Optimal pricing and capacity choice for a public service under risk of interruption

Transcription

1 Otimal ricing and caacity choice for a ublic service under risk of interrution Fred Schroyen y Adekola Oyenuga z November 6, 2. We are grateful to Kåre Petter Hagen, Nicolas Treich and Louis Eeckhoudt for detailed comments on an earlier version. Also thanks to Krisztina Molnar for a useful discussion. An earlier version of this aer was titled "Pricing of an Interrutible Service with Financial Comensation and Rational Exectations". The resent version was comleted while Schroyen visited the hositable environment of CORE (Louvain-la-Neuve, Belgium). It has bene ted from seminar resentations at the University of Antwer and at CORE. y Deartment of Economics, Norwegian School of Economics and Business Administration, Helleveien 3, N-545 Bergen, fred.schroyen@nhh.no. z Deartment of Economics, Norwegian School of Economics and Business Administration, Helleveien 3, N-545 Bergen, adekola_oyenuga@yahoo.no.

2 Abstract We develo rules for ricing and caacity choice for an interrutible service that recognise the interdeendence between consumers ercetions of system reliability and their market behaviour. Consumers ost ex ante demands, based on their exectations on aggregate demand. Posted demands are met if ex ost suly caacity is su cient. However, if suly is inadequate all ex ante demands are roortionally interruted. Consumers exectations of aggregate demand are assumed to be rational. Under reasonable values for the consumer s degrees of relative risk aversion and rudence, demand is decreasing in suly reliability. We derive oerational exressions for the otimal ricing rule and the caacity exansion rule. We show that the otimal rice under uncertainty consists of the otimal rice under certainty lus a marku that ositively deends on the degrees of relative risk aversion, relative rudence and system reliability. We also show that any reliability enhancing investment though lowering the oerating surlus of the ublic utility is socially desirable as long as it covers the cost of investment. JEL code: D, D24, D45, H42, Q25. Keywords: service interrution, rationing, system reliability, secondbest ricing, caacity choice, rudence. 2

3 Introduction. Motivation The San Francisco Public Utilities Commission (SFPUC) rovides water to the citizens of San Francisco and under contractual agreement with 29 wholesale water agencies to.6 million additional customers within three Bay Area counties. Overall, 2.4 million eole receive 26 million gallons (over billion liter) of water er day. Most of the water is imorted from the Sierra Nevada, delivered through the Hetch Hetchy aquaducts. The most serious threat to the water suly of the Bay Area is a drought, through its imact on the Sierra Mountains snowack which feeds the Hetch Hetchy water suly system. How should the SFPUC, or any other ublic utility roviding a service with uncertain caacity, set its rice? And how should it evaluate the desirability of its infrastructure investments? Our advice is: (i) calculate the rice according to a standard inverse elasticity rule, corrected for the marginal cost of ublic funds (eq (4.8)) and add to it a mark-u that deends ositively on the degrees of relative risk aversion and rudence, and negatively on the degree of system reliability (eq (4.9)); and (ii) exand caacity if the ensuing total reliability imrovement, when multilied with the degree of relative risk aversion and the marginal cost of roduction, exceeds the cost of investment (eq (4.4)). In general, the suly of services by certain ublic utilities electricity, gas, water is characterized by an inherent uncertainty: ower generating caacities are subject to temorary failures, the variability of surface water levels are not entirely redictable, and likewise for the in ow to water and gas reservoirs. In addition, the quality of the delivered service may not be fully controllable (risk of ollution). Insulating consumers of these services for any suly risk, that is, guaranteeing a % service reliability, would in most cases require investments in infrastructure that are rohibitively costly. The rst best solution is a set of contingency contracts clear agreements for the delivery of units of service under well de ned contingencies, which are aid in advance. By buying rights to such contingency deliveries, a consumer can should he wish secure himself a certain delivery in the future. Although new markets have been develoed in recent years to insure against variations in temerature, reciitation, and events like drought, fall freeze, etc., such markets remain often closed to the general ublic. An alternative is the let consumers face sot rices that balance demand and suly at any Similarly, demand for these services dislays a variability that may only be described statistically. Often, factors that cause a dro in suly caacity, such as a drought, will also cause demand to rise. In this aer, we abstract from any demand variability. 3

4 time. There are two roblems with such an arrangement. First, it may be very costly to inform consumers in real time about the governing rice level. Second, it may imose on consumers a considerable rice risk. The rice elasticity for residential water demand in the Bay Area is :76 (SFPUC, 27: 2). A % suly reduction would require rices to rise by 57%. 2 For the Gironde area in France, Nauges and Reynaud (2) estimate the short run demand elasticity for domestic water use at :8. With such an inelastic demand, a % suly reduction would require a rice increase of 25%! 3 Politicians are reluctant to allow the service rice of ublic utilities swing that much. Irresective of whether rice stability is an objective or a constraint, the balancing of a variable suly with demand calls for quantity rationing. In this aer, we are interested in the otimal olicies for ricing and caacity choice of an uncertain suly under two conditions: (i) that rices are be ket stable, and (ii) that in the event of excess demand, the service is rationed in roortion to notional demands the demands that would ensue at the (stable) rice. First, roortional rationing is frequently racticed. In the case of the Bay Area water suly, the master contract between the SFPUC and its wholesale customers exlicitly stiulates a roortional rationing rule (SFPUC, 27: 7). But even when a roortional rationing rule is not literally ractised, it is not uncommon that good or loyal customers are being rioritized by the service sulier in case of excess demand. The rationing rule is then said to be maniulable (in the sense of Benassy, 977: 52) because customers can in uence their share of the scarce suly by signalling a larger demand. A roortional rationing rule is the rime examle of a maniulable allocation rule, and its study is useful to get insight in other situations where customers can exert in uence on the amounts of the service nally allocated to them. Though our focus on a roortional rationing rule is motivated on these ositive grounds, we note that there exist normative reasons for such a rule (see Moulin, 2)..2 Relation to the literature Our aer is closely related to the literature on eak-load ricing under demand and suly uncertainty. The main resumtion of this literature is 2 Ignoring scale e ects on the willingness to ay. 3 Using a data set covering 42 large industrial and commercial customers in Northern California, Borenstein (27) calculated customer bills under time-invariant, time-of-use, and real-time ricing (RTP) schemes and found that, after adjusting for seasonal variation, the coe cient of variation of a customer s bill is on average nearly ve time larger under RTP than under the time-of-use structure that they tyically face. 4

