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On the optimal design of demand response policies

Abstract

We characterize the optimal regulatory policy to promote efficient demand response (DR) in the electricity sector. DR arises when consumers reduce their purchases of electricity below historic levels at times when the utility’s marginal cost of supplying electricity is relatively high. The US Federal Energy Regulatory Commission (FERC) advocates compensation for DR that reflects the utility’s marginal cost. We show that the optimal policy often provides less generous compensation, and demonstrate that implementation of the FERC’s policy can reduce welfare well below the level secured by the optimal DR policy.

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Notes

  1. 1.

    §1252(f) of the Energy Policy Act of 2005 (Pub. L. No. 109-58, 119 STAT. 966 (2005)) states that “It is the policy of the United States that time-based pricing and other forms of demand response, whereby electricity customers are provided with electricity price signals and the ability to benefit by responding to them, shall be encouraged.”

  2. 2.

    The U.S. Department of Energy (2006) defines DR to encompass “Changes in electric usage by end-use customers from their normal consumption patterns in response to changes in the price of electricity over time, or to incentive payments designed to induce lower electricity use at times of high wholesale market prices or when system reliability is jeopardized.”

  3. 3.

    Order 745 states that a “demand response resource must be compensated for the service it provides to the energy market at the market price for energy, referred to as the locational marginal price (LMP)” (Federal Energy Regulatory Commission 2011, ¶2).

  4. 4.

    The FERC’s authority to implement this compensation policy also has been challenged. The US Court of Appeals for the District of Columbia (2014) vacated FERC Order 745 in May 2014. However, in January 2016, the Supreme Court overturned the decision of the Appeals Court, thereby reinstating Order 745 (US Supreme Court 2016).

  5. 5.

    Bushnell et al. (2009), Borlick (2010), and Borlick et al. (2012), among others, offer corresponding conclusions.

  6. 6.

    Chen et al. (2010) and Li et al. (2011) document the optimality of setting the price of electricity equal to its instantaneous marginal cost of production and propose an iterative algorithm to achieve the optimal outcome in the presence of limited information.

  7. 7.

    Formally, \(C^{\prime }(X)>0 \) and \( C^{\prime \prime }(X)\ge 0 \) for all \(X>0.\)

  8. 8.

    In practice, a utility’s production costs may increase discontinuously at output levels where less efficient auxiliary generating units are brought on line. We assume \(C(\cdot )\) is continuously differentiable for analytic tractability. This assumption does not alter our primary qualitative conclusions. Our model also can be viewed as one in which the utility is a distribution company that purchases electricity from competitive suppliers at increasing marginal cost.

  9. 9.

    A consumer’s choice of on-site production technology might be affected by such factors as his status as a commercial or residential customer, his projected consumption of electricity, the characteristics of his commercial/residential property (including the available space or the rooftop slope and exposure to the sun), and local zoning ordinances, for example. These considerations and others may lead some consumers to refrain from any investment in on-site production capabilities. For expositional ease, we abstract from the possibility that a consumer might invest in multiple distinct production technologies.

  10. 10.

    Each consumer is assumed to consume all of the electricity he generates on-site, thereby abstracting from the possibility that a consumer might supply electricity to other consumers or sell electricity to the regulated utility.

  11. 11.

    DNV GL (2014) reports that solar capacity represents the major component of distributed generation (DG) capacity in eight of the ten US states with the most DG capacity. CHP units powered by natural gas account for the majority of DG capacity in Connecticut and New York.

  12. 12.

    Section 4.1 considers the setting where the regulator is not permitted to set a fixed charge (R), perhaps because of concerns about the financial burden that a substantial fixed charge can impose on individuals with limited wealth who consume little electricity. Section 4.5 considers the setting where the unit retail price of electricity (r) can vary with the realized state.

  13. 13.

    In principle, a consumer might be penalized for purchasing more than the established baseline level of electricity, in which case \(x_{i}^{d}\) might be negative. We follow industry practice in abstracting from this possibility.

  14. 14.

    We thereby abstract initially from the possibility that, as in Chao (2009, 2011) and Chao and DePillis (2012), a consumer’s choice of \(x_{i}^{u} \) in one period might affect the value of \( \underline{x}_{i}\) that is established in future periods. Section 4.3 considers the possibility that consumers might be able to influence their baseline consumption levels.

  15. 15.

    The analysis in Sect. 4.5 admits state-specific retail prices, \(r(\theta ),\) that can function like peak load prices. In practice, peak load prices often are designed to generate sufficient revenue to cover the utility’s capacity costs (e.g., Crew et al. 1995). The fixed retail charge (R) can play this role in our model. Section 4.1 considers the optimal design of r and \(m(\theta )\) when fixed retail charges are not feasible.

  16. 16.

    The utility’s profit is zero under the optimal regulatory policy in all of the settings we analyze. Section 4.4 considers a setting where social welfare includes the losses from environmental externalities associated with electricity production.

  17. 17.

    The “\( \cdot \)” here denotes factors other than \(\theta \) that affect consumers’ electricity production and consumption. These factors can include r and \(m(\theta ).\)

  18. 18.

    Formally, \(\Omega _{i}^{D}\,(\Omega _{i}^{-D}\)) is the set of \(\theta \in [\underline{\theta },\,\overline{\theta }]\) for which \(\frac{\partial V_{i}(x_{i}^{u}+x_{i}^{o},\,\theta )}{\partial x_{i}^{u}}|_{x_{i}^{u}=\underline{x}_{i}}<\,({\ge })\,r+m(\theta )\) at the solution to [RP].

  19. 19.

    Formally, unless otherwise noted, we assume \(\Omega _{i}^{D}\ne \{\varnothing \}\) for some \(i\in \{ 1,\ldots ,N\}.\) For expositional simplicity, we also assume that \(x_{i}^{u}(\cdot ,\,\theta )>0\) for all \(\theta \in [\underline{\theta },\,\overline{\theta }],\) for \(i=1,\ldots ,N.\)

  20. 20.

    Ramsey (1927) and Baumol and Bradford (1970) characterize Ramsey prices. Joskow and Tirole (2007) identify conditions under which optimal retail prices for electricity reflect Ramsey principles.

  21. 21.

    The deviation of \(m(\theta )\) from marginal cost here does not reflect the deviation of price from marginal cost that commonly arises under peak load pricing to ensure revenue that matches operating costs (e.g., Crew et al. 1995). The regulator can choose the fixed charge (R) to ensure the utility’s financial solvency in the basic setting analyzed here.

  22. 22.

    To illustrate, two of the three major electric utilities in California (Pacific Electric and Gas and San Diego Gas and Electric) impose no fixed retail charge. The third utility (Southern California Edison) imposes a monthly fixed charge of only $0.99 (Borenstein 2014).

  23. 23.

    Borlick (2011) notes that the marginal-cost compensation for DR advised by the FERC requires consumers who do not provide DR to subsidize those who do.

  24. 24.

