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Econometric analysis of residential time-of-use electricity pricing experiments
Journal
of Econometrics
14 (1980) 2877306. North-Holland
Publishing
Company
ECONOMETRIC ANALYSIS OF RESIDENTIAL TIME-OF-USE ELECTRICITY PRICING EXPERIMENTS Douglas
W. CAVES and Laurits University
of Wisconsin, Madison,
Received January
1980, final version
R. CHRISTENSEN* WI 53706, USA received
May 1980
This paper adapts the two-stage neo-classical model of consumer behavior to the analysis of time-of-use pricing of electricity. Emphasis is placed upon the relationship between partial elasticities, which can be accurately estimated from the first stage, and total elasticities, which can be estimated only by using less reliable information to estimate the second stage. Three functional forms are implemented with data from the Wisconsin Pricing Experiment. Results indicate that (1) the CES and generalized Leontief functional forms are preferred, (2) price elasticities vary substantially with price, and (3) peak and off-peak electricity are partial substitutes but total complements.
1. Introduction Residential electricity pricing experiments in several states are yielding a wealth of data for utility rate-makers, regulators, and economists who wish to analyze customer responses to time-of-use rate schedules. It would be desirable to be able .to summarize concisely the information from any to facilitate comparisons among the particular experiment in order experiments. A convenient method of summarizing customer responses is through the use of elasticities of substitution and price and income elasticities. These elasticities can provide a compact but powerful framework for the analysis of consumer responsiveness to time-of-use pricing. In this paper we present an econometric methodology for estimating elasticities for residential electricity consumption by time-of-use. Our approach is based on the neoclassical theory of consumer behavior. This approach assures us that the estimating equations which are specified will be consistent with utility maximization under a budget constraint; hence the estimated elasticities will be internally consistent. These estimated elasticities are required for carrying out cost-benefit analyses of time-of-use pricing, as well as for determining optimal time-of-use rate schedules. *The authors are grateful for the cooperation and assistance of all the members of the team involved in designing and analyzing the Wisconsin experiment. Thanks are also due to Beth Van Zummeren. Kathy Kuester and Joe Herriges for proficient research assistance and Richard Parks for helpful comments on a previous draft of this paper, which was presented at the Electric Power Research Institute Conference on Modelling of Electricity Demand by Time-ofDay. San Diego, June 1978.
288
D.19: Caves and L.R. Christensen,
Residential
electricity
pricing experiments
In addition to internal consistency, the use of estimating equations derived from a utility function has an additional advantage which we exploit in this paper. By placing certain restrictions on the form of the utility function we can pursue our analysis in two stages. This is important because electricity is only one of a great number of commodities purchased by utility customers. Thus, in addition to allocating electricity purchases among different time periods, the consumer must also allocate total income between electricity and other goods and services. Each of these levels of allocation can be summarized by a set of elasticities. It is desirable to analyze the allocation of consumer expenditures in stages because the time-of-use experiments provide extensive data on electricity use but almost no data on the consumption of other goods. Typically all that is available is a crude estimate of income, which can be used to indicate the size of the consumer’s total budget. If we include this income estimate in our analysis, we run the risk of introducing serious measurement error into a set of data which is otherwise accurate. However, if we disregard income and focus only on electricity use, without considering the total consumer budget, then the elasticities which we measure are only partial elasticities. The twostage analysis offers a resolution to this dilemma. By partitioning our analysis into two stages we can clearly distinguish those components of price and income elasticities which are based on sound experimental data from those components which must necessarily be based on less reliable data, or obtained from extraneous sources. The key to conducting a two-stage analysis is the assumption of a separable utility function. Such a utility function is consistent with consumer budgeting proceeding in two stages. At the first stage of the analysis we characterize the allocation of electricity consumption by time-of-day. The estimation at this stage utilizes only experimental time-of-use data, with a resulting high level of reliability. The elasticities from this stage are the partial price elasticities and the partial elasticities of substitution. Although partial elasticities are conditional on the allocation of income between electricity and other goods, they have a number of useful interpretations, which we indicate below. At the second stage of the analysis we make use of less reliable information to convert the partial elasticities into total price elasticities and total elasticities of substitution. In order to implement our methodology it is necessary to adopt a specific functional representation of consumer preferences. We discuss the attractiveness of three particular representations - the CES, the translog, and the generalized Leontief. We demonstrate the use of all three of these functional forms with data from the Wisconsin Residential Time-of-Use Electricity Pricing Experiment. We find that the CES and generalized Leontief forms are well-suited to the estimation of electricity elasticities by time-of-use, but that the translog form is inappropriate. Finally, we use the
D.W Caves and L.R. Christensen,
Residential
partial electricity elasticities in combination to obtain total electricity elasticities.
electricity
pricing experiments
with some extraneous
289
estimates
2. Methodology Our methodology is based on the neoclassical theory of consumer behavior. The traditional starting point for this theory is the specification of a direct utility function which indicates the level of welfare for the individual consumer as a function of the consumption of all commodities available, U=U(X,,X, The consumer
)...)
X,).
(1)
is viewed as maximizing
(1) subject
to a budget
constraint,
(2)
Y=w,x,+w,x,+...+w,x”,
where Yis total income.’ For empirical analysis a more convenient specification of an indirect utility function, V=V(W,/y,
w,/I:...,
point
of departure
is the
W”/Y).
(3)
Implicit in the indirect utility function is the constrained maximization of (1). That is, (3) indicates the maximum utility which can be achieved, given any set of prices and income. The convenience of (3) follows from the fact that empirically estimable demand equations are easily derived using Roy’s (1943) identity,
xi=
Y(av/a~)
(av/awj)wj.
i 1 j=
1
The basic characteristics of consumer preferences can be summarized set of price and income elasticities. The price elasticities are defined as qij = (dXi/dWj)( Wj/Xi). The income
elasticities
are defined
(5) as
Viy=(axilay)(y/Xi)’
‘Throughout purchase.
this paper
by a
(6)
we include
‘saving’ in the list of commodities
which
individuals
can
290
D.W Caves and L.R. Christensen,
The relation between (1915) equation, rlij
=
Sj~ij
Residential
the price and income
-
electricity
elasticities
pricing experiments
is given by the Slutsky
(7)
Sjyli~,
where sj= WjXjiY The oij are the Allen elasticities of substitution.’ They are useful in the analysis of consumer preferences because they are symmetric (aij =aji) and because restrictions on utility functions and estimating equations can be translated into restrictions on the oij. In addition the product, sjcij =qfi, is the income-compensated price elasticity, which shows the percent change in Xi resulting from a one percent increase in Wj if income is allowed to vary so that utility is held constant. In principle, the estimation of the system of equations (4) would provide a complete characterization of consumer preferences. If the list of commodities includes electricity consumption during different time periods, then the elasticities most important for electricity can be obtained from the electricity equations. However, in the analysis of consumer preferences for electricity by time-of-use, lack of data on consumption of commodities other than electricity makes it impossible to estimate (4) as specified. Even the subset of equations in (4) which are specific to electricity cannot be estimated without data on income and prices of all commodities. While it might be justifiable to assume that prices of non-electricity goods do not vary in a cross section, there would still be a problem with the measurement of income: the measures of income which are typically available for the time-of-use pricing experiments are not sufficiently accurate to make the estimation of (4) attractive. It is desirable, therefore, to seek a model which yields estimating equations requiring data only on electricity prices and consumption by time of day. To derive such a model we assume that the utility function (1) can be written in the separable form,
U=U(E(E,,...,E,),G,,...,G,),
(8)
where E is homogeneous of degree one in the Ei, each of which indicates consumption of electricity during a different time period. The Gi are consumption levels of non-electricity commodities. With this structure we can interpret E as a quantity index of electricity consumption. The indirect utility function corresponding to (8) can be written as
~=~W(P,lK...,P,IY), ‘See Allen (1938). For Goldberger (1967).
a discussion
Qt/X...,Q,/Y),
of additional
relationships
among
these elasticities,
see
D.W Caues and L.R. Christensen,
Residential
electricity
pricing experiments
291
where Pi is the price of Ei and Qi is the price of Gi. The function H is homogeneous of degree minus one in Y and degree one in the Pi. Thus we can rewrite (9) as
Q,/K...,Q,/Y),
~=~W’,,...~Pq)/~
(9’)
where P can now be interpreted as a price index for E. We can now derive estimating equations by applying Roy’s identity to (9’). The estimating equation for Ei is given by Ei= ((ay/aP)(aP/iiPi)
. Y)/D,
(10)
where
k=l
k=l
Total electricity
expenditures
M = i
M can be written
as
(11)
P,E,.
k=l
Substituting
(10) into (11) we find i
(~S/~P)(I~P/~P,)P,Y
D,
(12)
k=l
which can be combined
with (10) to obtain
Ei= (aP/dP,)M
(13) k=l
For estimation to obtain
it may be convenient
zi= (dP/dPi)Pi
to multiply
both sides of (13) by Pi/M
(14) k=l
where zi =P,E,/M is the share of electricity expenditure during period i in total expenditure on electricity. The P function does not depend on either income or prices of commodities other than electricity; thus eq. (14) can be estimated using only data on electricity price and consumption by time of use. This makes (14) especially
292
D.W Caws and L.R. Christensen,
Residential
electricity pricing experiments
attractive for the analysis of electricity consumption in cases where data other commodities are either missing or measured inaccurately. Estimation (14) yields estimates of partial price elasticities and partial elasticities substitution. We denote the partial price elasticity as gj=
(aEi/?Pj)(Pj/E,)
lM,
on of of
(15)
which is conditional on the level of electricity expenditures. This is analogous to qij in (5) which is conditional on total income. The elasticity of Ei with respect to electricity expenditures is lie=
(?EJaM)(M/E,).
