Why “Energy Price Brakes” Encourage Moral Hazard, Raise Energy Prices, and Reinforce Energy Savings

Dertwinkel‐Kalt, Markus; Wey, Christian Article — Published Version Why “Energy Price Brakes” Encourage Moral Hazard, Raise Energy Prices, and Reinforce Energy Savings The RAND Journal of Economics Provided in Cooperation with: John Wiley & Sons Suggested Citation: Dertwinkel‐Kalt, Markus; Wey, Christian (2025) : Why “Energy Price Brakes” Encourage Moral Hazard, Raise Energy Prices, and Reinforce Energy Savings, The RAND Journal of Economics, ISSN 1756-2171, Wiley, Hoboken, NJ, Vol. 56, Iss. 2, pp. 129-144, https://doi.org/10.1111/1756-2171.12489 This Version is available at: https://hdl.handle.net/10419/323872 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ The RAND Journal of Economics ARTICLE Why “Energy Price Brakes” Encourage Moral Hazard, Raise Energy Prices, and Reinforce Energy Savings Markus Dertwinkel-Kalt1Christian Wey2 1Max Planck Institute for Research on Collective Goods, University of Münster 2Düsseldorf Institute for Competition Economics (DICE), Heinrich Heine University Düsseldorf Correspondence: Markus Dertwinkel-Kalt ([email protected].de) Accepted: 17 January 2025 Funding: This work was financially support by the German Research Foundation (DFG project 490725816, Markus Dertwinkel-Kalt; 235577387/GRK 1974, Christian Wey). Keywords: energy crisis | energy price brake | energy price policies | energy saving ABSTRACT To help households and firms with exploding energy costs in the aftermath of the Ukraine war, a new policy called the “energy price brake” was implemented. A unique feature of this relief measure is that it provides a transfer that increases in the consumer’s contractual per-unit price of energy. In a formal model, we show that this policy creates incentives for moral hazard of energy providers to raise per-unit prices. Whereas this moral hazard problem increases the policy’s fiscal costs, it also reinforces energy savings. Whether the policy’s main beneficiaries are consumers or firms depends on the market structure. JEL Classification: D04, L12, Q48, K33 1 Introduction In the aftermath of the Russian invasion of Ukraine in February 2022, European governments have implemented various energy price relief programs for households and firms to address the skyrocketing energy costs. These programs, including lump-sum transfers, energy price subsidies, energy tax cuts, and price caps, aim to alleviate the financial burden for consumers caused by the crisis. Avoiding a shortage of energy—especially, natural gas—has become a top priority for policymakers, particularly in Germany, where natural gas is the major energy source for both large firms and households.1Whereas conventional measures like price caps or energy subsidies provide immediate relief to consumers, such approaches risk causing a breakdown in the energy market by creating excess demand. To address these challenges, a novel policy instrument called the “energy price brake” was developed and implemented in Germany starting from January 1, 2023, with a substantial budget of up to 200 billion euros being allocated to it. The energy price brake applies to both natural gas and electricity in Germany, and could also become a policy instrument in the EU in future energy crises (see EU 2023). Thus, an in-depth analysis of the effects of the energy price brake is warranted. The energy price brake serves two primary objectives: (i) incentivizing energy savings among consumers and (ii) providing financial relief and protection against excessive energy prices. It operates through a transfer scheme defined by the following equation: “transfer =(contractual per-unit price −guaranteed per-unit price) ×quota.” (1) This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). The RAND Journal of Economics published by Wiley Periodicals LLC on behalf of The RAND Corporation. The RAND Journal of Economics, 2025; 56:129–144 https://doi.org/10.1111/1756-2171.12489 129 Here, the “contractual per-unit price” refers to the consumer’s current contractual price per unit of energy (i.e., per kWh) that the energy provider determines, the “guaranteed per-unit price” refers to an energy price per kWh that the government sets upon implementing the energy price brake, and the “quota” refers to a fixed energy quantity that the consumer cannot influence after the announcement of the energy price brake (it could, for instance, refer to a percentage of the respective consumer’s previous consumption level or a percentage of the median consumer’s previous consumption level).2The transfer scheme behind the energy price brake is distinguishably different from other transfer schemes as it involves a lump-sum transfer that varies in the consumer’s current contractual per-unit price. In this paper, we provide a formal analysis of the effects of the energy price brake on energy suppliers and consumers, which could be households or energy-consuming firms. For that, we build on a model of supplier–consumer contracting. First, suppose a monopolistic energy supplier (“he”) offers a twopart tariff contract contingent on the presence of the energy price brake, and a consumer (“she”) decides whether to accept the contract and determines her energy consumption level.3 We demonstrate that the energy price brake creates a moral hazard problem on the supplier’s side, driven by the fact that the joint surplus of the supplier and the consumer increases with the contractual per-unit price. Because the supplier can extract the whole surplus via the fixed fee (keeping the consumer indifferent between accepting and rejecting the contract offer), he has an incentive to raise the per-unit price and therefore also the transfer opportunistically. By increasing the per-unit price, the energy price brake not only sustains but also strengthens the incentives for energy conservation. This, however, goes along with two drawbacks: First, given a monopoly supplier, the energy price brake fails to financially relieve consumers as the supplier pockets the entire additional surplus. Second, as the governmental transfer depends on the contractual prices, the fiscal costs of the energy price brake will be much higher than expected if policymakers do not take the moral hazard problem into account. We next show that competition does not resolve the moral hazard problem. The intuition is that a consumer will choose the contract that gives her the highest utility. The consumer benefits from a higher per-unit price as this also raises her transfer. Thus, unlike under a monopoly, consumers are the beneficiaries of the transfer scheme. Also under competition, the transfer scheme is exploited, but here it helps to achieve both policy objectives (i.e., incentivizing energy savings and providing consumer relief).4 After this general analysis of the implications of an energy price brake, we next look into the effects of the regulatory constraints instituted in Germany’s recent legislation of the energy price brakes (EWPBG 2022;StromPBG2022). Therefore, we add these regulatory constraints—which aim to ban the misuse of the transfer scheme—successively to our model. The first constraint regards the energy contract’s per-unit price, the second