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An assessment of interfuel substitution by electric utilities
An assessment of interfuel substitutionby electric utilities Noel D. Uri* Federal Energy Administration, O&e qfCoal, Nuclear Washington D.C. 20461. USA (Received 8 October 1976; revised 15 February 1977)
and Electric
Power Analysis,
This paper applies a translog model to a pooled sample in order to measure the extent of regional interfuel substitution effects in the electric power industry. The results obtained indicate that relative changes in fuel prices both regionally and nationally have significant effects on fossil fuel consumption. This, in turn, has important implications for public policy. In particular, the market system appears better able to deal with exogenous shifts in energy supplies than has frequently been assumed in the formulation of public policies toward the energy crisis. Further, compared to aggregate United States time series estimates, a more elastic fuel price response is found thus questioning whether full long-run adjustment is being measured in the pure time series estimates. Given the importance of the latter in energy tax policy analysis for example, the question is indeed more than academic.
The extensive use of energy is a fundamental characteristic of the United States economy. The satisfaction of much of the energy requirements results primarily from the utilization of fossil fuels--coal, oil, and natural gas. These fuels, in addition to being consumed directly, are used in the generation of electrical energy. While the fossil fuels may not be substitutable for each other in all uses, they can, with existing technology, be converted into electrical energy thus, in an indirect sense, making all of them substitutes. Simply because it is technologically feasible to convert primary fuels to electrical energy does not mean, however, that one can conclude that utilities look upon these fuels as competitive, in an economic sense, with one another. This study is concerned with estimating the extent to which this competition exists by measuring interfuel substitution in the generation of electrical energy. Further, because of the wide regional disparity in the availability of fossil fuels, the analysis will be conducted using regionally disaggregated data.
*The author is Administration: not necessarily Administration staff members.
an economist with the Federal Energy The views expressed are those of the author and do represent the policies of the Federal Energy or the views of other Federal Energy Administration
The present study estimates a normalized restricted profit function (capital and labour are restricted to be fixed inputs ex post) using regional data to estimate the substitution relationships that exist. A translog restricted profit function is estimated which is a ‘flexible’ functional form in that it does not place a priori restrictions on the extent of interfuel substitution, scale bias, or the degree of returns to scale. Additionally, the translog restricted profit function provides a second-order local approximation to an arbitrary twice-differentiable restricted profit function. Finally, even though labour and capital are restricted ex post, interfuel substitution is allowed; no restrictive assumptions such as putty-clay technology are maintained.
Model and restrictions The quantity of electrical energy generated in a region is assumed to be a function of variable and fixed inputs: Q = F(X, Z) = F(Xr ,..., X,;
Z1 . . . . . Z,)
(1)
where Q is generation and X and Z are vectors of variable and fixed inputs, respectively. Rather than estimate the production function directly. the existence
Appl.
Math. Modelling,
1977, Vol 1, June
253
Assessment of interfuel substitution: N. D. Uri of interfuel substitution is ascertained by the estimation of a normalized restricted profit function l I. The properties of the underlying production structure (e.g., substitutability of inputs) can then be derived from the normalized, restricted profit function. Following Atkinson and Halvorsen’ normalized restricted profit is defined as: F(X, Z) -
5 PiXi
(2)
i=l
where Pi is the price of input output. Let: 7r = F(x*,z)
-
i
i divided
by the price of
pix;
where the identification conditions ?i’ij= g,, and +jj = 4ji are maintained. This usage differs from that given in reference 5 where such conditions are termed symmetry restrictions. The explanation for the distinction is found below. By Hotelling’s lemma; reference 5 p 42-55:
where XT is the profit maximizing amount of input i. Using this lemma, logarithmic differentiation of the restricted profit function yields demand functions for each variable input:
(3)
i=l
-am71 ~~ = _!Y.pi zln Pi c?Pi n
where the asterisk indicates that the quantities of X are chosen to maximize normalized restricted profit. Hence:
of the restricted
profit function
71= n(p~,P,,P,;Z,,Z,,Z,),
is
(5)
where PI = price of coal; P2 = price of oil; P, = price of gas; Zi = the quantity of labour; Z2 = quantity of capital, and Z, = vintage of capital and all prices are normalized by dividing by the price of output. The basic advantage of estimating a restricted profit function rather than a production function or cost function is that the arguments of a restricted profit function are exogenous so that ordinary least squares produces consistent estimates. Fuel prices are determined in relatively competitive markets. and the price of output is determined by regulatory agencies. This advantage is not shared by production functions whose arguments are quantities of endogenous inputs, or cost functions whose arguments are prices of inputs and output, which may be considered endogenous. In addition, the estimation of a restricted profit function rather than a production function avoids the difficult task of modelling the profit-maximizing choice of capital stock. In particular, capital decisions are affected by regulatory constraints and by peak-load demand. The use of a restricted profit function avoids modelling these difficult decisions by permitting analysis of profit-maximizing decisions on ftiel choice. with fixed capital and labour. Finally, estimation of an unrestricted profit or cost function requires the burdensome task of calculating the user cost of capital, since the prices of all variable inputs are arguments of these functions. A number of flexible functional forms of the normalized restricted profit function could be estimated. The one used here is the translog2.5.s.” given as: ln7r = Mg + t 3
+ C i=l
254
Appl.