5 that both sot ricing and contingency contracts are allocation mechanisms that are too costly to imlement and therefore that recourse has to be taken to other rationing mechanisms to bring demand in line with available suly. For examle, Brown and Johnson (969), Turvey (97), Visscher (973), Meyer (975), Carlton (977); and Crew and Kleindorfer (978) have focused on demand uncertainty while Chao (983), Fakhraei, Narayanan, Hughes (984), Coate and Panzar (989), and Kleindorfer and Fernando (993) have extended the analysis to include suly uncertainty. Tyical for this art of the literature is that rationing takes lace on the basis of characteristics that the are assumed to be observable to the rovider, such as outage costs and/or willingness to ay. Parallel to it, a literature has develoed where rationing takes lace on the basis of unobservable but revealed characteristics of consumers. Examles are self rationing (Panzar and Sibley, 978, Woo, 99, Doucet and Roland, 993) and riority servicing (Marchand, 974, Chao and Wilson, 987, and Wilson, 989a,b). For a detailed survey of both strands of literature, see Crew, Fernando and Kleindorfer (995). 4 In this literature, there are very few aers that exlicitly model how consumers formulate their demand based on the erceived reliability of that service. One excetion is the aer by Coate and Panzar (989) on random service rationing. 5 Insired by Rees (98), they let risk neutral rms decide on their caital equiment before knowing whether electricity will be available for roduction. In doing so, rms assign a robability to being blacked out (the comlement of the exected system reliability). Once caital is installed, demand for electricity follows from short run ro t maximization. Ex ost, if actual electricity suly falls short of aggregate demand, a fraction of rms is blacked out in a random way. The model is then closed by requiring that the mathematical exectation of the actual black out robability equals the anticiated black out robability (rational exectations). The authors show that electricity demand ositively deends on service reliability and characterise the otimal ricing and caacity choice for the ublic utility. 4 The literature often focusses on consumers and roducers. A recent literature discusses the role of interrutible service contracts on deregulated ower markets as instruments for hedging the wholesale market exosure of retail suliers to a volatile sot rice; see Gerda and Varaiya (993), Kamat and Oren (22), Baldick et al. (26), and Rocha and Siddiqui (28). 5 In his model on ublic utility ricing with uncertain demand, Tschirhart (98) adds system reliability as an exlanatory variable to the mean demand equation, and assumes that higher reliability leads to higher mean demand. He does not derive the demand schedule on the basis of erceived reliability. 5

6 .3 The role of the degree of relative rudence In the resent aer, we study the otimal ricing and caacity choice by a ublic utility when inadequate suly is allocated across risk averse consumers according to a roortional rationing rule. Consumers ost ex ante demands for a designated consumtion eriod. These demands will be met if ex ost suly caacity is su cient. If not, all ex ante demands will be roortionally interruted. The anticiated system reliability is determined by anticiated aggregate demand. As Coate and Panzar, we assume rational exectations: anticiated demand coincides with aggregate ex ante demand. Unlike with random rationing, however, the roortional rationing rule is maniulable in that a single consumer may in uence his ersonal service reliability by over/understating his demand. When risk-averse consumers can in uence their own service reliability, how they resond to system reliability is an imortant factor in de ning otimal rice and caacity levels. In this resect, the concet of recautionary behaviour lays a crucial role. Such behaviour follows from a ositive third derivative of the consumer s concave utility function, a roerty coined rudence by Kimball (99). The original discussion of recautionary behaviour is in terms of a consumer s savings decision when facing increased uncertainty with resect to future labour income or the return to savings see Leland (968), Sandmo (97) and Kimball (99). This discussion was recently closed by Eeckhoudt and Schlesinger (28). In articular, they show that whenever relative risk aversion exceeds, consumers will save more when the rate of return distribution undergoes a rst degree stochastically dominating shift (i.e., lower rates of return become more likely); and that whenever relative rudence exceeds 2, consumers will save more when the rate of return distribution undergoes a 2nd degree increase in risk (a.k.a. a mean reserving sread). We show that these conditions on relative risk aversion and relative rudence will also govern the consumer s resonses to increased uncertainty in our model. Su ciently risk averse and rudent consumers will react to increased uncertainty by exanding ex ante demand. In fact, whenever the erceived reliability of the system goes down, ex ante demand goes u exactly the oosite reaction as in Coate and Panzar s (989) model. This result underscores the interdeendence of system reliability, rationing rule and demand behaviour. We show that the otimal rice is one that internalises the external e ect of individual demands on system reliability. We also show that although infrastructure investments which increase the reliability of suly will reduce roducer oerating surlus, they tyical roduce 6

