Optimal Procurement of Distributed Energy Resources

Transcription

1 Optimal Procurement of Distributed Energy Resources by David P. Brown* and David E. M. Sappington** Abstract We analyze the optimal design of policies to motivate electricity distribution companies to adopt e cient distributed energy resources (DER) and manage associated project costs. The optimal policy often entails a bias against new DER projects and implements cost sharing when DER projects are undertaken in order to foster cost containment while limiting excessive pro t for the utility. Failure to adequately tailor the degree of cost sharing to the prevailing environment can raise procurement costs substantially. The distribution company may optimally be awarded more than the cost saving it achieves. Keywords: JEL Codes: distributed energy resources, procurement, regulation L51, L94 * Department of Economics, University of Alberta, Edmonton, Alberta T6G 2H4 Canada (dpbrown@ualberta.ca). ** Department of Economics, University of Florida, Gainesville, FL USA USA (sapping@u.edu). We thank the editor and three anonymous reviewers for very helpful comments and suggestions. Support from the Government of Canada s Canada First Research Excellence Fund under the Future Energy Systems Research Initiative is gratefully acknowledged.

2 1 Introduction The role of electricity distribution companies has been changing in recent years and continues to evolve. Historically, these companies primarily arranged for the transport of electricity from a central generation site to remote customer locations. Today, electricity often is produced at sites other than the primary generation point, and distributed generation in the form of rooftop solar panels is expanding rapidly. Remote storage of electricity also exhibits ever-increasing potential to complement or substitute for distributed and centralized generation. In addition, improving ability to manage the demand for electricity can reduce the need for increased network transmission and distribution capacity (e.g., Ruester et al., 2014; Jenkins and Perez-Arriaga, 2017). Advances in distributed generation, distributed storage, energy e ciency, and demand management o er the prospect of less costly and potentially more reliable electricity supply. 1 To take full advantage of new opportunities as they emerge, regulators would like to motivate distribution utilities to rst determine which new projects that entail distributed energy resources (DER) are superior to more traditional modes of operation, and then work diligently to integrate the most promising new DER projects into utility operations in a cost-e ective manner. 2 Such motivation can be di cult to provide, though, because regulators typically have limited knowledge of both the potential gains from new DER projects and the utility s ability to manage project costs (e.g., CPUC, 2016a). If regulators do not structure nancial incentives appropriately, utilities may not implement the most promising projects and may not work diligently to manage the costs of projects that are implemented. 1 Biggar (2017, p. 12) notes that as much as 20% of the capacity of a distribution network may only be used for a few hours each year. Consequently, DER projects that curtail or help to satisfy peak demand for electricity have the potential to reduce network capacity costs substantially. A demand-side management program in Rhode Island is estimated to have reduced costs by approximately 24% below the level associated with traditional network investment between 2012 and 2017 (National Grid, 2017). NEEP (2017) describes corresponding projects in several U.S. states, including Connecticut, Maine, New Hampshire, and New Jersey. St. John (2016) and RevConnect (2017) discuss related projects in California and New York. 2 California has adopted a pilot program that requires the state s major utilities to implement DER alternatives to traditional capital investment (California Public Utilities Commission (CPUC), 2016b). Proposed legislation in the Commonwealth of Massachussets (2017) would require utilities to consider non-wire alternatives before undertaking new investments in the electricity distribution grid. 1

3 Regulators have adopted di erent policies to compensate utilities for the DER projects they pursue. Some regulators tend to follow the principles that underlie standard rate of return regulation, sometimes permitting a relatively high return on DER projects to encourage their adoption despite the uncertainties they introduce (e.g., California Public Service Commission, 2016a). Other regulators have implemented policies that e ectively require the utility and its customers to share unanticipated cost savings or cost overruns on DER projects (e.g., Ofgem, 2009; NYPSC, 2015, 2016). The purpose of this research is to analyze the optimal design of regulatory policy in a stylized setting where the distribution utility can choose between a core project and a noncore DER project. 3 The core project can be viewed as standard infrastructure investment that increases the utility s output. 4 The non-core DER project is an alternative, less standard means to achieve the same outcome. 5 The cost of implementing the core project is known to all parties. In contrast, the utility has privileged information about the likely cost of the non-core project and about the e cacy of the utility s e ort to manage project costs. We characterize the regulatory policy that minimizes the expected cost of securing the required increase in output. 6 We focus on determining how the optimal policy varies with the characteristics of the potential DER projects and with the regulator s (limited) knowledge of these projects. We show, for example, that the optimal policy often will entail less cost sharing when the DER project is an internal project (i.e., one designed and managed by the utility) than when it is an external project (i.e., one designed and managed by an una liated third-party entity). 7 We also show that expected procurement cost can increase 3 We focus on a wires-only utility or the wires portion of a vertically integrated utility for expositional ease. The key ndings reported below hold more generally. 4 The output can be viewed as the amount of electricity or the reliability of the electricity supply that the distribution utility delivers to a speci c geographic location during a period of peak demand, for example. 5 For instance, the non-core project might entail the temporary remote storage of electricity produced during an o -peak period and subsequent delivery of the electricity during a period of peak demand. 6 Other regulatory goals are discussed in Section 8. 7 A DER can be owned, installed, and operated by the utility. For instance, in 2014, Arizona Public Service was approved to own, install, and operate 10 MWs of rooftop solar capacity (ACC, 2014). Alternatively, a utility can procure a DER project (e.g., rooftop solar capacity) from an external third-party vendor that is not a liated with the utility. External projects often are encouraged to stimulate the development of 2