the contract’s fixed payment, and the third the overall transfer of the energy price brake to the consumer. Interestingly, we find that these regulatory constraints do not fundamentally alter our core insights. First, the contractual per-unit price that a supplier can charge is constrained (but could well be above marginal costs). The legislation on the energy price brakes states that energy providers must not increase the price beyond “an objectively justified” amount (see §27 in EWPBG 2022 and §39 in StromPBG 2022); otherwise, the Federal Cartel Office could intervene. As it is questionable whether the Federal Cartel Office has sufficient capacity to monitor price increases of all energy providers in the market,5firms’ discretion to raise prices beyond what is objectively justified is arguably substantial, though not unlimited. We show that with this constraint in place the main message of our preceding analysis stays valid: the per-unit price is raised above the equilibrium level that would prevail in the absence of the transfer scheme. Second, energy price brake regulations require that the fixed payment must reflect costs, and that bonus payments to consumers for signing a contract are virtually eliminated (see §4 (1) of EWPBG 2022). Given those restrictions, the supplier can effectively only set the contractual per-unit price. Then, the moral hazard problem will again arise under a monopoly and under competition, provided that consumer utility increases in the contractual per-unit price. Intuitively, a consumer benefits from a higher per-unit price when it increases the transfer by more than the energy bill. With linear contracts, there is no fixed fee via which surplus can be shifted between the consumer and the supplier, and thus a consumer does not unequivocally prefer high contractual prices. Whether a consumer wants to sign a high-price contract now depends on her demand curve. If the consumer’s optimal consumption level lies above the quota, the consumer prefers a low-price contract. If it lies below the quota, however, she is willing to sign a high-price contract and the moral hazard problem arises. Third, the transfer of the energy price brake itself is capped in such a way that a consumer cannot pay less than zero for her annual energy consumption. Without this constraint, the energy price brake would allow some extreme savers to lower their annual energy bill not only to zero but also below zero. This transfer cap could increase consumption up to a level that the bill becomes zero. Thus, it could increase an “extreme saver’s” energy consumption, which is, as we show, the more likely the higher the contractual per-unit price. Nevertheless, the mass of such extreme savers is arguably negligible. In addition, we consider also a scenario that is conceivable, but that is not part of the German legislation on the energy price brakes, namely, that the government implements a cost-based price regulation that strictly constrains the contractual prices suppliers can charge. Such a regulation of prices also does not solve the moral hazard problem. A consumer could have the incentive to sign a high-price contract as this ensures a higher transfer. Consumers with lower equilibrium energy consumption are more inclined to favor a high-price contract. With regulated prices, energy suppliers could respond to the demand for high-price contracts with the choice of higher-cost wholesalers. Finally, we discuss other possible solutions to the moral hazard problem arising from the energy price brake. Here we also offer a remedy which entails imposing a limit on the extent to which the transfer can increase in response to a higher contractual 130 The RAND Journal of Economics,2025 per-unit price. Once this maximum level is reached, any further increase in the contractual price does not result in a higher transfer. This regulation allows the policymaker to constrain the milking incentives without spoiling saving incentives or suppliers’ profitability. 1.1 Related Literature We contribute to the literature dealing with the energy crisis and the resulting energy policies (Bachmann et al. 2022; Kesternich, Von Graevenitz, and Wambach 2022, and Kruse-Andersen 2023) and, more generally, to the literature that evaluates energy savings policies (e.g., Reiss and White 2008,Ito2015,orFraser2022). To the best of our knowledge, we are the first to investigate the energy price brake theoretically. We are also unaware of any work investigating a transfer scheme similar to the proposed price brake. Price caps and lump-sum transfers analyzed in the literature do not share the novel and distinguishing feature of the price brake that the joint surplus of firms and consumers increases in price (given the price is not below marginal cost). Whereas an alternative approach would be to start with the principal’s optimization problem and derive optimal transfer schemes (see Laffont and Tirole 1993; Viscusi, Harrington, and Sappington 2018), our goal is to assess the effectiveness of the existing energy price brake and compare it to alternative instruments. We contribute to the current policy debate on policies in the energy crisis (see Amaglobeli et al. 2023;Fabra2023; Haan and Schinkel 2023; Sirin et al. 2023). Noteworthy in our context is the (informal) policy brief on the energy price brake (Atayev and Hillenbrand 2022) that points to the fact that this measure reduces consumers’ incentives to search for better deals, which mitigates price competition and might raise prices. Our paper is organized as follows. In Section 2, we provide a graphical illustration of the incentives arising from the energy price brake (in comparison to alternative policies). In Section 3, we first present the basic setup and the benchmark analysis, where no relief program is in place (Section 3.1). Here, we derive the market outcomes under a monopoly and under competition when suppliers offer two-part tariff contracts. In Section 3.2, we introduce the energy price brake and show how it could be exploited with no constraints on the energy prices being in place, which highlights the incentives for moral hazard. In Section 3.3, we analyze the (arguably more realistic) case where the contractual per-unit price is restricted by legal constraints. In Section 4, we provide extensions on linear tariffs (Section 4.1), capped transfers (Section 4.2), regulated prices (Section 4.3), and potential solutions to the moral hazard problem (Section 4.4), before we conclude in Section 5. All proofs are relegated to the Appendix. 