m _V . _ _ Anna“... _EL . L 1.... _._ . . _ mm Much“. .T... An» 7. _ Ezfiu . R. . . .. . Manna N W. , mu A, nHu . .- l. i... : Jr; «ix. in: 313 This is to certify that the thesis entitled AN ECONOMETRIC APPROACH TO TECHNOLOGICAL CHANGE AND RETURNS TO SCALE IN STEAM-ELECTR l C GENERATION presented by JEFFREY A. ROTH has been accepted towards fulfillment of the requirements for Ph. D. degreein Economics Date May 2|, |97| 0-7639 ABSTRACT AN ECONOMBTRIC APPROACH TO TECHNOLOGICAL CHANGE AND RETURNS TO SCALE IN STEAM-ELECTRIC GENERATION by Jeffrey A. Roth Three null hypotheses were tested concerning steam- electric generation at the plant level, over the period 19u8 to 1965. These null hypotheses are that: (l) Steam-electric generation is subject to constant returns to scale. (2) Technological change in steam-electric generation did not significantly affect plant efficiency between 19u8 and 1965. (3) The relationship between plant output of electricity and inputs of capital, fuel, and labor is independent of the num- ber of turbogenerator units per plant. These null hypotheses were tested using a three-stage procedure. First, an appropriate model of the generating plant was selected from a number of alternative models, on the basis of freedom from specification error. Second, using an independent data sample, the appropriateness of the chosen Jeffrey A. Roth model was confirmed. Third, the null hypotheses were tested using estimates of the parameters of the chosen model. The most apprOpriate model was found to be a fixed- relative-factor-proportions model with output exogenously determined. The generating process was found to be subject to increasing returns to scale. Technological change did not significantly improve plant efficiency, except insofar as it made scale increases possible. Ceteris paribus, increases in the number of turbogenerator units per plant were found to necessitate increases in total installed plant capacity for a given level of output. AN ECONOMETRIC APPROACH TO TECHNOLOGICAL CHANGE AND RETURNS TO SCALE IN STEAM-ELECTRIC GENERATION by \\ \ Jeffrey At Roth A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1971 ACKNOWLEDGEMENTS "All, all are gone, and the temples, libraries, And schools of Castalia are no more. At rest Amid the ruins, the glass beads in his hand, Those hieroglyphs once so significant That now are only colored bits of glass, He lets them roll until their force is spent, And silently they vanish in the sand." --Hermann Hesse, "The Last Glass Bead Game Player" At various points in the dissertation process, the student-apprentice may find himself identifying with Sisyphus, the mythical king of Corinth whose sentence was a life of pushing a rock to the top of a hill, only to watch it roll back to the bottom. An important distinction is that while Sisyphus was left to his perpetual task in solitude, the thesis writer is blessed with the support--technical, finan- cial, and spiritual--of others around him. In particular, I wish to acknowledge the efforts of my guidance committee in my behalf. Professor Bruce Allen gra- ciously accomodated his schedule to my needs, in order to provide helpful comments concerning exposition and the reasons for professional interest in my results. Professor Jan Kmenta initially stimulated my interest in econometrics and has provided helpful advice and encouragement throughout my entire graduate career. More attention to his suggestions at an ii early stage of the dissertation process would have saved me much unnecessary grief. My chairman, Professor James Ramsey, made me aware of many of the difficulties and limita- tions of econometric research. Furthermore, he helped me to discover the need for advance planning in a research project. The final product benefitted greatly from the comments of all three men. I also appreciated the computer assistance given willingly by Art Havenner, Steve Scheer, and Ron Tracy. For financial assistance, I am grateful to Michigan State University's Institute of Public Utilities for a dis- sertation research grant, and to the Economics Department for patient renewal of my assistantship. Thanks are due to several others for help at various stages: a number of Evans Scholars, particularly Fred Locke and Jim Ferguson, who assisted in the data processing; Barbara Schieffer and Cynthia Oakes for the drawings; and Mrs. Rosalie Moyer for the final typing. Finally, I would be remiss if I failed to acknowledge the moral support of a congenial group of fellow graduate students, too numerous to mention, with whom I frequently quenched the thirst for knowledge. iii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . ii LIST OF TABLES . . . . . . . . . . . . . . . . . . . .viii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . ix CHAPTER I. INTRODUCTION A. Scale, Technology, and the Thermal Electric Production Function . . . . . . . . l B. Existing Literature on Thermal Electric Production and Cost Functions . . . . . . . . 8 1. Direct Methods . . . . . . . . . . . . . 9 2. Profit-Maximization Models with Factor Substitution . . . . . . . . . . . 16 3. Non-Substitution Models . . . . . . . . . 2” H. Other Approaches . . . . . . . . . . . . H2 II. CONSIDERATIONS RELATED TO POSTWAR TECHNOLOGICAL CHANGE . . . . . . . . . . . . . . H8 A. The Productive Process in Electrical Generation . . . . . . . . . . . . . . . . . 49 l. The Furnace . . . . . . . . . . . . . . . 50 2. The Steam Generator . . . . . . . . . . . 51 3. The Turbogenerator . . . . . . . . . . . 53 4. Steam Cycles . . . . . . . . . . . . . . 5H B. Postwar Innovations in Steam—Electric Generation . . . . . . . . . . . . . . . . . 59 l. Innovations in Combustion . . . . . . . . 59 2. Innovations in Steam Generation . . . . . 62 iv TABLE OF CONTENTS (cont'd.) 3. Innovations in Generator Cooling . . . . . 66 4. Summary of Innovations in Plant Design . . . . . . . . . . . . . . . . . . 68 5. The Methodology of Innovations in Plant Design . . . . . . . . . . . . . 69 C. The Stratification of Generating Plants . . . 71 III. THEORETICAL STRUCTURE AND METHODOLOGY . . . . . . 77 A. Recent Results in Production Theory . . . . . 78 1. Alternative Models of the Firm . . . . . . 79 2. Generalized Production Functions . . . . . 8” B. Generalizations of the ZKD Results . . . . . . 86 1. The Generalized n-factor Cobb~Douglas Function . . . . . . . . . . 87 2. The Generalized n—factor CES Function . . 91 C. The Alternative Models to Be Considered . . . 95 1. Applications of the Generalized ZKD Model . . . . . . . . . . . . . . . . 95 a. Cobb-Douglas . . . . . . . . . . . . . 95 b. CES . . . . . . . . . . . . . . . . . 96 c. GPF's based on the Cobb— Douglas and CBS functions . . . . . . 98 2. Models Used by Previous Researchers . . . 105 a. The Hart and Chawla Model . . . . . . 105 b. A Modification of the Ling Model . . . 109 3. Models Involving the Number of Units per plant . . . . . . . . . . . . . . . . 110 D. Methodology of the Present Study . . . . . . . 11H 1. Data for the Present Study . . . . . . . . 114 V TABLE OF CONTENTS (cont'd.) 2. Stage One--Restriction of the Maintained Hypothesis 3. Stage Two--Confirmation of the Restricted Maintained Hypothesis 9. Stage Three--Inference Concerning Scale, Technology, and the Number of Units . . . . . IV. EMPIRICAL RESULTS A. Stage One--Choice of a Model B. Stage Two—-Confirmation of the Chosen Model C. Stage Three--Results Concerning Scale, Technology, and Machine Mix 1. Scale Effects 2. Effects of Technological Change 3. Significance of the Number of Units per Plant 4. Consistency of the Present Results with Previous Results V. CONCLUSIONS FROM THIS STUDY A. Summary of the Thesis B. Implications of the Results C. Limitations of the Study and Suggestions for Future Research 1. Procedural Limitations 2. Limitations of Scope 3. Suggested Further Research APPENDICES I. APPENDIX A: RESULTS OF STAGE ONE II. APPENDIX B: THE RESULTS OF STAGE TWO III. APPENDIX C: DATA USED IN THE STUDY vi 116 122 125 133 1w 199 152 154 161 165 171 175 177 177 179 182 185 193 196 TABLE OF CONTENTS (cont'd.) IV. APPENDIX D: SUMMARY OF MODELS CONSIDERED . . . . 226 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . 229 vii Table 10. 11. 12. 13. 19. 15. LIST OF TABLES Coal Furnace Type Planned in Samples of Plants Under Construction . . . . Number of Turbine Bleed points for Samples of Units Under Construction Generator Cooling Method in Sample of Units Under Construction Definition of Cells for Classification of Generating Plants Correlation Coefficients Among Factors, by Cell Estimation Results for LEHCC/N, LEHCF/N, and LEHCL/N . . Results of Tests of the Constant Returns to Scale Hypothesis Results of Tests for Equality of F2 across Cells . . . . . . . . Results of Tests for Equality of P1 across Cells . . . . Results of Tests that 620 Results of Stage One Results of Stage Two List of Plants in Sample, by File Number Data . . . . . . . . . . . . . . . . . . Alternative Models of the Steam-Electric Generating Plant . . . . . . . . viii 61 61 68 73 173 150 153 156 160 162 186 194 200 20” 226 Figure 1. LIST OF FIGURES Schematic Diagram of Steam-Electric Generating Plant . . . . . . Schematic Diagram of a Watertube Boiler Graphic Representation of the Rankine Steam Cycle . . . . . . . . . . Graphic Representation of the Reheat- Regenerative Steam Cycle ix 49 52 55 57 Chapter I. Introduction This chapter has two major objectives. In Section A., the problem to be considered will be defined and its methodo- logical and empirical importance examined. In Section B., a critical discussion of previous studies of the present prob- lem and related issues will be presented. I.A. Scale, Technology, and the Thermal Electric Production Function In recent decades there have been numerous innovations in fossil-fueled steam-electric generation. Examples, to be discussed in detail in Chapter II.A., include additional reheat and regenerative preheat cycles, hollow—conductor generator cooling systems, and use of steam in its super- critical state. Simultaneously, the size of turbogenerator units, as measured by output rating of the generator, has been increasing. For example, according to the Modern Plant Design Surveys compiled by Power magazine, the capacity of the largest central-station turbogenerator unit under con- struction increased from 150 electrical megawatts (mw) in 19u9 to 715 mw in 1965.1 Thus, the possible causes of any observed improvement in the efficiency of the generation 1These Surveys appear on a fairly regular annual basis in special issues of Power magazine. They contain detailed technical information on the characteristics of turbogenera- tor units under construction at the time of publication. 1 process include both technological change and changes in scale of plant. The problem addressed in this dissertation is to estimate the separate effects on the steam-electric generation process of changes in technology and scale. This will be done in the following manner. First, technologically homogeneous populations of turbogenerator units will be defined. Then, using sample production data from each population, the production function for each population will be estimated. By comparing the estimated efficiency parameters of the production functions across populations and testing the hypothesis of constant returns to scale within each population, one has a means of examining the effects of changes in scale and technology on the production function for electricity. The problem is of interest for two reasons, one of which may be classified as "applied," the other as "methodological." An example of an applied problem stems from a contro- versy in public utility regulation related to the "fair rate of return principle." A "fair return" has been defined as the "entire excess in operating revenues, over and above cur- rent operating deductions, for which a commission will make provision in a rate case as a component of the company's 2 annual revenue requirements." Furthermore, ". . . the allowed-for return is arrived at as a multiple of two factors: k‘ 2J. C. Bonbright, Principles of Public Utility Rates (New York: Columbia UnIVersityTPress, 19617} 1M9. the rate base, and the . . . 'fair' rate of return thereon. The rate base . . . represents the total quantum of invested capital. . . ."3 Briefly, the controversy centers on whether regulation under this principle leads to inefficiency in the regulated industry. Two different arguments have been advanced to show that this is the case. The first is that regulation according to the fair return principle "creates an environ- ment in which incompetence is rewarded and efficiency is penalized because the determination of total revenue require- ments on a cost-plus basis assures the company that all expenses will be covered, while at the same time eliminating the possibility that any gains from greater productivity can be retained."u Statements alleging that this is in fact the case have been made by economists J. S. Bain, G. C. Means, and H. M. Trebing, and, perhaps not unexpectedly, by American Telephone and Telegraph.5 In contrast to this argument that regulation alters the regulated firm's incentive to maximize profits, Averch 3Bonbright, 22f cit., lu9-150. uH. M. Trebing, "Toward an Incentive System of Regula- tion," Public Utilities Fortnightly, 72 (July 18, 1963), 22. 5J. S. Bain, Industrial Organization (New York: Wiley, 1959), 599; G. C. Means, Pricing Power and the Public Interest (New York: Harper, 1962), 191; H. M. Trebing, "A Critique of the Planning Function in Regulation," Public Utilities Fort- ni htl , 79 (March 16, 1967), 26ff; American—Telephone and jeIegraph Co., Profits, Performance, and Progress (1959), 91. and Johnson6 concluded that even a profit-maximizing firm, regulated to earn a fair rate of return on capital, has an incentive to adjust to the regulatory constraint by the uneconomic substitution of capital for other factors, so that factors are not combined optimally from the social point of view. Thus, the argument that regulation leads to allo- cative inefficiency has been made both with and without reference to the assumption of profit-maximization. On the other hand, J. C. Bonbright claims at least a plausible case for the thesis that "what has saved regulation from being a critical influence in the direction of mediocrity and tardy technological progress has been its very 'defi- ciencies' in the form of regulatory lags and . . . acquies- cence by commissions in fairly prolonged periods of theoreti- cally 'excessive' earnings on the part of companies."7 Definitive empirical resolution of this controversy is of course difficult since one cannot observe what a regulated utility's record of innovation would have been in the absence of regulation during a given period. But empirical observa- tion of a series of innovations that did in fact improve the efficiency of steam-electric generation certainly strengthens the presumption against the thesis that regulation stifles technological progress. On the other hand, if one observes 6H. Averch and L. L. Johnson, "Behavior of the Firm under Regulatory Constraints," American Economic Review 52 7Bonbright, 92. cit., 262. a long series of capital-embodied innovations, each the result of expensive research and development, without a cor- responding improvement in generating efficiency, the credi- bility of the Averch—Johnson hypothesis is strengthened. A study of the profitability of various innovations by electric utilities is beyond the scope of the present study. However, comparison of the parameters of the production func- tion across cells defined by the innovations embodied in the members of those cells should give a tentative indication whether or not these innovations have contributed to effi- ciency. This comparison will be made as a part of the present study. The application in this study of three fairly recent theoretical developments lends methodological interest to the work. The first, a set of tests developed by J. B. Ramsey for specification error in least squares regression analysis, is useful in the problem of choice between alter- native forms of a given maintained hypothesis.8 One may apply the tests to a regression estimated from sample data to determine whether that regression is subject to one or more types of specification error, such as omitted variables, incorrect functional form, simultaneous equation problems, or heteroskedasticity. Previous applications of the tests include a study of the U. S. aggregate production function ‘1 8J. B. Ramsey, "Tests for Specification Error in Classi- cal Linear Least-squares Regression Analysis," Journal of the figwal Statistical Society, Ser. B., 31 (1969), 3503371. by Ramsey and Zarembka, a study of the demand function for money by R. F. Gilbert, a study by T. w. Murray involving international trade data, and a study of the term structure of interest rates by F. Bonello.9 A second recent theoretical development being applied in the present study is the notion of the generalized produc- tion function (GPF), developed by Zellner and Revankar.10 Since one feature of the GPF is a returns-to-scale parameter which varies over the range of output, use of such a trans- formation of the production function should provide an im- provement over previous studies of returns to scale in steam- electric generation, which were constrained by constancy of this parameter. In the review of the literature in Section B of this chapter, it will be pointed out that both Nerlove and Dhrymes and Kurz found indication of such variability.11 k 9J. B. Ramsey and P. Zarembka, "Alternative Functional Forms and the Aggregate Production Function," Michigan State University Econometrics WorkshOp Paper #6705; R. F. Gilbert, IE? Demand for Money: an Analysis of Specification Error, unpublished Ph.D. dissertation, Michigan State University (1969); T. W. Murray, Data Errors and Economic Parameter £§timation= A Case Study of InternationaI Trade Data, unpub- lished Ph.D. dissertation, Michigan State University (1969); F. Bonello, The Term Structure of Interest Rates, the Expec- Egtions Hypothesis, and the FormuIatIEn of Expected Interest Rates, unpublished Ph.D. dissertation, MiEhigan State—Univer- Slty (1968). 10A. Zellner and N. S. Revankar, "Generalized Production gunctions," Review of Economic Studies 36 (2), (April 1969), 91-250. llM. Nerlove, "Returns to Scale in Electricity Supply," ed by C. Christ, Measurement in Economics-Studies in Mathe- flgtical Economics and EconometriCs in Memory of Yehuda Grun- feld (Stanford: StanFOrd UniverSity PreSSRI963) 1674198;’P. . hrymes and M. Kurz, "Technology and Scale in Electricity Generation," Econometrica 32 (July 196a), 287-315. However, lacking the means to parameterize it, they were unable to test its significance. Thus, there is prior evi- dence that use of GPF's will be helpful in examining the steam-electric generation process. To this author's know- ledge, the literature contains only two instances of estimation of a GPF, the first being a production function for the U. S. transportation equipment industry in the Zellner and Revankar paper cited above, and the second a U. S. aggregate production function in the Ramsey and Zarembka study, also cited above. The third recent theoretical result being applied in the present study, due to Zellner, Kmenta, and Dreze (ZKD), is a production model in which ordinary least squares estimates 0f the parameters of a production function are shown to be consistent.12 Since use of the specification error tests is limited to the single-equation case, the ZKD results are im- POPtant to the methodology of the present study. In turn, use of the tests will be helpful in testing the aPProPriateness of the ZKD model to the steam-electric gener- ating plant. One of the underlying assumptions of this model is that output is a stochastic function of the inputs employed by a firm attempting to maximize expected profit. As will become clear in the review of the literature in Section B., this assumption is directly incompatible with the assumption, made by many previous investigators of the industry, that °Utput of the generating plant is exogenously determined. \- 12A. Zellner, J. Kmenta and G. Dréze, "specification and EStimation of Cobb-Douglas Production Function Models," lchiometrica 314 (October 1966) 7814-795. Comparison of the specification error test results for both types of models will assist in determining which assumption is appropriate. In summary, it is clear that the problem addressed in the present study--separation and quantification of the effects of returns to scale and technological innovation in steam-electric generation--is of interest because of its bearing on the problem of regulation of public utilities. Furthermore, the study is of methodological interest because of the application of several recent theoretical developments to the problem. I.B. Existing Literature on Thermal Electric Production and Cost Functions Previous economic studies of thermal electricity genera- tion may be classified into four groups. First, there are those in which no explicit consideration is made of optimiza- tion by the firm, so that estimation of a cost or production relation is carried out by ordinary least squares, without benefit of an explicitly formulated model. On the other hand, those studies which do consider the implications of optimization by the firm may be grouped into two sub-classes. In the first sub-class are those using models that allow for factor substitution in the neoclassical tradition, so that simultaneous equation estimation is usually required for consistency of the estimates. In the second sub-class are those in which substitution of factors is not permitted, the production coefficients being either fixed relative proportions of output or functions of variables other than factor prices or output. Estimation of these models has been carried out by ordinary least squares, using inputs as dependent variables, with output and the other variables as regressors. Finally, there are several studies which do not neatly fit into any of the first three types. An example is a study by Ling, in which a synthesis of engineering and economic principles leads to the estimation of a rather unusual cost relation.13 Each of these studies will be discussed in turn. Spe- cial attention will be given in the following discussion to the problem addressed, the specification of the model, the treatment of the technologically heterogeneous nature of generating plants, and the level of aggregation at which the study is carried out. Direct Methods The first postwar economic study of thermal electrical generation is the ordinary least squares estimation by Nordin of the relation between the total fuel cost and output as a percentage of capacity.11+ 5H1 observations were made on a single turbogenerator unit in 19u1. Each observation con- sists of the approximate output and fuel input during one eight-hour shift. The functional form estimated was quadratic l38. Ling, Economies of Scale in the Steam-Electric Power Generating Industry (Amsterdam: North-Holland, 196”). luJ. A. Nordin, "Note on a Light Plant's Cost Curves," Econometrica 15 (3), (July, 19u7), 231-235. 10 in output; a cubic equation was also estimated, but it did not significantly change the goodness of fit. Two comments, the first previously pointed out by 15 First, Galatin, should be made regarding this study. Nordin has explicitly considered the instantaneous nature of the generation process, using hourly observations on output as an approximation to instantaneous output, then summing the hourly outputs to approximate the output for an eight-hour shift. Second, the absence of consideration of optimization by the firm was no doubt necessitated by the lack of appropriate econometric techniques in 19u7, the year of publication. However, since the observed turbogenerator was the only one in the plant, there was no possibility of short-run reallocation of labor among machines, as would be possible in a multi-unit plant. Therefore, it seems quite plausible that no short-run optimization by the plant manage- ment was possible. Thus, this study is an example of the appropriate use of a simple estimation technique. Another estimation of the cost function at the plant 2.16 The data consisted level was published by Lomax in 195 of a cross-section of thermal plants in two regions of Great Britain, each plant having operated more than 6600 hours during a one-year period overlapping 19u7-19u8. A log-linear 15M. Galatin, Economies of Scale and Technological Changg énThermal Power Generation (Amsterdam: NorthJHolIand, 1968), 3. 16K. S. Lomax, "Cost Curves for Electricity Generation," Economica 19 (1952), 193-197. 11 relationship was estimated between average working costs (total cost excluding cost of land and equipment) as the dependent variable, and installed capacity and load factor 17 No other functional forms were as independent variables. considered. The estimated results indicate that ceteris paribus, average costs fall as either installed capacity or load factor increases. Assuming that capital is the fixed fac- tor, one may draw the inference that plants in the sample are experiencing decreasing average costs. However, unless one assumes that the prices of variable factors are identi- cal for all plants, one does not know whether the source of decreasing costs lies in factor markets or the production process. Further, even if factor prices are assumed to be identical for all plants, the fact that no attempt was made to stratify observed plants on the basis of embodied tech— nology makes it impossible to tell whether the regression re- sults with respect to installed capacity are phenomena of scale or of technology. Presumably the larger plants employ more modern technology, so that stratification by embodied technology is necessary to determine whether, independently of technology, larger plants generate electricity at lower average cost. 17 . . . . Load factor 18 a measure of intenSity of capital utilization, defined as total actual output for some period divided by the potential output if the turbogenerator had been operated at capacity during the period. 12 Another attempt to estimate cost functions without explicitly considering the implications of optimization was 18 The study consists of time-series esti- made by Johnston. mation of the relation between total working costs and output for 17 "short-run firms"--plants whose installed capacity did not change during the period of observation; and for 23 "long- period firms"--plants whose installed capacity did change during the period. Linear, quadratic, and cubic relations were estimated for all plants, the "best" form being chosen on the basis of "fit," as measured by R2 and R2. In no case did the cubic term significantly improve the fit. Criticisms of this work made by Galatin center on John- ston's supposed confusion about the assumptions necessary to consider the estimated relations as cost functions.19 Johnston reasons that if a plant with variable capacity size over time is assumed to operate on the long-run total cost envelope to the short-run total cost curves, then a regres- sion of cost on output will be an estimate of the long-run cost function for the plant. Galatin claims that this is so only_if ". . . in each year of observation, the firm operates at that output having minimum average cost on the particular short-run total cost function relating to the scale of plant "20 in that year. Since that does not hold, Galatin continues, 18J. Johnston, Statistical Cost Analysis (New York: McGraw-Hill, 1958) un.7u. lgGalatin, 22° cit., 5M-6l. 201bid., 57. 13 ". . . it is difficult to give meaning to Johnston's ."21 Galatin's error here has a history dating 22 results. . back to Jacob Viner in 1931. The necessary condition is in fact that the firm choose its scale of plant so that the given output cannot be produced at lower average cost by using a different scale of plant.23 While the plausibility of this assumption for Johnston's sample is as yet unresolved, this is at least the correct necessary condition. The same question led Galatin to question why Johnston used "a measure of output rather than capacity of plant as the independent variable for estimating capital costs" in a 27 Although Galatin is correct for later part of his study. the short run in stating that capital costs are independent of the level of plant operation, in the long run they are a function of the scale of plant, which is in turn a function of output for a cost-minimizing firm. Thus, output becomes a valid explanatory variable for capital costs. Regressing capital expenditures on installed capacity as Galatin sug- gests would have led Johnston into the error of regressing capacity on itself, except for the effect of variations in the price of capital and in the amount of ancillary equip- ment installed with new turbogenerators. 