(16)
and is identically equal to unity due to the homogeneity of P. Corresponding to the Slutsky relationship (7), which links elasticities, is a Slutsky relationship that links the partial elasticities,
vgj= zjcgj-
the
c Zj?yim.
total
(17)
The gtj are the partial elasticities of substitution. The term zjayj is the partial compensated price elasticity and shows the percent change in Ei resulting from a one percent change in Pj if electricity expenditures are allowed to vary so that utility derived from electricity commodities [i.e., E(E,, . .,I?,)] is constant. The partial price elasticities (15) and partial elasticities of substitution in (17) are sufficient to analyze many of the policy issues regarding the impact it is useful to have of time-of-use pricing. However, for some purposes estimates of the total price elasticities and total elasticities of substitution. These elasticities are not directly available from the estimated electricity equations (13) or (14); therefore, we now consider the additional information required to estimate the total elasticities. For any Ei and Pj we can use eq. (13) to compute the total price elasticity, yij= (%/?Pj)(Pj/Ei)=M(c;?fli/?Pj)(Pj/Ei)+
h’,(GM/?Pj)(Pj/Ei),
(18)
where we have used the notation ni to represent (ZP/,?P,)/(~fi=, (ZP/?P,)P,). Since Ei = HiM [from (13)] we can eliminate Ei from the right-hand side of (18), obtaining qij = (ani/?Pj)(Pj/ni)
+ (?M/2Pj)(Pj/M).
(19)
The first term on the right-hand side of (19) is simply the formula for the partial price elasticity, qTj of eq. (15). The second term on the right-hand side
D.W Caves and L.R. Christensen,
Residential
electricity
pricing experiments
of (19) is equal to zj(l +qgE), where qEE= (?E/8P)(P/E) is the elasticity demand for aggregate electricity.3 Thus (19) can be written as rlij=rl:j+Zj(l
+rEE).
293
of
(20)
Eq. (20) partitions the total elasticity, qij, into two components. The first component, yTj, depends only on the first stage of the model and represents the reaction of Ei to a change in Pj with electricity expenditures fixed. The second component represents the further reaction in Ei due to the fact that a change in Pj causes the price index, P, to change (in an amount equal to zj) which in turn causes a change in aggregate electricity consumption, E. By combining eq. (20) with eqs. (7) and (17) we can derive the total elasticity of substitution as oij
=
w, ’
(a:j
+
(21)
)?EE) + YEY>
where W, is the share of electricity in the total budget (M/Y) and qEy is the income elasticity of aggregate electricity, (ZE/?Y)(Y/E). If Ofj is non-negative, the electricity commodities Ei and Ej are partial substitutes. Eq. (21) demonstrates that arj>O does not imply oij>O; therefore, commodities which are partial substitutes may well be total complements.4
3. Functional forms for the estimation
of electricity elasticities
Estimation of the electricity demand equations (13) or the electricity expenditure share equations (14) requires the specification of a functional form for P which is homogeneous of degree one in the Pi. We consider three alternative forms for P the CES, the translog, and the generalized Leontief. The CES (constant elasticity of substitution) form was proposed by Arrow, Chenery, Minhas and Solow (1961). The CES form for P can be written as P=
( i
-l/P
s,p,:p
i=l
where xi 6, = 1. Application of equations:
1
,
of Roy’s identity
(22) to (22) yields the following
set
(23) jNote that (?M/ZPJ)(Pj/M)= (a.bf/aP)(P/bf)(LlP/?P,)(P,/P)= (1 + (?E/SP)(P/E))(SP/?P,)(P,/P). Since the P function 1s homogeneous of degree one, (?P/SP,)(P,/P)= (8P/?Pj)P,/x(?P/?Pk)Pk = zj, where the last equality results from Roy’s identity. 4This result is analogous to that used by Berndt and Wood (1979) in the context of producer theory to explain conflicting findings with respect to energy-capital substitutability.
294
D.N! Caves and L.R. Christensen,
Residential
electricity
pricing experiments
The most convenient way to obtain estimates of the parameters in (22) is to take logarithmic differences of pairs of the equations in (23); for example, ln(Ei/Ek) = ln(6,/6,) + ( - 1 - p) ln(Pi/Pk).
(24)
The translog functional form has been proposed by Christensen, and Lau (1971, 1975). The homogeneous of degree one translog can be written as
fnP=
i i=
zilnPi++ 1
i
i
i=l
j=l
Jorgenson form for P
p;jlnPilnPj,
(25)
where
Application expenditure
of Roy’s identity share equations:
Zi =CYi
+
i
Pij
to
(24)
yields
the
following
electricity
(26)
ln(Pj/Pk).
j=l
ifk
The generalized The homogeneous written as
Leontief form has been proposed by Diewert (1971, 1974). of degree one generalized Leontief form for P can be
P=CC i
YijP)P),
(27)
j
where yij=yji
Application expenditure
and
cyij=l.
of Roy’s identity share equations :
j$l
to
(27)
?ij(pi/pk)“(pj/pk)f i’i
Diewert substitution
yields
f. i=l
i j=l
the
following
electricity
-,j(pi/pk)i(pj!pk)‘I.
(1974, p. 131) provides a formula for computing the elasticities from an indirect utility function. For the electricity function
(28
1
of
P
D.W Caves and L.R. Christensen,
this formula
can be written, Oij=
c
Residential
electricity
k
c -
((dP/;iPi)(dP/dPj))
+
!I
(2P/GPj)
CPk(c3P2/6Pi dP,) k
1
(
)I
(dP/dPi)
P, (l?P/dP, iiP,)
k
)I
CPJdP/?P,)
Pk(d2P/ZPk dP,)P,
cc
m
295
for i # j,
(d2P/dPi dPj)CPk(ZP/dPk)
-
pricing experiments
(29 )
n
k
Applying Diewert’s formula to (21), (24), and (26), we obtain the following expressions for the cross-elasticities of substitution for electricity, for i fj, CES oij = 1 + p, Translog aij
Generalized
=
l +
(30)
(J3ij,/ZiZj),
Leontief
aij’~{~ij(Pi/Pk)‘(Pj/Pk)‘~/zizj
CCyij(pi/pk)t(Pj/Pk)~ i j
.
We see that the CES form has the restrictive feature that the elasticities of substitution between electricity consumption in any pair of time periods are constrained to be equal. There is no such restriction imposed by the translog or generalized Leontief forms, which is why they have been dubbed ‘flexible’ functional forms. Only in the case where 4 = 2 is the CES form as flexible as the translog and generalized Leontief forms. In this case there are only two free parameters to be estimated for all three forms and only a single crosselasticity of substitution. For the case of two commodities all three forms have the same local properties, but their global properties are quite different. The CES is attractive since it can provide a model which is globally valid for any non-negative value of the elasticity of substitution. The translog and generalized Leontief forms do not have this property. Their relative attractiveness depends on the magnitudes of the true elasticities of substitution which are being modelled. Roughly speaking, the translog form is more suitable for modelling ‘large’ elasticities and the generalized Leontief form is more suitable for modelling ‘small’ elasticities.’ ‘See Caves and Christensen
(1980) for further
discussion
296
D.W Caves and L.R. Christensen,
Residential
electricity
pricing experiments
4. Estimation of the partial elasticity of substitution and the partial price elasticities for peak and off-peak electricity from the Wisconsin experiment Of the many electricity pricing experiments from which data are currently available, the Wisconsin experiment is the only one which has included both mandatory participation and several different pricing treatments. Mandatory participation permitted the selection of a random sample of customers, which will greatly enhance the reliability of inferences regarding customer responses in the population. The customers in the experiment have been placed on four different price ratios, which permits econometric estimation of the partial price and substitution elasticities for electricity. In essence the Wisconsin experiment consists of approximately six hundred customers facing ten different pricing treatments. There is a control group which faces the same price for electricity regardless of the time of day. There are nine test cells resulting from the specification of three different peak to off peak price ratios (2:1, 4:1, and 8:l) and three different lengths of the peak pricing period (6 hours, 9 hours, and 12 hours in duration) which are in of the Wisconsin effect only for non-holiday weekdays. 6 The major limitation experiment for our analysis is that each customer faces only two distinct pricing periods a peak or high-priced period and an off-peak or lowpriced period. Thus direct application of our methodology allows estimation of only a single elasticity of substitution between peak and off-peak electricity consumption. The analysis which we present below draws upon data from the summer months of July and August, 1977. Since the Wisconsin Public Service Corporation is summer peaking, the effect of time-of-use rates during these months is of most interest. Our analysis is based upon monthly aggregates. We compute each customer’s total usage for the month for the peak and offpeak periods. We note that the control group (1 :l price ratio) can be included in the analysis of any of the three lengths of peak period since the full day can be arbitrarily divided into sub-periods. We begin our statistical analysis by estimating the CES form. With only two pricing periods the set of estimating equations (23) reduces to a single equation with two independent parameters, which can be estimated by a least squares regression. The estimating form (23) is particularly convenient
‘The summer
peak periods
are:
Six hour peak 9 a.m.-12 a.m. Nine hour peak 8 a.m.- 5 p.m. Twelve hour peak 8 a.m.- 8 p.m. The winter peak periods
and
1 p.m.4p.m
and and
5 p.m.-8 4 pm-9
are:
Six hour peak 9 a.m.-12 a.m. Nine hour peak 8 a.m.-12 a.m. Twelve hour peak 8 a.m.- 8 p.m.
p.m. p.m.