    The regulator seeks to maximize the relevant weighted average of the expected welfare of the two types of consumers while ensuring non-negative profit for the regulated utility. The proof of Proposition 3 includes a formal statement of [RP-d].

  25. 25.

    We further assume that, for all \(i=1,\ldots ,N,\) consumer i’s expected welfare is a strictly concave function of \(a_{i}\) and consumer i chooses \(a_{i}>0.\)

  26. 26.

    Consumer i’s welfare now includes both the personal cost of action \(a_{i}\) and the impact of this action on \( \underline{x}_{i}.\)

  27. 27.

    The proof of Proposition 4 includes a formal statement of [RP-a].

  28. 28.

    See the proof of Proposition  4.

  29. 29.

    This linear structure for the losses from externalities due to electricity production by consumers is adopted for analytic and expositional simplicity. The key qualitative conclusions drawn below persist under nonlinear structures.

  30. 30.

    e(X) is an increasing function. For simplicity, we abstract from the possibility that the social loss from externalities due to production by the utility might vary with the amount of electricity that consumers produce.

  31. 31.

    As noted above, the utility can be viewed as a distribution company that purchases electricity from competitive suppliers. If government policies (e.g., emissions taxes) compel electricity suppliers to internalize the social losses from environmental externalities, then the utility’s marginal cost of procuring electricity will reflect both the physical marginal cost of generating electricity and the associated marginal social losses from externalities. (Fabra and Reguant 2014 find that a large fraction of emissions costs are passed on to consumers in the form of higher retail prices for electricity.) The optimal unit compensation for DR in this setting would reflect the difference between the utility’s marginal cost of procuring electricity and the prevailing unit retail price of electricity.

  32. 32.

    Recall from Lemma 1 that \(\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )}<\left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\right| \) for all \(i=1,\ldots ,N.\) Therefore, \(e^{\prime }(X^{u})-\frac{\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )}}{\left| \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} \right| }>0\) when \(e^{\prime }(X^{u})=e_{i}=\underline{e},\) a constant, for all \(i=1,\ldots ,N.\) Consequently, Eq. (5) implies that \(m(\cdot )\) is optimally increased above \(C^{\prime }(\cdot )-r\) when the marginal social loss from externalities is constant and identical for all sources of electricity production. The increase in \(m(\theta )\) serves to reduce social losses from externalities because the increase in the amount of electricity consumers produce on-site as their DR increases is less than the amount of their DR.

  33. 33.

    For simplicity, we assume here that \(\frac{\partial V_{i}(x_{i}^{u},\,\theta )}{\partial x_{i}^{o}}>C_{i}^{\prime }(0)+e_{i}\) for all \(x_{i}^{u}\ge 0,\) for \(i=1,\ldots ,N.\)

  34. 34.

    The proof of Proposition 6 provides a formal statement of [RP-s].

  35. 35.

    This conclusion reflects the maintained assumption that the regulator can set a fixed charge (R) that does not affect electricity consumption.

  36. 36.

    Chao (2011, p. 79) observes that “In the special case where the [retail price of electricity] equals the wholesale price, the optimal demand response payment would be zero. Therefore, for consumers on dynamic retail pricing, there is no longer any reason to pay then for demand reduction.”

  37. 37.

    As is evident from the proof of Proposition 6, the optimal policy also typically does not induce efficient levels of consumption and DR in the presence of nontrivial externalities from on-site production.

  38. 38.

    See, for example, King and Chatterjee (2003), Espey and Espey (2004), Narayan and Smyth (2005), Taylor et al. (2005), Wade (2005), Bernstein and James Griffin (2006), and Paul et al. (2009). It is readily verified that consumer i’s price elasticity of demand for electricity in this setting is \(\frac{1}{\alpha _{i}}.\)

  39. 39.

    This formulation reflects a common approach to capturing changes in building energy use due to ambient temperature variation (e.g., Eto 1988).

  40. 40.

    PJM Interconnection is the “regional transmission organization (RTO) that coordinates the movement of wholesale electricity in all or parts of Delaware, Illinois, Indiana, Kentucky, Maryland, Michigan, New Jersey, North Carolina, Ohio, Pennsylvania, Tennessee, Virginia, West Virginia and the District of Columbia” (www.pjm.com/about-pjm/who-we-are.aspx).

  41. 41.

    ISO New England is “the independent, not-for-profit corporation responsible for keeping electricity flowing across the six New England states and ensuring that the region has reliable, competitively priced wholesale electricity” (www.iso-ne.com/about). We investigate potential outcomes in the California, ISO New England, and PJM Interconnection regions because Bushnell (2007) provides estimates of the cost parameters a and b in these three regions. We focus on outcomes in the PJM Interconnection region here for brevity and because this region is the largest and the most populous of the three regions.

  42. 42.

    The data reveal that the distribution of \(\theta \) is also approximated reasonably well by a generalized extreme value (GEV) distribution with parameters (\(\mu ,\,\sigma ,\,\xi ) = (18.460,\,10.928,\,{-}0.029\)). The key qualitative conclusions reported below are unchanged when this GEV distribution is employed instead of the identified gamma distribution.

  43. 43.

    The optimal regulatory policy in the absence of a DR policy is characterized in Brown and Sappington (2016).

  44. 44.

    This average hourly load, 90,314 MW, is total annual consumption (791,152,262 MWh) in the PJM Interconnection region in 2013 divided by 8760, the number of hours in a year (Pennsylvania New Jersey Maryland 2014).

  45. 45.

    ISO-NE (2006) and Thomas et al. (2014) estimate that variable energy production costs constitute between 48 and \(60\,\%\) (an average of \(54\,\%\)) of ratepayer revenue. Revenue is calculated as the product of the average retail rate for electricity and the total load in the PJM Interconnection region in 2013 (Pennsylvania New Jersey Maryland 2014).

  46. 46.

    Thus, \(r=83.19\) denotes a price of approximately \(\$0.083\) per kWh.

  47. 47.

    Thus, \(R=299.96\) represents a monthly fixed charge of approximately \(\$25.\)

  48. 48.

    \(E \{ m(\theta )\} =\int _{\theta _{m}}^{\overline{\theta }}m(\theta ) dG(\theta ),\) where \(\theta _{m}=42.5\) is the smallest realization of \(\theta \) for which DR is provided both at the solution to [RP] and under the optimal FERC policy in the benchmark setting. The qualitative conclusions drawn below are robust to alternative plausible definitions of peak-load production costs.

  49. 49.

    Formally, \(E \{C^{P}(\theta )\} =\int _{\theta _{m}}^{\overline{\theta }}C(\cdot ) dG(\theta ).\)

  50. 50.

    \(E\{ W\} =\sum \nolimits _{i=1}^{N}\int _{\underline{\theta }}^{\overline{\theta }}[V_{i}(x_{i}^{u}(\cdot ,\,\theta ),\,\theta )-rx_{i}^{u}(\cdot ,\,\theta )+m(\theta )x_{i}^{d}(\cdot ,\,\theta )]dG(\theta )-NR,\) reflecting Eq. (1).