21Galatin, 22. cit., 57. 22Jacob Viner, "Cost Curves and Supply Curves," Zeit- schrift ffir Nationalekonomie, III (1931), 23-u6. 23See, for example, R. H. Leftwich, The Price System and Resource Allocation, Nth Ed. (Hinsdale, Illinois: Dryden Press, 19707, 161ff. 21‘Galatin, 22. cit., 160. 1n One of Galatin's objections regarding Johnston's work does seem valid, however. Galatin correctly points out that a long-run total cost function is theoretically derived from 25 Johnston's time-series in- a fixed production function. eludes observations on plants containing units of different vintages. The different technologies embodied in these units may change the form of the production function over time. Therefore, it may be difficult to define Johnston's regres- sion as an estimate of the long-run cost function. Stratifi- cation of the plants into technologically homogeneous cells and estimation of a separate function for each cell would alleviate this problem. A more fundamental question arises from Johnston's behavioral assumptions about the firms in his sample. In order to justify single-equation estimation, Johnston points out that since the firms in the study "were all generating electricity . . . at such times and in such quantities as directed by the British Central Electricity Board, . . . they were in no wise . . . adjusting output in the search for ."26 Leaving aside the question of maximum profits. . whether it is reasonable to assume cost-minimization by a firm that is not attempting to maximize profits, the assump- tion of cost minimization usually indicates the presence of a simultaneous equation problem. It may be that Johnston is 25Galatin, 92. cit., 57. 26Johnston, 22. cit., SH. 15 using an ordinary least squares estimator where it may not be appropriate. A second questionable technique is Johnston's reliance on significance tests and goodness-of—fit measures in his choice of functional form.27 As explained in Chapter III.D., this approach to choosing the Specific functional form of a general maintained hypothesis does not lead one to unbiased tests of hypotheses about the parameters of the relationship. A procedure less conducive to pretesting bias is first to choose a functional form using some criterion such as that discussed in Chapter III.D., basing this choice on informa- tion gained from one data sample. Then the parameters of the chosen function may be estimated using data from a second sample, drawn independently of the first. Despite these shortcomings, the equations estimated by Johnston yielded high values of R2. For twelve of the seven- teen "short-period firms," the equation chosen on the basis of'R by Johnston contained only a linear term in output, indi- cating a horizontal marginal cost curve. A significantly positive coefficient for an index of time suggested that such factors as plant obsolescence are shifting the cost function upward over time. Serial correlation statistics calculated for the residuals of six of the estimated equation did not indicate significant non-randomness of the disturbances. 27Johnston, 32. cit., Suff. 16 The cost equations estimated for the 23 "long-period firms" were quite similar to those for the "short-period firms." Again, the values of R2 were greater than .95, and the total cost functions chosen on the basis of R were predominately linear in output. Since the estimated func- tions had significantly positive intercepts, the indication is that long-run average cost decreases over small values of output, approaching 2 mills per kilowatt-hour asymptot- ically. The fourth investigation of influences on the cost of thermal electricity generation without consideration of the problems of optimization was conducted by Iulo.28 In this purely empirical study, the problem was to describe the relationship between average cost for a firm and various "historical," "operating," and "market" characteristics such as the size of producing units, construction cost, and consumption per residential customer. No attempt is made to explain by means of theory the relations observed in Iulo's sample and no production or cost function is esti- mated. Profit-Maximization Models with Factor Substitution The first attempt to study thermal electricity generation in a neoclassical framework was the estimation by Nerlove of . . 9 . . . . a firm's cost function.2 His estimated function was derived 28W. Iulo, Electric Utilities--Costs and Performance (Pullman, Washington: Washington State University, 1961). 29Nerlove, gp. cit. 17 from cost-minimization by a firm generating electricity according to a Cobb-Douglas production function. The study is carried out at the firm level since Nerlove is concerned with the implications of the form of the cost function for the regulation of the firm. No consideration is given to the fact that Nerlove's sample of firms operating in 1955 contains machines of different vintages which are likely to incorporate varying technologies. Thus, it is difficult to separate scale effects from the effects of technological change, particularly since a search of the engineering liter- ature as well as evidence later collected by Dhrymes and Kurz reveals that larger units are likely to be newer and tech- nologically more advanced.30 Nerlove's procedure is to solve simultaneously the three—factor Cobb-Douglas production function and the familiar marginal productivity conditions resulting from cost minimi- zation with fixed factor prices. The reduced form thus derived yields a cost function, linear in the logarithms of output and the factor prices: 1 a1 a2 a3 (1.1) C + K + -—Y + —-—-Pl + ——P2 + _p3 7’ V I" P P I" where C, Y, P1’ P2, and P3 represent logarithms of cost, output, and the prices of labor, capital, and fuel, respec- tively. K is the constant term and ai, i = 1,2,3, is the 30Dhrymes and Kurz, 22° cit. 18 exponent of each of the respective factors in the Cobb- Douglas function r; the returns-to-scale parameter, is equal to the sum of the ai, i = 1,2,3. V is assumed to be distributed stochastic disturbance. In order to incorporate the restriction that the price coefficients must sum to unity, Nerlove divides both sides of the cost function by the price of fuel, pointing out that division by either of the other prices would have been equivalent. This yields one of his models to be estimated, called Model A: 1 a1 a2 (I-2) c-P3 =K+—Y+-——(Pl-P3)+—(P2-P3)+V r r r where all variables have been defined above. In order to avoid data problems associated with the price of capital, Nerlove makes the assumption that the price of capital is the same for all firms, and derives an alterna- tive model to be estimated, Model B: 1 a1 a3 (I-3) C=K'+—Y+—Pl+_P3+v r r r where K' = K + (a2/r) P2. Estimation of Models A and B by ordinary least squares gave generally plausible results, with decreasing long-run average costs indicated. Rather than merely accepting these results, Nerlove went on to plot the regression residuals, and found strong evidence that the relationship is not in fact linear in logarithms. Hypothesizing that this result 19 may arise from an inverse relation between the degree of returns to scale and output, he re-arranged his observations by increasing order of output, divided them into quintiles, and estimated a separate cost function for each quintile. This procedure provided strong empirical evidence for the variable returns to scale hypothesis. In order to allow for this cause of non-linearity of the cost function, Nerlove added a quadratic term in Y and re-estimated Models A and B. The most striking result of this modification was an increase in the estimated returns to scale parameter for firms in the three largest size groups.31 Nerlove's results have two important implications for the present study. The first is the importance of analysis of residuals to the choice of one model from a number of alternatives. Nerlove's original Models A and B met the cri- teria employed by most other investigators of steam-electric generation, namely a high proportion of "explained" variance as measured byR2 or R2, and significant coefficient estimates having the theoretically expected signs. Had he not employed additional criteria concerning the distribution of residuals, he would not have become aware of significant non-linearity in the costeoutput relationship and taken steps to explain it theoretically. In the present study, advantage is taken of new techniques for the analysis of residuals. The second implication of Nerlove's results for the present study is strong empirical evidence of a variable returns to scale 31Nerlove, ER- cit., l8uff. 20 parameter. Since publication of Nerlove's work, a means of parameterizing variable returns to scale as a function of output in a neoclassical production function has been dis- covered by Zellner and Revankar.32 Use of their technique is discussed in Chapter III.A. below. A second approach in the neo-classical tradition to the study of thermal electricity generation is a l96u study by 33 In this article, plants were initially Dhrymes and Kurz. assumed to minimize the cost of producing an exogenously determined output, produced according to a three-factor CES production function. The sample consisted of 362 new plants constructed between 1937 and 1959, each plant being observed only once, in its first year of operation. The observations were divided into 16 cells, according to vintage of construc- tion and size, as measured by the nameplate-rated installed capacity. A separate empirical analysis was carried out for each cell. The first step in the empirical analysis was to estimate the input demand equations resulting from simultaneous solu- tion of the constrained cost-minimization conditions. The explanatory power of the labor equation was quite weak, and the coefficients measuring the sensitivity of the labor input to changes in relative factor prices were for the most part statistically insignificant. Explaining this result by 32Zellner and Revankar, 22. cit. 33Dhrymes and Kurz, EB: cit. 21 engineering features of the production process that allow one to treat the labor input as a constant proportion of output, Dhrymes and Kurz revised their production function to a mixed Leontief-CES production, given by: . 8f 8K i. (I-M) Q = min[g(L), A(afF + aKK ) Y] where Q represents output, and L, F, and K the inputs of labor, capital, and fuel, respectively. Cost is defined by: (I—S) C = pKK + pFF + pLL where pK and pF represent the prices of capital and fuel, respectively. Minimization of cost with respect to K and F under the constraint of the production function (I-u) yields input demand functions. The capital demand function was approximated by: P (I-6) logK = a + a _E + a log Q o 1P 2 F and estimated by ordinary least squares. Then, as the second step of a two-step least squares procedure, the fitted values of log K obtained from estimation of equation (I-6) were sub- stituted into the fuel demand equation, which in turn was estimated by restricted least squares, where the restriction is that aFaK = 1. This procedure gave estimates aF’aK’éF’ and 8K of four of the production function parameters “F’GK’BF and BK. Estimates of the remaining parameters of the production 22 function, A and y, were obtained by applying ordinary least squares to: A+vlogQ (I-7) log Z (D) 03) 7< where Z = 8FF + a K . The results obtained appear quite satisfactory, in the sense that the proportion of variance explained by the regressors in the estimated fuel input equation is high in all cells. With one exception all estimated coefficients are significant, with the theoretically expected signs. Comparison of estimated parameters across cells yields the conclusions that technological progress has operated to in- crease the efficiency of electricity generation and that in- creasing returns to scale is the rule in the industry. For six of the thirteen cells for which the number of observations was sufficient for estimation, the hypothesis that BK = BF was accepted, indicating homogeneity of the production function. Further, for all but one of the remain- ing six cells, a simulation-estimation experiment provided strong indications that the degree of returns to scale tends to fall with increases in plant size, as measured by name- plate rating of installed capacity. Dhrymes and Kurz point out that no probability content may be attached to these results with regard to the degree of returns to scale. Two comments are in order regarding the method used by Dhrymes and Kurz to stratify the plants contained in the 23 sample. A two-way classification was employed, based on technological period, i.e., vintage of the plant, and on size of plant as measured by nameplate-rated capacity.37 In the first place, as is discussed in Chapter II, inspection of a source of information on technical speci- fications of turbogenerators such as the Pgwgg_Modern Plant Design Surveys reveals that even among units of approximately the same size, the technology employed in units constructed during a four-to-five year period is likely to vary consid- 35 Inferences about the impact of technology on the erably. production function based on technologically heterogeneous cells are not likely to be as precise as they would have been had technologically homogeneous cells been employed. In the second place, inferences based on linear regres- sion are usually made under the maintained hypothesis that the dependent variable has a conditional mean which is some linear function of the hypothesized regressors and that its conditional distribution is normal with constant variance over the sample. Use of nameplate-rated capacity as both a criterion for stratification of observations and as the dependent variable in the capital input equation will, in general, cause violation of this maintained hypothesis. The conditional mean of the dependent variable within a cell becomes a function of the intervals chosen to define cells auDhrymes and Kurz, 22. cit., 297. 35See Note 1, p. l. 24 rather than of the hypothesized regressors. Furthermore, the conditional distribution of the dependent variable can- not be normal since it is truncated at the end points of the cell, while the normal density function extends from -~ to +w. Thus, for purposes of inference, an alternative method of stratifying the sample would appear to be preferable.36 Another aspect of the work of Dhrymes and Kurz that relates to the present study is that as mentioned above they, like Nerlove, found a strong empirical suggestion that the degree of returns to scale in steam-electric generation tends to fall as output increases. However, Dhrymes and Kurz were not able to make statements with probability content about this phenomenon. Since the publication by Dhrymes and Kurz, the generalized production function developed by Zellner and Revankar has given one a means of parameterizing variability of returns to scale, so that inferences can be made about its significance. NOn-Substitution Models The first study to make use of a non-substitution model to investigate technological change in steam-electric genera- 37 tion was published by Komiya. His sample of 235 newly- constructed plants was divided into four vintage cells, using 36For a discussion of this issue, see J. B. Ramsey, Tests for Specification Error in Classical Linear Least Squares Re- gression Analysis, unpublished Ph.D. dissertation, University of Wisconsin, 1968. 37R. Komiya, "Technological Progress and the Production Function in the United States Steam Power Industry," Review of Economics and Statistics uuz2 (May, 1962), 156-166. 25 vintage as a proxy for technology. Each of the cells was further subdivided by the type of fuel used--coal or non- coal. Two models were considered and estimated in this study. The first was a Cobb-Douglas type production model, in which the elasticity of substitution between any pair of three inputs--labor, capital, and fuel--is constrained to unity. This model was rejected because the estimated re- gression coefficients were generally insignificant and did not have the 2.2riori expected signs. The conclusion was that the substitution model did not explain the input-output relationship of power generation satisfactorily, although acknowledgement was made of the possible existence of simul- taneous equation problems. Some general criticisms of this use of coefficient estimates as a criterion for the choice between several alternative models are made in Chapter III.D. Factor substitution was dismissed on the grounds of these empirical results, and attention was focused on a "limitational model," which allowed no such substitution. Following Komiya's notation, the following set of input func- tions was estimated: (I-8) log Yf log Af + 8f log X2 (I-9) log Yc log AC + 8 log X1 + uc log X2 C (I-IO) log Y2 log A2 + 82 log Xl + u2 log X2. The variables are defined as follows: Yf is fuel input in Btu's when the unit is operated at capacity. YC is the capital cost of equipment per generating unit, measured in constant dollars. Y2 is the average number of employees per 26 generating unit, X the average size of generating unit in l _ megawatts, and X2 the number of units in the plant. If 8c = 82 = 1 and "c = u2 = 0, the model becomes one of fixed proportions. Using dummy variables, various versions of this model were estimated, some allowing the slope and intercept para- meters to vary across the cells in the sample, others incor- porating various restrictions on these parameters across cells. Komiya's regression results indicate that with respect to both fuel and labor, economies of scale are important. The results also indicate a long-run trend of substitution of capital for labor. Several remarks regarding Komiya's results are in order. First, as explained above in the discussion of the work by Dhrymes and Kurz, vintage is a rather poor proxy for tech- nology since at any given period, design engineers may be experimenting with several different technologies. Second, measuring the capital input by capital cost per unit as Komiya did, it seems difficult to impute the "scale effect" with respect to capital entirely to improved efficiency of the productive process. Quantity discounts on turbogenera- tor units would also give rise to a decrease in per-unit capital costs as the number of machines increased. Third, rejection of the possibility of factor substitu- tion on the basis of the poor regression results obtained from the Cobb-Douglas model seems a bit premature. Such results could be accounted for by simultaneous equation 27 problems, as suggested by Komiya, or by the restriction of the pair-wise elasticity of substitution to unity implicit in the use of the Cobb-Douglas function. Furthermore, Komiya's inclusion of each plant only once, when new, does not allow one to observe long-run capital-fuel substitution in a single plant, in the form of installation of technologically more advanced machines that make more efficient use of fuel. For these reasons, Komiya's results must be viewed with some reservations. These problems could be corrected by making the following procedural changes: an alternative classifica- tion scheme that more precisely reflected technological change; an alternative measure of capital input; experimentation with alternative functional forms that allow factor substitution at non-unitary elasticities of substitution; and incorporation of successive observations on plants in the sample. Another attempt to examine changes in productivity in steam—electric generation was made by Barzel.38 He began by stating two difficulties with estimated production functions that cast doubt on their validity in studies of technological change over time. First, one does not usually know the func- tional form relating inputs to output and is ordinarily compelled to derive it from the same set of data used for purposes of inference. Second, one does not know whether or not shifts in the production function are neutral. Barzel 38Y. Barzel, "Productivity in the Electric Power Industry, 1929-1955," Review of Economics and Statistics an (2), (May 1962), 156-166. 28 alone among previous investigators seems to have been aware of the first problem; but unfortunately, he did not have available statistical techniques to solve it correctly. Instead, he derived an index of productivity change between year 1 and year 2, so that he might examine its path over time. The index was derived under the following assumptions: constant returns to scale, perfect competitiOn in all markets, and constant marginal products of all inputs over the two years being compared. Barzel then tested the appropriateness of these assumptions. His first test of the hypothesis of constant returns to scale was estimation of a Cobb-Douglas relation between out- put and the three inputs, labor, capital, and fuel. The sum of the estimated factor exponents was significantly less than unity indicating decreasing returns to scale. Barzel ignored this result as a "statistical twist" and attempted to measure economies of scale using the following approach. Defining productivity as output per unit of cost, size as the installed generating capacity of each plant, and load factor as the number of Kwh produced during 1959 per in- stalled kw capacity, Barzel estimated a log-linear relation using productivity as the dependent variable, with size and load factor as regressors. This relation may be written: Q B (I—ll) _ AK°(LF) , c where Q, C, K, and LF represent output, total cost, capital, and load factor, respectively. Barzel did not complete the 29 stochastic specification, but one may assume a multiplica- tive lognormally distributed random disturbance, eui. Barzel's results from estimation of equation (I-11) on a 1959 cross-section of plants that first operated between 1953 and 1955 were interpreted to mean that a 10% increase in plant size causes a 1.09% increase in productivity; while a 10% increase in load factor causes a 3.73% increase in productivity. Applying these cross-section estimates to time-series observations between 1929 and 1955 on produc- tivity as defined by Barzel, installed capacity, and load factor, Barzel accounts for the 2.83-fold increase in pro- ductivity over the period as the product of four factors: 1.98, due to the increase in quantity of electricity pur- chased annually per customer; 1.23, due to increased average plant size; 1.18, due to higher average load factor; and 1.32, the residual, which Barzel attributes to technological prog- ress during the period. Several doubts arise when one considers Barzel's methodology carefully. First, load factor is used to mean output per installed kilowatt of capacity, so that the product of load factor and capacity is output. Since the right-hand side of equation (I-ll) estimated by Barzel involves that product, one sees that both sides of the equa- tion are functions of output. Thus, the explanatory power of equation (I-ll) would not appear to be great. Second, application of the 1959 estimates of the para- meters of equation (I-11) to time-series data on the variables 30 in equation (I-ll) is implausible since the true values of the parameters are likely to have varied as technology changed between 1939 and 1955. As explained in Chapter II of the present study, many of the technological innovations observed during the postwar period have operated in the direction of making larger-scale plants technologically feasible. Thus, Barzel's assumption of a stable relation between productivity and scale over more than a quarter-century seems a bit rash. By ignoring the fact that technological change has brought about the use of larger-scale plants, Barzel may have ser- iously underestimated its impact. In a second paper Barzel attempted to consider the effect of scale of plant on the quantities used of each of the fac- 39 Assuming tors of production in steam-electric generation. output to be a Cobb-Douglas type function of the inputs, utilization factor, and size of unit, Barzel showed that this formulation led to an identity relation between dependent and independent variables in the function--the identity whose existence he ignored in equation (I-ll), which was used in his previous paper. Therefore, he rejected all models incor- porating an explicit production function in favor of models examining the demand for factors. The form selected for estimation of the fuel input func- tion is given by: 39Y. Barzel, "The Production Function and Technical Change in the Steam-Power Industry," Journal of Political Economy 72 (April, 1969). 31 (I-l2) log yf = [bi log Xi, i = 0, . . ., 19, l where yf represents fuel input and the independent variables are defined as follows: unity plant size measured by the number of installed kilowatts anticipated average load of plant, measured by the observed load factor in first full year of operation a within-plant index of x2 over time anticipated average input price ratio measured as the average price of fuel per Btu to the average labor cost per man-year in the first year of opera- tion of the plant a within plant index of x” over time age of plant defined as accumulated number of hours of operation x define a set of dummy variables, where x has a value of 10 for plants that began operation in 19u3 and 1 for other years, and the other x. are similarly defined. The reader will note that the logarithmic transformation converts these binary variables into the familiar values of l and 0. Equation (I-12) was estimated using data on 220 plants first operated between 19u1 and 1959. Coefficients of x1, x2, and x3 are .896, .898, and .893, respectively, all sig- nificantly different from unity. These results indicate that fuel input rises less than proportionately to changes in those variables. Coefficients of the factor price ratio variables are also significant, with the theoretically expec- ted signs. Coefficients of the year dummies, which represent 32 the effect of technological change if all other variables affecting demand for fuel are included in equation (I—12), indicate sporadic decreases in fuel requirements over the period of the sample. Labor input was also used in another model, with a vector of independent variables identical to that of equa- tion (I-12). Although serious potential measurement errors are acknowledged in the values of labor input, size, anti- cipated load factor, and the load index coefficients, the results indicate similar but more pronounced scale and load effects than were evident in the fuel equation. Results similar to those of the fuel equation were obtained with respect to the other independent variables. An input equation for capital, as measured by the undeflated value of plant and equipment, was also estimated. The estimated coefficients indicated the following: a size effect similar to that obtained in the other regressions, substitutability of capital for the other two factor prices, and a slight positive relation between capital input and load factor. Shifts indicated by the coefficients of the dummy variables indicate a rapid increase in capital input until 1951, with no clear pattern thereafter. Two comments seem appropriate regarding Barzel's meth- odology. First, rejection of all explicit production func- tion models because the modified Cobb-Douglas function produced an identity seems a bit premature. Although the 33 particular function used by Barzel produced an identity, other familiar production functions are not subject to that problem. Second, there is evidence to the effect that newer, more advanced technology makes possible the use of larger 90 Thus, rather than allowing technological scales of plant. change to shift only the intercepts of the input functions, one would be interested in exploring the interaction between technological change and the scale effect. Along the same lines, in order to separate scale and load effects from the effects of technology, one would wish to see whether the coefficients of installed capacity size and load factor vary across technologically homogeneous cells. Finally, one would be interested in seeing whether input functions of the Barzel type give evidence of variable returns to scale similar to that found by Nerlove and by Dhrymes and Kurz. Another approach to the problem of identifying the effects of economies of scale and technological change in ”1 His steam-electric generation was employed by Galatin. discussion emphasized the problems caused by aggregation of instantaneous data on individual turbogenerator units into annual data on entire plants when production is studied at the plant level. He began by developing a model of a multi- unit plant, for which the objective is to minimize the cost ”OLing, 22. cit., p. 20. l”Galatin, 22. cit. 314 of generating some exogenously determined level of output. Economies of scale and technological change were defined in terms of this model. Unit heat rate, a measure of efficiency defined as fuel input to a turbogenerator unit in Btu's per kilowatt of out- put was assumed to be a function of the capacity of the unit and the degree of capacity utilization. For empirical use, Galatin developed a measure of capacity utilization, PF*, which is essentially the published plant factor, corrected for the fact that a given machine may not operate "hot and connected to load"72 at all times during the year. A similar correction to unit heat rate was also made. Galatin went on to point out that the instantaneous production period for electricity may be approximated by w finite intervals of length t, say one hour, while data are usually collected over longer intervals denoted by T, e.g., one year. Choosing a Cobb-Douglas function as an example, Galatin showed that when a t-period stochastic relation is aggregated into T-period intervals, ordinary least squares estimation is generally not appropriate for estimation of the T-period relation. However, under certain assumptions, among them that all units within a given plant are the same size, ordinary least squares is appropriate for estimation of the following pair of alternative functions: "2i.e., actually generating electrcity for transmission to customers. (I-13) a. * ’1 + + it a(PF )it + BXi Y V. K it - . -l -l (I 1n) alt a(PF*)it + 8(XiK) + y + v. , it where ait is Galatin's corrected unit heat rate, PF* is the corrected plant factor, and XiK is the nameplate rating of the i-th turbogenerator unit.Ll3 These models were estimated using a sample of 812 obser- vations on 158 different plants in which all units were of the same capacity. The sample data were divided into cells by unit vintage. Equation (I-1u) was chosen as the "better fit," presumably on the basis of R2, significance of estimated coefficients, and consistency of the estimated relationship with g 2riori expectations. Estimates 3 and 8 of coefficients a and B were positive in all cells, indicating that increases in either plant factor or unit size lead to decreases in cor- rected heat rate. This result was taken as evidence of economies of scale; comparison of parameter estimates across cells provided evidence of decreasing unit heat rates due to improved technology. To investigate the effects of scale and technological change on the capital input, Galatin assumed capital cost per turbogenerator unit to be a polynomial function of the number of units in the plant and the capacity of the units, given by: C C 2 T E 3 (I-lS) l—— or __ = a N + a N + a N + BX , N N 1 2 3 K ”3Ga1atin, 22. cit., 103. 