Optimal regulation of energy network expansion when costs are stochastic

University of Groningen

Optimal regulation of energy network expansion when costs are stochastic

Zwart, Gijsbert

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Journal of Economic Dynamics and Control

DOI:

10.1016/j.jedc.2020.103945

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Zwart, G. (2021). Optimal regulation of energy network expansion when costs are stochastic. Journal of

Economic Dynamics and Control, 126, [103945]. https://doi.org/10.1016/j.jedc.2020.103945

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ContentslistsavailableatScienceDirect

Journal

of

Economic

Dynamics

&

Control

journalhomepage:www.elsevier.com/locate/jedc

Optimal

regulation

of

energy

network

expansion

when

costs

are

stochastic

R

Gijsbert

Zwart

∗

University of Groningen, Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 11 September 2019 Revised 29 May 2020 Accepted 12 June 2020 Available online 17 June 2020

JEL classification: D86 G31 L51 Keywords: Adverse Selection Dynamic contracting Real option Investment timing Principal-Agent model Regulation

a

b

s

t

r

a

c

t

Weanalyzeoptimalregulationofthegradual investmentsinenergynetworks necessary toaccommodatethe energytransition.We focusonarealoption problemwhere costs ofnew networktechnologyarestochasticand notobservabletothe regulator.Wesolve fortheregulatoryschemethatoptimallybalancestimelyinvestmentswithrentextraction inthisdynamicagency context.Wethenapplythismethodologytoasituationinwhich investmentcanbeeitherintraditionalnetworktechnology,withobservablecosts,orusing aninnovativenetworktechnologyforwhichthereisasymmetricinformationoncosts.The optimal choicetradesoff the potential benefitsofcheaper expansionwith thecosts of overcominginformationfrictions.

© 2020TheAuthor(s).PublishedbyElsevierB.V. ThisisanopenaccessarticleundertheCCBYlicense. (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

The transitionto a more environmentally friendly energymarket requires large investments, not only by competitive energy producers, but also by monopolistic firms owning and operating power and gas grids. For example, to accom-modatelarge amounts ofsmall-scalerenewable electricity productionatconsumers’ homes(suchassolar panels, micro-CHPs), system balancing functions may have to be decentralized in smart grids, that require deployment of new tech-nologies.Bertoliniet al.(2018), forexample,studyhow smart grids facilitateinvestment indomestic solar panels,while

Sidhuetal.(2018)explorethecostsandbenefitsofintroducingenergystorageinthegrid,allowingbetteraccommodation ofdistributedgeneration.Likewise,achoiceforcarboncaptureandstoragewillrequireinvestmentsinpipeline infrastruc-turestotransportthegreenhousegasfromitsindustrialsourcestostoragelocations.Andlargescaledeploymentofelectric vehiclesnecessitatesmoreelaborateinfrastructureforroadsidecharging.

Acommoncharacteristicofthesenewinfrastructures isthatthey arestillunderdevelopment. Asinallnew technolo-gies,costs fordeploymentareexpectedtofall inthelongerrunasmoreexperienceisgained,buttheirevolutionwill be

R I have benefited from useful comments from seminar participants at the Brescia workshop on Investments, Energy, and Green Economy, in April 2019.

I also thank two anonymous referees for their insightful comments.

∗ _{Corresponding author.} E-mail address: g.t.j.zwart@rug.nl https://doi.org/10.1016/j.jedc.2020.103945

0165-1889/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

uncertainintheshorterrun(EPRI,2011;Sidhuetal.,2018).Also,investmentsinthenewenergysystemswillbemade pro-gressively,startingatthoselocationswithhighbenefits,andexpandingascostsarereduced.Optimalinvestmentwilloccur gradually,takingintoaccounttheoptionvalueofwaitingforimprovedinformationoncosts.Suchinvestmentproblemsare welldescribedbyrealoptions,inparticularwithongoingexpansionopportunitiessuchaspioneeredine.g.Pindyck(1988). The speed atwhich network firms will adopt such new technologies will,however, not only be guided by changing costsandsocietalbenefits.Theenergyandnetworkownersarenaturalmonopolies,andtheyaretypicallyregulatedbythe government.The regulatoryframeworks that thesefirmsoperateinwill determinetheir allowed revenues, andtherefore willalsoaffecttheirinvestmentbehavior(Cambinietal.,2016).

Inthispaper,weanalyzetheproblemofoptimalregulationoflong-runinfrastructureinvestmentprocessesinthe pres-enceofasymmetricinformationoncosts.Todoso,weconsidera realoptionmodelwithcontinuousinvestmentina dy-namicprincipal-agentsetting.The networkfirm,who istheagent, privatelyobservesthe stochasticinvestmentcosts, and decides onitsinvestments.The governmentregulator,whoisthe principal,doesnot observethe stochasticcosts,butcan observecapacitybuiltby thenetwork. Theregulator will determinetheallowed revenues(consistingof asetof capacity dependent fees),whichwilldrivethenetwork’s investmentbehavior. Weaskwhat setoffeestheregulator shouldsetto elicitsocialsurplusmaximizinginvestmentbehaviorastheseunobservedcostsfallstochastically.1

Intheabsenceofinformationasymmetry,therealoptionproblemoffindingsurplusmaximizinginvestmentwasstudied byPindyck(1988)(seealsoDixitandPindyck,1994,chapter11),whoshowedthat capacityshouldbeexpandedwhenever costs dropbelowa capacitydependent threshold:thehighercapacity,the lowercosts needtofalltoincrease that capac-ityfurther. Dobbs(2004)considered the regulatoryproblemof incentivizinginvestment by amonopolist usingpricecap regulation,butdidnotspecifytheasymmetryofinformationthatnecessitatessuchregulation.Morettoetal.(2008) inves-tigatehowvariousregulatoryschemescompareinrealoptioninvestmenttiming.Guthrie(2006)providesasurveyofsuch applicationsofrealoptionsinvestmentinregulationcontexts.

Regulationofdiscrete,lumpyinvestmentwasstudiedfromamechanismdesignperspective inBroerandZwart(2013), whoassumedstaticasymmetricinformationonaconstant costlevel.There,thestochasticvariable,demand,wasassumed tobe observablebyboth regulatorandthemonopolist.Anextension tocontinuousinvestmentinnetworkexpansion was consideredinWillemsandZwart(2018),whoinadditionlookedatafinancingconstraintrequiringtheinvestmentstobe paidoutofusagefeesonanongoingbasis.

Our firstcontribution here isto address the regulatory problemwith dynamic asymmetric information: the regulator cannot observecosts whichare continuously evolving. Weusethe frameworkfordealingwithsuch dynamicadverse se-lectionincontinuous-time realoptionsmodels developedinArve andZwart (2016).Related techniqueswere exploredin

BergemannandStrack(2015);Maeland(2006).Inpreviousrealoptionsapplications,discreteinvestmentswereconsidered. Inthispaperwedemonstratehowthemethodologycanbeexpandedtodealwithcontinuousinvestmentmodels.