  51. 51.

    Larger percentage increases in expected welfare arise in the settings analyzed in Brown and Sappington (2016).

  52. 52.

    Reported percentage changes may not reflect the entries in Table 1 exactly because these entries are rounded.

  53. 53.

    Systematic increases in the marginal cost of production (i.e., increases in a) also enhance the welfare gains generated by an optimal DR policy. To illustrate, suppose a increases from 0 to 20, while all other parameters are held constant at their levels in the benchmark setting. (The average value of a in the settings considered in Brown and Sappington 2016 is approximately 23.) The increase in expected welfare that the optimal DR policy generates in this case (relative to no DR policy) rises to \(33.6\,\%\) (from the \(17.4\,\%\) generated in the benchmark setting). Bushnell’s (2007) estimate of \(a=0\) in the PJM region reflects in part substantial supply by nuclear generators. Some of these generators are scheduled for retirement in the near future, which will tend to increase a. However, increased supply of energy from renewable sources may reduce a.

  54. 54.

    A value of b substantially below Bushnell’s (2007) estimate might arise, for example, from pronounced reductions in the price of natural gas, which often is employed to power peak-load production units. The US experienced sharp reductions in the price of natural gas between 2007 and 2009 (www.infomine.com/investment/metal-prices/natural-gas/all/). The ongoing replacement of (low cost) coal generation by natural gas generation in the PJM region can introduce a countervailing effect on b.

  55. 55.

    The regulator might also be permitted to specify the terms under which consumers must “buy” their assigned baselines (e.g., in a day-ahead market) before they are eligible to sell demand reduction (e.g., in a real-time spot market) (Bushnell et al. 2009).

  56. 56.

    Our key qualitative conclusions hold for any specified (exogenous) values of \(\underline{x}_{i},\) and so will hold for the optimal (endogenous) such levels.

  57. 57.

    Future research might also characterize the optimal DR policy in settings with richer intertemporal structures. In practice, consumers may secure additional benefit from a DR program as their stochastic demand for electricity naturally falls below the established baseline level at various times, or as they intentionally substitute electricity consumption in other periods for consumption foregone while supplying DR (e.g., Graff Zivin et al. 2014).

References

  1. Baumol, W., & Bradford, D. (1970). Optimal departures from marginal cost pricing. American Economic Review, 60, 265–283.

    Google Scholar 

  2. Bernstein, M., & James Griffin, J. (2006). Regional differences in the price-elasticity of demand for energy. Rand Corporation Report NREL/SR-620-39512. www.nrel.gov/docs/fy06osti/39512.

  3. Borenstein, S. (2014). Whats so great about fixed charges? Haas: Energy Institute. http://energyathaas.wordpress.com/2014/11/03/whats-so-great-about-fixed-charges.

  4. Borenstein, S., & Holland, S. (2005). On the efficiency of competitive electricity markets with time-invariant retail prices. Rand Journal of Economics, 36, 469–493.

    Google Scholar 

  5. Borlick, R. (2010). Pricing negawatts: DR design flaws create perverse incentives. Public Utilities Fortnightly, 148, 14–19.

    Google Scholar 

  6. Borlick, R. (2011). Paying for demand-side response at the wholesale level: The small consumers’ perspective. The Electricity Journal, 24, 8–19.

    Article  Google Scholar 

  7. Borlick, R., et al. (2012). Brief as amici curiae in support of petitioners. Filed in the United States Court of Appeals for the District of Columbia, Electric Power Supply Association et al. v. Federal Energy Regulatory Commission et al., USCA Case #11-1486. www.hks.harvard.edu/fs/whogan/Economists%20amicus%20brief_061312.

  8. Brown, D., & Sappington, D. (2016). Technical appendix to accompany ‘On the optimal design of demand response policies.’ http://uofa.ualberta.ca/arts/about/people-collection/david-brown.

  9. Bushnell, J. (2007). Oligopoly equilibria in electricity contract markets. Journal of Regulatory Economics, 32, 225–245.

    Article  Google Scholar 

  10. Bushnell, J., Hobbs, B., & Wolak, F. (2009). When it comes to demand response, is FERC its own worst enemy? The Electricity Journal, 22, 9–18.

    Article  Google Scholar 

  11. Chao, H. (2009). An economic framework of demand response in restructured electricity markets. ISO New England Discussion Paper. www.hks.harvard.edu/hepg/Papers/2009/Demand%20Response%20in%20Restructured%20Markets%2002-08-09.

  12. Chao, H., & DePillis, M. (2012). Incentive effects and net benefits of demand response regulation in wholesale electricity market. ISO New England Working Paper.

  13. Chao, H. (2011). Demand response in wholesale electricity markets: The choice of the consumer baseline. Journal of Regulatory Economics, 39, 68–88.

    Article  Google Scholar 

  14. Chen, L., Li, N., Low, S., & Doyle, J. (2010). Two market models for demand response in power networks. In IEEE SmartGridComm Proceedings (pp. 4569–4574).

  15. Crew, M., Fernando, C., & Kleindorfer, P. (1995). The theory of peak-load pricing: A survey. Journal of Regulatory Economics, 8, 215–248.

    Article  Google Scholar 

  16. DNV GL Energy. (2014). A review of distributed energy resources. Prepared by DNV GL Energy for the New York Independent System Operator. www.nyiso.com/public/webdocs/media_room/publications_presentations/Other_Reports/Other_Reports/A_Review_of_Distributed_Energy_Resources_September_2014.

  17. Energy Information Administration. (2014a). Retail sales of electricity by state by sector by provider. EIA Form 861. www.eia.gov/electricity/data/state/.

  18. Espey, J., & Espey, M. (2004). Turning on the lights: A meta-analysis of residential electricity demand elasticities. Journal of Agricultural and Applied Economics, 36, 65–81.

    Article  Google Scholar 

  19. Eto, J. (1988). On using degree-days to account for the effects of weather on annual energy use in office-buildings. Energy and Buildings, 12, 113–127.

    Article  Google Scholar 

  20. Fabra, N., & Reguant, M. (2014). Pass-through of emissions costs in electricity markets. American Economic Review, 104, 2872–2899.

    Article  Google Scholar 

  21. Federal Energy Regulatory Commission. (2011). Demand response compensation in organized wholesale energy markets. Docket No. RM10-17-000, Order No. 745, 18 CFR Part 35, 134 FERC, 61,187.

  22. Graff Zivin, J., Kotchen, M., & Mansur, E. (2014). Spatial and temporal heterogeneity of marginal emissions: Implications for electric cars and other electricity-shifting policies. Journal of Economic Behavior and Organization, 107, 248–268.

    Article  Google Scholar 

  23. Hogan, W. (2009). Providing incentives for efficient demand response. Prepared for the Electric Power Supply Association, Comments on PJM demand response proposals. Federal Energy Regulatory Commission, Docket No. EL09-68-000.