D.W. Caves and L.R. Christensen,
Residential
electricity
pricing experiments
291
in that the regression coefficient on the logarithm of the price ratio is the negative of the eleasticity of substitution (- 1 -p = -$,0) between peak and off-peak electricity consumption. If the estimated a;, is non-negative, then the implied consumer preferences are well-behaved globally. The intercept term in the regression provides an estimate of ln(6,/6,). Both 6, and 6, are identified since 6, + 6, = 1. We estimate a separate CES model for each month and each length of peak period and present the results in table 1. All the estimates of qPO are positive, significantly greater than zero, and fall in the range from 0.09 to 0.17. These bounds on the partial elasticities of substitution imply that increasing the price ratio from 1 :l to 8:l would result in changes of the logarithmic ratio of peak to off-peak consumption of 0.18 and 0.35, respectively.’ With only two electricity commodities our model is a valid representation of consumer preferences if the partial elasticity of substitution is nonnegative. The CES functional form is capable of providing a valid model of consumer preferences at all price ratios, because the estimate of the eleasticity of substitution from the CES model is independent of the price ratio. The translog and generalized Leontief models do not possess this capability. Caves and Christensen (1980) have shown that the range of price ratios over which the translog can provide a valid representation of consumer preferences depends upon the degree of substitution in the preferences being represented. In the two-commodity case if the elasticity of substitution is very low, the range over which the translog can closely approximate consumer preferences is very small. The range of acceptable approximation expands as substitution possibilities increase. The generalized Leontief form, on the other hand, is capable of approximating preferences characterized by low substitution possibilities over the full range of price ratios. In fact the generalized Leontief, like the CES, can exactly represent a fixed coefficient (a,,=O) preference function. For large values of the elasticity of substitution, however, the range over which the generalized Leontief function can closely approximate consumer preferences decreases. The magnitude of elasticity of substitution estimated from the CES model suggests that the generalized Leontief model is more suitable than the translog model for this application. Nonetheless, for the sake of illustration, we obtain parameter estimates for both of these models.* The estimates are presented in table 1. We compute the partial elasticities of substitution implied by these translog and generalized Leontief estimates and present them in table 2 along with the CES estimates. We find that the estimates ‘These figures can be interpreted approximately as 18 ‘7, and 35 Y;, changes in the consumption ratio. ‘For the two-commodity case, estimation of the parameters from either of the two share equations will yield identical results. We have estimated the translog share equation using linear least squares and the generalized Leonteif share equation using non-linear least squares,
D.W. Caves and L.R. Christensen,
298
Residential Table
electricity
pricing
experiments
1
Estimates of the CES, translog and generalized Leontief models of consumer preferences for peak and off-peak electricity; 6, 9, and 12 hour peaks in July and August 1977 (standard errors in parentheses).
August
July Parameter 6
9
12
6
9
12
CES model ~ eq. (23) In (fi,&)
$,=(I
RL
- 1.537
+P)
- 0.999
-0.519
- 1.396
- 0.848
-0.275
(0.036)
(0.030)
(0.029)
(0.036)
(0.035)
0.158
0.092
0.094
0.165
0.097
0.124
(0.030)
(0.025)
(0.025)
(0.029)
(0.029)
(0.024)
0.108
0.06 1
0.065
0.119
0.054
0.130
Trunsfog
(0.027)
model ~~ ey. (25)
0.175
0.271
0.379
0.195
0.303
0.437
(0.007)
(0.006)
(0.006)
(0.008)
(0.008)
(0.006)
0.175
0.208
0.207
0.182
0.207
0.195
(0.006)
(0.005)
(0.005)
(0.006)
(0.006)
(0.005)
0.792
0.885
0.895
0.774
0.844
0.885
Generalized
Leonriej model ~
q.
(27)
0.122
0.225
0.328
0.138
0.249
0.374
(0.006)
(0.007)
(0.009)
(0.007)
(0.010)
(0.010)
0.061
0.048
(0.011)
(0.01
0.79s
0.885
I)
0.047
0.064
0.054
0.057
(0.012)
(0.012)
(0.014)
(0.013)
0.894
0.777
0.844
0.889
from the translog are not well-behaved for the full range of price ratios. For each month and peak length the partial elasticity of substitution is estimated to be negative at the lowest or highest price ratio. On the other hand, the generalized Leontief estimates are well-behaved for all price ratios, and the estimated elasticities of substitution are similar to those of the CES. Given the poor behavior of the translog, we employ only the CES and the generalized Leontief models for the remainder of our empirical analysis. To obtain estimates of the partial price elasticities from the Allen elasticities in table 2, we can use eq. (17) along with the fitted shares from the estimating equations. In table 3 we present the partial price elasticities which result from the estimated generalized Leontief model for July. The results for August and for the CES model are similar.
D.W Caves and L.R. Christensen,
Residential
elecfricity
pricing experiments
299
Table 2
Allen elasticities
of substitution CES, (2) translog
between peak and off-peak electricity and (3) generalized Leontief models.
9 hour peak
6 hour peak
for the (1)
12 hour peak
Price ratio
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
-0.290
0.115
July 8:l
0.158
0.294 0.168
0.092
0.001 0.118
0.094
4:l
0.158
0.279 0.169
0.092
0.155 0.110
0.094
2:l
0.158
1 :l
0.158
8:l
0.158 0.181 -0.218
0.092
0.143 0.111 -0.053
0.121
0.069 0.103
0.094
0.170 0.097
0.094
0.120 0.099
0.203
0.092
0.165
0.256 0.169
0.097
0.132
0.124
4:l
0.165
0.264 0.168
0.097
0.146 0.121
0.124
0.060 0.127
2:t
0.165
0.166 O.t78
0.097
0.164 0.120
0.124
0.205 0.117
1:l
0.165
0.097
0.022 0.128
0.124
0.209 0.117
August
5. Estimation
-0.158
0.198
of the total elasticity
-0.056
-0.394
0.145
of substitution and total price elasticities
Up to this point we have modelled only the first stage of our two-stage model of consumer budgeting. Since there are only two time periods (peak and off-peak) in our model, consumption during those two periods must be partial substitutes ($,>O). This does not, however, rule out the possibility that peak and off-peak electricity are total compliments (dPOCO). To assess this possibility we refer to eq. (21) which shows how to obtain the total elasticity of substitution between peak and off-peak electricity. To evaluate the total elasticity we must have estimates of the price and income elasticities of aggregate electricity, along with an estimate of the share of total electricity expenditures in the total budget. These three estimates must come from the second stage of the model of consumer behavior. We could use the Wisconsin experimental data to estimate the second stage. To do so we would use our first-stage estimates to obtain estimates of the electricity price index, P, corresponding to each of the experimental price ratios. These estimates of P along with data on total income and total electricity expenditures could then be used to model the second stage. However, the desirability of this procedure is limited by the design of the Wisconsin experiment. To minimize the impact on customer bills the peak and off-peak rates were chosen so that the experiment would have no impact
300
D.W Cuces and L.R. Christensen,
Residential
electricity
pricing
experiments
Table 3 Partial price elasticities and expenditure share for peak and offpeak electricity from the generalized Leontief model for July 1917.
Partial
Rate
Peak expenditure share
price elasticity
qip 6 hrn1r peak
8:l
- 0.629
-0.539
-0.371
-0.461
0.554
4:l
~0.510
- 0.659
- 0.490
-0.341
0.410
2:l
-0.412
- 0.769
-0.588
-0.231
0.282
1:l
PO.349
- 0.854
-0.651
-0.146
0.183
Y hour peuk 8:l
-0.739
~ 0.380
~0.261
- 0.620
0.704
4:l
-0.610
- 0.500
- 0.390
- 0.500
0.562
2:l
- 0.475
- 0.636
- 0.525
PO.364
0.409
1:l
PO.361
- 0.760
- 0.639
- 0.240
0.273
8:l
-0.812
PO.303
-0.188
- 0.697
0.788
4:l
-0.710
- 0.393
- 0.290
- 0.607
0.577
2:l
-0.574
~ 0.523
- 0.426
- 0.477
0.529
1:l
- 0.437
- 0.662
PO.563
- 0.338
0.375
12 hour peak
on company revenues if customers did not shift their consumption toward the off-peak period. This offsetting feature of peak and off-peak price changes, in conjunction with the low elasticity of substitution found above, implies that the electricity price index, P, does not change substantially with the experimental pricing treatments. The small amount of variation in P across pricing treatments makes it difficult to obtain reliable measures of qEE from the experimental data. For these reasons we turn to other studies which have made use of the variation in electricity prices among individuals or regions or over time to measure qEE. These studies have typically found that demand for electricity is inelastic in the short-run, though there is wide disparity among the estimates.’ We have chosen -0.5 as a representative value. Although the Wisconsin data are not well suited to estimate qTEE,we can utilize these data to estimate the income elasticity, qEY, and the share of ‘See Taylor
(1975, 1976).
D.W Caces and L.R. Christensen,
Residential
electricity
pricing experiments
301
electricity in the total budget, WE. We have used data on income and electricity expenditures for all customers during the same months of the year prior to the year that the test rates were applied. The equation we have estimated is lnM=a+hln
Y
(31)
where Y is average monthly income as reported on the questionnaire. The estimate of a is 1.531 with standard error 0.179. The estimated income elasticity, given by the parameter h, is 0.229 with standard error 0.024. At the estimated average income level for the residential customer population ($1,500/ month), eq. (3) implies W,=O.O16. We now have the necessary information to convert the partial elasticity of substitution estimates of table 2 into total elasticity of substitution estimates. The six CES partial elasticities in table 2 have an average value of 0.122. Using our estimates of qEE, qer and WE, a partial elasticity of 0.122 implies a total elasticity of -23.4. This indicates that peak and off-peak electricity display strong complementarity, even though they are partial substitutes. Our estimate of the total elasticity of substitution between peak and off-peak electricity is not precise since we have taken our estimate of qEE= -0.5 from extraneous sources. We cannot attach any of the usual measures of sampling error to our estimates to assess whether our finding of total complementarity is statistically significant. We can, however, compute the range of values of qEE which are consistent with total complementarity, conditional on our estimates of gPO, qEy and W,. We find that total complementarity will prevail as long as qEE lies below -0.13. We conclude that it is very likely that peak and off-peak consumption are total complements.” We now use eq. (20) to estimate the total price elasticities which correspond to the partial price elasticities of table 4. We illustrate this computation by making use of our representative value for qEE of -0.5. The estimated elasticities are presented in table 4.