2 A Graphical Illustration of the Price Brake vs. Other Energy Policies The energy price brake and its relation to other financial relief programs can be illustrated at the hand of the household’s budget line. Suppose the household can spend her income 𝑚 FIGURE 1 The thin line gives the household’s budget line without the energy price brake, and the thick line gives the budget line with the energy price brake. on energy consumption 𝑥(measured in kWh) and on other goods 𝐶(measured in euros).6Let 𝑝be the contractual energy price measured in euros per kWh, and suppose it exceeds the guaranteed price of the energy price brake. In this section, we abstract from the fixed payment, as here we focus on the opportunity costs of energy consumption, which are independent of the fixed payment included in a two-part tariff. Under the energy price brake, the household faces the budget line 𝑝𝑥 +𝐶=𝑚+𝑇(𝑝), (2) where 𝑇(𝑝) >0is the transfer specified in Equation (1). Figure 1depicts the budget line of a household with and without an energy price brake in place. The horizontal axis represents the energy consumption level 𝑥, and the vertical axis the consumption expenditures on other goods 𝐶. In the absence of the energy price brake, the budget line is given by the thin line connecting the points (0, 𝑚) and (𝑚∕𝑝, 0) with a slope of 𝑑𝐶∕𝑑𝑥 =−𝑝, which reflects the opportunity cost of energy consumption in terms of foregone expenditures on other consumption goods 𝐶. With the introduction of the energy price brake, the household receives a transfer 𝑇(𝑝), which is independent of its energy consumption in the current period. For a given price 𝑝, the transfer is, therefore, just like a fixed transfer payment to the household and does not affect the opportunity costs of energy consumption. Thus, the slope of the budget line is again given by −𝑝, exactly as in the absence of such a transfer scheme (see the thick line in Figure 1). Moreover, the other elements of the energy price brake—namely, the guaranteed price and the quota—only affect the amount of the transfer and therefore do also not affect the opportunity costs of energy consumption. Figure 2elucidates the implications of the fact that the transfer of an energy price brake depends on the contractual energy price 𝑝. It shows how the household’s budget line is affected when the contractual per-unit price increases from 𝑝(solid line) to 𝑝′ (dashed line). Both budget lines intersect at the quota because then the household effectively pays the guaranteed price (as 131 FIGURE 2 Budget lines with energy price brake transfers for 𝑝 (solid line) and 𝑝′>𝑝(dashed line). specified by the energy price brake) for the consumed quota. For all consumption levels below the quota, the budget line for 𝑝′ lies above the one for 𝑝. This reveals the incentive for households with an equilibrium consumption level below the quota to choose energy contracts with high contractual prices.7If consuming less than the quota, the household effectively pays the guaranteed price for energy consumption but also benefits from the higher transfer for a higher contractual price; this relaxes her constraint on consuming other goods. In the following, we compare the energy price brake to various other policies and measures implemented in the energy crisis. One obvious measure is a general cut on energy taxes, which was observed in several countries during the 2022/23 energy crisis (Sgaravatti et al. 2023). Such a tax cut reduces the relative costs of energy and turns the thin budget line in Figure 1outward around the vertical intercept. Furthermore, it is instructive to compare the energy price brake with a price-cap regulation such as the Dutch energy price ceiling system (for details see Haan and Schinkel 2023), where an energy price cap (of, for instance, 40 euro cents per kWh for electricity in 2023 in the Netherlands) applies to a certain quota.8Only for energy consumption that exceeds this quota, the contractual perunit price applies.9 Figure 3shows how such a price-cap regulation affects the household’s budget line. The thin solid line represents the budget line without any intervention. The (kinked) dashed line depicts the budget line under a price-cap regulation with a quota. For consumption levels below the quota, the opportunity costs of energy consumption are given by the capped price 𝑝CAP, which is smaller than the contractual price 𝑝. Only for consumption levels above the quota (as specified in the price-cap regulation), opportunity costs are given by 𝑝. The thick solid line, on the contrary, represents the budget line under the energy price brake, where the opportunity costs of energy consumption are always given by 𝑝. Thus, ceteris paribus, the energy price brake should induce higher energy saving incentives than under a price-cap regulation. FIGURE 3 Budget line without any intervention (thin solid line), with a price cap (dashed line), and with a price brake (thick solid line). Relatedly, Austria has implemented an energy price subsidy, where the government subsidizes the electricity price by paying a certain percentage of the energy price up to some consumption quota, but maximally 30 cent per kWh. 10 For a given contractual price, the Austrian relief program gives rise to a budget constraint that resembles the one under the Dutch energy price ceiling system, as prices per unit are dampened up to some quota. Energy vouchers reduce saving incentives by even more than the preceding policies. Countries like Croatia, France, and Portugal have implemented energy vouchers for some demanding groups (Sgaravatti et al. 2023), so that up to some certain quantity energy is free. This policy can be represented by the dashed budget line in Figure 3when this is flat up to the kink. Altogether, the discussed alternatives to the price brake (namely, price caps, energy price subsidies, energy vouchers, and tax cuts) reduce the opportunity costs of energy consumption (at least up to some quota) and could therefore aggravate the energy crisis in the form of a possible market breakdown. The energy price brake (like a fixed transfer scheme) does not suffer from this problem, so that this scheme is particularly attractive when the risk of a market breakdown could pose a real problem. In fact, our analysis below shows that the energy price brake even tends to raise energy prices, so that energy saving incentives are reinforced by this policy. 3Model and Analysis 3.1 Benchmark (Without Transfer Scheme) Suppose a monopolist supplier offers a two-part tariff contract with a contractual per-unit price 𝑝≥0andafixedpayment𝐹. The consumer’s overall utility is 𝐶𝑆 ={𝑈(𝑥) −𝑝𝑥 −𝐹if the contract is accepted 𝑅if the contract is rejected, (3) where 𝑈(𝑥) is the utility of consuming energy quantity 𝑥≥0, and 𝑅≥0represents the consumer’s reservation utility (i.e, the maximal utility the consumer can obtain when switching to the best alternative). 