Tariff regulation with energy efficiency goals

Energy Economics 49 (2015) 122–131

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Energy Economics journal homepage: www.elsevier.com/locate/eneco

Tariff regulation with energy efficiency goals☆ Laura Abrardi a, Carlo Cambini b,c,d,⁎ a

Politecnico di Torino, DIGEP, Italy Politecnico di Torino, DIGEP, Italy c IEFE-Bocconi, Milan, Italy d EUI — Florence School of Regulation, Italy b

a r t i c l e

i n f o

Article history: Received 25 March 2014 Received in revised form 8 December 2014 Accepted 9 January 2015 Available online 19 February 2015 JEL classification: L51 Keywords: Energy efficiency Demand-side regulation Decoupling Price-cap

a b s t r a c t We study the optimal tariff structure that could induce a regulated utility to promote energy efficiency by its customers given that it is privately informed about the effectiveness of its effort on demand reduction. The regulator should optimally offer a menu of incentive compatible two-part tariffs. If the firm's energy efficiency activities have a high impact on demand reduction, the consumer should pay a high fixed fee but a low per unit price, approximating the tariff structure to a decoupling policy, which strengthens the firm's incentives to pursue energy conservation. Instead, if the firm's effort to adopt energy efficiency actions is scarcely effective, the tariff is characterized by a low fixed fee but a high price per unit of energy consumed, thus shifting the incentives for energy conservation on consumers. The optimal tariff structure also depends on the cost of the consumer's effort (in case the consumer can also adopt energy efficiency measures) and on the degree of substitutability between the consumer's and the firm's efforts. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In recent years, energy conservation has risen to the attention of the public and policymakers due to the increased sensibility towards environmental issues, suggesting the adoption of new incentives for demand-side management and a revision of the current regulatory framework. Traditional regulation methods, such as rate of return regulation as well as incentive regulation methods (Armstrong and Sappington, 2006; Joskow, 2008), are not designed to provide incentive to utilities for promoting energy conservation and energy efficiency projects. However, energy conservation and the improvement of the level of efficiency in the use of energy are important energy policy goals. Indeed, alternative policy instruments to pursue energy conservation,

☆ We would like to thank Massimo Filippini, Alberto Iozzi and David E.M. Sappington as well as the participants to the 41st Annual Conference of the European Association for Research in Industrial Economics — EARIE (Milan, 2014) and to the Research Unit on Complexity and Economic — UECE (Lisbon, 2014), for the valuable discussion and suggestions to previous versions of the paper. Moreover, we would like to thank the Editor, Richard E.J. Tol, and two anonymous reviewers for their useful comments. ⁎ Corresponding author. E-mail addresses: [email protected] (L. Abrardi), [email protected] (C. Cambini).

http://dx.doi.org/10.1016/j.eneco.2015.01.017 0140-9883/© 2015 Elsevier B.V. All rights reserved.

such as the adoption of white certificates,1 have been only partially successful in promoting energy efficiency, meaning that some additional regulatory interventions are needed.2 The addition of energy efficiency goals to the social planner's objectives calls for a revision of the standard regulatory schemes. Indeed, incentive-based regulation performs badly in promoting energy conservation, as the reduction in consumption (ultimate goal of the energy efficiency programmes) and the consequent decrease in revenues and profits constitute a disincentive for the adoption of energy efficiency measures by the utilities (Brennan, 2010a,b; Carter, 2001; Gillingham et al., 2009). A regulatory intervention that compensates the utility for the costs of quantity reduction by raising prices has an ambivalent effect: on the one hand, it creates a distortion in the utility's optimal decisions by raising the opportunity cost of lost sales. On the other hand, higher prices increase the consumers' incentive to ration their consumption. Any policy intervention for incentivizing energy efficiency

1 White certificates are tradable permits representing a certain amount of energy saving to achieve a quantified target of energy reduction. 2 The implementation of white certificates is sensitive to the specific market structure and regulatory framework of the national context of application, thus requiring countryspecific adaptations. Moreover, white certificates target energy efficiency, and only a few measures target behavioural change of consumption patterns, thus representing only a partial response to energy conservation objectives. For more details, see Mundaca and Neij (2009), Bertoldi et al. (2010) and Giraudet et al. (2012).

L. Abrardi, C. Cambini / Energy Economics 49 (2015) 122–131

should then reconsider the price structure of traditional models, considering the more articulated effects that an intervention on prices could have over the goal of reducing energy demand. Our contribution aims at analysing the interplay between tariff regulation and energy efficiency goals and provides guidance to regulators with respect to policy design. Regulators have addressed the issue of adopting new, more environmentally sustainable, regulatory schemes in a number of ways. A widely adopted solution is decoupling,3 which guarantees to utilities constant revenues and profits regardless of how much energy they deliver (Brennan, 2013). An obvious flaw of decoupling policies is that, while they do not discourage utilities to adopt energy conservation programmes, they neither provide incentives to their efficient realization. This weakness becomes especially critical in the presence of information asymmetries about the cost or effectiveness of the programmes. An alternative to decoupling policies consists of monetary incentives based on the utilities' performance with respect to the energy efficiency goals. However, incentive schemes present drawbacks as well. First, they present a not negligible implementation complexity and require significant regulatory capabilities, both in the design of the incentive measures and in the monitoring of the utilities' performance. An even more fundamental problem is related to the scarcity of formal guidance at the theoretical level, which can result in suboptimal and overlapping energy-efficiency incentives, possibly conflicting with the incentives towards productive efficiency intrinsic in different tariff mechanisms. Although several authors have provided useful policy discussion (e.g., Brennan, 2010a; Eto et al., 1998; Moskovitz, 1989; Stoft and Gilbert, 1994), rigorous theoretical work in the field is recent and sparse, which could explain the notable variety of the measures adopted by regulators at the implementation level (see Dixon et al., 2010, for a survey of energy conservation policies in the United States and Tanaka, 2011, for other countries). The few existing theoretical studies range from moral hazard models (Chu and Sappington, 2013; Eom and Sweeney, 2009), that assume an unobservable effort in demand-reducing activities, to adverse selection studies (Chu and Sappington, 2012), which instead assume the utility's superior information on the effectiveness and cost of energy conservation activities, to models in which the consumers' preferences cannot be perfectly observed by the regulator (Lewis and Sappington, 1992). Moreover, a quite substantial literature is available on gain sharing plans (Chu and Sappington, 2012; Eom, 2009). Gain sharing plans provide energy firms incentives towards energy conservation by specifying in advance how the realized benefits of the programmes will be divided between the utility and consumers. One important limit of all these works is that, by studying how to promote energy efficiency (EE hereafter) activities by the utilities, they restrict the attention to the supply side of the market. However, by doing this, they overlook an important instrument of demand side management, namely the price. As a matter of fact, the firm's effort in promoting EE activities has certainly an important effect on consumers' demand, but we cannot disregard the primary role of the energy price on demand management. In practice, utilities are usually regulated through a tariff, whose function is twofold: it not only provides the necessary revenues to the firm, but it also presents itself as an instrument of demand management. A complete analysis cannot discount the latter, and should thus include the price in the set of incentives towards energy conservation. Actually, the need to formally include the consumers' response to the policy adopted has already been suggested (Chu and Sappington, 20124) but little work has been done on this issue. The optimal use of

3 Decoupling has been effectively implemented in the US in the states of Oregon and California and, on a more limited basis, in Maryland, New Jersey, North Carolina, Utah and Ohio (Kushler et al., 2006). 4 The authors claim that “in settings where consumers can undertake energy conservation activities, energy prices might also be structured to influence these activities”.

