Industrial Electricity Demand: New Results Using Bootstrapping Techniques S. M. Khalid Nainar Research Associate Public Utility Research Center University of Florida February 1985 Acknowledgements. I am grateful to Professor Sanford Berg for suggesting this topic to me and encouraging me along the way. I also thank Professors Stephen Cosslett and Kim Sawyer for enlightening me on various methodological points. The usual disclaimer is invoked, however.
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Industrial Electricity Demand:
New Results Using Bootstrapping Techniques
S. M. Khalid NainarResearch Associate
Public Utility Research CenterUniversity of Florida
February 1985
Acknowledgements. I am grateful to Professor Sanford Berg forsuggesting this topic to me and encouraging me along the way. I alsothank Professors Stephen Cosslett and Kim Sawyer for enlightening me onvarious methodological points. The usual disclaimer is invoked,however.
Abstract
This study examines the responsiveness of large industrial customers to time-of-use rate structures. The study set out to achieve twopurposes: (1) to replicate the Hirschberg-Aigner (H-A) analysis forFlorida data and (2) to introduce a correction in the H-A methodologywith regard to simultaniety bias between the output proxy and economicdemands, and to derive correct standard errors for estimates of variousprice elasticities.
The analysis corroborates the H-A analysis of Southern Californiadata, although the price elasticity estimates in our study are relatively higher. The analysis indicates that simultaniety bias did notsignificantly change the estimates.
When deriving price elasticity estimates from estimated cost-shareelasticities in a translog formulation, the variance of the priceelasticity estimates consists of two components: (1) variance due tovariations in observation and (2) variance due to the derivation of aprice elasticity estimate based on the estimated value of the cost shareelasticity. H-A analysis derives various price elasticity estimatestaking into account only the first component of the variance. Consequently, H-A underestimate the variance of their price elasticityestimates and, correspondingly, overestimate the significance of theseprice elasticity estimates. When we correct this analysis by takinginto account the second component of the variance of price elasticityestimates, some estimat~s that were significant earl ier becomeinsignificant.
INDUSTRIAL ELECTRICITY DEMAND:
New Results Using Bootstrapping Techniques
Introduction
Over the last few years, and particularly since the passage of PURPA,
electric utilities have shown interest in time-of-use (TOU) price
structures. Such pricing tries to mirror the opportunity costs of
producing electricity, which varies with the time of day, the day of the
week, and across seasons.
Many empirical studies have tested the responsiveness of industrial
customers to the new TOU rate structures. Notable among these are
Henderson [13] and Hirschberg and Aigner (H-A) [14]. H-A proposed a new
methodology to capture responsiveness to demand or capacity charges; they
found that the various price elasticities for different SrCtwo-diqit
classified group industries (except SIC 35 -- machinery except electrical)
were not significantly different from zero.
The purpose of this study is to apply t~e same type of analysis to
data for Florida provided by FP&L, to see whether we obtain the same
2
resul ts. H-A had assumed that a simul tanei ty bi as between tota1 energy
(used as output proxy) with economic demands Xi was insignificant, ensuring
consistency of estimates. We suggest a refinement to the H-A methodology
affecting how cost share elasticities translate into the relevant price
elasticities. Furthermore, we also allow for simultaneity bias.
The following section outlines the specification of the econometric
model used in our study. Next, we describe the data base. Then we present
the results, note the limitations of this study, and outline directions for
future research.
