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The energy crisis and its impact on the stability of the demand for factors of production
Applied Energy 13 (1983) 281-294
The Energy Crisis and Its Impact on the Stability of the Demand for Factors of Production N o e l D. Uri Department of Economics and Business, The Catholic University of America, Washington, DC 20064 (USA)
SUMMARY This paper addresses the question of the stability of the demand for the Jaetors ojproduction at the aggregate let*elJ'or United States manufacturing over the period 1947 to 1976. The results are conclusive. The demand Jot capital and labour has remained virtually constant over the period of investigation whilst the relatire importance of the Jactors affecting the demandjor energy has changed in a statisticalO' sign(tic'ant fashion. "
I N T R O D U C T I ON Existing models of the d e m a n d for factors of production can be viewed as either based on static optimisation with instantaneous adjustment or as myopic optimisation with constant marginal costs of adjustment. Such models inherently have several significant limitations. First, it is unlikely that adjustment of firms to the desired factor input levels is an instantaneous process or that marginal costs of adjustment are constant (i.e. that supply curves of the factors of production are perfectly inelastic). Given these considerations, recent years have seen a renewed interest in the study (both theoretical and empirical) of production functions. A large number of studies exist suggesting one or another production function that will mitigate the limitations alluded to above. (See, for example, Berndt and W o o d , 4'5 Berndt e t a l . , a Field and Grebenstein, ~2 Halvorsen, ~a Hudson and Jorgenson, 15 N o r m a n and Russell, 2° etc.) One 281
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issue, however, that has been ignored completely in these is whether the structures that are being posited and analysed are stable. (Stability is defined in the statistical sense of the estimated coefficients of the explanatory variables remaining constant over time.) In general, this is unacceptable. There is no reason to believe a p r i o r i that in the dynamic environment of, say, the United States economy, over the past three decades, the relative importance of the factors of production in the aggregate production function has remained unchanged. One reason for the absence of empirical investigations on this issue is the fact that the usual procedures for testing whether a regression relationship is constant over time in the statistical sense, namely the use of the F-test (Chow 7) or d u m m y variables (KmentalS), requires a p r i o r i knowledge of the point in time when the change in the estimated coefficients is suspected. Except for obvious dramatic changes such as war, this knowledge may not be readily available. The question of the stability of the functional relationship underlying the demand for factors of production is of critical importance. Policy inferences (e.g. relating to the impact of policy initiatives on demand for energy) are made on the basis of past behaviour. If this functional relationship has been subject to change, then, necessarily, the inferences will be, at least in part, unsatisfactory. One must be reasonably certain that the production function has not changed structurally. The purpose of this paper, then, is to examine the existence of a stable production function and attendant demand for the factors of production utilising a statistical test developed by Brown e t al. 6 that is designed to detect shifts in a regression relationship. This approach is adopted in deference to those mentioned previously because it does not require prior knowledge of the shifts but rather tests for the presence of such occurrences over the sample period. Before doing this, however, it is useful to review the theoretical rationale underlying the production process and provide a general discussion of the demand for factors of production.
THEORETICAL CONSIDERATIONS Consider the following example. Assume that a cost minimising competitive firm is initially in equilibrium with output at some equilibrium level and that the firm possesses a positive, twice differentiable strictly concave production function with four inputs (capital, K; labour, L;
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energy, E and materials, M). Next, assume that the price of one of the factor inputs changes relative to the others (e.g. the price of energy rises). The total effect of this on the d e m a n d for the other factors can be broken down into two parts. First, there will be the gross substitution effect resulting in a reduction in the use of the factor with the higher relative price and an increase in the use of the other factors (note that this will always be the case--less of the input whose price increased and more of the other inputs (Henderson and Quandt,14 p. 70)). Secondly, there will be an expansion effect which is just the difference between the total effect and the gross substitution effect where the total effect of such a price changes results when the output of the production function is held constant and all inputs adjust to their new cost-minimising levels. The magnitude of the expansion effect is dependent, of course, upon the marginal products of the factors of production (Johansen16). It is possible for the expansion effect to dominate the gross substitution effect with the result that one actually observes an increase in the use of the factor having the initial price increase. That is, it is possible that factors of production are not gross substitutes but net complements. Whether net substitutability or complementarity exists depends on whether the gross substitution effect or the expansion effect is dominant. This is an empirical issue. What is of relevance now is the extent to which one can sort out these effects. A myriad of techniques exists to allow for such a determination. In other words, a large number of specifications are available that permit the estimation of elasticities of substitution for the factors of production. These specifications necessarily presuppose that the assumptions concerning the nature of the production function are correct. There is no theurgical rationale for accepting one set of assumptions over another. It need not be the case, for example, that the factors of production are substitutable in the fashion assumed because the technology involved in the production process is prohibitive. The best that can be done is to accept Eisner's ~° admonition (although in a slightly different context) that estimation of production functions is a tricky and difficult business and the best posture to take is one of humility.
TEST O F STABILITY As previously noted, statistical tests made to ascertain whether a regression relationship over the sample horizon is stable have been limited
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to the use of d u m m y variables when a change in the relationship is suspected or dividing the same period at that point and performing a Chow test. Both these techniques require a p r i o r i knowledge of the point in time when the function shifts. This information might not be available. A method for determining whether a regression relationship is constant over a given period based on an examination of the residuals of the relationship has been developed by Brown et al. 6 This procedure will be used to test the stability of the demand for the factors of production (and implicitly, the production function) as previously discussed. To give an appreciation of the test, it is briefly discussed here. Essentially, a way of investigating the time-variation of a regression coefficient is to fit the regression on a short segment of n successive observations and to move this segment along the series. A significance test for constancy based on this approach is derived from the results of regression based on nonoverlapping time segments. The method is based on a test statistic which equals the difference between the sum of squared residuals of the entire sample less the cumulative sum of squared residuals of the nonoverlapping segments divided by the cumulative sum of squared residuals of the non-overlapping segments. The null hypothesis that the regression relationship is constant over time implies that the value of the test statistic is distributed as F. Consider the basic regression model: yt = x't]], + ~t
t=l
.... ,T
(1)
where Yt is a vector of observations on the dependent variable, x t is a column vector of observations on k regressors, ~t is a vector of regression coefficients (the subscript t implying that t h e / f s may not be constant over time) and G is a vector of normally and independently distributed variables with mean zero and variances 0.2, t = 1,2 . . . . . T. The first element in x t will be taken to equal unity for all values of t if the model contains a constant. The hypothesis of constancy over time, which is denoted by H 0, is 31 ~ f12 . . . . .
f i t = fi
0_2 ~ 0.2 . . . . .