36 where CT is total capital cost of a plant including land, structures, and equipment; CE is total equipment cost, N the number of machines, and X the size of each machine. K The results of estimation suggested that cost per machine falls for plants containing one to three machines and rises thereafter; not surprisingly, per-unit capital cost increased with the size of unit. Galatin acknowledged that unlike the fuel input equation, which he considered a pure g§_22§£ pro— duction function, the capital input equation combined effects from changes in the production process with effects from changes in the price of capital. Finally, a function was estimated relating labor input to the number and size of machines, as well as intensity of machine use. The estimated function was: (I-16) L = a + BIN = yXK + 6PF* where L is the average number of employees of the plant, PF* is the adjusted plant factor discussed above, and the other variables are as defined for equation (I-lS). The estimated results indicated that labor input increases with respect to all three independent variables; however, the rate of increase is less than proportional to increases in the number of mach- ines, indicating the presence of economies of scale. Further- more, comparison of the coefficients across cells indicated that technological innovation has accounted for at least part of a significant reduction in labor requirements over the period covered in this study. 37 Galatin's methodology gives rise to some doubts about his results. His recognition of the problems inherent in aggregation of hourly data into annual data is commendable. However, his concentration on that one type of mis-specifi- cation caused him to ignore other problems in model construc- tion. For instance, although Galatin presents a model of the multi-unit plant in his Chapter II, that model provides no ,explicit justification in terms of either the objectives of the plant managers or the technology of the industry for his maintained hypothesis in Chapter III that average fuel input is related to both corrected plant factor and size of machine.”” This hypothesis is the basis for the estimation of equations (I-l3) and (I-lu). Similarly, no theoretical justification is presented for the estimated labor and capital input functions. Even if the maintained hypothesis stated by Galatin in his Chapter III is accepted, the choice of the specific functional form of the relation is undetermined. Equations (I-l3) and (I-19) are constructed so as to avoid aggregation problems; but no consideration is given to alternative func- tional forms that might satisfy that requirement. Further- more, no attempt is made to determine on grounds other than goodness of fit and significance of the coefficients whether either of these is in fact an appropriate form of the u“Galatin, 22. cit. 35. 38 relation. The reader is referred to the discussion above of the paper by Nerlove, in which analysis of the residuals led to modifications of a model which gave satisfactory results on the same grounds used by Galatin, and to Chapter III.D. below, in which the deficiencies of Galatin's choice of procedure are explained. Another question raised by Galatin's specification of the model to be estimated arises from his exclusion from the sample of plants which consisted of units of several differ- ent capacities. This restriction was made to ensure homo- skedasticity, but no test was made to determine whether the disturbances of equations (I-13) and (I-lu) as estimated were in fact homoskedastic or whether estimation with an unrestricted sample would have yielded homoskedastic dis- turbances. A criticism of the capital input equation (I-15), originally made by Somermeyer, is discussed by Galatin in ”5 Somermeyer pointed out that since no restric- an appendix. tions were placed on the coefficients, the estimated equation implied negative average capital expenditures for some fairly common combinations of number and size of machines. A func- tional form was suggested that would avoid this problem. Finally, the comments made above in the discussion of the Dhrymes and Kurz article regarding the use of vintage as a usGalatin, 22. cit., 139-191. 39 proxy for technology are also applicable to the same procedure as used by Galatin. In another study, following similar methodology to that of Komiya, Hart and Chawla recently made a comparison of pro- duction functions for steam-electric generation between Britain and the U. 8.76 Under the assumption that two non-substituta- ble factors are used in the generation of electricity, the production function is given by: (I-l7) Y = min{K1Xl,K2X2}, where Y denotes the output of electricity, Xl the input of capital, and X the input of fuel. 2 This formulation assumes constant returns to scale and allows for the possibility that one of the factors may not be fully used. However, Hart and Chawla wish to relax the assumption of constant returns to scale. Furthermore, their measure of capital is corrected for unused capacity, and they assume that the fuel input is fully used. Therefore, they restate the production function as a pair of equations, one for each factor: (I-18) Y = tixg, i = 1, 2. The interpretation of this formulation is that the expansion path of the firm is a ray through the origin with slope Kl/Kz, and that the function is homogeneous of degree 8. P. E. Hart and R. K. Chawla, "An International Compari- son of Production Functions: The Coal-Fired Electricity Generating Industry," Economica 37 (May, 1970), lea-177. 140 This model may be termed a "fixed-relative-factor-proportions" model. Hart and Chawla explain that if both capital and fuel are fully used as the firm proceeds along its expansion path, the estimates of B from both component equations of the model should be equal. If the estimates are significantly differ- ent, ". . . either . . . the . . . production function . . . is inappropriate, or . . . there is a multiplicative measure— ment error, Presumably in X1, which takes the value XEl-b2," where b1 and b2 are the estimates of B in equation (I-18) obtained when i = 1 and 2, respectively.‘47 An example of this type of measurement error, which would make (bl-b2) positive, would be failure to correct completely for inac- curacies in the measurement of capital for unused capacity. Having explained equation (I-18), Hart and Chawla then decide to make their work comparable with that of Komiya. Toward this end, they assume output to be the exogenous variable and rewrite the equation as: (I-l9) xi = x§Y3*, i = 1, 2. They then proceed to estimate equation (I-19). Two problems, one notational and the other substantive, arise in the transformation from equation (I-l8) to (I-l9). Both are overlooked by Hart and Chawla. First, identifying the parameters of equation (I-19) by asterisks, it is clear 1 that .3 = tit? and 3* = %~for i = 1, 2. This should have been pointed out to avoid confusion. 1”Hart and Chawla, 22. cit., 171. u1 Second, Hart and Chawla ignore a substantive problem because their discussion does not include the stochastic specification of their model. If one assumes that equation (I-18) is characterized by a multiplicative disturbance term eui, where the ui are i.i.d. N(@,021), then the transforma- tion indicated by equation (I-19) contains a multiplicative disturbance term evi, where vi = ui/B. Obviously, the vi 2 are i.i.d. N(O,EEI). But if in fact the vi are i.i.d. B N(6,021), then conversely the u. are i.i.d. N(¢,8202I). Hart 1 and Chawla apparently implicitly assume the distributions of both ui and vi to be N(O,02I). Since the hypothesis tests employed by Hart and Chawla require certain distributional assumptions about the disturbances, this change in distribu- tional properties caused by the transformation from equation (I-18) to (I-19) should be made clear. Incorporating the stochastic specifications of equations (I-18) and (I-19), the equations may be written: (I-18)' Y. = r-XB euij, i = 1, 2; j = 1, . . ., n J l ij (I-19)' x.. = ntY?*eVij i = 1 2- ° = 1 n where j 13 13 s as] coo-99 refers to the observed plant, uij’ Vij are each i.i.d. N(0,02I), and other terms are defined as above. The data used by Hart and Chawla were from the five-year period 1959 to 1963 with data from each country being divided into two groups--"old vintage," installed before 195k; and "new vintage," installed during or after that date. N2 Cross-sectional regressions were run for each of the five years for each group of plants. Estimates of the scale parameter 8 were uniformly less than unity for all regres- sions, indicating increasing returns to scale. For the U. K. the new vintage plants were shown to have higher returns to scale than the old. The opposite result, obtained for the U. 8. sample, was attributed to measurement errors. While old vintage American plants appeared to be more effi- cient than their British counterparts, no significant cor- responding difference was noted for new vintage plants. The study might have been improved by the use of available information on technical characteristics of installed generating equipment in order to stratify the plants in the sample. As was pointed out in discussions of other previous studies, technological homogeneity within cells can be guaranteed with more certainty in this way. By do doing, one would know whether the observed differences in efficiency between British and American old vintage plants were due to British delays in implementing new tech- nological advances or to some other factor causing ineffi- ciency in British plants of comparable design to the American plants. Other Approaches A substantially different approach to the problem of economies of scale at the firm level in steam-electric #3 generation was employed by Ling.”8 Rejecting a purely econometric approach, Ling used an "analytical" approach, designed to incorporate the principles of economic opti- mization within the constraints known to engineers. Con- cerning himself with economies of scale at the firm level, Ling started by stating and demonstrating, with frequent reference to the engineering literature, three principles that govern economic considerations:“9 (1) The forced outage (breakdown) rate of generating units is independent of size, so that the use of larger generating units implies a larger required reserve capacity to keep the probability of system failure constant. (2) Larger generating units can be installed at a lower cost per kilowatt of capacity than smaller ones. (3) Larger generating units are necessary to use effectively the more advanced steam conditions (primarily higher temperatures and pressures), which result in lower fuel cost due to lower heat rates, usually at higher investment cost. Furthermore, reference was made by Ling to the results of Hegetschweiler and Bartlett on fractional load performance uaLing, 22. cit. ”gIbid., 20. nu which indicate decreasing heat rate as unit load factor 50 Heat rate is the ratio of fuel input (Btu) increases. to output (mw). After stating these assumptions, Ling postulated a hypothetical generating plant. Values of parameters such as heat rate, probability of turbine failure, etc., were chosen for the hypothetical plant so as to be plausible from an engineer's point of view. Under these assumptions, Ling simulated long-run plant expansion according to an econo- mically efficient plan first derived by Kirchmayer, g£_§$,51 Under this plan, the first unit added to a plant facing increasing demand for its output should have a capacity equal to 10% of the initial plant capacity. Subsequent additional units should have capacities equal to successively smaller proportions of existing capacity until the 7% level is reached. The capacity of each unit added after this point should have a capacity equal to 7% of the existing capacity lnefore the addition. Using load duration curves to explain efficient alloca- tion of load among existing units in the short run, Ling calculated the value of average cost for the hypothetical 50H. Hegetschweiler and R. L. Bartlett, "Predicting Per- formance of Large Steam Turbine-Generator Units for Central Stations," ASME Transactions 79 (1957) 1085. 51L. K. Kirchmayer, A. G. Mellor, J. F. O'Hara and J. R. Stevenson, "An Investigation of the Economic Size of Steam- Electric Generating Units," AIEE Transactions 7n (III), (1955), 600-609. NS firm before any additions, assuming efficient allocation of load. A correction was made to the cost data to allow for the effect of different intensities of capacity utilization, as measured by the plant factor. To generate a long-run cost function from this short-run function, Ling calculated average cost for his hypothetical firm after each successive addition made according to the economically efficient plan just ex- plained. This calculation was made for each value of installed capacity, under various assumed values of the system load factor. The final step in the derivation of an analytical long- run cost function was the choice of a functional form to fit the data generated according to the above process. A Cobb- Douglas type average cost function was rejected due to indications of non-linearity and unsatisfactory marginal cost functions derived from the function.52 Instead, the following non-linear form was adopted: (1-19) ca = kSnnm + P 1“ n where Ca represents annual average generating cost in mills per kilowatt-hour, n represents system load factor, S system installed capacity, and k, n, m, and p parameters to be estimated. Fitting equation (I-l9) to the data computed from the analytical model yielded a value for R of 0.999, and indicated average generating cost to be a decreasing function of system capacity and system load factor. 52Ling, 22. cit., H7. us As the final step in his analysis, Ling fitted equation (I-19) to a sample of data on four large independent steam-gen- erating stations, the data having been collected by the Federal Power Commission between 1938 and 1958. When applied to real data, the non-linear form given by equation (I-19) yielded only a slight improvement over the Cobb-Douglas form, in terms of the value of'R-2 and significance of the coefficients. Both functions performed quite well, given various data limitations cited by Ling. Among the limitations was a very narrow range of variation in the value of system load factor, making inferences based on the estimated coef- ficient of that variable suspect. A problem cited by Ling points up what may be a sig- nificant oversight on his part. All the assumptions about efficiency of components, forced outage rate, etc., under- lying the analytical model are themselves based on some implicitly postulated state of technology, while the tech- nology represented in the data varied as new plants were 53 One would be added during the period of the sample. interested in seeing whether stratification to hold the state of technology constant would change the empirical results. Although on the basis of his data, Ling's conclusion that the effect of technological advance on efficiency has been con- tinuous and gradual seems plausible, any conclusion about the relative importance to efficiency of technological change 53Ling, 92. cit. 28. ”7 r O I I O O J” and economies of scale must await such stratification. However, with these reservations, Ling appears to have inte- grated successfully a number of engineering and economic principles. This concludes the review of previous studies of the steam- electric generation process. It should be remembered by the reader that most of the criticism of the prior work centered on the implications of the methods used to classify plants or units on the basis of some proxy for technology, the attention paid to the stochastic specifications of the models, and the extent to which possible alternative specifi- cations were considered. It was also pointed out that in the only two studies that considered the possibility of variable returns to scale over various outputs, strong indi- cations thereof were found. However, since previous authors had no means of representing this variability as a production function parameter, no statistical inferences concerning this phenomenon could be made. It is the purpose of the present study to use some recent theoretical developments to solve some of these difficulties. 5L*Ling, 22. cit., 71. Chapter II. Considerations Related To Postwar Technological Change In Chapter I.B. the point was made that proxies used by most investigators for technological change in steam- electric generation had serious deficiencies. Specifically, it was explained that the use of nameplate-rated capacity by Dhrymes and Kurz led to pre-testing bias. Further, the use of machine vintage, it was claimed, did not reflect accurately the pattern of technological change in the industry because at any given time several different technologies are being used in newly constructed plants. Thus, heterogeneous gener- ating equipment may be lumped together within a vintage cell while similar equipment may be treated as being different because of an arbitrary decision based on date of construction. The present chapter is written in the belief that in order to make meaningful statements about the economic effects of technological change in this industry, one must identify the specific technological changes that have actually affected the physical process of generating electricity. Section A. of this chapter is devoted to a brief summary of the genera- tion process and to a consideration of the major innovations in generating equipment during the period of this study, l9u8 to 1965. Based on this discussion, a classification of generating plants intended to group together technologically homogeneous plants is proposed in Section B. 98 II.A. The Productive Process In Electrical Generation The process of fossil-fueled electricity generation is conveniently broken down into three stages: fuel combustion, which takes place in the furnace; steam generation, which is the function of the boiler and several other heat transfer devices that improve efficiency; and electricity generation, which is accomplished by the turbogenerator unit. The output of each prior stage is the input to the next stage. Specifically, fuel is the input to the furnace, which produces heat. The heat is transferred to the boiler, where it is used to convert water into steam. The steam is con- ducted to the turbine, where its expansion causes the turbine to rotate. The work output of the rotating turbine shaft is used to run an alternating current generator. The steam used in the turbine is condensed and re-used in the boiler. The process is described in Figure 1: Figure l.--Schematic Diagram of Steam- Electric Generating Plant Fuel i Steam Work 1. BOILER TURBINE ‘JNGENERATOR Heat URNA E F C Water Steam \\‘1CONDBNSER ‘ \7 Electricity H9 50 Each of the three productive stages--production of heat, production of steam, and generation of electricity--will be discussed in order before turning to an examination of the entire system. The Furnace The functions of a furnace may be listed as follows: to provide heat by burning fuel, to transfer the heat to part of the water and steam by radiation, and to facilitate the proper circulation of water and steam within the boiler.1 Conventional furnace fuels are vaporized oil, vaporized natural gas, and either crushed or pulverized coal.2 Modern furnaces may burn all three, either separately or simul- taneously, allowing firms to take advantage of fluctuations in relative prices and qualities of the fuels. Other fac- tors which affect the choice of fuel are handling cost, cost of disposal of the by-products of combustion, operating labor required, and maintenance of the fuel-processing equipment and furnace.3 Gas, oil, and pulverized coal are burned "in suspension"; that is, air and atomized fuel are mixed and ignited in a burner and blown into the furnace, where the mixture burns 1A. H. Zerban and E. P. Nye, Power Plants (Scranton: International Textbook, 1956), 122. 2Pulverized coal has been ground to a dust-like consis- tency. Crushed coal is in the form of small chunks. Atomic energy is not considered in the present study. 3B. G. A. Skrotski and w. A. Vopat, Power Station Engineering and Economy (New York: McGraw-Hill, 1960), 25ff. suspended in mid-air by large fans. Crushed coal is ordi- narily burned lying on a grate called a fuel bed, as air is blown through it to increase the combustion rate and to carry the heat to transfer surfaces, where the heat raises the temperature of the boiler water. In either case, a technologically efficient furnace pro- vides conditions for continuous combustion by keeping the fuel in constant contact with air after it is ignited and by providing conditions suitable for complete combustion. The latter function requires time and space for the carbon to burn completely in suspension so that the colder surfaces surrounding the furnace don't chill the mixture below kin- dling temperature, thereby ending combustion. Artificially induced air turbulence within the furnace increases the available time for combustion, improves fuel- air mixing, and keeps fuel and air passing each other at high speeds, so that combustion products are swept away and unburned carbon exposed.u Therefore, induced turbulence in the furnace chamber reduces air pollution and improves plant efficiency since less available heat is lost in the form of unburned carbon, which either goes up the smokestack as flyash or collects and hardens on the sides of the furnace chamber as slag. The Steam Generator The function of the steam generator is to provide for 1“B. G. A. Skrotski, Electric Generation--Steam Stations (New York: McGraw-Hill, 1956), 88ff. 52 efficient vaporization of water. At one time the only essential component of a steam generator was a drum, in which vaporization took place. But since about 1900, the more complex watertube boiler has been standard for cen- 5 tral-station plants. This type contains a large number of water circuits of the type illustrated in Figure 2. Figure 2--Schematic Diagram of a Watertube Boiler ;:}v~rg, Steam Water-Steam 9‘: jig/Z45; Drum M1Xture 2%2%21’-——--—Feedwater Riser ’ A Water HotrGases,:::::a' Downcomer om f Furnace z”/~ Feedwater enters the boiler drum below water level and mixes with the water already circulating in the unit. The heat transferred from the furnace to the riser at the left in Fig— ure 2 forms steam bubbles in the riser, which are of course less dense than the water. Therefore, upon re-entering the drum, the steam collects at the top, from where it can be drawn off at the throttle. Since the mixture in the riser is less dense than the water in the downcomer, gravity induces continuous circulation.6 The moisture content of steam decreases as the temperature at which water-to-steam conversion takes place increases. This 5Zerban and Nye, 22. cit., 160. 6Skrotski, 22. cit., 112. 53 ' in turn temperature, called the "saturation temperature,’ increases as the pressure at conversion increases. Moisture content in steam greater than 12% increases the likelihood of the formation of droplets in the steam at the low-pressure end of the turbine. These droplets cause erosion of the turbine blades. In order not to exceed the 12% limit, steam produced in the boiler may be still further pressurized and heated in a device called a "superheater" before entering the turbine. Superheaters became standard equipment for 7 steam generators before l9u8, the first year of this study. The Turbggenerator Once the steam has been produced and pressurized, it is used in the turbine to provide work output to the generator. Since the generator is usually mounted directly on the turbine shaft, the turbine and generator are considered a single unit, the "turbogenerator." Allowing the steam to flow through a nozzle to a lower pressure in the turbine converts part of the internal energy of the steam into kinetic jet energy. Within a large turbine, this process is repeated at multiple stages of nozzles so that the turbine shaft is driven at several points. Of course, due to the expansion of the steam at previous stages, the pressure at each stage is lower than at the preceding stage. Since steam flows from high- to low-pressure locations in the turbine, efficiency is increased by lowering the steam pressure at the exhaust, known as backpressure. Backpressure 7W. J. Kearton, Steam Turbine Theory and Practice (London: Pitman and Sons, 19317 21. 59 can be reduced either by increasing the area of the exhaust annulus or by increasing the surface area of the condenser, a device which uses cold water from an external source to condense the steam into reusable water.8 The purpose of condensation is to remove as much internal energy as possi- ble from the steam.9 Each turbine in a plant drives an a-c generator, which actually produces electricity for transmission to consumers. Unfortunately, generation of electricity produces heat as a by-product in the generator. Since the necessary electrical insulation is also effective thermal insulation, this heat must be removed artificially. Consequently, all large gener- ators have some sort of cooling system for the windings. Steam Cycles With an understanding of the operation of the individual components of a generating unit, one can discuss the theory of Operation of a complete unit. Using this theory to define efficiency in terms of steam cycles, one is then able to assess the contribution to efficiency of various technological innovations. As an introduction to the theory of steam-electric generation, several energy forms should be defined. Kinetic energy, due to velocity, and potential energy, due to ‘ 8W. A. Wilson and L. G. Malouf, "An Approach to the IAXDnomic Problem of Matching Condenser Surface with Exhaust- IhUiulus Area," ASME Transactions 1956, 73, 135. 