Load management programs, cross-subsidies and transaction costs: the case of self-rationing

Resource and Energy Economics 22 Ž2000. 161–188 www.elsevier.nlrlocaterECONbase

Load management programs, cross-subsidies and transaction costs: the case of self-rationing Jean-Thomas Bernard ) , Michel Roland Departement d’economique, UniÕersite´ LaÕal, Quebec, QC, Canada G1K 7P4 ´ ´ ´ Received 1 October 1996; received in revised form 21 September 1998; accepted 6 May 1999

Abstract Load management programs are used by electric utilities to decrease peak consumption. Although they are generally offered simultaneously with regular service, economic models of their allocative efficiency are based on the implicit assumption that they are the only service available. We present a model in which participation to a particular load management program, called self-rationing, is optional. We show that, under a break-even constraint, welfare-maximizing prices involve a subsidy from the self-rationing program to regular service whenever peak demand is less elastic than base demand. If cross-subsidization is precluded, regular service is viable only if there exist transaction costs to participate in the self-rationing program. q 2000 Elsevier Science B.V. All rights reserved. JEL classification: D45; L51 Keywords: Cross-subsidies; Electricity pricing; Rationing

1. Introduction The electricity market restructuring which is taking place in many countries has unbundled the vertically integrated electric utilities. Generally, competition is introduced at the generation stage and electricity output is sold on spot markets. Transmission and distribution ŽT & D. networks continue to be regulated because of their natural monopoly characteristics. However, details of T & D regulation )

Corresponding author. Tel.: q1-418-656-5123; fax: q1-418-656-7412; E-mail: [email protected]

0928-7655r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 8 - 7 6 5 5 Ž 9 9 . 0 0 0 1 8 - 4

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vary among jurisdictions. A number of possibilities, such as nodal pricing, tradable transmission rights or more conventional average cost pricing, are either under consideration or implemented for the pricing of transmission services. At the distribution level, there exist two alternative models for the supply of electricity to end-users. The ‘‘wholesale competition’’ model corresponds to traditional public utility regulation where one company is in charge of bringing both ‘‘wires’’ and power to the consumer. For such a company, market restructuring only means that it buys electricity on competitive wholesale markets instead of obtaining it from a regulated public utility. 1 The ‘‘retail competition’’ model unbundles wires and power and allows end-users to get their power from the supplier of their choice Žcompeting distribution companies or power marketers. or even directly on the spot market. 2 However, even in the case that end-users could make spot market transactions, most of them are expected to get their electricity under contracts which establish price and conditions of delivery. This is due to the characteristics of the commodity: because electricity cannot be stored economically and its demand is subject to cyclical and random fluctuations, spot prices can vary widely over a short span. There then exist at least two reasons for which it is efficient to shield end-users from spot price fluctuations. First, end-users are presumably more exposed to risk than power marketers, since the latter can to some extent diversify risk from non-correlated demands among consumers. 3 Second, frequent price changes lead to substantial transaction and adjustment costs for the consumers, mainly due to the need to get information on price changes and to react rapidly to them. These costs are likely to be substantially lower for marketers whose job is precisely to be informed about the evolution of the market. 4 Under traditional public utility regulation, protection against price fluctuations has been virtually complete. Regular services of public utilities gave the right to consumers to use whatever quantity they wanted according to a pre-determined

1 The use of price cap regulation is then often proposed in order to give the distribution company incentives to obtain electricity at least cost on the wholesale markets. 2 See Joskow Ž1997. for a comprehensive survey of electricity market restructuring in the United States. Bohi and Palmer Ž1996. discusses more specifically comparative advantages of wholesale competition and retail competition models in the distribution sector. 3 By assuming coincident peak demands and consumer risk neutrality, we exclude such a consideration in this paper. Note that since electricity demand is in practice highly dependent on common shocks, such as temperature or general economic activity, the opportunity to diversify risk is probably limited. 4 Bohi and Palmer Ž1996. mention that ‘‘because of the fixed cost of acquiring information, the larger distribution companies will be able to spread those wtransactionx costs over more units of electricity than the smaller marketing companies’’. Of course, the argument stands a fortiori for the individual consumer.

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tariff structure. 5 Because of the fixed prices, public utilities had to install enough capacity to meet peak demand with a given reliability standard. With time, however, increasing capacity costs have made such standards increasingly expensive to meet. To avoid building new power generation facilities, utilities have devised rationing plans, called load management programs, that gave them some leeway to curtail loads of a pre-determined group of consumers. 6 Participation in the programs were voluntary and induced by price incentives. 7 Whatever is the model chosen for the distribution sector, load management programs are likely to expand as the electricity industry is restructured. This is because significant variations in marginal costs and heterogeneity in consumer preferences make product differentiation a natural outcome for electricity markets. According to ŽBohi and Palmer, 1996, p. 17., ‘‘marketing companies in the retail model may be expected to offer a range of options that trade off quality and price of services in order to serve the diversity of consumer preferences. In designing these trade-offs, marketers will be particularly interested in cutting consumption during periods when the spot price is high.’’ 8 Similarly, when there is only wholesale competition, reducing peak period consumption would allow the distribution company to reduce its electricity purchase costs. To the extent that price cap regulation replaces rate-of-return regulation, so that cost reductions are done to the benefit of the distribution company, incentives to create IrC programs increase. Three electricity rationing programs which can be considered as IrC service has been studied in the economic literature. Under ‘‘self-rationing’’, 9 the con-

5

Although the pricing structure could be a multi-part tariff or could include time-of-day fluctuations, it was typically not responsive to random factors: price changes had to be approved by regulatory commission hearings and the implied time lags excluded any possibility to deal with a temporary situation. 6 Although load management programs exist in different forms, the most common is interruptiblercurtailable ŽIrC. service for industrial customers. With an IrC program, the consumer determines a power level Žthe firm power. above which all consumption may be curtailed upon notification by the utility. 7 Voluntary participation is an essential feature of load management programs; they would otherwise be equivalent to a reduction of quality unilaterally implemented by the utility. Regulators would have opposed such reduction in quality. 8 The on-going experience with competition in natural gas markets can be instructive in this manner. For instance, the production stage of natural gas industry has been deregulated in Canada since 1985, while transmission remains regulated by the National Energy Board, a federal agency, and distribution is regulated by provinces. In the case of the province of Quebec, there is retail competition but Gaz ´ metropolitain, the LDC which owns and maintains the distribution network, can participate in retail ´ markets. In 1995, interruptible sales accounted for 25% of the gas delivered by Gaz metropolitain. ´ 9 Self-rationing was first proposed by Panzar and Sibley Ž1978.. Generalizations of the Panzar and Sibley model can be found in Schwarz and Taylor Ž1987., Woo Ž1990., Lee Ž1993., and Doucet and Roland Ž1993.. Roland Ž1992. gives conditions under which self-rationing is equivalent to IrC service as described in the preceding paragraph.

