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Measuring the impact of regulation in a dynamic context : an application to Bell Canada Patry, Michel 1987

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MEASURING THE IMPACT OF REGULATION IN A DYNAMIC CONTEXT: AN APPLICATION TO BELL CANADA By MICHEL PATRY B.A.A., Ecole des Hautes Etudes Commerciales de Montreal, 1978 M.Sc, Ecole des Hautes Etudes Commerciales de Montreal, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Economics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1987 © M i c h e l Patry, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Economics The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 1987.12.15 ABSTRACT In t h i s t h esis, a model of producer behavior for a regulated u t i l i t y that f u l l y takes into account the dynamic nature of the c a p i t a l accumulation process of the firm i s developed and empirically implemented using recent data on B e l l Canada. On the basis of t h i s model of producer behavior, loss formulae that approximate the value of foregone output due to imperfect regulation i n a dynamic context are derived and estimates of the deadweight loss i n the case of B e l l are provided. The estimation results indicate the importance of dynamic elements, such as expectations and adjustment costs of investment, i n modeling the behavior of B e l l . They also suggest that rate of return regulation may have affected the investment decisions of the u t i l i t y . TABLE OF CONTENTS i i i Abstract ... i i L i s t of Tables ... v Acknowledgement . . . v i i Chapter 1: Introduction ... 1 Chapter 2: A Dynamic Model of a Rate-Regulated Firm ... 16 2.0 Introduction ... 16 2.1 Some re s u l t s on the behavior of a regulated firm i n a dynamic context with quasi-fixed inputs ... 21 2.2 Concluding remarks ... 38 Chapter 3: Regulation and Firm Behavior: An Econometric Model for B e l l Canada ... 40 3.0 Introduction ... 40 3.1 A model of profit-maximization with exogenous outputs ... 45 3.2 A model of profit-maximization with endogenous output ... 62 3.3 Data section ... 67 IV Chapter 4: Estimation Results ... 82 4.0 Introduction ... 82 4.1 Empirical results for the constrained maximization model with exogenous outputs ... 83 4.2 Empirical results for the constrained maximization model with endogenous output ... 92 4.3 Concluding remarks on the estimation ... 105 Chapter 5: A Producer Prices Approach to Measuring the Loss of Output Due to Imperfect Regulation i n a Dynamic Environment ... 109 5.0 Introduction ... 109 5.1 A one-sector dynamic loss measure ... 114 5.2 Tentative results about the deadweight loss due to i n e f f i c i e n t regulation ... 127 5.3 A two-sector dynamic deadweight loss measure due to regulation ... 131 Chapter 6: Conclusion ... 141 Bibliography ... 149 Appendix A ... 159 Appendix B ... 171 LIST OF TABLES V Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Nomenclature of the variables i n the constrained maximization model B e l l data set: summary s t a t i s t i c s Gross user cost of c a p i t a l and allowed return on c a p i t a l for B e l l : 1952-1980 Excess allowed return on c a p i t a l for B e l l : 1952-1980 Parameter estimates: model with exogenous outputs Summary s t a t i s t i c s for the model with exogenous outputs Monotonicity and curvature properties on the cost function: model with exogenous outputs Summary s t a t i s t i c s on B e l l ' s technology: average values Parameter estimates: model with endogenous output (CAPM) Parameter estimates: model with endogenous output (DCF) Summary s t a t i s t i c s for the model with endogenous output Likelihood r a t i o t e s t on 8 .. 51 .. 73 .. 74 . . 75 .. 84 . . 85 . . 87 . . 90 . . 94 .. 95 . . 96 . . 98 Table 4.9 Estimated long-run t o l l e l a s t i c i t y of demand for B e l l , various studies Table 4.10 Monotonicity and curvature properties on the cost function: model with endogenous output Table 4.11 Summary s t a t i s t i c s on B e l l ' s technology: average values Table 5.1 Estimates of the deadweight loss due to i n e f f i c i e n t c a p i t a l accumulation Table A . l Data sources for the gross service price of c a p i t a l Table A.2 Data sources, various series ACKNOWLEDGEMENTS This d i s s e r t a t i o n has benefitted immensely from the comments, advice, and encouragement of the members of my supervisory committee, Hugh Neary (Chairman), Margaret Slade, and Erwin Diewert. Each of them w i l l immediately recognize the mark of his or her influence on t h i s d i s s e r t a t i o n . I want to express my gratitude to them. My studies at the University of B r i t i s h Columbia were made possible by the f i n a n c i a l support of the Ecole des Hautes Etudes Commerciales de Montreal and of the S o c i a l Sciences and Humanities Research Council of Canada. The department of Economics of UBC and the I n s t i t u t d'economie appliquee of the Ecole des HEC provided an excellent environment to study and undertake research i n economics. F i n a l l y , I would l i k e to thank my parents for teaching me the value of education and for t h e i r support, and my wife, Carole Hennessey, for her help, unending support and love during a l l those years. Without her continued encouragement, t h i s d i s s e r t a t i o n would not have been completed. I t i s to her that t h i s d i s s e r t a t i o n i s dedicated. CHAPTER 1 INTRODUCTION 1.1 In the l a s t decade or so the l i t e r a t u r e on the economics of regulation has become increasingly concerned with the c r i t i c a l evaluation of ex i s t i n g regulatory p r a c t i c e s . Deregulation has become a p o l i t i c a l topic and some regulatory reforms have been undertaken i n the United States and Great B r i t a i n , notably i n the a i r transportation and telecommunications sectors. In Canada the restructuring of regulatory i n s t i t u t i o n s has become an issue and studies on the eventual impact of deregulation have mu l t i p l i e d . The growing disenchantment with the performance of the regulatory system i n general has one of i t s roots i n the perceived ineffectiveness of many regulatory regimes i n fo s t e r i n g economic e f f i c i e n c y i n the production and d i s t r i b u t i o n of resources. As a r e s u l t , the case for more market competition and for new methods of regulating business practices has gained i n popularity. But the task of choosing among various i n s t i t u t i o n a l frameworks supposes that one could actually achieve a d e l i c a t e balance, since the costs of any system of regulation must be weighted against the costs associated with a l t e r n a t i v e regimes. As Demsetz (1969) warns, one must compare the actual or predicted performance of various e x i s t i n g or implementable schemes, rather than comparing the imperfect functioning of a given system to some t h e o r e t i c a l optimum. The fact that a given set of i n s t i t u t i o n s does not achieve a " f i r s t - b e s t " a l l o c a t i o n does not necessarily warrant i t s being therefore relinquished. This exercise, i n turn, requires that policy-makers have some information about the impact of d i f f e r e n t regulatory practices on the behavior of enterprises, and about the r e l a t i v e costs and benefits of al t e r n a t i v e regulatory schemes. The provision of such information has not been the focus of most of the l i t e r a t u r e on regulation. This l i t e r a t u r e has largely concentrated on s t a t i c (and sometimes dynamic) models of the behavior of regulated enterprises, seeking to determine whether the predicted behavior i s " e f f i c i e n t " or not. The most studied type of regulation i s the control of natural monopolies through "rate of return regulation". I t i s with t h i s type of regulation that the present thesis i s concerned. More p r e c i s e l y , the aim of t h i s d i s s e r t a t i o n i s the assessment of the impact of rate-regulation on the largest telecommunications enterprise i n Canada, B e l l Canada, i n a dynamic context. The dynamic character of the analysis here i s very important since rate-regulation i s generally perceived as a f f e c t i n g the investment decisions of regulated firms, thus i n t e r f e r i n g with the enterprises' intertemporal a l l o c a t i o n of resources. 1.2 The goals of t h i s thesis are: (i) the development of a t h e o r e t i c a l model of producer behavior for a regulated u t i l i t y that takes f u l l y into account the dynamic character of the c a p i t a l accumulation process of the firm; ( i i ) the empirical application of the model of producer behavior to B e l l Canada i n order to determine the basic c h a r a c t e r i s t i c s of the firm's production structure, to ascertain the importance of expectations and adjustment costs i n the c a p i t a l accumulation decisions of the company, and to i d e n t i f y the impact of regulation on the firm; ( i i i ) the d e r i v a t i o n of loss formulae that approximate the value of foregone output due to less than perfect (rate of return) regulation; and f i n a l l y , (iv) the measurement of some losses due to regulation. The t h e o r e t i c a l and empirical l i t e r a t u r e on the regulation of natural monopolies i s b r i e f l y reviewed, and the contribution t h i s thesis makes i s c l a r i f i e d i n the following paragraphs. 1.3 Under rate of return regulation, a firm must submit i t s p r i c e schedule to a regulatory commission or board (the Canadian Radio-Television and Telecommunications Commission i n the case of B e l l Canada) for approval-'-. The commission i s generally taken to set (or approve) prices that w i l l provide the u t i l i t y with a " f a i r rate of return" on i t s invested c a p i t a l . More pre c i s e l y , three quantities have to be determined by the regulatory authority: the rate base, the allowed rate of return and the allowed operating expenses. The rate base consists of the amount of c a p i t a l actually embodied i n the u t i l i t y ' s plants, and i s measured by the value of the firm's assets minus depreciation. The allowed rate of return i s , i n p r i n c i p l e , the rate of return that the u t i l i t y i s permitted to earn on i t s .rate base; t h i s should enable the firm to a t t r a c t new c a p i t a l and should constitute a f a i r reward to investors. In p r a c t i c e , the determination of the allowed rate of return i s one of the most complicated and c o n t r o v e r s i a l issues i n the regulatory process. The allowed operating expenditures, f i n a l l y , include a l l non-base-input costs that are deemed reasonable. Once these quantities are known to the regulators, the allowed rate of return i s applied to the rate base and the allowed operating expenditures are added to t h i s amount to obtain the firm's required revenues. A decision on a price vector that w i l l generate these revenues i s then arrived at. 5 The s t a t i c equilibrium of t h i s process has been captured i n an abstract model of the regulated firm proposed by Averch and Johnson (1962). In t h e i r seminal a r t i c l e , the authors derived two important propositions: regulated u t i l i t i e s w i l l overinvest i n c a p i t a l whenever the allowed rate of return exceeds the cost of c a p i t a l to the u t i l i t i e s and, under the same circumstances, regulated monopolies w i l l f i n d i t p r o f i t a b l e to "invade" other markets and cross-subsidize some of t h e i r a c t i v i t i e s . Numerous extensions, s p e c i a l i z a t i o n s and c r i t i c i s m s of the Averch-Johnson ("A.-J.") model have been worked out subsequently and constitute an important l i t e r a t u r e . The f i r s t A.-J. proposition has been tested empirically a number of times but the jury i s s t i l l out: a clear verdict has not yet been rendered because of c o n f l i c t i n g evidence. Takayama (1969), Zajac (1970, 1972), Baumol and Klevorick (1970), Sheshinski (1971), Bailey (1973) and McNicol (1973) developed, refined and corrected on a few points the o r i g i n a l model of Averch and Johnson. But the fundamental A.-J. r e s u l t , that i s the incentive that a regulated firm may have to use more c a p i t a l than a competitive ( e f f i c i e n t ) producer would use i n producing the regulated firm's output remained. In order to prevent any misunderstanding i t i s worth i n s i s t i n g on the d e f i n i t i o n of o v e r c a p i t a l i z a t i o n t h i s thesis adopts. Ov e r c a p i t a l i z a t i o n here means the incentive a regulated producer has to use more c a p i t a l than i t should to minimize i t s t o t a l production cost. I t does not mean that the regulated enterprise uses more c a p i t a l than an unregulated firm would i n producing a d i f f e r e n t output v e c t o r . 2 The s t a t i c theory of the regulated monopolist was expanded when the d u a l i t y of the model was developed by Fuss and Waverman (1977), Cowing (1978), Diewert (1981a) and Fare and Logan (1984). Sheshinski and Bailey also pioneered the welfare analysis of rate-regulation. But more w i l l be said about t h i s l a t e r . 1.4 The e f f e c t of uncertainty on the A.-J. model of firm behavior has been studied i n papers by Perrakis (1976 a,b), Peles and Stein (1976), Das (1980), Bawa and Sibley (1980), Burnes, Montgomery and Quirk (1980) and Braeutigam and Quirk (1984). The model elaborated and estimated i n t h i s thesis does not include a stochastic demand side but i t nevertheless retains some features of the models developed by these authors. In t h i s thesis as i n most of these papers, the firm, i n maximizing the expected discounted value of i t s p r o f i t s , has to make a decision on the l e v e l of i t s stock of c a p i t a l that depends on the r e a l i z a t i o n s of a number of variables whose values are not known with c e r t a i n t y at the decision point. Expectations thus need to be formed for some variables. The model of producer behavior presented i n Chapter 2 overlooks t h i s question altogether but the estimating model of Chapter 3 deals with i t and introduces r a t i o n a l expectations i n the estimation. Since investment decisions generally a f f e c t a business' p r o f i t a b i l i t y for a long period of time, c a p i t a l accumulation decisions are taken i n a forward-looking fashion. Myopic behavioral models of the firm i n which the c a p i t a l stock i s f r e e l y chosen i n each period, as are a l l other inputs, may s e r i o u s l y f a i l to capture the essence of the decision-making process of the enterprise. The seriousness of the shortcomings of such s t a t i c models depends mostly on the longevity of c a p i t a l goods, on the role played by expectations i n the decison-making process and on the existence of adjustment costs of investment. Recent advances i n the theory of the firm have focused on the dynamic elements that play a c r u c i a l role i n the determination of a firm's investment p o l i c y . These issues are p a r t i c u l a r l y relevant to the analysis of the behavior of regulated u t i l i t i e s since these firms are remarkably c a p i t a l intensive and, p r i n c i p a l l y , because a great deal of the debate about the e f f i c i e n c y of rate of return regulation has centered around the A.-J. o v e r c a p i t a l i z a t i o n r e s u l t . 8 Major departures from the s t a t i c t h e o r e t i c a l model of a firm's behavior i n a regulated environment can f i r s t be found i n Joskow (1972, 1973, 1974) and Klevorick (1973) who stressed the importance of regulatory lags, hence introducing an element of dynamics into t h e i r analysis. Adjustment costs of investment of the Lucas (1967), Treadway (1971) and Mortensen (1973) type were introduced i n the t h e o r e t i c a l model of the regulated firm by A. Marino (1978a, 1979), E l - H o d i r i and Takayama (1981), and Dechert (1984). This i s a s i g n i f i c a n t development since these costs have a d i r e c t impact on the investment decisions of the firm. Whereas E l - H o d i r i and Takayama and Dechert s p e c i f i e d the producer's problem as that of the maximization of discounted p r o f i t subject to a series of regulatory constraints, Marino assumed that regulation only required that the discounted sum of p r o f i t does not exceed a given percentage of the discounted sum of c a p i t a l cost. This i s the working hypothesis that t h i s thesis w i l l make. Hence, the form taken by the regulatory constraint i n t h i s d i s s e r t a t i o n relaxes the hypothesis that regulation i s binding i n each and every period; rather the regulatory constraint i s assumed to bind "on average" over a number of time periods-*. While Marino and E l - H o d i r i and Takayama found that the regulated u t i l i t y w i l l invest more than an unregulated concern, Dechert demonstrated that such a r e s u l t may or may not occur i n the steady state depending on the importance of scale economies (hence on the concavity of the u t i l i t y ' s revenue function). Notice however that those papers d i d not focus on the notion of o v e r c a p i t a l i z a t i o n retained i n t h i s thesis. This thesis w i l l show that a weak form of the o r i g i n a l A.-J. e f f e c t holds i n the presence of convex adjustment costs and under a regulatory constraint defined over the planning horizon of the firm (as i n Marino , 1978a, 1979; and Gollop and Karlson, 1980) i n a context of continuous planning by the firm: that i s when c a p i t a l i s treated as a quasi-fixed input. Other propositions concerning the bounds on the Lagrange m u l t i p l i e r and other aspects of the behavior of a regulated monopolist are also derived. Most of those t h e o r e t i c a l developments have not however found t h e i r way into the empirical l i t e r a t u r e on regulated u t i l i t i e s ; t h i s l i t e r a t u r e has been cast primarily i n a s t r i c t s t a t i c framework. I t i s to t h i s l i t e r a t u r e that the next section i s dedicated. 1.5 The f i r s t attempts to test empirically the v a l i d i t y of the A.-J. model are i n Spann (1974) and Co u r v i l l e (1974). Neither could r e j e c t the hypothesis of an A.-J. bias. Petersen (1975), Cowing (1978), Hayashi and Trapani (1978) and Pescatrice and Trapani (1980) also obtained r e s u l t s l a r g e l y i n favor of the A.-J. model, while Boyes (1976), Smithson (1978), Gollop and Karlson (1980), and Fuss and Waverman (1981) f a i l e d to confirm the basic propositions of the A.-J. model. But these papers do not reproduce the same tes t time and again. On the contrary, the reader i s presented with a wide d i v e r s i t y of estimation strategies and problem formulations. Each paper follows a d i f f e r e n t path to test the theory. For instance, while most studies estimated the cost function, C o u r v i l l e and Boyes aimed d i r e c t l y at the production function. A l l contributions, except those of Smithson and Gollop and Karlson, are based on s t a t i c (long-run) models of the firm. And a l l , except Fuss and Waverman, deal with a cross-section of firms operating i n the American e l e c t r i c i t y generation industry. This thesis borrows from many of these contributions on some points and departs from a l l of them on others. Following Smithson and Gollop and Karlson, a dynamic approach i s taken i n t h i s t h e s i s . Smithson i s the f i r s t who looked at c a p i t a l accumulation i n a non-static manner, doing so by introducing the assumption of a p a r t i a l adjustment mechanism for a l l inputs i n a b a s i c a l l y s t a t i c model. Gollop and Karlson, on the other hand, developed a dynamic, intertemporal model of choice which they estimated for t h i r t y - n i n e u t i l i t i e s over a five-year period. In t h e i r model, i t i s the longevity of the c a p i t a l goods that impels the firm to look farther ahead, and they s p e c i f i c a l l y abstracted from uncertainty, lags i n regulation or adjustment costs. In contrast, the estimated model of producer behavior i n Chapter 3 i s characterized by a forward-looking firm which has r a t i o n a l expectations about the r e a l i z a t i o n of unknown variables and faces convex costs of adjustment. Lags i n the adjustment process of (regulated) prices are also considered i n the empirical analysis. While previous empirical work has mostly centered on one single industry (with the ensuing r i s k that the evaluation of A.-J. models w i l l be based on t h e i r a p p l i c a b i l i t y to one p a r t i c u l a r sector of the economy), the empirical model i n t h i s thesis i s applied to the behavior of B e l l Canada, which operates i n the Canadian telecommunications industry. This thesis thus enlarges an already r i c h l i t e r a t u r e on the telecommunications industry i n Canada-*, and complements i n p a r t i c u l a r the study by Fuss and Waverman on the regulation of B e l l . In contrast to what i s done here, Fuss and Waverman used a s t a t i c model of the firm and s p e c i f i e d the regulatory constraint i n such a way that i t i s assumed to be revised on a yearly basis. F i n a l l y , i t should be pointed out that the estimated capital-accumulation equation of Chapter 3 transposes to a dynamic context the estimation strategy found i n Pescatrice and Trapani (1980) and also suggested i n Diewert (1981a). 1.6 Turning now to the welfare analysis of rate-regulation, one can go back to Wilcox (1966), Wein (1968) and Kahn (1968) for a preliminary q u a l i t a t i v e assessment of the costs and benefits of regulation. Already contained i n these papers i s the idea that, even i f monopoly i s an " e v i l " , i t does not l o g i c a l l y e n t a i l the d e s i r a b i l i t y of regulation. And conversely, any costs found to be induced by regulation do not by themselves invalidate the i n s t i t u t i o n . For as Schmalensee (1974) put i t : "there are no shortcuts, rules of thumb, or general theorems that the analyst can employ; detailed quantitative forecasts must be generated." Klevorick (1971) captured the e s s e n t i a l ingredients of a l l welfare analysis of rate-regulation when he suggested that society's problem may be that of choosing an "optimal" f a i r rate of return which need not coincide with the u t i l i t y ' s cost of c a p i t a l . A s i m i l a r re s u l t i s due to Sheshinski (1971) who demonstrated that, i n a one-consumer economy, some regulation i s always worthwhile. Later, Callen, Matthewson and Mohring (1976) used the framework developed by Klevorick and Sheshinski to examine numerically the e f f e c t a rate of return constraint has on outputs, costs, c a p i t a l i n t e n s i t y and welfare for various parameter values for Cobb-Douglas production and demand functions. Recently, Diewert (1981a) developed a producer price approach to measure the loss of e f f i c i e n c y due to regulation. His treatment, although p a r t i a l - e q u i l i b r i u m i n nature and r e s t r i c t e d to the production sector of the economy alone, attempts to capture the general equilibrium impact of rate-regulation. This impact, as Bailey (1973) had pointed out, involves not only the loss of production e f f i c i e n c y within the regulated sector but also the loss i n exchange e f f i c i e n c y : the loss of e f f i c i e n c y induced by the d i s t o r t i o n s between regulated and non regulated goods. Those sources of loss are taken into account by Diewert but the losses to J consumers are ignored: d i s t r i b u t i o n a l issues are not addressed i n t h i s framework. This thesis w i l l use the methodology developed by Diewert to develop one-sector and two-sector loss measures due to regulation and, by estimating the model of producer behavior developed i n Chapters 2 and 3, i t w i l l provide estimates of the loss of output due to the i n e f f i c i e n t regulation of B e l l Canada. This i s the f i r s t time that the deadweight loss of rate-regulation has been estimated using a model of welfare analysis. This thesis thus takes a step i n the d i r e c t i o n indicated by Schmalensee. 