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The economics of seemingly abundant resource : efficient water pricing in Vancouver, Canada Renzetti, Steven 1990

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THE ECONOMICS OF A SEEMINGLY ABUNDANT RESOURCE : EFFICIENT WATER PRICING IN VANCOUVER, CANADA by STEVEN J. RENZETTI B.A.(Honours), The University of Toronto, 1982 M.A The University of British Columbia, 1983 A THESIS SUBMITTED LN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Economics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1990 © Steven J. Renzetti, 1990 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 ^ d 0 AJ 0 M t (L X The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT Current North American water pricing practices are inefficient because they are based on average utility expenditures rather than marginal costs and because they typically neglect factors such as the cyclicity of demands, the time of consumption and the value of the water resource. Despite strong criticisms of these practices (Hirshleifer, DeHaven and Milliman, 1960; Pearse, 1985) and the presence of well articulated theoretical models of efficient pricing alternatives (eg., peak-load pricing) no empirical study has been done to document the magnitude of the efficiency gains from altering water prices. A simulation program computes the impact upon a representative water utility's output and deficit and upon aggregate consumer surplus of a move from current practice to efficient prices. The program is based on the estimated costs of supply and demand for water for the city of Vancouver, Canada. A time series of quarterly observations for the period 1975-1986 is used to estimate short and long run marginal costs. The estimated cost structure of the utility is also used to test for optimal employment of its fixed factors: water in storage and capital. Cross-sectional data sets are used to estimate market demands for residential and industrial users. The estimation results indicate that long run marginal cost exceeds short run costs by a large margin and that there is some evidence of over-capitalization by the utility. Water demands are seen to be inelastic for indoor and outdoor residential consumption but are elastic for industrial consumption. Simulation results show that a move to seasonally differentiated pricing (with an annual charge calculated to i i recoup the resulting deficit) raises aggregate surplus by approximately 4%. Conversely, a move from current practice to Ramsey prices leads to a decrease in aggregate consumer surplus of approximately 13%. iii T A B L E O F C O N T E N T S ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENT I. INTRODUCTION 1 A. OVERVIEW 1 B. HYPOTHESES TO BE TESTED 5 C. SUMMARY OF RESULTS 6 II. WATER PRICING: THEORY AND PRACTICE 9 A. INTRODUCTION 9 B. THE THEORY OF EFFICIENT PUBLIC UTILITY PRICING 10 C. CURRENT PRICING PRACTICES 22 D. EVALUATION OF WATER PRICING PRACTICES 25 iv III . S T R U C T U R E O F T H E C O S T S O F S U P P L Y A N D D E M A N D 31 A . I N T R O D U C T I O N 31 B . T H E S T R U C T U R E O F W A T E R S U P P L Y C O S T S 31 C . E S T I M A T E S O F A G G R E G A T E W A T E R D E M A N D S 5 7 D . C O N C L U S I O N 73 I V . M O V I N G T O E F F I C I E N T P R I C E S : S I M U L A T I O N R E S U L T S 92 A . I N T R O D U C T I O N 92 B . T H E O R E T I C A L I S S U E S 92 C . M E T H O D 100 D . S I M U L A T I O N R E S U L T S 115 E . C O N C L U S I O N S 124 V . C O N C L U S I O N S 136 A . I N T R O D U C T I O N 136 B . S U M M A R Y 136 C . F U T U R E R E S E A R C H 141 B I B L I O G R A P H Y 144 A P P E N D I X 1 : D A T A 157 A . G V W D D A T A 157 B . V W W D A T A 163 C . R E S I D E N T I A L W A T E R D E M A N D S 165 v D. INCOME DISTRIBUTION 168 E. INDUSTRIAL WATER DEMANDS 168 APPENDIX 2 : TESTS 177 vi LIST OF TABLES Table 3.1:--GVWD Restricted Cost Function Parameter Estimates 75 Table 3.2.--GVWD Input Price Elasticities 76 Table 3 .3 : - -GVWD Fixed Factor S ta t i s t i c s : Average Values 77 Table 3.4:~GVWD Fixed Factor Statistics: 1986 Values 78 Table 3.5:--VWW Short-Run Cost Function Parameter Estimates 79 T a b l e 3 . 6 : - - V W W R e s t r i c t e d C o s t I n p u t P r i c e Elasticities 80 Table 3.7:-VWW Fixed Factor Statistics: 1986 Values 81 Table 3.8:--VWW Long-Run Cost Function Parameter Estimates 82 Table 3.9:-VWW Long-Run Cost Input Price Elasticities 83 Table 3.10:--Long-Run Marg ina l Costs, VWW (1986, $/1000m3) 84 Table 3 .11 : - -Res iden t ia l Water Demands Es t ima t ed Coefficients 85 Table 3 .12 : - -Res iden t i a l Water Demands E l a s t i c i t y Values 86 Table 3.13:~Industrial Water Demands 87 Table 4.1:~Pricing Rules for the Simulation Program 127 vii Table 4.2:-Equilibrium Prices and Quantities 129 Table 4.3:--Changes in Aggregate Surplus 132 Table 4.4:--Change in Net Surplus, Net of Metering Cost 133 Table of GVWD Cost Data 174 Table A1.2:-Summary of VWW Cost Data 175 Table A1.3:-Summary of Demand Data 176 Table A2.1:~Summary of Tests on the Cost Functions 179 Table A2.2:-Testing for Constant Marginal Cost 180 viii LIST OF FIGURES F igure 3 .1 : - -GVWD's L R M C From Engineer ing Cost Estimates 88 Figure 4.1:--Effect of a Pricing Change for A Water Utility 94 ix ACKNOWLEDGEMENT In doing the research for this thesis I have accumulated several intellectual debts which I am pleased to acknowledge. The members of my thesis committee (Phil Neher, Paul Bradley, Margaret Slade and Bill Schworm) provided valuable comments throughout the project. Several classmates at UBC (Shelley Phipps, Peter Burton and Sergio Gai) critically assessed many of my ideas. The demand estimation could not have been done without the assistance of two Environment Canada economists (Don Tate and Roger McNeill) who supplied me with the raw data and with insights into water management practices in Canada. George Bratton of the Vancouver Water Works explained many of that utility's engineering features to me. Finally, I wish to thank my wife, Diane Dupont. She has given me a tremendous amount of love and support while I was doing this research and has also taught me a great deal about doing applied economics. It is to her that I dedicate this thesis. All responsibility for errors and omissions remains my own. x 1 I . INTRODUCTION A. OVERVIEW The purpose of this thesis is to evaluate the potential for efficiency gains from moving from the pricing practices currently employed by North American water utilities and toward efficient prices. While the economics literature has been consistent in its criticism of water utilities' pricing practices (Hirshleifer, DeHaven, and Milliman, 1960; Martin et. al.. 1984; Pearse, 1985) it does not yet contain empirical estimates of the magnitude of the gains to be had from altering those practices. A case study of the operations of a particular water utility (the Vancouver Water Works) is undertaken in order to derive estimates of these potential gains. Most North American water utilities set prices using a variant of average cost pricing, with costs each year defined as the sum of operating and debt retirement expenditures. On the one hand, these prices typically exceed the short-run marginal cost of output because they are calculated by including expenditures on fixed inputs (eg., capital maintenance). On the other hand, these prices also can be expected to fall short of the long-run marginal cost of output since the full costs of capital are usually not included. This is because most utilities value their capital stock at its historic acquisition cost rather than its replacement cost. Furthermore, prices charged by municipal water utilities discriminate across different user groups, but are invariant with respect to distance and time. These observations form the basis for the contention that current water utility pricing practices are inefficient. In addition to its concern with examining water prices there are several other 2 related issues that this thesis addresses. The first of these issues concerns the structure of water supply costs. The principal interest here is in establishing the form of the marginal costs of supply. The second issue relates to the structure of water demands and, in particular, to establishing the degree of price responsiveness that these demands exhibit. The third is related to evaluating a representative utility's decision to change its pricing practices and adopt efficient prices. While economic theory would suggest a priori that such a move should be supported on efficiency grounds, there are several reasons for studying the issue further. First, efficient prices may lead the utility to earn a deficit. If the utility is required to be financially self-sufficient then these prices may need to be altered. Second, many municipal water utilities do not meter the consumption rates of residential consumers.1 It may be argued that the costs associated with the installation and maintenance of flow meters would exceed the efficiency gains from changing pricing regimes. Finally, also of concern in the evaluation of changing pricing rules is the distributional impact of these changes. In the evaluation of price changes if one assumes that the utility holds a particular degree of aversion to inequality, then one can examine how this position influences the analysis of the impact of price changes. The order in which these issues are addressed establishes the structure of this thesis. In Chapter Two, formal models of the pricing behaviour of public utilities are reviewed. A model of a welfare maximizing utility is presented. After establishing the optimal pricing rule for this utility four extensions to the model which have been pursued in the theoretical literature are considered. These extensions examine the following circumstances: increasing returns to scale in the utility's technology, cyclicity 3 in the temporal structure of demands, the presence of uncertainty, and the incorporation of specific distributional aims into the utility's pricing policies. This set of extensions is not exhaustive. However, it does represent the areas where much of the work done in public sector pricing has been carried out. Furthermore, these extensions are closely related to the issues relating to the reform of water utility pricing. Once the results of these models are established, the pricing behaviour observed for most North American municipal water utilities is summarized and evaluated. The principal conclusion from this chapter is that North American water pricing practices diverge significantly from efficient pricing rules. The next chapter reports on the estimation undertaken to derive the parameter values and economic relationships necessary to evaluate a move from current water pricing practices to efficient practices. This analysis is based on data drawn from the operations of two water utilities operating in British Columbia. Restricted cost functions are estimated for the Greater Vancouver Water District (GVWD) and the Vancouver Water Works (VWW). The former is a regional wholesaler of water. It captures water in three artificial lakes and distributes it to the local municipal water utilities. The VWW is responsible for the delivery of potable water to consumers in the city of Vancouver. In each case the estimation is conducted with quarterly time-series data for the period 1975-1986. Explanatory variables include input prices, the levels of output and fixed factors. The relatively small amount of attention paid to water utilities means that there are little data regarding their operations. Given the difficulty of compiling a data set for the water utility industry, it was decided to adopt a case study approach. This approach allows for a detailed analysis of the utilities' operations but runs the 4 risk of being unrepresentative. The latter point, however, may be an advantage. If it can be demonstrated that reforming water prices in Vancouver (where water shortages are only experienced during summer months) leads to aggregate welfare gains then one would expect that these gains will be larger for cities facing more severe water shortages (eg., Calgary or Tucson). Chapter Three also reports on the estimation of water demand equations. These equations are estimated for residential and industrial user groups using flexible functional forms and data sets of observations on prices, aggregate consumption, and other explanatory variables. In the case of the residential group, the data are pooled time-series, cross-sectional observations from Victoria, British Columbia. Industrial water demands are estimated using a cross-sectional data set drawn from observations on manufacturing water users throughout British Columbia. Chapter Four is concerned with estabHshing a method to evaluate the impact of changing water pricing rules and reporting the results of that analysis. A computer program is constructed to simulate the introduction of efficient pricing rules and to compute the impact on aggregate consumption, the utility's budget deficit, and aggregate surplus of this change.2 The program incorporates parameters from the estimation results. The program is employed to consider moving to alternative forms of efficient prices and also to establish the sensitivity of each pricing rule's results to changing parameter values. The alternative efficient prices are drawn from the review of the theoretical pricing literature in Chapter Two. Chapter Four also presents a discussion of the use of aggregate Marshal1!an consumer surplus measures as indicators of welfare change and of the problems of interpretation in the presence of second-best constraints. 5 Chapter Five concludes the thesis. In this chapter the results of the research are summarized and evaluated relative to the goals of this project. Suggestions for future research are also presented. The thesis is completed by a bibliography and two appendices. The first of these details the data sources and methods used to construct the data used in estimation and simulation programs. The second appendix reports the results of tests conducted on the estimated cost equations. B. HYPOTHESES TO BE TESTED Several of the questions which this thesis seeks to address can be framed as testable hypotheses. These hypotheses are the following: 1. The prices charged by the VWW and GVWD during the period of observation (1986) are significantly different from their respective marginal costs of supply. 2. The following parameter values are significantly different from zero: a. Short-run marginal cost of supply (for VWW and GVWD). b. Price elasticity of demand (for aggregate residential and industrial demands). 3. The change in pricing rules from the VWWs current practice to seasonally-differentiated peak-load prices will increase the aggregate surplus from consumption. This will be true in the presence of a break-even constraint and the need to install flow metering devices. 6 4. Peak-load prices will generate larger aggregate surpluses than will average-cost pricing. 5. The change in pricing rules from the VWWs current practice to Ramsey prices will increase aggregate surplus. C. SUMMARY OF RESULTS With respect to the costs of supply, it is found that there is some seasonal variation across quarterly short-run marginal costs. More significantly, these values are substantially smaller than the estimated long-run marginal costs. For example, the average values for the short run and long run marginal costs are $6.5 and $66.5, respectively (where the figures are expressed in 1986 dollars per thousand cubic metres of output). In contrast, output prices charged by the VWW range from $0 (residential) to $108.40 (commercial). The principal conclusion from the estimation of the water demand equations is that both user groups' demand equations demonstrate a degree of price responsiveness. The estimated average values of the residential summer (outdoor) and winter (indoor) price elasticities are -0.649 and -0.014, respectively. In contrast, industrial water demands demonstrate an average price elasticity of -1.913. Finally, with respect to the simulation results, it is found that the move from the VWWs current pricing practices to efficient prices yields positive net gains to aggregate surplus under a variety of parameter values. For example, moving to seasonal pricing (with no discrimination across consumers) raises residential prices 7 but lowers prices facing industrial and commercial users while keeping the utility at a zero profit level. As a result, there is little change predicted for aggregate consumption. It is demonstrated, however, that households' welfare is lowered by the direct impact of the price increases and is raised by the indirect impact of firms passing on the reductions in water prices in the form of lowered output prices. The net impact of these two offsetting effects is predicted to be positive, indicating that the move to seasonal pricing may raise aggregate welfare by approximately 4 %. In contrast, the utility's simulated move to a form of Ramsey pricing leads all consumers to experience increases in per unit prices (although all annual charges are removed). This set of pricing changes is predicted to lead to a decrease in aggregate welfare by approximately 13 %. 8 Endnotes 1. These include Toronto, Vancouver and Montreal. 2. The impact of price changes for water supply on the costs of operating sewage treatment facilities is neglected in the simulation program due to data limitations. 9 II. WATER PRICING: THEORY AND PRACTICE A. INTRODUCTION This chapter has three objectives: to review the economic theory of public utility pricing,1 to illustrate current North American water pricing practices, and to evaluate the latter in light of the former. The next section reviews the basic models and results of the public utility literature and considers some extensions that are relevant to the operation of water utilities. Section C presents an outline of the philosophy and guidelines followed by water utilities in their pricing decisions. The final section evaluates current pricing practices. Several conclusions are reached. First, the theory of efficient public sector pricing has evolved to generate pricing prescriptions relevant to a variety of 'real-world' circumstances. Second, North American water utilities follow pricing and capacity expansion rules which diverge significantly from the prescriptions contained in the economics literature. Third, in theory there exist potential efficiency gains from reforming the pricing practices of water utilities. However, there are no empirical studies which have estimated the size of these potential gains. 10 B. THE THEORY OF EFFICIENT PUBLIC UTILITY PRICING The purpose of this section is to review the theoretical economic literature concerned with public utility pricing.2 The most important question addressed in this literature concerns the structure of prices that will result from a utility's efforts to maximize some measure of the net surplus from the consumption of its output. In this review a simple model is first used to answer this question. Subsequently, several extensions are examined: the presence of increasing returns to scale or scope, cyclicity in demands, incorporation of distributional considerations, and allowance for the presence of uncertainty. This list is certainly not exhaustive, however, it does encompass some of the most important strands of the theoretical public utility pricing literature and also the portions of that literature most relevant to the consideration of water utility pricing. It should be noted that discussion of two issues indirectly related to this literature is delayed until Chapter Four. These issues are the use of Marshallian consumer surplus as an index of welfare change and the use of partial analysis when considering optimal public pricing rules. Consider a public utility producing a single output, Q, and facing an uncompensated market demand P = P(Q) and costs of production C = C(Q). The utility is assumed to choose its level of output to maximize the net aggregate surplus associated with consuming its output. Hence, the utility wishes to maximize the sum of consumer and producer surplus through choice of output: (2.1) aJJ* / P(Q)dQ - C(Q) 11 which leads to the first order necessary condition: (2.2) p' - C'(Q') where Q* denotes the optimal level of output, p* is the associated price and the prime indicates the partial derivative. Thus, maximizing aggregate surplus leads to the familiar "price equal to marginal cost" result (Hotelling, 1938). The list of assumptions required to achieve this result, however, is lengthy. These assumptions include the following: demand for the utility's output must be a function of no price but its own; is known with certainty by the utility; and is stable over time. Furthermore, the utility is assumed to be indifferent to the distributional consequences of its decisions and also to the possibility that its pricing policy may not be financially self-sustaining. The model may be extended by relaxing the assumptions upon which it is based. The first extension to be considered addresses the implications of allowing for increasing returns. If the utility's production technology is characterized by increasing returns to scale (ie., C'(Q) < C(Q)/Q, for Q £ Q*), then the marginal cost pricing rule will cause the utility to earn a deficit. If the utility is constrained to remain financially self-supporting (ie., the use of lump-sum taxes by government to subsidize the utility's operations is ruled out), then it must reform its pricing strategy. The first approach to solving this second-best problem stems from Ramsey (1927). The optimization problem in (2.1) may be amended to include the following 'break-even' constraint: (2.3) P(Q)'Q - C(Q) fc 0 12 In this simple version of the problem where only one output is considered, solving the constrained maximization problem leads to the well-known 'inverse-elasticity' rule for pricing (Brown and Sibley, 1986, ch. 3): (2.4) r - cm m p* T] 1 + ijr where *F is the (unobserved) multiplier associated with the constraint and T | is the absolute value of the price elasticity of market demand. Thus, the optimal price deviates from marginal cost in this second-best case and the magnitude of the deviation is inversely related to the price elasticity of market demand. An alternative approach to this problem stems from Coase (1946). Here, rather than choose a price away from marginal cost, Coase proposes that a 'two-part' pricing strategy be followed. A per unit consumption charge is set at the marginal cost of the predicted level of output and any resulting deficit is made up by charging a lump-sum fee to each consumer. Thus, if there are H consumers, then the optimal fee, A*, is given in equation (2.5): The fee is essentially a head tax and is not directly related to the level of consumption of any one consumer. Following the Coase prescription, a first-best solution is achieved so long as the condition in (2.6) is satisfied (assuming all H consumers are identical^ 13 Q* _ _ { PiQ)dQ - PiQ')Q' (2.6) A' * -2 The expression on the right hand side of (2.6) is just the average consumer surplus at the optimal price P(Q*). If this condition is not met, then some consumer may be excluded from the market whose marginal willingness-to-pay for the good exceeds C'(Q*). This would be inefficient. Several authors have noted the restrictiveness of the identical consumers assumption in the Coase model and have recast the choice of (p,A) assuming that there is a distribution of consumers with differing willingnesses-to-pay for the utility's output. In this case, the market size (ie., the number of consumers) will depend on (p,A) and, thus, the utility must optimize with respect to both variables. This gives rise to an optimal two-part tariff (Oi, 1972; Ng and Weissner, 1974). To see this, suppose that consumers are distributed according to some characteristic y and that this distribution may be represented by a probability density function Ky). Then, each consumer's demand for the utility's output is Ph=Pk(Qh,Yh) ^ total market demand facing the utility is the following: (2.7) P(Q) - H- { Ph(Qh, y) • AY) <*Y to 14 where Yo arid Yi are the minimum and maximum values for the characteristic variable and H is the number of consumers. If the utility faces a constant unit cost (v) of 'connecting' one more consumer to the market, then it can be shown that the optimal two-part tariff has the following structure (Brown and Sibley, 1986, chapter 4): (2.8) a) P* - C/(g'> - A • [ 1 " Q ( P ' T o ) 1 P* Qu „ F L . C P * - c '«n) q(p, Yo) + - v) e (2.8) O) - — A* e where q(p, Yo) is the demand of the marginal consumer (ie., whose consumer surplus just equals the connection charge), QM is the average consumption level, v is the marginal cost of adding a consumer to the distribution network and e is the market's price elasticity with respect to A. An interesting feature of the optimal prices in equation (2.8) is that they are similar in structure to Ramsey prices: the optimal two-part tariff is concerned both with the market's response to changes in p and its response to changes in A. Thus, by incorporating information concerning the distribution of demands and by endogenizing the size of the market, the optimal two-part tariff represents a synthesis of the Coase and Ramsey streams of public utility pricing. Most recently, this synthesis has been extended even further with the analysis of the structure of optimal non-linear price schedules (Goldman, Leland, and Sibley, 1984; Brown and Sibley, 1986, chapter 5). In these models the logic of Ramsey pricing 15 is applied to demonstrate that the slope of the optimal nonuniform price schedule depends on the behaviour of the market price elasticity as well as the behaviour of marginal costs. For example, suppose marginal costs are constant but that consumers with larger values of the characteristic variable also have demands which exhibit larger price elasticities. Then, the optimal price schedule will have a negative slope because Ramsey pricing dictates that the margin by which price exceeds marginal costs should fall as the market demand's price elasticity rises. The second extension to the basic model arises from a desire to incorporate the effects on optimal pricing decisions of cyclical demands. Many public utilities (eg., gas, telephone, and water) face aggregate demands which consistently (if irregularly) rise during certain 'peak' time periods. Given that most public utilities have production technologies which are capital-intensive and that they also produce outputs which frequently cannot be stored3, then in the short-run the utility may be constrained by the extent to which it can respond to peak levels of demand. This raises the question of how the utility should price its output for a given level of capacity when demand fluctuates cyclically.4 Boiteaux (1960) and Steiner (1957) consider a public utility which produces an output, Qt, in each of two time periods of equal length. Once capacity is chosen, the utility's output in each period is constrained not to exceed capacity Q. It is assumed that it costs "b" to produce one more unit of output and "p" to expand the level of capacity by one unit. Short-run and long-run marginal costs are assumed to be constant and total costs are separable as shown in equation (2.10): _ 2 _ (2.10) c«?„ P<? 1-1 16 Furthermore, it is assumed that the utility seeks to maximize the aggregate net surplus (undiscounted) over the two periods given the market demand curve Pt(Qt) prevailing in each period In the short run, the utility is also constrained by existing capacity, Q t £ Q. The short-run objective function for the utility is then given in equation (2.11). ( 2 . 