4 substantially if the optimal compensation policy for an internal DER project is implemented for an external DER project (or vice versa). In some jurisdictions, regulators o er utilities a menu of optional compensation structures in an attempt to induce the utilities to truthfully reveal their capabilities and their investment needs. 8 We compare optimal procurement policies and industry outcomes when the regulator can, and cannot, o er such a menu of compensation structures. Under some conditions, the restriction to a single compensation structure is not limiting for the regulator. When the restriction is limiting, it often does not greatly increase procurement cost. However, the restriction can increase the utility s pro t dramatically. The substantial increase in pro t arises when the single structure (optimally) awards the utility a large fraction of the cost reduction it secures in order to motivate substantial cost-containment e ort. 9 Even when the regulator can implement a menu of optional compensation structures, she may optimally award the utility more than the full amount of the cost reduction it achieves. 10 The large reward for cost reduction (and corresponding relatively limited compensation when a cost reduction is not achieved) can help to secure low project cost while limiting the utility s incentive to exaggerate the potential cost savings from the non-core project. The utility s incentive to misrepresent these savings also can be ameliorated by introducing a bias toward the core project, i.e., by inducing the utility to implement the core project even when it is more costly than the non-core project. 11 a competitive supply of DER projects and to limit the ability of utilities to exercise any market power that they might possess. Regulators in California and New York require nearly all DER projects to be external (NYPSC, 2014, 2015, 2016; CPUC, 2016b). All DER projects will be external in jurisdictions where distribution utilities are unbundled from the generation and retail sectors (as is the case in many regions of the European Union). 8 See Crouch (2006), Joskow (2008), Ofgem (2009), Cossent and Gomez (2013), and Jenkins and Perez-Arriaga (2017), for example. 9 The substantial pro t implies that the prices the utility charges for its services may substantially exceed cost. Although such pricing distortions introduce ex post allocative ine ciency, the distortions minimize ex ante expected procurement cost by promoting valuable cost-containment activity. 10 Such a reward may be optimal if the utility s cost-containment e ort is particularly e ective at reducing the nal cost of the non-core project when the project s innate cost (i.e., its cost in the absence of costcontainment e ort) is high. 11 In isolation, this bias against the low-cost project tends to increase procurement cost. However, the bias limits the rent the utility secures when it undertakes the non-core project, which serves to reduce 3

5 Our study is not the rst to analyze the design of regulatory policy to promote e cient network improvements, potentially including DER projects. 12 Costa et al. (2017), for example, document a utility s reluctance to undertake investments that do not expand its rate base in the presence of a generous allowed rate of return on the authorized rate base. We abstract from di erent rate-base treatments of investment in order to focus on the di culties created by asymmetric information about likely project costs and the e cacy of the utility s cost-containment e orts. Crouch (2006) analyzes the menus of optional cost-sharing plans that the UK electricity regulator, Ofgem, employs to induce utilities to reveal their investment needs (potentially including DER projects) and to control ensuing costs. Cossent and Gomez (2013) extend Crouch (2006) s analysis of Ofgem s policy and demonstrate how the policy s principles might be employed to improve regulatory policy in the Spanish electricity distribution sector. Jenkins and Perez-Arriaga (2017) augment Cossent and Gomez (2013) s analysis by providing a simulation of a large-scale urban electricity distribution network that regulators can employ to better assess the utility s likely investment needs and production costs. The authors observe (p. 85) that under policies like Ofgem s, the regulator has signi cant exibility and discretion to... balance the fundamental regulatory trade-o s between allocative e ciency (extracting rents from the utility) and productive e ciency (providing incentives for cost savings). We build upon this observation by determining how the regulator optimally employs this exibility and discretion to minimize expected procurement costs when she has limited knowledge of key innate properties of the non-core DER project and the utility s ability to reduce project costs. 13 The basic elements of our analysis have been studied in the information economics literaprocurement cost. On balance, the bias reduces expected procurement cost. 12 Many studies (e.g., Couture and Gagnon, 2010; Yamamoto, 2012; Poullikkas, 2013; Brown and Sappington, 2017) examine the impact of policies like net metering and feed-in tari s on DER investment. Our distinct focus is on how to motivate utilities to integrate and manage DER projects e ciently. 13 The subtleties involved in characterizing optimal regulatory policy compel us to analyze a model that is far more streamlined model than the sophisticated resource planning models of Jenkins and Perez-Arriaga (2014, 2017). 4

6 ture. 14 However, the conclusions drawn in this literature do not seem to be widely recognized in discussions of how to encourage the e cient deployment of DER projects in the electricity sector. One purpose of this research is to clarify the implications of earlier theoretical research for these important discussions. We also contribute to the literature by: (i) determining how the optimal regulatory policy di ers according to whether the DER project is internal or external; (ii) specifying the changes that arise in the optimal policy as key elements of the prevailing environment change; (iii) identifying conditions under which the regulator is not hindered by employing a single compensation structure rather than a menu of optional compensation structures; and (iv) demonstrating the potential merit of awarding the utility even more than the cost reduction it secures in certain settings. We also complement earlier studies by exploring the losses that can arise when the regulator s policy instruments are limited and when the regulator implements in one setting (e.g., where the non-core project is an internal project) a policy that is optimal in a di erent setting (e.g., where the non-core project is an external project). The analysis proceeds as follows. Section 2 describes the key elements of the model. Section 3 identi es the optimal procurement policy in benchmark settings. Section 4 characterizes the optimal policy in the setting of primary interest when the regulator can a ord the utility a choice among compensation structures. Section 5 provides the corresponding characterization when such a choice is not feasible. Section 6 examines the potential gains from introducing distinct compensation structures for internal and external projects. Section 7 considers an extension of the basic model in which it can be optimal to award the utility more than the cost reduction it secures. Section 8 reviews our key ndings and discusses additional extensions of our analysis Related models that include both ex ante information asymmetry and ex post unobserved cost-containment e ort date back at least to Sappington (1982). La ont and Tirole (1993) and La ont and Martimort (2002), among others, provide important additional modeling e orts and comprehensive reviews of related literature. In light of these studies and others (e.g., Joskow, 2014; Jenkins and Perez-Arriaga, 2017), the basic trade-o s and the initial ndings reported below (Conclusions 1 and 2 in particular) are well known. 15 Brown and Sappington (2018) provides the proofs of all formal conclusions in the text. 5