11 132 The RAND Journal of Economics,2025 Let 𝑈(𝑥) be at least twice continuously differentiable over ℝ≥0. We impose further standard assumptions; namely, 𝑈(0) = 0,𝑈′∶=𝜕𝑈(𝑥)∕𝜕𝑥 >0,and𝑈′′ ∶=𝜕2𝑈(𝑥)∕𝜕𝑥2<0. Thus, the utility from zero consumption is set to zero, utility is strictly increasing in the amount of energy consumed, and the marginal utility decreases with higher energy consumption levels. Note that we do not impose restrictions on any other higher-order derivative of 𝑈(𝑥). In particular, the third derivative can be positive or negative and may also alternate its sign along 𝑥≥0. We can easily relate our setup to the graphical exposition in Section 2(see Figure 1) if we assume that the overall consumer utility depends not only on the utility of energy consumption 𝑈(𝑥) but also linearly on expenditures on other goods 𝐶(i.e., the overall consumer utility is quasi-linear and equal to 𝑈(𝑥) +𝐶). For a two-part tariff, the budget constraint is given by 𝑝𝑥 +𝐹+ 𝐶≤𝑚and thus differs from (2), where we assumed a linear energy price. As the consumer will exhaust her budget, we get 𝐶𝑆 =𝑈(𝑥) −𝑝𝑥 −𝐹+𝑚, which differs from (3) only in the constant 𝑚that does not affect any of our results.12 For the sake of brevity, we work with (3), which omits 𝐶and the associated budget constraint. The supplier’s profit is given by 𝜋∶=(𝑝 −𝑐)𝑥 +𝐹, (4) where 𝑐≥0gives the marginal cost of energy supply. The joint surplus of the supplier and the consumer is then given by 𝐶𝑆 +𝜋=𝑈(𝑥) −𝑐𝑥. (5) Let 𝑘∶=lim𝑥→0+𝑈′be the choke price, that is, the lowest price at which demand is zero. To obtain a non-trivial solution in our following analysis, we impose the following assumption on the choke price and the joint surplus. Assumption 1. The choke price 𝑘satisfies 𝑐<𝑘<∞.In addition, there exists 𝑥>0so that there is scope for Pareto-improving trade, that is,𝑈(𝑥) −𝑐𝑥 >𝑅. The contracting game proceeds in two stages. In the first stage, the supplier (“he”) offers a two-part tariff contract; in the second stage, the consumer (“she”) accepts or rejects the offered contract. If she accepts, she determines her energy consumption level 𝑥. If she rejects, she realizes her reservation value 𝑅. We solve this game for subgame-perfect Nash equilibria. If the consumer accepts the contract, she solves max𝑥≥0𝑈(𝑥) − 𝑝𝑥 −𝐹. Her energy demand 𝑥(𝑝)then follows from the first-order condition 𝑈′−𝑝≤0, (6) which holds as an equality if the solution is strictly positive (𝑥(𝑝) >0), in which case 𝑑𝑥(𝑝)∕𝑑𝑝 =1∕𝑈′′ <0holds (i.e., demand is downward sloping). If 𝑈′<𝑝for all 𝑥>0,then𝑥(𝑝) = 0. The next lemma summarizes our results on the consumer’s demand function. Lemma 1 (Energy demand). Suppose the consumer has accepted a two-part tariff contract with a contractual per-unit price 𝑝≥0. Then, her demand 𝑥(𝑝) follows from (6) and depends on 𝑝 as follows: i) If 𝑝∈[0,𝑘), then 𝑥(𝑝) >0and 𝑑𝑥(𝑝)∕𝑑𝑝 =1∕𝑈′′ <0,as well as lim𝑝→𝑘−𝑥(𝑝) =0. ii) If 𝑝≥𝑘, then 𝑥(𝑝) =0. The consumer accepts a contract offer if her participation constraint 𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) −𝐹≥𝑅(7) is satisfied, with 𝑥(𝑝)being characterized in Lemma 1. If 𝑥(𝑝) =0, then an acceptable contract must satisfy 𝐹≤0with |𝐹|≥𝑅;that is, a fixed payment is made from the supplier to the consumer. If (7) is violated, the consumer rejects the contract offer and realizes 𝑅. In the first stage of the game, the supplier anticipates the consumer’s demand function 𝑥(𝑝) as well as her participation constraint (7), and sets a two-part tariff contract that solves max 𝐹,𝑝 (𝑝 −𝑐)𝑥(𝑝) +𝐹s.t. (7). In equilibrium, the consumer’s participation constraint must bind. Substituting this into the supplier’s profit function, the supplier solves max 𝑝≥0𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) −𝑅, (8) which gives the first-order condition 𝑈′−𝑐=0. By Assumption 1, there exists a unique profit-maximizing two-part tariff contract (𝐹∗,𝑝∗)with 𝑝∗=𝑐,sothat𝑥∗∶=𝑥(𝑐) >0. This solution maximizes the joint surplus (5), because any price 𝑝larger or smaller than 𝑐decreases the joint surplus (for a proof see Lemma 2 in the Appendix). With the fixed payment 𝐹∗, the monopoly supplier extracts the entire joint surplus net of the consumer’s reservation utility, so that 𝐹∗=𝑈(𝑥∗)−𝑝∗𝑥∗−𝑅>0holds. We proceed with the analysis of competition, where at least two suppliers with marginal costs 𝑐compete for the consumer. The contracting game under competition is as follows. In the first stage, suppliers simultaneously offer two-part tariff contracts to the consumer; in the second stage, the consumer accepts one of the contracts or rejects all offers. If the consumer accepts one of the contracts, she determines her energy consumption level 𝑥.If the consumer rejects all contracts, she realizes 𝑅. Again, we solve this game for subgame-perfect Nash equilibria. If the consumer accepts one of the contracts, her demand is given by Lemma 1. When facing more than one acceptable contract, the consumer selects the contract with the highest overall utility; in case of indifference, the consumer selects each of the contracts with a strictly positive probability. Under competition, all firms make zero profit, so that 𝐹∗∗ =0and 𝑝∗∗ =𝑐hold. In this case, the consumer pockets the entire joint 133 surplus, which is maximal as in the monopoly case. Proposition 1 summarizes the benchmark results. Proposition 1 (Benchmark result). Assume that a supplier offers a two-part tariff contract. Then, the equilibrium both under monopoly and competition implements the joint surplus maximizing solution: i) The contractual energy price per unit is equal to marginal costs: 𝑝∗=𝑝∗∗ =𝑐. ii) The consumer’s energy consumption is 𝑥∗=𝑥(𝑐) >0. Moreover, under a monopoly the fixed payment is 𝐹∗=𝑈(𝑥∗)− 𝑐𝑥∗−𝑅>0and the consumer realizes overall utility of 𝐶𝑆 =𝑅, whereas under competition the fixed payment is 𝐹∗∗ =0and the consumer realizes 𝐶𝑆 =𝑈(𝑥∗)−𝑐𝑥∗>𝑅. We can now analyze the most basic energy relief measure, namely, an unconditional fixed transfer to consumers, 𝑇>0. This transfer is not part of the joint surplus of the supplier and the consumer net of her reservation utility, so that 𝑇does not affect the participation constraint (7), but only increases the consumer’s overall utility by 𝑇. This holds obviously both under monopoly and under competition. Corollary 1 (Unconditional fixed transfer). An unconditional fixed transfer 𝑇>0from the government to the consumer only increases the consumer’s overall utility by 𝑇, and does not affect the market outcome under monopoly or competition. 3.2 Energy Price Brake: Unconstrained Contractual Per-Unit Price Suppose that prior to the contracting game, the government implements an energy price brake with a transfer 𝑇(𝑝) defined by 𝑇(𝑝) ∶=max{(𝑝 −𝑠)𝛼𝑥,0}, (9) where 𝑝is the contractual per-unit price that applies when the transfer scheme is in place, 𝑠>0is the guaranteed per-unit price, 𝑥>0gives a reference energy consumption level, and 𝛼∈ (0, 1) gives some share of the reference consumption level; we call 𝛼𝑥the “quota.” Note that 𝑝is set by the supplier, whereas the government sets 𝑠and 𝛼.13 The transfer 𝑇(𝑝) is therefore a lump-sum payment that increases linearly in the price set by the supplier, 𝜕𝑇(𝑝)∕𝜕𝑝 =𝛼𝑥>0,aslongas𝑝>𝑠. By this, the consumer’s opportunity cost of any unit of gas consumption is left unchanged and given by the current contractual energy price 𝑝. We assume that 𝑠∈(0,𝑐), so that the transfer is strictly positive for all 𝑝≥𝑐. Thus, given the benchmark equilibrium outcome (see Proposition 1), the transfer scheme, ceteris paribus, offers financial relief for consumers in the form of a transfer payment 𝑇(𝑝∗), which is independent of the market structure (monopoly or competition). With a transfer scheme 𝑇(𝑝) in place, overall consumer utility is given by 𝐶𝑆 ={𝑈(𝑥) −𝑝𝑥 −𝐹+𝑇(𝑝), if the contract is accepted 𝑅, if the contract is rejected. (10) It follows that the introduction of 𝑇(𝑝) does not affect energy demand 𝑥(𝑝) (as given by Lemma 1) because the transfer does not depend on the energy consumption level 𝑥. Critically, the transfer scheme affects the consumer’s utility from accepting a certain contract, because the transfer depends directly on the contractual per-unit price. It follows that the consumer cannot realize the transfer without accepting the respective contract. Consequently, the