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price as an incentive to energy conservation is the aim of the present work. Our focus is primarily on regulated markets, with a special emphasis on the European scenario.5 However, the present analysis also relates to markets where deregulation has been more widely adopted, provided that the retail price is affected by some sort of regulatory interventions.6 In our model, an energy utility can exert effort to promote energy conservation by consumers and induce them to reduce their demand. These activities may be highly effective, and allow consumers to maintain the same level of comfort (i.e., the lumen-hours of light or the heating-degree hours) with significantly lower consumption, or instead have a lower effectiveness. For example, the installation on the consumers' premises of energy-saving devices allows consumers to save energy and yet benefit from the same levels of usage of the appliances. The firm is privately informed about the effectiveness of energy conservation activities. The regulator can merely observe the results of the firm's effort, namely the quantity of energy consumed. In this environment, the regulator must motivate the utility to deliver the effort by providing explicit financial rewards for observed levels of energy consumption. The firm's revenues come from a two-part tariff, with a unit price on the energy consumed and a fixed fee. In this setting, a firm that knows that its EE activities are highly effective gains a rent by underexerting the effort and explaining its low performance in terms on demand reduction with an unresponsive demand. Our purpose is to find the optimal menu of tariffs, which induces the firms to correctly self-select and at the same time allows the optimal trade-off between rent extraction and inefficiency. In this regard, our approach differs substantially from Chu and Sappington (2013), who study policies that allow the first best outcome, abstracting from the goal of rent reduction. We find that the tariff structure is indeed an instrument for energy conservation. When demand is less responsive to EE activities, the most efficient tariff has a strong variable component, with a high price paid per unit of energy consumed covering all the firm's costs: due to the low impact of EE effort on demand, it is more cost-effective trying to reduce demand through a higher price rather than through EE activities. When instead EE activities are highly effective, the regulator should base its tariff structure to a larger extent on the fixed fee, so as to replicate a decoupling policy and incentivize the utility to exert the effort; the energy price in this case is low so as to reduce the marginal loss from any unsold quantity, thus minimizing the incentives not to invest in EE activities. The presence of information asymmetries only strengthens this result, with the regulator offering a menu of tariffs where the one designed for the firm in the highly effective setting is polarized towards a fixed revenue solution, while the contract designed for the other type of firm is polarized towards a standard price regulation policy.

5 The most recent report by the Agency for the Cooperation of Energy Regulators (ACER/ CEER, 2014; pages 87–89) points out that in a minority of EU member states (e.g. Great Britain, Germany, the Netherlands, the Czech Republic, Slovenia and the Nordic countries) retail prices for residential consumers are fully liberalised, and there is no government intervention apart from social security policies. In the majority of EU countries end-users price regulation still exists with the national regulator setting the level or methodology of the regulated price, while in others (Belgium, France, Greece, Hungary and Spain), these are set by the government, with the regulator providing only an opinion. All in all, as of 31 December 2013, household end-user price regulation existed in 15 countries (out of 27). Full price deregulation is expected to be implemented only since 2017 onwards in most of the member states. Electricity household end user prices are subject to ex ante regulation also in Australia and New Zealand (ACCC/AER, 2013). 6 In the US, for example, the end-user prices are largely deregulated, but not in all states. For example, the Californian Public Utilities Commission (CPUC) sets the rates that electricity utilities may charge Californian consumers. Sappington et al. (2001) provide an overview of the regulatory regimes in place in the US electric utility industry, up to the beginning of the 2000. More recent data (Brattle Group, 2010) shows that 5 US states employed incentive regulation (such as earning/revenue sharing plans or price cap) in 2010, while the others rely on rate freeze/cost plus mechanism. Moreover, incentive regulation mechanism is currently adopted in several states for specific regulatory targets, such as quality of service or environmental sustainability.

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We then extend our setting assuming that not only firms but also consumers can exert an effort in energy conservation activities. In this case, the regulator should decide how to incentivize efforts from both parties and which tariff structure is preferable to allocate efforts between consumers and firms. The main problem in this framework is that the price qualifies at the same time as the loss of the firm for any lost sale and the gain for the consumer for any reduction of demand. It follows that a high price is both a disincentive to the firm to exert effort, and an incentive for the consumer. The regulator should then find the optimal trade-off between the firm's and the consumer's incentives. This optimal allocation of incentives obviously depends on the relative costs of the firm's and consumer's effort, but also on the degree of substitutability between them. When efforts are complement, meaning that the effort of say the firm generates a positive effect on the consumers' effort, then the regulator may prefer to reduce the optimal regulated price and increase the fixed component, hence moving towards a decoupling policy, to promote energy efficiency activities. On the contrary, when efforts are substitute, the increase in effort by one party negatively affects the other, hence the regulator may prefer to increase the regulated price and decrease the fixed component in order to reduce overall consumption given that efforts in energy conservation activities are reduced. Our analysis proceeds as follows. Section 2 describes the core elements of the model. Section 3 finds the socially optimal solution in the perfect information benchmark. Section 4 provides the optimal policy under imperfect information. Section 5 presents an extension where consumers can also exert effort in energy-saving activities. Section 6 concludes. All longer proofs are reported in the Appendix A.

The regulator grants the firm a constant price p on sales and a fixed transfer F. The firm chooses the level of effort in order to maximize its profits π(q, e) = pq + F − C(q) − ψ(e). Note that a tariff structure in which the firm's revenues are entirely made up by the fixed fee and are thus uncorrelated with the quantity sold is, in our framework, equivalent to a decoupling policy.9 Conversely, a fixed fee equal to zero defines a standard price-cap or rate of return policy, in which revenues are entirely proportional to sales. The presence of a regulator is crucial for our analysis. Indeed, in the absence of regulation, the firm would implement the monopolistic outcome, curtailing the costly and sales-reducing effort. Incentives to energy efficiency would entirely fall on consumers, whose surplus would ultimately be reduced. The regulator seeks to maximize the welfare W, defined as the consumer's net surplus minus the social cost of the production plus a share α b 1 of the firm's profits: W = V(x) − pq − F − D(q) + α π(q, e). The introduction of the firm's profits into the regulator's objective function replicates Baron and Myerson (1982) approach and constitutes a generalization of the situation analysed by Chu and Sappington (2012, 2013), in which the regulator's objective is focused only on the consumer's surplus. Given that π(q, e) = pq + F − C(q) − ψ(e), the welfare can be rewritten as

2. The model

max V ðxÞ−pq−F;

An energy-selling firm can devote some effort e to promote energy efficiency (EE) programmes by its customers. The energy efficiency programmes could be, for instance, the installation of energy-efficient appliances at the customers' premises or the promotion of energysaving behaviour that does not sacrifice the consumer's comfort. In choosing their level of consumption, customers seek to maximize their utility V(x), that is a function of the comfort7 x, such that V(0) = 0, Vx N 0, Vxx b 0. The customer's comfort is a function of the quantity q of energy consumed and of the effort e devoted by the firm to energy efficiency activities: x = q + θe, where θ N 0 is the marginal impact of EE effort on comfort.8 The parameter θ can be interpreted as the responsiveness of the demand to the EE effort, or alternatively as the effectiveness of EE activities. We assume that θ can take on two possible values, i.e. θ ∈ {θH, θL}, with θH N θL. If θ = θH, the demand is highly responsive (we denote this framework with the H subscript) to EE activities and one unit of effort by the firm allows customers to significantly increase their comfort given the same level of consumption or, similarly, to strongly reduce their consumption and yet maintain the same levels of comfort. Conversely, if θ = θL, EE activities have lower marginal impact on consumers' comfort (we will refer to this state as L). The production of quantity q entails for the firm a total production cost C(q), with C′ N 0 and C″ N 0, which includes fixed costs that do not depend on the scale of the production. Moreover, the production of q entails a social cost (e.g., the environmental loss) equal to D(q), with D′ N 0 and D′ N 0. The firm incurs also in the cost ψ(e) for delivering effort e, with ψ(0) = 0, ψ′ N 0 and ψ″ N 0. 7 For a similar approach based on the consumer's comfort, see also Chu and Sappington (2013). 8 Similarly to Chu and Sappington (2013), we assume that the level of comfort is positively correlated with the quantity consumed q, i.e., ∂x/∂q N 0 for all e. Moreover, given a level of consumption q, the energy efficiency effort e is assumed to be comfortenhancing, i.e., ∂x/∂e = θ N 0 for all q. Note that we assume a linear relationship to simplify the analysis and provide a clear intuition of the results. The outcome remains unaffected if we generalize the comfort function as x = f(q, θ, e) with ∂x/∂(⋅) N 0.