1.7 The plan of the d i s s e r t a t i o n i s thus the following: a t h e o r e t i c a l model of a regulated u t i l i t y i s developed i n Chapter 2; a stochastic" s p e c i f i c a t i o n of t h i s intertemporal model of producer behavior i s provided i n Chapter 3; the empirical results and their implications concerning the e f f e c t of regulation and the role played by the dynamic elements of the model are discussed i n Chapter 4 ; and, f i n a l l y , a welfare analysis based on a producer p r i c e approach to the measurement of the deadweight loss due to regulation i s presented i n Chapter 5. NOTES TO CHAPTER 1 See Waverman (1982). Many authors do compare the c a p i t a l stock used by a regulated producer to that employed by an unregulated firm that does not necessarily produce the same output l e v e l . The confusion created by the varying usages of "o v e r c a p i t a l i z a t i o n " i n the l i t e r a t u r e i s apparent i n the Presman and Carol (1971), E l - H o d i r i and Takayama (1973) and Presman and Carol (1973) exchange of views. Sheshinski (1971) and Marino (1978, 1979) also seem to adopt the second view of o v e r c a p i t a l i z a t i o n . Gollop and Karlson (1980) also used a sim i l a r s p e c i f i c a t i o n for the regulatory constraint. See Breslaw and Smith (1982b), Denny et a l . (1981a), Fuss and Waverman (1977, 1981), Kiss et a l . (1981) and Bernstein (1986, 1987). CHAPTER 2 A DYNAMIC MODEL OF A RATE-REGULATED FIRM 2.0 INTRODUCTION A very general intertemporal p r o f i t maximization model of a rate-regulated monopoly i s developed i n t h i s chapter. An informal description of the model i s given i n t h i s introduction and a formal presentation follows i n section 2.1. The remainder of the chapter deals with s p e c i f i c r e s u l t s pertaining to the e f f e c t of regulation on the c a p i t a l accumulation and output decisions of the firm. The model of producer behavior used here belongs to the family of "dynamic temporary equilibrium" or "intertemporal p r o f i t maximization with quasi-fixed inputs" models. Hicks (1939, ch. 15) was the f i r s t to use such a model. Malinvaud (1953), Dorfman, Samuelson and Solow (1958), and Diewert (1977) have developed Hicks' approach further. In Hicks (1939), producers are assumed i n every period to make production plans, specifying the input-output quantities they plan to trade over the following t'-period planning horizon ( f , the horizon length, i s a behavioral parameter treated as exogenous). The p a r t i c u l a r feature of these plans i s that they include the stocks of c a p i t a l that remain with the firms a l l through the planning horizon 1, 2, t ' . The stream of revenues expected by each firm consists of the net revenues (pr o f i t ) associated with each period of the horizon and the value of the stocks l e f t i n t ' . These stocks are distinguished from the flow variables on the basis that there are well defined market c l e a r i n g prices for every flow variable i n each period, whereas no such prices e x i s t for the stocks. In other words, the l e v e l s of the stocks at the beginning of each period constrain the producers for the length of the period, so that stocks are f i x e d i n the short-run. These stocks can, however, be varied i n the long-run since producers can use them more or less i n t e n s i v e l y and speed up or slow down th e i r depreciation by upgrading (investment a c t i v i t i e s ) or maintaining them (maintenance and repair a c t i v i t i e s ) . Hence the "quasi-fixed" nature of these inputs. To maximize the i r p r o f i t s , producers w i l l choose a p a r t i c u l a r time path for these stocks. The temporary nature of the model comes from the f a c t that producers plan for t' periods but carry out t h e i r buying and s e l l i n g decisions for period one only. At the end of period one, they can change the i r expectations about future business conditions and adjust stocks i n consequence. C l e a r l y , however, they cannot change the l e v e l of the stocks they have inherited from l a s t period's decisions and which are t h i s period's beginning stock l e v e l s . The dynamic aspect of t h i s model i s c l e a r : since no period i s self-contained, the decisions taken i n " t " w i l l a f f e c t next-period p o s s i b i l i t i e s , and i n turn depend on period "t-1" decisions as well as on pr i c e expectations for the future. The existence of flow variables having w e l l defined prices i s not d i f f i c u l t to accept. On the other hand, what do stocks consist of? Hicks (1939) refers to "plant s i z e " but Diewert (1981b) and Diewert and Lewis (1982) suggest these stocks may include: (i) inventories of goods i n process; ( i i ) bolted down units of various types and vintages of c a p i t a l stocks (as opposed to u n i n s t a l l e d pieces of c a p i t a l which can be thought of as flow v a r i a b l e s ) ; and ( i i i ) various reserves of renewable or unrenewable resources. As noted above, stocks are f i r m - s p e c i f i c assets that have no well-defined market p r i c e . The model i s thus f a i r l y general for i t endogenizes the depreciation rate of the stocks, allows for adjustment costs i n varying the stock l e v e l s and encompasses the "vintage" models of investment. If the depreciation rate i s assumed f i x e d and exogenous and "vintages" are ignored, i t reduces to a Jorgensonian model of investment with (possibly) adjustment costs. To f i x notation and background, suppose there are I outputs ( i = 1, — , I ) , J variable inputs Xj (j = 1, , J ) , and N types of stocks or c a p i t a l goods (which can t h e o r e t i c a l l y be distinguished according to t h e i r vintages and other c h a r a c t e r i s t i c s ) s n (n = 1, N). Let y f c, x f c and s f c be the I, J and N dimensional column vectors^- of outputs, inputs and stocks i n period t. Also define the J -dimensional (column) vector of prices for the flow inputs i n period t : wfc = (wj, w$) T, the N-dimensional vector of regulatory (excess return) variables e f c = (e£, . . . , e J j ) T and the inverse demand functions i n t, cp t(Y t) = (cpftyt), . . . ^ { ( y t ) ) 1 = p f c = (pj, P J ) T where <P^(yfc) i s the i t h output p r i c e 2 i n period t . I t i s further assumed that x f c > Oj, y f c > Oj, s f c > C J J , wfc >> Oj, <pt(yt) = Pfc » Oj and that: Vycptfyt) = (Vycpjtyt), ... ,Vyq>f (y f c) ) T f with Vycpjty11) = tacpj/-ayj, (^pj/ ayj, a<pJ/ay{]T e x i s t . F i n a l l y , {x t}, {y*-}, {s t} and (e t> refer to the sequences of vectors (x^, x 2, x t ' ) , and so on; and denote the row vectors ( x 1 T , x 2 T , x t , T ) , ( y 1 T , y 2 T , y * 1 ' 1 ) , ( s 1 T , s 2 T , s t , T ) and (e 1*, e 2 T , e t , T ) by x T, y T , s T , and e T respectively ( r e c a l l that x T i s the transpose of x; hence x, y, s, and e are column vectors). Following Malinvaud (1953), Diewert (1981 b, c ) , and Diewert and Lewis (1982), write the one period technology set-* of the firm as S R = {(y, x, s^, s^)} where s^ > 0^ i s the beginning of the period and s 1 > 0 N i s the end of the period stock vector. S R, which could be indexed according to time, describes the set of tradeoffs open to the firm: i t can, given s^, produce more outputs or use less inputs at the cost of depleting or running down s^ - or, on the other hand, i t can use more x to produce less y and obtain a larger end-period stock of c a p i t a l s^ -. See Malinvaud (1953) for more d e t a i l . For given output levels y, stocks s^ and s^ and for w >> Oj, l e t the one period variable cost function C(t) be defined by: I min {w . x: (y, x, s°, s 1) e S R} I i f there i s such a vector i n S R; C(y, w, s°, s 1) = < I «>, otherwise. ...(2.1) C(t) i s a variable cost function with factors ( s u , si ) f i x e d i n the short-run. I t i s conceptually similar to the variable p r o f i t function discussed by Diewert (1981b) and Diewert and Lewis (1982). I t follows that the usual properties of a variable cost function apply. In p a r t i c u l a r , i t can be shown that C(y, w, s^, s^) i s a concave and p o s i t i v e l y l i n e a r l y homogeneous function i n w for fix e d (s^, s l ) 4 ; and i f d i f f e r e n t i a b l e , Shephard's lemma implies: x ( y , w , s ° , s 1 ) = V w C ( y , w , s ° , s 1 ) , . . (2.2) w h e r e x ( y , w , s u , s1) i s t h e v e c t o r o f c o n d i t i o n a l d e m a n d f u n c t i o n s w h i c h m i n i m i z e C ( • ) g i v e n t h e v e c t o r ( y , w , s ^ , s 1 ) , a n d V w C ( y , w , s ° , s 1 ) = [ac/awlf dC/aw 2, c > C / d W j ] T . A f o r m a l m o d e l o f a r e g u l a t e d m o n o p o l i s t f a c i n g t h e o n e p e r i o d t e c h n o l o g y s e t s { S § } a n d p l a n n i n g f o r t ' p e r i o d i s d e v e l o p e d i n t h e n e x t s e c t i o n . 2.1 SOME R E S U L T S ON THE B E H A V I O R OF A R E G U L A T E D F I R M I N A DYNAMIC CONTEXT WITH Q U A S I - F I X E D I N P U T S I n t h i s s e c t i o n , a n i n t e r t e m p o r a l m o d e l f o r a m u l t i p l e - o u t p u t , m u l t i p l e - i n p u t m o n o p o l i s t r e g u l a t e d t h r o u g h a n i n t e r t e m p o r a l r a t e o f r e t u r n c o n s t r a i n t i s d e v e l o p e d . I t i s a s s u m e d t h a t : ( A l ) t h e f i r m w a n t s t o m a x i m i z e i t s n e t p r e s e n t v a l u e a n d c h o o s e s a p l a n f o r t h e c h o i c e v a r i a b l e s x f c , y f c , s*-a c c o r d i n g l y . I n e a c h p e r i o d t h e e n t e r p r i s e i s s u p p o s e d t o p l a n f o r t h e f o l l o w i n g t ' p e r i o d s . T h a t i s , t h e r e i s c o n t i n u a l p l a n n i n g r e v i s i o n , with t' as the ( f i n i t e ) length of the planning horizon (see Hicks, 1939; Diewert, 1981b). This section w i l l consider the non-stochastic version of the model only. A stochastic s p e c i f i c a t i o n i s provided i n Chapter 3. The monopolist i s facing three types of constraint: (i ) the firm must choose (x f c, y f c, s f c) that are f e a s i b l e given S§; ( i i ) i t faces a common-carrier o b l i g a t i o n which means that i t must service a l l comers at the regulated p r i c e ; and ( i i i ) i t i s l i m i t e d to (x t, y f c, s t) that provide i t with at most a f a i r (or allowed) rate of return on i t s c a p i t a l stock. Formally translated, these constraints imply: (x f c, y f c f s f c) e S§, ...(2.3) pt = q>t(yt ) f ...(2 . 4 ) D(Tt) < f (x, y, s, e) , ... (2.5) where D(n.) i s the discounted sum of p r o f i t s and f ( y , x, s, e) i s the regulatory constraint function. To specify the form of rate-regulation, the following functional form for f i s imposed: (A2) f(yT, x T f S T # eT, = E ^ R f O , t) e* s t _ 1 ; e t > 0, ... (2.6) where R(0,t) i s the present value i n period zero of a d o l l a r i n period t, e f c = (e£, e^, e^, e ^ ) T i s the vector of excess return on the various components of the c a p i t a l stock, and the allowed excess return which i s assumed non negative by d e f i n i t i o n applies to the stocks at the beginning of each period. (2.6) corresponds to the "cycle constraint" s p e c i f i e d i n Marino (1978, 1979) and c l o s e l y resembles the constraint s p e c i f i c a t i o n of Gollop and Karlson (1981).^ The import of (A2) i s that regulation constrains the firm to earn, over some horizon, no more than an allowed return on i t s c a p i t a l . But t h i s allowed return i s not less than the u t i l i t y ' s cost of c a p i t a l . The excess return on c a p i t a l , e, i s thus p o s i t i v e or n i l . Writing the producer's problem using the one period v a r i a b l e cost function and (2.3)-(2.6), the regulated monopolist behaves as i f solving: Max (1-u) {E£' R(0,t) [ <p t(y t)y t - C ( y t , w t , s t " 1 , s t ) ] t t {y r},{s r} + R(0,t') Q s t'} + u [ R(0,t) e t s t - 1 ] , ...(2.7) / where u i s the Lagrange m u l t i p l i e r associated with the regulatory constraint and Q i s the scrap value of stocks at t' . I t w i l l prove useful to introduce i n the analysis the p o s s i b i l i t y that the firm indulges i n some "rate-base padding" or "gold-plating" a c t i v i t i e s . To do so i t i s convenient to define the following quantities: R = [R(0,1), R(0,2), R(0,t')] ; ...(2.8a) V(y,R) = R(0,t) cp t(y t) yt ; ...(2.8b) C(y, w, s, R) = [R(0,t) C(t)] - R ( 0 , f ) s^Q ; ...(2.8c) zt = (zl[ , z^ , Zy z j ) T > Oj ; ...(2.8d) vt = ( v j , v^ , ..., v£ , v^ ) T > 0 N ; ...(2.8e) I T 2T t- 1 T T z = ( z x l , z Z i , 1 P ; ...(2.8f) I T ? T t'T T v = ( v x , v z l , v r 1 r . ...(2.8g) Respectively, (2.8) describes a discount factor vector, discounted t o t a l revenues, discounted t o t a l costs minus the market scrap values of the f i n a l period stocks, vectors of unused (from a productive point of view) units of variable inputs (z) and vectors of unused units of c a p i t a l stocks (v). To deal with the p o s s i b i l i t y that the u t i l i t y may f i n d i t advantageous to buy unused factors of production i n order to manipulate the regulatory process, (2.7) can temporarily be modified i n the following way: Max L' = (1-u) [V(y,R) - C(y, w, s, R) iYt ), (s f c } (z f c }, {vfc } - 3^11(0, t) (wfc z t) - G(v,R)] + u E ^ R(0,t) e fc(s t _ 1 + v t - 1 ) , y f c > 0, s f c > 0, z f c > 0, v f c > 0, u > 0 ; ...(2.9) where G(v,R) i s the discounted cost of buying and i n s t a l l i n g unused c a p i t a l stocks. These unproductive units of c a p i t a l can be conceived of as slack variables. And the degree of slack, which may vary according to time and type of c a p i t a l , i s measured here by [ vjj / (sjj + v^ ) ]. I f the optimal l e v e l of slack i s zero, there i s no rate-padding, gold-p l a t i n g or X - i n e f f i c i e n c y going on. On the contrary, i f [ v£ / (sJi + vjj ) ] i s p o s i t i v e , i t i s because the u t i l i t y i s wasting resources. Assume that: (A3) every function i n (2.7) and (2.9) i s once continuously d i f f e r e n t i a b l e ; and that (A4) OG/avg) > 0. Padding (holding unused capi t a l ) i s therefore not costless by d e f i n i t i o n of G(v,R). As a r e s u l t , a s o l u t i o n (y, s, z, v, u ) to (2.9), i f i t e x i s t s , must s a t i s f y the following necessary ( f i r s t - o r d e r ) Kuhn-Tucker conditions^: 27 Conditions M f (1-u) [ 3V/"&yJ - 3C/3yJ ] < 0, Y\ J ( 1 -U ) y^ [ 3V/ayJ - ac/ay£ ] = 0, ... (2.10a) ... (2.10b) ( i = 1, ..., I) and (t = 1, t ' ) . ^ (1-u) [ -dC/as£ ] + u R(0,t+1) e £ + 1 < 0, .(2.10c) < (1-u) [ -3C/3s^ ] + s£ u R(0,t+1) e ^ + 1 = 0, n x n n n n ,..(2.10d) : (n = 1, ...,N) and (t = 1, t ' - l ) . ^ (i-u) [ -ac/as£'] < o, ^ J n \ * * 4 - I ~ * 4 - I (i-u) s^ [ -ac/as^ ] = o, : n = 1, , N. (2.10e) (2.10f) / * t ' (1-u) [ -R(0,t) wV ] < 0, (1-u) z* [ -R(0,t) ] = 0, (2.10g) (2.10h) : (j = 1, J) and (t = 1, t') V n f * t- * t-+i ' (1-u) [-dG/dv^ ] + u R(0,t+1) L < 0, ...(2.10i) * * f r *t * t- + 1 (1-u) [ -dG/av^ ] + v£ u R(0,t+1) e £ + 1 = 0, (n = 1, N) and (t =1, t ' - l ) . ...(2.10J) f 'n (1-u) [ -9G/Bv^' ] < 0, (1-u) v£ [ -dG/dv^ ] = 0, : ( n = 1, . . . , N) . . . . (2.10k) ...(2.101) u ( t ~V(y,R) + C(y, w, s, R) + E ^ R l C t ) (wfc z f c ) * t - 1 t- * t - - 1 * t - - 1 + G(z,R) + E^ = 1R(0,t) ( 1 + 1 ) ] > 0, ( 2 (2.10m) u [ -V(y,R) + C(y, w, s, R) + E ^ R ( 0 f t ) (wfc zt ) ^+ G(z,R) + E^R(0,t) et ( s ^ 1 + v*" 1 ) ] = 0 , > > ( 2 > 1 Q n ) * * + - * + - * 4 - * + -u > 0; Y ^ > 0 ; s n - 0 > Z j ^ O ; v n ~ ° ..(2.10o) Conditions M are sim i l a r to those i n Marino (1978), i n which i d l e factors of production are not allowed, and p a r a l l e l those i n Bailey (1973) who dealt with a purely s t a t i c problem. Establishing bounds for u i s important since t h i s Lagrange m u l t i p l i e r plays a key role i n the i n t e r p r e t a t i o n of the f i r s t - o r d e r conditions (2.10) and, i n p a r t i c u l a r , i n assessing the impact of regulation on the firm. But the endogeneity of u renders other proofs on i t s bounds inappropriate. Hence the r e s u l t s i n Bailey (1973), Marino (1978) and Diewert (1981a) do not necessarily extend to t h i s context. Thus the f i r s t proposition derived from Conditions M establishes bounds on u. Proposition 2.1 Assuming G(v,R) i s an increasing function * of v (hence using (A4)), 0 < u < 1 i f at l e a s t one e^ > 0. In general, u < Min (OG/av^ )/[OG/av; ) + e ^ 1 ]}, {n,t} n n n where e£ = R(0,t) e£. Proof Using (2.101): (1-u) [cK3/av^ ] > u e^ , and using (2.10o): u > 0. Thus, 0 < u < [aG/av^ ] < 1 i f e l 1 > 0. OG/av 1 ] + e t + 1 n n Q.E.D. Notice that proving Proposition 2.1 makes e x p l i c i t use of the assumption (dG/avJj) > 0. T h e f o l l o w i n g p r o p o s i t i o n w i l l p r o v e u s e f u l when t h e i s s u e o f o v e r c a p i t a l i z a t i o n i s c o n s i d e r e d . I t e s t a b l i s h e s i n a s t r a i g h t f o r w a r d m a n n e r a s i g n r e s t r i c t i o n o n [ ^ C / ^ s J j ] , f o r a l l n . R e c a l l t h a t , f o r c o n v e n i e n c e , e*- = R ( 0 , t ) e t . P r o p o s i t i o n 2.2 V t C ( y , w , s , R)=[dC/ds^, dC/ds^ ] >0N f o r t = 1, t ' - l , a n d e £ > 0. P r o o f U s i n g (2.10c) a n d P r o p o s i t i o n 2.1: u e 1" 1 < (1 - u ) V t C ( y , w , s , R) ; s r e a r r a n g i n g , u s i n g (A2) a n d P r o p o s i t i o n 2.1 a g a i n g i v e s : V t C ( y , w , s , R) > u (1 - u ) 1 e r + 1 > 0 N . s T o s e e how t h i s r e s u l t h e l p s i n d e t e r m i n i n g t h e i m p a c t o f r e g u l a t i o n o n t h e c a p i t a l a c c u m u l a t i o n d e c i s i o n s o f t h e f i r m , c o n s i d e r t h e f i r s t - o r d e r c o n d i t i o n (2.10d). W e r e u = 0, s o t h a t t h e f i r m i s i n e f f e c t i v e l y r e g u l a t e d , t h e n d C / d s ^ = 0, a n d t h e e f f i c i e n t ( c o s t - m i n i m i z i n g ) p r o d u c e r c h o o s e s t o i n c r e a s e s j j u p t o t h e p o i n t w h e r e t h e m a r g i n a l c o s t o f d o i n g s o i s j u s t o f f s e t b y t h e m a r g i n a l g a i n s a n i n c r e a s e i n s £ w i l l b r i n g i n (t+1) t h r o u g h r e d u c e d v a r i a b l e c o s t s . Remember t h a t i n c r e a s i n g t h e s t o c k l e v e l i n t means a m o r e i n t e n s i v e u s a g e o f v a r i a b l e i n p u t s o r t h e b u y i n g o f m o r e f l o w v a r i a b l e s w h i l e , a t t h e same t i m e , i t w i l l r e d u c e t h e variable cost of producing any output l e v e l next period. By contrast, when u > 0 the firm i s e f f e c t i v e l y regulated and Proposition 2.2 implies that 3C/ds£ > 0. Thus, the regulated producer increases s£ up to the point where a marginal p o s i t i v e change i n t h i s stock pushes up today's variable costs more than i t reduces tomorrow's. A common assumption i n the l i t e r a t u r e i s that C(y, w, s, R) i s convex i n s (see Marino, 1979; E l - H o d i r i and Takayama, 1981). Then c l e a r l y stocks are b u i l t up i n t more than they would be under e f f i c i e n t production circumstances. Proposition 2.4 below w i l l formalize t h i s l i n e of thought. Before that, the optimal l e v e l of slack i s shown to be zero. Proposition 2.3 i s based on the following assumption, which should appear quite r e a l i s t i c a f t e r the above discussion: (A5) V fcC(y, w, s, R) << V .G(v,R), for a l l y,s. s v That i s , the increase i n t o t a l costs brought about by a per-unit increase i n a component of s i s smaller than the incremental cost associated with a per-unit increase i n a component of v. The reason i s i n t u i t i v e . Augmenting the l e v e l of the productive factor s£ contributes to reduce variable costs i n the future, c e t e r i s paribus. On the other hand, increasing the l e v e l of unused stock v£ implies immediate additional costs but no gain i n the future by d e f i n i t i o n of v£. * t * t Proposition 2.3 Assuming ( A l ) , (A5), y >> 0, s >> 0 N and e f c >> 0 for a l l t, then the optimal * t l e v e l of slack i s zero. Hence, z = 0 T u * t and v = 0XT for a l l t. N * t Proof (2.10d) and (2.10i) y i e l d , given s >> 0 N: (1-u) [-V tC(y, w, s, R)] + u e t + 1 = 0 N > s (1-u) [-V G(v,R)] + u e t + 1 . v Using (A5), (l-u)t-V fcG(v,R) ] + ue^ •L<<0N. And from (2.10j), i t follows that v f c = 0 N for a l l t < t ' . For t = t ' , (2.10e) and (2.10k) together with the assumption above imply (2.10k) holds as a s t r i c t inequality. And, by (2.101), vfc'= 0^. z f c = 0Jf for a l l t, follows from (2.10h), Proposition 2.1 and w^  >> 0 T. Q . E . D . This proof extends to the dynamic context the important r e s u l t o r i g i n a l l y due to Bailey (1973) i n the s t a t i c case about the irrelevance of X - i n e f f i c i e n c y , rate-base padding, etc... under r e a l i s t i c assumptions. An immediate implication of Proposition 1.3 i s that regulated producers can be taken to operate on t h e i r production f r o n t i e r . This fact i s important and makes i t possible i n the next chapters to focus on the a l l o c a t i v e e f f i c i e n c y costs of regulation and to neglect resource costs due to technical i n e f f i c i e n c y per-se. I t should be remembered, however, that t h i s r e s u l t c r u c i a l l y depends upon ( A l ) . The common b e l i e f i n the presence of rate-padding and gold-plating may be compatible with other objective functions. Using t h i s r e s u l t , and assuming an i n t e r i o r solution obtains, the f i r s t - o r d e r conditions (2.10) can be rewritten as the following set of necessary conditions associated with the s o l u t i o n to problem (2.7): R(0 , t ) ( l - u ) [ ( p r ( y ) + V cp^Y) Y ~ V C f y S w r , s t _ i , sZ)] = O j , t = 1, ..., t ' ; ...(2.11a) (1-u) [ R(0,t) (-V ^ ( y * , wfc, s t _ 1 , s*1 ) + R(0,t+1) s *t-+1 t- + l *t *t + l * (-V ^C(y , w t + i, s C , s r + i ) ) ] + u R(0,t+1) e t + i = 0 , t = 1, ..., t'-1; ...(2.11b) 34 * * 4 - I •(- I * 4 - I _1 * 4 - I (1-u) [R(0,t') (-V ,C(yz , w\ s r , s ) + Q] = 0 . s ...(2.11c) Now define: mfc E m (y f c) = -7 <pt(yt) y f c , t = 1, ..., t' ; ...(2.12a) PL = [u R(0 ft+1) e l A / ( l - u ) ] > 0 N . ...(2.12b) Substituting (2.12) into (2.11) y i e l d s the following * t * t set of necessary conditions when s >> 0N , y >> 0^. Conditions R: (1-u) R(0,t) - mfc - 7 yC(y t , w f c , S t _ 1 , tt)] = 0 I , t = 1, ..., t 1 ; ...(2.13a) [R(0,t) (-V ^ ( y * 1 , wfc, s t - 1 , tt)) + R(0,t+1) (-7 t C ( y t + 1 , s s 4-J-1 * 4 - + 1 f w , S ^ , s " 1 ) ) ] + u = 0 N , t = 1, . . . , f ; . . .(2.13b) (1-u) [R(0,t") (-7 ,C{y^ , w , s r x , s c) + Q)] = 0„ . s c ...