7 2 The Model We consider a setting where the utility can secure a speci ed level of output by implementing either a traditional ( core ) project or a non-traditional ( non-core ) project. Output might constitute an increase in the amount and/or the reliability of electricity delivered to a speci c geographic location during a speci ed time period, for instance. The core project might entail expanding the utility s distribution facilities to relieve local network congestion. The non-core project might entail storing electricity generated in one (o -peak) period and delivering the electricity in a di erent (peak) period or initiating programs to curtail the peak demand for electricity. Because the core project entails relatively familiar activities, the cost of the project (c 0 ) is assumed to be known and deterministic. 16 In contrast, the nal cost of the non-core project is not known with certainty at the time it is undertaken. For simplicity, the nal cost of the non-core project is assumed to be either low (c) or high (c), where c < c 0 < c. The probability that the realized cost is low, p(e; ) 2 (0; 1), varies with the innate project potential, 2 f 1 ; 2 g, and with the utility s cost-containment e ort, e 0. For any given level of e 0, the low cost realization (c) is more likely under the non-core project when it has high potential ( 2 ) than when it has low potential ( 1 ). Formally, p(e; 2 ) > p(e; 1 ) for all e 0. Furthermore, unless otherwise noted, the marginal impact of cost-containment e ort is assumed to be more pronounced under the high-potential non-core project than under its low-potential counterpart, i.e., p e (e; 2 ) p e (e; 1 ) > 0 for all e In addition, diminishing returns to cost-containment e ort are less pronounced when = 2 than when = Thus, innate expected cost (i.e., expected cost in the absence of any cost-containment e ort) is lower and the potential to contain costs is more pronounced under the non-core project when = 2 than when = This cost includes the expenses associated with installing, maintaining, and operating e ciently relevant additions to the utility s infrastructure. 17 Here and throughout the ensuing analysis, the subscript e denotes the partial derivative with respect to e. Section 7 discusses the changes that can arise when p e (e; 2 ) < p e (e; 1 ) for all e Formally, p ee (e; i ) 0 for i = 1; 2 and p ee (e; 2 ) p ee (e; 1 ) for all e 0, where the subscript ee denotes the second partial derivative with respect to e. 6

8 The regulator and the utility both know how e and a ect the likelihood of securing the low cost realization under the non-core project (i.e., the functional form of p(e; ) is common knowledge). However, only the utility observes the realization of. The regulator knows only that = i with probability i 2 (0; 1) for i = 1; 2, where = 1. D(e) is the personal cost the utility incurs when it delivers cost-containment e ort e. D() includes the opportunity cost the utility s executives and managers incur when they devote their scarce time and energy to increasing the likelihood of the low cost realization under the non-core project. D() is an increasing, convex function of e. 19 The regulator and the utility both know the functional form of D(). However, the regulator cannot observe the level of e ort the utility supplies or the corresponding e ort cost the utility incurs. Consequently, the regulator cannot elicit cost-containment e ort by reimbursing the utility directly for the e ort it supplies. Instead, the regulator must motivate the utility to deliver e by promising the utility a higher level of pro t when c is realized than when c is realized under the non-core project. The utility s pro t is the di erence between: (i) the payment (r) it receives from the regulator; and (ii) the sum of the realized production cost and any relevant e ort cost. r 0 denotes the payment the regulator delivers to the utility when it implements the core project. Corresponding payments when the utility implements the non-core project can vary with the realized project cost. These payments also may vary with the particular compensation structure the utility selects. We initially consider a setting where the regulator can design two distinct compensation structures and assign a structure to the utility based on its report of the non-core project s potential. Formally, when the utility reports = j, it is instructed to undertake the non-core project with probability j 2 f0; 1g and the core project with probability 1 j. 20 If it undertakes the non-core project after reporting = j, the utility 19 Formally, D(0) = 0, D 0 (e) 0, and D 00 (e) > 0 for all e 0, where the rst inequality holds strictly for all e > If 1 = 2 = 1 (0), the regulator e ectively requires the utility to implement the non-core project (the core project). If 1 6= 2, the utility e ectively chooses the project it will undertake by choosing its report of. To illustrate, when 1 = 0 and 2 = 1, the utility chooses to implement the core project if it reports 7

