Bousquet, Alain

Ladoux, Norbert

English

Research Papers in Economics

Fuel-Switching Capability
Alain Bousquet￿and Norbert Ladouxy
University of Toulouse, IDEI and CEA
November 30, 2004
Abstract
Taking into account the link between energy demand and equip-
ment choice, leads to a new model of energy demand where the key
variables are energy prices, price volatility and the costs of equipment.
In particular we show that adjustment to changes in relative energy
prices are generally asymmetric, as the adjustment involves adaptation
of equipment costs. As a consequence, the optimal equipment choice
in the long run could imply short-term allocative ine¢ ciency in en-
ergy use. This model can be seen as a mix or a generalization of the
Putty-Putty versus Putty-Clay models of energy use. Depending on
the technology and in particular costs of energy equipment, the model
gives drastically di⁄erent predictions about the e⁄ect of energy prices
and uncertainty on energy demand.
Key Words : Energy Use, Equipment choice, Investment
JEL Classi￿cation : 022, 640, 723
We are grateful to the Institut Fran￿ais de l￿ Energie for its ￿nancial
research support.
￿University of Toulouse, CEA and LEERNA, 21 allØe de Brienne, 31000 Toulouse
CØdex, tel: (33) 5 61 12 85 91, fax: (33) 5 61 12 86 37, Email : bousquet@cict.fr.
yUniversity of Toulouse, IDEI and LEERNA, 21 allØe de Brienne, 31000 Toulouse
CØdex, tel: (33) 5 61 12 85 90, fax: (33) 5 61 12 86 37, Email : ladoux@cict.fr.
11 Introduction
This paper provides a micro-economic foundation of the concept of fuel-
switching capability. Fuel switching is the extend to witch a producer can
reduce the use of one type of energy and uptake of another source of energy in
its place. We develop a micro-economic model where energy consumption
and energy equipment capacity are both endogenous. In many situations
the amount of input that will be used in the future is necessarily less than
an upper limit determined by the installed equipment capacity. Energy
demand and electricity generation are good examples of such a situation.
Once the ￿rm has chosen an equipment, there is a maximal amount of
energy use, determined by the power and energy e¢ ciency characteristics of
the equipment. As a direct consequence the amount of one kind of energy
that could be substituted by another kind of energy is necessarily limited.
This paper is a mix of previous model of ￿rm behavior under input
rationing and uncertainty. The main feature of our analysis is that a ￿rm
has to decide on its machine characteristics determining a maximum level
of use for some input before the demand conditions are completely known.
In short, the producer trade-o⁄ is to choose ex ante the optimal level of
￿ exibility of the technology through di⁄erent input capacities allowing to
reach ex post e¢ ciency given market conditions. In our paper we extend
the analysis of the ￿rm behavior under price uncertainty when input use is
subject to endogenous rationing.
Uncertainty and risk play a huge role in energy supply as well as in
energy demand, especially on equipment choice. Understanding the e⁄ects
of price uncertainty on ￿rm￿ s production and input choice decisions has long
been an important issue in economics (see for example Sandmo (1971), Abel
(1983) or Dixit and Pindyck (1994)). Real options approach to investment
identi￿es three characteristics of most investment decisions, ￿rst uncertainty
over future pro￿t streams, second irreversibility and ￿nally the choice of
timing.
Most of existing models consider and study a situation where capital
is ￿xed or quasi-￿xed factor and other inputs are variable factors. Plant
capacity is then de￿ned by the maximum amount that can be produced per
unit of time with existing plant and equipment, i.e. given the level of capital.
In the energy sector, as long as energy could not be used or generated
without some equipment, we are in a situation where variable factors are
linked to ￿xed factors. This is precisely what we want to develop in this
paper. We consider that input use for variable factors could be constrained
by ￿xed factors, according to the size and more generally the characteristics
of the equipment.
22 The model
This section provides a simple model of a competitive ￿rm with input price
uncertainty. The time horizon is two periods. In our model the production
decision is made in two stages : an ex ante plan and an ex post plan. Ex ante,
the price of input is unknown. The ￿rm makes an investment and chooses an
equipment size which introduces an upper limit in the total amount of input
that could be used in second period. Ex post, the equipment and therefore
the maximal amount of input that could be used is given and the price of
input is known. The ￿rm cannot adjust the size of the equipment but can
choose a low rate of use of its equipment if the realized price of the input is
too high. The two-stage production decision problem is solved backwards.
2.1 Unrestricted cost function
We solve our model in the particular case of a CES production function.
Part of the following results could be derived from a more general production
function. Nevertheless, as we will focus mainly on the role of substitution
possibility between inputs, and show its great importance in the choice of
equipment capacity, the entire model is solved for this particular functional
form, which is
y = ￿ [￿x
￿
1 + (1 ￿ ￿)x
￿
2]
1=￿ ;
where y represents output, and xi for i = 1;2 represents input use.