(2.13c) These "Conditions R" have a nice economic in t e r p r e t a t i o n . A regulated firm constrained by a f a i r rate of return c e i l i n g on i t s p r o f i t s w i l l pick a vector of outputs such that marginal cost i s less than p r i c e . If mfc i s thought of as a markup vector i n period t, (2.13a) emphasizes that non-zero mfc leads the firm to act as i f i t were a competitive firm facing prices (p f c - mt) instead of p f c. Thus, i n t u i t i o n suggests that a regulated firm w i l l produce "too l i t t l e output". A similar reasoning applies to (2.13b): instead of choosing {s t} that minimizes the cost of producing {y t} at {w*-}, the firm's behavior implies i t chooses the end-period stocks i n t i n such a way that the marginal cost of adding one unit of c a p i t a l at the end of t i s larger than the incremental savings an addit i o n a l u n i t w i l l bring i n (t+1): the firm "overshoots" the optimal l e v e l of stocks. In general, then, i n t u i t i o n suggests that "too much" c a p i t a l w i l l be used by a regulated firm. In order to gain a better understanding of t h i s l a t t e r phenomenon, imagine that a competitive or e f f i c i e n t firm i s asked to produce the outputs chosen by the regulated firm, {y*-}, and the l a s t period stock vectors, s^' . The c a p i t a l stocks that would solve the e f f i c i e n t producer's problem, say {s t}, would s a t i s f y : Min. { s r } R(0,t) C(y * t t t-1 . . . (2.14) w s The necessary f i r s t - o r d e r conditions for t h i s problem are: *r t t--i - t- *t-+i t-+i R(0,t) V t C ( y c , wr, s c i , s c ) + R(0,t+1) V t C ( Y r , w r + 1, s s i s t + 1 ) = 0 N , t = 1, . . . , t ' - l . ...(2.15) These are also s u f f i c i e n t i f the matrix of second order 1 t' -1 der i v a t i v e s of (2.15) with respect to (s-1-, s ) i s p o s i t i v e d e f i n i t e at {s t}. Denote t h i s matrix by H. But a stronger characterization i s necessary to obtain a cle a r o v e r c a p i t a l i z a t i o n r e s u l t . Therefore, assume that * t t t-1 t (A6) C(y , w , s , s ) i s convex i n the stocks * and consider the system of equations (2.16) i n (y,w,s,T,u.) * t - t- t - - i t- * t - + i +1 t-R(0,t) V .C(y , wc, s r \ s c ) + R(0,t+1) V tC(y t v , s r , s c s^ s t + 1 ) + x u f c = 0 N , t = 1, ..., t ' - l . ...(2.16) When x = 1, (2.16) i s equivalent to (2.13b) and when = 0, i t reduces to (2.15). Therefore, as x increases from 0 to 1, (2.16) transforms the f i r s t - o r d e r conditions for an e f f i c i e n t firm into the "dist o r t e d " f i r s t - o r d e r conditions (2.13b)'. Now, by (A6), H i s posit i v e d e f i n i t e . The nonsingularity of H allows one to use the i m p l i c i t function theorem to express the s-solution to (2.16) as functions o f the (given) variables y, w, u., x (hence taking u, as a parameter vector) i n a close neighborhood of {§*•}. The question of in t e r e s t regarding the use of c a p i t a l i s whether** s(y, w, u,, 0) < s(y , w, u., 1). Proposition 2.4 demonstrates that t h i s i s the case for s£ i f the following extra assumption i s made: (A7) l e t Un > 0 a n d Urn = °» f o r a 1 1 9 * t a n d m * n -Proposition 2.4 I f (A6) and (A7) hold, and treating u. as a vector of parameters, then the introduction of rate-regulation w i l l lead to o v e r c a p i t a l i z a t i o n i n s£ i f e^ +^ > 0. Proof Let be a ( ( f x N) - N) row vector with zeroes everywhere except i n the ( ( t - l ) N + n ) t h place and [ u. ] T ® E £ be the Kronecker product. Then, I/ds£/dT] = [ u ] T ® E ^ H - 1 , where H - 1 i s the inverse of H and i s po s i t i v e d e f i n i t e by (A6). Since y.^  i s po s i t i v e by (A7), the sign of the above derivative i s po s i t i v e . Now, using the mean value 38 theorem: s£(l) - s£(0) = [3sg(a) /ax ] > 0, where a e [ 0,1 ]. Q.E.D. 2.2 CONCLUDING REMARKS This concludes the b r i e f exploration of the dynamics of the model of producer behavior introduced i n t h i s chapter. The necessary conditions for a p r o f i t maximizing p o s i t i o n were derived and shown to be d i s t i n c t from that of an e f f i c i e n t firm (conditions R). Four propositions were derived concerning (i) bounds on the Lagrange m u l t i p l i e r , ( i i ) bounds on the f i r s t - o r d e r conditions, ( i i i ) the use of unproductive units of c a p i t a l stocks and flow services (which were shown never to be pr o f i t a b l e ) and (iv) an A.-J. e f f e c t . 39 NOTES TO CHAPTER 2 1. Notation: x 1 means the transpose of vector x. w x, where both w and x are vectors, i s the dot product: w x = Ej w-j X j . And w >> Oj means W j > 0 for a l l j while w > Oj means W j > 0 for a l l j but W j > 0 for some j . 2. Notice that t h i s formulation allows for interdependent demand functions since the i t " inverse demand function <pt(yt) has the whole y t vector for argument. 3. For the remainder of t h i s section, time superscripts w i l l be omitted. 4. I t can also be shown that, i n general, C(t) i s nonincreasing i n the components of s^ and nondecreasing i n the components of s^ i f S R s a t i s f i e s free disposal i n the stocks. See Diewert and Lewis (1982) for the proof i n the context of a p r o f i t function. 5. This i s a very p a r t i c u l a r way of specifying the regulatory constraint. Most authors use a period-by-period constraint. This l a s t hypothesis appears u n r e a l i s t i c for i t implies a continuous adjustment i n the regulatory process. In addition, (A2) i s the most trac t a b l e s p e c i f i c a t i o n for estimation purposes and for convenience i s maintained through t h i s t h e o r e t i c a l section. See Gollop and Karlson (1981) for a t h e o r e t i c a l , development of a shorter period constraint; note that they move to a s p e c i f i c a t i o n l i k e (A2) for estimation purposes. 6. By (A2) and (A4), [ e ^ + 1 + (dG /avt ) ] > o, for a l l n and t and w!j- > 0, for a l l t and j . Thus the Fromovitz-Mangasarian constraint q u a l i f i c a t i o n i s met and conditions (2.10) are necessary. 7. ^ i s interpretable as a regulation-induced d i s t o r t i o n i n the shadow value of c a p i t a l perceived by the firm. Treat t h i s as a parameter. Then for any x e [0,1] equation (2.16) i s the relevant f i r s t - o r d e r condition for a firm to minimize cost when perceiving a shadow value that includes the d i s t o r t i o n -cu>. 8. Let u T = ( u 1 T , u t , _ 1 T ) . CHAPTER 3 REGULATION AND FIRM BEHAVIOR; AN ECONOMETRIC  MODEL FOR BELL CANADA 3.0 INTRODUCTION In t h i s chapter, econometric models of the behavior of B e l l Canada are presented. The object of the empirical implementation of these models i s manyfold. In the f i r s t place, i t w i l l provide an econometric foundation to the model of producer behavior developed i n Chapter 2. Secondly, i t w i l l generate additional information on the production structure of B e l l Canada. Thirdly, the empirical r e s u l t s w i l l help to ascertain the importance of dynamics and the impact of regulation on the u t i l i t y ' s decisions. F i n a l l y , the estimated models w i l l allow the computation of the loss of output due to less than perfect regulation. B e l l i s the most important enterprise i n the Canadian telephone industry. I t operates i n a l l of Ontario and Quebec as well as i n other parts of eastern Canada and accounts for roughly 60% of the whole industry's output, labor force and equipment. Considered a "natural monopoly" by the federal government, the enterprise i s regulated by the Canadian Radio-Television and Telecommunications Commission (CRTC) through a procedure which c l o s e l y resembles the regulatory framework analyzed i n the f i r s t chapter of t h i s thesis 1. Because of i t s sheer importance i n terms of employment, output, etc... and because of the recent debate 2 concerning the structure of regulation i n t h i s p a r t i c u l a r industry, the Canadian telephone sector appears to be an i d e a l candidate to implement empirically the model of Chapter 2 and evaluate the loss measures to be developed i n Chapter 5. The s e l e c t i o n of B e l l Canada i s j u s t i f i e d by two c r i t e r i a : i t s importance and the fact that r e l i a b l e , firm-s p e c i f i c , and not t y p i c a l l y "accounting" data are available for nearly a t h i r t y - y e a r period. The cost structure of B e l l Canada has often been investigated with rather sophisticated econometric methods. But, as was pointed out i n the Introduction, none of the published studies has yet incorporated the two e s s e n t i a l l y dynamic aspects of the c a p i t a l accumulation process that are expectations and the presence of adjustment costs i n a model of a regulated firm. U n t i l very recently, a l l empirical studies of B e l l ' s behavior and technology have postulated a s t a t i c framework i n which c a p i t a l i s a variable factor of production and prices are known with c e r t a i n t y to the u t i l i t y . Examples are Kiss et a l . (1981), Fuss and Waverman (1977, 1981), Denny et a l . (1981a), and Breslaw and Smith (1982b, 1983). Also, a l l but one study abstracts from any e f f e c t regulation might have had on the firm's decisions. Fuss and Waverman (1981) attempt to incorporate the e f f e c t of regulation through an "A.-J. e f f e c t " and to t h i s end develop a very comprehensive model of a rate-regulated firm. But t h e i r empirical estimation of both a short-run and a long-run version of t h i s model performed "poorly", i n the words of the authors (see p. X and pp.136-141) though a standard p r o f i t -maximizing model led to very good r e s u l t s . M u l t i c o l l i n e a r i t y was suspected by the authors as the primary factor responsible for the disappointing empirical results (see bottom of p. 141). More recently Bernstein (1986, 1987) estimated one-output and two-output dynamic systems of input demand equations for B e l l Canada that are characterized by r a t i o n a l expectations and convex costs of adjustment (but ignoring regulation). This contributes somewhat to bridging the gaps i n the l i t e r a t u r e . This thesis departs from those studies, and more generally from the e x i s t i n g l i t e r a t u r e on the estimation of the technology of regulated monopolies, i n the following ways: (i ) regulation i s allowed to have an impact on the investment and output decisions of B e l l i n a model that incorporates three e s s e n t i a l l y dynamic features: r a t i o n a l expectations, convex adjustment costs and lagged adjustments of prices to t h e i r desired l e v e l s ; ( i i ) the regulatory constraint i s s p e c i f i e d i n a s l i g h t l y d i f f e r e n t way than i n most t h e o r e t i c a l or empirical work; and, ( i i i ) two d i s t i n c t measures of the user cost of c a p i t a l are used i n the estimation i n order to check the robustness of the r e s u l t s . Two basic models are a c t u a l l y developed and estimated, each using two d i f f e r e n t c a p i t a l cost variables. The f i r s t i s a constrained model of profit-maximization i n which B e l l does not control the prices and levels of i t s outputs and the second i s a profit-maximization model i n which one output i s endogenously determined. Both models, i t should be noted, allow the i d e n t i f i c a t i o n of a l l the relevant parameters necessary to accomplish the purposes of the empirical work defined i n the f i r s t paragraph of t h i s chapter. Moreover, since profit-maximization implies cost-minimization, there i s one main advantage and one major drawback to maintaining the hypothesis of endogenous output. On the p o s i t i v e side, the added f i r s t - o r d e r condition may produce a gain i n e f f i c i e n c y i n terms of parameter estimates but, on the other hand, biased estimates may r e s u l t i f the u t i l i t y i s cost minimizing but not p r o f i t maximizing with respect to some of i t s outputs. The l a t t e r p o s s i b i l i t y i s probably remote for a standard business operation but not for B e l l . In f a c t , i t i s a matter of debate whether the u t i l i t y r e a l l y does control a l l of i t s prices and outputs. Since i t i s generally admitted that the prices of " l o c a l services" are set by regulators, the output levels of these services can then s a f e l y be regarded as exogenous to the firm since B e l l has the "common c a r r i e r " o b l i g a t i o n to service a l l comers at the regulated p r i c e s . T o l l prices, on the other hand, can be regarded as determined by the firm and approved by regulators, or as merely influenced by the u t i l i t y and b a s i c a l l y exogenous to i t . Fuss and Waverman (1977, 1981) have argued for the former hypothesis and b u i l t s t a t i c models based on i t . Kiss et a l . (1981) and Bernstein (1986, 1987) have opted for the second a l t e r n a t i v e . Since i t i s very d i f f i c u l t to determine which i s the best al t e r n a t i v e a p r i o r i , both models are estimated and the l i k e l i h o o d of the hypotheses i s judged by the o v e r a l l performance of the estimated models. 45 The structure of each model, i t s stochastic s p e c i f i c a t i o n , as well as the estimation strategy are presented i n the next two sections. A data section closes t h i s chapter. The empirical results are presented and discussed i n Chapter 4. 3.1 A MODEL OF PROFIT-MAXIMIZATION WITH EXOGENOUS OUTPUTS In t h i s two-output, three-input model of B e l l Canada, the prices and l e v e l s of the l o c a l and t o l l outputs are assumed exogenous to the decision making process of the firm. The model developed here captures many dynamic aspects of the c a p i t a l accumulation process that are absent from most of the l i t e r a t u r e . I t also takes into account the impact of regulation and, by using a f l e x i b l e functional form, imposes as few a - p r i o r i r e s t r i c t i o n s on the data as possible. But a number of s i m p l i f y i n g assumptions must s t i l l be made. These and the e s s e n t i a l c h a r a c t e r i s t i c s of the estimated intertemporal model of producer behavior are outlined below. (A8) The producer i s assumed to choose, i n each period, a plan (that i s , a vector of optimal l e v e l s for a l l the decision variables) that maximizes the net present value of the stream of p r o f i t s associated with i t . This i s i n accordance with the Hicks-Malinvaud-Diewert framework of Chapter 2. Generally, however, the producer w i l l carry on the execution of that plan only for the f i r s t period since a new plan w i l l be drawn next period that takes into account a l l the information then available. (A9) This maximization i s constrained by a c e i l i n g on the net present value of p r o f i t s imposed by the CRTC. This constraint i s known with certainty to the u t i l i t y and does not change with the passage of time. The intertemporal regulatory constraint thus takes the form i t has i n Chapter 2, which i s consistent with Marino (1978a, 1979) and Gollop and Karlson (1980). (A10)Bell's outputs are aggregated into two variables: l o c a l output and t o l l output. Although a much f i n e r disaggregation of output revenues i s availab l e for B e l l , econometric t r a c t a b i l i t y requires that a few aggregates be defined. Econometric studies of B e l l Canada have a l t e r n a t i v e l y used s p e c i f i c a t i o n s with one, two or three outputs. M u l t i c o l l i n e a r i t y problems would render the estimation of a three-output r e s t r i c t e d cost function probably very arduous since the output variables are highly c o l l i n e a r and, together with the technological change proxy and the c a p i t a l stock, would put the number of highly c o r r e l a t e d variables appearing on the right-hand side to f i v e . (All)Labor and materials are considered variable inputs, i . e . inputs whose le v e l s are being chosen i n each period given a complete knowledge of current p r i c e s . No costs beyond the purchase price of the inputs are incurred when the firm adjusts the levels of these inputs. (A12)A c a p i t a l aggregate i s assumed to e x i s t for B e l l ; i t i s treated as a quasi-fixed input with an exogenously determined rate of decay. Hence any regulation induced e f f e c t on the use of c a p i t a l i s hypothesised to a f f e c t the investment pattern d i r e c t l y since the f i r m does not control the r e a l rate of decay of i t s c a p i t a l . This, of course, i s a strong but standard assumption-*. Using the notation of Chapter 2, (A12) implies that s l = s 0 ( i _ 5 ) + j l t where s^ i s the beginning of the period stock l e v e l , the end of the period l e v e l , 6 the rate of depreciation or decay of the stock and I i s gross investment. A large number of production processes are compatible with (All)-(A12): the l e v e l of output i n any period t can be made to depend on s u , or on s^ -, for example; one can consider the existence of external or i n t e r n a l adjustment costs defined either over net or gross investment; and there may e x i s t delays i n the i n s t a l l a t i o n of the investment goods; etc... The s p e c i f i c a t i o n chosen i n t h i s section i s thus only one of a host of sensible representations of the technology set S R. Cle a r l y , there are many sp e c i f i c a t i o n s along the above l i n e s that are possible and that might be tested against the data. Some experimentation was ac t u a l l y done but there i s no guarantee that the adopted s p e c i f i c a t i o n i s the most appropriate. In p a r t i c u l a r , i t i s assumed that: (A13) the c a p i t a l goods i n s t a l l e d i n any period become immediately productive. This means that B e l l i n h e r i t s i n period t a stock of c a p i t a l from period t-1 and combines t h i s stock with the net investment i n the stock accruing i n year t to obtain i t s productive c a p i t a l stock, K>, that w i l l determine together with the variable inputs u t i l i z e d the output production l e v e l s i n t. There are no delivery lags or gestation period but, (A14)the accumulation or decumulation of the stock of c a p i t a l i s assumed to be subject to convex costs of adjustment. Furthermore, these costs are assumed to be strongly separable from the rest of the technology; that i s , the cost-minimizing variable-input demands are independent of the l e v e l of adjustment costs. This assumption i s made for the sake of econometric t r a c t a b i l i t y . Other s p e c i f i c a t i o n s were t r i e d but proved i n f e r i o r ^ . The o r i g i n s of those adjustment costs can be e i t h e r : (i) the costs associated with the reorganization of production, retooling and r e t r a i n i n g implied by the i n s t a l l a t i o n of the new equipment (i n which case these costs are " i n t e r n a l " to the firm) or, ( i i ) the costs associated with the need to r a i s e new c a p i t a l , or with the higher prices paid when the firm orders large quantities of c a p i t a l goods (i n which case these costs are said to be "external" to the firm). The l a t t e r source of costs i s more i n l i n e with the s e p a r a b i l i t y of those costs from the rest of the technology. To complete the characterization of the model two more assumptions have to be made: (A15)Bell i s assumed to be price-taker i n a l l input markets , and (A16)the l e v e l s of both t o l l and l o c a l output prices are assumed to be exogenously determined by the regulatory commission. The nomenclature of the variables entering t h i s model of producer behavior i s given i n Table 3.1, a complete de s c r i p t i o n of a l l variables used i n the econometric work i s given i n section 3.3, below, and i n Appendix A. TABLE 3.1 NOMENCLATURE OF THE VARIABLES  IN THE CONSTRAINED MAXIMIZATION MODEL l o c a l output quantity t o l l output quantity labor input quantity materials input quantity c a p i t a l stock price of l o c a l output price of t o l l output price of labor price of materials user cost of c a p i t a l (i=l,2) Lagrange m u l t i p l i e r allowed excess return on R t present value i n T of one $ i n t expectations taken i n x firm's technology set i n t proxy for technological change wfc / in* w t L f c + Mfc = Cfc / m t (A8)-(A16) imply that B e l l attempts ,in any year x, to solve the following problem: Ma£ E x E ^ ( l - u ) R(x,t) ( P L Y L + P T Y T " C ( Y L ' Y T ' m t ' (K. } - vt Kfc - 0.5 B (Kfc - K t _ 1 ) 2 } + (1-U) R(x,t') K t'Q t' + u { R(x,t) e f c K t _ 1 }, ...(3.1) where B i s a cost of adjustment parameter and u i s the Lagrange m u l t i p l i e r associated with the u t i l i t y ' s regulatory constraint^. Notice that adjustment costs are defined over net investment and assumed to be strongly separable from the res t of the technology. A more general s p e c i f i c a t i o n with i n t e r n a l adjustment costs could be conceived but, unless a-p r i o r i r e s t r i c t i o n s are imposed on the estimation, t h i s introduces too many parameters (see note 4 ) . The r e s t r i c t e d cost function appearing i n (3.1), C ( t ) , i s found by solving the following constrained minimization problem: Min { wfc L f c + mfc Mfc : (yj ,Y! ,Lfc ,Mt e S t }, ...(3.2) t t L , M where Sfc i s the technology set of the firm i n year t. The " s h i f t s " occuring i n the technology are captured by the technological change proxy i n (3.1). The r e s t r i c t e d cost function C(t) = C( y£ ,y{ , wfc , mfc ,Kt , Ft ) i s monotonically nondecreasing i n the input prices and the 53 outputs, nonincreasing i n K r, l i n e a r l y homogeneous and concave i n the input prices (w,m). The cost function i s also assumed to be twice continuously d i f f e r e n t i a b l e with respect to i t s arguments. the l e v e l of the variable inputs i n each period t given the p r i c e vector (w,m), the output levels and the stock of c a p i t a l . This solution, however, i s not independent of the dynamic elements i n the producer's problem since i t depends on Kfc. In f a c t , assumption (All) allows the producer's problem to be broken down into two stages: the f i r s t stage i s that of the short-run problem of choosing the l e v e l s of the v a r i a b l e inputs, whereas the-second stage corresponds to the long-run optimization problem described by (3.1). The maximization of (3.1) and Shephard's lemma y i e l d s the following f i r s t - o r d e r conditions at any year x: C(t) thus II solves" the firm's problem of choosing L x = a c(x ) / a w T , ... (3.3) MT = dC ( T)/dm x, ... (3.4) (1-u) {-3C (T)/dK T - v T - B(K T - K T~ 1)} + E {(1-u) R ( T , T + 1 ) B ( K X + 1 - K T )} + u R(x,t+1) e T + 1 = 0 . . .(3.5) 54 (1-u) {-dC(t' )/dKt' - v f c' - B(K t f- K^'"1)} + Qfc' = 0 . ...(3.6) Where Q stands for the scrap value of the firm's stock of c a p i t a l at the end of period t ' , hence (3.6) i s an end-point condition. In addition to the r e g u l a r i t y conditions on C ( t ) , a s u f f i c i e n t condition for a constrained optimum implied by (3.3)-(3.6) i s that ( 9 2 C / 9 K 2 ) > 0 and B > 0 or, more generally, that the objective function i n (3.1) be concave i n K. One way of looking at (3.3)-(3.6) i s to consider these equations as determining the optimal paths for {Lfc , Mfc , Kfc} given a l l future prices and output lev e l s and to t r y to solve for these paths. By assuming that a l l input prices and output lev e l s are known with certainty and expected to remain s t a t i c over time (actually the hypothesis of stationary expectations with respect to r e l a t i v e prices i s s u f f i c i e n t : t h i s implies that a l l prices and the discount rate change at a constant rate) and by using s p e c i f i c functional forms (such as a quadratic cost function; see Berndt, Morrison and Watkins, (1981)) one can solve e x p l i c i t l y the optimal control problem for the inputs t r a j e c t o r i e s ^ . In such a case, the long-run solution i s characterized by the equilibrium condition that the c a p i t a l stock remains constant from period to period. Once an expression for the long-run (steady state) c a p i t a l stock K** i s obtained, an approximation to the c a p i t a l accumulation equation i n the neighborhood of K** can be derived and estimated. An a l t e r n a t i v e way of looking at (3.3) -(3.5) i s suggested i n the papers by Pindyck and Rotemberg (1983a, b) and consists i n t r e a t i n g the f i r s t - o r d e r conditions as estimating equations. Notice that these equations hold n e c e s s a r i l y at every period T even i f the producer plans ahead up to period t' but r e a l i z e s i t s plan for only one period. Therefore, foregoing an e x p l i c i t solution for the optimal t r a j e c t o r i e s , one can look at (3.3)-(3.5) as regression equations once an operational d e f i n i t i o n of E T i s given: that i s , once an expectations formation process i s posited. This avenue offers the p o s s i b i l i t y of retaining both the generality of f l e x i b l e functional forms and the r a t i o n a l expectations hypothesis (RE). Experimentation with both a normalized quadratic r e s t r i c t e d cost function (as i n Denny, Fuss and Waverman, 1981b), and with a r e s t r i c t e d translog cost function (as i n Pindyck and Rotemberg, 1983b) led to the s e l e c t i o n of the second f u n c t i o n a l form. Therefore, the following r e s t r i c t e d translog cost function normalized by the price of materials i s s p e c i f i e d i n which c i s t o t a l variable cost normalized by the price of materials and w i s the normalized price of labor: l n c T = a 0 Q + a Q 1 l n wT + a Q 2 l n y x + a ^ l n y x + a 0 4 l n K T + a Q 5 F T + 0.5 [ a u ( l n wT ) 2 .