9Skrotski and Vopat, 22. cit., 272. 55 position, are well known. Internal heat energy of steam is the amount of heat that has been transferred from the fur- nace to the steam in the boiler. Another energy form, which exists only so long as a flow is present, is called flow energy, and is evaluated as the product of the pressure and volume of a vapor at a given temperature. The sum of flow energy and internal heat energy is defined as enthalpy (h) and is measured in British thermal units per pound (Btu/1b.). Thus, the enthalpy of a given volume of steam may be increased by raising either its temperature or its pressure. The simplest operational power cycle using vapor as the working medium is the Rankine power cycle, which is illustra- ted in Figure 3: Figure 3--Graphic Representation of the Rankine Steam Cycle 1.; \ T 3 Work to Generator Boiler . {—Condenser l Pum s Lid Work efficiency is defined as the ratio of net output of work to input of work. In the case of transformation of energy from heat to work, cycle efficiency Et is defined by: (II-l) E : work OUtPUt - work input t heat input ' 56 Due to the "law of conservation of energy," the work output of the turbine is (h3 - h”), where hi represents enthalpy at point i in Figure 3. Since work input to the boiler is (h2 - hl) and heat input is (h3 - h2), one may write: (h3 - hu) - (h2 - hl) (II-2) Et — h3“hz which is a fundamental definition in thermodynamics.10 Prior to 1920, metallurgical conditions limited steam temperatures and pressures to about 600°F and 250 pounds per square inch (psi). As mentioned above, water droplets, which are likely to form from such low-temperature steam when pressure drops over the turbine stages, will damage turbine blades. One method of preventing this is known as the "reheat cycle." In this cycle, instead of passing the steam straight through all stages of the turbine, it is bled off at an intermediate stage, before the pressure drop is sufficient to cause the formation of droplets. The steam is removed, reheated, and returned to the turbine at the fol- lowing stage. The process of reheating, in addition to raising the maximum tolerable pressure, increases thermal effi- ciency because the steam is reintroduced at a higher enthalpy than that at which it was removed.11 10This discussion is summarized from several basic engi- neering texts. See Zerban and Nye, 22.‘2£:., Ch. 2, Skrotski and Vopat, o . 223,, Ch. 3, Skrotski, 22. cit., Ch. 3, and C. M. Leonard, undamentals of Thermodynamics_TEhglewood Cliffs: Prentice-Hall, 1958). llSkrotski and Vopat, 22. cit., 62ff. 57 Attempts to improve efficiency by raising the average temperature of the medium in the cycle led to "regenerative preheating" of boiler feedwater. In a regenerative cycle steam is bled from the turbine at an intermediate stage as in the reheat cycle; but instead of being reheated, the steam is used to heat feedwater in a feedwater heater. Several regenerative cycles may be combined in a single boiler-turbogenerator system.12 The combined reheat-regenerative cycle is described in Figure u, using three regenerative preheaters as an example.13 Figure H--Graphic Representation of The Reheat4Regenerat1ve Steam Cycle Turbine dork to Generator High Pressure Heat Stage Sup- Heat f ’Rejection rom Furnace Each unit marked P in Figure u is a feedwater pump. Each unit designated R is a regenerative preheater. The num- bers represent points at which enthalpy is different from its value at the immediately preceding point. lzskrotski and Vopat, 22. cit., 64767. 13Figure u and the discussion following were adopted from Skrotski and Vopat, Ibid., 67ff. 58 Using the thermal cycle illustrated in Figure u, one notes that the formula for thermal efficiency of such a cycle is: E _ (h8-hg)+(1-m1)(hg-h10)+(1-ml-m2)(hlo-hll) t - EB-h7 (II—3) (l-m1-m2-m3)(hll-h12) - AW + h8_h7 : 2, and m3 are the masses of steam removed at each bleedpoint; and AW is the total energy demanded by all where ml, m auxiliary power needs. Equation (II-3) requires some elaboration. Heuris- tically, (h8-h9) represents the work done in the high-pressure section of the turbine by one unit of steam. Then if the mass ml of that steam is removed at point 9, (l-ml) of it loses enthalpy in the amount (hG-hIO)’ the work output of one unit of steam between points 9 and 10. Similarly, (l-mlem2) units of steam produces work equal to (hlo-hll) per unit, and so on. Obviously, additional regenerative cycles add positive terms of the form (1-ml-...) (hk-hk+l) to the numerator, increasing Bt' Each successive term is smaller than the one preceding it since an additional value of mi is being sub- tracted from the multiplier. In other words, regenerative preheaters exhibit diminishing marginal productivity. Both reheat and regenerative cycles were well known by the beginning of the present survey period, 19N8-l965, and regenerative preheaters were standard equipment by l9u8. 59 However, according to Power magazine's annual Modern Plant Design Survey, temperatures and pressures were not gener- ally high enough to justify the additional cost of installing reheat cycles until the early 1950's, at which time they, too, became standard.lq II.B. Postwar Innovations In Steam-Electric Generation The rather tedious discussion just concluded is a necessary prerequisite to understanding the importance of various innovations that were introduced into steam-electric generation during the period of this study, 1998-1965. This understanding, in turn, will form the basis for the classi- fication of turbogenerator units into technologically homo- geneous cells. With these considerations in mind, the discussion now turns to the postwar innovations, their con- tribution to efficiency, and their economic implications. Innovations in Combustion Two new types of coal furnaces appeared during the period of the present study, l9u8-1965: the cyclone furnace and the pressurized furnace. In order to burn lower-grade coal, which produces more flyash, plants in some areas built shortly after World War II 15 were equipped with cyclone furnaces. These receive crushed 1”'See Chapter I, note 1, of this manuscript. 15Skrotski and Vopat, 22. cit., inn. 50 coal in a whirling stream of high-velocity air, which throws the coal to the rim of the burner by centrifugal force, where the high-speed air promotes a high combustion rate. The ash melts to form slag, which drains into a secondary furnace carrying burning coal particles with it. Combustion is continued in the secondary furnace so that the 16 An additional amount of flyash is substantially reduced. advantage of the cyclone furnace is that the coal crusher requires less auxiliary power than the pulverizer used in suspension furnaces. Even though cyclone furnaces require additional forced-draft fans, it is often less costly to substitute expenditure on such capital equipment for expen- diture on low-grade fuel, even when the cost of auxiliary 17 That is, the power to run the fans is taken into account. potential use of such equipment exemplifies the possibility of capital-fuel substitution in a generating plant at the design stage. During the late l9u0's the American Gas and Electric System was developing the pressurized furnace, in which com- bustion and heat transfer take place under no to 50 psi pres- sure, provided by forced-draft fans. These units require special airtight gaskets surrounding the burning chamber. However, units with such gaskets can also be used in a con- ventional induced-draft operation. 16Skrotski, 22. cit, 105ff. l7Skrotski and Vopat, 22. cit., luuff. 61 The main advantages of the pressurized furnace are: reduced air leakage, which improves both efficiency and con- trol of the furnace; a reduction in heat loss, because of turbulence; and improved safety, because no adjustment is needed during start-up and low-load periods. Its disad- vantages include difficulty in locating leaks in the inner shell and limited access for manual slag removal. Further- more, since auxiliary indueed-draft fans are usually required, the initial cost is greater than that of conventional furnaces. After the first pressurized unit was placed in operation in lane, there was a "wait-and-see" period of four to five years during which ordinary pulverized coal and cyclone furnaces were used extensively, before pressurized furnaces gained general acceptance for large plants. Table 1 shows the post- war trend in furnace design for plants under construction, as reported in Power magazine's annual Modern Plant Design Sur- veys of 1998, 1957 and 1966. Table 1 Coal Furnace Type Planned in Samples of Plants Under Construction1 113 19118 1957 1955 Pulverized coal, induced draft 100% 92% 2u% Cyclone 0 8% 0 Pressurized 0 0 76% 18Entries were calculated from data in the 19u8, 1957, and 1966 Modern Plant Design Surveys of Power Magazine (New York: McGraw-Hill, monthly, 19u8-l965). 62 The data in Table l are consistent with the hypothesis that rising fuel costs led to use of the more efficient cy- clone furnace, which was itself replaced in large plants by the still more efficient pressurized furnace. Innovations in Steam Generation The most remarkable improvements in the efficiency of electricity generation have resulted from innovations in steam generators. Innovations have taken place in the tech- nologies of metallurgy, welding, and feedwater pumps, all of which aided engineers in their attempts to raise design temperatures and pressures. The effect of higher design temperatures and pressures on cycle efficiency may be discussed in the context of equa- tion (II-2). It will be recalled that heat input during the cycle is represented by (h3 - h2) in that equation. It may be shown that for any increase 5 in (h3 - h2), (h3‘hu)-(h2-hl) (h3+E-hu)-(h2"hl) ”C(hu7'hl) h3—h2 - h3+e-h2 = (hBJhé77+e(h3-h2) . Since h3 > h2 for a positive input of heat during the cycle, and since h is non-negative throughout the cycle, this dif- ference must be negative. Therefore, at constant values of h“, h2, and hl’ a larger input of heat (h3 - h2) during the cycle decreases cycle efficiency, as defined by equation (II-2) One method of decreasing the necessary heat input during the cycle, thereby increasing cycle efficiency, is to raise the average steam temperature over the entire cycle. 63 This may be accomplished by means of superheating and/or raising boiler pressure, both of which make more efficient use of a given heat input (h3 - h2). Consequently, as improved welding and metallurgical techniques and more powerful feedwater pumps permitted, engineers steadily raised design temperatures.and pressures between l9u8 and 1957.19 For example, the proportion of central—station plants under construction designed for operation above 1000°F/1500 psi rose from 2% in 19u7 to 52% in 1958.20 With a furnace of given efficiency, the decrease in heat input for a given work output due to the higher design pressure made possible by metallurgical progress implies a decrease in fuel input for the given work output. These technical advances are clear examples of capital-fuel substi-_ tution. Empirical analysis of this substitution is simplified by the fact that the joint effect of the improvements in welding, metallurgy, and feedwater pumps is embodied in higher design temperatures and pressures. Therefore, in lieu of detailed separate studies of these three technologies, one can summarize their effects in terms of temperature and pres- sure. Another example of capital-fuel substitution is the addition of regenerative preheaters to turbogenerators. Over 19Skrotski and Vopat, 22. cit., 60ff. 20See the 19k? and 19u8 Modern Plant Designs of Power magazine, 22. cit. 64 the period of the present study, 19u8 to 1965, the average number of regenerative preheaters per unit increased steadily, as shown in Table 2. In engineering terminology, the addition of preheaters lowers the heat rate of a turbo- generator unit, i.e., it lowers the fuel input required for a given output. Table 2 Number of Turbine Bleedpoints for Samples of Units Under Construction No. Bleedpoints Percent of Plants Under Construction 19u8 1957 1966 0 1 1 0 l 8 1 0 2 8 2 3 3 18 6 0 u 38 13 6 5 20 30 6 6 5 20 26 7 2 21 3M 8 0 6 22 11 0 0 3 100% 100% 100% This particular form of capital-fuel substitution appears particularly well suited to large plants. This becomes appar- ent when one notes that in 1966 the five units under construc- tion with less than six bleedpoints were the five smallest units, in terms of generator rating in kilowatts. Incidentally, the data in Table 2 offer strong evidence for the contention made in Chapter I.B. that in any given year new plants embody 21Power, 22. cit. 65 various technologies, so that vintage of unit is not an ade- quate proxy for technology. As mentioned above, the use of higher steam pressures required thicker boiler drum walls made of stronger alloys. Furthermore, units had to be taller to make the gravitational force of falling water sufficient to maintain circulation. These rising capital costs, together with ever increasing fuel costs, spurred the Ohio Power Company to announce, in 1953, plans for the world's first steam generator to produce steam in its supercritical state.22 Stated simply, conventional sub-critical boilers heat water under pressure and circulate it through tubes to the boiler drum where the lower pressure allows the water to flash to steam. On the other hand, supercritical generators heat the water at a pressure above 3206 psi, the critical pressure, so that at 705.N°F water vaporizes without boiling. Since the water never flashes to steam, supercritical units eliminate the cost of a drum while taking advantage of the greater efficiency afforded by the higher pressures.23 Immediately after plans for Ohio's Philo supercritical unit were announced, four other supercritical units were ordered by other firms. But after this initial flurry other power companies waited for the feedback of experience before 22C. R. Earle, "Power Engineers Take Long Step to usoo- psi, 1150-F Steam," Power Engineering 58 (1), (Jan. 195M), 58-61. 23 Skrotski and Vopat, 22. cit., 391ff. 66 building supercritical units, so that in 1957, the year the first five began operation, no others were under construction.27 However, by 1966, 26% of all units under construction incor- 25 which has been called porated this technological advance, "the most widely and rapidly accepted innovation in the steam power plant design in the long history of technological development in the industry."26 Just as increasing pressures had compelled the introduc- tion of the reheat cycle before World War II, so supercritical pressures demanded the introduction of double-reheat cycles to protect turbine blades. In 1966 all supercritical plants under construction incorporated double reheat and up to eleven regenerative cycles. Innovations in Generator Cooling The standard procedure for cooling generators at the beginning of the survey period was to force air or hydrogen through the generator passages in order to absorb the heat losses. In a conventional air-cooled system, the air passed over the tube and fin surfaces of a cooler before returning to the generator passages to be recirculated. However, more efficient means of cooling generators have been developed, the relative advantages and disadvantages of which are 21+J. Tillinghast, R. H. Pechstein, and w. s. Morgan, "Design and Operating Experience with a Supercritical Pres- sure Unit--Part 1," Power Engineering 70 (3), (March 1966), 58.61. 25Power Modern Plant Design Survey, 1966. 26Tillinghast, Pechstein, and Morgan, 22. cit., 58. 67 briefly discussed in the next few paragraphs. Since hydrogen is so much less dense than air, energy losses due to friction of the generator rotor against the gas in which it is turning, known as "windage losses," are only 10% as high in hydrogen as they are in air. In addi- tion, the higher thermal conductivity of hydrogen raises the level of potential output for a given amount of heat loss. Finally, insulation is much more durable when it is not in contact with oxygen.27 Thus, in l9u8 half the units under construction listed in the Power survey had hydrogen-cooled generators; ten years later, hydrogen was the planned medium for all units under construction. The fact that hydrogen was not used as a cooling agent before the end of World War II may be due to wartime rationing of hydrogen. Prior to 1952, the cooling system design required that heat from the conductor pass through the electrical insula- tion before reaching the cooling medium.28 Since that year many new generators have been built with hollow conductors, so that the heat is transferred directly from the metal to the cooling medium. Hollow-conductor cooling is so much more efficient than tube-and-fin cooling that the upper limit to maximum generating, which used to be set by thermal stress is now determined purely by mechanical efficiency con- siderations. 27Skrotski, 22. cit., 351. 28Ibid., 351-352. 68 Since the pumping load for a given amount of heat loss is much less for a liquid than for a gas, so that the heat transfer rate is much higher, modern large plants have begun using liquid hollow-conductor cooling systems.29 Table 3 does not indicate the full strength of the trend toward liquid hollow-conductor cooling for all plants unless one realizes that the new cooling methods completely predominate among the largest units under construction. Table 3 Generator Cooling Method in Sample of Units Under Construction Type 19H8 1957 1966 Hydrogen 69% 100% 58% Air 31 0 0 Liquid Hydrogen 0 0 27 Other Liquid 0 0 3 Liquid Hydrogen, Hollow Conductor 0 0 12 Summary of Innovations in Plant Design From the discussion just concluded, it is apparent that power plant design between l9u8 and 1965 was a dynamic pro- cess with frequent changes in the state of technology. Be- cause of these changes, it is to be expected that newer plants are more efficient than older ones. In furnace design, the stoker furnace common in the 1990's was replaced during the 1950's by pressurized and cyclone furnaces designed to burn fuel more efficiently. Metallurgical advances 29Skrotski, 22. cit., 352. 69 allowed rather steady increases in design temperatures and pressures so that the 900 psi — 900°F boiler of 19u8 was superseded by giant 2u00 psi - 1100°F boilers a decade later. These in turn became obsolete for large plants when the boiler was replaced by the supercritical steam generator, which reduced capital costs as it provided high-temperature, high—pressure steam which is conducive to efficient turbine Operation. In turbine technology, the higher temperatures and pressures were accompanied by increases in the number of reheat and regenerative cycles included in new turbogenera- tors. As pointed out above in the discussion of steam cycles, these improve efficiency by raising the average steam tem- perature over the whole generating cycle. Finally, as the old forced-draft air cooling systems for generators were replaced with hollow-conductor systems, often with hydrogen as the cooling medium, a given-size turbine was enabled to drive increasingly higher rated generators. The Methodology of Innovation in Plant Design Considering these examples of technological innovation, several comments about the nature of technological change in this industry seem appropriate. The approach of plant design engineers appears to be inductive and experimental rather than deductive. For example, it was pointed out above that after supercritical steam generators were first intro- duced, there was a period of several years during which engineers could observe supercritical units in operation 70 before the method gained widespread acceptance. Even after that time, according to the surveys of modern plant design conducted by Power magazine during the period, there was extensive experimentation with pressures in the neighborhood of 5000 psi. These pressures have been largely abandoned, with 3500 psi being the most common supercritical pressure at present. Examination of the engineering literature provides additional evidence for the contention that power plant design is an inductive, experimental science. The literature is replete with examples of case studies giving detailed descriptions of innovations incorporated in new units and evaluations of their success.30 On the other hand, even the few studies that might be termed theoretical in nature rely on polynomial approximations to empirically observed non- linear relations rather than on functional forms deduced from theoretical relations. This is due in part to the experimental nature of technological change in steam-electric generation.31 However, it should also be noted that the use of steam at high and supercritical pressures is not relevant for low-capacity generating units. This becomes evident from 30See for example: "Etiwanda, Study in Station Economy," Power 96 (6), (June 1952), 100ff.; W. Greacen, III, "Reheat Unit Extends Goudey Capacity," Power 97 (2), (Feb. 1953), 71- 75; "Power Takes You Through New Centra Costs Plant," Power 96 (2), (Feb. 1952), 75-79. 31See for example, C. W. Elston and P. H. Knowlton, Jr., "Comparative Efficiencies of Central—Station Reheat and 71 the study of Power magazine's annual survey of modern plant design for any year; the explanation is discussed by Ling 32 For these reasons the discussion and in Chapter IV below. of innovations just concluded will form the basis of the stratification method for turbogenerator units to be explained in Chapter II.C. Hopefully, this method will prove superior to the vintage-size proxies used by previous authors to represent technological change. II.C. The Stratification Of Generating Plants The discussion of postwar innovations just concluded emphasized the contribution of each development to the tech- nical efficiency of the generating process. By relating these innovations to observable design characteristics of newly-installed turbogenerators, one can construct a classi- fication system defining technologically homogeneous cells for turbogenerator units. An entire plant may be said to belong to a cell if all its units belong to that cell. Classes of units were defined with respect to the fol- lowing physical characteristics: temperature and pressure at the turbine throttle, furnace type, number of bleedpoints for Non-Reheat Steam Turbing-Generator Units," ASME Transactions 7a (1952), 1389; H. Hegetschweiler and R. L. Bartléft, 22. cit.; G. B. Warren and P. H. Knowlton, Jr., "Relative Engine iciencies Realizable from Large Modern Steam Turbing- Generator Units," ASME Transactions 63 (l9ul), 125. 32 Ling, 22. cit., 20. 72 regenerative feedwater heaters, number of reheat cycles, and generator cooling apparatus. The definitions were made with the following considerations in mind. First, insofar as possible, each change in any of these characteristics that might be expected to produce an observable improvement in plant efficiency should be recognized in the cell definitions. Second, where a large majority of plants incorporating a given innovation are similar with respect to all other design characteristics, units incorporating the innovation but dif- fering with respect to the other characteristics should not be included in any cell. Third, since rates of progress in the various components of the generating process are not equal, the cell definitions should not preclude the possi- bility of overlapping with respect to some characteristics. For example 6 or 7 bleedpoints were installed on many turbines using 1800-2050 psi steam, and on many in the 2&00-3206 psi range. Therefore, if those pressure ranges are the most important differentiating characteristics between two cells, the acceptable bleedpoint ranges for both cells should include 6 and 7. Fourth, insofar as possible the cell division should provide a number of observations within each cell sufficient for empirical analysis. The classification resulting from these considerations is presented in Table H. The units in Cell I were constructed with generators rated in the no to 80 megawatt (mw) range, for the most part. ms oomcnsm oonwpdwwosm I mmopm Huoo onaepo>asa . on noposocoo soaaom - on: Hao no\occ new I o so\e o vwsqu u HA cowoncxm I N: oaoao>o I >0 ”waonazm cum .Na\wm Leased o so\w w .Hq . m \Hma mONm NIH m1: .wmopm .om ma HHH> moooaanoooa o so\w w cum \ana ocNmuooHN H $10 .so .nnosa .oa Ha H> moooaanOOOH o so\w w Na .Nm \Hoa oonaccHN a ens .so .awona .on No > N hoomoanoooa o so\w w m \aoa emONnecsH H s-m .so .unosa .om cAN >H N mooooa o so\w w m \ann cmmauemaa H s-m .nnosa .oa sea HHH N mooooauoom m .sa< \awo aaaauoee H-c sue names .so me HH woommnoom o po\m w Nm .na< \ana easucee e ale .oa .noxopm NHN H mamaooo assumpomaoa moaozu Hmonom mucwoacooam Hos» can mcowvu>aomno Haoo ROthocum whammonm mo amass: mo soAEdz maze oomchsm mo aonssz messam wcwumnoaoo mo :owvmowmwmmMHu now : maAMB waaoo mo cowuwcwwoo 71+ These are the lowest ratings represented in the data sample for the present study. The majority of the Cell I units in the sample were constructed between 19u8 and 1950 although some units of this type were being installed through the mid-1950's. It is anticipated that these units will be among the least efficient in the sample. The units placed in Cell II differ from those in Cell I primarily in their use of cyclone and pressurized furnaces although some were designed to operate at slightly higher temperatures and pressures. These units were common in plants constructed between l9u8 and 1952. The units in Cells III and IV were the most common type installed during the early 1950's although their construction continued to some extent throughout the decade. The two cells differ from each other primarily in design pressure. The nameplate rating of attached generators varied between HO and 125 mw in Cell III, and between 100 and 160 mw in Cell IV. It will be noted that the higher design temperatures and pres- sures of these units made additional regenerative preheaters efficient, as shown by the larger number of bleedpoints. In éujdition, it can be seen from Table u that the higher tempera- ttires and pressures also justified the additional cost of a lPeeheat cycle. Cells V and VI represent the bulk of generating equipment itIstalled in central-station plants during the late 1950's and eEarly 1960's. The wide range of generator ratings for these 75 units, 130 to 570 mw, indicates a high degree of versatility. They differ from units in the lower-numbered cells primarily in having higher design temperatures and pressures. They differ from each other solely in the type of generator cooling system installed. Cell VII includes all supercritical units constructed between the year 1957, when the first such unit began opera- tions, and the year 1965. The generator range in this class is 600 to 850 mw. All these units incorporated pressurized furnaces and at least four bleedpoints. Problems of turbine pitting due to condensation of the steam made use of at least one reheat cycle mandatory and designers of a number of these units experimented with a second one. Liquid hydrogen was introduced as a generator cooling medium in some of the units and all such generators employed hollow conductors. Unfortunately, very few new plants were built before 1965 containing only supercritical units. Since a plant is included in a cell only if all units in the plant are in the same cell, only a limited sample of data on supercritical units is available. The reader will note from the discussion of the pre- Ceading few paragraphs that there is a great deal of over- léipping of technologies across both time and generator Péitings. Thus, it becomes apparent that the common practice <31? grouping units by either or both of these two criteria 76 conceals great disparities in technology among members of a given group. Specific examples of this practice are cited in Chapter I.B. Chapter III. Theoretical Structure and Methodology In Chapter I, the problem being considered was stated to be the separation and quantification of the effects of tech- nological change and returns to scale in steam—electric genera- ‘tion. Previous approaches to the problem were also discussed :in that chapter. It was argued that those studies lacked a ssuitable proxy for technological change, and that insufficient czonsideration was given to alternative specifications of the IDPOdUCtion model for electricity generation, particularly to sst>ecifications permitting variable returns to scale in the ggeeneration process. In Chapter II, an alternative classifi- czéiticn system for power plants was proposed, with the objective c>1F’ defining technologically homogeneous cells. The focus of this chapter is on the specification of 13T1€2 production model for steam-electric generation. In Part A, summaries of theoretical results of Zellner, Kmenta, and l The IDI‘EEJZe (ZKD) and of Zellner and Revankar are presented. former study is a production function model in which ordinary let‘El-sst:squares estimates of the parameters of a Cobb-Douglas pr’OCiuction function are shown to be consistent. The latter .