Ž .
Energy Economics 22 2000 649]666
Heating technology and energy use: a discretercontinuous choice approach to
Norwegian household energy demand
Kjell Vaage
U Department of Economics, Uni
¨
ersity of Bergen, Fosswinckelsg. 6, 5007 Bergen, Norway
Abstract
The aim of this paper is to describe the structure of the household’s energy demand as a discretercontinuous choice and, on this basis, establish an econometric model suitable for
the data available in the Norwegian Energy Sur
¨
eys . The discrete appliance choice is
Ž specified as a multinomial logit model, with a mixture of appliance attributes operating
. Ž
. costs and individual characteristics income, housing unit characteristics, etc. as explana-
tory variables. In the next step the continuous choice of energy use is modelled conditional on the appliance choice. The energy prices turn out to be significant both when estimating
the appliance choice and the conditional energy demand. The estimated price elasticity for energy exceeds minus unity. The paper discusses how this relatively strong price response
should be interpreted in the context of other econometric analysis with no explicit appliance dependence. Finally, the significance of the many household characteristics at both stages of
the model signals a high degree of heterogeneity within the households, which justifies the use of detailed micro-data in the modelling of the energy demand. Q 2000 Elsevier Science
B.V. All rights reserved.
JEL classifications: C25; D12
Keywords: Residential energy demand; Discretercontinuous choice
1. Introduction
A salient feature of the demand for energy is its inherent dependence on indivisible household durables. Energy is not used directly, rather, it is used to
U
Tel.: q47-55-58-92-06; fax: q47-55-58-92-10. Ž
. E-mail address:
kjell.vaageecon.uib.no K. Vaage . 0140-9883r00r - see front matter Q 2000 Elsevier Science B.V. All rights reserved.
Ž .
PII: S 0 1 4 0 - 9 8 8 3 0 0 0 0 0 5 3 - 0
K. Vaage r Energy Economics 22 2000 649]666 650
power appliances that produce services such as heatingrcooling, cooking, lighting, etc. This has motivated the search for dynamic models in analysis based on time
series data, since a household’s response to changes in, e.g. relative prices, is different in the short-run, when the demand is limited by the given stock of
installations, than in the long-run, when the appliance has been optimally adapted to new conditions. Econometric analyses of residential energy demand based on
aggregated time series data are numerous; Norwegian studies include Rødseth and
Ž .
Ž .
Ž .
Strøm 1976 , Bjerkholt and Rinde 1983 , Vaage 1995 . Cross-sectional micro data for Norway are, however, available through three
extensive surveys carried out by the Central Bureau of Statistics in 1980, 1983, and Ž
. Ž
. Ž
. 1989, which are documented in Hem 1983 , Ljones 1984 , Ljones et al. 1992 ,
respectively. Of course, cross-sectional data do not allow a true study of dynamics. They do, however, allow a focus on the relationship between energy demand and
the stock of energy using appliances. This relationship is as follows.
The first decision involved in energy consumption is the choice of energy appliance. Then, conditional on this choice, the household decides how to utilise
the given stock. Since these are related decisions, the two choices must be modelled jointly to avoid biased and inconsistent parameter estimates. The econo-
metric analysis of the utilisation decision can be treated in a conventional manner in that the demand variable can be assumed to vary continuously with its explana-
tory variables. The appliance choice, on the other hand, is a typical example of a discrete variable, since the households have to choose among a limited number of
appliances. The primary aim of this paper is, accordingly, to describe the structure of the household’s energy demand as a discretercontinuous choice, and, on this
basis, establish an econometric model suitable for the data available in the Energy Sur
¨
eys . Due to data limitations we are restricted to concentrate on the choice of
heating technology. A unifying theoretical exposition of the discretercontinuous choice approach is
Ž .
found in Hanemann 1984 . Based on the same approach Dubin and McFadden Ž
. 1984 , present an econometric analysis of residential electric appliance holdings in
Ž .
the US, while Nesbakken and Strøm 1993 apply the 1990 Energy Survey in a discretercontinuous model for the energy demand in Norwegian households.
Ž .
Dagsvik et al. 1987 present an extension to a dynamic discretercontinuous choice model, and apply their model in an analysis of gas demand in the residential sector
in western Europe. In Section 2 we describe the structure of the discretercontinuous choice model,
and show how the model can be represented in a form that allows econometric estimation. Section 3 presents the Energy Sur
¨
ey data, while Section 4 reports and
discusses the results from the econometric analysis. Section 5 provides some concluding remarks.
2. The model
Ž .