+ a 2 2 ( l n y x ) 2 + a 3 3 ( l n ) 2 + ( l n K x ) 2 ] + a 1 2 l n wx l n y x + a 1 3 l n wx l n y x + a 1 4 l n wx l n K x + a 1 5 l n wx F x + a 2 3 l n y x l n y x + a 2 4 l n y x l n K x + a 2 5 l n y j F x + a 3 4 l n y x l n K x + a 3 5 l n y x F x + a 4 5 l n K x F x . ... (3.7) With t h i s s p e c i f i c a t i o n , the estimated f i r s t - o r d e r condition (3.3) can be expressed as i n (3.8), while the c a p i t a l accumulation equation (3.5) can be rewritten as (3.9). (wx L x ) / c x = a Q 1 + o t 1 1 l n wx + a 1 2 l n y j + a 1 3 l n y x + a 1 4 l n K x + a 1 5 F x . ...(3.8) 0 = [ a Q 4 + a 1 4 l n wx + a 2 4 l n y x + a 3 4 l n y x + a 4 4 l n K x + a 4 5 F X] (C X/ K X) + v X + B( K X - K x _ 1 ) - R ( T , T+l) E T [ B ( K X + 1 - K X) + P e X + 1 ], ...(3.9) where f5 = u/(l-u) i s an estimated parameter, u i s thus treated as a parameter, as i n Spann (1974), C o u r v i l l e (1974), Boyes (1976), Pescatrice and Trapani (1980) and Gollop and Karlson (1980). A more appropriate s p e c i f i c a t i o n would be a " r o l l i n g constraint" that was f u l l y consistent with the form of regulatory constraint used i n (A2), Chapter 2. This would require estimation of one regulatory parameter per period. The data however proved unable to y i e l d convincing estimates when more than one such parameter was used. Before discussing the estimation strategy and the nature of the expectations formation process, note that the hypothesis of li n e a r homogeneity of C(t) i n the input prices i s maintained through the normalization rule i n (3.7) as i s the assumption that C(t) has a symmetric Hessian matrix of pric e d e r i v a t i v e s . But the monotonicity and curvature conditions are not imposed and can be checked at a l l observation points. Also notice that the demand for materials i s replaced i n the estimation by equation (3.7), the translog cost function, to avoid s i n g u l a r i t y of the residuals covariance matrix. (3.7) and (3.8) correspond to the f i r s t stage of the producer's problem and involve variables whose values are known with c e r t a i n t y i n x. This i s not the case for equation (3.9) since expectations need to be taken. This l a s t condition simply says that the net e f f e c t on p r o f i t of an extra unit of c a p i t a l i s zero. This net e f f e c t i s made up of four components: the savings i n variable costs r e s u l t i n g from an a d d i t i o n a l u n i t of c a p i t a l , the current cost of adjustment, the expected savings i n future adjustment costs discounted to x, and the contribution to allowed excess p r o f i t s an ad d i t i o n a l unit of c a p i t a l makes. To estimate equations of t h i s sort while maintaining the RE hypothesis, Hansen (1982) and Hansen and Singleton (1982) suggest a generalized method of moments estimator. Their idea consists i n using an instrumental variables procedure that minimizes the c o r r e l a t i o n between any variable known at x and the residuals of (3.9). These residuals, which can be interpreted as expectational errors, are computed using the actual values of K T + 1 on the left-hand side. Moreover, as shown i n Hansen (1982), i f these residuals are assumed to be homoscedastic, the procedure reduces to nonlinear three-stage l e a s t squares. The estimation strategy thus consists, as i n Pindyck and Rotemberg (1983a, b), i n using nonlinear three-stage l e a s t squares to estimate the system of equations (3.7)-(3.9) with K T +1 as the dependent variable i n the c a p i t a l accumulation equation.^ As pointed out i n Pindyck and Rotemberg (1983b), using any variable known at x as instrument could be j u s t i f i e d only i f equations (3.7) and (3.8) held exactly, without errors. For i f these equations contain error terms because of technological shocks, measurement or optimization errors, these error terms are l i k e l y to be correlated with some variables i n the c a p i t a l accumulation equation. Hence, as i s suggested i n Pindyck and Rotemberg (1983b), the conditioning set does not include current variables appearing i n the cost, labor share or c a p i t a l accumulation equation. The set of chosen instruments include: the lagged (by one period) values of w , p L , L, M, y L , and q (the price of investment goods). The endogenous variables i n the system of estimating equations are C t / Kfc, (K*- - K t -^) and e f c. The nonlinear algorithm of SHAZAM (version 5.1) i s used to generate estimates of the parameters i n (3.7)-(3.9) once intruments have been substituted for the endogenous variables i n the system. A convergence c r i t e r i o n of 0.00001 i s employed to produce the f i n a l r e s u l t s . The i n c l u s i o n of e f c i n the set of endogenous variables deserves further discussion. Even i f t h i s variable i s t h e o r e t i c a l l y exogenous, i t cannot be treated as such i n the empirical investigation for the following reasons. As w i l l be made clearer i n the data section at the end of t h i s chapter, e f c i s defined for estimation purposes as the difference between the firm's actual return on c a p i t a l and i t s cost of c a p i t a l . Since r e a l i z e d p r o f i t s are d e f i n i t e l y endogenous, so i s the actual return on c a p i t a l and hence e t . I t i s possible to construct an exogenous indicator of the firm's "allowed return" on c a p i t a l by using the l e v e l of p r o f i t s approved by the regulators i n rate cases. This s o l u t i o n i s adopted i n Fuss and Waverman (1981) i n t h e i r s t a t i c model of a regulated u t i l i t y . However, the l e v e l of p r o f i t s approved by the regulatory commission does not appear to r e f l e c t t r u l y the r e a l permissiveness or stringency of the regulators' control, nor does the l a t t e r solution preclude the p o s s i b i l i t y that the firm influences the commission i n i t s s e t t i n g of the allowed rate of return**. Since t h i s thesis takes a long-run view of the regulatory constraint and assumes that the p r o f i t c e i l i n g i s defined over a long horizon, the actual l e v e l of B e l l ' s p r o f i t a b i l i t y over t h i s horizon seems to correspond more c l o s e l y to what regulation allows the firm to earn. Moreover, the allowed rate of return used i n Fuss and Waverman does not appear to be binding: the u t i l i t y sometimes earns more, sometimes less than the set rate of return. F i n a l l y , note that the practice of using the actual return on c a p i t a l as a proxy for the allowed return i s common i n the empirical l i t e r a t u r e on regulation and can be found, among others, i n C o u r v i l l e (1974), Spann (1974), Hayas"hi and Trapani (1976), Pescatrice and Trapani (1980), Gollop and Karlson (1980), Cowing (1980) and Nelson and Wohar (1983). Therefore, to minimize the r i s k of an endogeneity bias i t i s decided to instrument for e f c. Now, appending j o i n t l y normally zero-mean random terms to each equation, and labeling these ( u c , u L , u K ) t , the following error structure i s posited: [ cov ( u J , U j ) = , i , j = c, L, K ; (A17) 1 \ cov ( u J , U j " " 1 ) = 0 , i , j = c, L, K. Also r e c a l l that u K i s interpreted as an expectational error whereas u c and U L are seen as representing optimization or measurement errors but not errors i n expectations. Notice that the parameter estimates w i l l be consistent even i f the assumption of an homoscedastic error structure i s violated, although the standard errors of the parameters w i l l then be i n v a l i d . F i n a l l y , two measures of the user cost of c a p i t a l and, as a r e s u l t , two d i f f e r e n t excess return variables are a l t e r n a t i v e l y used i n the estimation. This step i s taken to check the robustness of the r e s u l t s to the construction of the cost of c a p i t a l variable. These variables are defined i n d e t a i l i n section 3.3. 3.2 A MODEL OF PROFIT-MAXIMIZATION WITH ENDOGENOUS OUTPUT The model of producer behavior described i n 3.1 can be transformed into a model i n which the firm i s a p r i c e -setter by adding a transformed f i r s t - o r d e r condition to the estimating equations, as shown i n Fuss and Waverman (1977) i n a s t a t i c context. B a s i c a l l y , the following hypothesis i s substituted for (A16): (A18) l o c a l price and output are exogenous to the firm but the price and quantity of t o l l output are chosen by B e l l so as to maximize i t s expected p r o f i t s . This i s the behavioral assumption underlying the papers by Denny et a l . (1981a, b) and Fuss and Waverman (1977, 1981). Under t h i s hypothesis, B e l l ' s constrained objective function becomes: E x (1-u) E*' R (x,t) { p£ y£ + <p (y£) y£ - C(t) - v f c Kfc - BfK*- - K t - 1 ) 2 } + (1 -U) R (T , t 1 ) Q Kfc'+ Max. < Yn Kfc } u { E ^ R (x,t) e f c K t _ 1 }, . . . (3.10) where cp(y^) i s the inverse demand function for B e l l ' s t o l l output. The necessary condition for a maximum of p r o f i t s associated with the choice of the optimal t o l l output l e v e l can be written as^: < p£ Y T )/ C T = (b x / ( l + ^ J ) [ a Q 3 + a 1 3 ln xw + a 2 3 l n YI + a 3 3 l n y x + a 3 4 K T + a 3 6 F T ], ...(3.11) i n which b^ i s the t o l l output e l a s t i c i t y of demand. By choosing a suitable s p e c i f i c a t i o n for the demand for t o l l output, a system of f i v e equations i s obtained i n place of the previous system of three estimating equations. Numerous attempts at estimating t h i s system were made. In most cases, the reg u l a r i t y conditions on the cost function were v i o l a t e d and the second-order conditions for a maximum of p r o f i t f a i l e d to be met. More s p e c i f i c a l l y , marginal cost was found to decrease more rapidly than marginal revenue almost everywhere. This was the case i n over twenty estimated models. To determine i f equation (3.11) i s indeed the source of those i r r e g u l a r i t i e s , the system of equations (3.7), (3.8) and (3.11) was estimated: t h i s imposes the profit-maximizing condition but ignores both regulation and adjustment costs. Again, the results were 6 4 disappointing. Because of the very poor re s u l t s no parametric t e s t of output endogeneity was done here. In comparison, Fuss and Waverman (1981) obtained very good r e s u l t s under an i d e n t i c a l maintained hypothesis. However, there are many major differences between t h e i r studies and t h i s one: Fuss and Waverman use three outputs instead of two, they treat c a p i t a l as a variable input, use a s p e c i f i c a t i o n i n which technical progress i s an output-augmenting process and estimate a hybrid translog variable cost function i n which the output variables are modified by the Box-Cox transformation. F i n a l l y , they ignore regulation and adjustment costs. Those considerations aside, i t may also be that the dynamic character of the cost function used here did not "mesh" very well with the s t a t i c formulation of the output determining equation. One p a r t i c u l a r l y strong aspect of the hypothesis contained i n (3.11) i s the implication that t o l l p r ices are adjusted continously so that the desired equality between marginal revenue and marginal cost i s obtained at each observation point. Even with yearly observations, t h i s may be an u n r e a l i s t i c assumption since rate hearings are held at i r r e g u l a r i n t e r v a l s and hardly once a year. Moreover, t h i s assumption implies that the price adjustments demanded by B e l l are systematically granted. A s p e c i f i c a t i o n i n which prices adjust slowly to the desired l e v e l may therefore more c l o s e l y r e f l e c t the sluggish way i n which B e l l can have i t s t o l l p r i c e adjusted. Above a l l , such a s p e c i f i c a t i o n can shed some l i g h t onto the p l a u s i b i l i t y of the profit-maximization hypothesis. A simple, and ad-hoc, process of adjustment i s the following variant of a Koyck p a r t i a l adjustment model for the pric e of t o l l services: p£ = p^" 1 + 9 (p£ - p^' 1 ), ...(3.12) where p£ i s the optimal (or desired) price l e v e l i n year t. If 8 = 1 , f u l l adjustment occurs every year and p£ = p£ : the f i r s t - o r d e r condition (3.11) then obtains. In contrast, for 0 < 8 < 1, B e l l gets only 8% of i t s desired adjustment i n any given year. The u t i l i t y ' s rule for choosing y T , analogous to (3.11), but considering the fact that p£ may be d i f f e r e n t from p£ i s : (pj Y T ) / C T = ( 8 b j / d + b x ) ) S y + (1-8) ( ( p X _ 1 y x )/ C X ) , ...(3.13) where Sy = [ ? l n C ( T ) /3 l n yi$ ]. As a r e s u l t , estimating (3.13), which has (3.11) as a spe c i a l case, makes i t possible to t e s t the hypothesis of instantaneous price adjustment against that of p a r t i a l adjustment. Since the e l a s t i c i t y of the demand for t o l l output enters the above f i r s t - o r d e r condition, i t i s preferable to complete the system of estimating equations by the demand for t o l l services. The double-loglO s p e c i f i c a t i o n i s chosen for the demand function: In y j = b Q + b 1 In (p£ / CPI X ) + b 2 In RPC T , ...(3.14) where CPI T i s the consumer price index and RPC T the r e a l per-capita income i n B e l l ' s t e r r i t o r y . The estimated model of producer behavior with endogenous output consists of equations (3.7), (3.8), (3.9), (3.13) and (3.14). The method of estimation i s nonlinear three-stage least squares and the set of instruments described i n 3.1 i s employed (notice however that yrj i s now an endogenous v a r i a b l e ) . The f i r s t - o r d e r conditions (3.5) and (3.13) are also s u f f i c i e n t i f the p r o f i t function i n (3.10) i s concave i n the choice variables (that condition ensures that the constraint i s convex). These conditions, along with the re g u l a r i t y conditions on C(x) are checked at each observation point. Both versions of the c a p i t a l cost variables are again used i n the estimation. 3.3 DATA SECTION The precise procedure for constructing the data seri e s used to estimate the models i n 3.1 and 3.2 i s outlined i n Appendix A and the f i n a l output and input price and quantity series are reported i n Appendix B. This section gives the d e f i n i t i o n s of a l l the variables along with summary s t a t i s t i c s on the data. The p r i n c i p a l source of the data i s a recent submission to the Canadian Radio-Television and Telecommunications Commission (CRTC) by B e l l Canada. The data are annual, covering the period 1952-1980. The two output variables are measured i n m i l l i o n s of constant 1976 d o l l a r revenues. The output price variables are D i v i s i a indexes of a l l l o c a l and t o l l categories of revenues normalized to 1.0 i n 1976. The quantity of labor i s m i l l i o n s of manhours and the average hourly wage rate i s used as the pr i c e of labor. The quantity of materials i s measured as the constant 1976 d o l l a r value of expenditures on materials, services, rents and supplies. And the price of materials i s given by an i m p l i c i t p r i c e index, normalized to 1.0 i n 1976. The quantity of c a p i t a l i s the constant 1976 d o l l a r t o t a l average net stock of c a p i t a l at reproduction cost. The determination of the user cost of c a p i t a l services (or cost of c a p i t a l ) for a firm i s not straightforward. This i s because the cost of c a p i t a l i s an opportunity cost and a forward-looking concept (Kolbe et a l . , 1984): investors look at the expected (marginal) return on t h e i r investments and compare t h i s return to the expected return on foregone investments. This d e f i n i t i o n of the cost of c a p i t a l implies that h i s t o r i c values about a firm's p o l i c y , r i s k i n e s s , c a p i t a l structure, etc... may be i r r e l e v a n t for i n f e r r i n g the cost of c a p i t a l of a marginal investment i n the firm. The case of regulated enterprises i s even more complex because i t may prove very d i f f i c u l t to ascertain the l e v e l of r i s k of a regulated concern. Moreover, i n the context of t h i s research, the measurement of the user cost of c a p i t a l turns out to be c r u c i a l since one of the important propositions of Chapter 2 implies that e f f e c t i v e regulation may induce o v e r c a p i t a l i z a t i o n by the regulated firms. This proposition can be tested against the data only i f a measure of the excess return on c a p i t a l i s ava i l a b l e . But t h i s excess return variable i s defined as the difference between the allowed return on c a p i t a l and the cost of c a p i t a l . Therefore, any measurement error involved i n the computations of either or both of those two v o l a t i l e variables w i l l immediately a f f e c t the chosen measure of the excess return on c a p i t a l and possibly bias the estimated parameters. The wide d i v e r s i t y of res u l t s that were obtained by d i f f e r e n t researchers while t r y i n g to tes t the o v e r c a p i t a l i z a t i o n e f f e c t i n a s t a t i c framework i n the U.S. e l e c t r i c u t i l i t y industry, for instance, may i n part be attributable to the choice of d i f f e r e n t measures for the cost of c a p i t a l and for the (gross) excess return on c a p i t a l services. While following the mainstream procedure for estimating the cost of c a p i t a l , t h i s thesis takes two precautionary steps against the possible bias induced by the choice of a p a r t i c u l a r user cost variable. F i r s t , the user cost formula retained does take into account the s p e c i f i c e f f e c t s of the f i s c a l regime, the existence of accelerated depreciation for tax purposes, the presence of a c e i l i n g constraint on the actual return to c a p i t a l and the fac t that c a p i t a l funds are raised from many sources. Secondly, two s l i g h t l y d i f f e r e n t measures of the cost of c a p i t a l services are derived using the two most common procedures for computing the cost of equity c a p i t a l : the discounted cash flow method (DCF) and the c a p i t a l asset p r i c i n g model (CAPM) method. These two user cost of c a p i t a l variables are then used a l t e r n a t i v e l y i n the econometric estimation i n order to te s t the robustness of the estimation to the choice of a p a r t i c u l a r c a p i t a l cost measure. 70 Most user cost of c a p i t a l formulae are i n the t r a d i t i o n developed by Christensen and Jorgenson (1969). The procedure adopted i n t h i s thesis l i e s i n the same t r a d i t i o n but i s based on a model of Boadway and Bruce (1979) i n which a consumer maximizes u t i l i t y over an intertemporal consumption stream. This maximization i s constrained by the consumer's a b i l i t y to borrow and the firm's a b i l i t y to generate d i s t r i b u t a b l e p r o f i t s . Fuss and Waverman (1981) have adapted t h i s model to the case of a regulated firm, and t h i s thesis further modifies t h e i r suggested measures of the cost of c a p i t a l and of the excess returns earned. Fuss and Waverman's user cost of c a p i t a l services i s given by: v 1 = q( 9 c R + ci (1-9) + 6 ) - ( a - f i ) g t q , i=l,2 ° * d - t ) (l-t)(a+g) ... (3.15) where q i s the asset price of c a p i t a l , 9 the f r a c t i o n of the firm's c a p i t a l financed by debt, 6 the economic depreciation rate, a the accelerated depreciation rate, t the tax rate on corporate income, g the treasury bond rate which i s used as a proxy for the personal borrowing rate, c B i s the cost of debt, and c£ i s the cost of equity c a p i t a l . This l a t t e r can be computed i n two ways, with c^ using the CAPM method and c£ using the DCF method. Notice that (3.15) defines a «gross» user cost of c a p i t a l , and that Cg and c j are accordingly a f t e r tax percentages. An i m p l i c i t assumption contained i n (3.15) i s that marginal investments do not a l t e r the c a p i t a l structure of the firm. This i s a standard assumption although i t may not be warranted. L a s t l y , note that Fuss and Waverman use exclusively a DCF method to compute (3.15), and that t h e i r s p e c i f i c a t i o n of the DCF model i s d i f f e r e n t from the s p e c i f i c a t i o n i n thi s thesis i n some minor respects. The two methods are described i n greater d e t a i l s i n the Appendix. Insofar as the procedures may ignore the e f f e c t s of c a p i t a l gains (or losses) due to appreciation of prices of c a p i t a l assets B e l l Canada owns, and of any neglected relevant investment tax c r e d i t s , the o v e r a l l e f f e c t would be to over-estimate the user cost of c a p i t a l and under-estimate the excess return. The resu l t i n g bias would favor r e j e c t i o n of the A . - J . hypothesis. Note also that use of the CAPM model may not be warranted i n the case of rate-regulation (see for example Brennan and Schwartz, 1982). The formula given i n Fuss and Waverman for the allowed gross return on c a p i t a l i s : s = q ( 6 c n + s„ (1-9) + 6 ), ...(3.16) B E d - t ) where s E i s the allowed (gross) rate of return on equity. s E i s assumed to be equal to the actual rate of return on equity. This solution i s also adopted by Spann (1974), Gollop and Karlson (1980), Pescatrice and Trapani (1980), Hayashi and Trapani (1976), Cowing (1978) and Nelson and Wohar (1983) among others. This i s consistent with the assumptions that regulation i s binding and that the firm maximizes i t s p r o f i t s . I t also implies that regulators allow the firm to earn a given rate of return when they do not react. F i n a l l y , the allowed (gross) excess return on c a p i t a l services can be obtained from (3.15) and (3.16): e 1 = (s-v 1) = q [ (s - c 1) (1-9) / (1-t) ] + (a - 6) [ ( q t g ) / (1 - t) (a + g) ] . ...(3.17) Summary s t a t i s t i c s on a l l variables appear i n Table 3.2. The values of v-1, v^ and s can be found i n Table 3.3. Table 3.4 l i s t s the e 1 values. Cursory examination of those tables reveals that, i n general, sg > c j and thus e 1 > 0. This i s consistent with the assumptions of Chapters 2 and 3 and opens the p o s s i b i l i t y of an A . - J . bias: hence that u > 0. Notice however that e 1 < 0 i n 1980 (using cj) and i n 1973, 74, 76, 77 and 1979 (using c |). Although these occurences are few, they are somewhat inconsistent with the the o r i z i n g i n t h i s thesis. Interestingly, Fuss and Waverman (1981) remark that regulation seems to have tightened for B e l l during the seventies: the r e l a t i v e l y TABLE 3.2  BELL DATA SET:  SUMMARY STATISTICS (1952-1980) Variable Name Mean Standard Deviation Minimum Maximum YL 0. 