9 is paid r j if the realized project cost is c, and it is paid r j if the realized cost is c. Therefore, the utility s expected pro t when = i and the utility reports = j is: j ( i ) = j n j ( i ) + 1 j [ r0 c 0 ], where (1) n j ( i ) max e 0 p(e; i ) r j c + [ 1 p(e; i ) ] [ r j c ] D(e). (2) Table 1 summarizes the key notation in the analysis. 21 [ Table 1 Here ] The regulator seeks to minimize the expected cost of procuring the requisite level of output. The regulator s formal problem in this setting, [RP], is: 2X Minimize i i f p(e i ; i ) r i + [ 1 p(e i ; i ) ] r i g + [ ] r 0 (3) r 0;, r i, r i, i 2 f0;1g i = 1 subject to, for i; j 2 f1; 2g (j 6= i): where e i arg max e 0 i ( i ) 0 and i ( i ) j ( i ), (4) f p(e; i ) [ r i c ] + [ 1 p(e; i ) ] [ r i c ] D(e) g. (5) Expression (3) re ects the regulator s objective to minimize expected procurement cost. The rst inequality in expression (4) ensures that the utility always secures at least its reservation level of expected pro t, which is normalized to zero. The second inequality in expression (4) ensures that the utility will report the realization of truthfully. 22 The interaction between the regulator and the utility proceeds as follows. First, the utility learns the realization of. Second, the regulator speci es the procurement policy, which consists of payments to the utility (r) and implementation probabilities ( ) that can vary with the utility s report of. Third, the utility reports and thereby e ectively chooses the project it will undertake and implements the associated project. Fourth, the = 1 whereas it chooses to implement the non-core project if it reports = The variables E c i and e i that appear in Table 1 are explained in more detail in Section The Revelation Principle (e.g., Myerson, 1979) ensures this formulation is without loss of generality. The value of r 0 could conceivably vary with the utility s report of. However, it can be shown that such variation is never strictly valuable for the regulator in the present setting where c 0 is common knowledge. 8

10 utility chooses its preferred level of cost-containment e ort. 23 Fifth, the project cost is realized and the regulator delivers the promised payment to the utility. 3 Benchmark Settings Before proceeding to characterize the solution to [RP], we consider the outcomes that would arise in four hypothetical benchmark settings. To do so, it is helpful to introduce the following de nitions: e i arg min f p(e; i ) c + [ 1 e 0 p(e; i ) ] c + D(e) g for i = 1; 2, and E c i p(e i ; i ) c + [ 1 p(e i ; i ) ] c + D(e i ). (6) In words, e i is the cost-minimizing ( e cient ) e ort supply under the non-core project when = i, and E c i is the corresponding ( e cient ) expected full cost (including e ort cost) of the project. We assume E c 2 < c 0, so the high-potential non-core project always o ers the prospect of lower expected cost than the core project. 24 In the hypothetical full-information setting, the regulator shares the utility s knowledge of from the outset of their relationship and can verify the level of cost-containment e ort the utility supplies under the non-core project. In the presence of such symmetric information, the regulator will implement the full-information outcome, under which: (i) the utility undertakes the non-core project when = i if and only if E c i c 0 ; (ii) the utility delivers the e cient level of cost-containment e ort when it implements the non-core project; and (iii) r 0 = c 0 and n 1( 1 ) = n 2( 2 ) = 0, so the utility secures no rent. 25 In the setting with known project potential, the regulator and the utility both observe the realization of at the outset of their interaction, but the regulator never observes the utility s supply of cost- 23 This e ort is normalized to 0 under the core project. Alternatively, this e ort can be taken to be e 0 > 0 and cost c 0 can be viewed as the sum of physical production cost and the utility s e ort cost D(e 0 ). 24 If this were not the case, the regulator would minimize expected procurement cost by setting r 0 = c 0 and requiring the utility to always undertake the core project (by setting 1 = 2 = 0). 25 E ort veri cation implies that the regulator can document conclusively the level of e ort the utility has supplied, and so can write a legally enforceable contract that links the utility s payment (r) to its e ort supply (e). 9

11 containment e ort. In the setting with veri able e ort, the regulator can verify the utility s supply of cost-containment e ort, but only the utility observes the realization of. In the setting with ex ante contracting, the regulator can specify the regulatory policy before the utility observes the realization of. The regulator never observes the realization of or the utility s e ort supply in this setting. 26 Lemma 1. The regulator can secure the full-information outcome in: (i) the setting with known project potential; (ii) the setting with veri able e ort; and (iii) the setting with ex ante contracting. Lemma 1 re ects the following considerations. 27 When the regulator observes the realization of i in the setting with known project potential, she can instruct the utility to undertake the project that entails the lowest e cient expected full cost and set payments r 0 = c 0 and r i = r i = E c i for i = 1; 2. The xed payment E c i induces the e cient e ort supply and eliminates the utility s rent under the non-core project when = i. When the regulator can verify the utility s e ort supply in the setting with veri able e ort, she can promise to reimburse the utility s observed full cost of production (including e ort costs). This policy always leaves the utility with zero pro t, so it is willing to undertake the project with the lowest e cient expected full cost and to deliver the e cient e ort supply. 28 When the regulator can commit to a policy when she shares the utility s imperfect knowledge of the prevailing environment (in the setting with ex ante contracting), she can o er the utility a xed payment equal to its e cient expected full cost. This payment eliminates the utility s expected rent and induces the utility to implement the project with the lowest e cient ex- 26 We assume that in all settings, if the utility is indi erent among multiple activities, it will undertake the activity preferred by the regulator. 27 As the ensuing discussion makes apparent, Lemma 1 holds regardless of the number of distinct realizations that can assume. 28 If, contrary to assumption, the utility might pursue the more costly project when it anticipates the same pro t under both projects, the regulator can deter such undesirable behavior at negligible cost if there is even the slightest chance that she can verify the actual realization of. The regulator can do so by threatening to penalize the utility (even slightly) if its observed behavior is ever determined not to minimize expected full cost. 10