  24. Hogan, W. (2010). Implications for consumers of the NOPR’s proposal to pay the LMP for all demand response. Prepared for the Electric Power Supply Association, Comments on demand response compensation in organized wholesale energy markets, Notice of Proposed Rulemaking. Federal Energy Regulatory Commission, Docket No. RM10-17-000.

  25. ISO-NE. (2006). Electricity costs white paper. ISO New England, Inc. www.iso-ne.com/pubs/whtpprs/elec_costs_wht_ppr.

  26. Joskow, P., & Tirole, J. (2007). Reliability and competitive electricity markets. Rand Journal of Economics, 38, 60–84.

    Article  Google Scholar 

  27. KEMA, Inc. (2011). PJM empirical analysis of demand response baseline methods. Report Prepared for the PJM Markets Implementation Committee. www.pjm.com/~/media/markets-ops/dsr/pjm-analysis-of-dr-baseline-methods-full-report.ashx.

  28. King, C., & Chatterjee, S. (2003). Predicting California demand response: How do consumers react to hourly price? Public Utilities Fortnightly, 141, 27–32.

    Google Scholar 

  29. Li, N., Chen, L., & Low, S. (2011). Optimal demand response based on utility maximization in power networks. In IEEE Power and Energy Society general meeting proceedings (pp. 1–8).

  30. London Economics International LLC. (2013). Estimating the value of lost load: Briefing paper prepared for the Electric Reliability Council of Texas. www.ercot.com/content/gridinfo/resource/2014/mktanalysis/ERCOT_ValueofLostLoad_LiteratureReviewandMacroeconomic.

  31. Narayan, P., & Smyth, R. (2005). The residential demand for electricity in Australia: An application of the bounds testing approach to cointegration. Energy Policy, 33, 467–474.

    Article  Google Scholar 

  32. National Oceanic and Atmospheric Administration, NOAA. (2014). Quality controlled local climatological data. Retrieved January 11, from www.ncdc.noaa.gov/data-access/land-based-station-data/land-based-datasets/quality-controlled-local-climatological-data-qclcd.

  33. Paul, A., Myers, E., & Palmer, K. (2009). A partial adjustment model of U.S. electricity demand by region, season, and sector. Resources for the Future Discussion Paper RFF DP 08-50. www.rff.org/documents/rff-dp-08-50.

  34. Pennsylvania New Jersey Maryland (PJM) Interconnection. (2014). Metered load data. www.pjm.com/markets-and-operations/ops-analysis/historical-load-data.aspx.

  35. Ramsey, F. (1927). A contribution to the theory of taxation. Economic Journal, 37, 47–61.

    Article  Google Scholar 

  36. Taylor, T., Schwarz, P., & Cochell, J. (2005). 24/7 hourly response to electricity real-time pricing with up to eight summers of experience. Journal of Regulatory Economics, 27, 235–262.

    Article  Google Scholar 

  37. Thomas, A., Lendel, I., & Park, S. (2014). Electricity markets in Ohio. Prepared for Ohio’s Manufacturers’ Association. Center for Economic Development and Energy Policy Center. http://urban.csuohio.edu/publications/center/center_for_economic_development/ElectricityMarketsInOhio.

  38. U.S. Court of Appeals for the District of Columbia Circuit. (2014). Electric Power Supply Association v. Federal Energy Regulatory Commission. No. 11-1486. www.ferc.gov/legal/court-cases/opinions/2014/11-1486.

  39. U.S. Department of Energy. (2006). Benefits of demand response in electricity markets and recommendations for achieving them. http://energy.gov/sites/prod/files/oeprod/DocumentsandMedia/DOE_Benefits_of_Demand_Response_in_Electricity_Markets_and_Recommendations_for_Achieving_Them_Report_to_Congress.

  40. U.S. Supreme Court. (2016). FERC v. EPSA et al. and ENERNOC et al. v. EPSA. Orders 14-840 and 14-841. www.supremecourt.gov/opinions/15pdf/14-840_k537.

  41. Wade, S. (2005). Price responsiveness in the AEO2003 NEMS residential and commercial buildings sector models. Prepared for the Energy Information Administration. www.eia.gov/oiaf/analysispaper/elasticity/pdf/buildings.

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Acknowledgments

We thank the Editor, Michael Crew, two anonymous referees, seminar participants, and Burcin Unel for helpful comments and observations.

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Correspondence to David P. Brown.

Appendix

Appendix

Proof of Lemma 1

(4) implies that when \(x_{i}^{d}>0,\) the value of \(x_{i}^{u}>0\) and the value of \(x_{i}^{o}>0\) produced using the dispatchable on-site technology are characterized by:

$$\begin{aligned} \frac{\partial V_{i}(\cdot )}{\partial x_{i}^{u}} = r+m(\theta ) =C_{i}^{\prime }(\cdot )\,\Rightarrow \,\frac{\partial ^{2}V_{i}(\cdot )}{\partial (x_{i}^{u}+x_{i}^{o})^{2}}\frac{d( x_{i}^{u}+x_{i}^{o})}{dm(\theta )} = 1 = C_{i}^{\prime }(\cdot )\frac{dx_{i}^{o}}{dm(\theta )}. \end{aligned}$$

Therefore, \(\frac{d( x_{i}^{u}+x_{i}^{o}) }{dm(\theta )}<0\) and \(\frac{dx_{i}^{o}}{dm(\theta )}>0,\) and so \(\frac{dx_{i}^{u}}{dm(\theta )}<0\) when consumer i employs the dispatchable technology.

Consumer i produces \(\overline{x}_{i}(\theta )\) units of electricity when he employs the non-dispatchable technology. Therefore, \(x_{i}^{o}\) and \(x_{i}^{u}\) are not affected by \(m(\theta ).\) \(\square \)

Proof of Proposition 1

The conclusions follow immediately from Proposition 5. \(\square \)

Proof of Corollary 1

First suppose \( x_{i}^{u} <\underline{x}_{i} \) for some \( i\in \{1,\ldots ,N\}.\) Then (4) and (5) imply that at the solution to [RP] identified in Proposition 1, \( x_{i}^{u} \) is determined by \(\frac{\partial V_{i}(x_{i}^{u}+x_{i}^{0},\,\theta )}{\partial x_{i}^{u}}=r+m(\theta )=C^{\prime }( X^{u}(\cdot ,\,\theta )).\) Therefore, given the consumption decisions of other consumers, the consumption and DR actions of consumer i are efficient.