7 a more than comensating increase in consumer surlus, a result we believe that calls for regulation of infrastructure decisions. 2 The model A ublic utility is the sole sulier of a good or service. The cost of roduction and delivery is a constant c er unit. There is a continuum of consumers with mass normalised to that all have a quasi-linear utility function over the consumtion of the service, w, and a numéraire commodity, Y : U(w; Y ) = u(w) + Y, where it is assumed that u >, u <, u > and lim w! u (w) = +. For future reference, we de ne the coe cients of relative risk aversion and relative rudence with resect to the service as R r (w) def = u (w)w and P u (w) r (w) def = u (w)w u (w) Two remarks are in lace. First, note that with constant relative risk aversion (CRRA), P r R r +. 6 Second, the above utility function imlies that the rice elasticity for the w-good equals R r(w). Table dislays tyical estimates for short run demand elasticities for gas, electricity and water. The low estimates imly values for R r well exceeding 2. This is useful to kee in mind when evaluating the results later on in the aer. Table. Demand elasticity estimates for selected utilities. residential commercial industrial electricity a :6 :28 :39 natural gas a :5 :28 :26 water :8 b :29 c a Lin et al. (987, 25): United States b Nauges and Reynaud (2, 8): Gironde (France) c Reynaud (23, 227) Gironde (France) Each consumer has an exogenous income at his disosal. In our model, consumers may be heterogeneous in terms of income as long as everybody s income is high enough. To simlify notation, however, we assume that incomes are identical and equal to m. If denotes the rice er unit, the reresentative consumer has Y = m w left for consumtion of the numéraire. 6 Pivotal values for R r and P r are and 2, resectively. Eeckhoudt et al. (29) show how using simle gambles one can elicit whether or not a resondent s degree of relative risk aversion and rudence exceeds these ivotal values. 7

8 Total suly is reresented by a random variable T with a commonly known cumulative distribution function F (T ). The realisation of this variable is exogenous to the consumer. Suly is thus uncertain and the extent to which aggregate demand X a exceeds realised suly is the level of suly inadequacy or excess demand. The consumer s ercetion of suly being adequate is given by Pr (T > ) = F ( ), where is the consumer s exectation regarding the aggregate demand. The assumed rationality of this exectation requires that = X a. It is commonly known that a ositive level of excess demand will result in consumtion being interruted or rationed o. In general, the realised consumtion of the service by erson i, w i, may be written as a function of the ex ante demands by all agents, (x i ; x i ), as well as the available caacity T, w i = f i (x i ; x i ; T ). We assume that the rationing functions f i () follow what is called a roortional rationing rule, i.e., w i = x i minf; T R x idi g: The scheme thus unfolds as follows. () The utility announces the rice in advance of the eriod. (2) Consumers choose their ex ante demand x. (3a) If the realised suly is adequate, then x will default as uninterruted consumtion. (3b) If the realised suly is inadequate, consumtion is curtailed to T x. (4) Consumers ay for the delivered ortion of x at the X a announced rice, and consumtion takes lace. 3 Consumer behaviour The rst question we answer is how the consumer behaves in choosing his ex ante demand and to what extent this demand would be in uenced by the rosect of him being interruted. The consumer s ex ante demand is the solution to the following utility maximisation roblem: max x V = Z T u X x + m TX x df (T ) e e + [u (x) + m x] [ F ( )] : (3.) If a situation with inadequate suly is exected with some ositive robability, then Pr (T ) < and F ( ) >. The demand ^x that solves 8

9 roblem (3.) must satisfy the necessary condition: u (^x) [ F ( )] + Z u T ^x T df (T ) = r F ( ); (3.2) where r F ( ) is de ned as the consumer s ercetion of the degree of system reliability: r F ( ) def T = F ( ) E X j T e Xe + [ F ( )] ; (3.3) with robability F ( ) there will be a suly shortage and only a fraction E T j T of total demand is exected to be satis ed, while with robability F ( ) no shortage occurs and demand is entirely met. Useful roerties of r F () are: Z r F ( ) = F (T ) dt; (3.4a) T rf ( ) = F ( ) E X j T e Xe <, and (3.4b) lim r F ( ) = : (3.4c)! Alying integration by arts on the left-hand side of (3.2) then allows us to rewrite this rst-order condition as (see aendix): Z u u T ^x T R X (^x) + e r ^x F (T ) dt = r F ( ): (3.5) We now comare this rst order condition with the case where suly is exected to be adequate in the sense that F (T ) = for all T. Then r F ( ) =, and the otimal order x must satisfy u (x ) = : (3.6) Comaring (3.5) with (3.6), shows that there are two reasons for ordering more under inadequate suly. First, the exected rice r F ( ) lies below the nominal rice. Second, the left-hand side of (3.5) contains a term that is not resent in (3.6). We call this term the marginal risk remium e ect. It accommodates for the utility consequences of a marginal ordered unit in those states of the world where suly is insu cient. Because the consumer faces a multilicative rather than additive risk, risk aversion alone is not su cient for a ositive risk remium. Only if relative risk aversion exceeds unity will a marginal order rovide a hedge against the consumtion risk. This is the second reason for osting a higher demand than under certainty. We write demand as ^x = ^x (; ; F ()). 9

10 Proosition If inadequate suly is exected with some ositive robability, and R r >, the consumer will ost a larger ex ante demand than when suly is deemed adequate. The second-order condition may be written as (see aendix): SOC^x = bu Rr b + X Z eu R bx e bu e r ( P e r 2)F (T )dt < ; (3.7) where ab above an exression means evaluation at bx, while ae means evaluation at bxt. A su ciently high relative rudence thus ensures that the second-order condition is veri ed. 3. Comarative statics at the individual level In this section, we investigate how the consumer who exects interrutions adjusts his ex ante order because of marginal changes in, and as well as marginal changes in the uncertainty surrounding the suly caacity. Simle comarative statics on (3.5) ( SOC^x) = r F ( ) < : (3.8) It is easy to show that with CRRA references, the rice elasticity is R r. A marginal rice increase will reduce maximal exected utility with the exected = bx r F ( ) < : (3.9) Proosition 2 The consumer s ex-ante demand is decreasing in the rice while a marginal rice increase reduces exected utility with the reliable art of the osted demand. How will the consumer s demand resond to a small change in the exected aggregate demand? Again taking comarative statics on ( SOC^x) = Z 2 eu err R e F (T ) e X rt dt e X e o +bu brr F ( ) rf ( ) :(3.) The last term on the right-hand side catures the e ect on the marginal exected outlay of an order. Since an increase in exected demand reduces