12 pected full cost and to deliver the e cient e ort supply whenever it implements the non-core project. In practice, regulators attempt to limit relevant information asymmetries in part by requiring utilities to provide detailed information about their network con guration and location-speci c capacities, 29 thereby enabling engineering models and optimization software to simulate the bene ts and costs of potential DER projects (MIT Energy Initiative, 2016; Jenkins and Perez-Arriaga, 2017). Lemma 1 implies that such e orts can be valuable. In practice, though, some information asymmetry will persist (Joskow, 2014) and it will generally be impossible to measure accurately the e ort a utility exerts to control the costs of non-core projects. Therefore, it is important to characterize the properties of optimal procurement policies when these two frictions prevail simultaneously. 4 Findings When Optional Cost-Sharing Plans are Feasible. Conclusion 1 reports that the regulator can sometimes secure the full-information outcome even when she cannot observe the potential of the non-core project () or verify the utility s cost-containment e ort (e). This will be the case when the cost of the core project is less than the e cient expected full cost of the low-potential non-core project (so c 0 < E c 1). In this case, the regulator can induce the utility to always implement the least-cost project by delivering to the utility: (i) the xed payment c 0 if it undertakes the core project; and (ii) the smaller xed payment, E c 2, if it undertakes the non-core project. These xed payments ensure that the non-core project is pro table for the utility if and only if = 2, and so induce the utility to always undertake the least-cost project while eliminating the utility s rent. Furthermore, because the xed payment E c 2 does not vary with realized cost, the utility is promised the full bene t of its cost-containment e ort. Consequently, the utility will deliver the e ort (e 2) that minimizes expected full cost when it undertakes the non-core project. 29 This is the case, for example, in California (CPUC, 2013), Hawaii (Hawaii PUC, 2014), and New York (NYPSC, 2014, 2015, 2016). 11

13 Conclusion 1. The regulator can secure the full-information outcome if only the highpotential non-core project is less costly than the core project. (Formally, 1 = 0, 2 = 1; r 0 = c 0, and r 2 = r 2 = E c 2 at the solution to [RP] if E c 2 < c 0 < E c 1.) The regulator s ability to secure the full-information outcome in the setting of Conclusion 1 is an artifact of the simplifying assumption that is binary. When can assume more than two values and when E c i < c 0 for at least two i realizations, the regulator will face a constraining trade-o. Speci cally, to secure the full-information outcome when E c 2 < E c 1 < c 0, the regulator would have to induce the utility to always undertake the non-core project (so 1 = 2 = 1) and promise the utility a xed payment (to induce the e cient level of cost-containment e ort) that eliminates its rent. However, the requisite xed payment for the low-potential project (E c 1) exceeds the corresponding payment for the high-potential project (E c 2), so the utility would always select the former payment and thereby secure rent when = 2. To understand the characteristics of the optimal procurement policy when the regulator cannot secure the rst-best outcome, it is important to consider n 1( 2 ) n 1( 1 ), the incremental pro t that accrues to the utility under payment structure (r 1 ; r 1 ) when is 2 rather than 1. Lemma 2 reports that this incremental pro t is positive and increases as 1 r 1 c (r 1 c), the utility s incremental payo when c is realized rather than c, increases. The incremental pro t is positive in part because the low cost (c) is more likely when = 2 than when = 1 for any given level of e. The incremental pro t increases with 1 because as 1 increases, the utility delivers more cost-reducing e ort, which is more e ective at reducing cost when = 2 than when = 1. Lemma 2. The utility secures more pro t under the (r 1 ; r 1 ) payment structure when = 2 than when = 1. Furthermore, the increment in pro t increases as 1, the incremental payo for realizing the low cost, increases. (Formally, n 1( 2 ) > n 1( 1 ) and 12

14 1 f n 1( 2 ) n 1( 1 ) g = p(e 12 ; 2 ) p(e 1 ; 1 ) > 0 when 1 > 0.) 30 Lemma 2 identi es the critical trade-o the regulator faces when the non-core project is always less costly than the core project. Reducing cost sharing, i.e., reducing the extent to which the utility shares the gain from a cost reduction with its customers by increasing 1 toward c c, increases the utility s cost-containment e ort toward its e cient level (e 1) and thereby reduces expected procurement cost when = 1. However, the diminished cost sharing increases the pro t the utility can secure under the (r 1 ; r 1 ) payment structure when the non-core project has high potential (i.e., when = 2 ), and thereby increases the payments (r 2 and/or r 2 ) the regulator must promise to the utility to induce it to select the (r 2 ; r 2 ) payment structure under the non-core project when = 2. Conclusion 2 reports that the regulator optimally resolves this trade-o by implementing some cost sharing when = 1, i.e., by setting r 1 below r 1, so 1 < c c. Doing so increases procurement cost when = 1 by reducing e 1 below e 1, but reduces overall expected procurement cost by reducing the rent that must be a orded the utility when = 2. Conclusion 2 also reports that the (r 2 ; r 2 ) payment structure will be the smallest xed payment (r 2 = r 2 ) required to convince the utility not to adopt the (r 1 ; r 1 ) cost-sharing payment structure when = 2 (so n 2( 2 ) = n 1( 2 ) > 0). The xed payment induces the utility to deliver the e cient e ort supply (e 2) when = 2. Conclusion 2. If the utility always implements the non-core project, the (r 1 ; r 1 ) payment structure will be a cost-sharing plan that provides zero expected pro t when = 1. The other payment structure will be a xed payment ( r 2 = r 2 ) equal to the pro t the utility could secure under the (r 1 ; r 1 ) payment structure when = 2. (Formally, if 1 = 2 = 1, then r 1 > r 1 (so 1 2 (0; c c)); n 1( 1 ) = 0, r 2 = r 2, and n 2( 2 ) = n 1( 2 ) > 0 at the solution to [RP].) 30 e 12 arg max f p(e; 2 ) [ r 1 c ] + [ 1 p(e; 2 ) ] [ r 1 c ] D(e) g is the level of e that maximizes the e 0 utility s expected pro t under the (r 1 ; r 1 ) payment structure when = 2. 13