Math.
jil ;jij In e In Pi
Zi In fi + k .i
i=l
I
1
3
1
(6)
6,,-ln~lnZj
j=l
Modelling,
1977,
Vol 1, June
Ei+
(4) i =
The specification chosen to be:
n
Mi
3 =-
1,
yij1n4+
C j=
71= rc(P,Z) = 7c(Pi ,. ., P,; z, ,. . ,Z,)
cX*
1
t j=
(8)
cYijlnZj 1
2, 3,
where Mi is the ratio of expenditures on input i to restricted profits. The own and cross-piece elasticities of demand for fuels are readily derived from the demand functions. The own-price elasticity is defined as: Vii
pi = ax. ap, XT
By Hotelling’s
lemma; ^
x:=-g
(10)
’
I
and
axj
P271
i7Pi
?P’
Therefore; q . i%/(?Pf Vii = -~ Similarly,
h/!?P
the cross-price
(11)
-M;
- Mi - 7ii
Mi elasticity equals;
(13
c?x* ‘1 ij =
Pj 4 * 2,/Z&?~ ,:pj ‘XT ii-n/i~. -
- MiMj - yij
(13)
Mi
Estimation procedure It is feasible to estimate the parameters of the normalized restricted profit function using ordinary least squares. This technique is certainly attractive from the point of view of simplicity. However, it neglects the additional information contained in the demand functions’, which are also easily estimable. Furthermore, even for a modest number of factors of production the normalized restricted profit function has a large number of regressors which do not vary greatly across regions. Hence multicollinearity may be a problem, resulting in imprecise parameter estimates. An alternative estimation procedure, and the approach used here, is to jointly estimate the normalized restricted profit function and the demand
Assessment
function as a multivariate regression system. Including the demand functions for each fuel in the estimation procedure adds many additional degrees of freedom without adding any unrestricted regression coefficients. This will result in more efficient parameter estimates than would be obtained by the application of ordinary least squares to the normalized restricted profit function alone. Additive disturbances are specified for each of the demand functions and for the normalized restricted profit function. Since the fuel demand functions are derived by differentiation, they do not contain the disturbance term from the normalized restricted profit function. It is assumed that the disturbances have a joint normal distribution. The parameter estimates are then obtained by the method of full-information maximum likelihood. The technique is described in reference 2.