Now suppose \(x_{i}^{u}(\cdot )> \underline{x}_{i}.\) Then (4) and (6) imply that at the solution to [RP] identified in Proposition 1, \( x_{i}^{u}\) is determined by:

$$\begin{aligned} \frac{\partial V_{i}(x_{i}^{u}+x_{i}^{0},\,\theta )}{\partial x_{i}^{u}} = r = \frac{\sum \nolimits _{i=1}^{N}\int _{\Omega _{i}^{-D}}C^{\prime }(X^{u}(\cdot ,\,\theta ))\frac{\partial x_{i}^{u}(\cdot )}{\partial r}dG(\theta )}{\sum \nolimits _{i=1}^{N}\int _{\Omega _{i}^{-D}}\frac{\partial x_{i}^{u}(\cdot )}{\partial r}dG(\theta )}. \end{aligned}$$

Therefore, given the actions of other consumers, the actions of consumer i are efficient if and only if, for all \(\theta \in [\underline{\theta },\,\overline{\theta }]{\text {:}}\)

$$\begin{aligned} \sum \limits _{i=1}^{N}\int _{\Omega _{i}^{-D}}C^{\prime }\left( X^{u}(\cdot ,\,\theta )\right) \frac{\partial x_{i}^{u}(\cdot )}{\partial r}dG(\theta )=\left[ \sum \limits _{i=1}^{N}\int _{\Omega _{i}^{-D}}\frac{\partial x_{i}^{u}(\cdot )}{\partial r}dG(\theta )\right] C^{\prime }\left( X^{u}(\cdot ,\,\theta )\right) . \end{aligned}$$

This equality typically will not hold because \(x_{i}^{u}(\cdot ,\,\theta ),\) and thus \(X^{u}(\cdot ,\,\theta ),\) vary with \(\theta .\) \(\square \)

Proof of Proposition 2

The proof parallels the proof of Proposition 5. \(\square \)

Proof of Proposition 3

Letting “  \( \widetilde{\cdot } \)  ” (“  \( \widehat{\cdot } \)  ”) denote variables for consumers who can (cannot) provide DR, expected weighted consumer welfare in this setting is:

$$\begin{aligned} E\left\{ U^{\alpha }(\cdot )\right\}= & {} \widetilde{\alpha }\left\{ \sum \limits _{i=1}^{\widetilde{N}} \int _{\underline{\theta }}^{\overline{\theta }}\left[ V_{i}\left( \widetilde{x}_{i}^{u}(r,\,m(\theta ),\,\theta )+\widetilde{x}_{i}^{o}(\cdot ),\,\theta \right) - r \widetilde{x}_{i}^{u}(\cdot )\right. \right. \nonumber \\&\quad + \left. \left. m(\theta )\widetilde{x}_{i}^{d}(\cdot )-C_{i}\left( \widetilde{x}_{i}^{o}(\cdot ),\,\theta \right) \right] d G(\theta )-\widetilde{N}R\right\} \nonumber \\&\quad +\, \widehat{\alpha }\left\{ \sum \limits _{i=1}^{\widehat{N}} \int _{\underline{\theta }}^{\overline{\theta }}\left[ V_{i}\left( \widehat{x}_{i}^{u}(r,\,\theta )- r \widehat{x}_{i}^{u}(\cdot )\right) \right] d G(\theta )-\widehat{N}R\right\} . \end{aligned}$$
(15)

The utility’s expected profit is:

$$\begin{aligned} E\left\{ \pi ^{\alpha }\right\}= & {} R[ \widetilde{N}+\widehat{N}] +\sum _{i=1}^{\widetilde{N}}\int _{\underline{\theta }}^{\overline{\theta }}\left[ r\widetilde{x}_{i}^{u}(r,\,m(\theta ),\,\theta )-m(\theta ) \widetilde{x}_{i}^{d}(\cdot )\right] dG(\theta )\nonumber \\&+\, \sum _{i=1}^{\widehat{N}}\int _{\underline{\theta }}^{\overline{\theta }}r\widehat{x}_{i}^{u}(r,\,\theta )d G(\theta )-\int _{\underline{\theta }}^{\overline{\theta }}C\left( \sum _{i=1}^{\widetilde{N}}\widetilde{x}_{i}^{u}(\cdot )+\sum _{i=1}^{\widehat{N}}\widehat{x}_{i}^{u}(\cdot )\right) d G(\theta ). \qquad \quad \end{aligned}$$
(16)

The regulator’s problem, [RP-d], is to choose \(\{ R,\, r,\, m(\theta )\} \) to maximize \(E\{ U^{\alpha }(\cdot )\} \) while securing non-negative expected profit for the utility. Let \(\lambda _{\alpha }\ge 0\) denote the Lagrange multiplier associated with the utility’s participation constraint (\(E \{ \pi ^{\alpha } \} \ge 0 \)). Then the Lagrangian function associated with [RP-d] is:

$$\begin{aligned} \pounds _{\alpha } = E\left\{ U^{\alpha }(\cdot )\right\} +\lambda _{\alpha }E\left\{ \pi ^{\alpha }\right\} . \end{aligned}$$
(17)

Because the value of R does not affect consumption decisions, differentiating (17) with respect to R,  using (15) and (16), provides \(\lambda _{\alpha }=\frac{\widetilde{\alpha }\widetilde{N}+\widehat{\alpha }\widehat{N}}{\widetilde{N}+\widehat{N}}.\)

Because \(\frac{\partial \widehat{x}_{i}^{u}(\cdot )}{\partial m(\theta )}=0\) for all \( i=1,\ldots ,\widehat{N},\) pointwise optimization of (17) with respect to \(m(\theta ),\) using (15), (16), Leibnitz’ rule, and the continuity of consumer welfare and profit (see the proof of Proposition 5) reveals that:

$$\begin{aligned} r+m(\theta )-C^{\prime }( \cdot ) = \frac{\widehat{N}[\widehat{\alpha }-\widetilde{\alpha }]\sum _{i=1}^{\widetilde{N}}\widetilde{x}_{i}^{d}(\cdot )}{[\widetilde{\alpha }\widetilde{N}+\widehat{\alpha }\widehat{N}]\sum _{i=1}^{\widetilde{N}}\frac{\partial \widetilde{x}_{i}^{u}(\cdot )}{\partial m(\theta )}}. \end{aligned}$$
(18)

(9) follows immediately from (18) because \(\frac{\partial \widetilde{x}_{i}^{u}(\cdot )}{\partial m(\theta )}<0\) when \(\widetilde{x}_{i}^{d}(\cdot )>0 \) and \(\frac{\partial \widetilde{x}_{i}^{u}(\cdot )}{\partial m(\theta )}\le 0\) when \(\widetilde{x}_{i}^{d}(\cdot )=0.\) \(\square \)

Proof of Proposition 4

Aggregate consumer welfare in this setting is:

$$\begin{aligned} E\left\{ U^{a}(\cdot )\right\} =\int \limits _{\underline{\theta }}^{\overline{\theta }}\sum \limits _{i=1}^{N}w_{i}(\theta )dG(\theta )-N R-D\left( a_{i}\right) . \end{aligned}$$
(19)

Because \(\sum \nolimits _{i=1}^{N}w_{i}(\theta )\) is continuous in \(\theta \) for all \(\theta \) (see the proof of Proposition 5), (19) and Leibnitz’ rule imply that \(a_{i}\) is determined by:

$$\begin{aligned} H_{i}\left( a_{i},\,r,\,m(\theta ),\,\theta \right) \equiv \int _{\underline{\theta }}^{\widetilde{\theta }_{i}}m(\theta ) \frac{\partial \underline{x}_{i}}{\partial a_{i}} d G(\theta )-D_{i}^{\prime }\left( a_{i}\right) = 0. \end{aligned}$$
(20)