62175 E+03 0.33356 E+03 0.19429 E+03 0.12837 E+04 PL 0. 85624 0.15998 0.71803 1.3205 y T 0. 43369 E+03 0.35352 E+03 75.968 0.12560 E+04 p T 0. 88829 0.11059 0.79112 1.1936 L 58 .693 6.63800 49.000 76.200 w 4. 9294 3.5350 1.7087 14.140 M 0. 19809 E+03 95.460 69.019 0.39935 E+03 m 0. 69356 0.26624 0.44626 1.4010 K 0. 43509 E+04 0.20972 E+04 0.12902 E+04 0.80055 E+04 v l 0. 10523 0.067605 0.045261 0.30311 V2 0. 11490 0.065915 0.058846 0.27418 e l 0. 015941 0.011781 -0.020947 0.039687 e 2 0. 006271 0.005078 -0.004383 0.013815 TABLE 3.3 GROSS USER COST OF CAPITAL  AND ALLOWED RETURN ON CAPITAL FOR BELL: 1952-1980 Year v x v z s 1952 0.0480 0.0754 0.0785 1953 0.0500 0.0725 0.0727 1954 0.0458 0.0630 0.0718 1955 0.0453 0.0588 0.0692 1956 0.0533 0.0599 0.0691 1957 0.0593 0.0615 0.0682 1958 0.0541 0.0640 0.0693 1959 0.0711 0.0720 0.0774 1960 0.0621 0.0698 0.0779 1961 0.0615 0.0673 0.0786 1962 0.0692 0.0730 0.0800 1963 0.0671 0.0692 0.0807 1964 0.0693 0.0174 0.0830 1965 0.0714 0.0740 0.0879 1966 0.0865 0.0792 0.0864 1967 0.0782 0.0849 0.0963 1968 0.0922 0.0922 0.1032 1969 0.1052 0.0976 0.1090 1970 0.1033 0.1106 0.1187 1971 0.0912 0.1146 0.1228 1972 0.0992 0.1297 0.1350 1973 0.1145 0.1426 0.1423 1974 0.1405 0.1567 0.1533 1975 0.1534 0.1865 0.1931 1976 0.1915 0.2061 0.2018 1977 0.1912 0.2108 0.2080 1978 0.2150 0.2297 0.2327 1979 0.2651 0.2675 0.2655 1980 0.3031 0.2742 0.2822 MEAN 0.1052 0.1149 0.1211 TABLE 3.4  EXCESS ALLOWED RETURN ON CAPITAL FOR BELL: 1952-1980 YEAR e x e 1952 0.0305 0.0313 1953 0.0227 0.0003 1954 0.0260 0.0088 1955 0.0240 0.0104 1956 0.0158 0.0091 1957 0.0090 0.0068 1958 0.0151 0.0052 1959 0.0063 0.0055 1960 0.0158 0.0081 1961 0.0171 0.0113 1962 0.0103 0.0092 1963 0.0136 0.0114 1964 0.0136 0.0115 1965 0.0165 0.0138 1966 0.0058 0.0072 1967 0.0181 0.0113 1968 0.0110 0.0110 1969 0.0037 0.0113 1970 0.0153 0.0098 1971 0.0317 0.0082 1972 0.0357 0.0053 1973 0.0278 -0.0002 1974 0.0128 -0.0034 1975 0.0397 0.0066 1976 0.0103 -0.0044 1977 0.0169 -0.0027 1978 0.0177 0.0030 1979 0.0004 -0.0020 1980 -0.0209 0.0086 MEAN 0.0159 0.0063 small and sometimes negative values for e 1 during t h i s period seem to support t h i s conjecture. F i n a l l y , an indicator of technological change has to be defined. Many measures of technological change e x i s t for B e l l Canada. Common indicators are the percentage of phones with access to d i r e c t distance d i a l i n g (A), the percentage of t o l l c a l l s using d i r e c t distance d i a l i n g (DDD), the percentage of phones connected to o f f i c e s with "modern" switching equipment (S), and various combinations of these three indicators ( Kiss et a l . , 1981, provide a l i s t of four of those i n d i c a t o r s ) . In addition, Fuss and Waverman experiment with a capital-augmenting indicator whereas Denny et a l . (1981a) and Fuss and Waverman (1981) use output-augmenting indica t o r s . These take the form: X e where X i s output or c a p i t a l , z i s one of the above indicators and a i s an estimated parameter. Studies of the US B e l l system have also employed indices based on past research and development expenditures. As pointed out i n Denny et a l . (1981a) and i n Bernstein (1987), i t i s widely believed that the single most important technological innovation of the l a s t t h i r t y years occured i n the s i x t i e s and consisted i n the development of modern switching equipment (electronic switchboards, e t c . . ) and the introduction of direct-distance d i a l i n g f a c i l i t i e s . A l l the above mentioned technological change indicators r e f l e c t t h i s pattern. After some experimentation, two indices were singled out for use i n a l l the estimations i n t h i s t h e s i s . These are: the percentage of phones with access to d i r e c t distance d i a l i n g (A), thi s i s used by Fuss and Waverman (1981); and one of the technological indicators i n Kiss et a l . (1981) defined as: T2 = FNEW [ h PDH + (1-h) A ], ...(3.18) where FNEW i s defined as one plus the percentage of crossbar and e l e c t r o n i c c e n t r a l o f f i c e s , PDH i s the percentage of d i a l phones and h = ( y L / ( y L + y T) ). The series defined by (24) ends i n 1978. A regression of known values on a constant and a time trend gave an R 2 of .997 . The f i t t e d values of the proxy variable for 1979 and 1980 were added to the series to complete i t . 78 NOTES TO CHAPTER 3 1. See Waverman (1982) and Green (1980). 2. See Chapter 5 of Economic Council of Canada (1981). 3. See Epstein and Denny (1980) for a short-run model of producer behavior in which the real rate of depreciation i s endogenously determined. 4. Alternate specifications in which costs of adjustment were made to depend ( linearly or logarithmically ) on input prices were used in the estimation but proved unsatisfactory: the regularity conditions on the cost function, including the sign restrictions on the adjustment cost coefficients, were generally violated. Also notice that the chosen specification i s consistent with the often imposed restriction that marginal adjustment costs vanish when net investment i s zero. 5. As i n Chapter 2, the i n i t i a l level of the capital stock i s given. 6. See Gould (1968), Brechling (1975) and Berndt, Fuss and Waverman (1980). Berndt, Morrisson and Watkins (1981) review the literature on the estimation of dynamic factor demands. 7. Specifically, taking K T + 1 to the L.H.S. and rewriting equation (3.9) gives the actual estimating equation: K x + 1 = [B R I T . T + I ) ] " 1 { [ a Q 4 + a 1 4 ln wx + a 2 4 In y x + a 3 4 ln y x + a 4 4 ln KT + o 4 5 F x ][C X/ K x] + v x + B(KX - K X _ 1) + B R(x ,T+l) KX } - ( 3 / B ) e X + 1 . ...(3.9') 79 8. See Waverman (1982) on the rate-setting process in the case of Be l l Canada and the factors having an impact on the regulatory outcome. Notice that some variables that appear to play a significant role in the determination of the allowed rate of return are indeed controlled by the u t i l i t y . 9 . Equation (3.11) is derived as follows: let S y = 3 In C/81n y T and C = dC/9y T, then marginal revenue i s given by MR = PT ( 1 + (1/bi) ) = C = S y ( C / Y T ) . ...(1) Hence: PT ((b x + D / b x ) = S y (C/y T), ...(2) and, ( P T Y T / ° ) " (bi/d+bi)) S Y . ...(3) Now consider the unrestricted and unnormalized variable translog cost function (where time superscripts have been omitted for simplicity): In C = a Q 0 + a Q 1 In w + a Q 2 In m + a Q 3 In y L + a ^ l n y ^ a Q 5 In K + a Q 6 F + 0 . 5 [ a ^ d n w ) 2 + a 2 2 ( l n m ) 2 + a 3 3 (In y L ) 2 + a 4 4 (In y T ) 2 + a 5 5 (In K) 2] + a 1 2 In w In m + a 1 3 In w In y L + a 1 4 In w In y T + a 1 5 In w In K + a l g In w F + a 2 3 In m In y L + a 2 4 In m In y T + a 2 5 In m In K + a 2 g In m F + A 3 4 L N Y L L N Y T + A 3 5 L N Y L L N K + A 3 6 L N Y L F +  A 4 5 L N Y T L N K + A 4 6 L N Y T F + A 5 6 L N K F « • • • ( 4 ) 80 Notice that the symmetry assumption is the only condition imposed on (4). The linear homogeneity of the cost function in input prices implies: a01 + a02 = 1 ' •••(5) a ^ + a^2 = a22 + a i 2 ~ ®' ...(6) EJL a^j = 0, i=l,2 and j=3,4,5,6. ...(7) Using (4), the e l a s t i c i t y of variable costs to t o l l output can be defined as: Sy = a Q 4 + a 1 4 In w + a 2 4 ln m + a 3 4 ln y L + a 4 4 l n y T + a 4 5 ln K + a 4 6 F . ...(8) When the linear homogeneity assumption (7) i s imposed, (8) becomes: S y = a04 + a14 l n ( w / m ) + a34 l n Y L + a44 l n yT + a 4 5 ln K + a 4 6 F. ...(9) Substituting (9) into (3) and renumbering the coefficients yields equation (3.11) in the text. 81 10. Other s p e c i f i c a t i o n s for the demand for t o l l output were t r i e d but proved i n f e r i o r to t h i s one. In p a r t i c u l a r , a s p e c i f i c a t i o n i n which the number of households i n B e l l ' s t e r r i t o r y i s included led to non s t a t i s t i c a l l y s i g n i f i c a n t p r i c e and income e l a s t i c i t i e s . A s p e c i f i c a t i o n that allows for a lag i n consumers' response to pr i c e changes was also rejected when the hypothesis of no .lags i n response could not be s t a t i s t i c a l l y rejected. Most empirical studies of B e l l ' s demands u t i l i z e a double-log functional form. Dobell (1972), Denny et a l . (1981a) and Fuss and Waverman (1981) are cases i n point. Experimentation with a semi-log s p e c i f i c a t i o n f o r the inverse demand function lead to disappointing r e s u l t s . CHAPTER 4 ESTIMATION RESULTS 4.0 INTRODUCTION Parameter estimates, for the two models of producer behavior of Chapter 3 and using the two a l t e r n a t i v e measures for the user cost of c a p i t a l , are presented and discussed i n 4.1 through 4.3 . The sample period i s 1953-1979 i n a l l cases. The f i r s t and l a s t observation of the data set have to be dropped because of the use of lagged and lead values of the c a p i t a l stock i n the c a p i t a l accumulation equation. In general, the estimation results do not prove very s e n s i t i v e to the choice between A and T2, the two technical change proxies. When discrepancies occured, they are reported. The best r e s u l t s , based on the value of the l i k e l i h o o d function and the r e g u l a r i t y conditions on C ( t ) , are chosen for each model. As a r e s u l t , the percentage of phones with access to d i r e c t distance d i a l i n g (A) i s used i n the estimation of the exogenous output models while the index T2, defined i n 3.3, i s used i n the estimated models with endogenous output. 4.1 EMPIRICAL RESULTS FOR THE CONSTRAINED MAXIMIZATION MODEL WITH EXOGENOUS OUTPUTS The estimated c o e f f i c i e n t s for the constrained model of profit-maximization described by the set of equations (3.7)-(3.9), i n Chapter 3, are shown i n Table 4.1 , and the goodness-of-fit s t a t i s t i c s are reported i n Table 4.2 . Note that non-normalized data have been used i n the estimation. Examination of Tables 4.1 and 4.2 reveals that the estimated models f i t the data rather well. Most parameters are s i g n i f i c a n t and the variance of the dependent variables i s well explained by the regression equations. The Durbin-Watson s t a t i s t i c s indicate that autocorrelation of the residuals does not seem to be a problem, except perhaps i n the c a p i t a l accumulation equation when v^ i s used. In addition, B, the adjustment cost parameter, i s s t a t i s t i c a l l y s i g n i f i c a n t i n both versions of the model. This s i g n i f i e s that B e l l i s not i n long-run equilibrium. Since P = u / ( l - u ) , the implied values for the Lagrange m u l t i p l i e r are 0.44 and 0.8. Although the hypothesis that the f i r s t value i s zero cannot be rejected, the second value i s s t a t i s t i c a l l y s i g n i f i c a n t and does f a l l within the th e o r e t i c a l range for u. This means that the hypothesis that regulation d i s t o r t s the investment decisions of B e l l cannot T A B L E 4.1 P A R A M E T E R E S T I M A T E S : M O D E L W I T H E X O G E N O U S O U T P U T S V-L ( C A P M ) v2 ( D C F ) coef. s t . dev. coef. st.dev. a00 46.559 50.322 -40.671 4.4980 a01 1.0343 0.2449 1.0326 0.0819 a02 -155.26 62.320 -28.462 3.2006 a03 63.380 30.736 4.4007 0.7286 a04 63.960 15.937 30.074 3.4050 a05 3.7625 5.5830 -4.5250 1.2659 a l l -0.0109 0.0303 -0.0430 0.0057 a22 86.985 26.903 22.661 2.8498 a33 18.768 7.3618 2.8738 0.4049 a44 -1.1978 1.3502 -0.3976 0.2102 a12 0.1418 0.1372 0.2012 0.0394 a13 -0.0122 0.0527 -0.0209 0.0137 a14 -0.1381 0.0758 -0.1705 0.0280 a15 -0.0499 0.0298 -0.0447 0.0159 a23 -40.253 14.119 -7.8379 1.0452 a24 -18.287 3.9493 -7.9155 1.0921 a25 -15.625 5.4697 -5.8876 0.7598 a34 9.6543 2.4202 3.4489 0.5713 a35 4.8227 2.3586 0.7348 0.2060 a45 8.0345 2.1755 4.4706 0.6248 B 0.0012 0.0003 0.0005 0.0001 P 0.7748 1.3319 5.3166 1.0664 85 TABLE 4.2 SUMMARY STATISTICS FOR THE MODEL WITH EXOGENOUS OUTPUTS Vl R 2 D.W. cost equation 0.9964 1.6841 labor share equation 0.8764 1.6253 c a p i t a l accumulation equation 0.9992 0.8236 Log of the l i k e l i h o o d function: 11.76 v 2 R 2 D.W. cost equation 0.9957 1.6209 labor share equation 0.8715 1.7295 c a p i t a l accumulation equation 0.9993 1.2859 Log of the l i k e l i h o o d function: 15.50 be rejected when V 2 i s used. The monotonicity and curvature properties on the cost function as well as the ( s u f f i c i e n t ) second-order conditions for a maximum of p r o f i t are checked and reported on i n Table 4.3. The estimated cost function appears well-behaved at most observation points; i t describes the behavior of B e l l Canada s a t i s f a c t o r i l y except for the pattern of the marginal cost of l o c a l output over a few years. The concavity of the objective function i n the c a p i t a l stock i s s u f f i c i e n t for a maximum of p r o f i t characterized by the estimated f i r s t - o r d e r conditions. This concavity condition i s v e r i f i e d and found to hold everywhere. Also note from the l a s t tables that the V 2 ~ s p e c i f i c a t i o n provides the more s a t i s f a c t o r y r e s u l t s : the maximized l i k e l i h o o d function i s greater, f a i l u r e s i n the monotonicity and curvature conditions scarcer and the residuals e x h i b i t more evidence of randomness i n the c a p i t a l accumulation equation. 87 TABLE 4.3 MONOTONICITY AND CURVATURE PROPERTIES ON THE COST FUNCTION: MODEL WITH EXOGENOUS OUTPUTS Monotonicity conditions: c a p i t a l l o c a l ouput t o l l output labor share 24/27 17/27 17/27 27/27 25/27 21/27 27/27 27/27 Curvature properties: concavity i n input prices 27/27 27/27 s u f f i c i e n t conditions fo r a maximum of p r o f i t 27/27 27/27 Additional information on the technology of B e l l Canada and the properties of the estimated equilibrium i s provided i n Table 4.4. I t can be shown by t o t a l l y d i f f e r e n t i a t i n g the normalized cost function that the desired factor price e l a s t i c i t i e s are: E L L = [ <x n / S L ] - [ 1 - S L ], ...(4.1) ELM = * ELL' ...(4.2) = [ a n / (1 - S L) ] - S L, ...(4.3) EKIL = -EWI' ...(4.4) where E-^j i s the c r o s s - e l a s t i c i t y of the demand of the i t n factor with respect to the price of the j t n factor ( i , j = L, M), and S L = ( ^ In C / 3 In w) i s the share of labor i n variable costs. Hence S L i s the e l a s t i c i t y of variable costs with respect to the price of labor. The e l a s t i c i t y of variable costs to l o c a l and t o l l outputs are defined i n a si m i l a r way. The scale e l a s t i c i t y i s defined as the growth i n t o t a l output as a l l inputs are scaled up at a common rate. This e l a s t i c i t y can be shown to be the inverse of the e f f e c t of output growth on the growth of t o t a l costs. Caves, Christensen and Swanson (1981) demonstrate that i n the case of a r e s t r i c t e d cost function the scale e l a s t i c i t y i s given by: SE = [ 1 - S R ] / E ± S i , ...(4.5) where S K i s the e l a s t i c i t y of variable costs to c a p i t a l and Sj_ i s the e l a s t i c i t y of variable costs to output i . F i n a l l y , the time s h i f t i n the variable cost function i s given by: 3 l n C/5t = [ B i n C/3F ] [^F/^t]. ...(4.6) The l a s t expression i s evaluated holding a l l variables other than F, the technological change proxy, constant at t h e i r mean value. The meaning of (4.6) i s immediate: i t represents the average annual rate at which the variable cost function i s " s h i f t i n g " through time because of technological change. One i n t e r e s t i n g feature of Table 4.4 i s the low s e n s i t i v i t y of the estimated c h a r a c t e r i s t i c s of B e l l ' s technology to the user cost s p e c i f i c a t i o n . Moreover, the data contained i n t h i s table generally confirm other studies' findings. As i n Bernstein (1986,1987), who also used a r e s t r i c t e d cost function approach, and i n Fuss and Waverman (1981) and Kiss et a l . (1981), who used a s t a t i c approach, the v a r i a b l e factor demands seem quite p r i c e i n e l a s t i c ^ . TABLE 4.4 SUMMARY STATISTICS ON  BELL'S TECHNOLOGY; AVERAGE VALUES V l E l a s t i c i t y of variable costs with respect to: l o c a l output 0.66 t o l l output 0.50 c a p i t a l -0.89 Scale e l a s t i c i t y 1.63 S h i f t i n the variable cost function -0.026 Own e l a s t i c i t y of input demand: labor ( E L L ) -0.35 materials ( E ^ ) -0.70 The estimated values for the technological s h i f t i n the cost function and the scale e l a s t i c i t y are extremely s e n s i t i v e to the econometric s p e c i f i c a t i o n , as noted i n Fuss and Waverman (1981; p.117) and Denny et a l . (1981a). The f i r s t of these studies reviews previous estimations of the scale e l a s t i c i t y of B e l l and finds a wide range of values, from 0.94 to 1.47 ( a l l estimates derived from long-run models of the u t i l i t y ) . Bernstein (1986,1987) arriv e s at values ranging from 1.13 to 1.84 i n a one-output model and averaging 1.5 i n a two-output model. Kiss et a l . (1981) report estimates i n the range 1.22-1.75 based upon the estimation of more than twenty one-two-and three-output models. In general then, the values found i n Table 4.4 seem i n l i n e with those i n the l i t e r a t u r e . As for the estimated "average downward s h i f t " i n the cost function, i t suggests that technological change alone i s responsible for an annual average reduction i n variable costs of something l i k e 2%. Using a very d i f f e r e n t s p e c i f i c a t i o n for technological change^, Bernstein (1987) finds an average value of 1.7%, a figure very close to the present r e s u l t . F i n a l l y , the reported results outperform those obtained when the al t e r n a t i v e technological change proxy, T2, i s used. The maximized l i k e l i h o o d values are greater i n both the V]_ and V 2 versions of the model (the values with T2 are 9.46 and 10.90); the regu l a r i t y conditions on the cost function are v i o l a t e d less often than when T2 i s u t i l i s e d ; otherwise, there are no s i g n i f i c a n t differences between the estimated properties of the cost function. In p a r t i c u l a r , the conclusions concerning the adjustment costs and regulatory parameters are upheld. 4.2 EMPIRICAL RESULTS FOR THE CONSTRAINED MAXIMIZATION MODEL WITH ENDOGENOUS OUTPUT The estimation results pertaining to the set of equations (3.7)-(3.9) and (3.13),(3.14) are presented i n Tables 4.5 to 4.7. The goodness-of-fit s t a t i s t i c s and the high percentage of s i g n i f i c a n t parameters i n both the v^ and V 2 versions of the model indicate that i n t h i s case also the estimated model f i t s the data rather well. The major conclusions arrived at i n the l a s t section regarding the e f f e c t of regulation and the importance of adjustment costs receive further empirical support. The estimated adjustment costs parameter i s again s t a t i s t i c a l l y s i g n i f i c a n t i n both the v^ and V 2 versions of the model and si m i l a r i n magnitude to that of 4.1. The regulatory parameter i s s t a t i s t i c a l l y s i g n i f i c a n t at the 0.1 and 0.025 l e v e l s of confidence depending on whether v^ or V 2 i s used i n the estimation. The implied Lagrange m u l t i p l i e r values are 0.71 and 0.88, well within the t h e o r e t i c a l range. The estimated © value, which t e l l s of the speed of the t o l l price adjustment process, i s highly s i g n i f i c a n t i n both versions of the model and close to 0.6, meaning that B e l l obtains on average 60% of i t s desired price adjustment i n any given year. Together, those observations on B, (3 and 0 strongly suggest that both dynamics (on the cost and demand sides) and regulation are important i n modelling a regulated u t i l i t y l i k e B e l l . In order to ascertain the relevance of lagged price response, a l i k e l i h o o d - r a t i o t e s t i s performed and the r e s u l t s are tabulated i n Table 4.8. The t e s t consists i n estimating each version of the model twice, once imposing the constraint that 0 equals one (instantaneous adjustment of t o l l prices) and then f r e e l y estimating 0. T h e i l (1971, p.397) demonstrates that: - 2 [ Ln (H 0) - Ln (H x) ], .(4.7) T A B L E 4.5 P A R A M E T E R E S T I M A T E S : MODEL WITH ENDOGENOUS OUTPUTS v x (CAPM) c o e f . s t . d e v . c o e f . s t . d e v . a 00 -69.555 10.116 e 0.5970 0.0611 a 01 1.1183 0.1751 ^0 -6.5609 1.3428 a 02 -14.208 7.4742 ^1 -1.3631 0.1643 a 03 -0.1191 0.1320 1.1840 0.1499 a 04 32.152 7.7635 a 05 -0.2920 0.0450 a l l -0.0930 0.0535 a 22 9.7729 3.0046 a 33 0.0007 0.0283 a 44 -1.9309 2.9237 a 12 0.2418 0.0823 a 13 0.0067 0.0309 a 14 -0.2143 0.0505 a 15 -0.0008 0.0005 a 23 0.0015 0.0588 a 24 -4.4188 2.8414 a 25 -0.1123 0.0244 a 34 0.0233 0.0298 a 35 0.0013 0.0005 a 45 0.1199 0.0230 B 0.0015 0.0005 P 2.4867 1.5553 TABLE 4.6 PARAMETER ESTIMATES: MODEL WITH ENDOGENOUS OUTPUTS v 2 (DCF) coef. s t . dev. coef. s t . dev. a 0 0 -66 .693 7.2798 e 0.6050 0.0944 a 0 1 1.1684 0.1754 -7 .4573 1.4547 a 0 2 -10 .756 3.3824 b l -1 .2395 0.1833 a 0 3 -0 .0280 0.1007 b 2 1.2833 0.1622 a 0 4 28.756 3.8823 a 0 5 -0 .2950 0.0360 a l l -0 .1095 0.0449 a 2 2 11.717 1.5907 a 3 3 -0 .0023 0.0381 a 4 4 -0 .0381 1.0121 a 1 2 0.2064 0.0575 a 1 3 0.0300 0.0166 a 1 4 -0 .2062 0.0468 a 1 5 -0 .0008 0.0005 a 2 3 -0 .0096 0.0795 a 2 4 -6 .3278 1.0914 a 2 5 -0 .1094 0.0179 a 3 4 0.0133 0.0429 a 3 5 0.0010 0.0007 a 4 5 0.1182 0.0170 B 0.0012 0.0003 P 7.0552 2.4224 96 TABLE 4.7 SUMMARY STATISTICS FOR THE MODEL WITH ENDOGENOUS OUTPUTS v l R 2 D.W. cost equation 0.9958 1.9423 labor share equation 0.8611 1.8108 c a p i t a l accumulation equation 0.9824 1.4947 t o l l output equation 0.9987 0.7975 t o l l demand equation 0.9723 1.9180 Log of the l i k e l i h o o d function: 133.23 v 2 R 2 D.W. cost equation 0.9956 1.7999 labor share equation 0.8536 1.7395 c a p i t a l accumulation equation 0.9831 1.4668 t o l l output equation 0.9990 1.1699 t o l l demand equation 0.9718 1.9518 Log of the l i k e l i h o o d function: 133.10 follows a Chi-square d i s t r i b u t i o n with r degrees of freedom where r i s the number of r e s t r i c t i o n s imposed, H Q i s the value of the l i k e l i h o o d function under the n u l l hypothesis and i s the corresponding value when the constraint i s relaxed. The hypothesis of instantaneous price adjustment can be rejected i n both versions of the model at the 0.01 l e v e l of confidence. This finding suggests that the f a i l u r e s i n estimating the standard profit-maximizing model with endogenous output may be ascribable to the unrealism of the assumption of f u l l p r ice adjustment i n one period. However, t h i s conclusion, and the results of t h i s model i n general, must be interpreted with some caution i n view of the ad-hoc nature of the posited adjustment process. Although i t seems reasonable and i s e a s i l y implemented econometrically, the lagged price response formulation i s not f u l l y r a t i o n a l i z e d by an underlying (constrained) optimizing behavior on the part of the u t i l i t y . Therefeore, a lagged price response i s estimated but not r e a l l y explained. Nevertheless, t h i s l a s t e f f o r t at introducing an addi t i o n a l dynamic element i n the modelling of a regulated u t i l i t y does indicate that the imposition of s t a t i c conditions for the choice of the output l e v e l may lead to a TABLE 4.8 LIKELIHOOD RATIO TEST ON 8 Test H Q : d.f. Chi-square Chi-square Computed Decision value (0.05) value (0.01) value v l 8 = 1 1 3.84 6.63 11.93 Reject v2 8 = 1 1 3.84 6.63 8.48 Reject Note: the maximized l i k e l i h o o d values under the n u l l hypothesis are 127.25 and 128.86 respectively. m i s p e c i f i c a t i o n problem, just as does the hypothesis that there are no marginal adjustment costs when the stock of c a p i t a l i s adjusted i n any period. The information contained i n Tables 4.5-4.7 attests that the estimated demand equation for t o l l output f i t s the data very well. As required by profit-maximization, the demand for t o l l output i s price e l a s t i c . Moreover, as can be seen i n Table 4.9, the estimated e l a s t i c i t i e s corroborate the findings of past studies. Tables 4.10 and 4.11 summarize the implications of the estimated model for B e l l ' s behavior and production structure. 