Renewable Technology Adoption Costs and Economic Growth

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Renewable Technology Adoption Costs and Economic Growth Bernardino Adao, Borghan Narajabad, and Ted Temzelides 2022-045 Please cite this paper as: Adao, Bernardino, Borghan Narajabad, and Ted Temzelides (2022). “Renewable Technology Adoption Costs and Economic Growth,” Finance and Economics Discussion Series 2022-045. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.045. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Renewable Technology Adoption Costs and Economic Growth∗ Bernardino Adao Borghan Narajabad Ted Temzelides Bank of Portugal Federal Reserve Board Rice University This Version: 2/22/2022 Abstract We develop a dynamic general equilibrium integrated assessment model that incorporates costs due to new technology adoption in renewable energy as well as externalities associated with carbon emissions and renewable technology spillovers. We use world economy data to calibrate our model and investigate theeffectsofthetechnologyadoptionchannelonrenewableenergyadoptionand ontheoptimalenergytransition. Ourcalibratedmodelimpliesseveralinterestingconnectionsbetweentechnologyadoptioncosts,thetwoexternalities,policy, andwelfare. Weinvestigatetherelativeeffectivenessoftwopolicyinstruments- Pigouviancarbontaxesandpoliciesthatinternalizespillovereffects-inisolation as well as in tandem. Our findings suggest that renewable technology adoption costs are of quantitative importance for the energy transition. We find that the two policy instruments are better thought of as complements rather than substitutes. JEL Codes: H21, O14, O33, Q54, Q55 Keywords: Technology adoption, scrapping, energy transition, climate, dynamic taxation ∗TheopinionsexpressedinthepaperdonotnecessarilyreflectthoseoftheBankofPortugal or the Federal Reserve System. We are grateful to Antonia Diaz, participants at the Yale UniversityCowlesFoundationSummerConferenceonMacroeconomicsandClimateChange, theViennaMacroConference,theCESifoAreaConferenceonEnergyandClimateEconomics, theASSAMeetings,andtheSEDMeetingsfortheircommentsonearlierversionsofthepaper. 1

1 Introduction We investigate the optimal transition from a primarily fossil-fueled world economy to a renewable-energy-fueled world economy. This transition depends on several components, including the relative costs and benefits of using different energy sources, the relative availability of fossil fuel, and the rate of technological progress in the energy sector. First, fossil fuel sources constitute an exhaustible resource. A second consideration involves environmental factors. As fossilfuelusegeneratesexternalitiesthroughincreasingthestockofgreenhouse gas (GHG) emissions, the need for a clean substitute becomes increasingly apparent. Inaddition,technologyspilloversimplytheneedtoconsiderthatinvestmentsinrenewabletechnologycreatepositiveexternalities. Ourmethodological contribution lies in exploring the quantitative significance of an additional, not well-studied factor in the process of new renewable technology adoption. In the presence of rapid technological progress and large capital costs, new technology adoptionofteninvolvescertainadditionalcosts. Thesecanincludecostsassociated with decommissioning, scrapping, or recycling old equipment; adjustment costs resulting from the switch; and legal and transaction costs associated with financing, selling, purchasing, and installing new equipment.1 The greater the speed of improvements, the higher the costs resulting from early adoption. Our modeling of this effect applies more generally, but we believe it is particularly relevantforenergyinvestments,whichtendtobecapitalintensive. Inaddition, renewableenergytechnologiesarenewandsubjecttorelativelyrapidtechnological progress as compared to fossil fuel. Our main contribution lies in exploring the quantitative significance of this channel in a dynamic general equilibrium integrated assessment model (IAM) that incorporates environmental and technological spillover externalities. Although our aggregate modeling approach abstracts from individual sectors, the following example might be instructive of the kind of effect we have in mind. Considerthemarketforelectricvehicles(EV).Batteriesareasignificant fractionofthecost,oftencloseto30percentto40percentofpurchasinganEV. By any measure, including increased energy density and reduced cost, batteries havebeensteadilyimprovinginrecentyears. Yet,incomparison,EVpurchases have remained relatively flat over the same time horizon. It is important to note that for virtually all EVs, the battery technology at the time of purchase is “embedded” in the vehicle. That is, it is difficult or impossible to upgrade it, unless the vehicle is replaced with a new one. It is reasonable to expect that, concerned about characteristics such as overall range, some consumers might prefer to wait until sufficient progress justifies an EV purchase. This would be 1Mauritzen (2012) discusses scrapping patterns for less productive wind turbines in Denmark. We should emphasize that the purpose of this paper is not to provide theoretical micro-foundations for scrapping and related costs. Rather, we are investigating the quantitativeimplicationsofconsideringsuchcostsfortheenergytransition. Costsassociatedwith new technology adoption are, of course, also relevant and have been studied in the context of fossil energy and stranded assets, see Rezai and van der Ploeg (2019).To our knowledge, our paper is the first to study the implications of such costs for optimal renewable energy investment. 2

in line with our reasoning: there are costs associated with early adoption.2 A second focus of our study concerns the effectiveness of different environmental policy instruments. While Pigouvian taxes on carbon emissions have several theoretical advantages and are generally favored by economists, they have proved difficult to implement in practice. As a partial substitute, many advocate policies that directly promote market penetration by renewables. We investigatethedegreeofsubstitutabilitybetweencarbontaxesandpoliciesthat promote renewable energy by internalizing spillovers. externalities. Inourmodel, energy, capital,andlaborareinputsintheproductionoffinal consumption goods. Energy can be produced from either fossil or renewable sources. Both require capital, which is also used in the production of the final good. At each point in time, productivity in the renewable energy production can increase as a result of adopting new capital. The actual improvement is subject to a spillover, as it depends on the aggregate investment in the renewable sector. We model the cost associated with replacing existing equipment as atemporaryadverseproductivityshocktotheproductionfunctionofrenewable energyfirms. Thesecostsareassumedtobeproportionaltothesizeofthecapital replacement in our analysis. Thus, our results are different from models of technologicalprogresswherethecostofadoptingandutilizingnewtechnologies is independent of the size of the existing stock of capital. An important modeling choice concerns the degree of substitutability between different forms of energy. We will distinguish between substitutability in production versus consumption of energy. We assume that, once produced, renewable and fossil-fuel-derived energy are perfect substitutes in the production of the final good. A high substitutability seems a reasonable benchmark when considering long-run effects, as we do in this paper. For example, substitutability is justified in the presence of energy storage.3 Importantly, fossil fuel and renewable forms of energy in our model are not perfect substitutes in terms of their required inputs, as they require different amounts of capital to produce. The evolving nature of renewable energy technologies will be a main focus, aswestudypropertiesoftheoptimalenergytransition. Whilewedonot consider technology advancements in fossil fuel in our benchmark case, we will investigate the robustness of our findings to introducing technological progress in the fossil fuel sector. We employ an IAM to characterize the optimal transition to a renewableenergy-fueled economy. The model is based on Golosov, Hassler, Krusell, and Tsyvinski (GHKT, 2014), who in turn build on Nordhaus’s pioneering work in climate economics. We decentralize the optimum using a Pigouvian tax on GHGemissionstogetherwitharevenue-neutralpolicyonrenewabletechnology 2FormoreonEVssee,forexample,https://www.eia.gov/todayinenergy/detail.php?id=36312#, https://www.researchgate.net/figure/Evolution-of-battery-energy-density-and-cost-Global- EV-outlook-2016 fig3 309313559, and https://www.statista.com/statistics/797638/batteryshare-of-large-electric-vehicle-cost/. EVpurchasesareoftensubsidized,whichweignoredfor thisdiscussion. 3Whilenotwidelyavailablecurrently,therearestrongindicationsthatsuchstoragetechnologieswillentercommercializationinthenextdecadeorso. 3

adoptionandfindthattheefficientenergymixinvolvesagradualdeclineinthe use of fossil fuel. We then calibrate the model using world economy data. The extent to which the world economy will use its available fossil fuel reserves depends on the assumption about the total endowment of accessible hydrocarbons. For realistic parametrizations, the transition is made well before exhaustionoftheseresources. IntheabsenceofaPigouviantaxoncarbonemissions, policies that incentivize penetration by renewables through internalizing spillovers may provide relatively small benefits and can even be detrimental to short-run growth. In addition, we find that the decrease in global temperatures withrespecttothestatusquoisnegligibleunderthispolicy. Incontrast,global temperatures decrease by about 10 percent if the optimal Pigouvian tax is in place. Similarly,thereductioninconsumptionoffossilfuelresultingfrominternalizingtechnologyspilloversissignificantlylargeriftheoptimalPigouviantax isalsoinplace. Whilethegainsfrominternalizingthespilloversalonearesmall, comparing the status quo to the scenario where both policies are implemented results in a consumption-equivalent welfare gain of 1.431 percent. We conclude that when it comes to social welfare, carbon taxes and policies that promote renewable energy by eliminating spillover externalities are best thought of as complements rather than substitutes. While theoretically desirable, Pigouvian taxes have generally proved difficult to implement in practice. For example, a short-lived government might choose a lower or no Pigouvian tax, as it is effectively more impatient than the representative agent. In the Appendix, we briefly discuss how the theoretical model can be extended to account for such considerations. The urgency of climate change has led to calls for action by several international organizations andgovernmentsaroundtheglobe. Asavarietyofmitigationmeasuresarecurrently under consideration, our findings suggest that, in the presence of rapid technological progress and the associated costs, such as scrapping costs, some caution might be warranted before we conclude that direct subsidies are always a suitable substitute in the absence of a carbon tax. Fourparameterswillbeimportantforourcalibrationstrategy: (1)thelevel of the spillover externality in renewable energy production, (2) the current “Pigouvian tax rate” (the ratio of the actual carbon tax relative to its optimalvalue),(3)theinitialresourcesoffossilfuel,and(4)theproductivityinthe renewable energy technology. To calibrate these parameters, we will use observations from the world economy, together with theoretical relationships derived in the context of our model. For example, we will employ the fact that the changeintherenewableenergyshareoftheenergyoutputdecreasesinthelevel of the spillover externality. Similarly, to get a handle on the current Pigouvian tax rate, we use the current level of fossil fuel consumption and the implication that an increase in the Pigouvian rate would result in a decrease in fossil fuel use. Importantly, the magnitude of the Pigouvian tax in our calibration should be interpreted as reflecting the difference between the opportunity cost of the average fossil fuel producer (which is rather low) and the average price paid by consumers of fossil fuel. Last, to calibrate the productivity in the renewable energysector,wewillusethecurrentshareofrenewablestogetherwiththefact 4

