Green investment under time-dependent subsidy retraction risk

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Journal of Economic Dynamics & Control

journalhomepage:www.elsevier.com/locate/jedc

Green investment under time-dependent subsidy retraction risk

Verena Hagspiel

^a,∗

, Cláudia Nunes

^b

, Carlos Oliveira

^c,d

, Manuel Portela

^b

aDepartment of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Trondheim 7491, Norway

bDepartment of Mathematics and CEMAT, Instituto Superior Té cnico, Av. Rovisco Pais, Lisboa 1049-001, Portugal

cISEG-School of Economics and Management, Universidade de Lisboa Rua do Quelhas 6, Lisboa 1200-781, Portugal

dREM-Research in Economics and Mathematics, CEMAPRE Portugal

a r t i c l e i n f o

Article history:

Received 31 October 2019 Revised 13 May 2020 Accepted 16 May 2020 Available online 28 May 2020 Keywords:

Real options Renewable energy Policy risk

Non-homogeneous poisson process Renewable energy support schemes Feed-in-tariffs

a b s t r a c t

Drivenbyambitioustargetstoreducegreenhousegasemissionsmanycountrieshavein- troducedsupport schemes to accelerateinvestments in renewableenergy. However, in recentyearsexperience showedthat,overtime,retractionofsupportschemes becomes morelikely.Thishasasevereeffectoninvestmentbehaviour.Inthispaperwestudythe effectofapotentialsubsidyretractionofafeed-intariff (FIT)oninvestmentinrenewable energycapacity,whereweexplicitlyaccountforthefactthatthelikelihoodofpolicyre- tractionmaychangeovertime.We showthatthe rangeofFITs,forwhichitisoptimal toinvestimmediately,decreasesthelongerasubsidyhasbeeninplace.IftheFIToffered istoosmalland/orthesubsidyhasbeeninstalledtoolongago,itisoptimalforthefirm towaitwithinvestmentuntil thesubsidyiseventuallyretractedand freemarketcondi- tionsprovetobeprofitableenough.Furthermore,weshowthatwhetherapolicymaker aimingatacceleratinginvestmentprefersinvestorstoconsiderretractionrisktobetime- dependent ornot,depends onhowmuch time haspassed since thesubsidy hasbeen introducedatthemomenttheinvestorconsidersinvestmentforthefirsttime.

© 2020TheAuthor(s).PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBYlicense.

(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Increasingtheshareofrenewableenergy(RE)productiontotheoverallenergymixisrecognizedascriticalinreaching ambitioustargetstoreducegreenhousegasemissions(EuropeanCommission,2015).Duetothederegulationofthemajor- ityofelectricity marketsworldwide, itisprivate investorswithan objectiveofmaximizingprofitthat decidewhetherRE projectsarebuiltornot(AbadieandChamorro,2014).Manycountrieshaveintroduceddifferenttypesofsupportschemes aimed at acceleratinginvestments inRE. Governments therewith, want to ensure competitivenessof RE production and encourage investments. By 2019 nearly all countries worldwide have employed RE support policies and targets (REN21, 2019). Feed-inpricingpolicies,i.e.feed-in tariffs(FITs) andfeed-in premiums (FIPs),havebeeninstrumental inencourag-

∗ Corresponding author.

E-mail address: verena.hagspiel@ntnu.no (V. Hagspiel).

https://doi.org/10.1016/j.jedc.2020.103936

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/ )

ing REprojectsworldwide, sincethey provideastableincometogenerators andhelpincrease thebankabilityofprojects (IRENAetal.,2018).By2019atleast111countrieshaveemployedsometypeoffeed-inpolicies(REN21,2019).

Policyuncertaintyintheformofsuddenrevisionsorretractionsofrenewablesupportschemes,however,hadadramatic impactoninvestmentsinREprojectsduringrecentyears.Thistypeofpolicyriskrepresentsasignificantchallengeforactors intheREsectorcurrently(Whiteetal.,2013)andhindersneededinvestments.Policyuncertaintyisfrequentlymentioned as(oneof)thekeyreasonsforthelackofinvestmentsinRE(Whiteetal.,2013,ClimateFinanceLeadershipInitiative,2019)¹ OnecountrywheretheeffectofpolicyriskrecentlyhaddramaticimpactonREinvestmentsisAustralia.In2019investment inREprojectsacrossthecountry“hassloweddramatically” afterarecord-breakingtwoyears” accordingtotheCleanEnergy Council,Australia’sREassociation.Investorswerecitingnationalpolicyuncertaintyasoneofthetwokeyfactorsforpulling backfrominvestments(TheSydneyMorningHerald,2019).AnotherrecentexamplecomesfromthesolarindustryinChina, whereonJune1,2018authoritiessuddenlyandveryunexpectedly,strictlylimitednewsolarinstallationsthatwouldqualify forFITs.Analystsestimated,thatthissuddenpolicychangesetastoptoatleast20GWofsolarprojectsthatwereexpected tobebuiltinChinain2018(TheEconomist,2018).

Fromaninvestor’sperspectiveexperienceshowsthatovertime,retractionorrevisionofsupportschemesbecomemore likely.Oneofthemainreasonsforthisistechnological development.Themainmotivationforpolicymakerstograntsup- port to renewable electricityprojects inthe formof subsidiesisto ensure their competitivenesscompared totraditional powersources,likecoalorgas.Asrenewableelectricitytechnologiesmature,investmentintheseresourcesbecomesprof- itableevenwithoutgeneroussupportfromgovernments.Therefore,firmsmayexpectthatgovernmentseventually wantto terminate thesesupportschemes orrevisethem inwaysthat make themlessgenerous. Arecentexample isChina, that significantlycutthetotalsizeofitsrenewablepowersubsidiesfor2020comparedtoprioryears².China’sgovernmentalso announcedto stopsubsidisingonshorewind capacityby 2021andisreportedlyconsidering toscaleback oreven abolish subsidiesfortheoffshorewindpowersectorin2022“asthecostsofrenewablescontinuetofallandtheindustrybecomes moreeconomicallycompetitive” (CXTech(2020)).Other reasonsforincreasinglikelihoodofpolicyretractionovertime are diminishingpublic funds,changesofgovernmentsorthefactthat REcapacitygoalshavebeenreached. Thereare several cases wherebudget constraints ledregulators to retract orsignificantly reduce provided FITs. Oneprominent exampleis Spain,wherea newgovernmentannounceda drasticretroactivelyimplementedcut ofsubsidiestoelectricity suppliersin 2013as a consequenceof a EUR26bn tariff deficit, which wasbesides the effects ofthe financial crisis alsoa result of a toogeneroussystemofsubsidiesprovidedbythe previousgovernment(FinancialTimes,2013).Similarly, Italyprovided generoussubsidiesfundedbypowerconsumersleadingtosignificantinvestmentinREfrom2005to2012buthadtocurtail theirrenewable supportschemessubstantiallyin2015 inordertorelieveconsumers,whowerestrugglingwithhighelec- tricitybillsintheaftermathoftheeconomiccrisis(TheWallStreetJournal,2014).AlsoregulatorsintheUKwereforcedto drasticallycuttoogeneroussubsidiesin2015(DepartmentofEnergy&ClimateChange,2015).

