- Email: [email protected]
Spot market pricing of water resources and efficient means of rationing water during scarcity (water pricing)
ELSEWER
Resource and Energy Economics
16 ( II9Y-I) 18Y--2 10
Spot market pricing of water resources and efficient means of rationing water during scarcity (water pricing) Jay Zarnikau * Crwter for Energy Studies, The Unit xmiy of Te..rusut Austin, 10100 Btmtet Roud Austin. TX 78758, USA
Abstract This paper advocates the application of the short-run marginal cost or spot market pricing principles to the pricing of water resources. The model developed here may provide some guidance in the design of more economically efficient water rates, in designing strategies to rationing water in times of drought or scarcity, and in the development of effective water utility planning strategies. Kq words: Water pricing; Marginal cost pricing; Spot market pricing JEL Classification:
025; L95
1. Introduction The introduction of mandatory water rationing in California and elsewhere in the early 1990s highlights the need for efficient means of rationing water during shortages. From an economic perspective, rationing schemes which arbitrarily establish limits to maximum water use fail to efficiently ration water resources. Adjusting price to optimal market clearing levels generally provides a more efficient solution, by rationing the resource to the highest value use, as revealed by the rate payers’ willingness to pay.
* The author is a Program Manager at the University of Texas Center for Energy Studies and the Director of Intcgratcd Rcsourcc Planning at Plancrgy, Inc. OYW7655/94/$07.00 0 1994 Elsevier Science B.V. All rights rcservcd SSDI 092%7655(94)00009-9
In the water utility industry, as in other utility industries, the marginal cost of supply varies temporally and spatially Technical limitations and costs of ~leter~~g and communications technology have traditionally been cited as impediments to the development of a water pricing system which would reflect the true marginal costs of supply. With recent advances in technology, these factcxs no longer pose insurmountable implementation problems, at least in applications involving large water uyers whose metering and communications equipment costs would appear small in relation to their total water hill. Rate design considerations and knowledge of customer elasticities have further posed implementation problems in the past. However, the literature addrebsing the pricing of electric power through spa t market pricing and priority pricing approachcs and the experiences of electric utilities in implementing such principles may suggest a number of clues to the design of more economically efficient water This paper provides both a formal theory of optimal spot market pricing for water resources and addresses a number of practical implementation issues. The following section describes the characteristics of water markets. Section 3 discusses present water utility rate design practice and reviews the literature on marginal cost pricing of water resources. Section 4 develops a mathematical model to demonstrate the determination of optimal water prices. Section 5 further discusses the calculation of optimal temporal and spatial water prices and contrasts the development of these optimal prices with current water utility rate design Fracticc. A number of practical implementation issues are addressed in Section 6, including metering and communications technology and partial implementations on large water users. The final section provides a summary of findings and recomme~7d;ltions.
2. Characteristics of water markets While there is great diversity among water supply systems, a number of economic characteristics appear to be pervasive. Generally regarded as natural monopolies, water utilities are typically government-owned or privately-owned and subject to regulation. Water utilities tend to he capital intensive, with large fixed costs relative to variable costs. Over one-third of a typical American water utility’s capital costs are related to water acquisition, including transmission and storage. Storage is often needed, since seasonal water demand tends to be negatively correlated with rainfall and stream flow. Another one-third of capital costs are related to the water distribution system. The utility’s cost of serving a customer tends to increase with the customer’s distance from storage and treatment facilities, since higher distribution system t’ incurred as distance increases.
While diseconomies may be related to distance, economies of scale may be related to increased system density. For example, the incremental cost of serving an additional customer or additional demand in an urban area with existing infrastructure tends to be lower than serving an additional customer in 2 sparsely inhabited rural area. Demand varies considerably over time. In the United States, daily peaks normally occur around 8 a.m. and 6 p.m. on weekdays. Seasonal variation in demand is evident as well, with seasonal demand patterns dependent upon climate. The availability of water resources to the water system varies over time, depending upon weather conditions and uses for water in alternative applications. Competing uses include recreational uses, irrigation uses, electricity generation water requirements, water transportation needs, and the requirements of other water utilities.
3. Retail pricing of water Today water is generally priced to retail customers based upon average embedded cost principles. The water utility’s revenue requirement or total cost of service is determined using historical accounting costs, adjusted for ‘known and measurable’ or ‘reasonably predictable’ changes. Uniform system-wide rates are normally established, which exhibit little variation over time. Present water prices fail to reflect the temporal and spatial variation in the costs of serving customers described in :he previous section. Marginal cost studies have found some limited application in water pricing through the development of time-of-use (primarily, seasonal) rates or usage block rate structures. Yet, less than one percent of the water rate payers in the United States are served under time-of-use rates. Marginal cost pricing principles have received limited attention in the water pricing literature. Following Vickery’s (1971) proposal for ‘flexible pricing’ of utility services, a number of authors sought to apply marginal cost pricing principles to water resource pricing. Papers advocating a marginal cost pricing approach include Hanke and Davis (1971, 1973), Sewell and Rouche (1974, Feldman (1975), Hanke (1981), Mann and Schlenger (1982), and Mann (1983, 1987). Among these papers there was a tendency to apply the ‘peak load pricing’ model, where peak load responsibility and long-run marginal costs provide a guide to rate design. Yet, little attention has previously been focused upon the determination of optimal real-time prices, recognizing the complexities of water utility system operations. The application of short-run marginal cost pricing or ‘spot market pricing’ principles was proposed by Bohn zt al. (1983) to describe the optimal pricing of utility services sold through networks. Under spot market pricing, tl by rate payers accurately re arginal costs of
192
J. Zurrrikula/ Rtwurce
und Entqy
Economics 16 (1994) 189-210
transporting the utility service. Since these costs change frequently over time as demand fluctuates and various combinations of resources with different operating costs are committed to meet the changing demand, daily or hourly price changes are often advocated. Geographical differentiation of prices is xrrggested under t theory as well. Theoretically, under such a system, economic efficiency would be achieved (provided revenue reconciliation problems could be efficiently addressed), with the price of the utility service equated with its value, and system expansion decisions determined appropriately. Spot market pricing may be contrasted with time-of-day pricing. Under time-ofday pricing, predetermined prices are established for predetermined time periods (various hours of the day or different seasons of the w*no*1 rLal /. Under s-Jot market pricing, prices are matched temporally and spatially to the actual myiginal cost experienced by the utility, rather than to historical or anticipated cost patterns. Spot market pricing principles have attained acceptance in the electric power industry. The theory of spot market pricing or real-time pricing of electricity has been refined and extended, a number of experimental tariffs have been introduced, and spot market pricing has been successfully implemented in a variety of applications involving bulk power transactions, sales of power from cog;enerators to utilities, and retail sales to large industrial customers. As demonstrated below, these same principles may be applied to water utility rate design.