0.2 ~- 0.2
The test is more concerned with detecting differences among the fl's than among the 0.'s.
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Energy and input ,[actor stability
Let b r be the regression estimate from the sample of the first r observations and let: y~-x'~b~ 1 Wr (2) N~ 1 -~ X t r t X ; _ l X r _ I ) - I X r wherer=k+l,k+2,...,TandX'r_l=[Xl,X2,...,Xr_l]. It is shown by Brown et al. 6 that these transformed residuals, Wk+ i, Wk+ 2. . . . , are normally distributed independent variables with zero means and finite variances. Equation (2) is just a generalisation of the Helmert orthogonal transformation. The variables I,V,can be obtained without repeated matrix inversion by the relationships: b, = br_ , + ( G X ~ ) - 'x',(y - x',b~_ l)
(3)
and: (X; ~ r ) '
1 -
~f~,
( X ; _ 1 ~ r - i ) - i XrXr(X; , , iXr - i) - 1
-i
=(,_,x,_,)
-
1+x;(X"
(4)
,x,_i)-'x,
Let Sr be the residual sum of squares after fitting the model to the first r observations assuming the null hypothesis is true. Thus: S~=S~
I+W 2
r=k+l
..... T
(5)
(These relationships are demonstrated to hold by Brown et al. 6) The quantities required for each new segment are computed by first adding a new observation to the segment just by using the relationships (3), (4) and (5) and then allowing for the effect of dropping an observation from the beginning by means of the following analogues of relationships (3), (4) and (5): bll = bn+ l - ( X n X n ) -
lXl(Yl -- x'bn+
(Xnfl('n) -1 = (Xn+ l X n + l ) -1 -q- ( Y n'+ i X n + l )
i)
(3(a))
-'
X1X1' ( x ~ + i X . + , ) - ' 1 - x'l(X'+ i X . + 1)- ix1
g . = s . + , - ( y , - x',~.)2/(1 + x , ( x ~ x . )
ix,)
(4(a))
(5(a))
where )?,, b. and ,~, are, respectively, the values of the regressor matrix, the coefficient vector and the residual sum of squares based on observations from t = 2 to n + 1. For ease of exposition, only the relationships for the first update are given here. Further relationships are, of course, analogous. (Proofs again are given by Brown et al. 6)
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The significance test for constancy, called the homogeneity test, is derived from the results of regressions based on non-overlapping time segments, using the analysis of variance. The time segments for a moving regression of length n, are {1,n}, { ( n + l ) , ( 2 n ) } . . . . , t ( p - 2 ) n + l , (p - 1)n}, {(p - 1)n + 1, TI, where p is the integral part of Tin and the variance ratio considered (i.e. the homogeneity statistic) is: ( T - kp) S(1, T ) - A -
(kp - k)
zX
(6)
where A = {S(1,n) + (S(n + 1),2n) + . - . + ( S ( p n - n + 1), T)} and where S(r, s) is the residual sum of squares from the regression calculated from observations from t = r to s inclusive. This is equivalent to the usual 'between groups over with groups' ratio of mean squares and, under Ho, is distributed as F(kp - k, T - kp). The next step is to estimate empirically the production function and examine its stability using the suggested test. Before this is accomplished, however, a discussion of the precise form of the production function and a delineation of the data are in order.
PRODUCTION FUNCTIONS AND THE DEMAND FOR FACTOR INPUTS There are many approaches to estimating the production function (see Johansen 16 for a summary of these). The one that has been capturing most attention recently is the transcendental logarithmic price possibility frontier or, more simply, the translog price possibility frontier. The price possibility frontier is a transcendental function of the logarithms of the prices of inputs. The translog price possibility frontier was introduced by Christensen et al.9 It distinguishes itself from earlier approaches in that it begins by positing, as an analogue to the production possibility frontier, a price possibility frontier of a general neoclassical specification. In conjunction with assumptions of perfect competition, demand for factor input equations are derived which must be estimated simultaneously to allow for the theoretically imposed restrictions on the parameters. The approach offers distinct advantages over other approaches to modelling the production function in that an explicit theoretical model serves as the basis for specification and reduces the problem of multicollinearity by decreasing the number of parameters to be estimated.
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Energy and input Jactor stability
Extensive use of this approach was made by Jorgenson and Berndt. 17 A model along the lines of that of Jorgenson and Berndt is developed and serves as the basis by which to assess the question of stability. The assumption, then, is that there exists a twice differentiable aggregate production function relating the flow of gross output Y to the services of four inputs: capital, XK; labour, XL; energy, X~ and all other intermediate materials, X~t. Further, the presumption is that production is characterised by constant returns to scale and that any technological change affecting K, L, E a n d M is Hicks-neutral. Corresponding to such a production function there exists a cost function reflecting the production technology. In general form, this cost function is written as:
P = E~(Y, PK, PL, PE, PM)
(7)
where P is the total cost and PK, PL, PE and PM are the input prices of K, L, E and M, respectively. As noted, for the purposes of estimation, the translog function is chosen as the specific functional form for P. It places no a priori restrictions on the Allen partial elasticities of substitution and can be interpreted as a second order approximation to an arbitrary twice differentiable cost function. This cost function with symmetry and constant returns to scale imposed is written log P = a 0 +
2
'22
7;; log Pi log Pj
a i log Pi + ~
i
(i,j = K, L, E, M )
j
(8)
where 7~j = 7j,.: the ~[s and y~j's are parameters to be estimated and the other elements are as defined previously. In order to correspond to a well behaved production function, the cost function must be homogeneous of degree one in prices; that is, for a fixed level of output, factor expenditures must increase proportionately when all factor prices increase proportionately. This implies the following relationships among the parameters:
~
cxi = 1
(9)
i
and:
(i,j = K, L, E, M ) i
j
"
j
(10)
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A convenient feature of the cost function employed is that the derived functions for the factor inputs (the examination of which is the objective of the present analysis) can be easily computed by partially differentiating eqn. (8) with respect to the factor prices; i.e.: (1 1)
P P / ~ P i = Xi
This result, known as Shephard's lemma, 21 is conveniently expressed in logarithmic form for the translog cost function as:
J where S; indicates the cost share of the ith factor of production. The cost share equations resulting from the cost function are then SK = ~r + 7rK log PK + ?'KLlog PL + 7rE log PE + 7KMlog PM
PE + ~';LM log PM
(13(b))
SE -~- (XE "-~ '~/KElog PK + ~LE log PL -~- )~EE log PL + ~/EM log PM
(13(c))
SL = ~L -~- 7KL log
PK +
S M = aM + "~'KMlog
)'LL log
PK +
PL +
( 13(a))
.... ' LE lo~
~/L.~I log PL + "~!EMlog P~: + 7M,~ log PM
(13(d))
Note the imposition of symmetry and the fact that the cost shares sum to unity. The characterisation of the structure of technology is accomplished by estimating the input demand equations (given by eqns (13(a)) to (13(d))) subject to the restrictions imposed by linear homogeneity in prices. Such an empirical implementation requires that the translog model (i.e. eqns (13(a)) to (13(d))) be embedded in a stochastic framework. It is presumed that deviations of the cost shares from the logarithmic derivatives of the translog cost function are the results of random errors in cost minimising behaviour, Consequently, an additive disturbance is appended to each of eqns (l 3(a)) to (13(d)). It is feasible to estimate the parameters of the cost function using ordinary least squares. This technique is certainly attractive from the point of view of simplicity, It neglects, however, the additional information contained in the share equations, which are also easily estimable. Furthermore, even for a modest number of factor prices, the translog cost function has many regressors which do not vary greatly. Hence multicollinearity may be a problem, resulting in imprecise parameter estimates.