1 1 ) "g* £ P,«?,)<*<?f - bQt - BO - X[Q - <?,] Optimizing with respect to Qt yields the following well-known first-order necessary conditions. p; - b - x; - o Q; - Q * o (2.12) V«?,* - Q> " 0 for t - 1,2 Thus, if consumption is less than capacity in either period, price is to be set at short-run marginal cost. If consumption is constrained by capacity, then price should rise above marginal cost until desired consumption just equals capacity (Rees, 1984). In the long-run, capacity is a choice variable for the utility. In this case, the 17 first order conditions are augmented by the following condition. (2.13) £ V - P Thus, suppose capacity is chosen so that there remains excess capacity in one period (the "off-peak") and no excess capacity in the other period (the "peak"). Then, efficiency requires that price equal short-run marginal cost in the off-peak period and long-run marginal cost in the peak period. An immediate problem with this result is that demand in the peak period may respond to the imposition of the peak-load price by falling to a level below that of demand in the off-peak period. This 'peak-shifting' phenomenon is anticipated by Steiner (1957) who uses the Samuelson technique of vertically summing market demands for a jointly consumed public good (here, utility capacity over the two periods) to determine the optimal level of output, Q*, common to both periods. The resulting prices satisfy the modified peak-load rule: (2.14) / y - D^Q'); P2' - D2(Q'); P* + P2* - 2* + p Thus, in contrast to the 'firm-peak' case, the price in each period may diverge from marginal cost. In both cases, however, profits are non-negative. Further extensions to the peak-load pricing rules are derived by relaxing other assumptions of the Steiner-Boiteaux model. Williamson (1966) allows the time periods to differ in length, while Gravelle (1976) assumes equal sized time periods, but introduces the possibility of storage. Both efforts at relaxing the assumptions in the 18 early model result in modified versions of the basic pricing rule. Finally, Panzar (1976) considers the Boiteaux-Steiner model and points out that the peak- versus off-peak distinction in prices is attributable not only to the cyclicity in demand, but also to the assumed separability of the cost function. Panzar demonstrates that, under a more general specification of costs, the utility's revenues in each period exceed its expenditures on variable inputs. Thus, while prices are highest in peak periods, consumers in each period make some contribution to fixed costs. The third extension to the basic model is related to incorporating distributional aims into the pricing decisions of the public utility. This topic has not been a major part of the theoretical public utility pricing literature. There are at least two reasons for this. The first is the general reluctance of most economists to diverge from efficiency considerations.6 The second reason stems from the view that assessing the distributional consequences of moving from inefficient to efficient prices is an issue separate from the establishment of the structure of efficient prices. The seminal work that incorporates distributional concerns into public pricing rules is Feldstein (1972). The motivation for Feldstein's approach is the regressive character of Coase's solution to the problem of deficits when the utility employs marginal cost pricing under increasing returns to scale. The lump-sum fee charged to each user is independent of the user's consumption level (or any other characteristic) and, as a result, is equivalent to a head tax. In the Feldstein model, the welfare changes of consumers with different income levels receive different social weightings and the implications for Coase two-part prices are considered. The Feldstein model assumes that the incomes of identical consumers are distributed according to a known density function, f(y), and that the public utility assigns a 19 different marginal social weighting u'(y) to each income level (where u'(y), the derivative of the social welfare function, falls as y rises). The utility then chooses its per unit price and annual fee to maximize the weighted aggregate Marshal li an consumer surplus subject to the requirement that the utility earn a zero profit. Feldstein shows that the optimal price exceeds marginal cost unless individual demands are independent of income or the same social weighting is applied to all income levels. The divergence of price from marginal cost and the efficiency losses arising from this distortion are outweighed by the equity gains from reducing the regressive annual fee. If f(y) is assumed to be lognormal (with mean and variance u and OF2, respectively), household demand is Cobb-Douglas (with price and income elasticities rjp and riy , respectively), marginal cost (mc) is constant and u'(y) has the form y p , then the optimal Feldstein price is the following: (2.15) p - — ^ • mc Tl, + 1 - (1 + where rv is the 'relative variance' of f(y) (ie-.o /^u2) and p is the parameter in the marginal social utility of income function which indicates the degree of inequality aversion held. Thus, increases in the average income elasticity, the dispersion of the income distribution or the degree of inequality aversion will all increase the divergence between price and marginal cost in the Feldstein model. The final extension to the basic public utility pricing model considers the impact of the utility not knowing the level of market demand with certainty. This situation may arise if some consumption is not observed (in the case of unmetered 20 water use) or if the level of aggregate demand is a function of stochastic forces (such as temperature). In this case, the utility will have an ex ante expectation regarding the level of demand, but the ex post realization of consumption may diverge from its predicted value. Furthermore, if the capacity of the utility is fixed in the short-run, the utility faces the possibility that demand may exceed capacity and that it must, therefore, ration output. Crew and Kleindorfer (1986) review the development of the public utility literature concerned with pricing under uncertainty. In this context market demand is usually modeled as having both deterministic and stochastic components as shown in equation (2.16): (2.16) P(Q, u) - P(Q) + « where u is a stochastic variable. In order to maximize the expected net surplus associated with consumption the utility must consider the probability distribution of the stochastic component of demand, as well as its own direct costs of production and the costs of rationing (should demand exceed capacity). The latter include added administration expenditures, expenditures to enforce rationing programs, and the value of resources consumed by consumers (including their time) to circumvent any rationing program. These rationing costs, r, are usually assumed to be a function of the difference between actual demand and capacity: 21 (2.17) HQ, Q) 0 if QiP) < Q - r(Q(.P) - Q) if QiP) ^ Q where Q(P) is the actual market demand. If the utility is risk-neutral, then it seeks to set its price to maximize the expected net surplus from consumption (ie., consumers' valuation net of production costs and rationing costs). The resulting pricing rules look very much like those of the basic model, with the exception that marginal cost is now the sum of the marginal costs of production and rationing (Crew and Kleindorfer, 1986, ch. 5; Rees, 1984, ch. 10). An important limitation of the uncertainty models, however, is that the specific forms of the optimal pricing rules appear to be tied to the specific forms of the market demand function (ie., whether there are additive or multiplicative stochastic terms) and the form of rationing cost functions assumed (Carleton, 1977; Visscher, 1973; Chao, 1983). The theoretical literature concerned with identifying the structure of optimal public utility prices has grown enormously since the writings of Ramsey (1927), Hotelling (1938), and Boiteaux (1960). While the spirit of the 'price equals marginal cost' solution remains in some cases, it has been supplanted by rules from models that incorporate concerns with peak-loads, demand distributions, deficits and uncertainty. As a result, this body of literature provides a relatively well developed standard against which the practices of North American water utilities may be judged. This task is undertaken in the next two sections. 22 C. CURRENT PRICING PRACTICES There are a variety of institutional and regulatory structures surrounding the operation of municipal and regional water utilities in North America. In Canada public ownership prevails with very few exceptions and most utilities are organized as part of either a municipal or regional government (Hanke and Fortin, 1985). In contrast, investor-owned utilities comprise approximately one quarter of the American industry (Feigenbaum and Teeples, 1983). In Canada the degree of regulatory control over water utilities' issuing of debt, cost accounting, and rate-setting practices varies across provinces. In most provinces there is no direct control exercised over rate-setting, but a requirement that new debt issues be approved by Cabinet is common. Only in Nova Scotia and Saskatchewan are there Public Utilities Commissions which review proposals for rate changes (Hanke and Fortin, 1985). Just as the institutional features of water utilities vary across Canada, so do details of their rate-setting practices. Most utilities, however, subscribe to the objectives of rate-setting as set out in the American Water Works Association (AWWA) Handbook (1972) and follow the methods outlined there. These procedures are summarized in the following quotation from the Handbook: Ordinarily, the development of water rates involves the following major areas of study: (1) determination of the level of annual revenues, or revenue requirements, necessary to provide for operations, maintenance, and development of the water system; (2) distribution of the annual revenue requirements, or costs of service, to basic cost functions, which in turn allows further distribution of these costs to customer groups or classes in accordance with respective class requirements for service; and (3) design of water rates that will, as nearly as practice, recover from each customer class the respective costs of providing services, (p. vi.) 23 This quotation contains an imphcit suggestion as to the objective of rate-setting: the principal role identified for prices is to earn an exogenously determined level of revenues. Thus, cost-recovery and not resource allocation is the guiding principle for rate-setting by most water utilities. The quotation also contains an outline of the procedure to be used to compute prices. The first step is to establish an estimate for projected annual expenditures. These are based on a forecasted level of output that is derived by applying an estimated growth rate to last period's output. Expected expenditures, then, are the sum of expenditures on operations, scheduled maintenance activities, and debt-retirement costs. Any expenditures related to system expansion are usually funded separately (from issues of new debentures) and not from current revenues. Once total expenditures or "revenue requirements" are determined, the utility seeks to distribute or allocate these expenditures in two ways. First, expenditures are allocated by 'function'. There are several ways to define these functions, but one of the most commonly used is the "Base-Extra Capacity" method (AWWA, Handbook. 1972, p. 12). Expenditures are allocated according to the provision of base capacity, peak capacity, and customer services (for example, customer billing). For example, the cost of repairs to a pump used two thirds of the day for base generation and one third of the day for meeting peak consumption would be allocated to base and peak capacity functions in a 2:1 ratio. Once expenditures are allocated to cost 'functions', then the unit cost per function for each customer class is computed by first assigning total function costs to each customer class in proportion to that class's share in that function's output (eg., peak-period output). Unit costs are then computed by dividing each function-customer class's total costs by the total output for which that customer 24 class is deemed responsible. Output prices may now be calculated. The computed unit costs are combined with information regarding the structure of each class's consumption structure (eg., the ratio of peak-day to average-day consumption) to generate the class-specific price or price schedule. For example, if the ratio of peak use to the average rate of consumption falls as consumption increases then this will result in a declining price structure. The usual results of the application of this set of procedures may be summarized (Hanke and Fortin, 1985). Residential consumers typically face either only an annual connection fee for service or a combination of annual fee and constant unit price if consumption is metered. In the latter case, a provision that some fixed quantity of output per month is provided at no cost is often encountered. Commercial, institutional, and industrial consumers usually face an annual fee and a price schedule characterized by a number of price 'blocks'. Here marginal prices take discrete jumps at different consumption levels and these marginal prices most frequently dechne with increases in consumption. Many of these users also face a variety of miscellaneous charges for fire protection, rental of water meters, etc. While these pricing rules imply that municipal water prices frequently differ across user groups it is rare for North American water utilities to discriminate with respect to price on the basis of distance from the supply centre or time of consumption. A small number of utilities have experimented with summer surcharges (Martin et al. 1984). The AWWA Handbook (p. 56) suggests setting these surcharges on the amount of consumption observed during peak periods in excess to that recorded during base periods. The primary benefit from such a charge is seen to be a more equitable 25 division of costs rather than influencing consumption decisions or system design. The Handbook concludes, however, that, "There are a number of unknowns regarding the effect of seasonal rates on peak system demands. Further experience and study is needed to determine whether or not and to what extent such rates do in fact moderate peak demands" (p. 57). Another dimension of water's characteristics over which prices do not vary is reliability of service. If a utility possesses a fixed pumping capacity and faces fluctuating demands, then system pressures can also be expected to fluctuate. This condition also occurs in the generation of electricity; power utilities frequently offer discounts to consumers willing to contract for interruptible power supplies. In contrast, water utilities have not pursued this form of price discrimination. This may be due in part to water's role as a tool to fight fires. Firms contracting to purchase interruptible water supplies (at lower prices) may experience an increase in their fire insurance premiums. D. EVALUATION OF WATER PRICING PRACTICES The cost accounting and pricing activities of water utilities have drawn criticisms from several authors (Hirshleifer, DeHaven, and Milliman, 1960; Grima, 1984; Martin et. al.. 1984; Pearse, Bertrand, and MacLaren, 1985; Tate, 1989). The arguments supporting these criticisms are summarized here. The broadest criticism of water utility pricing is directed at the objectives 26 which underlie the entire process of rate-setting. The A WW A Handbook identifies clearly that "revenue-recovery" is the primary motivating factor in the design of water prices. There is little recognition of the role played by prices to signal resource scarcity. Consumer demands are seen as exogenously determined 'requirements' and, as a result, there is no attempt to maximize the surplus arising from consumption through the choice of appropriate prices. Furthermore, since consumer demands are considered exogenous, no attempt is made to measure the market's valuation of output. Without this information, it is difficult to see how decision-making regarding a system's capacity or the pricing of output will yield a social optimum. There are several other criticisms that may be levelled against the specific procedures used by water utilities to compute prices. Annual recorded expenditures are assumed to represent the full economic costs of service. There are at least three reasons why this may not be the case. First, unless water is purchased from a regional wholesaler, it is typically assigned no value. Thus, when raw water supplies are not priced, the price facing consumers represents only the cost of treating, storing, and delivering water, rather than including the value of the water itself. Second, annual capital costs are measured as the scheduled expenditures to retire debt rather than the user cost of capital. In fact, some Canadian utilities are prohibited from including depreciation . as a cost of production and are required to value their capital stock at its historic acquisition cost (Hanke and Fortin, 1985). Furthermore, the costs of capacity expansion undertaken in any period are not incorporated into that period's prices. This is because these expansions are funded by issuing new debentures rather than from the current period's revenues. 27 Third, prices may reflect the costs of delivery, but they usually do not incorporate the costs of water treatment once consumption has taken place. Sewage treatment is usually administered by a separate public utility or municipal department. As a result, the information regarding the full social (and environmental) costs of water use is not contained in a single price. Rather, sewage treatment costs are usually recovered through property taxes and not through user charges (except in the case of large industrial water users (Hanke and Fortin, 1985)). These three factors suggest that utilities' recorded annual expenditures will not accurately measure the full costs of service. The A WW A Handbook devotes a large amount of space to the proper 'allocation' of costs. While the principle underlying this approach may be laudable ~ that prices should be tied as closely as practicable to the costs of providing particular services — the methods proposed contain little economic content. Because most utility costs are joint there is no way to separate them in an economically meaningful way. An attempt to divide these costs across 'functions' or customer classes results in a chimera of differential unit costs associated with output at different times. In addition to this problem of allocating joint costs, there is the separate issue of what should be the basis for consumer prices. What is relevant is not the average cost of service (however defined) but rather the behaviour of total costs when output changes incrementally. The basis for most utilities' prices, however, is the unit cost at the predicted level of output. The prices that result from the water industry's method of cost accounting are likely to be discriminatory. This is because price differences across consumer groups are not tied to differences in the marginal costs of service. In addition, water prices 28 neglect factors that would be expected to influence marginal costs. Factors such as distance from the supply centre and the time of consumption do not directly enter into the pricing formula. Thus, whether an individual's consumption occurs when excess capacity exists or when aggregate consumption is constrained by system capacity is irrelevant to the price of output for most water utilities. To conclude, there is little reason to believe municipal water prices are efficient. They are derived from an accounting system that emphasizes historical rather than economic costs and they are based on an artificial set of unit costs. The marginal costs of supply and factors that may be expected to influence them are neglected in the rate-setting exercise. It is interesting to note, however, that, while the economics literature is quite consistent in its criticisms of water pricing by municipal utilities, it contains few studies that document the magnitude of the gains to be expected from moving to efficient prices. Thus, while there are a number of studies that argue convincingly that water prices are inefficient, there is not a corresponding set of articles which demonstrates that welfare levels will be significantly higher at efficient prices. This situation contrasts strongly with the exhaustive analysis of the potential welfare gains to be had from reforming the pricing practices of other types of public utilities. For example, Mitchell (1978) examines the telecommunications industry and Dimopoulos (1981) and Protti and McRae (1980) examine the electric power industry. There is a need for this type of analysis in the water industry. It may be that the efficiency gains predicted by theory are smaller than the costs associated with implementing a new pricing scheme (these costs would most likely include the costs of installing and maintaining flow meters for residential consumers). Alternatively, 29 the aggregate net welfare gains may be positive but the distributional consequences of the pricing change may be objectionable. For at least these two reasons, an empirical study documenting the magnitude of the efficiency gains from altering water prices is required. 30 Endnotes 1. I am concentrating on the behaviour of public sector utilities in this analysis. Thus, I am not considering how privately-owned utilities may differ from public sector utilities. 2. This field already contains several surveys: Bos (1985), Rees (1984), Crew and Kleindorfer (1979), Brown and Sibley (1986). 3. This constraint also applies to water utilities. While most water utilities are able to store raw (ie., untreated) water in lakes, reservoirs, or aquifers, their ability to store potable water is constrained. Because the chlorine added to drinking water is chemically unstable, it will dissolve and leave water in a relatively short period, ie., less than twenty-four hours. This implies that the capacity of the utility to treat water and distribute treated water is its most significant short-run constraint (Twort, Law and Crowley, 1985). 4. It also raises the question of how large a distribution system should be installed. See Williamson (1986) and Rees (1984) for discussions of this problem. 5. A.A. Walters is quoted by Bos (1985, p. 212) as saving, "Why do you want to favour poor people simply because they go by railway? If you regard them as poor, give them money." 31 ELI. STRUCTURE OF THE COSTS OF SUPPLY AND DEMAND A. INTRODUCTION This chapter presents the methods used to estimate the structure of supply costs for the water utilities under study and the structure of demands for their output. The chapter also discusses the results of these procedures. The principal results relate to the utilities' quarterly marginal costs, the shadow values of their quasi-fixed inputs, and the price elasticities of each user group's aggregate demand. A detailed discussion of the data is contained in Appendix 1 and the results of statistical tests performed on the cost functions are presented in Appendix 2. B. THE STRUCTURE OF WATER SUPPLY COSTS 1. Review of Previous Studies There have been a number of attempts to characterize empirically the structure of water utility costs. These studies have been concerned primarily with either assessing the extent of scale economies in the technology of water delivery or identifying the efficiency differential that exists between publicly administered water departments and privately owned water utilities. There are no extant studies whose stated purpose is to derive marginal cost estimates that could be used in studying the efficiency of water prices. The largest part of the water cost literature is directed at obtaining econometric estimates of the degree of scale economies. One strand of this literature adopts a relatively simple econometric model and regresses total recorded 32 expenditures against output (Hines, 1969; Sewell and Roueche, 1974; Clark and Stevie, 1981; Fraas and Munley, 1984; Hayes, 1987). Input prices are implicitly assumed to be constant across all observations and all utilities are assumed to be in long-run equilibrium. Most of these studies find that the representative utility enjoys increasing returns over a fairly wide range of output levels. Hayes (1987), for example, uses a cross-sectional sample of utilities operating in 1976 and finds that returns to scale (defined as the inverse of the elasticity of total costs with respect to output) range from 1.901 for small utilities to 0.950 for very large utilities (in excess of 110 million cubic metres/year).1 Economies of scope follow the same pattern in the case where utilities supply both wholesale water and retail water. Hayes, however, does not report any standard errors with his parameter estimates. Kim (1987) presents a more sophisticated attempt to measure economies of scope. In Kim's econometric model, long-run costs are assumed to be a function of input prices, the levels of outputs and an index of capacity utilization. A translog form is chosen and price indices for labour, capital, and energy are generated. Each utility's total output is divided into residential and non-residential output. The cross-sectional data set consists of firm-level observations for 60 American water utilities operating in 1973. Capital prices are calculated as each utility's average interest rate on long-term debt plus a two percent depreciation rate. This sum is then multiplied by a regional construction price index. No price for the water input is included. Kim finds that the marginal cost for residential output is positive and increasing, while the marginal cost for non-residential output is lower than that for residential output and is declining over the sample range of output. Overall scale economies are present, but decline with the size of the utility: large utilities have a 33 scale elasticity of 0.875. Again, standard errors are not reported for the scale estimates. These studies appear to be consistent in their findings with respect to the presence of scale economies. The value of these studies as a basis for an analysis of pricing practices, however, is mitigated by two observations. First, all use annual observations and do not differentiate between short-run and long-run marginal costs. Only Sewell and Roueche (1974) attempt to estimate short- and long-run marginal costs, but they do so by arbitrarily allocating different fractions of capital costs to outputs at different periods. The second common feature of this literature is that it assumes the water utilities are in long-run equilibrium. This assumption is important because all of the studies are concerned with measuring scale economies. Much of the evidence concerning water utility planning, however, suggests that this assumption may be invalid. First, the failure of water utilities to employ time-differentiated (ie., peak-load) pricing can be expected to induce the utility to overexpand relative to the socially optimal level of capacity. Because water utilities establish their capacity to meet peak water 'requirements' this capacity will be larger than would be necessary if peak demands had been constrained by peak-load pricing.2 Second, water utilities frequently expand capacity in anticipation of future consumption growth (Martin et. al„ 1984). This practice is, in part, a response to the discreteness of increments to capacity and to the expected presence of increasing returns to these expansions.3 The implication of this practice is that the utility may appear over-capitalized (relative to the observed level of output), if it is examined immediately after an expansion of the system has occurred. 34 The second set of applied studies examining the costs of water supply is primarily concerned with testing for efficiency differences between publicly- and privately-owned utilities. The most recent attempts are contained in Teeples and Glyer (1987a, 1987b). They present an econometric model of the costs of water supply which contains an hedonic specification for output. They show that previous work by Feigenbaum and Teeples (1983) and Grain and Zardkoobi (1978) reduce to special cases of the more general form of the cost function. As such, the Teeples and Glyer work is the most general econometric specification of water supply costs in this part of the literature. Teeples and Glyer (1987a, 1987b) adopt a long-run specification and assume costs are a function of output, operating characteristics and input prices (labour, energy, raw water and a capital-materials composite input). Output is proxied by an hedonic sub-function in which a scalar measure of total output is regressed against measured output flows and output characteristics (eg., the ratio of wholesale to retail output). A translog form is used and the data come from a cross-sectional survey of annual price and quantity observations from firm-level responses to a survey conducted by the authors. Ownership effects are captured through the use of shift and slope dummy variables. Degrees of freedom constraints do not allow for the estimation of separate cost functions for private and public utilities. Teeples and Glyer report that ownership effects are negligible in the most general form of the estimation model. That is, most ownership dummy variables have very small and insignificant coefficients. The authors are also able to demonstrate that the magnitude of the ownership-dummy variables' coefficients diminish as the estimation model is generalized from that used by previous studies to the most 35 general form used by the authors. However, because the Teeples and Glyer estimation model uses a long-run specification and is based on annual observations, the comments made previously about the appropriateness of this assumption apply. The literature concerned with estimating the cost structure for municipal and regional water utilities has grown in recent years. Because of the aims of the studies, however, their methods are not directly applicable here. There are two reasons for this conclusion. First, the use of annual observations precludes the analysis of the seasonality of costs. Second, the maintained assumption of optimal input use prohibits testing for over-capitalization. Given the reasons cited above, it would seem more appropriate to adopt an estimation procedure which allows the analyst to test whether the utility is over-capitalized. The estimation method should also establish whether costs are seasonal. The procedure reported in the next section accomplishes these goals. 2. Econometric Model of the Wholesaler's Cost Structure a. Background The purpose of this section is to establish the method used to estimate the cost structure of a regional water utility. A wholesale, or regional, water utility is responsible for capturing, storing, and delivering water to smaller municipal water utilities. These responsibilities typically require a system of impoundment lakes (if the utility relies on surface water), pumping and treatment stations and a system of distribution mains. The production process is highly capital-intensive with labour being employed largely to monitor and perform maintenance on an automated system. 