Data The normalized restricted profit function with the fuel input demand functions are estimated with pooled annual data compiled by Census’ region* for the period 1960-74. The specific data sources are discussed in the appendix. A dummy variable was appended to each demand function to reflect the implementation of sulphur emission controls starting in 1969. The dummy variable is defined as 1 for 1969, 2 for 1970, etc. The model parameter estimates were robust with regard to the specification of this variable. Regional coefficients were not included in the estimating equations because the estimation procedure captures interregional variation. through the variance
Empirical results The condition that the restricted profit function be decreasing in the prices of variable inputs implies that the fitted expenditure ratios should be positive. All of the fitted ratios are positive. A second condition for a well-behaved restricted profit function is that the partial derivatives of restricted profit with respect to fixed inputs be positive. The calculated values of the derivatives with respect to capital and with respect to labour are all positive. Convexity of the restricted profit function is checked by determining if the values of the principal minors of the estimated Hessian are positive. This was found to be the case for each observation. * The regional classification is as follows: (1) New England, (Maine, New Hampshire, Vermont, Massachusetts. Rhode Island. Connecticut); (2) Middle Atlantic, (New York, New Jersey. Pennsylvania); (3) East North Central. (Ohio. Indiana, Illinois, Michigan, Wisconsin); (4) West North Central (Minnesota, Iowa, Missouri, North Dakota, South Dakota, Nebraska, Kansas); (5) South Atlantic. (Delaware, Maryland and D.C.. Virginia, West Virginia, North Carolina, Georgia. Florida); (6) East South Central, (Arkansas, Louisiana, Oklahoma. Texas); (8) Mountain, (Montana. Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah, Nevada); and (9) Pacific, (Washington, Oregon, California),
Table 1
Demand
of interfuel
equation
parameter
substitution: estimates Equation
-0.1977 (0.0975)
Price of coal
0.0889
for
Oil
Coal
Variable
IV. D. Uri
0.0889 (0.0407)
Natural
gas
0.1088 (0.0593)
(0.0407)
-0.2419 (0.0798)
0.1530 (0.0687)
Price of natural gas
0.1088 (0.0593)
0.1530 (0.0687)
-0.2618 (0.6946)
Quantity
of labour
1.2483 (0.5496)
1.5371 (0.7666)
1.3021 (0.7411)
Quantity
of capital
0.1688 (0.0497)
0.0992 (0.0512)
0.1000 (0.0499)
0.0001 (0.0876)
- 0.0376 (0.1747)
0.1329 (0.1975)
- 0.0899 (0.0429)
0.0517 (0.0368)
- 0.0292 (0.0339)
0.5559 (0.0313)
0.2205 (0.0341)
0.2236 (0.0313)
Price of oil
Vintage
of capital
Sulphur emissions control dummy Intercept
* Standard
errors of estimates
in parentheses
Since only the parameter estimates on the demand equations are of concern, in the interest of brevity only these estimates are presented (see Table I). All of the estimates of the coefficients on the fuel price terms are significantly different from zero at the 95% level leading to the conclusion that price elasticities are not zero. The significance of the vintage of the capital is marginal. The coefficients on the dummy variable which were included to account for sulphur emissions controls indicate a distinct, non-price induced shift away from the use of coal but not necessarily toward natural gas and oil. In fact, the deficit was being made up primarily by nuclear and internal combustion generation. Estimates of regional price elasticities of demand based on the ratios of fuel expenditures to restricted profits for 1974 are presented in Table 2. Own-price elasticities should be negative and cross-price elasticities should be positive. This is exactly what is observed. The computed elasticities are representative of the past few years given the relative stability of the fuel expenditure/restricted profit ratio. A wide variation in regional price elasticities of demand is observed in Table 2. The differences are attributable to the fuel share composition within each region. For example, the New England region with the highest fuel expenditures to normalized profit ratio for oil has the most inelastic demand for oil while the East South Central region with the lowest ratio has the largest price elasticity. These results follow from the properties of equation (12) and the negative estimated coefficients for yo, yoo, and yan.* Coal, oil, and natural gas are all in competition with each other to varying degrees, as a fuel for the generation of electrical energy and every indication from the results is that utilities do respond to relative price changes when making their fuel input selection.
*To avoid confusion at this point, the subscripts c, o, and g replace the subscripts I. 2, and 3 used earlier in the paper.
APP~.
Math.
Modelling,
1977,
Vol 1,
June
255
Assessment Table 2
of interfuel
Regional
substitution:
price elasticities
N. D. Uri
of demand
Region
for 1974 1,<
rl.0
1,0
vu
vog
%
* + +
* * *
*
1.04
”
0.64 0.22
0.45 * *
*
*
*
* 0.50 * 1.32
* 0.54 * 0.41
* 0.73 * 1.41
1.51 0.80 0.37 0.51
New England Middle Atlantic East North Central
-4.01 -1.22 -0.38
~ 0.34 -0.75 - 3.01
x * *
4 5 6
West North Central South Atlantic East South Central
-0.64 -0.89 -0.38
* - 1.07 -3.04
-1.64
*
* *
7 8 9
West South Central Mountain Pacific Total for US
x - 1.04
-1.95 PI.54 -0 51 -1.15
-0.55 -1.63 -2.24 -2.95
f
2.30 0.89 0.22
* Estimates of the elasticities 1960-74. It is a maintained
are omitted since the particular fuel represents hypothesis that the estimates are questionable
Coal plants ticity.