We then apply this methodology to studyhow optimal regulation should drive investment in a greenfield situation, wherethereisachoicebetweenatraditionaltechnology,withpubliclyobservablecosts,andaninnovativetechnologywith asymmetricinformationonstochasticcosts. Ineithercase,aninitial investmentwillbemade;afterthatthenetworkwill begraduallyexpandedascosts(whichfortheinnovativetechnologyareunobservabletotheregulator)declinefurther.

Thechoicebetweentraditionalandinnovativetechnologyisinpartdrivenby thepotential costandqualitybenefitsof theinnovative technology:both initialcosts andthereal-optionvalue ofexpansion mightbemorefavorableforthe new technology.Informationasymmetry,ontheotherhand,distortsdecisionsinthedirectionofthetraditionaltechnology,for whichtheregulatorcanincentivizeinvestmentwithouttheneedforpayingeconomicrentstothefirm.Wefindthatinthe optimalregulation,withasymmetricinformation,networkswithlowcostswillalwaysbeincentivizedtopursuethe innova-tivetechnology,providedthat thisalternativealsodominateswithoutinformationasymmetry.Forhighercostrealizations, ontheotherhand,theinformationasymmetrymakesithardertooutperformthetraditionaltechnology.Evenifthehighest possiblecost-levelwoulddominatethetraditionaltechnologyinasituationwithoutinformationasymmetry,optimal regu-lationmaycallforresortingtoinvestmentintraditionaltechnologyinthosecasesinstead,toenabletheregulatortolimit theallowedrevenuestothenetwork.

Inthe following,we willfirst introduce themodel.In section 3we solvethe modelinthe first-bestreal option situ-ation, along the linesof Pindyck (1988).Then we turn to theeffects of informationasymmetry. Ifthe regulator (whois theprincipal)cannotobservecosts, butthenetwork (theagent)can,we firstshowthat still,theregulatorcould designa setof feesthatinduces the agentto followthesame,first-bestreal-optioninvestment rule.Thisiscostly, though,asthe agentreceivesinformationrentsfromitsadvantagedposition.Theprincipalcandoabetterbalancingactbetweenefficient investmentandrent extractionbysetting feesthat delayinvestment, comparedto thereal-optionbenchmark.This delay isgreater foran agenthavinghigherex-antecost levels.In section5, weturnto theapplication. Fora specific,constant elasticitywelfare function,we explore thetrade-offsin termsofoptionvalues andinformationrentsbetweentraditional andinnovativetechnologies.Weconcludewithadiscussionoftheresultsandtheassumptionofregulatorycommitment.

1 We assume commitment on fees for the regulator. If the regulator could not commit, additional distortions would be introduced. The agent would have

incentives not to respond truthfully to protect future rents, resulting in a ratchet effect ( Laffont and Tirole, 1988 ). We also assume that there is no danger of renegotiation. Renegotiation typically increases agency costs ( Battaglini, 20 07; Bester and Strausz, 20 01; Hart and Tirole, 1988; Laffont and Tirole, 1990 ).

2. Model

Letusconsideracontinuoustime investmentmodelfora firmwithinitialcapacityQ0 andtheopportunity to contin-uouslyincreasethiscapacityby makingirreversibleinvestments.ThemarginalinvestmentcostsCt arestochastic,evolving accordingtoageometricBrownianmotion,

dCt=

μ

Ctdt+

σ

Ctdz.

Wehaveinmindthecaseinwhichthedrift

μ

<0,sothatcostsareexpectedtofall,butfluctuatearoundthatexpectedpath withvolatility

σ

.Thesocial benefitsofhavingcapacityQare measuredbyaflowofsurplus,w(Q)dt,withw(Q) increasing butconcave.Marginalbenefitsdecreasetozero,reachedatsomelevelQ∗whichmaybeinfinite.

Irreversibleinvestmentwill becarried out,on behalf ofa principal(whois theregulator),by an agent(theregulated network)who has private information on the stochastic costs Ct.The principal will contract withthe agent att=0 on capacityinvestments,capacityQiscontractable.2_{Theprincipal’sprioroncostsC}

0attime0isthatthesearedistributedon [CL,CH]accordingtocumulativedistributionF(C0),withdensity f=F.AsinArveandZwart(2016),weadoptthetechnical assumptionthat thedistributionof thelogarithmofC0 hasamonotone hazard rate.3 The principaldoesnot observethe evolutionofCt, buttheparameters ofthestochastic process,

μ

and

σ

,are assumedtobe commonknowledge, asisthe discountrater.

The principalcan use a setof capacity-dependentfees toencourage optimalbehavior. In generality, thesefees might consistofp(Q),acapacity-dependentflowfee,

φ

(Q) acontributiontowardsinvestmentcosts wheneverQisincreased,and anupfrontlumpsumcontractfee

φ

0.4Theprincipalisassumedtomaximizethepresentvalueofsurplusw(Q),minusthe paymentsmadetotheagent:foragivenchoiceoffees(p,

φ

,

φ

0),theprincipal’sexpectedsurplusisgivenby

W=E



_∞

t=0e

−rt

₍

_w

₍

_Qt

₎

_{− p}

₍

_Qt

₎₎

_{dt− e}−rt

_φ

₍

_Qt

₎

_dQt−

_φ

₀



_{. (1)}

Thefirsttermencapsulatestheexpectedbenefitsofnetworkcapacity,anditsexpansionovertime ascosts fall.Theother partsarethepresentvalueoftheexpectedfees:thecontinuousflowp(Q),theadditionalfeespaidoutwhenQisincreased bytheagent, andtheupfrontlumpsum

φ

0.Theexpectation hereisnot onlyontheevolution ofcosts Ct overtime,but alsoovertheinitialvalueofcostsC0,thelatterknowntotheagentbutunknowntotheprincipal.Here,dQtistheprocess ofinvestmentsincapacityexpansion.

Thisprocessofcapacityexpansionischosenbytheagent,whodeterminesaprivatelyoptimalthresholdcostlevelC¯

(

Q

)

. Theagentwillexpandcapacitywhenevercosts dropbelowthiscapacity-dependentthresholdlevel.Theoptimalthreshold fortheagentdependsonthestructureoftheexpostfees,p(Q)and

φ

(Q).Theagentchooses investmentsoastooptimize hisexpectedutilityU(Q0,C0),i.e.thesumoftheexantefee

φ

0,andthecontinuationutilityV(Q0,C0),

U

(

Q0,C0

)

=

φ

0+V

(

Q0,C0

)

=

φ

0+EC0



_∞ t=0e −rt_p

₍

_Q t

)

dt+e−rt

(

φ

(

Qt

)

− Ct

)

dQt



, (2)

withdQt governedbytheagent’soptimalchoiceofinvestmentthresholdC¯

(

Q

)

.Theexpectationishereconditionalonthe stochasticprocessstartingatC0.Theagent’sparticipationconstraintisU(Q0,C0)≥ 0foranyC0∈[CL,CH].