6. Allowing preferences to differ among customers In arriving at the preceeding estimates we have assumed that all customers have the same preferences with respect to use of peak and off-peak electricity. It is not necessary to maintain this assumption, however, because the Wisconsin experiment provides a large amount of information about the characteristics of the individual customers. This information was obtained “We have tested the robustness of our conclusions by estimating alternative equations which allow the income elasticity to vary, by utilizing data from another year, and by evaluating WE at income levels other than the sample mean. Our conclusions are not sensitive to these modifications.
302
D.W Cures
and LX.
Christensen,
Resinential
electricit!:
pricing
evperimenrs
Table 4 Total price leasticities for peak and off-peak electricity from the generalized Leontief model for July 1977.
Price elasticities
R:l
-0.352
-0.316
-0.148
-0.184
4:l
~ 0.305
- 0.364
-0.195
-0.136
3:l
-0.271
-0.410
-0.229
~ 0.090
1:1
-0.257
- 0.445
- 0.242
- 0.054
-0.113
- 0.268
Y hour /x’“k 8:l
-0.387
-0.232
4:l
- 0.329
- 0.28
2:l
- 0.270
1:l
- 0.224
8:l
-0.418
-0.197
- 0.082
- 0.303
4:l
-0.371
-0.231
PO.128
- 0.268
2:l
- 0.309
- 0.287
-0.190
PO.212
1 :1
- 0.249
- 0.349
-- 0.250
-0.150
I
-0.171
-0.219
- 0.340
- 0.229
PO.159
-0.396
PO.275
-0.103
l-7 hour pcwk
Table 5 F-values for testing the hypothesis that no significant difference exists between customers with and without air conditioning.
July 1977
Length
of
peak
August
Generalized CES
Leontief
1977
Generalized CES
Leontief
6 hr.
0.253
0.429
2.093
2.005
Y hr.
0.034
0.027
0.642
0.434
12 hr.
0.389
0.154
4.873”
4.512
“Significant
at 5 “,> level
D.W Cuves und L.R. Christensen,
Rrsidentiul
electricity
pricing
303
experiments
from a detailed questionnaire which was completed by virtually all participants shortly before the experiment began. Using data on customer characteristics, it is feasible to estimate distinct partial elasticities of substitution for different classes of customers. We explore the degree to which the partial elasticity of substitution varies over customer classes by employing two customer characteristics of particular interest. The first is the ownership of air conditioners, and the second is the level of total electricity usage. Air conditioning is thought to be an important determinant of residential customer load shapes during the summer months. The level of total usage is important since benefitcost analysis may indicate that time-ofday pricing is appropriate only for customers with high usage of electricity.
Table 6 Effect of (1) air conditioning predicted percentage
and (2) no air conditioning on partial elasticity of substitution of kWh on-peak for 12 hour peak customer for August 1977.
CES
Generalized
and
Leontief
“,,kWh on-peak
Allen cross-elas.
“,, kWh on-peak
Allen cross-elas.
(1)
(2)
(1)
(2)
(1)
(1)
34.9
37.9
0.131
0.130
35.3
37.8
0.138
0.151
37.2
40.0
0.122
0.133
Rate
8:l
(2)
(2)
4:l
36.8
39.9
0.131
0.130
2:1
38.9
42.1
0.131
0.130
39.1
42.1
0.114
0.123
1:l
41.1
44.3
0.131
0.130
41.0
44.1
0.115
0.122
For the analysis of air conditioning we estimate CES and generalized Leontief models in which all parameters are allowed to differ according to the presence or absence of air conditioning. In table 5 we report the computed F-statistics which can be used to test the hypothesis that the parameters differ significantly for customers with and without air conditioning. The only significant difference we find is for the 12 hour peak period in August. In all other cases there is no significant difference between the usage patterns of customers with and without air conditioning. In table 6 we present the estimated percent on peak and the partial elasticities of substitution for air conditioning and non-air-conditioning customers. The differences in the elasticities of substitution are minor. It appears that customers with air conditioning can substitute as well as other customers even when the peak period lasts 12 hours. The main impact of the difference due to air conditioning is in the percent of kWh used on peak. Here we find that
304
D.W Caves and L.R. Christensen,
Residential
electricity
pricing experiments
custmers with air conditioning use a lower percentage on peak, regardless of the relative prices of peak and off-peak consumption. Our analysis of the impact of consumption level is similar to the analysis of air conditioning. We have used information on average 1975 consumption to group customers into three consumption categories: low less than 600 kWh/month; medium - between 600 and 1200 kWh/month; and high over 1200 kWh/month. We estimate CES and generalized Leontief models which allow all parameters to differ according to consumption level. In table 7 we present F-statistics for testing the hypothesis that no significant differences exist among consumption levels. In all cases the data are consistent with the hypothesis that relative usage patterns do not differ among the three consumption levels.
Table 7 F-values for difference
testing the exists among
hypothesis that no significant different consumption levels.”
July 1977 Length
August
1977
of
peak
Generalized CES
Leontief
Generalized CES
Leontief
6 hr.
0.891
1.183
1.257
1.171
9 hr.
1.399
1.987
1.959
1.967
12 hr.
0.784
0.930
1.019
0.447
“None of the F-values
is significant
at the 5 “/, level.
7. Concluding remarks methodology whereby the neoclassical theory of We have proposed consumer behavior can be employed to analyze the responses of consumers to experimental time-of-use electricity prices. It is convenient to summarize customer responses by elasticities of substitution, rather than price elasticities. As tables 3 and 4 indicate, both partial and total price elasticities can be very different depending on where they are evaluated, even when the partial elasticity of substitution is constant. With a small partial elasticity of substitution, it is especially important to indicate the prices at which the price elasticities are evaluated. The partial price elasticities for any price ratio can be readily computed from the partial elasticity of substitution and the share at that price ration [see eq. (17)]. Comparisons of results from the
D.W Caces and L.R. Christensen,
Residential
electricity
pricing experiments
305
various time-of-use experiments would be facilitated by presentation of estimated partial elasticities of substitution. We have illustrated the proposed methodology by making use of three alternative representations of consumer preferences - the CES, translog, and generalized Leontief models. These illustrations indicate that the partial elasticity of substitution between peak and off-peak electricity consumption is sufficiently small that the translog model is unsuitable for this type of analysis. The CES and generalized Leontief models both behave well and are clearly suitable for analysis with two periods. Because of its simplicity the CES model appears to be the preferred model for two periods. However, when there are more than two time periods, the CES model is unsuitable because of its lack of flexibility, and the generalized Leontief becomes the preferred model. The estimated partial elasticities of substitution from the Wisconsin data are small, but they are statistically significant for peak periods of 6, 9, and 12 hours in length. This implies that electricity consumption can be shifted away from these peak periods. Further research is required to determine the costs and benefits of such shifting, and thus the optimal time-of-use rate structure for the Wisconsin Public Service Corporation’s residential electricity customers. Although we have found that there are significant substitution possibilities between peak and off-peak electricity consumption, we have also found these two commodities to be complements with respect to the consumer’s total income. This finding holds provided that the short-run elasticity of demand for total electricity usage exceeds 0.13 in absolute value, i.e., provided that demand is not extremely inelastic. Our finding that peak and off-peak electricity are partial substitutes and total complements underlines the importance of clearly specifying elasticity estimates as ‘partial’ or ‘total’.
References Allen, R.G.D., 1938, Mathematical analysis for economics (Macmillan, London) 503-509. Arrow, K.J., H.B. Chenery, B.S. Minhas and R.M. Solow, 1961, Capital labor substitution and economic efficiency, Review of Economics and Statistics 63, 2255250. Berndt, E.R. and D.O. Wood, 1979, Engineering and econometric interpretations of energycapital complementarity, American Economic Review 69, June, 342-354. 1980, Global properties of flexible functional forms, Caves, D.W. and L.R. Christensen, American Economic Review 70, June, 422432. Christensen, L.R., D.W. Jorgenson and L.J. Lau, 1971, Conjugate duality and the transcendental logarithmic production function, Econometrica 39, 255-256. Christensen, L.R., D.W. Jorgenson and L.J. Lau, 1975, Transcendental logarithmic utility functions, American Economic Review 65, June, 367-383. Diewert. W.E., 1971, An application of the Shephard duality theorem: A generalized Leontief production function, Journal of Political Economy 79, 481-507. Diewert, W.E., 1974, Applications of duality theory, in: M.D. Intriligator and D.A. Kendrick, eds., Frontiers of quantitative economics, Vol. 2 (North-Holland, Amsterdam).