By assumption:

$$\begin{aligned} \frac{\partial H_{i}(\cdot )}{\partial a_{i}} = \frac{d \widetilde{\theta }_{i}(\cdot )}{da_{i}} m\left( \widetilde{\theta }_{i}\right) \frac{\partial \underline{x}_{i}}{\partial a_{i}} g\left( \widetilde{\theta }_{i}\right) +\int _{\underline{\theta }}^{\widetilde{\theta }_{i}}m(\theta )\frac{\partial ^{2}\underline{x}_{i}}{\partial ( a_{i}) ^{2}} dG(\theta )-D_{i}^{\prime \prime }\left( a_{i}\right) < 0. \end{aligned}$$
(21)

(20) implies:

$$\begin{aligned} \frac{\partial H_{i}(\cdot )}{\partial m(\theta )} = \left\{ \begin{array}{ll} \frac{\partial \underline{x}_{i}}{\partial a_{i}}g(\theta ) >0&{} \quad \text {if}\quad \theta \in \Omega _{i}^{D}, \\ 0&{} \quad \text {otherwise}.\end{array}\right. \end{aligned}$$
(22)

(20), (21), and (22) imply:

$$\begin{aligned} \frac{\partial a_{i}}{\partial m(\theta )} ={-}\frac{\partial H_{i}/\partial m(\theta )}{\partial H_{i}/\partial a_{i}} \ge 0. \end{aligned}$$
(23)

The regulator’s problem, [RP-a], is to choose \(\{ R,\, r,\, m(\theta )\} \) to maximize \(E\{ U^{a}(\cdot )\}\) while securing non-negative expected profit for the utility. Let \(\lambda _{a}\ge 0\) denote the Lagrange multiplier associated with the utility’s participation constraint (\(E \{ \pi ^{a} \} \ge 0 \)). Then the Lagrangian function associated with [RP] is:

$$\begin{aligned} \pounds _{a} = E\left\{ U^{a}(\cdot )\right\} + \lambda _{a}E\left\{ \pi ^{a}\right\} . \end{aligned}$$
(24)

Let \(\frac{dx_{i}^{j}(\cdot )}{dm(\theta )}=\frac{\partial x_{i}^{j}(\cdot )}{\partial m(\theta )}+\frac{\partial x_{i}^{j}(\cdot )}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )} \) for \( j\in \{u,\,d,\,o\}.\) For the reasons identified in the proof of Proposition 5, expected consumer welfare and the firm’s expected profit are both continuous functions of \(\theta .\) Consequently, Leibnitz’ rule implies that pointwise optimization of (24) with respect to \(m(\theta )\) provides:

$$\begin{aligned}&\left[ 1-\lambda _{a}\right] \sum _{i=1}^{N}x_{i}^{d}(r,\,m(\theta ),\,\theta )g(\theta )-e^{\prime }\left( X^{u}\right) \sum _{i=1}^{N}\frac{dx_{i}^{u}(\cdot )}{dm(\theta )}g(\theta )-\sum _{i=1}^{N}e_{i} \frac{dx_{i}^{o}}{dm(\theta )}g(\theta )\nonumber \\&\quad -\lambda _{a} C^{\prime }\left( X^{u}\right) \sum _{i=1}^{N}\frac{dx_{i}^{u}(\cdot )}{dm(\theta )}g(\theta )+\lambda _{a}\sum _{i=1}^{N}\left[ r\frac{dx_{i}^{u}(\cdot )}{dm(\theta )}-m(\theta )\frac{dx_{i}^{d}(\cdot )}{dm(\theta )}\right] g(\theta ) = 0. \nonumber \\ \end{aligned}$$
(25)

Because the value of R does not affect consumption decisions, differentiating (24) with respect to R provides \({-}N+\lambda _{a}N=0\,\Rightarrow \,\lambda _{a}=1.\) Therefore, (25) can be written as:

$$\begin{aligned} \left[ r-C^{\prime }\left( X^{u}\right) \right] \sum _{i=1}^{N}\frac{dx_{i}^{u}(\cdot )}{dm(\theta )} =m(\theta )\sum _{i=1}^{N}\frac{dx_{i}^{d}(\cdot )}{dm(\theta )}. \end{aligned}$$
(26)

\(\frac{\partial x_{i}^{d}(\cdot )}{\partial m(\theta )}={-}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}>0\) because \(\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}=0\) if \(x_{i}^{u}(\cdot )>\underline{x}_{i}.\) Also, (4) implies that \( x_{i}^{u}(\cdot )\) does not vary with \( \underline{x}_{i},\) given r and \(m(\theta ).\) Therefore:

$$\begin{aligned}&\frac{dx_{i}^{u}(\cdot )}{dm(\theta )} = \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\quad \text {and}\quad \frac{\partial x_{i}^{d}(\cdot )}{\partial a_{i}} = \left\{ \begin{array}{ll} \frac{\partial \underline{x}_{i}}{\partial a_{i}}&{} \quad \text {if}\quad x_{i}^{u}(\cdot )\le \underline{x}_{i}, \\ 0&{}\quad \text {if}\quad x_{i}^{u}(\cdot )>\underline{x}_{i},\end{array}\right. \end{aligned}$$
(27)
$$\begin{aligned}&\Rightarrow \frac{dx_{i}^{d}(\cdot )}{dm(\theta )} =\left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\right| +\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )} > 0. \end{aligned}$$
(28)

(10) follows from (26), (27), and (28). \(\square \)

Proof of Corollary 2

Equation (4) Implies that \( x_{i}^{u} < \underline{x}_{i} \) at the solution to [RP-a] identified in Proposition 4 is determined by:

$$\begin{aligned} \frac{\partial V_{i}(x_{i}^{u}+x_{i}^{o},\,\theta )}{\partial x_{i}^{u}} = r+m(\theta ) = \frac{C^{\prime }( X^{u}(\cdot ,\,\theta )) \sum _{i=1}^{N}\left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} \right| +r\sum _{i=1}^{N}\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )}}{\sum _{i=1}^{N}\left\{ \left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\right| +\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )}\right\} }. \end{aligned}$$

Therefore, given the actions of other consumers, consumer i’s actions are efficient only if:

$$\begin{aligned}&\frac{C^{\prime }( X^{u}(\cdot ,\,\theta )) \sum _{i=1}^{N}\left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} \right| +r\sum _{i=1}^{N}\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )}}{\sum _{i=1}^{N}\left\{ \left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\right| +\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )}\right\} } =C^{\prime }\left( X^{u}(\cdot ,\,\theta )\right) \nonumber \\&\quad \Leftrightarrow \left[ r-C^{\prime }\left( X^{u}(\cdot ,\,\theta )\right) \right] \left[ \frac{\sum _{i=1}^{N}\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )}}{\sum _{i=1}^{N}\left\{ \left| \frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\right| +\delta _{i\theta }\frac{\partial \underline{x}_{i}}{\partial a_{i}}\frac{\partial a_{i}}{\partial m(\theta )}\right\} }\right] = 0. \end{aligned}$$
(29)

