Subsidy free pricing of interruptible service contracts

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UTTERWORTH I N E M A

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Energy Economics, Vol. 17, No. 1, pp. 53-58, 1995 Copyright ( 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0140-9883/95 $10.00 + 0.00

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Subsidy free pricing of interruptible service contracts T Randolph Beard, George H Sweeney and Daniel M Gropper This article anah'zes the issue o f cross-subsidy #7 the pricin9 structure o / a utility ~?[~erin9 interruptible service contracts. Utilizin 9 a s#nple theoretical model in which available capacit)' is subject to random shocks, we derive a set o f su[-[icient conditions [br the presence o / subsidies in the pricin 9 ~t~ priority service. We find that low priority customers receive no subsidies whenever they pay prices in excess o / variable costs. A test.[or subsidies to hiyh priority custonwrs is derived that can be implemented utiliziny available information. Extensions o / the analysis are discussed. Kevwords: Subsidy, Public utility prices, Interruptible service

While the rationing of scarce goods without recourse to spot markets is a very old issue in economics, recent years have seen greatly heightened interest in sophisticated pricing strategies intended to resolve transitory supply shortfalls by ex ante self-selection among customers. In the public utility area, 'priority' or, more commonly, 'interruptible service' contracts became quite common in the 1980s in response to elevated capacity costs, concerns over nuclear power safety and other motives for resistance to capacity expansion. By 1992 virtually all large investor owned utilities in the USA utilized some form of interruptible pricing contracts. 1 In a typical interruptible service system, customers (primarily larger industrial users) are offered different priorities of service from which they self-select. Customers selecting lower priority receive reduced rates in exchange for their acceptance of service discontinuance in the event of a supply shortfall due, for example, to unexpected loss in on-line capacity. Hence, in exchange for reduced service reliability, low priority users enjoy diminished costs in those instances when sufficient power is available. T Randolph Beard and Daniel M Gropper are with Auburn University; George H Sweeneyis with Vanderbilt University. We thank David Kaserman, Andy Barnett, Gibson Lanier, Harris Schlesinger, participants in the Southeastern Economic Theory Conference, University of Alabama, 1992, and an a n o n y m o u s referee for helpful comments. All errors are the authors'. Beard and Gropper acknowledge financial support from the Auburn Utilities Research Center, Auburn University. Final manuscript received 23 August 1994.

Em'r~ly E~mlomics 1995 Volume 17 Number 1

Most formal economic research on priority or interruptible service to date has focused on the efficiency and feasibility of such schemes. The path breaking analyses of Chao and Wilson [4], Wilson [! I], Doane and Spulber [5] and others have established several important conclusions. In general, a cogently designed priority of service scheme involving customer self-selection and correctly specified prices can obtain spot market (first best) allocative efficiency. Further, many of the benefits of such a system can be appropriated by utilizing only a small number of customer priority classes. These properties argue for the expansion of priority of service pricing systems in future years. An issue not addressed in previous research, although of potentially great practical (ie 'political') significance, is the question of subsidy in interruptible service pricing systems. Since the various priority classes of service will involve perhaps substantial tariff differentials, the problem of determining when a payment differential is large enough to imply a cross-subsidy in the pricing system of the utility naturally arises. This paper initiates the formal analysis of crosssubsidy in interruptible service price systems. Our goals are two. First, we wish to lay a foundation for further research by identifying some of the important issues that must arise in any analyses of this type.

aWe are indebted to M r Gibson Lanier, Alabama Power Company, for extensive information on the practical aspects of interruptible service provision a m o n g US utilities. See also EPRI [6].

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Subsidy free pricin9 o f interruptible service contracts: T R Beard et al

Second, we want to derive a feasible test for the presence of a subsidy (in the usual cooperative, game theoretic sense) that can be calculated using only readily observed magnitudes. While any sensible definition of subsidy must involve an element of the hypothetical, we wish, as far as possible, to construct a test for subsidy that relies on observable magnitudes. Such a requirement greatly increases the usefulness of our results in a practical setting. Further, we avoid any assumptions about whether the pricing system being analyzed is efficient, so that our conclusions are applicable even to poorly conceived systems. After some preliminary analysis, we derive two major results. First, we show that the high priority users never subsidize the low priority users whenever the low priority users pay at least their variable costs. Hence, the main subsidy issues will involve subsidies benefiting high priority users. Second, we derive a sufficient condition for subsidy utilizing plausible assumptions on the form of the supply uncertainty, and discuss this result. A concluding section that considers complications, extensions, and future research completes the paper.