36 Energy expenditures (primarily electricity for pumping stations) can be a significant fraction of variable costs, especially if the utility relies upon groundwater. Materials expenditures are comprised mostly of chemicals for water testing and treatment (e.g., chlorine, alum, etc.). Estimating the structure of wholesale costs will allow the marginal cost of output to be established. This is significant because the arguments made in Chapter Two indicate that the wholesale price may not reflect the marginal cost of supply. When the cost structure for the municipal water utility is estimated, it will be the estimated marginal wholesale cost and not the recorded output price that is employed to estimate marginal cost at the retail level. For reasons given below, it is possible to estimate the short-run cost structure econometrically, but this procedure is seen to be inappropriate for an examination of long-run costs. As a result, an alternative procedure is presented to compute long-run marginal costs. The utility under study is the Greater Vancouver Water District (GVWD). The GVWD is the regional water wholesaler for the Lower Mainland in Southwestern British Columbia and serves Vancouver and the dozen municipalities surrounding it. The GVWD captures water in three man-made lakes on the North Shore mountains above Vancouver. After filtration and treatment, the water is distributed to each municipality's water works and ultimately to consumers. In relative terms the GVWD is a large wholesaler. In 1986 its recorded output was 345 million cubic metres.4 This translates into an average daily output of 945,000 m3/day, while the ma-rimum recorded daily flow is 3,350,000 m3/day. 37 6. The Short-Run Cost Function It is assumed that the structure of the GVWD's productive technology may be established by estimating a cost function which is dual to the GVWD's underlying production function. This cost function is derived by first assuming that the utility chooses its inputs to minimize the cost of producing a given amount of output: (3.1) CJp, Q, Q) - " J 1 (pa I x e 7{x), Q *Q) where x is a vector of inputs; p » 0, a vector of input prices; Q , the exogenously determined level of output; and T(x) the utility's production technology set. Implicit to the specification of the cost function in (3.1) is the assumption that the utility may optimize over the entire set of inputs. In the short-run, however, this is unlikely to be true for any water utility. In the short-run, it would be prohibitively expensive, if not physically impossible, to alter the capital stock owned by the utility. In addition, the issues raised earlier in Chapter Two regarding the stochastic nature of water demands and the most frequently used pricing and capacity expansion rules suggest strongly that a utility may not be employing the cost minimizing level of capital at any time. For these reasons, modeling the utility to be in long-run equilibrium and basing an estimation model upon this assumption would seem unwise. As an alternative it is assumed that the utility may optimize over a subset of inputs. This approach to modeling the utility's cost structure assumes that the utility is in "partial static equihbrium" with respect to its variable inputs (cf. Brown and Christensen, 1979; Kulatilaka, 1987). This allows a restricted cost function to be derived: 38 (3.2) CR (p, Q,Q,Z,Z)- ™ ipMxM I xR e 71[x); Q *Q,Z*Z) where % is the vector of variable inputs (x R has lower dimensionality than x) and PR its corresponding price vector. Z is a vector of inputs whose quantities are fixed in the short-run to be no greater than Z. Total costs are then established as the sum of restricted costs and expenditures on fixed factors. If the utility is in long-run equilibrium then we can relate the cost functions in (3.1) and (3.2) by equation (3.3): where q is vector of user costs associated with the vector of fixed factors. Estimation of the restricted cost function CR(0 is expected to yield several results, particularly regarding the effects of incremental changes in the fixed factors and output on variable costs. An expression for the non-normalized change in variable costs attributable to an incremental change in output, ie., the short-run marginal cost, is derived by partially differentiating (3.2) with respect to Q : (3.3) CT(p,Q) - CR (PjpQ.Z) + q-Z (3.4) M C 5 (P, <?, 2) dCR(p, Q, Z) dQ 39 For any one element of the fixed factor vector (Z), taking the partial derivative yields the negative of the short-run shadow price associated with that factor ( - R j ) . This interpretation follows directly from applying Hotelling's Lemma (Kulatilaka, 1987). The shadow value measures the decrease in restricted costs that arises from an incremental increase in the stock of the jth fixed factor. Because the restricted cost function is assumed to be convex and non-increasing in the levels of the fixed factors, it follows that the estimate of ( -R j ) should be non-positive and, hence, the shadow price of the jth fixed factor should be positive (Varian, 1985): Partially differentiating (3.3) with respect to Z j and applying (3.5) yields the following result: (3.5) -Rj (p, Q, Z) dCR(p, Q, Z) (3.6) dCJp, Q) -Rfp, Q, Z) + qj where cjj is the jth fixed factor's user cost. Since the left hand side of (3.6) is zero as Z is not an argument of CT(P,Q), then: (3.7) R/p.Q.Z) - q, 40 Thus, a necessary condition for optimal long-run employment of each of the utility's fixed factors is that its short-run shadow value equal its user cost (Kulatilaka, 1987). If Rj < then the utility may have overemployed the jth fixed factor.6 The estimated restricted cost function is also expected to yield information regarding the response of input use to changes in relative prices. The non-normalized cross- and own-input price elasticities are obtained as the second partial derivative of the cost function with respect to the ith and jth prices: (3.8) ff Cjfp, Q, Z) _ dxfp, Q, Z) and dx.(p, Q, Z) p, (3-9) e„ - • ^ ' * 7  xi where the first equality is established using Shephard's Lemma and x -^) is the conditional demand for input L Note that the Ey are not necessarily symmetric. That is, in general, * Finally, in order for the restricted cost function to be dual to the production function, it must satisfy several regularity conditions. These conditions are: monotonicity, concavity in prices, linear homogeneity in prices, and symmetry (Diewert, 1974). Appendix 2 details how these conditions are examined using the 41 estimated restricted cost function. A translog functional form given as equation (3.10) is adopted to estimate the restricted cost function in (3.2) along with the restricted cost input share equations, derived using Shephard's Lemma (3.11). The translog restricted cost function for the GVWD is reported in (3.10). In each equation the index t represents a time series of quarterly observations for the period 1975:1 to 1986:4. (3.10) In CR, - 0 T\ U I K 2 i - i k-i 2 / - l M - 1 + E £ Y v W f c biZfi i-1 J-1 dlnCR, i J (3.11) SU - —-2 - a, + £ i t t f . ' l T ^ 42 Prices are included for the variable inputs electricity (E), labour (L), purchased water (S) and materials (M). The level of output is Z^t and is assumed to be exogenous because of the contractual agreements between the GVWD and its municipal customers. The GVWD is assumed to use two factors whose quantities are exogenously determined in the short-run: its capital stock (Z^ and the stock of water in storage (Z^). T is a time-trend representing an index of Hicks-neutral technological change. The variable D t measures the proportion of GVWD output delivered to the set of municipalities furthest from the GVWD's storage reservoirs. Because the number and location of the GVWD's customers has changed very little over the sample period, D t is included in an attempt to test for the effect of the changing spatial composition of total output on the costs of supply. Details regarding the construction of each variable are provided in Appendix 1. Prices and restricted cost are measured in nominal dollars and all variables are normalized to their respective mean values. The restricted cost function and three of the four share equations are estimated as a system of simultaneous equations using an iterative three-stage least squares procedure. A linear error term is appended to each equation and the errors are assumed to be independently and identically distributed. The following restrictions on the system's coefficients are imposed prior to estimation to account for symmetry and linear homogeneity in prices. (3.12) ott - a u , all i* - EJUSM £ Y „ - £ T, - 0 t-i j~i 43 Applying equations (3.4) through (3.7) to (3.10) leads to closed-form expressions for the marginal cost of output (3.13) and the shadow values of the fixed factors (3.14): dCjSp, Q, Z) (3.13) MCa - * ' ZQ, m*Q t-l The first term on the right hand side of (3.13) is the average cost of output and the second is the elasticity of restricted costs with respect to output (e6). The product of these two terms gives the marginal cost. BCJp, Q, Z) (3.14) -JL - _ ' 3 Z* Zp m*J M aiij - wys: Finally, applying (3.8) and (3.9) to (3.10) yields expressions for the variable inputs' cross- and own-price elasticities: (3.15) tA * i *** 44 where S i t is the actual (rather than the fitted) restricted cost share for the ith variable input. If actual shares are used then the elasticities can be shown to be asymptotically normally distributed (Anderson and Thursby, 1986). If fitted shares are used then the elasticities will not necessarily exhibit this property. c. Estimated Coefficients, Elasticities and Shadow Values Table 3.1 reports the estimated coefficients for the translog restricted cost function. Each coefficient's standard error is included and equation statistics are provided. As the system of equations is estimated with symmetry and linear homogeneity (in prices) imposed, it remains to examine the monotonicity and curvature conditions. The results of these calculations are reported in Appendix 2. In addition, because of the use of quarterly data, there is a possibility of the presence of autocorrelation. The results of applying Wallis's (1972) test for fourth order autocorrelation are reported in Appendix 2. While the untransformed translog coefficients are usually rather difficult to interpret, the coefficients on the time trend and distance-proportion variables can be examined for information regarding the effects of these variables on costs. The positive (and insignificant) coefficient on the time trend suggests an absence of cost-saving innovations. This is somewhat surprising as the GVWD, like other North American water utilities, has been evolving toward increased use of automated monitoring of its systems. The fundamental technology of water treatment and distribution, however, would seem to have remained static over the sample period. The distance variable is defined as the ratio of 'near' output to total output. The negative coefficient on the variable lnD t and the positive estimate of the 45 coefficient, fJQD, indicates that the effect of the changing spatial composition of output is not clear cut. If the restricted cost function (3.10) is differentiated with respect to the lnD t variable, the elasticity of restricted costs with respect to this proportion term may be calculated: ( 3 1 6 ) "adr " P* + P ^ At the mean value of Zqt this elasticity has an estimated value of 0.0404 and a standard error of 0.3234. Thus, increases in the proportion of total output going to 'far' municipalities are estimated to lead to increases in restricted costs, but this relationship is not statistically significant. Table 3.2 reports the estimated non-normalized own- and cross-price elasticities for the utility's variable inputs. These elasticities are calculated at the mean of the data set using (3.15). Each elasticity's asymptotic standard error is also reported. The estimated price elasticities are interesting for several reasons. All of the own-price elasticities are negative and significant. In addition, the cross-price elasticities demonstrate that the technology of the GVWD possesses a degree of substitutability between input pairs. Indeed, the only complementary relationship is between the materials input and purchased raw water. The estimated restricted cost function also provides information regarding the impact on operating costs of changes in the levels of output and each of the fixed factors. Because of the importance of these statistics to subsequent analysis, they are reported in two ways. Table 3.3 provides values for Ej\ M C J , and Rj at the mean of the data set. Table 3.4 reports the values for MCj and Rj calculated using the data for the 46 four quarters of 1986. Both tables provide standard errors for the estimates. One striking feature of Table 3.3 is seen in the relative magnitudes of the GVWD's short-run average and marginal costs. It is clear that the marginal cost of output is substantially smaller than average cost.6 One potential factor contributing to this result is the declining block structure for the price of electricity that the GVWD consumes. However, electricity's share in restricted cost is only 15 % at the mean of the data set. On the other hand, it is possible that average variable costs are overestimated because of the way in which the labour input is defined. GVWD accounting procedures do not allow for a finer disaggregation than 'total labour employed'. Some portion of this quantity of labour is engaged in administration and capital maintenance - both of which are unrelated to the level of output. Accounting for this bias may lower average costs without necessarily affecting marginal costs. The impact on restricted costs of the fixed factors is also of interest. From Table 3.3 it can be seen that restricted costs rise with increases in the capital stock and fall with increases in the stock of water in storage.7 The former implies that the estimated shadow value on the capital stock is negative, although insignificantly different from zero. Following Kulatilaka (1987), this result suggests that the GVWD is over-capitalized by a substantial margin. There are reasons to believe that the estimate of capital's shadow price accurately indicates over-capitalization. Because most water utilities favour time-invariant pricing rules, peak-period consumption can be expected to exceed the levels that would be optimal under peak-load pricing. Building to meet these excessive demands would lead to over-capitalization relative to the level of capacity needed with optimal pricing. Furthermore, in the instances where a capacity constraint is met, 47 public complaints regarding the rationing of output may compel utility planners to increase capacity. A factor strengthening this tendency stems from the fact that the debt incurred to expand capacity in the current period is pooled with all of the utility's remaining debt from previous expansions. The costs of current expansions (which can be expected to be higher in real terms than previous expansion, if the utility exploits sources of supply in the order of their relative cost to develop) will not be reflected in the current period's prices. A final factor is institutional. Canadian municipal water utilities are run almost exclusively by regional or municipal governments. As a result, they have not received the same regulatory scrutiny that private investor-owned utilities receive. If there is a tradeoff between the implicit rate of return on installed capital and the probabihty of experiencing insufficient capacity during peak periods, the unregulated utility might be expected to choose a level of capacity which reflects a heavier weighting of the latter concern than the former. On the other hand, there are several reasons why the conclusion of overcapacity may not be valid in the case of the operations of a water utility. The first stems from the lumpiness' of the increments to capacity which water utilities install. If capital must be installed in discrete units (the capacity of which might be large relative to the utility's total capacity), then the utility will periodically appear over-capitalized.8 A variant of this argument relates to the cyclical and stochastic character of water demands. If a utility is required to add to its distribution network in order to meet peak flow requirements (including some excess margin for safety), then the system will naturally be underutilized for much of the time. The final explanation acknowledges that water utilities, in fact, supply jointly 48 two distinct outputs: water for fire fighting and water for consumption by households, firms, etc. Water utilities are legally required to have at all times the capacity to supply water for fire fighting purposes (at a specified rate of flow) in excess of expected flows for other purposes. Thus, a utility can have the ex ante optimal level of capacity (based on the regulations concerning fire fighting capacity) but can also appear over-capitalized ex post if the demand for water for fire-fighting is lower than predicted. This regulatory constraint is particularly significant because the reserve capacity must be able to supply water for fire fighting at pressures much higher than those normally required for household or industrial use. This line of reasoning is weakened somewhat by the finding that the estimated value of capital's shadow price is found to be negative over the entire sample period, 1975 to 1986. The shadow value associated with water in storage is found to be positive as predicted and is significantly different from zero. This result suggests that changes in the scarcity of water affect short run costs. The estimated cost elasticity indicates that the one percent fall in the level of stored water necessary to increase variable costs by 2.5 % would represent a two metre drop in lake levels. It has been observed that over the course of an average summer the GVWD's storage lake levels can fall by six to eight metres. This would suggest that short run costs may rise by as much as 10% due only to the fluctuations in water availability. Table 3.4 demonstrates the seasonality of short-run marginal cost and shadow values for each quarter in 1986. Interestingly, the marginal cost of output displays relatively little seasonal variation; it is lowest when the output is lowest (winter), but does not rise significantly during the peak period (summer). In contrast, the shadow value of water in storage rises in the summer quarter as would be expected. 49 Somewhat surprisingly, the shadow value on capital decreases (ie., gets more negative) in the summer period. d. Calculating Long-Run Costs If a peak-load pricing scheme as described in Chapter Two is to be considered, the estimates of short-run marginal costs must be supplemented with information on long-run marginal costs. If the utility is assumed to be in long-run equilibrium, then the long-run marginal cost can be obtained by estimating a long-run cost function and taking its derivative with respect to output. There are two problems with this approach. First, the definition of long-run marginal cost is ambiguous when investment takes place infrequently in discrete units (Mann, Saunders, and Warford, 1980; Protti and McRae, 1980; Bernard and Chatel, 1984). Furthermore, estimating a utility's long-run costs is made difficult by the common industry practice of valuing capital at its historic acquisition costs, rather than its replacement cost. Finally, if the utility overestimates future output growth or follows the industry practice of building capacity in anticipation of future demand growth, then estimates of long-run marginal costs may be biased when a long-run equilibrium is assumed. The second objection to pursuing the neoclassical approach to estimating long-run marginal costs stems from the estimates of the GVWD's short-run shadow value on its capital stock. Even given the complicating factors arising from uncertainty and cyclicity of consumption, the negative shadow value (or more accurately, the shadow value estimate not being different from zero) provides a strong suggestion of over-capitalization. The assumption that the GVWD, or any water utility that has faced 50 recent periods of demand and capacity growth, is in long-run equilibrium may be untenable. In this situation, an alternative procedure may be used to compute a local approximation to a utility's true long run marginal costs (Turvey, 1976). This method measures the total cost of a utility's next planned expansion to its capacity, determines the expected increase to total output attributable to the project and computes the marginal capital cost as the ratio of the two. What is calculated is a point estimate to the average cost of a marginal increase to the utility's productive capacity. The long-run marginal cost that is calculated is the sum of computed marginal capacity costs and the estimated short-run marginal costs at the observed level of output. The most serious shortcoming of this approach is that it requires the analyst to assume that the utility has considered the menu of alternative ways of meeting the forecasted growth in demand and has chosen the cost minimizing form and timing of the expansion.9 This alternative approach uses engineering cost estimates for future capacity expansion projects. It is common for water utilities to conduct long range forecasts of consumption growth and to use them as a basis for their capital spending plans. These plans are then supplemented with five-year spending plans which are prepared in detail for approval by the utilities' directors or regulators. The estimates of the marginal capacity cost presented here are based on the GVWD's five-year capital spending plan for the period 1986-1990. The first step is to compute the present value of the cost of projects that contribute to system expansion. These costs are then annualized at a given discount rate and predicted life of the plant. The average cost of these projects is computed by 51 taking the ratio of the sum of their annualized costs to the annual increase in output attributable to them over their productive life. The application of these steps to the GVWD is summarized in Figure 3.1. The estimates of marginal capacity cost presented in Figure 3.1 appear quite high. Depending on the discount rate chosen, they range from $80 to $133 per 1000m3 of installed capacity (in 1986 dollars). This is in contrast to an estimated short-run marginal cost of approximately $5 per 1000m3 and an output price of $34.8 per 1000 m3. It would appear that the output price represents a midway point between the short run cost of treating and transporting one thousand cubic metres of water and the long run costs of installing the capacity for that output. It should be noted, however, that the seemingly large gap between the GVWD's output price and this estimate of marginal capacity cost is in part due to its policy of valuing capital at its historic acquisition cost, rather than its replacement cost. This section has reported on the estimation of the cost structure of the GVWD. Several interesting features have been found. First, short-run marginal costs are small, especially relative to the price of output and also relative to long-run marginal costs. Second, the restricted technology displays a significant degree of substitutability among variable inputs. Finally, the shadow price estimated for the stock of capital is negative (although insignificantly different from zero). This would suggest that the GVWD is over-capitalized. 3. Econometric Model of the Retailer's Cost Structure The purpose of this section is to report on the methods used to estimate the cost structure of a municipal water utility. Of particular interest are the estimates 52 of the short- and long-run marginal costs of supply. The Vancouver Water Works (VWW) is the municipal water utility whose costs are estimated. The VWW purchases water from the GVWD and delivers it to residential, industrial, commercial and institutional customers. The VWW is also responsible for providing water for fire-fighting purposes. a. The Short-Run Restricted Cost Function It is assumed that the VWW seeks to minimize the costs of supplying a given output level to a given number of consumers. Because the arguments underlying the econometric model of the VWWs costs are the same as those presented in the previous section, they are not repeated here. The VWW is assumed to choose the quantities of variable inputs to minimize costs subject to the constraints imposed by its technology, the level of output and the number of consumers. A translog functional form is used to estimate the VWWs restricted cost function. The form of the cost function and its associated share equations is the same as (3.10) and (3.11). Again, t indexes a time-series of quarterly observations from 1975:1 to 1986:4. The VWW, however, is assumed to choose the quantities of the following variable inputs: supervisory labour (L,), operations labour (LJ and purchased water (W). The VWW is constrained in the short-run by the capacity of its stock of capital (K), the level of output (QJ and the number of connections (QJ. Thus, the estimation model includes the prices of the variable factors, the quantities of the fixed factor and output levels and a time-trend which proxies technological progress. Details regarding the construction of each variable are contained in Appendix 1. 53 b. Results of Restricted Cost Function Estimation Prices and restricted costs are again measured in nominal dollars and all variables are indexed to their representative average values. The VWW restricted cost function and two of the three share equations are estimated as a simultaneous system of equations. An iterative three-stage least squares estimation procedure is employed once additive errors are appended to each equation. The symmetry and linear homogeneity restrictions in (3.12) are imposed prior to estimation. The estimated coefficients and their respective standard errors from the VWWs restricted cost function are reported in Table 3.5. The results of statistical tests that examine the structure of the estimated cost function are reported in Appendix 2. It is interesting to note that the time trend's coefficient is estimated to be positive (although it is insignificantly different from zero). As with the GVWD, there is no evidence of the VWW lowering its variable costs through technological innovation, at least of the sort that can be captured by a Hicks neutral time trend. Table 3.6 reports the own- and cross-price elasticities for the VWWs variable inputs. These are computed using (3.9). The own-price elasticities are all negative and all of the cross-price elasticities are positive (with the exception of purchased water and supervisory labour). Most of the VWWs variable input pairs are substitutes. Table 3.7 reports the estimated quarterly restricted marginal costs and shadow value of capital. The reported values (and their respective standard errors) are calculated using the values of input prices and expenditure levels for 1986. Marginal cost with respect to both output and the number of customers rises in the summer quarter. Furthermore, the restricted or short-run marginal costs associated with 54 output changes are larger than those calculated for the GVWD (in Table 3.4). Table 3.7 provides another contrast to the results reported in the previous section. It can be seen here that the estimated shadow price on the stock of capital is positive. If it could be shown that these estimated values were close to the utility's user cost of capital, then it could be argued that there is less reason to believe the VWW is over-capitalized. Thus, in order to test this hypothesis, a user cost of capital must be calculated. Jorgenson (1963) derives the following formula that may be used to calculate the user cost of capital:10 j (3.17) qK - (r + o)-P r It can be seen that the annual price of capital goods is a function of three parameters: the annual discount rate (r), the annual rate of depreciation (5), and the acquisition cost (PK). It is necessary to obtain estimates for the 1986 values of these parameters in order to consider whether the VWW is over-capitalized. The VWWs records indicate that in 1986 the only major capital expenditure is for a storage reservoir with a rated capacity of 4.8 million gallons at a total cost of $2.2 million. This translates into an acquisition cost of approximately $286.83 / 1000m3 of storage capacity. It is assumed that the interest rate facing the VWW may be proxied by the average yield on long-term debt held by the government of Canada. Finally, average depreciation rates for water utilities' capital structures may be obtained from Statistics Canada (Statistics Canada Catalogue 13-568). These data may be used in equation (3.17) to compute annual user cost figures and then may be converted into quarterly values. The last row in Table 3.7 reports these values. 55 A test may be conducted to determine whether to reject the hypothesis that the estimated shadow value equals the calculated price of capital. If a t-test is used then the estimated shadow values and their standard errors imply that at the 95% confidence level the hypothesis of overcapitalization may be rejected for periods 2 and 4 but not rejected for periods 1 and 3. It would appear, then, that the evidence indicates that the GVWD is overcapitalized while the VWW is not (or not to the same degree). Over the sample period the Greater Vancouver Region has experienced substantial extensive growth while the City of Vancouver grew at a much slower rate. The different rates of growth implied different rates of expansion for the two utilities. The regional utility added considerably to its storage, treatment and distributive capacity while the municipal utility did not. This observation may explain the difference in results and also suggest that the distortions introduced into capacity decision making by current utility pricing practices are most significant for utilities facing rapid growth in demand. c. The Long-Run Cost Function Under the assumption that the VWW is in long-run equilibrium with respect to the employment of its factors of production, a long-run cost function may be specified and estimated. This function will be identical to the translog restricted cost function outlined above in equation (3.10) except for three features. First, the price of capital, rather than its quantity, is included as an explanatory variable. Second, the dependent variables are the natural log of expenditures on labour, purchased water and capital. Finally, a capacity utilization variable is included as an explanatory variable. This variable is added because it is expected that the VWW will 56 be at capacity on average but that it will experience short-run cyclical fluctuations in capacity utilization. Details regarding the construction of these variables are found in Appendix 1. The restrictions of the estimated system of equations (the cost function and three of its four share equations) and the estimation procedures are as discussed earlier. Table 3.8 reports the estimated coefficients of the translog long-run or unrestricted cost function. The time index remains positive an insignificant, suggesting a lack of cost-reducing technological innovation over the period. Table 3.9 reports the own- and cross-price elasticities for the utility's inputs. Values are calculated at the means of the data set. All of the own-price elasticities are negative and significant, while most of the cross-price elasticities indicate that the inputs are substitutes. Capital and purchased water, however, are complements. Table 3.10 provides the calculated estimates of the VWWs unrestricted marginal costs with respect to output and the number of connections. The values are reported for each quarter of 1986. The marginal costs of output changes are higher than the values reported in Table 3.7. However, both sets of marginal costs display the same seasonal variation. Furthermore, the VWW long run marginal costs appear to be somewhat smaller than those calculated for the GVWD. This section has reported on the procedures used to investigate the cost structure of two representative water utilities. Most importantly, estimates of the utilities' marginal costs are derived. These estimates will be used to simulate the impact of moving to efficient prices as discussed in Chapter Four. C. ESTIMATES OF AGGREGATE WATER DEMANDS 57 The purpose of this section is to report on the methods used to estimate the structure of demand for potable water by each of the major water using groups. For most municipalities it is conventional to classify consumers as residential, industrial, or commercial/institutional; that scheme is followed here.11 In estimating each group's demand for water it is expected that two things will be learned. First, the estimated demand functions indicate the value which users assign to consumption. Second, the demand equations show the responsiveness of each group's desired consumption level to changes in the level and structure of prices. 