100 TABLE 4.9 ESTIMATED LONG-RUN TOLL ELASTICITY OF DEMAND FOR BELL, VARIOUS STUDIES Source E l a s t i c i t y value Breslaw and Smith (1982) chosen range for s e n s i t i v i t y analysis -1.2 to -1.8 Fuss and Waverman (1981) "competitive t o l l " -1.39 "monopoly t o l l " -2.05 Denny et a l . (1981a) "competitive t o l l " -1.44 "monopoly t o l l " -1.64 Dobell et a l . (1972) r e s i d e n t i a l demand -1.2 business demand -1.3 This d i s s e r t a t i o n v x -1.36 v 2 -1.24 101 TABLE 4.10 MONOTONICITY AND CURVATURE PROPERTIES ON THE COST FUNCTION: MODEL WITH ENDOGENOUS OUTPUTS Monotonicity conditions: c a p i t a l l o c a l output t o l l output labor share 22/27 21/27 27/27 27/27 21/27 21/27 27/27 27/27 Curvature properties: concavity i n input prices s u f f i c i e n t conditions for a maximum of p r o f i t 27/27 13/27 27/27 21/27 TABLE 4.11 SUMMARY STATISTICS ON BELL'S TECHNOLOGY: AVERAGE VALUES v l E l a s t i c i t y of variable costs with respect to: l o c a l output 0.80 t o l l output 0.21 c a p i t a l -0.69 Scale e l a s t i c i t y 1.67 S h i f t i n the variable cost function -0.01 Own e l a s t i c i t y of input demand: labor ( E L L ) -0.47 materials ( E ^ ) -0.95 103 Overall, the estimated cost function displays the desired properties. Although they are met less overwhelmingly than i n the exogenous outputs case, the monotonicity conditions on the cost function are found to hold at a large majority of observation points. The behavior of the cost e l a s t i c i t y with respect to l o c a l output i s again perverse over the same six-year period as previously; s i m i l a r l y , monotonicity of costs i n the l e v e l of c a p i t a l stock f a i l s i n the f i r s t years of the sample. On the other hand, the t o l l output and labor share monotonicity conditions, and the concavity of the cost function i n input p r i c e s , hold everywhere. The concavity of the objective function i n t o l l output and c a p i t a l i s checked at each observation point. This would make the estimated necessary conditions also s u f f i c i e n t for a maximum of p r o f i t . This indeed i s the case at about one half and two-thirds of the years i n the sample-*. In addition, the marginal revenue function i s found declining more rapidly than the marginal cost function everywhere. The opposite r e s u l t held for the estimation of a standard profit-maximizing model (0 = 1 ) , thus v i o l a t i n g a necessary second-order condition. This i s add i t i o n a l evidence for the f r u i t f u l n e s s of the dynamic-demand approach. Now comparing the values i n Table 4.11 to those i n 4.4, one s t r i k i n g fact to emerge i s that many estimated 104 features of B e l l ' s technology are not very sensitive to the objective function s p e c i f i c a t i o n . For instance, the scale e l a s t i c i t y and the input demand e l a s t i c i t i e s show l i t t l e v a r i a t i o n (although the l a t t e r appear somewhat more e l a s t i c i n the endogenous output case). Likewise for the e l a s t i c i t y of costs with respect to the stock of c a p i t a l , which goes to -0.7 from -0.8 or -0.9. In contrast, the e l a s t i c i t i e s of cost with respect to t o l l and l o c a l outputs and the estimated tim e - s h i f t i n the cost function are affected. The differences between the two estimated models are greatest for the t o l l e l a s t i c i t y of costs. The estimated marginal cost of t o l l output i s , on the whole, half what i t was i n 4.1 while that of l o c a l output i s some t h i r t y percent higher. Moreover, i n the endogenous output case, the marginal cost of t o l l output i s almost constant over the twenty-seven year period covered by the estimation: i t slowly climbs from a value of 0.2 i n 1953, with an output price of 0.83, to 0.33 i n 1979, when the p r i c e of t o l l output reaches 1.17 . 105 4.3 CONCLUDING REMARKS ON THE ESTIMATION A few remarks are i n order before concluding the analysis of the empirical r e s u l t s . F i r s t , i t appears that the o v e r a l l performance of the estimation i s s a t i s f a c t o r y although i t could be improved on a few points. For instance, i t i s s t i l l unclear why the l o c a l output monotonicity conditions f a i l to obtain i n a few years. One possible explanation l i e s i n the high c o r r e l a t i o n that e x i s t s between the l o c a l output quantity and the technological change proxies. In both the exogenous and endogenous-output cases, the c o e f f i c i e n t on the cross-term i n l o c a l output and the proxy for technological change i s negative, and i s large i n absolute value when compared to that on t o l l output and the proxy. I t may be that the chosen proxies, being more c l o s e l y correlated to the growth i n l o c a l output than i n t o l l output, ascribe more of the reduction i n costs to l o c a l output than a "perfect" index would, thus weighting down the e l a s t i c i t y of costs to l o c a l output. This point draws attention to the way i n which technical change enters the estimation. There are many plausible s p e c i f i c a t i o n s for capturing the e f f e c t of technical change and other s p e c i f i c a t i o n s may produce better r e s u l t s . The s e n s i t i v i t y of the results to the choice of a p a r t i c u l a r proxy i s another i n t e r e s t i n g issue that has been 106 mostly ignored i n t h i s work. While the exogenous output model was successfully estimated with two d i s t i n c t technical change proxies that generated si m i l a r r e s u l t s , the estimation of the endogenous output model using the percentage of phones with access to d i r e c t distance d i a l i n g proved much i n f e r i o r to that with the alternative technical change indicator, T2. For t h i s reason the results corresponding to the f i r s t s p e c i f i c a t i o n are not discussed i n 4.2. Further attempts at estimating t h i s l a t t e r model may yet lead to a better f i t because the models estimated are highly nonlinear. Thus, there i s a p o s s i b i l i t y that the obtained estimates corresponded to a l o c a l rather than a global maximum. Inc i d e n t a l l y , i t should be pointed out that a l l reported r e s u l t s were checked by re-estimating the models with d i f f e r e n t sets of s t a r t i n g values for the parameters, to ensure that convergence was to a global maximum of the l i k e l i h o o d function. Second, note that i n general the v^ and v 2 s p e c i f i c a t i o n s lead to the same q u a l i t a t i v e conclusions concerning the importance of regulation, adjustment costs, lags, etc... although some parameter values are sensitive to the user cost s p e c i f i c a t i o n , as was pointed out i n the discussion of the r e s u l t s . 107 F i n a l l y , the exogenous output model, and p a r t i c u l a r l y i t s V 2 _ s p e c i f i c a t i o n , i s singled out as the best d e s c r i p t i o n of B e l l ' s behavior. This conclusion i s based on the o v e r a l l performance of the estimation, on the fact that the s u f f i c i e n t conditions for optimization are met g l o b a l l y and l a s t l y , on t h e o r e t i c a l grounds: the profit-maximization model with lagged price adjustments i s to be seen as a f i r s t approximation to a dynamic decision rule for output choice that has not been thoroughly investigated. This does not mean that the empirical r e s u l t s i n 4.2 are unreliable. I t simply r e f l e c t s the greater confidence t h i s author has i n the robustness and a p p l i c a b i l i t y of the f i r s t model of cost minimization. NOTES TO CHAPTER 4 Fuss and Waverman (1981) estimate the labor demand e l a s t i c i t y and the materials demand e l a s t i c i t y at -0.437 and -0.371, but impose a zero c a p i t a l demand e l a s t i c i t y on the estimation. Without t h i s r e s t r i c t i o n , they obtain a p o s i t i v e , but non s i g n i f i c a n t , (long-run) c a p i t a l demand e l a s t i c i t y . This may r e s u l t from a m i s p e c i f i c a t i o n of the producer's problem that ignores both the existence of adjustment costs and regulation. Denny et a l . (1981a) also obtain a p o s i t i v e demand pr i c e e l a s t i c i t y for c a p i t a l of 0.019 i n a very s i m i l a r model. In Fuss and Waverman (1977), however, the the long-run c a p i t a l demand price e l a s t i c i t y i s estimated at -0.671. In Bernstein (1987), the technological change proxy i s a binary variable which takes the value 1 between 1958 and 1971, the years i n which most innovations were introduced at B e l l . This allows the author to obtain very s a t i s f y i n g empirical results but leaves open the question of the reasonableness of t h i s s p e c i f i c a t i o n which implies that costs decreased at once i n 1958 and, more troubling, increased at once at the end of the period because of technological change. Incidentally, the concavity of the objective function i s s u f f i c i e n t but not necessary for a maximum of p r o f i t . Hence the estimated set of f i r s t - o r d e r conditions could s t i l l r e s u l t from maximizing behavior even i f concavity g l o b a l l y f a i l e d . 109 CHAPTER 5 A PRODUCER PRICES APPROACH TO MEASURING THE LOSS  OF OUTPUT DUE TO IMPERFECT REGULATION IN A DYNAMIC ENVIRONMENT 5.0 INTRODUCTION Measuring the waste of resources induced by regulation i s necessary because present regulatory regimes do not succeed i n implementing an optimal a l l o c a t i o n of resources. The purpose of t h i s chapter i s to derive approximations to the deadweight loss of regulation. This loss i s the cost imposed on society by the deviations from the desired a l l o c a t i o n of resources that are brought about by the process of regulation i n a dynamic context. Losses i n e f f i c i e n c y , following Debreu (1951), are of three kinds: (a) the waste of resources due to the u n d e r u t i l i z a t i o n or underemployment of the factors of production of society; (b) the e f f i c i e n c y loss due to the f a i l u r e s , on the part of producers, to obtain the maximal output from a given set of u t i l i z e d resources; and, (c) the loss i n e f f i c i e n c y when inputs and outputs are not allocated i n a way that maximizes a c e r t a i n notion of welfare, such as the Pareto c r i t e r i o n . 110 The f i r s t type of waste can be ascribed to the economic i n s t i t u t i o n s of a society and to the management of c e r t a i n macro-variables and i s ignored i n t h i s thesis. The second type i s c l o s e l y related to the notion of X-i n e f f i c i e n c y or operations " o f f " the production f r o n t i e r . As shown i n Proposition 2.3, such misuse of resources i s never p r o f i t a b l e under rate of return regulation as long as p r o f i t maximization i s a maintained hypothesis. Hence t h i s kind of resource cost w i l l also be ignored i n the remainder of t h i s t h e s i s . This leaves the t h i r d kind of resource cost, usually c a l l e d " a l l o c a t i v e i n e f f i c i e n c y " . This misallocation of resources, which i s the focus of t h i s chapter, occurs whenever d i f f e r e n t producers or consumers face d i f f e r e n t p r i c e s for the same goods or whenever the private and s o c i a l p r i c e s d i f f e r . The measurement of t h i s l a s t type of resource cost has received considerable attention i n the past and focused p r i m a r i l y on the d i s t o r t i o n s induced by taxation and monopolistic p r i c i n g . There are b a s i c a l l y two methodologies for the measurement of waste: general and p a r t i a l equilibrium. Hotelling (1938), Hicks (1978), Boiteux (1951), Debreu(1951), Allais(1973) and Diewert (1981e) present t h e o r e t i c a l general equilibrium analyses. In recent years many attempts have been made to implement econometrically general equilibrium models of the economy and to I l l compute deadweight losses. Shoven and Whalley (1984) survey the l i t e r a u r e on applied general equilibrium models. Harris and Cox (1983) i s a recent application of t h i s methodology. This thesis does not adopt the general equilibrium approach because i t s information requirements are simply too high. To use i t , one needs to estimate the technologies of a l l sectors of the economy as well as the preference structure of a l l consumer groups. In addition, since t h i s chapter deals with the intertemporal loss of e f f i c i e n c y , a correct parametrization of consumers' preferences would normally require that the p o s s i b i l i t y of change i n tastes be incorporated into the estimation. F i n a l l y , even i f these information requirements could be met, the estimation would most l i k e l y proceed with simple functional forms to save degrees of freedom and for the sake of t r a c t a b i l i t y . Instead, a p a r t i a l equilibrium approach i s chosen i n t h i s t h e s i s . More p r e c i s e l y , a producer prices approach i n which only the revenues and costs of producers need to be estimated i s taken. The essence of t h i s approach i s the following: given a vector of "optimal" or "reference" p r i c e s , maximize the productive sector's net value of output and compare t h i s value to the distorted equilibrium net output vector, evaluating a l l inputs and ouputs at the "optimal" or reference p r i c e s . 112 This price-approach to the measurement of productive i n e f f i c i e n c y can be found i n Hicks (1941-42) and i n a number of papers by Diewert (1981a, b, c, 1985b). I t can be opposed to the quantity-approach underlying the Allais-Debreu methodology i n which one good or basket of goods i s used as a "reference good". But i n both cases, p r i c e and quantity approaches, the same kind of question i s being asked: "how much more output (evaluated at the reference p r i c e s i n the price-approach, or of the reference good i n the quantity-approach) can be obtained i f the d i s t o r t i o n s characterizing the i n e f f i c i e n t a l l o c a t i o n are removed?". In p r i n c i p l e , the removal of these d i s t o r t i o n s could a f f e c t very many prices and the welfare of many d i f f e r e n t consumers as the equilibrium conditions i n one market afte r another are affected by the change. Thus a general equilibrium approach i s required on t h e o r e t i c a l grounds; as noted, however, the p r a c t i c a l implementation of such a model i s extremely d i f f i c u l t , so i t i s decided to focus e x c l u s i v e l y on the productive side of the economy. In Diewert (1981a) t h i s ( p a r t i a l equilibrium) producer prices approach i s put to work to obtain a quadratic approximation to the loss of output due to (imperfect) rate of return regulation i n a s t a t i c context. As was pointed out i n the Introduction to t h i s thesis, Diewert's i s one of only a handful of papers that deal with the evaluation of monopoly regulation. The task of th i s chapter i s to extend the analysis to the case of a dynamic economy i n which the c a p i t a l accumulation decisions of producers are f u l l y endogenous. A one-sector measure i s derived i n the next section. The estimated model of producer behavior of Chapter 4 i s then used to generate estimates of the loss of output due to the A.-J. e f f e c t . As i t turns out, however, the existence of important non-convexities i n B e l l ' s technology renders the computation of the loss of output due to monopolistic p r i c i n g and i n e f f i c i e n t c a p i t a l accumulation impossible with the derived loss formula. A two-sector planning model of the economy i s developed i n the clos i n g section of t h i s chapter and used to arr i v e at a more general loss formula that should, i n p r i n c i p l e , allow one to overcome the problem associated with the presence of important non-convexities . 5.1 A ONE-SECTOR DYNAMIC DEADWEIGHT LOSS MEASURE In t h i s section, the loss of output r e s u l t i n g from the p r i c i n g and investment decisions of a rate-regulated u t i l i t y i s evaluated i n a one-sector model using the producer pr i c e approach outlined i n 5.0. Suppose the technology of one or more regulated producers can be defined and described as i n Chapter 2 and that a s o c i a l planner wishes to maximize the net present value of the regulated sector's production (the value of outputs minus that of inputs) using the reference (or "optimal", more w i l l be said below about that) prices {p t}, {wfc}, Q , and R over the horizon t = 1, t ' . Formally, the planner's problem i s to choose {y t} and {s*-} i n order to maximize the s o c i a l objective function (5.1): Max R(0,t) [ p u y s f c )] + R ( 0 , t ' ) Q S . . . (5.1) If a s o l u t i o n to (5.1) exists and i f : (Al^Cfy*-, w ,t t-1 , s ) i s twice continuously s d i f f e r e n t i a b l e i n i t s arguments; then, the following f i r s t - o r d e r conditions are necessary at the unconstrained maximum of (5.1). Conditions Wl R(0,t) [ p t - V t C ( y \ w\ s^1, s f c )] = Oj t ~ 1^  * * */ t j • ••(5*2ci) -R(0,t) V .C(y t, w1, s t _ 1 , s f c ) - ' s R(0,t+1) V C ( y t + 1 , w t + 1, s*, s t + 1 ) = 0 , s t = 1, ..., t ' - l ; ...(5.2b) R ( 0 , f ) [-V t C ( y t ' ; wt'/ s t _ 1 , s t') + Q ] = 0 . ...(5.2c) s Notice the "~" over yt and s t , which indicates that (5.2a)-(5.2c) hold at a s o c i a l optimum. Also assume that the (strong) second-order s u f f i c i e n t conditions for an unconstrained maximum are s a t i s f i e d at {y t}, {s*1}. This implies that: (A20) the matrix of second-order derivatives of (5.1) with respect to the components of y*- and s f c evaluated at {y c}, is } i s negative d e f i n i t e ( c a l l that matrix A). The essence of the producer prices approach to evaluating the loss of output due to the existence of any d i s t o r t i o n i s to compare the value of net production at {y t}, {st} to that at the d i s t o r t e d equilibrium {yt}, {|t}. By the d e f i n i t i o n of {yt}, {st}: E ^ i R(0,t) [p f c yt - C(y t, w\ s ^ 1 , s t )] > E ^ R(0,t) [ p f c yt - Cty 1 1, wfc, s t _ 1 , s f c ) ], ...(5.3) and, i n p a r t i c u l a r , E ^ R(0,t) [p f c y f c - C(ytt wfc, s t _ 1 , s t )] > E^ = 1R(0,t) [ y - C(y , w\ s r A , s c ) ], ...(5.4) where {yt}, {st} are the quantities which solve the regulated monopolist's problem i n Chapter 2. The producer prices measure of loss of output i s simply the difference between the two terms i n (5.4). Therefore, i f a complete characterization of the technology of the producers i n the regulated sector and {pt}, {wfc} were available to the welfare analyst, the computation 117 of the producer prices loss of output would involve solving equations (5.2) for {y*-}, { s ^ , computing the net value of output at t h i s s o c i a l optimum and comparing that value to the value of the observed, di s t o r t e d net production vector evaluated at {p*-}, {wt>. It i s because t h i s information i s not a v a i l a b l e i n general that the need arises to use approximations to t h i s loss of output. That the t(Y t>, {s^ -} ]-solution w i l l i n general d i f f e r from the d i s t o r t e d equilibrium [{y*1}, {s t}] can be deduced from a comparison of the f i r s t - o r d e r conditions of the two problems. Recall Conditions R of Chapter 2: Conditions R: (1 - u ) R(0,t) [ p f c - m f c - V C(y f c , w f c , I t _ 1 , tt)] = 0 , Y t = 1, ..., t ' ; ...(5.5a) * t t- * t - 1 * + [R(0,t) (-V t C ( y \ wr, s r \ s c ) ) + s *t-+l t + 1 *t *t+l t R(0,t+1) (-V t C ( y t + t w t + | s ) s + u r = 0 N, s t = 1, . . . , t' ; . . . (5.5b) 118 (1-u) [R(0,t') (-V t ,C(y , wr, s r 7 s ) + Q)] = 0 N s (5.5c) As before, l e t mt = -V y(p t(y t U-1^ { R ( 0 , t + l ) e t + 1 [u/(l-u)]}, and define d T E (p f c - p f c ). Also l e t R(0,t) = R(0,t), wfc = w** , and Q = Q. Then, following Diewert (1981a, b,c, d; 1985b), the following z-equilibrium can be defined where z e [0,1] can be thought of as a scalar of d i s t o r t i o n . Conditions DI R(0,t) [ p 1 + z (d* - mfc ) - V y C{ytt w1, s t _ 1 , st )] = Oj, t = 1, ..., t'; ...(5.6a) -R(0,t) V tC(ytf wfc , s*" 1, s f c ) -s R(0,t+1) V t C ( y t + 1 , w t + 1, st, s t + 1 ) + u f c z = 0, s t = 1, ..., t'-1; ...(5.6b) R(0,t') [ -V t C ( y t ! wfcl s t , _ 7 s t') + Q ] = 0N- ...(5.6c) 119 Consider the set of equations (5.6), when z = 0 Conditions DI reduce to Conditions Wl. When z = 1, the f i r s t - o r d e r Conditions R obtain. In general then, unless (d*-- m*-) = 0j_ and u*- = 0 N, Conditions Wl and Conditions M w i l l d i f f e r . Regard equations (5.6) as a system of t' x (I + N) equations i n {y t} and {s^} where p f c, wfc, Q, (d^ - m*-) and u t are f i x e d : these are the exogenously determined "optimal" or reference prices and the vectors of d i s t o r t i o n s d, m and u , assumed fix e d for convenience . (A20) and the i m p l i c i t function theorem guarantee that such functions e x i s t . Further assume that: (A21) the reference price vector i s : (P , W, Q, R) = (p, w, Q, R). This means that the loss of output due to regulation i s to be evaluated using the observed, d i s t o r t e d prices that p r e v a i l i n the regulated equilibrium. Hence, the question being asked i s : "how much more output, evaluated at the actual (observed) prices, can society get i f the d i s t o r t i o n s a f f e c t i n g the regulated sector's decisions are removed?". Assumption (A21) i s somewhat a r b i t r a r y . In fa c t , there i s some arb i t r a r i n e s s i n choosing any reference price vector (P, w, Q, R). In addition, doing away with (A21) would require that the "exogenously" determined reference prices be 120 computed, a task which would require that a general equilibrium model be estimated. This chapter therefore aims at answering the more lim i t e d question above, as i s done i n Diewert (1981a) i n a s t a t i c context. This procedure i s a c t u a l l y very si m i l a r to that suggested by Harberger (1971).! The strategy developed by Diewert to derive loss formulae that approximate the difference i n value between the two programs [{y*-}, {st}] and [{yt}, {st}] i s to express welfare as a function of z, which i s done i n (5.6), and to use a Taylor series approximation to the second-order around z = 0 to evaluate the change i n welfare. Using (5.1), W(z) = R(0,t) [ p t y(z) - C(y(z), w, s t ' 1 ( z ) , s t ( z ) ] + R ( 0 , f ) Q s t ( z ) . ...(5.7) A second-order approximation of the change i n welfare i s : W(l) - W(0) = W'(0) (1-0) + (1/2) W"(0) (1-0) 2. ...(5.8) This requires that W'(0) = [3W(z)/dz] z = 0 and W'(0) = [ d 2 W ( z ) / 3 z 2 ] z = 0 be evaluated. W'(0) = E ^ R ( 0 , t ) [ V Z y t(0) (p f c - V Y C(t)] + E ^ " 1 V z s t ( 0 ) [ - R ( 0 , t ) V t C ( t ) - R(0,t+1) V t C ( t + l ) ] + s s V , S t'(0) [ - R ( 0 , f ) V ,.,C(f ) + R ( 0 , t ' ) Q ], ...(5.9) z s^ where C(t) = C ( y t ( z ) , wfc , s t - 1 ( z ) , s t ( z ) ) . Thus, using Conditions DI and evaluating at z = 0: W'(0) = z { [ E * ^ R ( 0 , t ) V Z y t(0) ] -[ s j : ^ 1 U f c V z s t(0) ] } + 0 = 0. ...