3. Firstbest

Asabenchmark,wefirst analyzethecasewithoutasymmetricinformation.Theprincipalwillinthatcaseinstructthe agenttoadoptthefirst-bestinvestmentrule,andremunerate theinvestmentcosts,leavingtheagentzerorents.The prin-cipal’sobjectivethenistoprescribeaninvestmentrulemaximizingexpectedtotalsurplus,

W

(

Q0,C0

)

=EC0



_∞ t=0e −rt_w

₍

_Q t

)

dt− e−rtCtdQt



, (3)

withthefirsttermrepresentingtheflowofgrosssurplus,andthesecond termthecapitaloutlaysascapacityisexpanded byanincrementdQt.Theoptimalruleis(Pindyck,1988)toinvestwhenevercostsCdroptoacapacity-dependentthreshold levelC¯

(

Q

)

.Theoptimizationconsistsoffindingthatoptimalthreshold.

2 There is no loss in generality in assuming all contracting to take place at t = 0 . Since the only verifiable information to the principal will be the capacity

additions by the agent, there is no benefit in postponing part of contracting to later stages, as any future contracting contingent on such capacity additions can already be specified at t = 0 . There could be a downside to later contracting, if this also means respecting the agent’s participation constraints at those later dates. This would impose additional constraints on the principal’s maximization problem.

3 Commonly, the monotone hazard rate assumption is made on the type itself, to avoid solutions involving bunching. In our case, due to the multiplicative

nature of the stochastic process, we need to make that assumption on the log of C 0 .

4 There is some redundancy in this formulation: both the investment contribution _{φ( Q ) and the lump sum φ}

0 may be subsumed, by suitable shifts, into

the flow fee p ( Q ), keeping the present value of the fees the same. Such a regulation with only a flow fee is more likely to be observed in practical cases, as we will explain in section 5 . We keep the general fee structure here for clarity of the theoretical exposition.

To maximize W(Q0, C0), we first evaluate the right-hand side using standard real optionmethods (see e.g. Dixit and

Pindyck, 1994).In theregion inwhich noinvestment occurs,C>C¯

(

Q

)

,total welfareW(Q,C) satisfiesa Hamilton-Jacobi-Bellmanequation, rW

(

Q,C

)

=w

(

Q

)

+

μ

CWC+ 1 2

σ

2_C2_W CC,

wheresubscriptsdenotepartialderivatives.Thisequationissolvedby5

W

(

Q,C

)

=w

(

_rQ

)

+g

(

Q

)

Cλ−_.

forsomefunction g(Q),and

λ

− thenegativerootofthecharacteristicequation r=

μλ

+1

2

σ

2

λ

(

λ

− 1

)

.Thefirsttermhere equalsthepresentvalueoffuturesurplusfromcurrentcapacity,whereasthesecondtermrepresentstherealoptionvalue ofexpandingcapacityfurtherwhenCtdrops.Tofindg(Q),weusetheboundaryconditionattheinvestmentthresholdC¯.At thethreshold,wehaveavalue-matching(continuity)condition,

WQ

(

Q,C¯

)

=C¯, or

wQ

r +gQ

(

Q

)

C¯

λ−_=C¯_.

stating that investment occurswhen marginal costs of addingcapacity, C¯, equal themarginal benefits, consisting ofthe presentvalueoftheadditionalsurplusflows,andthechangeinthevalueoftheoptiontoexpandcapacityevenfurtherin thefuture.

To determine g(Q), we use the boundary condition at large capacity, where there is no value to further investment,

g

(

Q∗

)

₌0_,sothat g

(

Q

)

= Q∗ Q



wQ

(

q

)

r − ¯C

(

q

)



¯ C

(

q

)

−λ−_dq.

Now,optimizingWoverthethresholdC¯

(

Q

)

isachievedbyoptimizingg(Q).Usingpointwisemaximizationoftheintegrand, wefindthefirst-ordercondition

−

λ

−wQ_r

(

Q

)

−

(

1−

λ

−

)

C¯

(

Q

)

=0, or6 ¯ C

(

Q

)

=

_λ

λ

− −− 1 wQ

(

Q

)

r . (4)

Since the factor

λ

₋/

(

λ

−− 1

)

<1, investment is delayed beyond the moment at which marginal investment costs equal marginalincrease inpresentvalueofthesurplusflow.Thedifferenceisaccountedforby theoptionvalue ofdelayingthe investment(McDonaldandSiegel,1986).

4. Agencyproblem

Now consider the case where the principal needs to delegate the capacityexpansion to an agent who has superior knowledgeabouttheevolvingcosts Ct.Theprincipalneedstoremuneratetheagentfortheinvestmentcostsincurred,but alsotoincentivizetheagenttoinvestatthecorrectcostthresholdsC¯

(

Q

)

.

Asoutlinedabove,weassumethattheprincipalcanuseasetofcapacity-dependentfeestoencourageoptimalbehavior:

p(Q), a capacity-dependentflow fee,

φ

(Q) a contribution towards investment costs, and an upfront contract fee

φ

0. The principal’sexpectedsurplusforagivenstructureofthosefees,W(Q),isgivenby(1).The principal’sproblem,therefore,is to choosea menu offees(p,

φ

,

φ

0), so astooptimize hisexpected surplus, subjectto aparticipation constraintforthe agent,andsubjecttotheagentchoosinghisprivatelyoptimalinvestmentthresholdC¯

(

Q

)

giventhosefees.

4.1. Incentivizingfirst-bestinvestment

As a preliminarystep,let usconsider ascheme that succeedsin incentivizinginvestment atthe first-bestinvestment thresholdC¯

(

Q

)

.Supposefirsttheprincipaloffersonlyasinglestreamoffeesp(Q),tobepaidtotheagent.Theagent,after havingacceptedthiscontract,willchoosetoinvestinawaythatmaximizesitsutility,

V

(

Q0,C0

)

=EC0



_∞ t=0e

−rt_p

₍

_Qt

₎

_{dt− e}−rt_CtdQt



_.

5 Using the boundary condition that as C → ∞ , W should not diverge. 6 The same result can also be found using a smooth pasting condition on W

Thisproblemisthesameastheoneencounteredinthesolutionofthefirst-bestproblem(4),exceptforthesubstitutionof

p(Q)insteadofw(Q).Wethereforehavethattheagent’sprivatelyoptimalthresholdinthiscasewillbegivenby ¯

C

(

Q

)

=

_λ

λ

−

−− 1 pQ

(

Q

)

r . (5)

Comparison withthe first-best investment threshold (4)shows that we need that the marginal benefit for the agentof increasingcapacityQcoincideswiththemarginalwelfare,orinother wordspQ=wQ.