that their productivity is increasing in their share. The calibrated model allows us to derive several interesting connections between new technology adoption, the two externalities, Pigouvian taxation, growth, and welfare. We investigate the quantitative importance and potential complementarity between the two policy instruments by studying their effects inisolationandintandem. Ourfindingssuggestthatcostsassociatedwithnew technologyadoptionareofquantitativeimportancealongtheenergytransition. We find that when spillover externalities are internalized, renewable firms take on capital at a smaller scale, as they need to replace more of their capital in order to adopt newer technologies faster. This is due to the crucial assumption that the cost of technology adoption is proportional to the firm’s capital stock. Our results also point to complementarities between carbon taxes and internalizing spillovers, with sizable welfare gains only in the case when both policies are present. This complementarity exists because the optimal Pigouvian tax reduces the initial negative effect from the optimal renewable adoption policy. Asarobustnesscheck, wealsocalibratethemodeltakingintoaccounttechnological developments in the fossil fuel sector. The magnitude of the Pigouvian tax remains virtually unchanged, but the value of the spillover externality is greater than in the benchmark case. We find that incorporating fossil fuel growth results in a faster transition to a fully renewable economy in the status quo case and a lesser welfare gain from switching to the optimal technology adoption, as the calibrated value is already closer to the optimal level. Importantly, the main mechanisms governing the interactions between Pigouvian taxation and renewable technology adoption, which are the focus of our paper, remain intact when technological progress in fossil fuel is incorporated. 1.1 Related Literature Our paper contributes to the growing literature that uses IAM to study energy transitions, innovation, andgrowth. Intheeconomicgrowthliterature, Parente (1994)studiedamodelinwhichfirmschoosetoadoptnewtechnologiesasthey gainspecificexpertisethroughlearning-by-doing. Heidentifiedconditionsunder which equilibria exhibit constant per capita output growth. As in most of the literature on innovation and growth, Parente abstracted from issues related to climate and energy, which are the focus of our study. Atkeson and Burstein (2015)studytheimpactofpolicy-inducedchangesininnovativeinvestmentand the implications for medium- and long-run innovation and growth. They, too, abstract from climate and energy considerations. Nordhaus(1994)pioneeredthestudyofclimatefactorsindynamiceconomic modeling. Traditionally,mosteconomicanalysisofenergyandenvironmentalissuesfocusesoncomputablegeneralequilibriummodelsthatoftenabstractfrom endogenous technological progress.4 Acemoglu et al. (2012) study a growth model that takes into consideration the environmental effects from operating “dirty”technologies. Theyconsiderpoliciesthattaxinnovationandproduction 4See,forexample,NordhausandBoyer(2000)andreferencestherein. 5

in the dirty sectors. They find that subsidizing research in the “clean” sectors can speed up environmentally friendly innovation without the corresponding slowdown in economic growth. Consequently, optimal behavior in their model impliesanimmediateincreaseincleanenergyresearchanddevelopment(R&D), followed by a complete switch toward the exclusive use of clean inputs in production. We view our paper as complementary to theirs. We do not model directed technical change; instead, we introduce the replacement-cost channel associated with new capital adoption. While we think that their main recommendationsarelikelytoremainvalid,ourquantitativefindingssuggestthatthe optimalratesofnewtechnologyadoptionmightbeaffectedifwetakesuchcosts into account. As mentioned earlier, our analysis builds on GHKT (2014). They develop a tractabledynamicgeneralequilibriummodelthatincorporatesthefeedbackbetweenenergyuseandtheresultingclimateconsequences. Theyderiveaformula and numerical values for the optimal tax on carbon emissions. They, however, abstract from the costs associated with endogenous technological progress. We will employ several elements from their work in what follows, including the tractable modeling of the environmental externality. Van der Ploeg and Rezai (2016) extend the model in GHKT in several ways. They allow for general fossil fuel extraction costs, a negative impact of climate change on growth, mean reversion in climate damages, labor-augmenting and green technology progress, and a direct effect of the emissions stock on welfare. They discuss the social optimum as well as the optimal carbon tax and renewable energy subsidies.5 Acemoglu, Akcigit, Hanley, and Kerr (2016) use the structure in GHKT to study questions related to the transition to clean technologies. They employ a “ladder” model to study technological progress in both the clean and the dirty sectors, and they estimate their model using R&D and patent data. They assume that increased representation of fossil fuel encourages further use, and that fossil fuel use stops after 200 years. They conclude that both Pigouvian taxation and renewable energy subsidies are needed in order to make the (optimal)transitionsoonerratherthanlater. Thereasonisthatsubsidiesencourage technological progress without overtaxing short-run future output. Our model includes an additional channel. When technology is embedded in the current capital stock, subsidizing renewable resources encourages additional use of capital using the current technology. This, in turn, might not be optimal in the presenceofreplacementcosts. Fried(2018)developsadynamicgeneralequilibrium model to investigate the effect of carbon taxes in inducing innovation in greentechnologies. Thecarbontaxischosenexogenouslytomatcha30percent reductioninemissionsin20years. Themodelintroducesendogenousinnovation 5Other related papers include Stokey (1998), who considers growth under environmental constraints; Goulder and Schneider (1999), who study endogenous innovations in abatement technologies; Van der Zwaan et al. (2002), who investigate the impact of environmental policies in a model with learning-by-doing; and Popp (2004), who considers innovation in the energy sector and the costs of environmental regulation. See also Hartley et al. (2016), who study technological progress and the optimal energy transition, and Van der Ploeg and Withagen(2011),whopointtothepossibilityofagreenparadoxinthiscontext. 6

in both dirty and green energy production as well as in the non-energy sectors andassumespositivespilloversbetweencleananddirtyenergytechnologies. As a result, the paper finds that abstracting from endogenous innovation results in overestimating the size of the carbon tax needed to attain the given reduction in emissions. Morerecently,Hassleretal. (2020)introduceamulti-regiongeneral-equilibrium IAM. Their focus is on sub-optimal climate policies that may vary across regions. They consider the uncertainty associated with evaluating the cost from climate change and find that the costs of underestimating climate change are an order of magnitude larger than the costs of overestimating it. Thus, when various energy sources are sufficiently substitutable, ambitious climate policies can be thought of as effective insurance against adverse climate shocks. Their analysis shows that the economic costs of achieving climate goals increase substantiallywhensomeregionsdonotparticipateinmitigation. Finally,theyfind thatwhenitcomestoclimategoals,greenenergysubsidiesareapoorsubstitute for a Pigouvian tax on fossil fuel, particularly on coal. Their model does not explicitly study endogenous technical change. Gowrisankaran and Rysman (2012) explore a similar argument to ours in a different context. They study optimal decisions by consumers choosing the timing of purchase among an expanding set of available camcorders. While consumers usually own only one camcorder at a time, they may substitute an old camcorderwith a new one. As prices, quality, and variety improve over time, waiting is valuable in their model. They argue that initial market share fordigitalcamcorderswasmodest,asforward-lookingconsumerswererationally expectingthatcheaperandbetterplayerswouldappearinthefuture. Arelated effect is explored in Manuelli and Seshadri (2014). They focus on technology diffusion of tractors in American agriculture during the first part of the 20th century. They argue that part of the reason for the slow rate of adoption was thattractorqualitykeptimprovingoverthatperiod. Asaresult, farmerschose to postpone their purchase, rather than investing in a tractor that would soon become obsolete. The main contribution of our paper is to explore this channel in the context of renewable energy technology adoption. The paper proceeds as follows. Section 2 introduces the model. Sections 3 and 4 discuss efficiency, equilibrium, and optimal policy. Section 5 outlines our calibration strategy. Section 6 presents our main quantitative findings, including those from an extension of the base model that accounts for productivity improvementsinthefossilfuelsector. Abriefconclusionfollows. TheAppendix contains technical material and some extensions. 2 The Base Model We build an IAM that incorporates a version of the neoclassical growth model, together with energy, technology, and climate factors. Time is discrete and the horizon is infinite, t=0,1,.... There is a single consumption good per period, and all markets are competitive. The economy is populated by a representative 7

