.5 and -1.8 in model (5), but
only between -.9 and -1.4 in model (8).
The rather large elasticity for food
in model (5) is somewhat surprising, but this probably does not reflect a true
price response, and is instead the result of food prices rising slightly with
food shares dropping considerably as consumers spend larger incomes on other goods.
This is supported by the lower estimate of nFF in model (8) where preferences can
change through time.
Cross-price elasticities are presented only for energy in
order to save space.
Most of these are near zero except for food and transporta-
tion.
The negative cross-price elasticities for energy and transportation are
surprising, but again may represent something other than a true price effect; as
energy became cheaper during the 1960's, the infrastructure grew that could make
possible expansion of the transportation and communication shares.
We turn next to the breakdown of energy expenditures.
estimated, and the results are shown in Table 9.
homothetic.
Ten models were
Of these models,
ine are
Again, we had considerable difficulty estimating non-homothetic
models, and the one that we were successful in estimating - model (5) - is
somewhat suspect, since although convergence of the nonlinear estimation procedure was reached, the resulting parameter estimates are rather large.
48
We must stress, however, that these tests are all conditional on the assumption of homotheticity, and we have not tested the homotheticity restriction
since we have been unable to estimate the non-homothetic model.
51
To
-o CNi
C',l %O
I
Lrl
,
oo
o
I
HrH
H
I
I * I.
f-...
I*I
1 1
N
U4
o
t.-
H H
,
0o
I* I
I *4
1I
*
-I ---
c
,-
I t.1
. H
I
1
II
*
1
) L
*
*
II
0O-
%.D -
oc n o o,
I
0
I
-CN Os
co
. * -a*
I I
n .' c'o
*
I
I
I
1
O
I i II
H H
II
H
1
HH
r
%O
I
H
I
,0
Ln U)
1 1
I
*
oH H
,1t- OO D
N
, % r-
0c
H
HH
cnH
o
I
M n
I
H-I ,
.
oo
0 cn
'
m
I
I I I I
H
C..4~
00
·
-I
eC-.
Lrn-
H
,-c,
Io
,--i
..- ION -.
H
L)
4J
(U
o0I
0
,.-I
r-q
cu
0
z
oJIJ
II
0
,-I
~4
-H
0
n~
CN
I
U)q
H
P4
l.
I
I
-o
,.. o'
HH
H
Iloo
o
o o -I
~. ,' oo
HI HI HH
I I
I
I
HH
II
HH
I
,-
I
mQ ,-I -Iq
I
I
I
I
I
I I
oo oo
1
Oc.
HH
·
HH
'o r-
C
I1 I1
I
I
oo
I
I
I
I
-
oo
*1a *%ts -
HH
HH
I I
I I
I
I
I
I
I
I
I
I
0I *0.II
- a o00Io
'
I* II
II
*-ao
,Ia o
co o
T ONcc
o,10
IO
m '~oo
IT ..i-I1Il
-,1
M~ c,~
m.
roo
I I
-ac-)H
OoOC HH <.O
HI7
II
·
-Im-I
.c
oOO
c,,.. O1-o
cn, Ln.-I
o o
o-L~ co
,,,
-I oo l
r- n
co 1l
@. cn -I l un
CI
-I2 LI I0I
I
'-. o~
U)
I I
'
I
r-M~
¢
U0
O
Lm
I
-C
H
.1-
U)
w
I
oqI 1-4
,4,4
I coI
cn
U)
a)
0
H
I I
r-.
Il
1-I XI
I r-II)
,-q
-4
w
hII
U)
0
H
1r-4
N
HH
I
.--.
11
bo
p
9-I
I
1 1
I
Ml
H-IH
D.o' O 00o 0 0
I oo
Cc
oo
-. co
II I I
I
I
'0
II
H
U)
0
FzJ
mH
0O'.
HHI I HH
I I
I I
I* I
1
.
1
HH
o0 o
I *4 I *
1
1
Ln H
L U
oIcs
oo 0
c
I *4 I
1
-a
I
I
1
N
9:)
c'l oO
CcX
riI
4
co
U)
Io I
%O
rV-
I
c7 a% 'Ioa
7
H
I
%O
T-
C-100 crco
C\00
ON0'ON '.0N0a'.0%0
H H~-- H H4 r H
I H
coo
cc co " co
D %0 o. o'D o CO oD %O0
CNJ00
H co
% %O a-lla.
H HIr- r-I
-I -H
-H
H-
I
9¢
p
rq
PF
5= r-7 97
E,
FH
9='
w
F
F
~~~~_r
%=
-¢
5=
~
S
5=~
4H
H
N
H
5:-
1:
52
0:
oo 0
-1-1
HH,
HH
I
I
I
H
Hr-
I
C'r <
I
O~0O
C!
I
I
- oo
c J
oCi
-O
H
e00
O
I
I
I
~-
I
-H
I
m
OO--o.O0,cm4\
Co<D
-4 r-
I
I
0
HH
I
I
I
I
cn 04
-, 0
0o
H
I
H IH
cn o
OC
I
oo r-
H
o
r-oo 0o oL
r-
>~
C"
c4 c1
I
CN
c; ooc oo
c; I : I C I C;
Io
C~JC~4
HH
CNl
lt~
I
I I
U
I
I-4iT -I I
f
00
C
,-4-I
00 0 0 0 0 00
.O *
*~ .
* *
I
.
I
I I
CN
o* Oo
o o
0i
V'%
HH- 0 o0 00o
ooQ
o
I
I
00I
*;V-4
* ?-.1
*00 *
00
* * ooo
*0 *
oo~~~
oo oo
0*0I
1-4
w
4.
r10
r r
-t
H*
co
*
1
i-a0
4t)
1
1
1
4 00
OM
MC4 JH r-*r
Ln
Lr I
II II II
1
00
c c
oH 00
oH
00
Co
C~CoC
HH
4 C
LI Ln
I
cn
,-4O om cao oo o0D coo0 c 40c0
c1 N c,
I I
I
u'r N
I'D '\ ,- HII )
*; *
0sW
-II
'Ii
0
z
o2
*-I
0
l-i
0
H. H.
1
c' D H'.0O
HH
.
HH
OH-r-0
O
-. r.
00r
o
0o
HI
I II
II
cN
,-- mo
H HHH
-I-4
OMa
,-i ci
II
III I
II
IIHH
r- %
O
C
4J
Cd C-
II
II
* * * 1
tt i.N
I
I
1,0 1~0
N C l -l C)
co
U1
cC r,H-
,
Ced4
H
EH
c00C
C*N1Cf) 00
Co 00
O-
O
HH-
-,Ct
Sf1
ON ON
c
0
C14
z
O H
--I
c
* .
'
00
%O ONcc
oO
OO 1.
Lt
0 o0
I I
e14
g-i
-OO 0
HH~1
O0
LLr)
')
cot
0
00
HH1CxCq
HO
i uI
--m
c~~~c~~
cC;
c c~
;c c;co
-
C'O'
He
Ico I
o'<
02
0
i-4
I-4
0
02
-4
0
.
H
p
C, ON0 H
w
4I4
w
0
1.1
E4
I II
HH
00 ,-I
I I00 I)
Io In ,-I
I Cl
I cIC4nC
H
I I
I
I
U)
H- r- 'OrC")
O
-H
HH
i I
~0
II
H
I
H
I
I
CYC
OP-I
H
I
H
"
00
I
I
q
I
I
cN c
0 0 00
' H m -- NO oH
O MOC)N C 4N H N
I HH
I4
I4I I00
III IHH
II I
I ,4
I IHHI IHH
0
~1)
0s o0-
ONC
-
cn ur
-
H0
-H
I
I
Lt1L
oC It
0
I
OH
I
I
0
I
I
o
L
n
H0H - C-N4
HH
HH
HH
HH
I
I
II
II
I
II
I
o of
00
I
I
HH
I
I
00
II
008
1
C
'- -I O -I
-t'
< r
6
II II
00C1
cr
CS
l I
N-.
,r.
lCq
c4',-4
cx
lr,.I
00
Ot
oH
I
O 00
c N
e14eN
c III
dd
H.H-C) Cy
cH Ho
H ,a C-N
I
If1
0
Lr L
000000q
I
II II II II II II
ll
H HII
1-4
Nc
Cs CS
* ** *
I
00
Io C;
I
* *
I
C)
00
i
r
*8
(o O
l
* *
e- tC'4
0o O O
I I
I-iII
-! e"
.0 C'4 N
HOt~r! I l-I
1l
to
p
oo~~~
oo oo~ ooo
I I
-,
C
* *; I* I *; I*I
*; *
O O
H 00O o
0
i
0
I
I I
0I,-0
0 00 0I 0I 00
II
-0
¢'0aoo 0oco0
coo
@oi
a)
c00 c c0 00
N
C >0
C O
c,, csd
*9
oo
9o0 09o
CN C'4
00o LLOo
00
i ICI 9 9
I-I !-I
00 o 00
OO
OO
*9
cHc
csa;
HeN
HH
C4 oo0
O
0000Or-
I uI
C;
.D Lfl
l0)
C)
00 rII)C)
*6
00o
oo oo
C',OO
%e
m"a
r-I --.I
1.D%,.
* 1
I
I I
I
C
OO
*d
00o
H
U)
1-i
400'IO
%co
HHaN
r- -I
Cd
co O% ", Cq I 0 ec
,4"
'IcO
HHN
H
HH
,.-P.
I v4--It-4
-ir-
C,oO%O
'IOX
HH,
r.4--i I
a0 oC oiC O
O ('400
C",
a0 %
C,, 00
%O
%~O
%D00
C'D
O
O
HH o,%H
HH
HH
H6%
N
rv- rq 4 ,-4r'
l
r" r-i V- -.V4 r- i
C",
oO
e.o % C',
eo~00
H ,N Na"
vrI r-q
l
-I r-i
"4all 'IO
' 'dD
''aHH
-- oO
HH
e, 00
r-a r-4
II
4c
cd
H-
I
%=
~ I 9n
e
roH
H=
w
w
r=
9
m
r-,
p
w
$=
PLI
7-po
E-4
ap
1:4
r-P,
p
w
9P
w
C?4
w
9:,
:p
53
HcTON
OHC'J "
01
'a
00
Oq0
0
I 000O0000
II
I C)OOOH
I
~ 0O'I
~ -~I
0rr-
0ONC4
.
00
gC)zt
00
o0
r4
0 It r.0-#
cOHrn LL~~~~~~000
r_ IOP
z
'0
C)
"
....
·.....
-**
0
I I
II
,.
S
-It u
C O
- -rI
c0Jr
C) -0
H 000
)1 "L qr-C
.
i
I
1
i
i
i
i
1
·
i
1
i
c000
uMco0
·H 4o 000
4
o
o oo
(V
00
·i
00
00C
O0
%OI
cxT 0 a-C)
) C' H
-H
I
O
I
U
tO
M
r -I
co 000
000
n rOm
0c 0
0 0cn0 1
0H
'IO
4i.J
a)
IC4
.-I
a)
H
co
U
~ r C) C
a)
Qr4)
'C14
-tH
%'
-r%-
*- -I000
- 000
C)
s-'
H
00
%n
0
CDC
C7U
H)
II
Ir- H I II
0 ;C;C
i0 0
i i C Ii;C;C
iii'.C)
iii
CNrILn
0-1t
0 C;
d00m0
OLn00
00cq-q0
ON
- C')
C~00
p..