Clearly,oneexampleofsuchaschemeincentivizingfirst-bestinvestmentwouldbeallocatingtheentirebenefitsofthe networktothe agent, p

(

Q

)

=w

(

Q

)

.Inthat casetheagentisresidualclaimantto allsocial benefitsandcosts,andhence chooses the totalwelfare maximizing investment strategy. Such a schemeon its own doesnot, of course, maximize the principal’sutility, asitleaves all surplustothe agent. Totransfer partof thatutility to the principal,the schemewould needtobecomplemented withafixed fee,eitherintheformofanexantefee

φ

0,ora fixedcash flowtobe subtracted fromthefee p(Q).This constructionamounts toa sell-outcontract,wheretheprincipal transfersthe entireprojecttothe agentinreturnforafixedpayment.

Thissell-out contractcan, however,not transfer all ofthesurplus tothe principal.The obstacle tothat is theagent’s participationconstraint.Totalsurplusinthefirst bestdependsontheinitialcost levelC0: thelowerthisinitialcostlevel, thefaster theagentwill investandbenefit fromlarger flows.Hence, low-cost agents earnhighersurplusthan high-cost agents.Sinceinitialcostsarenotobservabletotheprincipal,however,theonlywaytomakethesell-outcontractacceptable toanyagenttype,istochargeanupfrontfee

φ

0(ortheequivalentconstantshiftinfeesp(Q))thatleavesthehighest-cost agent,whostartsatCH,withzerorents:

φ

0=−W

(

Q0,CH

)

,

sincewiththesell-outcontract,p

(

Q

)

=w

(

Q

)

,totalrentsfromtheflowfeealoneareequaltototalwelfareforthis highest-costtype.Withthisupfrontpayment,lower-costagentsearnarentW

(

Q0,C0

)

− W

(

Q0,CH

)

≥ 0.

Addingapayment

φ

(Q),afeepaidperunitofnewinvestment,doesnotalterthisanalysismaterially.Itisstraightforward tocheckthatthesameconclusionsfollow,butnowwithpQ

(

Q

)

+r

φ

(

Q

)

substitutedforpQ.

Theprincipalcandobetterbydroppingtherequirementthatinvestmentisatthefirst-besttiming.Theoptimalscheme consistsofamenuthatdifferentiates amonginitialagenttypesC0.Itdelaysinvestmentthresholdsforhigh-costagents,to reducetheinformationalrentsenjoyedbylower-costagents.Wenextturntotheanalysisofthatoptimalscheme.

4.2.Theoptimalscheme

The optimal scheme can be expressed in terms of a menu of fees, {p(Q; C0),

φ

0(C0)|C0 ∈[CL, CH]}, that is incentive compatible:anagentofinitialcostsC0 willmaximize itsprivateutilityU(Q0,C0)bychoosing,att=0,thecombinationof feesp(Q; C0)and

φ

0(C0)intended forthat initialcost levelC0.Forsimplicity,andwithout lossofgenerality, let usagain assumetheremunerationforcapacityinvestment

φ

(Q)iszeroforallcontractsinthemenu.It isstraightforwardto check thatasintheprevioussubsection,anynon-zero

φ

(Q)isequivalenttoachangeinpQ.

Theprincipal’sproblemisthentomaximize itsownutility,theexpectedvalueofflowofwelfarew(Q)minus the pay-mentsmadetotheagent,takingintoaccountthattheagentchoosestheinvestmentthresholdC¯

(

Q

)

inresponsetothefees itisoffered, maximizingits utility U(Q0,C0).The principal’soptimizationover menusissubjectto theincentive compat-ibility (i.e.out of themenu ofsets offees onoffer, agents preferto opt forthe setof feesintended fortheir individual initialcostlevelC0)andtotheagent’sparticipationconstraint,U(Q0,C0)≥ 0foranyinitialcostlevelC0.Bysubstitutingthe expressionfortheagent’scontinuationutility,(2),intheprincipal’ssurplus,equation(1),wehave

W= max {p(Q;C0),φ0(C0)} CH CL



EC0



 _∞ t=0e −rt_w

₍

_Q

₎

_{dt− e}−rt_C tdQt



− U

(

Q0,C0

)



dF

(

C0

)

, (6)

s.t.incentivecompatibilityandparticipationconstraints.

Notethat,again,expectationisnotonlytakenoverthefuturepathofcostsCt(capturedbytheexpectationoperatorE),but alsoexplicitlyoveritsinitialvalueC0,distributedaccordingtoF(C0).

Toanalyzethisproblem,wefirstrephrasethequestionintermsoffindingtheoptimalthresholdC¯

(

Q;C0

)

foreachinitial costlevel.Wealreadyknow,fromequation(5),thatthechoiceofathresholdisequivalenttothechoiceofafeep(Q)upto anintegrationconstant.Inaddition,weneedtoexpresstheagent’scontinuationutility,V(Q,C0),intermsofthatthreshold. Todoso,wenotethat,foranyC,theagent’scontinuationutilityV(Q,C)satisfiesanHJBequation,withvaluematchingand smooth-pastingdeterminingtheboundaryconditionsatC¯

(

Q

)

_,analogoustotheequationssolvedbytotalwelfareWinthe first-bestanalysisinsection3(smooth-pastingholdsbecausetheagentchoosesthethresholdC¯

(

Q

)

tooptimizeitsvalue):

rV

(

Q,C

)

=p

(

Q

)

+

μ

CVC+ 1 2

σ

2_C2_V CC (7) VQ

(

Q,C¯

(

Q

))

=C¯

(

Q

)

(8) VQC

(

Q,C¯

(

Q

))

=1 (9)

FromthisHJBequation,wecanderivethefollowingrelationlinkingvalueV(Q,C)andthethreshold:7

Lemma1. PrivatelyoptimalinvestmentfortheagentmakessurethatitscontinuationutilityV(Q,C)satisfiesthecondition

VQC

(

Q,C

)

=



C ¯ C

(

Q

)



λ−−1 (10) forallC_<C¯

(

Q

)

.

We can nowturnto the principal’sex anteproblemof specifyinga thresholddesignedforeach agenttype, C¯

(

Q;C0

)

. Onceweknowthatthreshold,wecanchooseaflowoffeesp(Q;C0)implementingthatthresholdaccordingtoequation(5).

φ

0(C0) isgoingtodothejobofextractingtherentsgeneratedfromtheexpostfees,upto aninformationrentnecessary toinducetheagenttorevealhisinitialcostlevelC0(i.e.,incentivecompatibility).

Restricting to direct incentive compatible mechanisms, we can rewrite the principal’s optimization (6) as choosing a menuofexantefeesandinvestmentthresholds

φ

0

(

C0

)

,C¯

(

Q;C0

)

,ratherthanthefeesp(Q;C0).