introduction of the transfer scheme affects the consumer’s participation constraint, which is now given by 𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) −𝐹+𝑇(𝑝) ≥𝑅, (11) with demand 𝑥(𝑝) following from Lemma 1. Next, we analyze the first stage of the contracting game, both for the monopoly case and the competition case. 3.2.1 Monopoly Anticipating the consumer’s decisions in the second stage of the game, the supplier solves max 𝐹,𝑝 𝜋=𝐹+(𝑝 −𝑐)𝑥(𝑝) s.t. (11). In the optimal solution, the participation constraint (11)must bind. This can be achieved by setting 𝐹=𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) + 𝑇(𝑝) −𝑅, which gives the reduced problem max 𝑝≥0ˆ 𝜋(𝑝) ∶=𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝) −𝑅. (12) The supplier’s maximization problem (12) depends on the sum of the joint surplus, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝), and the transfer, 𝑇(𝑝), which gives the “new” joint surplus of the supplier and the consumer under the energy price brake. Taking the derivative of ˆ 𝜋(𝑝) with respect to 𝑝gives 𝜕ˆ 𝜋(𝑝) 𝜕𝑝 =𝑈′−𝑐 𝑈′′ +𝜕𝑇(𝑝) 𝜕𝑝 for 𝑥(𝑝) >0and 𝜕ˆ 𝜋(𝑝) 𝜕𝑝 =𝜕𝑇(𝑝) 𝜕𝑝 for 𝑥(𝑝) =0. (13) Without a transfer 𝑇(𝑝), the optimal price would be the joint surplus maximizing price 𝑝∗=𝑐with 𝑥∗>0(see Proposition 1). Introducing the transfer scheme creates an incentive to raise the contractual per-unit price above 𝑐, because now 𝜕ˆ 𝜋(𝑝)∕𝜕𝑝 = 𝜕𝑇(𝑝)∕𝜕𝑝 >0holds at 𝑝=𝑐. Thus, the supplier raises the contractual per-unit price above the joint surplus maximizing price 𝑝∗=𝑐(see Proposition 1), so that energy consumption is reduced (according to Lemma 1) below the socially optimal level 𝑥∗. 134 The RAND Journal of Economics,2025 Equation (13) uncovers the fundamental drawback of an energy price brake that is not protected by supplementary regulations against misuse. Inspecting the derivatives in (13) reveals the incentive to arbitrarily inflate the contractual per-unit price. For prices 𝑝∈(𝑐,𝑘), the supplier’s profit function could, in general, take many forms depending on the higher-order derivatives of 𝑈(𝑥) (see Lemma 3 in the Appendix). However, in this region, it is clearly bounded from above, because the joint surplus 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) is strictly decreasing in 𝑝and becomes zero for 𝑝→𝑘 −(see Lemma 2 in the Appendix), whereas 𝑇(𝑝) is linearly increasing in 𝑝. As the transfer payment of the energy price brake increases even for contractual prices above the choke price, the supplier’s profit ˆ 𝜋(𝑝) is unbounded in 𝑝. Thus, by raising 𝑝above 𝑘, the supplier can realize a higher profit than for any price below 𝑘. The supplier can therefore realize an arbitrarily large profit by extracting the transfer from the energy price brake with the fixed payment, while allowing the consumer an overall utility of at least 𝑅. Proposition 2 (Unconstrained contractual energy price under monopoly). Suppose a transfer scheme 𝑇(𝑝) given by (9), and suppose that a monopoly supplier offers a two-part tariff contract to the consumer such that the consumer’s participation constraint (11) holds. Then, the supplier can increase his profit by any amount by raising the contractual price per unit by sufficiently much. Thus, we have 𝑝→∞,𝑇(𝑝) → ∞,and𝜋→∞, while 𝑥=0. 3.2.2 Competition Under competition, the consumer selects the contract which offers the highest overall utility. Because the energy price brake, as defined above, allows for milking the transfer scheme by any amount, it follows that 𝑝→∞and 𝑇(𝑝) → ∞,sothat𝑥=0 must hold. The main difference to the monopoly case is that competition forces firms to offer the most attractive contract to the consumer as, otherwise, she would not buy. Competition for the consumer’s contract acceptance, therefore, inevitably induces firms to inflate the contractual energy price as this makes the contract offer most attractive. We summarize those results as follows. Proposition 3 (Unconstrained contractual energy price under competition). Suppose a transfer scheme 𝑇(𝑝) given by (9), and suppose that at least two suppliers offer two-part tariff contracts to the consumer, such that firms make nonnegative profits. Then, 𝑝→∞,𝑇(𝑝) → ∞,and𝐶𝑆 → ∞, while 𝑥=0. Proposition 3 shows that competition leads essentially to the same outcome as a monopoly. Yet, in the monopoly case, it is the supplier who benefits from raising the contractual per-unit price above marginal costs because this increases his profit directly (given some consumer utility such that the consumer’s participation constraint is satisfied); under competition, it is the consumer’s decision rule to select the most attractive contract which induces firms to raise the contractual energy price (given some non-negative profit level if they sell). 3.3 Energy Price Brake: Constrained Contractual Per-Unit Price Our results on unconstrained milking of the energy price brake illustrate the incentives arising from this policy instrument, but unconstrained milking represents an obvious misuse of the energy relief scheme.14 Hence, assuming that the contractual price is constrained by some price 𝑝, which ensures a strictly positive energy consumption level, is reasonable. Thus, in the following, we impose a maximum contractual energy price per unit, 𝑝,with𝑝∈(𝑐,𝑘),sothat𝑥(𝑝) >0.15 Note that this price constraint would never be binding in the benchmark case (see Section 3.1), where 𝑝∗=𝑝∗∗ =𝑐holds; that is, the purpose of 𝑝 is to constrain potential misuse of the transfer scheme but not to lower regular energy prices. 3.3.1 Monopoly The energy price constraint 𝑝≤𝑝effectively constrains the monopolist’s ability to take advantage of the transfer scheme. Given 𝑝≤𝑝, the following proposition specifies the equilibrium contract and its properties. Proposition 4 (Constrained contractual per-unit price under monopoly). Suppose a transfer scheme 𝑇(𝑝) given by (9), and suppose the additional constraint 𝑝≤𝑝∈(𝑐,𝑘)holds. Then, the monopoly supplier’s equilibrium (two-part tariff) contract offer fulfills either (i) or (ii): i) Interior solution:𝑝fulfills (𝑈′−𝑐)∕𝑈′′ +𝜕𝑇(𝑝)∕𝜕𝑝 =0,so that 𝑝∗<𝑝≤𝑝. ii) Corner solution:𝑝fulfills 𝑝∗<𝑝=𝑝. Moreover, the supplier’s profit is 𝜋=𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝) −𝑅 and the consumer gets 𝑅, while 𝑇(𝑝) >𝑇(𝑝∗)and 0<𝑥(𝑝) < 𝑥(𝑝∗)always hold. Proposition 4 shows that the introduction of the transfer scheme raises the per-unit price also when it is effectively constrained. It raises both the equilibrium price and the transfer beyond the levels that would prevail without the energy price brake. The optimal price either fulfills the condition of a local maximum, or it is obtained at 𝑝. In any case, energy saving incentives are always reinforced by the introduction of the energy price brake (relative to the efficient energy consumption level 𝑥∗) but consumers are not relieved; they receive their reservation utility 𝑅and are indifferent to the situation without the transfer scheme. 