Energy and input Jactor stability
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An alternative estimation procedure, and the approach used here, is to estimate jointly the cost share equations in a multivariate regression system. This procedure is satisfactory since the cost share equations include all the parameters of the cost function except the constant and no information is lost by not including the price possibility frontier in the estimation procedure. One potential problem arises in attempting to estimate empirically the parameters in the cost share equations using an efficient estimating technique (e.g. that of Zellner22). The estimated disturbance covariance matrix required to implement Zellner's procedure is singular because the disturbance share equations must sum to zero. The Zellner procedure can be made operational by deleting one of the share equations from the system. The estimates so obtained, however, will not be invariant to which equation is deleted. Barten I has demonstrated that maximum likelihood estimates of a system of share equations with one equation deleted are invariant to which equation is dropped. Kmenta and Gilbert ~9 have shown that iteration of the Zellner estimation procedure until convergence results in maximum likelihood estimates. Iterating the Zellner procedure is a computationally efficient method for obtaining maximum likelihood estimates and it is the procedure employed here.
DATA In selecting time series data on factor quantities and prices there is a large number of data sets employing a variety of measurement techniques that could be used. One that has received wide use, and the one employed here, is that developed by Berndt and Wood 4 for four factors of production (capital, labour, energy and materials) covering the period 1947-1976. Procedures outlined by Christensen and Jorgenson s are used to construct the rental price of capital services from non-residential structures and producers' durable equipment, taking account of variations in effective tax rates and rates of return, depreciation and capital gains. Quantity indexes of capital are constructed by Divisia aggregation of capital services from non-residential structures and producers' durable equipment. Finally, the value of capital services is computed as the product of the capital quantity index and the rental price of capital. A
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more detailed discussion of procedures used in constructing these capital price and quantity indexes is found in Berndt and Christensen. 2 Since data on labour compensation are readily available, estimates of the price of labour are obtained by first concentrating on the measure of the quantity of labour. The measure of labour services is constructed as a Divisia index of production Cblue collar') and non-production ('white collar') labour man-hours, and adjusted for quality changes using the education attainment indexes of Christensen and Jorgenson. 8 The measure of the value of labour services is total compensation to employees in United States manufacturing adjusted for the earnings of proprietors. The price of labour is then computed as adjusted total labour compensation divided by the quantity of labour services. A more detailed discussion of methodology and data sources used in the construction of the labour price and quantity indexes is presented in Berndt and Christensen. 12 Annual price and quantity indexes for energy and other intermediate materials in the United States manufacturing 1947-1976 are constructed. Annual inter-industry flow Tables measuring flows of goods and services from 25 producing sectors to l0 consuming sectors and five categories of final demand, in both current and constant dollars, are presented by Faucett Associates. 1~ Based on these Tables, annual quantity indexes of energy are constructed as Divisia quantity indexes of coal, crude petroleum, refined petroleum products, natural gas and electrical energy purchased by establishments in United States manufacturing. The value of energy purchases is then computed as the sum of current dollar purchases of these five energy types. Finally, the price index of energy is formed as the value of total energy purchases divided by the quantity of energy. Annual quantity indexes of materials are constructed from the Faucett inter-industry flow Tables as Divisia quantity indexes of non-energy intermediate goods purchased by establishments in United States manufacturing from agriculture, non-fuel mining, construction, manufacturing excluding petroleum products, transportation, communications, trade, water and sanitary services, and foreign countries (imports). The value of total non-energy intermediate goods purchased is then computed as the sum of c,rrent dollar purchases of all these non-energy intermediate goods. Finally, the price index of materials is formed as the value of total non-energy intermediate goods purchased divided by the quantity of materials.
Energy and input factor stability
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TESTING FOR THE STABILITY OF D E M A N D FOR FACTORS OF P R O D U C T I O N Relying on the previous discussion, the objective is to explicitly test for the stability of the demand for the factors of production in United States manufacturing over the period 1947-1976. Dividing the data into five equal length intervals (i.e. P = 5 and n -- 6) allows for the computation of the test statistic for each of the share equations. As noted, however, one of the equations must be deleted. The decision was arbitrarily made to eliminate the materials share equation. Linear homogeneity in factor prices (i.e. constraints (9) and (10)) have been imposed. Additional regularity conditions which the cost function must satisfy in order to correspond to a well behaved production structure are monotonicity and convexity in the factor prices. Sufficient conditions for these are positive fitted cost shares and negative definiteness of the bordered Hessian matrix of the cost function. These conditions (upon empirical implementation) are met in general, allowing one to conclude that the estimated cost function represents a well behaved production structure. (The coefficient estimates, elasticities of substitution and price elasticity, for the sake of brevity, are not reproduced here. They are available from the author upon request.) The computed values of ~ (via eqn. (6)), the test statistic, for each of three share equations (capital, labour and energy) are given in Table 1. The results are quite conclusive. Neither the equation for the demand for capital nor the equation for the demand for labour were unstable for US manufacturing over the period 1947-1976. Such is not the case, however, for the demand for energy. The structure of energy demand has altered and quite significantly at both the 5 per cent level and the 1 per cent level. Furthermore, a sequential examination of the computed test statistic TABLE I C o m p u t e d Value of the Test Statistic
Share equation
1. Capital 2. L a b o u r 3. Energy * T h a t is, Foo s (20, 6).
Computed ralue ~[ ~
Tabulated critical t'alue*
0-2539 0.4152 9.544 3
3-87 3-87 3-87
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Noel D. Uri
shows that the interval 1971 to 1976 gave rise to the instability in the demand for energy. (That is, the sum of squared residuals over this period were of sufficient magnitude to raise the value of the test statistic on the energy share equation from one of indicating no structural shift to one evincing a significant shift.)