-I-0
C1
Ue
4..I
r.l
4-)
r'
00~'0
!....IL
~~~~-
O-4
rV)
e' J'C; C;.C;
N- C. -* H '
-~I
0 0 tf 00 cn a% 00H
III I I-I1I%
co I
·
in00 ON00000000000000
I
I
II
'C-Jr
C;
0COo
0
'J-c'
~ -4 It
)S )
-4
N
N
O0
C'
0 -tr000
Hz
0 ~rC N1re'P
000 C
)CDC
; qr- CDC
III
C')
cq
)C>0C
I
I
.C)C)
C)
54
Table 9
8
7
Homothetic,
stationary,
count.dum.vrbls
Homoth.,stat.,
additive,count.
dummy variables
-0.3630
0.0283
-0.3308
-0.0368
-0.2560
-0.1142
-0.0624
0.0860
-0.0335
0.0304
-0.2632
0.0717
0.0384
-0.1214
-0.0522
-0.0030
-0.2310
-0.0303
-0.2611
-0.2192
-0.3262
-0.3631
-0.2450
-0.0501
-0.3231
-0.3101
-0.1085
-0.4062
-0.5458
-0.4146
-0.6386
-0.3777
-0.7836
-0.6115
-0.5450
-0.8277
0.3098
-0.1226
-0.0563
-0.1309
-0.1226
0.1874
-0.0259
-0.0389
-0.0563
-0.0259
0.0262
0.0560
-0.1309
-0.0389
0.0560
0.1138
-0.4537
-0.0157
-0.3461
-0.1585
-0.2586
-0.1743
-0.3011
0.0000
-0.1209
-0.1370
-0.3689
-0.1201
-0.1239
-0.3005
-0.1330
-0.0824
-0.3335
-0.1967
-0.1715
-0.1674
-0.2524
-0.2654
-0.1794
-0.0027
-0.2188
-0.2454
-0.0204
-0.2378
-0.4480
-0.2814
-0.2378
-0.2613
-0.6899
-0.3978
-0.4211
-0.6620
0
0
0
0
0
0
0
62T
-0.9116
-0.3367
-0.8078
-0.5990
-0.7065
-0.4047
-0.5678
-0.3417
-0.5453
0.0293
-0.2522
0.0693
0.0379
-0.1247
-0.0343
-0.0150
-0.2240
-0.0331
-0.2329
-0.2016
-0.3022
-0.3342
-0.2220
-0.0372
-0.2987
-0.2884
-0.0823
0.1152
-0.2095
0.0407
-0.1047
0.0531
-0.5238
-0.1485
-0.1459
-0.3393
0.0791
-0.1125
-0.0429
-0.0763
-0.1125
0.1759
-0.0292
-0.0343
-0.0429
-0.0292
0.0270
0.0451
0.0763
-0.0343
0.0451
-0.0871
0.0104
-0.0013
B3T
-0 0005
a4r
0.0086
alD2
cXID
cx1 D12
aiD 3
ax JLD4
a1 D 5
cxlD 6
( I D7
aiD e
aid 9
c2D
(x 2D 2
cxD 3
cx!D4
ca2D
A,25
aID
2 3
aO2D
a2D8
a2lD6
a2D
a2D3 9
aa(3 D
D4
3
2
CxD
a D6
3 7
a3D4
a 3D?
a3D e
a 3 D9
a4 D 1
a4 D 2
a4D
ca4 D 3
a4D 5
c 4D 6
a4D7
a 4 D8
cx~D9
all
$12
813
31 4
a21
822
823
a24
53i
332
a33
83~
a1
a42
43
a4 4
_
(.0898)
(.0574)
(.0810)
(.0920)
(.0769)
(.0481)
(.0814)
(.0666)
(.0850)
(.0770)
(.0480)
(.0726)
(.0782)
(.0689)
(.0393)
(.0626)
(.0547)
(.0737)
(.0326)
(.0209)
(.0298)
(.0334)
(.0282)
(.0184)
(.0301)
(.0239)
(.0310)
(.0423)
(.0286)
(.0137)
(.0347)
(.0120)
(.0074)
(.0016)
(.0013)
(.0006)
(.0554)
(.0307)
(.0502)
(.0557)
(.0485)
(.0240)
(.0433)
(.0369)
(.0522)
(.0731)
(.0404)
(.0700)
(.0735)
(.0667)
(.0290)
(.0501)
(.0483)
(.0706)
(.0269)
(.0146)
(.0251)
(.0272)
(.0243)
(.0111)
(.0198)
(.0179)
(.0259)
(.0317)
(.0235)
(.0100)
(.0348)
(.0117)
(.0200)
(.0210)
(.0210)
(.0170)
(.0148)
(.0162)
(.0138)
(.0138)
(.0138)
(.0145)
(.0153)
(.0153)
(.0122)
(.0108)
(.0118)
(.0053)
(.0053)
(.0053)
(.0055)
(.0058)
(.0058)
(.0047)
(.0041)
(.0045)
(.0069)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(.0190)
(.0190)
(.0190)
0
0
0
10
Homoth.addit.,
country dummy
variables
9
Homothetic,
Country,dummy
variables
Parameter
.
Cont.
.
-0.6452
-0.2191
-0.5375
-0.3440
-0.4382
(.0224)
(.0233)
(.0224)
(.0224)
(.0224)
-0.3777
(.0242)
-0.5104
-0.1855
-0.2884
-0.0735
-0.3015
-0.0567
-0.0624
-0.2410
-0.0656
-0.0130
-0.2720
-0.1411
-0.1434
-0.1376
-0.2242
-0.2382
-0.1531
0.0272
-0.1880
-0.2181
0.0042
-0.1380
-0.3419
-0.1816
-0.3554
-0.1677
-0.5839
-0.2885
-0.3243
-0.5746
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0120
-0.0040
-0.0018
0.0062
(.0230)
(.0204)
(.0195)
(.0220)
(.0229)
(.0220)
(.0220)
(.0220)
(.0238)
(.0226)
(.0200)
(.0191)
(.0082)
(.0086)
(.0082)
(.0082)
(.0082)
(.0089)
(.0084)
(.0076)
(.0071)
(.0011)
(.0011)
(.0004)
55
In the first six models all of the coefficients are constrained to
be the same across countries.
is homotheticity.
In model (1) the only additional restriction
In model (2) we add the restriction of stationarity.
Model (3) allows time-varying preferences, but adds the restriction of
explicit additivity.
Model (4) includes both stationarity and explicit
additivity restrictions.
Models (2) and (3) are tested against model (1),
and model (4) is tested against model (1).
The computed test statistics for
models (2), (3), and (4) are 10.44, 45.36, and 44.64.
These are all above the
critical 5% levels, so that we cannot accept the hypotheses of stationarity
or explicit additivity.
Model (5) is non-homothetic, but the stationarity restriction is imposed.
To test for homotheticity, we compare this model to model (2) (so that the test
is conditional on stationarity and on the ai's being the same for each country).
The calculated test statistic is 61.92, which is well above the critical 5%
level of 9.5 (four degrees of freedom), so that we cannot accept the hypothesis
of homotheticity based on this test.
Nonetheless, because of the problems
involved in estimating non-homothetic models, we continue to impose the restriction of homotheticity.
Model (6) has the same specification as model (1), but it is estimated
using only four years of data: 1960 (or the first available year, if later than
1960), 1964, 1968, and 1972 (or the last available year).
We can expect that
the use of data at four year intervals will result in a model that is more
representative of the long-run.
Then, by comparing elasticities derived from
model (6) with those derived from model (1), we can determine whether model (1) and all of our other static models - are more representative of the long-run or
the short-run.
56
Models (7)-(10) are homothetic, but country dummy variables are introduced
so that the first-order coefficients (i's)
can differ across countries. Model
(7) has no additional restrictions, model (8) is stationary, model (9) is
stationary and explicitly additive, and model (10) is additive but non-stationary.
We first test the hypothesis that the ai's are the same across countries by
comparing model (7) with model (1), model (8) with model (2), model (9) with (4),
The calculated test statistics are 900.0, 860.4,
and model(10) with model (3).
813.6, and 889.2 respectively, and these are all far above the critical 5% level
of 46 (32 degrees of freedom).
of country dummy variables.
This provides strong support for the retention
Furthermore, it leaves us with some doubt about
(Unfortunately we were not able
the validity of the test for homotheticity.
to successfully estimate a non-homothetic model that included country dummy
variables.)
We next test stationarity and additivity by comparing models (8) and (7),
models (10)and (7), models (9) and (8), and models ()
and (10). The calculated
test statistics are 50.76, 55.44, 90.0, and 85.68 respectively.
These are all
well above the critical 5% levels, so that the hypotheses of stationarity and
additivity cannot be accepted.
Thus whether or not we permit the ai's to
differ across countries, the data do not support the assumption of constant
fuel shares for each country that are independent of price.
It is interesting to examine the elasticities implied by some of the nonadditive models that have been estimated.
Own price elasticities for models (1)
and (6) and own price elasticities and income elasticities for model (5) are shown
in Tables 10 and 11. Note that all of the own price elasticities are close to unity
for oil, gas, and electricity, but are much larger for solid fuel.
Removing the
homotheticity restriction results in price elasticities that are more uniformly
close to -1, and results in particularly large changes in the elasticities for
oil.
Given the results of the likelihood ratio tests, however, we could consider
differences in the oil and gas price elasticities from -1 as noise.
The solid
57
0J2 r
;:
% C4
eO
00
,I I
I
I
1
HH
I
1
H
I
I
C.11 H H H H HH
I I
I I
I
IO
I m...I-0
I
On
It
r-
I
cd 4-ia)
r -I"
0000
I
I I
I
I
ONO
Cl 0oNH0or-
r·.(i
I
iK
4Cx
HI HI
HH
I I
HHo
II
II
OH
C)I r-I
l l
I HI
C1
C Cl
HI HI
Cl
-)
Hd
Uoo
4-i
%o
c0 o0 IH
o
I'o
0
c0
HC'C
Oe~
a)
4
0
*I
* .
o,
a,
,-4
,-HI
E1 o1 II HIHH
00o
mn
4-0
. I
I. .
L~
0
r
0
z
,-!
N
I C.I
I I
o C
H
z
H0
1£ o7
H
I
Lt)O Co l
OY -
r--0
f-r
IKI
H4
HH
osO
Ur
F-=C
O
-I O <'
o- H
C) , H
I. o'
. c4 0 0
c oI
II
I II
Q)
0 -a)
E-
z
I I
I I
04 -i
'Cr4
U0
Lr r-.
He
c~ o
~ o~ HH
cl ~- 00
o0 ~1 HH
I
I
ci-,-4-)
* -K HI HI
P4
- oC4 00c
14
tL-
I
~4-
Ho
OO
IIl
I
I
oda)r--!
4-)
0 c-cW
11 <i0
a)
a-)
0
r.14..