eal-time pricing of water Optimal real-time prices, reflecting the short-run marginal cost of providing water to consumers. may be determined through the dual solution to the familiar industry-level constrained optimization model. The objective function is the standard welfare criterion of maximum producer and consumer surplus. A sufficient supply of water must be available to the system. Water balance constraints must hold throughout the water system. Capacity constraints must hold for storage and treatment facilities and pipelines. To simplify this exposition, capacity expansion opportunities will not be explicitly modeled ‘rere. However, binding capacity constraints may reflect the need for, and value of, additional storage, treatment, wastewater, or pipeline capacity. In this exposition, each of the ,’ variables represent the quantities of water or wastewater transported in some time period. The f parameters represent initial stocks of water. Similarly. the i parameters represent the capacity limit on the amount of water transported between facilities within a time period, while the k parameters designate a limit on the stock of water at a facility. The model developed below is designed primarily to address pricing and ptimal capacity xpansion planning. orizon considere (from t,, to T) will
be limited to less than five years. This will also permit us to neg cash flows. The water expected to be available from all sources to all reservoirs or storage tanks from the present time to a future period must be sufficie projected water demand on the utility system over that period:
A source of water (e.g., a river, reservoir, lake, or aquifer) is represented by IN. The initial stock of water at the source of raw water, I&G,,, plus cumulative additions (or reductions) from rainfall and other sources, C&,O~,Il; minus the cumulative amou!t of water taken from reservoir or source nl to storage facility j in time t, cZ,Z;Y*j,, cannot fall below the minimum desired reservoir or supply level b,. Additions to the raw water at source m, Ynl,, are likely to be exogenous, i.e., outside of the system operator’s control. The amount of water stored in each storage tank cannot exceed the capacity of that facility. The storage tank capacity constraints may thus be represented as:
c: c,,,$ represents the initial stock of water at storage facility j. ci z;,ymjl repreients tie cumulative additions of water from source m transported to ‘itorage facility j in time t, while &~,&I$ reports cumulative withdrawals of water from the storage facility. kj represents the capacity of storage facility j. Capacity constraints must hold for water treatment facilities:
$,,, represents the quantity of water transported from storage facility j to water treatment facility 12 in time C, and k, represents the capacity of water treatment facility n. It is assumed that the storage capacity of the water treatment facility is negligible. A set of water balance constraints requires that the amount of water transported from a storage facility cannot exceed the amount of water in that storage facility in each time period: t
t
104
9. Znrnikau / Resource arrd Energy Econcmic.~ I6 (I 994) 189-210
The capacity of the distribution system to transport water from the treatment facilities to ultimate customers may not be exceeded I:
t,,, represents the quantity of water transported from water treatment facility M to customer i in time t and k,,, represents the capacity of the water distribution system for moving water from treatment facilities to ultimate customers. uire that the quantity of water A second set of water balance constraints ent site is not less than the transported from all storage facilities to a tre: quantity transported from that treatment facility IO all water consumers:
The capacity of the wastcwater system may IW be exceeded ‘:
I$,, is the quantity of wastewater transported from customer i to wastewater treatment facility p in time t and ki, represents the capacity of the wastewater pipeline water system in moving wastewater from customer i to wastewater facility p. It wifl be assumed here that the quantity of wastewater transported to wastewater treatment facilities by each consumer will be less than or equal to the quantity of water used by the consumer in time t:
Alternatively, wastewater, Y&,, could be represented as some function of water usage by ultimate consumers, Y,,;[. Finally, the capacity of the wastewater treatme at facility may not be exceeded:
k,, represents the capacity of wastewater treatment facility p, The demand for water will depend upon its rice, the time of the day, the weather, and other variables. The utility’s cons ers are assumed to be profit
’ If desired, the domain of p”,, and k,, could be restricted to the mapping of feasible water distribution paths bctwccn treatment facilities and ultimate customers. ’ Ef desired, the domain of )i,,, and k ,p could bc rcstrictcd to the mapping of feasible wastewater distribution and transmission paths hetwccn ultimate custcrnxr i and wastcwatcr treatment facility p.
maximizers. Customer j’s profit, minus the cost of all nonwater variable inputs, yields the consumer’s short-run value-added fu
where Di, is the level of demand chosen by customer i at time I, and wI represents exogenous economic and weather-related factors. DIf, in turn, is a function of IV, and the price of water, plr. Fi( D,I, w,) is a monotonically decreasing function of the interior variables. Thus, the customer’s value-added function may be rewritten as:
Water supply must equal demand plus losses at all times to prevent pressure reductions, and other concerns:
Water losses resulting from leaks in the water transmission ar,ti distribution systems and evaporation, L,,ilTare assumed to be a function of water demand. The operating costs associated with providing water to the ultimate consumer will be divided into a number of components. The first component is the cost of acquiring water. A,, represents the cost of acquiring an ado::ional gallon of water from source m in time 1. The second component is the cost of water treatment. A,,, represents the cost of treating an additional gallon of water in treatment facility 12in time t. The third component of marginal cost will include the Cr
p in time period period I:
t J,
and customer prices, qr. to maximize welfare in each ti
I
j
n' I
tI
I1 I
subject to: Water Supply Constraints:
(1) Capacity Constraints for Storage Facilities:
Capacity Constraints for Water Treatment Facilities:
(3) Water Balance Constraints for Reservoir to Storage Water Transportation: QJ.l t
,I
m
I ,I
(4)
r’
istribution System Capacity Constraints: (5) Water alance Constraints Consumers:
i
for Water Transported
from Treatment
Sites to
(f9 1
astewater System Pipeline Constraints:
(7) alance Constraints:
(8)
astewater Treatmellt Facility Constraints:
Demand Constraints:
Note that all variables fm,I, & <,,I, and I$ must be non-negative. The solution to this constrained optimization problem provides optimal prices, as well as information regarding the value of additional capacity. The hgrangian to be maximized over all water and wastewater quantity variables f”,], , I$, t,,,, and )i,,, and prices, pr, is:
t
i
-
t
i
ti
t
+ CCYnt I1
\
t,,
/
t
\
t,,
m
m
(cy,,,, i -i
capacity constraints. These will be zero, cts the scarcity va ss a water supply
t
i
p
( __33 )
c,;I,,,- Q, + CL,,( 4,) I1
( 24),
?I
To derive optimal pricing conditions, it will first be demonstrated that the shadow price on the demand constraint O,,, is equal to the sum of the relevant marginai operating costs (the h terms) and the relevant shadow prices associated s). This must hold for each ion will be partially onship between price er conditions of welfare
.A
- P,pr - Ppt =
0 Y.p.t
( 28)
In order to derive the optimal water prices for customer i. the shacbw price variables require further development. The marginal costs and facilities associated with serving customer i in time t must be defineii: marginal operating cost of bringing water to the water utility system in time t, the system’s marginal operating cost of treating water in time t, the marginal operating cost associated with transporting water from the treatment facility from which incremental water would be acquired by customer i in time t, the marginal cost associated with the facility which would treat additional wastewater from customer i in time t, the shadow price that would emerge in the event that the system’s water treatment capacity was insufficient to meet the system’s needs, the shadow price that would emerge in the event that constraints on the amount of treated water transported to customer i in time l were binding, the shadow price that would emerge in the event that constraints on the amount of wastewater transported from customer i to the marginal treatment facility serving customer i were binding, the shadow price associated with a binding capacity constraint for the wastewater treatment facility serving customer i in tn Using the new shadow price variables defined specifically with rekrence LO customer i. the argi~al value of wdter to c the sum of the marginal operating costs and that customer. This is proven by co possible since eat This yields: ,, =
A:“* + A:‘* + A$*+
I
.I. Zarn iknu / Rtwww
Z(H)
WIJ Entvgy
Economics
16 C1994)
I X9-21
0
Thus, #,,, the shadow price variab e associated with the demand cmstmint, is equivalent to the social cost or. addi onal water consumption by c~st~~e~ i i time t. rice of water, pa emand, L),, is also a variable, since it is affecte Partial differentiation of the Lagrangian expression with respect to D,, yields:
Setting this derivative equal to zero yields:
(31) Assuming that customer i acts optimally, function of price. Thus:
the customer’s demand will be a
(32) c
Therefore:
Or. using the previous definition of O,,:
ustomer i is the sum of the ng the customer plus the sum of the
stomr
i’s water
s presented above suggest that water should be priced at short-r marginal cost, which should include charges to reflect the .osts of satisfying any supply system constraints, as well as the mar iated with prov water to the ultimate water consumer. This sectio es further econ interpretation of the results obtaine above, a discusses the calculation of optimal spot market water prices. In the absence of any binding capacity or water resource scarcity constraint associated with serving a particular customer, the optimal spot market price to the customer is simply the sum of the marginal operating costs associated with serving that customer, adjusted by water losses:
In the event that any capacity constraint becomes binding (e.g., the capacity of the water treatment facility serving custuAmer i is met), an additional charge will be added to the price of water quoted to the customer, thus raising the total price of water to the customer. In the pricing literature, such prices are often referred to as curtailment premiums, since they serve to ration the available water to those customers with the highest willingness to pay, which in turn is often an indication of the value of water to the customer at that particular time. As developed above, some of the curtailment premiums would be assessed to all of the customers of the water utility, while other curtailment premiums would be customer-specific. A water resource supply shortage, a shortage of water storage capacity, or a shortage of wastewater treatment facility capacity are assumed to affect all of the utility’s customers, while a bottleneck in the water transmission system might be a more localized problem, impacting only select customers. Should a shortage of water be anticipated, a further charge will be added to the price to reflect the scarcity value of water resources. This charge, as reflected in q* will be assessed upon all of the utility’s customers, to efficientlv ration available water and encourage water conservation during periods of water scarcity. The optimal spot market water prices will vary temporally and spatially. Over time, each of the components of the optimal s operating cost components, curtailment premium are subject to change. Geographically, within t marTina water tran premiums may be as the incremental wate cte tur
and the integrity of the water transmission customer.
and distribution
system serving the
The establishment of a system of temporally and spat&$ ~63::r.,latiated prices based upon calculations of short-run marginal costs requi?s r’+*r~ i%- raticn of a number of practical problems. Eight practical implemen~arlo:r : .c.xs will be discussed here: * Frequency of price changes. * Calculation of prices. * Price quotes. * Communications technology. * Revenue reconciliation. * Metering technology. * Customer responsiveness. * Revenue sources and uses. This section illuminates a variety of practical problems. The appropriate solution to these problems may prove utility specific, dependent upon the degree of variability in the utility’s costs, the sophistication of the utility’s customers, and the state (and cost) of available technology.