While FITs have long been considered to be the most effective scheme for accelerating development of RE sources (Coutureand Gagnon, 2010, del Rioand Mir-Artigues, 2012, Ritzenhofen andSpinler, 2016), we currently can observe a shiftfromtariff-basedinstrumentsto competitiveauctions (REN21,2019).Thismaylead investorstoperceivean increase intheriskofretractionofcurrentlyinstalledFIT schemes.Thereforeweproposetostudytheeffectofapotentialsubsidy retractiononinvestmentinREcapacityassumingthatthelikelihoodofpolicyretractioninnotconstantovertime.

WedevelopamodelinwhichaprofitmaximisinginvestorhastheoptiontoinvestinanREproject.Thecurrentsubsidy schemeprovides investorsin REprojects witha fixedFIT foreach unit produced, therewithshielding them frommarket uncertainty.However,investmentsarethreatenedbyapotentialsubsidyretraction,whichisassumedtobecomemorelikely thelongerthesubsidyschemehasbeenprovidedtoinvestors.Aftersubsidyretractionelectricityproducedmustbesoldon thespotorfuturesmarketatapricethatvariesovertime.Someoftheexamplesofsuchretroactivesubsidyrevisionshave occurredinseveralcountriesinthelast decade(REN21, 2015). ExamplesareSpain, Belgium, theCzechRepublic, Bulgaria andGreece(BoomsmaandLinnerud,2015).Inordertorelateourresultstoarealisticcaseofgreeninvestmentwepresent acasestudyconsideringinvestmentinanonshorewindprojectusingthemostrecentavailabledataforthecaseofEurope.

FITsandelectricitypricedataistakenfromGermany,wheredifferenttypesofFITsupportschemeshavebeenappliedsince 1991³Finally,we present extensionsto our model,that account for differenttypes of limits forthe subsidy period.We first considerthecasethattheregulatorannouncesthesubsidytobeprovidedforalimitedtimeperiod.Second,weanalysethe case,wheresubsidyisannouncedtoberetractedincaseelectricitymarketpriceraisestoacertainlevel.

Applying arealoptions approachallows usto accountforimportantcharacteristics ofREinvestments.RE investments entaillargesunkinvestmentcosts,whichareoftenspecifictotheconsideredproject.Second,theprojectvaluedependson uncertainfutureframeworkconditions,likefluctuatingelectricitypricesorchangingsupportschemes.Third,investorshave theoptiontopostponeinvestmentsifcurrentframeworkconditionsdonotjustifyimmediateinvestment.

OurresultsshowthattherangeofFITs,forwhichitisoptimaltoinvestimmediately,decreasesthelongerasubsidyhas beeninplace.IftheFITofferedistoosmalland/orthesubsidyhasbeeninstalledfortoolong,itisoptimalforthefirmto

1Recent examples are, for instance, The Climate Finance Leadership Initiative, a group of banks and asset managers assembled by Michael Bloomberg, that released a report outlining that one of the problems of securing investment for low-carbon energy in emerging-market countries is the unpredictability of government policies ( Climate Finance Leadership Initiative, 2019 ).

2https://oilprice.com/Latest- Energy- News/World- News/China- Slashes- Renewable- Subsidies.html .

3The German Electricity Feed-in Act (1991) was, in fact, the first green electricity feed-in tariff scheme in the world.

waitwithinvestmentuntilthesubsidyhasbeenretractedandfreemarketconditionsareprofitableenough.Specifically,we derivetheoptimalelectricitymarket pricethresholdthattriggers investmentinthiscase.Wefindthat increasing market pricevolatility(drift)discourages(encourages)investment,alsoforthecasesthatinvestmentismadewhenthesubsidyis stillprovided.

Furthermore,weshowthatconsideringsubsidyretractionrisktobetime-dependentcansignificantlyaffecttheoptimal investment strategy compared to the casewhen subsidy retraction risk isconsidered constant over time. While we can confirm earlierresearch inthat policy risk discourages investment (see, e.g., Dalby etal., 2018, Boomsma andLinnerud, 2015 orFuss etal., 2008 forsimilar results), itis not straightforward to concludewhethera policy maker, who aims to accelerateinvestment,wouldprefer investorstoconsidersubsidyretraction riskto betime-dependentornot.In factthis dependsontheFITprovidedandthetimepassedsincethesubsidyhasbeenintroduced.

Finally,weconcludethat settingalimit forthe periodduringwhichthesubsidyisprovided bothinthe formoftime oraspecificmarketprice,discouragesinvestment.Thebestfromapolicymakerspointofviewwiththemainobjectiveto accelerateinvestmentistoannounceasupportschemeforanunlimitedperiodoftime.

Realoptionstheoryhasbeenincreasinglyappliedtoinvestmentproblemsintheenergysectorundermarketuncertainty andpolicy changein recentyears.Early contributions towork on policy andinvestmentfrom areal options perspective havefocused ontaxpolicyuncertainty((Dixit andPindyck,1994,Chapter9),HassettandMetcalf,1999,PawlinaandKort, 2005).Anothergroup ofpapersstudied theeffectclimatechangepolicy uncertaintyrepresentedasexogenous eventthat createsuncertaintyinthecarbon priceoninvestmentinRE sources(Yang etal.,2008, Fussetal.,2008). Aratherrecent strandofliteraturehasanalysedtheeffectofpolicyuncertaintyintheformofrandomprovision,revisionorretractionofa subsidy(see,e.g.,Boomsmaetal.,2012,BoomsmaandLinnerud,2015,RitzenhofenandSpinler,2016,EryilmazandHomans, 2016andChronopoulosetal.,2016).Similartoourcase(RitzenhofenandSpinler,2016)and(BoomsmaandLinnerud,2015) studytheeffectofapotentialfutureretroactivelyappliedsubsidyretraction.RitzenhofenandSpinler(2016)consideracase whereregulatorsmaydecidetoswitchfromafixed-priceFITtoafree-marketregime.Theyfindthatuncertaintyregarding futureregulatoryregimesdelaysorevenreducesinvestmentactivityforFIT levelsnearelectricitymarketpricesandhigh probabilitiesofanimminentregimeswitch.Similarly,BoomsmaandLinnerud(2015)concludethattheprospectsofsubsidy terminationslowsdowninvestmentsifsubsidyitisretroactivelyapplied,butspeedsupinvestmentsifitisnot.Weconfirm theirresultthatpotentialretroactivesubsidyretractionreducesinvestmentactivity.