Preliminaries We turn now to a formal analysis of the subsidy problem. For simplicity, we focus on the case of a utility serving only two classes of customers with unvarying demands. 2 Only a random fraction of utility capacity is available at any given time, a circumstance that necessitates interruption of service to customers in some cases. For any given shortfall in capacity, units of service are interrupted to class 2 first and, if necessary, to class 1 as well. Within classes, no priority distinction is made among units of service, implying random rationing within classes. The class priority of service system is assumed to work in the sense that high priority (class 1) customers are never rationed unless no low priority units are provided to lower priority users. (It should be noted, however, that both classes may contain many of the same customers who hold both interruptible (class 2) and non-interruptible (class 1) contracts. 3) Our goal is to establish an implementable test for the presence of a subsidy in the pricing structure of the utility. 4 ZA cogent theoretical efficiency analysis of a variety of demand management policies with stochastic demands is in Doane and Spulber I-5]. For most utilities, contractual industrial loads are far less variable than residential loads, and interruptible service options are typically offered only to large industrial users. 3This practice is, in fact, widespread. '*In particular, we are interested in whether one product can be said to subsidize another. The issues of the customers' identities, and hence of anonymous equity in the pricing system, are topics of ongoing research. See Spulber [10] for an overview.

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Formally, we have: Di, i = l , 2 Q~, i-- 1, 2 Ko

tO

tOK o

= t h e units of service demanded by customers in class i; D~____0 = the units of service actually provided to customers in class i, so that Q~< D~ = the initial maximum capacity (in total deliverable units) of the utility; K o > 0 = a random variable where tOt[0, 1], with smooth marginal and cumulative densities f(tO) and F(tO), and where f(tO)>0 for all tOt[0, 1] =actually available capacity, so that 0 < tOKo < Ko 5

T~(tO,Di), i = l, 2 = t o t a l tariffs paid net of constant variable costs for customers in class i r = t h e uniform cost of capacity per potentially deliverable unit, so that capacity costs are r K o We assume in what follows that: (i) initial capacity K o is sufficient to meet demands in the best case scenario: Ko>~D 1 +D2; (ii) the demands D1 and D2 are those induced by the price system and the implied reliabilities of service; (iii) variable costs are constant per unit and can be taken as equal to zero (with customer payments, if any, taken as representing utility income in excess of variable costs); (iv) zero expected economic profits are earned by the utility. The tariffs paid by the customer classes T~(®,Di) represent contractual payments made by these customers that could be contingent on available capacity tO (or, equivalently, delivered units Qi) and demand Di. These contracts may take the form of delivery contingent uniform prices, two part tariffs, or any of a host of myriad alternatives. For our purposes all that matters is that these payment schedules have well-defined expected values. The reliabilities of service to the customer classes are defined as the probabilities with which a demanded unit in a class is actually delivered, as this is the relevant issue for the customer. For class 2 (low priority), actual consumption is Q2--D2

if D I + D z < t O K o

Q2=tOKo-Dx

if D I + D 2 > = ~ ) K o > D 1

Q2=0

if

tOKo
(1)

5This specification, which is common in analyses of this kind (see eg, Chao and Wilson [4]), is consistent with the view that generating capacity is composed of numerous independent machines operating in parallel rather than in sequence. The independence of the distribution of ® from K implies a kind of 'constant returns to reliability', an issue we return to below.

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Subsidy/i'ee pricing of interruptible service contracts: T R Beard et al For the high priority class l, actual consumption is

QI=D1

if OKo>D 1

QI=OKo if OKo
(2)

We define the reliability of service to class i, denoted Pi, as

p, = E(Q,)/D,

i = 1,2

DI/K,,

p~=-I-F(D,/Ko)+ 1 f OKof(O)dO D1

(4)

0

Pz-- 1 - F((Dx + D2)/Ko) (Ol + D~_)/K,,

(OKo-DOf(O) dO

Dz

Result 1: class 1 does not subsidize class 2 if ~Tz(O, D1)f(O)dO>~O, ie if the expected tariff for service to class 2 equals or exceeds the variable costs of service to class 2.

(3)

Performing the necessary calculations, taking expectations, and simplifying yields

+ ---

reliability equivalent to the integrated system must still buy capacity of K o. Hence, we have proven:

(5)

DI/K.

It is easy to show that l__>pt>=pz>0. We note that, in general, p~ is a function of the demand quantities D~ and D e, the initial capacity K o, and depends on the form of the distribution of the capacity shock O. The behavior of the functions p~ will be crucial in the analysis to follow.

