P. Soderholm r Energy Economics 23 2001 77]97
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83
the peak power.
11
It can thus be expressed as: Q
Ž . LF s
1 8760 PL
Ž .
where LF is the annual system load factor, Q is the total generation in GWh of Ž
. electric power during the year, and PL denotes annual peak load in GW . We
would expect that an increase in the system load factor from one year to another Ž
. should have a positive effect on the cost share for primarily base loaded fossil
fuels, i.e. preferably coal. That is, when the load factor increases it means there is less need for peaking power and consequently base loaded power would be more
fully utilized. Gas, on the other hand, and oil in particular are both heavily used for peak load and their cost shares ought to be negatively affected by an increase in
12
Ž .
the system load factor. Alternatively, an increase in peak load PL , i.e. a fall in
LF , causes cost share increases for primarily peak-loaded fuels. The empirical
assessment of these hypotheses follows in Section 5, but first Sections 3 and 4 outline the methodological approach of the paper.
3. The restricted translog model
In this paper we basically follow the Marshallian tradition and assume the Ž
. existence of a short-run restricted cost function, in which the capital input is fixed
at a level other than its full-equilibrium value. In other words, we presume that fossil fueled power production technology can be represented by the following
general variable cost function;
Ž Ž
. .
Ž . VC s VC P
P , P , P
, K , P , P ,Q, LF ,t 2
E C
O G
L M
where VC is the variable cost of producing fossil fueled electricity. P , P , P , P
C O
G L
and P are the input prices of the respective factor inputs; coal, oil, gas, labor and
M
material inputs. P is thus a function that aggregates the different fossil fuel
E
prices, that is an aggregate price index of energy inputs. The input prices are assumed to be exogenously determined, i.e. we assume competitive factor markets.
The effects of technical changes are denoted by a time trend, t, and LF represents the system load factor. Q and K denote fossil-fueled electricity output and power
generating capacities, respectively. Thus, in this short-run formulation the capital stock, K, is accounted for but changes in it are not explained.
11
The load factor should not be confused with the capacity factor, which relates the energy generated by a system over a period of time to the energy generated had the system operated continually at its
maximum capacity during the same period.
12
The recent increase in gas-fired CCGT has partly changed this picture, as these tend to be used for Ž
. base-load as well. Still, overall during the period under study 1984]1994 , gas has primarily been used
for peak load purposes.
P. Soderholm r Energy Economics 23 2001 77]97
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Due to lack of reliable labor and material price data it is assumed that energy and capital inputs are as a group weakly separable from labor and material inputs.
Essentially this means that, given fixed capital, the mix of fossil fuel inputs is Ž
. assumed to be independent of the non-energy inputs labor and material . The
weak separability assumption allows us to analyze a separate cost function for the energy and capital inputs alone. We assume that the conditions hold for the
Ž .
existence of an optimum cost function VC for fossil-fueled electricity production
E
of the following general form; Ž
. Ž .
VC s VC P
, P , P , K ,Q, LF ,t 3
E E
C O
G
For estimation purposes we are also interested in the cost-minimizing factor cost Ž .
Ž . shares S . These can be derived by differentiating Eq. 3 logarithmically with
i
respect to the fossil fuel prices, and by applying Shephard’s lemma. This yields the following general cost share equations;
lnVC
VC
P P E
E E
i i
i
Ž . S s
s s
for i s C,O,G 4
i
ln P
P
VC VC
i i
E E
where E , represents power generation use of the ith fuel. Before these cost share
i
equations can be estimated we have to specify the cost function. The translog cost Ž
. function was proposed by Christensen et al. 1971, 1973 and differs from earlier
representations of producer behavior in that it allows substitution between input factors to be unrestricted. It is obtained by a second-order Taylor expansion of the
logarithm of an arbitrary twice-differentiable cost function and can for our pur- poses be written as;
3 3
3
1 1
2
Ž .
lnVC s a q b lnQ q a
ln P q b
lnQ q
a ln P ln P
Ý Ý Ý
E Q
i i
QQ i j
i j
2 2
is 1
is 1 js1
3 3
1
2
q b
lnQln P q d t q d
t q d t lnQ q
d t
ln P
Ý Ý
Q i i
t t t
t Q t i
i
2
is 1
is 1
3
1
2
Ž .
q b
ln K q b
ln K q d lnQln K q
b ln K ln P q d t ln K
Ý
K K K
Q K K i
i t k
2
is 1
3
1 qb
ln LF q d
ln LF q d
ln LF ln P q d ln LF lnQ
Ý
L F L F L F
L F i i
L F Q
2
is 1
Ž . qd
t ln LF q d
ln LF ln K 5
t L F L F K
Ž . Differentiating Eq. 5 logarithmically and applying Shephard’s lemma yields
P. Soderholm r Energy Economics 23 2001 77]97
¨
85
three fuel cost share equations as linear functions of the logarithms of the fuel prices and the other independent variables so that;
3
S s a q b lnQ q b ln K q
a ln P q d
ln LF q d t
Ý
i i
Q i K i
i j j
L F i t i
js 1
Ž . for i , j s C,O,G
6 Ž .