H
02
r-H
4-i
I
I
HC'
0
H
I co
I
I I
I I
I
Hi-I 0 - 0
mIOI
I
It
E
HH
I
I
I
I
I I
It~00C
HH4
A c4o- o0 o-
r-
1-I I-4
I IH:
O
r--,-C)ON ~l r-O r-0
-.
_o
I
I
'.O'-I
r-
ro~
1
'h
C')
0Go
w-I
,-I
Xf~
U 4-ie.I
CJo
Ho
q4 ) rq 0
zH
-H
-W
.1
1
H
H.-I
1II
IHHI 0II0
0 1
1
01
2
0a)4U
I-i "~so)00
H
e0 0-)
>4
-H
III IrII
r.H
HH
% Cl
rl
0
00
Cl l
m
C)i02,'C)
oI %D
c,
%D
I
HH
I
0
0
1-4
0
H--
o,Ilo I
CC vJ
OJ o)
O C'2
I
t_I Ioo a,I oo
i::(14
C1
C
I I
csoO
3c
EH
HH H H O _O- HH
)s
COl_1=
H:
r-..I -I
.
r-I -- I
-I,
I00 I 0 H
Il r-I
r-.
0II
HH
0,_
1 .
C\2
ooHH
4H
H
CCOCN~
H e-c
HHoo
C CHH
Lf<Q
FI
Oi
IIHH
-q
OO
HH
l
-
Ir!
'.
H
E.i
O
4i
N
0r.;
'-4
I
2
0
,-4
I
0
t-4
:t
lII 0 *-I, C
CI co
o.
o0
c,
c10
HH
HI
\."
C*
O O
CI"
C
H
02t
0
I
D
HHo
CY C
O
l
4-i
co
co
C')
O
('24
-
l:
l::
CV
a
'C
H
0a 00
H
n
a
:-
4-i
H
H0
X
58
00
:
,HH
* I
K *
* *
0 LC
e~%O cn cn
CN ,- Ln'-It,-4 ,-I
I I
I I
" n
ecn
I
I
1
I
I I
*C
*
KK4¢
I I
O
H
0o,ao
M
I
* *,*, a
* 1;
I
1
r- 00
r-r-Ho o H rH.
... ..
CqJ H r- H- I
II
Cm
Lr-
--T oo
n <
I'D
0 1,
-t .,T
" C"c 00oo00
e[~~i)ci
H-r-I4
HHq
c
.-I ."C,4
, Ln Un
o cn
o Hr 0
vDor-
CN
C,4 O
I II I I I I I
C
1
0
.4-J
ctS
4-)
1
C1 1
1
1
1
%OCn
* *
1
0
00 :
0
[0
" Cq
4
%DLn
r- 0
*. *
-1
0z
0
,cu
§0
Iz
00
oo4r--
* *
H
100'
aI
I I
IIL O0
I I
HH
I-
INr
I
HH.
*4 *
u r-
cyO'
C)cn
e
Kf
co
CUn
a)
U
z
4-i
CU
H
U3
I
U
.1-1
,i-J Cd
Q
(2)
c
.rq
rz
U)
H-
IO
I
\0c0
*
1.
O
vo
I II
HHr
.r1
I-J
oC
*qr-
0"~00
'-
$
co
I I
p
-4 14
U)
'-H
1 I
l
*~
9
,-I ,-I
I
0
I
CY)c
rl ,-I
li
,-I
I
I
I
00
eo.o
99
,-I
C1
4-i
--I
U
C4 4
4-i
-H-
.,-I
o
-
p
4i
U
-H
0
Ud
U)
* 00,-I
1 00
I
1-JH4
*~~~~~~~~~~
1
H
H-
r-
H
0oo
rll r--i
HH
r-4
U)
CU
PL
00 O
1-4C
II
r-c
H
,0
I II
r-I
00
oH
U
IC *
-H
II
I
I,--I
r00
r-0
o00
10-4
I
· . 1jH
1010 r- r-i
I
*
*I *
I
Lr
*S
O r-
II
co 00
l
-II
Co 4-t
* 14
1- CL 00
*
1
00 0
11D
*
C14 C1
-S
00
bo
11
M
It 00
I
HH
IIl
C-400
pC
HH
II
r-HHir
%D
10 -
0 0
HsooH H
l
l
c0l
-',
%,O r- r-I Ce')
COC"1
-4'
00oo 1-bH
L)00
C N"4
H
- r-I
'I H HH
li
-400
14 00
0
CU
1-H"q
H r-D
0 r-oo
OH r- roH
,)
0:
11
CS
H
0
o
%O
% 'O1
10
O
.t
4U)
U)
:>
c1 00
O
,-
c1400
ID
,
cr o
c
o
I'D
aS oO
,-
-I
o400
m
mE
,-- ,-I
4-i
CU
H-
0O
r- 14
1=
04
04
e'J
"
In
.
t
:t
CU
H
rx-
z
.1
HH
11
r-
U)
-I
r-
0;
o
(1
4)
En
59
fuel elasticity is considerably greater than -1, and the electricity elasticity
is stable across the models, so that we may reasonably take these o1;ticittep
to be in the vicinity of -1.4 and -1.2 respectively.
The income elasticities
are, of course, all 1.0 in the homothetic model, but differ considerably from
1 (at least for three out of the four fuels)
in the non-homothetic
model
so as
to suggest re-estimating that model, perhaps under alternative restrictions.
Some of the income elasticities (e.g. for solid fuel) are highly unstable, so
that for the time being we have no grounds to assert that these elasticities
differ from 1.
The reader should compare the elasticities of models (1) and (6) in Table 10.
Note that these elasticities are quite similar, and usually do not differ by more
than 20%.
This leads us to infer that our static translog models are probably
more reflective of long-run behavior than short-run behavior.
This is comforting,
since most of our estimated own-price elasticities (for the consumption model as
well as the fuel model) are greater than unity.
Of course the final determina-
tion of whether these elasticities are long-run or short-run must await the
estimation of a dynamic model.
In Table 12 we present partial price elasticities for model (8).
contains country dummy variables that allow the
This model
's to differ across countries,
and although these parameters do not enter into the calculation of elasticities,
permitting them to vary would yield better estimates of the Bij's.
In Table 13
we present partial price elasticities for model (7), which is the same as model
(8), except that it does not contain the restriction of stationarity. (Recall that
using our likelihood test we rejected stationarity.)
By comparing Tables 12 and
13 we can determine the effect of imposing the unwarranted stationarity restriction.
60
Note that the elasticities in Table 12 are generally much larger than
those in Table 13, and vary considerably across countries in ways that in
some cases cannot be explained on economic grounds.
This variation across
countries - and across time - is reduced considerably in Table 13.
In view
of these elasticities, and in view of our likelihood test results, we retain
model (7) as our "preferred" model.
Note that the own price elasticities differ considerably from those for
model (1) (see Table 10), particularly for solid fuel and electricity.
The
oil price elasticities are surprising because they are so large, exceeding -2
for some countries.
Comparing the estimated values of
22 for models (1) and
(7), we see that the model (1) estimate is insignificant, while the model (7)
estimate has a t-statistic of over 5, so that the data strongly support the
larger elasticities.
In fact the prices of fuel oil fell over the period
1962-1970 for most countries in our sample, and the size of the oil shares
increased considerably - usually at the expense of the solid fuel shares. Part
of this shift in shares was probably due to changing availabilities of fuelburning equipment, but (over the long term) much of the increase in oil shares
could indeed be attributed directly to falling oil prices.
Note that the cross-price elasticities between gas and electricity, and
between solid fuel and electricity, are negative.
It is doubtful that these
fuels are truly complements in the residential sector; these elasticities are
probably the result of the installation of new gas lines and electric power lines
at the same time during the 1960's in many European countries, together with a
diminishing use of coal.
Note also that cross-price elasticities with gas (and
the own price elasticity of gas) are very large for West Germany.
This is the
result of an extremely limited availability of gas for residential use in that
country - the share of gas has been about .02.
61
r-
I
0
-N- LCO-
00 D
II
II
* *
.1 -
.
.- .
ln
Lrsev
oqcl
co)
H4
-q
NH
CN
I I
HHo
I wI
OH
CoNO
,
I-I
r-4 -4 4Ar4
oor--
t"N
N,"
H-4
HH
I
I
I
cn
I I
I
I
I
I
I
·c -IC
-* C
H,--
T4
.Lr)
-
-
H
C1-
Ci
II
CA 0
r-,--
4IC-ICH H
~0
-
cq
0 0
NO
-4 A
HH
L
N-- HH
'
'
-c
'
''
" tH
,-IN
,c'
iC .
C' ,
N-H
o)
L"
HN
I
en ON NO
N
H.H
ulf eq
Cr-- t
o.
· ·
ID OO
O
HO
CN
'-o o
0 0
cI
c O
" cN cqC
i
'
-K .
Ci
NU00
t Ln
N
Ien
N
CMN HH ,-- O CN
C
0C
O
I I'
N
I I
a: oO c1
, c1 -- o 0o
00 ,-HH
NIN H
OI
0
C
0
C0)C
,,-I 00 No O
O
C
II
.- '
O O - H
I''
OO
HO
H-I -I
II
-)
O
0
z
0
00
~ -N
C CS
Ii
C
-I0 N"
I I
,-IH,rl ,-I
oo
cN
IIt~l
I 00
O
r-0
Crr-iLn 0'
M -I
I I
I
o
I) I I I
I
CN
N
N-~ (X
II
I
I
r
H-r. %DO
U en HO H-H - C'O
.
H O
HH
I
un
-I
CNN-
N
H O
*
00H
*' *
HH HH
I
*
CD*H
.H-I'
* jI'
I I
%C,-ALn'
Ln
* .
O
t %O cn -i -~ c
q
-I - N
0c)
HN
I I
c-I
-ic-Ic
0
HO
I
,-
H
-tU~
HH
icc
HH HH
Lu oo
r- r
I
I
LnCo 0 o
o0, O D O
~HIHI
II
nN
I I
II
N--
-i
H-
CNN
I
H
I
oo N
N -
C)
I
I
I
%C
%O
HH
% I'DO%O
N O:
,--r-I
1-I --
I
I
,- y-
-I
HHf
N
0 CO
PI
N uLn L
HN
U,
Ci
N
c-
I-_
H- -I
. *
CN
T '0
-.
Ln
I I
*
m
m'
t
*
r- q
O
- -
-I
cN "N
*
N
Hr-H
-I
N
c· H
i - CN
· CNNCI
CN
e
HO
HH
CNH
'c
* **
*H
·*
-cic ic
-ic-K L
Oq Cn
O O
,- * H *c*
·
·
cq 0
C4O0
%DCY
'IO 'IC'D
9-1 0 HH
-cn
r
N Cn
r
N-N- OH
0
HOH i
.
Nc
.
-.. *
*
·
'IO'O
HHa%
l
r
I
On D
*
IH
* *
N
-
N
cn
-C
. .
l ,- O %O
N H - TN4cV')
r- r-
.
nN
un -I
0 CN
NO o
*
'0
, t
HH
. .
N--
H4H
ON00
'C M
Cn
.-n m CN O O
. .
. . .