Subsidy tests We turn now to the derivation of some tests for the presence of a subsidy in the pricing structure of the utility offering interruptible service. We adopt the standard notion that a subsidy exists if one class could provide itself with service, fully equivalent to that which they obtain in the integrated system, in a stand alone system at lower cost, the so-called incremental cost test of Faulhaber [7]. In the integrated (two class) system outlined in the previous section, the absence of economic rents to the utility implies that the pricing structure must satisfy

f TI(O, DOf(O)dO + f Tz(O, D2)f(O)dO-rKo=O

(6) so that, on average, customer payments in excess of variable costs equal the costs of capacity, rK o. An examination of Equation (4), however, allows us to derive the following result immediately. If, as postulated, the priority service system is actually implemented, then the presence of class 2 imposes no reliability burden whatsoever on class 1, as Pl is not a function of D2. Hence, changes in the size of the low priority class do not effect the welfare of class 1. A stand alone system composed solely of class 1 offering

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Result l, which is fairly transparent on reflection, illustrates the fact that, as a general proposition, the interesting subsidy issues in interruptible service primarily involve subsidies from the low priority to the high priority class; class 2 enjoys no subsidy so long as it pays for the specific costs of producing and delivering the units it actually consumes) The existence or non-existence of a subsidy from the low to the high priority class is a complex issue, insofar as any determination of subsidy necessarily involves a calculation of the capacity necessary to provide class 2 with a prespecified level of service in a stand alone system. Such a calculation requires at least some information on the distribution of capacity shocks. Nevertheless, we are able to arrive at a strong sufficient condition for the existence of a subsidy utilizing plausible assumptions on the distribution f(.). We turn now to this problem. To begin, let pl and P2 in the integrated system be denoted as p] and p~, indicating that both are dependent on the initial capacity level K o. Define K: as that level of capacity such that the reliability of service to class 2 customers in a stand alone system with capacity K~ will equal p~ ie: l -- F((D x + D2)/Ko) (DI + D~/K.)

+ -D2

(OKo-

DOf(O)

dO

DI/K,, D:/KI

if

= 1 -F(D2/K,)+ ~

OK,f(O)dO

(7)

o

(Note that, in a stand alone system with only one class of customer, the reliability of service is that of the high priority class given in (4).) The first step in determining the presence of a subsidy from class 2 to class 1 is to obtain useful upper bounds on the necessary capacity K~ of the stand alone system. This is achieved by examining the behavior of the reliability function Pl in (4). We note 6Such a conclusion would have to be modified if the selection of initial capacity reflected binding constraints on the reliabilities of service to different classes. Our analysis, however, merely takes the existing levels of reliability as given. It is easy to show, though, that, for any number of classes and class reliability constraints, only one constraint is typically binding.

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Subsidy free pricing of interruptible service contracts." T R Beard et al

first that Pl can be written as a function of the ratio of units demanded to system capacity alone. Letting x=D/K, we write Pl as Pl(X): 0 Pl

X

o.

o

where x is interpreted as the ratio of demand to maximal capacity. Differentiating with respect to x, we obtain:

b

0

02

°_

~l(x)

e¢

x t~

dO1/3x = - (1/x 2 ) / O f ( O ) dO < 0

c

(9)

~d

o

Noting that p°2<=p], Equations (8) and (9) allow us to conclude that (D2/KO>(Dt/Ko), so that

Kx <(D2/DOKo. While the relation K~ <(D2/DOKo does provide an upper bound of K~, this result is too weak to be useful. Derivation of a stronger test for subsidy requires some information on the behavior of the distribution of capacity shocks f(O). Fortunately, only relatively plausible assumptions are necessary. Differentiating pl(x) twice with respect to x (=D/K) yields: x go

O2p 1lax 2 = ( 2 / x a ) t O f ( ® ) dO - - ( f ( x ) / x ) Q/

o X

o

so that the sign ofc92p~fl?x2depends only on the sign of x

f of(®)dO f(x)(x2/2)

(1 1)

o

The sign of the expression in (11) can be determined unambiguously wheneverf(O) is a monotonic function over the interval [0, x]. We have the following lemma:

Lemma (i) if f ' ( O ) > 0 for Oe[0,x], then 02pl/OX2 0

Proof For part (i), suppose thatf'(O) > 0 for ® e [0, x]. Then

f(O)<=f(x) for all ®e[0,x], and therefore x

X0

(=OllkoI

X 1 (D21kl)

X = demand/capacity Figure I. Proof of the strong inequality test for subsidy when Pl (x)is concave: (~-~)2+(~)2 <(~d)2.