306
D.W
Caces und L.R. Christrnsrn,
Residentiul
rlectricify
pricing
exprriments
Goldberger. AS., 1967, Functional form and utility: A review of consumer demand theory, Social Systems Research Institute workshop paper SFM 6703 (University of Wisconsin. Madison, WI). Roy, R.. 1943, De I’utilitt-: Contribution li la thtorie des choix (Hermann et Cie., Paris). Slutsky. E.E., 1915, On the theory of the budget of the consumer, Giornale degli Economisti 51, July, l-26. Taylor, L.D., 1975, The demand for electriuty: A survey, Bell Journal of Economics 6, Spring, 74-l 10. Taylor, L.D., 1976. The demand for energy: A survey of price and income elasticities, Mimeo. for the National Academy of Sciences Committee on Nuclear and Alternative Energy Systems, April.
of Econometrics
14 (1980) 2877306. North-Holland
Publishing
Company
ECONOMETRIC ANALYSIS OF RESIDENTIAL TIME-OF-USE ELECTRICITY PRICING EXPERIMENTS Douglas
W. CAVES and Laurits University
of Wisconsin, Madison,
Received January
1980, final version
R. CHRISTENSEN* WI 53706, USA received
May 1980
This paper adapts the two-stage neo-classical model of consumer behavior to the analysis of time-of-use pricing of electricity. Emphasis is placed upon the relationship between partial elasticities, which can be accurately estimated from the first stage, and total elasticities, which can be estimated only by using less reliable information to estimate the second stage. Three functional forms are implemented with data from the Wisconsin Pricing Experiment. Results indicate that (1) the CES and generalized Leontief functional forms are preferred, (2) price elasticities vary substantially with price, and (3) peak and off-peak electricity are partial substitutes but total complements.
1. Introduction Residential electricity pricing experiments in several states are yielding a wealth of data for utility rate-makers, regulators, and economists who wish to analyze customer responses to time-of-use rate schedules. It would be desirable to be able .to summarize concisely the information from any to facilitate comparisons among the particular experiment in order experiments. A convenient method of summarizing customer responses is through the use of elasticities of substitution and price and income elasticities. These elasticities can provide a compact but powerful framework for the analysis of consumer responsiveness to time-of-use pricing. In this paper we present an econometric methodology for estimating elasticities for residential electricity consumption by time-of-use. Our approach is based on the neoclassical theory of consumer behavior. This approach assures us that the estimating equations which are specified will be consistent with utility maximization under a budget constraint; hence the estimated elasticities will be internally consistent. These estimated elasticities are required for carrying out cost-benefit analyses of time-of-use pricing, as well as for determining optimal time-of-use rate schedules. *The authors are grateful for the cooperation and assistance of all the members of the team involved in designing and analyzing the Wisconsin experiment. Thanks are also due to Beth Van Zummeren. Kathy Kuester and Joe Herriges for proficient research assistance and Richard Parks for helpful comments on a previous draft of this paper, which was presented at the Electric Power Research Institute Conference on Modelling of Electricity Demand by Time-ofDay. San Diego, June 1978.
288
D.19: Caves and L.R. Christensen,
Residential
electricity
pricing experiments
In addition to internal consistency, the use of estimating equations derived from a utility function has an additional advantage which we exploit in this paper. By placing certain restrictions on the form of the utility function we can pursue our analysis in two stages. This is important because electricity is only one of a great number of commodities purchased by utility customers. Thus, in addition to allocating electricity purchases among different time periods, the consumer must also allocate total income between electricity and other goods and services. Each of these levels of allocation can be summarized by a set of elasticities. It is desirable to analyze the allocation of consumer expenditures in stages because the time-of-use experiments provide extensive data on electricity use but almost no data on the consumption of other goods. Typically all that is available is a crude estimate of income, which can be used to indicate the size of the consumer’s total budget. If we include this income estimate in our analysis, we run the risk of introducing serious measurement error into a set of data which is otherwise accurate. However, if we disregard income and focus only on electricity use, without considering the total consumer budget, then the elasticities which we measure are only partial elasticities. The twostage analysis offers a resolution to this dilemma. By partitioning our analysis into two stages we can clearly distinguish those components of price and income elasticities which are based on sound experimental data from those components which must necessarily be based on less reliable data, or obtained from extraneous sources. The key to conducting a two-stage analysis is the assumption of a separable utility function. Such a utility function is consistent with consumer budgeting proceeding in two stages. At the first stage of the analysis we characterize the allocation of electricity consumption by time-of-day. The estimation at this stage utilizes only experimental time-of-use data, with a resulting high level of reliability. The elasticities from this stage are the partial price elasticities and the partial elasticities of substitution. Although partial elasticities are conditional on the allocation of income between electricity and other goods, they have a number of useful interpretations, which we indicate below. At the second stage of the analysis we make use of less reliable information to convert the partial elasticities into total price elasticities and total elasticities of substitution. In order to implement our methodology it is necessary to adopt a specific functional representation of consumer preferences. We discuss the attractiveness of three particular representations - the CES, the translog, and the generalized Leontief. We demonstrate the use of all three of these functional forms with data from the Wisconsin Residential Time-of-Use Electricity Pricing Experiment. We find that the CES and generalized Leontief forms are well-suited to the estimation of electricity elasticities by time-of-use, but that the translog form is inappropriate. Finally, we use the
D.W Caves and L.R. Christensen,
Residential
partial electricity elasticities in combination to obtain total electricity elasticities.
electricity
pricing experiments
with some extraneous
289
estimates
2. Methodology Our methodology is based on the neoclassical theory of consumer behavior. The traditional starting point for this theory is the specification of a direct utility function which indicates the level of welfare for the individual consumer as a function of the consumption of all commodities available, U=U(X,,X, The consumer
)...)
X,).
(1)
is viewed as maximizing
(1) subject
to a budget
constraint,
(2)
Y=w,x,+w,x,+...+w,x”,
where Yis total income.’ For empirical analysis a more convenient specification of an indirect utility function, V=V(W,/y,
w,/I:...,
point
of departure
is the
W”/Y).
(3)
Implicit in the indirect utility function is the constrained maximization of (1). That is, (3) indicates the maximum utility which can be achieved, given any set of prices and income. The convenience of (3) follows from the fact that empirically estimable demand equations are easily derived using Roy’s (1943) identity,
xi=
Y(av/a~)
(av/awj)wj.
i 1 j=
1
The basic characteristics of consumer preferences can be summarized set of price and income elasticities. The price elasticities are defined as qij = (dXi/dWj)( Wj/Xi). The income
elasticities
are defined
(5) as
Viy=(axilay)(y/Xi)’
‘Throughout purchase.
this paper
by a
(6)
we include
‘saving’ in the list of commodities
which
individuals
can
290
D.W Caves and L.R. Christensen,
The relation between (1915) equation, rlij
=
Sj~ij
Residential
the price and income
-
electricity
elasticities
pricing experiments
is given by the Slutsky
(7)
Sjyli~,
where sj= WjXjiY The oij are the Allen elasticities of substitution.’ They are useful in the analysis of consumer preferences because they are symmetric (aij =aji) and because restrictions on utility functions and estimating equations can be translated into restrictions on the oij. In addition the product, sjcij =qfi, is the income-compensated price elasticity, which shows the percent change in Xi resulting from a one percent increase in Wj if income is allowed to vary so that utility is held constant. In principle, the estimation of the system of equations (4) would provide a complete characterization of consumer preferences. If the list of commodities includes electricity consumption during different time periods, then the elasticities most important for electricity can be obtained from the electricity equations. However, in the analysis of consumer preferences for electricity by time-of-use, lack of data on consumption of commodities other than electricity makes it impossible to estimate (4) as specified. Even the subset of equations in (4) which are specific to electricity cannot be estimated without data on income and prices of all commodities. While it might be justifiable to assume that prices of non-electricity goods do not vary in a cross section, there would still be a problem with the measurement of income: the measures of income which are typically available for the time-of-use pricing experiments are not sufficiently accurate to make the estimation of (4) attractive. It is desirable, therefore, to seek a model which yields estimating equations requiring data only on electricity prices and consumption by time of day. To derive such a model we assume that the utility function (1) can be written in the separable form,
U=U(E(E,,...,E,),G,,...,G,),
(8)
where E is homogeneous of degree one in the Ei, each of which indicates consumption of electricity during a different time period. The Gi are consumption levels of non-electricity commodities. With this structure we can interpret E as a quantity index of electricity consumption. The indirect utility function corresponding to (8) can be written as
~=~W(P,lK...,P,IY), ‘See Allen (1938). For Goldberger (1967).
a discussion
Qt/X...,Q,/Y),
of additional
relationships
among
these elasticities,
see
D.W Caues and L.R. Christensen,
Residential
electricity
pricing experiments
291
where Pi is the price of Ei and Qi is the price of Gi. The function H is homogeneous of degree minus one in Y and degree one in the Pi. Thus we can rewrite (9) as
Q,/K...,Q,/Y),
~=~W’,,...~Pq)/~
(9’)
where P can now be interpreted as a price index for E. We can now derive estimating equations by applying Roy’s identity to (9’). The estimating equation for Ei is given by Ei= ((ay/aP)(aP/iiPi)
. Y)/D,
(10)
where
k=l
k=l
Total electricity
expenditures
M = i
M can be written
as
(11)
P,E,.
k=l
Substituting
(10) into (11) we find i
(~S/~P)(I~P/~P,)P,Y
D,
(12)
k=l
which can be combined
with (10) to obtain
Ei= (aP/dP,)M
(13) k=l
For estimation to obtain
it may be convenient
zi= (dP/dPi)Pi
to multiply
both sides of (13) by Pi/M
(14) k=l
where zi =P,E,/M is the share of electricity expenditure during period i in total expenditure on electricity. The P function does not depend on either income or prices of commodities other than electricity; thus eq. (14) can be estimated using only data on electricity price and consumption by time of use. This makes (14) especially
292
D.W Caws and L.R. Christensen,
Residential
electricity pricing experiments
attractive for the analysis of electricity consumption in cases where data other commodities are either missing or measured inaccurately. Estimation (14) yields estimates of partial price elasticities and partial elasticities substitution. We denote the partial price elasticity as gj=
(aEi/?Pj)(Pj/E,)
lM,
on of of
(15)
which is conditional on the level of electricity expenditures. This is analogous to qij in (5) which is conditional on total income. The elasticity of Ei with respect to electricity expenditures is lie=
(?EJaM)(M/E,).