________ 733-55 1A. Zellner, J. Kmenta, and J. Dreze, 22. cit., 78H- ;. A. Zellner and N. S. Revankar, 22. cit., 2M1-250. 77 78 derives a method of generalizing a neoclassical production function with given values of the elasticity of substitution and returns-to-scale parameters to a function with the same elasticity of substitution, but with returns to scale a pre- specified function of output. In Part B, the model will be applied to two generalized production functions (GPF's), one a generalization of an n-factor Cobb-Douglas function, the other a generalization of an n-factor CES function. In Part C, specific models to be considered on the basis of the results of Part B will be presented along with some specific rnodels of previous investigators which were discussed in (Shapter I.B. Estimation techniques for the models will be (discussed. In Part D. a methodology for testing three 113(potheses regarding steam-electric generation is presented. lira the next chapter, this method is applied to the models 1):?tesented in Part C of this chapter. III.A. Recent Results in Production Theory This section introduces two recent developments in pro- dL1<2tion theory, both of which are relevant to the present E’tTLlcly. First, the ZKD production model of the firm is intro- ‘311<:=€3d and compared to the traditional model of Marschak and Andbews.2 Second, a brief explanation will be given of the I‘C>'t:li.on of GPF's as developed by Zellner and Revanker. ______2___ E<1 2J. Marschak and W. J. Andrews, "Random Simultaneous (:1::;*Eltions and the Theory of Production," Econometrica 12 14-H), 1u3-205. 79 Alternative Models of the Firm Following the notation of Marschak and Andrews, assume that the i-th firm produces output according to a two—factor Cobb-Douglas production function given by: “1 a2 . (III-l) Y1 = AXliXQi, 1 = 1,. . ., n, where Y, X1, and X2 represent flows of output, labor, and capital, respectively, per unit of time. Profit, denoted by n, is defined by: (III-2) n = pY - lel - w2X2, vahere p, wl, and w2 represent the prices of output, labor, 51nd capital, respectively. Conditions (III-3) and (III-H) are necessary for maxi- nij.zation of profits by the i-th firm: 3fl° (Ila-w 1'=0, ax1i Sui CIIIAU zm 3X2i SJLITILJltaneous solution of equations (III-3), (III-u), and (III—1), and the addition of error terms to be justified below, yields the following system of equations: (113::I:-5) log Yi - oi log Xli - a2 log X2i = A0 + VOi’ (III—.6) log Yi - log Xli = A1 + vli’ (133E.3EZ-—7) log Yi - log X2i = 12 + v2i, i == w :1, . . .n, where A0 = log A, A1 = log , and P91 x w 23 ==: log . Each of Voi, vli’ and V2i is a stochastic P92 80 disturbance term assumed to be i.i.d. normal with zero mean and equal variance for all firms.3 Each disturbance is the sum of two random variables, the first of which depends on "the firm's 'economic efficiency' and on the degree of com- petition between its customers, or workers, or creditors"; while the second "is assumed to be the combined effect of a larger number of causes (not necessarily independent of each other), of which none has an influence considerably surpassing the influence of each of the other causes."u The reduced form of equations (III-5), (III-6), and (III-7) is given by: (III-8) log Yi = , -(l-a1-a2) A + v - - A (l-a ) - a A0 (III-9) 10g xli : 0 01 l 1 2 L) -(1-al-a2) A +V0i - a A - (l-a )A (III-10) log x2i = O l 1 l 2. -(l-al-a2) Examination of (III—9) and (III-10) shows immediately that 't}1<3 factor inputs Xli and X2i are functions of VOi’ the pro- du<2‘trzion function disturbance. Consequently, ordinary least Squ-Etres estimates of the production function will, in general, \ tea 3Irving Hoch, in "Simultaneous Equation Bias in the Con- 3<712 of the Cobb-Douglas Production Function," Econometrica §:: 1958), 566-578, multiplied the antilogs of A1 and A by irlzr‘éalrneters R and R to allow for the possibility that irms Sal_t:"srlne sample may efihibit systematic errors with respect to :1-=sfying the first-order conditions. ”Marschak and Andrews, 93. cit., 157 and 155. 81 This is the well-known problem "5 be biased and inconsistent. of "simultaneous equation bias. However, it has been shown by Zellner, Kmenta, and Dreze (ZKD) that under certain conditions, ordinary least squares estimates of two-factor Cobb-Douglas production function parameters are consistent. Their production model involves the assumption that output is a stochastic function of the inputs, the disturbance being generated by such factors as, "weather, unpredictable variations in machine or labor per- Consequently, profit is stochastic, ZKD 7 :fOrmance, and so on." 51nd firms are assumed to maximize its expected value. LlSed the following method to prove that these assumptions ianly that ordinary least squares estimates of the Cobb- I)c>uglas parameters are consistent. Following the ZKD notation, the production function is written: a a u - _ l 2 01 (III-ll) Yi - AxliXZie Where the u0i are assumed to be normally distributed with zero nleeéaau and constant variance; other variables are as defined above. Expected profit is defined by: a a 000 2 7 + -w Xl-w2X2, _ ‘I' . (III-.12) Eu.) = p*B-wlxl--v.w‘;x2 - p AXllx2 e H‘+ \ 5 . ( See, for example, A. S. Goldberger, Econometric Theory, New York: Wiley and Sons, 1961+), 288—290. 6Zellner, Kmenta, and Dreze, op. cit. 7Ibid., 787. 82 where 000 is the variance of the production function dis- turbance, and +'s indicate the expected values of the prices of output and the inputs. Necessary conditions for the maximization of expected profit are given by: (III-l3) EELEL = o, BXli 8E(n) 3X (III-1H) = 0. 2i Solving equations (III-l3), (III—1H), and the production function equation (III-ll) simultaneously, the reduced form equations of this model are: [co-a k a k+(1- -a -a )uO “a u --a u .] (III-15) log Yi l 1 2k 2 l 2 0i 1 ll 2 21 , (l-al-O‘Z) [GO-(02-1)kl-a2k2+(a2-1)uli-a u -] 2 21 (III-15) log Xli , (l-al-aZ) [“0+“1k1'(“1 1)k2 “1“11‘(“1 1)u2i] ( 1:11-17) log x2i , (1'01-02) ulfiere kl = log (wl ) - “00 and k2 = log (33—) - 0””. uli and Pal 2 P02 2 L1:2;L are stochastic disturbances resulting from deviations of fa . . . . + <2tor prices p, wl, and w2 from their ant1c1pated values p , + 2 clear that Xli and x2i are independent of uOi’ the production + “'JL 3 and w From equations (III-16) and (III-l7), it is function disturbance. Consequently, under the ZKD assumptions, Eilil 0 for 0 E f < w. Under these conditions, as pointed out by Zellner and Revankar, equation (III-20) has all the properties of a neoclassical production function. 12Dhrymes and Kurz, op. cit., 308. 86 It should be noted that if equation (III-19) is a production function, then equation (III-20) is not itself a production function but a transformation of one since the units in which V is measured will not in general be those in which Y is measured. The particular form of the transformation function g depends upon the nature of the pre-specified relation between the degree of returns to scale and the value of V. Speci- fically, Zellner and Revankar prove the following theorem for two inputs: "Let f(L,K) be a neoclassical production function homogeneous of degree af, a constant. Then a production function [sic], V = g(f), with preassigned returns to scale function (V), can be obtained by solving the following dif- ferential equations: (III-21) dV - V a(V) "13 They point out that generalization of this theorem to 1” Some cases involving more than two inputs is direct. specific returns-to-scale functions and their corresponding GPF's are discussed in the next section of this chapter. III.B. Generalizations of the ZKD Results In this section, the ZKD model is applied to generalized production functions (GPF's) based on n-factor Cobb-Douglas and CES functions. The result that ordinary least squares l3Zellner and Revankar, op. cit., 2H2. luIbid., Note 2, 2u1. 87 estimates of the parameters of the function being gen- eralized are consistent is shown to hold for these cases. Non-generalized n-factor Cobb-Douglas and CES functions are cited as special cases of these generalizations. The Generalized n-factor Cobb-Douglas Function Following the approach of ZKD, assume that a plant is generating electricity under a production function of the form:15 n . (III-22) Yi : A H X -e , i : l, o o o, m, r=l where Yi represents output of the i-th plant; X r = l, ri’ . . ., n, represents the rate of use of the r-th input for the i-th plant, and the u0i are i.i.d. normal stochastic disturbances having zero mean. Further, assume that the prices of output and of the n inputs, denoted by p, wl, . . ., w respectively, are distributed independent of the n, production function disturbance and have expected values p+, + + . wl, . . ., wn, respect1vely. In the general context of GPF's an alternative expres- sion for equation (III-22) is given by: n a u . -1 _ P 01 (III-23) g (Vi) - Arthrie , where Vi is a transformation ofothowrate of output as definedfl w by-Zellner and Revankar, and g'l(-) is the inverse of a trans- formation function having the properties listed immediately 15Zellner, Kmenta, and Dreze, op. cit., 786-789. 88 after equation (III-20). The expected value of profit is defined by E(n), where: n (III-2H) E(n) = p+E(Y)- 2 w;xr. r=l Since the production function disturbance u is nor- 0i mally distributed, the expected value of output is given by: n a 0' (III-25) E(Y) = Etg‘1(V)] = A n ere r=l 00 2 a where 000 is the variance of the production function dis- turbance uOi' A proof of this result is given in Parzen.16 A necessary condition for maximization of E(n) is given by: (III-26) M = 0, I‘ :I 1, o o o, no 3X r Two distinct types of stochastic disturbances will keep conditions (III—26) from being fulfilled exactly. The first, denoted by ugi, is said by ZKD to result from managerial errors. The second, denoted by u+ is attributed to devia- ri’ tions between expected and realized prices. Assume that dis— turbances of the latter type are randomly distributed over plants so that: + + (III-27) log (3%.) = log (32) + uri. p p 16E. Parzen, Modern Probability Theory and its Applica- tions (New York: Wiley, 1960) 3u8. 89 Then, substituting equation (III-25) into equation (III-2Q), and differentiating with respect to Xri’ con- dition (III-26) implies n equations of the form: °00 . “7" (III-28) 35‘") - paryle w - o - u.-I‘-° axri Xrie Oi Rearranging terms, taking logarithms, recalling equation (III-27), and using the definition Yi E g’1(Vi), one obtains n equations of the form: (III-29) Gi-Xri : kri+uri+uOi, I‘ = l, o o o, n, wr 0DO “51’ 2 where Gi = log [g'l(Vi)], Xri = log xri’ and kri = log Further, taking logarithms of both sides of equation (III-23), one obtains: n (III-30) G- - Z a X where k0i = log A. In matrix notation, the system consisting of the (n+1) equations (III-29) and (III-30) may be written: (III-31) Ax = k + Bu, where (1 A ' A ll 12 ° A = [' ' f " ']’ A11 ‘ In’ A12 ' ' A21 , A22 i (nXI), A21 = [Oman-1° . .02al](lxn)’ A22 = l, 90 r--x . l 'k . l L ' ‘ ru . L I'll I'll ' I'll - . ' xn-l,1 kn-1,i In .1 ' un-l,l O O '0 . x = . K = . B = '. U = t . O '0 . ' O — . . ' x21 k2]. _____ 'i— _ Ll 2i I -Xli kli 0 o o 0 0'1 u 1i 7 Gi kOi . L J’ L ’ L , a Luoi J . Applying a well-known formula, one may invert the parti- tioned matrix A, under the conditions that All and A22 are non-singular, as in the present case.17 The solution for X is given by: (III-32) x = A'lK + A-lsu. The elements of A"1 are not of importance to the result being derived. A‘lB may be partitioned as: r t l I i i In , ”(nXI) A’lsz ' - ---------------- '- - — - - an _ an-l o o 0- a1 ' l n n n ' L l- 2 a l- 2 a l- 2 a ' J r= P r=l P r=l r ' The (nXl) null vector in A-lB implies that the x l, ri’ P = . . ., n, are independent of the uOi’ the production function 17See, for example, C. G. Cullen, Matrices and Linear Transformations (Reading, Mass.: AddisonJWESIey,’I966), HO. 91 disturbances. Therefore, ordinary least squares estimators for the parameters of the model given in equation (III-22) are unbiased and consistent. For the special case in which Vi s g(Yi) = Yi, the derivation just completed proves the same result for an n-factor Cobb-Douglas production func- tion which has not been generalizes according to the Zellner and Revankar procedure. The Generalized n-factor CES Function Generalization of the Hodges results to a generalized n-factor CES production function is similar to the deriva- tion just completed.18 To begin, assume that the i-th plant is generating electricity according to the following pro- duction function: n ‘3. (III-33) Yi = A( z arxgi)peuo1, r=l where Yi and Xri’ r = l, . . ., n, have been defined above, and ar, r=l, . . ., n, A, v, and p are the distribution, efficiency, scale, and substitution parameters, respectively. It should be noted that an implicit assumption in this ver- sion of the generalized CES function is that the elasticity of substitution between pairs of inputs is equal for all pairs of inputs. This restriction is made to simplify esti- mation. In the context of GPF's, equation (III-33) may be 18Hodges, op. cit. 92 re-written as: -l o a: u0i GII-3H) g (Vi) = A(P:laPXPi)e , where Vi is a generalized production function as defined by Zellner and Revankar, and g'l(-) is the inverse of a trans- formation function having the properties listed immediately after equation (III-20). Under the ZKD assumptions regarding prices of the output and of the n inputs, the expected value of profit E(w) is defined by equation (III-2H). A necessary condition for the maximization of E(w) is given by equation (III-26). Since the disturbance term u0i in the production func- tion equation (III-33) is assumed to be normally distributed, the expected value of output is given by: (III-35) E(Y) = EEg'1(V)] = A( z arxgi)fie 2 = AZpe 2 r= l where 000 is the variance of the production function distur- n _ o bance, and Z - 2 arxri. r=l Substituting equation (III-3H) into equation (III-2H) and applying condition (III-26) for maximization of E(w), one obtains n equations of the form: -l v_ , v (III-36) 3 (Vi) F'lx p-1 : w;oA(FT2) euoi pi “ - 006 p+a vs 2 I" 93 These equations are first-order conditions for maximization of E(w) by the i-th firm with respect to each of the n fac- tors. The reader will note that before substitution, equa- tion (III-3H) was divided through by euOi so that both sides of equation (III-36) would be non-stochastic. + :‘: Applying the definitions of uri’ uP i’ and u given ri above and taking logarithms of both sides of equation (III-36), the conditions for maximization of E(u) imply n input demand functions of the form: V -1 _. ’ V (III-37) (3-1)ln[g (Vi)]+(D-l)lnxri - Cri+(;-l)u0i+uri’ I . = n where cPl l V wppA<3‘?)]_ 000 2 pvar So far, the derivation has proceeded in a manner precisely analogous to the derivation of the equation system (III-29) in the proof of the ZKD result for the n-factor Cobb-Douglas production function. However, the non—linear nature of the CES function necessitates a different approach to the solution for the input demand equations. The n equations represented by (III-37) plus the production function given by equation (III-3H) form a system of (n+1) equations which may be solved simultaneously in the following manner for the (n+1) unknowns Xri’ r = 1, . . ., n, and Yi' Any pair of the n input equations (III-37), e.g., the r-th and the s-th, ris, may be solved simultaneously for XSi 9a in terms of Xri' In general, (III-38) xsi = e (p-l) xri Equation (III-38) may be used to express all inputs except the r-th in terms of the r-th. In this way equation (III-3H) may be re-written so that no input except Xr. appears in the 1 production function, which is now given by: (Cri'csifuri'usii’ 2.u . (III-39) g'l(Vi) = A{xgi[ar+ 2 use ‘Tb-l) ]}°e 01. s¢r An equivalent logarithmic transformation of the production function is given by: _ '1 . : (III 39a) ln[g (V1)] lnA+vlnXPi ;,_ ’. ._ . v (Cri C81+uP1 uSl)p +31n[ar+ z e (0-1) ]+u sir 0i° Substituting equation (III-39a) into equation (III-37), one obtains the following solution for Iani: X p X -1 ’ (III-HO) 1nxPi = [(1.§)v+h:0. defined by equation (III-20). cf is the degree of homogeneity of f, the neoclassical production function to be generalized; a and h are constant parameters. Inspection of equation (III-H6) reveals that returns to scale are (af+h) when V = 0 and approach the limit (cf-h) as V approaches infinity. To apply Zellner and Revankar's procedure for defining generalized production functions (GPF's), one substitutes equation (III-MB) into the differential equation (III-21). Solution of the resulting equation defines the following function: 21Zellner and Revankar, op. cit., 2uu and 2u6ff. 100 1 2r (III-Q7) YI:F[(1+r)a+(l-r)Y]1'r2 = kf where k is a constant of integration and r = 2.. For the CD 3 “f and CBS models, af = Z a As Zellner and Revankar point out, i=1 testing the hypothesis that r = 0 is a test of the hypothesis i. that returns to scale are independent of output. Applying this transformation to the three-factor version of the stochastic Cobb—Douglas function presented above, one has: .3— 2? a a O. u . GII-ue) Yi+P[(1+r)a+(1-r~)Y.]1'P2 = FK.1F.2L-3e 01, 1 1 1 l where P = ky, and all other variables and parameters are as defined above. Taking natural logs of both sides of equation (III-#8), one obtains the function: (III-1+9) 1%.? 1n Yi+-i—3—:-Y 1n [(1+P)a+(l-P)Yi] : 1n F+al 1n Ki + a2 1n Fi+a3 1n Li +u0i. Estimation of equation (III-H91 which may be called CD1, is carried out according to the following method, which Ramsey and Zarembka call "two-stage maximum likelihood."22 If one knew the values of a and r, one could estimate the remaining parameters of equation (III-#9) by using the entire left-hand side as the dependent variable. Although knowledge of a and r is not available, consistent estimates of them may be found using the following approach. These estimates 22Ramsey and Zarembka, op. cit. 101 will be maximum-likelihood under the usual assumptions about the distribution of the uOi' Substitute for y, al, a2, and a3 in the equation (III-kg) their maximum-likelihood esti- mators as functions of the left-hand side, so that the like- lihood function appears as a function of only the data and the unknown parameters a and r. One then chooses estimates a and 5 such that the likelihood function so derived is maximized. Estimates of In P and the ai, i - 1,2,3, con- ditional on the choice of a and i are obtained by OLS estima- tion of the following: 1 2r ~ ~ - — . + - O = + . (III M9a) +- 1n Yl+ ~2 1n [(1 r)a+(l r)Yl] In F a] 1n KJ +a2 1n Fi +a3 ln Li+u0i' Applying the results of Box and Cox, confidence intervals for a and 5 may be obtained using the fact that -2 ln 2, where 2 is the likelihood function, is asymptotically distributed as Chi—square with k degrees of freedom, where k is the num- 23 This information may be ber of parameters to be estimated. used to test for constancy of the returns to scale parameter over the observed range of output. Comparison of estimates of In P across technologically homogeneous cells will provide an indication of the effects on the production function of technological change. Returns to scale functions may be cal- culated and compared across cells using equation (III-H6) and 23G. E. P. Box and D. R. Cox, "An Analysis of Transfor- mations," Journal of the Royal Statistical Society, Ser. B, 26 (196u), 21u-219. 102 the parameter estimates obtained for CD1. A similar procedure may be used to examine a CES func- tion, generalized by applying the specific transformation defined by equation (III-M7). Using the Taylor series approximation discussed above for the three-factor CES function, and Ramsey and Zarembka's "two-stage maximum likelihood" extimation procedure, the stochastic specifi- cation to be used is (III-50) 1 _23... IT? 1n Yi+1-%2 1n [(1+§)a+(1-%>Yi] = 80+81 1n Ki+82 ln Fi +83 ln Li +Bu(ln Ki - ln Pi)z +85(ln Ki - 1n Li)2 +86(ln Fi- 1n Li)2+ui, where all variables and parameters are as defined above, and the ui are assumed to be i.i.d. as N(0,o2I). This model may be called CESl. A second generalized function examined by Zellner and Revankar is derived from a returns-to-scale function defined by: .. “f (III-51) 0(V) - T;§V9 where cf) 0 is the degree of homogeneity of the function being generalized. For a > 0, returns to scale fall from af at V = 0, to zero as V+w. Applying the Zellner and Revankar method of generalizing neoclassical production 103 functions, denoted by f, the GPF with returns to scale behaving according to equation (III-51) is given by: BY h (III-52) Ye = chf , where c and h are constants. Applying the transformation defined by equation (III-52) to a three-factor Cobb-Douglas function, adding a multi- plicative random disturbance normally distributed with zero mean, and taking logarithms, one obtains: (III-53) ln Yi+eYi = 1n y +al ln Ki+a ln Fi+a ln Li+u 2 3 0i’ where i denotes the number of the observation, and all the other terms are defined as previously. Under the ZKD assump— tions the logarithm of the likelihood function, 1n 2, is given by: A s - ~ -N 2 N 1 (III-5H) 1n 1 = const. -ln 0 + E In (1+6Y.)-——[zo(e)- 1n 7 2 i=1 1 202 1 -a1 1n Ki '(13 1n IJi]20 where N is the number of observations and zi(6) = 1n Yi+eYi. Maximizing 1n 2 with respect to 02, and substituting the con- ditional maximum-likelihood estimate of 02 thus obtained into equation (III-53), one obtains: n (III-55) ln 2* = const.:%-1n { z [zi(6)- 1n y-al 1n Ki i=1 -02 1n Fi-a3 ln L132} N + z ln(1+6Yi). i=1 10% For 6 = 60, it is evident from equation (III-55) that maximization of 1n 2* is equivalent to estimation of the following regression by ordinary least squares: (III-56) ln Yi+80Yi = ln y+al ln Ki+a2 1n Fi+a3 ln Li+u0i. This model may be denoted as CD2. By repeating this procedure for various values of 6, one can find the estimates of the parameters 6, y, al, a2, and a3 associated with the global maximum of the likelihood function. These estimates will be maximum-likelihood if the u i”N(0,OZI). Zellner and 0 Revankar attribute this "maximum-likelihood search procedure" to Box and Cox.2” Similarly, under the ZKD assumptions about the stochas- tic nature of the productive process, maximum-likelihood estimates are available for the parameters of an approxi- mation to a generalized three-factor CES function, under the same generalization as employed in equation (III-56). This approximation is given by: (III-57) 1n Yi+60Yi = BO+Bl ln Ki+32 1n Fi+33 ln Li +8q(ln Ki- 1n Fi)2+35(ln Ki- ln Li)2 +86(ln Fi— ln Li)2+ui, where all variables and parameters are as defined above. This function may be called CES2. 2”Box and Cox, op. cit. 105 As in the case of the generalization considered in equa- tion (III-u9a), a test is available for the hypothesis that the returns to scale parameter is constant over the observed range of output. Such a test may be based on the fact that -2 1n 2 is distributed as Chi-square, where 1 is defined by equation (III-5M). Degrees of freedom for the Chi-square test are 5 for CD2 and 8 for CES2. Models Used by Previous Researchers In addition to the CD, CES, CD1, CESl, CD2, and CES2 models discussed above, alternative versions of the models estimated by Hart and Chawla and by Ling will also be con- sidered in this thesis. A discussion of each of these models follows: The Hart and Chawla Model As explained in the review of the literature in Chapter I.B., the model used by Hart and Chawla is a fixed-relative- factor-proportions model. The reader will recall that under the assumptions that all inputs are fully utilized and that inputs are combined in fixed relative proportions, the ex- ponents of all factors are equal and determine the degree of homogeneity of the production function. No matter which com- ponent of equation (I-l8)' or (I-19)' is considered, the value of 8 will be the same. Two characteristics of the three-factor approach employed in the present study preclude use of their model as presented. 106 First, the measurement of capacity has not been converted for unused capacity. Second, it seems unrealistic to assume that labor is fully utilized. Therefore, the empirical analysis of the fixed-relative-proportions function will be carried out under different assumptions. Assuming a multiplicative stochastic disturbance term and using the notation of the present study, the following equation will be considered: 8 u . 83 u 8 u1i 2 2 3i (III-58) Y = minEYlKile ’Y2Fi e l,y3Li e 1 Under this production function the inputs are assumed to be combined in fixed proportions, determined by the pro- portional values of 71’ 72, and Y3. If factors are combined in these proportions, Y is given by the minimum of the three components of equation (III-58), which may be called the "relevant component." To examine the degree of homogeneity of equation (III-57), assume that 82 #03303.) < T2 3 - 1 j " where A is an arbitrary constant net less than unity, and all other variables are as defined above. Following a derivation similar to that of inequality (III-70), (III-71) may be replaced by: (III-72) plim {Qai-&j|: k/Sg.+sg.} < 3:5: 1 J - |a._ . for any arbitrary k. Noting that [T2] = =-&-J-—3 it is 2 2 s +s. A “i~'“' apparent that rejecting the null hypothesis if T% :_20 provides W a large-sample two-tailed test of the null hypothesis at the 5% level, when one makes no distributional assumptions about the stochastic disturbances for the estimated production function models. 