infinite-lived household that discounts the future at rate β ∈ (0,1) and values period t consumption, c , through a utility function u(c ). We assume that u t t is smooth, strictly increasing, and strictly concave, and that the usual Inada conditionshold. Thelaborendowmentisnormalizedto1, andlaborissupplied inelastically. Therearethreedifferenttypesoffirms,allownedbythehousehold: thefinalgoodfirmaswellastwotypesofintermediategoodfirmsthatproduce energyfromfossilfuelandfromrenewablesources. Ineachperiod,thehousehold chooses how much capital, k , to rent at rate r and receives profits resulting t t from the firms’ activities. All capital depreciates at rate δ ∈(0,1). Therepresentativefinal-good-producingfirmproducesoutput,y ,usingcapt ital, kc, labor, l , and energy, e . Production can be affected by environmental t t t quality,indexedbyΓ ,whichreflectsthetotalstockofGHGsintheatmosphere. t Ignoringenvironmentaldamages,thefinalgoodproductionfunctionisgivenby y ≤F (kc,l ,e ,Γ ). (1) t t t t t We assume that environmental quality, Γ , affects output through a damage t function D (Γ ), and that damages are multiplicative. Thus, the final good t t production function becomes F (k t c,l t ,e t ,Γ t )=(1−D t (Γ t ))F(cid:101)(k t c,l t ,e t ), (2) (cid:2) (cid:0) (cid:1)(cid:3) where 1−D (Γ ) = exp −π Γ −Γ . Here, Γ represents the pre-industrial t t t t GHG concentration in the atmosphere, and π is a variable that parametrizes t the effect of higher GHG concentrations on damages. The function D captures the mapping from the stock of GHG, Γ , to economic damages measured as a t percentage of output. We assume that F(cid:101)(k t c,l t ,e t ) has a Cobb-Douglas form: F(cid:101)(k t c,l t ,e t )=A(cid:101)t (k t c)θk(l t )θl(e t )1−θ, (3) where A(cid:101) is a productivity parameter, while θ, θ k , θ l ∈ (0,1), and θ k +θ l = θ. Thus, the final good production function can be rewritten as y t ≤A t (k t c)θk(l t )θl(e t )1−θ, (4) where A t ≡(1−D t (Γ))A(cid:101)t . The level of GHG evolves according to t−T (cid:88) Γ −Γ= (1−d )f , t≥T, (5) t n t−n n=0 whered ∈[0,1],andf indicatestheanthropogenicGHGemissionsinperiod n t−n t−n. The variable 1−d represents the amount of carbon that remains in the n atmospherenperiodsintothefuture,andT definesthestartofindustrialization. The depreciation structure in (5) is characterized by three parameters. It is assumed that a fraction ϕ of emitted carbon stays in the atmosphere forever, L whileafraction(1−ϕ )oftheremainingemissionsexitintothebiosphere. The 0 remaining part decays at geometric rate ϕ. Thus, 1−d =ϕ +(1−ϕ )ϕ (1−ϕ)n. (6) n L L 0 8

The level of the GHG concentration can be then decomposed to a permanent part, Γp, and a decaying part, Γd: t t Γ =Γp+Γd, (7) t t t where Γp =Γp +ϕ f , (8) t t−1 L t Γd =(1−ϕ)Γd +(1−ϕ )ϕ f . (9) t t−1 L 0 t Energycanbeproducedbyusingfossilorrenewablesources. Wewilldistinguish between substitutability in production versus substitutability in consumption betweenthetwoformsofenergy. Moreprecisely,weassumethatenergyderived from fossil fuel and that derived from renewable sources are perfect substitutes in the production of the final good.6 As we measure fossil fuel use in units of carbon content, the flow of anthropogenic GHG emissions equals f , the fossil t fuel used in energy production in period t. Let (cid:36) denote the available stock of t fossil fuel in period t. Given (cid:36) , the law of motion for (cid:36) is 0 t (cid:36) ≤(cid:36) −f . (10) t+1 t t We assume imperfect substitutability when capital is used in the production of different forms of energy. Fossil-fuel-derived energy production uses fossil fuel and capital as inputs according to production function ef t ≤A f (f t )1−αf (cid:16) k t f (cid:17)αf , (11) where A > 0 and α ∈ (0,1). This specification captures that, by using f f additional capital, more energy can be extracted from the remaining fossil fuel reserves. This specification also allows for using more capital, higher kf, or t technologicalprogress, higherA , inthefossilfuelsectorleadingtoareduction f in emission intensity through improvements in fuel efficiency. Weassumeameasureoneofcompetitiverenewable-energy-producingfirms. Constant returns to scale in capital and productivity imply that the number of firms does not affect the aggregate industry variables. Thus, issues related to free entry are not relevant for our analysis. As the firms are heterogenous, we need to keep track of the identity, j, of each individual firm. The renewable energy output of firm j is given by er ≤Ψ(i )(E )1−αr (cid:0) kr (cid:1)αr, (12) j,t j,t j,t j,t where E is firm j’s productivity parameter, E is given for all j, and α ∈ j,t j,0 r (0,1). We interpret i as the new technology adoption rate by firm j in period j,t t. Adopting newer technologies boosts future productivity, but implies a cost in terms of a contemporaneous output loss. This cost is increasing in the adoptionrateofnewtechnology. Moreover,thecostisproportionaltotherenewable 6Hassleretal. (2020)allowforimperfectoutputsubstitutabilityinenergyandconcentrate onmedium-runquestions. 9

firm’s current production. If a renewable firm decides to not adopt a newer technology, then it can fully utilize the stock of its current capital with the current embedded technology without any disruption. However, if a renewable firm chooses to further improve the technology embedded in next period’s capital, then during the current period it needs to replace its current capital with capital that uses an improved technology. Thus, the productivity improvement will result in a temporary disruption to the firm’s current production.7 It is naturaltoassumethattheoutputlosswillbeproportionaltothefirm’scurrent production.8 Moreprecisely,weassumethatthenewtechnologyadoptionrate,i,reduces firmj’scurrentoutputbyafactorΨ(i )≥0,whereΨ(·)issuchthatΨ(0)=1, j,t Ψ(cid:48)(·)<0, Ψ(cid:48)(cid:48)(·)<0, and Ψ (cid:0) i (cid:1) =0, for some i. We interpret this as the total cost associated with the replacement of current equipment.9 This implies that capital replacement costs are independent of the depreciation rate. We will consider the possibility of a spillover effect, where aggregate technology adoption also affects the productivity of each individual firm. Put differently, as more firms adopt new technologies, the benefits affect the entire renewable energy sector. This creates an externality, leading to a discrepancy betweenequilibriumanddesirablelevelsofnewcapitaladoption.10 Weconsider this effect to be especially relevant, as investments in the energy sector tend to be capital intensive. Thus, if innovators do not expect to capture the resulting returns, under-adoption of new technologies relative to the optimum is likely to 7One could think of this as a “time-to-build” constraint in adopting newer technologies, whichisakin,yetdifferent,fromthestandard“time-to-build”constraintforphysicalcapital. 8Alternatively one could introduce different vintages of renewable capital with stochastic depreciation,sothatolderandnewervintagescancoexistandcontributetotheproduction. Thenin ordertoimprove theoverallproductivity oneneedsto replaceoldervintagesbefore their full depreciation with more productive new ones. Therefore, the older vintages are removedbeforetheirnaturaldepreciationandcannotyieldtheirfulloutput. Thisalternative wayofmodelingwouldnaturallyleadtoadistributionofvintageswithinandacrossfirms. To avoidtheresultingcomplexity,andsincedistributionalissuesarenotthefocusofourstudy, weadopteda“reduced-form”approachtomodelingthiseffect. Thereisextensiveliteratureon dynamic vintage-capital-related models. See, for example, Benhabib and Rustichini (1991), Chari and Hopenhayn (1991), Greenwood, Hercowitz, and Krusell (1997), and Jovanovic (2012). Boucekkine,DeLaCroix,andLicandro(2017)providearecentreview. 9Admittedly, important innovation also takes place in the fossil fuel sector. Mainly for simplicity,wewillconcentrateontechnologicalprogressintherenewablesectorinourbaseline model. Wewilllaterextendourmodeltoaccountfortechnologicalprogressinfossilfuel. This leads to some quantitative differences, but the qualitative features of our results will remain unchanged. 10The infant industry argument is sometimes used to justify subsidising the production of renewable energy. According to this argument, since fossil fuel technologies are more mature, renewable technologies cannot compete on an equal basis since scale and benefits fromlearning-by-doingcanonlybeachievedunderalargermarketshare. 10

occur.11 More precisely, the productivity of firm j evolves according to (cid:18)(cid:90) 1 (cid:90) 1 (cid:19) lnE ≤ξi +(1−ξ) i kr dj/ kr dj +lnE , (13) j,t+1 j,t j,t j,t j,t j,t 0 0 where0≤ξ ≤1parametrizesthestrengthofthespillovereffect. Thecasewhere ξ =1correspondstonospillovers,whileξ =0correspondstotheotherextreme, where productivity is entirely determined by spillovers. In order to abstract from any size-dependent advantage to firms, the above expression normalizes each firm’s technology adoption by its capital stock. Eachperiod,theproductionfactorsareallocatedfreelyacrosssectors. Total capital used in the economy cannot exceed the total supply; i.e., for all t, (cid:90) 1 kc+kf + kr dj ≤k . (14) t t j,t t 0 In addition, the energy used in the production of the final good cannot exceed the total supply of energy: (cid:90) 1 e ≤ef + er dj. (15) t t j,t 0 The next section discusses desirable allocations for our model economy. 3 Efficiency We begin by characterizing allocation efficiency in terms of some key relationships. Wewilllatercompareefficientoutcomestomarketallocations. Thesocial plannerchoosesasequence{c ,kc,kf,f ,ef,Γp,Γd,{i ,kr , E ,er } }∞ , t t t t t t t j,t+1 j,t j,t+1 j,t j t=0 j ∈[0,1], to solve the following problem: ∞ (cid:88) max βtu(c ), t t=0 subject to (8)-(15) and (cid:104) (cid:105) c t +k t+1 ≤(1−D t (Γp t +Γd t )) A(cid:101)t (k t c)θk(l t )θl(e t )1−θ +(1−δ)k t , (16) as well as non-negativity constraints and given the initial values for the stock variables. We let µ denote the Lagrange multiplier on the production constraint in F the fossil fuel sector (equation (11)) and µj be the multiplier on the production r 11Bosettia et al. (2008) argue that international knowledge spillovers tend to increase the incentivetofree-ride,thusdecreasinginvestmentsinenergyR&D.Braunetal. (2009)perform an empirical study of spillovers in renewable energy. They document significant domestic and international knowledge spillovers in solar technology innovation as well as significant internationalspilloversinwind. 11