W ¼ V ðxÞ−DðqÞ−α ðC ðqÞ þ ψðeÞÞ−ð1−α Þðpq þ F Þ:

ð1Þ

Notice that, from the definition of the welfare in Eq. (1), the firm's revenues have a social cost equal to (1 − α). Consumers' demand is given by the solution to the following problem:

q

ð2Þ

that is V x ¼ p:

ð3Þ

Vx, on the left-handside of Eq. (3), is a function of both q and θe. For a x x ∂x ¼ ∂V ¼ V xx  1 b 0 : from Eq. (3), the quantity given level of θe; ∂V ∂q ∂x ∂q

demanded is a decreasing function of the price. It also seems important to point out the role of θe in determining the x x ∂x consumer's demand function. For a given level of q; ∂∂V ¼ ∂V ¼ V xx  ðθeÞ ∂x ∂ðθeÞ

1 b 0. Hence, Vx is a decreasing function of θe. In other words, an increase of θe shifts the demand function: consumers reduce their demand for any p when θe increases, as the effort exerted by the firm allows customers to achieve the same level of comfort with lower consumptions. We first find the optimal policy under perfect information. Then, we will assume that θ and e are private information of the firm and the regulator can only observe the quantity consumed. 3. Perfect information benchmark In perfect information, the regulator contracts over the tariff (p, F) and the effort e. It is worth noting that setting a price equal to p and an effort e, indirectly determines the consumer's demand q through the demand function defined by Eq. (3). It follows that the regulator can equivalently contract over the price or the quantity. The regulator solves       max V x j −p j q j −F j −D q j þ απ j q j ; e j qi ;ei

9

For an analysis of decoupling policies, see, e.g., Brennan (2013).

L. Abrardi, C. Cambini / Energy Economics 49 (2015) 122–131

s.t.       π j q j ; e j ¼ p j q j þ F j −C q j −ψ e j ≥ 0

ð4Þ

Vx ¼ pj

ð5Þ

for all j ∈ {H, L}. Constraint (4) represents the firm's participation constraint of at least zero profits, while constraint (5) is the consumer's demand function expressed by Eq. (3), namely Vx = p. As rents are costly for the regulator, in the optimal solution constraint (4) is binding. The first best solution is defined by the following Proposition. Proposition 1. Under perfect information, the regulator offers to the firm a contract (p∗j , Fj∗, e∗j ) if θ = θj, with the following characteristics: 

0

q j : V x ðqÞ ¼ C ′ ðqÞ þ D ðqÞ 

0

ð6Þ

e j : V x ðeÞ ¼ ψ ðeÞ=θ j

ð7Þ

     pj ¼ Vx qj; ej

ð8Þ

         F j ¼ C q j þ ψ e j −p j q j

ð9Þ

for all j ∈ {H, L}. From Eq. (6) of Proposition 1, the optimal quantity is such that the marginal benefit of one additional unit of consumption (Vx) is equal to its marginal cost, namely the marginal production cost C′ and the marginal social cost D′. Moreover, from Eq. (5), the regulator sets the price equal to the sum of marginal production and social costs, thus leading consumers to choose the first best level of consumption. From Eq. (7), the optimal level of the firm's effort is such that the marginal utility of effort (namely, the increase of the consumer's surplus by Vxθj) is equal to its marginal cost, ψ′. Finally, the fixed transfer is such that the firm's profits are zero. Before analysing the first best levels of q, e and p, let us study the demand function, defined by the value of θe. The following Lemma defines the contribution of the firm's effort in determining the consumer's comfort in the two cases, H and L. Lemma 1. In perfect information, it holds θH e∗H N θLe∗L, implying a lower demand function when EE activities are highly effective. Lemma 1 implies that, for a given price p, the demand for energy is lower when the consumers' comfort is strongly affected by the firm's EE activities. From another perspective, the consumers' comfort depends on the combined effect of the firm's effort e and of the effectiveness θ of EE activities. Given the same quantity, consumers are able to achieve a higher comfort when EE activities are highly effective (i.e., θ = θH), though Lemma 1 doesn't elaborate whether this is true only because of the higher θ or also because of higher e. Proposition 2 will provide further clarification. The pair of contracts (p∗H, FH∗ , e∗H) and (p∗L, F∗L, e∗L) satisfying conditions (6), (7), (8) and (9) are characterized by the following Proposition. Proposition 2. In perfect information, q∗H b q∗L, e∗H N e∗L, p∗H b p∗L, x∗H N x∗L, FH∗ N FL∗. Fig. 1 provides a graphical representation of the results of Proposition 2 in perfect information. On the left, the figure represents condition (6) defining the optimal quantity. Given that θHe∗H N θLe∗L from Lemma 1, it follows that Vx(q; θHe∗H) b Vx(q; θLe∗L) for all q, as shown in Fig. 1.a with the function Vx(q; θHe∗H) on the left of Vx(q; θLe∗L). From Eq. (6), it follows that q∗H b q∗L. The vertical axis shows the price, as provided by the optimality condition (8) that defines the demand function. Given that Vx(q∗H; θHe∗H) b Vx(q∗L; θLe∗L), we obtain p∗H b p∗L. In Fig. 1.b we derive the optimal value of e by exploiting condition (7). As q∗H b q∗L,

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it must be that Vx(e; q∗H) N Vx(e; q∗L) for all e. Moreover, as p∗H b p∗L, the only possible solution is the one showed in the figure, with e∗H N e∗L. Indirectly, the figure also provides information about the final level of comfort achieved by the customer. Indeed, since p∗H b p∗L and given that Vx = p by Eq. (5), it follows that V xH b V xL , thus implying x∗H N x∗L. The results of Proposition 2 have a straightforward interpretation. As consumption is socially costly, it is in the regulator's interest trying to achieve the customers' comfort by investing in EE activities rather than through consumption. Indeed, an additional unit of effort in EE activities entails a marginal reduction of the consumers' demand equal to θ. This implies that the marginal effect of effort on demand reduction is stronger when θ = θH rather than when θ = θL. Given the higher effectiveness of EE activities in the H case, the optimal solution must entail e∗H N e∗L and q∗H b q∗L. In other words, when EE activities are highly effective, the optimal regulatory policy is to spur the EE effort, curbing consumption, as opposed to the situation in which EE activities have low effectiveness. Incentives for exerting high effort can be provided by reducing the marginal loss from any unsold quantity, that is the price. Indeed, p∗H b p∗L. To compensate the firm in the H case for the lower revenues and higher effort cost, a larger fixed component must be included in the tariff, i.e. F∗H N F∗L. Note that, when EE activities are highly effective, the two-part tariff offered to the firm entails a larger fixed component and a lower price than in case EE activities are less effective. Thus, we can conclude that, even in the first best situation, the contract offered to the firm is closer to a decoupling policy in case EE activities are highly effective, while it is more similar to a standard per unit regulated tariff if EE activities have lower effectiveness. The adoption of a two-part tariff with a not-negligible fixed component exposes the regulator to the risk of exclusion of the most vulnerable customers. A regulator who is completely averse to such possibility has two options. The first is to subsidize a share of the fixed part of the tariff through the use of public funds. This would reduce the cost of energy charged to consumers. The second consists in trying to reduce the fixed transfer to the firm. Given the direct correlation between the fixed transfer and the effort (from Proposition 2), the reduction of the transfer is achieved by a downward distortion of the effort with respect to the first best level. It follows that the regulator has to trade-off the cost of the use of public funds and the cost of the inefficiency related to the departure from the first best outcome. In either case, the outcome is socially suboptimal, and the solution chosen by the regulator depends on the relative cost of the use of public funds and of the cost of inefficiency. 4. Asymmetric information Assume that the value of the parameter θ is the firm's private information, although its probability distribution is common knowledge. Let us denote with λ the probability that θ = θH, and with 1 − λ the probability that θ = θL. While the effort e and the effectiveness of EE activities θ are unobservable by the regulator, the quantity consumed q can be observed and verified.10 In the previous section, we showed that perfect information allows the full rent extraction from firms. However, if the same first best contracts are offered under asymmetric information, firm H earns a positive profit by choosing the contract designed in perfect information for firm 10 Note that this approach is completely equivalent to the one adopted by Chu and Sappington (2013), which assumes that the comfort x is observable and verifiable. The focus on consumer's comfort as an incentive for utilities to energy conservation was first suggested by Sant (1980) as part of the Energy Savings Model, according to which utilities should sell customers the actual services for which they use energy, such as light and heat, rather than the quantity of energy consumed. While Fox-Penner (2010) notices that the adoption of smart meters may eventually facilitate an accurate measure of comfort, this approach could be difficult to implement in practice. For this reason we use, as observable variable, the quantity consumed. Moreover, our approach presents an additional advantage, in that it allows identifying the optimal solution even in case comfort cannot be verified.