I M P L I C A T I O N OF T H E RESULTS The implications of the results for estimating an aggregate production function are clear. Events of the decade of the 1970's have affected the demand for--and hence the use of--energy but have virtually left unchanged the demand for capital and labour. That is, for these latter two factors, the relative importance of the price of capital, the price of labour, the price of energy and the price of materials in influencing the share of total expenditures (and hence demand) has remained constant. The anomaly is that the relative importance of the factor prices in influencing the demand for energy has not emulated the behaviour of the demand for capital and labour. One must be careful, however, to avoid inferring that the quantity demanded of capital and labour has remained unchanged. (The factor complementarity versus substitutability has been dealt with extensively elsewhere (see Berndt and Wood's 5 summary).) The estimation results clearly show the price of all of the factors influencing each expenditure share. Thus, an increase in the price of energy does lead to an increase in the quantity of capital used in the production process. The magnitude of this response for the capital and labour share equations for each of the factors of production in the aggregate (i.e. considering, for example, the capital share equation with the prices of capital, labour, energy and materials) remained unaltered over the sample period. Another way of expressing this is that the share elasticities for the capital and labour equations did not vary. The share elasticities for the energy demand equation did change due primarily to the precipitous energy price increases following the oil embargo in October, 1973. The precise nature and size of these increases is unfortunately not easily assessable. Due to various econometric problems using a very short time series based solely on the interval 1971 to 1977 (e.g. robustness of the estimates), there is no way to obtain reliable estimates of the share equation coefficients (and hence the elasticities of
Energy and input.['actor stability
293
substitution and the price elasticity). This does not vitiate, however, the results of the statistical test employed for checking on the stability question.
CONCLUSION The foregoing analysis has been presented in an attempt to address the question of the stability of the d e m a n d for the factors of production at aggregate level for US manufacturing. The results are conclusive that the demand for capital and labour have remained virtually constant over the period of investigation (1947 1976) while the relative importance of the factors affecting the d e m a n d for energy has changed in a statistically significant fashion.
REFERENCES 1. A. P. Barten, Maximum likelihood estimation of a complete system of demand equations, European Economic Review, 1 (1969), pp. 7 73. 2. E. R. Berndt and k. R. Christensen, The translog function and the substitution of equipment, structures, and labor in US manufacturing, 1929 1968, Journal of Econometrics (March, 1973), pp. 81 114. 3. E. R. Berndt, M. A. Fuss and L. Waverman, Dynamic Models of the Industrial Demand For Energy, Electric Power Research Institute, Palo Alto, 1977. 4. E.R. Berndt and D. O. Wood, Technology, prices, and the derived demand for energy, Reciew of Economics and Statistics, LVll (August, 1975), pp. 259 68. 5. E. R. Berndt and D. O. Wood, Engineering and economic interpretation of energy Capital complementarity, American Economic Review, 69 (June, 197~), pp. 342- 54. 6. R. L. Brown, J. Durbin and J. M. Evans, Techniques for testing the constancy of regression relationships over time, Journal of the Royal Statistical Society, 37(2) (1975), pp. 149 63. 7. G. Chow, Tests of equality between two sets of coefficients in two linear regressions, Econometrica, 28 (July, 1960), pp. 591-605. 8. L.R. Christensen and D. W. Jorgenson, The measurement of US real capital input, 1929 1967, Retiew of Income and Wealth (December, 1969), pp. 293 320. 9. L. R. Christensen, D. W. Jorgenson and L. J. Lau, Transcendental logarithmic production frontiers, The Ret'iew of Economics and Statistics, 55 (1973), pp. 28-45.
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10. R. Eisner, Econometric studies of investment behavior: A comment, Economic Inquiry, 12 (1974), pp. 91 103. 11. Faucett Associates, Data det'elopment for the I-0 energy model, Jack Faucett Associates, Chevy Chase, Maryland, 1973. 12. B. C. Field and G. Grebenstein, Capital-energy substitution in US manufacturing, The Review of Economics and Statistics, 62 (1980), pp. 207-12. 13. R. Halvorsen, Econometric models ~[" US energy demand, D.C. Heath and Company, Lexington, 1978. 14. J. Henderson and R. E. Quandt, Microeconomic theory (2nd edn), McGrawHill Book Company, New York, 1971. 15. E.A. Hudson and D. W. Jorgenson, US energy policy and economic growth, The Bell Journal of Economics and Management Science, 5(2) (Autumn, 1974), pp. 461 514. 16. L. Johansen, Production junctions, North Holland Publishing Company, Amsterdam, 1972. 17. D. W. Jorgenson and E. R. Berndt, Production structure, D. W. Jorgenson and H. S. Houthakker (Eds), Energy resources and economic growth, Data Resources Inc., Lexington, Massachusetts, 1973. 18. J. Kmenta, Elements of econometrics, The Macmillan Company, New York, 1971. 19. J. Kmenta and R. F. Gilbert, Small sample properties of alternative estimates of seemingly unrelated regressions, Journal of the American Statistical Association, 63 (1968), pp. 1180 -1200. 20. M. R. Norman and R. R. Russell, Detelopment ofmethodsJbrJorecasting the national industrial demandJor energy, Electric Power Research Institute, Palo Alto, USA, 1976. 21. R. W. Sheppard, Cost and production functions, Princeton University Press, Princeton, 1953. 22. A. Zellner, An efficient method of estimating seemingly unrelated regressions and tests for aggregates bias, Journal of the American Statistical Association, 57 (1962), pp. 348-68.