(16)
and is identically equal to unity due to the homogeneity of P. Corresponding to the Slutsky relationship (7), which links elasticities, is a Slutsky relationship that links the partial elasticities,
vgj= zjcgj-
the
c Zj?yim.
total
(17)
The gtj are the partial elasticities of substitution. The term zjayj is the partial compensated price elasticity and shows the percent change in Ei resulting from a one percent change in Pj if electricity expenditures are allowed to vary so that utility derived from electricity commodities [i.e., E(E,, . .,I?,)] is constant. The partial price elasticities (15) and partial elasticities of substitution in (17) are sufficient to analyze many of the policy issues regarding the impact it is useful to have of time-of-use pricing. However, for some purposes estimates of the total price elasticities and total elasticities of substitution. These elasticities are not directly available from the estimated electricity equations (13) or (14); therefore, we now consider the additional information required to estimate the total elasticities. For any Ei and Pj we can use eq. (13) to compute the total price elasticity, yij= (%/?Pj)(Pj/Ei)=M(c;?fli/?Pj)(Pj/Ei)+
h’,(GM/?Pj)(Pj/Ei),
(18)
where we have used the notation ni to represent (ZP/,?P,)/(~fi=, (ZP/?P,)P,). Since Ei = HiM [from (13)] we can eliminate Ei from the right-hand side of (18), obtaining qij = (ani/?Pj)(Pj/ni)
+ (?M/2Pj)(Pj/M).
(19)
The first term on the right-hand side of (19) is simply the formula for the partial price elasticity, qTj of eq. (15). The second term on the right-hand side
D.W Caves and L.R. Christensen,
Residential
electricity
pricing experiments
of (19) is equal to zj(l +qgE), where qEE= (?E/8P)(P/E) is the elasticity demand for aggregate electricity.3 Thus (19) can be written as rlij=rl:j+Zj(l
+rEE).
293
of
(20)
Eq. (20) partitions the total elasticity, qij, into two components. The first component, yTj, depends only on the first stage of the model and represents the reaction of Ei to a change in Pj with electricity expenditures fixed. The second component represents the further reaction in Ei due to the fact that a change in Pj causes the price index, P, to change (in an amount equal to zj) which in turn causes a change in aggregate electricity consumption, E. By combining eq. (20) with eqs. (7) and (17) we can derive the total elasticity of substitution as oij
=
w, ’
(a:j
+
(21)
)?EE) + YEY>
where W, is the share of electricity in the total budget (M/Y) and qEy is the income elasticity of aggregate electricity, (ZE/?Y)(Y/E). If Ofj is non-negative, the electricity commodities Ei and Ej are partial substitutes. Eq. (21) demonstrates that arj>O does not imply oij>O; therefore, commodities which are partial substitutes may well be total complements.4
3. Functional forms for the estimation
of electricity elasticities
Estimation of the electricity demand equations (13) or the electricity expenditure share equations (14) requires the specification of a functional form for P which is homogeneous of degree one in the Pi. We consider three alternative forms for P the CES, the translog, and the generalized Leontief. The CES (constant elasticity of substitution) form was proposed by Arrow, Chenery, Minhas and Solow (1961). The CES form for P can be written as P=
( i
-l/P
s,p,:p
i=l
where xi 6, = 1. Application of equations:
1
,
of Roy’s identity
(22) to (22) yields the following
set
(23) jNote that (?M/ZPJ)(Pj/M)= (a.bf/aP)(P/bf)(LlP/?P,)(P,/P)= (1 + (?E/SP)(P/E))(SP/?P,)(P,/P). Since the P function 1s homogeneous of degree one, (?P/SP,)(P,/P)= (8P/?Pj)P,/x(?P/?Pk)Pk = zj, where the last equality results from Roy’s identity. 4This result is analogous to that used by Berndt and Wood (1979) in the context of producer theory to explain conflicting findings with respect to energy-capital substitutability.
294
D.N! Caves and L.R. Christensen,
Residential
electricity
pricing experiments
The most convenient way to obtain estimates of the parameters in (22) is to take logarithmic differences of pairs of the equations in (23); for example, ln(Ei/Ek) = ln(6,/6,) + ( - 1 - p) ln(Pi/Pk).
(24)
The translog functional form has been proposed by Christensen, and Lau (1971, 1975). The homogeneous of degree one translog can be written as
fnP=
i i=
zilnPi++ 1
i
i
i=l
j=l
Jorgenson form for P
p;jlnPilnPj,
(25)
where
Application expenditure
of Roy’s identity share equations:
Zi =CYi
+
i
Pij
to
(24)
yields
the
following
electricity
(26)
ln(Pj/Pk).
j=l
ifk
The generalized The homogeneous written as
Leontief form has been proposed by Diewert (1971, 1974). of degree one generalized Leontief form for P can be
P=CC i
YijP)P),
(27)
j
where yij=yji
Application expenditure
and
cyij=l.
of Roy’s identity share equations :
j$l
to
(27)
?ij(pi/pk)“(pj/pk)f i’i
Diewert substitution
yields
f. i=l
i j=l
the
following
electricity
-,j(pi/pk)i(pj!pk)‘I.
(1974, p. 131) provides a formula for computing the elasticities from an indirect utility function. For the electricity function
(28
1
of
P
D.W Caves and L.R. Christensen,
this formula
can be written, Oij=
c
Residential
electricity
k
c -
((dP/;iPi)(dP/dPj))
+
!I
(2P/GPj)
CPk(c3P2/6Pi dP,) k
1
(
)I
(dP/dPi)
P, (l?P/dP, iiP,)
k
)I
CPJdP/?P,)
Pk(d2P/ZPk dP,)P,
cc
m
295
for i # j,
(d2P/dPi dPj)CPk(ZP/dPk)
-
pricing experiments
(29 )
n
k
Applying Diewert’s formula to (21), (24), and (26), we obtain the following expressions for the cross-elasticities of substitution for electricity, for i fj, CES oij = 1 + p, Translog aij
Generalized
=
l +
(30)
(J3ij,/ZiZj),
Leontief
aij’~{~ij(Pi/Pk)‘(Pj/Pk)‘~/zizj
CCyij(pi/pk)t(Pj/Pk)~ i j
.
We see that the CES form has the restrictive feature that the elasticities of substitution between electricity consumption in any pair of time periods are constrained to be equal. There is no such restriction imposed by the translog or generalized Leontief forms, which is why they have been dubbed ‘flexible’ functional forms. Only in the case where 4 = 2 is the CES form as flexible as the translog and generalized Leontief forms. In this case there are only two free parameters to be estimated for all three forms and only a single crosselasticity of substitution. For the case of two commodities all three forms have the same local properties, but their global properties are quite different. The CES is attractive since it can provide a model which is globally valid for any non-negative value of the elasticity of substitution. The translog and generalized Leontief forms do not have this property. Their relative attractiveness depends on the magnitudes of the true elasticities of substitution which are being modelled. Roughly speaking, the translog form is more suitable for modelling ‘large’ elasticities and the generalized Leontief form is more suitable for modelling ‘small’ elasticities.’ ‘See Caves and Christensen
(1980) for further
discussion
296
D.W Caves and L.R. Christensen,
Residential
electricity
pricing experiments
4. Estimation of the partial elasticity of substitution and the partial price elasticities for peak and off-peak electricity from the Wisconsin experiment Of the many electricity pricing experiments from which data are currently available, the Wisconsin experiment is the only one which has included both mandatory participation and several different pricing treatments. Mandatory participation permitted the selection of a random sample of customers, which will greatly enhance the reliability of inferences regarding customer responses in the population. The customers in the experiment have been placed on four different price ratios, which permits econometric estimation of the partial price and substitution elasticities for electricity. In essence the Wisconsin experiment consists of approximately six hundred customers facing ten different pricing treatments. There is a control group which faces the same price for electricity regardless of the time of day. There are nine test cells resulting from the specification of three different peak to off peak price ratios (2:1, 4:1, and 8:l) and three different lengths of the peak pricing period (6 hours, 9 hours, and 12 hours in duration) which are in of the Wisconsin effect only for non-holiday weekdays. 6 The major limitation experiment for our analysis is that each customer faces only two distinct pricing periods a peak or high-priced period and an off-peak or lowpriced period. Thus direct application of our methodology allows estimation of only a single elasticity of substitution between peak and off-peak electricity consumption. The analysis which we present below draws upon data from the summer months of July and August, 1977. Since the Wisconsin Public Service Corporation is summer peaking, the effect of time-of-use rates during these months is of most interest. Our analysis is based upon monthly aggregates. We compute each customer’s total usage for the month for the peak and offpeak periods. We note that the control group (1 :l price ratio) can be included in the analysis of any of the three lengths of peak period since the full day can be arbitrarily divided into sub-periods. We begin our statistical analysis by estimating the CES form. With only two pricing periods the set of estimating equations (23) reduces to a single equation with two independent parameters, which can be estimated by a least squares regression. The estimating form (23) is particularly convenient
‘The summer
peak periods
are:
Six hour peak 9 a.m.-12 a.m. Nine hour peak 8 a.m.- 5 p.m. Twelve hour peak 8 a.m.- 8 p.m. The winter peak periods
and
1 p.m.4p.m
and and
5 p.m.-8 4 pm-9
are:
Six hour peak 9 a.m.-12 a.m. Nine hour peak 8 a.m.-12 a.m. Twelve hour peak 8 a.m.- 8 p.m.
p.m. p.m.