I*
00 00
n a
C4 r-
II
NONOo
HH
"H0H
N oo
ON I D
O NNO
0O o N N
N %O
NC
-- -- 0%C
H-
C-,
t
_-
cv)
_
-
C)
C
c-I
HH -
4C-
HHir-
C"
(C)
N
H
00
H
I
I
m
H ,I I
I
,O') r-1 00
0 I
I
CoH1
I ' -T CNH CNN
I I
4i)
Hwz
. .
LncN c r-
L()H
4.44
u)
1
*I
NCO
NO
cnN
*· *
*·
I
0OO n
-q O O O O
* .
. . I'I
I. I
H;
4-A
4
- K-K
*
-K
* *
I
O
0 -I
t C~ NO
H
O
- *
-*-!
OH
H
I
H
Cd
- %N
H
i
H'CH H
r4,
Ni -H
°
D
N-* -
LI'~
Q
0 o'P
t r- rO O
Ln
I
I
N O N~ONON NO N~O
-'-i0-t oo-0
£OD-OD
,%HO- 0£O O@0
.1'
en
Cl,
*
'
*T
5=:
Cv
-
C7:
62
lY
c oC, 0 L
D "C r~-D
%O%D
0%
,D
00(
II
;3
I Il
II4
I I4
I II I
cn
W)
*n U)T
*
*II
:0
0
r-
o
H
LnH
r
I
I I,I.,
r-
o o
0%MC %O 0 r- o ,4
* r-* M D.
-I u'%
00
0 oHo
c H
4 c8c~
c~
I
en
t
H H HUAO
' OH
00 . 00 ..
I""
HH
Lun o0'o
*C
.1cC C C 00
ICK
* - * ... ...
C
e4J
Dco
0
C~r
4H
- -I
0 0 00
00I
c; c;
c8c
00 00
O00
HO
O c00
cAO
*8
I 6I
Co
II II
00
HC4 N
II
HH HH 00
o o o0o HH~
-o
00 00 00 o --I oIIro
II
II
00
II
C L
r-%
I.<1 ,D C1
HH
I
I
C
*CIC
COH
HHc; HH
HH
H
*.M
CU
%
C0%
0C
%' O
L I4
OH
4C %-0 HH
;c;0
H I
HH 00
II II IIll
0
00
Cn
.,
n
COLA4
4I
HH
II
0
?O HO
cn r-
COHC
l
I H- O
O 1'
0 I0I
C
o o
H,-
r
LAna
cc; HH;
c; HH
4 4
0
00 c0IIl
004
Ln00
COO 00
OH
OCO
0
*
*
OH
%LA
00
%?O
o rCe
%?'DC-iH
H-
c;;
0 c
0
0
0 O
-
0 n 00
*
8c;
N CA
N
o~4Co
a00 c J0
0
-It O
.-0
c8c;
ICC
* C;
o o
II
c;
O0 C4l
H- -
1Cl
1
1
I I0
I0 I
*
Cs O
H*r
C
Un n
P-
%C
00C
oo;C00CC
o0o
1
1
I
I
*- M
o o4 o3o
0
z
-W
'44
4-i
H
-
HA n
I I
I I
I
oIo
I
o
'0o 00
0
0O
OH
I I
L
CD
CN4H
HH
00 00
00
*I*IHH*I**
CO
CM o
'
?H
1%--.? I. C
%?'J H
HO C4C4 HH0 0 0 * 0* 0
I* 68 68 68 66 C;C00
II
o
LA
M
r-
1-4
OH1
I
r-- c
CO1 I
- O
-H
%?
LuAO
COCD cH
O
HH
t-i
HH
H
H
C4
C'I
C'
L
C.IL
-N~ I
I
H
H- I HH
4c
II
?
0I 0I
-4
r- i
I
CON
I
i I
00
or
00OHI
O0
I
II
rT4
COC
HO O
cC4 LA
O-Nc %0 N r -% O 00 O C4H
P
,-N c .
HO cn'c0
HH 'ON
· D c H -tc HOH H-- H-t
H cON
C* *
H00 00 I1 I1 HO 0000000000 o O q q oIIo
44
0
I Iv; I I 4;c
I4I c;I c;
I
-tH ; 0'-:i
c;
c;
I ;
I
00H
I'd
II II88
r~
8 C;8
8 88
0
'T Ir-I ~
~t O~
L.,
o W
O,-;-Il
· '1' ,-,-A~ OH
**
*I
-H
* *
ICC
**
0'0
oH
0000
o00
c;
00o
00
rNLA
iC00'Nfl
HH
00
a N
00 D
I I
Io
II
I
00**
CON
00 0 I 0C
I
.4
-H
,-.I
ul LACC;oO C
co
H<eH
40lK sc*r-H.. L.n I .O
?HO N
I I
I I
I I
0c"
? H-
c4cC'C
cnc Ifc;c
* O**H * H*I 1
p-4
I I
c4LE 00N.
cOH ,-I r-I
0c c
00
*
C;
COC 0"O
C'J%
c
C JL
0 a%
co4
C400
co0 C4 0
0 cN 0 CO
N 0 e0
40 C 00 C 0
'OO ON
'O'DO
%O
'O
0'0 0'O
0'ooN
0'ON
O O 'O %O 'o %O 'O O %O O 'O O 'O O 'OO
'%O
ON0
0'0
0
0'ON
0'0a
O
0O
0'0
0
0'0'00
ON'Ol
HH~ID. H,-I H r-I HH4
rqIH HHf HHr41
'hO O
HH
HH
- -f94rHH
0 0
0'
HH~1,
-I
C~4I C C I~
00 0 c; O00 co
i- 1c;c
II
I I
CN40
W
04
HH
WHr-I
4
-4
N
q
C"
C"
o0
C
CN00
0'
0'
'.O O
HH4 H-I
I'DIO
HHfrq
CA
00
I I
aI !-
4-i
Wc
i-4
,t
&
C
r')
(
=
r
.
_
-
C7
o4
1=
c)
..
-4
N
9=
,-~~
-
N
°
gr J=
~
~ ~
~
*
9=
~')
('~
4
:
(
¢~
=
..1-
-
*
5=
,.1.~~~~~~~~
4.
63
I II
Un I
,_
4
_; 1
C4
Lnr-ir l-
oo
I CNI
C CI
co r-
o o
-H
O r-
0 U3
0
HOI
H
~-~o
r, ro0 0
~--~-
I I
I
KK
*
Cn H
O r-
00
00
0. 0
cN
I
1Ci
C1C
* *
o'c
0c
'r-
r-.
o
r3
,
rI
r rA ~r--. i .
co
%D
c4
I
--,
I
r- r-_
Coo r-ocr-
C.-,
I .
r- -4
* c0
o o
HH
HH00o
a r-
oq
o
-.-CuI
:T Ln
me',
* C
D U'nLn
Inr-r-4 -T
H-H C, r-i
ILn v-I un-t
co
I
I
00
r-4
-0
HrIcq
n - o
~I I
I I
' .C
-'O
o o
Oq '4
- r- *~
*
o CI
I
I
-
--iL
O r-
0-
00
I
o
N N
O
Ln
*~
8 00
008 uo
c'Qc
I
-- -1
00
00
I
I
o o oo
00
o c
00Ir 0 0
I I
oo1
~-oo j
go
C; (J
I I
0 0,
0 o
I
00
00
C' c'17
'1)
~..oo oI
I
I
c
·-I 0
Hc
I
HO
00CC
.
I r-
0
z
HO
oo
(O
O'
r-.r'-
-.
II.
I
Ln
O
r-4
-4
I I
C'-)0
(
N 4 C)
-
00
II
I
o cO
o'001
* *- *
-*
-ic
I I
I I
o,-
o
oo 'o£I OI
xOD
.H
o
O o
0oo
o
ur
0
0
z
I
I
CL0 HCo
- "
co
00 COO
00
I I 00i
CHOC)
I
I
4I-J
.i
Or-
C*i-I
I
I
I
I
I II
'H --
CY) -
C0
I
0C r- r-II
I
cn a%,-o
Hmo
o3o
Olo
Lfe'
-oH
0
II
I II
II
J
cd
u3
'lc °°q T°,
oc
d..
I
r--O
o'-o
Ldc~ c~4tc; r
~o, o3 o~ 03Ch
. d.
0
m
00o
O
c-i o -0
oo
O Ol
-- C14¢,, CY)oq
C,1¢,, ,-- I
go
O C
* r-.
gg
I
,
I
II II
I I
C* ' . Lo
.c)
o. .,I) .N
00 00
J~~c;
gg6
:
co
coo
.J.
I
I
'00 00 00 00 00 00
~4
c~'u3 ~,1o
0~0~
-IC 14 :-
0
-4
UA
,-o
HeN
H-i
I
o o
-
0
4-i
I
I
-H
I
oN
00
I
I
K
.c
.
.
I I
I I
M-
N
I
.I
.
* ciC
I I
L
HH<
H H
II
eNe
l
(N o
¢,4oq es N
uLoc
eNeN
r- -o
HH~
CQN H
-
CV
.
oeq
IIll
KK C ICC
-H
HO
o c; r-C; c;
~C) 00
00 a00
0000 00
c
c)
I
HcoH
nH aH'r-H
.H
'-0r
,1
u
1 i.1' 0' c,
o"1
*
00
O0
OH
00ul)
00O
, -.
v ~
ooa
C1 C1
001o 0o4II 0I4 0-I 0-I
0 0 00
I I
HH
00o
I
I
CN
r-I
I
I
U)
,0
Os
Hr-
0
fr4
c r
O
HH
I
I
H
coo ,H3o
o uL)
III H HII H 0I I 0
r-o C oo
oC0O
II
io
o
u
o~o
o
-(v,-O~c;~Q
¢,1
o c~c;
.rc;
eN~C~)
0
r-
00 v3
- r-i 'N
O ~0
n N
r-
r-
I II I ~
~Hgo;e
eH
r1
H
-g
N*-g
NNN o
0
'-4
C' o
C)
oH
e-o D
Ha
4
Co
H-
'qC4oo '0i
'cDD
O
cs
$:7
OH
(
m
F
,i 0
O
H
-t.t
:
t
F=
c
oO Cq o
NC0
D
C'IO
HH
o
-
=
1
HH
~~~~~~~
-4
H
C100
,
ClC0!0
eNCO %D D
H oH
Y
4
.1
F'
1=
-
HO
.t-4
N
Cq 0
¢,
%
eNCD DO
H aH
-
I
C-i
004
9=
F
C1 0
CN co
CNCX
% I'D V.
HH
ci
04
C1
00
%DO
C
0
%O%
H H CHH aa aH
H HH
c.
N
0)
0)
et
r7
S=
cn
Z
_
,0)~
C14 00
%D
a H
C'
00
%0%
%
H HN
~ ~~~~~~~~~~~~~~~o
-t
9
-I
64
These price elasticities are "partial" elasticities, in the sense that
they do not account for the effect of fuel price changes on total energy
expenditures.
In Table 14 we present total price elasticities for our two
Recall from
preferred models - fuel model (7) and consumption model (8).
Table 8b that the price elasticity of energy, as calculated from our consumption breakdown model, is close to 1.
This explains only the total fuel price
elasticities are nearly the same as the partial elasticities.
5.3
Dynamic Translog Models
(to be added)
5.4
Logit Models of Fuel Choice
We estimate a number of static and dynamic logit models to describe
the dependence of fuel shares on prices, income, and temperature.