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Fig. 1. First best outcome.

L. This result is a direct consequence of the fact that, being EE activities highly effective, the H firm can achieve the L level of comfort and quantity with lower effort cost than the L firm. When firms have an informative advantage on the value of θ, the regulator offers a menu of incentive compatible contracts. Each contract includes the specification of the tariff in terms of price p and fixed component F, and the level of observable quantity q that the consumer demands, given the price and the firm's effort e. Formally, the regulator offers the menu of contracts (pH, FH, qH) and (pL, FL, qL). In the optimal solution, the firm will self-select and choose contract (pH, FH, qH) if θ = θH, or (pL, FL, qL) if θ = θL. The correct self-selection of firms is ensured by the incentive compatibility of the menu of contracts. To this aim, let us consider the effort exerted by firm H. If it chooses the contract designed for firm L, the effort exerted to achieve sales equal to qL is eHL ¼

xL −qL ; θH

ð10Þ

where the subscript HL indicates the effort that a firm of type H needs to exert to show sales equal to qL. Given that xL = qL + θLeL, expression (10) becomes: eHL ¼

θL e: θH L

Indeed, as θL b θH firm H needs to exert a lower effort than firm L to sell the same quantity qL, given price pL. Similarly, the effort exerted by firm L to sell quantity qH is: eLH ¼

 pH qH þ F H −C ðqH Þ−ψðeH Þ ≥pL qL þ F L −C ðqL Þ−ψðeL Þ þ ψðeL Þ−ψ

  θ pH qH þ F H −C ðqH Þ−ψðeH Þ ≥ ψðeL Þ−ψ L eL : θH

condition (17) is always strictly positive. It follows that condition (13) is not binding. It is also worth noting that Eq. (17) expresses the rents that must be paid to firm H to ensure the incentive compatibility of the menu of contracts. In particular, Eq. (17) implies that the informational rent paid to firm H is a decreasing function of firm L's effort. The solution of the regulator's problem is defined in Proposition 3. Proposition 3. Optimal regulation under imperfect information is characterized by the offer of the menu of contracts (pH, FH, qH) and (pL, FL, qL) such that 0

0



qH : V x ðqH Þ ¼ C ðqH Þ þ D ðqH Þ; or qH ¼ qH

eH : V x ðeH Þ ¼

max λ½V ðxH Þ−DðqH Þ−α ðC ðqH Þ þ ψðeH ÞÞ−ð1−α ÞðpH qH þ F H Þ þ

qH ;eH ;qL ;eL

ð17Þ

Given that, by assumption, ψ′ N 0 and eL N θθHL eL , the left hand-side of

0

qL : V x ðqL Þ ¼ C ðqL Þ þ D ðqL Þ

The regulator solves

 θL eL ; θH

that is

0

xH −qH θH ¼ e : θL θL H

þð1−λÞ½V ðxL Þ−DðqL Þ−α ðC ðqL Þ þ ψðeL ÞÞ−ð1−α ÞðpL qL þ F L Þ

Constraint (12) represents the demand function from Eq. (5). Constraints (13) and (14) constitute the participation constraints of firms H and L respectively. Finally, constraints (15) and (16) ensure the incentive compatibility of the contracts. Let us focus on the incentive compatibility constraint for the H firm. If Eq. (14) is binding, constraint (15) becomes:

eL : V x ðeL Þ ¼

ψ0 ðeH Þ  ; or eH ¼ eH θH

   0 ψ ðeL Þ λ θ : 1þ ð1−α Þ 1− L θH θL 1−λ

ð18Þ ð19Þ ð20Þ

ð21Þ

ð11Þ

s.t. V xi ¼ pi ∀i

ð12Þ

pH qH þ F H −C ðqH Þ−ψðeH Þ ≥ 0

ð13Þ

pL qL þ F L −C ðqL Þ−ψðeL Þ ≥ 0

ð14Þ

pH qH þ F H −C ðqH Þ−ψðeH Þ ≥ pL qL þ F L −C ðqL Þ−ψðeHL Þ

ð15Þ

pL qL þ F L −C ðqL Þ−ψðeL Þ ≥ pH qH þ F H −C ðqH Þ−ψðeLH Þ

ð16Þ

Comparing the results of Proposition 3 with the ones of Proposition 1 in perfect information, we observe that in an asymmetric information setting, the regulator imposes first best effort and quantities (and consequently prices) to the H firm (no distortion at the top), so that eH = e∗H, qH = q∗H and pH = p∗H. The L firm's effort is distorted downward by the   λ coefficient 1 þ 1−λ ð1−α Þ 1− θθHL N1 in Eq. (21), implying eL b e∗L. The undereffort of the L firm with respect to the first best increases the demand function in the L setting: Vx(qL; eLθL) b Vx(qL; e∗LθL). This in turn increases the quantity consumed when the environment is L: qL N q∗L. Moreover, from Eqs. (12) and (19), the price in the L environment is higher than in the first best: pL N p∗L. Finally, since rents are costly to

L. Abrardi, C. Cambini / Energy Economics 49 (2015) 122–131

the regulator, the L firm obtains no rent (constraint (14) is binding in the optimum), i.e. FL = C(qL) + ψ(eL) − pLqL b F∗L. Conversely, the H   firm obtains a rent equal to ψðeL Þ−ψ θθHL eL , so that the H firm's fixed

transfer is higher than the one in the first best: F H ¼ C qH þ ψ eH −   pH qH þ ψðeL Þ−ψ θθHL eL NF H . The result of Proposition 3 presents two typical features of the literature on regulation under imperfect information: i) an efficient level of effort and positive rents for type H; ii) undereffort and no rents for type L. The incentive compatibility of the menu of contracts is ensured by the provision to the H firm of informational rents. These rents are increasing with the effort exerted by the L firm. As a consequence, the reduction of rents requires the downward distortion of eL. The optimal trade-off between rent extraction and efficiency can be intuitively derived as follows. Reducing eL by one unit has three effects. First, the marginal benefit on the consumer's comfort in the L environment is reduced by θLVx. Second, the effort cost of the L firm is reduced by ψ′. Third, type H's rent decreases by ψ0 − θθHL ψ0 . The unit social cost of the firm's rents is