1. Residential Water Demands Households use water for a variety of purposes. These uses may be categorized as either outdoor uses (lawn watering, car-washing) or indoor uses (cleaning, cooking, waste removal). North American households use large amounts of water. The average North American household uses 1.7 m3/day while the average European household uses only 0.5 m3/day (Hanke and de Mare, 1984). Another characteristic of residential water use is its marked cyclicity. These cycles are daily (with peak consumption rates in the early morning and early evening) and annual (with peak consumption during summer months). The large majority of residential water consumers are supplied by a public water utility. As such, the decision of whether to connect to a public supply network or to remain self-supplied is relevant only for a very small number of rural consumers. Water related expenditures vary significantly across Canada (Wasny, 1986), but are 58 usually composed of an annual connection charge (unrelated to the level of consumption) and in some instances a second expenditure related to the quantity consumed. Per unit prices are usually constant, but in some instances decrease with the quantity consumed. In the last twenty years, a relatively large number of studies have attempted to characterize the structure of residential water demands. While these studies have used different types of data and different specifications, their common goal has been to obtain estimates of the price and income elasticities of demand. Howe and Linaweaver (1967) represent a good example of early attempts to estimate residential water demands. Aggregate cross-sectional observations are used to estimate single equation demand functions. The dependent variable is the quantity of annual consumption by all residential consumers in a given municipality divided by the number of households. Explanatory variables include average price, average income per household, environmental variables and a vector of variables identifying the households' socio-economic characteristics (for example, number of residents per household). The price variable is constructed as the average over the marginal price blocks contained in each utility's price schedule. Howe and Linaweaver report that the average price and income elasticities are estimated to be inelastic, but are significantly different from zero. Taylor (1975) argues that using an average price specification when consumers face a non-linear price schedule will lead to a simultaneity bias being introduced into the estimation procedure. This is because the computed price depends upon the level of observed consumption. As a result, the price variable may be correlated with the equation's error term. This may lead to biased coefficient estimates (Judge et. al. 59 1980, p. 531). As an alternative, Taylor suggests a two-part specification for price: the marginal price and a difference variable defined to equal the difference between total expenditure on water and the level of expenditure that would have prevailed if only the marginal price had been charged. This two-part price is meant to capture the structure of the price schedule. Taylor predicts that the difference variable will have an estimated coefficient equal, but opposite in sign, to the coefficient on the income variable. Following the Taylor proposal a large number of papers attempted to test its predictions (Foster and Beattie, 1979; Billings and Agthe, 1980; Terzas and Welch, 1982; Polzin, 1984; Williams, 1985). While these studies use different price specifications they have several common features. They typically use aggregate data. Further, they assume residential water demands to have either linear or log-linear forms. Despite these difficulties and the differences in the specification of the price variable, the studies are consistent in finding estimated average price and income elasticities with values between zero and one in absolute value.12 However, just as consistently the Taylor difference variable's coefficient fails to have either the predicted sign or magnitude. Some authors have used other econometric methods to cope with the presence of simultaneity bias. Jones and Morris (1984) suggest an instrument be constructed from the information contained in the sample's price schedules and that this variable be used to proxy the price of water. Alternatively, Chiccoine, Deller, and Ramamurthy (1986) implement a three stage least squares (3SLS) method where the structure of the price schedule and the demand equation are simultaneously estimated in a system of equations. Interestingly, the 3SLS and OLS techniques yield little difference in 60 coefficient estimates. Despite the recent extensions in econometric technique, the quality of the available data sets remains the most significant constraint facing researchers. Few time-series data sets contain monthly household-level observations and, as a result, little is known about the relative size of the price elasticities of summer (outdoor) and winter (indoor) water use. Furthermore, most price variation observed in these data sets stems from either deflating nominally rigid prices or from using regional prices. With these caveats in mind, the weight of evidence supports the conclusion that residential water demands are characterized by long-run price and income elasticities with average values between zero and one in absolute value. One other feature of the empirical water demand literature is significant. While a substantial amount of effort has been extended to estimate price elasticities, little has been done to use the estimated demand equations to determine consumer valuation of water.18 This information would obviously be critical to any analysis of the effects and desirability of reforming water prices. There are two objectives for the econometric model of water demands developed in this thesis. First, it must yield estimates of the market demand's responsiveness to price changes (including possibly peak-load prices). Second, it must provide information regarding the market's valuation of consumption in the range of observed prices. The development of such a model is discussed next. It is assumed that aggregate residential water demand per time period is a function of the number of households and the demand for water by each household. Each household is seen to consume water, Wh, and a composite commodity, representing all other goods. Tastes are assumed to be identical across households 61 but household characteristics (summarized by the vector of attributes, ZJ differ. Thus, each household seeks to maximize utility given in equation (3.18) (3.18) subject to exogenously determined prices and level of household income, Y h . Thus, differences in imply that households will enjoy different levels of utility even if W h and Xh are the same. Maximizing household utility subject to the budget constraint yields the household's uncompensated demand for water: where P w and P x are the prices for water and the composite commodity, respectively. As the function W(0 is expected to be homogeneous of degree zero in prices and income, its arguments may be normalized by dividing by Px. It is assumed that the distribution of household water demands is identical to the distribution of household incomes." Thus, if H is the number of households and f(Yh) is the probabihty distribution of household incomes, then aggregate residential water demand, Q ,^, is given by the following expression16: (3.19) wh - ** r» z*) (320) 62 (3.21) Qw - <?„(P„ P„ f¥ fh, 6) - " • / 8) Z |>^; 0 * * z where 6 is a vector of parameters entering in the income density function. If the impact of alternative pricing rules is to be considered, it is necessary to estimate the functions W(0 and f(Yh, 0). Given the interests of this research, an ideal data set for estimating the demand relationship (3.21) would provide quarterly household-level observations on consumption, price, income and other relevant variables. Unfortunately, such a data set does not exist and would be very difficult to construct. The data set that is employed here, however, does have several attractive features. The VWW charges its residential consumers only an annual charge for water use. As a result, data from Vancouver cannot be used to estimate residential water demands. As an alternative, data are collected for residential consumption in several cities close to Vancouver. In 1987 Environment Canada compiled a data set concerning residential water consumption in three British Columbia municipalities: Victoria, Saanich, and Oak Bay (Gai, 1988). While residential water use is metered in these cities, the method of billing that the local water utilities employ implies that reconstructing household-level water purchase records would have been prohibitively expensive. As a result, the data set contains observations for a "representative" household which are constructed simply by dividing aggregate recorded residential water purchases per time period by the number of residential customers. The data set, however, does attempt to distinguish between indoor and outdoor 63 water use. This is done by assuming that all residential water used between October and April is for indoor use. Total water use during the period May through September is composed of indoor water use (assumed to continue at a rate equal to the average rate over the October-April period) and outdoor water use (defined as the difference between observed purchases and imputed indoor use). Thus, for each municipality in a given year, the data set provides two observations on residential water consumption: the average monthly indoor water use of a "representative" household between October and April and the average monthly outdoor water use for the household between May and September. The data set also contains information regarding the explanatory variables entering the demand equation. Observations on the nominal marginal price schedule of water, nominal average household income, average number of residents per household, and two environmental variables (precipitation and temperature) are included. The data set then represents a pooled, time-series, cross-sectional survey on residential water consumption and the factors affecting it. The construction of individual variables and the distribution of observations across the three municipalities are detailed in Appendix 1. In order to estimate the summer and winter residential water demand equations, a specific functional form must be chosen. One approach is to assume that the unobservable utility function (3.18) has a specific functional form and to derive the implied form for the demand equation. This approach ensures that the demand equation satisfies the integrability condition, but also implies that if the functional form chosen for the utility function is flexible (in the sense defined by Diewert (1974)), 64 then the derived form of the demand equation to be estimated will not be flexible. An alternative approach, proposed by Hausmann (1981), is to choose a flexible functional form for the demand equation. This ensures that the functional form used to estimate the demand equation is general enough to produce a local second order approximation. Furthermore, the analyst can test whether the estimated demand equation satisfies the necessary integrability conditions. A translog functional form is adopted for the estimation of the summer and winter residential water demands. Both equations have the following form: 3 (322) In Wh - o0 + aplnP + aylnYh+ £ a, lnZy 3 + anlnPlnYh + £ InPlnZ^ + E E The dependent variable is the log of water consumption per household per month. The term is a 3-dimensional vector of explanatory variables: temperature, precipitation, and number of people per household. The variable P is an instrumental variable representing the price schedule for residential water and Y h is average after-tax income per household. The method used to construct the instrumental variable is detailed in Appendix 1. Both P and Y h are normalized by a regional price index and, thus, are measured in real dollars. As a result of this normalization, equation (3.22) 65 is estimated with homogeneity of degree zero imposed. Finally, linear additive error terms are appended to (3.22) prior to estimation. Both summer and winter demand equations are estimated using ordinary least squares (OLS). Table 3.11 reports the estimated coefficients for both demand equations and each equation's statistics. The equation statistics suggest that each equation possesses relatively good explanatory power. It is, however, rather difficult to interpret the estimated coefficients in their original form. By taking the log derivative of (3.22) with respect to price and income, expressions for the price and income elasticities are derived: (3.24) I r -dlnWk dlnYu - a Table 3.12 reports the values of these elasticities (and their respective standard errors) for the summer and winter demand equations. The elasticities are computed by substituting average values of the variables into (3.23) and (3.24), along with the estimated coefficient values. The elasticities' values conform to prior expectations. The price and income elasticities are negative and positive, respectively, and are between zero and one in absolute value. Furthermore, the summer elasticities are larger in absolute value than the winter elasticities. This finding parallels that of Howe and Ldnaweaver 66 (1967). Once the water demands of the representative household have been estimated, it remains to establish the structure of the distribution of those demands. Recall from equation (3.21) that the distribution of household demands is assumed to be identical to the distribution of household incomes, flyh,0). It is necessary to choose a specific functional form for this distribution and to estimate the parameters of the distribution. The lognormal probabihty distribution function is chosen to represent the distribution of household incomes in Vancouver. This distribution has been used frequently in past studies of income distribution (McDonald and Ransom, 1979; Harrison, 1982). Its primary advantages are that it is skewed to the right for most parameter values and, unlike the Pareto distribution, it is not constrained to be downward sloping over all income levels. Assume that yh is a random variable distributed lognormally with mean a and variance <|>2. Then, define Y h = log(yh) for y h > 0. It can be shown (Aitchinson and Brown, 1957) that Y h is normally distributed with mean u and variance o2. The density function for yh is defined by the following formula: (325) /Cy*) _ ; yhy 0 and the mean (a) and variance (<J>2) of this distribution may be related to the parameters u, o 2 as follows: 67 (3.26) meaniyj - a - exp (a + ^-o2) w(yj) - 4>2 - a V , v2 - «p(o2} - 1 The parameters p and o2 can be estimated from data on the distribution of household income levels in Vancouver. The 1981 Population Census conducted by Statistics Canada provides the necessary data. For a given set of income ranges (for example, $10,000 - $14,999; $15,000 - $19,999; etc.) the number of households and the average household income in each income range is provided.16 Estimates of p and o2 are obtained using the following formulae: (3.27) fl - £ yr •/„ (3.28) d 2 - £ (A - Jf r-1 where r indexes the income range, yr is the average income in each range and f, is the number of households in the rth range relative to the total number of households. A A The computed estimates for p and o2 are 10.542 and 0.4329, respectively. A A The estimated parameters p and o2 are then used to yield estimates of a and <t>2. These parameter values determine completely the estimated distribution of household incomes in Vancouver.17 Aggregate residential water demands are then obtained by applying the estimated water demands for the representative household 68 and the estimated distribution of household incomes to equation (3.21). 2. Industrial Water Demands Firms use water for a variety of purposes. These include cooling intermediate inputs, producing steam, moving intermediate inputs and inclusion in the firms' final output (eg., in beer production).18 Water using firms may be supplied either by a public utility or self-supplied (that is, relying on their own pumping facilities to withdraw water from the environment). The distribution of water use across industries is quite uneven. A small number of industries with water-intensive production processes (eg., petro-chemicals, pulp and paper and metal refining) account for most recorded industrial water use in Canada.19 Most firms using these water-intensive processes are self-supplied. For a manufacturing firm connected to a public utility there are several sources of expenditures associated with water use. There are the utility's annual fees and per unit charges that must be paid for water delivery and sewage removal. There are also internal expenditures associated with any treatment of the water prior to use (for example, minerals must be removed if the water is used to produce high pressure steam) and any treatment of water prior to discharge. However, few firms in North America use purchased water more than once, and, as a result, recycling expenditures tend to be very small. Finally, expenditures may be required for maintaining the capital stock associated with water use. Despite the ubiquity of industrial water use there are surprisingly few studies of the structure of firms' water demands. This situation stands in marked contrast to the exhaustive analysis applied to investigating industrial employment of capital, 69 labour, and energy inputs. There are two recently published studies of water use in American manufacturing. Grebenstein and Field (1979) estimate a single translog cost function for the American manufacturing industry using state-level cross-sectional observations from 1973. Water is included as an input and two different price estimates are used.20 Water's cost share is estimated to be between 1.2 % and 1.9 % and, depending on which price for water is employed, the average value for water's own (output constant) price elasticity is estimated as -0.326 or -0.801. Water and labour are found to be substitutes, while water and capital are complements. Babin, Willis, and Allen (1982) estimate translog cost functions for several U.S. manufacturing industries while including water in their estimation model. Data are again a cross-section of state level observations. Water's estimated mean cost share exceeds 1 % in only 4 of the 8 industries studied and water's own-price elasticity ranges from 0.14 (Food and Kindred Industries) to -0.66 (Paper). In contrast, Williams and Suh (1986) represent a recent attempt to examine industrial water demands from a different perspective. Single equations are estimated to represent aggregate industrial water demands for firms connected to public water supplies. The data are a cross-sectional survey where each observation represents aggregate industrial water use in one municipality. Aggregate industrial demands are modeled to depend on water's price, each municipality's aggregate value added, and the number of industrial water connections in each city. The demand equations are estimated using OLS and a variety of proxies for water's price (eg., the average over the utility's marginal price blocks, the first marginal price, etc.). Depending upon the specification of the price variable, the price elasticity's average 70 values range from -0.4376 through to -0.9735 and the demand elasticity with respect to value added ranges from 0.1762 to 0.2962. There is one principal shortcoming that emerges from these studies of industrial water demands. The fact that industrial water prices are almost always structured as declining price schedules is not considered. This suggests that may be a problem of identification surrounding the estimation of the equation: it is not clear whether a single demand curve is traced out over alternative price schedules or the estimated negative coefficient on price is attributable to the negative slope of the industrial price schedule. In order to assess the impact on the industrial sector of the VWW changing its pricing rules, it is necessary to estimate an aggregate water demand curve for that sector. If the industrial sector minimizes the costs of producing a given level of aggregate output, then its technology may be represented by a cost function of the form: (3.29) C, - C,(P, <?,) where P is a vector of input prices and Qj is the sector's level of output. Furthermore, if it is assumed that industry's technology is separable in its water use from its other inputs, then the conditional aggregate industrial water demand function will be of the general form (Blackorby, Primont and Russell, 1978): (3.30) Qm - /(P*. <?,) 71 where Q w is the aggregate quantity of water desired by industrial customers, P w is the price of water (or price schedule) and Qj is the level of aggregate output produced by industrial customers. It is assumed that (3.30) can be represented by a translog functional form. The reason for choosing a specific functional form for the demand equation (rather than specifying the form of the aggregate cost function and deriving the resulting demand system) follows Hausman (1981), as discussed in the previous section. The data set used to estimate the aggregate industrial water demand stems from a survey conducted by Environment Canada in 1982. The survey asked approximately 3000 Canadian manufacturing firms for information regarding their water use and water-related expenditures in 1982. The responses from British Columbia were extracted from the survey and were used to generate aggregate regional observations on industrial water use. Appendix 1 describes the methods for constructing these aggregate consumption and price observations. While this data set has the advantage of being based on firm-level consumption and price observations, it does have two shortcomings. First, it has no information regarding input use other than water. As a result, the relationships between water and other inputs cannot be established. Second, the observations are annual. This is a drawback because it was hoped to use the estimated demands to predict the effect of moving to seasonal pricing. There is, however, some evidence that industrial water use displays little seasonal variation. Tate and Scharf (1985), in reporting on the results of the Environment Canada survey, indicate that total water use by manufacturing firms in 1982 was divided across the four quarters of the year in the following proportions: 24.5 %, 25.8 %, 25.0 %, and 24.6 %. Hence, there seems to be 72 no seasonal pattern to aggregate industrial water use. The industrial demand equation is estimated with three principal arguments: the price of the water input (P), the level of aggregate output (V) and a proportion variable (F). The last variable measures the proportion of publicly-supplied water consumption for each aggregate observation (the remainder of the firms being self-supplied). The form of the translog water demand equation to be estimated is the following where the subscript i denotes the ith regional observation: + apvInP/nVl + aPflnP/nFi + ay^nV^nFt Where Qj is defined as the aggregate water consumption by the manufacturing sector in the ith region. Once an additive error term is appended, the aggregate industrial water demand (3.31) is estimated using OLS. 2 1 The price variable is represented by an instrument. Details regarding the construction of the instrument are in Appendix 1. The results of the estimation are reported in Table 3.13. The price and output elasticities are derived from the estimated form of (3.31) by taking the log derivatives with respect to the price and output terms, respectively: (3.31) lnQt - o0 + ajtnPt + a}JinVi + arlnFi (3.32) *1|P dlnQ, dlnPt dbiQ, dlnVt ay + *yJnVt + aFvtnPi + aYJjtnFi 73 The average values of these elasticities and the standard errors of these average values are also reported in Table 3.13. Both elasticity estimates possess the predicted sign and are significantly different from zero. The magnitude of the price elasticity is relatively large given water's small cost share in most manufacturing processes and given the results of other researchers. A possible explanation for the size of the estimated price elasticity stems from the composition of the data set. The Industrial Water Use Survey collected responses only from firms using in excess of 4545 cubic metres (one million gallons) of water per year. Hence, the proportion of large water users (i.e., those firms with water-intensive production technologies) most likely exceeds the proportion found in the population of firms. This is significant as it would be expected that firms with water-intensive technologies (and relatively large water cost shares) would, ceteris paribus, exhibit larger demand elasticities. Finally, the elasticity of aggregate water demand with respect to the proportion variable is not reported in the Table, but is found to have an average value of-0.6044 and a standard error of 0.1352. This suggests that firms connected to a public water supply system tend to be smaller water users than self supplied firms. D. CONCLUSION This chapter is concerned with studying the structure of the costs of water supply and the structure of water demands. For each of these topics, after the extant literature is reviewed, the econometric models are established and data are discussed. Results are then presented and evaluated. The most important sets of results are the 74 estimated marginal costs of supply and price elasticities of demand. These estimated parameters play critical roles in the simulation program that evaluates alternative pricing rules. This program is discussed in the next chapter. 75 Table 3.1:--GVWD Restricted Cost Function Parameter Estimates Var. Coeff. St Err. Var. Coeff. St Err. 495.0400 1119.5000 Pww 43.3470 95.7470 4.6018 3.5952 P Q K 0.1778 0.3617 -1.7590 2.6868 P Q W -0.6153 5.7327 ds** -1.4348 0.6883 P K W 1.1831 3.7511 a M -0.4081 1.7721 P D -4.0295 6.1507 -0.0729 0.0286 P Q D 0.8251 1.2583 < * L L 0.0109 0.0196 Y E Q 0.1654 0.0512 Oss 0.0466 0.0121 Y E K 0.0240 0.0423 A M M 0.0148 0.0267 Y E W -1.1795 0.7452 0.0066 0.0180 Y L Q -0.1842 0.0384 U E S 0.0135 0.0072 Y L K -0.1705 0.0347 O E M * * 0.0528 0.0185 Y L W 0.8785 0.5566 « L S -0.0047 0.0109 Y M Q -0.0358 0.0253 A L M -0.0128 0.0288 Y M K 0.1028 0.0274 <W* -0.0549 0.0135 Y M W 0.0672 0.3677 P Q 3.7067 27.728 Y S Q 0.0546 0.0099 P K -9.4391 17.9470 Y S K 0.0436 0.0114 Pw -204.5900 462.2100 Ysw 0.2338 0.1428 PQQ -0.2027 0.4142 Pr 0.0046 0.0027 P K K 0.7316 0.8579 Notes: *• Variable inputs are electricity (E), labour (L), purchased water (S) and materials (M). Fixed inputs are capital (K) and water in storage (W). Other variables are output (Q), Distance (D) and Time (T). Estimates statistically significant at cc=0.10 are denoted by * and those significant at a=0.05 by **. 2 The cost function is estimated as part of a system of equations. The R 2 values for each of the system's equations are the following: cost function, 0.9572; material cost share, 0.6136; energy cost share, 0.5502; labour cost share, 0.7806. Table 3.2:-GVWD Input Price Elasticities 76 E L S M E -1.629" 0.670" 0.221" 0.738" (0.271) (0.169) (0.069) (0.174) L 0.127** -0.431** 0.071 0.223" (0.031) (0.035) (0.197) (0.051) S 0.282** 0.481" -0.334" -0.126 (0.089) (0.132) (0.150) (0.519) M 0.293** 0.498" -0.404 -0.669** (0.071) (0.109) (0.614) (0.102) Notes: 1 Elasticities are calculated using average values for input prices and actual shares. The variable inputs are electricity (E), labour (L), purchased water (S) and materials (M). Standard errors are reported in parentheses. 2- Estimates statistically significant at ct=0.10 are denoted by * and those significant at a=0.05 are denoted by ". 77 Table 3.3:-GVWD Fixed Factor Statistics: Average Values Cost Average R j 1 Elasticity Cost Output 0.321" 7.509 2.414" (0.103) (0.772) Capital 0.447 43.998 -19.661 (0.411) (19.288) Water in Storage -2.488* 913.200 2272.500* (1.522) (1390.100) Proportion2 0.0404 (0.3234) Notes: *• Rj is the first derivative of the restricted cost function with respect to each of the variables, j = Q,K,W. For output, R Q measures the short-run marginal cost; for the two fixed inputs (water in storage and capital), R ^ and R g measure the shadow value of an additional unit of the fixed input. 2 Because the Proportion variable is calculated as an index, it is not possible to compute its average and marginal costs. 3- Estimates statistically significant at a=0.10 are denoted by * and those significant at a=0.05 are denoted by ". 78 Table 3.4:--GVWD Fixed Factor Statistics: 1986 Values M C Q RK Rw Winter 4.193" -37.34* 3029.2 (1.320) (22.55) (2745.0) Spring 5.883" -36.14* 2117.8 (1.714) (23.21) (2898.2) Summer 3.021" -51.72* 9647.8" (1.940) (35.27) (5565.3) Fall 4.492" -40.57* 5434.2* (2.199) (27.62) (3837.0) Notes: l- All variables are measured in 1986$ and are calculated using 1986 values for the explantory variables. M C Q is the short-run marginal cost and Rx and Rw are the shadow values of capital and water in storage. 