15 Lemma 3 speci es the conditions under which the regulator will always induce the utility to undertake the non-core project. Lemma 3. The regulator will always induce the utility to undertake the non-core project when = 2. She will also induce the utility to undertake the non-core project when = 1 if the associated potential cost saving from implementing the project is su ciently pronounced. (Formally, at the solution to [RP]: (i) 2 = 1; and (ii) 1 = 1 if E c 1 < 1 c E c 2.) The regulator may induce the utility to undertake the core project when = 1 even when c 0 is considerably higher than Ec 1. Doing so can reduce the rent that the regulator must cede to the utility when = 2. (See Part A of the Appendix.) When the regulator always induces the utility to undertake the non-core project, the incremental payo she awards the utility for realizing cost c when = 1 (i.e., the value of 1 ) varies with the prevailing environment. Conclusion 3 reports that the regulator will increase 1 (i.e., reduce the extent of cost sharing) as she becomes more certain that = The reduced cost sharing increases the utility s cost-containment e ort toward its e cient level and thereby reduces expected procurement cost when = 1. In contrast, the reduced cost sharing increases procurement cost when = 2 by increasing the xed payment (r 2 = r 2 ) that must be awarded the utility to induce it not to select the (r 1 ; r 1 ) payment structure. However, the expected loss from the increased procurement cost when = 2 declines as 2 declines. Conclusion 3. The regulator optimally increases the incremental reward for successful cost reduction (i.e., reduces the extent of cost sharing) under the (r 1 ; r 1 ) payment structure as she becomes more certain that the non-core project has low potential. (Formally, d 1 d 1 > 0 at the solution to [RP] when 1 = 1.) 31 Conclusion 3, like Conclusions 4 7 below, holds when j D 000 (e) j and j p eee (e; i ) j are su ciently small for all e 0, for i = 1; 2. These limits on the magnitudes of relevant third derivatives help to ensure that the regulator s objective in [RP] is a concave function of 1. These limits constitute su cient conditions for the Conclusions to hold, but generally are not necessary conditions (as the proofs of the Conclusions reveal). 14

16 Conclusion 4 identi es conditions under which the regulator will increase the extent of cost sharing under the (r 1 ; r 1 ) payment structure (i.e., reduce 1 ) as 2 increases and 1 declines, so the prevailing information asymmetry ( 2 1 ) becomes more pronounced. Conclusion 4. Suppose: (i) the non-core project is su ciently likely to have high potential; or (ii) diminishing returns to the utility s cost-containment e ort are su ciently limited under the low-potential non-core project. Then the regulator optimally increases the extent of cost sharing under the (r 1 ; r 1 ) payment structure as the prevailing information asymmetry ( 2 1 ) becomes more pronounced. (Formally, when 1 = 1 at the solution to [RP]: (i) d 1 d 2 < 0; and (ii) d 1 d 1 > 0 so su ciently small for all e 0.) d 1 d ( 2 1 ) < 0 if 2 is su ciently large or jp ee (e; 1 )j is An increase in 2 (which increases 2 1 ) increases the rate at which n 1( 2 ) n 1( 1 ), the utility s incremental pro t under the (r 1 ; r 1 ) payment structure when is 2 rather than 1, increases with 1. Therefore, as 2 increases, the regulator reduces 1 in order to reduce expected procurement cost by reducing the utility s rent when the non-core project has high potential (i.e., when = 2 ). 32 An increase in 1 (which reduces 2 1 ) reduces the rate at which an increase in 1 increases n 1( 2 ) n 1( 1 ). The corresponding reduced concern with the utility s rent when = 2 alone would lead the regulator to increase 1. If the increase in 1 also increases the rate at which an increase in 1 increases e 1 toward e 1, then the regulator will increase 1. However, an increase in 1 can reduce the rate at which an increase in 1 increases e 1 if jp ee (e; 1 )j is large. If this e ect is su ciently pronounced and if 1 is su ciently likely, the regulator could conceivably reduce 1 as 1 increases. Conclusion 4 identi es conditions under which these latter considerations are outweighed by the regulator s reduced concern with the incremental rent that an increase in 1 generates as 1 increases. 32 Other authors have noted in related but distinct settings the gains from increased cost sharing as the information asymmetry between the regulator and the utility becomes more pronounced. See Schmalensee (1989) and Jenkins and Perez-Arriaga (2017), for example. 15

17 Conclusion 5 reports that the regulator will also increase the extent of cost sharing under the (r 1 ; r 1 ) payment structure (i.e., reduce 1 ) as the maximum potential cost reduction under the non-core project ( c c) increases. An increase in c c increases the nancial bene t of a cost reduction and thereby increases the rent the utility derives from the (r 1 ; r 1 ) payment structure when the non-core project has high potential (i.e., when = 2 ). The regulator reduces this rent and the corresponding procurement cost by reducing r 1 r 1, so 1 optimally increases by less than c c increases. Conclusion 5. The regulator optimally increases the extent of cost sharing under the (r 1 ; r 1 ) payment structure (i.e., reduces 1 ) as the magnitude of the potential cost reduction under the non-core project increases. (Formally, the solution to [RP].) d( r 1 r 1 ) d( c c ) < 0 and d 1 d( c c ) < 1 when 1 = 1 at To understand how the optimal procurement policy changes as it becomes more onerous for the utility to deliver cost-containment e ort, it is helpful to consider the hypothetical setting where the utility s e ort cost, D(e), can vary with. Speci cally, suppose D(e; i ) is the utility s cost of delivering e ort e when = i. i > 0 D(e; i > 0, so an increase in i corresponds to an increase in the total and marginal cost of delivering e when = i, which induces the utility to reduce e i, ceteris paribus. Conclusion 6. When the non-core project is su ciently likely to have low potential, the regulator optimally reduces the extent of cost sharing under the (r 1 ; r 1 ) payment structure (i.e., increases 1 ) as cost-containment e ort becomes more onerous for the utility to deliver. (Formally, when 1 = 1 at the solution to [RP]: (i) d 1 d 2 > 0 ; and (ii) d 1 d 1 su ciently large, whereas d 1 d 1 < 0 when 2 is su ciently large. Therefore, d 1 d 1 is su ciently large, where = 1 = 2.) > 0 when 1 is > 0 when Conclusion 6 reports that as cost-containment e ort becomes more onerous for the utility to supply when = 2, the utility secures less rent from the (r 1 ; r 1 ) cost-sharing contract 16