131 The third and final null hypothesis which may be tested in Stage Three is that the value of Ni’ the number of turbo- generators in the i-th generating plant, does not affect the plant production function. The reader will note that poten- tially this null hypothesis may be impossible to test under the maintained hypothesis restricted for the purpose of inference. Only if the maintained hypothesis is given by CD/N, CES/N, or {LEHCC/N, LEHCF/N, LEHCL/N} is this hypothe- sis relevant. A test of the null hypothesis is simply a test of the significance of 5, the coefficient of Ni under these potential maintained hypotheses. Let 3 denote the OLS estimate of 6, and 33 the stan- dard error of 3. Under the assumption that the least squares disturbances for the estimated model satisfy the full ideal conditions, T3 = Sgis distributed as "Student's" t with (N-K) degrees of freedom. Therefore, under the normality assumption, a test of the null hypothesis may be carried out at any desired level of significance. If the normality assumption is not made about the least- squares disturbances, a test of the null hypothesis may be based on the Tchebycheff Inequality. Under the null hypothe- sis that 6 = 0, the Inequality states that: (III-73) Pr{|3| 3_ x03} : —i_2., where as denotes the population variance of 6, and A is an arbitrary constant not less than unity. Following a deriva- tion similar to the two just completed, inequality (III-73) 132 may be replaced by: . . . l (III-71+) plim {|5| _>_ ksé} _<_ k—z- for any arbitrary k. Noting that IT 0’) 3l = s that rejection of the null hypothesis when T3 :’/20 provides 9 it is clear 0!) a distribution-free, large-sample, two-tailed test of the null hypothesis. In this section of Chapter III, a procedure has been proposed which is carried out in three stages. In the first stage, a maintained hypothesis consisting of a number of potential alternative models of steam-electric generation is restricted to one or more specific models. In the second stage, an independent test of the restricted maintained hypothesis is carried out for the purpose of independent con- firmation. Also, tests are used to examine the skewness and kurtosis of stochastic disturbances under the restricted maintained hypothesis. In the third stage, tests are pro- posed for three null hypotheses concerning steam-electric generation at the plant level. The specific tests to be used depend on the restricted maintained hypothesis employed and on the distributional assumptions warranted by the results of Stage Two. The results of applying all three stages to the data sample used in the present study are dis- cussed in Chapter IV. Chapter IV. Empirical Results The objective of Chapter IV is to report the results of the empirical procedure explained in Chapter III. D. The results are reported stage by stage, in the order of their exposition in the previous chapter. The reader is reminded that the alternative models are summarized, by name, in Table 5, page 113. IV. A. Stage One--Choice of a Model The reader will recall that at the end of Chapter III. C. 18 equations representing 12 different models of the steam—electric generating plant were listed. It was men- tioned in Chapter III. D. that the problem to be addressed in Stage One of the empirical analysis was to choose one or more of the alternatives as a restricted maintained hypothesis under which to carry out statistical inference concerning the generation process. To accomplish this, each equation was tested for specification error using data samples from Cells I through IV. The results of the specification error tests RESET, WSET, and BAMSET, together with the value of R2, are reported for each model tested in Appendix A. The most striking result of the specification error tests is that all functions considered were rejected at the 1% level for non-normality of disturbances by WSET. 133 134 It is well-known that non-normality of disturbances affects only the maximum-likelihood property of OLS estimates. Therefore, it was decided to use the results of WSET as one of several criteria for the choice of statistical tests to be used in Stage Three, rather than as a criterion for the choice of a model. Considering only the results of RESET and BAMSET, the three-factor functions based on generalizations of the ZKD model were not found to be appropriate, for the most part. At the 1% level CD was rejected by BAMSET in Cells I and III, while CES was rejected by at least one of the two tests in all cells except IV. The possibility that misspecifica- tion associated with these models might be the result of variable returns to scale did not appear plausible, since generalization of these functions according to the Zellner and Revankar procedure did not appreciably affect the test statistics. Further, all the estimates of r, a, and 6, the parameters of the returns to scale functions in CD1, CD2, CESl, and CESZ, were not significantly different from zero. Therefore, the hypothesis of a constant returns-to- scale parameter over the observed range of output could not be rejected, even at the 5% level. The reader will recall that two versions of a fixed- relative-proportions model were considered. The model con- sisting of the components LUHCC, LUHCF, and LUHCL, in which 135 output was assumed to be a stochastic function of inputs, was not found to be free of specification error. At least one of the components was rejected by at least one of the tests in each cell examined. Due to problems associated with duplicate observations, neither LUHCC nor LUHCL was tested in Cell III, and LUHCL was not tested in Cell IV. Similar test results were obtained for the model consisting of LEHCC, LEHCF, and LEHCL, in which output was assumed to be ex0genously determined. Addition of a term in N, the number of turbogenerator units in the plant, to the CD, CES, and fixed-relative- pr0portions functions improved the test results somewhat. CD/N was rejected at the 1% level by RESET in Cell II, and CES/N was rejected by BAMSET in Cells I and II. But with these exceptions, the null hypothesis of no specifi- cation error except for non-normal disturbances was not rejected for either model in any of the cells examined. Values of R2 were greater than .99 for both models in all four cells. Unfortunately, both estimated functions exhibited a number of coefficients with negative signs. Other esti- mates were not significantly different from zero. Only in Cells III and IV were coefficients of any factors in CD/N except fuel significant at the 5% level, and in both those cells the estimated capital coefficients were negative. 136 Similar results were obtained for CES/N except in Cell II, where the coefficients of both ln K and N were significantly negative. It was suspected that the large standard errors of the estimated coefficients might result from a high degree of multicollinearity among the factors. The stan- dard error of the i-th coefficient, sai, is given by Vszxxll, where 32 is the mean squared error of the esti— 11 is the i-th diagonal element of mated regression and x [X'XJ'1, X being the observed regressor matrix. By de- creasing the value of IX'XI, multicollinearity increases the values of the xii and, in turn, the values of the Sdi’ To examine the extent to which multicollinearity caused the large observed standard errors, the xii were compared to the values of 82 for a few cells. The value of xii/s2 ranged from 76.39 for the labor input in Cell I to abour 3.5 billion for the fuel input in Cell III. Even when these ratios are deflated by their respective means, their values are approximately 10.2 and 218 million. These figures provided some support for the presumption of a high degree of multicollinearity among the inputs. Further evidence that a high degree of multicollinea- rity was in fact present is apparent from Table 5, in which the gairwise correlation coefficents for the factors are presented. 137 Table 5 Correlation Coefficients Among Factors, by Cell Cell Factor Pair I II III IV F,K .906 .896 .955 .962 F,L .829 .609 .788 .755 K,L .779 .699 .793 .801 The final model to be considered was the group con— sisting of LEHCC/N, LEHCF/N, and LEHCL/N. These equations are the components of a fixed-relative-proportions model with output exogenous, modified by inclusion of a term in N, the number of units per plant. With the exception of Cell II, this set of equations performed fairly well, dis- regarding the problems of non-normality. As can be seen from Appendix A, at the 1% level RESET rejected only the fuel input equation in Cell I and the labor input equation in Cell III. But neither of those tests rejected any input function in Cell IV. All SIOpe coefficients, representing elasticities of inputs with respect to output, lay between 0 and l, as would be expected under the increasing returns to scale hypothesis suggested by most earlier writers dis- cussed in Chapter I. B. The values of R2 calculated for each input equation are not surprising to a student of the steam-electric generation process. For the fuel equations, R2 ranged 138 from .9858 to .9969 across cells, indicating that output is an important determinant of the fuel input. This is in agreement with the findings of many engineers who have examined that relationship under various steam conditions.1 A smaller pr0portion of the variation in capital input was explained by output and the number of machines, pre- sumably because differences in plant factor across plants account for some variation across plants in the installed capacity required to generate a given annual output with a given number of units. The value of R2 for the capital input equation ranged from .8063 to .9300 across cells. Finally, relatively low values of R2, between .5098 and .7182, were obtained for the labor input equation in the various cells. This is not surprising in light of the observation made by Dhrymes and Kurz that the operations performed by labor "are more or less supplementary to the major operations of turning fuel into electricity."2 The fact that the results obtained in Cell II differed so markedly from the results obtained in the other cells led to the suspicion that some plants in the sample from that cell might not be representative of the cell. There- fore, plots of the data for Cell II were made and examined lLing, pp. cit., pp. 28-35. 2Dhrymes and Kurz, pp. cit., pp. 287-315. 139 for the presence of outliers. Three outliers were found. Further investigation of the outlying observations shed some light on the causes. One plant, which produced an outlier in its second year of operation, had a plant fac- tor of only 7% and a heat rate 30 to 90% higher than that of other plants operated by the same firm. The guess was made that as a new plant, it was still being tested and adjusted. The other two outliers were apparently subject to some sort of reporting error, since data for the year of observation of the outlier on average production cost and plant heat rate in each case differed substantially from the corresponding data for the same plant, for pre— ceding and subsequent years. The fact that the sample in Cell II did in fact contain outliers indicated that the specification error tests had successfully identified a sample for which the fixed-relative-proportions model was not appropriate. However, since the inappropriateness could be explained by undesirable properties of the sample, it was impossible to determine whether in addition the model was inappropriate for the population defined by Cell II. Clearly none of these models, which comprise the set of alternatives under the maintained hypothesis, was found to be appropriate for all cells. However, several of the models could immediately be eliminated from further con- sideration for some cells. The CES model was strongly 190 rejected in all cells except IV, as was the CD model in all cells except II and IV. Generalization of these models to parameterize variability in returns to scale over the observed range of output did not significantly alter the results. However, inclusion of a term in N to incor- porate the effects of variations in machine mix noticeably improved both the specification error tests results and the observed value of R2 in most cells. Similarly, addi- tion of this term to the equations LEHCC, LEHCF, and LEHCL raised the calculated values of R2 without introducing specification error. Therefore, several models were imme- diately eliminated from further consideration, either because of statistically significant specification error, because another alternative produced a higher value of R2 without statistically significant specification error, or because a model involving fewer parameters yielded both equal values of R2 and similar specification error test results. The models immediately eliminated were CD in Cells I, III, and IV; CDl and CD2 in all cells; CES, CESl, and CES2 in all cells; the model consisting of LUHCC, LUHCF, and LUHCL in-all cells; the model consisting of LEHCC, LEHCF, and LEHCL in all cells; CD/N in Cell II; and CES/N in Cells I and II. Therefore, the specification error tests were able to eliminate most, but not all, of the potential alternatives. 191 After these models were rejected, the following possibilities remained: Cell I--{LEHCC/N, LEHCF/N, LEHCL/N}, CD/N Cell II--CD, {LEHCC/N, LEHCF/N, LEHCL/N} Cell III-—CD/N, CES/N, {LEHCC/N, LEHCF/N, LEHCL/N} Cell IV--CD/N, CES/N, {LEHCC/N, LEHCF/N, LEHCL/N} The reader will note that for each cell, one three- equation model and at least one one-equation model remained. Furthermore, by reference to Appendix A, it is obvious that some of the components of the three—equation model were actually rejected by RESET or BAMSET at the 1% level. Because of the following considerations, the model was not rejected out of hand despite these test results. To date, no rigorous method is available for com- paring alternative models according to the presence of specification error when the alternatives consist of different numbers of equations. Two considerations are relevant to this problem. First, it seems reasonable to assume that if the tests are applied, using the same data, to a one—equation model and to each equation of a three- equation model at the same nominal a-level, the probability of rejecting one component of the three-equation model is greater than the probability of rejecting the one- equation model, under the full ideal conditions. This should be kept in mind in comparing models consisting of 1H2 different numbers of equations. Second, both a knowledge of the technology of steam-electric generation and an ex- amination of Table 5 indicates high positive pairwise correlations between the factors. It is well-known that this phenomenon is a cause of multicollinearity when all three factors are used as explanatory variables in one regression. In turn, large coefficient standard errors are associated with multicollinearity. Consequently, operationally significant relationships between output and the individual factors may appear statistically insig- nificant when all three factors appear as regressors in the same equation. A_fortiori, such a situation will lower the power of the tests. For these reasons, it may be desi- rable to accept provisionally the hypothesis that the three-equation model is the relevant model, subject to further testing in Stage Two. Another consideration relates to one of the overall objectives of this study--namely, inference concerning the effects of technological change on the production function, to be carried out in Stage Three. For this objective, it is helpful to be able to express technological change parametrically, a relatively simple task if the same model is used for all cells. Thus, one would like to use the same functions for as many cells as possible without com- pletely violating the criteria for acceptance of a model. 143 The three-equation model consisting of LEHCC/N, LEHCF/N, and LEHCL/N is helpful in meeting this objective. Considering all these factors, and recalling that the purpose in Stage One of the analysis is merely to generate a restricted hypothesis, to be tested in Stage Two, it was decided to relax the specification error test criteria somewhat. Specifically, it was decided to accept as a tentative hypothesis the three-equation model even if one of its components was rejected at the 1% level by either RESET or BAMSET but not both. On this modified criterion, the model {LEHCC/N, LEHCF/N, LEHCL/N} is tenta- tively acceptable for all cells except Cell II. As men- tioned above, the fact that this rejection was due at least partially to the presence of outliers in the Cell II sample makes evaluation of the appropriateness of the model to the Cell II population impossible. Use of this model for all cells as a tentatively maintained hypothesis has two advantages over the other models for the inferences to be carried out in Stage Three. First, the problem of multicollinearity due to high pairwise correlation coef- ficients between factors is avoided. Second, a model with comparable coefficients over all cells is available. Neither of these conditions is met by any other model not rejected immediately on the basis of the RESET and BAMSET results. 199 Therefore, the following model was tentatively accep- ted as a maintained hypothesis for all four cells: (III-66a) 1n Ki 1n r1 + alln Yi + 51 Ni + V11 (LEHCC/N) (III-66b) 1n Pi 1n r2 + a2 1n Yi+ 62 Ni + v2i (LEHCF/N) (III-66c) 1n Li = ln r3 + a3 1n Yi+ 53 Ni + v3i (LEHCL/N) The terms Vli’ V2i’ and V3i are assumed to be i.i.d. stochastic disturbances, with zero means and constant variances. The assumption of normality was not warranted by the results of WSET. Having tentatively chosen this maintained hypothesis, an attempt at independent confirma- tion of it was made in Stage Two. IV.B. Stage Two-—Confirmation of the Chosen Model The next task is to use a different data sample to test the model given in equations (III-66) for specifica- tion error. In this way one can obtain independent evi- dence regarding its appropriateness for each of those cells. Another task, as explained in Chapter III.D., is to test whether the distribution of disturbances for this model, despite being non-normal, has skewness and kurtosis coefficients sufficiently close to those of the normal distribution to warrant the assumption of normality for the inferences to be carried out in Stage Three, below. The results of both these tests are discussed in the present section. 145 For the purposes of Stage Two, a second data sample of 50 was randomly drawn without replacement from the available data in Cells I-IV. Using these samples, the chosen mode1--LEHCC/N, LEHCF/N, and LEHCL/N-—was re- estimated and tested for specification error using RESET, WSET, and BAMSET. Furthermore, skewness and kurtosis statistics were calculated for the distribution of Theil residuals for each equation in each cell. The values of all these statistics, as well as the corresponding values of R2, are presented in Appendix B. The null hypothesis being tested with the specification error test statistics is that the disturbances are i.i.d. N(¢,021). The null hypothesis being tested with the aid of /B; and g2 is that the skewness and kurtosis parameters are zero and 3, res- pectively, as they would be if the true distribution of disturbances were normally distributed. For those equa- tions and cells for which the value of a statistic indi- cates rejection of the null hypothesis, the level of sig- nificance of the rejection is indicated with asterisks-- 1 for the 10% level, 2 for the 5% level, and 3 for the 1% level. The results are summarized in Appendix B. The reader will recall that in Stage One, the fixed- relative-proportions model was accepted if no more than one of the two tests RESET and BAMSET rejected no more than one of the three component equations at the 1% level. 146 Using the same standard in Stage Two, the model was con- firmed for Cells I, III, and IV, as is evident from Appen- dix B. As in Stage One, the rejection in Stage Two of the model for Cell II was apparently due to the presence of outliers in the data sample. The fact that the results of Stage Two so closely duplicated the results of Stage One was taken as independent confirmation of the results obtained in the first stage. However, in basing inferences on this model, the possibility of mis-specification should be borne in mind. The second problem in Stage Two, investigation of skewness and kurtosis, was less clear-cut. Based on the statistics /EI and g2 calculated from the Theil residuals for each equation, the null hypothesis of normal skewness was rejected at the 1% level for all equations in Cell II and for the capital equation in Cell III. Also at the 1% level, the null hypothesis of normal kurtosis was rejected for the fuel equation in Cell I and all equations in Cell II. The choice of test procedures for Stage Three, based on the results of these skewness and kurtosis tests, depends heavily on the interpretation of the fixed-relative- prOportions production model as it was stated in Chapter III.C. For this reason, the reader is reminded that accor- ding to that discussion the degree of homogeneity of the 147 production function is given by the smallest of the three factor exponents, i.e., the degree of homogeneity of what what was called the relevant component term in Chapter III.C. Of course, if the three exponents are not signifi- cantly different from each other, any of the three may yield the degree of homogeneity of the production function. For the sake of consistency, the statistical tests to be used in Stage Three should be based on the relevant component equation. Therefore, it is evident that if the relevant component of the model in Cell I is LEHCF/N, for which non-normal kurtosis is indicated, the tests for Stage Three involving Cell I should not be based on the normality assumption. Similarly, tests involving Cell II should not be based on normality if LEHCC/N is the relevant component. In both these cases tests should be based on the Tcheby- cheff Inequality, as explained in Chapter III.D. Similarly, all tests involving Cell II should be based on the Tcheby- cheff Inequality. In all other situations involving Cells I-IV, tests may be constructed that incorporate the nor- mality assumption for the regression disturbances. As explained in Chapter III.D., the available numbers of observations for Cells V, VI, and VII were too small to allow the three—stage procedure to be used. Therefore, as was explained earlier, the restricted maintained hypo- thesis for those cells was to be the same as that for 148 Cell IV. Extending that rule to include the stochastic part of the maintained hypothesis, tests involving Cells V, VI, and VII are based on the assumption of normality. Concisely stated, the maintained hypothesis for Stage Three is given as follows: Yi = min {leglealNieu1i, yZKEZeGZNieuZi, 73K§3e63Nieu3i} This model is to be estimated in separate components-- (III—66a), (III-66b), and (III-66c), denoted as LEHCC/N, LEHCF/N, and LEHCL/N. respectively. ‘ l, 2, 3, i = l, . . ., n, are assumed to be i.i.d. as follows: Cell I: uli’ u3i N N(¢, 021) u2i’ m (¢, 021) Cell II: uli, uzi, u3i m (¢, 021) Cell III: u2i, u3i m N(¢, 021) uli’ W (C, 021) Cells IV-VII: uli, uzi, u3i m N(¢, 021). The form of estimation is as separate equations-- (III-66a), (III-66b), and (III-66c). It should be noted that this formulation, also called {LEHCC/N, LEHCF/N, LEHCL/N}, includes the implicit assumption that output is exogenously determined, with inputs used as the dependent variables. Although the model is to be estimated by or- dinary least squares, the specification error test results dictate caution in the interpretation of the estimated re- sults, which are presented in the next section of this Chapter. 149 IV. C. Stage Three--Results Concerning Scale, Technology, and Machine Mix As the first step in Stage Three of the empirical analysis, a third sample of 50 was drawn from each of the available data pools in Cells I-IV. The data base for Cells V, VI, and VII consisted of all available data from each of these cells. For each cell, the model consisting of the three equations LEHCC/N, LEHCF/N, and LEHCL/N was estimated, using the data sample for that cell. The esti- mated results are presented in Table 5. For each of the three components for each cell, estimates of the three estimated coefficients F, a, and 6 appear immediately above their respective standard errors. The value of R2 for each equation also is given. The first item of concern in interpreting the results is determination of the relevant component of the three- equation model. The reader will recall from the discussion in Chapter III.C that the relevant component is the one including the factor that “limits" output, in the sense that increases in output require proportionately larger increases in the "limiting" factor than in the others. Estimates of the proportionate increases in each input required for a given change in output, i.e., the output elasticities of the inputs, are given by $1, i = l,...,3. From Table 6, it is evident that the relevant component 150 mam. mam. «we. 0mm. 0mm. mam. see. we moo.o mmo.o oao.o eoo.o «Ho.o emo.o oeo.o New eoo.o- muo.o- Hmo.o mmo.o Noo.o- mao.o emo.o NM Nae.o :eo.o mmo.o :Ho.e omo.o mao.o Nao.o «we eam.o mmm.o mee.o oam.a emm.o mmm.o Nam.o aw omo.o Hom.o e~:.e Hmo.e HNH.o aeo.o meo.o New omm.m me~.m mme.oH emm.m mom.m Hmm.oe Ham.m am daemapmum z\eomma HH> H> > >H HHH HH H .02 Heme mam. ass. ems. emm. cam. mom. was. we emo.o mmo.o e:o.o «mo.o emo.o mmo.o eeo.o ems mem.o mma.o os~.o Hee.o mee.o Hmo.o eeo.o- Hm meo.o eHH.o aeo.o Heo.e emo.o smo.o mmo.o awe Nee.o mea.o moa.e ome.o Ham.o mae.o mmm.o aw :mm.o mee.o mm:.o oa:.e mem.o me~.o :om.o Hmm mam.a mm~.~ Hmm.~ mmo.o Nem.o mma.o- mom.o- Hm oaumaumpm z\oozma HH> H> > >H HHH HH H .02 same ”deco: z\aomma dad .z\momma wz\oomm4 mom muddwmm coapmewpmm w CHAMH 151 000. 000. 000. :00. 000. 00:. 0H0. 000.0 H00.0 000.0 000.0 H00.0 000.0 000.0 001.0: H:H.0 000.0: 000.0 000.0 00H.0 000.0: m 030.0 H:H.0 000.0 030.0 000.0 0:0.0 000.0 000 H00.0 000.0 0H0.0 000.0 000.0 000.0 000.0 0m 000.0 030.0 000.0 000.0 000.0 000.0 000.0 0mm 003.0 000.0 :0:.: 0:0.: H00.0 000.0 300.0 0m OHPmH#Mpm z\qommq HH> H> > >H HHH HH H .02 Haoo A.U.#coov 0 mHQMH 152 in each cell over the observed range of output in each cell is the fuel equation, LEHCF/N. In all cells, 82 significantly exceeded 61 and 83 at the 5% level. For this reason, an estimate of the degree of homOgeneity of the production function in each cell is given by 82 in 1 each cell, where 82: 32. Scale Effects Having chosen the relevant component equation, one may proceed to test the hypotheses stated in Chapter III.D. concerning the effects on the production function of changes in scale, technology, and machine mix. The null hypothesis concerning scale is that at the plant level, the steam-electric generation process exhibits constant returns to scale. Under the null hypothesis, a2 = 1. Under the alternative of increasing returns to scale, 02 < l. The null hypothesis was tested against this one-tailed alterna- tive at the 5% level, using the statistic Tl, as defined in Chapter III.D. For Cells I and II, for which the nor- mality assumption was not warranted for LEHCF/N, the null hypothesis was rejected if Ti > 20, as explained by the Tchebycheff Inequality. For the remaining cells, a test at the 5% level was based on the fact that T1 is distributed as "Student's" t with (N-2) degrees of freedom. The results of these tests are presented in Table 7. The null hypothesis 153 was rejected in favor of the alternative of increasing returns to scale in all cells except Cell VI. Table 7 Results of Tests of the Constant Returns to Scale Hypothesis* I: T§ = 92.25* II: TE = 86.68* III: T1 = -2.1* . - _ * IV. Tl - 4.3 . = _ a V. T1 3.7 VI: T1 = —.095 VII: T1 = -u.u* *indicates increasing returns to scale significant at the 5% level. It is interesting to note that the plants in Cell VI employed units with the largest nameplate-rated generators and most advanced furnaces, turbines, and generator cooling systems of any subcritical steam generators. Furthermore, the supercritical steam generators in Cell VII exhibited statistically significant increasing returns to scale. It appears reasonable to infer from these results that the introduction of supercritical generators in 1958 was the result of implicit recOgnition by engineers that the ad- vanced units in Cell VI exhausted all technOIOgical scale economies possible using subcritical steam generators. 154 The introduction of supercritical steam generators removed this barrier to growth of electricity generators, by res- toring the condition of increasing returns to scale. In this connection, it is significant that the largest unit installed in a Cell VI plant by 1965 was rated at 570 mw; while the largest unit installed in a supercritical plant by that date was rated at 850 mw. These figures are reported in Chapter II.B. By 1968, according to £23333 magazine's Modern Plant Design Survey, the corresponding figures were 725 mw and 1300 mw. Apparently, the size of the largest unit in supercritical plants grew by 53% between 1965 and 1968 while the corresponding size grew by only 39% during the same period for Cell VI plants because of the superior potential economies of scale associated with supercritical steam generation. Effects of Technological Change The second question to be considered concerning the results presented in Table 6 is whether technological change as represented by the cell classification scheme of Table 4 has affected plant efficiency independently of scale effects. The results presented in Table 7 indicate that variations in output require significantly less than pro- portional variations in fuel input, one form of economy of 3Power, pp. cit., October, 1968. 155 scale. One method of isolating changes across cells in fuel efficiency independently of scale or output effects is to compare across cells the fuel input necessary to pro- duce one megawatt—hour of electricity with one turbgenerator unit. Examining equation (III-66b), LEHCF/N, it is seen that when Yi = Ni = 1, Pi = r, e62 egi. As will be explained below, 62 was not significantly different from zero for any cell except Cell IV. Therefore, the expected value of F- 000 1 when Yi = N- = l is r2 e 2 , Therefore, under the assump- 1 tion that 000 is equal for all cells, and under the null hypothesis that technolOgical change has not affected efficiency independently of scale effects, r2 is equal for all cells. Under the alternative that a given class of plants is mor efficient than another, I, in the former . cell is less than r2 in the latter. This null hypothesis! was tested for each pair of cells, using the test statis- tic T2, as defined in Chapter III.D. Under the stochastic specification of the maintained hypothesis, the appropriate test when either Cell I or II is a member of the pair being considered is to reject the null hypothesis when T3 > 20. For pairs not involving either of those cells, one may construct a test using the fact that T2 is distri— buted as "Student's" t with 2(N-K) degrees of freedom. The results of these tests are presented in Table 8. In calculating T2, F2 for the higher-numbered cell was 00. u a Hm pcmowmwcwfim onMOHUC0 « 0+0 0+0 000.0 N 00 00. u 00 000.