The model presented above suggested that the price quoted to the utility customer in time t be equated to the short-run marginal cost of serving that customer in time t, yet the duration of I was not defined. The shorter the duration of t, the better will be the match achieved between prices and changing marginal costs. Yet, greater frequency in prices changes may reduce customer satisfaction in the pricing system and may complicate billing, metering, and communications requirements. In applications of spot market pricing in the electric power industry, prices often vary on an hourly basis. However, the typical water utility probably experiences less short-run volatility in its marginal costs over time than a typical electric utility, thus reducing the need to change prices with such frequency. As a first step in the implementation of such a pricing system, perhaps it will prove sufficient to change prices twice per day. The day time price for that particular day, reflecting the anticipated average marginal cost of providing water y time hours for that particular day, would presumarged during the night time hours, generally a period wer water use a lower costs of providing water. This price differential consumer to s ift consumption from the
6.2. Calculation of prices
ot
arket pricing principies to water pricing will require water system operators to estimate time and location specific marginal costs for the water utility system, possibly for one day in advance. It may be particularly difficult to estimate appropriate curtailment premiums and scarcity value components of thl: optimal spot market price. If any curtailment premiums or scarcity value charges are required due to insufficient capacity relative to demand or an anticipated shortage of water resources, they should be set at levels sufficient to discourage enough demand to permit the system’s constraints to be satisfied, yet no higher. Thus, calculating the appropriate levels of such charges requires an understanding of the utility customers’ price elasticities of demand. The utility would presumably obtain such knowledge through experience. 6.3. Price quotes While the model presented earlier suggested that the prices quoted to customers should be the actual marginal costs present at that time, in practice it may be necessary to provide the customer with price quotes or projections at least a day in advance to provide the customer with an opportunity to react to the prices by modifying water consumption appropriately. As the notification time for the prices increases, the customer’s responsiveness is likely to increase. 6.3. Communications Under spot market pricing, water rate payers must be notified of price changes. In applications of spot market pricing in the electric power industry, a variety of communications technologies have been employed. Often rate payers receive quotes of the prices to be in effect for the next day through a computer network or cable television station. Fax machines, newspaper advertisements, or transmission of price quotes to a decoder at the customer’s site through radio signals or telephone wires might work as well, at least for customers subscribing to such services. Similar communications means could be applied in water pricing programs. 6.5. Rer lenue reconciliation As well-recognized in the utility pricing literature, there is little assurance that marginal cost pricing schemes will yield revenues to the utility exactly equal to the utility’s revenue requirement (the amount of revenue that the utility woul provided a reasonable opportunity t regulatory and pricing practices).
revenue requirement amount if its average costs exceed its marginal costs (a situation common in capital intensive industries with sufficient capacity). A variety of revenue recolrsiliation measures have been suggested as means of altering prices to better match revenues with the utility’s revenue requirements. These means include: * Adders. Adding (or subtracting) a fixed charge to each customer’s bill. This is analogous to a customer charge under traditional pricing. * Multipliers. Multiplying marginal cost based prices by a fixed facor to adjust the utility’s revenues. * Ramsey pricing. Adjusting prices in inverse proportion to the customer’s price elasticity of demand. It should be noted that each of these revenue reconciliation means will result in some loss of economic efficiency, yet the resulting pricing will still likely result in a better match of prices to marginal costs than traditional rate making approaches.
Under current water utility pricing, taking monthly readings from water meters which record cumulative usage provides sufficient information to enable the utility to calculate the customer’s total monthly water bill. Under spot market pricing, more frequent measurement of usage would be required to match consumption with the periods in which various prices were in effect. Advances in remote meter reading technology may permit water to utilities to read meters with greater frequency. For example, Indiana Bell provides a service to permit utilities to read electric, gas, and water meters via telephone. A special reading unit installed in the customer’s home automatically routes the information from the meter over telephone lines to the utility’s computer, which then processes the bill. The customer’s access to his or her telephone is not interrupted. Thus, if two different prices were in effect during a given day, two daily meter readings could be taken at the end of each pricing period. Cable television systems have also been tested as a means of remote meter reading. Further, ‘time and demand meters’ which record water demand against chronological time are expected to be commercially available by the year 24iOO4O 6.7. Customer responsir ~e~tess Spot market pricing provides accurate price signals to encourage water users to shift water consumption from relatively high price periods to lower price periods. The water rate payers’ responsiveness will determine the benefits which might
’ Personal communications 1991.
with marketing
rcprcscntatives
from Schlumberger
and ABB Kent, July
accrue to the rate payer (in the form of lower water utility bills) and to the utility (in the form of higher load factors and lower operating costs). The water consumer’s responsiveness will be constrained by a variety of factors, including lifestyle (in the case of residential customers) and production requirements (in the case of industry). Yet, advances in technology may facilitate responsiveness. For example, programmable sprinkling systems could be set to satisfy weekly yard watering demands during those periods of the week in which water prices were expected to be lowest. 6.8. Assessing the expected benefits versus implementation costs In the mathematical model presented earlier, the implementation costs associated with the transition from traditional water ratemaking to a spot market pricing system were ignored, as were the additional communications, metering, and consumer education costs imposed upon the utility and its rate payers. For such a pricing change to prove economical, the benefits realized must exceed these implementation costs. It has been the experience of electric utilities with new pricing strategies with spot market pricing features that the benefits of a well-designed pricing program may outweigh the costs for large power users. For such customers, the additional communications, metering, and consumer education costs may prove insignificant in relation to the customer’s total power bill and the savings that may be realized from even small changes in consumption. For smaller customers, implementation costs are likely to exceed the benefits that could be potentially realized. A similar result is likely to hold for water utility pricing. Consequently, it might prove advantageous to apply a spot market pricing system to large water users (industrial and commercial users, including golf courses, etc.?, while serving residential customers under a traditional pricing system ‘. 6.9. Revenue sources and uses In theory, the revenue collected through the h charges provides revenue to permit the utility to cover its operating costs ‘. The curtailment premiums provide
5 It should be noted that the application of two different types of pricing systems by the same utility will introduce a number of complications to this analysis. For example, the customers served under traditional pricing will not rcceivc the proper price signals and thus will not respond efficiently when system capacity constraints arc approached. Thus, it may hc necessary to overstate the curtailment premiums assessed upon the customers served under spot market pricing (to compensate for the lack of response from the customers served under traditional pricing) to elicit a sufficient response to assure that capacity constraints are satisfied. ’ Additional revenue beyond the amounts required to cover operating expenses will be available provided marginal operating costs exceed average operating costs.
revenue for system enhancements. The amounts collected and information regarding which customers are willing (and able) to pay the charges indicates the lever of w-vice vatu~d kq* varims cu:;tmners. and the locations at which syste ment is required. In practice. however, matching system expansion and enhanccn customers valuing various levels of service may be very difficult. Investment projects tend to be lumpy, making it difficult to match the revenues collected from a specific type of charge to a specific pPoject. It may be difficult to different customers served through the same treatment facilities and pipelines with the different levels of service suggested by their consumption behavior.