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sumer chooses the maximum power he may use. When his desired consumption exceeds this ceiling, the part above this level represents the interrupted service. ‘‘Priority services’’ 10 consist of servicing customers in a given order until capacity is met. This order is pre-determined by the self-selection of customers into classes which are differentiated by their service priority and the price of electricity. Finally, under ‘‘pro-rated services’’, consumers choose a baseload level of service which serves as the basis to determine payment and allocation under different contingencies. Spulber Ž1992a, b. presents this approach and shows its relationships with self-rationing and priority services. These models share the characteristics that the rationing scheme represents the only type of service available to consumers. Unless a user foregoes electricity consumption altogether, he has no choice but to participate in the rationing plan. 11 The absence of so-called regular services can be justified on efficiency grounds. Traditional methods of rationing within regular services, such as brown-outs and rolling black-outs, are random and therefore less efficient than the rationing methods proposed in the models. IrC services is thus shown to be Pareto-superior to regular service, provided monetary transfers are allowed. This eliminates any rationale for the supply of regular services. In practice, all IrC programs that were available under traditional public utility regulation were offered as alternatives to regular services. There exist at least two reasons for this fact. First, there is the traditional argument that withdrawal of existing services would occur at the expense of some consumers, since monetary transfers generally do not take place. Second, although less important than under spot pricing, there are significant transaction costs associated with IrC programs which are ignored in the theoretical models and which raise doubt about the allocative efficiency of imposing IrC service to all clients. 12 The same two arguments are often presented in most deregulation undertakings to insure, at least

10

Cf. Marchand Ž1974., Chao and Wilson Ž1987., Chao et al. Ž1986., Viswanathan and Tse Ž1989., Oren and Doucet Ž1990., Wilson Ž1989a, b, 1993., and Doucet and Oren Ž1991.. 11 Wilson Ž1989a, p. 24. considers partial implementation of priority service in the sense that priority service would not be offered to some market segment. But this segment is then treated as a special priority class whose service order is based on the average marginal willingness to pay within the class. In the event that such practice results in a lower reliability, it could be interpreted as a regulation imposed by the electric utility. Again, this would have been forbidden in traditional public utility regulation. We argue below that minimum reliability standards will be set by regulatory agencies in most restructured markets as well. 12 For instance, IrC service often requires the installation of a new meter. More importantly, it forces the consumer to re-arrange his activities when curtailment occurs. Again, existence of such transaction costs can be suspected from Gaz metropolitain’s experience with its IrC program Žsee footnote 8.. This ´ program reaches only 216 customers out of 150,195 in total even though interruptible sales represent 25% of total deliveries. This shows that interruptible services are attractive primarily to large customers, suggesting that the existence of fixed transaction costs make them uneconomic before consumption reaches a significant threshold.

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for a transitional period, universal access to electricity services which meet minimum quality standards. Assuming that these standards remain about the same as those that existed under public utility regulation, IrC services will in fact continue to coexist with traditional regular services and the consequences on allocative efficiency of such coexistence has not been analyzed. This leaves unanswered questions such as whether consumers will pay more or less for regular services and whether these services can be sustained without cross-subsidization or not. This paper presents a model in which one particular type of IrC service program, self-rationing, is offered to consumers as an alternative to regular service. Regular service is justified by the existence of a fixed cost of participation in the self-rationing program. The choice of self-rationing for the characterization of IrC services is made because of the relative simplicity of the rationing scheme.13 The institutional framework under which we model the pricing of electricity is one where a regulatory agency seeks to maximize welfare under a non-negative profit constraint 14 for a monopolistic electric utility. Although this can appear at odds with the on-going deregulation, we use this approach only to determine what is the efficient Žsecond-best. allocation when both regular and self-rationing coexist. Under conditions for which this allocation is subsidy-free, enough competition in the markets should lead to it without requiring prices to be fixed by a regulator. However, when there are cross-subsidies, the efficient allocation is not sustainable under competitive conditions. We then look at the properties of a ‘‘third-best’’ solution where a subsidy-free constraint is added to the non-negative profit constraint. The results are as follows. When cross-subsidization is allowed, prices of both regular and self-rationing services can change relative to the situation where only one type of service is offered. Participants in the self-rationing program are likely to subsidize regular service users. When cross-subsidies are not allowed by a regulatory agency or because of competitive pressure, the pricing structure of each service is not modified by the presence of the other service: regular service is priced on the base of average cost, while pricing in the self-rationing program is based on marginal cost. However, the price leÕel of regular service is likely to increase following the introduction of self-rationing.

13

However, it is well known that self-rationing as proposed by Panzar and Sibley Ž1978. can lead to inefficient allocations: the most striking inefficiency is the possibility of curtailing some consumers while capacity is not fully utilized. Although a number of extensions were made to improve allocative efficiency of self-rationing Žsee Schwarz and Taylor, 1987; Woo, 1990; Doucet and Roland, 1993., we do not include them in our model because they would complicate the analysis without modifying the pricing principles that we analyze. 14 In the context of competition, a non-negative Žlong run. profit constraint can be seen as a ‘‘participation’’ or ‘‘individual rationality’’ constraint that arises from the behavior of producers.

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Section 2 presents the notation with respect to cost and demand functions, and defines formally regular service and self-rationing. The co-existence of a self-rationing program and regular service is analyzed in Section 3. Section 4 introduces a constraint on cross-subsidies. Section 5 presents the concluding remarks.

2. The Model 2.1. Cost function We follow Panzar and Sibley Ž1978. and assume that the utility operates only one type of generation plant which has constant operating cost b per unit of output and constant capital cost b per unit of capacity. If output is represented by y and capacity by K, the cost function is then defined on the domain Ž y, K .< y F K 4 by C Ž y ; K . s by q b K .

Ž 1.

The utility faces a random load and it has the mandate to satisfy this fluctuating load. There is no supply uncertainty. As a result, capacity must simply be set at the highest level that load can reach. 15,16 Letting Q be the random load variable, which takes values in the interval w Q L ,Q H x with an expected value of Q, we obtain for the expected cost: EC Ž Q;Q H . s E Ž bQ q b Q H . s bQ q b Q H s C Ž Q;Q H . .

Ž 2.

The expected cost of serving the random load reduces to the cost of serving expected load. For this reason, we can use expected load as an ex ante or long-term measure of the utility’s output. As a result, the Žlong-term. average cost is given by: C Ž Q;Q H . Q

15

sbqb

QH Q

.

Ž 3.