The Energy Crisis and Its Impact on the Stability of the Demand for Factors of Production N o e l D. Uri Department of Economics and Business, The Catholic University of America, Washington, DC 20064 (USA)
SUMMARY This paper addresses the question of the stability of the demand for the Jaetors ojproduction at the aggregate let*elJ'or United States manufacturing over the period 1947 to 1976. The results are conclusive. The demand Jot capital and labour has remained virtually constant over the period of investigation whilst the relatire importance of the Jactors affecting the demandjor energy has changed in a statisticalO' sign(tic'ant fashion. "
I N T R O D U C T I ON Existing models of the d e m a n d for factors of production can be viewed as either based on static optimisation with instantaneous adjustment or as myopic optimisation with constant marginal costs of adjustment. Such models inherently have several significant limitations. First, it is unlikely that adjustment of firms to the desired factor input levels is an instantaneous process or that marginal costs of adjustment are constant (i.e. that supply curves of the factors of production are perfectly inelastic). Given these considerations, recent years have seen a renewed interest in the study (both theoretical and empirical) of production functions. A large number of studies exist suggesting one or another production function that will mitigate the limitations alluded to above. (See, for example, Berndt and W o o d , 4'5 Berndt e t a l . , a Field and Grebenstein, ~2 Halvorsen, ~a Hudson and Jorgenson, 15 N o r m a n and Russell, 2° etc.) One 281
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282
Noel D. Uri
issue, however, that has been ignored completely in these is whether the structures that are being posited and analysed are stable. (Stability is defined in the statistical sense of the estimated coefficients of the explanatory variables remaining constant over time.) In general, this is unacceptable. There is no reason to believe a p r i o r i that in the dynamic environment of, say, the United States economy, over the past three decades, the relative importance of the factors of production in the aggregate production function has remained unchanged. One reason for the absence of empirical investigations on this issue is the fact that the usual procedures for testing whether a regression relationship is constant over time in the statistical sense, namely the use of the F-test (Chow 7) or d u m m y variables (KmentalS), requires a p r i o r i knowledge of the point in time when the change in the estimated coefficients is suspected. Except for obvious dramatic changes such as war, this knowledge may not be readily available. The question of the stability of the functional relationship underlying the demand for factors of production is of critical importance. Policy inferences (e.g. relating to the impact of policy initiatives on demand for energy) are made on the basis of past behaviour. If this functional relationship has been subject to change, then, necessarily, the inferences will be, at least in part, unsatisfactory. One must be reasonably certain that the production function has not changed structurally. The purpose of this paper, then, is to examine the existence of a stable production function and attendant demand for the factors of production utilising a statistical test developed by Brown e t al. 6 that is designed to detect shifts in a regression relationship. This approach is adopted in deference to those mentioned previously because it does not require prior knowledge of the shifts but rather tests for the presence of such occurrences over the sample period. Before doing this, however, it is useful to review the theoretical rationale underlying the production process and provide a general discussion of the demand for factors of production.
THEORETICAL CONSIDERATIONS Consider the following example. Assume that a cost minimising competitive firm is initially in equilibrium with output at some equilibrium level and that the firm possesses a positive, twice differentiable strictly concave production function with four inputs (capital, K; labour, L;
Energy and input factor stability
283
energy, E and materials, M). Next, assume that the price of one of the factor inputs changes relative to the others (e.g. the price of energy rises). The total effect of this on the d e m a n d for the other factors can be broken down into two parts. First, there will be the gross substitution effect resulting in a reduction in the use of the factor with the higher relative price and an increase in the use of the other factors (note that this will always be the case--less of the input whose price increased and more of the other inputs (Henderson and Quandt,14 p. 70)). Secondly, there will be an expansion effect which is just the difference between the total effect and the gross substitution effect where the total effect of such a price changes results when the output of the production function is held constant and all inputs adjust to their new cost-minimising levels. The magnitude of the expansion effect is dependent, of course, upon the marginal products of the factors of production (Johansen16). It is possible for the expansion effect to dominate the gross substitution effect with the result that one actually observes an increase in the use of the factor having the initial price increase. That is, it is possible that factors of production are not gross substitutes but net complements. Whether net substitutability or complementarity exists depends on whether the gross substitution effect or the expansion effect is dominant. This is an empirical issue. What is of relevance now is the extent to which one can sort out these effects. A myriad of techniques exists to allow for such a determination. In other words, a large number of specifications are available that permit the estimation of elasticities of substitution for the factors of production. These specifications necessarily presuppose that the assumptions concerning the nature of the production function are correct. There is no theurgical rationale for accepting one set of assumptions over another. It need not be the case, for example, that the factors of production are substitutable in the fashion assumed because the technology involved in the production process is prohibitive. The best that can be done is to accept Eisner's ~° admonition (although in a slightly different context) that estimation of production functions is a tricky and difficult business and the best posture to take is one of humility.
TEST O F STABILITY As previously noted, statistical tests made to ascertain whether a regression relationship over the sample horizon is stable have been limited
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to the use of d u m m y variables when a change in the relationship is suspected or dividing the same period at that point and performing a Chow test. Both these techniques require a p r i o r i knowledge of the point in time when the function shifts. This information might not be available. A method for determining whether a regression relationship is constant over a given period based on an examination of the residuals of the relationship has been developed by Brown et al. 6 This procedure will be used to test the stability of the demand for the factors of production (and implicitly, the production function) as previously discussed. To give an appreciation of the test, it is briefly discussed here. Essentially, a way of investigating the time-variation of a regression coefficient is to fit the regression on a short segment of n successive observations and to move this segment along the series. A significance test for constancy based on this approach is derived from the results of regression based on nonoverlapping time segments. The method is based on a test statistic which equals the difference between the sum of squared residuals of the entire sample less the cumulative sum of squared residuals of the nonoverlapping segments divided by the cumulative sum of squared residuals of the non-overlapping segments. The null hypothesis that the regression relationship is constant over time implies that the value of the test statistic is distributed as F. Consider the basic regression model: yt = x't]], + ~t
t=l
.... ,T
(1)
where Yt is a vector of observations on the dependent variable, x t is a column vector of observations on k regressors, ~t is a vector of regression coefficients (the subscript t implying that t h e / f s may not be constant over time) and G is a vector of normally and independently distributed variables with mean zero and variances 0.2, t = 1,2 . . . . . T. The first element in x t will be taken to equal unity for all values of t if the model contains a constant. The hypothesis of constancy over time, which is denoted by H 0, is 31 ~ f12 . . . . .
f i t = fi
0_2 ~ 0.2 . . . . .