3.3.2 Competition Competition does not affect the contractual per-unit price and the energy consumption level as derived for the monopoly case. Note first that a consumer who faces more than one acceptable contract offer chooses the contract that yields the highest overall utility (10). Under competition, firms offer two-part tariff contracts that, again, maximize the joint surplus, but they make zero profits. Hence, the fixed payment must be negative with 𝐹=−(𝑝 − 𝑐)𝑥(𝑝).Thus,ontopof𝑈(𝑥(𝑝)), the consumer also fully pockets 135 the transfer 𝑇(𝑝) as well as the supplier’s profit margin (𝑝 − 𝑐)𝑥(𝑝), and her expenses are 𝑝𝑥(𝑝). Proposition 5 (Constrained contractual per-unit price under competition). Suppose a transfer scheme 𝑇(𝑝) given by (9), and suppose the additional constraint 𝑝≤𝑝∈(𝑐,𝑘)holds. Then, under competition, 𝑝and 𝑥(𝑝) are the same as under a monopoly (i.e., they are as specified in Proposition 4), suppliers realize 𝜋=0and consumers get 𝐶𝑆 =𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝) > 𝑅. Proposition 5 again shows that competition leads to the same market outcome as a monopoly. In both cases, the contractual per-unit price is increased above the socially efficient level (i.e., 𝑐). As in the monopoly case, the suppliers maximize the joint surplus (5) plus the transfer of the energy price brake 𝑇(𝑝). However, competition between suppliers drives profits to zero, which implies a negative fixed payment. Propositions 4 and 5 show that introducing an energy price brake reduces energy consumption compared to the benchmark situation with no transfer scheme (or when compared with a fixed transfer payment; see Corollary 1). This result is a direct result of the facts that (i) the contractual energy price increases above 𝑐in the presence of the energy price brake, and (ii) energy demand is downward sloping (see Lemma 1). Thus, the energy price brake achieves the objective of reducing energy demand. Moreover, Propositions 4 and 5 show that whether the objective of the energy price brake to relieve consumers financially is reached depends on the market structure. This objective can be achieved with competition among suppliers because consumers then fully pocket the energy price brake transfers. On the contrary, consumers are not relieved in the presence of a monopolistic supplier that fully pockets the transfers from the energy price brake. 4Extensions In the following, we analyze how additional restrictions on energy supply contracts affect the equilibrium outcome under an energy price brake. First, we suppose that a supplier can only freely choose the contractual per-unit price (i.e., we have a regime of “linear energy contracts”); second, we examine the case of “capped transfers” (as specified in Germany’s energy price brake legislation whereby a consumer’s energy bill cannot be negative); and third, we consider (cost-based) “regulated energy prices.” Those restrictions could limit the moral hazard problem induced by the energy price brake, but they cannot eliminate it entirely. Finally, in a fourth extension, we discuss potential solutions to the moral hazard problem. 4.1 Linear Energy Contracts If the supplier can set a two-part tariff contract, he can maximize the joint surplus with the per-unit price 𝑝, and share it efficiently with the fixed payment 𝐹. Energy market regulations, however, could constrain the providers’ ability to set or alter the fixed payment.16 How do our results change when the supplier can only set linear energy contracts, that is, the per-unit price? As in the previous section, we assume some maximal price 𝑝that the contractual per-unit price 𝑝cannot surpass; that is, 𝑝≤𝑝,with 𝑝∈(𝑐,𝑘). 4.1.1 Monopoly Assume a monopoly supplier and the same two-stage contracting game as before, with the only difference that the supplier can now only set a “linear” energy price 𝑝. Consumer utility is given by (3), with 𝐹=0. If a consumer accepts the offer, her demand 𝑥(𝑝) is given by Lemma 1. For a given transfer scheme 𝑇(𝑝) and energy demand 𝑥(𝑝),the participation constraint of the consumer is given by 𝐶𝑆(𝑝) ∶=𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) +𝑇(𝑝) ≥𝑅, (14) where the left-hand side of the inequality is the overall utility from accepting a contract with price 𝑝<𝑘. Anticipating energy demand 𝑥(𝑝), the supplier solves max 𝑝𝜋(𝑝) ∶=(𝑝 −𝑐)𝑥(𝑝) s.t. (14). (15) Unlike the two-part tariff case, where the transfer scheme 𝑇(𝑝) can render an outcome with 𝑥(𝑝) =0profitable (see Proposition 1), with linear prices, the profit is always zero when the per-unit price surpasses the choke price. The profit function 𝜋(𝑝) has at least one local maximum.17 To proceed parsimoniously, we impose the standard assumption that the marginal profit function changes its sign only once so that there is only one local maximum. In addition, we assume that 𝑝 does not restrict the attainability of the unique local maximum (in fact, we can think of 𝑝being close to 𝑘). Hence, as before, the maximal price 𝑝is not used as some form of price-cap regulation to restrict the monopoly supplier’s price-setting behavior in the absence of the energy price brake; however, it could restrict the exploitation of the transfer payment when an energy price brake is in place.18 Assumption 2. The supplier’s marginal profit 𝜕𝜋(𝑝)∕𝜕𝑝 has at most one zero over (𝑐, 𝑘),which we denote 𝑝𝐼.Moreover,𝑝𝐼<𝑝. The analysis of the effects of the transfer scheme 𝑇(𝑝) depends on the contracting outcome in the absence of it. Given energy demand (Lemma 1), we have to distinguish two cases depending on whether or not the participation constraint 𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) ≥𝑅(16) is binding. Case I (Participation constraint (16)not binding).The unique local maximum 𝑝𝐼gives the monopoly solution, and follows from the first-order condition 𝜕𝜋(𝑝) 𝜕𝑝 =𝑥(𝑝) +𝑝−𝑐 𝑈′′ =0. (17) 136 The RAND Journal of Economics,2025 By (6), 𝑈′−𝑐 𝑈′′ > = <0⇔𝑝< = >𝑐. Note also that lim𝑝→𝑘−[𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝)]=0 follows from lim𝑝→𝑘−𝑥(𝑝) =0and 𝑈(0) =0, and part ii follows from 𝑥(𝑝) =0for all 𝑝≥𝑘(Lemma 1). Thus, the joint surplus has a unique maximum at 𝑝∗=𝑐.□ Part i of the proposition for the monopoly case follows immediately from Lemma 2 because the solution to (8)mustbethesameasthesolution to the maximization of the joint surplus. Part ii for the monopoly case follows from Assumption 1, so that the energy consumption level is strictly positive and the socially optimally one. In equilibrium, the consumer’s participation constraint (7) must bind, so that the fixed fee is given by the maximal joint surplus net of the consumer’s outside option utility; that is, 𝐹=𝑈(𝑥∗)−𝑐𝑥∗−𝑅, which is also the supplier’s equilibrium profit. The consumer then obtains 𝐶𝑆 =𝑅. Under competition, suppliers are perfectly substitutable from the consumer’s perspective (as there is no product differentiation; moreover, marginal costs are constant and the same for all suppliers), so that the consumer always chooses the contract offer with the highest overall utility 𝐶𝑆. It is then straightforward to see that the equilibrium contract offer must maximize the consumer’s overall utility (i.e., the contractual perunit price is set to marginal costs to maximize the joint surplus according to Lemma 1), whereas no supplier can realize strictly positive profits with 𝐹>0(as the supplier could be undercut by 𝐹−𝜀,with𝜀>0). It follows that 𝑝∗=𝑐and 𝐹=0must hold under competition, so that any supplier’s profit is zero and the consumer’s overall utility is equal to the maximal joint surplus; that is, 