Economic Model and Econometric Specification
Fo11 owi ng H-A, we assume that a fi rm has the fo 11 owi ng type of
production function:
Y = f(K, L, NE, E), (1)
where Y is output, K is capital, L is labour, E is energy, and NE is other
non-energy input (materi a1s) . Energy coul d be subdi vi ded (fo11 owi ng H-A)
as consisting of electric and non-electric. Electric energy is further
subclassified according to time-of-day. The electricity input can be
broken into demand (KW) and energy (KWH); both are assumed to be sensitive
to time of use. The KW demand is defined as maximum instantaneous load
within the period under consideration, while the KWH energy is tt'E total
electricity IlS'ed within the same period. Formally, KW demand = max KW(t)
where KW(t) is the usual load curve and
KWH energy =
t 2f KW( t)dt.t 1
(2)
3
Assuming weak separability in the electricity inputs, we can write
Y = g(H(X), 8). (3)
where H is the electricity input function and 8 is a vector of all other
inputs and X is the vector of time-differentiated electricity inputs. This
characterization reflects a two-step optimization procedure. First, the
firm is assumed to determine its total electricity cost as a function of
the level of output and prices of all other inputs (both energy and
non-energy). Next, it allocates its electricity consumption (KWH) and
demand (KW) by time-of-use as a function of total electricity cost and
time-of-use pricing structure.
Further, given the duality of cost and production functions, we have
C = C(P,Z,E) (4)
where C is total electricity cost; P is the vector of prices of the various
electricity inputs; Z is a vector of exogenous factors, such as weather;
and E is the tota1 energy consumed, wh i ch is used as a proxy for total
output. Using Shepard I s Lemma, we get the input demand functions for
various electricity inputs:
dC.dP ~ = Xi and i s I
1
(5)
where Xi is the cost minimizing level of electric input i and I is the set
of all electric inputs.
4
The functional form that is used is the translog function
1nC = a + L: a,. ln P. + t L: L: S·· ln P. ln P.. , , ..,J , J, , J
+ L: y. 1n P. ln E + 1/J ln E (6). , ,where
,
r a,. = 1 L: S··· = (}\Lib b.. = O¥j L: y. = a [3.. = SJ.'. -\Li rj ( 7),. 1 . lJ ,.'J ,. 1 '.]J .
Equation (7) assumes that the cost function, (6) is positive linear
homogeneous in the elements of P (the electricity input price vector). The
underlying cost function (3) is assumed to be twice differentiable.
Although the input demand functions obtained from the translog cost
function are not linear in parameters, usingShephard1s Lemma (4) we have
·alnC . ac PiX ia1nP"": =(l/C) Pi ap. = -c-·_. :: Mi, . • , .
(8)
where Mi is the cost share of the i th electric input. After some
manipulation, we obtain:
(9)
With this specification, we have a problem with "simultaneity bias" between
E (output proxy) and "demands" Xi. H-A assume that because of highly
nonlinear relationships, the estimates obtained w'@lultlbe consistent1. This
assumption is testable, so later we consider a simultaneous equation
formulation to see if the bias is significant.
5
Next, the own-price and cross-price elasticities are estimated from
the parameters of the cost share equations in the following way:
(3 ••
Own price elasticity: ni ; =M~~~ + Mi - 111
Cross-price elasticity of i th input with respect to the
jth input price is:(3 ••
- 1J M 1· -J. J.n·· - M +. T1J i J i, j E: I
(10)
(11)
Since the price elasticities are functions of cost shares, they are
variable over observations. H-A assume normal distribution and obtain
standard errors that are not correct. We do bootstrapping for n iiestimates (see Efron [8, 9, 10, 11]) and generate a distribution for n iiand compute the mean value for 11 i i and the "ri ght" standard errors. The
bas i c problem is, when we get n'i from (3i values, we have to correct for
sample size in the sense that we are getting values of n based on one value
of 8. So, to control for that variance, we run our system regression
several times (say 50 times), after constructing residuals and artificial Y
(dependent variable vector). From these regressions we compute (3 and n
each time.- From this we will have 50 values for n i.e., n1' n2' n50 ·
Next, we compute the mean n as usual and the right standard error as:
50s . d = L
n i=l(12)
We should note here that we did not impose any distribution on n; i.e., it
was truly non-parametric. Even if we a·ssume a normal distribution for n as
H-A did, we should technically still construct residuals and use a random
normal generator on these residuals; and then get the artificial Y the
6
dependent variable vector and run system regressions several times and then
get the various n's and their standard errors. That way, we would have
controlled for the sample size. 2
As noted ea rl i er, H-A proposed an improved methodology of defi ni ng
demand for each good in terms of its characteristics, following Lancaster
[18]. We look at the results obtained by this approach. While earlier
studies, notably Chung and Aigner [5J defined inputs according to prices
paid for them, this analysis defines inputs with intrinsic characteristics
and varying prices. With that change in definition of a "good", the prices
of these goods also have to change. Thus,
Cost of energy = (KWH) [$/KWH + ~~~W]
where KWH is total energy used in the particular rating period, $/KWH is
the energy price, $/KW is demand charge; and #hr. are the hours in the
rating period under consideration. The term in brackets is the usual
effective energy price minus the capital charge plus the conventional
energy charge. The remaining portion of total electricity costs are called
potential energy consumption (PEC) and
Cost of PEC = [max KW - ~~~] [$/KW].