18 when = 2. The regulator s corresponding reduced concern with the utility s rent when = 2 leads her to reduce the extent of cost sharing (i.e., increase 1 toward c c) in order to reduce expected procurement cost when = 1 by inducing the utility to increase e 1 toward e 1. As cost-containment e ort becomes more onerous for the utility to supply when = 1 (i.e., as 1 increases), the utility secures more rent from the (r 1 ; r 1 ) cost-sharing contract when = 2. The regulator s corresponding increased concern with the utility s rent when = 2 leads her to reduce 1, particularly when she believes the non-core project is likely to have high potential (i.e., when 2 is relatively large). However, an increase in 1 leads the utility to reduce e 1. To ensure e 1 does not decline unduly, the regulator may increase 1, particularly when she believes the non-core project is likely to have low potential (i.e., when 1 is relatively large). Together, these considerations imply that as cost-containment e ort becomes systematically more onerous for the utility to supply, the regulator often will increase 1, particularly when 1 is large. The cost sharing that the regulator optimally implements when = 1 also varies with the e cacy of the utility s cost-containment e ort. Conclusion 7 characterizes the relationship between 1 and e ort e cacy () in the special case where does not vary with e or. Conclusion 7. The regulator optimally reduces the extent of cost sharing under the (r 1 ; r 1 ) payment structure (i.e., increases 1 ) as the utility s cost-containment e ort becomes more e ective at securing a cost reduction under the non-core project. (Formally, d 1 d 1 = 1 at the solution to [RP] if p e (e; 1 ) = p e (e; 2 ) = > 0 for all e 0.) > 0 when Conclusion 7 re ects the fact that as the utility s cost-containment e ort becomes more e ective at securing cost c under the non-core project, the regulator will induce more e ort by reducing the extent of cost sharing (i.e., by increasing 1 ). 17

19 5 Findings When Optional Cost-Sharing Plans are Not Feasible. In practice, it is relatively uncommon for regulators to o er utilities an explicit choice among payment structures. 33 Consequently, it is important to determine how the optimal procurement policy changes when the regulator can only specify a single payment structure for the utility when it implements the non-core policy. Let [RP1] denote the regulator s problem in this setting. 34 Also in this setting, let: (i) r and r, respectively, denote the payments delivered to the utility when costs c and c are realized under the non-core project; (ii) n ( i ) max e 0 f p(e; i) [ r c ] + [ 1 p(e; i ) ] [ r c ] D(e) g denote the utility s expected pro t under the non-core project when = i ; and (iii) r c (r c) denote the incremental pro t the utility receives when cost c, rather than cost c, is realized under the non-core project. Conclusion 1 identi es the optimal procurement policy in this setting when the e cient expected full cost of the low-potential non-core project, E c 1, is su ciently large that the regulator only induces the utility to undertake the non-core project when the project has high potential ( 2 ). In this case, the regulator o ers the utility a choice between the core project with payment r 0 = c 0 and the non-core project with xed payment E c 2. This xed payment induces the utility to supply the e cient level of cost-containment e ort (e 2) and eliminates the utility s rent when = 2. Because E c 2 < E c 1, the E c 2 xed payment renders the low-potential non-core project unpro table for the utility, so the utility implements the core project when = 1. Now consider the more interesting setting where E c 1 is su ciently far below c 0 that the regulator always induces the utility to implement the non-core project. 35 Conclusion 8 reports that the regulator optimally achieves this outcome with a cost sharing plan ( 2 33 The U.S. Federal Communications Commission a orded suppliers of telecommunications services a choice among payment structures in the 1990s (Sappington and Weisman, 1996, pp ). Ofgem has been o ering electricity distribution companies a choice among compensation arrangements for capital expenditures since 2005 (e.g., Crouch, 2006; Joskow, 2014). 34 [RP1] is [RP] with the additional restriction that r 1 = r 2 and r 1 = r The remainder of the discussion in this section considers the setting where the regulator always induces the utility to implement the non-core project. 18