(28) implies that (29) holds if and only if \(r = C^{\prime }( X^{u}(\cdot ,\,\theta )) \) for each \(\theta \in [\underline{\theta },\,\overline{\theta }].\) These inequalities typically will not all hold because \(x_{i}^{u}(\cdot ,\,\theta ),\) and thus \(X^{u}(\cdot ,\,\theta ),\) vary with \(\theta .\) \(\square \)

Proof of Proposition 5

Let \(\lambda \ge 0\) denote the Lagrange multiplier associated with the utility’s participation constraint (\(E \{ \pi \} \ge 0 \)). Then the Lagrangian function associated with [RP-e] is:

$$\begin{aligned} \pounds = E\{U(\cdot )\} -E\{ L(\cdot )\} + \lambda E\{\pi \}. \end{aligned}$$
(30)

Pointwise optimization of (30) with respect to \(m(\theta ),\) using (1), (2), (11), and the envelope theorem provides:

$$\begin{aligned}&[1-\lambda ]\sum _{i=1}^{N}x_{i}^{d}( r,\,m(\theta ),\,\theta )g(\theta )-e^{\prime }\left( X^{u}\right) \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} g(\theta )-\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}}{\partial m(\theta )}g(\theta )\nonumber \\&\quad -\lambda C^{\prime }\left( X^{u}\right) \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} g(\theta )+\lambda \sum _{i=1}^{N}\left[ r\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}-m(\theta )\frac{\partial x_{i}^{d}(\cdot )}{\partial m(\theta )}\right] g(\theta ) = 0. \end{aligned}$$
(31)

Because the value of R does not affect consumption decisions, differentiating (30) with respect to R provides \({-}N+\lambda N=0\,\Rightarrow \,\lambda =1.\) Therefore, (7) holds. Also, \(\frac{\partial x_{i}^{d}(\cdot )}{\partial m(\theta )}={-}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\) because \(\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}=0\) if \(x_{i}^{u}(\cdot )>\underline{x}_{i}.\) Therefore, (31) can be written as:

$$\begin{aligned} \left[ r+m(\theta )-e^{\prime }\left( X^{u}\right) -C^{\prime }\left( X^{u}\right) \right] \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}-\sum _{i=1}^{N}e_{i} \frac{\partial x_{i}^{o}}{\partial m(\theta )} = 0. \end{aligned}$$
(32)

\(\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}<0\) because \(\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}< 0\) when \(x_{i}^{d}(\cdot ) >0 \) and \(\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\le 0\) when \(x_{i}^{d}(\cdot ) =0.\) Therefore, (12) follows from (32).

Let \(\Omega _{i}^{=}\) denote the set of \(\theta \in [\underline{\theta },\,\overline{\theta }]\) for which \(\frac{\partial V_{i}(x_{i}^{u}+x_{i}^{o},\,\theta )}{\partial x_{i}^{u}}|_{x_{i}^{u}=\underline{x}_{i}}=r+m(\theta )\) at the solution to [RP-e]. Observe that:

$$\begin{aligned}&V_{i}\left( x_{i}^{u}(r,\,m(\theta ),\,\theta )+x_{i}^{o}(\cdot ),\,\theta \right) -r x_{i}^{u}(r,\,m(\theta ),\,\theta )+m(\theta )\left[ \underline{x}_{i}-x_{i}^{u}(r,\,m(\theta ),\,\theta )\right] \nonumber \\&\quad = V_{i}\left( x_{i}^{u}(r,\,\theta )+x_{i}^{o}(\cdot ),\,\theta \right) -r x_{i}^{u}(r,\,\theta )\quad \text {for all}\,\theta \in \Omega _{i}^{=}. \end{aligned}$$
(33)

Further observe that (1) can be written as:

$$\begin{aligned}&E\{ U(\cdot )\}=\int \limits _{\underline{\theta }}^{\overline{\theta }}\sum \limits _{i=1}^{N}w_{i}(\theta )dG(\theta )-N R\quad \text {where}\,w_{i}(\theta )\equiv \left\{ \begin{array}{ll} w_{i}^{D}(\theta )&{}\text {if}\,\theta \in \Omega _{i}^{D},\\ w_{i}^{-D}(\theta )&{} \text {if}\,\theta \in \Omega _{i}^{-D},\end{array}\right. \nonumber \\&w_{i}^{D}(\theta ) \equiv V_{i}\left( x_{i}^{u}(r,\,m(\theta ),\,\theta )+x_{i}^{o}(\cdot ),\,\theta \right) -r x_{i}^{u}(r,\,m(\theta ),\,\theta )\nonumber \\&\qquad \qquad \quad +\,m(\theta )\left[ \underline{x}_{i}-x_{i}^{u}(r,\,m(\theta ),\,\theta )\right] -C_{i}\left( x_{i}^{o}(\cdot ),\,\theta \right) ,\quad \text {and}\nonumber \\&w_{i}^{-D}(\theta ) \equiv V_{i}\left( x_{i}^{u}(r,\,\theta )+x_{i}^{o}(\cdot ),\,\theta \right) -r x_{i}^{u}(r,\,\theta )-C_{i}\left( x_{i}^{o}(\cdot ),\,\theta \right) . \end{aligned}$$
(34)

Equation (33) Implies that for any \(\widehat{\theta }\in \Omega _{i}^{=},\,\lim _{\theta \,\rightarrow \,\widehat{\theta }^{-}}\sum _{i=1}^{N}w_{i}^{D}(\theta )=\lim _{\theta \,\rightarrow \,\widehat{\theta }^{+}}\sum _{i=1}^{N}w_{i}^{-D}(\theta )\) and \(\lim _{\theta \,\rightarrow \,\widehat{\theta }^{-}}\sum _{i=1}^{N}w_{i}^{-D}(\theta )=\lim _{\theta \,\rightarrow \,\widehat{\theta }^{+}}\sum _{i=1}^{N}w_{i}^{D}(\theta ).\) Consequently, \(\sum \nolimits _{i=1}^{N}w_{i}(\theta )\) is continuous in \(\theta \) for all \(\theta .\) Corresponding arguments reveal that \(\sum \nolimits _{i=1}^{N}\widetilde{\pi }_{i}(\theta )\) is continuous in \(\theta \) for all \(\theta .\) The established continuity and Leibnitz’ rule ensure that differentiation of (30) with respect to r provides:

$$\begin{aligned}&\sum \limits _{i=1}^{N}\int _{\Omega _{i}^{D}}\left\{ \left[ r+m(\theta )-C^{\prime }\left( X^{u}\right) -e^{\prime }\left( X^{u}\right) \right] \frac{\partial x_{i}^{u}(\cdot )}{\partial r}-e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial r}\right\} d G(\theta )\nonumber \\&\quad +\sum \limits _{i=1}^{N}\int _{\Omega _{i}^{-D}}\left\{ \left[ r-C^{\prime }\left( X^{u}\right) -e^{\prime }\left( X^{u}\right) \right] \frac{\partial x_{i}^{u}(\cdot )}{\partial r}-e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial r}\right\} d G(\theta ) = 0. \qquad \end{aligned}$$
(35)