If there exists a rationing program, the highest level that load can reach is the aggregate demand by consumers at the posted price less the quantity that the utility has the right to curtail. As a result, reliability to consumers who adopt the rationing program can be less than 100% even though there is no supply uncertainty. The interest of a rationing program comes from the fact that the expected marginal willingness to pay of some customers for the highest possible demand level can be less than the cost of supplying this unit. It thus increases welfare having the consumer to reduce his use. 16 Lee Ž1993. allows the utility to purchase electricity from third-party suppliers and consequently, to own production capacity which is lower than peak load. This leads to more complex pricing formulas. We avoid such complexities in order to focus on the optional nature of self-rationing and its impact on pricing.

J.-T. Bernard, M. Rolandr Resource and Energy Economics 22 (2000) 161–188

Considering changes in the distribution load levels, Žlong-term. marginal cost as: dC dQ

sbqb

dQ H

17

we can also define the

.

dQ

167

Ž 4.

Even though both operating and capital costs are linear, the utility can experience decreasing, constant or increasing returns to scale depending on whether dQrdQ H is less than, equal to, or greater than QrQ H . In the latter case, uniform marginal cost pricing would not allow the utility to break even. Eq. Ž3. displays a fundamental feature of electricity production, i.e., average production cost depends on the Žinverse. ratio of average to peak load. This ratio is the so-called load factor and it receives special attention from utilities. Increasing the load factor is often stated as being the main objective of load management programs. 2.2. Demand function and indiÕidual load factors Each consumer’s quantity demanded is an increasing function of the random variable t, which varies over the interval w t L ,t H x according to cumulative distribution function G. For simplicity, t is called temperature. Consumers differ with respect to their demand for power: for a given uniform price p, some consumers desire more power than others, whatever is the temperature. Therefore, individual demands do not cross. Consumers can be indexed by a variable, u , according to the increasing order of their quantity demanded. The demand function q is also assumed to be twice differentiable in all its variables. The above assumptions translate into the following derivatives: q p Ž p,t , u . - 0;

qt Ž p,t , u . ) 0;

qu Ž p,t , u . ) 0; ; Ž p,t , u . .

Ž 5.

By inverting q ŽP,t, u . with respect to p, we get the marginal willingness to pay function P ŽP,t, u .. Assumption Ž5. then implies that Pq - 0, Pt ) 0 and Pu ) 0, ;Ž q,t, u .. 18 Furthermore, we assume that absolute values of demand elasticities with respect to prices, ´ Ž p,t, u ., and with respect to the type parameter, j Ž p,t, u ., are decreasing with temperature, i.e.,: E

´ t Ž p,t , u . s p

17

Et

ž

q p Ž p,t , u . q Ž p,t , u .

/

)0

Ž 6.

Later on, such changes will be brought either by changes in prices or by entry or exit of consumers into the various services which are offered. In Section 2.2, it will be assumed that the consumer type distribution is continuous and that the demand function is also continuous with respect to consumer type; changes in peak and average loads are well defined under these assumptions. 18 These assumptions were originally made by Panzar and Sibley Ž1978. and they have been adopted by all authors listed in footnote 9.

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E

j t Ž p,t , u . s u

Et

ž

qu Ž p,t , u . q Ž p,t , u .

/

- 0.

Ž 7.

For illustrative purposes, let us consider that the type parameter corresponds to income. The above assumptions mean that the impacts of price and income on consumption become relatively less important as temperature increases. This can reflect a situation where electricity uses which are sensitive to temperature, such as air conditioning, constitute essential goods or services. 19 For the electric utility, the load factor is the demand property of interest because it determines average production cost. For reasons that are related to the functioning of self-rationing and which will become apparent in the proof of Lemma 2, we define a more general concept, the load factor conditional on the temperature being below a given level. Let q Ž p,t, u . be the expected quantity demanded by a type-u consumer at price p, conditional on temperature being below t, i.e.,: t

q Ž p,t , u . '

Ht q Ž p, x ,u . dG Ž x . L

GŽ t .

.

Ž 8.

The individual load factor Žconditional on temperature being below t . is then written as: l Ž p,t , u . '

q Ž p,t , u . q Ž p,t , u .

.

Ž 9.

The assumptions with respect to demand elasticities imply the following properties for the load factor. Lemma 1. l p (p,t,u ) - 0, lu (p,t,u ) ) 0, ;(p,t,u ). Proof. Partial differentiation of l Ž p,t, u . with respect to p yields: l p Ž p,t , u . 1 s

1 s GŽ t . 19

t

q Ž p,t , u . q p Ž p, x , u . y q p Ž p,t , u . q Ž p, x , u .

L

q Ž p,t , u .

½ H½

H GŽ t . t

t

q Ž p, x , u .

q p Ž p, x , u .

tL

q Ž p,t , u .

q Ž p, x , u .

y

2

q p Ž p,t , u . q Ž p,t , u .

5

5

dG Ž x .

dG Ž x . .

Ž 10 .

This would certainly be the case for regions which experience extreme weather conditions, such as Quebec, where electricity is used for heating, or Florida, where it is used for air conditioning. ´

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169

From assumption Ž6., the term in square brackets is negative, ; x g w t L ,t x. Hence, l p Ž p,t, u . ) 0, ;Ž p,t, u .. The proof of lu Ž p,t, u . ) 0, ;Ž p,t, u ., follows from assumption Ž7. and a similar reasoning. B Lemma 1 implies that an increase of average consumption Žfor temperature levels below t ., which is induced either by a price reduction or by stronger preferences for electricity, raises the load factor. This is intuitively explained by the fact that such an increase, which is unrelated to temperature, enlarges the share of electricity consumption which is not sensitive to temperature, and thus flattens the consumption pattern. 2.3. Regular serÕice Regular service is modeled in the following way: whatever is the temperature, the consumer can purchase as much power as he desires at unit price p. For simplicity, we set the reliability of regular service at 100%. 20 As a result, the producer must install enough capacity to meet peak demand for regular service. When temperature is t and price is p, a regular service user of type u purchases quantity q Ž p,t, u .. His load factor is: tH

R

l Ž p, u . ' l Ž p,t H , u . s

Ht

q Ž p,t , u . dG

L

q Ž p,t H , u .

Ž 11 .

and the expected surplus is: ES R Ž p, u . '

tH

Ht H0q p ,t ,u Ž

.

P Ž qX ,t , u . y p d qX dG.

Ž 12 .

L

2.4. Self-rationing Under self-rationing, ‘‘each consumer subscribes to a particular level of capacity before the state of nature is revealed. He pays a capacity charge for the 20 The absence of supply uncertainty allows us to assume such a reliability level. Reliability level of less than 100% as well as supply uncertainty can be incorporated by having the producer to install a capacity level equal to average load plus an appropriate multiple of the standard deviation. The optimal regular service price should then take into account the marginal benefit of capacity in terms of the avoided welfare loss from rationing Ži.e., rationing cost.. This only involves an additional term in the optimal pricing formulas but does not change the principles behind these formulas as well as their economic meaning. Fixing the reliability level at 100% is thus made for ease of presentation. An appendix showing how to include simultaneously imperfect regular service and supply uncertainty is available from the authors upon request.