constraint for renewable energy firm j (equation (12)). Similarly, we let µj be E themultiplierfortheevolutionoffirmj’sproductivity(equation(13)). Finally, µ and µ are the multipliers associated with the distribution of the capital K E stock across sectors and with the supply of energy (equations (14) and (15)), respectively. The first-order condition (FOC) with respect to er gives jt (cid:90) 1 µj dj =µ . (17) r,t E,t 0 Moreover, the marginal utility from producing an extra infinitesimal amount of renewable energy should be equal across firms; i.e., µj =µh , for any two firms j and h. (18) r,t r,t The FOC with respect to kr gives j,t (cid:18) i −i (cid:19)(cid:90) 1 (cid:32) E (cid:33)1−αr (1−ξ) j,t t µj dj+Ψ(i )α µj j,t =µ , (19) kr E,t j,t r r,t kr K,t t 0 j,t where kr = (cid:82)1 kr dj, and i = (cid:82)1 i kr dj/ (cid:82)1 kr dj. Since (17) and (18) give t 0 j,t t 0 j,t j,t 0 j,t thatµj =µ ,equation(19)impliesthattheonlynon-aggregatevariablethat r,t E,t influences i is Ej,t. j,t kr j,t The FOC with respect to i gives j,t (cid:32) E (cid:33)1−αr µj (cid:82)1 µj dj −µj Ψ(cid:48)(i ) j,t =ξ E,t +(1−ξ) 0 E,t . (20) r,t j,t kr kr kr j,t j,t t Finally, the FOC with respect to ef gives t µ =µ . (21) E,t F,t Since µj = µ = µ , and i is a function of Ej,t, we have that µj E,t is also r,t E,t F,t j,t kr kr j,t j,t a function of Ej,t. The following result greatly simplifies our analysis. It asserts kr j,t that if E and kr are proportional to the initial values of E , then i = i , j,t j,t j,0 j,t t for all j and t. In other words, although renewable-energy-producing firms are heterogeneous, efficiency implies that they choose identical levels of i . t Proposition 1 In an efficient allocation, E k j j r , , t t = k E t r t and i j,t =i t , for all j. Proof. ForanyinitialvaluesofE ,thereisasolutionsuchthatE ,kr ,µj , j,0 j,t j,t E,t and µj are proportional to the initial values of E . Then (20) implies that r,t j,0 i =i , for all j ∈[0,1]. From (19), Ej,t is a function of i only. As i =i , j,t t kr j,t j,t t j,t we have Ej,t = Et. kr kr j,t t 12

4 Equilibrium and Optimal Policy WederivethecompetitiveequilibriumFOCforconsumersandfirmsintheAppendix. Using these, we first characterize the equilibrium choice of investment in the renewable technology. In what follows, we let Φ (i ) stand for the govt j,t ernment policy conditional on a renewable firm’s investment. Provided that ξ < 1, in the absence of government policy this investment will be lower than optimal. Of course, the magnitude of the distortion depends on the level of the externality, ξ. Proposition 2 InacompetitiveequilibriumwithΦ (i )=0, i islowerthan t j,t j,t optimal when ξ <1. The proof is given in the Appendix. Next, we discuss optimal policy. This needs to take into account two distortions. First, there is under-investment in i due to the spillover effects. The second distortion is due to the social t costs associated with the environmental externality from GHG emissions. The next Proposition demonstrates that both distortions can be fully accommodated through the use of two instruments. First, a policy that taxes firms in proportion to their under-investment in i restores optimal investment by makt ing firms indifferent between paying a “penalty” or pursuing the optimal level ofinvestment. Second,aPigouviantaxinternalizestheexternalityfromcarbon emissions. AsinGHKT(2014),underthespecialassumptionsoflogutilityand 100percentdepreciationofcapital,thePigouviantaximposedonthefossilfuel firms does not depend on the growth rate of the economy. Proposition 3 (1) The optimal allocation can be supported by a combination of a revenue-neutral policy, Φj(i )=−(1−ξ)peΨ(cid:48)(i∗) (cid:16) e∗ j, r t (cid:17) (i −i∗), imt j,t t t Ψ(i∗ ) j,t t j,t posed on renewable firms, together with a Pigouvian tax on fossil fuel use, τf = t (cid:80)∞ βju(cid:48)(c∗ t+j ) π y∗ (1−d ), where pe is the price of energy, {c∗,y∗,i∗}∞ j=0 u(cid:48)(c∗) t+j t+j j t t t t t=0 t is the solution to the planner’s problem, and 1−d = ϕ +(1−ϕ )ϕ (1− j L L 0 ϕ)j. (2) If u(c) = log(c), α = α = α, π = π, all t, and δ = 1, τf = r f t t (cid:104) (cid:105) y π ϕL + (1−ϕL)ϕ0 does not depend on the growth rate of the economy. t 1−β 1−(1−ϕ)β The proof is given in the Appendix. The optimal policy in our model has several interesting implications. First, the policy on renewable energy firms generates no revenue, but it reduces the household’s current profits from the renewable sector, as a result of inducing additional innovation compared to the competitive equilibrium. Second, the Pigouvian tax reduces the household’s profits from the fossil fuel sector. However, the household receives a lump-sum transfer of equal magnitude; thus, its budget constraint remains unchanged. Finally, there is a separation between the two schemes, as the total effect on the household’s budget is the same as the resource cost of innovation in the planner’s problem. For the remainder of the paper we will assume that u(c) = log(c) and will set δ = 1. Moreover, we will assume that the stock of fossil fuel is large 13

enough so that consumption of fossil fuel is not constrained. In the Appendix we solve the constrained planner’s problem backward, from a final state, where only renewable energy is used, and show that the consumption of fossil fuel is endogenously bounded. In other words, the full transition to the renewable energyregimecantakeplacepriortotheexhaustionoffossilfuelresources. This isduetothegrowingproductivityintherenewablesectoreventuallysurpassing athresholdthatmakesusingfossilfuelthelessefficientsource. Whileallocating additionalcapitaltothefossilfuelsectorincreasestheproductionofenergyper unit of fossil fuel, the present value of the marginal environmental damages limits the overall benefit from fossil fuel use. In the next section we calibrate our model in order to study the optimal timing of the transition to a renewable energy regime, as well as the effects of the GHG accumulation prior to this transition. This will allow us to explore the quantitative significance of the technology adoption effect on optimal policy and welfare. 5 Calibration In this section we calibrate our model in order to study the transition from the current, predominantly fossil fuel economy, to an economy that fully relies on renewable energy. We use the calibrated model to evaluate the interaction between the two policy instruments: (1) the Pigouvian taxation on carbon emissions,and(2)thetechnologyadoptiontargetingforrenewableenergyfirms. Moreprecisely,weevaluatehowthetwopolicieswouldaffecttheshareofrenewable energy, the accumulation of GHG, global temperatures, economic growth, andwelfare,firstinisolationandthenintandem. Thiswillallowustoquantify the significance of the technology adoption effect and to explore the potential substitutability between the two policy tools. Themodel’sparameterscanbedividedintofourcategoriesrelatedtopreferences,technology,environmentaldamages,andthecurrent(statusquo)policies in place. We will use a log utility function and a benchmark annual discount rateof4percent,whichgivesβ =0.9610,asaperiodiscalibratedto10years.12 Given the length of the period, there is some justification in considering the benchmark case of full depreciation of capital, δ =1. Turning to the aggregate Cobb-Douglas production function, we set the share of capital and labor, respectively, to θ = (1/3)×0.95 and θ = (2/3)×0.95, which imply an energy k l share of 1−θ = 1−(θ +θ ) = 0.05. We set the productivity growth rate in k l the final good sector so that the balanced growth rate is 2 percent, while the long-run population growth rate is set to zero. The production functions in therenewableenergysectorandfossilfuelenergysectorareconstantreturnsto scaleCobb-Douglasfunctionswithcapitalshareequalto0.5. Wewishtoclarify that the effect of costs associated with technology adoption, such as scrapping costs, remains intact even under 100 percent capital depreciation. The best analogy would be in the context of a continuous-time model. Although capital 12Mostmacroeconomicstudiesuseyearlydiscountratesbetween2percentand5percent. 14

depreciates fully after one period (10 years), in a continuous time model, technology adoption costs, such as scrapping costs, would affect the productivity of renewableenergyintheinterim. Asweworkindiscretetime,thecostsimposed in the beginning of a period in our setup can be interpreted as capturing the averageofsuchcostsovertheten-yearperiodwhenthiscapitalisused. Wealso discuss a continuous time representation of these costs in the Appendix. We assume the following form for the renewable technology adoption cost function, Ψ: (cid:32) (cid:18) i (cid:19)ψ (cid:33)1/ψ Ψ(i)= 1− . i This functional form satisfies the earlier assumptions that Ψ(0)=1, Ψ(cid:48)(·)<0, Ψ(cid:48)(cid:48)(·) < 0, and Ψ(i) = 0, for i = i. Moreover, the elasticity of the technology adoption cost with respect to the adoption rate is given by Ψ(cid:48)(i) 1 (i/¯ı)ψ − = × . (22) Ψ(i) i 1−(i/¯ı)ψ AsshownintheAppendix,thiselasticityplaysanimportantroleindetermining boththelong-runandthetransitionaltechnologyadoptionrateintherenewable sector. The parameter ψ provides us with a degree of freedom to match a long-run adoption rate that is consistent with the long-run growth rate of the economy. To calibrate Ψ, we need to assign values to two parameters: i (the highest possible technology adoption rate in the renewable sector) and ψ. We will use the relationship between Ψ(·) and the optimal asymptotic long-run growth rate of il, the productivity in the renewable sector. As we show in the Appendix,ifthespilloverexternalityisfullyinternalized,thelong-runil isgiven by Ψ(cid:48)(il) β − = (1−α). Ψ(il) 1−β Combining the above equation with (22) gives (cid:32) (cid:33)1/ψ 1 i= 1+ ×il. (23) ( β (1−α)il) (1−β) Thus, for a given il, a lower ψ implies a higher i. In our baseline calibration, we set i to its maximum attainable level, which corresponds to setting ψ to its lowest possible level; i.e., by setting ψ =1/(1−α)=2.13 To determine il, note that an asymptotically balanced growth path requires equal asymptotic growth rates between the renewable energy sector (which is 13If ψ = 1/(1−α) < 2, then Ψ(i)a− 1 α is no longer a concave function and the cost of technologicaladoptionisnolongerincreasing. 15