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), per capita
income Y, temperature T, and, in the case of the dynamic models, lagged shares.
Recall
from equation
(37) that
this leads
to a set of three
equations
that must
be estimated simultaneously, since certain coefficients are constrained to be
49
the same across equations.
We therefore use iterative Zellner estimation to
estimate all of the models.
49
Note that even without cross-equation coefficient constraints, simultaneous
equation estimation is desirable in that insofar as errors are correlated
across equations, it yields more efficient parameter estimates.
65
Our static logit models are of the general form
a
log(S
4) =
where ai4k =ik-a4k
i~k
I ai4kDk + biPi -b4P
k=l
ci
4
+ Ci4Y + di4T, i=1,2,3.
ci-c4 , and d=di-d4,
4
ik-4k'
ci4'ci_'4'
i4
4k
(59)
as in equation (351. The Dk are
country dummy variables (countries are ordered alphabetically), and the fuels are
ordered (1) liquid, (2) solid, (3) gas, and (4) electricity.
Parameter estimates
for four of the more promising static models are shown in Table 15 (t-statistics
are in parentheses, and the R2 for each equation is shown).
Model (1) is linear in prices and per capita income.
the intercept dummy variables are highly significant.
Note that most of
The price coefficients,
on the other hand, are insignificant for solid fuel and electricity, and significant but with the wrong sign for natural gas.
The income variables are all
significant; as expected, increasing incomes lead to greater use of oil and
electricity and less use of solid fuel (we have no a priori expectation regarding the income effect on gas).
Temperature variables are added to model (2), but these are all insignificant,
and they provide no improvement in the price coefficients, so they are dropped
from the remaining models.
Model (3) is the same as model (1), except that price
dummy variables areintroduced
for solid fuel in Canada and the U.S. (CNSD, USSD),
and for gas in Norway and West Germany (NRGD, WGGD).
There is virtually no solid
fuel used in the residential sectors of Canada and the U.S., and little or no gas
used in the residential sectors of Norway
and West Germany.
This is not because
prices are too high, but because in Canada and the U.S. other fuels are readily
available that are cleaner and more convenient, and in Norway and West Germany
the extremely limited supplies of gas are not made available to residential
consumers.
Note that the price dummy variables are indeed highly significant.
66
Table 15 - Parameter Estimates for Static Logit Models
2
1
Linear in
prices, income
Parameter
al
0.3153 (0.56)
3.5200 (7.55)
-0.0818 (-0.20)
0.9231 (3.12)
-3.2645(-14.94)
-0.8303 (-4.86)
-0.3529 (-2.44)
-0.4702 (-3.29)
0.1918 (3.02)
-0.6383 (-4.12)
-2.4808(-18.61)
-0.3018 (-5.57)
0.8470 (5.29)
-1.0693 (-6.13)
-0.1109-(-1.44)
0.0375 (0.09)
-1.8056 (-6.44)
-6.1332(-27.96)
-0.5859 (-2.30)
-0.6565 (-2.74)
-0.8341 (-4.78)
0.6726 (2.71)
-9.0474(-48.38)
-0.2165 (-1.73)
-0.6366 (-4.84)
-2.5981(-22.63)
-3.2036(-64.17)
-0.000 (-3.92)
a 24 1
a341
a142
a242
a342
a 43
a243
a3m3
a144
a244
a 344
al45
a2 4
a 3 45
al 46
a 24 6
a 346
a 14 7
a2 4 7
a3 4 7
al
48
a 2 48
a
488
a 33 m
a 14 9
a2 49
a 3 49
bl
b2
-3.985Xl10- 5
(-0.89)
5.841x10- 5 (11.46)
1.990X10- 6
(0.27)
b3
b4
CNSD
USSD
NRGD
WGGD
0.0002 (1.71)
-0.0014(-12.00)
-0.00013(-2.79)
C14
C24
c34
dl4
d 24
d3 4
I
Eqn(1)R
Eqn(2)RA'
Eqn(3)R
4I
Linear in
Prices, income,
temperature
0.6937
(0.73)
3.2409
(3.88)
-0.3132
(-0.73)
0.8080
(2.16)
-3.2060 (-10.72)
-0.7675
(-4.42)
-0.3581
(-2.05)
-0.5571
(-3.15)
0.1167
(1.51)
0.6105
(-2.38)
-2.6112 (-11.42)
-0.4263
(-4.61)
0.8150
(5.08)
-1.1026
(6.29)
-0.1303
(-1.69)
-0.0999
(-0.22)
-1.7827
(-5.24)
-6.0818 (-27.72)
-0.6553
(-2.44)
-0.7667
(-3.08)
-0.9432
(-5.13)
0.6398
(52.46)
-9.1359 (-46.23)
-0.3062
(-2.28)
-0.6672
(-4.16)
-2.5580 (-18.39)
-3.1633 (-57.23)
-0.0004
(-4.02)
-3.335X10-5(-0.72)
5.678X10-5(11.09)
5.670X10-6(0.75)
0.000173(1.58)
-0.00141 (-11.93)
-0.00015 (-3.17)
-0.00385 (-0.19)
0.01185 (0.66)
0.01138 (1.64)
.7844
.7868
.9938
.9961
.9938
II
4
3
Linear in
prices, income;
dummy variables
Linear in logs of
prices and income
dummy variables
0.1270 (0.220)
15.3070 (3.73)
2.5185 (5.140)
5.3262 (1.37)
-0.5301 (-1.22)
-0.1918 (-0.05)
1.4187 (4.53)
1.3114 (4.08)
1.6660 (1.42)
42.4883 (3.92)
-0.3672 (-1.90)
-0.4862 (-2.17)
-0.2941 (-1.98)
-0.3534 (-2.47)
-0.4241 (-3.20)
-0.4367 (-3.33)
0.2643 ( 4.06)
0.2179 (3.63)
-0.6262 ((-3.97)
-0.6724 (-4.36)
-2.3730 (-19.25)
-2.3873(-19.62)
-0.2805 (-5.26)
-0.2950 (-5.03)
0.9534 ( 5.78)
0.8988 ( 5.68)
-0.9705 (-5.95)
-0.9788 (-6.11)
-0.0339 (-0.43)
-0.1001 (-1.36)
0.7692 ( 1.79)
0.4054 (0.91)
-1.5148 (-5.26)
-1.7199 (-4.79)
-5.8846 (-23.72)
-12.9588(-10.76)
-0.1635 (-0.60)
-0.2765 (-0.97)
-0.3621 (-1.48)
-0.6554 (-2.46)
-0.0889 (-0.35)
-0.1535 (-0.52)
1.0376 ( 4.00)
1.0249 (3.95)
-4.5508 (-2.85)
32.9861 (2.30)
0.0869 ( 0.62)
-0.0058 (-0.04)
-0.5879 (-4.37)
-0.6306 (-4.83)
-2.5431 (-24.08)
-2.5653(-24.78)
-4.1449 (-16.28)
-10.1876 (-4.70)
-0.00047 (-5.17)
-2.0431 (-5.88)
-2.033X10- 5 (-0.50) -0.3358 (-1.10)
7.414X10-6(0.33)
-0.0343 (-0.23)
-9.118x10-6(-1.15) -0.0380 (-0.13)
-0.00059 (-4.13)
-5.0851 (-4.22)
-0.00063 (-2.75)
-4.7594 (-2.93)
5.333X10-5(2.36)
0.8221 (5.37)
0.00057 (3.80)
0.9395 (3.23)
0.00021 (1.86)
0.00013 (1.17)
-0.00118(-10.12)
-0.000119(-10.41)
-0.00016(-3.38)
-0.00013 (-2.86)
.7856
.9949
.9957
.7759
.9949
.9962
.9962
I
.-
67
The price coefficients are improved in that three of them have the correct
sign, and the fourth, while positive, is insignificant.
Model (4) is the same as model (3) except that it is logarithmic in
prices.
This results in no improvement in the estimated coefficients.
We
therefore retain model (3) as our "preferred" static model.
Our dynamic logit 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 prices and income.
The dependence on past shares is intended to
incorporate both habit formation and stock adjustment effects.
It leads to
equations of the form
9
log(SiS4=
ai4 kDk + biPi - b 4 P4 + ci4 Y +
iSitl-4S4,t-l
(60)
i=1,2,3
Note that this is not a Koyck adjustment model.
The coefficients
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.
We present estimation results for three dynamic models in Table 16.
The
first model is identical to static model (3), except that it contains lagged
share terms.
The second model is identical to static model (4), but with lagged
share terms.
The third model is the same as the first, but all of the intercept
coefficients ai4k are constrained to be the same across countries.
Model (1) is our "preferred" dynamic model.
Note that many of the inter-
cept dummy variables are significant, so that inter-country differences must be
68
Table 16 - Parameter Estimates for Dynamic Logit Models
1
Parameter
2
Linear in
prices, income
3
Linear in
logs of prices
Linear in prices,
income; no intercept
dummy variables
i
a 1 41
a2 4 1
a3 4 1
a1 4 2
a2 4 2
a 3 42
a 1 43
a2 4 3
a 3 43
a
1 44
a2 4 4
a3 4 4
a 14 5
a2 4
a
3 4
5
a 1 46
a 2 46
a3 4 6
a1 47
a2 4 7
a 34 7
a1 48
a2 4 8
a3 4 8
a1 4
a2 4 9
a3 4 9
bi
b2
b3
b4
CNSD
USSD
NRGD
WGGD
c14
C24
C34
1
2
3
4
Eqn(1)R2
Eqn(2)R2
Eqn(3)R2
-0.9339
1.1268
-1.0223
0.1902
2.8482
0.2822
-0.0600
-0.2086
0.0600
-0.1934
-1.2759
-0.1720
0.0935
-0.6289
0.0416
0.8205
-0.1717
-4.2357
0.0412
0.2821
0.5868
0.0673
1.7224
0.2219
0.0543
-1.1047
-2.8778
-0.00018
(-2.09)
3. 3997
( 1.86)
(-2.60)
( 0.65)
(2.31)
(1.44)
(-0.52)
(-1.66)
( 0.79)
(-1.37)
(-5.65)
(-1.65)
( 0.60)
(-3.54)
-0.9392
(-0.57)
( 2.18)
(-0.55)
(-13.92)
(0.18)
(1.21)
(2.25)
( 0.29)
5
3.612x10
-3.967xlO- 5
-1.317xlO
5
-0.00061
-0.00090
(-1.10)
( 1.56)
(-0.31)
(-4.70)
(-10.29)
(-2.88)
( 0.93)
(-1.94)
(-1.91)
(-4.30)
(-4.20)
9.740x10
5
0.00068
6.993xlO 5
-0.00092
-4.552xlO 5
5.1800
1.6636
3.9691
1.9629
.8785
(4.77)
(6.01)
(0.80)
(-8.01)
(-1.04)
(11.14)
(3.23)
(6.29)
(6.23)
(1.03)
(-0.25)
(0.02)
(-0.40)
(11.25)
(0.44)
(-1.21)
0. 0586
-0.1237
46.5229
0.0941
0.1357
-0.2573
-0.0169
-0.2713
-1.2910
-0.2430
0.0086
-0.6728
-0.0579
0.2468
-0.5274
-12.2412
-0.2719
0.0162
0.3140
-0.1421
51.1490
0.0805
-0.1848
-1.1784
-10.7211
-0.6379
0.2183
-0.1956
-0.0849
-5.4310
-6.7050
.9963
.9971
.9968
_
i~~~
i
-0.3446
0.0295
-2.1596
(-1.36)
( 0.10)
(-6.92)
(-1.94)
(-5.53)
(-2.29)
(0.05)
(-3.84)
(-0.84)
(0.61)
(-1.44)
(-12.00)
(-1.12)
(0.06)
(1.18)
(-0.59)
(3.76)
(0.56)
(-1.08)
(-4.92)
(-6.00)
(-2.40)
(0.71)
(-1.59)
(-0.34)
(-4.37)
(-4.38)
(7.50)
1.1992
1.636x10-5
-0.00093
5.0881
1.8127
4.1353
1.7554
.8820
.9962
i
(-2.05)
(-0.22)
0. 9452
-8.872x10
i
5
(5.07)
(0.19)
(-8.10)
(-1.96)
(10.22)
(3.27)
(5.78)
(5.65)
-7.062x10- 5 (-2.09)
4.969x10- 6 ( 0.22)
5.200x10-5 ( 3.02)
1. 859x10 -6 ( 0.66)
-0.00020
(-11.48)
-0.00107
(-50.76)
-0.00013
(-6.88)
9.049x10S ( 0.69)
-7.703x10- 5 (-1.07)
-0.00047
(-5.05)
-0.00017
(-1.92)
4.9224
(15.70)
4.3937
(14.55)
12.2985
(12.49)
2.2369
(13.51)
.8354
.9934
.9335
69
accounted for in estimating elasticities.