(1 − α), as they reduce the consumers' surplus, whose weight in the welfare function is 1, but increase the firm's profits, which weight α. The probability of type H is λ, so that the expected social gain in rent re  duction rents is λð1−α Þ ψ0 − θθHL ψ0 . Similarly, the expected reduction of

the consumers' comfort is (1 − λ)θLVx and the expected reduction of effort cost is (1 − λ)ψ′. The marginal cost in terms of reduction of consumers' utility is equal to the marginal benefit in terms of rent reduction and decrease of effort cost when   θ 0 0 0 ð1−λÞθL V x ¼ ð1−λÞψ þ λð1−α Þ ψ − L ψ ; θH which leads to the optimal condition (21). The introduction of the firm's profits into the regulator's objective function reduces the distortion of the effort of the inefficient firm. In the limit case in which the firm's profits have the same weight than the consumer's surplus (i.e., α = 1), the optimal result with asymmetric information replicates the first best outcome in terms of effort. Indeed, the distortion of the firm's effort is necessary to reduce its rents. However, if the rents increase the welfare, the need to reduce them is less pressing. Conversely, the effort eL is subjected to the highest downward distortion when the firm's profits have no consequence on the social welfare, i.e. α = 0. The tariff structure under imperfect information has important policy implications. Given that FH N FH∗ N FL∗ N FL and that pL N p∗L N p∗H = pH, asymmetric information, compared to the perfect information setting, strengthens the pressure towards a decoupling policy for the H firm and towards standard regulation for the L firm. In perfect information, the effort e∗L of the L firm is already lower than e∗H due to its lower marginal benefit in terms of demand reduction. When asymmetric information issues are introduced, the effort of the L firm must be further distorted downward with the objective of rent reduction. This further polarizes the menu offered towards the extreme policies of a standard per unit regulated tariff and decoupling. This result contributes to the debate of whether decoupling or standard policies are to be adopted in order to induce energy efficiency. A decoupling policy is socially preferable if EE activities are highly effective. In contrast, standard regulation is to be preferred if EE activities have a small impact on demand. 5. Extension: the consumer's effort

direction. However, the customer's responsibility cannot be discounted. Up until now, we examined how the customer can be induced to energy conservation simply through the instrument of the price. This kind of approach overlooks the fact that EE actions can be undertaken not only by firms, but by customers themselves. For example, a customer may decide to purchase energy saving appliances or shift as much as possible of her consumption during off-peak hours. If not only firms, but also customers can exert effort in energy conservation activities, the regulator's problem becomes that of allocating in the most efficient way the right amount of effort on the two parties. Ideally, a EE initiative should be adopted by the one for whom it is less costly. In this framework, the price has an additional role, as it can help the regulator to induce customers to exert energy-saving effort. Indeed, it is a common occurrence for a customer to trade-off the extra-expenditure of investing in energy-efficiency with a higher expenditure on the energy bill. To best analyze this case, we abstract from asymmetric information issues, so as to focus on the optimal allocation of EE activities between customers and firms and the tariff that can achieve that allocation. Nonetheless, intuitions about the effects of asymmetric information in this context will be provided at the end of this section. Suppose that the consumer's comfort x is defined by x = q + θ(ef + e c + βe f e c), where e f and e c are the efforts exerted by the firm and the consumer respectively and q is the quantity consumed. The parameter β indicates the degree of substitutability between the activities undertaken by firms and customers: the activities are substitutes (negative externality) for β b 0, as the effort exerted by the consumer reduces the firm's incentive to do the same because of the negative component βe f ec , or can be complement (positive externality) for β N 0. The effort for the consumer has a cost K(ec), with K(0) = 0, K′ N 0, K″ N 0. Consumers choose the quantity q and the effort ec that maximize their net surplus V(x) − pq − F − K(ec). It follows that the effort exerted by the consumers is such that Vxθ(1 + βef) = K′(ec). This optimality condition offers interesting insights in case the market is unregulated. We have already noted how, in the absence of regulation, the firm exerts no effort (ef = 0) due to its cost and its demand-reducing effects. In this extended framework, the fact that ef = 0 affects the consumer's decision to invest in EE activities. In particular, from the optimality condition Vxθ(1 + βef) = K′(ec) with ef = 0, ec is such that Vx = K′(ec)/θ. This level of effort corresponds to the socially optimal level when only consumers can exert effort. Indeed, condition Vx = K′(ec)/θ is similar the first best condition in Eq. (7). In other words, when the firm does not cooperate and offers no contribution in terms of EE activities, the consumer assumes full responsibility of the socially optimal effort. In the first best situation we are studying, the regulator solves   max V ðxÞ−pq−F−K ðec Þ−DðqÞ þ α π q; e f q;e

s.t.   πðq; eÞ ¼ pq þ F−C ðqÞ−ψ e f ≥ 0

ð22Þ

Vx ¼ p

ð23Þ 0

K ðec Þ : Vx ¼  θ 1 þ βe f

ð24Þ

Eq. (22) represents the firm's participation constraint, Eq. (23) is the demand function and Eq. (24) describes the consumer's choice of effort ec. The first best quantity q∗ and efforts e∗f and e∗c are defined as follows: 

When energy conservation issues are considered, the firm has certainly an important role, as it can for example promote a more energyconscious behaviour by customers or adopt initiatives that go in that

127

0

0

q : V x ðqÞ ¼ C ðqÞ þ  DðqÞ 0   ψ ef  ef : Vx ef ¼ θð1 þ βec Þ

ð25Þ ð26Þ

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L. Abrardi, C. Cambini / Energy Economics 49 (2015) 122–131

K 0 ðec Þ   ec : V x ðec Þ ¼  θ 1 þ βe f       p ¼ V x q ; e f ; ec






F ¼C q

     þ ψ e f −p q :

ð27Þ

ð28Þ

ð29Þ

Notice that the regulator's preferred level of effort e∗c coincides with the consumer's autonomous choice of effort. This result is no coincidence, and it follows directly from primary weight that the consumer's surplus has into the regulator's objective, which fully aligns the objectives between regulator and consumers. A first immediate result from the optimal conditions from Eqs. (25) to (29) is that, if K(⋅) = ψ(⋅), from Eqs. (26) and (27) we obtain that e∗f = e∗c. The reason for this result is straightforward. Given that the weight of the firm's profits is lower than that of the consumer's surplus in the regulator's objective, the first best outcome implies zero profits for the firm, and the welfare function can be rewritten as V(x) − K(ec) − C(q) − ψ(ef) − D(q). The same weight of the consumer's and firm's effort cost in the welfare function explains why, if these cost functions are identical, the first best outcome entails the same effort by both consumers and firm. To fully understand the implications of the first best outcome, we now present some comparative statics, whose results are summarized in the following Proposition for the case characterized by the absence of externalities (β = 0). Proposition 4. Ceteris paribus, if β = 0, both the optimal quantity q∗ and the price p∗ increase and both optimal level of comfort x∗ and the total effort e∗ = e∗f + e∗c decrease when: a. the consumer's effort costs K(⋅) increase; in this case, e∗f increases and e∗c decreases; b. the firm's effort costs ψ(⋅) increase; in this case, e∗c increases and e∗f decreases; c. the effectiveness θ of EE effort decreases; in this case, both e∗c and e∗f decrease. Let us focus on the first point of Proposition 4 and denote the initial functions of the effort cost as ψðÞ and K ðÞ, with ψðÞ ¼ K ðÞ, and the relative effort and price as, respectively, ē∗ and p ¼ V x ðq ; e ; e Þ. If K(⋅) increases (i.e., K ðÞNψðÞ), then it is optimal to distort both efforts, in such a way that e∗c b ē∗ b e∗f . The reason for which e∗c is reduced is obvious, and it is related to the increase of its direct costs. To explain the increase of the firm's effort, we must take into account that, given that the consumer's contribution is now lower in terms of effort, the marginal benefit of an additional unit of the firms' effort is higher. From Eq. (26), the optimal level of the firm's effort is when its marginal cost (ψ′(ef)) is equal to its marginal benefit (θVx(ef; ec, q)); since its marginal benefits have now increased, it is optimal to increase the level e∗f of the firm's effort with respect to ē∗. In other words, the reduction of e∗c due to its higher costs is partially compensated by an increase of e∗f . Moreover, when the cost of the consumer's effort increases, the price is set at a higher level: p Np . This occurs because, due to the higher costs of the EE activities, the consumer tends to invest less in them, and to rely more on consumption to achieve her comfort. This is counterproductive in terms of energy-conservation goals. To induce the consumer to curb the quantity of energy consumed, the regulator increases the price. Notice that in a perfect information framework, the price loses its role of incentive for the firm, as the firm's effort can be perfectly observed and contracted upon. In this case, the increase of the price only reduces the fixed fee, so that the firm's profit remains zero, with no retro-effect on the firm's effort. The increase of the consumer's effort