However,all oftheaforementionedpapersanalysingpolicy uncertaintyintheformofsubsidyretractionconsiderthat thelikelihoodofsubsidyrevisionsisconstantovertime.Wecontributetothisstrandofliteraturebyallowingtheprobability ofsubsidyretractiontodependonthetimesincethesupportschemewasoriginallyintroduced.Specifically,welookatthe casethatinvestorsperceivesubsidyretractiontobecomemorelikelythelongersubsidyhasbeenprovided.

There is also a growing body of empirical work studying policy risk in the power sector. The majority of this work focuses on analyzing how policy changes and/or policy uncertainty impact investments in renewables (see, for exam- ple,Eyraud et al.(2011), LüthiandWüstenhagen (2012), Linnerudet al.(2014), Karneyeva andWüstenhagen (2017)and Sendstadetal.(2020)).Here,Sendstadetal.(2020)specifically,investigatestheeffectofretroactive policychangesonthe investmentdecisions inrenewable energyandfinds that suddenunexpected policy changes deterfurtherinvestment ac- tivityinrenewables.Weare,however,not awareofanypaperthat attemptstomeasureinvestors’perceivedpolicy riskin thepower sector nor attempts to measure policy risk forthe power sector in general. A recentstrand of literature,ini- tiated by the work of(Baker et al., 2016), attempts to measure policy uncertainty in a more general sense referring to the probability of political decisions, events, or conditions significantly affecting current and future business conditions.

Bakeretal.(2016)developedanewindexofeconomicpolicyuncertaintybasedonnewspapercoverage frequencyforthe UnitedStates. Using firm-level data they findthat policy uncertainty isassociated withgreater stockpricevolatility and reducedinvestmentandemployment inpolicy-sensitive sectorslike defense,health care,finance, andinfrastructurecon- struction. We are however, not aware ofany work usingthis methodology to measure policy uncertainty forthe power sector.

Fromamoretechnicalpointofview,policyuncertaintyisusuallymodelledintheliteraturebyahomogeneousstochastic process withdifferent states,each one representing a differentlevel of subsidy (or the retraction ofit). Technically, our contributiontothestateoftheartistoconsiderannon-homogeneousstochasticprocesstomodelpolicyuncertainty.Inthis casetheinvestmentproblemreliesontheresolutionofatime-dependentoptimalstoppingproblemwheredifferentregimes areconsidered.Fromapuremathematicalpointofview,thistypeofoptimalstoppingproblemsarestudiedbyOliveiraand Perkowski(2019).Inthistheoreticalpaper,thesystemofHamilton-Jacobi-Bellman(HJB)equationsisderived andtechnical conditionssatisfiedbyvaluefunctionsarepresented.Sinceoursetup fitsintheonepresentedintheprevious paper,we take intoaccount some ofthe technicalresults ofOliveira andPerkowski (2019). Ourpaper contributesto theliterature bypresentingthespecificsolution ofsuch astoppingproblemandderivesexplicitexpressions forthevalue functionand thresholdcurvesfortheinvestmentproblemconsidered.Thishas,tothebestofourknowledge,notbeendonebefore.

Theremainderofthispaperisorganisedasfollows.Section2introducesabenchmarkmodel,forwhichwepresentthe solutionprocedureinSection3.Comparativestaticsresultsforthebenchmarkmodelare presentedinSection 4.We then checktherobustnessofourmodelinSection5,relaxingakeyassumptionwemakeinthebenchmarkmodel.Thesolution forthe robustnessmodel ispresented in Section 5.1, andthe results are analysed inSection 5.3, basedon a casestudy introducedinSection5.2.WefinallypresentrelevantextensionsinSection6,andconcludeinSection7.

2. Model

WestudyaprofitmaximisingfirmthatconsiderstheoptiontoinvestinaREproject.Inordertoaccelerateinvestment innewREprojectsregulatorsarecurrentlyprovidingasubsidyintheformofafixedfeed-in-tariff (FIT)thatoffersasubsidy paymentof(^p+F)^perMWhofelectricityproducedduringthecontract.Investorsareawareofthepossibilitythat,atsome random pointin time,

ν

^{,regulatorsmayretract the currentsubsidyscheme,which would,asa}consequence, exposethe firm tomarketrisk. SimilartoBoomsma etal.(2012),BoomsmaandLinnerud(2015),Ritzenhofen andSpinler(2016) and Dalbyetal.(2018),weareconsideringasinglesubsidyrevision.Afterthesubsidyhasbeenretractedthefirmwillsellthe electricityatmarketpriceperMWhofelectricityproduced.

Weconsiderthatretractionofthesubsidysupportschemewilloccuratan exponentiallydistributedrandomtime,de- noted by

ν

^{, withintensity}

λ

^{(t), that is assumedto be increasing withtime. Therefore, retractionof the subsidysupport}

schemebecomesmorelikelyastimegoesby.Forillustrationpurposeswe assumethefollowingtime-dependentintensity

λ

(^s)=

λ

0s.Then

F_ν

(

^u

)

^{=1−e−u∞λ0sds=1−e−λ0u2/2.}

Mostofthequalitativeresultsthat weshowinthepapercarryoverforotherintensityfunctions,aslongastheyarenon- decreasingint.

In what follows, the stochastic process

θ

^{provides usthe} information regarding the state of the subsidy. Specifically,

θ

=1whilethesubsidyisstillactiveand

θ

=0,otherwise.Therefore,

θ

^s=

1, s<

ν

0, s≥

ν

Duetothestructureof

ν

^,

θ

^{isnot,apriori,aMarkovprocess,unlessonetakesintoaccount the}informationregardingthe momentwhen thesupport schemewasimplementedandthecurrenttime. Therefore,fromnowon,we considerthe bi- dimensionalpair(t,

θ

^{),wheretrepresentsthetimeelapsedsincethemomentwhenthesupportschemewas}implemented.

Thisnewprocessis,infactaMarkovprocess.

Inlightoftheinformationintroducedinthepreviousparagraph,onecanupdatethedistributionof

ν

^{takingintoaccount}

that

ν

> t. We therefore, introduce a new random variable, represented by

ν

^{t. The} distribution function for

ν

^{t can be}

computedastheconditionaldistributionoftherandomvariable

ν

^{,asweshowinthenextequation:}

P

( ν

^t>t+s

)

=P

( ν

>s+t

| ν

>t

)

=e^{−tt+sλ0udu}.