2 Estimates statistically significant at a=0.10 are denoted by * and those significant at oc=0.05 are denoted by ". 79 Table 3.5:--VWW Short-Run Cost Function Parameter Estimates Var. Coeff. St Err. Var. Coeff. St Err. -364.3000 630.6100 P K K * 6.0906 1.4480 «w 0.9845 1.5443 P Q W Q C -2.2547 4.0414 -1.4293 1.8979 P Q W K 0.1497 0.2985 ** 1.4448 0.4833 P Q C K -22.0400 10.9190 0.0742 0.0069 YwQw 0.2508 0.0119 •• a L o L o -0.0929 0.0077 Y W Q C -0.2360 0.3586 0.0187 0.0074 Y W K -0.1118 0.0321 •* OwLo -0.0928 0.0041 YLOQW -0.2073 0.0145 Owl* 0.0187 0.0074 YLSQC 0.3900 0.4409 *• «LoLs 0.0743 0.0056 ** Y L B K 0.2109 0.0392 P Q W 9.3892 17.4630 *« YloQw -0.0435 0.0036 P Q C 73.8880 297.7300 YLOQC -0.1541 0.1122 P K * 71.6440 44.2020 •* YLOK -0.0991 0.0110 P Q W Q W 0.1916 0.0995 P T -0.49E-04 0.0065 P Q C Q C 9.0847 71.6990 Notes: 1. The subscripts Lo, Ls, and W denote the prices of variable inputs operations labour, supervisory labour and purchased water, respectively. The subscripts Qw, Qc, and K denote output, number of connections and capital stock, respectively. T is a time trend. Estimates statistically significant at a=0.10 are denoted by * and those significant at ct=0.05 are denoted by ** 2 The cost function is estimated as part of a system of equations. The R 2 values for each of the system's equations are the following: cost function, 0.9685; operations labour cost share, 0.8535; water cost share, 0.9292. 80 Table 3.6:--VWW Restricted Cost Input Price Elasticities L 0 L, W L 0 -0.820** 1.251** 0.445*' (0.189) (0.636) (0.147) L, 0.270** -0.700** -0.096 (0.137) (0.084) (0.082) W 0.549** -0.549 -0.349*' (0.182) (0.469) (0.136) Notes: *' These are variable input price elasticities and are calculated using average values for input prices and actual shares.The variable inputs are operations labour (Lo), supervisory labour (Ls), and purchased water (W). Standard errors are reported in parentheses. ** Estimates statistically significant at a=0.10 are denoted by * and those significant at ct=0.05 are denoted by ** 81 Table 3.7:~VWW Fixed Factor Statistics: 1986 Values VariableXQuarter1 Winter Spring Summer Fall SRMC (QW) 4 . 3 5 " 6 . 6 1 " 8.94" 6 .22" (2.73) (3.48) (2.79) (2.61) SRMC (QC) 62.05 82.73 324.86 152.19 (256.61) (273.72) (432.23) (310.64) RK 13 .33" 17 .12" 10.93 16 .17" (6.68) (7.87) (10.62) (7.40) P K 2 9.42 8.89 8.81 8.87 Notes: 1 ' All figures are measured in 1986 dollars. Standard errors are in parentheses. SRMC (QW) is the short run marginal cost with respect to output; SRMC (QC) is the short run marginal cost with respect to connections; RK is the shadow value on the stock of capital. *• P K is the quarterly user cost of capital. s- Estimates statistically significant at a=0.10 are denoted by * and those significant at a=0.05 are denoted by " 82 Table 3.8:--VWW Lone-Run Cost Function Parameter Estimates Var. Coeff. St Err. Var. Coeff. St Err. aO" 479.0100 235.4500 PQW -5.4687 8.6850 aW 0.3587 2.1137 PQC** -203.8900 96.8890 au -1.3123 1.6112 PQWQW -0.1055 0.2775 •* 0.4407 0.2552 PQCQC 43.2240 19.9020 O K 1.5129 1.3603 PQWQC 1.3150 2.0843 *• 0.1806 0.0072 YwQw 0.1441 0.0233 " L o U * * -0.1370 0.0055 YWQC -0.1701 0.4589 ** 0.0166 0.0027 •* TLOQW -0.0200 0.0024 ** A K K -0.0584 0.0099 YLOQC -0.0634 0.0553 a W L o 0.0023 0.0030 YLsQw -0.0360 0.0190 *• -0.0217 0.0081 YLBQC 0.3860 0.3499 *• -0.1613 0.0049 YKQW -0.0880 0.0149 OLoLe" -0.0409 0.0055 YKQC -0.1525 0.2951 0.0220 0.0030 reu -0.3948 0.8956 ** <*UK 0.1977 0.0088 O Q U 0.1330 0.1763 Pr 0.0010 0.0015 Notes: 1. The subscripts Lo, Ls, W,and K denote the prices of variable inputs operations labour, supervisory labour, purchased water and capital, respectively. The subscripts Qw and Qc denote output and number of connections. T is a time trend. Estimates statistically significant at oc=0.10 are denoted by * and those significant at a=0.05 are denoted by** . 2 The cost function is estimated as part of a system of equations. The R2 values for each of the system's equations are the following: cost function, 0.6613; supervisory labour cost share, 0.7223; water cost share, 0.4800; capital cost share, 0.6269. 83 Table 3.9:--VWW Long-Run Cost Input Price Elasticities L e L . K W L o -1.334" -0.549" 0.722" 0.167" (0.229) (0.105) (0.216) (0.027) L . -0.119" -0.629" 0.106* 0.059 (0.023) (0.052) (0.073) (0.084) K 1.236" 0.833* -0.733" -0.094" (0.369) (0.572) (0.241) (0.012) W 0.209" 0.344 -0.129 -0.097" (0.034) (0.484) (0.107) (0.024) Notes: 1 Elasticities are calculated using average values for input prices and actual shares. The variable inputs are operations labour (Lo), supervisory labour (Ls), capital (K), and purchased water (W). Standard errors are reported in parentheses. 2 Estimates statistically significant at a=0.10 are denoted by * and those significant at a=0.05 are denoted by ** 84 Table 3.10:-Long-Run Marginal Costs. VWW (1986. $/1000m8) VariableXQuarter Winter Spring Slimmer Fall Output 53.89** 71.05" 85.83" 55.15" (15.85) (19.07) (12.95) (14.16) Connections 471.36* 383.51 591.88* 541.72" (320.43) (293.39) (365.17) (320.45) Note: 1 Estimates statistically significant at a=0.10 are denoted by * and those significant at a=0.05 are denoted by ", 85 Table 3.11:~Residential Water Demands Estimated Coefficients Coefficient Summer Winter Est. St. Er Est. St. Err Op -44.746 97.375 -18.048 17.432 -48.472 83.022 -11.007 28.736 % -0.254 184.110 42.416 62.359 OR 11.042 26.532 3.705 10.552 Or 360.390 1152.800 94.800 222.0080 CCpp -9.801** 6.687 0.715 2.131 3.611 4.821 1.287 2.818 -25.517 33.729 5.521 19.071 ORB -1.056 0.353 0.425 0.590 O n . -99.498 275.320 -31.298 60.897 OpY 3.832** 1.879 -2.119** 1.237 C P N 8.975 13.840 -2.474 5.960 OPE -2.074** 1.145 -0.305 0.420 CCpr 10.601 22.149 10.225** 4.467 « Y N -3.476 5.285 8.367** 4.891 «YE 0.854** 0.497 -0.510* 0.359 -0.548 11.594 0.189 3.085 1.669 1.400 0.566 0.885 OJJ T 4.216 40.013 -30.903** 9.418 Ct H T -2.896 6.459 0.540 2.522 Or 0.013 0.018 0.075** 0.026 Oo -395.260 2508.500 -106.160 409.570 Dx -1.119** -0.385 1.017 0.868 D 2 -0.432** -0.203 0.675* 0.420 R 2 0.7941 0.7465 F 577.7810 2042.7710 Notes: *• The subscripts P, Y, N , R, T denote price, income, number of people per household, rainfall, and temperature, respectively. D x and D 2 are dummy variables which identify observations coming from Oak Bay and Victoria, respectively. 2- Estimates statistically significant at a=0.10 are denoted by * and those significant at a=0.05 are denoted by **; 86 Table 3.12:--Residential Water Demands Elasticity Values Summer Winter Est. St. Err Est. St. En-Price -0.6487 1.4390 -0.0137 0.6360 Income 0.9097 1.2931 0.5515 0.8354 Notes: 1 These elasticities are calculated by substituting the average value of each explanatory into equations (3.23) and (3.24). 2 Estimates statistically significant at a=0.10 are denoted by * and those significant at oc=0.05 are denoted by ** Table 3.13:--Industrial Water Demands 87 Coefficient Estimate Standard Error a P S-pv So 3.217 9.201 -3.134 7.657 2.594 2.001 -4.495 2.003 0.023 0.371 -0.158 0.106 0.739 0.832 -0.500 0.192 -0.071 0.108 31.785 82.404 R 2 = 0.547 F = 139.12 Estimated Elasticities Elasticity Estimate Standard Error Price -1.913** 0.644 Output 0.785** 0.278 Notes: L The subscripts P, V, F denote the variables price, output and proportion, respectively. The elasticities are calculated using the average values of the data. 2 Estimates statistically significant at a=0.10 are denoted by * and those significant at cc=0.05 are denoted by **: 88 Figure 3.1:-GVWD's LRMC From Enrineering Cost Estimates 1. Present Value of Capacity Expansion Projects. 1986-1990 PV K (i=0.10) = $23,005 Million PV K (i=0.05) = $25,498 Million 2. Annualized Capital Costs Projects are assumed to have a 50 year lifetime. A K (N=50,i=0.10) = $2,319 Million A K (N=50,i=0.05) = $1,403 MiUion 3u Expected Addition to Output Forecasted growth in output is 1.2 % per year over 1986-1990. Annual Increment to total output over life of plant AQ = 17.48 Million m s 4^  Average Cost of Marginal Expansion of System MCC (i=0.10) = Ag/Aq = $0.133/m3 MCC (i=0.05) = AR/AQ = $0.080/m8 Note: l- All dollar figures are measured in constant 1986 dollars. m s = one cubic metre, i = discount rate, N = the life of the project. 89 Endnotes 1. One cubic metre equals approximately 220 Imperial gallons. 2. Employing efficient prices may be expected to influence the average utility's capacity in other ways. Using prices that are spatially differentiated may influence the pattern of development in a city (eg., slower growth in the suburbs) and, thus, influence the pattern of expansion for the distribution system. Furthermore, the composition of aggregate demand may be influenced by a change in relative prices (eg., residential versus industrial) and this might have some effect on system design. 3. Swallow and Marin (1988) develop rules for optimal capacity expansion when additions to capacity must occur in discrete units. 4. This rate of output is roughly equivalent to filling the British Columbia Place football stadium twice a day every day of the year. 5. This conclusion neglects many factors. If the utility may expand its capacity incrementally, if it has no uncertainty regarding future demand growth and if the demand for the utility's output is not cyclical, then the result holds. On the other hand, if demand is cyclical or grows stochastically or, if capital can be added in discrete units, then conditions for the optimal employment of this factor are much more difficult to establish (Chao, 1983). 6. The costs are measured in nominal dollars per 1000m3 of output. In 1986, the GVWD's output price was $34.8 / 1000m3. 7. An estimate of capital's shadow price may be calculated for each observation using equation (3.14). These results are not reported here, but it is found that the estimated shadow price is negative for every observation. 8. This can be seen clearly in the results of the optimal expansion rules derived by Swallow and Marin (1989). 9. Rather than use Turvey's method of computing the long-run marginal cost, it would have been possible to follow a procedure suggested by Kulatilaka (1987). Note that equation (3.7) can be inverted so that the optimal capital stock is a function of its market price, the prices of variable inputs and the levels of output and the stock of water. Once an estimate of the optimal capital stock is derived, it could be substituted into (3.14), thus allowing that derivative to be interpreted as the long-run marginal cost of output. The primary difficulty with this approach is that the estimated short-run cost function is only a local approximation to the unknown cost function. Using the estimated cost function would likely lead to an inaccurate estimate of the optimal capital stock given the magnitude by which the market price of capital is seen to differ from the estimated shadow price of capital. 90 10. This form of the Jorgenson user cost of capital assumes that there are no capital gains and no tax-related distortions. With respect to the latter, the Canadian federal government has occasionally provided grants to municipal water utilities for the extension or up-grading of infrastructure. If these grants are in the form of lump-sum payments, then they should not affect the user cost calculations. Conversely, if the grants follow a cost-sharing formula, then they may reduce the acquisition cost of capital, and, thus, its user cost. For a review of recent federal grant programs, see MacLaren (1985). 11. No water demand equation is estimated for the commercial sector due to a lack of data. This omission is dealt with in Chapter Four. 12. The empirical results of these models are summarized in Danielson (1979) and William and Suh (1986). 13. A recent study of the value of water consumption is Gibbons (1986). 14. This is a commonly used assumption in the applied public utility literature (Dimopoulos,1981; Protti and McRae, 1980). Mitchell (1978) is an exception in that household demand for telephone services is assumed to follow a joint distribution over household income and a taste parameter. 15. This approach to generating an aggregate residential demand function is chosen for two reasons. Firstly, it means that the market demand's price elasticity is a weighted average of the price elasticities at different income levels, where the weights are the relative frequency of different income levels. Secondly, the structure of equation (3.21) allows different social weights to be assigned to different income levels. Thus, the utility may be modeled as assigning different weights to the change in consumer surplus at different income levels and the significance of this weighting may be investigated. This issue is discussed further in Chapter Four. 16. This data comes from a special compilation of Census data done by Statistics Canada for Lewis Soroka of Brock University. Professor Soroka kindly allowed me to use this data set. 17. When predicted frequencies from flyh) are compared to actual frequencies, the estimated distribution appears to fit well for low and middle income levels; for higher income levels (eg., yh > $50,000 in 1981 dollars) the estimated pdf performs poorly • primarily due to the lack of observations in the data set for these income levels. 18. Some very large firms also use water for power generation through hydro-electric faculties. 19. Tate and Scharf (1985) report that the following four industries account for 91 % of recorded water use by Canadian manufacturing firms in 1982: petro-chemicals, pulp and paper, metal refining, and chemicals. 91 20. One price series is constructed from the American Water Works Association's index of average public utility prices and the second is derived by dividing total recorded expenditures on water inputs by the total volume of water purchased. 21. White's (1980) correction for the presence of heteroskedastic errors is used in the estimation. 92 IV. MOVING TO EFFICIENT PRICES : SIMULATION RESULTS A. INTRODUCTION The principal issue that this thesis seeks to address is the impact of changes in the pricing rules used by a representative water utility. In Chapter Two, it is argued that there are strong efficiency-based reasons for supporting a move toward a form of marginal cost pricing. Chapter Three details the procedures undertaken to establish the structure of costs and aggregate demands. This chapter draws on material from the previous chapters and presents the results of a series of numerical simulations that investigate the impact on consumers and on the utility of adopting efficient prices. This impact is examined by measuring the change in aggregate consumer surplus. The rest of the chapter is divided into three sections. The next section addresses several important theoretical issues surrounding the evaluation of pricing changes. The third section details the method used to conduct the numerical simulations. The final section presents the results of the simulations. B. THEORETICAL ISSUES The purpose of this section is to consider some of the theoretical issues surrounding the evaluation of changes in a representative water utility's pricing rules. This discussion is meant to illustrate the significance of the assumptions made in the 93 simulation program. The evidence presented in the preceding chapters demonstrates that the pricing rules used by North American water utilities are inefficient. It, unfortunately, does not necessarily follow that compelling these utilities to move to efficient prices will lead to an increase in aggregate welfare. In order to understand this paradox, consider first a simple example where such a move appears to lead to a potential Pareto improvement. In Figure 4.1 a water utility with constant marginal cost, (b+P), sells a single output Q and faces a market demand curve as shown. The utility initially sets its unit price, p, and annual fee, A, as follows: p0 = 0, AQ = ((b+P)-Qo)/H where H is the number of consumers (assumed exogenous). Aggregate consumer surplus is CS0 = OaQo - Ao-H and the utility earns zero profits. As an alternative, assume the utility pursues marginal cost pricing (i.e., px = (b+p), A<, = 0). Then, at the new level of aggregate consumption, the utility still earns zero profits, but aggregate consumer surplus is CSj = a(b+p)e. It is easily seen that the change in the pricing rule increases aggregate consumer surplus by CSX - CS0 = ecQo- It would appear, then, that moving to efficient prices leads to a potential Pareto improvement and an increase in aggregate welfare.1 There are several reasons why this conclusion may not be warranted. The first concerns the use of a change in the aggregate Marshallian consumer surplus as an index of welfare change. The second arises from the partial nature of the analysis and the possibility that it ignores distortions which exist in other markets. The third is based on the presence of commercial consumers of the utility's output and the associated difficulty of predicting how water pricing changes will change the output 95 prices of these agents. These three theoretical considerations suggest that the evaluation of the proposed alterations in water pricing practices is not a straightforward exercise and each is discussed briefly here. The use of the aggregate Marshallian consumer surplus as an index of a population's welfare change is a common procedure in applied welfare studies (Mitchell, 1978; Protti and McRae, 1980; Dimopoulos, 1981). Nonetheless, it is well understood that the change in the welfare of a heterogeneous population is not perfectly indexed by the change in aggregate Marshallian consumer surplus (Boadway, 1974). There are three reasons for this. First, even if only one price changes and the welfare of a single consumer is considered, the Marshallian surplus will index welfare changes only if income effects are absent. Second, if more than one price changes (in the case of a single consumer), then there is no single correct order in which to assess the impact of the price changes. Thus, the measure of welfare change is 'path-dependent' in that it depends upon the order in which price changes are considered. Third, when many heterogeneous consumers occupy the market, it is not generally true that the surplus measure generated from the aggregate demand curve is equivalent to the sum of the surpluses generated from individual demand curves. That is, it is not always possible to represent the preferences of a heterogeneous population with the preferences of an aggregate consumer (Varian, 1984). For these reasons, the aggregate Marshallian consumer surplus is not, in general, an index of welfare change when consumers are heterogeneous and more than one price changes. In the instance where only one price changes, there are two separate lines of defence supporting the use of the Marshallian surplus measure as an index of welfare 96 change. The first is empirical and is summarized by Willig (1976). Willig demonstrates that bounds for the discrepancy between the Marshal li an consumer surplus and the compensating and equivalent variations are, in the case of a single price change, computable from observable parameters. Furthermore, in many cases, the size of these discrepancies will be smaller than the errors arising from the estimation of demand parameters. Thus, the spirit of the Willig argument is that in many applied studies, the substitution of the Marshallian surplus measure for either the compensating or equivalent variation will introduce a source of error which is easily computed and which is unlikely to be the most significant source of error.2 The second approach seeks to identify the set of conditions that must be satisfied for the Marshallian surplus measure to act as a consistent index of welfare change. In order to identify these conditions, Blackorby and Donaldson (1985) assume that the Marshallian consumer surplus of each of H consumers ( C S j , C S H ) is known and that a social decision rule states that a project is at least as good to society as the status quo if and only if HCSj CSH) £ 0. Then, it is asked whether there exists a Bergson-Samuelson social welfare function that ranks the project consistently with IXCSj CSH). Blackorby and Donaldson demonstrate that the answer is affirmative if the following conditions are satisfied: 1) individual preferences are homothetic (but may differ) 2) the decision rule is linear, i.e., H IXCSi CSH) £ 0 *+ ZbkCS^OO^are weights) h=i 3) the social welfare function is Cobb-Douglas in real incomes Thus, the use of the Marshallian consumer surplus measure may be defended by 97 appealing either to empirical grounds such as the errors inherent in variable measurement and estimation or by assuming preferences are restricted and limiting the analysis to adopt only those social decision rules that are known to be consistent. The second issue concerns the validity of the partial analysis typically used in applied studies of utility pricing. There are two reasons to be concerned with the use of partial analysis. The first is that the price change in one industry may generate significant price changes in other markets. A complete welfare analysis would incorporate these indirect effects. As already indicated, however, there is no single correct way to measure welfare change when several prices change. The second basis for concern with partial analysis stems from the presence of distortions elsewhere in the economy. It is well known that, if an industry is distorted (e.g., it is a monopoly supplier), then there are instances when efficiency dictates prices diverge from marginal cost in other industries (Boadway and Bruce, 1984). This is just an example of the problem of the second-best. Thus, if there is an industry that is closely related to the municipal water market and that possesses an immutable distortion, then the use of partial analysis and the identification of optimal public prices in that context may be misleading. As in the case of Marshallian consumer surplus measures, the defence of partial analysis rests on either empirical or theoretical grounds. Concerning the former, Lancaster (1979) argues that an individual public utility or utility industry is likely to be small relative to the economy as a whole. As a result, it would be very difficult to use the utility's prices to counterbalance distortions elsewhere. Hence, Lancaster asserts that it is reasonable to ask an individual utility or utility industry to optimize in isolation of possible distortions that exist elsewhere unless those 98 distortions are easily identified and well understood. While Lancaster's contention may be invalid for the larger public utilities (e.g., telecom muni cations, electrical generation), it would seem reasonable for municipal water utilities. The theoretical grounds for neglecting second-best concerns rest on identifying the restrictions on technology and preferences that permit a partial analysis. Jewitt (1981) employs a simple general equilibrium model of an economy with a distorted sector and an undistorted sector. The conditions under which optimal prices for the un distorted sector may be identified independently of conditions in the distorted sector are established. These conditions turn out to be quite restrictive; consumer preferences must be 'psuedo-separable' (Gorman, 1976). That is, compensated demands for the output of each sector must be a function only of the own sector prices, the own sector expenditures, and the level of the consumer's utility. The third issue arises from the observation that most public utilities sell their output to both residential and commercial consumers. As a result, a change in the utility's pricing structure can be expected to influence households in two distinct ways. Households will be directly affected by the change in the price of the utility's residential output. Households will also be indirectly affected by changes in the prices of the commercial sector's outputs that are brought on by the change in the utility's output price. In principle, this heterogeneity of the utility's customers complicates the analysis, but does not change it qualitatively. Two additional forms of information are required. First, the cost structures of the commercial customers and the structures of the markets in which they compete must be ascertained. This will allow a prediction to be made of the expected change in commercial output prices when the 99 public utility's price changes. Second, a more general model of household preferences must be obtained so that the welfare implications of changes in commercial prices may be assessed. The data requirements for this extended analysis are daunting. As a result, an alternative approach is to identify the conditions under which it may be assumed that the total change in households' aggregate consumer surplus is the sum of the change in consumer surplus attributable to the change in the utility's residential price and the change in the consumer surplus of the commercial sector attributable to the change in the utility's commercial price. Brown and Sibley (1986) consider the problem of direct and indirect impacts3 and attempt to determine when 'myopic' optimal prices (i.e., those that ignore their indirect impact on consumer welfare) will be the same as those that take into account both effects. It is assumed that the utility seeks to maximize the sum of producer and aggregate consumer surplus subject to a break-even constraint. In this case, the change in total surplus attributable to the myopic Ramsey prices will closely approximate the change in total surplus attributable to non-myopic Ramsey prices when the following conditions are met: households' demands for the outputs of the utility and the commercial sector are independent, and the commercial sector is either perfectly competitive or faces a linear market demand curve. Intuitively, the Brown and Sibley result indicates that if the commercial sector is competitive then any cost savings arising from a decrease in the utility's prices will bypassed along entirely to households in the form of lower output prices. This discussion concerning the theoretical issues which surround the numerical simulation of alternative pricing rules suggests the following conclusion. Assessing 100 the welfare implications of any price change in the face of heterogeneous consumers and market distortions is a difficult task. Only under a restrictive set of assumptions regarding the structure of consumer preferences and the structure of markets may changes to aggregate Marshallian consumer surplus be computed and used to gauge changes in aggregate welfare. C. METHOD 1. Introduction The purpose of the simulation program developed here is to establish the approximate costs and benefits which can be expected to arise as a result of moving to alternative water pricing rules. The previous section discusses the theoretical issues that surround this type of analysis and this section outlines the method that is used to conduct the analysis. This section is organized in the following fashion. Subsection Two examines the general structure of simulation programs and reviews other attempts to examine the impact of alternative utility pricing rules. Subsection Three details the assumptions underlying, and structure of, the simulation program used here. The last subsection enumerates the pricing policies that are to be considered as alternatives to the current practice. 