were
in New
England
rapidly
phased
out in the late 1960s
The estimates in Table 2 provide a basis for computing average elasticities for all regions. Long-run own-price elasticities for the aggregate United States are -0.98, - 1.15, and -2.95 for coal, oil, and gas respectively. They indicate a somewhat larger elasticity than found in other studies. If the results here are true, the policy implications are significant. Time series analyses in particular have tended to find oil demand to be very inelastic. Hudson and Jorgenson’, for example, when applying a translog approach to the energy subsector of the electric utility industry estimated price elasticities for 1969 of -0.45, - 0.67, and - 0.59 for coal, oil, and gas. The comparison of the results here and those of Hudson and Jorgenson is interesting since it tends to confirm the suspicion that the earlier pure time series approach captured only a portion of the long-run price response in industries which have long periods for capital turnovers as in electric utilities. It is instructive to consider the magnitude of the cross price elasticities in order to determine the main channels of interfuel substitution. Table 2 reports their elasticities. The effects of higher oil prices, say, will create an approximately equal stimulus to both coal and gas consumption. In view of the minor share of gas in most regions together with the much larger share of coal, the primary alternative to oil will be coal. There is considerable regional variation in this observation but in the aggregate it is expected to hold. The study by Hudson and Jorgenson for the electric utility sector provides a basis for comparison of crossprice elasticities as well. Based on 1969 estimates. they find the elasticity of the demand foregone with respect to the price of oil to be 0.20 while the elasticity of demand for coal with respect to the price of oil to be 0.43. The elasticity of the demand for coal with respect to the price of gas is -0.20. indicating complementarity. While the standard errors of their estimates are not given, it seems unrealistic to assume the latter result is statistically significant. In all three cases, the results obtained here indicate comparatively similar interfuel substitution effects.
Conclusion The results obtained in-the foregoing analysis indicate that relative changes in fuel prices both regionally and nationally have significant effects on fossil fuel consumption. This, in turn, has important implications for public policy. In particular, the market system
256
Appl.
Math.
Modelling,
1977,
Vol 1, June
rl.c 0.16 0.49 1.63 * 0.69 1.64
1 2 3
-0.98
‘?CP * * *
less than 5% of total in such situations.
due to non-economic
fuel
reasons
expenditures
resulting
*
*
* 0.81 * 1.47
v9. + l
*
*
0.40 0.83 1.63 1.71
over the period
in a large own-price
elas-
appears better able to deal with exogenous shifts in energy supplies than has frequently been assumed in the formulation of public policies toward the energy crisis. Further, compared to aggregate United States time series estimates, a more elastic fuel price response is found thus questioning whether full long-run adjustment is being measured in the pure time series estimates. Given the importance of the latter, in energy tax policy analysis for example’, the question is indeed more than academic.
References 1 2 3
Atkinson, S. and Halvorsen. R. J. P&t. &on.. 1976. 84. 959 Berdnt, E. K. et al. Am. Econ. SIX. Measurement. 1974, 3. 653 Bureau of Labor Statistics, Employment and Earnings, States and Areas. US. Government Printing Office. Washington D.C.: 1975 Christensen, L. R.. et a/. Econometrica, 1971. 39, 255 Idem. ‘Transcendental Logarithmic Production Frontiers.‘ Rev. Econ. Stat., 1973. 55, 28 Diewert. W. E.. ‘Applications of duality theory,’ Paper No. 16. Strategic Planning and Research Division, Department of Manpower and Immigration. Canada, 1973 Edison Electric Institute. ‘Statistical Year Book of the Electric Utility Industry’ Edison Electric Institute, New York: annual Hudson. E. H. and Jorgenson, D. W. Bell J. Econ. Manage. Pi.. 1974, 5, 461 Idem, Discussion Paper 395. Harvard Institute of Economic Research, January 1975 Lau, L. J. and Yotopoulos, P. A. Am. Econ. Rec. 1971. 61, 94 National Coal Association. ‘Steam Electric Plant Factors’, National Coal Association. Washington: annual Samuelson. P. A. Rer. Econ. Stud. 1953. 21, I
Appendix Normalized fuel prices are calculated by dividing cost per million Btu by the price of output. Data on expenditure per million Btu for each type of fuel are from Lau and Yotopoulos”. Data on the price of output are from Diewert6 and reflect average prices computed by dividing revenue in each region by sales for each year. The average number of people employed by electric utilities for each region are from data given in reference 3. The quantity of capital is equal to installed generating capacity, which is measured by the manufacturer’s nameplate rating”. The vintage of the capital is measured by a time proxy (1960 = 1, 196 1 = 2. etc). Restricted profits are calculated as net generation, obtained from reference 10, minus normalized expenditures on fuels.