The proof for part (ii) is analogous. QED The lemma provides us with sufficient information to produce a useful inequality test for the presence of a subsidy whenever it can be assumed that the marginal density of demand shocks is monotonic over a relevant interval. While we will argue below that this monotonicity is quite plausible for a utility offering good service quality, we note here that the concavity or convexity of Pl in x does not actually depend on f(®) being monotonic over the internal [0, x]. Iff(®) is not monotonic over the required interval, however, it is difficult to give an economic interpretation to this curvature. Hence, while this monotonicity is not actually required for the results below, we will proceed as if this condition is met. 7 Recall that, given any maximal capacity K, ® represents the 'percentage of available capacity' while f(®) gives the likelihoods (suggestively, but incorrectly, the 'probabilities') attached to different levels of available capacity. Since the capacity stock K presumably represents the combined maximal output of numerous independent machines, f(®) is (approximately) unimodal and exhibits an interior maximum. For all levels of® to the left of the maximum,if(O) > 0, and, for points to the right, f ' ( ® ) < 0 , so that f(O) is monotonic over all intervals that do not contain the maximum as an interior point. Hence, so long as the relevant interval [0, x] does not contain argmaxf(®),

x

~®f(O) dON~ ;®f(x)dO= f(~'~)(X2/2) o o 56

0

VNote that this requirement, which is sufficient but not necessary, applies only to a specified interval, and not to the entire support off(O).

Energy Economics 1995 Volume 17 Number I

Suhsi~lv ./i'ee pricin,q ~?/ interruptible service contracts." T R Beard et al

monotonicity would appear highly likely. We will return to this theme below. We now derive the inequality result assuming monotonicity of f(®) over the relevant interval or, more weakly, concavity or convexity ofpa in x. Figure 1 illustrates the function pl(x) under the condition that f ' ( ® ) > 0 , so that pl(x) is concave over the interval [0, xl] where xl =(D2/KO, Xo=(D1/Ko), and xl>Xo. The reliabilities of service in the initial, integrated system are indicated on the vertical axis. The points a,/9 and c have coordinates (Xo, P t), (Xo, P2), and (x 1, P z), respectively, and form a right triangle. The tangent to p~(x) at x~ is drawn, and the point d has coordinates (xo-- P2 + ~Pl/CX(Xl)(Xo- xl)). Hence we must have (ab) 2 +(bc)2<(cd) z for the Euclidean distances of the line segments ab, bc, and cd. Expanding the relevant terms, calculating distances and simplifying this inequality yields the relationship (x 1 --.%) > (p~ - - p ° z ) / ( - - ? p l / ~ x )

(12)

where ?p~/L~ is evaluated at xt. Applying the definitions of x® and x~, evaluating Opl/Ox at x 1, and noting that I):,K I

f of(O)dO<(Oz/K~)F(O2/K 0

(13)

0

allows us to obtain the result that (KI/D2)<(Ko/D 1) (1-(p°x-p°2) ). A parallel argument for the case in which f ' ( O ) < 0 over the relevant range [0,x~] leads to the corresponding result that (KI/Dz)<(Ko/D 0 (1 +(p] -p~)) ~. Hence, we obtain an inequality result for the required capacity of the stand alone system:

Result 2 (i) i f f ' ( O ) > 0 over the interval [0, xl], then

(K1/D2)<(KolD1)(1 --(p] --p~)) (ii) i f f ' ( O ) < 0 over the interval [0, x~], then (K,/D 2) < (K o/D ~)(1 + (p] - p~))- 1 Result 2, combined with the zero expected profit constraints for the integrated and stand alone systems, gives us the strong inequality tests for a subsidy from low to high priority consumers.

Result 3 (i) I f f ' ( ® ) > 0 over the interval [0, xa], then class 2 subsidizes class 1 if D2 Tz(O, D2)f(®) dO > rk o~ - (1 - (p] - p~)) wl

Ener~,ly Economics 1995 Volume 17 Number 1

(ii) I f f ' ( O ) < 0 over the interval [0,Xl] then class 2 subsidizes class 1 if