Therandomvariable

ν

^{t denotesthe(random)timeremaining untilretractionofthesubsidy,giventhatithasbeenactive}

inthelasttunitsoftime.Additionally,onecannoticethattheexpectedtimeof

ν

^{t isequalto}

E[

ν

^t]=

√2

π

1−

t

λ

⁰

e^λ0t2/2

λ

^{0 , (1)}

where^{denotes the}distribution functionofa normaldistribution,withmean0andvariance 1.Wenote that E[

ν

^{t] isa}

decreasingfunctionoftandverifiesE[

ν

^t]→0.Thishighlightsthatthesubsidyretractionislikelytooccursoonerwhent increases.

We furthermore assume that the electricity market price, hereafter denoted by P=

{

^Ps:s≥0

}

, follows the following stochasticdifferentialequation:

dPs=

μ ( θ

^s

)

^Psds+

σ ( θ

^s

)

^PsdWs, P0=p (2)

where

μ ( θ )

=

0,

θ

⁼¹

μ

^,

θ

^=0,

σ ( θ )

=

0,

θ

⁼¹

σ

^,

θ

^{=0 (3)}

andW=

{

^Ws:s≥0

}

^{isa Brownianmotion.By}construction,theprocess P incorporatestheinformationregardingthe ac- tivenessofthesubsidy.Therefore,aslongastheFITisprovided,theunitpriceisfixedandequaltop.Inthisperiodthefirm earns p+F foreachunit producedby unitoftime.We interpretthisFIT asa premiumFofferedontopoftheelectricity priceatthebeginningoftheplanninghorizon,i.e.p+F.Uponretractionofthesubsidy,thefirmearnstheelectricitymar- ketprice,whichisassumedtofollowageometricBrownianmotion(asin,forexample,Fletenetal.(2007),Ritzenhofenand Spinler (2016),Boomsma andLinnerud(2015) andChronopoulos etal.(2016)), withdrift

μ

^{andvolatility}

σ

^{.We further-}

more,assumethattheprojectissufficientlysmallnottoaffectlong-termelectricityprices,i.e.theproducerisaprice-taker, followingamongothers(Boomsmaetal.,2012),(RitzenhofenandSpinler,2016)and(BoomsmaandLinnerud,2015).

Giventhedefinitionof

μ

⁽

θ

^{) and}

σ

⁽

θ

^{), wemake the}simplifyingassumptionthat atthemomentofsubsidyretraction, themarketpriceisequaltop,independentlyofthe timeelapsedsincethebeginning oftheplanninghorizon.Inpractice however,once the subsidy isretracted,the company will sellthe electricity atthemarket price, which is not knownto thefirm beforehand(asthe marketpriceevolvesrandomlyover time).Therefore,inthat case, thepriceoftheelectricity

atthe momentoftheinvestmentwillbe arandomvariable,withadensityfunctionthat we,ingeneral, cannotcompute explicitly.Thismeansthatwecannotfindclosed-formexpressionsforthewaitingregionandthevaluefunction.Makingthis simplifyingassumptionallowsustoderiveanalyticalresultsfortheproblem,whichwillserveasanimportantbenchmark case.

InSection5werelaxthisassumption,allowingtheelectricitymarketpricetoevolveaccordingtoageometricBrownian motionfromthebeginningoftheconsideredplanninghorizonon.Inthiscase,weareonlyabletofindthewaitingregion andthevaluefunctionconditionalonthemarketpriceattheretractiontime.Then,tosolvethe(unconditionaloptimisation) problem,one needs to compute theexpectedvalue ofthe value function,whichis ahighlynon-linear function.Forthat reason,thesecomputationsare,ingeneral,onlypossibletodofromanumericalpointofview.However,wewillshowthat thequalitativeresultsofthebenchmarkmodelarerobusttothisassumption.

Furthermore,inthespecial casethat the intensityofretractionis constant,the involvedexpressions andexpectations turnouttobesimpler,andweareabletoprovideananalyticalresultforthevaluefunctionevenwhenthepriceisevolving fromthebeginning(seeSection5.1forthederivationsofthiscase).

Theprofitfunctionofthefirm,herebydenoted_θ^{(.),isgivenby}

θ

(

^Ps

)

⁼

K

(

^p+F

)

−C

(

^K

)

,

θ

=1

KPs−C

(

^K

)

,

θ

^{=0 (4)}

whereK>0representstheannualproduction,C(.)denotestheproductioncosts,andp+F representstheFIT.Uponinvest- mentthefirmneedstopayasunkcostofI.

Theinvestmentproblemofthefirmcanthenbeformulatedasthefollowingoptimalstoppingproblem:

V_θ

(

p,t

)

=sup τ E_θ,p,t

∞

τ e^−rs

_θ

(

^Ps

)

^ds−e^−rτI

(

^K

)

, (5)

whereweusetheindex

θ

∈{0,1}todenotethecurrentstateofthesubsidy(non-activeoractive,respectively),andristhe exogenouslygivendiscountrate.Furthermore,duetothetime-dependenceoftheintensityrate

λ

0t,atwhichthesubsidyis retracted,weneedtotaketimeintoaccountintheoptimisationproblem(5).Forthatreason,V_θ hasanexplicitdependency onthevariablet,whichweincludenotation-wise.

Fromthemathematicalpointofview,thisisanoptimalstoppingproblem,whereboththeprofitfunctionandtheprice dynamicsareaffectedbyaswitchingregime.In(OliveiraandPerkowski,2019),theauthorsconsideramoregeneralset-up, wherethey deduce thesystemof Hamilton-Jacobi-Bellman(HJB)equations that characterise thevalue function andthey proveexistenceanduniqueness ofsolutiontotheequations.Sinceinthatcasetheauthorsdealwithan exitproblem,we re-writeourproblem(5)inamoreconvenientwayasfollows

V_θ

(

p,t

)

=sup τ E_θ,p,t

_∞

0

e^−rs

_θ

(

^Ps

)

^ds− _τ

0

e^−rs

_θ

(

^Ps

)

^ds−e^−rτI

(

^K

)

(6)

=E_θ,p,t

∞ 0

e^−rs

θ

(

^Ps

)

^{ds +sup}

τ E_θ,p,t

_τ

0

e^−rs

(

⁻

θ

(

^Ps

))

^{ds−e−rτI}

(

^K

)

.