and Electric
Power Analysis,
This paper applies a translog model to a pooled sample in order to measure the extent of regional interfuel substitution effects in the electric power industry. The results obtained indicate that relative changes in fuel prices both regionally and nationally have significant effects on fossil fuel consumption. This, in turn, has important implications for public policy. In particular, the market system appears better able to deal with exogenous shifts in energy supplies than has frequently been assumed in the formulation of public policies toward the energy crisis. Further, compared to aggregate United States time series estimates, a more elastic fuel price response is found thus questioning whether full long-run adjustment is being measured in the pure time series estimates. Given the importance of the latter in energy tax policy analysis for example, the question is indeed more than academic.
The extensive use of energy is a fundamental characteristic of the United States economy. The satisfaction of much of the energy requirements results primarily from the utilization of fossil fuels--coal, oil, and natural gas. These fuels, in addition to being consumed directly, are used in the generation of electrical energy. While the fossil fuels may not be substitutable for each other in all uses, they can, with existing technology, be converted into electrical energy thus, in an indirect sense, making all of them substitutes. Simply because it is technologically feasible to convert primary fuels to electrical energy does not mean, however, that one can conclude that utilities look upon these fuels as competitive, in an economic sense, with one another. This study is concerned with estimating the extent to which this competition exists by measuring interfuel substitution in the generation of electrical energy. Further, because of the wide regional disparity in the availability of fossil fuels, the analysis will be conducted using regionally disaggregated data.
*The author is Administration: not necessarily Administration staff members.
an economist with the Federal Energy The views expressed are those of the author and do represent the policies of the Federal Energy or the views of other Federal Energy Administration
The present study estimates a normalized restricted profit function (capital and labour are restricted to be fixed inputs ex post) using regional data to estimate the substitution relationships that exist. A translog restricted profit function is estimated which is a ‘flexible’ functional form in that it does not place a priori restrictions on the extent of interfuel substitution, scale bias, or the degree of returns to scale. Additionally, the translog restricted profit function provides a second-order local approximation to an arbitrary twice-differentiable restricted profit function. Finally, even though labour and capital are restricted ex post, interfuel substitution is allowed; no restrictive assumptions such as putty-clay technology are maintained.
Model and restrictions The quantity of electrical energy generated in a region is assumed to be a function of variable and fixed inputs: Q = F(X, Z) = F(Xr ,..., X,;
Z1 . . . . . Z,)
(1)
where Q is generation and X and Z are vectors of variable and fixed inputs, respectively. Rather than estimate the production function directly. the existence
Appl.
Math. Modelling,
1977, Vol 1, June
253
Assessment of interfuel substitution: N. D. Uri of interfuel substitution is ascertained by the estimation of a normalized restricted profit function l I. The properties of the underlying production structure (e.g., substitutability of inputs) can then be derived from the normalized, restricted profit function. Following Atkinson and Halvorsen’ normalized restricted profit is defined as: F(X, Z) -
5 PiXi
(2)
i=l
where Pi is the price of input output. Let: 7r = F(x*,z)
-
i
i divided
by the price of
pix;
where the identification conditions ?i’ij= g,, and +jj = 4ji are maintained. This usage differs from that given in reference 5 where such conditions are termed symmetry restrictions. The explanation for the distinction is found below. By Hotelling’s lemma; reference 5 p 42-55:
where XT is the profit maximizing amount of input i. Using this lemma, logarithmic differentiation of the restricted profit function yields demand functions for each variable input:
(3)
i=l
-am71 ~~ = _!Y.pi zln Pi c?Pi n
where the asterisk indicates that the quantities of X are chosen to maximize normalized restricted profit. Hence:
of the restricted
profit function
71= n(p~,P,,P,;Z,,Z,,Z,),
is
(5)
where PI = price of coal; P2 = price of oil; P, = price of gas; Zi = the quantity of labour; Z2 = quantity of capital, and Z, = vintage of capital and all prices are normalized by dividing by the price of output. The basic advantage of estimating a restricted profit function rather than a production function or cost function is that the arguments of a restricted profit function are exogenous so that ordinary least squares produces consistent estimates. Fuel prices are determined in relatively competitive markets. and the price of output is determined by regulatory agencies. This advantage is not shared by production functions whose arguments are quantities of endogenous inputs, or cost functions whose arguments are prices of inputs and output, which may be considered endogenous. In addition, the estimation of a restricted profit function rather than a production function avoids the difficult task of modelling the profit-maximizing choice of capital stock. In particular, capital decisions are affected by regulatory constraints and by peak-load demand. The use of a restricted profit function avoids modelling these difficult decisions by permitting analysis of profit-maximizing decisions on ftiel choice. with fixed capital and labour. Finally, estimation of an unrestricted profit or cost function requires the burdensome task of calculating the user cost of capital, since the prices of all variable inputs are arguments of these functions. A number of flexible functional forms of the normalized restricted profit function could be estimated. The one used here is the translog2.5.s.” given as: ln7r = Mg + t 3
+ C i=l
254
Appl.