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Fig. 1. Consumption path with respect to temperature.

amount he subscribes to, and a usage charge for each unit actually consumed. If Žand only if. his usage exceeds his subscribed capacity, a circuit breaker, or fuse, activates, curtailing further consumption.’’ 21 The capacity level a chosen by the consumer is called subscribed power. Each unit of subscribed power is sold at price k. The constraint imposed on consumption by subscribed power is made acceptable by a price reduction on energy use, i.e., the consumer obtains electricity at price r - p. Consumption at temperature t corresponds to the minimum of a and the quantity which the consumer desires, q Ž r,t, u .. As the partial derivative qt exists and is positive for all Ž r,t, u ., it follows that, for given values of a , r and u , there ˆ u . s a . 22 Below this threshold, the exists one and only one tˆ such that q Ž r,t, consumer may buy the desired quantity at price r,q Ž r,t, u .. Above this threshold, consumption is restricted to a . Fig. 1 represents the consumption path for an individual of type u when subscribed power is equal to a .

21

Panzar and Sibley Ž1978, p. 888.. Italics in the original. As shown by Panzar and Sibley Ž1978., a positive price for subscribed power Ž k ) 0. ensures that tˆ- t H , i.e., there exist temperature levels for which the consumer is limited by his predetermined choice of subscribed power. 22

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171

ˆ u . s a . The consumer’s Let tˆŽ r, a , u . be the implicit function defined by q Ž r,t, problem consists in choosing the amount of subscribed power that maximizes his expected surplus: 23 a

q r ,t , u tˆ r , a , u Ž P y r . d q dG q H H H H aG0 t 0 tˆŽ r , a , u . 0 Ž

max

.

Ž

.

tH

X

Ž P y r . d qX dG y k a y d

L

Ž 13 . where d is the fixed transaction cost associated with participation in the program. The optimal value for this objective function is denoted by ES S Ž r,k, u .. A solution a S s a S Ž r,k, u . to this problem must satisfy the following Kuhn and Tucker condition: tH

HtˆŽ r , a ,u . S

P Ž a S ,t , u . y r dG y k F 0;

tH

½H

tˆŽ r , a S , u .

5

P Ž a S ,t , u . y r dG y k a S s 0; a S G 0.

Ž 14 .

A consumer who purchases a positive quantity of subscribed power chooses the amount such that the expectation of the marginal willingness to pay Žnet of usage charge., which represents the expected benefit of an extra unit of subscribed power, is equal to the cost of this unit. For a S ) 0, it is easy to show 24 from the Kuhn and Tucker condition Ž14. that auS ) 0 and that:

a rS s

1 y G Ž tˆ. tH

Htˆ

s 1 y G Ž tˆ. a kS - 0

Ž 15 .

Pq Ž q,t , u . dG

where arguments of functions a S and tˆ are omitted for clarity. For a self-rationing user of type u , the load factor is given by:

l S Ž r ,k , u . '

Ht tˆq Ž r ,t ,u . dG q

1 y G Ž tˆ. a S

L

. Ž 16 . aS The objective of increasing the load factor through IrC program is met under ˆ the consumer faces a lower price assumption Ž6.: for any temperature below t, under the self-rationing program than under regular service and, from Lemma 1, ˆ i.e., l Ž r,t,ˆ u . ) this implies a higher load factor for temperature levels below t, ˆ u .; when temperature is above t,ˆ consumption is at its peak, implying a load l Ž p,t, factor of 1 for temperature levels between tˆ and t H . Globally, then, participation in the self-rationing program increases the load factor. 23 24

The arguments of P are omitted for simplicity. See Panzar and Sibley Ž1978, p. 890..

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172

Lemma 2. l S(r,k,u ) ) l R (p,u ), ; r - p, ;u . Proof. See Appendix A. The consumer’s choice between regular service and the self-rationing program is analyzed in the next section. 3. Optional self-rationing 3.1. Consumer’s choice The consumer participates in the self-rationing program to the extent that ES S Ž r,k, u . ) ES R Ž p, u .. We define Q S and Q R as the sets of consumer types choosing self-rationing and regular service, respectively, i.e., Q S Ž p,r ,k . s  u g w u L , u H x < ES S Ž r ,k , u . ) ES R Ž p, u . 4 Ž 17 . and Q RŽ p,r,k . is the complement of Q S Ž p,r,k . relative to w u L , u H x. Henceforth, we omit arguments Ž p,r,k . to simplify notation. Since ESuR ) 0, ;Ž p, u . and ESuS ) 0, ;Ž r,k, u ., the sets Q R and Q S are not necessarily convex: the conditions imposed thus far does not prevent successive intervals on w u L , u H x to belong either to set Q R or to set Q S . However, it is clear that the measure of Q S is inversely related to fixed transaction cost d because as d increases the number of consumers willing to participate in the self-rationing program decreases. Moreover, revealed preference arguments can be used to derive some necessary conditions on the choice of regular service or self-rationing. Lemma 3. If a consumer of type u chooses regular serÕice, then k d p-rq R q . t l Ž p, u . H q Ž p,t , u . dG

Ž 18 .

Ht

L

Proof. If a consumer chooses regular service, it must be true that: p

tH

Ht

q Ž p,t , u . dG - r

L

tH

Ht

q Ž p,t , u . dG q kq Ž p,t H , u . q d .

Ž 19 .

L

Otherwise, the consumer could have exactly the same consumption opportunities under self-rationing but at a lesser or equal cost. If the same consumption path is strictly less expensive under self-rationing, choosing regular service is clearly not optimal. In the case where regular service consumption path would cost exactly the same under self-rationing, welfare could be improved under self-rationing by reducing subscribed power, 25 implying that choosing regular service is still not optimal. From definition Ž11., Eq. Ž19. implies Eq. Ž18.. B 25

Cf. footnote 22.

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Of course, choosing the self-rationing program implies that transaction cost does not outweigh the gain for the consumer to participate in the program. For this reason, we can extend Lemma 3 to show that higher type consumers tend to prefer self-rationing. Corollary. If there exists a consumer type u U such that

dF Ž pyr .

tH

Ht

q Ž p,t , u U . dG y kq Ž p,t H , u U .

Ž 20 .

L

then [u U , u H ] ; Q S . Proof. Eq. Ž20. implies that pGrq

d

k

Ž 21 .

q

U

R

l Ž p, u .

tH

Ht

U

q Ž p,t , u . dG

L

so, from Lemma 3, a consumer of type u U does not use regular service and, consequently, participates in the self-rationing program. Now, from Lemma 1 and the fact that demand is increasing with respect to u , rq

d

k

-rq

q

R

l Ž p, u .

tH

Ht

q Ž p,t , u . dG

k l Ž p, u U . R

L

d

F p, ;u ) u U .

q tH

Ht

Ž 22 .