(5.10) D i f f e r e n t i a t i n g again with respect to z, and evaluating at z = 0 gives: W " ( 0 ) = E ^ [ R ( 0 , t ) mt y f c (0) ] -E ] : ^ 1 u f c s t ( 0 ) . ...(5.11) 122 Therefore, W" (0) = [ m T , -u T ] / V z y(0)\ \ V , s(0) . . . (5.12) where m = [R(0,1) m , R(0,t') m ] and a T= iu 1 T , F i n a l l y , using (A19) and (A20), the derivatives of y(z) and s(z) with respect to z around z = 0 can be computed as: \ / -\ = A " . . ..(5.13) v z y(0) V z s(0) V -u Using (5.10), (5.12) and (5.13), (5.8) can be written as: •L 1 = W(l) - W(0) = (0.5) W'(0) (0.5) [ m T , - i l T ] A - 1 [m T , -u T ] T < 0. ...(5.14) The i n e q u a l i t y i n (5.14). follows from the negative definiteness of A at z = 0 while LT_ i s the (positive) deadweight loss due to imperfect regulation. Notice that the information needed to compute i s quite l i m i t e d : knowledge of the d i s t o r t i o n vectors m and u and l o c a l knowledge of the Hessian matrix of the regulated sector's cost function with respect to {y t} and {s t} evaluated at z = 0. There i s however one drawback i n the computation of L^: the matrix A~^ " i s defined at the unobserved "optimal" a l l o c a t i o n of resources. Two possible ways of dealing with t h i s d i f f i c u l t y are: ( i ) the use of a quadratic approximation to the cost function i n applied work, such as the normalized r e s t r i c t e d quadratic cost function (see Lau, 1976; and Denny et a l . , 1981b), which has the nice property that the Hessian of C(t) -1 * - l i s a matrix of constants; or ( i i ) approximate A by A , where the l a t t e r i s the Hessian of C(t) evaluated at the observed (distorted) equilibrium. The loss formula (5.14) can also be s p e c i a l i z e d to handle a number of s p e c i f i c s i t u a t i o n s . For instance, i f the regulated producers are price-takers and constrained to supply any f e a s i b l e quantity at the regulated p r i c e , t h e i r revenue (and outputs) become exogenous. The regulated producers' objective i s then to maximize, under the regulatory constraint, t h e i r expected p r o f i t by choosing {s t}. In t h i s case, the matrix A refers to the Hessian of C(t) with respect to {s t}, and the loss of output becomes: - L 2 = (1/2) [ -u T ] A _ 1 [ -u T ] < 0. ...(5.15) S i m i l a r l y , i f the regulated producers are constrained to earn exactly the competitive rate of return, jit = 0 N and L i i s made to depend only on m*~. Inspection of and L2 immediately reveals that the deadweight loss i s zero i f and only i f a l l d i s t o r t i o n s vanish; that i s , i f : ffi = °(t'x I) a n d * = ° ( f x N)* I f (mT, J T T , 0 T T , X N ) } > 0T f c, x ( I + 2 N ) ) , where a > 0 implies a^ > 0 but a^ ^ 0 for a l l i , then the deadweight loss approximations are always s t r i c l y p o s i t i v e . One may conjecture that the loss formulae generally increase with the size of the d i s t o r t i o n s . It i s possible to demonstrate that t h i s i s unambiguously the case i f (i) only one d i s t o r t i o n e x i s t s or, ( i i ) i f a l l d i s t o r t i o n s are scaled up. Those two cases are examined and discussed further below. Suppose there i s a unique whereas m^  = 0, for a l l other 9 Then, reduces to: 14= (-0.5) (0, 0, m^T , 0, 0) A - 1 ( 0 , , 0, m ^, 0, , 0) > 0. (5.16) D i f f e r e n t i a t i n g the quadratic function L| with respect to m ^  gives: d i s t o r t i o n , say m£ > 0 and j , and u = °Jt*x N)• dL^/dmJ; = -(0, 0, 1, 0, 0) 125 A - 1 (0, 0, m£, 0, 0) > 0. . . .(5.17) A analogous r e s u l t n a t u r a l l y holds f o r (dL £ / d u £ ) , and s i m i l a r l y f o r L 2 w i t h a p p r o p r i a t e m o d i f i c a t i o n s . T h i s shows t h a t , i n the extreme case of a unique d i s t o r t i o n , w e l f a r e i s i n v e r s e l y r e l a t e d t o the magnitude of m| or Consider now the change i n w e l f a r e t h a t would r e s u l t from s c a l i n g up or down a l l d i s t o r t i o n s . L e t k be a p o s i t i v e s c a l a r , then i t i s e a s i l y shown t h a t the deadweight l o s s approximations are m u l t i p l i e d by the square of t h i s s c a l a r . F o r m a l l y , the l o s s formulae are homogeneous f u n c t i o n s o f degree two i n the d i s t o r t i o n s . T h i s can e a s i l y be v e r i f i e d : L (km T , k ( -u T ) , k ( 0 ^ t , x N ) ) ) = (1/2) (km T , -ku T , 0^ t, x N ) ) A" 1 (km T , -ku T , N ) ) T 126 (1/2) k 2 (m T , -u T , 0 ^ t I x N ) ) - 1 - T - T T A ( m T , -u T , 0 ( t , x N ) ) k 2 L l (m , -u \ 0^ t, x N ) ) . . (5.18) Those two l a s t results are proved i n Diewert (1981a) for the s t a t i c measure of loss along with some other propositions. Since the structure of matrix A i s si m i l a r to the corresponding matrix i n Diewert (1981a), most of the propositions proved i n the l a t t e r hold i n t h i s context as well. Since i t has long been known that "removing" only one d i s t o r t i o n from a non-optimal state of the economy may not increase welfare (this i s the t y p i c a l second-best r e s u l t ) , the previous properties of the loss formulae should not come as a surprise. 5.2 127 TENTATIVE RESULTS ABOUT THE DEADWEIGHT LOSS DUE TO INEFFICIENT REGULATION The computation of the deadweight loss formulae L]_ and L 2 requires that the following quantities be estimated: (i) the vectors of deviations from optimal prices, and ( i i ) the inverse of matrix A, whose elements are the Hessian of the cost function evaluated at each period with respect to output and c a p i t a l . This inverse should i n p r i n c i p l e be evaluated at the s o c i a l welfare optimum but the strategy taken i n t h i s chapter consists i n using the Hessian at the di s t o r t e d equilibrium as an approximation to ca l c u l a t e the "true" A" 1. This d i s s e r t a t i o n has produced information on the vectors of d i s t o r t i o n s and the Hessian matrix of B e l l ' s cost function. However, there remains one major d i f f i c u l t y : the matrix A" 1 must be negative d e f i n i t e . In the p a r t i c u l a r case of B e l l Canada, the cost function i s not convex i n c a p i t a l and output, whether the Hessian of the cost function i s estimated using either the exogenous or the endogenous output model. Even i f economies of scale are compatible with negative definiteness of the matrix A ~ l , the very large scale economies estimated i n 4.1 and 4.2 re s u l t i n the marginal cost function for t o l l output declining almost everywhere; 128 t h i s makes the cost function d e f i n i t e l y non convex and A~^ res o l u t e l y non negative d e f i n i t e . Maybe another approximation of A ~ l could be used. A l t e r n a t i v e l y , an estimate of the welfare loss due to i n e f f i c i e n t c a p i t a l accumulation alone can be computed since the firm's objective function i s concave i n the c a p i t a l stock. This approximation to the deadweight loss i s but a f r a c t i o n of the t o t a l loss of output since i t does not capture the losses due to i n e f f i c i e n t output production. But i t can serve to assess the losses implied by o v e r c a p i t a l i z a t i o n . This l a s t a l t e r n a t i v e i s selected and the deadweight loss due to i n e f f i c i e n t c a p i t a l accumulation i s computed using equation (5.15). The matrix A i s estimated using the re s u l t s of the model with exogenous outputs. The present value of the stream of foregone output evaluated at the actual prices i s computed and the (present value of the) average yearly loss i s divided into the average value of (actualized) variable costs to convey a better idea of the magnitude of the losses. The res u l t s are reported i n Table 5.1. Even though the regulatory parameter i s not s t a t i s t i c a l l y s i g n i f i c a n t when v^ i s used, the losses are computed for the two versions of the model. Moreover, since the v a r i a b i l i t y i n the excess return variables i s quite high, 129 the losses are computed for four d i f f e r e n t time periods: 1953-60, 1961-66, 1967-72 and 1973-79 2. There are three s a l i e n t features of the numbers i n Table 5.1. F i r s t , the estimated losses are extremely se n s i t i v e to the choice of the user cost of c a p i t a l variable. I t i s easy to see why: the parameter estimates of the c a p i t a l - r e l a t e d c o e f f i c i e n t s and the excess return variables are very sensitive to the s p e c i f i c a t i o n of the user cost variable as i s evident from Tables 4.1 and 3 .4. Second, the estimated losses are v i r t u a l l y n i l when v^ i s used. Remember that the estimated Lagrange m u l t i p l i e r i s not s i g n i f i c a n t under t h i s s p e c i f i c a t i o n . The v ^ - s p e c i f i c a t i o n , although performing somewhat less well than the V 2 ~ s p e c i f i c a t i o n , does seem to indicate that rate of return regulation has very l i t t l e impact on the investment decisions of the firm and that losses, i f any, are n e g l i g i b l e . This r e s u l t may depend on the assumption that 3 i s a constant. Third, the estimated losses under the V 2 ~ s p e c i f i c a t i o n are rather small but are not n e g l i g i b l e . The losses represented nearly four percent of t o t a l cost i n the early s i x t i e s . The losses p r a c t i c a l l y vanish at the end of the period as the excess return on c a p i t a l tends towards zero (see Table 3 . 4 ) . But the preceding table should he handled with care! The reasons are simply the s e n s i t i v i t y of the res u l t s to the 130 TABLE 5.1 ESTIMATES OF THE DEADWEIGHT LOSS DUE TO  INEFFICIENT CAPITAL ACCUMULATION Percentage of average yearly variable costs represented by the losses: v l 1953-60 1961-66 1967-72 1973-79 0.15 0.06 0.09 0.05 1953-60 1.35 1961-66 3.72 v2 1967-72 1.18 1973-79 0.04 choice of a p a r t i c u l a r user cost variable, and the i n t r i n s i c l i m i t a t i o n s of the loss formula estimated: the figures i n the l a s t table take into account only the losses due to  "ov e r c a p i t a l i z a t i o n ". The loss i n e f f i c i e n c y a r i s i n g from the m - di s t o r t i o n s are completely ignored. 5.3 A TWO-SECTOR DYNAMIC DEADWEIGHT LOSS MEASURE DUE TO REGULATION F i n a l l y , for completeness another approach i s developed here to deal with some of the conceptual problems involved i n single-sector measures of deadweight loss. The very c a p i t a l intensive nature of regulated monopolies suggest that i f o v e r c a p i t a l i z a t i o n does i n fact occur, i t w i l l e n t a i l a reduction i n the stock of c a p i t a l available for other uses. In a dynamic context, the endogeneity of the c a p i t a l formation process i s of c r u c i a l importance. But the impact of regulation on that process i s not f u l l y captured by the loss measure L^. In t h i s section, a "competitive sector" i s brought into the analysis and linked to the regulated sector i n the following way: each sector produces an intermediate input that the other sector uses. The planner's problem consists i n maximizing the net present value of the economy's production. The derived measure of the loss of output approximates the difference, i n value, between two plans for the (two-sector) economy: given a vector of reference producer p r i c e s , the net discounted value of outputs for the two sectors i s maximized and compared to the net (discounted) value of outputs generated i n the imperfectly regulated economy. The model of c a p i t a l accumulation that was used i n Chapter 2 i s again u t i l i z e d to describe the technology and behavior of the competitive producers. The production p o s s i b i l i t i e s open to these producers are described by one-period technology sets {S^}. Each element of these sets i s an (J +1 + 2N) tuple {x, y, s^, s^ -}, where X j ( j = 1, J) are inputs ( i f negative) or outputs ( i f positive) used or produced by the competitive producers, y i s a vector of inputs produced by the regulated sector, and s^ and s^ are the beginning-and-end-period N-dimensional stock vectors. The prices corresponding to the x-vector are w*- = (wj, w 2 j ) > > a n o - assumed competitively determined and exogenous to the producers. Those x^'s are the same that are (possibly) used by the regulated sector's producers which are also assumed to be price-takers with respect to them. Assume the planner allocates to the competitive producers a given set of quasi-fixed inputs ( y t , ) at the beginning of each period and that those producers aim ( or 133 are instructed ) to maximize the current value of gross p r o f i t s , w x , given the l e v e l of the quasi-fixed stocks. As a r e s u l t , a one-period r e s t r i c t e d p r o f i t function Ti(w t; y t ; s*-"!, s^ -) can be defined, similar to that i n Diewert and Lewis (1982). Now assume the competitive producers are the net suppliers of the c a p i t a l goods to the rest of the economy, hence to the regulated sector. Let s t = (§*- - s f c) = ( t o t a l stocks i n the economy - stocks allocated to the regulated sector) = competitive sector's stocks. This means that the competitive sector produces a l l the investment goods i n the economy and transfers some of them to the regulated sector. The planner's problem consists i n selecting {y t}, {s^ -} and {s^} (the time path of outputs produced and stocks used i n the two sectors) i n order to maximize the discounted net value of outputs using the reference prices {pfc = p f c}, { W t = w t } f = R} a n d Q = Q . Let C(t) = C{Yt, wfc, s t _ 1 , s f c ) and u ( t ) = T i ( w t ; y f c, s t - 1 , s t)= TttwS Yt,s t - 1 - s t - 1 r i fc - s*- ) The s o c i a l valuation function to be maximized i s : Max E*' R(0,t) [ T i(t) - C(t) ] (Y > f {s r}, {s r} t _ i " ° I - f s t * ° N + R(0,t') s Q. ...(5.19) 134 Using the d i f f e r e n t i a b i l i t y of C(t) and n ( t ) , the necessary conditions (5.20) obtain: Conditions W2 R(0,t) [ V T i(t) - V C(t) ] = 0 Z, t = 1, t ' ; ...(5.20a) R ( 0 , t ) [-V C ( t ) - V T i ( t ) ] i + R(0,t+1) [-V tC(t+l) - V tn(t+l) ] = 0 N , S i t = 1, ..., t'-1; ...(5.20b) R ( 0 , t ' ) [-V , C ( t ' ) - V t , T i ( t ' ) ] = 0 N ; ...(5.20c) R(0,t) [ V t K ( t ) ] + R(0,t+1) [ 7 t T i ( t + l ) ] = 0 N, t = 1, . . . , t ' - 1 ; ...(5.20d) R ( 0 , t ' ) [ V t , T t ( t ' ) + Q ] = 0 N. ...(5.20e) 135 These Conditions W2, which are t' x (I + 2N) i n number i n the [ f x (J + 1) + 3N] exogenous variables {wfc}, (R(0,t)}, s°, s° and Q, state that the value of the marginal product of each quasi-fixed factor of production must be the same i n both sectors. Also assume that: (A22) the (strong) second-order s u f f i c i e n t conditions for an unconstrained ( i n t e r i o r ) maximum of (5.16) hold at {yt}, {s^}. {it}. This implies, as i n 5.1, that the matrix of derivatives of (5.20) with respect to the choice variables evaluated at the s o c i a l l y optimal a l l o c a t i o n i s negative d e f i n i t e . Let t h i s matrix be M. I t can be v e r i f i e d that M i s a symmetric, f x (I + 2N) by f x (I + 2N) matrix which has the Hessian matrices of C(t) and n(t) with respect to {yfc}» (st}, and {§t} as elements. In a context i n which the competitive producers pay for the regulated outputs at prices p > p and the regulated producers use the distorted-by-regulation user cost of c a p i t a l , the i m p l i c i t shadow prices of these inputs should diverge from t h e i r s o c i a l cost by m = p - V vu and u.. Hence the same strategy as that pursued i n 5.1 can be u t i l i z e d to generate a z-equilibrium and Conditions D2. Again, when z=0 a s o c i a l optimum obtains i n which the shadow prices of a l l intermediate inputs are equalized everywhere. On the other hand, when z=l, a d i s t o r t e d system obtains. Conditions D2 R(0,t) [ V y T i ( t ) - mfc z - V yC(t) ] = 0Z , R(0,t) [-V t c ( t ) - V t n ( t ) ] . £ + R(0,t+1) [-V tC(t+l) - V t R ( t + l ) ] + u f cz = 0 N , s £ t = 1, ..., t ' - l ; ...(5.21b) R(0,t') [ V t,C(t') + V t , T t ( t ' ) ] = 0 N ; ...(5.21c) £ R(0,t) [ V t R ( t ) ] + R(0,t+1) [ V t R ( t + l ) ] = 0 N ; t = 1, ..., t'-1; ...(5.21d) 137 R(0,t' ) [ V . , T x ( f ) + Q ] = 0. N (5.21e) As before, the negative definiteness of M (see (A22)) guarantees that the [{yt}, {st}, {§t}]-solution to (5.21) can be expressed as functions of the exogenous variables {wfc}, (R(0,t)}, {mt}, {yt}, Q a n c j z around z = 0. Using the i m p l i c i t function theorem, the gradient of {yt(z)}, {st(z)} and {§t(z)} with respect to z at z = 0 can be computed as: / v z y(0)\ V 2 s(0) \ v z i ( 0 ) / - T T T - 1 = (m , -y T , 0 ^ t I x N ) ) M ...(5.22) Defining welfare as a function of the scalar z and taking the f i r s t and second derivatives of W(z) with respect to z at z=0 gives: W'(0) = z [ E ^ V* y t(0) m fc + V* s t(0) y t + S t = l V z S t ( 0 N } ] = 0 ' ...(5.23) and 138 W"(0) = [ E ^ V* y t(0) m t - E ^ V* s t(0) ii t ] = y(0), s(0), V* s(0)] [m , -u , 0 | t , x N ) ] T ...(5.24) Using (5.22), the approximate change i n welfare can be shown to be equal to: W(l) - W(0) = W(0) + 0.5 W 1 ( 0 ) = (0.5) [m*, U T , 0 T t , x N ) ] M" 1 [«T f ^  , 0 T t , x N ) ] ...(5.25) The deadweight loss approximation (5.25) has two in t e r e s t i n g features: (i) i t takes into account the impact on the unregulated sector of the economy of the monopolistic character of the p r i c i n g decisions i n the regulated sector as well as the e f f e c t of o v e r c a p i t a l i z a t i o n on other production; ( i i ) since each element of M i s made up of elements of the Hessians of n(t) and C( t ) , even i f C(t) i s not p o s i t i v e d e f i n i t e i n y and s, M can s t i l l be negative d e f i n i t e . For instance, i n the case of a single regulated output, i t can e a s i l y be v e r i f i e d that the f i r s t element of M i s equal to the sum of [ 3 2 n / 3 y 2 ] and [-d2C/ay2]. Thus even i f the short-run marginal cost function of producing y i s f l a t or s l i g h t l y decreasing, which would make the matrix A entering non negative d e f i n i t e , the value of the marginal product of y i n the competitive sector may decline fas t enough so that matrix M w i l l s t i l l be well behaved and assumption (A22) maintained. The range of technologies over which the loss formula (5.25) i s well defined i s thus larger than that for the measures of loss derived i n 5.1. NOTES TO CHAPTER 5 Notice that, were p given, a l l the developments that follow would s t i l l be correct. A l l that would be required i s set t i n g d*1 ? 0 j . The small negative values i n the e^ series are set equal to zero i n the computation of the losses. 141 CHAPTER 6  CONCLUSION This thesis has introduced a number of dynamic elements into the t h e o r e t i c a l and empirical analysis of the behavior of a monopolist facing rate of return regulation. Expectations, adjustments costs, an intertemporal regulatory constraint and lagged price adjustments have been the focus of the analysis. The accomplishments and l i m i t a t i o n s of the research can be most e a s i l y reviewed under two sets of observations: the f i r s t deals with the t h e o r e t i c a l developments and the second deals with the empirical models and the res u l t s presented i n Chapters 3 and 4. 6.1 The aim of the th e o r e t i c a l part of t h i s thesis has been ( i ) the development of a dynamic model of a rate-regulated firm, and ( i i ) the derivation of loss formulae that allow the computation of the deadweight loss due to i n e f f i c i e n t regulation. Although one can f i n d many models for a rate-regulated u t i l i t y i n the l i t e r a t u r e , very few are cast i n a dynamic framework. The model presented i n t h i s thesis i s a very general model of c a p i t a l accumulation under an intertemporal p r o f i t constraint. This model i s then used to generate propositions about the behavior of the u t i l i t y . Some of these propositions can be found elsewhere i n the l i t e r a t u r e but they are derived i n contexts that d i f f e r on one or many points from that i n t h i s d i s s e r t a t i o n . Chapter 2 can i n f a c t be regarded as the basis upon which the empirical analysis of Chapters 3 and 4 and the t h e o r e t i c a l work of Chapter 5 are b u i l t . The major accomplishments of t h i s thesis on the t h e o r e t i c a l front are (i) the derivation of an A.-J. e f f e c t i n a very general intertemporal framework and, ( i i ) the re s u l t s of the l a s t chapter where approximations to a dynamic deadweight loss are worked out. Those extend the work of Diewert (1981a) into a dynamic environment. However, the the o r e t i c a l work i n t h i s thesis suffers many shortcomings. Among them are the p a r t i a l equilibrium nature of the loss formulae, which ignore consumers' losses. The approximations of Chapter 5 could possibly be modified i n future research by building up a consumer side. Equally important would be the development of formulae for the computation of deadweight loss i n cases where the u t i l i t y ' s technology exhibits serious non convexities (as i s the case with B e l l Canada). Another l i m i t a t i o n of the analysis i n Chapter 5 i s the reliance on exogenously determined reference p r i c e s . The producer side of the economy could also be developed to encompass a l l production u n i t s . This would require that c l e a r i n g conditions i n intermediate input markets be taken into consideration, and would give the analysis a more general-equilibrium flavour. In short, there i s much work to be done i n the l i t e r a t u r e on the measurement of waste i n a regulated environment where output and input prices are d i s t o r t e d by market or regulatory f a i l u r e s . 6.2 Turning to the empirical work now, the most important accomplishment here i s found i n the s p e c i f i c a t i o n of a model of producer behavior incorporating both the impact of regulation and dynamic elements such as adjustment costs, r a t i o n a l expectations and lags i n the adjustment of the p r i c e l e v e l . When work on the empirical section of t h i s thesis began, no papers existed that incorporated adjustment costs i n the empirical analysis of a rate-regulated u t i l i t y . Since then, the papers by Bernstein (1986, 1987) introduced them i n the analysis of telecommunications i n Canada. But the i n s e r t i o n of these costs i n a model of a regulated u t i l i t y i s a novelty, as i s the e f f o r t made to determine the s e n s i t i v i t y of the conclusions concerning o v e r c a p i t a l i z a t i o n to the choice of d i f f e r e n t user cost variables. 144 I t i s useful at this point to sum up the major conclusions of the estimation. In the f i r s t place, t h i s d i s s e r t a t i o n makes i t quite clear that dynamic elements play a c r u c i a l role i n the c a p i t a l accumulation decisions of B e l l . In p a r t i c u l a r , remember that adjustment costs play a s i g n i f i c a n t role i n a l l estimated models, be i t with exogenous or endogenous outputs and regardless of the user cost of c a p i t a l s p e c i f i c a t i o n . Consequently, previous models of B e l l ' s behavior that postulate a long-run equilibrium may lead to erroneous conclusions, and p a r t i c u l a r l y so when these concern the e f f e c t regulation has on the investment decisions of B e l l . In addition, the estimation r e s u l t s indicate that the Averch-Johnson hypothesis cannot be rejected i n the case of B e l l since at l e a s t i n two out of four estimated models the regulatory parameter appears s i g n i f i c a n t , has the proper sign and f a l l s within the t h e o r e t i c a l range defined i n Chapter 2. Moreover, the s t a t i s t i c a l r esults suggest that the user cost of c a p i t a l s p e c i f i c a t i o n has an impact on the conclusions reached about the A.