The CES production function is de￿ned for ￿ 2]￿1;1], and 0 ￿ ￿ ￿ 1.
Moreover we know that the CES production function leads to the
Leontief production function, as ￿ ! ￿1, the Cobb-Douglas production
function as ￿ = 0, and the linear production function, as ￿ = 1.
We denote by ￿ = 1
1￿￿ the substitution elasticity between the two inputs.
We start from the usual cost minimization program of the ￿rm and we
will introduce later on exogenous and then endogenous rationing on input
use.
The optimal input demand are the solution of the following uncon-
strained cost minimization program,
Min
x1;x2
p1x1 + p2x2;
Subject to y = ￿ [￿x
￿
1 + (1 ￿ ￿)x
￿
2]
1=￿
;
3and could be expressed as
x1 =
y
￿
"
￿ + (1 ￿ ￿)
￿
(1 ￿ ￿)p1
￿p2
￿ ￿
1￿￿
# ￿1
￿
;
x2 =
y
￿
"
(1 ￿ ￿) + ￿
￿
(1 ￿ ￿)p1
￿p2
￿ ￿￿
1￿￿
# ￿1
￿
:
2.2 Restricted cost function
Consider now the ￿rm behavior under exogenous rationing, as in Lee and
Pitt (1987) and Squires (1994). We limit ourself to the case where only
input 1 is subject to a quantity constraint, but the model can be generalized
when input 2 is also subject to some quantity constraint. We consider and
denote the constraint on input 1 as, follows:
x1 ￿ x1:
The cost minimization program is
Min
x1;x2
p1x1 + p2x2;
Subject to y = ￿ [￿x
￿
1 + (1 ￿ ￿)x
￿
2]
1=￿ , and x1 ￿ x1:
Optimal input demand are
8
> > <
> > :
x1
x2
if p1 ￿ ￿1;
x￿
1(p1;p2;y)
x￿
2(p1;p2;y)
if p1 > ￿1:
Where ￿1 is the virtual price of input 1 (see Heckman (1974) and Neary
and Roberts (1980) for a complete treatment of this concept) at which the
unconstrained demand for input 1 is exactly equal to x1, and
￿1 = p2
￿￿
y
￿x1
￿￿
￿ ￿
￿1￿￿
￿
(1 ￿ ￿)
￿1
￿ ￿: (1)
Moreover when the constraint on x1 is binding we have
x2 =
2
4
￿
y
￿
￿￿
￿ ￿x1
￿
1 ￿ ￿
3
5
1
￿
:
42.3 Optimal input capacity
Suppose now that the threshold x1 is endogenously determined. Assume
￿rst that some equipment is necessary to use input 1. Second, assume that
the characteristics of the equipment, in particular the size of this equipment,
induce a constraint on the maximal amount of input such that x1 ￿ x1. The
cost of the equipment is a function of x1, and consider the case where the
cost of the equipment could be written as c1x1. Here, c1 is the constant
marginal cost of the equipment capacity. Finally assume that the ￿rm faces
an uncertain price for input 1. So p1 is assumed to be a random variable
with density function ￿(p1), cumulative density function ￿(p1) and with
p1 2 [0;+1[.
We assume that the ￿rm has to decide about the level of the capacity x1
prior to the knowledge of the input price. For each possible value of x1, we
know that there exist a price threshold, denoted by ￿1 such that if p1 ￿ ￿1
the equipment capacity constraint will be binding while for p1 > ￿1 input
use is such that x￿
1 < x1.
The ex ante problem corresponding to the choice of the input capacity
constraint x1, for a risk neutral ￿rm is,
Min
x1
Z ￿1
0
(p1x1 + p2x2)￿(p1)dp1 +
Z +1
￿1
(p1x￿
1 + p2x￿
2)￿(p1)dp1 + c1x1:
The ￿rst order condition is
Z ￿1
0
(p1 ￿ ￿1)￿(p1)dp1 + c1 = 0:
In general it is not possible to obtain an analytical solution for x1
￿, this
will depend on the shape of the statistical distribution ￿.
However, remark that it is possible to solve the last FOC with respect to
the price threshold and denote the solution by ￿￿
1, which is independent of
the technology characteristics. As a consequence, given ￿￿
1, the parameters
of the technology plays now a role in the optimal capacity level which is
determined, according equation (1), by
x1
￿ =
y
￿
"
￿ + (1 ￿ ￿)
￿
(1 ￿ ￿)￿￿
1
￿p2
￿ ￿
1￿￿
# ￿1
￿
.