The cost share equations in Eq. 6 form the basis of the empirical investigation, and they allow us not only to estimate the degree of price-induced fuel substitu-
tion, but also to test two interesting hypotheses with regard to the electricity production technology. First, we can test whether changes in the system load factor
Ž .
LF tend to have fuel using or fuel saving effects. Following the discussion in
Section 2, a priori we would expect that; Ž .
d d
- d
- 7
L F C L F O
L F G
We also estimate a constrained version of the translog fuel demand system, i.e. a model in which d
s 0. This permits us to perform a likelihood ratio test of the
L F i
null hypothesis that the system load factor has no significant effect on overall fossil fuel demand. Second, since the parameters. d , also appear in the cost share
t i
equations we are able to analyze in what way technical change affects fossil fuel Ž
. Ž
choice. A positive negative value of d , indicates that technical changes over
t i
Ž .
time have been ith fuel using saving . Traditionally economists have employed a simple time trend to represent technical developments over the sample period.
Although it is difficult to determine what it actually represents, the time trend tends to play an important role in ‘determining’ factor demand patterns in the
Ž .
Ž .
power sector e.g. Nelson, 1986 . This study follows Matsukawa et al. 1993 who assume that ‘the effects of technical changes on interfuel substitution may be
interpreted as the cumulative effect of fuel-switching promotional policies on fossil Ž
. fuels,’ p. 44 . These policies include, for example, tax deductions and favorable
loans for utilities converting old oil-fired plants to coal burning. Of course, other impacts are also caught by the time trend and ultimately it is difficult to resolve
exactly what the coefficients show. Still, it is reasonable to expect that the time trend in our case has been oil saving and coal using in response to the oil crises in
the 1970s. The time trend effect on gas use, however, is harder to project. In the mid-1980s gas was still seen as very scarce resource that should not be ‘wasted’ for
power generation purposes. However, in the late 1980s technical changes in the design and performance of the CCGT, and a more relaxed attitude towards
Ž depletion issues led to a substantial increase in gas use. This trend was at least in
. the UK reinforced by the decision to deregulate electricity markets. Thus, the
average effect of the time trend on gas use in power generation remains an empirical question, which is pursued in Section 5.
Several restrictions must be satisfied in order for the translog model to represent a well-behaved cost function. The following parameter restrictions are imposed
P. Soderholm r Energy Economics 23 2001 77]97
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86
since the fuel cost shares must add to one and the cost function must be homogenous of degree one
13
in prices;
3
a s 1
Ý
i
is 1
Ž . 8
3 3
3 3
3 3
a s a s
b s b
s d
s d s
Ý Ý
Ý Ý
Ý Ý
i j ji
Q
i
K i L F i
t i
is 1
js 1
is 1
is 1
is 1
is 1
Ž . In addition, partial differentiation of Eq. 5 with respect to prices reveals the
symmetry condition a s a . Finally, when using aggregate data it is common to
i j ji
Ž .
assume constant returns to scale CRS since it reduces the problem of multi- Ž
. collinearity. Brown and Christensen 1981 show that CRS requires that the
following restrictions on the share equations be imposed;
14
Ž . b
q b s
9
Q i K i
To obtain a measure of the degree of interfuel substitution we calculate the Ž
. cross-price elasticities of fuel demand. Following Uzawa 1962 , Berndt and Wood
Ž .
Ž 1975 show that, for the translog model, the cross- and own-price elasticities h
i j
. Ž .
and h are relatively simple to compute once the parameters in Eq. 6 have been
i i
estimated. We have; a q S S
a q S
2
y S
i j i
j i i
i i
Ž .
h s ,i j and h
for i , j s C,O,G 10
i j i i
S S
i i
These own- and cross-price elasticities will differ at every data point, and Ž
normally they are computed at the means of the data sample assuming that this .
value is on the regression line . It is important to note that these elasticities are only partial price elasticities, i.e. they account for the substitution between the
fossil fuels under the constraint that the aggregate quantity of fossil fuels con- sumed remains constant. Also, the short-run elasticities are valid only for the levels
of capital at which they are evaluated, and they do not provide any information about the substitution between capital equipment and the different fossil fuels.
Still, they allow us to estimate the degree of short-run interfuel substitution in fossil-fueled power plants in west Europe, and as was discussed above, within the
proposed cost share model we can also analyze the impact of system load factors on fossil fuel choice.
13
That is, variable cost must increase proportionally when all fuel prices increase proportionally, output held fixed. The concavity and monotonicity conditions of the cost model are not imposed, but are
instead checked by examining the estimated parameters of the model.
14
The additional conditions for CRS that have to be imposed on the translog cost function are Ž
. reviewed in Brown and Christensen 1981 . In contrast to nuclear power, a number of empirical studies
have confirmed that constant returns to scale is a reasonable assumption in the case of fossil-fueled Ž
. power generation e.g. Kamerschen and Thompson, 1993 .
P. Soderholm r Energy Economics 23 2001 77]97
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4. Model estimation and data issues