-J. e f f e c t . Which s p e c i f i c a t i o n , and conclusion, i s the more appropriate i s not d e f i n i t e l y established. However, the estimated models under the V 2 ~ s p e c i f i c a t i o n seem i n general to outperform the v^-145 s p e c i f i c a t i o n : the DCF method may do a better job at tracking B e l l ' s cost of c a p i t a l than the CAPM model. In t h i s case, the A.-J. e f f e c t would seem to be a supported hypothesis. The r e s u l t s presented i n Chapter 4 suggest that neglecting both adjustment costs (and expectations) and regulation leaves out two s i g n i f i c a n t influences on the u t i l i t y ' s investment decisions, but two influences that work i n opposite d i r e c t i o n s . This l a t t e r follows because the presence of convex costs of adjustment slows down the rate at which a firm builds up i t s c a p i t a l stock, whereas tying the firm's p r o f i t a b i l i t y to i t s c a p i t a l stock induces i t to "ove r c a p i t a l i z e " . I t i s therefore d i f f i c u l t to determine i f previous studies biased the marginal cost of c a p i t a l to the u t i l i t y upwards or downwards, since most ignored both of these e f f e c t s . The lagged price responses introduced here are novel i n the l i t e r a t u r e on econometric models of the regulated u t i l i t y . The estimates from the endogenous output models, despite t h e i r t h e o r e t i c a l shortcomings, do indicate that the hypothesis of instantaneous price adjustments must be rejected, and they stress the importance of dynamic regulatory features that have been l e f t out of empirical analyses to date (although t h e o r e t i c a l work on the subject can be found i n Klevorick 1973, 1974; more on t h i s point l a t e r ) . F i n a l l y , the estimated models have allowed the computation of the welfare losses imputable to i n e f f i c i e n t c a p i t a l accumulation. Those losses are small but not n e g l i g i b l e . I t would be desirable to obtain the information about the competitive s e c t o r 1 s technology that i s necessary to implement the more general loss formulae of Chapter 5. But notwithstanding i t s obvious l i m i t a t i o n s , the e f f o r t undertaken i n 5.2 i s a novel attempt to use second-order approximations a c t u a l l y to estimate the deadweight loss due to rate of return regulation. Further attempts are c r i t i c a l l y needed i f a p r a c t i c a l appraisal of the magnitude of the costs of regulation i s ever to be obtained. The tentative r e s u l t s of Chapter 5 o f f e r not only indications as to the siz e of the losses involved but point out some computational problems that hinder t h e i r estimation. There are a number of weaknesses where immediate improvements are possible. The modeling of technological change i s one case i n point. I t i s possible that the chosen s p e c i f i c a t i o n for technological change i s responsible for the (few) f a i l u r e s i n the e l a s t i c i t y of costs with respect to l o c a l output. I t would also be i n t e r e s t i n g to determine just 147 how s e n s i t i v e the estimation i s to the s p e c i f i c a t i o n of technological change and to the choice of a p a r t i c u l a r proxy. Another possible extension that i s suggested by the empirical results would be to endogenize the " s t i c k i n e s s " of prices by developing a choice model for a regulated monopolist with costs of adjustment defined over p r i c e changes. These costs are c e r t a i n l y not n e g l i g i b l e i n the case of a regulated enterprise, which needs to j u s t i f y i t s "required price increases" at rate hearings. Or a model could be formulated i n which prices could be adjusted only at s p e c i f i e d i n t e r v a l s , or i n which marginal costs are perceived only with a lag. At any rate, a sounder t h e o r e t i c a l basis to the model of lagged price adjustments estimated i n Chapter 4 i s d esirable. This i s one d i r e c t i o n future research could f r u i t f u l l y look into. F i n a l l y , i t should be noted that the conclusions concerning the importance of adjustment costs are a r r i v e d at under the maintained hypothesis of r a t i o n a l expectations on the part of the u t i l i t y . Even though t h i s seems a very reasonable hypothesis, any t e s t concerning the adjustment cost parameter i s i n f a c t a t e s t of the j o i n t hypothesis of adjustment costs and r a t i o n a l expectations. 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"Lagrange M u l t i p l i e r " Values at Constrained Optima." Journal of Economic Theory 4:125-131. 159 APPENDIX A The data base used i n t h i s thesis was assembled from a v a r i e t y of sources. The bulk of the data was taken from recent submissions to the Canadian Radio-Television and Telecommunications Commission (CRTC) by B e l l Canada. The following o r i g i n a l sources were tapped: (1) B e l l Canada, Information Requested by National Anti-Poverty Organization (NAPO), 30 March 81-612 CRTC. (2) F i n a n c i a l S t a t i s t i c s on Canadian Telecommunication Common Car r i e r s , Department of Communications. (3) The F i n a n c i a l Post Corporation Service, Maclean Hunter Ltd., various years and companies. (4) Gestion Financiere, Lustzig, Schwab and Charest, 1983. (5) The Regulation of Telecommunications i n Canada, M. Fuss and L. Waverman, Economic Council of Canada, 1981. 160 (6) S t a t i s t i c s Canada. A . l Output and variable input series for B e l l (1952-1980) Outputs Two output price variables were constructed for B e l l corresponding to two output quantity varia b l e s : l o c a l and t o l l outputs. The price variables are D i v i s i a price indexes normalized to 1.0 i n 1976 and the output quantity variables are constant 1976$ of revenues i n m i l l i o n s of $. (1) breaks down B e l l ' s revenues into 10 categories of output: l o c a l service revenues, message t o l l service revenues ( I n t r a - B e l l ; Trans-Canada and Adjacent Members; US and Overseas), other t o l l service revenues (WATS; TWX; private l i n e s ; miscellaneous other t o l l ) , d i r e c t o r y advertising revenues and miscellaneous revenues. For each category of output both current $ figures and constant 1967$ figures are available. The l o c a l output price index i s a D i v i s i a price index using the constant 1967$ figures of l o c a l service revenues, d i r e c t o r y advertising and miscellaneous 161 revenues as output figures and the i m p l i c i t prices derived by d i v i d i n g the constant $ figures into the current $ figures. This index i s then normalized to 1.0 i n 1976 and divided into the sum t o t a l of those three sources of current revenues to give a l o c a l output quantity variable i n m i l l i o n s of constant 1976$. A similar procedure i s employed to obtain the price and quantity of t o l l output. The t o l l output price index i s a normalized D i v i s i a p r i c e index of the seven remaining categories of revenues for B e l l : i n t r a - B e l l , Trans-Canada, US and overseas, WATS, TWX,PL, miscellaneous other t o l l . The t o l l output quantity index i s obtained by d i v i d i n g t h i s price index into the corresponding t o t a l current $ revenues s e r i e s . Labor The quantity of labor i s m i l l i o n s of manhours unadjusted for qu a l i t y change. The price of labor i s the average hourly wage rate which i s equal to t o t a l labor compensation i n 162 m i l l i o n s of current $ divided by the quantity of labor. Materials The cost of materials, services, rents and supplies i n current $ i s divided by the corresponding figure i n constant 1967$ to get a price index of materials. This price index i s then renormalized to 1.0 i n 1976 and divided into the current $ cost figures to obtain a constant 1976$ quantity index of materials i n m i l l i o n s of $. A.2 Ca p i t a l input prices and quantities series A.2.1 Ca p i t a l stock series The quantity of c a p i t a l i s the constant 1976$ t o t a l average gross or net stock of c a p i t a l (at reproduction cost). F i r s t , an asset price index of c a p i t a l i s obtained by di v i d i n g the current $ values of the stock of c a p i t a l by the constant $ values and by renormalizing t h i s series to 1.0 i n 1976. Then 163 t h i s series, i s divided into the current $ value of the average gross stock of physical c a p i t a l to obtain a gross quantity of c a p i t a l i n mi l l i o n s of constant 1976$. The quantity of net c a p i t a l i s KN = KG (1-6) where KG i s the quantity of gross c a p i t a l and 6 the (economic) depreciation rate. 6 i s estimated by taking the r a t i o of the value of depreciation expenses i n constant 1976$ over the value of the gross stock of physical c a p i t a l i n constant 1976$. A.2.2 D e f i n i t i o n of the user cost of c a p i t a l services and the allowed gross return on c a p i t a l services. Remember that Fuss and Waverman's user cost of c a p i t a l services i s given by: v = q( 9 c R + cb, (1-6) + 6 ) - (a-6) g t q , i=l,2 a * d - t ) (l-t)(a+g) ...(Al) where q i s the asset p r i c e of c a p i t a l , 8 the f r a c t i o n of the firm's c a p i t a l financed by debt, 6 i s the economic depreciation rate ("EDEP" i n Appendix B), a the accelerated depreciation rate ("ADEP" i n Appendix B), t the tax rate on corporate income, g the treasury bond rate which i s used as a proxy for the personal borrwing rate, c B i s the cost of debt, c^i i s the cost of equity c a p i t a l using the CAPM method and c£. i s the cost of equity c a p i t a l using the DCF method. The DCF method i s probably the most widely used to compute c E . I t r e l i e s on the equivalence of the market pr i c e of a stock (MV) and the present value of the cash flows investors expect from the stock. By making the assumptions (i) that the discount rate w i l l remain constant; ( i i ) that a l l relevant cash flows are dividends; and ( i i i ) that the dividends are expected to grow at a constant rate x, the market value of a stock can be written as a perpetuity: MV = D/(r+x) , ...(A2) where D stands for the dividend. Solving for r, the discount rate required by the investors, gives: c g = r = ( D / M V ) + x . ...(A3) Using (A3) to forecast backwards what the cost of equity c a p i t a l was for B e l l Canada, the actual values of D and MV can be used on the assumption that investors expected the dividends that were act u a l l y paid. The d e f i n i t i o n of x, however, i s more problematic. The "sustainable" growth rate method i s retained i n t h i s thesis. I t consists i n using that rate x which could be sustained by the growth i n the firm's earnings. The expected growth rate of the dividends i s then 165 measured as the rate of return on book equity times the proportion of earnings that are not d i s t r i b u t e d to the shareholders. This gives the following formula for x: x = (EPS/BV) [ (EPS - D) / EPS ], ...(A4) where EPS i s earnings per share and BV i s the book value of equity. (A3) and (A4) complete the DCF model to compute c^ for B e l l . The CAPM i s based on a theory of c a p i t a l market equilibrium which predicts that investors w i l l hold only e f f i c i e n t p o r t f o l i o s : that i s , p o r t f o l i o s with the highest return for a given r i s k l e v e l . To induce an investor to hold an investment which i s more (less) r i s k y than a p o r t f o l i o containing a l l the stocks i n the market ("the market p o r t f o l i o " ) , one should give her a higher (lower) return than the return on a l l stocks. The competitive nature of the c a p i t a l market leads to a "ri s k - r e t u r n l i n e " that gives the required rate of return by investors for any l e v e l of r i s k . The CAPM also holds that a l l the information about a firm's r i s k i n e s s can be compounded into a single c o e f f i c i e n t , c a l l e d the beta c o e f f i c i e n t . A firm's beta measures the v o l a t i l i t y of i t s returns and the c o r r e l a t i o n of those returns with other assets. Let r m be the rate of return of the market 166 p o r t f o l i o and r j be the rate of return on asset j or firm j . Then, the beta of firm j i s defined as: b j E ajm/ am > ...(A5) where G j m i s the covariance between r j and r m and i s the variance of r m . A beta value of one means that the return on asset j , on average, moves up or down by the same amount as the market return: both are equally r i s k y . More pre c i s e l y , i t can be shown that: b j = Pjm (°"j/am)' ... (A6) where p j m i s the c o r r e l a t i o n between r j and r m (A6) gives a means to determine just "how r i s k y " a firm i s . The CAPM solves the problem of how to compensate investors for r i s k y projects by posing that the r i s k - r e t u r n l i n e i s (Kolbe et a l . , 1984; p. 70): E(r-j) = r f + {bj ( E ( r m ) - r f)} , ...(A7) where E ( r j ) i s the expected required return on asset j , r f i s a r i s k - f r e e rate of return and E ( r m ) i s the expected rate of return on the market. By assuming that r j can be approximated by the return on short term Canadian treasury b i l l s and that [ E ( r m ) - r f ] , the r i s k premium, i s stable and can be estimated, and by estimating b^ by an ordinary l e a s t -squares regression of (r-j-rf) on ( r m - r f ) , (A7) can be used to generate the expected required rate of return on any firm's equity. This i s done i n Lustzig et a l . (1983) for B e l l : the estimated b-value i s 0.2483 while the r i s k premium for the Canadian stock market ( based on the performance of stocks at the TSE ) i s found to be 0.045. The l a s t quantity which needs to be defined i s the allowed (gross) return on c a p i t a l . The formula given i n Fuss and Waverman i s : s = q (8 c R + s„ (1-6) + 6 ), ...(A8) B E (1-t) where s E i s the allowed (gross) rate of return on equity. s E i s assumed to be equal to the actual rate of return on equity ("ROR" i n Appendix B) as defined i n Fuss and Waverman (1981). F i n a l l y , the allowed (gross) excess return on c a p i t a l services can be obtained from (Al) and (A8): e 1 = (s-v 1) = q [ (s - c 1 ) (1-6) / (1-t) ] + ( a - 6 ) [ ( q t g ) / (1 - t) (a + g)] . ...(A9) The sources of the variables entering the user cost of c a p i t a l computations appear i n Table A . l . The values of 168 TABLE A . l DATA SOURCES FOR THE GROSS SERVICE PRICE OF CAPITAL Series Name q (asset price) (1) DEBT (3) EQUITY (3) (a-6)* (5) t (3) g (6) DIV (3) MVS (3) BVS (3) EPS (3) 6 (1) b (4) RP (4) Source : see above see above The values for a for 1979, 1980 are not included i n (5), the value 0.156124 was used for 1978 through 1980. 169 v-*-, v 2 , s and e^ can be found i n Tables 3.3 and 3.4. A.3 Other variables sources. A number of other, non-company related, variables were used i n the empirical section of t h i s research. Whenever necessary the series were converted into m i l l i o n s of constant 1976$ or renormalized to 1976 = 1.0. Table A.2 indicates the source of each vari a b l e . 170 TABLE A.2 DATA SOURCES, VARIOUS SERIES Canadian GNP (millions of current $) Stat. Can. 13 -213/531 GNP of Quebec and Ontario (millions of current $) Stat. Can. 13 -213/531 Consumer Price Index Stat. Can. 13 -004 Population-Canada Stat. Can. 91 -201 Population Quebec Stat. Can. 91 -201 Number of phones i n service Stat. Can. 56 -002 Number of households i n Quebec and Ontario Stat. Can. 93 -801 171 APPENDIX B Y E A R Y L P L Y T P T 1952 178. 5779 0.7122940 69.95073 0. 8234367 1953 194. 2879 0.7185213 75.96753 0. 8266689 1954 211. 1320 0.7180341 82.48059 0. 8268611 1955 230. 2288 0.7218907 95.71648 0. 8274437 1956 254. 6142 0.7226621 109.5865 0. 8267443 1957 282. 1068 0.7249027 120.7473 0. 8232069 1958 306. 0608 0.7312272 127.7318 0. 8306466 1959 329. 7831 0.7793001 140.1886 0. 8624096 1960 353. 7957 0.7815246 148.9939 0. 8718478 1961 379. 6783 0.7819778 159.9414 0. 8646920 1962 407. 6671 0.7827466 187.3890 0. 8207524 1963 431. 3107 0.7880629 200.1987 0. 8256795 1964 452. 4994 0.7880673 228.3193 0. 8251603 1965 486. 7681 0.7880549 257.5972 0. 8233784 1966 526. 6242 0.7878483 290.0707 0. 8042866 1967 566. 8604 0.7878483 324.9560 0. 7970310 1968 605. 0646 0.7885108 359.9938 0. 7911246 1969 652. 0020 0.7929423 413.4762 0. 7961764 1970 698. 3586 0.8028826 450.3413 0. 8486897 1971 740. 9561 0.8323030 470.6530 0. 8643311 1972 776. 4966 0.8557668 531.6042 0. 8745227 1973 827. 7736 0.8791051 617.5275 0. 8940492 1974 900. 6324 0.8979246 701.8339 0. 9084769 1975 978. 2822 0.9417528 799.1926 0. 9429517 1976 1045 .100 1.000000 867.7000 1 .000000 1977 1104 .039 1.062010 940.2824 1 .032031 1978 1169 .326 1.160155 1048.921 1 .098843 1979 1213 .960 1.238262 1136.082 1 .170514 1980 1283 .651 1.320530 1255.959 1 .193590 APPENDIX B YEAR L w l M w2 1952 48. 40000 1. 570682 64. 20717 0. 4469906 1953 49. 00000 1. 708735 69. 01855 0. 4462568 1954 51. 80000 1. 763900 77. 14815 0. 4557275 1955 56. 10000 1. 828414 88. 43002 0. 4557275 1956 60. 20000 1. 871146 103 .5278 0. 4733026 1957 62. 60000 1. 949904 104 .3574 0. 4829557 1958 61. 30000 2. 091582 114 .8097 0. 4903766 1959 57. 60000 2. 290729 120 .7825 0. 5000725 1960 55. 10000 2. 464083 126 .2575 0. 5061085 1961 51. 80000 2. 662471 131 .7325 0. 5078472 1962 51. 60000 2. 781085 141 .1894 0. 5149111 1963 53. 20000 2. 850667 148 .6554 0. 5247036 1964 54. 10000 2. 922810 148 .9872 0. 5376301 1965 55. 50000 3. 011189 162 .5918 0. 5547635 1966 58. 30000 3. 166364 169 .0623 0. 5796680 1967 56. 60000 3. 460724 165 .2464 0. 6027364 1968 54. 60000 3. 817894 172 .8782 0. 6229819 1969 55. 50000 4. 151423 206 .0602 0. 6493248 1970 56. 10000 4. 636506 205 .8943 0. 6741323 1971 55. 20000 5. 003822 244 .5513 0. 6996486 1972 55. 10000 5. 640980 250 .3582 0. 7229642 1973 57. 80000 6. 079481 265 .1242 0. 7573810 1974 61. 60000 6. 792906 280 .2220 0. 8336248 1975 61. 30000 8. 171729 277 .5674 0. 9190559 1976 64. 30000 9. 208647 299 .3016 0. 9999945 1977 66. 60000 10 .23515 335 .3041 1. 084091 1978 71. 20000 10 .83087 367 .8225 1. 163061 1979 73. 10000 12 .48873 370 .3111 1. 261642 1980 76. 20000 14 .14047 399 .3454 1. 401043 A P P E N D I X B Y E A R % K G <3G 1952 1170 .983 1704. 822 0. 4886494 0.4914883 1953 1290 .225 1865. 836 0. 4885213 0.4813392 1954 1409 .998 2028. 989 0. 4765253 0.4785141 1955 1577 .965 2243. 140 0. 4778940 0.4792835 1956 1765 .067 2474. 408 0. 4885933 0.4896929 1957 1975 .380 2731. 531 0. 4972208 0.4974134 1958 2204 .473 3017. 540 0. 5039298 0.5031581 1959 2432 .858 3305. 867 0. 5067290 0.5055859 1960 2669 .571 3600. 435 0. 5071976 0.5056055 1961 2890 .692 3885. 731 0. 5062802 0.5043581 1962 3106 .851 4200. 270 0. 5065901 0.5001583 1963 3340 .729 4465. 952 0. 5113854 0.5087829 1964 3567 .874 4777. 816 0. 5131628 0.5103168 1965 3791 .829 5096. 278 0. 5164262 0.5134924 1966 4038 .109 5460. 565 0. 5327246 0.5294506 1967 4292 .717 5865. 151 0. 5643978 0.5608211 1968 4539 .174 6279. 543 0. 5960996 0.5920017 1969 4804 .944 6719. 255 0. 6220259 0.6175833 1970 5061 .501 7149. 339 0. 6582830 0.6529974 1971 5338 .079 7617. 403 0. 6962617 0.6901302 1972 5635 .387 8117. 028 0. 7295151 0.7230725 1973 5898 .145 8581. 882 0. 7726328 0.7641913 1974 6200 .414 9103. 617 0. 8444920 0.8330206 1975 6568 .948 9754. 627 0. 9282764 0.9219727 1976 6928 .801 10443 .44 0. 9999998 1.000006 1977 7278 .731 11064 .85 1 .062589 1.066856 1978 7511 .191 11554 .13 1 .145863 1.148611 1979 7699 .002 12050 .55 1 .262488 1.259901 1980 8005 .524 12674 .45 1 .391439 1.391129 174 APPENDIX B YEAR DEP ADEP v l v 2 1952 0. 5870000E-01 0. 5882400E-01 0. 4802051E-01 0 .7540052E-01 1953 0. 5860000E-01 0. 5874500E-01 0. 5001744E-01 0 .7248212E-01 1954 0. 5810000E-01 0. 9353900E-01 0. 5580000E-01 0 .9744100E-01 1955 0. 5580000E-01 0. 9744100E-01 0. 4526142E-01 0 .5884605E-01 1956 0. 5550000E-01 0. 9653900E-01 0. 5329865E-01 0 .5991736E-01 1957 0. 5850000E-01 0. 9835900E-01 0. 5926170E-01 0 .6145431E-01 1958 0. 5840000E-01 0. 6556100E-01 0. 5411610E-01 0 .6403612E-01 1959 0. 6020000E-01 0. 6680800E-01 0. 7111114E-01 0 .7195336E-01 1960 0. 5980000E-01 0. 6578800E-01 0. 6213568E-01 0 .6983341E-01 1961 0. 5980000E-01 0. 6520700E-01 0. 6146279E-01 0 .6725296E-01 1962 0. 6030000E-01 0. 6531200E-01 0. 6919845E-01 0 .7029698E-01 1963 0. 6190000E-01 0. 6753800E-01 0. 6711682E-01 0 .6925920E-01 1964 0. 6260000E-01 0. 6795900E-01 0. 6926456E-01 0 .7137297E-01 1965 0. 6390000E-01 0. 6896800E-01 0. 7139041E-01 0 .7404131E-01 1966 0. 6510000E-01 0. 7003400E-01 0. 8065371E-01 0 .7920803E-01 1967 0. 6560000E-01 0. 9951600E-01 0. 7815917E-01 0 .8493052E-01 1968 0. 6680000E-01 0. 9980800E-01 0. 9221525E-01 0 .9215649E-01 1969 0. 6900000E-01 0. 1001910 0. 1052161 0 .9763033E-01 1970 0. 6920000E-01 0. 1083220 0. 1033216 0 .1106156 1971 0. 6970000E-01 0. 1326050 0. 9116820E-01 0 .1146339 1972 0. 7390000E-01 0. 1385210 0. 9923898E-01 0 .1297081 1973 0. 7720000E-01 0. 1466110 0. 1145049 0 .1425785 1974 0. 7950000E-01 0. 1452860 0. 1404638 0 .1566471 1975 0. 8400000E-01 0. 1632080 0. 1534073 0 .1865399 1976 0. 8670000E-01 0. 1593450 0. 1914557 0 .2061271 1977 0. 8690000E-01 0. 1702510 0. 1911813 0 .2107956 1978 0. 8860000E-01 0. 1561240 0. 2149607 0 .2296701 1979 0. 9080000E-01 0. 1561240 0. 2651202 0 .2674904 1980 0. 9160000E-01 0. 1561240 0. 3031141 0 .2741821 175 APPENDIX B YEAR ROR T 2 1952 0. 7365500E-01 0. 2184000E-01 0. 6834100E-01 48. 39000 1953 0. 7119700E-01 0. 2826600E-01 0. 7070500E-01 49. 30000 1954 0. 7365300E-01 0. 2543200E-•01 0. 5958900E-01 50. 28000 1955 0. 6923100E-01 0. 2733100E-•01 0. 5332700E-01 50. 19000 1956 0. 6520300E-01 0. 4042400E-01 0. 5275200E-01 52. 39000 1957 0. 5790800E-01 0. 4877400E-01 0. 5274300E-01 55. 47000 1958 0. 6161100E-01 0. 3371600E-01 0. 5278300E-01 61. 46000 1959 0. 6846900E-01 0. 5927300E-01 0. 6075100E-01 65. 41000 1960 0. 7123100E-01 0. 4319100E-01 0. 5759400E-01 71. 85000 1961 0. 6801900E-01 0. 3929100E-01 0. 4942600E-01 75. 91000 1962 0. 6823000E-01 0. 5169800E-01 0. 5360400E-01 79. 14000 1963 0. 6960200E-01 0. 4679100E-01 0. 5061800E-01 86. 92000 1964 0. 7078500E-01 0. 4866600E-01 0. 5230100E-01 94. 70000 1965 0. 7873200E-01 0. 5101500E-01 0. 5570200E-01 96. 84000 1.966 0. 6941400E-01 0. 6112400E-01 0. 5866000E-01 99. 02000 1967 0. 7922600E-01 0. 5758200E-01 0. 6896900E-01 100 .0000 1968 0. 8084400E-01 0. 7384900E-01 0. 7375300E-01 104 .1200 1969 0. 7755300E-01 0. 8309900E-01 0. 7123600E-01 106 .7600 1970 0. 8075400E-01 0. 7108100E-01 0. 8154000E-01 108 .9500 1971 0. 8183200E-01 0. 4679900E-01 0. 8100400E-01 110 .3400 1972 0. 8820800E-01 0. 4678200E-01 0. 9125900E-01 111 .7600 1973 0. 8967900E-01 0. 6587300E-01 0. 1069200 112 .8400 1974 0. 8412900E-01 0. 8941600E-01 0. 1110400 114 .7400 1975 0. 1229500 0. 8512400E-01 0. 1320300 116 .4900 1976 0. 8920400E-01 0. 9984900E-01 0. 1158500 117 .5500 1977 0. 8099500E-01 0. 8449100E-01 0. 1049600 118 .3800 1978 0. 9396900E-01 0. 9793200E-01 0. 1129400 121 .5700 1979 0. 1050500 0. 1280500 0. 1303500 122 .5300 1980 0. 9341600E-01 0. 1391000 0. 1137800 124 .4000 H H H H H H H H H H H H H H H H H H H H I - ' l - ' H - ' H H t - ' l - ' l - ' M HHHHh , Hh J Hh , h J l - 'h J l - , H WtON WW OOOlWMOvl V O V D V O V D 0 0 00 00 00 0O U l N J O O O O l i U W O ' • • I D 0 0 O - J v J O H W O O ^ k D ( I \ H m 0 0 U - J ( » W > J H O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 00 00 -^ J ~J -~J -J OJ M v£> oo - J a\ U I O O M J K I O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 00 00 00 00 00 00 00 00 00 00 00 0 0 - J - J ^ J ^ J - J C T > U 1 0 ^ i t . 0 J W O O O O O O f > ^ J i t ^ ^ ^ ^ i t > N ) N W ( B N W N H I - ' ^ 0 1 0 U W m u O O O O OKJimOlNHDHtOOWHUUlHffUCNUDOOMWvlUHiUDOOOO o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 

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