11 the reliability rate (cf (3.4b)), so does the marginal exected outlay, and this encourages a higher ex ante order. The rst two terms in curly brackets account for the e ect on the marginal risk remium. Two sets of su cient conditions are identi able for this e ect to be ositive. The rst is that relative risk aversion is larger than but falling (the latter being equivalent to + R r < P r ). The second is revealed by noting that the term in square brackets may be rewritten as + epr 3 err (see aendix). Therefore, a relative rudence larger than 3 and a relative risk aversion exceeding again ensure >. With the range for r mentioned earlier, and P r ' R r +, these conditions will be veri ed. The e ect of a small increase in anticiated aggregate demand on maximal exected = bx [u (bx) ] ( F ( ): (3.) Since u (x ) =, this e ect is negative under the same conditions that give bx > x. This suggests that there is a negative demand externality: higher exectations about aggregate demand boost individual demand and reduce individual welfare. This externality will lay an imortant role in the otimal ricing rule to be derived in Section 4. Proosition 3 When either R r exceeds but is falling, or when P r and R r exceed 3 and, resectively, the consumer s ex-ante demand is increasing in the exected aggregate demand. The e ect of a marginal increase in the exected aggregate demand on the consumer s maximal exected utility is negative under the same conditions that give bx > x. Finally, we examine the e ect of marginal changes in the suly distribution. For this urose, we rede ne F () as a weighted average of two robability distributions, G() and H(): F (T; ) def = ( ) G (T ) + H (T ). If G rst degree stochastically dominates H, then d > can be thought of as a FSD-deteriorating shift and the condition is that On the other hand, if Z z F (T; ) dt = F (T; ) = H(T ) G(T ) ; 8 T 2 [; ) : (3.2) Z z [H(T ) G(T )] dt ; for all z 2 [; ) ; (3.3)

12 then G second order stochastically dominates H, and d > can be thought of as a SSD-deteriorating shift. 7 Using (3.4a), the e ect of d on r F () F ( = Z F (T; )dt = r H ( ) r G ( ): (3.4) Thus both a rst and second order dominance deteriorating shift reduce the reliability of the system. The imortant observation is that the consumer s net marginal utility behaves asymmetrically around T = where it dislays a kink. 8 This is intuitive, as in situations with T >, no interrution occurs, and the net marginal utility is indeendent of the degree of excess caacity. However, with T <, the degree of caacity shortage will a ect the net marginal utility of the ex ante order. This is resented in Figure, where it is assumed that u T ^x T T is falling in T. For this to haen, it is su cient that R r ( T ^x) exceeds for all T. The net marginal utility function is now non-increasing in T, and its exected value will increase due to an FSD-deteriorating shift. 9 For T <, convexity of the net marginal utility is equivalent to P r ( T ^x) 2. Since the maximum of two convex functions is convex, the exected value of the net marginal utility function will increase due to an SSD-deteriorating shift. Formally, we may di erentiate (3.5) comletely to ( SOC^x) F ( ) + eu F err (T; ) dt {z@ } X {z e } marginal outlay e ect marginal risk remium e ect (3.5) The marginal outlay e ect is clearly negative. To see the e ect of an FSDdeteriorating shift, it follows immediately from (3.5) that a relative risk R 7 If in addition G and H have the same mean (i.e., F (T; )dt = R [H(T ) G(T )]dt = ), then d > may be considered as a mean-reserving sread (cf. Rothschild and Stiglitz, 97), also called a 2nd degree increase in risk. 8 The consequences kinks in the ayo function for the e ects of SSD-shifts on otimal decisons were rst discussed by Kanbur (982). 9 It can also be shown that R r ( T X bx) > is a necessary condition for the exected e marginal utility to increase for any arbitrary FSD-deteriorating shift. It can also be shown that R r ( T X bx) > and P e r ( T X bx) > 2 together are a necessary set of e conditions for the exected marginal utility to increase for any arbitrary SSD-deteriorating shift. 2

13 Figure : Net marginal utility as a function of available suly In- aversion exceeding is su cient to give rise to a higher ex ante order. tegration by arts allows us to rewrite the marginal risk remium as bu b ( Z h i eu Rr epr 2 dt: Thus a relative risk aversion exceeding and relative rudence exceeding 2 are jointly su cient conditions for the marginal risk remium e ect to be ositive for any shift in distribution that lowers reliability at all levels. Grahically, these conditions ensure that the function drawn in Figure is convex and therefore that its exected value will raise above due to such shifts. The otimal resonse is to bring this exected value down to zero again by increasing the ex ante order. The condition on relative rudence is reminiscent of the analysis of recautionary savings behaviour: if the rate of return to savings becomes more risky, the consumer will increase the amount saved if and only if his relative rudence exceeds 2 (this result dates back to Leland (968); a modern account is found in Eeckhoudt and Schlesinger, 28). Prudence needs to be high enough to lace a higher order because on the one hand a more risky distribution makes the uncertain consumtion of the service or good less attractive comared with the certain consumtion of the numéraire (the substitution e ect), but on the other hand, the increase in risk makes the consumer more cautious (the recautionary motive e ect). Here, we need in addition a condition on relative risk aversion. Grahically, this is easy to understand. If bu + R b r where negative, the net marginal bene t function would cease to be convex in the neighbourhood of the kink in Figure. A relative risk aversion exceeding at ^x rules this ossibility out. As we will show in the next section, stability in a rational exectations equilibrium requires recisely that bu + R b r >. 3