This paper proposes the design of retail water rates based upon short-run marginal cost or spot market pricing principles. Present retail water pricing, with its dependence upon historical average costs and rates which remain fixed as short-run marginal costs fluctuate, fails to efficiently ration water when water sources are insufficient or when reservoir, pipeline, and wastewater are approached, and fail!: to provide price signals conveying the true costs of providing water. Spot market pricing may provide an efficient solution to the water pricing problem. provided the practical considerations identified here are effectively addressed.
To illustrate and verify the result3 derived in this paper, a simple numerical example was constructed. This non-linear programming model - representing a hypothetical water utility system - was solved under a variety of assumed states and optimal water prices were calculated ‘. In this hypothetical water utility system, there are two water sources (a lake and a river). two storage facilities (JA and JB), two treatment facilities (TA and TB), two wastewater treatment facilities (WA and WB), and two customers (CA and ). Water losses were assumed to be negligible. Fig. 1 depicts this hypothetical system.
S CC& written mdcl
ha\ hccn construr‘tcd,
for thih cxamplc
i\ i~vi~ilahlc from t’nc author upon rcqucst. A dynamic
2s well 3s the static model dc5crihcd
hclow.
as:
4, = % + and the value of v,
ction. was assume function yields the customers’ value added func-
t
tion: D,$ - ( %/rl)D,, F,, = O-5( l/N (37) The equations developed earlier for water supply constraints, deman straints, and constraints for storage, treatment, and distri ution were retained. The following parameter values were assumed: ?ml (Initial Water Levels): Lake 1000 River 1000 h (Minimum Reservoir Level): 550 k j (Storage Capacity): JA 200 JB 350 i,ri (Distribution System Capacity):
TA TB
CA
CB
400 200
400 200
k, (Capacity of Wastewater Treatment Facilities): WA WB
400 400
D,. (Projected Customer Demand j: 325 100
CA CB cquiring Water): Lake River
0.01 0.02 -
A,, (Cost of Water Treatment): TA TB
m = LAKE or L
/
-
0 (cu,wu,WsIrwatcr
Cwlomcr
Trcnlmrnl
WD
Cll
m = RIVER or R
Fig.
1. Hypothetical
water utility
system.
h,, (Cost of Wastewater Treatment) ’ \lI
vv
0.04 0.04
A r1
WB A,,, (Water Distribution Costs): CA
CB
TA
0.00 1
T _--_
0.002
0.004 0.005
In a (static framework (Le., solving the model for a single ti e period only), three palssible states were analyzed ‘: I. NO binding constraints. 2. Insufficient water treatment capacity (k-,.,, was reduced to 323. 3. An increase in demand levels, necessitating the acquisition of raw water from a more expensive source (II,.,, was increased). The model was solved using the General Algebraic Modeling ystem (G,\MS) and the NOSS non-linear pr~~~rarnrnjn~ solver.
Calculation
of optimal prrces for t STATE 1 No birding
STATE 2
TATE 3
lnsufficicnt
Higher Icvcl
water
of demand
Ireatment capacity
rices for Customer A: (Marginal (Marginal (Marginal (Marginal (Shadow (Shadow Treatment (Shadow Treatment
Cost of Raw Water) Water Treatment Cost 1 Water Distribution Cost) Wastewater Treatment Cost) Price for Insufficient Storage Capacity) Price for Insufficient Water Capacity) Price for Insufficient Wastewatcr Capacity)
O,, (Shadow Price for Demand Constraint) p (Optimal Price)
0.0 I O.(l~,c O.OO1 0.036 0 0 0
0.01 0.05 0.00 1 0.036 0 15.008 0
0.02 0.05 0.002 0.036 0 0 0
0.097 0.097
IS.105 15.105
0. I08 0.108
0.0 1 0.05 0.004 0.036
0.01 0.05 0.004 0.036
0.02 0.05 0.00s 0.036
Optimal Prices for Customer B: (Marginal Cost of Raw Water) (Marginal Water Treatment Cost) (Marginal Water Distribution Cost) (Shadow Price fur Insufficient Wastewater Treatment Capacity) (Shadow Price for Insufficient (Shadow Price for Insufficient Treatment Capacity) (Shadow Price for Insufficient Treatment
Storage Capacity) Water
0 0
Wastewatcr
0
i; 15.008 0
0 (J 0
Capacity)
O,, (Shadow Price for Demand Constraint) p (Optimal Price)
0.100 0.100
15.108 15.108
0.111 0.111
The results of this analysis are presented in Table 1. *r th;* first state, the optimal price is simply the sum of the marginal operating cocts. pd is reflected in the optimal price when insufficient water treatment capacity prevails. Th under State 2 conditions reflects marginal operati price associated with the wat marginal cost of water under &e 3 - attribut necessitates reliance upon a water prices faced by the
Bohn, R., M. Caramanis nct%rorkh. Working
and F. Schwcppc,
Bohn, R., M. (‘;~r;inlanis time, Rand Journal Brown,
G. and C.B.
pricing of QUhlic Utility
198.3. Optimal
paper HBS X3-3 I (Hanpad University. and F.
19X-l. Optimal
Schu~ppc.
Camhridgc,
pricing
of Economics 15(3), 360-376. 1967, A socially optimum McG~i:c,
in chxtrical
pricing
SCn’iCcS sold t
MA).
policy
nctw’ork\
oven s
for a public
water
agency,
beaux Rcsourccs Research 3( I). 33-43. Feldman,
S., 1975. Peak load pricing through demand mctcring.
Association Hammer,
(Scptcmhcr)
Journal of tho American
Water Works
490-494.
M., 19X6, Water and wastcwatcr
technology
(John Wiley
and Sons Publishers.
New York.
NY). S. and R. Davis,
Hankc,
American
1971, Demand
WaterWcrks
Hanke. S. and R. Davis,
Association 1973. Potential
managcmcnt
through
(Scptemher)
555-560.
for marginal
cost pricing
rcsponsivc
pricing,
Journal
of the
in water rcsourcc managcmcnt,
Water
Rcsourccs Rcscarch O(4). X0%--83. Hanks,
S..
1981,
(Fchruary).
Oq
the
marginal
K;rhn, A., 19SX. The economics hd.. Lllllnn. P., t98? \?;j+dr* IL, pricing (January), Mann,
of
supply,
water
Water
Engineering
and
Managcmcnt
of rcgulatron
(The MIT
Press. Cambridge.
rcscarch needs, Journa!
of the American
MA). Water Works
Association
ii-27.
P. and D. Schlcngcr,
,\mcrican Mann,
cost
6,O--63.
1987, Marginal
Water Works Association
P., 1987.
Associ,ition Organixtion
Reform
(March),
in costing
cost and scasonai pricing
(March).
of water service, Journal of the
h- 1 1.
and pricing
water,
Journal
of
the American
Water
Works
43-45.
for Economic
Cooperation
and Dcvelopmcnt,
lY87,
Pricing
of water scrviccs
(OECD.
Paris, France). Rafts,lis. (i.. 1989, Arthur (‘hclsca,
I’oung
guide to water an d wastcwatcr
finance and pricing (Lewis
Publishers,
Ml ).
Rllc>,. J. and C. Schcrcr,
1979, Optimal
water pricing
and storage w;th cyclical
supply
and demand.
Water Rcsourccs Rcscarch, 23.3--23Y. Schwcppc.
F.. M. C‘aramanis.
Academic Siddiqi,
Publishers,
S. and M.L.
R. ‘I’ah(~rs and R. Bohn,
Baughman,
199 2, Rcliahility
diffcrcntiatcd
qZ-WM-1 IS-6 IEEE I992 Winter Power Meeting, Vickrcy.
W..
Managcmcnt Ziirnikau,
19x8. Spot pricing
of clcctricity
(Kluwcr
Boston, MA).
1971, Rcsponsivc
pricing
and public
real-time
New York, utility
pricing
NY, January
scrviccs,
Bell
of electricity
(Paper
26-30).
Journal
of Economics
and
Science 2, 337-34~.