U

q Ž p,t , u . dG

L

Then necessary condition Ž18. for choosing regular service is not met and consumers of type u ) u U participate in the self-rationing program. B Note that condition Ž20. is sufficient but not necessary for a consumer to choose self-rationing. Consequently, the Corollary does not imply that set Q S is convex; we could observe a consumer of type u for whom condition Ž20. does not hold but nevertheless elects to self-ration, while a higher type consumer chooses regular service. A necessary condition for participation in the self-rationing program is given in the following Lemma. 26 26

Lemmas 3 and 4 potentially have an interesting spin-off for the empirical estimation of the otherwise elusive transaction cost; since price, load factors and individual consumptions are data simple to gather for an electric utility, these lemmas suggest a way to derive upper and lower bounds for transaction cost.

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Lemma 4. If a consumer of type u chooses self-rationing, then p)rq

d

k

.

q

S

l Ž r ,k , u .

tˆ

Ht q Ž r ,t ,u . dG q

Ž 23 .

1 y G Ž tˆ. a S

L

Proof. Duplication of the proof of Lemma 3 leads to the result.

B

It is of course possible for the transaction cost to be so high that necessary condition Ž23. is never met, thus precluding any participation in the self-rationing program. There would be no point to offer self-rationing under these conditions. 27 This case is obviously ruled out here. 3.2. Additional notation In order to facilitate the presentation and the interpretation of the producer’s problem optimality conditions derived in Section 3.3, additional notation related to the demand function is introduced. First, expected individual consumption under regular service and under self-rationing are written q R and q S , i.e.,: q R Ž p, u . '

tH

Ht

q Ž p,t , u . dG;

L

q S Ž r ,k , u . '

Ht tˆq Ž r ,t ,u . dG q

1 q G Ž tˆ. a S .

Ž 24 .

L

Second, capital letters mean that aggregation over consumers is made on corresponding small letter variables. Accordingly, Q i Ž p,r ,k . '

i

HQ q d F , i s R ,S; t

Q H Ž p,r ,k . '

HQ q Ž p,t

H ,u

.dF;

R

A Ž p,r ,k . '

HQ a

S

d F.

Ž 25 .

S

27 Note also that condition Ž23. implies that prices must be such that p) r q k. Otherwise, no consumer would choose self-rationing: each consumption unit could be obtained at lesser or equal cost under regular service without constraint on consumption.

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When it does not create confusion, arguments of functions defined in Eqs. Ž24. and Ž25. are omitted. 3.3. Producer’s problem The utility must determine the price of regular service, p, the self-rationing usage charge, r, the price of subscribed power, k, and the capacity K. Its objective is to maximize consumers’ and producer’s aggregate expected surplus. It is constrained to break even and to install capacity sufficient to meet both peak consumption of regular service and total subscribed power. The producer’s problem is thus written as: max

AS Ž p,r ,k . q PS Ž p,r ,k , K .

pG0, rG0, kG0, KG0

s.t. PS Ž p,r ,k , K . G 0

Ž 26 .

QH q A F K where AS is the aggregate expected consumers’ surplus and PS is the expected producer’s surplus, i.e., AS Ž p,r ,k . s

HQ ES R

R

Ž p,u . d F q H ES S Ž r ,k ,u . d F

Ž 27 .

QS

PS Ž p,r ,k , K . s Ž p y b . Q R q Ž r y b . Q S q kA y b K .

Ž 28 .

Let f s AS q Ž1 q l.PS q m Ž K y Q H y A. be the Lagrangian function, where m G 0 and l G 0 are multipliers. Using the consumers’ optimality condition, partial derivatives of f with respect to decision variables can be expressed as follows: 28

f p s Ž 1 q l . Ž p y b . Q Rp y m Q pH q lQ R 28

Ž 29 .

Note that a change in one of the prices p, r or k, modifies the sets Q R and Q S . We should therefore consider the impact of a modification of those sets on the Lagrangian function. Let uˆ be a marginal consumer type for whom ES R Ž p, uˆ . s ES S Ž r,k, uˆ .; the fact that uˆ-consumers may ‘‘switch’’ sets does not change the expected aggregate consumers’ surplus, precisely because ES R Ž p, uˆ . s ES S Ž r,k, uˆ .. The switch may however modify their expected consumption, and therefore have an effect on the revenues they bring to the utility and on the level of capacity installed to meet their needs. We neglect these effects by assuming that the consumer type density function f is such that f Ž u . is very small, whatever is u . We therefore make an ‘‘atomistic’’ assumption: consumers of any type are small in number relative to the whole population, so that the single impact of their choices on capacity and revenues can be neglected.

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fr s Ž 1 q l . k y m A r q lQ S q Ž r y b . Ž 1 q l . Q rS fK s Ž k y m q l k . A k q Ž 1 q l. Ž r y b .

HQ

Ž 30 .

1 y G Ž tˆ. a kS d F q l A Ž 31 . S

f K s yb q m y bl .

Ž 32 .

Assuming an interior solution, 29 partial derivatives f p , fr , f k , and f K must equal zero at the optimal solution. We obtain from Eq. Ž32.:

m s b Ž 1 q l . ) 0.

Ž 33 .

For optimal prices, there are two cases to be discussed.

Case 1. The profit constraint is not binding (constant or decreasing returns to scale). Under this condition, l s 0 and it is easy to verify that a solution satisfying necessary conditions Ž29. – Ž32. is the following: psbqb

Q pH

ž / Q Rp

Ž 34 .

rsb

Ž 35 .

ksmsb.

Ž 36 .

It is clear from Eq. Ž29. that any optimal solution from the regular service price satisfies implicit form Ž34.. Furthermore, it is shown in Appendix B that the above explicit prices for the self-rationing service are the unique optimal prices for this service. To highlight the economic intuition behind condition Ž34., let us define g as the additional expected consumption that is made possible by installing an extra unit

29

The solution K s 0 can be obtained when production costs are so high that all consumption becomes socially undesirable. For instance, this would occur if P Ž0,t H , u H ., which is the highest willingness to pay possible, were inferior to the cost of supplying the first unit of electricity, bq b . Obviously, this case is without interest. If K ) 0, ps 0 results in f p ) 0, in contradiction with the Kuhn and Tucker condition that f p be non-negative at the optimum. Similarly, r s 0 and k s 0 implies that fr ) 0 and f k ) 0, respectively.

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of capacity allocated to regular service, i.e., the marginal productivity of capacity for regular service. This coefficient may be expressed as: 30

gs

Q Rp

Ž 37 .