14 Figure 2: Utility as a function of available suly. Proosition 4 If R r >, an FSD-deteriorating shift in the distribution of available caacity will result in a higher ex ante demand. If R r > and P r > 2, any shift in the distribution of available caacity that reduces the reliability at all levels will result in a higher ex-ante demand. The e ect of a erturbation of the caacity distribution function on the maximal exected utility is = Z ( eu ) F (T )dt; (3.7) = ( bu F ( bu Z eu bu F R r dt; where the second equality follows after integration by arts. Insection of either (3.7) or (3.8) shows that neither an FSD-deteriorating shift, nor an SSD-deteriorating shift need result in a fall in exected utility. The reason is that utility as a function of T, u bx minf; T g + m bxf; T g, will not be monotonically increasing (FSD-deteriorating shift) nor concave (SSD-deteriorating shift) whenever ^x > x. This is illustrated in Figure Comarative statics under rational exectations Previously, we treated the anticiated aggregate demand as an exogenously de ned variable. We now roceed by imosing rational exectations (RE), so 4

15 that this anticiation is con rmed in equilibrium, viz., = ^x (; ; F ()). 2 This means that everywhere in the analysis, we can relace by ^x. For exectations to be imlicitly de ned by the model, j^x= 6=. In addition, for the RE equilibrium to be stable under eductive learning, we need @ j^x= <. Working out by means of and (3.7) j^x= = bu + b R r br r + R ^x eu e R epr bu r 2 ( F (^x)) F (T j^x= >, the stability condition thus amounts to: which we assume to hold from now on. ^x dt : (3.9) bu + b R r > ; (3.2) The equilibrium e ects on demand from changes in outut rice or caacity uncertainty j = ^x r F (^x) h i h bu F b + b i < ; (3.2) R bu j = bx + R bx eu err bu (T ) dt i h h F b + b i > : (3.22) R bu F bu Note that the stability assumtion ensures that the denominator is ositive. Therefore, the same conditions that guarantee the exected sign at the individual level, will do so in equilibrium. 2 The simlicity with which the rational exectations equilibrium can be de ned in our model is due to our assumtion that consumers have identical references. If references where heterogeneous, i.e., U(x; Y ; ) = u(x) + Y where has cdf () on suort, ^x (; ; F (); ) solves Z T max u x minf; x g T + m x minf; g df (T ) and the rational exectations condition becomes = R 2 ^x ; Xe ; F (); d ( ). (This is the Bayesian Nash equilibrium concet in a game with incomlete information about tyes, but with common knowledge that tyes are drawn from the distribution () on.) 5

16 We can now deduce the equilibrium e ects of changes in rice and suly uncertainty on consumer welfare. The former e ect is dv d j j j RE; = bx br F b Rr bu + b R r : (3.23) Since bx > x, the denominator will exceed R b r. Thus the welfare e ect of a rice increase is less detrimental in equilibrium than at the individual level. This is because a rice increase will reduce aggregate demand which in turn reduces the likelihood of being rationed o and hence imroves welfare (cf (3.)). The e ect of suly uncertainty on consumer welfare is given by dv d j @X j j RE: (3.24) In the aendix, we show this can be written as R bx eu dv d j RE = bu bu e R r h brr i e bu Pr bu + b R dt : (3.25) Recall that at the individual level, a shift in the distribution function of either an FSD or SSD tye, could a ect exected utility in both in a ositive or negative way. Equation (3.25) shows that any shift that reduces reliability at all exectations levels will reduce exected consumer welfare under relative risk aversion and with relative rudence larger than. We have now all the ingredients for carrying out an analysis of the otimal ricing and investment olicy. 4 Welfare maximising ricing and investment In this section, we study the otimal ricing olicy, and the welfare e ects of changes in the caacity distribution. For this urose, we de ne social welfare as the sum of exected consumer surlus V and exected ro t, while accounting for the fact that any loss that the ublic rm makes has to be nanced through distortionary taxation on other economic activities (cf La ont and Tirole, 993: 24). 6

17 Denoting the shadow cost of ublic funds by >, the roblem of the regulator is then: max W def = V + ( + )(E K); (4.) where V is the consumer s exected utility from (3.) with x =, E is the sulier s exected oerating surlus, and K denotes xed costs. In the aendix, we show that: E = ( c) r F (bx)bx; (4.2) where we remind the reader that c is the marginal cost of roduction. 3 then have and also de d j RE = r F (bx)bx + ( c)( j RE, = r F (bx)bx de d (bx) RE = ( = ( c) R bx We c + R b bu r + R b bu r ; (4.3) bx + r j RE, (er G er H ) dt : (4.4) eu bu e R r epr 2 bu + b R r An increase in (whether an FSD- or SSD-deterioration) has two oosite e ects on exected oerating surlus. On the one hand, it reduces reliability, while under the other hand it increases ex ante demand (under the conditions mentioned in section 3.3). Exression (4.4) shows that when consumers are su ciently rudent, the net e ect will be ositive. The manager of the utility has therefore little incentive to enhance reliability. 4. The otimal ricing rule When consumers rationally exect a reliability rate below %, the ex ante demand will satisfy (3.5) with = ^x. Using (3.23) and (4.3), the otimal 3 Problem (4.) reduces to ro t maximisation when!. Alternatively, we could formulate the roblem as a utility maximisation roblem, subject to the constraint that the oerating surlus (together with any exogenous subsidies) should cover the xed costs. Under this alternative, ( + ) becomes the endogenous Lagrange multilier to the break even constraint E K. Analytically, both aroaches are equivalent. 7