Onemaynoticethat thecomputations regardingthefirst expectedvaluein thesecond lineof(6)are straightforward sinceitdoesnotdependonthestoppingtime

τ

^{.Thereforewearelefttofindthesolutionofthesecondterminthesecond}

lineof(6),whichwedenoteby

v

_θ

(

p,t

)

=sup τ E_θ,p,t

_τ

0

e^−rs

(

−

(

^Ps

))

^ds−e^−rτI

(

^K

)

. (7)

WenextpresenttheHamilton-Jacobi-Bellman(HJB)equationsthatcharacterisev_θ(p,t).Inthiscase,aswehavetworegimes (thatcorrespondtotwopossiblestatesofthesubsidy:activeorinactive)thatinfluencethedynamicsofthepriceandprofit functionofthefirm,weneedtosolvetwosetsofHJBequations:

min

(

^r

v

0

(

^p,t

)

^−L0

v

0

(

^p,t

)

⁺

0

(

^p

)

^,

v

0

(

^p,t

)

^+I

(

^K

) )

^{=0, (8)}

min

(

^r

v

1

(

^p,t

)

^−L1

v

1

(

^p,t

)

⁺

¹

(

^p

)

^,

v

1

(

^p,t

)

^+I

(

^K

) )

^{=0, (9)}

whereLi(fori=0,1)istheinfinitesimalgeneratorofthebi-variateprocess(

θ

,P)=

{

(

θ

s,Ps),s>0

}

^{.Inviewoftheassump-}

tionsabout

θ

^{andP,thefollowingholds:}

L0

v

0

(

^p,t

)

⁼

μ

^p

∂ v

0

(

^p,t

)

∂

^{p +}

1

2

σ

^2p2

∂

²

v

0

(

^p,t

)

∂

^{p2 ,}

L1

v

1

(

p,t

)

=

∂ v

1

(

p,t

)

∂

^{t +}

λ (

^t

) ( v

0

(

p,t

)

−

v

1

(

p,t

) )

^.

Finally,wenotethatinOliveiraandPerkowski(2019)theauthorsdonotconsideraspecificprofitfunctionnorspecificprice dynamics.Therefore,allthecomputationsandtheanalysisnecessarytoobtainthevaluefunctionswillbepresentedinthe nextsections.

3. Modelsolution

Inthissectionwe presentthevalue functionofthefirm dependingonwhetherthesubsidyisstill presentornot.We firstderivethevaluefunctionforthecasethatthesubsidyhasbeenretractedalready(seeSection3.1)andthencontinue withderivingthevaluefunctionforthecasethatthesubsidyisstillactive(seeSection 3.2).Wethen presenttheoptimal investmentstrategyofthefirm.

3.1. Solutionwhenthesubsidyhasbeenretracted

Incasethesubsidyhasbeenretracted,wehaveastandardinvestmentproblem,forwhichthesolutionisalreadyknown;

see,forinstance,DixitandPindyck(1994).Moreover,we knowthatforsuchacasetheinvestmentdecisionisathreshold decision,meaningthatinvestmentisoptimalforlargevaluesofprice(p≥p^∗₀)while,forsmallvaluesofprice(p<p^∗₀),itis optimaltowait.Uponinvestment,thefirmsearnsaperpetualvaluegivenby:

E_θ,p,t

∞ 0

e^−rs

(

^KPs−C

(

^K

) )

^{ds −I}

(

^K

)

= K r−

μ

^p−

C

(

^K

)

r −I

(

^K

)

.

Before investment (ie., p<p^∗₀), thefirm does not earnanyprofits.Therefore, thevalue of thefirm isequal tothe value of the optionto invest. Furthermore,asthe subsidy is not active, there isno longer the need forthe time dependency.

Therefore,wemaydroptheexplicitdependencyontofthevaluefunctionV₀.

In view oftheseconsiderations, andapplicationofresults providedby Dixitand Pindyck(1994),we endup withthe resultpresentedinthefollowingproposition.Theproofsallpropositionscanbefoundintheappendix.

Proposition1. Thevaluefunctionwhenthesubsidyisnolongeravailable,i.e.V₀(p),isgivenby:

V₀

(

^p,t

)

≡V₀

(

^p

)

=

Ap^d1, p≤p^∗₀

K

r−μp−^C(_r^K)−I

(

^K

)

, p>p^∗₀ where

p^∗₀= d1

d1−1

(

^I

(

^K

)

+C

(

^K

)

r

)

×r−

μ

K , (10)

A= K r−

μ

^d11

p^∗₀^1−d1, (11)

withd₁=^σ2−2μ+√

4μ²−4μσ²+σ⁴+8rσ²

2σ² >1.

3.2. Solutionwhenthesubsidyisstillactive

When thesubsidy isstill active,the firm’srevenueswouldbe constantupon investment.Giventhe electricityprice p atthe momentthe firm decidesto invest,the firm willreceiveby unit ofelectricity produced pwitha top-up payment ofanamountFduringthelife ofthesubsidy.Oncethesubsidywillberetracted,thefirm willbe exposedtooutputprice uncertainty.Thus,weneedtotakeintoaccountthepossiblechangesinthestateprocess

θ

^.

Asdiscussedabove,inordertofindV₁onehastocomputethefunctionv₁,whichsatisfies r

v

1

(

p,t

)

−

∂ v

1

(

p,t

)

∂

^{t −}

λ

0t

( v

0

(

^p

)

−

v

1

(

p,t

) )

^+K

(

^p+F

)

−C

(

^K

)

=0,

inthecontinuationregion.Hereweomitthetimedependencyofv₀ont,astheproblemistime-homogeneousinthiscase (seeSection3.1).

Thesolutionofthisproblemisnotstraightforward.Wepresentthesolutioninthefollowingproposition.

Proposition2. AssumethatFissuchthatitsatisfiesthefollowingcondition F≡ −

σ

²

2 1 K

C

(

^K

)

+rI

(

^K

)

r d1<F< 1 K

rI

(

^K

)

+C

(

^K

)

1−E[e^−rν0] ≡F.

Thenthevaluefunctionfortheoptimalstoppingproblempresentedin(6)isgivenby:

V1

(

p,t

)

=

⎧ ⎪

⎨

⎪ ⎩

V_1,1

(

^p,t

)

, p<c^∗₁

V1,2

(

^p,t

)

^,

(

^p,t

)

^∈[c∗1,p^∗₀

)

^×[t1^∗

(

^p

)

^,∞

)

V1,3

(

p,t

)

,

(

p,t

)

∈[c^∗₁,p^∗₀

)

×[0,t₁^∗

(

^p

))

V1,4

(

p,t

)

, p>p^∗₀

where,

V1,1

(

p,t

)

=V1,2

(

p,t

)

=Ap^d1E

e^−rνt

,

Fig. 1. Investment and continuation regions in the ( p, t )-plane.