Tosolvetheconstrainedoptimization,we usethestandard(Mirrlees)observationthatoptimalityoftruthfullyrevealing

C0impliesthat dU dC0

(

Q,C0

)

=

∂

V

∂

C0

(

Q,C0

)

. (11)

Moreover, the resulting optimalinvestment thresholdshould be decreasing in the ex ante type C0. We will ignore this monotonicityconditionfornow,andcheckthat itholdsexpost, dueto ourmonotonicityassumptiononthehazard rate. We proceed by using(11),in conjunctionwitha binding participation constraintforthe highestcost type, to writeU= −C_H

C₀ VCdC.Pluggingthisintotheprincipal’sobjectivefunction(6)anddoingapartialintegrationthenyields

W = CH CL



EC0



_∞ t=0e −rt_w

₍

_Q

₎

_{dt− e}−rt_C¯

₍

_Q;C0

₎

_dQt



₊F f

(

C0

)

VC

(

Q,C0

)



dF

(

C0

)

. (12)

WeneedtooptimizethisexpressionbychoosinganappropriateinvestmentthresholdC¯

(

Q;C0

)

foreachvalueofinitialcost levelC0.Todoso,firstweevaluatetheexpectedcontinuationutility,thefirstexpectedvaluetermundertheintegral,fora giveninitialcostlevelC0andagivenchoiceofthresholdC¯

(

Q

)

.(Thisisthesamecomputationasinsolvingthefirstbestin

section3.) EC0



_∞ t=0e −rt_w

₍

_Q

₎

_{dt− e}−rt

₍

_C¯

₍

_Q

₎₎

_dQt



₌ w

(

Q

)

r + Q∗ Q



C0 ¯ C

(

q

)



λ−



wQ r − ¯C

(

q

)



dq. Forthesecondtermintheintegrandin(12)wecanuselemma1towrite

VC0

(

Q,C0

)

=−  Q∗ Q VQC0dq=− Q∗ Q



C0 ¯ C

(

q

)



λ−−1 dq.

Combiningthesetwoexpressions,itisstraightforwardtocomputetheoptimalthresholdC¯

(

Q_;C0

)

foreachvalueofC0∈[CL,

CH],8

Proposition1. Theoptimalmenuresults inadifferentinvestmentthresholddependingontheagent’sinitialcostlevelC0,given

by ¯ C

(

Q;C0

)



1+ 1 C0 F f

(

C0

)



=

λ

−

λ

−− 1 wQ r . (13)

Comparingwiththefirst-bestthreshold(4),weseeasimilarexpressionexceptfortheappearanceofthefactorinvolving thedistributionoftheinitialcostsF(C0).Forthelowesttype,C0=CL,F

(

CL

)

=0sowegetnodistortionthere,

¯

C

(

Q;CL

)

=

λ

−

λ

−− 1 wQ

r .

Higher cost types,in contrast,get a distortion,leadingto lower thresholds thanthe first-best ones. Thismeans that the optimalregulatory schemeinduces theagentto delayinvestment comparedto the first-bestreal-optionbenchmark. The reason is that lower cost typeshave to be awarded an information rent to reveal their true exante type; the delay in investmentreducestherentsobtainedby claiminghigherthanactualcosts initially.Thisinturnreducestherequiredrent toinduceagentstoreporttheirinitialcoststruthfully.

7 proof in the appendix

8 and verify that it satisfies the required monotonicity in C 0

Itremains tocompute theactualmenuofcontracts,includingtheupfrontfee

φ

0 andtheex-postfeep(Q).Thisfeecan simplybechosenaccordingtoequation(5),leadingto



1+ 1 C0 F f

(

C0

)



pQ=wQ, (14)

makingtheagentresidualclaimanttothetotalutilitygeneratedmoduloacorrectionfactor.Thecorrespondingupfrontfee

φ

0isthensetsoastomakesurethattotalutilityequalstherequiredinformationrentU=−

CH

C0 VCdC.

5. Application:traditionalorinnovativeinfrastructure?

Letusnowapply theseresultstoanswerthefollowingquestion. Considera greenfieldsituationinwhichthe network firmhastobuildanewnetworkinfrastructure,say,connectinganewly builtneighborhoodtothegrid.Suppose thereare twoalternativetechnologiestodoso.

Thefirst involveschoosing atraditional,well-understood, technology,forwhich thereisno asymmetry ofinformation betweenthe networkfirmandtheregulator.The secondalternativeis anewtechnology,forwhichthenetworkfirm has superiorinformation,andwhichthereforesuffersfromagencyfrictions.

Bothtechnologieshavestochasticallyfallingcosts,Cold

t andCtnew,respectively.Theseevolveaccordingtogeometric Brow-nianmotions,with

μ

old_,

_μ

new_{≤ 0,andlikewisevolatilitiesthatmaybetechnology-specific.}9_{Attime zero,thetechnology}

ischosen.Foreithertechnology,dependingontheinitialvalueC₀old,new_,thefirmwillfirstbuildaninitialcapacityQ₀old,new_,

andwillsubsequentlykeeponexpandingcapacity,usingthatsametechnology,ascostsfall.

Thequestionfacingtheregulatoris,whenitshouldaskthefirmtobuildinfrastructureusingthetraditional,wellknown technology;andwhen itshould insteadencourage thefirm to startbuildingthe newgrid technology,which mighthave lower cost, or higherlikelihood of future cost declines, butwhich may necessitaterents left to the firm in view of the informationasymmetrythatneedstobeovercome.

Weanalyzetheregulator’schoiceinthespecificcasewhenthesocialsurplusfloww(Q)isofconstant-elasticitytype, w

(

Q

)

=w· Q1−γ

1−

γ

with0_<

γ

_< 1 theinverseelasticity(asin Dobbs,2004; WillemsandZwart, 2018). Thefactor wmeasuresquality,and couldbe differentforthetwo technologies,wold _versuswnew_{.Forinstance,thenewtechnologymightbemoreconducive} toaccommodatingenvironmentallyfriendlygenerationinsidethenetwork,makingwnew_>wold_.

Foranalyzing theoptimalinvestmentstrategylet usfirst,asabenchmark, establishtheoptimalinvestmentrules,and totalwelfare,inthefirst-bestcaseswithoutinformationasymmetry,foreithertechnology.

Lemma2. Inthegreenfieldinvestmentprojectwithconstantelasticitysocialsurplus,optimalinvestmentintherealoptioncase withcurrentcostlevelC0involvesbuildinginitialcapacityQ0,

w· Q0−γ =rC0

λ

−− 1

λ

− ,

andexpandingcapacitywhenthethresholdC¯

(

Q

)

₌ λ−

λ−−1

w·Q−γ

r iscrossed. Total(expected)surplus (includingthecostsC0Q0 of

theinitialinvestment)is

W

(

C0

)

=₁₋

γ

_γ

·

_γ

₍

γ

_λ

(

λ

−− 1

)

−− 1

)

+1· C0Q0.