Q pH

Since a unit decrease of average consumption increases operating costs by b and requires the installation of 1rg units of capacity, whose cost is b per unit, b q brg simply represents the expected marginal cost of production of regular service. As a result, to the extent that it allows the utility to break even, marginal cost pricing of both regular and self-rationing services is optimal. We derive below a necessary condition for a slack profit constraint. Although possible, empirical studies suggest that this condition is unlikely to be met in practice. As a result, the tight profit constraint case seems more interesting both in theory and in practice. Case 2. The profit constraint is binding (increasing returns to scale). Under this condition, g ) 0 and replacing m by b Ž1 q l. in Eq. Ž29. yields:

Ž p y b . Ž 1 q l . Q Rp s b Ž 1 q l . Q pH y lQ R .

Ž 38 .

Using the definition of g , this equation can be rewritten to obtain the following implicit solution for the regular service price:

b pyby

g sy

p

l

QR

1ql

pQ Rp

l sy

1

1ql E

Ž 39 .

where E is the elasticity of expected consumption of regular service, i.e., E'

30

pQ Rp QR

.

Ž 40 .

Since Q H is the capacity allocated to regular service, dQ R EQ R rE p g' s H , H dQ EQ rE p which is Eq. Ž37.. When the profit constraint is not binding at the optimum, g must be less than 1 because of the capacity constraint Žas will be seen in Eq. Ž45. below.. We thus have p) bq b s r q k, a result that was required by incentive compatibility Žsee footnote 27..

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The price of regular service is therefore established according to Ramsey’s rule: for a given l, the difference between price and Žexpected. marginal cost is lower the more elastic is the Žexpected. demand. The value of l is adjusted so that the utility breaks even. The conditions for the optimal self-rationing prices can also be expressed as a Ramsey rule if one notices that two commodities are being sold under self-rationing: expected consumption, at price r, and subscribed power, at price k. Demands for these two commodities are complementary and consequently, we can define their price and cross elasticities: the price elasticity of average consumption is given by

hQ r s

rQ rS QS

.

Ž 41 .

The price elasticity of subscribed power, hA k , and cross elasticities hA r and hQ k , are defined in the same manner. We use these definitions of elasticities to express Eq. Ž30. in the following way: ryb

ž / r

hQ r q

ž

kyb k

kA

/ž / rQ

S

hA r s y

ž

l 1ql

/

.

Ž 42 .

Similarly, condition Ž31. on subscribed power pricing becomes:

ž

kyb k

/

hA k q

ryb

rQ S

r

kA

ž /ž /

hQ k s y

ž

l 1ql

/

.

Ž 43 .

As b represents the marginal cost of subscribed power and b is the marginal cost of expected consumption, Eqs. Ž42. and Ž43. are in effect Ramsey prices for interdependent demands. 31 Before providing an interpretation for the optimality conditions, we present a necessary condition, which is empirically testable, for a slack profit constraint.

31 The general form of the Ramsey rule for M commodities having interdependent demands reads as Žcf., for instance, Brown and Sibley, 1986, p.195.: EC Pj y M EQ j Pj Q j l hji sy is1,2, . . . , M P P Q 1q l j i i js 1

Ý

 0

where Pi is the price of good i, Q i , its quantity, and hji , the elasticity of good j with respect to the price of good i. C is the cost function. Note that if cross elasticities are zero, we obtain the usual Inverse Elasticity Rule.

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Proposition 1. If an optimal solution inÕolÕes a non-binding profit constraint, then regular serÕice peak demand is more elastic than regular serÕice aÕerage demand at the optimal prices. Proof. If the profit constraint is not binding, we obtain from conditions Ž34. – Ž36. and the capacity constraint,

b

Q pH

ž /

´

Q Rp

Q pH QH

QR y b QH G 0

F

Q Rp QR

´< EH
H

is the regular service peak demand elasticity.

Ž 44 . Ž 45 . Ž 46 . B

A number of empirical studies 32 suggest that peak demand is less elastic than base demand and consequently, than average demand. 33 This conforms to intuition, since peak demand is related to more essential uses of electricity and is less likely to be sensitive to price. In practice, then, the profit constraint is likely to be tight. The upshot is a second-best result that has been neglected in the literature of load management programs in general and of self-rationing in particular: if the regular service already offered by the utility cannot be priced at marginal cost, it is not optimal to price a newly offered program at marginal cost. As expected under these conditions, both services should rather be priced following a Ramsey rule. With constant marginal operating and capital costs, the only link between the prices of each service is the profit constraint Lagrange multiplier set to meet this constraint in a least cost fashion; no price interdependency exists because of consumption externality from one service to the other. Since subscribed power and energy consumption are complementary goods under self-rationing, a mark-down for one of these goods Ži.e., r - b or k - b . is 32

See, for instance, Filippini Ž1995. for Switzerland, Bernard and Chatel Ž1985. for Quebec, or ´ Caves and Christensen Ž1980, Table 4 for the 1:1 peak and off-peak price ratio. for Wisconsin. These studies find base Žor off-peak. demand elasticities that are higher than peak demand elasticities by a factor of 1.5, approximately. For Duke Power in North Carolina, Taylor and Schwarz Ž1986, p. 148. state that ‘‘The computed demand elasticity for hot weather is y0.210, approximately 88% of the average value of y0.239.’’ 33 Note that assumption Ž6., stipulating that the individual demand price elasticities decrease Žin absolute value. with temperature, does not imply that the aggregate peak demand is less elastic than aggregate expected demand. This is due to the fact that the aggregate demand elasticity is a weighted average of the individual demand elasticities, where the weights are each consumer’s share in total demand: partial derivatives of the shares with respect to temperature cannot be monotonic in consumer index, since the sum of these derivatives over the consumer set must be zero.

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possible under the tight profit case. For this to occur, however, we must either have the own-price elasticity of subscribed power Ževaluated at optimal prices. dominated by the average consumption cross-elasticity with respect to subscribed power price k or the own-price elasticity of average consumption dominated by the subscribed power cross-elasticity with respect to the price of energy. Proposition 2. If there exists a price mark-down on either r or k, then
HQ a

S

Ž r U ,kU ,u . d F.

Ž 47 .