The following exposition is largely based on Hanemann 1984 , but while his
K. Vaage r Energy Economics 22 2000 649]666 651
main concern is the modelling of demand for different brands of a commodity, we adapt the same approach to the residential energy demand.
Hanemann formulates a model where both the discrete and the continuous consumer choices are derived from the same utility maximisation problem. Broadly
speaking, this is a question of finding a random indirect utility function which, when the random component is specified, can be used to derive a model for the
estimation of choice probabilities for different heating technologies. Conditional on the chosen technology, energy demand functions are derived by applying Roy’s
identity to the same indirect utility function. This will now be demonstrated step-by-step.
Suppose x is a continuous vector measuring the consumption of N different energy sources in a given household. We start out with the following utility
function:
1
Ž .
Ž . u s u x
,b, z,s,« 1
where x is a vector of goods and z is a numeraire, b is a vector of characteristics of x
, s is a vector of characteristics of the individual, and « is a random component. The discrete choice is to decide whether or not x is to be zero, while the
i
continuous choice is to decide how much x to use, given that x 0 in the first
i i
step. As in all other works referred to in this article we restrict our household to
2
Ž .
choose only one heating alternative. Hanemann 1984 does this by choosing
utility functions that restrict the indifference curves to be linear or concave and thereby ensure corner solution.
3
Ž Suppose that a consumer decides to choose good i with corresponding tech-
. nology . Given this choice we get the conditional utility function:
Ž .
Ž . u s u
x ,b , z,s,«
2
i i
, i
i
that is maximised subject to p x q z F y and x , z G 0. We then define the
i i
i
conditional indirect utility function as: Ž
. w
Ž .
Ž .
x Ž .
¨
p ,b ,s,« u x p ,b , y,s,« ,b , z p ,b , y,s,« ,s,«
3
i i
i i
i i
i i
i i
By Roy’s identity the continuous-choice demand for energy, conditional on the choice of technology i, can be derived as:
1
The household index is omitted for ease of exposition.
2
The majority of the households participating in the Energy Sur
¨
ey have installed heating technology
that can combine or switch between electricityrsolid fuelroil. This must not be confused with multiple appliance choice; still the households are grouped according to single choices like electricity, oil,
oil q electricity, etc.
3
This is tantamount to assuming that the different alternatives are perfect substitutes.
K. Vaage r Energy Economics 22 2000 649]666 652
Ž .
¨
p ,b , y,s,« rp
i i
i i
Ž .
Ž . x
p ,b , y,s,« s y
4
i i
i
Ž .
¨
p ,b , y,s,« r y
i i
i
Ž . To find econometric models for choice probability and energy use, we have to i
describe the random component « and specify how it enters the utility function
i
Ž . and ii choose a specific form of the utility function’s non-random components.
We start with the former part: A household chooses appliance irconsumption of energy good x if the utility
i
from this alternative exceeds the utility from all other choice alternatives. If the Ž
. indirect utility is decomposed into a non-random part, w s w p ,b , y,s , and a
i i
i i
random part, « , the choice probability, P , may be expressed as:
i i
Ž . P s Pr w q « G w q «
, all i j s Pr « y « - w y w , all i j . 5
4 4
i i
i j
j i
j i
j
Assuming that « is independent and identical type I extreme value distributed,
i
Ž .
4
the choice probability can be expressed by the multinomial logit model MNL :
w
i
e Ž .
P s 6
i
N
w
j
e
Ý
js 1
Ž .
As for the randomness of the utility function, Hanemann 1984 assumes that it is introduced through the econometrician’s limited possibility of observing the
household’s evaluation of the quality of the different goods. Instead of letting b, the characteristics of x, enter the utility function directly, the quality index c is
i
Ž .
constructed as a function of K non-random properties of good i, b , and the
i k
random component « :
5 i
K
Ž .
Ž . c
b ,« s exp a q
g b q «
7
Ý
i i
i i
k i k
i
ž
ks 1
It is now the quality index that enters the utility function. By assuming that the consumers maximise the product c x and letting all N arguments be entered
i i
additively:
N
U
Ž .
Ž . u x
,c, z,s s u c
x , z,s
8
Ý
i i
ž
is 1
4
Ž .
See, e.g. McFadden 1974 .
5
Ž .
Alternative specifications are discussed in Hanemann 1984 , p. 547. Since economic theory can hardly give any guidance on this question we choose the multiplicative form on the basis of pragmatic
considerations.
K. Vaage r Energy Economics 22 2000 649]666 653
maximisation subject to the budget constraint will result in a corner solution.
6
The restriction of energy goods as perfect substitutes is thereby taken care of.
Given the choice of goodrtechnology i the conditional utility function, u , will
i
be given by:
U
Ž .