- u 00 «00.30 u my oo.0 n We 00> 0+0 0+0 .00.0 n 00 00. u 00 00. u 00 00.0 n m0 00.0 n we H> 0-0 0-0 «00.0 n 00 000.0- N 00 00.0 n m0 00.0 u we > 0+0 0 TL 0 a 00.- u 00 «00.00 H 00 00.: n 00 >0 0+0 0 0+0 0 000.30 u 00 00.0 n 00 000 To 0 00.00 H 00 00 .02 0000 > >0 000 00 0 .02 0000 mHHmo mmOhom 00 mo hpwamswm how wvmmh mo mpadwmm 0 0.2090 156 157 subtracted from r2 for the lower-numbered cell. Therefore, if the latter cell includes more efficient plants than the former, the difference is positive. For those pairs for which the appropriate test was based on the Tchebycheff Inequality, so that the test statistic is T2, the sign of the difference appears in parentheses in the corresponding entry in Table 8. From Table 8, a mixed impression emerges. As expected, plants in Cell II appear significantly less efficient than plants in Cells III, IV, and VII, independently of scale effects. Similarly, plants in Cell V appear significantly less efficient than plants in Cells VI and VII. These results are in accordance with 3 priori expectations. However, the result that plants in Cell V are less effi- cient than plants in Cells III and IV is surprising. The reader will recall that the only technological difference between Cells V and VI is that generators in Cell VI employed conventional liquid and hydrogen cooling systems. Steam generators in both these cells operated at higher temperatures and pressures than those of Cells III and IV; but more importantly, generators in Cells V and VI had higher nameplate ratings than did those in Cells III and IV. The implication of the results in Table 8 concerning Cell V seems to be that heat losses of these large genera- tors using conventional tube-and-fin cooling systems 158 decreased plant efficiency substantially. Only when hollow- conductor cooling was introduced, in Cell VI, did the in- crease in efficiency from higher temperatures and pressures more than compensate for the heat loss incurred with the higher-rated generators. Differences in r2 between all other pairs of cells were not significant. For these cells, more advanced technology did not improve plant efficiency at levels of output approaching one megawatt—hour. Therefore, the ad- vantage to technological change in these cells appears to have been to allow the construction of larger plants. These large—scale plants were able to operate more effi- ciently than their less advanced predecessors only at higher rates of output. This would appear to explain the fact, discussed in Chapter II, that at any point in time, only plants with the most highly-rated generators incor- porate the most advanced technology. In this connection, a comparison of the values of P1, the intercept of the capital component, LEHCC/N, across cells is revealing. This comparison indicated that to generate one megawatt—hour of electricity with one turbogenerator, plants in the technologically more advanced cells require that genera- tor to be of greater installed capacity. Following an argu- ment similar to that which justified comparison of P2 across cells, if Tl increases as technology advances, higher levels of technology require larger turbogenerators 159 to be equally efficient at low outputs. To test whether such a phenomenon actually occurred, the null hypothesis that P1 is equal for all cells was tested against the alternative that for any pair of cells P1 is larger in the higher-numbered cell of the pair. As with the test for differences in r2 across cells, Pl for the higher- numbered cell was subtracted from F1 for the lower-numbered cell. Therefore, negative differences support the conten- tion that technolOgical change has made the process more capital-intensive. As shown in Table 9, with two exceptions--III and IV, and V and VI—-the sign of the difference indicated that for each pair of cells, the one involving the more advanced technology required a more highly rated unit to generate one megawatt-hour of electricity than did the other. The difference was statistically significant at the 5% level for the following pairs of cells: I-V, I-VI, I-VII, II—V, II-VII, III-VII, IV—V, IV—VI, IV-VII, V-VII, and VI-VII. Since P1 and P2 did not increase similarly, and since in this model inputs are combined proportionately to r1, P2 and F3, technological change appears to have made the industry more capital-intensive. These results provide additional support for the practice of engineers to use the most advanced technology only in construction of the most highly-rated units, at a given point in time. .Hu>oa 00 03p pm vacuowwwcmwm mwumowocw « 0 Any 0 alv 0 00.0- n 00 «00.0- 0 00 «00.0- n 0 «000.00 0 00 «000.00 0 00 000.0: 0 00 00> 0 0-0 0 0-0 0 0 000. u a «00.0 n 00 000.0 0 00 000.0 0 00 «01.0- n a 0> 010 010 000.0- 0 00 010.0 0 m0 «000.00 0 We «00.0- n 00 > 0+0 0-0 000. 0 m0 000. 0 m0 00.0: n 00 >0 0-0 0 0-0 0 000.0 0 0 00.0 n 0 000 0 0 0:0 0 000. u 00 00 .oz 0000 0> > >0 000 00 0 .oz 0000 wHHmo mmOfiom Ha Mo huHHMSvm how mpmwa mo mvadmwm 0 00000 160 161 Significance of the Number of Units per Plant The third area involving the results in Stage 3 is the effect of machine-mix, as represented by N, the num- ber of turbogenerator units in a plant of given installed capacity. The relevant estimated coefficient is 6, and arguments were advanced in Chapter III.D for both positive and negative signs for 6. Only the value of 62, in LEHCF/N, is relevant from the standpoint of plant efficiency, since fuel is the "limiting" factor. However, the sign and statistical significance of 6 were considered for all three equations, to explore possible effects of machine-mix on the inputs of all three factors. The null hypotheses that 61, 62, 63, = 0 were tested against two-tailed alternatives at the 5% level, using the test statistic T3, defined in Chapter III.D. Depending on the equation and the cell, tests were based on either the Tchebycheff Inequality OP "Student's" t-statistic, whichever was appropriate under the maintained hypothesis. The results of these tests are presented in Table 10. For those tests for which the Tchebycheff Inequality was used, so that T3 is the appro- priate test statistic, the sign of 6 is given in parentheses. As is clear from examination of Table 10, 6 was not significantly non-zero for any equation in Cells I and II. In the capital equation, LEHCC/N, 61 was significantly positive in Cells IV, V, VI, and VII. This indicates that 162 Table 10 Results of Tests that 6 =0 Function LEHCC/N LEHCF/N LEHCL/N Cell No I T = —1.uu T7’= 19.52 T = -l.67 3 3 3 (+) II Tg = .0925 T3 = 1.2633 T§ = 3.71 (+) (+) (+) III T2 = 8.86 T = -.l894 T = 2.58* 3 3 3 (+) ‘ = * : * IV T3 _ 3.u3* T3 3.57 T3 3.30 .. i: - -_ :3: v T3 — 5.07 T3 _ 1.99 T3 - 3.16 : 7" :.., :, VI T3 3.30 T3 69 T3 1 98 - ”c 3.. :— VII T3 - 8.96a T3 1.12 T3 5.15 forva plant containing more units, a given increase in Output requires a proportionately greater increase in installed capacity. Given the previous finding that technologically advanced units are inefficient at low outputs, this result is not surprising. In the fuel equation, LEHCF/N, 62 was significantly positive in Cell IV, indicating that for that cell, the elasticity of fuel input with respect to output increases as the number of units in the plant in- creases. For other cells, 62 was not significant. The results for LEHCL/N are mixed, with 63 being significantly negative in Cells V and VII, but positive in Cells III and IV. 163 The only consistent pattern in these results is the group of significantly positive values of 61 in LEHCC/N for Cells IV-VII. This indicates that for the most tech- nologically advanced plants, the amount of installed plant capacity necessary to generate a given output increases as the plant capacity is divided among more turbogenerator units. The value of N, the number of units, did not sig- nificantly affect plant efficiency, which is measured by LEHCF/N. Therefore, it appears that the loss of plant efficiency one would expect from division of a given plant capacity among more units was approximately compensated for by the gains in efficiency from additional possibilities of substitution among units made possible by the addition of units to the plant. The empirical results reported here suggest the following conclusions: (a) Fuel is the "limiting" factor in steam—electric genera- tion for all cells in the sense that an increase in output requires a proportionately greater increase in the input of fuel than of the other two factors, capital and labor. (b) Increasing returns to scale is characteristic of steam- electric generation at the plant level, since an increase in output requires less than proportional increases in all three inputs. 164 (c) Technological change has not significantly affected plant efficiency independently of scale, since the fuel input function did not shift significantly across cells as defined in Table 4. However, by enabling firms to construct plants incorporating more highly-rated turbogenerators, technological advances have made plants more efficient through economies of scale. (d) Technological change in the industry has operated in the form of substitution of capital for the other factors, since only the capital input function intercept increased significantly between 1948 and 1965. (e) The most technologically advanced plants are less suitable than other plants for low levels of output. Evidence for this is that at low levels of output, the newer plants make no more efficient use of fuel than the older plants, while requiring prOportionately higher levels of installed plant capacity. (f) Variability of the returns-to-scale parameter is not statistically significant for any single technologically homogeneous group of generating plants. However, the most modern subcritical plants exhausted virtually all technical potential economies of scale using subcritical generation. Plants incorporating supercritical generators are subject to increasing returns to scale. 165 (g) The number of turbogenerator units in a plant of given installed capacity is not consistently related to plant efficiency, except in the following manner: for the most modern plants in the sample, allocation of plant capa- city among more units raises the plant capacity required to produce any given level of output. Given these conclusions, it appears that the methodo- logy of the present study, particularly the technolocial classification scheme and the three-equation model, has made possible more precise statements than those of previous authors. A brief discussion of the consistency of the present findings with earlier findings is presented below. Consistency of the Present Results with Previous Results For the most part, the results obtained in the present study tend to confirm the results of previous authors. Since Komiya” and Hart and Chawla5 estimated models con- ceptually similar to the group consisting of LEHCC/N, LEHCF/N, and LEHCL/N, direct numerical comparison with their results is possible. The comparison indicates a fair degree of consistency among the three studies. Less directly, the 1+Komiya, R., pp. cit., pp. 156-166. 5Hart, P.E., and Chawla, R.K., pp. cit., pp. 164-177. 166 present study tends to support and to explain most of the conclusions of Nerlove,6 Dhrymes and Kurz,7 and Galatin.8 In his fuel input equation, Komiya used average size of generating unit as a measure of the size of plant, as opposed to installed capacity in the present study. None— theless, the present results may be consistent With Komiya'a conclusion that the “improvement in the thermal efficiency can be explained by the increase in the scale of production rather than by the shift of the function."9 In both studies, the conclusions were reached that increases in the scale of plant operations require less than prOportional increases in fuel input; while technological change has not signifi- cantly shifted the intercept of the fuel input equation. These conclusions were also apparent in the Hart and Chawla results. Although shifts in Komiya's estimated capital equation across vintage cells were not significant, they were of the same direction as those of the present study, indicating increases over time in the amount of capital used in plants of a given size. The lack of significance may result from his use of vintage as a proxy for technology. 5Nerlove, M., pp. cit., pp. 167-198. 7Dhrymes and Kurz, pp. cit. 8Galatin, M., pp. cit. 9Komiya, pp. cit., p. 161. 167 It has been stressed throughout this thesis that this prac- tice groups technologically dissimilar plants together, which tends to conceal differences between the production function parameters of the technologically dissimilar plants. This effect could account for the absence of significant shifts of the production function in both Komiya's study and the present one. Hart and Chawla did not consider the signifi- cance of the shift of the capital intercept across their two cells, but it appears consistent with Komiya's results. As in the present study, Komiya's labor input equation performed less satisfactorily, with lower values of R2 and non-systematic changes in the coefficients across cells. In neither his study nor the present one were any conclusions regarding the labor input justified by the results. Hart and Chawla did not consider the labor input. Considering studies in other situations, less specific comparisons are possible. The present study tends to con- firm the general conclusion of Galatin, Nerlove, and Dhrymes and Kurz that increasing returns to scale is the prevailing condition in steam-electric generation. However, several more specific conclusions of those authors were not directly confirmed. For example, testing of generalized production functions in the present study provided no support for the suggestions by Nerlove and Dhrymes and Kurz that returns 168 to scale are a variable function of output. However, Nerlove's data sample, a cross-section of plants operating in 1955, and Dhrymes and Kurz's vintage samples both in- clude technologically dissimilar plants. Therefore, what both these studies took for decreases in the returns-to- scale parameter due to increases in output may in fact be due to technological change in subcritical generating plants. The reader will recall that for the plants in Cell VI re- turns to scale were found to be approximately constant in the present study. Any Cell VI plants observed by previous authors producing large outputs under constant returns to scale could be interpreted as a decrease in the returns—to- scale parameter if plants were not carefully differentiated. Another apparent discrepancy between the present study and previous one concerns the effect of technological change on plant efficiency. Both Galatin and Dhrymes and Kurz claim to have shown that technological change has im- proved plant efficiency independently of scale effects, while the present results indicate that technological change improved plant efficiency only by allowing firms to achieve economies of scale. The apparent contradiction with Galatin's result is easily resolved when one realizes that his conclusion that technological change has signifi- cantly shifted the production is based on the entire period of his study, 1920-1953. However, the results of the 169 present study are comparable with only the last two of Galatin's vintage cells, 1945-50 and 1951-53, between which he reports no significant shift. The source of the contradiction with the results of Dhrymes and Kurz is less apparent. Their conclusion that technological change has increased plant efficiency inde- pendently of scale effects is based on the fact that in their study the estimated CES production function intercept A increases monotonically across vintage cells for three of four capacity cells. No standard errors for A are presented, so that one is unable to evaluate the signifi- cance of this shift. In a footnote, the authors point out that "comparison of A across size groups, for a given tech- nological period, is not very meaningful since we are estimating segments of a production function by a suitable approximation which does not have to yield the same constant, A, for each segmentJJO However, since technologically more advanced plants are used to produce larger outputs than are less advanced ones, as explained in Chapter II of this thesis, production functions for different cells are also estimated over different segments. Since plants in all cells but one exhibited increasing returns to scale in the 10Khrymes and Kurz, pp. Cit., p. 309. 170 present study, the observed differences between cells in plant efficiency observed by Dhrymes and Kurz appear to be the joint result of technology and scale changes. Recog- nizing that technological change has frequently enabled firms to increase their scales of plant, the present con- clusions are not inconsistent with those of Dhrymes and Kurz. Taken as a whole, the results of the present study appear to confirm the work of previous authors and to make their conclusions somewhat more specific. Some implications of these results are discussed in Chapter V. Chapter V. Conclusions from this Stugy_ V.A. Summary of the Thesis The problem explicitly addressed at the outset of this study was to separate the postwar effects of technolo- gical change and increases in scale of production on the efficiency of stem-electric generation. It was pointed out in Chapter I that this question might be of interest to both regulatory agencies and regulated electric power companies. Furthermore, it was pointed out that previous studies of this question were generally subject to two methodological problems. First, such proxies as vintage were used for indicating technological change, rather than considering specific innovations. In this way, important technological differences between generating plants con- structed contemporaneously were often concealed. Thus, spurious differences were attributed to plants constructed at different periods, even though they may have incorporated identical technology. Second, the production models used as maintained hypotheses under which to study the question were simply assumed, with no attention paid to their apprOpriateness for the generating plant. For this reason, the models were often unwittingly estimated under undesir- able constraints or assumptions, e.g., restrictions on 171 172 the elasticity of substitution, which destroyed the opti- mal pr0perties of the estimation technique employed. Furthermore, failure to examine the appropriateness of the stochastic specifications of the estimated models intro- duced the possibility of unwarranted inferences. Hypothe- sis tests based on unfulfilled normality assumptions may have raised the probability of Type II errors above the nominal a-levels. In Chapters II and III, a procedure was prOposed to alleviate these problems. It was thought that adequate indicators of technological change could not be defined without a basic understanding of the mechanical process of converting fuel into electricity. Following a dis- cussion of this subject, what appeared to be the most im- portant postwar technological innovations in steam-electric generation were used as defining characteristics for tech- nologically homogeneous cells of generating plants. These cells were thought to represent technological change more adequately than the vintage cells used by previous authors. To insure that inferences concerning the various cells were carried out under appropriate maintained hypotheses, a three-stage procedure was proposed. Under the assump- tion that an inappropriate model would be subject to statis- tically significant specification error, the first stage was to estimate a number of alternative models for each 173 cell and to apply three specification error tests to each estimated model. Among others, the potential alternatives included generalizations of the ZKD model in which output is a stochastic function of the inputs and firms are assumed to maximize expected profit, generalized production functions to allow for variability of returns to scale, and various fixed-relative-factor-proportions functions. The choice of a model for each cell was based on the specification error test results, relative values of R2 for the alterna- tives, agreement of signs of estimated coefficients with prior expectations, and freedom from such difficulties as multicollinearity. The model selected in Stage One was a fixed-relative-proportions model, with output and the num- ber of turbogenerators per plant assumed to be exogenous at the plant level. Similar models had been applied pre- viously to this industry by Komiya and by Hart and Chawla. In Stage Two, the chosen model was re-estimated, using a different data sample for each cell, and tested again for specification error, as independent confirmation of the choice made in Stage One. Furthermore, in each cell a test was made of the null hypothesis that the distribu- tion of disturbances for each component of the three- equation model had skewness and kurtosis parameters equal to those of the normal distribution. For those components and cells for which either hypothesis was rejected, it _._-‘0 Eff ' 174 was decided to use the Tchebycheff Inequality for hypothe- sis tests in Stage Three, since that inequality requires no distributional assumptions. In this way, it was hoped that "Student's" t-test would be used only in those cases where the normality assumption was warranted. In Stage Three, the three components of the fixed- relativeefactor-pr0portions model were again estimated, using a third data sample from each cell. The following null hypotheses were tested, using the estimated results: a. Steam—electric plants are subject to constant returns to scale. This hypothesis was rejected for most cells. b. TechnOIOgical innovations have not affected the pro- duction function parameters across cells, except as they have allowed generating plants to capture economies of scale. This null hypothesis was not rejected for most cells, with one exception. For the most modern plants in the sample, technological change has taken the form of substitution of capital for the other factors. c. The number of"turbogenerator"units in a plant does not significantly affect either factor proportions or re- turns to scale. This null hypothesis was not rejected for most cells. In addition to the results of the hypothesis tests, the estimated results indicated some other facts about steam-electric generation. First, in the context of a ‘13. . ’l.‘ 3‘ ‘Inn.&~ I 175 fixed-relative-proportions model, fuel may be considered the "limiting" factor, in the sense that increases in out- put require proportionately greater increases in the fuel input than in the other inputs. Since despite this fact, inputs are assumed to be combined in fixed proportions, 0 one concludes that capital and labor are underutilized in .1 the generating process. Second, plants incorporating the most modern technology are least suitable for generating low annual outputs. This helps to explain why all plants :J being constructed at a given time do not incorporate the most modern technology available. Third, variability of returns to scale does not appear statistically significant within any technological cell. However, the most advanced plants to use subcritical steam generators appear to have exhausted all potential economies of scale. For the most part, these conclusions seem consistent with the results of previous investigators. In the con- cluding section of this chapter, a short discussion of the limitations of this study is presented, along with some implications for policy and for further research. V.B. Implications of the Results The fundamental conclusion of the present study may be stated as follows. The-steam—electric generation pro— cess is subject to increasing returns to scale at the plant 176 level. Some postwar technological innovations, notably those that increased turbogenerator temperatures and pressures, have improved plant efficiency by allowing plants to capture the potential economies of scale. No specific postwar innovation appears to have increased plant efficiency independently of scale effects. Moreover, the most modern subcritical plants appear to have exhausted all potential scale economies using that method. This conclusion speaks well for the influence of re- gulatory agencies in the power industry. The fact that generating plants have grown larger and more efficient under regulation appears to invalidate the arguments cited in Chapter I contending that regulation leads to inefficiency of the regulated industry. In fact, considering the state— ment in Chapter II that innovation in this industry appears to be an inductive, trial-and-error process, fair-rate-of— return regulation may have stimulated innovation by re- lieving firms of part of the potential loss from expensive unsuccessful experiments. Of course, any conclusion of this sort would require more detailed study of the his- torical patterns of innovation and rate schedules of specific firms. A second implication of the conclusions of this study is that the trend toward concentration of generating equip- ment in fewer, larger generating plants is both explicable 177 and desirable economically. As emphasized in the discus- sion of results, the more advanced levels of technology appear to improve Operating efficiency only in plants pro- ducing relatively large levels of output. Of course, this is due to the fact that successful innovations have im- proved efficiency by making economies of scale accessible through higher design temperatures and pressures. It is not surprising that a regulated profit-maximizing firm will increase its scale of plant to achieve economies of scale as these become technologically attainable. Furthermore, the improved efficiency in the use of limited supplies of fossil fuels is undeniably a benefit to society of this concentration. V.C. Limitations of the Study and Suggestions for Future Research Procedural Limitations The results of this study are conditional on several details of procedure, which should be discussed explicitly. The first of these details concerns the relationship be- tween technological change and the capital input. Not only is capital equipment in a given generating plant heterogeneous, but in a time—series study of the present type, the array of potential kinds of capital equipment changes as technology advances. An example of this type of secular change in the concept of capital is the fact 178 that the term "boiler", common in 1948, has since been replaced by the term "steam generator“, so that supercri- tical turbogenerator units may be adequately described. For these reasons, no single physical measure of capital was available for the present study. Measures based on the monetary value of capital, being calculated for the benefit of tax accountants rather than observing economists, appear to be unsatisfactory also. In the present study, the proxy chosen for capital was installed plant capacity. Within any of the technolo- gically homogeneous cells, this proxy is likely to be fairly satisfactory, since the type and quantity of ancillary equipment--preheaters, superheaters, etc.-~is likely to be roughly uniform for all plants in a given cell. However, it should be pointed out that in this framework, such sub- stitution of capital for other factors as plant automation will not be recorded. Therefore, it is entirely possible that with some alternative measure of capital or classifi- cation system for generating plants, the fixed-relative- proportions model employed in the present study would have been completely untenable. To the extent allowed by avail- able data, significant examples of this type of factor substitution were incorporated in the cell definitions, with the results that alternative measures of the capital input might have caused more significant changes in the intercept of the capital input equation across cells. 