V1,3

(

p,t

)

=V1,4

(

p,t

)

=

K

(

^p+F

)

r −C

(

^K

)

r −I

(

^K

)

+

K p r−

μ

⁻

K

(

^p+F

)

r

E[e^−rνt], andp^∗₀isgivenby(11).Finallyt₁^∗(^p)^{isuniquelydefinedbytheequation:}

Ap^d1E

e^{−rνt∗1(p)}

=

K

(

^p+F

)

r −C

(

^K

)

r −I

(

^K

)

+

K p r−

μ

⁻

K

(

^p+F

)

r

E[e^{−rνt∗1(p)}]. (12)

Additionally,c^∗₁≡c^∗₁(^F)^{verifiesthefollowing}conditions:

(

ⁱ

)

^c∗1

(

^F

)

^∈

(

^0,p∗0

)

^{and t}1^∗

(

^c∗1

)

^{=0, if F<F<F}

(

ⁱⁱ

)

^lim

FF

c^∗₁

(

^F

)

=0, and lim

FFc₁^∗

(

^F

)

=p^∗₀.

TheresultspresentedinProposition2showthatwehavetodistinguishbetweenfourregionsforthevaluefunctionin casethesubsidy isstill active.Fig. 1illustrates the shapesofthe differentregions, assuming that the intensityfunction,

λ

^{(t), atwhich the subsidyis retrievedis} non-decreasing int. Ifthe initial value ofthe process P isstrictly smaller than athresholdc^∗₁ thefirm isin thecontinuation region.We notethat thethreshold c^∗₁ isdefinedimplicitlyby theequation t₁^∗(^c∗₁)=0,asstatedinProposition2.Weindicatethiscontinuationregionby“Wait(1)” inFig.1.Forsuchlowvaluesofpit isinfactneveroptimalforthefirmtoinvest,neitherbeforenorafterthesubsidyisretracted,independentlyofhowmuch timehaspassedsincethesubsidyhasbeeninstalled.Oncethesubsidyhasbeenretractedthefirmwill investassoonas thepricehitsthelevelp^∗₀.Thevaluefunctioninthisregion,V_1,1(.,.),isthereforeequaltothevalueoftheoptiontoinvest atp^∗₀,scaledbythediscountfactorE[e^−rνt],whichaccountsfortheexpectedtimeittakesuntilthesubsidyisretracted,for agivent.

IfthecurrentvalueoftheprocessPislargeenough,specifically p≥p^∗₀,itisoptimalforthefirmtoinvestimmediately independenton whetherthe subsidy isstill provided ornot. We indicate thisregion by “Invest (4)” in Fig.1. Thevalue functioninthisregion,V_1,4(.,.)consistsoftwo terms.The firsttermrepresentsthediscountedexpectedfuturecashflows oftheprojectincasethefirmwouldreceivetheFITforeverminustheinvestmentcost.Thesecondtermaccountsforthe additionalrevenuesearned inthe open marketassoon asthe subsidyis retracted,scaled by afactor accountingforthe expectedtimeuntilthesubsidyisretracted.

Forintermediatevaluesofpbetweenc^∗₁andp^∗₀,whetherthefirmisinthecontinuationregionorinthestoppingregion dependsonhowmuchtime haspassedsincethesubsidywasinstalled(represented byt).Here wedistinguishtwocases.

Ifp is such that [c^∗₁,p^∗₀] and sufficientlymuch time has passedsince the introduction ofthe subsidy, i.e.t>t₁^∗(^p),it is neveroptimalforthefirmtoinvestbeforethesubsidyhasbeenretracted.Oncethesubsidyhasbeenretracteditisoptimal forthefirmtoinvestatthemomentthepricehitsthelevelp^∗₀.Thisissimilartothecaseinthefirstcontinuationregion (Wait(1)).Thevaluefunctioninthiscontinuationregion,indicatedby“Wait(2)” inFig.1,denotedbyV_1,2(.,.),istherefore equaltothevalueoftheoptiontoinvestoncethesubsidyhasbeenretractedscaledbyadiscountfactoraccountingforthe expectedtime untilsubsidyretraction.If, however,relatively littletime haspassedsincethesubsidyhasbeenintroduced, specificallyt≤t₁^∗(^p),itisoptimalforthefirmtoinvestimmediately.Thevaluefunctionforthiscase,V_1,3(.,.),isequaltothe valueintheother investmentregion.Thisinvestmentregionisindicatedby “Invest(3)” inFig.1.Notethat theboundary curveseparatingtheinvestmentandcontinuationregionsinthiscaseincreaseswiththevalueofp,aswellasthetimet,for whichthesubsidyhasbeenactivealready.Theparticularshapeoftheboundarycurve dependsontheparticularintensity

Fig. 2. Investment and continuation regions for the case that the subsidy is active (left graph) and that the subsidy has been retracted already (right graph).

rateatwhichthesubsidyisexpectedtoberetrieved.However,aslongas

λ

^(.)isanon-decreasingfunction,theretractionof thesubsidybecomesmorelikelyastimepasses, andtherefore,one mayexpectthat,alsointhiscase,theboundarycurve isupwardslopinginthe(p,t)-plane⁴

Notethatthe investmentstrategy(giventhesubsidyisstill present)dependson howmuchtimehaspassedsincethe subsidyintroductionatthemomentthefirmconsiderstheinvestmentproblemforthefirsttime.Incasethefirmconsiders investmentatthemomentthesubsidyhasbeenintroduced,i.e.t=0,thenthehorizontallineatt=0inFig.1represents thedecisionruleasafunctionofp.Fort=0thefirmwillinvestifp>c^∗₁.Ifthefirmconsiderstheinvestmentproblemfor thefirsttimewhent>0timehasalreadypassedsincethesubsidyintroduction,thenthethresholdcurvet₁^∗(^p)^determines theinvestmentstrategy. Fort=t^∗,forexample,thefirminvestsimmediatelyifp>p^∗,asindicatedinFig.1.Otherwise,it isoptimalforthefirmtowaitwithinvestmentuntilthesubsidyhasbeenretractedandthepricehasreachedthethreshold p^∗₀.

Fig.2illustratestheinvestmentandwaitingregionforbothcases.Theleftgraphshowsthedifferentregionsforthecase thatsubsidyisactiveandtherightgraphillustratestheinvestmentandwaitingregionsforthecasethatsubsidyhasbeen alreadyretracted.Considerthepoint(tˆ,pˆ)indicatedintheleftgraphofFig.2.Ifthefirmconsidersinvestmentforthefirst timeattˆforavalueofp=pˆ,thenitisoptimalforthefirmtoinvestimmediately.Thetimeperiodittakesuntilsubsidyis eventuallyretractedthenonlyaffectstheprojectrevenuesearnedbutnottheinvestmentstrategy.Now considerthepoint (tˆˆ,p)ˆˆ insidethecontinuationregion.Ifthefirmconsidersinvestmentforthefirsttimeaftertˆˆtimeunitshavepassedsince theintroductionofthesubsidy,theoptimaldecisionofthefirmistowaitwithinvestment.Infact,thefirm willfirstwait untilthesubsidyhasbeenretracted.Thismovementintimeisindicatedbytheverticalarrowtowardstheretractiontime v.Oncethesubsidyhasbeenretractedtheoptimalinvestmentstrategy isillustratedintherightgraphofFig.2.Thefirm willthencontinuewaitinguntilthepriceprocessP(s)hitstheinvestmentthresholdp^∗₀,atwhichitisoptimaltoinvest.