0.2 ~- 0.2
The test is more concerned with detecting differences among the fl's than among the 0.'s.
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Energy and input ,[actor stability
Let b r be the regression estimate from the sample of the first r observations and let: y~-x'~b~ 1 Wr (2) N~ 1 -~ X t r t X ; _ l X r _ I ) - I X r wherer=k+l,k+2,...,TandX'r_l=[Xl,X2,...,Xr_l]. It is shown by Brown et al. 6 that these transformed residuals, Wk+ i, Wk+ 2. . . . , are normally distributed independent variables with zero means and finite variances. Equation (2) is just a generalisation of the Helmert orthogonal transformation. The variables I,V,can be obtained without repeated matrix inversion by the relationships: b, = br_ , + ( G X ~ ) - 'x',(y - x',b~_ l)
(3)
and: (X; ~ r ) '
1 -
~f~,
( X ; _ 1 ~ r - i ) - i XrXr(X; , , iXr - i) - 1
-i
=(,_,x,_,)
-
1+x;(X"
(4)
,x,_i)-'x,
Let Sr be the residual sum of squares after fitting the model to the first r observations assuming the null hypothesis is true. Thus: S~=S~
I+W 2
r=k+l
..... T
(5)
(These relationships are demonstrated to hold by Brown et al. 6) The quantities required for each new segment are computed by first adding a new observation to the segment just by using the relationships (3), (4) and (5) and then allowing for the effect of dropping an observation from the beginning by means of the following analogues of relationships (3), (4) and (5): bll = bn+ l - ( X n X n ) -
lXl(Yl -- x'bn+
(Xnfl('n) -1 = (Xn+ l X n + l ) -1 -q- ( Y n'+ i X n + l )
i)
(3(a))
-'
X1X1' ( x ~ + i X . + , ) - ' 1 - x'l(X'+ i X . + 1)- ix1
g . = s . + , - ( y , - x',~.)2/(1 + x , ( x ~ x . )
ix,)
(4(a))
(5(a))
where )?,, b. and ,~, are, respectively, the values of the regressor matrix, the coefficient vector and the residual sum of squares based on observations from t = 2 to n + 1. For ease of exposition, only the relationships for the first update are given here. Further relationships are, of course, analogous. (Proofs again are given by Brown et al. 6)
Noel D. Uri
286
The significance test for constancy, called the homogeneity test, is derived from the results of regressions based on non-overlapping time segments, using the analysis of variance. The time segments for a moving regression of length n, are {1,n}, { ( n + l ) , ( 2 n ) } . . . . , t ( p - 2 ) n + l , (p - 1)n}, {(p - 1)n + 1, TI, where p is the integral part of Tin and the variance ratio considered (i.e. the homogeneity statistic) is: ( T - kp) S(1, T ) - A -
(kp - k)
zX
(6)
where A = {S(1,n) + (S(n + 1),2n) + . - . + ( S ( p n - n + 1), T)} and where S(r, s) is the residual sum of squares from the regression calculated from observations from t = r to s inclusive. This is equivalent to the usual 'between groups over with groups' ratio of mean squares and, under Ho, is distributed as F(kp - k, T - kp). The next step is to estimate empirically the production function and examine its stability using the suggested test. Before this is accomplished, however, a discussion of the precise form of the production function and a delineation of the data are in order.
PRODUCTION FUNCTIONS AND THE DEMAND FOR FACTOR INPUTS There are many approaches to estimating the production function (see Johansen 16 for a summary of these). The one that has been capturing most attention recently is the transcendental logarithmic price possibility frontier or, more simply, the translog price possibility frontier. The price possibility frontier is a transcendental function of the logarithms of the prices of inputs. The translog price possibility frontier was introduced by Christensen et al.9 It distinguishes itself from earlier approaches in that it begins by positing, as an analogue to the production possibility frontier, a price possibility frontier of a general neoclassical specification. In conjunction with assumptions of perfect competition, demand for factor input equations are derived which must be estimated simultaneously to allow for the theoretically imposed restrictions on the parameters. The approach offers distinct advantages over other approaches to modelling the production function in that an explicit theoretical model serves as the basis for specification and reduces the problem of multicollinearity by decreasing the number of parameters to be estimated.
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Energy and input Jactor stability
Extensive use of this approach was made by Jorgenson and Berndt. 17 A model along the lines of that of Jorgenson and Berndt is developed and serves as the basis by which to assess the question of stability. The assumption, then, is that there exists a twice differentiable aggregate production function relating the flow of gross output Y to the services of four inputs: capital, XK; labour, XL; energy, X~ and all other intermediate materials, X~t. Further, the presumption is that production is characterised by constant returns to scale and that any technological change affecting K, L, E a n d M is Hicks-neutral. Corresponding to such a production function there exists a cost function reflecting the production technology. In general form, this cost function is written as:
P = E~(Y, PK, PL, PE, PM)
(7)
where P is the total cost and PK, PL, PE and PM are the input prices of K, L, E and M, respectively. As noted, for the purposes of estimation, the translog function is chosen as the specific functional form for P. It places no a priori restrictions on the Allen partial elasticities of substitution and can be interpreted as a second order approximation to an arbitrary twice differentiable cost function. This cost function with symmetry and constant returns to scale imposed is written log P = a 0 +
2
'22
7;; log Pi log Pj
a i log Pi + ~
i
(i,j = K, L, E, M )
j
(8)
where 7~j = 7j,.: the ~[s and y~j's are parameters to be estimated and the other elements are as defined previously. In order to correspond to a well behaved production function, the cost function must be homogeneous of degree one in prices; that is, for a fixed level of output, factor expenditures must increase proportionately when all factor prices increase proportionately. This implies the following relationships among the parameters:
~
cxi = 1
(9)
i
and:
(i,j = K, L, E, M ) i
j
"
j
(10)
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Noel D. Uri
A convenient feature of the cost function employed is that the derived functions for the factor inputs (the examination of which is the objective of the present analysis) can be easily computed by partially differentiating eqn. (8) with respect to the factor prices; i.e.: (1 1)
P P / ~ P i = Xi
This result, known as Shephard's lemma, 21 is conveniently expressed in logarithmic form for the translog cost function as:
J where S; indicates the cost share of the ith factor of production. The cost share equations resulting from the cost function are then SK = ~r + 7rK log PK + ?'KLlog PL + 7rE log PE + 7KMlog PM
PE + ~';LM log PM
(13(b))
SE -~- (XE "-~ '~/KElog PK + ~LE log PL -~- )~EE log PL + ~/EM log PM
(13(c))
SL = ~L -~- 7KL log
PK +
S M = aM + "~'KMlog
)'LL log
PK +
PL +
( 13(a))
.... ' LE lo~
~/L.~I log PL + "~!EMlog P~: + 7M,~ log PM
(13(d))
Note the imposition of symmetry and the fact that the cost shares sum to unity. The characterisation of the structure of technology is accomplished by estimating the input demand equations (given by eqns (13(a)) to (13(d))) subject to the restrictions imposed by linear homogeneity in prices. Such an empirical implementation requires that the translog model (i.e. eqns (13(a)) to (13(d))) be embedded in a stochastic framework. It is presumed that deviations of the cost shares from the logarithmic derivatives of the translog cost function are the results of random errors in cost minimising behaviour, Consequently, an additive disturbance is appended to each of eqns (l 3(a)) to (13(d)). It is feasible to estimate the parameters of the cost function using ordinary least squares. This technique is certainly attractive from the point of view of simplicity, It neglects, however, the additional information contained in the share equations, which are also easily estimable. Furthermore, even for a modest number of factor prices, the translog cost function has many regressors which do not vary greatly. Hence multicollinearity may be a problem, resulting in imprecise parameter estimates.