Math.
jil ;jij In e In Pi
Zi In fi + k .i
i=l
I
1
3
1
(6)
6,,-ln~lnZj
j=l
Modelling,
1977,
Vol 1, June
Ei+
(4) i =
The specification chosen to be:
n
Mi
3 =-
1,
yij1n4+
C j=
71= rc(P,Z) = 7c(Pi ,. ., P,; z, ,. . ,Z,)
cX*
1
t j=
(8)
cYijlnZj 1
2, 3,
where Mi is the ratio of expenditures on input i to restricted profits. The own and cross-piece elasticities of demand for fuels are readily derived from the demand functions. The own-price elasticity is defined as: Vii
pi = ax. ap, XT
By Hotelling’s
lemma; ^
x:=-g
(10)
’
I
and
axj
P271
i7Pi
?P’
Therefore; q . i%/(?Pf Vii = -~ Similarly,
h/!?P
the cross-price
(11)
-M;
- Mi - 7ii
Mi elasticity equals;
(13
c?x* ‘1 ij =
Pj 4 * 2,/Z&?~ ,:pj ‘XT ii-n/i~. -
- MiMj - yij
(13)
Mi
Estimation procedure It is feasible to estimate the parameters of the normalized restricted profit function using ordinary least squares. This technique is certainly attractive from the point of view of simplicity. However, it neglects the additional information contained in the demand functions’, which are also easily estimable. Furthermore, even for a modest number of factors of production the normalized restricted profit function has a large number of regressors which do not vary greatly across regions. Hence multicollinearity may be a problem, resulting in imprecise parameter estimates. An alternative estimation procedure, and the approach used here, is to jointly estimate the normalized restricted profit function and the demand
Assessment
function as a multivariate regression system. Including the demand functions for each fuel in the estimation procedure adds many additional degrees of freedom without adding any unrestricted regression coefficients. This will result in more efficient parameter estimates than would be obtained by the application of ordinary least squares to the normalized restricted profit function alone. Additive disturbances are specified for each of the demand functions and for the normalized restricted profit function. Since the fuel demand functions are derived by differentiation, they do not contain the disturbance term from the normalized restricted profit function. It is assumed that the disturbances have a joint normal distribution. The parameter estimates are then obtained by the method of full-information maximum likelihood. The technique is described in reference 2.
Data The normalized restricted profit function with the fuel input demand functions are estimated with pooled annual data compiled by Census’ region* for the period 1960-74. The specific data sources are discussed in the appendix. A dummy variable was appended to each demand function to reflect the implementation of sulphur emission controls starting in 1969. The dummy variable is defined as 1 for 1969, 2 for 1970, etc. The model parameter estimates were robust with regard to the specification of this variable. Regional coefficients were not included in the estimating equations because the estimation procedure captures interregional variation. through the variance
Empirical results The condition that the restricted profit function be decreasing in the prices of variable inputs implies that the fitted expenditure ratios should be positive. All of the fitted ratios are positive. A second condition for a well-behaved restricted profit function is that the partial derivatives of restricted profit with respect to fixed inputs be positive. The calculated values of the derivatives with respect to capital and with respect to labour are all positive. Convexity of the restricted profit function is checked by determining if the values of the principal minors of the estimated Hessian are positive. This was found to be the case for each observation. * The regional classification is as follows: (1) New England, (Maine, New Hampshire, Vermont, Massachusetts. Rhode Island. Connecticut); (2) Middle Atlantic, (New York, New Jersey. Pennsylvania); (3) East North Central. (Ohio. Indiana, Illinois, Michigan, Wisconsin); (4) West North Central (Minnesota, Iowa, Missouri, North Dakota, South Dakota, Nebraska, Kansas); (5) South Atlantic. (Delaware, Maryland and D.C.. Virginia, West Virginia, North Carolina, Georgia. Florida); (6) East South Central, (Arkansas, Louisiana, Oklahoma. Texas); (8) Mountain, (Montana. Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah, Nevada); and (9) Pacific, (Washington, Oregon, California),
