Price Regulation and the Incentives to Pursue Energy Efficiency by Minimizing Network Losses

This paper examines the incentives embedded in different regulatory regimes for investment by utilities in energy efficiency programs that aim to reduce network losses. In our model, a monopolist chooses whether to undertake an investment in energy efficiency, which is not observable by the regulator. We show that, in equilibrium, the monopolist chooses to exert positive effort more often under price cap regulation than under no regulation or mandated target regulation and that she exerts no effort under rate of return regulation. This result contrasts with an extensive literature that focuses on end-user energy conservation and shows that price caps are ineffective for achieving energy efficiency as utilities have an incentive to maximize sales volume. Thus, policies that are designed to promote demand-side energy conservation may diminish the utilities’ incentives to pursue supply-side energy efficiency through minimizing network losses.


INTRODUCTION
Energy efficiency programs have returned to the forefront of public policy. 1Such public interest in energy efficiency programs has not been seen since their introduction in the 1970s as a response to the oil shocks.Energy efficiency is seen today as a cost-effective approach to sustainable energy use and greenhouse gas emission reductions.For instance, Metz Bert and Intergovernmental Panel on Climate ChangeWorking Group III (2001) argue that energy efficiency improvement could potentially contribute to half of greenhouse gas emission reduction by 2020.
Despite their high profile, energy efficiency opportunities have by and large not materialized (Interlaboratory Working Group, 2000).This is particularly the case in the electricity sector where between 20 and 60 percent of total electricity used could be conserved by cost-effective initiatives (see, for example, Rosenfeld 1993).For a critical review of the extent to which there is such an energy efficiency gap, see Greenstone and Allcott (2012).This paper distinguishes between two types of approaches that can be used by utilities to pursue energy efficiency.The first approach focuses on the consumer-end and the second approach on the utility's internal efficiencies such as minimizing network losses.While considerable attention has been paid to the first approach, the pursuit of internal efficiencies by the utilities has not attracted the same degree of attention despite its potential impact on energy conservation.This paper focuses on this second approach.
The first approach includes demand side management (DSM) programs. 2While such programs may be able to reduce consumption at a cost per kilowatt-hour lower than the cost of supplying that electricity, they are still not widely used (Freeman et al., 2010).
There are several reasons why this may be the case.First, electricity suppliers (especially network service providers) need to make a range of infrastructure investments that are required to integrate distributed energy resources, including energy storage technologies, the digital hardware and software for improving transmission and distribution system reliability and security, and the supply-side and customer-side systems needed for full customer connectivity (Lester and Hart,  2012).Second, these firms operate in a regulated environment where prices are usually set by a regulatory agency or government department that may not reward such investments by not considering them prudent or efficient.Finally, investment in energy efficiency may not be recouped if demand for electricity falls in the future or if the regulatory regime changes.
Indeed, creating incentives for suppliers/distributors to undertake energy efficiency initiatives at the consumer end can be complex as these incentives interact with the form of price regulation.For example, in a context where firms are allowed to recover, via prices, all the costs of supplying electricity to consumers, regulated firms may have little incentive to undertake energy efficiency investment that promotes end-user energy conservation.Indeed, there is a large literature (see, for example, Wirl, 1995; Eto et al., 1997; Brennan, 2010; Sullivan et al., 2011) that points out that price cap regulation is ineffective at inducing the electricity supplier to promote end-user conservation and, instead, provides incentives for suppliers to maximize sales as long as sale prices are above marginal cost.This had led to proposals to decouple allowed revenue from sales quantity, which would tend to favour revenue cap or rate of return regulation. 3n contrast, this paper analyses the incentives embedded in different regulatory regimesrate of return, price cap and mandated target regulation-for the electricity supplier to undertake energy conservation programs to minimize network losses.
Network losses are significant.Estimates of the overall losses between the power plant and final consumers are in the range of 8 to 15% (with half of that attributed to distribution losses) according to the International Electrotechnical Commission (International Electrotechnical Commission, 2007). 4For a survey, general background and comparative analysis of the available techniques to reduce network losses, see Kalambe and Agnihotri (2014).
While the interaction between network losses and price regulation is an important policy issue, this is the first paper to the best of our knowledge that focuses on incentives to pursue energy efficiency at the supply-side rather than at the end-user side. 56.The existence of asymmetric information is a real world problem faced by regulators everywhere.An example of the existence of asymmetric information considered in this paper is that of a manager who faces a choice between investing time and effort (not measured in any systematic way) to find a cost effective way to reduce network losses or devoting his or her time to other tasks.A regulator is unable to measure and to reward the manager's effort and can only reward actual outcomes.
7. This result was originally established by Marino and Sicilian (1988).8.This is consistent with recent research that shows that efficiency standards are often substantially inferior to pricing policies to achieve energy efficiency.See Parry et al. (2014).
We pursue this by building a theoretical model of a monopolist who can choose whether or not to undertake an investment in energy efficiency.The investment is not observable by the regulator who can only determine whether the investment has been successful in terms of the level of energy efficiency achieved.More specifically, the firm's choice of effort affects the probability of a successful outcome with a higher effort resulting in a higher probability of achieving a better energy efficiency outcome. 6n this setting, regulatory regimes cannot explicitly compensate the firm for the effort it has put into energy efficiency.We explore how different existing regulatory regimes perform in terms of the expected amount of energy efficiency and total welfare.In particular, we show that the monopolist more often chooses to exert effort under price cap regulation than under no regulation and that she exerts no effort under rate of return regulation. 7owever, the comparison, in terms of the embedded incentives to pursue energy efficiency, between price cap and rate of return regulation, and between rate of return regulation and an unregulated monopolist, is ambiguous and complex as demonstrated in the paper.In contrast, we show unambiguously that mandated target regulation is always dominated by both price cap and rate of return regulation in terms of expected welfare although a mandated target regulatory regime can do better than an unregulated monopolist.The key reason is that mandated target regulation is too coarse and the trade off between providing incentives to invest in energy efficiency and rent extraction is less pronounced than under existing regulatory regimes such as price cap and rate of return regulation. 8More generally, we provide a full characterization of the optimal effort, optimal prices (regulated or unregulated) and expected welfare for the different regimes and show the tradeoffs between rent extraction and incentives.
It is important to note that our results are different from the familiar corollary to the leading theme in regulatory economics over the last three decades, namely that price-caps restore incentives to control costs that are diminished or eliminated by rate of return regulation.Instead, our analysis highlights that policies that encourage utilities to promote end-user energy conservation (e.g., by switching from price cap to rate of return regulation) may reduce the incentives for the electricity suppliers to improve energy efficiency through the minimization of network losses.
This paper is organized as follows.Section 2 describes the key elements of the model.Section 3 develops the benchmark case of an unregulated monopolist, and describes the monopolist's choice of optimal effort and profit-maximizing quantity and price.In Section 4, we characterize outcomes under three distinct regulatory regimes, namely, rate of return, price cap and mandated target regulations.Section 5 compares the welfare under the different regulatory regimes, and Section 6 concludes.