2. Background Most attempts to simulate the impact of changed parameter values on the equilibrium values of economic variables share a common objective. The motivation for using this type of analysis usually stems from an inability to sign or determine the 101 magnitude of comparative statics results from a theoretical model. For example, the impact of the imposition of a wage tax on optimal labour supply is not something which can be predicted from a theoretical model. As a result, the sign and magnitude of this effect must be established empirically. In addition to sharing a common objective, many simulation efforts exhibit a common format. The form of the equihbrium value of a variable or set of variables is first established. This value is usually a function of the parameters of behaviourial equations (e.g., supply and demand equations). A 'base-case' equilibrium value for the variable of interest is then computed by providing the model's coefficients or parameters with specific values. This step often also involves 'cahbrating' the model so that base-case predicted values equal observed values. The value of any parameter or coefficient may be perturbed and the resulting impact on the equilibrium value of the variable of interest may be computed. Moreover, a sensitivity analysis may be conducted to examine how the magnitude or sign of the measured impact is influenced by changes in the model's other coefficients' and parameters' assumed values. The use of simulation methods to supplement theoretical modelling can yield several benefits. The most important of these is likely to be the ability to establish the sign and magnitude of comparative statics results. In addition, the influence of key parameters on these results may be investigated through sensitivity analysis. There are, however, some significant limitations to this type of analysis. First, it is usually not feasible to establish numerical estimates of the standard errors of computed comparative statics results. For example, it may be found that the average elasticity of labour supply with respect to a wage tax is -1.0, but it is usually quite difficult to establish whether this estimate is statistically significant. Second, most 102 simulation programs are based on parameters and coefficients which are constants. This feature has two implications. The first implication is that the possible endogeneity of some behaviourial parameters may be neglected. The second implication is that the accuracy of the results may be inversely related to how far the analysis moves the equilibrium away from its base case. This is because some parameter values (e.g., elasticities of substitution among inputs) may only be local approximations. There have been a number of attempts to simulate the impact of alternative pricing regimes on the output and profitability of public utilities and on the welfare of their customers (Mitchell, 1978; Protti and McRae, 1980). Representative of this type of analysis is Dimopoulos (1981). Dimopoulos (1981) examines the welfare implications of implementing different pricing rules for a regulated electrical utility. The pricing rules considered are: average cost pricing, a two-part declining block tariff (with two marginal prices), and an increasing block tariff. Dimopoulos assumes that the utility seeks to maximize the sum of producer and aggregate consumer surplus subject to remaining financially viable. The utility knows the electricity demands of the representative household and knows the structure of the distribution of demands. The utility possesses a degree of inequality aversion summarized by its marginal social utility of income function. This function assigns different weights to the consumer surplus enjoyed by families with different income levels. In order to compute the surplus levels associated with each pricing rule, Dimopoulos parameterizes the model. The utility's technology is constant returns to scale and household's demands have constant price and income elasticities. The 103 income distxibution is a hybrid linear-Pareto probabihty density function4 and the marginal social utility function has a constant elasticity form. Thus, the most important parameters of the model are: marginal cost, income and price elasticities of demand, the elasticity of income distribution and the elasticity of the marginal utility of income functions. Values for the parameters are not estimated but rather are taken from the extant literature. Dimopoulos is able to compute the surplus per household for the different pricing rules and for different parameter values. Interestingly, for most sets of parameter values, the values of surplus per household show very little variation across pricing rules. Conversely, the household surplus values vary significantly across sets of parameter values under a given pricing rule. For example, the highest welfare levels occur when the utility generates the aggregate surplus without imposing different weights to different income levels. In this case, only efficiency matters; the constancy of marginal costs (combined with an exogenous fixed cost) implies that the Coasian two-part tariff is optimal. Crew and Kleindorfer (1986) also consider the welfare implications of alternative pricing rules but do so for a water utility which faces fluctuating demands and supplies of raw water. The utility is constrained by its fixed reservoir size and seeks to maximize the present value of discounted surplus over twelve months through its choice of monthly prices. Using the optimal control model of Riley and Scherer (1979), Crew and Kleindorfer establish that the optimal output price is the shadow value on the stock of water in the reservoir and investigate the welfare costs of using either time-invariant prices or prices that may change at discrete intervals. It is assumed that the market demand for water has a constant price elasticity 104 and follows a known cycle through a one year period (rising in summer, falling in winter). Marginal costs are constant. Three specific pricing rules are considered: average cost pricing that is time-invariant; a myopic rule that sets price equal to marginal cost and allows the price to change in the next sub-period (month) if actual storage diverges from optimal storage; and a myopic pricing rule that allows prices to change only if actual storage diverges from the optimum by a given magnitude and that also constrains the size of price changes to pre-set limits. After parameterizing the model, Crew and Kleindorfer simulate the impacts of each pricing rule. As Dimopoulos, they find that aggregate welfare levels differ very little across pricing rules (here, by less than one percent). However, Crew and Kleindorfer do not investigate whether different parameter values would lead to different results. These efforts to simulate the welfare impacts of alternative pricing rules point out the benefits and costs of employing this approach. On the benefit side, the sign and magnitude of comparative static results may often be established. The cost of the approach lies in the loss of generality associated with the choice of specific functional forms and parameter values. 3. Structure of the Simulation Program The purpose for constructing the simulation program is to assess the impact of a representative utility's decision to alter its pricing format. Of primary interest is the impact on the utility's revenues and costs and on consumers' aggregate surplus. Specifically, the program is used to compute the change in surplus arising from the change in pricing rules and to compare the efficiency gain to the cost of implementing 105 the change. A related interest is directed toward identifying the role played by several important parameters in determining the final results. Before outhning the structure of the program it is worthwhile listing the assumptions that are maintained during its operation. The utility is assumed to know the structure of the aggregate non-residential demands, the demand for a representative household and the distribution of residential water demands. The utility seeks to maximize the sum of consumers' surpluses over the year in which it operates (assumed to be 1986) while maintaining a balanced budget. The year is divided into four periods of equal length, the third of which ("summer") corresponds to the peak season. The utility ignores the impacts that its pricing decisions have on other markets and it assumes that the necessary conditions for the 'flow-through' of its pricing changes to final consumers are satisfied. The implication of this assumption is that industrial and commercial consumers are expected by the utility to pass on any cost changes (arisng from water price changes) completely to households in the form of changes in their output prices. Finally, the market responses to price changes (as summarized by the estimated demand equations) are assumed to be instantaneous. There are two important benefits that can be expected to accrue from rationalizing prices, but that are not incorporated into the simulation program. The first stems from an indirect benefit of introducing universal metering of consumption (a prerequisite to unit pricing). The installation of flow meters would aid significantly in leak detection and thus reduce the loss of water between pumping station and final use. The second benefit would arise if the water utility's total output were to decrease as a result of the pricing change. In this case, it can be expected that total sewage 106 treatment costs would decrease because of decreased flows of water (Strudler and Strand, 1983; Fraas and Munley, 1984). Neither of these effects is incorporated due to data limitations. The goal of the program is to obtain equilibrium values for the following variables: the aggregate consumption levels for the residential, industrial and commercial groups, the utility's revenues and costs and the level of consumers' surplus for each of the three consumer groups. Equilibrium values for these variables are computed for each quarter and then summed to give annual values. The overall structure of the program may be summarized as follows. First, base-case values are established. The estimated demand and cost equations are scaled so that the predicted values of the dependent variables are equal to observed values for 1986.6 Second, a new pricing rule is established. This rule specifies the structure of unit prices for each user group. These prices may differ across quarters, as in the case of peak-load pricing. Third, given the new prices, new levels of aggregate desired consumption are calculated. At these consumption levels, the utility's costs and revenues and the levels of aggregate consumers' surplus are recalculated. Fourth, the resulting vector of quarterly equilibrium values for consumption, revenues, costs and surplus are summed to give annual values. If a deficit is earned by the utility, then the annual fee necessary to erase the deficit is calculated. Finally, three different values for the change in aggregate consumers' surplus are calculated: the first simply sums the surplus measures of the three consumer groups and subtracts the corresponding base-case value; the second takes the first value and nets out any change in annual fees; and the third takes the second and nets out the costs assumed to be incurred when moving to the new pricing regime. 107 Having outlined the general workings of the simulation program, it is worthwhile to examine it in some detail. This will allow for the identification of the roles played by specific assumptions and the significance of particular parameters. The demand side of the program is based on the estimated demand equations that were reported in Chapter Three. The predicted water demand for the representative household is derived from the estimated form of equation (3.22) once 1986 values of all explanatory variables other than price and income are inserted. For 'off-peak' quarters, household demand is taken to be indoor water demand and, for the peak quarter, household demand is the sum of indoor and outdoor water demands. In order to obtain the aggregate quarterly residential demand, equation (3.21) is employed. The estimated industrial water demands reported in Chapter Three provide predicted annual consumption levels. Values of non-price explanatory variables for 1986 are substituted into the estimated form of (3.30) and then the predicted quarterly value of the dependent variable is computed by dividing by four.6 The remaining group of water consumers is the commercial-institutional sector. Almost nothing is known about the structure of demands for this sector. In Vancouver, the commercial sector represents approximately 24 % of total consumption in 1986. In order to include this sector's predicted consumption levels in the simulation program, an aggregate demand curve is constructed from the available information. VWW records provide quarterly consumption levels for 1986 and an estimate of this sector's price elasticity is taken from Williams and Suh (1986). The resulting quarterly aggregate water demand function for the commercial-institutional sector has the following form: (41) °cm " 3 5 1 1 4 . 1 - ( 6 9 . 9 3 5 ) Pc 108 where P e is the average price charged by the VWW to commercial customers. Once prices are set, the total quarterly demand for water may be established as the sum of the predicted aggregate quarterly demands of the residential, industrial and commercial sectors. Total annual consumption is obtained by summing over the four quarters. The next part of the program computes the predicted revenues and costs of the utility. The VWWs predicted revenue is computed as the sum of two parts. The first is the product of per unit price and predicted aggregate consumption levels for each user group. In the base case where industrial and commercial users face a multi-part declining block price structure, the average of the marginal price blocks is used. The second part of the utility's revenue is a function of the number of customers and their characteristics rather than the level of their consumption. This component of revenues is further divided in the program into two parts. The first is the product of the number of customers and the annual charge for connecting to the distribution system levied against each customer. The level of the annual charge is endogenously determined by the program. The second is the sum of a set of miscellaneous charges made for the rental of oversized water meters, inspections and other services provided by the utility. In the program these revenues are assumed to be exogenously given. Thus, the 'variable' portion of revenues is computed once prices are set and predicted aggregate demands are determined. The endogenous portion of annual customer charges is computed only after predicted costs are known and is calculated as the difference between predicted total costs and predicted revenues with the 109 difference being divided by the total number of customers. This manner of calculating the annual customer charge ensures that the utility has a balanced budget. It is important to note that this calculation assumes that the number of customers is exogenous to the level of the annual customer charge. This would be a questionable assumption in the case of the pricing of telecommunications services (Mitchell, 1978), but it does seem reasonable in the case of municipal water supplies. The total quarterly costs of the utility are computed using the estimated VWW cost functions. During off-peak periods, total cost is predicted by the VWWs restricted cost function. During the peak period, total cost is predicted by the VWWs unrestricted cost function. In both cases, 1986 values for the exogenous variables (other than output) are substituted into the estimated cost functions. The price of water purchased by the VWW from the GVWD, however, is represented by the latter's estimated short-run marginal cost for periods 1,2, and 4, and by its long-run marginal cost for period 3. For the base case calculations, the estimated variable cost function is scaled so that estimated total costs equal observed total expenditures which in turn equal recorded total revenues for 1986. The last endogenous variables to be calculated are the levels of aggregate consumers' surplus enjoyed by each user group. The program approximates the residential user group's consumers' surplus by its Marshallian surplus. This surplus is calculated in two stages. The first calculates the Marshallian surplus for the representative household in each quarter: 110 A (4.2) CSfrJ - / w 1 (gh, yh, Zh)dqh - -± - pR-qh where w*H0 is the inverse of the estimated household demand equation; yh is household income; q*h is the predicted level of desired consumption of the representative household; Zj, is a vector of socioeconomic characteristics; is the annual customer charge; and PR is the price facing residential consumers.7 Aggregate consumer surplus is then computed by aggregating over income groups and applying differential social weights to the surpluses earned by households with different income levels. (4.3) CSR - HR- / ftyh, 6) • p ^ • CSk(yj dyh where H B is the number of residential customers and f(yh) is the probability density function of household income in Vancouver. This probability density function is assumed to be lognormal and its estimated parameters are reported in the previous chapter. The function p(yh) is the social marginal utility of income and, following Feldstein (1972) and Dimopoulos (1981), it is assumed to have the constant elasticity form given in equation (4.4). (*4) - y" I l l where yh is household income and p is a parameter reflecting the utility's aversion to income inequality. The annual value for the residential group's aggregate consumer surplus is obtained by summing over the four quarters. Calculating the surpluses of the industrial and commercial user groups is more straightforward as their demands are aggregate and they predict annual rather than quarterly desired consumption. In each case, the aggregate Marshallian surplus is calculated as the area beneath the aggregate demand (up to the predicted level of consumption) net of expenditures on water purchases and on customer charges.8 Once the vector of output prices is specified, then the aggregate annual consumer surplus is computed as the sum of the surpluses earned by each user group. (4.5) CST - CSR + CSj + CSC This value is computed at base case prices and again after new prices are specified. The change in total surplus as a result of altering the pricing regime is then computed as:9 (4.6) A CST - CST(new case) - CST(base case) - Metcost(HR) where Metcost(NE) is a cost function that computes the added costs that the utility must undertake in order to adopt the new pricing rule.10 These costs are assumed to be a linear function of the number of residential customers because the principal cost to be incurred involves the installation, monitoring and maintenance of flow meters in private households. Data made available by the VWW staff from an 112 internal report allowed this function to be parameterized as follows: (4.7) Metcost(HR) - a M C H R - ($21.30) H R The figure $21.30 represents the VWWs estimate of the annualized cost of residential metering. It is the sum of annual reading and billing costs and the amortized11 installation and annual maintenance costs. To summarize, the structure of the program is as follows. The base case values for aggregate consumption, the utility's costs and revenues, and consumers' surplus are calculated using quarterly data from 1986 (including the VWWs actual pricing structure). New prices are set. Demands are recalculated. The utility's new revenues and costs are recalculated, including an annual customer charge if a deficit is anticipated. The new levels of aggregate consumer surplus are calculated. The quarterly values are then summed to give annual values. The change in aggregate consumer surplus is computed and the added costs of implementing the pricing scheme are netted out. The results are then reported. This procedure is repeated for alternative pricing rules and, for each pricing rule, for alternative parameter values. The review of the public utility pricing literature that is presented in Chapter Two suggests that a water utility could reform its pricing structure in several directions. Because of differences in the models underlying alternative efficient pricing schemes, however, it is not always clear which pricing rules will dominate. For example, will a move to Ramsey prices lead to a greater gain in net aggregate surplus than a move to peak-load pricing? By considering alternative pricing rules, the simulation program can be used to address this type of question. The pricing rules considered here are summarized in Table 4.1. The first set 113 of pricing rules represents the VWWs actual (in 1986) practice. Residential consumers face only an annual customer charge while commercial and industrial users face both per unit prices and annual customer charges. There is no discrimination according to time of use. The first option to the base case maintains its time-invariant quality but does not discriminate among user groups. All users are charged the long-run average cost of production and annual customer charges are removed. The second option is peak-load pricing. There are two forms of this pricing rule considered. The first may be thought of as long-run peak-load pricing'. Each user is charged the predicted short-run marginal cost during the three off-peak quarters and the predicted long-run marginal cost during the peak quarter. Again, all users face an annual fee equal to their share of the deficit. The second form of peak-load pricing may be thought of as 'short-run peak-load pricing' because it takes the short-run capacity constraint presented by the VWW production technology into account. Thus, in off-peak periods, price is set equal to short-run marginal cost. During the peak period, price is set at a level which generates a desired aggregate demand equal to capacity.12 This pricing rule is also supplemented with an annual charge to recover any deficit. The annual charge is common across users. A variant of the seasonal pricing rules compels commercial and industrial users to face peak-load prices but maintains the base case assumption of a price for residential consumers of zero. The reason for considering this 'half-way' pricing rule is that it would be implemented with almost no additional cost (all commercial and industrial consumers are already metered). It may, therefore, yield a larger net benefit than complete peak-load pricing that requires the installation of residential 114 flow meters. The final two options differ in form from marginal-cost based pricing. Option three follows Feldstein's (1972) prescription for distributionally sensitive public pricing. The formulae for the per unit price and annual customer charge are given by equation (2.15). While these prices do not vary across users, they are given a 'peak' flavour by allowing the marginal cost term (p) to vary across seasons. During off-peak quarters this term is equal to short-run marginal cost and during the peak quarter, it is set equal to long-run marginal cost. The fourth alternative to the base case is an hybrid peak-load Ramsey pricing rule. The peak-load character of the rule is demonstrated by the differentiation between short- and long-run marginal costs. The Ramsey feature of the rule comes from the fact that there is discrimination across user groups according to their respective price elasticities of demand. Thus, each user's price is scaled up over marginal cost in inverse proportion to his/her demand elasticity. The result is a pricing rule that discriminates across users and seasons, but does not require any annual customer charges to be collected. The impact of moving to each of the alternative pricing rules is a function of the estimated demand, cost and other parameters. The most significant of these are the following: 1. p, a: the mean and standard deviation of the income probability density function. 2. p: the inequality aversion parameter in the marginal social utility of income function. 3. ctjjc: the metering cost parameter. 4. TjB,T|i,Tlc: th e price elasticities for each of the three user groups. 5. 115 SRMC,LRMC: the short-run marginal and long-run marginal costs for the VWW. As indicated in the next section each of these parameters is given a base case value. Using these values the simulation program compares the impact of each pricing rule. The parameter values are then altered and the pricing rules are again compared. D. SIMULATION RESULTS This section reports and discusses the numerical results obtained from simulating the effects of a change in pricing policy for a representative water utility. The primary interest in conducting these simulations is to identify the conditions under which a pricing change will give rise to a potential improvement in aggregate welfare. While several pricing rules are considered as alternatives to the VWWs current pricing rule, they are not seen to be equally important. The peak-load pricing rules have been identified by economists for at least three decades as the welfare-maximizing solution to the problem of a utility with fixed capacity serving cyclically fluctuating demands and, as such, are considered in the greatest detail. In addition to the base-case pricing rule currently used by the VWW, six alternative pricing rules are considered. These rules are described in the previous section and are the following: long-run and short-run peak-load or seasonal pricing, seasonal pricing with unchanged residential prices, average cost pricing, Feldstein pricing, and a hybrid of seasonal and Ramsey pricing. In all cases, except those of 116 average cost and Ramsey pricing, the per unit prices are supplemented with annual connection charges. Each comparison of the alternative pricing rules is a function of the assumed parameter values. One particular set of parameter values is assumed to be the 'standard case'. The parameter values in the standard case are the following: Standard Case Parameter Values (1) Inequality aversion parameter: p = 0.0 (2) Income Probability Density Function a) Mean and Variance: p = 10.542 and o2 = 0.187 b) Limits of Integration: y mia = $1,000; y n u u = $1,000,000 (3) Metering Costs: (1986 $/year/household) = 21.30 (4) Marginal Costs of Supply13 (a) Short-run Marginal Cost (1986 $ / 1000 m 3 / quarter) Season b Winter 4.95 Spring 6.61 Summer 8.94 Fall 6.22 b) Long-run Marginal Cost (1986 $ /1000 m3 / quarter) Season |J Winter 53.89 117 Spring 71.05 Slimmer 85.83 Fall 55.15 In addition to these particular parameters, the results for each pricing rule are also dependent upon the estimated parameters of the utility's cost functions and of each user group's aggregate demand function. Table 4.2 reports the equilibrium quarterly prices and quantities under each of the pricing rules. The peak-load pricing rules lead to significant price decreases for industrial and commercial consumers (compared to the base-case prices) during off-peak quarters. Residential users experience the largest increase in per unit prices but, like the other cases, see a decrease in annual fees. Because prices increase for residential users and decrease for industrial and commercial users, aggregate consumption in both the peak and off-peak periods differs little from the base-case. In both the short-run and long-run peak-load pricing rules, consumption rises by approximately 15 % over the base case. The average cost pricing simulation yields a constant unit price which is invariant across seasons and which implies no annual user fee. Aggregate output is marginally lower than the base-case. The Feldstein pricing simulation yields results identical to those of the peak-load pricing rule. This is because the assumed value of the marginal social utility of income function's parameter, p, is zero and, in this case, Feldstein pricing reduces to marginal cost pricing. (See equation (2.15) in Chapter The Ramsey-peak-load pricing simulation yields results that differ significantly Two.) 118 from the others. Aggregate output is 18 % less than in the base-case. This is largely attributable to the 80 % and 44 % decreases in third quarter consumption by residential and industrial consumers, respectively. These changes in aggregate consumption result from substantial price increases. The requirement that there be no annual charges implies that prices must rise above marginal costs the most for the two user groups whose demand elasticities are smallest: residential and commercial. This simulation is the only one (with the exception of the case of peak-load pricing with residential users exempted) where the utility employs price discrimination. Prices for commercial users exceed those for industrial consumers by a factor of 3.5 and for residential users by a factor of 2.5. The current pricing policy used by the VWW (represented here by the base-case simulation) is characterized by significant price discrimination in both unit prices and annual charges. Consumption is cyclical with output rising during the third quarter. The alternative pricing rules lead to moderate changes in aggregate consumption because these rules typically result in different directions of price changes for different users. Nonetheless, the magnitudes of the price changes are significant both for per unit prices and annual charges. Residential consumers see per unit price increases in all simulations. Industrial and commercial users see prices in off-peak quarters decreasing and peak-prices increasing for most simulations. The magnitude and complexity of these price changes suggest that their welfare impacts will be significant, but difficult to predict a priori. Table 4.3 summarizes the changes in aggregate surplus for alternative pricing rules assuming that all parameters are at their respective standard values. In each case, the change in surplus is computed as the difference between the consumer 119 surplus under the alternative pricing rule and the surplus computed using the base-case pricing rule. The first column reports the change in residential users's aggregate surplus that results from considering only the direct impact of the change in their water price. This change is computed gross of any change in the residential annual charge. The second column repeats this calculation but nets out any change in the residential annual change. It also nets out the added costs undertaken by the utility to install and maintain flow metering devices in all residential households. The third column adds the change in surplus for industrial and commercial water users (net of changes in annual fees) to the second column. Thus, the third column represents the sum of the direct and indirect changes to households's aggregate Marshallian surplus which arise from the change in pricing undertaken by the municipal water utility. All figures are annual values and are expressed in 1986 dollars. With the exception of the simulation that keeps residential prices unchanged, all of the alternative pricing rules lead to a decrease in households' aggregate Marshallian surplus because of the direct impact of raising residential water prices. In contrast, when both the direct and indirect effects of the changes in the utility prices are taken into account, the peak-load pricing scenarios show positive changes in aggregate surplus. For example, the move from current pricing practices to a policy of short-run peak-load pricing (common across users) is predicted to generate an increase in aggregate surplus of $2.40 million. This figure is net of any changes to users' annual connections fees made to ensure that the utility's budget remains balanced and it is net of the annualized cost of the utility's installing and maintaining residential flow metering devices.14 The two peak-load pricing rules differ in their impacts. The short-run pricing 120 rule results in a lower peak-period price (calculated to ensure that aggregate demand equals the utility's capacity during the peak period) and a higher annual fee. The combination of these two differences implies that the short-run pricing rule leads to a higher consumption level in the peak period and, as a result, a slightly larger gain in aggregate consumer surplus. The predicted impact of moving to Ramsey pricing is somewhat surprising. Table 4.3 indicates that such a move would lead to relatively large surplus losses. By comparing the values in columns two and three it can be seen that the direct loss to residential consumers outweighs the indirect gains that emanate from industrial and commercial firms passing on any cuts in water prices. The reason for this result stems from recognizing that, in contrast to the utility using annual connection charges to recoup the deficit arising from marginal cost pricing, raising unit prices above marginal cost is a relatively inefficient way to cover the deficit. For example, if unit prices rise from zero to P x and aggregate consumption falls from Qo to Qx, then the utility may apply P ^ revenue against any deficit but consumers experience a surplus loss of approximately P&x + 1/2APAQ. The second term is simply the Harberger triangle of deadweight loss. For residential consumers, its value for the summer quarter alone is approximately $1.7 million. It is interesting to note that the simulation which changes industrial and commercial prices but maintains the zero price for residential consumers leads to a smaller surplus gain than the other peak-load rules. This is somewhat counterintuitive as this simulation requires no new meters to be installed (as all industrial and commercial consumers are already metered). However, the fact that residential prices do not rise means that there is no efficiency gain arising from this 121 segment of the market. This is important because residential consumption accounts for over 60 % of market demand. The absolute magnitude of the surplus changes recorded in Table 4.3 is somewhat difficult to assess. The VWW has approximately 93,000 connections and its 1986 recorded output is 114 million cubic metres. Thus, the surplus gains from a move to short-run peak-load pricing are approximately $25.8 / connection per year or $0,021 / cubic metre per year. The latter figure can be compared to the price paid for water by the VWW to the GVWD of $0.0384 / cubic metre. Thus, while the surplus gain appears fairly small when computed per connection, it appears fairly large when compared to the wholesale price for water. Another way to assess the significance of the entries in Table 4.3 is to consider their size relative to the computed base-case levels of aggregate surplus. Thus, below each entry the size of the change is expressed as a percentage of the base-case level. These percentage figures appear in parentheses. With the exception of the Ramsey pricing results, the changes in aggregate surplus are small in relative terms. The most likely reason for these small values is that, for each user group, per unit prices are moving in opposite directions from the changes in annual fees. The welfare impacts of these two separate effects tend to offset, leaving only the net efficiency gain or loss. The size of the surplus changes may also be compared to the results found in other studies considering the welfare effects of public utility pricing changes (Dimopoulos 1981; Mitchell, 1978; Crew and Kleindorfer, 1986). In contrast to the findings of Dimopoulos (1981) and Crew and Kleindorfer (1986), it is seen in this thesis that the size of the surplus gain does vary significantly across pricing rules. 