Inthesymmetricinformationsetting,theregulator’soptimalchoicebetweentraditionalornewtechnologyisthenclear: pickthealternativeyieldingthelargestexpectedsurplus.Inthespecialcasewhenbothtechnologies’costsfollowthesame stochasticprocesses,sothat

λ

old

− =

λ

new− ,wehavethat Wnew≥ Wold _{isequivalentto} wnew

wold



Cold 0 Cnew 0

1

−γ ≥ 1.

Clearly,intheabsenceofdifferencesinquality,wnew_=wold_{,itisoptimaltochoosethelowestcosttechnology,forwhich} initialinvestmentishigher.Modificationsfordifferencesinthestochasticparametersarestraightforward,andreflect poten-tialdifferencesintheexpansionoptionvalueofeithertechnology.

Letusnow add asymmetryofinformation onthecostsCnew

t forthenewtechnology,where theregulator onlyknows thedistributionofinitialcosts,F

(

Cnew

0

)

,boundedbetweenlowestandhighestlevelsCLandCH,respectively.Thetraditional technologyiswellknown,anddoesnotsufferfrominformationalasymmetry.Wecanuseourresultsfromproposition1for

computingtheoptimalregulationifweonlyfocusonthenewtechnology:afirmofinitialcostsCnew

0 isincentivizedtoinvest asifitscostswerelargerbyafactor

α

(

Cnew₀

)

≡ 1+ 1 Cnew 0 F f

(

C new 0

)

.

In our setting withconstant elasticity surplus, this means that initial capacity is distorted downwards compared to its symmetricinformationcounterpartinlemma2,to

wnew_Q˜−γ

0 =r

α

(

C0new

)

C0new

λ

−− 1

λ

− ,

andtheregulator’sexpectedsurplus,whichisnowreducedbecauseoftheinformationrentslefttothefirm,becomes,using theanalogouscalculationasinlemma2

¯ Wnew₌

γ

1−

γ

·

γ λ

γ

(

−

λ

+−− 11−

)

γ

CH CL

α

(

Cnew 0

)

C0newQ˜0dF

(

C0new

)

.

ComparedtoasymmetricinformationaverageoverthedifferentinitialcostslevelsCnew

0 ,theintegrandhereisreducedby afactor

α

(

Cnew

0

)

1−1

γ _<1(notetheextrafactorof

α

comingfromthedistortedcapacitychoice,Q˜0).

A simple first attempt to making the trade off betweenusing the traditional established technology versus the new technology withitsinformation rents,isto comparethe two expectedsurpluses,Wold _andW¯new_{,anddirect thefirm to} choosewhichevertechnologygiveshighestsurplus.

Theregulatorcandobetter,however,byallowingthefirmtomakethattechnologychoiceitself,usingitssuperior infor-mationontheactual valueofCnew

0 .Providedthat thelowest costlevelforthenewtechnology,CL,givesabetteroutcome thanusingthetraditionaltechnology,orWnew

₍

_C

L

)

>Wold,theoptimalregulationwillinvolveatleastsomeprobability of deploymentofthenewtechnology.

Proposition2. Inthegreenfieldinvestmentprojectwithconstantelasticitysocialsurplus,optimalregulationinvolvesinvestment inthenewtechnologyforlow-costtypes,andtraditionalinvestmentforhighercostslevelsofthenewtechnology.Thetransition occursataninitialcostlevelC˜forthenewtechnologyforwhich

Wnew

₍

_α

₍

_C˜

₎

_C˜

₎

_=Wold

₍

_Cold 0

)

.

Comparedtothefirst-besttrade-off,inthepresenceofinformationasymmetry,weneedtoswitch betweentraditional andnew technologiesguided by the virtual costs of the newtechnology, which are inflated by the factor

α

(

Cnew

0

)

. This means thatthetraditional technologywillbe selectedmorefrequentlythan withsymmetric information.The intuitionis thatoptingforthetraditionaltechnologyisoptimalnotonlyifitallowsbuildingmorecapacityatlowercosts (efficiency); havingthetraditionaltechnologyasan outsideoptionalsoallowstheregulator tocutdownontheinformationrentsthat needtobepaidtothefirmincaseswherenewtechnologycostsarelow,andthenewtechnologyisselected.

Sincethemark-upfactor

α

isequaltooneforthelowestcosttype,Cnew

0 =CL,andincreasesforhighercostrealizations, wehavethefollowingcorollary.

Corollary 1. IfforCnew

0 =CL,thenew technologydominatesthetraditionaltechnology,thenalsowithinformationasymmetry,

optimalregulationwillimplementthattechnologyforlowcostrealizations.Incontrast,evenifforhighinitialcostsCH thenew

technologyismoreefficientthantheoldone,optimalregulationmaycallfortraditionaltechnologyinthatsituation.

Lowcostsdonotleadtodistortionsintheoptimalscheme,andwehavetheefficientoutcome.Decisionsforhighcosts aredistorted,ontheotherhand,andaretakenasifcostsareevenhigherbyafactor

α

(CH)>1.Whatmattersforchoiceof oldversusnewtechnologyisnotCH,but

α

(CH)CH ascomparedtotheoldtechnology’scostlevelC0old.10

Howcouldtheoptimalregulationschemebeimplementedinpractice?Inmanycountries,regulationinvolvessettingan allowed levelofrevenuesforthefirmtocollectfromitsconsumersthroughitscumulative tariffsforallservicesitoffers. Thislevelmaybecomputedbymultiplyingaregulatoryrateofreturnwiththefirm’sregulatoryassetbase.11_Inthecontext

of the currentapplication, we may envisagean existing network firm that is allowed regulated revenues fromcharging tariffsforallitsservicestoallitscustomers.Wecanthenincorporatetheoptimalschemeforthecurrentgreenfieldproject intosuchacountry’sregulatoryschemebydeterminingthecontributiontothefirm’sallowedrevenuesfromthisparticular greenfieldproject.

If the firm chooses investing in traditional technology, accordingto the optimal investment trajectory Qold

t , it should receiveadditionalallowed revenuesequalto rtimesitsinvestmentoutlays,making goodits capitalcosts,butearningno rentsontop.If,ontheotherhand,itoptsforthenewtechnology,theproject’scontributiontoallowedrevenueswilldepend

10 Of course, taking into account potential differences in stochastic process for either technology (embodied in _λ

−), as well as quality differences, the w -factor.

onthecapacityinvestmentQnew

0 .DependingonthechoiceofQ0new,contributionstothefirm’sallowedfuturerevenueswill besettothesumofaQ-dependentcontribution

p

(

Q

)

=

_α

₍

wnew· Q1−γ Qnew

0

)(

1−

γ

)

, andafixed contribution, φ0(Qnew0 )

r .Here,

α

(

Q0new

)

isdetermined tobe themark-up factorforinitial costsC0new consistent withtheoptimalchoiceofinitialcapacityQnew

0 ,i.e.solving Cnew 0 =

_λ

λ

− −− 1 wnew_·

₍

_Qnew 0

)

−γ

α

(

Cnew 0

)

r .