The price dummy variables for
solid fuel in Canada and the U.S. and for gas in Norway and West Germany
are highly significant, and are retained.
Three of the four price coefficients
are significant and have the correct sign; the fourth (for solid fuel) has the
wrong sign but is insignificant.
This is not very surprising - solid fuel shares
are low (and falling) in some countries with low solid fuel prices because cleaner
and more convenient fuels have become more readily available.
Finally, note
that all of the lagged share coefficients are positive and highly significant,
so that dynamic adjustments seen to play an important role in the determination
of fuel shares.
Model (2) is logarithmic in prices, and the result is a loss of statistical
significance for two of the four price coefficients, with no noticeable improvement in any of the other coefficients.
Model (3) is linear in prices, but
contains no intercept dummy variables.
The result is that three of the four
price coefficients have the wrong sing.
We therefore use model (1) to calculate
dynamic elasticities, and to forecast changes in shares.
In Table 17 we present price and income fuel share elasticities for our
preferred" static model for two different years, and in Table 18 we present
short-run and long-run price and income fuel share elasticities for our "preferred"
50
dynamic model.50 (We must stress that the solid fuel elasticities for Canada and
the U.S., and the natural gas elasticities for Norway and West Germany are of
dubious meaning since the respective fuel shares are so close to zero.)
50It is straightforward to calculate income and price elasticities of shares
for the static logit model. From equation (35) we can obtain the income
elasticity for a model with n shares as follows:
dSk/Sk - dSn/S
Since
n
= (ck-cn)dY
dSk = 0, we have that
n-l
I dSk
k=l
n-l
t
=
k=l
Sk
S
n
dSn + Sk(ck-cn )dY} = -dS
,
or,
(cont.)+
70
%\IC00 Cq
,-i -I
I
r-4
00'
H
Ln
CD
"00
-i I-4
c r_
o
q
I
I
I
I
u
o
r O
I' I'
1
I
I
O-4H
-Mc
H- O
<4
c/3
-iI
I
I
O0
" cq
oo
*
r- r-H
*.
HH
oo
* I
*-W
nH
mJ
·L
.
i
I I
10
.jctO
u
.
I
·
I
" o 'oO
4-0-t 0
II H
I I
Hei
3
O
4-I
z0
I
I
I
0coc
I
'I0r-q
-I
r-4
co
H·
\0
HFI*41I
e * uIH
. CoX
o o -H
l
cn r-
(-I
c 00o
O-'
*-~
*oO * e
rI.
.r-I
I
-J
I
.r4
4q0
I
En
-4
¢0
-H
4-J
o r-
I
p
w
40
I
+
H H o'-~
c~4CN
00
I
1
C'
I'"
z
U]
C-
r-4
rI
0
.-I
H
oe
-4
rI rI
I
I
o
1p
I
E4-
.4
H
.H
~C3
-H
-D
0n
Cd
C~)
I
C-
CqCI
*
I
HO
H-I
,-*
-* -
.
.
0
· '
· oo
H r-.
Ii
qCY
I
C4* H
un
co
·e
s
·
I
rI
ILO
0
cn
I
·
o
O,
0aJ
O
q0
uI
H
%1
C14-I
02
-H
Cnn - 0
00
I
HH
OH
II
I. -I.I
I
I
I'
OH
O O
I1 '00
Ln
I'I
coo r-4,-4
I-
*4
0
foo
0O
or0 00o
IH
1-.
"o0
"o
0
HH
aN cO
'IO
H
ON aN
c'Co
HHO
aN O
0
4
0
" C)
0
0
4c
0w
0
-i
I
%OHr
rI
o
·
·t
Ho
rI
cw
H0 O
II
00 D
- -
O O
I
0-4
00
0
C 0o
r-i
w
*
...C
0
I
w
w
U
cd
4-
I
*1-4
oq
u
0
0
raI
u
oo
c
00c ,',
I
I
-H
4I
-H
.r4
cn
o Ln
cq
·
c
*' ' -I
4-4
il
CO
C
O
'Dor-r4r4 rq
.Or--4
c io
w_
O
O
CHo
ao
oro
.- I4
-H
u
H4
0-
-H
-'
H
02
02
0n
do
U
H0
.I0
~o
HH
--4
,-4
0
,-I
co
0n
bc
v
Hl
,-Iet
--
71
rU
.
oo
Lr o
:3
cn
Cq
O~*
oo0<
o*- C" oo r
H
I' 0'oi C'oI
I
,I
0
I
C4
N
CN%
I l_
IC
Io
I
I
:3:
ILC')
,
(I'FO %0r-
H
,I 00
,-4I I
CoI -
I'
I'
1-O m
H'
4I
I'
Y CSl
*
-It.
or- H'fr
r-
-
n
r-
In
.-HI
.-4
rc- oo
I Ln
-'O
r- %O
.
%O
Lt
* .V I.
I* C
I
,o c,,
I
,-I
0
u
.H
0w
cl
I
I
w
I
Hq
oI
I I'
OCN
HI
I
I
I
,-.{
I
HI CJ
04
I
In
I
z
C-)
w
cU
0
4
ci
-H
0
1-1
a
OC 00co 0 0
(4 4 c4
,4 0(
- I I
I I
co
an
00o'
I
I
r.)
N
0
0
0(")
9
" i~O
I u
P.4
I
I.-
O
4
I
'-I
Cn
M
H
I'
I' C
I
H
riHC
0
u
H
4~
r-
I I
r-O0
oI
I
esHI
0,,.0
lz
0
4-
1-4
OH
li
'
"4
.1-
(t
-H
"r4
"4J
(N-1H(
'q I'
0
I
0
C-)
4.
Ie,
* -
CU
O N
I
0H Ur
oO
,ICS
uI
*
H O
ooo
OOL(r
H0a
I I
I
I
I* I*
*n14
"
4.J
H
,'H
4I OC
I
( N
o NI
1 1
1
1
O O
o4u'
(NLIr
H
I
.H
u'n
H-l Ir1.
-H
*0
O'
a
0c
'O' O
-Io*O* C4* %D
C0 I '
I' ,I
,-
0
I I
-e
C.
PLU
I
.co
-I
* t*
O'.o
. HDk
r4
I
I
Hq
H-
CU H'40
N
%D 01
ON
a
I I
IIo
I I
,- r-
o
oo'
Hc
(N
,I
H-iN
II
o
D
I'ON
II
I
-I
It
00oo
cU
II'
(
C-)
I
a'N
Ia
00i-
o* H
1-
Or-
P,I
o
Oq H
cn
I
l I
p p
00
In
w GI
".
-H
0x
n
O
C
~0 '
Lfn
M Ln
~o.a1- ~C
HO'4 I
,
H-HO
- O (N O M I
I
O- U
L'
l
C:
C mH
4
1
'
I
I
I
4JJ 00Q
11
U1 i
En 1.
um 1.
cn
1.
II11 0-4
II U)0
al
En F4
*
U2
u
".-
-H
-. 4
0
CU
0
0'
-H
*1.
-
(1
C0
bO
02
O
.
.
U
0
H
0
_
CU
72
Note that with the exception of liquid fuel, the own price elasticities
(These are share elasticities, but for
in the static model are quite small.
very small changes in price, they will approximately equal the quantity elasThe larger elasticities for liquid fuel are supportive of the
ticities.)
results of our static translog model with country dummy variables.
Nonetheless
it is surprising that these elasticities are so much smaller than those obtained
from the static translog model - and in fact are unreasonably small for solid
fuel and electricity, and have the wrong sign for gas.
We assume that this is
a result of the fact that the logit model is more restrictive than the translog
model, and carries restrictions that do not yield a valid representation of fuel
demands.
The income elasticities for the static model are less disconcerting.
It is reasonable that solid fuel has a negative income elasticity (note that
unless all income elasticities are zero, at least one must be negative), since
coal is indeed the "inferior" fuel for residential use.
As European per capita
incomes have increased, more and more homes have had oil, gas, and electric
n-l
50(cont.)
dSn/Sn = cndY(l-Sn ) - d
CkSk
k=l
After some manipulation, this reduces to
dS /Si
n-l
' (ci-cn)
niY = dY/Y
-
I
(ck-cn)SkY
k=l
To obtain the own price elasticities of shares, note that
dSk/S
- dSn/S
Then using the fact that
S
nii
dSi/Si
dPi/Pi
i
i
= bkdPk
dSk = 0 and ISk =1,
we can obtain:
bi(l-Si)Pi , i=l, ..., n-l
biSiPi , i=n
We do not derive analytical expressions for long-term and short-term dynamic
elasticities. Instead, these elasticities are calculated by simulating the
dynamic model.
73
heating equipment installed, and of these latter three fuels, oil has
usually been preferred.
Price and income elasticities calculated for the dynamic model are
extremely large in the long run in many cases.
If all of the price elasti-
cities had the correct sign, we could interpret this as indicating that in
the long run fuels are nearly perfect substitutes, so that if the relative
price of a particular fuel increased from its "equilibrium" level, the share
of that fuel would become close to zero after enough time had elapsed.
This
is indeed reasonable, particularly if the "long run" is long enough (we use ten
years for the long run in claculating elasticities).
However some of the price
elasticities have the wrong sign, even though three of the four estimated price
coefficients have the correct sing.
In addition, the signs of both the price
and income elasticities change from country to country, and in some cases the
sign of the long run income elasticity is the opposite of that for the shortrun elasticity.
In addition, if fuels were perfect substitutes, long-run income
elasticities would be close to zero, which they certainly are not.