D.W. Caves and L.R. Christensen,
Residential
electricity
pricing experiments
291
in that the regression coefficient on the logarithm of the price ratio is the negative of the eleasticity of substitution (- 1 -p = -$,0) between peak and off-peak electricity consumption. If the estimated a;, is non-negative, then the implied consumer preferences are well-behaved globally. The intercept term in the regression provides an estimate of ln(6,/6,). Both 6, and 6, are identified since 6, + 6, = 1. We estimate a separate CES model for each month and each length of peak period and present the results in table 1. All the estimates of qPO are positive, significantly greater than zero, and fall in the range from 0.09 to 0.17. These bounds on the partial elasticities of substitution imply that increasing the price ratio from 1 :l to 8:l would result in changes of the logarithmic ratio of peak to off-peak consumption of 0.18 and 0.35, respectively.’ With only two electricity commodities our model is a valid representation of consumer preferences if the partial elasticity of substitution is nonnegative. The CES functional form is capable of providing a valid model of consumer preferences at all price ratios, because the estimate of the eleasticity of substitution from the CES model is independent of the price ratio. The translog and generalized Leontief models do not possess this capability. Caves and Christensen (1980) have shown that the range of price ratios over which the translog can provide a valid representation of consumer preferences depends upon the degree of substitution in the preferences being represented. In the two-commodity case if the elasticity of substitution is very low, the range over which the translog can closely approximate consumer preferences is very small. The range of acceptable approximation expands as substitution possibilities increase. The generalized Leontief form, on the other hand, is capable of approximating preferences characterized by low substitution possibilities over the full range of price ratios. In fact the generalized Leontief, like the CES, can exactly represent a fixed coefficient (a,,=O) preference function. For large values of the elasticity of substitution, however, the range over which the generalized Leontief function can closely approximate consumer preferences decreases. The magnitude of elasticity of substitution estimated from the CES model suggests that the generalized Leontief model is more suitable than the translog model for this application. Nonetheless, for the sake of illustration, we obtain parameter estimates for both of these models.* The estimates are presented in table 1. We compute the partial elasticities of substitution implied by these translog and generalized Leontief estimates and present them in table 2 along with the CES estimates. We find that the estimates ‘These figures can be interpreted approximately as 18 ‘7, and 35 Y;, changes in the consumption ratio. ‘For the two-commodity case, estimation of the parameters from either of the two share equations will yield identical results. We have estimated the translog share equation using linear least squares and the generalized Leonteif share equation using non-linear least squares,
D.W. Caves and L.R. Christensen,
298
Residential Table
electricity
pricing
experiments
1
Estimates of the CES, translog and generalized Leontief models of consumer preferences for peak and off-peak electricity; 6, 9, and 12 hour peaks in July and August 1977 (standard errors in parentheses).
August
July Parameter 6
9
12
6
9
12
CES model ~ eq. (23) In (fi,&)
$,=(I
RL
- 1.537
+P)
- 0.999
-0.519
- 1.396
- 0.848
-0.275
(0.036)
(0.030)
(0.029)
(0.036)
(0.035)
0.158
0.092
0.094
0.165
0.097
0.124
(0.030)
(0.025)
(0.025)
(0.029)
(0.029)
(0.024)
0.108
0.06 1
0.065
0.119
0.054
0.130
Trunsfog
(0.027)
model ~~ ey. (25)
0.175
0.271
0.379
0.195
0.303
0.437
(0.007)
(0.006)
(0.006)
(0.008)
(0.008)
(0.006)
0.175
0.208
0.207
0.182
0.207
0.195
(0.006)
(0.005)
(0.005)
(0.006)
(0.006)
(0.005)
0.792
0.885
0.895
0.774
0.844
0.885
Generalized
Leonriej model ~
q.
(27)
0.122
0.225
0.328
0.138
0.249
0.374
(0.006)
(0.007)
(0.009)
(0.007)
(0.010)
(0.010)
0.061
0.048
(0.011)
(0.01
0.79s
0.885
I)
0.047
0.064
0.054
0.057
(0.012)
(0.012)
(0.014)
(0.013)
0.894
0.777
0.844
0.889
from the translog are not well-behaved for the full range of price ratios. For each month and peak length the partial elasticity of substitution is estimated to be negative at the lowest or highest price ratio. On the other hand, the generalized Leontief estimates are well-behaved for all price ratios, and the estimated elasticities of substitution are similar to those of the CES. Given the poor behavior of the translog, we employ only the CES and the generalized Leontief models for the remainder of our empirical analysis. To obtain estimates of the partial price elasticities from the Allen elasticities in table 2, we can use eq. (17) along with the fitted shares from the estimating equations. In table 3 we present the partial price elasticities which result from the estimated generalized Leontief model for July. The results for August and for the CES model are similar.
D.W Caves and L.R. Christensen,
Residential
elecfricity
pricing experiments
299
Table 2
Allen elasticities
of substitution CES, (2) translog
between peak and off-peak electricity and (3) generalized Leontief models.
9 hour peak
6 hour peak
for the (1)
12 hour peak
Price ratio
(1)
(2)
(3)
(1)
(2)
(3)
(1)
(2)
(3)
-0.290
0.115
July 8:l
0.158
0.294 0.168
0.092
0.001 0.118
0.094
4:l
0.158
0.279 0.169
0.092
0.155 0.110
0.094
2:l
0.158
1 :l
0.158
8:l
0.158 0.181 -0.218
0.092
0.143 0.111 -0.053
0.121
0.069 0.103
0.094
0.170 0.097
0.094
0.120 0.099
0.203
0.092
0.165
0.256 0.169
0.097
0.132
0.124
4:l
0.165
0.264 0.168
0.097
0.146 0.121
0.124
0.060 0.127
2:t
0.165
0.166 O.t78
0.097
0.164 0.120
0.124
0.205 0.117
1:l
0.165
0.097
0.022 0.128
0.124
0.209 0.117
August
5. Estimation
-0.158
0.198
of the total elasticity
-0.056
-0.394
0.145
of substitution and total price elasticities
Up to this point we have modelled only the first stage of our two-stage model of consumer budgeting. Since there are only two time periods (peak and off-peak) in our model, consumption during those two periods must be partial substitutes ($,>O). This does not, however, rule out the possibility that peak and off-peak electricity are total compliments (dPOCO). To assess this possibility we refer to eq. (21) which shows how to obtain the total elasticity of substitution between peak and off-peak electricity. To evaluate the total elasticity we must have estimates of the price and income elasticities of aggregate electricity, along with an estimate of the share of total electricity expenditures in the total budget. These three estimates must come from the second stage of the model of consumer behavior. We could use the Wisconsin experimental data to estimate the second stage. To do so we would use our first-stage estimates to obtain estimates of the electricity price index, P, corresponding to each of the experimental price ratios. These estimates of P along with data on total income and total electricity expenditures could then be used to model the second stage. However, the desirability of this procedure is limited by the design of the Wisconsin experiment. To minimize the impact on customer bills the peak and off-peak rates were chosen so that the experiment would have no impact
300
D.W Cuces and L.R. Christensen,
Residential
electricity
pricing
experiments
Table 3 Partial price elasticities and expenditure share for peak and offpeak electricity from the generalized Leontief model for July 1917.
Partial
Rate
Peak expenditure share
price elasticity
qip 6 hrn1r peak
8:l
- 0.629
-0.539
-0.371
-0.461
0.554
4:l
~0.510
- 0.659
- 0.490
-0.341
0.410
2:l
-0.412
- 0.769
-0.588
-0.231
0.282
1:l
PO.349
- 0.854
-0.651
-0.146
0.183
Y hour peuk 8:l
-0.739
~ 0.380
~0.261
- 0.620
0.704
4:l
-0.610
- 0.500
- 0.390
- 0.500
0.562
2:l
- 0.475
- 0.636
- 0.525
PO.364
0.409
1:l
PO.361
- 0.760
- 0.639
- 0.240
0.273
8:l
-0.812
PO.303
-0.188
- 0.697
0.788
4:l
-0.710
- 0.393
- 0.290
- 0.607
0.577
2:l
-0.574
~ 0.523
- 0.426
- 0.477
0.529
1:l
- 0.437
- 0.662
PO.563
- 0.338
0.375
12 hour peak
on company revenues if customers did not shift their consumption toward the off-peak period. This offsetting feature of peak and off-peak price changes, in conjunction with the low elasticity of substitution found above, implies that the electricity price index, P, does not change substantially with the experimental pricing treatments. The small amount of variation in P across pricing treatments makes it difficult to obtain reliable measures of qEE from the experimental data. For these reasons we turn to other studies which have made use of the variation in electricity prices among individuals or regions or over time to measure qEE. These studies have typically found that demand for electricity is inelastic in the short-run, though there is wide disparity among the estimates.’ We have chosen -0.5 as a representative value. Although the Wisconsin data are not well suited to estimate qTEE,we can utilize these data to estimate the income elasticity, qEY, and the share of ‘See Taylor
(1975, 1976).
D.W Caces and L.R. Christensen,
Residential
electricity
pricing experiments
301
electricity in the total budget, WE. We have used data on income and electricity expenditures for all customers during the same months of the year prior to the year that the test rates were applied. The equation we have estimated is lnM=a+hln
Y
(31)
where Y is average monthly income as reported on the questionnaire. The estimate of a is 1.531 with standard error 0.179. The estimated income elasticity, given by the parameter h, is 0.229 with standard error 0.024. At the estimated average income level for the residential customer population ($1,500/ month), eq. (3) implies W,=O.O16. We now have the necessary information to convert the partial elasticity of substitution estimates of table 2 into total elasticity of substitution estimates. The six CES partial elasticities in table 2 have an average value of 0.122. Using our estimates of qEE, qer and WE, a partial elasticity of 0.122 implies a total elasticity of -23.4. This indicates that peak and off-peak electricity display strong complementarity, even though they are partial substitutes. Our estimate of the total elasticity of substitution between peak and off-peak electricity is not precise since we have taken our estimate of qEE= -0.5 from extraneous sources. We cannot attach any of the usual measures of sampling error to our estimates to assess whether our finding of total complementarity is statistically significant. We can, however, compute the range of values of qEE which are consistent with total complementarity, conditional on our estimates of gPO, qEy and W,. We find that total complementarity will prevail as long as qEE lies below -0.13. We conclude that it is very likely that peak and off-peak consumption are total complements.” We now use eq. (20) to estimate the total price elasticities which correspond to the partial price elasticities of table 4. We illustrate this computation by making use of our representative value for qEE of -0.5. The estimated elasticities are presented in table 4.