20 (0; c c)) that eliminates the utility s rent if and only if = 1. Conclusion 8. When the cost of the core project su ciently exceeds the e cient expected full cost of the non-core project with low potential, the regulator will induce the utility to always implement the non-core project. She will do so with a cost-sharing contract that provides the utility with zero expected pro t when the non-core project has low potential. (Formally, if c 0 E c 1 is su ciently large, the regulator will induce the utility to always implement the non-core project at the solution to [RP1] with a cost-sharing contract in which r > r (so 2 (0; c c) ), where n ( 2 ) > n ( 1 ) = 0.) The cost sharing identi ed in Conclusion 8 reduces expected procurement cost relative to the xed payment r = r = E c 1. This is the case because for a given level of costcontainment e ort, the low cost realization c is more likely to arise when the non-core project has high potential ( 2 ) than when it has low potential ( 1 ). Consequently, by reducing r below r, the regulator e ectively secures for consumers a portion of the reduction in expected cost that arises when the non-core project has high potential ( 2 ) rather than low potential ( 1 ). At least for small levels of cost sharing, the corresponding reduction in expected procurement cost when = 2 outweighs the increase in expected procurement cost that arises when the cost sharing induces the utility to reduce e i below e i for i = 1; Conclusion 9 reports that the regulator optimally implements less cost sharing (i.e., increases ) when she can only o er a single compensation structure under the non-core project. This is the case because cost sharing in this instance reduces the utility s costcontainment e ort below its e cient level under the non-core project both when it has low potential and when it has high potential. In contrast, when the regulator can o er the utility a choice between compensation structures under the non-core project, the cost sharing that is implemented under the (r 1 ; r 1 ) payment structure does not reduce the utility s cost- 36 Observe that d d f p(e 1; 1 ) c + [ 1 p(e 1 ; 1 ) ] c + D(e) g = c c = [ p e (e 1; 1 ) ( c c ) + D 0 (e 1) ] de 1 0. d = 19

21 containment e ort below its e cient level when the non-core project has high potential. 37 Conclusion 9 refers to and 1, which are the values of r c (r c) at the solution to [RP1] and 1 r 1 c (r 1 c) at the solution to [RP], respectively. Conclusion 9. When the regulator always induces the utility to undertake the non-core project, she will implement less cost sharing under the non-core project with low potential when she cannot o er multiple optional compensation structures. (Formally, > 1.) The magnitude of 1 can be viewed as the reduction in the extent of cost sharing that is optimally implemented when = 1 if the regulator can only o er a single compensation structure under the non-core project. The magnitude of the reduced cost sharing can be substantial. To demonstrate this fact, consider the following baseline setting. Baseline Setting: e p(e; ) = +, D(e) = 1 + e 2 e2, c 0 = 100; 000, c = 60; 000, c = 120; 000, 1 = 0:25, 2 = 0:75, = 150, = = 1 = 0:5. The baseline setting adopts simple, tractable functional forms for the p(e; ) and D(e) functions. The separability of e and in the p() function introduces the simplifying assumption that the e cient level of cost-containment e ort does not vary with the potential of the non-core project. 38 The values of c and c imply that although the non-core project could result in costs that are 20% above their level under the core project (c 0 ), the non-core project could also reduce costs by twice this amount (i.e., by 40% below c 0 ). The e cient expected full cost in the baseline setting is: (i) 89; 366 (approximately 90% of the cost of the core project, c 0 ) under the low-potential non-core project; and (ii) 74; 366 (approximately 75% of c 0 ) under the high-potential non-core project. Table 2 reports that is approximately 60% larger than 1 in the baseline setting Recall from Conclusion 2 that the utility will choose the xed payment r 2 = r 2 when = 2 under the optimal procurement policy when the regulator can o er two distinct compensation structures under the non-core project. Consequently, the utility will always set e 2 = e 2, regardless of the magnitude of e i is de ned by p e(e i ; ) [ c c ] = D0 (e i ), [ c c ] [ 1 + e i ] 2 = e i, e i [ 1 + e i ]2 = 200, e i 5:2. 39 In the baseline setting: (i) r 1 = 66; 266 and r 1 = 115; 055 at the solution to [RP]; and (ii) r = 69; 694 and 20

22 Thus, the regulator implements substantially less cost sharing when she can only o er a single compensation structure under the non-core project. The reduction in cost sharing a ords the utility more than a 60% increase in expected pro t (Efg) under the non-core project, which can lead to a substantial reduction in allocative e ciency. However, the reduced cost sharing induces the utility to increase its cost-containment e ort substantially, 40 so expected procurement cost (Efrg) increases by only 1:1% in this setting. 41 [ Table 2 Here ] The key forces that in uence the optimal magnitude of 1 when optional cost-sharing plans are feasible continue to in uence the optimal magnitude of when optional plans are not feasible. One additional e ect also arises in this latter setting. Now, in addition to in uencing the utility s cost-containment e ort when = 1 and the utility s rent when = 2, the value of a ects the utility s cost containment e ort when = 2. This additional role for could conceivably alter some of the qualitative conclusions reported in Conclusions 3 7. However, it generally does not do so under plausible conditions. To illustrate, observe that as 2 becomes more likely (so 2 increases), the regulator might conceivably increase to induce the utility to increase e 2 toward e 2 when = 2. However, the regulator often will reduce instead in order to reduce the utility s rent when = 2, just as in the setting of Conclusion 3. This will be the case, for instance, when the e cacy of the utility s cost-containment e ort under the non-core project is not much larger when = 2 than when = 1. Conclusion 12 in the Appendix provides the formal details. The Appendix also reports that the direct counterparts to Conclusions 5 7 r = 111; 643 at the solution to [RP1]. 40 In the baseline setting, e 1 = 2:71 at the solution to [RP] whereas e 1 is 21:4% higher (e 1 = 3:29) at the solution to [RP1]. 41 The qualitative conclusions re ected in Table 2 are quite robust to variation in model parameters. (See Brown and Sappington (2018) for details.) The nding that a contract designer (here, the regulator) does not experience a major reduction in expected welfare when she is limited to o ering a single compensation structure in the presence of adverse selection (here, incomplete information about ) is consistent with the ndings of Reichelstein (1992), Bower (1993), and Rogerson (2003). Chu and Sappington (2007) identify a setting in which the expected welfare loss can be pronounced, depending on the distribution of the relevant environmental parameter (here, ). 21