cost also entails a reduction of the optimal level of comfort x∗ and the increase of the quantity q∗ consumed — from Eq. (25). The opposite outcome in terms of efforts can be achieved when ψ(⋅) increases, as in this case it results e∗f b ē∗ b e∗c and p Np . The intuition is analogous to the one just provided. Given that ψðÞNK ðÞ ¼ K ðÞ, the regulator requires less effort from firms. The lower effort by firms raises the marginal benefit of one unit of consumer's effort, so that, even if the cost of the consumer's effort remains unvaried, it is optimal to increase e∗c above the initial level ē∗. The decrease of the total effort (e∗f + e∗c b 2ē∗) increases the consumer's demand. An increase in price is then necessary to induce consumers to reduce their consumption and partially oppose the increase of demand produced by the lower efforts. Conversely, starting from a situation in which K(⋅) = ψ(⋅), a decrease of K(⋅) entails e∗f b ē∗ b e∗c and a decrease of the price p Np . The effort costs are not the only parameters with an impact on the first best outcome. The effectiveness of the EE activities, embodied by the parameter θ, has an influence as well. The higher is θ, the higher is the marginal benefit produced by the efforts – from Eqs. (26) and (27) – and, consequently, the higher are both e∗f and e∗c, so as to best exploit their higher effectiveness. Due to the increase of the efforts, the consumer is able to reduce the quantity consumed q∗. The reduction of demand reduces also the social cost of its provision, and this is reflected by the decrease of the price, so that the higher is θ, the lower is the price. Moreover, thanks to the higher effectiveness of EE activities, the consumer is able to increase her comfort despite lower consumptions, because of the higher effort exerted by both herself and the firm and their high effectiveness. This result confirms what we obtained in the previous sections, as the policy with highly effective EE activities entails higher comfort and lower consumption in the H scenario. If β ≠ 0, the optimal policy must take into account the externality generated by the EE activities of consumers and firms. We thus have the following result. Proposition 5. Ceteris paribus, if β increases, both the optimal quantity q∗ and the price p∗ decrease and both optimal level of comfort x∗ and the total effort e∗ = e∗f + e∗c increase. The result of Proposition 5 has a straightforward interpretation. If β increases, a positive externality arises and, to exploit it, the efforts are increased. The increase of efforts allows consumers to reduce the quantity consumed — more precisely, it reduces the marginal benefit of one additional unit of consumption, which in turn reduces the quantity consumed in the optimum. The reduction of the optimal quantity implies lower marginal social costs (C′(⋅) + D′(⋅)). The price must then be diminished, so as to reflect the lower social costs and thus induce the consumer to choose the optimal quantity. When efforts are complements, meaning that the effort of say the firm generates a positive effect also on the consumers' effort, then the regulator may prefer to set a lower price and a larger fixed component, hence moving towards a decoupling policy, to promote energy efficiency activities. On the contrary, when efforts are substitutes, the regulator may prefer to increase the regulated price and decrease the fixed component in order to reduce the overall consumption given that efforts in energy conservation activities are reduced. The fact that β is different from zero has interesting implications when a variation of the degree of substitutability is combined with the increase in effort costs, whose effects are described in Proposition 4. In Proposition 4, we showed that, when β = 0, an increase of K(⋅) reduces the consumer's effort and increases the firm's. It may no longer be so (or not at the same extent) when β N 0. Indeed, when β N 0, the reduction of e∗c has a negative impact on the total effort ef + ec + βefec, which is amplified by the shrink of the additional component βefec. The fact that the consumer's and the firm's efforts are complement activities implies that the reduction of e∗c cannot be as pronounced as when β = 0, so as to exploit the effect of the positive component βefec. Moreover, the decrease

L. Abrardi, C. Cambini / Energy Economics 49 (2015) 122–131

of e∗c implies also a lower increase of e∗f , due to the complementarity between them. The higher is β, the smaller are both the decrease of e∗c and the increase of e∗f . Furthermore, if externalities are present, the total effort, after the increase of its cost, is not reduced as much as when externalities are absent. Indeed, the increase of the effort cost would normally induce the regulator to give up pursuing some EE goals, and rely more on consumption to provide the consumer's comfort. However, when EE activities present a positive externality (but the same reasoning is valid when they are highly effective — i.e., θ is high), the regulator prefers to continue stimulating the effort in EE activities and keep consumption low, even when the cost of effort increases. To a smaller decrease of the total effort in turn corresponds a smaller increase of the quantity consumed and consequently a smaller increase of the price, reflecting the smaller increase of the marginal social cost. It is important to remark that the sign of β is relevant for these results. When β b 0, indicating substitute efforts, the reduction of e∗c and increase of e∗f caused by the increase of the consumer's effort costs K(⋅) is amplified with respect to the case in which β = 0. Indeed, due to the substitutability between the two efforts, the decrease of e∗c strengthens the increase of e∗f , and in turn the increase of e∗f strengthens the decrease of e∗f . This happens because the decrease of the effort exerted by the consumer's reduces the negative component βefec, and this increases the firm's incentive to exert its own effort e∗f . When β b 0, an increase of θ does not produce a definite outcome in terms of effort and price. The reason is that a higher effectiveness of the EE effort is a favourable circumstance only if the total effort increases. However, when the negative externality is very strong (β is sufficiently large), the increase of the individual efforts produces a decrease of the total effort due to the large negative component βefec. In this case, to best exploit the higher effectiveness of EE activities, the individual efforts must be reduced. A reduction of the total effort implies in turn higher quantities and higher prices. However, the outcome in this case strongly depends on the extent of the variations of the parameters and on how large is the negative externality. When asymmetric information is introduced into the picture, the outcome becomes of more complex interpretation. If we adopt the extended definition of comfort x = q + θ(ef + ec + βefec) into the asymmetric information model presented in Section 4, the optimality conditions become: 0

0



qH : V x ðqH Þ ¼ C ðqH Þ þ D ðqH Þ; or qH ¼ qH 0

0

qL : V x ðqL Þ ¼ C ðqL Þ þ D ðqL Þ

efH

    ψ0 e f  : V x e f ð1 þ βecH Þ ¼ ; or e f H ¼ e f H θH

      ψ0 e f  λ θ 1þ ð1−α Þ 1− L efL : V x e f ð1 þ βecL Þ ¼ θH θL 1−λ ecH : V x ðec Þ ¼