PEC is then interpreted by H-A as a measure of an average load curve shape
and hence of variations in the load. So the elasticity of PEC provides a
good indication of the load-levelling effect of demand charges. Again, the
PEC can be time-differentiated as we did for the energy input. The cost
7
share equations now become:
p -M,-t = a,- + L: B- - ln (-pJ)t + y- ln Et + e,-
, jfO' J 0 '
i = A, B, C, D t = 1, ... T. (13)
We consider only 3 equations, because of add-up restrictions; otherwise, we
will have a singular equation system, on account of the above add-up
restrictions, noted in (7), earlier in the paper. We estimate by Zellner's
SURE [22J and N3SLS [12J methods. We check for autoregression in errors,
as we did in the other approach. Following the others, we do not allow for
substitution across nonadjacent time-periods. Next, the various prices
were tri ed as numera ire pri ce and we selected the one that gave best
estimates -- implicitly minimizing condition index for relevant matrix. As
H-A note, this choice does not affect the properties of our estimators.
Data
The data used were provided by Load Research Group at Florida Power &
Light, Miami, Florida. The overall data base included about 22 companies
and the data on their electricity consumption. The companies belonged to
GSLDT-3 and CST-3 groups. These are companies with over 4,000 KW demands.
Due to data problems with some companies we included only 6 companies and
restricted ourselves only to GSLDT-3 class for this study. This still
provided 96 observations of which we left out about 6 observations on
account of some missing components. Also for some components (e.g., KWH)
the data were from rate systems while others were from load research
systems. To the extent that there are discrepancies of this sort, our
results will be affected. Also, exact start dates and closing dates of
8
bi 11 i ng peri ods for each company were not known and to that extent a
further errors-in-variables problem creeps into our results. The exact
tariff structure presented in table form is appended at the end of the
paper. While there were several price variations in energy charges, we had
only one price variation in the demand charges. This point also has to be
noted when we consider the results. The time series used was from
September 1982 to December 1983.
Results
The resul ts are presented in two sets: Fi rst we present the pri ce
elasticities computed with reference to commodities as usually defined
(i.e., on basis of price). Next, we present the price elasticities result
computed with reference to commodities defined on the basis of
characteristics [18J. The various commodities defined by the Lancaster's
characteristics approach [18J are illustrated in Figures 1 and 2. As can
be observed in Figure 2, A refers to peak energy commodity; D represents
off-peak PEC commodity; C represents on-peak PEC commodity, and B refers to
off-peak energy commodity. The values for elasticities with respect to B
off-peak energy commodity can be easily obtained from cost-share elastici
ties for other commodities using the add-up restrictions alluded to
earlier. It was found that there was no significant autocorrelation, hence
we did not use Zellner's SURE with AR method [4J.
The results are generally similar to the results obtained by other
authors [19, 15J, although the specifics are quite different and notable.
All elasticities are significantly positive in sign, a finding that
corroborates the H-A study. The key difference in our results is that the
values are relatively quite high. As H-A point out, this situation might
KWmaxkw
(AREA x no. of days x demand charge)= PEe cost
TIME OF DAY 24(AREA x no. of days x energy charge =ENERGY cost
Figure I
Distinction between energy and PEe fora firm subject to a demand and anenergy -chorg-e,._(nc>tfjQlrvoryln-gprices)
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