THE MODEL
This Section develops a model of a regulated monopolist that supplies electricity to final consumers.The model shares many features of the large literature on moral hazard and incentives for cost reduction by a regulated monopolist.See, for example, Laffont and Tirole (1993).In particular, a monopolist chooses how much effort to exert in order to reduce costs via the minimization of network losses.The effort entails a cost, borne by the monopolist.The regulator observes the total cost (net of network losses) but not the effort.
Our approach differs from the standard model in two ways.First, while in the standard model the regulator has access to two instruments (a fixed payment and a price), we instead focus on the case where the regulator has only one instrument, namely the price that the regulated firm is allowed to charge consumers.As discussed in the 2015 Nobel Prize citation for Jean Tirole, regulators in practice do not have access to fixed payments (transfers). 9Thus, a key point of departure from the standard framework is that we consider the incentives faced by the regulated firm under two stylized forms of price regulation that are used in practice, rather than considering what would be the optimal mechanism to minimize network losses.This is an important distinction as we take a positive rather than a normative approach to regulatory economics in the sense that we recognize that the optimal mechanism, which will involve a price and an optimal transfer, may not be implementable.
Second, while the standard model focuses on a price/transfer mechanism to incentivize the regulated firm to exert efforts to reduce the marginal cost of production, here the focus is the incentives embedded in different price mechanisms (with no transfers) to reduce costs via minimizing network losses only.While, in principle, a more general formulation could allow the firm to choose amongst different types of actions to reduce costs, perhaps each entailing different costs, the emphasis here is exclusively on network losses.
We now turn to the particulars of the model.For simplicity, the demand function for electricity in the market is assumed to be linear, and the inverse demand function is given by: where denotes the amount of electricity that could be consumed by end users, and is the unit Q P retail price of electricity. 10The monopolist can purchase wholesale electricity at a fixed price cϾ and faces no fixed costs.In order to supply units of electricity to consumers the monopolist 0 Q needs to purchase units in the wholesale market.In general, due to network losses, exerting effort the monopolist can improve energy efficiency by optimizing the operation E ∈{0,e} of the network. 11In particular, we denote by the energy efficiency parameter, which is Q s a function of effort and can take two values .We assume that positive effort implies 0ϽUϽUϽ1 a higher probability of achieving high energy efficiency as detailed below.
We also assume that the cost of exerting effort is equal to .Thus, the total cost of E E acquiring units in the wholesale market and exerting effort is given by: 12.This model is necessarily a simplification of the complex decision-making process of a utility.The aim of this paper is not to provide a general model of decision making by a utility but rather to show that regulatory regimes that are known to provide utilities with incentives to pursue demand-side energy efficiency can have the opposite effect on supply-side energy efficiency.
It follows that the monopolist's profit from selling units to final consumers is equal to 12 Energy efficiency is not a deterministic function of effort and instead is determined according to the probabilities defined in Table 1, where denotes the probability that the low energy efficiency m parameter eventuates when low effort is exerted. U