179 A second procedural limitation of the present study was the limitation of the range of potential alternative models to those that could be estimated by a single-equation technique. This limitation was caused by a lack of ade- quate factor price data and the fact that the specification error test statistics used in Stages One and Two have been developed only for single—equation ordinary least squares estimation. This limitation precluded direct consideration of models of the Dhrymes and Kurz variety, in which plants are assumed to combine substitutable inputs in an effort to minimize the cost of generating an exogenously deter- mined level of output. However, the effect of failure to incorporate the cost-minimization conditions into an equa- tion system to be estimated, known as simultaneous equa- tion bias, did not appear to be statistically significant in the model estimated for the purpose of statistical in- ference. The specification error tests are designed to reject models in which the simultaneous-equation problem is statistically significant. Limitations of Scppe Three limitations were placed on the sc0pe of the present study for reasons of convenience, the time period of the data sample, and the lack of quantifying social costs and benefits. While these limitations in no way in- validate the results, they do suggest caution in their application. 180 First, for the sake of convenience no consideration was given to the process of transmission of electricity after generation. The present results strongly indicate that productive efficiency is increased if generating faci- lities are concentrated in large units and plants, which are then able to achieve economies of scale. But it should be borne in mind that in areas of low population density, this concentration of generating capacity is likely to raise transmission costs substantially. Other conditions equal, transmission losses increase with the distance be- tween the source of electricity and the site of consump- tion. In recent years, transmission losses have been reduced by increasing the voltage at which the electricity is sent from the generating plant. But in turn, high- voltage transmission requires a greater investment in transmission equipment--heavier wires, stronger towers, and larger transformers. Therefore, in any managerial decision concerning the construction of a new plant, the gains in generating efficiency must be weighed against possible resulting increases in transmission costs. To the author's knowledge, no economic investigation of electricity transmission has been carried out to date. Another limitation in the scope of the present study is exclusive concentration on fossil-fueled thermal elec- tricity generation. Hydroelectric plants were excluded 181 from the study for the sake of convenience, and nuclear plants were excluded due to the fragmentary nature of re- ported experience with nuclear generation that existed at the time data were collected for the present study. Since that time, production data on nuclear-fueled plants have become available, as have more extensive data on plants using exclusively supercritical steam generators. The present study could be extended fairly easily to include these data. The third and most implicit limitation on the SCOpe of the present study is the exclusive concentration of the production function analysis on those inputs which are pur— chased in market transactions—-capital, fuel, and labor. In recent years concern has mounted over pollution of air with the by-products of combustion and thermal pollution of the source of cooling water with hot water from steam generators. Looked at another way, these types of pollu- tion may be viewed as the use of two factors not purchased in the market--clean air, and water of a suitable tempera- ture to support aquatic life. Concentration of generating facilities, which the present study has shown to improve the efficiency of fuel use, tends to concentrate the use of these two factors in fewer areas. The implications of this fact should be considered. 182 On the one hand, this concentration of generating equipment redistributes the cost of generating the nation's electricity from society at large to those living near the large plants. This inequitable redistribution of social cost may be considered undesirable. But on the other hand, the devices used in large plants to improve combustion efficiency simultaneously reduce the emission of unwanted by-products; while SOphisticated condensing equipment used in large plants raises steam cycle efficiency as it reduces the temperature of used water returned to its sources. Thus, it is not clear that efficient use of factors pur- chased in the market is incompatible with general reduction of the social costs of electricity generation. No considera- tion was given to these questions in the present study, due to a lack of data and adequate techniques for measuring social cost. Suggested Further Research Some related research is suggested immediately by a consideration of the limitations of the present study. For example, by simply updating the study from 1965 one could broaden it to include more observations on plants incorporating supercritical steam generators, as well as the class of plants using nuclear fuel. 183 The functions developed in the present study relate inputs of three factors to the output of electricity, for plants incorporating various levels of generation technology. A similar approach could be used to relate plant output to available electricity for users, under various types of transmission technology. In this way, a link could be established between factor inputs and the usable output of electricity in various technologically homogeneous cells, with the cells defined in terms of both generation and transmission equipment. With this type of link, one would be better equipped to examine the trade-off between gains of generation efficiency and losses of transmission efficiency resulting from concentration of generating equipment in fewer large plants. In a similar way, the output of undesirable by-products of combustion could be related to the input of fuel in plants of various well-defined technological types. If one had apprOpriate data, it would then be possible to examine the impact of technological change and economies of scale on air pollution by steam-electric generating plants. In conclusion, the present study appears to have demon- strated a viable approach to problems of technological change and returns to scale in steam-electric generation. 184 The results of the study are of course conditional on that approach, but they appear consistent with the results of previous studies carried out using different approaches. It appears that the method may be extended to more recent technological changes, and to a broader concept of plant efficiency, in which transmission costs and the social costs of electricity generation are included. APPENDICES Appendix A Results of Stage One In this Appendix, the results are presented for Stage One of the empirical analysis, in which a model was tenta- tively selected as a restricted maintained hypothesis for each of Cells I through IV. The results are presented in Table 12 in the following manner. For each model considered, and for each of the cells used in Stage One, the values of R2 and three specification error test statistics are pre- sented. The statistic produced by RESET is F, that of WSET is W, and that of BAMSET is M. In addition, if a model was rejected by a test, that is indicated by one or more asterisks following the test statistic. One asterisk represents rejection at the 10% level, two represent re— jection at the 5% level, and three represent rejection at the 1% level. All statistics presented are the output of DATGEN, a program designed to carry out the specification error tests. 185 186 Table 11 Results of Stage One Eq. (III-41a): CD Cell No. I II III IV Stat:::::“‘:\_ ' F 1.6071 u.o777** 3.5501** 3.6739* w .5711*** .6989*** .0685*** .7464*** M 18.3359*** 4.9286* 51.2817*** 3.2072 R2 .9890 .9902 .9589 .9978 Eq. (III-45): CES Cell No. I II III IV Statistiz“\~,l P 3.7787** 4.5784*** 4.3156*** 3.7309** W .5233*** .7323*** .0986*** .7655*** M 17.6391*** 2.7689 44.4970*** 4.2259 R2 1.0000 .9991 1.0000 .9906 187 Table 11 (cont'd.) Eq. (III-49): CDl . Cell No. I II III IV St;:I;:iZ“~0,b F 1.6002 4.0776** 4.0662** .6739 w .5710*** .6989*** .0596*** .7ueu*** M 18.33l3*** 4.9296* 56.4919*** .2067 R2 .9890 .9902 .95u9 .9978 Eq. (III-50): CESl Cell No. I II III IV Sta::::lz“"‘-0,_: F 3.u932** 4.5785*** 4.7836*** 3.7160** w .5178*** .7323*** .093u*** .7657*** M 18.1493*** 2.7689 48.0752*** 9.1975 R2 1.0000 .9950 .9909 .9853 Eq. (III-56): CD2 Cell No. I II III IV StatiZIIZ“‘~!.E F 1.3603 u.0777** 4.0706** 3.6606** w .5687*** .6989*** .0596*** .7463*** M 18.1744*** 4.9287* 56.5339*** 3.1823*** R2 .9892 .9902 .9550 .9978 ‘ 188 Table 11 (cont'd.) Eq. (III-57): CESZ ~.Cell No. I II III IV Statistic F 3.9932** 4.5785*** 4.7836*** 3.7160** w .5178*** .7323*** .0939*** .7657*** M 18.1493*** 2.7689 48.0752*** 9.1975 R2 1.0000 .9950 .9909 .9853 Eq. (III-60a): LUHCC Cell No. I II III IV Statk F .7390 1.3095 N/A 2.2357 w .7211*** .9752*** N/A .3959*** M 9.5762 9.9807*** N/A l3.6106*** R2 .8719 .7996 .8217 .8558 Eq. (III-60b): LUHCF Cell No. I II III IV Stati::i;“~!.= F 1.0353 4.1498** 2.7376* 4.0156** W .4365*** .7016*** .0700*** .6946*** M 16.6246*** 9.6867*** 48.5665 2.9456 R2 .9861 .9900 .9568 .9966 189 Table 11 (cont'd.) Eq. (III-60c): LUHCL Cell No. I II III IV Stati::ié\s F 5.3996*** .9789 N/A N/A w .7599*** .6540*** N/A N/A M 9.0251** .9663 N/A N/A R2 .9929 .9312 .5619 .7171 Eq. (III-61a): LEHCC Cell No. I II III IV Stati;::;\\‘p F .0135 .2999*** 9.095599 .9237*** w .7933*** .7652*** .8138*** .7157*** M 1.0905 .0280 6.1490** .2170 R2 .8719 .7996 .8217 .8558 Eq. (III-61b): LEHCF Cell No. I II III IV StatiEIEEFNSE F 5.0075*** .5063*** 125.5156*** 3.9265** w .4548*** .7064*** .8042*** .6981*** M 20.5522*** .5929 6.0813** .9928 R2 .9861 .9900 .9568 .9966 190 Table 11 (cont'd.) Eq. (III-61c): LEHCL Cell No. I II III IV Statik F 1.5355 3.1933** 11.9313*** 1.1927 w .7321*** .71860*** .8151*** .7618*** M .0539 .5792 1.5208 7.1331** R2 .9929 .9312 .5619 .7171 Eq. (III-63): LING Cell No. I II III IV Sta;:::13\\k F 4.8002*** 2.8242** .3638 1.2839 w .6970*** .7296*** .7236*** .7952*** M 2.6565 3.0592 1.6176 .1623 R2 .3012 .7191 .3750 1.0000 Eq. (III-64): CD/N Cell NO. I II III IV Sta;:;:I3‘\E F 1.1290 4.5113*** .8350 .9672 W - .6359*** '.6769*** .7100*** .6368*** M 8.0869** 2.7919 1.3358 7.3137** R2 .9964 .9907 .9998 .9982 191 Table 11 (cont'd.) Eq. (III-65): CES/N ' Cell No. I II III IV Stati::IC\5p F 1.1980 .2323 .9285 1.1515 w .6154*** .6651*** .6270*** .5829*** M 11.5573*** 12.3708*** 3.1220 1.5599 R2 .9956 .9992 .9998 .9968 Eq. (III-66a): LEHCC/N Cell No. I II III IV StatIZIIE“. F 2.1190 6.9671*** 1.3297 3.1780** w .7888*** .7990*** .7705*** .7519*** M 2.9883 2.8539 5.2865* 1.8188 R2 .8967 .8063 .9109 .9300 Eq. (III-66b): LEHCF/N Cell No. I II III IV Statistic F 7.6798*** 2.7097 .2296 1.1163 w .7654*** .6471*** .7578*** .7597*** M 1.3912 18.8550*** 1.1927 4.6826* R2 .9996 .9858 .9962 .9969 192 Table 11 (cont'd.) Eq. (III-66c): LEHCL/N Cell No. I Stat:::?b§L F 1.1909 5.1355*** 9.2395*** 2.3192* w .6511*** .7991*** .8308*** .7239*** M .5919 7.6183** .9808 7.2869 R2 .7182 .5098 .6830 .6893 : *A‘ Appendix B The Results of Stage Two In Table 13 are the values of R2, /BI, g2, and the results of the specification error tests RESET, WSET, and BAMSET obtained when the model {LEHCC/N, LEHCF/N, LEHCL/N} was re-estimated, using a second data sample. The null hypothesis being tested with the specification error test statistics is that the disturbances are i.i.d. N(¢, 02I). The null hypothesis being tested with the aid of /EE and g2 is that the skewness and kurtosis parameters are zero and 3, respectively, as they would be if the true distri- bution of disturbances were normally distributed. For those equations and cells for which the value of a statistic indicates rejection of the null hypothesis, the level of significance of the rejection is indicated with asterisks-- l for the 10% level, 2 for the 5% level, and 3 for the 1% level. 193 194 Table 12 Results of Stage Two Eq. (III-66a): LEHCC/N . Cell No. I II III IV Stati::;:r\‘\‘_ F 1.2116 9.1878*** 1.2205 1.6070 w .7578*** .8216*** .7835*** .6390*** M 2.6792 9.5109*** .8732 11.2815*** R2 .8988 .8660 .8970 .9631 /Ei .7048** 1.0055*** 1.0095*** -.3907 g2 3.9025 5.17399“ 3.9976 9.710989 Eq. (III-66b): LEHCF/N Cell No. I II III IV Statistic F 4.1869** 2.7781* 1.1988 3.799999 w .3905*** .6333*** .6966*** .7810*** M 15.4880*** 5.0655* 1.9300 .9919*** R2 .9900 .9929 .9957 .9981 “31 -.2960 2.6979*** .9525 .0169 g2 1.8786*** 15.2708*** 2.6582 2.3119 195 Table 12 (cont'd.) Eq. (III-66c): LEHCL/N Cell No. I StatIZII;\\3EP F .0990 6.3280*** 3.6512** .9876 w .07577*** .6674*** .8088*** .6933*** M 9.7150* 10.698l*** 2.1001 9.1019 R2 .6356 .6186 .6110 .7937 /51 .3952 l.5459*** .2870 -.8979 g2 3.0017 6.0484*** 2.7377 2.9765 Appendix C Data Used in the Study The data used in the present study were collected for 175 privately-owned central-station steam-generating plants which first began operations during the period 1948 to 1965, inclusively. These data were stored on magnetic tape for computer use, one tape file being allotted to each plant included in the sample. These plants and the firms by which they were owned are listed by file number in Table 14. An attempt was made to include all plants constructed during the 1948-1965 survey period; however, a few plants, for which the necessary data were not complete, were drOpped from the sample. Data on design characteristics of newly installed turbogenerator units were collected from the Modern Plant Design Surveys published annually by Ppppp magazine.l Specifically, data were collected for each new unit on the furnace type and fuel used, number of bleedpoints and number of reheat cycles for the turbine, operating tempera- ture and pressure of the steam generator, and both the lPower,New York: McGraw-Hill. Monthly, (1948—1965). 196 197 system and medium used to cool the generator. Data were also collected on the degree of plant automation. Un- fortunately, these data were not sufficiently complete to be useful. On the basis of design characteristics of the original plant equipment, each plant was placed in a cell according to the definitions presented in Table 4 of Chap- ter II. If none of those cell definitions was apprOpriate, the plant was drOpped from the sample. For each plant in each cell, a time-series of annual production data was begun with the first year of plant operation. The production data were collected from the annual supplements to Steam-Electric Plant Construction Cost and Annual Production Expensesz, published by the Federal Power Commission. Each time-series was continued until either 1965 or the first year in which a turbo- generator unit was installed whose design characteristics did not match those of the cell in which the plant had been placed originally. The specific data that formed the sample for this study are listed by cell in Table 14 at the end of this appendix by plant file number. Using Table 13, the file number may be used to identify each plant and the firm by which it is owned. Output data are simply annual obser- vations on net generation of electricity, measured in 2Federal Power Commission, Steam-Electric Plant Construction Costs and Annual Production Expenses. Wash— ington, D. C. Published annually. 0" r _. 198 thousands of megawatt-hours.3 The capacity variable is net continuous plant capability when not limited by con- denser water. This figure, which is usually slightly larger than the nameplate rating suggested by the manufac- turer, is the absolute maximum output at which a turbo- generator may be safely Operated. Fuel input, measured in Btu's, is calculated as the sum over all fuel types of the products of the fuel input in tons (coal), barrels (oil), or cubic feet (gas) by the Btu content per ton, barrel, or cubic foot, whichever is appropriate. The labor input, measured in man-hours, is calculated as the product of average number of employees, as reported by the Federal Power Commission for each plant, by the average weekly working hours of power plant employees, as reported by the U.S. Department of Labor. It may be argued correctly that these data do not reflect the number of shifts of operation of the various plants. However, no superior data were available. Plant factor, reported by the Federal Power Commission, is designed to measure the ratio of ac- tual output divided by 8760 times the nameplate rating of installed turbogenerator units. An observed plant 3Ibid. 199 factor greater than 100 indicates that the plant for which it is observed Operated at an output greater than name- plate rating at some period during the year. Plant factor is reported as zero for all plants not in production a full year or those plants adding or deleting a unit during the year. The number of turbogenerator units operated for the entire year is reported for each year of observation. Finally, heat rate, a measure of plant efficiency used by engineers, is calculated as the quotient of fuel input in Btu's divided by output in kilowatt-hours. 200 Table 13 List of Plants in Sample, by File Number File «3000101014:me Firm Alabama Power Co. Alabama Power Co. Alabama Power Co. Southern Electric Generating Co. Arizona Public Service Co. Arizona Public Service Co. Arizona Public Service Co. San Diego Gas 6 Electric Co. San Diego Gas 6 Electric Co. Southern California Edison Southern California Edison Southern California Edison Southern California Edison Southern California Edison Southern California Edison Southern California Edison Southern California Edison Public Service Co. Public Service Co. Public Service Co. Connecticut Light Hartford Electric of Colorado of Colorado of Colorado 8 Power Co. Light Co. United Illuminating Co. Delaware Power 6 Light Co. Potomac Electric Power Co. Potomac Electric Power Co. Florida Power Corp. Florida Power Corp. Florida Power Corp. Florida Power 6 Light Co. Florida Power 6 Light Co. Florida Power 6 Light Co. Florida Power 6 Light Co. Florida Power 6 Light Co. Gulf Power Co. Gulf Power Co. Tampa Electric Co. Tampa Electric Co. Georgia Power Co. Georgia Power Co. Georgia Power Co. Georgia Power Co. Plant Barry Gadsden Greene County Gaston Cholla Four Corners Ocotillo Encina South Bay Alamitas Cool Water El Segundo Etiwanda Huntington Beach Mandalay Beach Redondo Beach San Bernardino Arapahoe Cameo Cherokee Norwalk Harbor Middletown Bridgeport Harbor Edge Moor Chalk Point Dickerson, Md. P.L. Barton Higgins Sewannee Cape Kennedy Ft. Meyers Palatka Cutler Port Everglades Scholz Lansing Smith J.J. Gannon Hooker's Point Harllee Branch Hammond Jack McDonough McManus 93 99 95 96 97 98 99 50 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 82 83 85 86 87 88 201 Table 13 (cont'd.) Georgia Power Co. Georgia Power Co. Central Illinois Light Co. Central Illinois Gas 6 Light Co. Central Illinois Public Service Co. Commonwealth Edison Co. Commonwealth Edison Co. Electric Energy, Inc. Illinois Power Co. Illinois Power Co. Indiana—Michigan Electric Co. Northern Indiana Public Service Co. Indianapolis Power 6 Light Co. Public Service Co. of Indiana Public Service Co. of Indiana Interstate Power Co. Interstate Power Co. Iowa Power 6 Light Co. Iowa Public Service Co. Iowa Southern Utilities Co. Central Kansas Power Co. Kansas Gas 6 Electric Western Power 6 Gas Co. Kentucky Utilities Co. Kentucky Utilities Co. Gulf States Utilities Co. Gulf States Utilities Co. Gulf States Utilities Co. Louisiana Power 6 Light Co. New Orleans Public Service, Inc. Southwestern Electric Power Co. Southwestern Electric Power Co. Southwestern Electric Power Co. Central Maine Power Co. ‘ Baltimore Gas 6 Electric Co. Holyoke Water Power Co. New England Power Co. Western Massachusetts Electric Co. Consumers Power Co. Consumers Power Co. Consumers Power Co. Consumers Power Co. Detroit Edison Detroit Edison Upper Peninsula Generating Co. Upper" Peninsula Power Co. Mitchell Yates E.D. Edwards Sabrooke Meredosia Ridgeland Will County JOppa Hennepin Vermillion Breed Bailly H.T. Pritchard Gallagher Wabash River Fox Lake Lansing Council Bluff Neal Bridgeport Colby Gordon Evans Cimarron River E.M. Brown Green River Roy S. Nelson Sabine Willow Glen Little Gypsy Michaud Knox Lee Lone Star Wilkes Walter F. Wyman Charles F. Crane Mt. Torn Brayton Point West Springfield J.H. Campbell B.C. Cobb Dan E. Karn J.R. Whiting River Rouge St. Clair Presque Isle Escambia 202 Table 13 (cont'd.) 89 90 91 92 93 94 95 96 97 98 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 Upper Peninsula Minnesota Power Minnesota Power Northern States Northern States Northern States Power Co. 6 Light Co. 6 Light Co. Power Co. Power Co. Power Co. Mississippi Power Co. Mississippi Power Co. Mississippi Power 6 Light Co. Mississippi Power 6 Light Co. Kansas City Power 6 Light Co. Kansas City Power 6 Light Co. Missouri Public Service Co. Missouri Power 6 Light Co. Union Electric Co. Nevada Power Co. Nevada Power Co. Nevada Power Co. Sierra Pacific Power Co. Public Service Co. of New Hampshire Atlantic City Electric Co. Public Service Electric 6 Gas Co. Public Service Electric 6 Gas Co. Public Service Co. of New Mexico Consolidated Edison Consolidated Edison Long Island Lighting Long Island Lighting New York State Electric 6 Gas Corp. Niagara Mohawk Power Corp. Niagara Mohawk Power Corp. Carolina Power 6 Light Co. Carolina Power 6 Light Co. Carolina Power 6 Light Co. Duke Power Co. Duke Power Co. Duke Power Co. Duke Power Co. Montana-Dakota Utilities Co. Otter Tail Power Co. Otter Tail Power Co. Cincinnati Gas 6 Electric Co. Cleveland Electric Illuminating Co. Ohio Edison Co. Toledo Edison Oklahoma Gas 6 Electric Co. J.H. Warden Aurora Clay Boswell Lawrence Wilmarth Bison Jack Watson - Sweatt Delta Natchez Hawthorn Montrose Ralph Green Mexico Meramer Clark ‘" Reid Gardner Sunrise Tracey Merrimack B.L. England Hudson Mercer Reeves Astoria Ravenswood E.F. Barrett Port Jefferson Milliken Albany Dunkirk Ashville H.B. Robinson Weatherspoon G.G. Allen Dan River W.S. Lee Marshall Heskett Crookston Ortenville W.J. Beckjard Eastlake Niles Bay Shore Arbuckle .3 203 Table 13 (cont'd.) 135 Oklahoma Gas 6 Electric Co. Mustang 136 Public Service Co. of Oklahoma Southwestern 137 Metropolitan Edison Co. Portland (Penn) 138 Pennsylvania Electric Co. Shawville 139 Pennsylvania Electric Co. Warren 140 Pennsylvania Power 6 Light Co. Brunner Island 141 Philadelphia Electric Co. Cromby 142 Philadelphia Electric Co. Eddystone 143 West Penn Power Co. Armstrong 144 West Penn Power Co. Mitchell 145 South Carolina Electric 6 Gas Co. Canadys 146 South Carolina Generating Co. Urquhart 147 Central Power 6 Gas Co. J.L. Bates 148 Houston Lighting 6 Power Co. Sam Bertran 149 Houston Lighting 6 Power Co. Webster 150 Houston Lighting 6 Power Co. T.H. Wharton 151 Southwestern Public Service Co. Cuningham 152 Southwestern Public Service Co. Nichols 153 Southwestern Public Service Co. Plant "X" 154 Texas Electric Service Co. Eagle Mountain 155 Texas Electric Service Co. Morgan Creek 156 Texas Electric Service Co. Permian Basin 157 Texas Power 6 Light Co. Collin County 158 Texas Power 6 Light Co. Lake Creek 159 Texas Power 6 Light Co. River Crest 160 Texas Power 6 Light Co. Stryker Creek 161 Texas Power 6 Light Co. Galley 162 West Texas Utilities Co. Oak Creek 163 West Texas Utilities Co. Paint Creek 164 Utah Power 6 Light Co. Carbon County Section II 165 Utah Power 6 Light Co. Gadsby Section I 166 Utah Power 6 Light Co. Gadsby Section II 167 Utah Power 6 Light Co. Gadsby Section III 168 Utah Power 6 Light Co. Naughton 169 Northern Virginia Power Co. Riverton 170 Virginia Electric 6 Power Co. Portsmouth 171 Virginia Electric 6 Power Co. Possum Point 172 Appalachian Power Co. Clinch River 173 Appalachian Power Co. Kanawha River 174 Winconsin Power 6 Light Co. Rock River 175 Wisconsin Power 6 Light Co. Nelson Dewey . 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Harv aum.w ua.m amuuucn.a «mow no u “once a,. v“ .:-ziuuon?:.nlgam».o::r:uunga.pnzsuoquqaq.oaillusulanam an » pa.»oKa:lua;»;sr- :9 v» gong «go.» “9.5 uopc»u>.m «am» am » poouu mpnznzsz:mosr :.L=m.uen:i|;;waseiieswanvow.:slsallltmu#w mm » .=.m~ “,2111‘-_- my Hana muc.a ua=.c vumuccu.o pcoo a u uopom ...4mu -,::q:uoa¢;;:guuaa.m.alxuuaaa.oiupq.auo~m.m.:nr||u|uoow +‘ m In puma» -cz...l mu .uonm naua.a. ua=.o mamampuu.o .mom uq . poo». .0: -s.- :umnueu:a,auw.c¢;:ilruom.o. .p.cu~o‘m ”mum ya » .oopu 31-4-i;.:- n» “our one.» acm.o apnowuu.~ waaa ea 3 ”ovum ; oo-- ;:-;uomc.-: -;um.o:ll:a24¢m.a,s_.avmmuo.o.xlzlnlapu~oz oo txmaaglrlz-- papou.:!u:;-z;- a» moor .oom.a Hom.o mcwmama.o ”gum on a pawn» o2-4 .-.:mcmw‘.-. wpm.w1t;II;wcn.o;lsmmmuuou.a||!;&Illuuoa a » .-2z;»o~u»-;llz;- -;. a: new» wag.» u:n.° . Mucouuu.a mow: up . mmoe um 1-;;uI¢cngit;zimn:.miil:tawma.o;2;moxcuwm.p :IIIvIImboa ya » “coca :a-,-iiu- s! . uu “0mm ogc.v H~=.o u¢umawn.o nan» om a wagon u» ;---»cnc-::--uao¢~;rli! away: ixnpbuwwm.wz:zlill-wwcq- uu;l, » .uaqou.-1:‘ :. - a» Home ooc.= H a.” qumaxu».u ”we“ on a “noun u~_liiznivo¢pellanmva.wq wma.91!lau~;ooo.w.IJllll:»~Nu ¢. u .pooom nurl!zn-1; NM go)». o)c.c pma.o uummgwo.m puma 09 u uuowm 0¢g:..n::+cmm-aaazeao.uilzxyxgug.:alaubaouun.u «.ma m m ooawaes-::-.-z:- an ”can umou.u a muo.o pwmmuocm.o anmc a a ammo ..ca.s‘;:iimpnagillumumabsllutt~ua.oitmuHauuwo¢m ppm» ow‘ u .omuo . ‘iz . - . m. germ m~m=.~ awm.o waummuom.a auuq «o a omao ,- mu-;s:vi.uonc:|q.napo.o:. 4~w.p;|~uwau»ouoo 5mm. up» u -loumo ...... I: 21J+ ma .wo Hru up» Hwo “we ”we ape “um ”aw fluw Hum flaw “um .u~ ~u~ Huw .uu Hum nan; I u U- Iva, III!» 'IIID Ali--.-. 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Auzo.ou;;:::z- , . --- ”tau uoflp.m .0m.o ~.m.»w»q.o wqwa cu . omum Auau vmu~.=.ltlivgcm.n Iuapupamm.o- . mam» aw I!uxvlllill!.uaoum :v:;; ...;. ‘-u “Vac “emu.o .cm.o pwuwo.;..o map“ oo . owmo -. ”9»; .: wqmw.otellt.aom.o ,»¢oqoflou.o.iiiix-;womas. om .s-:?::.4:;..o.om--..f-. ‘ e 9.99 paom.m‘ aca.p ».~uoo-.~ mmum o . aqua -=»omu .mew.ms ouaw.a:-»~uouu»u.~ am~m mm .zlzle!;:lvzou»u ‘1 we)“ na.m.p ua».° ~u.ao¢w..o .~.. g» . o..u Homa.: umpa.w--tllruum.o-‘Numuppmo.uzii-ii!nmuuw am - -.Vo.uu.:tea -..f He»m no»..o uwm.o ~moum.»..u .aoo a» . om»m H~au ~u~m.~anr:--fioa.ou;HuP-oma.o-a m.ao. ‘bps: .i.::: n 1:.--1 . omuu .; “can ~uou.& “03.9 puwauuma.o «was ac n oguu -ucaw 1..»aoe.ozi!n§:poa.o -uummuomm.o iiII!I:.#w9 pa. u -.omow :u...%; ..ma P~mo.~ ~o..o puuoou~n.° .uoo a u oop~ I .ponm ;: -wg.»-zlz;¢oou.o -pgmmguoa..nlnl!eu;.~»«:1 mm .mmz--n:-!a:y emu. flewo gapu.o Mc..= puaouo¢m.m .»o~ no» u omam ”03‘ ..-nuqm.u szr, mou.o -Hupwmmma.o.l:g-:l .-ar:! ow -eiu;-a|1:.-; owe» ;+ 3 Home nc.~.o Ma..a Ho».ompu.a .»m~ am » coco we»; ppm~.~,a----~»~.c p».uo-o.olnnltulr~mou- I_c. -atni;..u‘xul::Irsg coco :-;‘ Hcmm “ou~.o Npm.c puw.gum».o «an. we m . oo~o i Boga #mgo.¢.;t-u-~pm.an u.om»-m.pnrn:llr:u.mu: m», m ooou 5:-ss-.p,: - ; .. “ea“ me‘.o ~uu.o ».c~mxum.m «mom a“ u com. g.~.»mi;::».uw.+ziiuz ~u9.a, p..9mm~a.o-.lzuct.uaom wm‘ u .. o~ao 1:- p¢mo Hpr.u ~uo.a uw-~aoo.v uuwo a~ » ueoao ;;:ui‘»o,o:s:u_uoa.u.||::zm»a.a -pMHoaomc.u;, -uuour 4‘ ma .Im uap.o.uxx.;a ‘s, .- we», gcow.o ~9m.a “cumuomm.o u»ao e u Macao z.,.g"ca~,i--p¢o~.@;e::--upu.aniwamumuwu.p|||||l|;uuu~ be M popuo-anzyas:ic-.:, page Awfiu.u . ~pu.a puapmuuu.. «moo mg u come .-rvzapo$¢zii-»pum.o- . 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Hx CHVCC+ HH CH mm+Hm CH Nm+Hx CH Hm+om n Hwoo+Hx CH ammo :oH HCmuHHHV wos+wq :H ma+wm :H Nu+Hx CH Ha+> :H u flwoo+flw :H mmo nod AmmIHHHV Show mEdz wmmm hmnEdz cowpmsvm pcwam mcwpmhwcww aflhpomamuemmpm way mo mHmUo: w>flpmcmmpa< “5239:001me manna. 228 Hm>+H2mC+Hw CH mu+mC CH C HH CH 2\HozmH NHH HommuHHHv H H H .N>+.zNH+Hw CH HC+HC CH C .C CH z\mo:mH NHH HnmouHHHv H H HH>+.zHC+HH CH HC+HC CH u .x CH z\oo:mH HHH HmmwuHHHv Egon mEmz mwwm amnesz cowpmsvm “swam wcwpmpmcmw UHLHUmHMIEmwpm may mo mamvoz w>wpmchmpa< CCCCHCCouuleCHCCB BIBLIOGRAPHY American Telephone and Telegraph Co. ."Profits, Performance, and Progress." New York. (1959). Anscombe, P., and Tukey, J. "The Examination and Analysis of Residuals." Technometrics. Vol. 5. (1963). pp. 191-160. Averch, B., and Johnson, L. L. "Behavior of the Firm under Regulatory Constraints." American Economic Review. Vol. 52. (December, 1962). pp. 1052—1069. Bain, J. S. Industrial Organization. New York: Wiley. ‘u .. (1959). ,, Barzel, Y. 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