J., 1990, (‘ustomcr
J l(4). 99- 1lb.
rcsponsivcncss
to real-time
pricing
of clcctricity,
The Energy
Journal
Resource and Energy Economics
16 ( II9Y-I) 18Y--2 10
Spot market pricing of water resources and efficient means of rationing water during scarcity (water pricing) Jay Zarnikau * Crwter for Energy Studies, The Unit xmiy of Te..rusut Austin, 10100 Btmtet Roud Austin. TX 78758, USA
Abstract This paper advocates the application of the short-run marginal cost or spot market pricing principles to the pricing of water resources. The model developed here may provide some guidance in the design of more economically efficient water rates, in designing strategies to rationing water in times of drought or scarcity, and in the development of effective water utility planning strategies. Kq words: Water pricing; Marginal cost pricing; Spot market pricing JEL Classification:
025; L95
1. Introduction The introduction of mandatory water rationing in California and elsewhere in the early 1990s highlights the need for efficient means of rationing water during shortages. From an economic perspective, rationing schemes which arbitrarily establish limits to maximum water use fail to efficiently ration water resources. Adjusting price to optimal market clearing levels generally provides a more efficient solution, by rationing the resource to the highest value use, as revealed by the rate payers’ willingness to pay.
* The author is a Program Manager at the University of Texas Center for Energy Studies and the Director of Intcgratcd Rcsourcc Planning at Plancrgy, Inc. OYW7655/94/$07.00 0 1994 Elsevier Science B.V. All rights rcservcd SSDI 092%7655(94)00009-9
In the water utility industry, as in other utility industries, the marginal cost of supply varies temporally and spatially Technical limitations and costs of ~leter~~g and communications technology have traditionally been cited as impediments to the development of a water pricing system which would reflect the true marginal costs of supply. With recent advances in technology, these factcxs no longer pose insurmountable implementation problems, at least in applications involving large water uyers whose metering and communications equipment costs would appear small in relation to their total water hill. Rate design considerations and knowledge of customer elasticities have further posed implementation problems in the past. However, the literature addrebsing the pricing of electric power through spa t market pricing and priority pricing approachcs and the experiences of electric utilities in implementing such principles may suggest a number of clues to the design of more economically efficient water This paper provides both a formal theory of optimal spot market pricing for water resources and addresses a number of practical implementation issues. The following section describes the characteristics of water markets. Section 3 discusses present water utility rate design practice and reviews the literature on marginal cost pricing of water resources. Section 4 develops a mathematical model to demonstrate the determination of optimal water prices. Section 5 further discusses the calculation of optimal temporal and spatial water prices and contrasts the development of these optimal prices with current water utility rate design Fracticc. A number of practical implementation issues are addressed in Section 6, including metering and communications technology and partial implementations on large water users. The final section provides a summary of findings and recomme~7d;ltions.
2. Characteristics of water markets While there is great diversity among water supply systems, a number of economic characteristics appear to be pervasive. Generally regarded as natural monopolies, water utilities are typically government-owned or privately-owned and subject to regulation. Water utilities tend to he capital intensive, with large fixed costs relative to variable costs. Over one-third of a typical American water utility’s capital costs are related to water acquisition, including transmission and storage. Storage is often needed, since seasonal water demand tends to be negatively correlated with rainfall and stream flow. Another one-third of capital costs are related to the water distribution system. The utility’s cost of serving a customer tends to increase with the customer’s distance from storage and treatment facilities, since higher distribution system t’ incurred as distance increases.
While diseconomies may be related to distance, economies of scale may be related to increased system density. For example, the incremental cost of serving an additional customer or additional demand in an urban area with existing infrastructure tends to be lower than serving an additional customer in 2 sparsely inhabited rural area. Demand varies considerably over time. In the United States, daily peaks normally occur around 8 a.m. and 6 p.m. on weekdays. Seasonal variation in demand is evident as well, with seasonal demand patterns dependent upon climate. The availability of water resources to the water system varies over time, depending upon weather conditions and uses for water in alternative applications. Competing uses include recreational uses, irrigation uses, electricity generation water requirements, water transportation needs, and the requirements of other water utilities.
3. Retail pricing of water Today water is generally priced to retail customers based upon average embedded cost principles. The water utility’s revenue requirement or total cost of service is determined using historical accounting costs, adjusted for ‘known and measurable’ or ‘reasonably predictable’ changes. Uniform system-wide rates are normally established, which exhibit little variation over time. Present water prices fail to reflect the temporal and spatial variation in the costs of serving customers described in :he previous section. Marginal cost studies have found some limited application in water pricing through the development of time-of-use (primarily, seasonal) rates or usage block rate structures. Yet, less than one percent of the water rate payers in the United States are served under time-of-use rates. Marginal cost pricing principles have received limited attention in the water pricing literature. Following Vickery’s (1971) proposal for ‘flexible pricing’ of utility services, a number of authors sought to apply marginal cost pricing principles to water resource pricing. Papers advocating a marginal cost pricing approach include Hanke and Davis (1971, 1973), Sewell and Rouche (1974, Feldman (1975), Hanke (1981), Mann and Schlenger (1982), and Mann (1983, 1987). Among these papers there was a tendency to apply the ‘peak load pricing’ model, where peak load responsibility and long-run marginal costs provide a guide to rate design. Yet, little attention has previously been focused upon the determination of optimal real-time prices, recognizing the complexities of water utility system operations. The application of short-run marginal cost pricing or ‘spot market pricing’ principles was proposed by Bohn zt al. (1983) to describe the optimal pricing of utility services sold through networks. Under spot market pricing, tl by rate payers accurately re arginal costs of
192
J. Zurrrikula/ Rtwurce
und Entqy
Economics 16 (1994) 189-210
transporting the utility service. Since these costs change frequently over time as demand fluctuates and various combinations of resources with different operating costs are committed to meet the changing demand, daily or hourly price changes are often advocated. Geographical differentiation of prices is xrrggested under t theory as well. Theoretically, under such a system, economic efficiency would be achieved (provided revenue reconciliation problems could be efficiently addressed), with the price of the utility service equated with its value, and system expansion decisions determined appropriately. Spot market pricing may be contrasted with time-of-day pricing. Under time-ofday pricing, predetermined prices are established for predetermined time periods (various hours of the day or different seasons of the w*no*1 rLal /. Under s-Jot market pricing, prices are matched temporally and spatially to the actual myiginal cost experienced by the utility, rather than to historical or anticipated cost patterns. Spot market pricing principles have attained acceptance in the electric power industry. The theory of spot market pricing or real-time pricing of electricity has been refined and extended, a number of experimental tariffs have been introduced, and spot market pricing has been successfully implemented in a variety of applications involving bulk power transactions, sales of power from cog;enerators to utilities, and retail sales to large industrial customers. As demonstrated below, these same principles may be applied to water utility rate design.