𝐶𝑆 =𝑈(𝑥∗)−𝑐𝑥∗. Proof of Corollary 1. An unconditional fixed transfer 𝑇>0neither affects the consumer’s participation constraint (7) nor the consumer’s energy demand according to Lemma 1. It, therefore, does also not affect the supplier’s maximization problem, so that the market equilibrium as described in Proposition 1 remains the same. □ Proof of Proposition 2. The supplier faces maximization problem (12). To understand its solution, it is helpful to examine how 𝑝affects the sum of the joint surplus (see Lemma 2) and the transfer 𝑇(𝑝); this then determines the marginal profit (see Equation (13)). Given energy demand 𝑥(𝑝) according to Lemma 1, the following lemma specifies the properties of the joint surplus, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) (see Lemma 2), augmented by the energy price brake 𝑇(𝑝), which we define by Π(𝑝) ∶=𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝). (A2) Lemma 3 (Joint surplus with energy price brake). Assume the government offers an unconstrained energy price brake (9) to the consumer. Then Π(𝑝), fulfills the following properties: i) It is continuous everywhere and it is differentiable for all 𝑝≥0except at 𝑝=𝑠and 𝑝=𝑘, where it has two kinks. ii) It is strictly increasing for all 𝑝∈[0,𝑐], obtains the value 𝑇(𝑘) at 𝑝= 𝑘, and it has the constant slope 𝜕Π(𝑝)∕𝜕𝑝 =𝜕𝑇(𝑝)∕𝜕𝑝 =𝛼𝑥>0for all 𝑝>𝑘. iii) It is bounded from above and from below on [𝑐, 𝑘]. iv) On [𝑐, 𝑘],Π(𝑝) has a maximum either at some 𝑝∈(𝑐,𝑘)(interior solution) or at 𝑝=𝑘(corner solution). Proof of Lemma 3. Note first that Π(𝑝) is the sum of 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) (which properties are given in Lemma 2) and 𝑇(𝑝). Note that 𝑇(𝑝) =0 for 𝑝≤𝑠and 𝑇(𝑝) linearly increasing with slope 0<𝜕𝑇(𝑝)∕𝜕𝑝 <∞for all 𝑝>𝑠. Part i). By Lemma 2, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) is continuous in 𝑝and has a kink at 𝑝=𝑘, and 𝑇(𝑝) is linear in 𝑝for 𝑝>𝑠.Moreover,𝑇(𝑝) is zero for 𝑝≤𝑠and linearly increasing for all 𝑝>𝑠, so that 𝑇(𝑝) is also continuous and has a kink at 𝑝=𝑠. It follows that Π(𝑝) is continuous in 𝑝with two kinks at 𝑝=𝑠and 𝑝=𝑘. Likewise, Π(𝑝) is differentiable everywhere except at points 𝑝=𝑠and 𝑝=𝑘. Part ii). By Lemma 2, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) is increasing in 𝑝for 0≤𝑝≤𝑐 (with a zero at 𝑝=𝑐), and 𝑇(𝑝) is strictly increasing in 𝑝for all 𝑝>𝑠,with𝑠<𝑐. Thus, Π(𝑝) is strictly increasing in 𝑝for all 0≤𝑝≤𝑐; and in particular, at 𝑝=𝑐. By Lemma 2, 𝑈(𝑥(𝑝)) − 𝑐𝑥(𝑝) is zero for 𝑝≥𝑘, and 𝑇(𝑝) is linear for all 𝑝>𝑠. Part iii). An upper bound is given by Π(𝑝) <Π(𝑐) +Π(𝑘) <∞, which follows from 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) strictly decreasing in 𝑝(see part ii of Lemma 2) and 𝑇(𝑝)linearly increasing in 𝑝.Alowerbound is given by Π(𝑝) >0, which follows from lim𝑝→𝑘−𝑈(𝑥(𝑝)) − 𝑐𝑥(𝑝) =0(Lemma 2) and 𝑇(𝑝) >0for all 𝑝∈[𝑐,𝑘]. Part iv). Because of part ii, there cannot be a maximum at 𝑝≤𝑐. Then, there are two possible cases: either 𝜕Π(𝑝)∕𝜕𝑝 >0for all 𝑐≤𝑝<𝑘,withlim𝑝→𝑘−Π(𝑝) =𝑇(𝑘), or there exists at least one price 𝑝∈(𝑐,𝑘), where the condition for a local maximum 𝜕Π(𝑝)∕𝜕𝑝 =0holds. In the former case, the unique maximum is reached at 𝑝=𝑘, and in the latter case, there are two possible candidates for a maximum: either at a price 𝑝∈(𝑐,𝑘),where 𝜕Π(𝑝)∕𝜕𝑝 =0holds, or at 𝑝=𝑘. The former solution gives the interior solution and the latter one the corner solution. □ Note that a solution to the supplier’s maximization problem (12)must also maximize Π(𝑝) as they differ only in the constant 𝑅. Thus, Lemma 3 allows us to characterize also the marginal profit (see Equation (13)) and therefore prove the proposition. First, 𝑝∗=𝑐cannot be a solution because at this point the firm’s marginal profit strictly increases (by part ii of Lemma 3). For 𝑐<𝑝≤𝑘there may exist an interior maximum (by parts iii and iv of Lemma 3) and/or a maximum at 𝑝=𝑘where the profit is given by 𝑇(𝑘). As the firm’s profit can be raised by any amount due to 𝑇(𝑝) for 𝑝>𝑘with 𝑥(𝑝) =0, the first part of the proposition follows. It is then obvious that there is also a large enough contractual energy price with 𝑇(𝑝) >𝑅, which allows the supplier to extract the arbitrarily large transfer which gives the last part of the proposition. □ Proof of Proposition 3. Suppose a firm offers a contract (𝑝, 𝐹) that is accepted by the consumer. Given the consumer’s energy demand, the firm’s profit is 𝜋=(𝑝 −𝑐)𝑥(𝑝) +𝐹, which gives 𝐹=𝜋−(𝑝 −𝑐)𝑥(𝑝). Substituting this into the consumer’s overall utility (3), we get 𝐶𝑆(𝑝,𝐹) ∶= Π(𝑝) −𝜋. Thus, when the consumer faces different contracts that satisfy the participation constraint (11), the consumer selects the contract with the highest overall utility. Firms thus compete in two-part tariffs (𝑝, 𝐹) to maximize the consumer’s overall utility which must lead to 𝑝→∞, because then the transfer from the energy price brake also becomes arbitrarily large. Subtracting a fixed profit level does not compromise the attractiveness of the contract. □ Proof of Proposition 4. The supplier solves (12) under the additional constraint 𝑝≤𝑝. The statements of the proposition then follow from Lemma 3. In particular, part iv of Lemma 3 also applies to the subinterval 𝑝∈[𝑐,𝑝]. Thus, Π(𝑝) either has maximum at some 𝑝∈(𝑐,𝑝], where the condition for a local maximum 𝜕𝜋 𝜕𝑝 =𝜕Π(𝑝) 𝜕𝑝 =𝑈′−𝑐 𝑈′′ +𝜕𝑇 𝜕𝑝 =0 holds (“interior solution”), or it obtains a global maximum at 𝑝=𝑝 (“corner solution”). Whereas Π(𝑝) is not differentiable at 𝑝=𝑘,itis differentiable at 𝑝, which implies that the condition for a local maximum could be satisfied at 𝑝=𝑝. By Assumption 1, both the interior and the corner solution can be implemented by the supplier with a fixed payment, which leaves an overall utility of 𝑅to the consumer. The supplier then realizes the profit as stated in the proposition. Both the interior and the corner solution increase the contractual energy price above 𝑝∗=𝑐, so that energy demand decreases with 𝑥(𝑝) <𝑥(𝑝∗). Clearly, the transfer of the energy price brake is also larger when compared with the transfer that would result from the price in the benchmark case where 𝑝∗=𝑐. This proves the proposition. □ Proof of Proposition 5. To prove this proposition, we can use Proposition 3. Again, firms compete in offering two-part tariff contracts, and the consumer selects the contract that gives her the maximal overall utility 143 𝐶𝑆(𝑝,𝐹) =Π(𝑝) −𝜋. It is then obvious that a firm cannot make a positive profit, and 𝑝is chosen to maximize Π(𝑝) (i.e., the sum of the joint surplus plus the transfer from the energy price brake). Only if at least two suppliers offer such a contract, there is no profitable unilateral deviation incentive; that is, we have reached a subgame-perfect equilibrium. Thus, the equilibrium price is either obtained as an interior solution or a corner solution. It then must also hold that energy consumption is strictly lower than in the benchmark case without an energy price brake. Finally, and in contrast to the monopoly outcome as described in Proposition 5, firms now make zero profits whereas the consumer fully pockets the joint surplus including the transfer of the energy price brake. Notably, here the fixed payment is strictly negative (i.e., the consumer gets a bonus payment), so that 𝐹=−(𝑝 −𝑐)𝑥(𝑝) holds. □ Proof of Proposition 