Table 1: Relationship between Effort Level and Energy Efficiency
U U We assume that so that, as is standard in moral hazard models, the high level of 1 mϾ 2 effort is more likely to lead to higher energy efficiency than the low level of effort.Note that the assumption that high energy efficiency can eventuate even under zero effort is a normalization to simplify the characterization of equilibrium behaviour.Without loss of generality, we assume a symmetric probability for energy efficiency under the two levels of effort.
Given Table 1, the quantity of electricity the monopolist needs to purchase from the wholesale market is an expected value determined by the electricity price and the effort P devoted to energy efficiency , which could be denoted as According to (2), the expected level of energy efficiency is defined as by assumption.We also assume that: Equation (4) provides a sufficient condition for an interior solution and also plays a role, as we will establish later, in the comparison between the different threshold values for the cost of effort associated with the regulatory regimes we consider.
Consumer surplus is denoted by and overall surplus by is the weight assigned by the regulator to the monopolist's expected profit.The 0Ͻ γϽ1 objective of the regulator is to maximize expected total surplus.As noted before, the regulator can observe the realization of the level of energy efficiency-for example, by monitoring both demand and the amount of electricity purchased by the supplier from the wholesale market, but not the level of effort.

THE UNREGULATED MONOPOLIST
From (1), the unregulated monopolist's expected profit can be rewritten as p(Q , E) = s .We can then calculate it by using ( 2) and (3): The first-order conditions (FOCs) are respectively These FOCs yield that the optimal amount of electricity the monopolist purchases in the wholesale market is a function of effort: The profit-maximizing prices are then: One can readily check that the second-order conditions are satisfied.We can substitute (6) into (5) to obtain the maximized expected profits as a function of effort: We stress that the monopolist chooses a single price prior to the realization of the energy efficiency.Although the energy efficiency outcome will be known by the time the electricity is used by consumers, we assume that the monopolist sells the electricity in the retail market at a fixed price determined prior to the delivery of electricity.

Optimal Choice of Effort
In order to ensure that the monopolist remains in the industry the following participation constraint needs to be satisfied: s For a positive choice of effort to be optimal, the following incentive compatibility constraint, obtained directly from ( 7), needs to be satisfied: * * ¯p -p Ͼ0.

h l
Combining the participation and incentive compatibility constraints, we can now determine the unregulated monopolist's optimal choice of effort in Table 2.
It is straightforward to show that follows from equation ( 4). e ˜Ͼ0 1 Note that given (7) and, therefore, the participation constraint is automatically * p у 0 l satisfied for both choices of effort.That is, if the cost of positive effort, , is greater than the benefit e of the effort (as measured by the difference in expected profits under the two efforts), then zero effort is optimal.Otherwise, the unregulated monopolist chooses to exert positive effort.This is summarized in the following result.The proof is straightforward and, therefore, omitted.