Theallowed revenuescomponentequalto p(Q) determines howallowed revenuesincrease asthe assetbaseQincreases, givingtherightincentivesforoptimalinvestmentincapacityexpansionasfuturecostsdecrease.Thefirm receivespositive expectedrents from thisQ-dependent component. These rents can be computed analogously to the computationof the first-bestsocialsurplus,andareequalto

V=

γ

γ

− 1

(

γ λ

−

γ λ

−

γ

−+1

)

wnew_·

₍

_Qnew 0

)

1−γ

α

(

Qnew 0

)

r .

The

φ

0/rcomponentto theallowedrevenuesdoesnot changeascapacityislaterincreased:itisacomponenttototal allowedrevenuesthatstaysfixed overtime.Itisusedtoshifttotalrentsforthefirm suchthatthey becomeequaltothe requiredinformation rents. In particular, forCnew

0 =C˜– the maximal level of costs above which thefirm should refrain frominvestinginthe newtechnology,butshould adoptthe traditionaltechnologyinstead – totalinformationrents need tobe zero.For thatcost-level, therefore,thiscomponentis negative,

φ

0=−V.Forlower C0new, there needto be positive informationrents,toensureincentivecompatibility,and

φ

0willbelargerthan−V.

6. Discussion:commitmentversuslearning

Thesolutiontotheoptimalregulationproblemmakesclearthatonlytheinitialcostlevel,C0,playsaroleinthesetting oftheinvestmentthresholdsC¯

(

Q

)

;subsequent updatesofthesecosts,asaresultofthestochasticprocess thatCfollows, donotaltertheoptimalpolicy.Indeed,wereC0commonknowledgeexante,aswesawinsection4.1,theregulator could achieve thefirst-best resultbyusing thesell-out contractwith pQ=wQ, andextractingallrents through anappropriate ex-antefixedfee(or,alternatively,afixedshiftinthestreamoffeesp).

Incontrast,theinformationadvantagethattheagentobtainsatlaterdates,asaresultofstochasticmovementsofCtafter

t=0,doesnotresultinanyadditionalrentstotheagent,andasaconsequencedoesnotnecessitatefurtherdistortionsin thethreshold.Theintuition forthis(see Baron andBesanko, 1984;Es˝o andSzentes,2017) isthat thoselater updatesare ex-antesymmetricinformation:the expectation value ofanyinformationrentsthat the agentmight obtaininthe future canalreadybeextractedinitiallywhensigningthecontract.

Acrucialassumptioninthecurrentmodelisthat theregulator isabletocommit,ex-ante, totheregulatoryschedule. Asiswell knownfromtheliteratureondynamicmechanismdesignandtheratcheteffectinregulationtheory(Baronand Besanko,1984;LaffontandTirole,1988),itiscrucialthat theregulatorcanalsocommitnotto usetheinformationlearnt fromthe agent’schoice ofcontract (the choice ofthreshold) to later adjustthresholds to some value closerto the first-best.12 _{Ofcourse,ex-post suchan adjustmentwouldbe optimal:once informationalasymmetry hasbeenresolved, there}

isnolonger anyneedfordistortedinvestment.From an ex-antepoint ofview,however,suchlearning, andupdating the incentiveschemeon the basis ofthat learning, would be detrimentalto the regulator’sobjective.The reasonis that the agentwillanticipatesuch regulatorybehavior,and, knowingthatfuturerentswillbe extractedbytheregulator basedon theagent’s earlierrevelation of private information, theagent willin response requirehigher exante transfers in order to truthfullyreveal its initial cost level. Byspreading distortions across periods andignoring potential updatesbased on informationrevealedthroughtheagent’schoices,totalwelfarecostsareminimized.

Regulatorycommitmenthasbeenwellappreciatedasbeingofhighimportanceintheenergysector. Intheabsenceof such commitment,firms wouldrefrain fromsinking investments,for fear ofex postregulatory expropriation. Inthe US, firmsareprotectedfromsuch‘regulatorytakings’throughlegalmeasuresfollowingtheSupremeCourt’sHopeGasruling.In Europe,CERRE(2013)arguesthatthesettingoftheregulatoryassetbasecanprovidesuchregulatorycommitment.13_Since,

aswesaw,ourschemeofregulatoryfeescan beimplementedintermsofcontributions tothisregulatoryassetbase,this providessomecomfortthatcommitmentcanbefeasibleinpractice.

12 Even aside from private information, the exogenous changes of the stochastic process could also give rise to commitment problems for the regulator.

See for instance Di Corato (2013) who in a real option setting studies the risk of expropriation of a company’s assets by a foreign host country, and the resulting distortions in profit sharing contracts this engenders.

13 In particular, “There is general agreement that RABs are an effective commitment device for natural monopoly elements of infrastructure companies –

ProofofLemma1.. WehavethatV(Q,C)satisfiestheHJBequation rV

(

Q,C

)

=p

(

Q

)

+

μ

CVC+ 1 2

σ

2_C2_V CC.

WecantakeQ-andC-derivativesfromthisequationandrewritetoobtain rCVQC

(

Q,C

)

=

μ

C

_∂

∂

C

(

CVQC

)

+ 1 2

σ

2_C2

∂

2

∂

C2

(

CVQC

)

,

sothefunctionCVQC itselfalsosatisfiesanHJBequationwithoutasourceterm. Furthermore,wehavethesmooth-pastingconditiononC¯

(

Q

)

,

VQC

(

Q,C¯

(

Q

))

=1

whichequivalentlycanbewrittenas ¯

C

(

Q

)

VQC

(

Q,C¯

(

Q

))

=C¯

(

Q

)

.

From thesetwo equations,we learn that, given Q, above the boundary, C>C¯

(

Q

)

, the function CVQC satisfies an HJB equation andon thatboundary,we knowitsvalue. Fromthat information,we canfindan explicitexpressionforCVQC:it shouldbeequalto CVQC

(

Q,C

)

=E

_¯ C

(

Q

)

e−rT

=_C¯

₍

_Q

₎



C ¯ C

(

Q

)



λ− fromwhichfollowstheexpressioninthelemma,

VQC

(

Q,C

)

=



C ¯ C

(

Q

)



λ−−1 . 

ProofofProposition1.. Theprincipaloptimizesitsobjectivefunctionwhichiswrittenasequation(12),

W = CH CL



EC0



_∞ t=0 e−rtw

(

Q

)

dt− e−rtC¯

(

Q;C0

)

dQt



+F f

(

C0

)

VC0

(

Q,C0

)



dF

(

C0

)

. Inthemaintext,wehaveexpressionsforthetwopartsinthisintegral,

EC0



_∞ t=0e −rt_w

₍

_Q

₎

_{dt− e}−rt

₍

C¯

(

Q

))

dQt



= w

(

Q

)

r + Q∗ Q



C0 ¯ C

(

q

)



λ−



wQ r − ¯C

(

q

)



dq. and VC0

(

Q,C0

)

=−  Q∗ Q VQC0dq=− Q∗ Q



C0 ¯ C

(

q

)



λ−−1 dq. Thecombinationofthesetwopartsleadsto

W = CH CL



w

(

Q

)

r +  Q∗ Q



C0 ¯ C

(

q

)



λ−



wQ r − ¯C

(

q

)

− ¯ C

(

q

)

C0 F f

(

C0

)



dq

dF

(

C0

)

. ThiswecanoptimizepointwiseforeachpossibleC0,leadingto

¯ C

(

Q;C0

)



1+ 1 C0 F f

(

C0

)



=

_λ

λ

− −− 1 wQ r .