the only source of energy in the long run) and the final good sector. This, in (cid:16) (cid:17) turn, implies il =log gl×(gc)θ 1 l =0.198. Using (23), we set i=0.489.14 We follow GHKT (2014) in our calibration of the environmental damage parametersandthecomputationofthePigouviancarbontax. Inparticular,we set π =2.379×10−5×10, ϕ=0.0228, ϕ =0.2, and ϕ =0.393. The optimal L 0 carbon tax follows from the last part of (83) in the Appendix and is given by (cid:20) (cid:21) ϕ (1−ϕ )ϕ T /y =π L + L 0 . (24) P (1−β) (1−β(1−ϕ)) Given our calibration, this equation implies that T /y =3.55×10−4, which is P equivalenttoataxof$24.9perton. Thisisbroadlyconsistentwiththeclimate economics literature given the assumed level of discounting.15 We chose 2015 as our base year. Four parameters related to the energy sector remain to be calibrated: (1) the current stock of fossil fuel, W ; (2) the 0 current productivity of the renewable sector, E ; (3) the current Pigouvian tax 0 level, τf; and (4) the spillover from the renewable technology adoption, ξ. For our baseline calibration we set W =666GtC.16 0 We set E , τf, and ξ to match three data moments: (1) the current share of 0 renewable energy in total energy production, s , (2) the current consumption 0 of fossil fuel, f ; and (3) the change in the share of renewable energy in the last 0 period (10 years), s −s . The current productivity of the renewable sector 0 −1 affects the renewable share in total energy production. In turn, the spillovers fromtherenewabletechnologyadoptionaffectthechangeintheproductivityof the renewable sector and, thus, the change in the share of renewable energy. In addition, the Pigouvian tax affects the use of fossil fuel and, consequently, the share of fossil fuel and renewable sources in total energy production. In what follows, we denote by τf the value of the Pigouvian tax as a percentage of its optimal level, τ∗. Setting E = 14.92, τf = 0.63·τ∗, and ξ = 0.54, our model 0 matches f =100 GtC,17 s =10.2 percent, and s −s =2.3 percent.18 0 0 0 −1 14For sensitivity, we also set i to levels corresponding to 10 percent and 20 percent slower than the maximum attainable renewable technology growth rate. These slower rates correspondtosettingito0.449and0.408,respectively. Thesetranslatetoamaximumattainable annual growth rates of 4.6 percent and 4.2 percent, respectively, instead of the 5.0 percent baseline. Inourrobustnessexercisewewillaccordinglysetψ equalto2.21and2.50,respectively,inorderforitosatisfy(23). 15SeeFigure2inGHKT(2014). 16SeeSection4.3inLi,Narajabad,andTemzelides(2016). 17See EPA: https://www.epa.gov/ghgemissions/global-greenhouse-gas-emissionsdata#Trends. 18Thisincludesallmodernplustraditionalrenewables(includingbiomass). Wecalculated an initial 10-year growth rate of 4.7 percent for renewables, with a corresponding rate of 2 percent for the entire energy sector (we excluded nuclear energy from this calculation). Seehttps://www.ren21.net/reports/global-status-report/. Weremarkthatwhile,withafew exceptions, an explicit carbon tax is largely absent in most countries, several uses of fossil fuelaretaxedatrelativelyhighrates. Gasolineandotherfuelrelatedtotransportationarea leadingexample. Theinitialvalueofτf shouldbeinterpretedinthatlight,asthedifference betweenthemarginalcosttofossilfuelproducersandtheaveragepricepaidbyconsumersof fossilfuel. Ofcourse,thefossilfuelmarketsharesmanycharacteristicsofanoligopoly. 16

6 Quantitative Findings Our calibration allows us to evaluate the quantitative significance of the technologyadoptioneffectaswellastheeffectsofthecarbontaxandtherenewable adoption policy in isolation and in tandem. We simulate our model considering different scenarios for the two policy parameters. Figure 1 below shows the paths for the share of renewable energy (top), accumulated fossil fuel consumption (middle), and global temperatures (bottom) in each respective policy scenario. The dotted, dashed, dot-dashed, and solid lines indicate the status quo benchmark (business as usual), optimal technology adoption, optimal Pigouvian tax, and combined optimal policies (full optimum), respectively. Clearly, the outcomes under either an optimal Pigouvian tax policy alone or the optimaltechnologyadoptionpolicyalonediffersignificantlyfromtheoutcomewhen bothpoliciesarepresent. Thisthoughtexperimenthelpsusunderstandhowthe two policies interact in the presence of the technology adoption channel. Next, we comment on each panel individually. Thefirstpanelgivestheshareofrenewablesinenergyproductionasafunction of time under the different policy scenarios. Note that the optimal technology adoption policy leads to a “rotation” of the status quo path, while the Pigouvian tax “shifts” the status quo path along the transition. As a result, setting technology adoption to its optimal level in the absence of the optimal Pigouvian tax reduces the share of renewables in the short run relative to the status quo. At the same time, the full switch to renewable energy production occurs somewhat earlier than in the benchmark case. In contrast, setting the Pigouvian tax to its optimal level in the absence of a policy inducing optimal technologyadoptionincreasestheshareofrenewableenergyimmediately. Inthe full optimum, setting both policies to their combined optimal levels reduces the short-runshareofrenewableenergy. However,thetransitiontoafullyrenewable global economy takes place by 2070, the earliest among the four scenarios.19 The second panel describes the evolution of cumulative fossil fuel consumption in the same four scenarios. Interestingly, absent a tax on GHG emissions, the cumulative fossil fuel consumption is initially somewhat more intense if the technology externality is internalized than in the status quo. This is because the faster growth in renewable energy productivity allows the economy to rely fully on renewable energy earlier. Similarly, when both the Pigouvian tax and the technology adoption are set to their optimal levels, the economy reaches the fully renewable energy state earlier and more fossil fuel is left unused. Consistent with the “green paradox,” this also implies a heavier use of fossil fuel initially than in the case where the Pigouvian tax is in place but the renewable policy is absent. By comparison, GHKT (2014) find that, while increasing in 19Note that the fossil fuel consumption drops to zero in finite time, not just asymptotically. The reason for this is that fossil fuel and renewable energy are assumed to be perfect substitutes in consumption. Thus, as the consumption of fossil fuel vanishes, its marginal productivity, which depends on the marginal productivity of energy, remains finite. Since damagesfromemissionsgrowproportionallytoGDP,thereisapointafterwhichtheproductivityofrenewablesbecomeshighenoughtomakefossilfuelobsolete. 17

1.00 0.75 0.50 0.25 0.00 2050 2100 2150 2200 erahs ygrene elbaweneR 600 400 200 0 2050 2100 2150 2200 )tnelaviuqe CtG( noitpmusnoc leuf lissof eitalumuC 2.8 2.4 2.0 1.6 2050 2100 2150 2200 Year )suisleC( level lairtsudnierp eht evoba erutarepmet labolG Full optimal policy Optimal renewable technology adoption Optimal Pigouvian tax Status quo Figure 1: Benchmark Calibration: Energy and Temperature 18

the status quo, optimal consumption of fossil fuel stays relatively flat. Thethirdpanelshowsthepathforglobaltemperaturesunderourfourpolicy scenarios. In order to map carbon concentrations into global temperatures, T, we use the following expression (see GHKT, 2014): (cid:18) (cid:19) S T(S )=3ln t /ln(2), t S where S is the pre-industrial level of atmospheric carbon concentration. Consistentwiththefossilfueluseinthetoppanel,theglobaltemperatureincreases under both the business-as-usual and the optimal technology adoption scenarios, reaching short of 2.8 degrees Celsius above the pre-industrial level. The temperature under the technology policy alone (in the absence of a Pigouvian tax) later falls slightly faster than in the benchmark case. Under the optimal Pigouviantaxandinthefullyoptimalcase,globaltemperaturespeakataround 2.2and2.0degreesCelsiusabovethepre-industriallevel,respectively,andthen decline over time. GHKT (2014) find a peak temperature of almost 10 degrees Celsius above thepre-industriallevelwithoutpolicyintervention. Inthecontextofthemodel, nopolicyinterventionmeansthatthepriceoffossilfuelisequaltothemarginal cost of fossil fuel, which makes fossil fuel consumption quite attractive. The high temperature increase is explained by the fact that GHKT (2014) assume a larger total endowment of fossil fuel, zero carbon taxes, and no endogenous growth in productivity of renewable energy. Instead, they assume that the extraction efficiency and the efficiency of green technologies grow both at the samerate(2percentperyear). Theequalgrowthinproductivitydoesnotallow a productivity gain of renewables over fossil fuel over time, and contributes to the large increase in the temperature in their analysis. Additionally, GHKT (2014)findthattheoptimaltaxwouldlimitheatingtoabout3degreesCelsius. This temperature rise is similar to the one obtained in the status quo of our calibrated model. This is due to the fact that in the status quo the renewable sector exhibits faster technological progress than the fossil fuel sector, and the carbontaxisalreadyalmost2/3ofthedamageresultingfromcarbonemissions. Inthestatusquocasetherenewableefficiencygrowssufficientlyfastsothatthe economy is fully renewable by 2120. In contrast, as can be seen in Figure 7 of GHKT (2014), they find that the economy continues to use fossil fuel even under the optimal case way beyond 2120. The three panels in Figure 2 describe the over-time contribution of certain key variables to related growth rates under our four policy scenarios. The top panelshowstheperiod-by-perioddifferenceinrelateddamagescausedbyGHG emissions. Of note, the implementation of the technology policy in the absence of a carbon tax has a greater negative effect on growth compared to the status quo. After the full transition takes place, the contribution to growth is positive (if small) in all four cases, due to the gradual decline in the stock of emissions. Thesecondpanelplotsthecontributionoftheenergysectortoeconomicgrowth. Thestatusquoscenarioresultsinasizablenegativecontributiontogrowth. This 19