eal-time pricing of water Optimal real-time prices, reflecting the short-run marginal cost of providing water to consumers. may be determined through the dual solution to the familiar industry-level constrained optimization model. The objective function is the standard welfare criterion of maximum producer and consumer surplus. A sufficient supply of water must be available to the system. Water balance constraints must hold throughout the water system. Capacity constraints must hold for storage and treatment facilities and pipelines. To simplify this exposition, capacity expansion opportunities will not be explicitly modeled ‘rere. However, binding capacity constraints may reflect the need for, and value of, additional storage, treatment, wastewater, or pipeline capacity. In this exposition, each of the ,’ variables represent the quantities of water or wastewater transported in some time period. The f parameters represent initial stocks of water. Similarly. the i parameters represent the capacity limit on the amount of water transported between facilities within a time period, while the k parameters designate a limit on the stock of water at a facility. The model developed below is designed primarily to address pricing and ptimal capacity xpansion planning. orizon considere (from t,, to T) will
be limited to less than five years. This will also permit us to neg cash flows. The water expected to be available from all sources to all reservoirs or storage tanks from the present time to a future period must be sufficie projected water demand on the utility system over that period:
A source of water (e.g., a river, reservoir, lake, or aquifer) is represented by IN. The initial stock of water at the source of raw water, I&G,,, plus cumulative additions (or reductions) from rainfall and other sources, C&,O~,Il; minus the cumulative amou!t of water taken from reservoir or source nl to storage facility j in time t, cZ,Z;Y*j,, cannot fall below the minimum desired reservoir or supply level b,. Additions to the raw water at source m, Ynl,, are likely to be exogenous, i.e., outside of the system operator’s control. The amount of water stored in each storage tank cannot exceed the capacity of that facility. The storage tank capacity constraints may thus be represented as:
c: c,,,$ represents the initial stock of water at storage facility j. ci z;,ymjl repreients tie cumulative additions of water from source m transported to ‘itorage facility j in time t, while &~,&I$ reports cumulative withdrawals of water from the storage facility. kj represents the capacity of storage facility j. Capacity constraints must hold for water treatment facilities:
$,,, represents the quantity of water transported from storage facility j to water treatment facility 12 in time C, and k, represents the capacity of water treatment facility n. It is assumed that the storage capacity of the water treatment facility is negligible. A set of water balance constraints requires that the amount of water transported from a storage facility cannot exceed the amount of water in that storage facility in each time period: t
t
104
9. Znrnikau / Resource arrd Energy Econcmic.~ I6 (I 994) 189-210
The capacity of the distribution system to transport water from the treatment facilities to ultimate customers may not be exceeded I:
t,,, represents the quantity of water transported from water treatment facility M to customer i in time t and k,,, represents the capacity of the water distribution system for moving water from treatment facilities to ultimate customers. uire that the quantity of water A second set of water balance constraints ent site is not less than the transported from all storage facilities to a tre: quantity transported from that treatment facility IO all water consumers:
The capacity of the wastcwater system may IW be exceeded ‘:
I$,, is the quantity of wastewater transported from customer i to wastewater treatment facility p in time t and ki, represents the capacity of the wastewater pipeline water system in moving wastewater from customer i to wastewater facility p. It wifl be assumed here that the quantity of wastewater transported to wastewater treatment facilities by each consumer will be less than or equal to the quantity of water used by the consumer in time t:
Alternatively, wastewater, Y&,, could be represented as some function of water usage by ultimate consumers, Y,,;[. Finally, the capacity of the wastewater treatme at facility may not be exceeded:
k,, represents the capacity of wastewater treatment facility p, The demand for water will depend upon its rice, the time of the day, the weather, and other variables. The utility’s cons ers are assumed to be profit
’ If desired, the domain of p”,, and k,, could be restricted to the mapping of feasible water distribution paths bctwccn treatment facilities and ultimate customers. ’ Ef desired, the domain of )i,,, and k ,p could bc rcstrictcd to the mapping of feasible wastewater distribution and transmission paths hetwccn ultimate custcrnxr i and wastcwatcr treatment facility p.
maximizers. Customer j’s profit, minus the cost of all nonwater variable inputs, yields the consumer’s short-run value-added fu
where Di, is the level of demand chosen by customer i at time I, and wI represents exogenous economic and weather-related factors. DIf, in turn, is a function of IV, and the price of water, plr. Fi( D,I, w,) is a monotonically decreasing function of the interior variables. Thus, the customer’s value-added function may be rewritten as:
Water supply must equal demand plus losses at all times to prevent pressure reductions, and other concerns:
Water losses resulting from leaks in the water transmission ar,ti distribution systems and evaporation, L,,ilTare assumed to be a function of water demand. The operating costs associated with providing water to the ultimate consumer will be divided into a number of components. The first component is the cost of acquiring water. A,, represents the cost of acquiring an ado::ional gallon of water from source m in time 1. The second component is the cost of water treatment. A,,, represents the cost of treating an additional gallon of water in treatment facility 12in time t. The third component of marginal cost will include the Cr
p in time period period I:
t J,
and customer prices, qr. to maximize welfare in each ti
I
j
n' I
tI
I1 I
subject to: Water Supply Constraints:
(1) Capacity Constraints for Storage Facilities:
Capacity Constraints for Water Treatment Facilities:
(3) Water Balance Constraints for Reservoir to Storage Water Transportation: QJ.l t
,I
m
I ,I
(4)
r’
istribution System Capacity Constraints: (5) Water alance Constraints Consumers:
i
for Water Transported
from Treatment
Sites to
(f9 1
astewater System Pipeline Constraints:
(7) alance Constraints:
(8)
astewater Treatmellt Facility Constraints:
Demand Constraints:
Note that all variables fm,I, & <,,I, and I$ must be non-negative. The solution to this constrained optimization problem provides optimal prices, as well as information regarding the value of additional capacity. The hgrangian to be maximized over all water and wastewater quantity variables f”,], , I$, t,,,, and )i,,, and prices, pr, is:
t
i
-
t
i
ti
t
+ CCYnt I1
\
t,,
/
t
\
t,,
m
m
(cy,,,, i -i
capacity constraints. These will be zero, cts the scarcity va ss a water supply
t
i
p
( __33 )
c,;I,,,- Q, + CL,,( 4,) I1
( 24),
?I
To derive optimal pricing conditions, it will first be demonstrated that the shadow price on the demand constraint O,,, is equal to the sum of the relevant marginai operating costs (the h terms) and the relevant shadow prices associated s). This must hold for each ion will be partially onship between price er conditions of welfare
.A
- P,pr - Ppt =
0 Y.p.t
( 28)
In order to derive the optimal water prices for customer i. the shacbw price variables require further development. The marginal costs and facilities associated with serving customer i in time t must be defineii: marginal operating cost of bringing water to the water utility system in time t, the system’s marginal operating cost of treating water in time t, the marginal operating cost associated with transporting water from the treatment facility from which incremental water would be acquired by customer i in time t, the marginal cost associated with the facility which would treat additional wastewater from customer i in time t, the shadow price that would emerge in the event that the system’s water treatment capacity was insufficient to meet the system’s needs, the shadow price that would emerge in the event that constraints on the amount of treated water transported to customer i in time l were binding, the shadow price that would emerge in the event that constraints on the amount of wastewater transported from customer i to the marginal treatment facility serving customer i were binding, the shadow price associated with a binding capacity constraint for the wastewater treatment facility serving customer i in tn Using the new shadow price variables defined specifically with rekrence LO customer i. the argi~al value of wdter to c the sum of the marginal operating costs and that customer. This is proven by co possible since eat This yields: ,, =
A:“* + A:‘* + A$*+
I
.I. Zarn iknu / Rtwww
Z(H)
WIJ Entvgy
Economics
16 C1994)
I X9-21
0
Thus, #,,, the shadow price variab e associated with the demand cmstmint, is equivalent to the social cost or. addi onal water consumption by c~st~~e~ i i time t. rice of water, pa emand, L),, is also a variable, since it is affecte Partial differentiation of the Lagrangian expression with respect to D,, yields:
Setting this derivative equal to zero yields:
(31) Assuming that customer i acts optimally, function of price. Thus:
the customer’s demand will be a
(32) c
Therefore:
Or. using the previous definition of O,,:
ustomer i is the sum of the ng the customer plus the sum of the
stomr
i’s water
s presented above suggest that water should be priced at short-r marginal cost, which should include charges to reflect the .osts of satisfying any supply system constraints, as well as the mar iated with prov water to the ultimate water consumer. This sectio es further econ interpretation of the results obtaine above, a discusses the calculation of optimal spot market water prices. In the absence of any binding capacity or water resource scarcity constraint associated with serving a particular customer, the optimal spot market price to the customer is simply the sum of the marginal operating costs associated with serving that customer, adjusted by water losses:
In the event that any capacity constraint becomes binding (e.g., the capacity of the water treatment facility serving custuAmer i is met), an additional charge will be added to the price of water quoted to the customer, thus raising the total price of water to the customer. In the pricing literature, such prices are often referred to as curtailment premiums, since they serve to ration the available water to those customers with the highest willingness to pay, which in turn is often an indication of the value of water to the customer at that particular time. As developed above, some of the curtailment premiums would be assessed to all of the customers of the water utility, while other curtailment premiums would be customer-specific. A water resource supply shortage, a shortage of water storage capacity, or a shortage of wastewater treatment facility capacity are assumed to affect all of the utility’s customers, while a bottleneck in the water transmission system might be a more localized problem, impacting only select customers. Should a shortage of water be anticipated, a further charge will be added to the price to reflect the scarcity value of water resources. This charge, as reflected in q* will be assessed upon all of the utility’s customers, to efficientlv ration available water and encourage water conservation during periods of water scarcity. The optimal spot market water prices will vary temporally and spatially. Over time, each of the components of the optimal s operating cost components, curtailment premium are subject to change. Geographically, within t marTina water tran premiums may be as the incremental wate cte tur
and the integrity of the water transmission customer.
and distribution
system serving the
The establishment of a system of temporally and spat&$ ~63::r.,latiated prices based upon calculations of short-run marginal costs requi?s r’+*r~ i%- raticn of a number of practical problems. Eight practical implemen~arlo:r : .c.xs will be discussed here: * Frequency of price changes. * Calculation of prices. * Price quotes. * Communications technology. * Revenue reconciliation. * Metering technology. * Customer responsiveness. * Revenue sources and uses. This section illuminates a variety of practical problems. The appropriate solution to these problems may prove utility specific, dependent upon the degree of variability in the utility’s costs, the sophistication of the utility’s customers, and the state (and cost) of available technology.