ThefollowingremarkcommentsontheinvestmentstrategythatresultswhenatoosmallortoolargeFITisofferedto thefirm.

Remark1.

(a) Ifthedecision-makeroffers aFIT equalto p+F foreach unit produced,then itholdsthat c₁^∗=0.This meansthat itisoptimalforthefirmtoinvestforanyinitialpricepgiveninvestmentisconsidered att=0.As aconsequence, theoptimalstrategyillustratedinFig.1changesinthesensethatthewaitingregion(1)disappears;Moreover,when F>F, the waiting region (1) will not exist eitherand the investment region (regions (3) plus (4)) will be larger thantheoneobtainedwhenF=F asforthiscaseitholdsthatt₁^∗(^p)>0forevery p∈[0,p^∗₀)^{.Thismeans,thatitis} optimalfora firm toinvestforevery pricepaslong aslessthant₁^∗(^p=0)^{time haspassedsince thesubsidy has} beenprovided.

(b) IfF=F,then both the waitingregion(2) andthe investmentregion (3)depicted inFig. 1will disappearbecause lim_FFc^∗(^F)=p^∗₀,whichimpliesthattheboundaryt₁^∗turnsintoaverticallinein p^∗₀.Inthiscase,thefirmdoesnot haveany additionalincentiveto makethe decisionearlier thanin thecasethat subsidy is not provided.It isalso importanttonotethatwhenF<F,theinvestmentdecisionwillbeoptimalonlyforpricesgreaterthan p^∗₀.

Remark1showsthatiftheFITofferedissufficientlysmallerthanthecurrentmarketprice,specifically,iftheFIToffered issmallerorequaltop+F (notethatF<0),thenitismoreprofitableforthefirmtoinvestinthefreeelectricitymarket

4Indeed, one may confirm that the proof of Proposition 2 in Appendix A.2 only rely on the monotony of the function λ( t ) and not on its specific expression.

wherenosubsidyisprovided.ThismeansthatpolicymakerscannotspeedupinvestmentbyofferingaFITbelowthislevel.

InthefollowingwewillfocusonanalysingthecasesthatFissetsuchthattheconditionpresentedinProposition2holds.

Inthenext proposition,we providetheexpectedtime toundertake theinvestmentdecisiongivenitisnot optimalto investimmediately.

Proposition3. Incasethecurrentpriceisp,thetimeelapsedsincetheintroductionofthesubsidyisequaltot,thesubsidy is stillactiveanditisnotoptimalforthefirmtoinvestimmediately,theexpectedtimetoundertaketheinvestmentdecision,which wedenotebyE_1,t[

τ

I],isgivenby

E1,t[

τ

I]=

E[

ν

t]+ ln^p_p^∗0

μ

⁻¹2

σ

²

1_{t>t+

1(p),μ>¹2σ²}, (13)

whereE[

ν

t]isgivenbyEquation(1),and1_A denotestheindicatorfunctionofpropositionA⁵

Theexpression(15)fortheexpectedinvestmenttimestatedinProposition3shouldbeunderstoodasfollows.Ifthetime twhen thefirm considers investmentissuch thatt<t₁^∗(^p), itisoptimalforthe firm toinvestimmediately. Thismeans thatinthiscasetheexpectedtimetoinvestmentisequaltozero.Incaset>t₁^∗(^p),however,itisnotoptimalforthefirm toinvest immediately. Then thefirm first waits until the subsidyis retracted(which willtake E_1,t[

ν

t] time, on average).

Oncethesubsidyisretractedthefirm willwait untilthemarketpriceincreasesup top^∗₀ (whichwilltake

ln^p_p^∗0 μ−¹₂σ²

time, onaverage),andthentaketheinvestmentdecision.Notethattheexpectedtimeforthepricetohit p^∗₀isfiniteifandonly if

μ

>¹2

σ

^{2(seeWillmottetal.,1995).}

4. Comparativestaticsresults

Inthissectionwe willfirstpresentsome analyticalresultsrelatedtothebehaviouroftheinvestmentboundaryinkey parameters.Wethencomparehowtheoptimalinvestmentstrategyandtheexpectedtimeofinvestmentdifferswhenthe firmconsiderstheretractionrisktobeconstantcomparedtotime-dependent.Inordertoallowcomparisonwealsopresent theanalyticalresultsforthecaseofconstantretractionrisk,i.e, weconsiderthecase

λ

(^t)=

λ

ˆ^{.Notethatweassumehere} that the intensityrate of retraction increaseslinearly withtime, i.e.,

λ

(^t)=

λ

^0t, order to derive the expressions for the relevantquantitiesexplicitly.

We note againthat for thecase that the subsidyis no longer active, theproblem isa standard one. Detailed results regardingthecomparativestaticsofthiscasecanbefoundintheliterature(see,forexample,DixitandPindyck(1994))and canbe easily derived analytically.Indeed,one can provethat p^∗₀ increaseswith

σ

^{anddecreases with}

μ

^{.Thismeans the}

higherthemarketpriceuncertaintythelargertheinvestmentthresholdasthevalue oftheoptiontoinvestincreases.The higherthetrend ofthepriceprocess, thesmaller theinvestmentthreshold asthevalue oftheproject increasesandthe firmismoreeagertoundergoinvestmentandtherewith,receivethe projectvalue. Inthefollowingwenow focusonthe resultsrelatedtotheinvestmentstrategywhenthesubsidyisstillactive.Proposition4presentsresultsonthebehaviorof theinvestmentthresholdcurveinkeyparameters.

Proposition4. Theinvestmentthresholdcurve,t₁^∗(^p),increaseswith

μ

^{andFanddecreaseswith}

σ

^and

λ

0,i.e.

∂

^t1^∗

(

^p

)

∂μ

^>0,

∂

^t1^∗

(

^p

)

∂σ

^<0,

∂

^t∗1

(

^p

)

∂λ

0 <0,

∂

^t1^∗

(

^p

)

∂

^{F >0.}

Theboundaryc^∗₁decreaseswith

μ

^{andFandincreaseswith}

σ

^and

λ

0,i.e.