Ž .
Ž . u x
,c , z,s s u c
x , z,s .
9
i i
i i
i i
Ž .
Muellbauer 1976 has shown that this gives the following conditional demand and indirect utility functions:
U
Ž .
Ž .
Ž .
x p
,c y,s s x p rc
, y,s rc 10
i i
i ,
i i
i i
U
Ž .
Ž .
Ž .
¨
p ,c , y,s s
¨
p rc , y,s .
11
i i
i i
i i
Hence, quality and quantity affect the consumer’s utility in identical ways; a doubling of quality, for example, will give the same increase in utility as a doubling
in quantity. If this is carried over to the demand and indirect utility functions, we obtain that quality improvements are equally evaluated as price reductions; the
consumers adapt, accordingly, to ‘quality per money’.
As noted earlier, alternative i is chosen if
U U
Ž .
Ž .
¨
p rc , y,s G
¨
p rc , y,s .
i i
i j
j j
In Hanemann’s setting income, y, and other individual characteristics, s, affect the indirect utility function, but not the choice probability.
7
This implies that the choice of alternative i depends entirely on p r c
Furthermore, in Hanemann’s
i i.
formulation the discrete choice does not depend on the parameterisation of p r c .
i i
Ž .
The choice probability may, therefore, be expressed in logarithmic form as: 4
Ž .
P s Pr log p y logc F log p y logc , all i j ,
12
i i
i j
j
Ž . and with Eq. 7 substituted for c we get:
i
K K
Ž .
P s Pr « q a q g
b y log p G « q a q
g b y
log p 13
Ý Ý
i i
i k
i k i
j j
k jk
j
½ 5
ks 1
ks 1
We will maintain Hanemann’s convenient representation of the p r c ratio,
8 i
i
but in addition we wish to allow choice probabilities to be influenced by income Ž
. and other individual characteristics. Hence, Eq. 13 is extended to:
6
Ž .
A formal treatment of this property is found in Deaton and Muellbauer 1980 .
7
In other words, it is assumed that individual characteristics have identical impact on each choice alternative.
8
As will be apparent in a short while, this is assumed to achieve tractable econometric models with identifiable parameters.
K. Vaage r Energy Economics 22 2000 649]666 654
M K
P s Pr « q a q b y q d
s q g
b y log p G « q a q b y
Ý Ý
i i
i i
i m m k
i k i
j j
j
½
ms 1
ks 1
M K
Ž .
q d
s q g
b y log p
14
Ý Ý
jm m k
jk j
5
ms 1
ks 1
where the b s and the d s are the respective variable coefficients. If all non-random components are collected in:
M K
Ž .
w s a q b y q d
s q g
b y log p
15
Ý Ý
i i
i i m m
k i k
i
ms 1
ks 1
Ž .
Eq. 14 simplifies to: 4
Ž .
P s Pr « q w G « q w .
16
i i
i j
j
Ž . From Eq. 6 we see that the choice probabilities can now be expressed as:
M
Ž .
Ž .
exp a rm q b rm y q d rm s
Ý
i i
i m m
½
ms 1
K
Ž .
Ž .
q g rm b y
1rm log p
Ý
k i k
i w r m
5
i
e
ks 1
Ž .
P s s
17
i
N N
M
w j rm
Ž .
Ž .
e exp a rm q b rm y q
d rm s
Ý Ý
Ý
j j
jm m
½
is 1
is 1
ms 1
K
Ž .
Ž .
q g rm b y
1rm log p
Ý
k jk
j
5
ks 1
which is the multinomial logit model. Note that all the parameters are scaled by m, which is a measure of the dispersion of the extreme value distribution.
9
Hence, the larger the dispersion, the less the effect from the explanatory variables. In other
words, if the ‘uncertainty parameter’ m is ‘large’, the observed non-random variables explain little, whereas random elements explain a lot.
Values of m should be positive a priori; firstly, because a price increase for alternative i, measured by y1rm, should lead to a lower probability of choosing
that alternative. Secondly, and even more fundamentally, negative m implies a negative dispersion of the EV distribution, which makes no sense.
As will be demonstrated below, an estimate of m is needed in the estimation of Ž
. the continuous choice of energy use. From what is explained above we now see
that through the specification of the price variable in the indirect utility function, m becomes identifiable as the inverse of the estimated price parameter from the
discrete choice model.
The next step is to derive a model for the continuous choice of utilisation of the
9
Ž . The scale parameter in 6 is, for ease of exposition, set to 1.
K. Vaage r Energy Economics 22 2000 649]666 655
appliance stock. This is done by applying Roy’s identity to the chosen indirect Ž
.