18 rice olicy then necessarily satis es the rst-order condition: dw d j Rr b RE = bx br F + R b bu r Rearranging then gives: u (bx) + ( + ) br F bx c + R b bu r + R b bu r = : (4.5) + R r (bx) = c; (4.6) which imlicitly de nes the welfare maximising level of the ex ante order bx. A necessary condition for a nite rice ^ to maximise ro ts is that the square bracket term be ositive. This imoses an uer bound on the coe cient of relative risk aversion given by +. If = :2 (:3), then R r must not exceed 6 (4 ). The corresonding SOC is 3 u (bx) + [P r (bx) ] < ; (4.7) requiring that P r (bx) < +2. If = :2 (:3), then P r must not exceed 7 (5 3 ). With CRRA references, P r R r +, and both uer bound conditions are equivalent. An imortant feature of (4.6) is that it is indeendent of the reliability rate r F (bx), and thus of the suly distribution F (). Therefore, the otimal ex ante order under uncertainty is identical to the otimal order under adequate suly, x. If we denote the otimal rice under suly adequacy by, then (4.6) shows that this rice must satisfy the familiar inverse elasticity rule c = + "(x ). (4.8) Earlier, we concluded that for a given rice, the consumer will lace a higher ex ante order when he exects inadequate suly, relative to when he deems suly to be adequate. It follows that the otimal rice under inadequate suly, b, needs to exceed to choke o the ex ante demand, and to equalise the demand in both cases. Intuitively, the absence of a market for contingent claims and the use of a roortional rationing rule introduces a negative externality among consumers. A single consumer neglects the fact that when lacing a higher order to hedge against an uncertain delivery, he reduces the exected reliability rate of the system, thereby harming everybody else. The otimal rice di erence b thus acts as Pigouvian tax to internalise this demand externality. 8

19 Relacing in the consumer s necessary condition (3.5) by b, bu by, and rearranging, we obtain the otimal mark-u of the rice under suly inadequacy relative to the otimal rice under adequacy: b = r F x Z x + eu err F (T ) dt: Using a rst order Taylor aroximation of the square bracket term around T = x, we may also write this as ^ ' R r (Pr ) r F rf 2 (P r 2) s F ; (4.9) rf where s def = F (x ) E ( T x ) 2 j T x + [ F (x )], i.e., the second moment of the degree of suly reliability. This mark-u rule has a straightforward oerational content, linking the size of the Pigouvian tax to the degrees of relative risk aversion and rudence, and summary statistics of system reliability. Clearly, the mark-u is increasing in risk aversion. To see the e ect of rudence, note r ^ r ' Rr s r 2 r = R r 2r F Z x 2 T df (T ) > : x Intuitively, a strong degree of rudence underscores the consumer s recautionary motive when lacing an order. This boosts the ex ante demand, and thus has to be mitigated through a higher rice. Finally, system reliability has the e ^ ' 2 R r [Pr + (Pr 2)s F ] F which is negative when P r > 2. Proosition 5 The otimal ex-ante demand is indeendent of whether suly caacity is exected to be adequate or not. The otimal rice when suly is regarded inadequate must rise above the corresonding otimal rice with adequate suly in roortion to the degree of relative risk aversion, to the extent consumers are rudent, and to the extent the system is unreliable. In the secial case of logarithmic utility, the mark-u reduces to b = r. Hence, a erceived reliability of 75% requires a rice exceeding the r base level by 33%. 4 This is a ceteris aribus result as P r and R r are related through P r R r + + d log R r(w) d log w. 9

20 4.2 Welfare e ect of a reliability imroving investment Let us now look at the welfare e ects of an investment that leads to a reliability imroving shift in the caacity distribution. We denote this shift as d. This amounts to a reduction in the arameter so that d = d. Since the otimal ex ante order is entirely governed by (4.6) and therefore indeendent of the caacity distribution F (), any change in this distribution will trigger a rice e ect to kee the ex ante order at x. In the aendix, it is shown that de d j x = R r + + Z x eu R epr r 2 dt: Thus, if P r 2, even under otimal ricing will a reliability imrovement have a negative imact on the exected oerating surlus of the utility. The imact on consumer welfare is given by dv d j x = R r and therefore ositive if P r. ) dk d Z x eu R epr r dt; The e ect on social welfare is then found as (4.)+( + )(4.) ( + dk, where is the marginal investment cost: d Z dw x d j x = eu e h R r epr dt ( + )dk d : (4.2) The rst term accounts for the welfare e ects of a changes in risk exosure due to d. The strength of this e ect deends on the degree of relative risk aversion. Its sign deends on the degree of relative rudence. A su cient condition for this welfare e ect to be ositive is that Pr < +2 (cf the SOC (4.7)) and that Pr. Proosition 6 If P r (T ) < +2 (all T < x ), the social welfare e ects of an FSD- or SSD-imroving shift in the suly distribution are always ositive (when ignoring the investment cost). Since the utility when concerned with maximising ro ts (or minimising losses) would never enact reliability imroving investments, the above roosition underscores to need for regulation of infrastructure choices. We conclude by investigating the investment rule (4.2) under the assumtion of CRRA references. Then R r and P r are constant and related 2

21 Figure 3: Density and reliability functions before (solid) and after (dashed) an FDS imrovement. as P r = R r +, and the square bracket term in (4.2) reduces to (+) R r. This term will be ositive i R r < +, which is exactly the earlier mentioned necessary condition for an interior otimal ricing olicy. Furthermore, we can make use of the otimality condition (4.6) to rewrite this term as ( + ) c. The welfare e ect of a reliability imroving investment then becomes Z dw x d j eu x = ( + ) F dt dk : (4.3) d The aearance of the marginal cost on the bene t side is not surrising: the otimal ricing rule tells us that the marginal cost exactly measures the marginal willingness to ay, discounted for the social cost of the risk remium. In Figure 3, the bell shaed curves deict the suly density function before (solid) and after (dashed) an FSD imroving shift. 5 The monotonically downward sloing lines are the corresonding reliability functions. Exression (4.3) suggests that we should comute the bene t of this shift as the area between the reliability functions, but weighted by the ratio u (T ). Since u (T ) = u (T ) u (x ) > because of risk aversion, (4.3) suggests the 5 The density function f(t ) corresonds to a transformed Beta distribution: f(t ) = Beta( T ; 4; 2), where = (solid) and = : (dashed). 2