isduepartlytodamagesandpartlytoscarcityandtheresultingincreaseinthe shadow price of fossil fuel. As the resource constraint on fossil fuel is far from binding under the fully optimal policy scenario, equation (68) in the Appendix implies that the net contribution of energy to growth is positive and increasing during the energy transition. Next, we turn our attention to welfare comparisons across these scenarios. Following Lucas (1987), we report the consumption-equivalent percentage welfare gain from these policies over the business-as-usual benchmark. Moving from the status quo to optimal technology adoption alone (in the absence of a Pigouviantax)wouldimplya0.251percentconsumption-equivalentgain,while the optimal Pigouvian tax alone would result in a gain of 1.023 percent, confirming the relative importance of the carbon tax. Comparing the status quo to the scenario where both policies are implemented results in a consumptionequivalent welfare gain of 1.431 percent. This is one of the main findings of our quantitative analysis. The difference between the welfare gain from applying either policy in isolation versus implementing both amounts to about a 0.157 percent increase in consumption, suggesting a sizeable complementary between the two policies. Thus, our model points to sizable welfare gains only when the policies are adopted in tandem. Tofurtherhighlighttheroleofthecapitalreplacementcostsinthesefindings, werunthemodelunderthesameparametrizationasbeforebutwiththesecosts “shutdown”; i.e., Ψ=1.Intheabsenceofsuchcosts, wesetthegrowthratein therenewabletechnologyequaltoitslong-runvalue,asimpliedbythebalanced growth path.20 We then target f = 100 GtC, which implies s −s = 2.1 0 0 −1 percent. The resulting Pigouvian tax rate is close to the one under technology adoption: τf =0.62·τ∗. Theimplieddynamicsfortheshareofrenewables,fossil fuel consumption, and global temperature, as well as the corresponding effects on growth, are reported in Figures 3 and 4. By comparing the cases with and withouttechnologyadoptioncosts,wenoticeanumberofimportantdifferences. Optimal penetration by renewables starts lower in the case with replacement costs, but it soon overtakes, and the transition to the fully renewable state occurs earlier in this case. Fossil fuel consumption and global temperatures demonstrate corresponding differences. In the case without replacement costs, the consumption-equivalent welfare gain from setting the Pigouvian tax to its optimal level (fully optimal policy) corresponds to a 1.05 percent increase in consumption. Weconcludethattherenewabletechnologyadoptioncostsplaya significantrolewhenwequantifytheoptimalenergytransition,asthedifference between the welfare gain from applying the fully optimal policy with versus withoutthesecostsamountstoanapproximate0.381%increaseinconsumption. The comparison between the benchmark case and the case without replacement costs also illustrates the complementarity between the two policies in our benchmark case. Because in the benchmark case the cost of renewable tech- 20Notethatifunlikeourbenchmarkcase,thecostofadoptingnewrenewabletechnologies or the improving productivity in this sector is not proportional to output, then the optimal solution for technological progress in the renewable sector would be similar to the case with long-runconstantgrowthintherenewablesectorwhichwestudyhere. 20

0.00 −0.05 −0.10 2050 2100 2150 2200 )tnecrep dezilaunna( htworg ot CHG ni egnahc fo noitubirtnoC 0.04 0.02 0.00 2050 2100 2150 2200 )tnecrep dezilaunna( htworg ot ygrene fo noitubirtnoC 0.05 0.00 −0.05 −0.10 2050 2100 2150 2200 Year )tnecrep dezilaunna( htworg ot ygrene dna CHG fo noitubirtnoC Full optimal policy Optimal renewable technology adoption Optimal Pigouvian tax Status quo Figure 2: Benchmark Calibration: Growth Contributions 21

nology improvements is proportional to the capital deployed to the renewable sector, it is more effective to improve the renewable technology faster when a smallershareofenergyisproducedbytherenewablesector. Thus,asweexplain before,undertheoptimalrenewabletechnologyadoption,itisoptimaltoreduce theshareofrenewableearlyonandcatchonlater. Ifinsteadthecostofimprovingrenewabletechnologywasindependentofthecapitaldeployedinthissector, the technological improvement rate in the renewable sector would be constant and independent of the share of renewables in energy production. Therefore, thecasewhereadoptioncostsarenotcommensuratewiththeamountofcapital usedintherenewablesectorissimilartothecasewithoutreplacementcoststhat we described above. In such a case, adopting the optimal Pigouvian tax would not affect the dynamics of improvement in the renewable technology. Thus, there would be no complementarity between the optimal renewable technology adoption policy and the optimal Pigouvian tax. In contrast, in our benchmark case where the cost of improvements in renewable technology is proportional to the capital used in the sector, the optimal Pigouvian tax reduces the initial negative effect from the optimal renewable adoption policy. In summary, the dynamics of the improvements in renewable energy resulting from capital replacement costs generate a complementarity between the optimal technology adoption policy and the optimal Pigouvian tax. 6.1 Productivity Growth in the Fossil Fuel Sector As a robustness check, here we explore how our findings are affected if we consider productivity growth in the fossil fuel sector. We calibrate this to grow at the same rate as it did between 1950 and 2000. We make two additional assumptions. First,weassumeaconstantshareofenergyintheglobaleconomy duringthesefivedecades. Second,weassumethatrenewableenergywasasmall share of total energy, and that almost the entire increase in energy production during that period was due to the rise in the use of fossil fuel. Between 1950 and 2000, the annualized growth rate of world GDP in constant prices was 1.68 percent. In the same period, the annualized growth rate of CO emissions 2 was 1.27 percent.21 Based on these observations, and assuming a constant carbon intensity, the productivity of the fossil fuel sector has grown at the annualized rate of 0.40 percent during that period. Incorporatingproductivitygrowthinfossilfuelchangesthecalibratedvalues of both the spillover parameter, ξ, and the Pigouvian tax rate, τf, relative to thebenchmarkcalibration. Thebiggestdifferenceisinthenewcalibratedvalue of ξ, which is closer to the optimal level at ξ = 0.69 compared to ξ = 0.54 in the benchmark calibration. This is because the renewable sector must now exhibithigherproductivitygrowthrelativeto thebenchmarkinorderto match thesamelevelofincreaseintheshareofrenewables. Incontrast, thecalibrated value for the Pigouvian tax rate remains at τf = 0.63·τ∗, almost the same as its benchmark calibrated level. 21See https://www.statista.com/statistics/264699/worldwide-co2-emissions/ and https://ourworldindata.org/grapher/world-gdp-over-the-last-two-millennia. 22

1.00 0.75 0.50 0.25 2050 2100 2150 2200 erahs ygrene elbaweneR 600 400 200 0 2050 2100 2150 2200 )tnelaviuqe CtG( noitpmusnoc leuf lissof eitalumuC 2.8 2.4 2.0 1.6 2050 2100 2150 2200 Year )suisleC( level lairtsudnierp eht evoba erutarepmet labolG Optimal Pigouvian tax Status quo Figure 3: No Scrapping: Energy and Temperature 23

0.00 −0.04 −0.08 −0.12 2050 2100 2150 2200 )tnecrep dezilaunna( htworg ot CHG ni egnahc fo noitubirtnoC 0.04 0.02 0.00 −0.02 2050 2100 2150 2200 )tnecrep dezilaunna( htworg ot ygrene fo noitubirtnoC 0.05 0.00 −0.05 −0.10 2050 2100 2150 2200 Year )tnecrep dezilaunna( htworg ot ygrene dna CHG fo noitubirtnoC Optimal Pigouvian tax Status quo Figure 4: No Scrapping: Growth Contribution 24

Figure 5 shows the paths for the share of renewable energy (top), accumulated fossil fuel consumption (middle), and global temperatures (bottom), under our four scenarios. Like in Figure 1, these scenarios correspond to the status quo (dotted line), optimal renewable technology adoption (dashed line), optimal Pigouvian carbon taxation (dot-dashed line), and combined optimal policies (solid line). AcomparisonbetweenFigures1and5revealsthattheeconomyreachesthe fullrenewablestageearlierwhenweincorporateproductivitygrowthinthefossil fuel sector to the model. This seemingly counter-intuitive result is due to the differenceinthecalibratedvalueofξ,asthespilloverfromrenewabletechnology adoption is higher when we incorporate productivity growth in fossil fuel. As a result,theproductivityoftherenewablesectorgrowsfasterandsurpassesearlier the level at which the fossil fuel sector is no longer competitive. In addition, as the middle panel of Figure 5 demonstrates, fossil fuel is not exhausted under the status quo path. Consequently, the maximum global temperature reaches 2.6 degrees Celsius above the pre-industrial level under the status quo (Figure 5 bottom panel), below the 2.8 degrees Celsius achieved under the status quo path of the benchmark calibration (Figure 1). The qualitative effects of switching from the status quo to the paths associated with optimal renewable technology adoption and optimal Pigouvian taxation, respectively, are similar to those in our benchmark calibration. As seen in the top panel of Figure 5, the optimal renewable technology adoption path results in an immediate lower renewable share. However, faster growth in therenewablesectorresultsinanearliertransitiontoafullyrenewableeconomy. In contrast, switching from the status quo to the optimal Pigouvian taxation path would result in an immediate rise in the renewable share, but the path for renewables in this case is more or less parallel to the status quo. Finally, the combinedoptimalpolicyleavesthecurrentlevelofrenewableenergysharemore or less unchanged, but the resulting path in this case would be parallel to the optional adoption path, thus reaching the fully renewable economy faster than in the other cases. The paths for fossil fuel consumption and for global temperatures (middle and bottom panels, respectively, of Figure 5) remain similar to those in the benchmark calibration (middle and bottom panels, respectively, of Figure 1). Moving from the status quo to optimal technology adoption alone (but no Pigouvian tax) would imply a 0.74 percent consumption-equivalent gain, while theoptimalPigouviantaxalonewouldresultinagainof1.09percent,confirming the relative importance of the carbon tax. Comparing the status quo to the situation where both policies are applied results in a consumption-equivalent welfare gain of 1.53 percent. These findings are broadly consistent with the welfare gains in our benchmark case. 25