Table 1
Demand
of interfuel
equation
parameter
substitution: estimates Equation
-0.1977 (0.0975)
Price of coal
0.0889
for
Oil
Coal
Variable
IV. D. Uri
0.0889 (0.0407)
Natural
gas
0.1088 (0.0593)
(0.0407)
-0.2419 (0.0798)
0.1530 (0.0687)
Price of natural gas
0.1088 (0.0593)
0.1530 (0.0687)
-0.2618 (0.6946)
Quantity
of labour
1.2483 (0.5496)
1.5371 (0.7666)
1.3021 (0.7411)
Quantity
of capital
0.1688 (0.0497)
0.0992 (0.0512)
0.1000 (0.0499)
0.0001 (0.0876)
- 0.0376 (0.1747)
0.1329 (0.1975)
- 0.0899 (0.0429)
0.0517 (0.0368)
- 0.0292 (0.0339)
0.5559 (0.0313)
0.2205 (0.0341)
0.2236 (0.0313)
Price of oil
Vintage
of capital
Sulphur emissions control dummy Intercept
* Standard
errors of estimates
in parentheses
Since only the parameter estimates on the demand equations are of concern, in the interest of brevity only these estimates are presented (see Table I). All of the estimates of the coefficients on the fuel price terms are significantly different from zero at the 95% level leading to the conclusion that price elasticities are not zero. The significance of the vintage of the capital is marginal. The coefficients on the dummy variable which were included to account for sulphur emissions controls indicate a distinct, non-price induced shift away from the use of coal but not necessarily toward natural gas and oil. In fact, the deficit was being made up primarily by nuclear and internal combustion generation. Estimates of regional price elasticities of demand based on the ratios of fuel expenditures to restricted profits for 1974 are presented in Table 2. Own-price elasticities should be negative and cross-price elasticities should be positive. This is exactly what is observed. The computed elasticities are representative of the past few years given the relative stability of the fuel expenditure/restricted profit ratio. A wide variation in regional price elasticities of demand is observed in Table 2. The differences are attributable to the fuel share composition within each region. For example, the New England region with the highest fuel expenditures to normalized profit ratio for oil has the most inelastic demand for oil while the East South Central region with the lowest ratio has the largest price elasticity. These results follow from the properties of equation (12) and the negative estimated coefficients for yo, yoo, and yan.* Coal, oil, and natural gas are all in competition with each other to varying degrees, as a fuel for the generation of electrical energy and every indication from the results is that utilities do respond to relative price changes when making their fuel input selection.
*To avoid confusion at this point, the subscripts c, o, and g replace the subscripts I. 2, and 3 used earlier in the paper.
APP~.
Math.
Modelling,
1977,
Vol 1,
June
255
Assessment Table 2
of interfuel
Regional
substitution:
price elasticities
N. D. Uri
of demand
Region
for 1974 1,<
rl.0
1,0
vu
vog
%
* + +
* * *
*
1.04
”
0.64 0.22
0.45 * *
*
*
*
* 0.50 * 1.32
* 0.54 * 0.41
* 0.73 * 1.41
1.51 0.80 0.37 0.51
New England Middle Atlantic East North Central
-4.01 -1.22 -0.38
~ 0.34 -0.75 - 3.01
x * *
4 5 6
West North Central South Atlantic East South Central
-0.64 -0.89 -0.38
* - 1.07 -3.04
-1.64
*
* *
7 8 9
West South Central Mountain Pacific Total for US
x - 1.04
-1.95 PI.54 -0 51 -1.15
-0.55 -1.63 -2.24 -2.95
f
2.30 0.89 0.22
* Estimates of the elasticities 1960-74. It is a maintained
are omitted since the particular fuel represents hypothesis that the estimates are questionable
Coal plants ticity.