22 following oerational su cient condition for caacity exansion: exand if cr r Z dk dt >. (4.4) d 5 Conclusion The objective of this aer has been to analyse questions of otimal ricing and caacity choice for an interrutible ublic service, while recognizing the interdeendence of system reliability and consumer demand. Overall, the analysis shows that ercetions of system reliability lay a signi cant role in the formation of consumer demand for a ublic service like electric ower, water, transort, gas, etc. Furthermore, the interdeendence between system reliability and demand must be taken into consideration when determining the service rice and caacity investment. In articular we have shown that when consumers are roortionally rationed in case suly falls short of aggregate demand, suly uncertainty tyically leads to larger ex ante orders to hedge against the uncertainty. This recautionary reaction is in the same sirit of a recautionary savings increase due to increased uncertainty about the rate if return on savings. A su cient condition for it to come about is that consumers are su ciently risk averse and rudent. In addition, we have shown that the welfare otimal rice under suly uncertainty is one that imlements the same consumtion level as under certainty. This means the rice must exceed the marginal willingness to ay for the socially otimal level by a mark-u that counteracts the recautionary motives for consuming more. Finally, we have shown that even though a reliability imroving investment will result in lower oerating surlus, this dro will be outweighed by an increase in consumer surlus. Our results are derived within a stylised model that could be extended in several dimensions. One is the introduction of more heterogeneity among consumers, in articular by allowing for di erences in references for the good or service in question. We exlained in footnote 2 how this should be done. A second extension is the introduction of risk aversion with resect to income. This would call for an extra instrument, to wit the use of monetary comensation in case a consumer gets rationed. In several countries, the regulatory authorities imose electricity suliers to hand out comensations in case of interrution. One may exect that the size of the comensation 22

23 (er unit undelivered) relative to the rice (er unit delivered) will hinge on the di erence in relative risk aversion with resect to consumtion of the articular good and the numéraire. References [] Baldick, R., Kolos, S. and Tomaidis, S. (26). Interrutible electricity contracts from an electricity retailer s oint of view: valuation and otimal interrution. Oerations Research, 54: [2] Benassy, J.-P. (977) "On quantity signals and the foundations of e ective demand," Scandinavian Journal of Economics, 79: [3] Borenstein, S. (27) "Customer risk from real-time retail electricity ricing: bill volatility and hedgability," Energy Journal, 28:-3. [4] Brown, G. and Johnson, B.M. (969). "Public utility ricing and outut under risk." American Economic Review, 59: [5] Carlton, D. (977). "Peak load ricing with stochastic demand." American Economic Review, 67: 6-. [6] Chao, H.-P. (983). "Peak-load ricing and caacity lanning with demand and suly uncertainty." Bell Journal of Economics, 4: 7-9. [7] Chao, H.-P. and Wilson, R. (987). "Priority service: ricing, investment and market organization." American Economic Review, 47: [8] Coate, S. and Panzar, J.C. (989). Public utility ricing and caacity choice under risk: a rational exectations aroach, Joumal of Regulatory Economics, : [9] Crew, M.A. and Kleindorfer P.R. (978). "Reliability and ublic utility ricing," American Economic Review, 68: 3-4. [] Crew, M.A., Fernando, C.P. and Kleindorfer, P.R. (995). Theory of eak-load ricing: a survey. Journal of Regulatory Economics, 8: [] Doucet, J. and Roland, M. (993). "E cient self-rationing of electricity revisited." Journal of Regulatory Economics, 5:

24 [2] Eeckhoudt, L., Etner, J. and Schroyen, F. (29), The values of relative risk aversion and rudence: a context-free interretation Mathematical social Sciences, 58: -7. [3] Eeckhoudt, L. and Schlesinger, H. (28). Changes in risk and the demand for saving. Journal of Monetary Economics, 55: [4] Fakhraei, S. H., Narayanan R., and Hughes T. (984). "Price rigidity and quantity rationing rules under stochastic water suly", Water Resources Research, 2: [5] Gerda, T.W. and Varaiya, P.P. (993). Markets for ricing interrutible electric ower. IEEE Transactions on Power Systems. [6] Kamat, R. and Oren.,S. (22). Exotic otions for interrutible electricity suly contracts. Oerations Research, 5: [7] Kanbur, S.M.R, (982). Increases in risk with kinked ayo functions, Journal of Economic Theory, 27: [8] Kimball, M.S. (99). Precautionary savings in the small and the large. Econometrica, 58: [9] Kleindorfer, P.R. and Fernando, C.P. (993). Peak-load ricing and reliability under uncertainty. Journal of Regulatory Economics, 5: 5-23 [2] La ont, J.J. and Tirole, J. (993) "A theory of incentives in rocurement and regulation" MIT Press, Cambridge, Mass. [2] Leland, H.E. (968). "Saving and uncertainty: the recautionary demand for saving" Quarterly Journal of Economics, 82: [22] Lin, W., Chen Y. and Chatov R. (987) "The demand for natural gas, electricity and heating oil in the United States", Resources and Energy, 9: [23] Marchand, M. (974). "Pricing ower sulied on an interrutible basis." Euroean Economic Review, 5: [24] Meyer, R. (975). "Monooly ricing and caacity choice under uncertainty." American Economic Review, 65: [25] Moulin, H. (2) "Priority rules and other asymmetric rationing methods" Econometrica 68 (2),