1.00 0.75 0.50 0.25 0.00 2050 2100 2150 2200 erahs ygrene elbaweneR 600 400 200 0 2050 2100 2150 2200 )tnelaviuqe CtG( noitpmusnoc leuf lissof eitalumuC 2.4 2.0 1.6 2050 2100 2150 2200 Year )suisleC( level lairtsudnierp eht evoba erutarepmet labolG Full optimal policy Optimal renewable technology adoption Optimal Pigouvian tax Status quo Figure 5: Long-run Economy with Productivity Growth in Fossil Fuel 26

7 Conclusion We incorporated costs associated with adopting new capital in the renewable energy sector in an IAM framework. As renewable technologies are still relatively new, and the sector is experiencing rapid growth, such advancements can be frequent, resulting in significant costs. The details of modeling these costs become less relevant on a balanced growth path, when they become the dominant energy source. We showed that in a capital-intensive industry like theenergysector,whentechnologicalprogressisembeddedinthecapitalstock, such costs can have sizable quantitative implications for the optimal share of renewableenergy,especiallyintheshortrun. Weinvestigatedtheirquantitative implicationsfortheoptimalenergytransition. InthecasewherePigouviancarbon taxes are infeasible, tax/subsidy policies that subsidize renewables-in our casebyinternalizingspillovereffects-mightnotbeasuitablesubstitute. Hassler et al. (2020) found that, when it comes to accomplishing climate-related goals, making renewable energy cheaper is not an effective substitute for making fossil fuel more expensive. Our model reaches a similar conclusion. It suggests that these two policies are better thought of as complements. The tax/subsidy scheme forces firms to incur a higher technology adoption costs, by for example scrapping a larger portion of their capital, than under the status quo path. Particularlyinearlyyears, whenthepolicyhasnotyetalteredtheproductivity oftherenewablesectorrelativetothestatusquo,thisdiscouragesheavyinvestment in renewable energy capital. As time passes, and the policy results in a significantly more productive renewable sector, it becomes more profitable to investinrenewables,resultingintheirsharemorequicklysurpassingthatoffossil fuel. This conclusion holds true when we incorporate technological progress in fossil fuel into the model. 27

8 Appendix 8.1 Optimization by Households and Firms The representative household owns the firms as well as the capital and fossil fuel stocks. It rents capital to firms and sells fossil fuel to the non-renewable sector. The representative household’s problem is given by ∞ (cid:88) max βtu(c ) t t=0 s.t. ∞ (cid:88) p [c +k −(1−δ)k ]≤ t t t+1 t t=0 ∞ (cid:20) (cid:90) 1 (cid:21) (cid:88) p r k +w l +pff + πr dj+z , t t t t t t t j,t t t=0 0 (cid:36) ≤(cid:36) −f , (25) t+1 t t where p is the Arrow-Debreu price of the period t final good, δ is the depret ciation rate of capital, r is the rental price of capital, w is the wage rate, pf t t t is the price of fossil fuel, (cid:82)1 πr dj stands for the profits of renewable firms, and 0 j,t z are lump-sum transfers from the government. The government collects taxes t from the fossil fuel energy sector and rebates them lump-sum to households, balancing its budget in every period. The FOCs, which are also sufficient for a maximum, imply βu(cid:48)(c ) 1−δ+r = t+1 (26) t+1 u(cid:48)(c ) t and βu(cid:48)(c ) p t+1 = t+1. (27) u(cid:48)(c ) p t t Equation(26)saysthattherentalpriceofcapitalplusthenon-depreciatedpart of capital must equal the marginal rate of substitution between consumption in two consecutive periods. Equation (27) says that the marginal rate of substitution between consumption in period t and consumption in period t+1 must equal the relative price of the respective consumption goods. The final good-producing firms rent capital, hire labor, and buy energy in competitive markets at prices w , r , and pe, respectively. The representative t t t firm in the final good sector solves (cid:104) (cid:105) max A t ·(k t c)θk(l t )θl(e t )1−θ−r t k t c−w t l t −pe t e t . The FOCs imply that the marginal input productivities equal their respective prices: θ A (kc)θk−1(l )θl(e )1−θ =r , (28) k t t t t t 28

θ A (kc)θk(l )θl−1(e )1−θ =w , (29) l t t t t t and (kc)θk(l )θl (1−θ)A t t =pe. (30) t eθ t t Firms in the fossil fuel sector rent capital and buy fossil fuel. Additionally, they pay a per unit tax on the GHG emission from fossil fuel use, τ . The t representative firm in this sector solves max (cid:104) peA (f )1−αf (cid:16) kf (cid:17)αf −r kf − (cid:16) pf +τ (cid:17) f (cid:105) . t f t t t t t t t TheFOCsimplythatthevalueofthemarginalinputproductivitiesequaltheir respective prices: (cid:32) (cid:33)1−αf f peα A t =r (31) t f f kf t t and (cid:32) kf (cid:33)αf (cid:16) (cid:17) pe(1−α )A t = pf +τ . (32) t f f f t t t The production function for the renewable energy firms is given by (12). It dependsonthefirm’sproductivity,thefirm’stechnologyadoptionrate,andthe capitalused. Thefirmsinthissectorrentcapitalandreceiveasubsidy, Φ(i ), j,t whichisafunctionofthefirm’stechnologyadoptionrate,i . WeallowΦ(i ) j,t j,t to be negative and assume it is differentiable. In each period t, the renewable firm j maximizes future discounted profits subject to (13): ∞ max (cid:88) βt+τu(cid:48)(c ) (cid:104) pe Ψ(i )(E )1−αr (cid:0) kr (cid:1)αr −r kr +Φ(i ) (cid:105) t+τ t+τ j,t+τ j,t+τ j,t+τ t+τ j,t+τ j,t+τ τ=0 (cid:18)(cid:90) 1 (cid:90) 1 (cid:19) s.t. lnEj ≤lnEj +ξi +(1−ξ) i kr dj/ kr dj t+1 t j,t j,t j,t j,t 0 0 i ≥0, and E given. (33) j,t 0 Letλj betheLagrangianmultiplierassociatedwithequation(13). TheFOCs E,t of this problem are (cid:32) (cid:33)1−αr E peα Ψ(i ) j,t =r , (34) t r j,t kr t j,t −βtu(cid:48)(c ) (cid:104) peΨ(cid:48)(i )(E )1−αr (cid:0) kr (cid:1)αr +Φ(cid:48)(i ) (cid:105) =ξλj , (35) t t j,t j,t j,t j,t E,t and λj +βt+1u(cid:48)(c )pe (1−α )er =λj . (36) E,t+1 t+1 t+1 r j,t+1 E,t 29

Equation(34)saysthatthevalueofthemarginalproductivityofcapitalshould be equal to its rental price. Equation (35) says that the cost of increasing the adoptionrate, whichisthelossinproductionplusthemarginalsubsidy, should equalthebenefitfromincreasingtheadoptionrate,whichcomesfromthevalue of having a higher level of productivity next period. Equation (36) says that the value this period from relaxing constraint (13) should be equal to the value fromrelaxingthatconstraintnextperiodplusthebenefitofhigherproductivity next period. 8.2 Proof of Proposition 2 Proposition 2 in the text states: Proposition 2: In a competitive equilibrium with Φ(i )=0,i is lower than j,t j,t optimal when ξ <1. Proof. From Proposition 1, the social planner chooses i j,t = i t and E k j j r , , t t = k E t r t . This, together with the FOC (20), implies that (cid:18) kr(cid:19)αr kr (cid:90) 1 −Ψ(cid:48)(i )E t µj =ξµj +(1−ξ) j,t µj dj. (37) t j,t E t r,t E,t k t r 0 E,t The FOCs of the social planer’s problem also give (kc)θk(L )θL βtu(cid:48)(c )(1−θ)A t t =µ =µj . (38) t t (e )θ E,t r,t t Equation (37) together with (38) and (30), implies (cid:18) kr(cid:19)αr kr (cid:90) 1 −βtu(cid:48)(c )peΨ(cid:48)(i )E t =ξµj +(1−ξ) j,t µj dj. (39) t t t j,t E t E,t k t r 0 E,t The FOC with respect to E is j,t+1 1 (cid:18)kr (cid:19)αr 1 µj +µj (1−α )Ψ(i ) j,t+1 =µj , (40) E,t+1E t j +1 r,t+1 r j,t+1 E j,t+1 E,tE t j +1 which can be rewritten using condition (12) as µj +βt+1u(cid:48)(c )pe (1−α )er =µj . E,t+1 t+1 t+1 r j,t+1 E,t Solving for µj , we obtain E,t ∞ (cid:88) µj = βt+τu(cid:48)(c )pe (1−α )er , if lim µj =0. (41) E,t t+τ t+τ r j,t+τ E,τ τ→∞ τ=1 30