T2(O, D 2 ) f ( O ) d O > r k o ~ ( 1 +(p] -p°2))Interpretation of result 3 is relatively straightforward. If, for example, case (i) applies (as, we will argue below, is quite likely), then the expected capital cost contribution of class 2 to the integrated system should not exceed the ratio of high priority to low priority demand, multiplied by the reliability deflated capacity cost ( 1 - (p] - p°2))rK o. If, for example, the low priority class represents 5% of total demand, and faces reliability 10 percentage points lower than the high priority class, then the low priority class should pay less than 4.7% of capacity costs on average. We return finally to the issue of the monotonicity of the marginal density of capacity shocks discussed earlier. As noted before, this monotonicity is sufficient (but not necessary) to obtain result 3. If, for example, one assumes that pa(x) is convex in x, then result 3 (ii) applies automatically, although this fairly natural assumption does imply some constraints (weaker than monotonicity), on the behavior of f(®). Somewhat surprisingly, however, it appears that the concavity of Pl (on [0,xl] ) may be likely when, as is typically the case, the utility offers highly reliable service. Informally, iff(®) is bell shaped and has an interior maximum at some point O, then f ' ( ® ) > 0 on [0, xl] so long as Xl < O. Yet the interpretation of the O = argmaxf(®) is that O is the 'most likely' level of available capacity. Hence, the assumption x~ < O actually amounts to the highly credible claim that, in the most likely state of capacity availability, rationing does not occur. Therefore, if service is fairly reliable, one typically expects f ' ( ® ) > 0 on [0,Xl], and result 3 (i) applies.

Conclusions and extensions As interruptible service arrangements become more widespread, questions of cross-subsidies are likely to arise with increasing frequency. With this in mind, the present paper seeks to initiate the formal analysis of subsidized pricing of priority service. Utilizing a simple theoretical model, we were able both to identify some of the relevant issues, and offer some simple conditions to test for the presence of a subsidy. We found that, as a general proposition, low priority users do not receive a subsidy so long as they pay at least the variable costs of their service. Further, under plausible assumptions on the form of the capacity uncertainty, high priority users enjoy a subsidy whenever low priority users pay an amount (in excess of variable costs) that exceeds a fraction of the integrated system's

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Subsidy,free pricin9 of interruptible service contracts: T R Beard et al

capacity costs determined by the relative size of low priority demand, and by the difference in actual service reliabilities between classes. Several issues remain unresolved by our initial analyses. First, as mentioned earlier, we focus on the case involving only two priority classes. While this is consistent with many actual priority systems, the case of many classes should be examined. Second, we focus on supply shortfalls attributable to production disruption rather than unanticipated demand peaks. Although interruptible service is typically offered in the USA only to contractual customers, the case of unanticipated demand shocks is potentially important. However, any such analysis must by necessity involve peak load issues, an important complication. Third, the stochastic specification adopted here (multiplicative capacity shocks), while a realistic model for a 'large' system utilizing multiple independent generation facilities, does imply a kind of'constant returns to scale' in providing service reliability. 8 Unfortunately, more general specifications are likely to involve considerable complexity in analysis. As our goal was primarily to identify the fundamental effect of service priority and reliability (rather than special techno-

SAn alternative would be to allow f(.) to depend on K so that, when K changes,the distributionf(.) undergoesstochasticdominance type shifts. Such a formulation would allow for 'scale economies' in the provision of reliability. We note, though, that the mere presenceof a priority systemgeneratesinterestingsubsidy problems even with a form of 'constant returns' in the technology.

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logical features) on the subsidy problem, we omit analysis of this issue here. Finally, our assumption of a smooth, unimodal capacity shock distribution is at best an approximation to the circumstances faced by many utilities. We therefore urge further investigation on these topics.

References 1 Baumol W, Panzar J and Willig R, Constestable Markets and the Theory of Indust O' Structure, revised edn, Harcourt, Brace, Jovanovich, Orlando, FL, (1988) 2 Berg S and Tschirhart J, Natural Monopoly Regulation, Cambridge University Press, Cambridge, (1988) 3 Brown S and Sibley D, The Theory of Public Utility Pricing, Cambridge University Press, Cambridge, (1986) 4 Chao H and Wilson R, 'Priority service: pricing, investment, and market organization', American Economic Review, 1987, 77, pp 899-916 5 Doane M and Spulber D, 'Design and implementation of electricity curtailment programs' paper prepared for the Journal of Regulato O" Economics, Editor's Conference, 1992 6 EPRI hmorative Rate Design Survey. 1986 Final Report, EM-5705, (1986) 7 Faulhaber G, 'Cross-subsidization: pricing in public enterprises', American Economic Review, 1975, 65, 966-977 8 Sharkey W, The Theol3' of Natural Monopoly, Cambridge University Press, Cambridge, (1982) 9 Sherman R, The Regulation of Monopoly, Cambridge University Press, Cambridge, (1989) 10 Spulber D, Reyulation and Markets The MIT Press, Cambridge, MA, (1989) 11 Wilson R, 'Efficient and competitive rationing', Econometrica, 1989, 57, 1~40

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