Moreover,fromtheassumedmonotonicityofthelogofthehazardrate,thisthresholdisdecreasinginC0. 

Proof of Lemma 2.. With w

(

Q

)

=w·Q1−γ

1−γ , we have wQ=w· Q−γ. Plugging this into the first-best cost-threshold from

section3,wehaveinvestmentthreshold(equation4) ¯

C

(

Q

)

=

_λ

λ

−

−− 1 w· Q−γ

r .

Totalwelfareattheinvestmentthresholdis W

(

Q,_C¯

₍

_Q

₎₎

₌ w· Q1−γ

(

1−

γ

)

r+g

(

Q

)

C¯ λ−_, with g

(

Q

)

= _∞ Q ¯ C

(

q

)

−λ−



wq−γ r − ¯C

(

q

)



dq

=



w

λ

₋ r

(

λ

−− 1

)

−

λ− w r

(

λ

−− 1

)

Qγ (λ−−1)+1

γ

(

λ

−− 1

)

+1.

We get total welfare for the greenfield project by subtracting construction costs to bring initial capacity to threshold, ¯

C

(

Q

)

Q₌ λ−

λ−−1

w·Q1−γ

r . 

ProofofProposition2.. TheprincipalnowhastheoutsideoptionofobtainingWold_{,thesocialwelfareforthetraditional} technology,andhastodeterminetheoptimalcostlevelC˜abovewhichshutdownofthenewtechnologywillbepreferred; thismeansoptimizing

C˜ CL ECnew 0



_∞ t=0e −rt_wnew

₍

_Q

₎

_{dt− e}−rt

_α

₍

_Cnew 0

)

C¯

(

Q;C0new

)

dQt



dF

(

Cnew 0

)

+

(

1− F

(

C˜

))

Wold overC¯aswellasC˜.Here,

α

(

Cnew

0

)

isthemark-up,1+ 1 Cnew

0

F

f

(

C0new

)

.Clearly,themaximumobtainsforC˜suchthatthevirtual welfare(i.e.,welfareasifcostlevelisinflatedby

α

(

Cnew

0

)

)isequaltoWold,or,fromlemma2

γ

1−

γ

·

γ

(

γ

λ

(

−

λ

− 1−− 1

)

+

)

1

α

(

˜ C

)

C˜Q˜=Wold_, withC˜= λ− λ−−1 wnew_{· ˜Q}−γ r .  References

Arve, M., Zwart, G., 2016. Optimal procurement and investment in new technologies under uncertainty. Mimeo.

Baron, D.P. , Besanko, D. , 1984. Regulation and information in a continuing relationship. Information Economics and Policy 1 (3), 267–302 .

Battaglini, M., 2007. Optimality and renegotiation in dynamic contracting. Games and Economic Behavior 60 (2), 213–246. doi: 10.1016/j.geb.2006.10.007 . Bergemann, D. , Strack, P. , 2015. Dynamic revenue maximization: A continuous time approach. Journal of Economic Theory 159, 819–853 .

Bertolini, M., D’Alpaos, C., Moretto, M., 2018. Do smart grids boost investments in domestic pv plants? Evidence from the italian electricity market. Energy 149, 890–902. doi: 10.1016/j.energy.2018.02.038 .

Bester, H. , Strausz, R. , 2001. Contracting with Imperfect Commitment and the Revelation Principle: The Single Agent Case. Econometrica 69 (4), 1077–1098 . Broer, P. , Zwart, G. , 2013. Optimal regulation of lumpy investments. Journal of Regulatory Economics 44, 177–196 .

Cambini, C., Meletiou, A., Bompard, E., Masera, M., 2016. Market and regulatory factors influencing smart-grid investment in europe: Evidence from pilot projects and implications for reform. Utilities Policy 40, 36–47. doi: 10.1016/j.jup.2016.03.003 .

CEER, 2017. Ceer report on investment conditions in european countries. Council of European Energy Regulators, C17-IRB-30-03. CERRE, 2013. Regulatory stability and the challenges of re-regulating. Centre on Regulation in Europe.

Di Corato, L. , 2013. Profit sharing under the threat of nationalization. Resource and Energy Economics 35 (3), 295–315 . Dixit, A. , Pindyck, R. , 1994. Investment under uncertainty. Princeton University Press .

Dobbs, I.M. , 2004. Intertemporal price cap regulation under uncertainty. The Economic Journal 114, 421–440 .

Dumas, B. , 1991. Super contact and related optimality conditions. Journal of Economic Dynamics and Control 15 (4), 675–685 . EPRI, 2011. Estimating the costs and benefits of the smart grid. Electric Power Research Institute Technical Report.

Es ˝o, P. , Szentes, B. , 2017. Dynamic contracting: An irrelevance theorem. Theoretical Economics 12 (1), 109–139 .

Guthrie, G. , 2006. Regulating infrastructure: The impact on risk and investment. Journal of Economic Literature 44, 925–972 .

Hart, O.D., Tirole, J., 1988. Contract Renegotiation and Coasian Dynamics. The Review of Economic Studies 55 (4), 509–540. doi: 10.2307/2297403 . Laffont, J.-J. , Tirole, J. , 1988. The dynamics of incentive contracts. Econometrica 56 (5), 1153–1175 .

Laffont, J.-J., Tirole, J., 1990. Adverse Selection and Renegotiation in Procurement. The Review of Economic Studies 57 (4), 597–625. doi: 10.2307/2298088 . Maeland, J., 2006. Valuation of an irreversible investment: Private information about a stochastic investment cost. Mimeo.

McDonald, R. , Siegel, D. , 1986. The Value of Waiting to Invest. Quarterly Journal of Economics 101 .

Moretto, M. , Panteghini, P.M. , Scarpa, C. , 2008. Profit sharing and investment by regulated utilities: A welfare analysis. Review of Financial Economics 17 (4), 315–337 .

Pindyck, R.S. , 1988. Irreversible investment, capacity choice, and the value of the firm. American Economic Review 78 (5), 969–985 .

Sidhu, A.S., Pollitt, M.G., Anaya, K.L., 2018. A social cost benefit analysis of grid-scale electrical energy storage projects: A case study. Applied Energy 212, 881–894. doi: 10.1016/j.apenergy.2017.12.085 .