Lemma 1 The unregulated monopolist chooses a positive level of effort whenever
Oth-0 р eϽ e ˜.The effort level chosen by the monopolist depends on the expected values of energy efficiency, the shape of the demand function (here this refers to the parameters and ) and the a b wholesale price.
13.In practice, however, regulators typically attempt to ensure that incurred costs are efficient.See, for example, Joskow  (1974) and Joskow et al. (1989).
14.Note that this captures an important feature of actual ROR regulatory regimes where some expenditures are simply not allowed.Allowing a fraction of the cost of effort to be recovered throught the ex-post prices would not change the incentives of the regulated monopolist to undertake zero effort.
We now compute the social welfare in the absence of regulation: * * where and Given equations ( 6) and ( 7), we can rewrite the expected total social welfare under an unregulated monopoly for different levels of effort as follows: In the next two sections we investigate whether a welfare-maximizing regulator may be able to improve upon (8) under different regulatory regimes.

REGULATION AND INCENTIVES
This section investigates the incentives for energy efficiency provided by three different types of regulatory regimes commonly observed around the world.

Rate of Return Regulation (ROR)
Under rate of return regulation, the regulator determines prices to cover the actual costs (including the cost of capital) incurred by the monopolist to supply electricity. 13In this paper, we consider a stylized, pure form of rate of return regulation where the regulator observes both the actual electricity consumed ( ) and the electricity purchased by the monopolist ( ).Therefore, Q Q s the regulator can infer the level of energy efficiency that has eventuated although she cannot observe the effort exerted by the monopolist.
The regulator cannot mandate a certain level of effort as it is not verifiable in court.As a result, we assume that the regulator, when setting ex-post prices, does not allow the regulated firm to recover the cost of effort.That is, the regulator sets ex-post prices, conditional on the realization of the energy efficiency outcome, such that profits not including the cost of effort equal zero.This implies that if the regulated monopolist were to undertake positive effort, this would lead to negative profits. 145.We attempt to model price cap regulation to closely resemble how regulation works in practice.This means that while the price is set ex-ante, the regulated firm has an opportunity to interact with the regulator.Of course, we do not We now turn our attention to determining the optimal ex-post regulated prices under rate of return regulation.As the weight given to the profits of the monopolist is less than one, the regulator will set so that profits are zero.That is: The next proposition summarizes the outcomes under rate of return regulation including social welfare which is equal to expected consumers' surplus given that the monopolist earns zero profits.

Proposition 1 Under rate of return regulation, the monopolist chooses
, and regulated prices E = 0 are given by ( 9), and expected social welfare is given by: 2 2 2bU 2bU

Price Cap Regulation
Under our stylized form of price cap regulation, the regulator sets an ex-ante price-that is, a fixed price that is not conditional on the realized quantum of energy efficiency-in order to maximize expected social surplus.In doing so, the regulator faces a trade-off between providing incentives for energy efficiency and reducing the monopolist's rent.In particular, the regulator solves the following problem This problem is solved in several steps.First, we identify the optimal price cap as a function of effort.We then determine the socially optimal level of effort under the optimally determined price cap.Finally, we combine these two steps to fully specify the solution to the problem above, that is, an ex-ante fixed price and the optimal level of effort, as a function of various parameters including the cost of effort. 15The first step is completed in the following Lemma.The proof is in the Appendix (which is available online beside the article on the IAEE website).
model this interactive process explicitly and instead, as short hand for this process, solve the game simultaneously.That is, when setting the ex-ante price, the regulator takes into account how the monopolist will react to the different prices.Then we compute the optimal choice of effort under the optimally chosen price caps.Generally speaking this approach may lead to a different solution from solving the game backwards, as one would do in a standard Stackelberg game.Solving the game backwards would involve calculating the monopolist's choice of effort as a function of a regulated price.Then the regulator would choose a regulated price to maximize total welfare.In our model, however, decreasing the price cap increases the incentives to exert positive effort and so the two approaches yield the same equilibrium.We thank an anonymous referee for raising this issue.

Lemma 3
The optimal price cap, as a function of effort, is given by: We now turn our attention to determining the optimal effort from a monopolist who, in equilibrium, will face one of the two fixed prices above.If the price cap is set at , the monopolist's c U l expected profit is equal to: * * * and, therefore, the incentive compatibility constraint can be written as: Similarly, if the price cap is set at , the monopolist's expected which implies that the incentive compatibility constraint, is always satisfied and the monopolist chooses to exert positive effort at this ex-ante price.