S

Assume further that each self-rationing consumer u g Q S receives the same subscribed power Ž a S Ž r U ,kU , u .. than the one chosen under integrated network and is restricted Žor is allowed. to have the same consumption path Žmin q Ž r U ,t, u ., a S Ž r U ,kU , u .4 ,; t .. If electricity consumption and subscribed power were charged at marginal cost, then the stand-alone system revenue would be bQ S q b A

Ž 48 . S

which is equal to the stand-alone cost CŽ Q ; A.. Marginal cost pricing of the production sold to self-rationing consumers would then permit the integrated network to recover the cost attributable to the self-rationing program. This means that self-rationing users cannot be responsible for the tight profit constraint. As a 34

Note, however, that Taylor and Schwarz Ž1990, p. 442. obtain from their econometric estimates that: ‘‘Contrary to expectations, the demand charge induces a greater reduction in peak energy consumption than in household maximum demand . . . ; that is, the indirect effect of the demand charge is stronger than its direct effect.’’ In our model, this corresponds to
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result, any divergence of the optimal solution from marginal cost pricing of the self-rationing service must be for the purpose of raising additional funds for the network. B In an article aimed at providing empirical tests for the presence of cross-subsidization between two reliability classes of interruptible services, Beard et al. Ž1995, p. 55. derived a similar result: ‘‘the interesting subsidy issues in interruptible service primarily involve subsidies from the low priority to the high priority class’’, the low and high priority classes representing self-rationing and regular consumers, respectively, in our context. Beard et al. used this result to investigate whether actual implementations of interruptible services involved cross-subsidization 35 and made no assumption on the way that utilities do or should fix their prices. In contrast, our model predicts that in the most likely case, i.e., when peak demand is less elastic than average demand, optimal implementation of a self-rationing service involves a subsidy from this service to regular service. Making use of different assumptions, Crew and Fernando Ž1994. obtain an opposite result. These authors look at the incentives given by interruptible service tariffs when consumers get a discount off the demand charge: if some ‘‘consumers’ individual peak demands do not coincide with the system peak, priority winterruptiblex service offers the potential for free ridership on the part of such consumers. They would receive the discount, but effectively would not face interruption.’’ 36 With non-coincident demands, this means that interruptible service attracts consumers with small load factors so that the probability of being interrupted is low. Interruptible service then proves to be a hidden subsidy to customers who choose this service. 37 The type of services Žself-rationing. that we consider specifically avoids the incentive problem that they bring up since the demand charge in the self-rationing program Žinterruptible service. is higher, and the energy price lower, than for regular service. 38 35

At this effect, they derive tests for cross-subsidization based on observable data. Crew and Fernando Ž1994, p. 136.. 37 A utility would willingly give such subsidies when it faces competition on the one hand, and is required by a regulatory agency to undergo hearings for price modifications on the other hand. Hidden subsidies through interruptible services are then useful to retain customers who would otherwise ‘‘bypass’’ the utility. 38 Note that because we do not consider the possibility of non-coincident peaks, the subsidy issue of Crew and Fernando was excluded from our model at the outset. However, the positive Žor the higher. demand charge and lower energy price of self-rationing would remain incentive compatible in presence of non-coincident peaks as Crew and Fernando mention: ‘‘This perverse behavior wof giving discount for no or little curtailmentx can be avoided by moving from a Chao–Wilson type approach of providing a discount on the service charge’’ Žp. 133.. Roland Ž1992. derives the optimal pricing of self-rationing in presence of non-coincident peaks. The demand charge is then equal to the marginal value of capacity Žwhich depends on aggregate demand. weighted by the probability that a consumer’s load coincide with peak demand. Again, this is incentive compatible and is the equivalent of the suggestion of Crew and Fernando to have priority service where the demand charge varies by time of day Žp. 136.. 36

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Proposition 3 implies that optimal prices are not sustainable when there is free entry in the self-rationing service market. Since interruptible services are generally offered to the industrial sector and this is precisely the sector most likely to be lured by non-utility generators, this result suggests that a utility’s interruptible services would be difficult to maintain in a competitive environment and that low-reliability service to industrial consumers would be taken over by non-utility generators, leaving high-reliability service to the regulated utility. The recent evolution of electricity markets seems to confirm this conjecture. Even when competition is precluded, regulatory agencies may wish to restrict cross-subsidies between different classes on equity grounds. In the following section, we show that the implementation of a self-rationing program can still have an effect on the welfare of users who choose to keep regular service when no cross-subsidies are allowed.

4. Constraint on cross-subsidization The producer’s problem is the same as above except that the profit constraint is replaced by two constraints specifying that expected revenue from each service must recover the associated cost. These constraints read as Ž p y b . Q R G b Q H for regular service, and Ž r y b . Q S q kA G b A, for the self-rationing program. This new problem is straightforward to solve. Since marginal cost pricing within the self-rationing program allows to cover costs, r s b and k s b represent optimal prices. Then, there are two possibilities for regular services. First, marginal cost pricing, i.e., p s b q brg , allows to recover the attributable cost to regular service and is thus optimal. This would represent exactly the same situation as in Case 1 of Section 3.3. Second, the cost recovery constraint is tight under marginal cost pricing, forcing the utility to price at average cost. If we define the load factor of regular service users as LR Ž p,Q R . '

QR Q

H

s

HQ

q Ž p,t H , u . R

QH

l R Ž p, u . d F

Ž 49 .

then the optimal price can be written in the following Žimplicit. form:

b psbq

R

L

Ž p,Q R Ž p,b, b . .

.

Ž 50 .

However, in order for this price to be consistent with consumers self-selection, there must exist some transaction cost. In the absence of transaction cost, there is no price that simultaneously satisfies Eq. Ž50. and retains consumers in regular service. In such a case, there would be no point to offer regular service.

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Proposition 4. If d s 0, then Q R s Ø. Proof. Ž1. Suppose first that we preclude cross-subsidization. From Lemma 3, with d s 0, if a type u consumer chooses regular service, then p

tH

Ht

q Ž p,t , u . dG - b

tH

Ht

L

q Ž p,t , u . dG q b q Ž p,t H , u .

Ž 51 .

L

where we use the fact that the optimal subsidy-free prices for self-rationing are r s b and k s b . Summing over the set of consumers who could choose regular service, we get

b p-bq

R

L

Ž p,Q R Ž p,b, b . .

.

Ž 52 .

But this is in contradiction with the optimal price for regular service given by Eq. Ž50.. As a result, there exists no equilibrium price for regular service, i.e., there exists no price which can attract consumers in regular service when subsidies are excluded. Ž2. At the optimal solution r s b and k s b , the cost recovery constraint for regular service is not binding. The irrelevance of regular service is thus independent of whether we a priori exclude cross-subsidization or not. B Proposition 4 shows that the practice of excluding the regular service option in self-rationing models is consistent with the Žimplicit. assumption contained in these models that there is no transaction cost: the choice of regular service would be dominated by self-rationing and offering this choice would be superfluous. However, the complete absence of transaction cost represents only a limiting or extreme case. At the other extreme, the transaction cost could be so high so as to preclude any participation in self-rationing, as it is clear from Lemma 4. Between these two polar cases stands the intermediate and more interesting one where the transaction cost is at such a level that both services coexist. With this intermediate case in mind, we see from Eq. Ž52. that the regular service price is related to consumers’ load factor while self-rationing prices Ž r s b and k s b . are independent of demand characteristics. Compared to an initial situation where only regular service is offered to consumers, introduction of self-rationing, by modifying the set of regular service users, can change the price of regular service. Corollary of Lemma 3 has already shown that if there existed a consumer type u U who, by choosing self-rationing over regular service, can reduce his electricity bill by an amount greater than the transaction cost, then consumers of types higher than u U would participate in the self-rationing program. For a sufficiently low transaction cost, the existence of such a u U is highly plausible. This means that the largest consumers Ž u ) u U . most likely participate in the self-rationing program. From Lemma 1, these consumers have the highest

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load factors when using regular service. Their participation in self-rationing thus tends to affect adversely those remaining on regular service.