6. Allowing preferences to differ among customers In arriving at the preceeding estimates we have assumed that all customers have the same preferences with respect to use of peak and off-peak electricity. It is not necessary to maintain this assumption, however, because the Wisconsin experiment provides a large amount of information about the characteristics of the individual customers. This information was obtained “We have tested the robustness of our conclusions by estimating alternative equations which allow the income elasticity to vary, by utilizing data from another year, and by evaluating WE at income levels other than the sample mean. Our conclusions are not sensitive to these modifications.
302
D.W Cures
and LX.
Christensen,
Resinential
electricit!:
pricing
evperimenrs
Table 4 Total price leasticities for peak and off-peak electricity from the generalized Leontief model for July 1977.
Price elasticities
R:l
-0.352
-0.316
-0.148
-0.184
4:l
~ 0.305
- 0.364
-0.195
-0.136
3:l
-0.271
-0.410
-0.229
~ 0.090
1:1
-0.257
- 0.445
- 0.242
- 0.054
-0.113
- 0.268
Y hour /x’“k 8:l
-0.387
-0.232
4:l
- 0.329
- 0.28
2:l
- 0.270
1:l
- 0.224
8:l
-0.418
-0.197
- 0.082
- 0.303
4:l
-0.371
-0.231
PO.128
- 0.268
2:l
- 0.309
- 0.287
-0.190
PO.212
1 :1
- 0.249
- 0.349
-- 0.250
-0.150
I
-0.171
-0.219
- 0.340
- 0.229
PO.159
-0.396
PO.275
-0.103
l-7 hour pcwk
Table 5 F-values for testing the hypothesis that no significant difference exists between customers with and without air conditioning.
July 1977
Length
of
peak
August
Generalized CES
Leontief
1977
Generalized CES
Leontief
6 hr.
0.253
0.429
2.093
2.005
Y hr.
0.034
0.027
0.642
0.434
12 hr.
0.389
0.154
4.873”
4.512
“Significant
at 5 “,> level
D.W Cuves und L.R. Christensen,
Rrsidentiul
electricity
pricing
303
experiments
from a detailed questionnaire which was completed by virtually all participants shortly before the experiment began. Using data on customer characteristics, it is feasible to estimate distinct partial elasticities of substitution for different classes of customers. We explore the degree to which the partial elasticity of substitution varies over customer classes by employing two customer characteristics of particular interest. The first is the ownership of air conditioners, and the second is the level of total electricity usage. Air conditioning is thought to be an important determinant of residential customer load shapes during the summer months. The level of total usage is important since benefitcost analysis may indicate that time-ofday pricing is appropriate only for customers with high usage of electricity.
Table 6 Effect of (1) air conditioning predicted percentage
and (2) no air conditioning on partial elasticity of substitution of kWh on-peak for 12 hour peak customer for August 1977.
CES
Generalized
and
Leontief
“,,kWh on-peak
Allen cross-elas.
“,, kWh on-peak
Allen cross-elas.
(1)
(2)
(1)
(2)
(1)
(1)
34.9
37.9
0.131
0.130
35.3
37.8
0.138
0.151
37.2
40.0
0.122
0.133
Rate
8:l
(2)
(2)
4:l
36.8
39.9
0.131
0.130
2:1
38.9
42.1
0.131
0.130
39.1
42.1
0.114
0.123
1:l
41.1
44.3
0.131
0.130
41.0
44.1
0.115
0.122
For the analysis of air conditioning we estimate CES and generalized Leontief models in which all parameters are allowed to differ according to the presence or absence of air conditioning. In table 5 we report the computed F-statistics which can be used to test the hypothesis that the parameters differ significantly for customers with and without air conditioning. The only significant difference we find is for the 12 hour peak period in August. In all other cases there is no significant difference between the usage patterns of customers with and without air conditioning. In table 6 we present the estimated percent on peak and the partial elasticities of substitution for air conditioning and non-air-conditioning customers. The differences in the elasticities of substitution are minor. It appears that customers with air conditioning can substitute as well as other customers even when the peak period lasts 12 hours. The main impact of the difference due to air conditioning is in the percent of kWh used on peak. Here we find that
304
D.W Caves and L.R. Christensen,
Residential
electricity
pricing experiments
custmers with air conditioning use a lower percentage on peak, regardless of the relative prices of peak and off-peak consumption. Our analysis of the impact of consumption level is similar to the analysis of air conditioning. We have used information on average 1975 consumption to group customers into three consumption categories: low less than 600 kWh/month; medium - between 600 and 1200 kWh/month; and high over 1200 kWh/month. We estimate CES and generalized Leontief models which allow all parameters to differ according to consumption level. In table 7 we present F-statistics for testing the hypothesis that no significant differences exist among consumption levels. In all cases the data are consistent with the hypothesis that relative usage patterns do not differ among the three consumption levels.
Table 7 F-values for difference
testing the exists among
hypothesis that no significant different consumption levels.”
July 1977 Length
August
1977
of
peak
Generalized CES
Leontief
Generalized CES
Leontief
6 hr.
0.891
1.183
1.257
1.171
9 hr.
1.399
1.987
1.959
1.967
12 hr.
0.784
0.930
1.019
0.447
“None of the F-values
is significant
at the 5 “/, level.
7. Concluding remarks methodology whereby the neoclassical theory of We have proposed consumer behavior can be employed to analyze the responses of consumers to experimental time-of-use electricity prices. It is convenient to summarize customer responses by elasticities of substitution, rather than price elasticities. As tables 3 and 4 indicate, both partial and total price elasticities can be very different depending on where they are evaluated, even when the partial elasticity of substitution is constant. With a small partial elasticity of substitution, it is especially important to indicate the prices at which the price elasticities are evaluated. The partial price elasticities for any price ratio can be readily computed from the partial elasticity of substitution and the share at that price ration [see eq. (17)]. Comparisons of results from the
D.W Caces and L.R. Christensen,
Residential
electricity
pricing experiments
305
various time-of-use experiments would be facilitated by presentation of estimated partial elasticities of substitution. We have illustrated the proposed methodology by making use of three alternative representations of consumer preferences - the CES, translog, and generalized Leontief models. These illustrations indicate that the partial elasticity of substitution between peak and off-peak electricity consumption is sufficiently small that the translog model is unsuitable for this type of analysis. The CES and generalized Leontief models both behave well and are clearly suitable for analysis with two periods. Because of its simplicity the CES model appears to be the preferred model for two periods. However, when there are more than two time periods, the CES model is unsuitable because of its lack of flexibility, and the generalized Leontief becomes the preferred model. The estimated partial elasticities of substitution from the Wisconsin data are small, but they are statistically significant for peak periods of 6, 9, and 12 hours in length. This implies that electricity consumption can be shifted away from these peak periods. Further research is required to determine the costs and benefits of such shifting, and thus the optimal time-of-use rate structure for the Wisconsin Public Service Corporation’s residential electricity customers. Although we have found that there are significant substitution possibilities between peak and off-peak electricity consumption, we have also found these two commodities to be complements with respect to the consumer’s total income. This finding holds provided that the short-run elasticity of demand for total electricity usage exceeds 0.13 in absolute value, i.e., provided that demand is not extremely inelastic. Our finding that peak and off-peak electricity are partial substitutes and total complements underlines the importance of clearly specifying elasticity estimates as ‘partial’ or ‘total’.
References Allen, R.G.D., 1938, Mathematical analysis for economics (Macmillan, London) 503-509. Arrow, K.J., H.B. Chenery, B.S. Minhas and R.M. Solow, 1961, Capital labor substitution and economic efficiency, Review of Economics and Statistics 63, 2255250. Berndt, E.R. and D.O. Wood, 1979, Engineering and econometric interpretations of energycapital complementarity, American Economic Review 69, June, 342-354. 1980, Global properties of flexible functional forms, Caves, D.W. and L.R. Christensen, American Economic Review 70, June, 422432. Christensen, L.R., D.W. Jorgenson and L.J. Lau, 1971, Conjugate duality and the transcendental logarithmic production function, Econometrica 39, 255-256. Christensen, L.R., D.W. Jorgenson and L.J. Lau, 1975, Transcendental logarithmic utility functions, American Economic Review 65, June, 367-383. Diewert. W.E., 1971, An application of the Shephard duality theorem: A generalized Leontief production function, Journal of Political Economy 79, 481-507. Diewert, W.E., 1974, Applications of duality theory, in: M.D. Intriligator and D.A. Kendrick, eds., Frontiers of quantitative economics, Vol. 2 (North-Holland, Amsterdam).
306
D.W
Caces und L.R. Christrnsrn,
Residentiul
rlectricify
pricing
exprriments
Goldberger. AS., 1967, Functional form and utility: A review of consumer demand theory, Social Systems Research Institute workshop paper SFM 6703 (University of Wisconsin. Madison, WI). Roy, R.. 1943, De I’utilitt-: Contribution li la thtorie des choix (Hermann et Cie., Paris). Slutsky. E.E., 1915, On the theory of the budget of the consumer, Giornale degli Economisti 51, July, l-26. Taylor, L.D., 1975, The demand for electriuty: A survey, Bell Journal of Economics 6, Spring, 74-l 10. Taylor, L.D., 1976. The demand for energy: A survey of price and income elasticities, Mimeo. for the National Academy of Sciences Committee on Nuclear and Alternative Energy Systems, April.