Proposition 2
The optimal price cap, level of effort and expected social welfare are fully characterized in Table 3.
Note that the threshold value under price cap regulation is higher than that in the absence of regulation, i.e., Hence, the regulated firm exerts high effort more often than under no regulation.The intuition for this is that under price cap regulation, the regulator sets a price ex-ante so that expected profits are zero.The regulator anticipates the optimal effort for the value of the cost of effort and ensures that the monopolist is incentivized to put in positive effort in situations where an unregulated monopolist would choose zero effort.Thus, the range of values of the cost of effort for which it is worthwhile from a society's perspective to engage in positive effort is larger under price cap regulation than under unregulated monopoly.
In the same vein as it can be checked that is also increasing in , decreasing in e ˜, e ˜a b 1 2 and either increasing or decreasing in depending on its value.c It is worth noting that in our setting there is no meaningful distinction between price cap and revenue cap regulation.The optimal revenue cap can be implemented by the optimal price cap, which will result in the same expected revenue and welfare given that the demand function is deterministic.For completeness, we spell out the optimal revenue cap in the next Lemma with associated calculations provided in the Appendix.

Lemma 4
The optimal revenue cap, as a function of effort, is given by:

Mandated Target Regulation
Mandated targets for achieving energy efficiency have become widespread.For example, in the European Union there is a target of a 20% improvement in energy efficiency (as well as a separate target of a 20% share of energy consumption from renewable resources) to be achieved 16.See http://europa.eu/legislation_summaries/energy/energy_efficiency/.http://europa.eu/legislation_summaries/energy/energy_efficiency/ 17.While mandated energy efficiency targets can be achieved through a range of initiatives (e.g., minimum standards for white goods, energy taxation, and metering), the exclusive focus on network losses is warranted for two key reasons.First, transport costs represent a significant component of final energy consumption (32% within the European Community).Second, the other policy levers, while important, are typically outside the control of economic regulators.
18.While it is conceivable that the regulator might set an unattainable target to raise social welfare via the penalty up to the point where the firm's participation constraint is binding, our key focus is on incentives and as such we focus on only two possible targets, namely the two potential values for the energy efficiency variable.We are thankful to an anonymous referee for raising this issue.by 2020. 16Under this approach, the regulator sets a minimum percentage of the energy demand that has to be met through energy efficiency programs.In our model we capture this regulatory approach by assuming that the regulator sets a target, The regulated firm needs to meet this mt U .target in order to avoid a penalty . 17d у 0 In this setting, the objective of the regulator is to set the target in order to maximize expected welfare subject to the firm's participation constraint: As the regulator values profits less than consumer surplus, the third term of the total welfare reflects the expected penalty if the target is not met.In our setting, there are only two possible meaningful targets, namely or 18 If the mandated target is set at , it is of course not binding and mt U U. U = U the monopolist's expected profit is identical to the case of an unregulated monopolist.If, however, the regulator sets as a target, the monopolist's expected profit is equal to: The first-order conditions (FOCs) are respectively Note that these are the same FOCs as faced by an unregulated monopolist.This follows as the relationship between effort and the probabilities of the energy efficiency outcome is exogenous.Thus, the regulated monopolist's expected profit under mandated target regulation is identical 19.Note that as it is possible to have We ignore this possibility as is subject to assumption to the unregulated monopolist's profit except for the penalty that is imposed under low energy efficiency: The threshold value of effort cost for energy efficiency is That is, the range of effort cost for which the monopolist finds it worthwhile to exert positive effort is larger than that for the unregulated monopolist.This is because by exerting positive effort, the monopolist may increase the likelihood that it avoids paying the penalty for not achieving the target.We also note that the following condition ensures that the monopolist's participation constraint is satisfied: The choice of the optimal target will depend on the cost of effort.We note that the sign of will depend on the value of the penalty.A straightforward calculation allows us to state the following lemma: Lemma 5 Assume that (10) holds.Then 19 Likewise, when the cost of effort is smaller than so that positive effort is exerted, it is e ˜4 straightforward to show that .Therefore, the regulator's optimal choice of mandated target is .This is summarized in the next proposition.4.