∂

^c∗1

∂μ

^<0,

∂

^c∗1

∂σ

^>0,

∂

^c∗1

∂λ

0 >0,

∂

^c∗1

∂

^{F <0.}

TheresultspresentedinProposition4show thatkeepingallother parameters constant,increasingthevolatility (drift) decreases(increases)theoptimalinvestmentboundary,therewithdecreasing(increasing)theinvestmentregionforthecase thatthesubsidyisactive.Thisisnotsurprisingashigherelectricitypriceuncertaintyincreasesthevalue ofthefirm’sin- vestmentopportunityasitwillreceivetheelectricitypriceoncethesubsidyhasbeenretracted.Therefore,thefirmdemands ahigherFIT(i.e.largerp,aswekeepFconstantforthiscase)toundergoinvestmentimmediately.Ahigherelectricityprice drift,however,increases thevalue of theproject.Therefore, forhigherdrift thefirm has alarger incentiveto investim- mediately,whichmeansthat therangeofpricespforwhichitisoptimaltoinvestimmediatelyatthefirsttimethefirm considersinvestment,islarger.IncreasingFandtherewith,theFIToffered,hasthesameeffect.Proposition4alsoshowsthat increasingthe likelihoodofpolicy retraction,i.e.increasing

λ

0,decreases theinvestment region,andtherewith therange ofFITsforwhichthefirmis willingtoinvestimmediately.The reasonforthat isthatincreasing

λ

0 decreases theproject value,asthefirm expectsto receivetheFIT over ashorterperiodoftime. Therefore,it demandshigherlevels oftheFIT

5If μ< ¹₂σ², then this expected time is infinite, as follows from standard results of stochastic processes; see, for instance, Kulkarni (2016) .

tojustifyimmediateinvestmentwhensubsidyis(still)provided.Fromapolicymakersperspective thataimstoaccelerate investment,policyriskisdamagingasitreducesthefirm’swillingnesstoinvest.Ifinvestorsperceivepolicyrisktobelarger, they willbemorehesitanttoinvest.Inthat caseregulatorswouldneedtoofferahigherFIT inorderto triggerthesame amountofinvestment.

WealsonotethattheresultofProposition4showsthatincasethepricepliesin(^c1^∗,∞)^and

μ

^{orFincrease,thefirm}

maystilldecidetoinvestevenifmoretimeelapsedsincetheintroductionofthesubsidies.Onthecontrary,if

σ

^increases,

itmaybetoolateforafirmtoinvestinthatrangeofprices.Inthatcasetheinvestmentwillonlyoccurafterthesubsidyis removedandthemarketpricereachesthevalue p^∗₀.Thismeansthatourresultsarecoherentwiththestandardrealoptions resultthatincreasingthevolatilitypostponestheinvestmentdecision.

We now analyze how the investment strategy differs when subsidy retraction risk is assumedto be time-dependent compared to the case that the firm assumes it to stay constant over time. Therefore, we provide the equivalent of Proposition 2forthecase

λ

(^t)=

λ

ˆ ^{inthe following}proposition. Thismeansthat theprobability that thesubsidy willbe retractedis assumedto beconstant witha rate

λ

ˆ^{.In thiscasethetime until retraction is}exponentially distributed,and we aredealingwithahomogeneous process,asoppositionofthetime-dependentcase,that leadstoa non-homogeneous process.

Proposition5. Assumethat

λ

(^t)=

λ

ˆ,

∀

^t,andFissuchthat

∼F≡C

(

^K

)

+rI

(

^K

)

K −p^∗₀<F <r+

λ

ˆ K

I

(

^K

)

+C

(

^K

)

r

≡F˜.

Then,thevaluefunctionfortheoptimalstoppingproblempresentedin(5)isgivenby:

V₁

(

^p

)

⁼

_Apd1ˆλ

r+λˆ p<p^∗

K(^p+F)

r+ˆλ −^C(_r^K)+_r−^{K p}_μr^ˆλ

+λˆ −I

(

^K

)

^p≥p^∗, wherep^∗isthesmallestsolutionofthefollowingequation:

A

(

^p∗

)

^d1ˆ

λ

r+ˆ

λ

⁼

K

(

^p∗+F

)

r+

λ

ˆ ⁻

C

(

^K

)

r + K p^∗

r−

μ λ

ˆ

r+ˆ

λ

^−I

⁽

^K

⁾

^{. (14)}

Theexpectedtimetotakethedecisiontoinvest,incasep<p^∗,isgivenby E1[

τ

^I]=

1

λ

ˆ ⁺ ln^p_p^∗

μ

−¹₂

σ

²

1_{μ>1 2σ²}.

ForthecasethatsubsidyhasbeenretractedalreadytheresultsarethesameaspresentedinProposition1.

Incasetheretractionprobabilityisconsideredtostayconstantovertime,itisoptimalforthefirmtoundergoinvestment immediatelythefirsttime itconsidersinvestment,ifpislargerorequalthanthethresholdp^∗.Otherwise,itisoptimalto waituntilthesubsidyisretractedandtheelectricitymarketpricehitsthethresholdp^∗₀(givenbyequation(11)).Theoptimal decisioninthiscaseisindependentofhowmuchtimehaspassedsincethesubsidyhasbeenintroduced.

Wenowcomparethecasesofconstantversustime-dependentretractionprobability.Notethatifthefollowingrelation betweentheintensityparameters

λ

0and

λ

ˆ ^holds:

λ

ˆ²

λ

^{0 =}

2

π

^,

then the expectedtime to retraction attime t=0is the sameforthe two cases,i.e. E[

ν

0]= ¹_ˆλ.As E[

ν

t] is a decreasing functionoft,itfollowsthat

E[

ν

^t]< 1

λ

ˆ^{, fort>0,}

which means that as time since the introduction of the subsidy passes, the expected time until retraction in the non- homogeneous case(i.e.,when

λ

(^t)=

λ

0t) decreases andthedifference betweenthissituation andthehomogeneous one increases.Forthefollowinganalysiswe willassumethattheexpectedtimetoretractionattimet=0isthesameforthe twocases.

Fig. 3illustrates how theinvestment strategies ofthe casesofconstant andtime varying retraction probability differ.

Thebluesolidlineillustratestheinvestmentboundaryt^∗₁(^p)^forthecaseoftime-dependentretractionrisk,whiletheblue dashedlinerepresentstheboundarybetweenthewaitingandinvestmentregionfortheconstantcase.Notethatournumer- icalresultsindicatethat c^∗₁<p^∗ alwaysholds.AsdepictedinFig.3,one maydefine t^∗≡t₁^∗(^p∗),whichrepresentsthelast momentsincesubsidyintroduction,atwhichitisstilloptimaltoinvestforaFITschemepaying p^∗+F perunitproduced ifthefirmconsidersretractionrisktobetime-dependent.

Wefindthatift≤t^∗therangeofFITs,p+F,forwhichthefirmwillinvestimmediatelyislargerifitconsidersretraction probability tobetime-dependent. Ifhowever,t> t^∗,theopposite holds.Afirmthat considersretractionprobability tobe