were
in New
England
rapidly
phased
out in the late 1960s
The estimates in Table 2 provide a basis for computing average elasticities for all regions. Long-run own-price elasticities for the aggregate United States are -0.98, - 1.15, and -2.95 for coal, oil, and gas respectively. They indicate a somewhat larger elasticity than found in other studies. If the results here are true, the policy implications are significant. Time series analyses in particular have tended to find oil demand to be very inelastic. Hudson and Jorgenson’, for example, when applying a translog approach to the energy subsector of the electric utility industry estimated price elasticities for 1969 of -0.45, - 0.67, and - 0.59 for coal, oil, and gas. The comparison of the results here and those of Hudson and Jorgenson is interesting since it tends to confirm the suspicion that the earlier pure time series approach captured only a portion of the long-run price response in industries which have long periods for capital turnovers as in electric utilities. It is instructive to consider the magnitude of the cross price elasticities in order to determine the main channels of interfuel substitution. Table 2 reports their elasticities. The effects of higher oil prices, say, will create an approximately equal stimulus to both coal and gas consumption. In view of the minor share of gas in most regions together with the much larger share of coal, the primary alternative to oil will be coal. There is considerable regional variation in this observation but in the aggregate it is expected to hold. The study by Hudson and Jorgenson for the electric utility sector provides a basis for comparison of crossprice elasticities as well. Based on 1969 estimates. they find the elasticity of the demand foregone with respect to the price of oil to be 0.20 while the elasticity of demand for coal with respect to the price of oil to be 0.43. The elasticity of the demand for coal with respect to the price of gas is -0.20. indicating complementarity. While the standard errors of their estimates are not given, it seems unrealistic to assume the latter result is statistically significant. In all three cases, the results obtained here indicate comparatively similar interfuel substitution effects.
Conclusion The results obtained in-the foregoing analysis indicate that relative changes in fuel prices both regionally and nationally have significant effects on fossil fuel consumption. This, in turn, has important implications for public policy. In particular, the market system
256
Appl.
Math.
Modelling,
1977,
Vol 1, June
rl.c 0.16 0.49 1.63 * 0.69 1.64
1 2 3
-0.98
‘?CP * * *
less than 5% of total in such situations.
due to non-economic
fuel
reasons
expenditures
resulting
*
*
* 0.81 * 1.47
v9. + l
*
*
0.40 0.83 1.63 1.71
over the period
in a large own-price
elas-
appears better able to deal with exogenous shifts in energy supplies than has frequently been assumed in the formulation of public policies toward the energy crisis. Further, compared to aggregate United States time series estimates, a more elastic fuel price response is found thus questioning whether full long-run adjustment is being measured in the pure time series estimates. Given the importance of the latter, in energy tax policy analysis for example’, the question is indeed more than academic.
References 1 2 3
Atkinson, S. and Halvorsen. R. J. P&t. &on.. 1976. 84. 959 Berdnt, E. K. et al. Am. Econ. SIX. Measurement. 1974, 3. 653 Bureau of Labor Statistics, Employment and Earnings, States and Areas. US. Government Printing Office. Washington D.C.: 1975 Christensen, L. R.. et a/. Econometrica, 1971. 39, 255 Idem. ‘Transcendental Logarithmic Production Frontiers.‘ Rev. Econ. Stat., 1973. 55, 28 Diewert. W. E.. ‘Applications of duality theory,’ Paper No. 16. Strategic Planning and Research Division, Department of Manpower and Immigration. Canada, 1973 Edison Electric Institute. ‘Statistical Year Book of the Electric Utility Industry’ Edison Electric Institute, New York: annual Hudson. E. H. and Jorgenson, D. W. Bell J. Econ. Manage. Pi.. 1974, 5, 461 Idem, Discussion Paper 395. Harvard Institute of Economic Research, January 1975 Lau, L. J. and Yotopoulos, P. A. Am. Econ. Rec. 1971. 61, 94 National Coal Association. ‘Steam Electric Plant Factors’, National Coal Association. Washington: annual Samuelson. P. A. Rer. Econ. Stud. 1953. 21, I
Appendix Normalized fuel prices are calculated by dividing cost per million Btu by the price of output. Data on expenditure per million Btu for each type of fuel are from Lau and Yotopoulos”. Data on the price of output are from Diewert6 and reflect average prices computed by dividing revenue in each region by sales for each year. The average number of people employed by electric utilities for each region are from data given in reference 3. The quantity of capital is equal to installed generating capacity, which is measured by the manufacturer’s nameplate rating”. The vintage of the capital is measured by a time proxy (1960 = 1, 196 1 = 2. etc). Restricted profits are calculated as net generation, obtained from reference 10, minus normalized expenditures on fuels.