25 [26] Nauges C and Reynaud A (2) "Estimation de la demande domestique d eau otable en France", Revue économique, 52: [27] Panzar, J. and Sibley, D. (978). "Public utility ricing under risk: the case of self-rationing." American Economic Review, 68: [28] Rees, R. (98). "Consumer choice and nonrice rationing in ublicutility ricing." In Proceedings of an International Symosium on Public Regulation and Public Enterrises, edited by Paul Kleindorfer and Bridger Mitchell. Lexington, MA: D.C. Heath. [29] Reynaud A (23) "An econometric estimation of industrial water demand in France", Environmental and Resource Economics 25: [3] Rocha, P. and Siddiqui, A. (28). Risk analysis of interrutible load contracts. (Retrieved May, 2 from htt:// les/rr296.df] [3] Rothschild, M. and Stiglitz, J. E. (97). Increasing risk I: a de nition, Journal of Economics Theory, 2: [32] Sandmo, A. (97). "The e ect of uncertainty on saving decisions," Review of Economic Studies, 37: [33] San Francisco Public utility Commission (SFPUC) (27) "Measures to reduce the economic imacts of a droughtinduced water shortage in the SF Bay Area," Bay Area Economic Forum, retrieved Setember, 2 from htt:// les/df/hetchhetchydroughtimacts- SFPUC-FINAL.df. [34] Tschirhart, J. (98). "On ublic utility ricing under stochastic demand," Scottish Journal of Political Economy, 27: [35] Turvey, R. (97). "Public utility ricing and outut under risk: comment." American Economic Review, 6: [36] Visscher, M. (973). Welfare maximizing rice and outut with stochastic demand: comment. American Economic Review, 63: [37] Wilson, R. (989a). "E cient and cometitive rationing," Econometrica, 57: -4. [38] Wilson, R. (989b). "Ramsey ricing of riority service," Journal of Regulatory Economics, :

26 [39] Woo, C.-K. (99). "E cient electricity ricing with rationing." Journal of Regulatory Economics, 2: Aendix Proerties of the reliability function r F (X) can be written as Z X Integration by arts gives r F (X) = [ T X F (T )]X T df (T ) + [ F (X)] : X Z X F (T ) dt + [ F (X)] ; X which gives (3.4a). (3.4b) follows straightforwardly by di erentiating the right-hand side of (3.4a) and rearranging. To rove (3.4c), start from ((3.4a) and use the mean value theorem for integrals: r F (X) = X F (Z)X = As X!, Z! even faster. F (Z), for some Z 2 [; X]: Rewriting the rst order condition using integration by arts Integration by arts the second left-hand side term in (3.2) gives: Z Z u T T u X ^x e X df (T ) = e u (^x) F ( ) T ^x " X u T ^x # e where it is assumed that limu T! (3.5) u T ^x T X ^x + e F (T ) dt; T ^x T =. Then (3.2) can be written as The second-order condition and comarative statics. Deriving (3.5) once again with resect to ^x yields the exression for the second-order condition: def SOC^x = u (^x) + Z + u T ^x R r u T ^x T R r T ^x T T X ^x F (T ) dt; e 26

27 which we may re-exress as: Because SOC^x = bu + R r = Z (6.) may be written as: SOC^x = bu + = bu + = bu bx Z Z br r + eu T u [u + u x eu T eu h i Rr e + R eu e F (T ) r dt: (6.) u x u u ] = u u [ P r + R r ]; (6.2) eu eu h e Rr i + ( ( eu ) T n o 2 e F (T ) Pr dt Z which is eq (3.7) in the text. eu bu e R epr F (T ) r 2 eu eu )[ Pr e + R e F (T ) r ] dt The e ect of a change in the exected aggregate demand u T ^x T R r ^x! h i eu err h i eu err + eu R e T ( ) 2 r ( ) ^x 2 eu nh i h i = err er err ( ) 2 r + R e o rex h i = eu 2 err R e ( ) rex ; 2 the derivative of the rst-order condition with resect to becomes: Z 2 eu err R e F (T ) X rex dt + bu brr F ( ) r ( ) ; e = dt ; Z 2 eu err R e F (T ) X rex dt e SOC^x X e o +bu brr F ( ) r ( ) : 27

28 Making use of (3.7), this becomes: 8 R >< 9 2 eu err bu R e F (T ) rex dt >= X e + brr F ( ) r ( ) = bu n e brr + R eu e o : R epr bx bu r 2 F (T ) dt which is exression (3.) in the text. The RE stability condition The RE stability condition is that in an equilibrium, when = ^x, j RE <. Z ^x eu br r + bu e R epr F (T ) r 2 ^x = br r + R ^x br j e RE = Z ^x dt eu bu + brr F (^x) r (^x)^x o bu Z ^x eu bu h err i F (T ) br r + R ^x 2 err R e dt eu bu e R epr F (T ) r 2 ^x dt + brr F (^x) ^x eu e R epr bu r 2 F (T ) F (T ) rt ^x dt r (^x)^x +c bu ; dt ^x where use is made of the fact that Rrx = R r [ P r +R r ] (cf (6.2)). Using the rst-order condition (3.5), the second term in the numerator may be relaced by +c ( r(^x)). Keeing in mind that r (^x)^x = F (^x) + r(^x), bu bu we + b j bu e RE = r ( F (^x)) br r + R ^x eu e : R epr bu r 2 F (T ) dt ^x which is exression (3.9) in the text. The utility e ects in equilibrium 28

29 dv d j j j RE = bx br F + (bu ) ( F b ^x r F (^x) ) h i h bu F b + b i R bu r = bx br F b Rr bu + b R r ; which is exression (3.23) in the text. dv d j j j j RE Z bx = ( bu ) [br G br H ]bx bu eu R bu e r [er G er H ]dt b 3 R bu r (br G br H ) R + ^x bx eu e 5 R epr + (bu bu ) ( F b )^x r 2 (er G er H ) dt i h h F b + b i R bu r 8 h + >< b i h R bu r ( bu ) [br G br H ]bx bu R bx eu e i 9 R bu r [er G er H ]dt (bu >: bu ) ^x 4 b 3 >= R r (br G br H ) R + ^x bx eu e 5 R epr bu = r 2 (er G er H ) dt >; + R b bu r ( R bx eu br e ) r R bu r [er G er H ]dt h R bx eu + e i R epr = bu bu bu r (er G er H ) dt + R b bu r eu e h R bu r br i r + e bu Pr [er G er H ]dt R bx = bu which is exression (3.25) in the text. bu + b R r The comonents of exected welfare and their derivatives 29