Energy and input Jactor stability
289
An alternative estimation procedure, and the approach used here, is to estimate jointly the cost share equations in a multivariate regression system. This procedure is satisfactory since the cost share equations include all the parameters of the cost function except the constant and no information is lost by not including the price possibility frontier in the estimation procedure. One potential problem arises in attempting to estimate empirically the parameters in the cost share equations using an efficient estimating technique (e.g. that of Zellner22). The estimated disturbance covariance matrix required to implement Zellner's procedure is singular because the disturbance share equations must sum to zero. The Zellner procedure can be made operational by deleting one of the share equations from the system. The estimates so obtained, however, will not be invariant to which equation is deleted. Barten I has demonstrated that maximum likelihood estimates of a system of share equations with one equation deleted are invariant to which equation is dropped. Kmenta and Gilbert ~9 have shown that iteration of the Zellner estimation procedure until convergence results in maximum likelihood estimates. Iterating the Zellner procedure is a computationally efficient method for obtaining maximum likelihood estimates and it is the procedure employed here.
DATA In selecting time series data on factor quantities and prices there is a large number of data sets employing a variety of measurement techniques that could be used. One that has received wide use, and the one employed here, is that developed by Berndt and Wood 4 for four factors of production (capital, labour, energy and materials) covering the period 1947-1976. Procedures outlined by Christensen and Jorgenson s are used to construct the rental price of capital services from non-residential structures and producers' durable equipment, taking account of variations in effective tax rates and rates of return, depreciation and capital gains. Quantity indexes of capital are constructed by Divisia aggregation of capital services from non-residential structures and producers' durable equipment. Finally, the value of capital services is computed as the product of the capital quantity index and the rental price of capital. A
290
Noel D. Uri
more detailed discussion of procedures used in constructing these capital price and quantity indexes is found in Berndt and Christensen. 2 Since data on labour compensation are readily available, estimates of the price of labour are obtained by first concentrating on the measure of the quantity of labour. The measure of labour services is constructed as a Divisia index of production Cblue collar') and non-production ('white collar') labour man-hours, and adjusted for quality changes using the education attainment indexes of Christensen and Jorgenson. 8 The measure of the value of labour services is total compensation to employees in United States manufacturing adjusted for the earnings of proprietors. The price of labour is then computed as adjusted total labour compensation divided by the quantity of labour services. A more detailed discussion of methodology and data sources used in the construction of the labour price and quantity indexes is presented in Berndt and Christensen. 12 Annual price and quantity indexes for energy and other intermediate materials in the United States manufacturing 1947-1976 are constructed. Annual inter-industry flow Tables measuring flows of goods and services from 25 producing sectors to l0 consuming sectors and five categories of final demand, in both current and constant dollars, are presented by Faucett Associates. 1~ Based on these Tables, annual quantity indexes of energy are constructed as Divisia quantity indexes of coal, crude petroleum, refined petroleum products, natural gas and electrical energy purchased by establishments in United States manufacturing. The value of energy purchases is then computed as the sum of current dollar purchases of these five energy types. Finally, the price index of energy is formed as the value of total energy purchases divided by the quantity of energy. Annual quantity indexes of materials are constructed from the Faucett inter-industry flow Tables as Divisia quantity indexes of non-energy intermediate goods purchased by establishments in United States manufacturing from agriculture, non-fuel mining, construction, manufacturing excluding petroleum products, transportation, communications, trade, water and sanitary services, and foreign countries (imports). The value of total non-energy intermediate goods purchased is then computed as the sum of c,rrent dollar purchases of all these non-energy intermediate goods. Finally, the price index of materials is formed as the value of total non-energy intermediate goods purchased divided by the quantity of materials.
Energy and input factor stability
291
TESTING FOR THE STABILITY OF D E M A N D FOR FACTORS OF P R O D U C T I O N Relying on the previous discussion, the objective is to explicitly test for the stability of the demand for the factors of production in United States manufacturing over the period 1947-1976. Dividing the data into five equal length intervals (i.e. P = 5 and n -- 6) allows for the computation of the test statistic for each of the share equations. As noted, however, one of the equations must be deleted. The decision was arbitrarily made to eliminate the materials share equation. Linear homogeneity in factor prices (i.e. constraints (9) and (10)) have been imposed. Additional regularity conditions which the cost function must satisfy in order to correspond to a well behaved production structure are monotonicity and convexity in the factor prices. Sufficient conditions for these are positive fitted cost shares and negative definiteness of the bordered Hessian matrix of the cost function. These conditions (upon empirical implementation) are met in general, allowing one to conclude that the estimated cost function represents a well behaved production structure. (The coefficient estimates, elasticities of substitution and price elasticity, for the sake of brevity, are not reproduced here. They are available from the author upon request.) The computed values of ~ (via eqn. (6)), the test statistic, for each of three share equations (capital, labour and energy) are given in Table 1. The results are quite conclusive. Neither the equation for the demand for capital nor the equation for the demand for labour were unstable for US manufacturing over the period 1947-1976. Such is not the case, however, for the demand for energy. The structure of energy demand has altered and quite significantly at both the 5 per cent level and the 1 per cent level. Furthermore, a sequential examination of the computed test statistic TABLE I C o m p u t e d Value of the Test Statistic
Share equation
1. Capital 2. L a b o u r 3. Energy * T h a t is, Foo s (20, 6).
Computed ralue ~[ ~
Tabulated critical t'alue*
0-2539 0.4152 9.544 3
3-87 3-87 3-87
292
Noel D. Uri
shows that the interval 1971 to 1976 gave rise to the instability in the demand for energy. (That is, the sum of squared residuals over this period were of sufficient magnitude to raise the value of the test statistic on the energy share equation from one of indicating no structural shift to one evincing a significant shift.)