It appears
that the dynamic model is unstable, with changes in fuel shares cascading so
that equilibrium adjustment to a price or income change is never reached.
The results of simulating both the static and dynamic models are shown in
Table 19.
Actual values of prices and incomes are used up to 1973, and extra-
polated relative prices and incomes are used after 1973, with the extrapolations
done using the 1960-1973 growth rates for each country.
The 1975 and 1980
simulated share values, then, show the projected effects of a continued change
in relative prices implied by each model; they do not represent our "forecast"
of future shares since there is good reason to expect that relative prices will
not continue to move as they have in the past.
74
Table 19:
Simulated and Actual Fuel Shares
(1) = Simulation of Static Model
(2) = Simulation of Dynamic Model (3) = Actual Values
BELGIUM
Liquid
Solid
Electricity
Gas
;
(1)
1962
1966
.09
.16
1970
1975
1980
.20
.26
.33
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
.09
.09
.14
.23
.48
.97
.14
.22
.53
.42
.31
.15
.06
.54
.45
.27
.08
.56
.43
.31
.16
.17
.20
.21
.20
.16
.17
.15
.18
.18
.21
.24
.29
.39
.41
.21
.24
.30
.19
.25
.28
.30
.02
-
.00
.20
.14
.01
CANADA
Liquid
Solid
Gas
Electricity
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
1962
1966
1970
1975
.37
.36
.41
.44
.4
.42
.67
.97
.3.38
.36
.35
-
.02
.01
.00
.00
.02
.01
.00
.00
.03
.01
.00
-
.17
.17
.16
.14
.15
.15
.09
.01
.16
.19
.16
-
.44
.46
.43
.42
.43
.41
.23
.02
.43
.44
.48
-
1980
.48
.97
.00
.00
-
.12
.01
-
.40
.02
-
-
FRANCE
Liquid
1962
1966
1970
1975
1980
Solid
Gas
Electricity
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
.09
.14
.15
.20
.26
.08
.13
.16
.18
.05
.08
.12
.20
-
.41
.32
.24
.14
.06
.43
.33
.19
.07
.01
.44
.31
.20
-
.24
.26
.28
.30
.29
.24
.25
.31
.49
.86
.24
.26
.27
-
.26
.29
.33
.36
.38
.25
.29
.33
.25
.08
.24
.30
.33
-
ITALY
Liquid
1962
1966
1970
1975
1980
Solid
Gas
Electricity
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
.11
.13
.14
.12
.12
.10
.13
.15
.12
.09
.08
.16
.24
-
.17
.13
.09
.06
.03
.18
.13
.09
.06
.03
.20
.12
.07
-
.27
.27
.27
.27
.26
.27
.28
.25
.20
.13
.26
.27
.25
-
.44
.46
.50
.54
.58
.44
.46
.52
.63
.75
.45
.46
.43
-
75
Table
19- Cont.
NETHERLANDS
Solid
Liauid
I
1962
1966
1968
1975
1980
_
_
Gas
._
-
.
l
Electricity
-
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
.30
.35
.29
.42
.46
.27
.28
.31
.86
.97
.24
.34
.39
-
.26
.18
.17
.06
.03
.26
.19
.14
.02
.34
.18
.20
.19
.12
.19
.21
.18
.16
.21
.22
.04
.01
.18
.19
.26
.28
.33
.34
.35
.27
.31
.33
.08
.02
.24
.28
.29
.00
.21
NORWAY
Liquid
Solid
Gas
Electricity
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
1962
1966
1970
1975
.09
.15
.17
.00
.11
.14
.15
.01
.13
.14
.14
-
.23
.16
.13
.14
.22
.15
.15
.18
.21
.17
.15
-
.01
.00
.00
.00
.01
.00
.00
.00
.01
.00
.00
-
.67
.68
.70
.85
.66
.71
.70
.80
.66
.69
.70
-
1980
.00
.00
-
.10
.17
-
.00
.00
-
.89
.82
-
USA
Liquid
1962
1966
1970
1975
1980
Solid
Gas
Electricity
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
.34
.34
.31
.28
.25
.32
.24
.11
.07
.05
.32
.34
.30
-
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
.00
-
.25
.24
.24
.24
.23
.27
.29
.32
.23
.12
.26
.25
.24
-
.40
.42
.45
.48
.52
.41
.46
.57
.70
.83
.41
.41
.45
-
U.K.
Liquid
Solid
Gas
Electricity
F
.
_
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
1962
1966
.06
.09
.38
.32
.29
.25
.20
.41
.31
.26
.17
.04
.39
.32
.24
.20
.21
.22
.24
.26
.17
.23
.28
.47
.86
.20
.36
.38
.39
.46
.50
.27
.10
.05
.04
.07
.06
.06
.18
1970
1975
1980
.13
.08
.07
.04
.36
.42
.44
1$i..
.01
mm
I I
.
I _
_
.
I
l
.25
.
.38
.38
.32
.08
$ II
W. GERMANY
Liquid
19t2
1966
1970
1975
1980
Solid
Gas
Electricity
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
.[
.26
.25
.29
.33
.
.22
.30
.83
.98
..23
.23
.29
-
.12
.08
.05
.03
.01
.11
.13
.09
.06
-
.02
.02
.02
.02
.02
.02
.02
.02
.01
.00
.
.08
.05
.02
.00
.o
.64
.68
.66
.64
.7
.68
.63
.14
.01
.63
.66
.63
-
.02
.02
-
76
Elasticity
Country
Estimate
US
-.12(S.R.),-.50(L.R.)
.16(S.R.),-.63(L.R.)
-.28
Source
Energy own price
US
US
US
(a)
(b)
(c)
-. 40
(d)
US
-.50(S.R.),-1.70(L.R.)
(e)
Canada
Norway
-. 33 to -. 56
(f)
-.30
-.35(S.R.),-.78(L.R.)
-.63(S.R.),-1.30(L.R.)
-.42(S.R.),-1.30(L.R.)
-.38(S.R.),-.42(L.R.)
(g)
W. Germany
Italy
Netherlands
U.K.
6 countries*
(e)
(e)
(e)
(e)
-.71
pooled
20 OECD countries-pooled
(e)
(h)
-0.42
Energy income
Fuels own price
(partial)
U.S.
U.S.
.10(S.R.),.60(L.R.)
.20(S.R.),.80(L.R.)
(a)
(b)
U.S.
Canada
.27
.83 to 1.26
(c)
(f)
Norway
1.08
(g)
6 countries*
1.09
(e)
pooled
20 OECD countries-pooled
151
(h)
Canada
Norway
20 OECD coun-
oCa
cP
r-4
Fuels own price
(total)
gas & oil:
-0.96
electricity: -0.34
electricity: -.22 to -.60
oil:
-.33
(f)
(f)
(g)
(h)
tries-pooled
gas:
Canada
coal:
-0.81
gas & oil:
1.24
electricity: 1.88
(h)
(f)
(f)
Norway
electricity:
(g)
U.S.
U.S.
electricity: -L0 to -1.2
gas: -. 15(S.R.), -. 01l(L.R.)
oil: -. 8(S.R.),-l.l(L.R.)
(i)
(a)
(a)
elec.: -.19(S.R.),-1.00(L.R.)
(a)
U.S.
gas:
oil:
-1.34
-1.89
(b)
(b)
electricity. -1.13
(b)
gas:
(j)
U.S.
-1.05
(h)
0 to 1.4
-1.28
to -1.77
electricity: -0.40
(j)
U.S.
gas:
-. 91
oil:
-.91
elec.:
-. 84
(Q)
(Q)
(Q)
Canada
gas:
(k)
20 OECD coun-
oil:
tries-pooled
gas:
-1.11
(h)
coal:
-. 98
(h)
-.20(S.R.),-1.3(L.R.)
-. 52
(h)
l_
SURCES:
(a) Joskow and Baughman [69]
(b) Baughman and Joskow [6]
(c)
Nelson
(c],
Jraan
[47]
cn
r
1S
(e) Nordhaus
[48]
(f) Fuss and Waverman [68]
(g) Rdseth and Str~m [56]
IMN AA--- -- A1^_ Aw
ri i
(i)
(i)
Halvorsen
[25]
Liew [41]
(k) Berndt and Watkins [10]
- __
I
t 0\ _
TT
_T
._ _
_-
r
,
77
The historical simulation performance of the static model is quite
good, with share errors less than 10% in most cases.
Note that the modPl
98).
projects solid fuel shares to drop to below 10% for most countries by
This is the result not of an extrapolated rise in the relative price of
solid fuel, but of an extrapolated rise in incomes.
The dynamic model shows,
for each country, the share of one particular fuel becoming close to 1.
Again,
this could be interpreted as near-perfect substitutability (in Belgium, for
example, the 1980 estrapolated price of liquid fuel falls relative to the prices
of the other fuels).
in the dynamic model.
However, it is more likely the result of a basic instability
Clearly further work needs to be done in specifying and
estimating dynamic logit models for fuel shares.
78
6.
Summar
and Conclusions
The results presented in this paper include a wide range of implied
elasticities for total energy demand, and for the demands for individual fuels.
For example, the elasticities for our "preferred" static translog fuel choice
model are considerably larger (about twice as large) as those for our
"preferred" static logit fuel choice model.
As we stressed in the beginning
of this paper, however, we view these results as very preliminary, and we
see their main value in suggesting where further work could be best directed.
The results for both the translog models and logit models must, at this
stage, be considered inconclusive.
Translog models must be estimated in
their non-homothetic form (perhaps using a different computer algorithm), so
that we can test conclusively whether the assumption of homotheticity can be
maintained.
This is essential, since all of our other tests (stationarity,
additivity, etc.) were based on this assumption of homotheticity.
In addition,
the imposition of the homotheticity restrictions greatly reduces the power of
the translog model.
Under homotheticity explicit seperability (additivity) of
any category implies that its own price elasticity is -1 and its cross elasticities are zero.
The calculated own price elasticity of energy in our consumption
breakdown model is -1.01, but a different number might have resulted had the
model been non-homothetic.
We also need to better identify the difference between short-run and longrun elasticities.
One way to do this will be to estimate static models across
time and across countries separately.
However, it is also necessary to obtain
better specifications and estimations of dynamic models.
79
In some ways the variations in estimated elasticities between our
different models is not surprising.
A survey of the existing literature
indicates a very wide range of published and unpublished energy elasticity
estimates.
Observe in Table 20 that long-run price elasticity estimates for
the United States range from -.33 to -.89 for oil, -1.01 to -1.77 for gas, and
-0.4 to -1.21 for electricity. 51 Own price elasticity estimates, for total
52
residential energy consumption in the U.S. range from -.28 to -.63.52
A
similar range of elasticity estimates exists for Canada and the European countries.5 3 Unfortunately we are unable, with our results at this point, to narrow
these "confidence intervals" on demand elasticities.
Hopefully, such a narrow-
ing will come about with further work.
51-
51SeeHalvorsen [25], Baughman and Joskow
and Cope [27].
52
See Baughman and Joskow [6]
[6]
,
Liew
41], and Hirst, Lin,
, Nelson [44], and Jorgenson [35].
53SeeAdams and Griffen [ 1], Berndt and Watkins
R6dseth and Strom [56], and Nordhaus [48].