122 However, the results here parallel those of Dimopoulos in that changes in parameter values affect the absolute magnitude of the efficiency gains under each pricing rule more than they do the ranking of alternative pricing rules. For example, Mitchell (1978), in a study of telephone pricing, demonstrates that welfare gains shrink as metering costs rise. In the case of no metering costs, Mitchell (1978) finds the welfare gains to range from 4 % to 13 %. In contrast, when metering costs are "high", welfare gains range from -3 % to 3 %. The results here also parallel those of Dimopoulos (1981) and Mitchell (1978) in finding that net efficiency changes are small relative to starting levels of observed consumer surplus. The results reported in Table 4.3 point out the significance of considering not only the direct effects of utility price changes on residential consumers but also their indirect effects. The results, however, also indicate that the assumption of complete flow-through of the utility's price changes to intermediate consumers is critically important. In some instances, inclusion of the indirect effects changes the sign of the surplus change, as in the case of both peak-load pricing simulations. Table 4.4 presents the results of running the alternative pricing simulations under differing assumptions regarding parameter values. Each column corresponds to a specific pricing rule and each row corresponds to a particular perturbation of a parameter's value. Each entry in the table measures the change in total net surplus from the level observed at the current pricing rules. Note that the first row in Table 4.4 repeats the third column of Table 4.3. There are several parameter simulations summarized in Table 4.4. The first two broaden the income distribution by increasing the standard deviation of the density function or by increasing the limits of integration (where Y,,^ is set at $500 123 and Ym„ at $2 million). The next two perturbations affect the value of the utility's income inequality aversion parameter. These simulations bracket the standard case by setting p at -0.25 and 0.25. The fourth simulation raises metering costs by 25 %. The last two simulations change the level of short-run and long-run marginal costs. These are increased and decreased by 20 % in the sixth and seventh simulations, respectively. One way to consider the results in Table 4.4 is to examine whether the ranking of alternative pricing rules (according to the size of the change in surplus) changes across different parameter values. In most cases, the short-run peak-load pricing and Ramsey pricing simulations are ranked first and last, respectively. The only exception is the case when the standard deviation of income (a) = 0.6, where Feldstein pricing yields the largest surplus. This dominance of the peak-load pricing rule would suggest that it leads to the largest surplus increase over a relatively wide range of parameter values. An alternative perspective on the results in Table 4.4 is gained by considering how the net surplus changes vary for a given pricing rule across parameter settings. The magnitudes of entries change dramatically across parameter values. For example, the change in aggregate surplus from moving to short-run peak-load pricing varies from -$1.39 million to $55.78 million depending on the parameter values assumed. Furthermore, in the case of both peak-load pricing rules, the change in surplus increases under the fohowing parameter values: o = 0.6; Broad Income; Unit Cost = -20 % and p = -0.25. Conversely, the change in surplus declines under peak-load pricing for parameter values: metering costs +20 % and p = 0.25. This pattern is repeated for the other pricing rules with the exception of Ramsey pricing. 124 Any parameter change which lowers costs or broadens the income distribution leads to an increase in the change in surplus from changing prices. Conversely, parameter changes which cause costs to rise or which discount surplus changes for high income households lead to smaller changes in aggregate surplus. With respect to this latter result, increasing the value of the utility's inequality aversion parameter leads to monotonic decreases in aggregate surplus. Within the context of this simulation program, then, there is a conflict between the utility's aversion to income inequality (reflected in its discounting of the surplus gains of higher income households) and its desire to maximize aggregate net surplus. This conflict is also present in the results reported by Dimopoulos (1981). Perhaps the most interesting results in Table 4.4 relate to peak-load and Ramsey pricing. The former leads to increases in aggregate net surplus in all cases except one. In contrast, Ramsey pricing never leads to an increase in aggregate net surplus. E. CONCLUSIONS The purpose of this chapter is to report on the numerical simulations undertaken to analyze the impact of a representative water utility reforming its pricing rules. There are at least two reasons why these simulations are necessary. First, the theoretical models that predict the welfare-improving properties of marginal cost based pricing typically neglect the difficulties and costs associated with administrative and monitoring activities. Second, reforming utility pricing frequently 125 requires not only changes in relative prices across user groups, but may also include price changes in different directions. Prior to reporting the numerical results of the simulations, the chapter outlines the structure of the computer simulation program and also reviews some of the theoretical issues surrounding these simulations. With respect to the latter, two significant concerns about the usefulness of conducting numerical simulations are identified. The first is the use of the change in the estimated aggregate Marshallian surplus as a consistent index of welfare change. The second arises from the observation that changes in utility pricing practices will affect households directly through changes in residential prices and indirectly through changes to the prices charged to commercial and industrial users. In both of these cases, the conditions necessary to support the proposed numerical analysis are presented and assessed. Several conclusions may be drawn from the numerical simulation results. First, the movement to alternative prices requires consideration of substantial price changes. Since prices are seen to move in opposite directions in several cases, the impact on aggregate consumption is relatively small. Second, the seasonal or peak-load pricing rules consistently generate the largest increase in aggregate consumer surplus (net of changes to annual fees and metering costs) over a variety of parameter values. This result must be qualified by pointing out that the direct impact on households of moving to seasonal pricing is predicted to be negative. This negative effect, however, is almost always outweighed by the positive impact of falling commercial and industrial output prices that are assumed to follow from decreases in the water prices facing these users. Third, in contrast to seasonal prices, Ramsey prices regularly lead to 126 substantial decreases in aggregate surplus. This result arises from the distortionary impact of raising prices above marginal costs as a means of recouping the utility's deficit. Fourth, changes to parameter values lead to changes in the predicted impacts of the alternative pricing rules. While these changes are quite significant in terms of the absolute magnitude of the predicted change in aggregate surplus, the directions in which the parameter perturbations cause the surplus measure to change conform to prior expectations. Finally, from Table 4.4 it can be seen that the pricing rule that moves industrial and commercial users to peak-load pricing and keeps residential users' per unit prices unchanged frequently leads to positive net surplus gains. The success of this 'partial' peak-load pricing rule is significant because in some senses it is the rule that would be simplest for the utility to adopt. This is because industrial and commercial users are already metered and, hence, may be made to face seasonal prices at little additional cost to the utility. 127 Table 4.1:--Pricing Rules for the Simulation Program 1} Current Practice (Base Case) Residential: p=0, annual charge -Commercial: p=pc, annual charge = Ac Industrial: p=p1; annual charge = Aj (From VWW data, no seasonal variation) 2} Average Cost Pricing p=TC/Q for residential, commercial and industrial users annual charge = 0 (no seasonal variation) 3) Peak-Load Pricing a) Long-run pP E A K=LRMC, P0FF=SRMC, for all users groups annual charge = DEFICIT/N b) Short-run pPEAK_p* w h e r e DT(p*) = capacity, for all user groups and where DT(p) is market demand P°FF=SRMC annual charge = DEFICIT/N 4} Partial Peak-Load Pricing same as (3) except residential pricing stays the same (although the residential annual charge may change since the same annual charge is charged to all users) 5) Ramsey Pricing pOFFK=SRMC.(eo/TiR), pPEAKB=LRMC.(ep/T|E) for residential users p0FFc=SRMC.(e0/nc), pPEAKc=LRMC-(8j/nc) for commercial users p^^SRMC^eo/rii), p^^URMC-CBp/ri!) for industrial users (no annual charge) (each user's mark-up over marginal cost is inversely related to absolute value of own demand elasticity) Table 4.1:--Pricing Rules for the Simulation Program (continued) Feldstein Pricing mc -11 ,p TU + 1 - (1 + O (This is equation (2.15) from Chapter 2.) A=DEFICIT/N (no variation across user groups) 129 Table 4.2:--EQuilibrium Prices and Quantities  (Standard Parameters Values) Base Case VRP VRQ VIP VIQ VCP VCQ 1 0.0 16894.10 87.30 3293.60 108.40 5979.92 2 0.0 17547.20 87.30 3639.20 108.40 6909.87 3 0.0 23158.80 87.30 4048.50 108.40 7986.76 4 0.0 14846.30 87.30 3311.80 108.40 6656.56 Annual Fee: A E = $60.14, A x = Ac = $118.61 Total Output: Q = 114272.61 thousand m 3 a| Peak-load Pricing VRPNEW VRQ Long-run VIPNEW VIQ 1 2 3 4 4.4 6.6 85.8 6.2 16853.36 17485.13 21693.97 14816.13 4.4 6.6 85.8 6.2 Annual Fee: A K = A, = Ac = $38.14 Total Output: Q = 125010.0 thousand m 3 2. b) Peak-load Pricing : Short-run VRPNEW VRQ VIPNEW 1 2 3 4 4.4 6.6 48.8 6.2 16894.10 17547.20 22325.73 14846.30 4.4 6.6 48.8 6.2 4004.90 4385.30 4056.70 3996.70 VIQ 4004.90 4385.30 6113.60 3996.70 Annual Fee: A& = A1 = AC = $46.77 Total Output: Q = 128777.6 thousand m3 3. Partial Peak-load Pricing (Except Residential) VRPNEW VRQ VIPNEW VIQ 1 2 3 4 0.0 0.0 0.0 0.0 16894.10 17547.20 23158.80 14846.30 4.4 6.6 85.8 6.2 4004.90 4385.30 4056.70 3996.70 Annual Fee: AR = A1 = AC = $76.06 Total Output: Q = 126607.9 thousand m 3 VCPNEW VCQ 4.4 6.6 85.8 6.2 VCPNEW 4.4 6.6 48.8 6.2 4.4 6.6 85.8 6.2 9018.80 10277.60 8497.80 9923.70 VCQ 9018.80 10277.60 9443.60 9923.70 VCPNEW VCQ 9018.80 10277.60 8497.80 9923.70 130 Table 4.2:--Equiribrium Prices and Quantities (continued) 4. Average Cost Pricing VRPNEW VRQ VTPNEW VIQ VCPNEW VCQ 1 89.9 16047.49 89.9 3282.10 89.9 6287.10 2 89.9 16700.54 89.9 3626.50 89.9 7264.90 3 89.9 21624.94 89.9 4034.30 89.9 8397.10 4 89.9 14027.96 89.9 3300.20 89.9 6998.50 Annual Fee: A B = ^ = Ac = 0 Total Output: Q = 111591.1 thousand m! 5. Feldstein Pricing Yields the same prices as Peak-load pricing when standard parameter values are used. VRPNEW VRQ VIPNEW VIQ VCPNEW VCQ 1 4.4 16853.10 4.4 4004.90 4.4 9018.80 2 6.6 17485.13 6.6 4385.30 6.6 10277.60 3 85.8 21693.97 85.8 4056.70 85.8 8497.80 4 6.2 14816.13 6.2 3996.70 6.2 9923.70 Annual Fee: A R : = Aj = Ac = $76.06 Total Output: Q = 125010.0 thousand m3 6. Ramsev Pricing VRPNEW VRQ VIPNEW VIQ VCPNEW VCQ 1 9.1 16420.30 7.1 5710.30 32.3 5341.40 2 13.9 17055.10 10.8 6091.00 49.1 6097.60 3 180.1 4646.90 140.1 2266.60 637.7 5298.50 4 13.1 14429.90 10.2 5575.80 46.3 5886.00 Annual Fee: Ag = Aj = Ac = 0 Total Output: Q = 94819.4 thousand m! Table 4.2:--Equilibrium Prices and Quantities (continued) 131 The seasons are: 1 equals winter, 2 equals spring, 3 equals summer, and 4 equals fall. VHP is the actual current residential price; VRQ is the associated quantity. VRPNEW is the new residential price under alternative pricing rules. VIP is the actual current industrial price; VIQ is the associated quantity. VIPNEW is the new industrial price under alternative pricing rules. VCP is the actual current commercial price; VIQ is the associated quantity. VCPNEW is the new commercial price under alternative pricing rules. 132 Table 4.3:-Changes in Aggregate Surplus1  Alternative Pricing Rules  (Standard Parameters Values) Pricing ACSR(gross)2 ACSE(net)s ACS^net)4 Rule Peak-Load Pricing -199.2 -1,332.9 2,091.0 (Long-run) (-0.5)6 (-3.5) (3.9) Peak-Load Pricing -113.7 -1,318.2 2,399.6 (Short-run) (-0.3) (-3.4) (4.5) Peak-Load Pricing (except 0 -1,506.2 1,377.2 residential) (0.0) (-4.0) (2.6) Average Cost Pricing -519.6 -1,328.4 -382.5 (-1.4) (-3.5) (-0.7) Feldstein Pricing -199.2 -1,332.9 2,091.0 (-0.5) (-3.5) (3.9) Ramsey Pricing -7,086.2 -8,770.1 -7,083.2 (-18.6) (-23.6) (-13.3) Notes: 1 - All values are in thousands of 1986 dollars. 2 ACSK(gross) = direct change in aggregate residential consumer surplus gross of changes in annual fees and gross of metering costs 3. ACSR(net) = ACS^gross) - (change in annual residential charge and metering costs) ACSr(net) = ACSR(net) + ACS/net) + ACSc(net) (where I, C, denote Industrial and Commercial user groups, respectively) Figures in parentheses are percentage changes of consumer's surplus from the base level. 133 Table 4.4:—Change in Net Surplus, Net of Metering Cost1 Pricing Rules/ PLP(LR) PLP(SR) PLP-R Average Feldstein Ramsey Parameter Values/ Base 2.09 2.40 1.38 -0.38 2.09 -7.08 o=0.6 2.40 2.69 1.55 0.30 76.05 -13.39 Broad Y Limits 3.40 3.67 2.11 4.21 3.40 -24.04 P= -0.25 51.56 55.79 41.48 26.47 51.59 -81.72 P= 0.25 -1.41 -1.39 -1.64 -1.54 -1.41 -2.07 Meter Costs +25% 1.67 1.98 0.96 -0.79 1.67 -7.66 Unit Cost +20% 0.21 0.60 -0.64 -1.70 0.21 -2.59 Unit Cost -20% 3.15 3.41 2.56 0.11 3.14 -7.08 Note: 1 - All values are in millions of 1986 dollars. 134 Endnotes 1. In the case of the water utility, the principal source of welfare gain arises from the delay in capacity expansion made possible by altering the price structure. 2. In order to compute the bounds on the error introduced by using the Marshallian surplus, information regarding the base income level, range of income elasticity values and the change in Marshallian consumer surplus (resulting from a change in the price of water) is required. Using the estimates of the income elasticities reported in Chapter Three, the change in surplus (to be reported below) and Wilhg's equation (Willig, 1976, p.589, equation (lb)), then the following bounds may be established: 0.19£-03 s  A ,"  E * 0.69£-03 where A is the change in the Marshallian consumer surplus and E is the equivalent variation. The narrowness of the computed bounds is not as strong a defence for the use of Marshallian surplus as it would appear for two reasons. First, Willig assumes that income is constant while in the simulations done here, changes in the annual charge imply changes in income. Second, the equivalent variation is derived from a complete model of the consumer's preference structure while the Marshallian surplus is derived from a single uncompensated demand curve. The Willig theorem, however, does not consider whether there is an additional source of error stemming from the partial character of the Marshallian measure. 3. Brown and Sibley refer to these indirect effects as "flow-through effects". 4. This is a probability density function which is linear for income levels between zero and some arbitrary income, y0. Above y0, the distribution is a Pareto function. 5. This procedure was conducted in two steps. First, values of the non-price explanatory variables specific to Vancouver were substituted into the estimated demand equations. Then, after making these substitutions the value of the estimated residual was increased or decreased so that the estimated dependent variable equalled its actual value. The size of the scaling factors varied, but in absolute value did not exceed 25 % of the observed value of the dependent variable. 6. The industrial water demands are estimated with cross-sectional data and the residential water demands are estimated with pooled data. It is assumed that the demand elasticities from each data set represent long-run responses to price changes. 7. Given that the demands are estimated in log form, it is not possible to set the lower bound of integration at zero. Instead, the lower bound is set at 5% of the base equilibrium quantity. 135 8. It is assumed here that the utility may discount the consumers' surplus accruing directly to high income residential consumers. It does the same for any benefits they receive from lower commercial and industrial output prices arising from the introduction of lower water prices to these sectors. This procedure assumes that households receive these indirect benefits in proportion to their income. 9. Recall that the utility is assumed to earn zero profits in the base case and the new case. 10. It is possible that increases in the price of water could induce households and firms to purchase water conserving devices (e.g., a firm could install capital equipment which would allow it to recycle water). In this case, the value of these expenditures would have to be subtracted from the change in net consumer surplus in equation (4.6). 11. Residential water meters have an expected fife of approximately 40 years. Installation and annual maintenance costs are amortized at 8 % over the life of the meter. 12. The quarterly capacity of the VWW was computed from data provided by VWW engineers regarding the utility's daily flow and storage capacities. 13. For the purposes of the simulations, marginal cost is assumed to be constant. With a translog cost function, the marginal cost of output is a function of variable input prices, quantities of fixed inputs, the level of output, and the estimated parameters representing the utility's technology. The rate of change of marginal costs (i.e., the second partial derivative of cost with respect to output) is a function of the level of output and the estimated parameters. The values of these second derivatives are calculated and tested using observations for each quarter of 1986. They are all found to be insignificantly different from zero. Details of these tests are presented in Appendix 2. 14. The Feldstein pricing rule yields equilibrium prices and quantities identical to those of the peak-load pricing rule. This is because the standard parameter set assumes that p equals zero (this parameter is the utility's inequality aversion parameter). In this case, Feldstein pricing reduces to marginal cost pricing. V. CONCLUSIONS 136 A, INTRODUCTION Three tasks are undertaken in this chapter. First, a summary of the purpose, methods and results of this thesis is presented. Second, the contributions of the thesis are outlined and evaluated. Finally, directions for future research are identified. B. SUMMARY The primary purpose of this thesis is to investigate whether reforming the pricing practices of North American water utilities can be expected to lead to an increase in efficiency. As related goals, this thesis seeks to investigate econometrically the structure of the costs of supply and of the demand for water in municipal markets. These estimates serve as the basis for measurements of the changes in aggregate consumer surplus when a representative water utility is assumed to adopt alternative pricing rules. There are several methods used to achieve these goals. The current accounting and pricing practices employed by the majority of North American municipal and regional water utilities are reviewed. An assessment of the efficiency of these practices is made. The literature concerned with developing efficient public sector pricing rules is then reviewed. Several pricing rules are drawn from this literature to serve as alternatives to water utilities' current practices. The selected pricing rules yield efficient consumption levels and share features that make them relevant to 137 water pricing. The operations of the regional and municipal water utilities in Vancouver, Canada are chosen to serve as the basis for a case study. Data regarding quarterly output, input prices and quantities, and expenditures are collected. These data are then used to estimate restricted cost functions for both utilities. These cost functions provide several important types of information: the restricted or short-run marginal costs of output, the shadow value associated with the levels of the fixed inputs and the conditional input price elasticities. In the case of the regional water supplier, the marginal cost of capacity expansion is computed from engineering data while, in the case of the municipal supplier, this parameter is obtained from an estimated long-run cost function. The estimated marginal cost of output for the regional supplier is used in the simulation program to represent the cost of obtaining one cubic metre of raw water input for the municipal utility. The regional wholesaler's recorded output price is not used as it is expected to measure the average cost of its output. The demand for water is also investigated. Quarterly data on average household income, consumption and characteristics yield estimated residential water demand functions for outdoor and indoor consumption. Establishment-level data on water purchases, output and prices allow industrial water demands to be estimated. In both cases, it is assumed that household and firm water demands are insensitive to changes in annual user charges. The efficient pricing rules are evaluated in a simulation program that is based on the estimated cost and demand equations. The program computes the equilibrium level of consumption for a given pricing rule and then computes how Marshallian 138 consumer surplus changes from the base case (i.e., the current pricing practices) to the alternative pricing rule. All of the simulations are designed to keep the utility's deficit unchanged and equal to zero. The change in aggregate consumer surplus is computed net of changes in annual user charges and also net of costs incurred by the utility as a result of moving to the efficient pricing rule. The most important example of these additional costs arises from installing and monitoring flow metering devices in private households. The results of these simulations are then tested for their sensitivity to changes in important parameter values. The results of these efforts can also be summarized. There exists a strong case to be made that the current accounting and pricing practices used by water utilities promote inefficient levels of consumption. Output prices do not reflect the marginal costs of supply and are insensitive to the time of consumption. Furthermore, the value of water itself is frequently not reflected in output prices. In terms of the estimated costs of supply, short-run marginal costs are relatively small and display some seasonal variation. Long-run marginal costs are significantly larger than short-run marginal costs - differing by a factor often in some cases. The shadow price of capital is not significantly different from zero for the regional water supplier. This suggests that the water utility is significantly over-capitalized. Possible explanations for this result are considered in Chapter Three. In contrast, the shadow price of water in storage is positive and varies positively with the demand for water. The short-run and long-run marginal costs of the retail utility also differ by a substantial margin. In this case, however, marginal costs vary procyclically. The estimated shadow price of the municipal utility's capital stock is positive and does not 139 differ greatly from an estimated user cost of capital. There is a difference in the results, then, with respect to the question of over-capitalization. It is argued in Chapter Two that the pricing behaviour of water utilities may give rise to over-capitalization. One important reason for this is that utilities do not include the current cost of capacity expansion in output prices. This distortion is likely to be largest for utilities serving rapidly growing populations. This conclusion is supported by a comparison of the population growth rates in the areas served by the GVWD and the VWW. Over the sample period the growth rate in the former was double that of the latter (British Columbia Ministry of Municipal Affairs, 1986). Both the estimated residential and industrial demands for water are found to be sensitive to price changes. For residential consumers, summer outdoor consumption is found to be more price sensitive than winter indoor use. Industrial water consumers are found to have elastic water demands despite water's share being, on average, less than 3 % of operating costs. The simulated substitution of efficient prices for current prices yields several interesting results. Seasonal or peak-load prices lead to increases in net aggregate Marshallian consumer surplus. In contrast, Ramsey prices lead to decreases in aggregate surplus. These results are robust to changes in most parameter values. To the extent that the predicted changes in aggregate Marshallian consumer's surplus may be used to guide pohcy prescriptions, these results provide qualified support for economists' long-standing criticisms of water utilities. Even in the case of Vancouver, where marginal costs of supply are relatively low, does the move to peak-load pricing imply an increase in aggregate welfare. In cities where marginal 140 costs are higher (e.g., where water must be pumped from underground sources) or where demand's cyclical fluctuations are more severe, these results would be strengthened. Furthermore, the finding of over-capitalization lends support to the economist's prediction about the inefficiency of pricing rules that are time-invariant and capacity expansion rules that treat demand growth as exogenous. The results presented here, however, do not provide unqualified support for reforming water prices. In some instances, the move to efficient prices leads to losses in aggregate welfare. In addition, some of the parameters and maintained assumptions are seen to play a crucial role in determining the final sign and magnitude of predicted welfare changes. Examples of these significant assumptions are the size of metering costs and the assumption regarding complete flow-through of pricing changes. The results reported in Chapters Three and Four may also be interpreted in light of the hypotheses presented in Chapter One. Of the five hypotheses, only the last (stating that the move to Ramsey pricing will yield a net increase in welfare) is rejected. Prices charged by both utilities do differ from their respective marginal costs. Both the marginal cost of supply and the price elasticity of demand differ from zero (although in the case of residential demands, not significantly so). Reform of the VWWs pricing rules does yield net increases in welfare. Having summarized the goals, methods and results of this project, it is worthwhile to enumerate what are its perceived contributions to the literature. Despite the fact that economists since the 1960's have criticised the practices of North American water utilities, this thesis is the first empirical assessment of reforming municipal water prices. While the pricing activities of other public utilities 141 have been exhaustively studied and other facets of water-related development projects have been assessed, the topic considered here has been unduly neglected. The development of an estimation model for the structure of the water utility's costs which is consistent with cost-minimizing behaviour follows recent developments in the literature. No other study, however, has employed quarterly time series data to estimate a restricted cost function and to use this function to derive estimates of both short-run marginal costs and the shadow prices of the fixed factors. Furthermore, no other study has attempted to test whether water utilities are over-capitalized. The modelling of water demands undertaken here has two significant features. The estimation of industrial water demands using a microeconomic data-set, a flexible form and a technique which corrects for simultaneity bias is novel. Furthermore, dktmguishing between outdoor and indoor residential demands (and price elasticities) has been done very infrequently in the extant literature. There certainly have been a large number of studies simulating the impact of utility pricing changes for aggregate consumption and consumer surplus. However, the separation of the effect on residential consumers into direct and indirect effects has rarely been attempted in the applied literature. C. FUTURE RESEARCH The preceding discussion of this project suggests some natural extensions and directions for future research. 