From (4), for \(i=1,\ldots ,N,\,\frac{\partial x_{i}^{u}(\cdot )}{\partial r}=\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\) for all \(\theta \in \Omega _{i}^{D}.\) Therefore, (32) and (35) imply:

$$\begin{aligned} r\sum \limits _{i=1}^{N}\int _{\Omega _{i}^{-D}}\frac{\partial x_{i}^{u}(\cdot )}{\partial r}dG(\theta )=\sum \limits _{i=1}^{N}\int _{\Omega _{i}^{-D}}\left\{ \left[ C^{\prime }\left( X^{u}\right) +e^{\prime }\left( X^{u}\right) \right] \frac{\partial x_{i}^{u}(\cdot )}{\partial r}+e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial r}\right\} dG(\theta ). \nonumber \\ \end{aligned}$$
(36)

(13) follows directly from (36). \(\square \)

Proof of Corollary 3

(4) Implies that \(\frac{\partial V_{i}(x_{i}^{u}+x_{i}^{o},\,\theta )}{\partial x_{i}^{o}}=C_{i}^{\prime }(x_{i}^{o},\,\theta )\) at the solution to [RP-e]. Therefore, \(\frac{\partial V_{i}(x_{i}^{u}+x_{i}^{o},\,\theta )}{\partial x_{i}^{o}}=C_{i}^{\prime }(x_{i}^{o},\,\theta )+e_{i}\) if and only if \(e_{i}=0.\) \(\square \)

Proof of Proposition 6

Expected social losses from externalities are:

$$\begin{aligned} E\left\{ L^{s}(\cdot )\right\} = \int _{\underline{\theta }}^{\overline{\theta }}\left[ \sum _{i=1}^{N}e_{i}x_{i}^{o}(\cdot )+e\left( \sum _{i=1}^{N}x_{i}^{u}(\cdot )\right) \right] dG(\theta ). \end{aligned}$$
(37)

Let \(\lambda _{s}\ge 0\) denote the Lagrange multiplier associated the utility’s participation constraint (\(E \{ \pi ^{s} \} \ge 0 \)). It is readily verified \(\lambda _{s}=1 \) at the solution to the regulator’s problem in this setting. Pointwise optimization of the relevant Lagrangian function with respect to \(m(\theta )\) provides:

$$\begin{aligned}&\left[ 1-\lambda _{s}\right] \sum _{i=1}^{N}x_{i}^{d}(r(\theta ),\,m(\theta ),\,\theta )g(\theta )-e^{\prime }\left( X^{u}\right) \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} g(\theta )-\sum _{i=1}^{N}e_{i} \frac{\partial x_{i}^{o}}{\partial m(\theta )}g(\theta )\nonumber \\&\quad -\,\lambda _{s} C^{\prime }\left( X^{u}\right) \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )} g(\theta )+\lambda _{s} \sum _{i=1}^{N}\left[ r(\theta )\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}-m(\theta ) \frac{\partial x_{i}^{d}(\cdot )}{\partial m(\theta )}\right] g(\theta ) = 0. \nonumber \\ \end{aligned}$$
(38)

Because \(\lambda _{s}=1,\) (38) can be written as:

$$\begin{aligned} m(\theta ) = C^{\prime }( \cdot ) -r(\theta )+e^{\prime }( \cdot )+\frac{\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )}}{ \sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}}. \end{aligned}$$
(39)

Corresponding pointwise optimization with respect to \(r(\theta )\) provides:

$$\begin{aligned} r(\theta ) = C^{\prime }( \cdot )+e^{\prime }( \cdot ) +m(\theta )\left[ \frac{\sum _{i=1}^{N}\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\cdot )}\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}{\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}\right] +\frac{\sum _{i=1}^{N}e_{i} \frac{\partial x_{i}^{o}}{\partial r(\theta )}}{\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}. \end{aligned}$$
(40)

Using (40), (39) can be written as:

$$\begin{aligned} m(\theta ) = \left[ \frac{\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}{\sum _{i=1}^{N}\left[ 1+\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\theta )}\right] \frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}\right] \left[ \frac{\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )}}{\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}}-\frac{\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial r(\theta )}}{\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}\right] . \end{aligned}$$
(41)

Using (41), (40) can be written as:

$$\begin{aligned} r(\theta )= & {} C^{\prime }\left( X^{u}\right) +e^{\prime }\left( X^{u}\right) +\frac{\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial r(\theta )}}{\sum _{i=1}^{N}\left[ 1+\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\theta )}\right] \frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}\nonumber \\&+\,\left[ \frac{\sum _{i=1}^{N}\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\cdot )}\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}{\sum _{i=1}^{N}\left[ 1+\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\cdot )}\right] \frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}}\right] \frac{\sum _{i=1}^{N}e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )}}{\sum _{i=1}^{N}\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}}. \end{aligned}$$
(42)

Conclusions (i) and (ii) of the proposition follow directly from (41) and (42) because \(e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )} =e_{i}\frac{\partial x_{i}^{o}(\cdot )}{\partial r(\theta )} = 0\) when consumers do not produce electricity or when their production entails no externalities. Conclusion (iii) of the proposition follows from (41) and (42) because \(\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\cdot )}={-}1,\,\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}=\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )},\) and \(\frac{\partial x_{i}^{o}(\cdot )}{\partial m(\theta )}=\frac{\partial x_{i}^{o}(\cdot )}{\partial r(\theta )}\) when \( x_{i}^{d}(\cdot ) > 0 \) for all \( \theta \in [\underline{\theta },\,\overline{\theta } ]\) and for all \( i=1,\ldots ,N.\) \(\square \)

Proof of Corollary 4

(14) follows immediately from (41) because \(\frac{\partial x_{i}^{u}(\cdot )}{\partial r(\theta )}<0,\,\frac{\partial x_{i}^{u}(\cdot )}{\partial m(\theta )}\le 0,\) and \(\frac{\partial x_{i}^{d}(\cdot )}{\partial x_{i}^{u}(\cdot )}\in \{0,\,-1\}.\) \(\square \)

Proof of Corollary 5

The proof parallels the proof of Corollary 3. \(\square \)

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Brown, D.P., Sappington, D.E.M. On the optimal design of demand response policies. J Regul Econ 49, 265–291 (2016). https://doi.org/10.1007/s11149-016-9297-3

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Keywords

  • Electricity pricing
  • Demand response
  • Regulation

JEL Classification

  • L51
  • L94