ecL : V x ðec Þ ¼



K 0 ðec Þ

θH 1 þ βe f H 

K 0 ðec Þ

θL 1 þ βefL



ð30Þ ð31Þ

ð32Þ

ð33Þ

ð34Þ

129

meaning that asymmetric information produces the same effects to the basic model as well as to this extended case. However, the analysis of the optimality conditions from Eqs. (30) to (35) under asymmetric information provides further insights. Let us first consider the case with β = 0. When the firm is privately informed about the effectiveness of its effort, it must be given a rent to induce the revelation of information. To reduce this rent, efL is decreased (from Eq. (33)). A lower effort by the firm reduces the comfort of the consumer. Lower levels of comfort are associated to a higher marginal surplus, thus increasing the consumer's incentive to increase her own effort ecL (from Eq. (35)). It follows that, since the firm's effort is socially costly, the responsibility of effort is shifted more heavily on the consumer. This has an interesting side effect. The higher consumer's effort reduces the marginal benefit of the firm's effort. For this reason, efL is subject to a further downsizing, thus compounding the initial reduction motivated by the desire to reduce the informative rents. In conclusion, the inefficient firm's effort shrinks for two reasons: first, to reduce the informative rent of the efficient firm; second, because of its lower marginal benefit due to the higher consumer's effort. When β ≠ 0, the impact of asymmetric information is further complicated by the interactions of complementarity or substitutability between efforts. Let us first examine the case in which efforts are complements (β N 0). When, as a consequence of asymmetric information, the L firm's effort is reduced, two effects arise. On the one hand, the lower efL reduces the level of comfort achievable (all other things being equal), thus increasing the consumer's marginal benefit of her effort and consequently providing incentives to increase her effort ecL. On the other hand, the marginal effect of consumer's effort on total effort ef + ec + βefec is given by 1 + βefL, which is lower due to the lower efL; due to the complementarity between efforts, the decrease of efL reduces the consumer's incentive to exert effort. The net effect on ecL is ambiguous, depending on the relative strength of the two forces. In turn, depending on the sign of the variation of ecL, the effect on efL may be positive or negative. In particular, if ecL increases, it is optimal to reduce the downward distortion of efL (with respect to the basic model in the absence of consumer's effort), so as to exploit the complementarities between efforts. Similar observations hold for the case in which efforts are substitutes (β b 0). With respect to the perfect information case, here efL must be reduced to decrease the informational rents to the H firm. The marginal effect of efL on total effort is 1 − |β|ecL, which may be positive or negative depending on the magnitude of |β|. In case the marginal effect is negative (the substitutability is very strong), the reduction of efL increases the total effort and thus consumer's comfort, thus decreasing the consumer's incentive to exert effort. On the other hand, the marginal impact of the consumer's effort on total effort is 1 − |β|efL: a lower value of efL induces the consumer to increase her effort in order to exploit its higher impact to the substitutability relationship. Depending on the sign of the variation of ecL, the effect on efL may be positive or negative. Intuitively, if ecL increases, it is optimal to further increase the downward distortion of efL (with respect to the basic model in the absence of consumer's effort), so as to exploit the substitutability between efforts. 6. Conclusions



ð35Þ

This set of conditions is, under some respects, analogous to conditions from Eqs. (25) to (27) of the extended model under perfect information. The only difference with that case is the downward distortion of   λ ð1−α Þ 1− θθHL , while the principle of no distorefL by the coefficient 1−λ tion at the top applies to the efficient firm. These results (downward distortion of the L firm's effort and no distortion at the top) perfectly mirror those obtained in the basic model of asymmetric information,

The tariff used to regulate a utility not only accounts for its revenues and profits, but it is also an important instrument of demand management. The latter aspect is especially relevant when energy conservation goals are introduced in the regulator's objectives. Quite surprisingly, the role of the price as an incentive to energy conservation has so far received scant attention by the literature, which has rather focused on the incentive schemes to be provided to the firms (decoupling policies or gain sharing plans), thus employing a supply-side perspective. The issue of inducing energy conservation through the price is a delicate matter, as an increase of the energy price on one side induces

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consumers to reduce their demand, but on the other side it increases the firm's marginal loss for any unit of energy unsold, thus reducing the utility's incentives to further promote energy conservation. The choice of price thus allocates the incentives to energy conservation between consumers and firms. Energy efficiency activities may have a variable impact on demand. For example, the simple promotion of a more responsible, energy-saving behaviour may be less effective – in terms of demand reduction – than the actual installation of energy-saving devices at the consumers' premises. We find that the higher is the effectiveness of the firm's energy efficiency effort, the more cost-efficient is to have firms bearing most of the responsibility for energy conservation. To this aim, the optimal tariff requires consumers to pay a significant fixed fee, but a relatively low price per unit of energy consumed. This solution, that approximates a decoupling policy, provides strong incentives to the firm to exert energy conservation effort – which is highly effective – , in two ways. First, because of the significant fixed component in the tariff, the firms' revenues are mostly independent from the volume sold. Second, the low price implies low opportunity costs for any quantity unsold. When instead the firm's energy efficiency effort has a low impact on demand reduction, consumers should pay a high per unit price but a low fixed fee. The reason, once again, is quite intuitive. As the firm's effort is costly but has a low potential in terms of demand reduction, the energy conservation goal is best achieved by operating directly on consumers, and specifically by reducing their demand through a high price. This high price reduces the incentives for the firm to promote EE activities, but due to their high cost this is consistent with a welfare maximizing solution. When the firm is privately informed about the effectiveness of the EE activities, the regulator has an additional problem, that is to extract the firm's information, given that a firm in the high effectiveness environment has always an incentive to underperform in effort and explain the high consumptions with a scarcely responsive demand. This hidden information problem can be solved by offering to the firm a menu of contracts, designed in such a way that the firm correctly self-selects on the basis of its information. The tariff designed for the high effectiveness environment coincides with the one offered in the perfect information benchmark — thus replicating the no-distortion at the top result of traditional regulation models. Conversely, the tariff designed for the low effectiveness environment distorts downward the firm's effort, further increasing the unit price. With imperfect information, a trade-off emerges between rent reduction and efficiency. The firm in the high effectiveness setting obtains a rent by pretending a low impact of its effort on demand. This rent is directly proportional to the effort exerted by the firm in low effectiveness environment. To lessen the informational rent of the former, it is necessary to reduce the effort of the latter. Although the assumption of consumers' perfect rationality and information provides a convenient way to model the consumers' behaviour, in reality several customer barriers are present, which prevent customers to make the socially optimal choice about consumption and investments in energy efficiency. Such barriers derive, for example, from a lack of information and of rationality. Thus, new regulatory instruments should be introduced that address the problem of customers' bounded rationality. Finally, a comprehensive model should also incorporate all the externalities of energy efficiency activities and should consider a more general setting where more than one policy tools might be available, for example not only regulated tariffs but also tradable permission. We leave these interesting extensions to future analysis.

Lagrangean multiplier. As it must be πj(qj, ej) = 0, the objective function becomes V(xj) − C(qj) − ψ(ej) − D(qj) which, derived by qj and ej, leads to the conditions in Proposition 1.

▪

Proof of Lemma 1. The proof will proceed by absurd. Suppose, instead, that e∗HθH b e∗LθL. Then, given that by assumption θH N θL, it must be e∗H b e∗L. Moreover, Vx(q) is a decreasing function of eθ for a given level of x x ∂x x ¼ ∂V ¼ V xx  1 b 0. Since e∗H θ H b e∗LθL by hypothesis, ∂∂V b 0 imq: ∂∂V ðeθÞ ∂x ∂ðeθÞ ðeθÞ

plies that Vx(q; e∗H θ H) N Vx(q; e∗LθL) for all q. The optimal quantity q∗j is the solution of condition (6). Since Vx(q; e∗H θ H) N Vx(q; e∗LθL) for all q, 2

then, from Eq. (6), q∗H N q∗L. Given that we assumed d C′(q∗H)

plies follows

+ D′(q∗H) N C′(q∗L) that p∗H N p∗L. Vx(e) is a

D′(q∗L).

ðCDÞ N0, dq2

q∗H N q∗L im-

+ From conditions (3) and (6), it decreasing function of q for a given level

x x ∂x x ¼ ∂V ¼ V xx  1 b 0. Since q∗H N q∗L, ∂V b 0 implies that Vx(e; q∗H) b of e: ∂V ∂q ∂x ∂q ∂q

Vx(e; q∗L) for all e. Moreover, ψ′(e)/θH b ψ′(e)/θL. The optimal effort e∗j is the solution of condition (7). Since Vx(e; q∗H) b Vx(e; q∗L) for all e, and ψ′(e)/θH b ψ′(e)/θL for all e, then Eqs. (3) and (7) imply that p∗H b p∗L, which is a contradiction.