The model presented above suggested that the price quoted to the utility customer in time t be equated to the short-run marginal cost of serving that customer in time t, yet the duration of I was not defined. The shorter the duration of t, the better will be the match achieved between prices and changing marginal costs. Yet, greater frequency in prices changes may reduce customer satisfaction in the pricing system and may complicate billing, metering, and communications requirements. In applications of spot market pricing in the electric power industry, prices often vary on an hourly basis. However, the typical water utility probably experiences less short-run volatility in its marginal costs over time than a typical electric utility, thus reducing the need to change prices with such frequency. As a first step in the implementation of such a pricing system, perhaps it will prove sufficient to change prices twice per day. The day time price for that particular day, reflecting the anticipated average marginal cost of providing water y time hours for that particular day, would presumarged during the night time hours, generally a period wer water use a lower costs of providing water. This price differential consumer to s ift consumption from the
6.2. Calculation of prices
ot
arket pricing principies to water pricing will require water system operators to estimate time and location specific marginal costs for the water utility system, possibly for one day in advance. It may be particularly difficult to estimate appropriate curtailment premiums and scarcity value components of thl: optimal spot market price. If any curtailment premiums or scarcity value charges are required due to insufficient capacity relative to demand or an anticipated shortage of water resources, they should be set at levels sufficient to discourage enough demand to permit the system’s constraints to be satisfied, yet no higher. Thus, calculating the appropriate levels of such charges requires an understanding of the utility customers’ price elasticities of demand. The utility would presumably obtain such knowledge through experience. 6.3. Price quotes While the model presented earlier suggested that the prices quoted to customers should be the actual marginal costs present at that time, in practice it may be necessary to provide the customer with price quotes or projections at least a day in advance to provide the customer with an opportunity to react to the prices by modifying water consumption appropriately. As the notification time for the prices increases, the customer’s responsiveness is likely to increase. 6.3. Communications Under spot market pricing, water rate payers must be notified of price changes. In applications of spot market pricing in the electric power industry, a variety of communications technologies have been employed. Often rate payers receive quotes of the prices to be in effect for the next day through a computer network or cable television station. Fax machines, newspaper advertisements, or transmission of price quotes to a decoder at the customer’s site through radio signals or telephone wires might work as well, at least for customers subscribing to such services. Similar communications means could be applied in water pricing programs. 6.5. Rer lenue reconciliation As well-recognized in the utility pricing literature, there is little assurance that marginal cost pricing schemes will yield revenues to the utility exactly equal to the utility’s revenue requirement (the amount of revenue that the utility woul provided a reasonable opportunity t regulatory and pricing practices).
revenue requirement amount if its average costs exceed its marginal costs (a situation common in capital intensive industries with sufficient capacity). A variety of revenue recolrsiliation measures have been suggested as means of altering prices to better match revenues with the utility’s revenue requirements. These means include: * Adders. Adding (or subtracting) a fixed charge to each customer’s bill. This is analogous to a customer charge under traditional pricing. * Multipliers. Multiplying marginal cost based prices by a fixed facor to adjust the utility’s revenues. * Ramsey pricing. Adjusting prices in inverse proportion to the customer’s price elasticity of demand. It should be noted that each of these revenue reconciliation means will result in some loss of economic efficiency, yet the resulting pricing will still likely result in a better match of prices to marginal costs than traditional rate making approaches.
Under current water utility pricing, taking monthly readings from water meters which record cumulative usage provides sufficient information to enable the utility to calculate the customer’s total monthly water bill. Under spot market pricing, more frequent measurement of usage would be required to match consumption with the periods in which various prices were in effect. Advances in remote meter reading technology may permit water to utilities to read meters with greater frequency. For example, Indiana Bell provides a service to permit utilities to read electric, gas, and water meters via telephone. A special reading unit installed in the customer’s home automatically routes the information from the meter over telephone lines to the utility’s computer, which then processes the bill. The customer’s access to his or her telephone is not interrupted. Thus, if two different prices were in effect during a given day, two daily meter readings could be taken at the end of each pricing period. Cable television systems have also been tested as a means of remote meter reading. Further, ‘time and demand meters’ which record water demand against chronological time are expected to be commercially available by the year 24iOO4O 6.7. Customer responsir ~e~tess Spot market pricing provides accurate price signals to encourage water users to shift water consumption from relatively high price periods to lower price periods. The water rate payers’ responsiveness will determine the benefits which might
’ Personal communications 1991.
with marketing
rcprcscntatives
from Schlumberger
and ABB Kent, July
accrue to the rate payer (in the form of lower water utility bills) and to the utility (in the form of higher load factors and lower operating costs). The water consumer’s responsiveness will be constrained by a variety of factors, including lifestyle (in the case of residential customers) and production requirements (in the case of industry). Yet, advances in technology may facilitate responsiveness. For example, programmable sprinkling systems could be set to satisfy weekly yard watering demands during those periods of the week in which water prices were expected to be lowest. 6.8. Assessing the expected benefits versus implementation costs In the mathematical model presented earlier, the implementation costs associated with the transition from traditional water ratemaking to a spot market pricing system were ignored, as were the additional communications, metering, and consumer education costs imposed upon the utility and its rate payers. For such a pricing change to prove economical, the benefits realized must exceed these implementation costs. It has been the experience of electric utilities with new pricing strategies with spot market pricing features that the benefits of a well-designed pricing program may outweigh the costs for large power users. For such customers, the additional communications, metering, and consumer education costs may prove insignificant in relation to the customer’s total power bill and the savings that may be realized from even small changes in consumption. For smaller customers, implementation costs are likely to exceed the benefits that could be potentially realized. A similar result is likely to hold for water utility pricing. Consequently, it might prove advantageous to apply a spot market pricing system to large water users (industrial and commercial users, including golf courses, etc.?, while serving residential customers under a traditional pricing system ‘. 6.9. Revenue sources and uses In theory, the revenue collected through the h charges provides revenue to permit the utility to cover its operating costs ‘. The curtailment premiums provide
5 It should be noted that the application of two different types of pricing systems by the same utility will introduce a number of complications to this analysis. For example, the customers served under traditional pricing will not rcceivc the proper price signals and thus will not respond efficiently when system capacity constraints arc approached. Thus, it may hc necessary to overstate the curtailment premiums assessed upon the customers served under spot market pricing (to compensate for the lack of response from the customers served under traditional pricing) to elicit a sufficient response to assure that capacity constraints are satisfied. ’ Additional revenue beyond the amounts required to cover operating expenses will be available provided marginal operating costs exceed average operating costs.
revenue for system enhancements. The amounts collected and information regarding which customers are willing (and able) to pay the charges indicates the lever of w-vice vatu~d kq* varims cu:;tmners. and the locations at which syste ment is required. In practice. however, matching system expansion and enhanccn customers valuing various levels of service may be very difficult. Investment projects tend to be lumpy, making it difficult to match the revenues collected from a specific type of charge to a specific pPoject. It may be difficult to different customers served through the same treatment facilities and pipelines with the different levels of service suggested by their consumption behavior.