EXPECTED WELFARE UNDER DIFFERENT REGULATORY REGIMES
This section compares the different regulatory regimes from an expected welfare perspective.The comparison is driven by the size of the cost of effort.In particular, for and in the e у e ˜2 low-penalty case, the monopolist chooses zero effort under all possible scenarios.In this case, price cap regulation always dominates an unregulated monopolist-both set prices ex-ante (that is, prior to the realization of the energy efficiency outcome) and prices are lower under price cap regulation.This particular ranking is of course true for any cost of effort as a regulator could always choose the unregulated monopolist's price if it were to increase welfare.
Moreover, rate of return regulation performs better than price cap regulation when zero effort is optimal.In this case, it is not necessary to incentivize the firm to exert effort in energy efficiency and rate of return regulation then ensures that profits are ex-post zero, whereas under price cap regulation profits are only zero ex-ante so prices are higher to ensure that the firm's participation constraint is satisfied.
At the other extreme, when the cost of effort is sufficiently low (that is, price cap eϽ e ˜), 4 regulation always dominates rate of return regulation as it is socially optimal to set an ex-ante price that, whilst ensuring that the firm's participation constraint is satisfied, might result in positive (expost) profits for the regulated firm.At this price the firm is incentivized to exert effort in energy efficiency.For intermediate values of the cost of effort, the comparison between rate of return and price cap regulation is more complex as characterized in proposition 4 below.
Finally, we have shown that mandated target regulation is always dominated by both price cap and rate of return regulation in terms of expected welfare although it can do better than an unregulated monopolist.The key reason is that mandated target regulation is too coarse and the trade off between providing incentives to invest in energy efficiency and rent extraction is less pronounced.These results are summarized in the next proposition and its proof is in the Appendix.
Proposition 4 Assume that ( 4) and ( 10) hold.The comparison of expected social welfare under different regulatory regimes is summarized in Table 5.
where and

CONCLUSION
This paper develops a theoretical model to investigate the relationship between a regulated firm's incentive to invest in energy efficiency to reduce network losses and the nature of the regulatory regime.In this paper, the reason that the regulated monopolist may not undertake investment in energy efficiency is not due to her desire to maximize quantity but rather due to the inability of a regulator to commit to reimburse the effort costs given that these are not directly observable.
In this light, our result is not simply a corollary to the leading theme in regulatory economics over the last three decades, namely that price regulation introduces incentives for cost reduction which are lacking under rate of return regulation.Instead, we provide another channel through which regulatory regimes can disincentivize regulated firms to invest in energy efficiency.Price cap, rate of return and mandated targets embed different incentives for the regulated firm to pursue energy efficiency at the network end.Rate of return provides no incentive to invest in this type of energy efficiency initiative.Even if the investment in energy efficiency is successful, the ex-post nature of rate of return regulation ensures that the firm earns zero economic profits.Price cap regulation, in contrast, provides incentives for investment in energy efficiency at the network end as the firm is able to capture, in the event that the investment is successful, some of the economic rents.This comparison between price cap and rate of return regulation is similar in spirit to the standard result in regulatory economics that the former induces greater effort to reduce marginal costs than the latter. 20The novelty here is the application of the standard framework to energy efficiency.In addition, the comparison with mandated target regulation, which provides incentives to invest in energy efficiency at the network end by penalizing the firm for not achieving its target, is also novel.
Our analysis suggests a key message that is of relevance to policy makers.That is, policies that are designed to encourage utilities to promote end-user energy efficiency (such as revenue decoupling) may reduce the incentives for utilities to pursue internal energy conservation.In particular, when the cost of effort to undertake energy efficiency investment at the network end is low-that is, when there are existing opportunities that can be pursued at low cost and that are likely to result in energy savings-a price cap regime is likely to perform better than a rate of return regulatory regime.Conversely, when the cost of effort is too high, rate of return regulation is welfare superior to price regulation as in this instance it is not optimal for the firm to invest in energy efficiency.Second, mandated target regulation is clearly an inferior policy to stimulate investment in energy efficiency at the supplier end of the network.The key reason is that mandated target regulation is too coarse an instrument to provide the appropriate incentives to the regulated firm.
1 erwise she chooses zero effort.The threshold value increases when the demand shifts outward e ˜1 (i.e., increases), or it becomes flatter/ less steep (i.e., decreases).The value of increases with a b e ˜1 for sufficiently low values of and decreases with for sufficient high values of .Explicitly,

Proposition 3
The optimal mandated target for energy efficiency is The characterization mt U = U. of mandated target regulation is summarized in Table