10
utility function. The following function is suggested by Hanemann 1984 , but
extended to include individual characteristics:
Ž .
1yr yŽry
1.
w x
u p
e b
y q d9 s
i i
i
Ž .
¨ s y
18
i
ž
ž
Ž .
r y 1
c r y
1 b
i i
Via Roy’s identity and some simplification this yields:
y 1 Žry1.w i Žry1.«
i
Ž .
x s u p e
e 19
i i
For convenience we apply the logarithmic transformation: Ž
. Ž
. Ž
. log x s logu y log p q r y 1 w q r y 1 « .
20
i i
i i
Ž .
Ž .
« in Eqs. 19 and 20 are the random components conditional on the choice of
i 11
good i, «
¨ G ¨
. The expected value of «
¨ G ¨
is:
i j
i j
y 1
Ž .
Ž .
E «
¨ G ¨ s
m log P
q 0.5772 ,
21
i i
j i
where 0.5772 is the Euler constant. The expected value of log x is therefore:
i
N
w x
Ž .
Ž .
E log x s logu y log p q r y 1 m log
q 0.5772
22
Ý
i i
ž
js 1
N
w rm
j
w Ž .x
The term m log e
q 0.5772
is referred to as the selection term, and
Ý
js 1
measures the extent to which the energy expenditures are influenced by the choice of heating technology. The interaction between the probability structure in the
discrete choice problem and the conditional demand structure takes place in this term. Obviously, if our assumption that appliance choice and appliance utilisation
are related decisions is correct, omission of this variable will introduce bias in the energy demand model. The use of appliance dummies in the demand equation, as
10
Hanemann proposes several conditional indirect utility functions that are suitable for the deriva- tion of conditional demand functions. His criteria are that they must belong to a class of utility
functions that have the desired properties of corner solution, etc., and that they must lead to tractable demand models.
11 y
1 y«
i
r m
Ž . The corresponding conditional
joint density of « ,...,«
is f
« s P
e exp
1 N
« ¨ G ¨ i
i
i i
j
Ž
y 1 y«
i
r m
. yP
e rm
. This is the formula for a univariate EV random variable with location parameter
i y
1
Ž .
m log P
and scale parameter m; see, for example, Ben-Akiva and Lerman 1985 , p. 104.
i
K. Vaage r Energy Economics 22 2000 649]666 656
is sometimes suggested, will not solve this problem. On the contrary; if there are components that influence both the energy demand and the appliance choice, the
appliance dummies would become correlated with the error term in the demand equation, resulting in parameter estimates that would be both biased and inconsis-
tent.
N
w rm
j
m is identified in the estimation of the discrete choice. So is
e , which is
Ý
js 1
the denominator in the multinomial logit expression. The variable in the curved bracket can, therefore, be calculated.
r y 1 tells us how much the energy demand is influenced by the choice of
Ž .
heating alternatives. Hanemann 1984 notes that a necessary and sufficient condition for good x to be essential with respect to the utility function is that
i
r - 1, i.e. the estimate of r y 1 should be negative a priori.
The energy prices in this model affect the technology choice through the operating costs. In addition, the price variable is allowed to have a direct effect on
the conditional demand model. With the chosen parameterisation this direct price
12
Ž .
elasticity is assumed to be minus one, and Hanemann 1984 , therefore, suggests
Ž .
the log of the energy expenditure as dependent variable. We will, however, allow the price parameter to be estimated freely, and use the parameter’s closeness to
minus one as a measure of the validity of the specification. Income and other individual characteristics affect the energy demand through
the selection term, but analogous to the price variable we want a specification where their potentially direct effect on the demand can be measured. This is
achieved by making u a function of these variables. Let b,d ,....,d be the impact
1 M
on log x from y,s ,...,s , respectively. For convenience we choose logu s a q
i 1
M
Ž .
b log y q d s q ... qd s
and the parameters b,d ,...,d interpreted as direct
1 1 M M
1 M
conditional demand elasticities. Accordingly, the estimating conditional demand equation becomes:
M
log x s a y a log p q blog y q d
log s
Ý
i 1
i m
m
ms 1
N
w r m
j
Ž .
Ž .
q r y 1 m log
e q
0.5772 q h
23
Ý
ž
is 1
h is a conventional error term where zero mean and constant variance are
Ž .
assumed, so Eq. 23 may be estimated by OLS. Note that the parameters are equal for all energy alternatives. The N demand equations are, therefore, pooled and
estimated simultaneously.
12
Our attempts to loosen this restriction led to highly non-linear demand functions, where some of the terms could not be identified.
K. Vaage r Energy Economics 22 2000 649]666 657
3. The data