This paper proposes the design of retail water rates based upon short-run marginal cost or spot market pricing principles. Present retail water pricing, with its dependence upon historical average costs and rates which remain fixed as short-run marginal costs fluctuate, fails to efficiently ration water when water sources are insufficient or when reservoir, pipeline, and wastewater are approached, and fail!: to provide price signals conveying the true costs of providing water. Spot market pricing may provide an efficient solution to the water pricing problem. provided the practical considerations identified here are effectively addressed.
To illustrate and verify the result3 derived in this paper, a simple numerical example was constructed. This non-linear programming model - representing a hypothetical water utility system - was solved under a variety of assumed states and optimal water prices were calculated ‘. In this hypothetical water utility system, there are two water sources (a lake and a river). two storage facilities (JA and JB), two treatment facilities (TA and TB), two wastewater treatment facilities (WA and WB), and two customers (CA and ). Water losses were assumed to be negligible. Fig. 1 depicts this hypothetical system.
S CC& written mdcl
ha\ hccn construr‘tcd,
for thih cxamplc
i\ i~vi~ilahlc from t’nc author upon rcqucst. A dynamic
2s well 3s the static model dc5crihcd
hclow.
as:
4, = % + and the value of v,
ction. was assume function yields the customers’ value added func-
t
tion: D,$ - ( %/rl)D,, F,, = O-5( l/N (37) The equations developed earlier for water supply constraints, deman straints, and constraints for storage, treatment, and distri ution were retained. The following parameter values were assumed: ?ml (Initial Water Levels): Lake 1000 River 1000 h (Minimum Reservoir Level): 550 k j (Storage Capacity): JA 200 JB 350 i,ri (Distribution System Capacity):
TA TB
CA
CB
400 200
400 200
k, (Capacity of Wastewater Treatment Facilities): WA WB
400 400
D,. (Projected Customer Demand j: 325 100
CA CB cquiring Water): Lake River
0.01 0.02 -
A,, (Cost of Water Treatment): TA TB
m = LAKE or L
/
-
0 (cu,wu,WsIrwatcr
Cwlomcr
Trcnlmrnl
WD
Cll
m = RIVER or R
Fig.
1. Hypothetical
water utility
system.
h,, (Cost of Wastewater Treatment) ’ \lI
vv
0.04 0.04
A r1
WB A,,, (Water Distribution Costs): CA
CB
TA
0.00 1
T _--_
0.002
0.004 0.005
In a (static framework (Le., solving the model for a single ti e period only), three palssible states were analyzed ‘: I. NO binding constraints. 2. Insufficient water treatment capacity (k-,.,, was reduced to 323. 3. An increase in demand levels, necessitating the acquisition of raw water from a more expensive source (II,.,, was increased). The model was solved using the General Algebraic Modeling ystem (G,\MS) and the NOSS non-linear pr~~~rarnrnjn~ solver.
Calculation
of optimal prrces for t STATE 1 No birding
STATE 2
TATE 3
lnsufficicnt
Higher Icvcl
water
of demand
Ireatment capacity
rices for Customer A: (Marginal (Marginal (Marginal (Marginal (Shadow (Shadow Treatment (Shadow Treatment
Cost of Raw Water) Water Treatment Cost 1 Water Distribution Cost) Wastewater Treatment Cost) Price for Insufficient Storage Capacity) Price for Insufficient Water Capacity) Price for Insufficient Wastewatcr Capacity)
O,, (Shadow Price for Demand Constraint) p (Optimal Price)
0.0 I O.(l~,c O.OO1 0.036 0 0 0
0.01 0.05 0.00 1 0.036 0 15.008 0
0.02 0.05 0.002 0.036 0 0 0
0.097 0.097
IS.105 15.105
0. I08 0.108
0.0 1 0.05 0.004 0.036
0.01 0.05 0.004 0.036
0.02 0.05 0.00s 0.036
Optimal Prices for Customer B: (Marginal Cost of Raw Water) (Marginal Water Treatment Cost) (Marginal Water Distribution Cost) (Shadow Price fur Insufficient Wastewater Treatment Capacity) (Shadow Price for Insufficient (Shadow Price for Insufficient Treatment Capacity) (Shadow Price for Insufficient Treatment
Storage Capacity) Water
0 0
Wastewatcr
0
i; 15.008 0
0 (J 0
Capacity)
O,, (Shadow Price for Demand Constraint) p (Optimal Price)
0.100 0.100
15.108 15.108
0.111 0.111
The results of this analysis are presented in Table 1. *r th;* first state, the optimal price is simply the sum of the marginal operating cocts. pd is reflected in the optimal price when insufficient water treatment capacity prevails. Th under State 2 conditions reflects marginal operati price associated with the wat marginal cost of water under &e 3 - attribut necessitates reliance upon a water prices faced by the
Bohn, R., M. Caramanis nct%rorkh. Working
and F. Schwcppc,
Bohn, R., M. (‘;~r;inlanis time, Rand Journal Brown,
G. and C.B.
pricing of QUhlic Utility
198.3. Optimal
paper HBS X3-3 I (Hanpad University. and F.
19X-l. Optimal
Schu~ppc.
Camhridgc,
pricing
of Economics 15(3), 360-376. 1967, A socially optimum McG~i:c,
in chxtrical
pricing
SCn’iCcS sold t
MA).
policy
nctw’ork\
oven s
for a public
water
agency,
beaux Rcsourccs Research 3( I). 33-43. Feldman,
S., 1975. Peak load pricing through demand mctcring.
Association Hammer,
(Scptcmhcr)
Journal of tho American
Water Works
490-494.
M., 19X6, Water and wastcwatcr
technology
(John Wiley
and Sons Publishers.
New York.
NY). S. and R. Davis,
Hankc,
American
1971, Demand
WaterWcrks
Hanke. S. and R. Davis,
Association 1973. Potential
managcmcnt
through
(Scptemher)
555-560.
for marginal
cost pricing
rcsponsivc
pricing,
Journal
of the
in water rcsourcc managcmcnt,
Water
Rcsourccs Rcscarch O(4). X0%--83. Hanks,
S..
1981,
(Fchruary).
Oq
the
marginal
K;rhn, A., 19SX. The economics hd.. Lllllnn. P., t98? \?;j+dr* IL, pricing (January), Mann,
of
supply,
water
Water
Engineering
and
Managcmcnt
of rcgulatron
(The MIT
Press. Cambridge.
rcscarch needs, Journa!
of the American
MA). Water Works
Association
ii-27.
P. and D. Schlcngcr,
,\mcrican Mann,
cost
6,O--63.
1987, Marginal
Water Works Association
P., 1987.
Associ,ition Organixtion
Reform
(March),
in costing
cost and scasonai pricing
(March).
of water service, Journal of the
h- 1 1.
and pricing
water,
Journal
of
the American
Water
Works
43-45.
for Economic
Cooperation
and Dcvelopmcnt,
lY87,
Pricing
of water scrviccs
(OECD.
Paris, France). Rafts,lis. (i.. 1989, Arthur (‘hclsca,
I’oung
guide to water an d wastcwatcr
finance and pricing (Lewis
Publishers,
Ml ).
Rllc>,. J. and C. Schcrcr,
1979, Optimal
water pricing
and storage w;th cyclical
supply
and demand.
Water Rcsourccs Rcscarch, 23.3--23Y. Schwcppc.
F.. M. C‘aramanis.
Academic Siddiqi,
Publishers,
S. and M.L.
R. ‘I’ah(~rs and R. Bohn,
Baughman,
199 2, Rcliahility
diffcrcntiatcd
qZ-WM-1 IS-6 IEEE I992 Winter Power Meeting, Vickrcy.
W..
Managcmcnt Ziirnikau,
19x8. Spot pricing
of clcctricity
(Kluwcr
Boston, MA).
1971, Rcsponsivc
pricing
and public
real-time
New York, utility
pricing
NY, January
scrviccs,
Bell
of electricity
(Paper
26-30).
Journal
of Economics
and
Science 2, 337-34~.
J., 1990, (‘ustomcr
J l(4). 99- 1lb.
rcsponsivcncss
to real-time
pricing
of clcctricity,
The Energy
Journal