▪

Proof of Proposition 2. From Lemma 1, e∗HθH N e∗LθL. Then, Vx(q; e∗HθH) b Vx(q; e∗LθL) for all q. It follows that, from Eq. (6), q∗H b q∗L. Moreover, from conditions (3) and (6), p∗H b p∗L. Given that q∗H b q∗L, it must be Vx(e; q∗H) N Vx(e; q∗L) for all e. Moreover, as θH N θL, ψ′(e)/θH b ψ′(e)/θL for all e. It follows that e∗H N e∗L. From Eq. (5), x∗H must be such that Vx(x) = p∗H and x∗L must be such that Vx(x) = p∗L. Since p∗H b p∗L and Vxx b 0, then x∗H N x∗L. F∗j = − p∗j q∗j + C(q∗j ) + ψ(e∗j ). Since production entails unvariant fixed costs, and q∗H b q∗L, p∗H b p∗L, then it must be that p∗Hq∗H − C(q∗H) b p∗Lq∗L − C(q∗L). Moreover, since e∗H N e∗L, then ψ(e∗H) N ψ(e∗L). Hence, FH∗ N FL∗.

▪

Proof of Proposition 3. Assuming that only constraints (14) and (15) – i.e., the participation constraint of the L firm and the incentive compatibility of the H firm – are binding in the optimum, and substituting them into the objective function, the regulator's maximizes:    θ λ V ðxH Þ−DðqH Þ−α ðC ðqH Þ þ ψðeH ÞÞ−ð1−α ÞðC ðqH Þ þ ψðeH Þ þ ψðeL Þ−ψ L eL þ θH þð1−λÞ½V ðxL Þ−DðqL Þ−α ðC ðqL Þ þ ψðeL ÞÞ−ð1−α ÞðC ðqL Þ þ ψðeL ÞÞ

which, derived by qH, eH, qL and eL, leads to the conditions in Proposition 3. We now verify that constraints (13) and (16) are not binding in the   optimum. From Eq. (17), pH qH þ F H −C ðqH Þ−ψðeH Þ≥ψðeL Þ−ψ θθHL eL N 0 as

θL θH

b 1 and ψ′ N 0. Moreover, Eq. (16) can be rewritten as

pLqL + FL − C(qL) − ψ(eL) ≥ pHqH + FH − C(qH) − ψ(eH) + ψ(eH) − ψ(eLH). If Eqs. (14) and (15) are binding, then it holds 0 ≥ψðeL Þ−ψ     θL θH θH eL þ ψðeH Þ−ψ θL eH which is always satisfied for eL b eH. It follows that Eq. (16) is not binding in the optimal solution.

▪

Proof of Proposition 4. a. The proof will proceed by absurd. Suppose, instead, that e∗ increases. Then, given that Vx(q, e∗) is a decreasing function of e∗ for a given x x ∂x ¼ ∂V ¼ V xx  θ b 0), Vx(q, e∗) must decrease for level of q (as ∂∂V ðe Þ ∂x ∂ðe Þ

all q. The optimal quantity q∗ is the solution of condition (25). Since Vx(q, e∗) decreases for all q, then, from Eq. (25), q∗ decreases. 2

Given that we assumed d

ðCDÞ dq2

N 0, a decrease of q∗ implies the de-

Appendix A

crease of C′(q∗) + D′(q∗). From Eqs. (25) and (28), it follows that p∗ decreases as well. The optimal effort e∗c is the solution of condi-

Proof of Proposition 1. The Lagrangean function is = V(xj) − pjqj − Fj − D(qj) + α πj(qj, ej) + λ(pjqj + Fj − C(qj) − ψ(ej)), where λ is the

tion (27). Since p∗ decreases and, from Eq. (27), p ¼ K ðθec Þ, then the assumed increase of K′(ec) implies a lower e∗c. Moreover, the optimal

0

L. Abrardi, C. Cambini / Energy Economics 49 (2015) 122–131

effort e∗f is the solution of condition (26). Since p∗ decreases and, 0 ψ ðe Þ from Eq. (26), p ¼ θ f , then e∗f decreases. But, if both e∗c and e∗f decrease, then the assumed increase of e∗ is a contradiction. So, when K(⋅) increases, e∗ decreases. Then, from Eq. (25), q∗ increases and, from Eq. (28), p∗ increases. Given that V′ N 0 and V′′ b 0, x∗ is re0 ψ ðe Þ duced. Since p∗ increases and, from Eq. (26), p ¼ θ f , then e∗f increases. Given that e∗ = e∗c + e∗f , and that e∗ decreases while e∗f increases, then e∗c is reduced. b. When ψ(⋅) increases, e∗ decreases (see proof by absurd in the previous point). Then, from Eq. (25), q∗ increases and, from Eq. (28), p∗ increases. Given that V′ N 0 and V′′ b 0, x∗ is reduced. Since p∗ increases 0

and, from Eq. (26), p ¼ K ðθec Þ , then e∗c increases. Given that e∗ = e∗c + e∗f , and that e∗ decreases while e∗c increases, then e∗f is reduced. c. The proof will proceed by absurd. Suppose, instead, that e∗ increases. Then, given that Vx(q, e∗) is a decreasing function of e∗ for a given x x ∂x ¼ ∂V ¼ V xx  θ b 0), Vx(q, e∗) must decrease for level of q (as ∂∂V ðe Þ ∂x ∂ðe Þ

all q. The optimal quantity q∗ is the solution of condition (25). Since Vx(q, e∗) decreases for all q, then, from Eq. (25), q∗ decreases. 2

Given that we assumed d ∗

∗

ðCDÞ dq2

N 0, a decrease of q∗ implies the de-

crease of C′(q ) + D′(q ). From Eqs. (25) and (28), it follows that p∗ decreases as well. The optimal effort e∗c is the solution of condi0 ψ0 ðe Þ tion (27). Since p∗ decreases and, from Eq. (27), p ¼ K ðθec Þ ¼ θ f , then the assumed decrease of θ implies lower e∗c and e∗f , which contradicts the initial hypothesis of the increase of e∗. So, when θ decreases, e∗ decreases. Then, from Eq. (25), q∗ increases and, from Eq. (28), p∗ increases. Given that V′ N 0 and V″ b 0, x∗ is re0 ψ0 ðe Þ duced. Since p∗ increases and, from Eq. (26), p ¼ θ f ¼ K ðθec Þ, then ∗ ∗ both ef and ec decrease.

▪

Proof of Proposition 5. The proof will proceed by absurd. Suppose, instead, that e∗ decreases. Then, given that Vx(q, e∗) is a decreasing function of e∗ for a given level of q, Vx(q, e∗) must increase for all q. The optimal quantity q ∗ is the solution of condition (25). Since Vx (q, e ∗) increases for all q, then, from Eq. (25), q ∗ increases. Given that we assumed ∗

∗

2

d ðCDÞ N0, dq2

an increase of q∗ implies the in-

crease of C′(q ) + D′(q ). From Eqs. (25) and (28), it follows that p∗ increases as well. The optimal efforts e∗f and e∗c are the solution of conditions (26) and (27). Since p∗ increases and, from Eq. (28), 0 ψ0 ðe Þ ðec Þ ¼ θð1þβef c Þ , then the assumed increase of β implies p ¼ θ K1þβe ð fÞ higher e∗c and e∗f , which contradicts the initial hypothesis of the decrease of e∗. So, when β increases, e∗ increases. Then, from Eq. (25), q∗ decreases and, from Eq. (28), p∗ decreases. Given that V′ N 0 and V″ b 0, x∗ is higher.

▪

131

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