The following ￿gure illustrates the solution. In the short term the max-
imum amount of input 1 the ￿rm can use is limited by x1 and the isoquant
is an arc with an extrema points
￿
x1;x2
￿
. For a given capacity level, the
ability to switch between the two inputs depends on the realized price and
is limited. We distinguish substitution possibilities and switching capacity
simply to keep in mind that the marginal rate of technical substitution is
5a local measure while the switching capacity is a global one and represents
the extent to which substitution possibilities may occur in the short run.
-
6
x2
x1
XXXXXX X
y = f(x1;x2)
x2
￿ x1
z
i s
0
￿￿1
p2
Figure 1 : Switching capability
3 Comparative statics
3.1 Technology and optimal capacity
From the theoretical model, it is possible to determine how the optimal
equipment capacity level, depends on the price distribution and production
function parameters. In this paper, we consider only the property of the
chosen capacity with respect to technology characteristics. It is possible to
explore with a CES production function how the elasticity of substitution
in￿ uences the optimal level of capacity x1.
First order condition shows that the optimal capacity coud be expressed
as the demand fo input 1 evaluated at a particular relative price
￿￿
1
p2 which
do not depend on the elasticity of substitution. As a consequence, com-
paring optimal capacities levels which follows from di⁄erent CES functions
distinguished by the values of the elasticity of substitution ￿ only, is rather
simple. Optimal capacity satisfy the standard equality between the MRTS
and the relative price
￿￿
1
p2,
MRTS =
dx2(x1)
dx1
= ￿
￿￿
1
p2
:
6Basic algebra for the CES production function with 2 factors, gives
x1
x2
=
￿
1 ￿ ￿
￿
￿￿
1
p2
￿￿￿
The sign of the derivative of the input capacity ratio with respect to the
substitution elasticity, could be stated simply.
@ x1
x2
@￿
= ￿ln
￿
(1 ￿ ￿)￿￿
1
￿p2
￿￿
(1 ￿ ￿)￿￿
1
￿p2
￿￿￿
It follows that the input ratio x1
x2 is increasing (decreasing) with respect
to ￿ when
￿￿
1
p2 is larger than (respectively lower) than ￿
1￿￿.
The relationship between the optimal level of the energy equipment ca-
pacity for input 1 and the parameters of the technology, is not straightfor-
ward in this case.
Deriving x1
￿ with respect to the substitution elasticity ￿ leads to the
following non-linear expression,
@x1
￿
@￿
=
x1
￿
1 ￿ ￿
￿
1
￿
ln(
￿x1
￿
y
) + ￿ ln(
￿￿￿
1
(1 ￿ ￿)p2
)(1 ￿ ￿(
￿x1
￿
y
)
￿￿1
￿
￿
;
It is easy to show that if
￿￿￿
1
(1￿￿)p2 = 1, then x1
￿ =
y
￿ which is independent
of ￿.
3.2 Consequences for energy demand modeling
The standard way to estimate the parameters of a CES production function,
consists of testing the following simple a linear regression :
ln(
x1
x2
) = ￿ ￿ ￿ ln(
p1
p2
);
where the estimated slope gives the elasticity of substitution ￿ = 1
1￿￿, and
the value of ￿ could be derived form the constant through the relation ￿ =
￿ ln( ￿
1￿￿).
Demand elasticity for input i, i = 1;2, is ￿(1 ￿ wi)￿, where wi is the
share of input i in total cost.
The estimation of a CES in the presence of equipment choice must be
derived from the estimation of the following Tobit model (Cragg (1971),
Blundell and Meghir (1987)), associated to the short run demand :
8
<
:
ln(x1
x2) = ￿ ￿ ￿ ln(
p1
p2) p1 > ￿1
ln(x1
x2) = ￿ ￿ ￿ ln(
￿1
p2) p1 ￿ ￿1
7Note that it is easy to verify the value of the threshold in the previous
censored regression. Using the fact that :
x1
x2
=
x1
"￿
y
￿
￿￿
￿￿x1
￿
1￿￿
# 1
￿
;
x1 =
y
￿
[A]
￿1
￿ ;
with A = ￿ + (1 ￿ ￿)
￿
(1 ￿ ￿)￿1
￿p2
￿ ￿
1￿￿
For a given set of parameters, ￿ = 0:5, y = 1, ￿ = 1, ￿ = ￿0:5, p2 = 1,
c1 = 1, and considering a Log normal distribution for the price of input
1 with parameter ￿ = 3 and ￿2 = 1, we estimate the corresponding CES
production function when the existence of energy equipment is ignored.