10], Fuss and Waverman [68],
80
REFERENCES
1.
Adams, F.G. and J.M. Griffin, "Energy and Fuel Substitution Elasticities: Results from an International Cross Section Study,"
unpublished, October, 1974.
2.
Adams, F.G. and P. Miovic, "On Relative Fuel Efficiency and the Output
Elasticity of Energy Consumption in Western Europe," Journal of
Industrial Economics, November, 1968.
3.
Allen, R.G.D., Index Numbers in Theory and Practice, Aldine Publishing
Co., Chicago, 1975.
4.
Balassa, B., "The Purchasing-Power Parity Doctrine: A Reappraisal,"
Journal of Political Economy, June, 1964.
5.
Barten, A.P., "Maximum Likelihood Estimation of a Complete System of
Demand Equations," European Economic Review, Fall, 1969, pp. 7-73.
6.
Baughman, M.L. and P.L. Joskow, "Interfuel Substitution in the Consumption of Energy in the United States," M.I.T. Energy Laboratory Working
Paper, May, 1974.
7.
Belsley, D.A., "Estimation of Systems of Simultaneous Equations, and
Computational Specifications of GREMLIN," Annals of Economic and
Social Measurement, October, 1974.
8.
Berndt, E.R., B.H. Hall, R.E. Hall, and J.A. Hausman, "Estimation and
Inference in Nonlinear Structural Models," Annals of Economic and
Social Measurement, October, 1974.
9.
Berndt, E.R. and N.E. Savin, "Estimation and Hypothesis Testing in Singular
Equation Systems with Autoregressive Disturbances," Econometrica,
September, 1975.
10.
Berndt, E.R. and G.C. Watkins, "A Residential and Commercial Natural Gas
Demand Model for Ontario and British Columbia," in The Economics of
Oil and Gas Self-Sufficiency in Canada, Conference Proceedings,
Department of Economics, University of Calgary, October, 1975.
11.
Brown, A. and H. Deaton, "Surveys in Applied Economics: Models of
Consumer Behavior," Economic Journal, December, 1972.
12.
Chenery, H., and M. Syrquin, Patterns of Development, 1950-1970, Oxford
University Press, London, 1975.
13.
Chow, G., "On the Computation of Full-Information Maximum Likelihood
Estimates for Nonlinear Equation Systems," Review of Economics and
Statistics, February, 1973.
81
14.
Christensen, L.R. and M.E. Manser, "Estimating U.S. Consumer Preferences
for Meat, 1947-71," unpublished, February, 1974.
15.
Christensen, L.R., D.W. Jorgenson, and L.J. Lau, "Transcendental Logarithmic
Utility Functions," American Economic Review, June, 1975.
16.
Cox, D.R., Analysis of Binary Data, Methuen & Co., London, 1970.
17.
Deaton, A.S., "The Measurement of Income and Price Elasticities,
European Economic Review, Vol. 6, 1975, pp. 261-273.
18.
Diewert, W.E., "Homogeneous Weak Separability and Exact Index Numbers,"
Technical Report No. 122, Institute for Mathematical Studies in the
Social Sciences, Stanford University, January, 1974.
19.
Domencich, T., and D. McFadden, Urban Travel Demand: A Behavioral Analysis,
North-Holland, Amsterdam, 1975.
20.
Eisner, M., and R.S. Pindyck, "A Generalized Approach to Estimation as
Implemented in the TROLL/1 System," Annals of Economic and Social
Measurement, January, 1973.
21.
Frisch, R., "A Complete Scheme for Computing All Direct and Cross Demand
Elasticities in a Model with Many Sectors," Econometrica, April, 1959.
22.
Gallant, W.R., "Seemingly Unrelated Nonlinear Regressions," Journal of
Econometrics, February, 1975.
23.
Goldberger, A.S. and T. Gamaletsos, "A Cross-Country Comparison of
Consumer Expenditure Patterns," European Economic Review, Vol. 1,
pp. 357-400.
24.
Gregory, P., and J.M. Griffen, "Secular and Cross-Section Industrialization
Patterns: Some Further Evidence on the Kuznets-Chenery Controversy,"
Review of Economics and Statistics, August, 1974.
25.
Halvorsen, R., "Residential Demand for Electric Energy," Review of Economics
and Statistics, February, 1975.
26.
Hausman, J. and D. Wood, "Using the Share Ratio (Logit) Specification for
Estimation of Price Elasticities in Demand Models," 1975.
27.
Hirst, E., W. Lin, and J. Cope, "An Engineering-Economic Model of
Residential Energy Use," Oak Ridge National Laboratory, Technical
Report #TM-5470, July, 1976.
28.
Hoel, M., "A Note on the Estimation of the Elasticity of the Marginal
Utility of Consumption," European Economic Review, Vol. 6, 1975, pp. 411-415.
82
29.
Hottel, H.C., and J.B. Howard, New Energy Technology: Some Facts and
Assessments, MIT Press, Cambridge, Mass., 1971.
30.
Houthakker, H.S., "Additive Preferences," Econometrica, Vol. 28, No. 2,
1960.
31.
Houthakker, H.S., "New Evidence on Demand Elasticities," Econometrica,
Vol. 33, No. 2, April, 1965.
32.
Houthakker, H.S. and L.D. Taylor, Consumer Demand in the United States
1929-1970, Harvard University Press, Cambridge, Mass., 1970.
33.
Hudson, E.A., and D.W. Jorgenson, "U.S. Energy Policy and Economic Growth,
1975-2000," The Bell Journal of Economics and Management Science,
Autumn, 1974.
34.
Hulten, "Divisia Index Numbers," Econometrica, November, 1973.
35.
Jorgenson, D.W., "Consumer Demand for Energy," Harvard Institute of
Economic Research, Discussion Paper 386, November, 1974.
36.
Jorgenson, D.W. and Z. Griliches, "Issues In Growth Accounting: A
Reply to Edward F. Denison," Survey of Current Business, May, 1972,
pp. 65-94.
37.
Jorgenson, D.W., and L.J. Lau, "The Structure of Consumer Preferences,"
Annals of Economic and Social Measurement, Vol. 4, Winter, 1975,
pp. 49-101.
38.
Kloek, T. and H. Theil, "International Comparisons of Prices and Quantities
Consumed," Econometrica, July, 1965.
39.
Kravis, I.B., Z. enessey, A. Heston, and R. Summers, A System of International Comparisons of Gross Product and Purchasing Power, Johns Hopkins
University Press, Baltimore, 1975.
40.
Lau, L.J., "Duality and the Structure of Utility Functions," Journal of
Economic Theory, Vl. 1, December, 1969.
41.
Liew, C.K., "Measuring the Substitutibility of Energy Consumption,"
unpublished, December, 1974.
42.
Lluch, C., and H. Powell, "International Comparisons of Expenditure Patterns,"
European Economic Review, Vol. 5, 1975, pp. 275-303.
43.
MacAvoy, P.W. and R.S. Pindyck, The Economics of the Natural Gas Shortage,
1960-1980, North-Holland Publishing Co., Amsterdam, 1975.
44.
Manser, M.E., "'TheTranslog Utility Function with Changing Tastes,"
Discussion Paper No. 333, S.U.N.Y. at Buffalo, Economic Research Group,
December, 1974.
83
45.
Manser, M.E. "Elasticities of Demand for Food: An Analysis Using Non-Additive
Utility Functions Allowing for Habit Formation," Discussion paper #357,
S.U.N.Y. at Buffalo, Economic Research Group, August, 1975.
46.
McFadden, D., "Conditional Logit Analysis of Qualitative Choice Behavior,"
in P. Zarembka (editor), Frontiers in Econometrics, Academic Press,
New York, 1973.
47.
Nelson, J.P., "The Demand for Space Heating Energy," Review of Economics
and Statistics, November, 1975.
48.
Nordhaus, W.D., "The Demand for Energy: An International Perspective,"
Unpublished, September, 1975.
49.
Oberhofer, W., and J. Kmenta, "A General Procedure for Obtaining Maximum
Likelihood Estimates in Generalized Regression Models," Econometrica,
May, 1974.
50.
Organization for Economic Cooperation and Development, Energy Prospects
to 1985, Paris, 1974.
51.
Park, R.E., "Prospects for Cable in the 100 Largest Television Markets,"
Bell Journal of Economics and Management Science, Spring, 1972.
52.
Philips, L., Applied Consumption Analysis, North-Holland, Amsterdam, 1974.
53.
Pindyck, R.S., and D.L. Rubinfeld, Econometric Models and Economic Forecasts,
McGraw-Hill, New York, 1976.
54.
Pollak, R.A., "Habit Formation and Dynamic Demand Functions," Journal of
Political Economy, Vol. 78, 1970.
55.
Pllak, R.A. and P.J. Wales, "Estimation of the Linear Expenditure System,"
Econometrics, Vol. 37, 1969.
56.
Rdseth, A. and S. StrSm, "The Demand for Energy in Norwegian Households
with Special Emphasis on the Demand for Electricity," University of Oslo,
Institute of Economics, Research Memorandum, April, 1976.
57.
Roy, R., De l'Utilite-Contribution a la Theorie des Choix, Hermann, Paris, 1942.
58.
Samuelson, P.A., "Analytical Notes on International Real Income Measures,"
Economic Journal, September, 1974.
59.
Statistiches Bundesamt, "Internationaler Vergleich der Preise fr die
Lebenshaltung," Preise Lohne Wirtschaftsrechsnungen, Reihe 10, various
years.
60.
Strotz, R.H., "The Empirical Implications of a Utility Tree," Econometrica,
Vol.
25, 1957.
84
61.
Taylor, L.D., "The Demand for Electricity: A Survey," The Bell Journal
of Economics and Management, Spring, 1975.
62.
Theil, H., "A Multinomial Extension of the Linear Logit Model," Inter--national Economic Review, Vol. 10, 1969, pp. 251-259.
63.
Theil, H., Theory and Measurement of Consumer Demand, North-Holland
Publishing Co., Amsterdam, 1975.
64.
Turvey, R. and A.R. Nobay, "On Measuring Energy Consumption," Economic
Journal, December, 1965.
65.
Wall, K.D., "Inter-Equation Constraints and the Specification of Dynamic
Structure," Working Paper No. 119, Computer Research Center, National
Bureau of Economic Research, January, 1976.
66.
World Oil Project, MIT, "A User's Guide to the MIT World Energy Demand
Data Base," MIT Energy Laboratory Working Paper #76-011WP, May, 1976.
67.
Zellner, A., "An Efficient Method of Estimating Seemingly Unrelated
Regressions and Tests for Aggregation Bias," Journal of the American
Statistical Association, June, 1962.
68.
Fuss, M. and L. Waverman, "The Demand for Energy in Canada," Working
Paper, Institute for Policy Analysis, University of Toronto, 1975.
69.
Joskow, P.L., and M.L. Baughman, "The Future of the U.S. Nuclear Energy
Industry," Bell Journal of Economics, Spring 1976.
70.
Berndt, E.R., M.N. Darrough, and W.E. Diewert, "Flexible Functional
Forms and Expenditure Distributions: An Application to Canadian Consumer Demand Functions," Discussion Paper 76-06, University of British
Columbia, January 1976.