I M P L I C A T I O N OF T H E RESULTS The implications of the results for estimating an aggregate production function are clear. Events of the decade of the 1970's have affected the demand for--and hence the use of--energy but have virtually left unchanged the demand for capital and labour. That is, for these latter two factors, the relative importance of the price of capital, the price of labour, the price of energy and the price of materials in influencing the share of total expenditures (and hence demand) has remained constant. The anomaly is that the relative importance of the factor prices in influencing the demand for energy has not emulated the behaviour of the demand for capital and labour. One must be careful, however, to avoid inferring that the quantity demanded of capital and labour has remained unchanged. (The factor complementarity versus substitutability has been dealt with extensively elsewhere (see Berndt and Wood's 5 summary).) The estimation results clearly show the price of all of the factors influencing each expenditure share. Thus, an increase in the price of energy does lead to an increase in the quantity of capital used in the production process. The magnitude of this response for the capital and labour share equations for each of the factors of production in the aggregate (i.e. considering, for example, the capital share equation with the prices of capital, labour, energy and materials) remained unaltered over the sample period. Another way of expressing this is that the share elasticities for the capital and labour equations did not vary. The share elasticities for the energy demand equation did change due primarily to the precipitous energy price increases following the oil embargo in October, 1973. The precise nature and size of these increases is unfortunately not easily assessable. Due to various econometric problems using a very short time series based solely on the interval 1971 to 1977 (e.g. robustness of the estimates), there is no way to obtain reliable estimates of the share equation coefficients (and hence the elasticities of
Energy and input.['actor stability
293
substitution and the price elasticity). This does not vitiate, however, the results of the statistical test employed for checking on the stability question.
CONCLUSION The foregoing analysis has been presented in an attempt to address the question of the stability of the d e m a n d for the factors of production at aggregate level for US manufacturing. The results are conclusive that the demand for capital and labour have remained virtually constant over the period of investigation (1947 1976) while the relative importance of the factors affecting the d e m a n d for energy has changed in a statistically significant fashion.
REFERENCES 1. A. P. Barten, Maximum likelihood estimation of a complete system of demand equations, European Economic Review, 1 (1969), pp. 7 73. 2. E. R. Berndt and k. R. Christensen, The translog function and the substitution of equipment, structures, and labor in US manufacturing, 1929 1968, Journal of Econometrics (March, 1973), pp. 81 114. 3. E. R. Berndt, M. A. Fuss and L. Waverman, Dynamic Models of the Industrial Demand For Energy, Electric Power Research Institute, Palo Alto, 1977. 4. E.R. Berndt and D. O. Wood, Technology, prices, and the derived demand for energy, Reciew of Economics and Statistics, LVll (August, 1975), pp. 259 68. 5. E. R. Berndt and D. O. Wood, Engineering and economic interpretation of energy Capital complementarity, American Economic Review, 69 (June, 197~), pp. 342- 54. 6. R. L. Brown, J. Durbin and J. M. Evans, Techniques for testing the constancy of regression relationships over time, Journal of the Royal Statistical Society, 37(2) (1975), pp. 149 63. 7. G. Chow, Tests of equality between two sets of coefficients in two linear regressions, Econometrica, 28 (July, 1960), pp. 591-605. 8. L.R. Christensen and D. W. Jorgenson, The measurement of US real capital input, 1929 1967, Retiew of Income and Wealth (December, 1969), pp. 293 320. 9. L. R. Christensen, D. W. Jorgenson and L. J. Lau, Transcendental logarithmic production frontiers, The Ret'iew of Economics and Statistics, 55 (1973), pp. 28-45.
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10. R. Eisner, Econometric studies of investment behavior: A comment, Economic Inquiry, 12 (1974), pp. 91 103. 11. Faucett Associates, Data det'elopment for the I-0 energy model, Jack Faucett Associates, Chevy Chase, Maryland, 1973. 12. B. C. Field and G. Grebenstein, Capital-energy substitution in US manufacturing, The Review of Economics and Statistics, 62 (1980), pp. 207-12. 13. R. Halvorsen, Econometric models ~[" US energy demand, D.C. Heath and Company, Lexington, 1978. 14. J. Henderson and R. E. Quandt, Microeconomic theory (2nd edn), McGrawHill Book Company, New York, 1971. 15. E.A. Hudson and D. W. Jorgenson, US energy policy and economic growth, The Bell Journal of Economics and Management Science, 5(2) (Autumn, 1974), pp. 461 514. 16. L. Johansen, Production junctions, North Holland Publishing Company, Amsterdam, 1972. 17. D. W. Jorgenson and E. R. Berndt, Production structure, D. W. Jorgenson and H. S. Houthakker (Eds), Energy resources and economic growth, Data Resources Inc., Lexington, Massachusetts, 1973. 18. J. Kmenta, Elements of econometrics, The Macmillan Company, New York, 1971. 19. J. Kmenta and R. F. Gilbert, Small sample properties of alternative estimates of seemingly unrelated regressions, Journal of the American Statistical Association, 63 (1968), pp. 1180 -1200. 20. M. R. Norman and R. R. Russell, Detelopment ofmethodsJbrJorecasting the national industrial demandJor energy, Electric Power Research Institute, Palo Alto, USA, 1976. 21. R. W. Sheppard, Cost and production functions, Princeton University Press, Princeton, 1953. 22. A. Zellner, An efficient method of estimating seemingly unrelated regressions and tests for aggregates bias, Journal of the American Statistical Association, 57 (1962), pp. 348-68.