142 One set of topics emanates from a desire to test explicitly some of the project's maintained assumptions. For example, is the distribution of household water demand identical to the distribution of household income? Also, is the elasticity of residential, commercial or industrial water demand with respect to the annual connection charge really equal to zero? If the demand for water of large firms is sensitive to the size of the utility's annual charge then efficiency dictates that the utility optimize over both its per unit price and its annual charge. If the utility fails to anticipate consumer responses to changes in the annual charge then it will likely err in its forecast of future demands. A second set of extensions would be made feasible by improved data sets or would attempt to extend the tests that gave inconclusive results. For example, the tests for over-capitalization may be extended to allow for demand uncertainty (cf. Rees, 1984, ch.10 ). Secondly, more information regarding the various dimensions of the output vector would allow a multi-output cost function to be estimated. Any resulting differences in the marginal costs of output could serve as the basis for welfare improving price (hscrimination across user groups. Finally, data regarding other benefits which might accrue from output reductions (e.g., reduced expenditures resulting from reduced flows to sewage treatment plants) could be incorporated into the welfare analysis. For example, at a recent conference on water pricing, engineers from an Ontario municipal water utility estimated that for every $1 spent to expand the utility's supply network in the near future, $9 would have to be spent to provide additional sewage treatment facilities (Tufgar et. al.. 1990). The last extension considers the assumption concerning what information the utility possesses regarding consumption levels and the distribution of demands. 143 Recent theoretical work suggests that increased information regarding demand distributions will allow the utility to design welfare-improving price schedules (Brown and Sibley, 1986). This suggests that there is a relationship between the form and amount of information possessed by the utility and the level of aggregate surplus it can generate through its pricing policies. This relationship merits further investigation. 144 BIBLIOGRAPHY Aitchison, J. and J. Brown The Lognormal Distribution. Cambridge: Cambridge University Press, 1957. 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APPENDIX 1: DATA 157 This appendix discusses the sources of information and the methods used to construct the data needed to estimate the Greater Vancouver Water District's (GVWD) restricted cost function and the Vancouver Water Work's (VWW) cost function. It also discusses the construction of the data sets used to estimate the residential and industrial water demands. A, GVWD DATA There are several sources of information. These include the GVWD's annual Operations and Maintenance Report (OMR), its annual Revenue and Expenditure  Budget (REB), and the annual Auditor's Report (AR). The GVWD also periodically updates and releases a five-year Capital Expenditure Plan which details planned capital spending and its anticipated impact on capacity. In addition to these published reports, GVWD staff provided some unpublished information. Most importantly, they supplied a set of monthly electricity billing records for the GVWD's six largest pumping stations (which account for approximately 90% of the GVWD's total electricity expenditures). For each variable included in the econometric model of the GVWD's costs, the source of the information and the actions taken to construct the variable will be described. (See Table A l . l for summary information about the data.) 1. Output Monthly recorded output to each municipality is available from OMR for 1975 158 to 1986. Observations are summed across municipalities to get total monthly output. The quarterly output variable is then constructed by summing across months in the following order: Winter (January, February, March); Spring (April, May, June); Summer (July, August, September), and Fall (October, November, December). 2. Proportion Variable This variable is constructed from the output data and is denned as the quarterly output to the municipalities furthest from the GVWD's source of supply (i.e., all municipalities with the exception of Vancouver, Burnaby, North Vancouver, and West Vancouver) divided by total quarterly output. 3. Water in Storage Daily observations on the levels of the three North Shore lakes (Capilano, Seymour, and Coquitlam) are available for the period 1978 to 1986. The recorded level of Capilano Lake is taken to be representative and the minimum recorded level in each quarter is used as the observation. Observations for this variable for the period 1975 to 1977 are obtained by regressing observed lake levels against current output and output lagged one quarter. The estimated equation is the following: 595.20 - 0.878E-3-G, + OAGZES-Q^ L * " (6.753) ( 0.117£-3) (C924JT-4) (41.1) R2 - 0.6531 F - 9713.2 159 Where 1^  is the lakelevel and Q is the recorded output at time t. This equation i s used to construct predicted lake levels when observations for output (1975-1977) are inserted. 4. C a p i t a l The GVWD AR provides data on the nominal value of gross additions to the capital stock and the beginning of period nominal book value of the existing stock. These observations are supplied for twelve subcomponents of the utility's capital plant and are available from 1961 to 1986. The total capital stock is represented by the sum of two of its subcomponents: distribution reservoirs and pumping stations. In order to construct a time series of the real value of the capital stock a perpetual-inventory method (Jorgenson, 1963) is used: (A12) - A + (i _ ^ . j ^ •** which implies that: (42.3) * „ . - X ; (1 - t i / ' A - + d - H , ) 7 ' ^ Where Kj t is the real value of the ith component of the stock in year t; I i t is the nominal value of gross investment in component i in year t; P i t is the price index for component i in year t and Uj is the (assumed constant) depreciation rate for the ith component. The depreciation rates are calculated as the inverse of the assumed life of stock 160 subcomponent. Distribution reservoirs are assumed to last 50 years and pumping stations are assumed to last 30 years. These estimates were provided by the GVWD engineering staff. The price index used is Statistics Canada's "Price Index for Capital Structures in the Water Systems Industry" (SC 13-568). All values are converted to 1981 dollars using this index. Starting values are taken to be the 1961 book values (converted to 1981 dollars). Finally, quarterly observations are constructed by finding the difference between each year's observation and apportioning one-quarter of the difference to each quarter. 5. Electricity Monthly records of electricity consumption and expenditures for the GVWD's six major pumping stations were provided by the GVWD staff. Quarterly observations were constructed by summing across stations and across months (in the same fashion as was used for the output variable). The British Columbia Public Utilities Commission supplied information regarding British Columbia Hydro's electricity rate structures for large industrial users over the period 1975 to 1986. Using the monthly pumping records a single price for electricity is computed as a weighted average across the marginal price blocks facing each pumping station. The weights used were each station's quarterly expenditure. This constructed price is a function of the level of consumption, however, and thus, an instrumental variable was constructed to avoid simultaneity bias in estimation (Judge et. al.. 1980) The instrument was computed by regressing the constructed price against two variables which are meant to summarize the information contained in the electricity rate schedules. The first (AVGPt) is the 161 average of the marginal price blocks prevailing at t and the second (AVGDt) is the average of the differences between marginal price blocks at t. The resulting equation (estimated using OLS) is the following: 0.598 + 0.068-ilVGP, + 0.059AVGD, n ' " (0.722) (0.061) (0.089) R2 - 0.8421 F - 586.8 6. Purchased Water The GVWD REB provides annual expenditures by the GVWD for provincial water use charges. As the rental fee is constant ($0.40/100,000 gallons of output per year in 1986) the annual expenditure is divided into quarterly observations in proportion to the GVWD's recorded output. The price of purchased water is taken from the Regulations to the British Columbia Water Act. A price specific to purchases by water utilities is provided in the regulations and this is the one used. 7. Materials There are three components to the materials input: chlorine; "materials, supplies, and other services"; and "gasoline, fuel, and lubricants". The GVWD annual expenditures on each of these three components are recorded in the REB over the period 1975 to 1986. For the last two components annual expenditures are converted to quarterly expenditures in proportion to recorded output. Chlorine annual expenditures are apportioned using the weights: Winter (0.20), Spring (0.20), Summer 162 (0.40), Fall (0.20). These were suggested by a GVWD engineer. A materials price index was constructed as a weighted average over the three components, with the weights being their respective quarterly expenditures. The price for chlorine was taken from an Industry trade newsletter, Chemical Marketing  Reporter, which records monthly wholesale price. Prices for the other two components were obtained from Statistics Canada price indices for gasoline and for equipment used by water and sewage utilities (SC 62-010). 8 . Labour The GVWD REB provides values for nominal annual expenditures on salaries and wages for 1975 to 1986. These observations are reduced by 25 percent to net out the costs of administrative labour (this proportion was suggested by the GVWD personnel staff) and then made quarterly by dividing by four. The GVWD was unable to supply wage information and, as a result, a quarterly time series on nominal wages was taken from Statistics Canada's time series on average weekly earnings for workers in Transportation, Communications, and Utilities industries in Vancouver (SC 72-002). 9 . Time Trend A variable set to equal 0.25 for the first observation (corresponding to the first quarter of 1975) and to 12 for the last observation (last quarter of 1986) was designated as a time trend variable. 163 B. VWW DATA There are several sources of information relating to the operations of the VWW. These include: the VWWs Engineering Department, the Canadian Union of Public Employees (Local 1004), and the City of Vancouver's Finance Department. For each variable included in the estimation model, the source of the information and the steps taken to construct the variable are described. (See Table A1.2 for summary information about the data.) 1. Output Total monthly output is available from the VWW Engineering Department for 1975 to 1986. Quarterly observations are constructed in the same fashion as the GVWD output. The VWW has recorded separately total output to metered and unmetered connections only since 1981 and, thus, only a single aggregate output measure could be used. 2. Connections Annual observations on the number of unmetered connections (primarily single family dwellings) and metered connections (commercial, industrial, and apartments) were supplied by the VWWs Finance Department for the period 1975 to 1986. These pnrmal values were converted into quarterly observations in the same fashion as the GVWD's capital stock variable. 164 8. Capital The British Columbia Ministry of Municipal Affairs' Municipal Statistics provides annual observations on the beginning of period nominal value of the VWWs capital stock and the nominal value of gross additions to it since 1971. The perpetual-inventory method was again used to construct a time series of annual observations on the real value of the capital stock. The starting point for the capital stock was taken as its value in 1971 (converted to 1981 dollars). The annual observations were converted to quarterly ones in the same fashion as the GVWD's capital stock. In the long-run specification of the VWW cost function, the price of capital is required, rather than an index of its real value. The price of capital used for the VWW estimation is the same time series as used for the test of the GVWD's overcapitalization. The construction of this variable is described in Chapter Three. 4. Purchased Water Quarterly observations on the quantity and price of water sold by the GVWD to the VWW are available from the GVWD's OMR. 5. Administrative Labour The VWW Finance Department's records provide the annual number of "Supervisory and Office Staff' and their total annual cost for the period 1975 to 1986. An average annual cost per administrative worker was computed by dividing the total cost by the number of workers. Quarterly observations were constructed by interpolating any change in annual observations equally over the four quarters. 165 6; Operations Labour The VWW Finance Department's records provide the annual number of "Operations Staff' and their total annual cost for the period 1975 to 1986. The Canadian Union of Public Employees provided information on labour agreements covering the VWW Operations Staff for the same period. These agreements specify wage levels for eight labour classifications. A time series of observations on a single wage rate was constructed as an weighted average over the eight classifications using information on the number of workers in each category from 1986. 7. Time Trend The time trend was the same as that used in the estimation of the GVWD cost function. C. RESIDENTIAL WATER DEMANDS The primary source of information is a data set on residential water consumption compiled by Environment Canada in 1987 (Gai, 1988). This data set is supplemented with information from Statistics Canada published sources. This section describes the manner of construction for each variable in the estimation model of residential water demands. (See Table A1.3 for summary information about the data.) 166 1. Water Consumption In 1987 Environment Canada compiled a data set concerning residential water consumption in Victoria, Oak Bay, and Saanich. The data set contains observations for consumption by a "representative household" which are constructed simply by dividing the recorded aggregate consumption by residential users by the number of residential connections. Total water use is divided into indoor and outdoor water uses. This is done by assuming that all water use between October and April is for indoor use. Total water use between May and September is composed of indoor uses (assumed to continue at a rate equal to the average rate over the October to April period) and outdoor uses (defined as the difference between observed purchases and imputed indoor uses). For each municipality in a given year, the data set provides two observations on consumption: the average monthly indoor water use for a representative household and the average monthly outdoor water use for the household between May and September. The total number of observations is as follows: Indoor Water Consumption # Observations Source Period 20 Victoria 1966-1985 10 Saanich 1976-1985 11 Oak Bay 1975-1985 167 Outdoor Water Consumption # Observations Source Period 21 Victoria 1966-1985 12 Saanich 1976-1985 11 Oak Bay 1975-1985 2. Price All of the municipal utilities charge a constant per unit price for residential consumption and, thus, there is no need to construct an instrument for the price variable. 3. Income The Environment Canada data set contains observations on the average nominal after-tax income per household. These estimates were obtained from Statistics Canada (Gai,1988). 4. Environmental Variables The data set also contains observations on average day-time temperature and average monthly precipitation. 5. Number of People Per Household Data on this variable are collected from Statistics Canada sources and included 168 in the Environment Canada data set (Gai,1988). D. INCOME DISTRIBUTION The distribution of household income in Vancouver (and, therefore, household water consumption) is assumed to follow a lognormal distribution (Spanos, 1985,339-341; Harrison, 1981). This distribution requires two parameters: the mean and standard deviation of the log of income. These were computed from a private compilation of 1980 Census data supplied by L. Soroka of Brock University. This compilation provides the number of households and the average income per household for 23 income classes. The income classes ranged from no income to income greater than $80,000. The mean was calculated as the weighted average of all classes' average income with weights being the number of households in each class. The standard deviation was calculated as the square root of the variance of mean income. E. INDUSTRIAL WATER DEMANDS The most important data source is Environment Canada's 1982 Industrial  Water Use Survey (IWUS). This is a national, cross-sectional survey of all manufacturing firms with significant water use (defined to be at least 4,500 m 3 per year). The survey received responses from approximately 3500 firms, but these make up approximately 90 percent of all industrial water use in Canada. Other data 169 sources include a survey conducted in 1980 of the water rate structures of all cities and towns in British Columbia. The survey was conducted by the British Columbia Water and Waste Association. The source and means of construction for each variable used in the industrial water demand estimation model is reported below. (See Table A1.3 for summary information about the data.) 1. Quantity of Water Consumed The British Columbia portion of Environment Canada's IWUS provides observations on the total annual water consumption for 372 manufacturing establishments. This water is used for a variety of purposes but water consumption for hydro-electric generation purposes is excluded. (Use of the word consumed may lead to confusion. In the water resources literature, water "use" is differentiated from water "consumption". The latter means that some of the water acquired for use is lost (eg., due to evaporation or incorporation in a final product). I will neglect this distinction here and follow the standard form in the economics literature regarding the definition of consumption.) The water intake for these firms represents 95 percent of all manufacturing water use recorded in 1981. In order to construct observations on aggregate water use by the entire industry the following steps were taken. First, the location of each establishment was determined. Then, aggregate values were constructed by summing all of the consumption observations in each census region. This procedure yielded 33 regional observations on water intake by the British Columbia manufacturing industry. 170 2. Price The total unit cost of water use is assumed to be composed of two parts. The first is "internal" and represents expenditure made after the water is acquired. The second is "external" and represents the costs of water acquisition. Data on both types of cost are available. The IWUS asked each firm for information on the annual "Operating and Maintenance" expenditures and the quantities of water used for the following categories: intake, water recirculation, water treatment prior to use, and water treatment prior to discharge. The "internal" average cost of water use for each firm is obtained by summing the average costs in each category. The "external" per unit cost of water use is established by noting the location of each firm and whether it indicated that it was self-supplied or reliant on a public water system. For self-supplied firms the marginal price of water use was determined by referring to the British Columbia government's rate schedule for industrial water withdrawals (listed in the Regulations to the British Columbia Water Act.) For publicly-supplied firms, the marginal price was determined from the industrial water rate schedule of the water utility supplying the firm. These rate schedules are listed in a survey of water utility operations conducted in 1980 by the British Columbia Water and Waste Association. The use of these marginal prices as explanatory variables in a demand equation might lead to the presence of a simultaneity bias because their values are correlated with the level of consumption. Jones and Morris (1984) and Schefter and David (1985) suggest a method of constructing an instrumental variable to avoid this problem. The procedure involves two steps. First, a single regional price observation 171 is generated as the weighted average of the firm-specific marginal prices (where the weights are firms' relative water expenditures). The regional prices are then regressed against variables constructed to represent information contained in the rate structures in each region. The resulting estimated equation which generates the instrument from the predicted values of the dependent variable is the following: j (41.5) AVGMPt - A 0 + £ AqMP- + B l C H A R G E i y-i where AVGMPj is the average marginal price in region i , MP y is the jth marginal price of the rate schedule in region i and CHARGE; is the connection fee charged under the rate schedule in region i . Equation (A1.5) is estimated using OLS. The resulting coefficients, their standard errors and the equation statistics are the following: 172 Coefficient Estimate Standard Error Ax 0.3335 0.1657 -0.1173 0.2954 A 3 0.1170 0.1683 A 4 0.1779 0.0871 CHARGE 0.0034 0.0074 Ao 1.8533 1.1781 R2=0.6782 F=73.084 The total per unit cost, or price, for each regional observation is the sum of the internal per unit cost and the external per unit cost (as represented by the instrumental variable). 3. Value Added The IWUS did not elicit information regarding each firm's level of production. As an alternative, observations on the annual level of value added generated by the total manufacturing industry in each census region of British Columbia was chosen to measure the rate of production at the aggregate level. The Statistics Canada publication, Manufacturing Industries in Canada : Sub-provincial Areas (SC 31-209) provides this information at the same level of aggregation as that used to construct the dependent variable. 173 4. Proportion Each aggregate observation contained both self-supplied and publicly-supplied firm. The proportion variable is defined as the ratio of publicly-supplied water intake relative to total observed water intake for each region. It should be noted that, for the purpose of conducting the pricing simulations, the data sets described here provided Vancouver-specific values for all variables. Table Al.l:-Summary of GVWD Cost Data 174 Variable Min Max Avg S.D. Output Restricted Cost Capital Stock Water in Storage Price of Labour Price of Materials Price of Energy Price of Water Proportion Notes: 44333.0 227230.0 19932.0 536.1 241.40 55.51 1.084 0.200 0.567 114410.0 1169700.0 35668.0 570.0 589.01 138.36 2.815 0.400 0.658 68000.0 511390.0 27735.0 561.7 441.85 97.63 1.931 0.283 0.620 15711.0 231550.0 4709.1 8.1 110.68 26.33 0.589 0.099 0.024 The units for these data are as follows: output is measured as 1000 mVqtr; capital stocks are measured in 1981 dollars; water in storage is measured in metres; costs are $/qtr; price of labour is $/week; Price of water is $/1000 ft8; price of materials is in $ per unit; price of energy is cents/kwh; price of capital is $/1000m3. All dollar figures are nominal. The proportion variable is a scalar. Table A1.2:--Summarv of VWW Cost Data 175 Variable Min Max Avg S.D. Output Connections Restricted Cost Total Cost Capital Stock Price of Operational Labour Price of Supervisory Labour Price of Water Price of Capital 16926.0 13404.0 543680.0 1861000.0 26594.0 6.55 12.31 24.4 17.56 39317.0 14315.0 1106100.0 42776.0 14.09 30.96 34.8 46.21 25693.0 14003.0 817270.0 4160400.0 2620200.0 32395.0 10.49 21.79 30.19 32.67 5397.7 167.0 193690.0 583198.0 4688.5 2.80 6.25 3.06 10.02 Notes: The units for these data are as follows: output is measured as 1000 ms/qtr; capital stocks are measured in 1981 dollars; water in storage is measured in metres; costs are $/qtr; price of labour is $/hour; price of water is $/1000 m8; price of capital is $/1000m8. All dollar figures are nominal. Connections are actual number of connections. Table Al.3:--Summary of Demand Data 176 Variable Min Max Avg S.D. (Industrial Data) Intake 52.81 1,030,800.0 599,750.0 16996.00 Value Added 25,976.00 945,690.0 179,650.0 170,430.00 Prop. 0.00 1.0 0.38 0.43 Price 2.2787 21.35 9.88 4.29 (Residential Data) Indoor Use 0.389 0.790 0.625 Income 17,191.00 35,674.00 26,582.00 # People 2.01 3.12 2.48 Rainfall 0.090 0.225 Outdoor Use 0.025 1.130 0.591 P r i c e 63.770 180.580 127.600 32.810 6673.50 0.32 Temperature -Summer 57.0 62.6 59.8 0.98 -Winter 41.5 46.1 43.8 1.31 Notes: The units for these data are as follows: intake is in 1000 m3/yr, whereas indoor and outdoor use are 1000 m3/quarter; value added is $1000 /yr, prices are in 1981 dollars /yr per 1000 m3, and annual income is in 1981 dollars. The number of people and the proportion variable are scalars, rainfall is in inches / quarter, and temperature is in Fahrenheit degrees. 177 APPENDIX 2 : TESTS The purpose of this appendix is to report on the tests undertaken regarding the structure of estimated costs. Each of the cost functions reported in the text are examined in four ways: 1. Curvature - whether the estimated function is concave in prices. 2. Monotonicity - whether the estimated function is increasing in prices. 3. Autocorrelation - whether the error structure exhibits fourth order autocorrelation. 4. Marginal Costs - whether marginal costs are constant. The first two consider whether the estimated cost functions conform to the predictions arising from the assumption of cost-minimizing behaviour. The third examines whether the estimating equation satisfies one of the assumptions of the classical regression model. The last considers whether marginal costs change with the level of output. Concavity in prices requires that a cost function's Hessian matrix of second partial derivatives (with respect to prices) be negative semi-definite. This condition is satisfied if the matrix's eigenvalues are all negative. In the case of the translog functional form, the elements of this matrix are functions of the data and thus must be computed at every data point. Monotonicity of the cost function indicates that costs are not decreasing in the levels of input prices. This condition is satisfied if the fitted cost shares are non-negative at every observation. The results of doing this calculation are summarized in Table A2.1. 178 There are several ways to test for fourth order autocorrelation (Judge et. al.. 1980). The method adopted here is that proposed by Wallis (1972). In order to test for fourth order autocorrelation Wallis demonstrates that the following test statistic, W, must be constructed from the estimated residuals (et). (A2A) W* - - £ L _ i-i Wallis also computes upper and lower bounds for the statistic W*. If W* < WL, then autocorrelation may not be rejected; if W* > W 0, autocorrelation may be rejected. If, however, WL < W* < W^ then the test is inconclusive. The results of the tests reported in Table A2.1 indicate that the cost functions uniformly satisfy monotonicity at all data points and curvature at most data points. The results with respect to the test for autocorrelation are mixed, however, the long-run VWW cost function produces a test statistic that does not allow rejection of the hypothesis that fourth order autocorrelation is present in the error structure. Table A2.2 presents the estimated values of the second derivative of the cost function with respect to output for each of the three estimated equations. The values are reported using data from the four periods of 1986. A t-test at a 5 % level of significance leads to the conclusion that the hypothesis that the second derivative is equal to zero cannot be rejected in every case. 179 Table A2.1:--Summary of Tests on the Cost Functions Test Function GVWD Short Run Cost VWW Short Run Cost VWW Long Run Cost Curvature1 192 eigenvalues 11 positive 144 eigenvalues 7 positive 192 eigenvalues 9 positive Monotonicity Fitted shares are positive at all observations Fitted shares are positive at all observations Fitted shares are positive at all observations Autocorrelation W*=1.12 reject auto correlation W*=1.303 inconclusive W*=0.669 cannot reject autocorrelation Note: Concavity requires that the eigenvalues of the Hessian matrix be negative at each data point. For the GVWD, the matrix of second order terms is a 4x4 and there are 48 observations, hence there are 192 eigenvalues calculated. The VWW cost functions are estimated with the same number of observations and the Hessian matrices for the short-run and long-run cost functions are 3x3 and 4x4, respectively. Table A2.2:-Testing for Constant Marginal Cost1 180 (1986 $/1000m3) Period GVWD VWW-SR VWW-LR Winter -1.2290 -0.1241 -0.9151 (1.3037)2 (0.2194) (0.7326) Spring -1.1510 -0.1227 -0.8400 (1.1060) (0.2479) (0.9535) Summer -1.4510 -0.1069 -0.3192 (1.3890) (0.1964) (0.4007) Fall -1.2728 -0.1233 -0.8632 (1.0550) (0.2147) (0.6884) Notes: I. s. Each entry is the value represents the estimated value of the second derivative of the cost function with respect to output. Standard errors of the estimates are in parentheses. 


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