6. We first analyze the equilibrium without the energy price brake 𝑇(𝑝).Given(16) holds, the monopoly supplier sets the price 𝑝𝐼according to (17), where Assumption 2 guarantees that 𝑝𝐼is unique and feasible (i.e., 𝑝𝐼<𝑝). Thus, the standard monopoly solution, 𝑝𝐼,is the equilibrium outcome whenever (16) is not binding at this price. If, to the contrary, (16)isviolatedat𝑝𝐼, then the monopolist sets the highest possible price 𝑝𝐼𝐼 where (16) holds as an equality. This follows from noticing that the left-hand side of (16)—namely, consumer utility—is strictly decreasing in 𝑝,with 𝜕 𝜕𝑝[𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝)]=−𝑥(𝑝) <0, (A3) where we used (6). Thus, there is a unique price 𝑝𝐼𝐼 <𝑝𝐼where the consumer’s participation constraint (16) holds as an equality (existence follows from Assumption 1). By Assumption 2, the supplier’s profit is strictly increasing in 𝑝for all 𝑝<𝑝𝐼, so that it is indeed optimal for the supplier to set the price 𝑝𝐼, where (16) holds as an equality. Next assume that an energy price brake 𝑇(𝑝) >0is in place. The introduction of 𝑇(𝑝) has the effect that it relaxes the consumer’s participation constraint (16), which is now given by (14). Consequently, when 𝑝𝐼 (according to Equation (17)) is the solution to (15)for𝑇(𝑝) =0(i.e., with no energy price brake in place), then this must also be the solution when 𝑇(𝑝) >0holds. Thus, the introduction of the energy price brake has no effect on the contractual energy price 𝑝𝐼, energy demand 𝑥(𝑝𝐼)>0,the monopolist’s profit 𝜋(𝑝𝐼), and it only increases the consumer’s utility by 𝑇(𝑝𝐼)from 𝑈(𝑥(𝑝𝐼)) −𝑝𝐼𝑥(𝑝𝐼)to 𝑈(𝑥(𝑝𝐼)) −𝑝𝐼𝑥(𝑝𝐼)+𝑇(𝑝𝐼). This proves part i of the proposition. Now suppose that (16) is binding in the profit maximizing solution to (15) for 𝑇(𝑝) =0, so that the monopolist sets 𝑝𝐼𝐼 <𝑝𝐼, where (16)holdsas an equality. Again, the only effect of the introduction of the energy price brake, with 𝑇(𝑝) >0, is to relax the consumer’s participation constraint (16), which is now given by (14). Clearly, the consumer’s participation constraint (14)mustbeslackat𝑝𝐼𝐼. As the supplier’s profit is strictly increasing in 𝑝at 𝑝𝐼𝐼 <𝑝𝐼(Assumption 2), he will always increase the price above 𝑝𝐼𝐼. This proves part ii.a of the proposition. As shown in the main text, the left-hand side of (14)(i.e.,𝐶𝑆(𝑝)) either increases or decreases at any 𝑝≤𝑝(according to Equation (20)), and it is strictly convex (see Equation (19)). Thus, if 𝜕𝐶𝑆(𝑝)∕𝜕𝑝 >0at 𝑝𝐼𝐼 (which holds if 𝛼𝑥>𝑥(𝑝𝐼𝐼)according to Equation (20)), then the supplier can profitably increase the price above 𝑝𝐼𝐼 up to 𝑝𝐼(the standard monopoly solution Equation (17)), because any such price increase must further relax the consumer’s participation constraint (14); that is, any increase in 𝑝also increases the consumer’s overall utility 𝐶𝑆(𝑝). Thus both the supplier and the consumer strictly benefit from the introduction of the energy price brake. This proves part ii.b of the proposition. The introduction of the energy price brake, therefore, always increases the contractual energy price, so that part ii.c of the proposition follows directly from Lemma 1. □ Proof of Proposition 7. Under competition, the consumer selects the contract which gives the highest utility (3), given that the utility is not smaller than 𝑅. If there is more than one such contract, then the consumer selects each of the contracts with a strictly positive probability. As energy is homogeneous, suppliers compete in Bertrand fashion. Therefore, without loss of generality, we consider the duopoly case with two suppliers, using the indices 𝑖and 𝑖′to represent a supplier’s identity. First, consider the case without a transfer scheme. Take firm 𝑖’s contract offer 𝑝𝑖. Suppose the consumer’s participation constraint (16) holds, then firm 𝑖’s profit function 𝜋𝑖is given by 𝜋𝑖(𝑝𝑖,𝑝 𝑖′)=⎧ ⎪ ⎨ ⎪ ⎩ (𝑝𝑖−𝑐)𝑥(𝑝𝑖), if 𝑝𝑖<𝑝𝑖′ 𝛽𝑖(𝑝𝑖−𝑐)𝑥(𝑝𝑖), if 𝑝𝑖=𝑝𝑖′ 0, if 𝑝𝑖>𝑝𝑖′, for 𝑖≠𝑖′ where 𝑥(𝑝𝑖)is the consumer’s energy demand (according to Lemma 1) and 𝛽𝑖∈ (0, 1) is the probability that the consumer selects firm 𝑖’s offer when 𝑝𝑖=𝑝𝑖′,with𝛽𝑖+𝛽𝑖′=1. Here, consumer utility is strictly decreasing in 𝑝(see (A3)). Thus, if 𝑝𝑖′=𝑐, then firm 𝑖cannot do better than also setting the price 𝑝𝑖=𝑐,in which case profits are zero. Clearly, all other prices cannot constitute an equilibrium, so that 𝑝∗=𝑐is the unique equilibrium price. Now, consider the introduction of a transfer scheme 𝑇(𝑝) >0so that the consumer’s participation constraint for a selected contract is given by (14). Facing two contract offers 𝑝𝑖and 𝑝𝑖′(both meeting the consumer’s participation constraint), the consumer selects the contract with a higher overall utility 𝐶𝑆(𝑝).If𝑝𝑖=𝑝𝑖′, so that 𝐶𝑆(𝑝𝑖)=𝐶𝑆(𝑝𝑖′),withoutlossof generality, 𝛽𝑖∈ (0, 1) gives the probability that the consumer selects firm 𝑖’s offer. Assume 𝜕𝐶𝑆(𝑝)∕𝜕𝑝 ≥0at 𝑝=𝑐(which holds if 𝛼𝑥≥𝑥(𝑐) according to Equation (20)). Firm 𝑖’s profit function for 𝑝𝑖,𝑝 𝑖′≤𝑝is then given by 𝜋𝑖(𝑝𝑖,𝑝 𝑖′)=⎧ ⎪ ⎨ ⎪ ⎩ (𝑝𝑖−𝑐)𝑥(𝑝𝑖), if 𝑝𝑖>𝑝𝑖′ 𝛽𝑖(𝑝𝑖−𝑐)𝑥(𝑝𝑖), if 𝑝𝑖=𝑝𝑖′ 0, if 𝑝𝑖<𝑝𝑖′, for 𝑖≠𝑖′ where 𝑥(𝑝𝑖)is the consumer’s energy demand (according to Lemma 1) and 𝛽𝑖∈ (0, 1) is the probability that the consumer selects firm 𝑖’s offer when 𝑝𝑖=𝑝𝑖′,with𝛽𝑖+𝛽𝑖′=1. Here, supplier 𝑖’s contract is selected for sure by the consumer whenever 𝑝𝑖>𝑝𝑖′holds, whereas it is selected with some positive probability 𝛽𝑖whenever 𝑝𝑖=𝑝𝑖′holds. It is then immediate to see that 𝑝is the unique equilibrium energy price. Clearly, any pair of prices with 𝑝𝑖<𝑝𝑖′≤𝑝, can be ruled out because then firm 𝑖has a strict incentive to increase its price to 𝑝𝑖′. Moreover, any pair of prices with 𝑝𝑖=𝑝𝑖′<𝑝can also be ruled out, because then firm 𝑖could increase its price by 𝜀>0to gain the entire market and thus to increase its profit. This proves part i of the proposition. In all other cases, that is, when 𝜕𝐶𝑆(𝑝)∕𝜕𝑝 <0at 𝑝=𝑐holds, then 𝐶𝑆(𝑝) either has a global maximum at 𝑝=𝑐or at 𝑝=𝑝, which follows from the strict convexity of 𝐶𝑆(𝑝) (see Equation (19)). If 𝐶𝑆(𝑝) ≥𝐶𝑆(𝑐), then 𝑝=𝑝is the unique equilibrium. If, to the contrary, 𝐶𝑆(𝑐) >𝐶𝑆(𝑝) ≥𝑅, then both 𝑝=𝑐and 𝑝=𝑝are both possible equilibria, and the latter one is strictly preferred from the firms’ perspective. By the same reasoning as above, any other price pair cannot be an equilibrium outcome. □ Proof of Corollary 2. Suppose the constraint 𝐹≥0. Then, the equilibrium as specified in Proposition 5 is not feasible, because here 𝐹<0.We show that under competition, 𝐹>0can be ruled out, so that 𝐹=0must hold in equilibrium. Suppose 𝐹>0. Both firms must make zero profits (otherwise, one firm could increase its profit by reducing 𝐹slightly to gain the entire market). Then, 𝐹=|(𝑝 −𝑐)𝑥(𝑝)|with 𝑝<𝑐must hold by the zero-profit condition. For 𝑝<𝑐, the joint surplus including the transfer, Π(𝑝), is increasing in 𝑝by part ii of Lemma 3. Thus, increasing 𝑝strictly increases the joint surplus, and by reducing the fixed payment in the right way, the additional joint surplus can be divided in a such way that the consumer is strictly better off and the firm can win the consumer, and also the firm is strictly better off. Thus, there cannot be an equilibrium with 𝐹>0. It then follows that 𝐹=0must hold, so that the equilibrium must be the same as in the linear case, which is given in Proposition 7. □ 144 The RAND Journal of Economics,2025