5. Conclusion If there are transaction costs associated with the participation to a load management program, social welfare may be enhanced by offering such a program as an option to existing regular services rather than making it compulsory. In order to attain maximum welfare, however, pricing policies must take into account the opportunities open to consumers. This can alter conclusions reached in the literature on the way load management programs should be priced. In the case of self-rationing, welfare-maximizing prices for self-rationing services depart from the marginal cost pricing rule obtained by Panzar and Sibley whenever peak demand is less elastic than average demand, a condition generally observed in reality. Prices for both regular and self-rationing follow Ramsey rules that involve cross-subsidization from self-rationing to regular service. This may create difficulties at the implementation stage either because regulatory agencies can disapprove such cross-subsidies or because competitors to the utility could take the opportunity to cream-skim the self-rationing segment of the market. If regulatory agencies or market pressure only allow subsidy-free prices, marginal cost pricing of self-rationing service would then be restored. The creation of a self-rationing program would, however, create an externality to consumers who choose not to participate in it by changing the price of regular service. Overall, the study of self-rationing as a stand-alone service was proven to be valid in the case where transaction costs are negligible. The introduction of transaction costs in the simplest way possible, i.e., via a fixed cost, already shows that implementation of load management programs involves phenomena, such as cross-subsidies or externalities, that have been neglected in the literature thus far. To the extent that load management programs were relatively rare or offered only to restricted customer classes, the impact of such phenomena may have been minor. However, considering the increasing popularity of these programs and the advent of market competition, this is likely to change. A better understanding of the impacts on pricing and allocative efficiency of various competing services is then called for.

Acknowledgements Financial support from the Government of Quebec ‘‘Fonds FCAR’’ is grate´ fully acknowledged. We thank J.A. Doucet and two anonymous referees for helpful comments. We alone are responsible for any remaining error.

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Appendix A. Proof of Lemma 2

ˆ u ., the self-rationing load factor can be written as: Since a S s q Ž r,t,

l S Ž r ,k , u .

s

Ht tˆq Ž r ,t ,u . dG L

q Ž r ,tˆ, u .

Ž definition of l S .

q Ž 1 y G Ž tˆ. .

s G Ž tˆ. l Ž r ,tˆ, u . q Ž 1 y G Ž tˆ. .

Ž definition of l .

) G Ž tˆ. l Ž p,tˆ, u . q Ž 1 y G Ž tˆ. . tH

) G Ž tˆ. l Ž p,tˆ, u . q

Htˆ

Ž from Lemma 1 and p ) r .

q Ž p,t , u . dG

q Ž p,t H , u .

Ž since q Ž p,t ,u . - q Ž p,t H ,u . ,; t .

Ht tˆq Ž p,t ,u . dG Ht q Ž p,t ,u . dG tˆ H

s

q Ž p,t H , u . q Ž p,tˆ, u .

L

q Ž p,t H , u .

q

q Ž p,t H , u .

Ž definition of l . ) l R Ž p,t , u . Ž since q Ž p,tˆ,u . - q Ž p,t H ,u . . .

B

Appendix B. Solution uniqueness for the self-rationing prices in the case of a non-binding profit constraint When l s 0 necessary conditions Ž29. – Ž32. become:

f p s Ž p y b . Q Rp y b Q pH s 0

Ž B.1 .

fr s Ž k y b . A r q Ž r y b . Q rS s 0

Ž B.2 .

fk s Ž k y b . Ak q Ž r y b . Ar s 0

Ž B.3 .

where we used Eq. Ž15. to replace HQ S w1 y GŽ tˆ.x a kS d F by A r in Eq. ŽB.3.. Ž1. Ž k y b . s 0 if and only if Ž r y b . s 0. Suppose that Ž k y b . s 0. Then, from Eq. ŽB.2., Ž r y b . s 0 since Q rS - 0. Similarly, if Ž r y b . s 0, then from Eq. ŽB.2., Ž k y b . s 0 since A r - 0. Ž2. There is no solution satisfying Eqs. ŽB.1., ŽB.2. and ŽB.3. with Ž k y b . / 0 and Ž r y b . / 0. 39 39

The following arguments are based on Panzar and Sibley Ž1978, p. 895..

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186

From Eq. ŽB.3., Ar

Ž k y b . s yŽ r y b .

Ak

.

Ž B.4 .

Substituting Eq. ŽB.4. into Eq. ŽB.2., we get:

Ž r y b . Q rS y

A2r Ak

s 0.

Ž B.5 .

Using Eq. Ž15. and definitions of A and Q S , we can expand the bracketed expression in the following way:

HQ Ht tˆq Ž r ,t ,u . dGd F r

S

L

2

HQ Ž1 y G Ž tˆ. .

2

a kS d F

S

q

HQ

a kS d F

y

S

HQ Ž1 y G Ž tˆ. . S

HQ a

a kS d F .

S k dF

S

Ž B.6 . The numerator of the second term is non-negative in virtue of the Cauchy–Schwarz inequality. The first term and the denominator of the second term being negative, this implies that the bracketed term in Eq. ŽB.5. is negative. As a result r s b is the only solution satisfying Eq. ŽB.5.. From Ž1., it follows that k s b is the unique solution for the subscribed power price. B

Appendix C Proof of Proposition 2 From Eqs. Ž42. and Ž43., we have:

ž

kyb k

Noting that Q kS s

/ž

hA k y

kA rQ

ryb

/ ž /ž

h s S Ar

r

hQ r y

rQ S kA

/

hQ k .

Ž C.1 .

40

HQ Ž1 y G Ž tˆ. . a

S k d F s Ar

Ž C.2 .

S

40

The first equality is obtained from partial differentiation of Q S with respect to k; the second, from Eq. Ž15..

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this can be written in the following way:

ž

kyb k

/

Ž hA k y hQ k . s

ryb

ž /Ž r

hQ r y hA r . .

Ž C.3 .

Suppose that the optimal solution involves a mark-down on either k or r. Then, sign Ž hA k y hQ k . / sign Ž hQ r y hA r . which implies that either hA k ) hQ k or hQ r ) hA r but not both.

Ž C.4 . B

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