In this example the estimated elasticity is b ￿ = 0:25, while the correct
value in this case is ￿ = 0:66. This very simple estimation show us that
the bias in the estimated elasticity could be very large when we do not
estimate the energy demand model derived from the CES, and according to
the demand threshold induced by the equipment capacity. This estimation
has been done only to illustrate the importance of a complete model of
energy demand including energy use and equipment choice. Unfortunately
we could not provide an empirical test of our model based on real data. At
least, we could use general results provided by the MECS (1991) and (1994)
to justify the importance of fuel switching capability in the short run and,
as a consequence, the relative importance of energy equipment and more
generally energy technology on the shape on energy demand.
4 Conclusion
Fuel costs are only one of several criteria that shape energy equipment de-
cisions. In this paper, we embed the micro-economic decisions associated
with investment under uncertainty, installed capacity, capacity utilization
and energy use. We show that the combination of input price uncertainty
and production technology, yields to a complex relationship between en-
ergy equipment purchasing behavior and energy demand. This model is
consistent with the empirical observation provided by the Manufacturing
Energy Consumption Survey (EIA 1994). In the electricity generation sec-
tor, inevitable trade-o⁄ between price level, price volatility and ￿xed costs
of power plants, leads to a mix of capacity over di⁄erent technologies and
short-run fuel ￿ exibility will be of a great importance in the future.
8From a theoretical point of view, our approach provides a very simple
and natural framework to understand asymmetric responses to energy price
changes and the existence of threshold e⁄ects of energy price changes. Con-
sidering the fact that investment in capital goods a⁄ects not only output,
but also input use for adjustable factors, our analysis contributes to enlarge
theoretical literature that identi￿es channels through which uncertainty may
in￿ uence investment.
95 Appendix
In the appendix, we show that the relationship between optimal equipment
capacity level and the substitution elasticity ￿, is highly non-linear. Figure 2
and 3, illustrates how the optimal input capacity level changes with respect
to ￿ and ￿. In this ￿gures, y and ￿ are set equal to 1 by convention. Here,
the ratio
￿￿
1
p2 is assumed to be equal to 1.
For ￿ = 0, the CES production function corresponds to the Leontief
production function with perfect complementaries between inputs. It is
easy to show that the optimal capacity is equal to
y
￿.
For large values of ￿, the CES production function corresponds to the
linear technology with perfect substitutability. It can be shown that when
￿ ! +1, the optimal capacity is equal to
y
￿￿ for ￿ > 1
2 and equal to 0 for
￿ < 1
2, as long as we assume in this simulation that,
￿￿
1
p2 = 1.
Figure 2:Technology characteristics and optimal capacity
The simulation shows clearly, that the optimal equipment capacity level
increase or decrease with respect to the substitution elasticity parameter
￿. Moreover the ￿gure illustrates that we could not expect a monotone
relationship in general between capacity and ￿, since for example when
￿ = 0:9 the capacity ￿rst increases with ￿ and then decreases. At least, this
illustrates the di¢ culty to determine the optimal equipment capacity in the
energy context where it is well known that the di⁄erent kinds of energy are
10more or less substitute. Moreover, since the model derived from the CES
production function leads to a complex relationship between capacity and
substitution elasticity, there is no chance to ￿nd simple results in the case
of a general production function.
Figure 3: Optimal capacity and substitution elasticity
6 References
Abel, A.B, 1983, ￿ Energy Price Uncertainty and Optimal Factor Intensity:
a mean-variance analysis￿ , Econometrica, 51, N￿6, 1839, 1845.
Blundell, R. W. and C. Meghir, 1987, ￿ Bivariate Alternatives to the
Tobit Model￿ , Journal of Econometrics, 34, 179-200
Cragg, J. G., 1971, ￿ Some Statistical Models for Limited Dependent
Variables with Application to the Demand for Durable Goods￿ , Economet-
rica, 39, 829-844.
Dixit, A., and R.S, Pindyck, 1994, Investment under Uncertainty, Prince-
ton University Press, Princeton, NJ.
Energy Information Administration, 1994, Manufacturing Energy Con-
sumption Survey of 1991, DOE/EIA-0512(91).
Heckman, J., 1974, ￿ Shadow Prices, Market Wages, and Labor Supply￿ ,
Econometrica, Vol 42, N￿4, 679-693.
11Lee, L.F. and M.M. Pitt, 1987, ￿ Microeconometric Models of Rationing,
Imperfect Markets, and Non-Negativity Constraints￿ , Journal of Economet-
rics, 36, 89-110.
Squires, D., 1994, ￿ Firm Behavior Under Input Rationing￿ , Journal of
Econometrics, 61, 235-257.
Neary, J. P. and K.W.S, Roberts, 1980, ￿ The theory of household be-
havior under rationing￿ , European Economic Review, 13, 25-42.
Sandmo, A., 1971, ￿ On the Theory of the Competitive Firm Under Price
Uncertainty￿ , American Economic Review, 78, N￿5, 65-73.
12

Fuel-Switching Capability

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