Replacing (41) in (39), we obtain −Ψ(cid:48)(i )E (cid:18) k t r(cid:19)αr = ξ (cid:88) ∞ βτ u(cid:48)(c t+τ )pe t+τ(1−α )er + (42) t j,t E u(cid:48)(c )pe r j,t+τ t τ=1 t t (1−ξ) (cid:88) ∞ βτ u(cid:48)(c t+τ )pe t+τ(1−α ) k j r ,t+τ (cid:90) 1 er dj. τ=1 u(cid:48)(c t )pe t r k t r +τ 0 j,t+τ Solving equation (36) for λj , we obtain E,t ∞ (cid:88) λj = βt+τu(cid:48)(c )pe (1−α )er , if lim λj =0. (43) E,t t+τ t+τ r j,t+τ E,τ τ→∞ τ=1 Finally, replacing (43) in equation (35) (with Φ(i )=0) gives j,t −Ψ(cid:48)(i )(E ) (cid:18)k j r ,t (cid:19)αr =ξ (cid:88) ∞ βτ u(cid:48)(c t+τ )pe t+τ(1−α )er . (44) j,t j,t E u(cid:48)(c )pe r j,t+τ j,t τ=1 t t It is straightforward to verify that the right-hand side of equation (42) is larger than the right-hand side of equation (44). Since −Ψ(cid:48)(i ) is increasing in i , j,t j,t everythingelsebeingequal,thevalueofi thatsatisfies(44)inthecompetitive j,t equilibriumequationislowerthanthei thatsatisfies(42)inthesocialplanner’s t FOC. 8.3 Proof of Proposition 3 Proposition 3 in the text states: Proposition3: (1)Theoptimalallocationcanbesupportedbyacombinationof arevenue-neutralpolicy, Φ(i )=−(1−ξ)peΨ(cid:48)(i∗) (cid:16) e∗ j, r t (cid:17) (i −i∗),imposed j,t t t Ψ(i∗ ) j,t t j,t on renewable firms, together with a Pigouvian tax on fossil fuel use, τf = (cid:80)∞ βju(cid:48)(c∗ t+j ) π y∗ (1−d ), t j=0 u(cid:48)(c∗) t+j t+j j where {c∗,y∗,i∗}∞t is the solution to the planner’s problem, and 1−d = t t t t=0 j ϕ +(1−ϕ )ϕ (1−ϕ)j. (2) If u(c)=log(c), α =α =α, π =π, all t, and L L 0 r f t (cid:104) (cid:105) δ = 1, τf = y π ϕL + (1−ϕL)ϕ0 does not depend on the growth rate of the t t 1−β 1−(1−ϕ)β economy. Proof. When Φ(i )=−(1−ξ)peΨ(cid:48)(i∗) (cid:16) e∗ j, r t (cid:17) (i −i∗), the firm j’s FOC j,t t t Ψ(i∗ ) j,t t j,t (35) is (cid:34) (cid:32) (cid:33)(cid:35) e∗r −βtu(cid:48)(c ) peΨ(cid:48)(i∗) j,t =λj . (45) t t t Ψ(i∗ ) E,t j,t Solving (36) for λj , we obtain E,t ∞ (cid:88) λj = βt+τu(cid:48)(c )pe (1−α )er , if lim λj =0. (46) E,t t+τ t+τ r j,t+τ E,τ τ→∞ τ=1 31

Combining this equation with (45) gives −Ψ(cid:48)(i∗)(E ) (cid:18) k t r(cid:19)αr = (cid:88) ∞ βτ u(cid:48)(c t+τ )pe t+τ(1−α )er . (47) t j,t E u(cid:48)(c )pe r j,t+τ t τ=1 t t The social planner’s problem gives rise to a similar condition, (42), which we repeat here: −Ψ(cid:48)(i )E (cid:18) k t r(cid:19)αr = ξ (cid:88) ∞ βτ u(cid:48)(c t+τ )pe t+τ(1−α )er +(1−ξ) t j,t E u(cid:48)(c )pe r j,t+τ t τ=1 t t (cid:88) ∞ βτ u(cid:48)(c t+τ )pe t+τ(1−α ) k j r ,t+τ (cid:90) 1 er dj.(48) τ=1 u(cid:48)(c t )pe t r k t r +τ 0 j,t+τ Toshowthattheseconditionsareidentical,thusimplyingthesamei ,itsuffices t to show that kr (cid:90) 1 j,t er dj =er . (49) kr j,t j,t t 0 This follows from kr (cid:90) 1 kr (cid:90) 1 (cid:18) kr(cid:19)αr−1 j,t er dj = j,t Ψ(i )kr t dj k t r 0 j,t (cid:82) 0 1 k j r ,t dj 0 t j,t E t kr (cid:18) kr(cid:19)αr−1(cid:90) 1 = j,t Ψ(i ) t kr dj (cid:82) 0 1 k j r ,t dj t E t 0 j,t (cid:18) kr(cid:19)αr−1 = kr Ψ(i ) t =er . (50) j,t t E j,t t Next, suppose that sellers of fossil fuel face a linear tax rate, τf = (cid:88) ∞ βj u(cid:48)(c∗ t+j ) π y∗ (1−d ), (51) t u(cid:48)(c∗) t+j t+j j j=0 t where{c∗,y∗}∞ solvestheplanner’sproblem,and1−d =ϕ +(1−ϕ )ϕ (1− t t t=0 j L L 0 ϕ)j. Underthistax,thefossilfuelproducers’optimalintertemporalsubstitution implies u(cid:48)(c )·pf =βu(cid:48)(c )·pf . (52) t t t+1 t+1 Using (32) for the price of fossil fuel, we obtain (cid:110) (cid:111) (cid:110) (cid:111) u(cid:48)(c ) MPF −τf =βu(cid:48)(c ) MPF −τf , t t t t+1 t+1 t+1 and using (51) for the tax, we obtain u(cid:48)(c ){MPF −π y∗(ϕ +(1−ϕ )ϕ )} t t t t L L 0 ∞ (cid:88) + βju(cid:48)(c∗ )π y∗ ((1−ϕ )ϕ (1−ϕ)j−1ϕ) t+j t+j t+j L 0 j=1 = βu(cid:48)(c ){MPF }, (53) t+1 t+1 32

where MPF is the period t marginal productivity of fossil fuel in units of the t y∗ final good. Clearly, the claim follows if t+j = χ, a constant. First, observe c∗ t+1 that ct = χ ⇔ k t g +1 = θkβ. This equation follows from the FOCs of the social yt yt planner, which include y y θkβy t = t+1 t. (54) c c kg t t+1 t+1 It remains to be shown that kf kr t+1 + t+1 =1−χ−θkβ ≡(cid:37), (55) y y t t where kr = (cid:82) kr dm. The social planner problem’s FOCs with respect to kr t t,m j,t implies (cid:32) E (cid:33)1−αr y (cid:32) er (cid:33) y y α Ψ(i ) j,t (1−θ) t =α j,t (1−θ) t =θk t (56) r t kr e r kr e kg j,t t j,t t t er 1 kr α (1−θ) j,t = j,t . (57) r e βy t t−1 The FOC with respect to kf implies t ef 1 kf α (1−θ) t = t . (58) f e βy t t−1 It is sufficient to show that (cid:32) (cid:33) α er +α ef β(1−θ) r t+1 f t+1 =(cid:37), (59) e t+1 which is true if α =α =α. r f 8.4 The Optimal Transition Herewecharacterizetheequilibriumallocationacrossthetransition,andwederivesomekeyexpressionsthatareusedinourcalibration. LetV(k;A,L,E,w;Γp,Γd) denote the value given k available units of capital and given that the aggregate productivityisA,thelaborsupplyisl,theproductivityintherenewableenergy sector is E, the stock of fossil fuel is w, and the stocks of permanent and depreciating emissions are Γp and Γd, respectively. We let g stand for the percentage productivity growth rate in the final good sector, while gl is the population growth rate. The optimal consumption and saving decision under log utility and full depreciationisgivenbyc=(1−βΘ)yandk(cid:48) =βΘy,whereΘ=θ +(1−θ −θ )α k k (cid:96) 33

is the marginal product of capital. The recursive formulation for V(·) is given by V (cid:0) k;A,L,E,w;Γp,Γd(cid:1) = max{ln((1−βΘ)y) i,f +βV(βΘy;gA,gll,eiE,w−f;Γp(cid:48),Γd(cid:48)) (cid:9) where y =e−π(Γp(cid:48)+Γd(cid:48)−Γ¯)ALθ(cid:96) (cid:16) f +Ψ(i)1− 1 αE (cid:17)(1−α)(1−θk−θ(cid:96)) kΘ Γp(cid:48) =Γp+ϕ f L Γd(cid:48) =(1−ϕ)Γd+(1−ϕ )ϕ f. (60) L 0 Utilizing the envelope theorem, we have V =Θ1 +βΘk(cid:48)V(cid:48) , which implies k k k k(cid:48) kV =Θ+βΘk(cid:48)V . (61) k k(cid:48) We guess that kV is a constant and we verify that k Θ 1 V = . (62) k 1−βΘk (cid:110) (cid:111) Using the same method, we have that V = 1 +β k(cid:48)V(cid:48) +gV(cid:48) , which, in A A A k(cid:48) A(cid:48) turn, implies (cid:26) (cid:27) Θ AV =1+β +(gA)V(cid:48) . (63) A 1−βΘ A(cid:48) Next, we guess that AV is a constant. As A(cid:48) =gA, this equation allows us to A verify that 1 1 V = . (64) A (1−β)(1−βΘ)A Similarly, we obtain θ 1 V = l , (65) L (1−β)(1−βΘ)L 1 V = (−π), (66) Γp (1−β)(1−βΘ) 1−ϕ V = (−π). (67) Γd (1−β(1−ϕ))(1−βΘ) The last expression reflects the depreciation rate of the temporary part of the emissions stock. Finally, the marginal value of stock of fossil fuel is given by V =β·V(cid:48) . w w−f The optimal choice of f on the equilibrium path implies   π·(ϕ +(1−ϕ )ϕ )(1+β·k(cid:48)V(cid:48) ) (1−α)(1−θ −θ )  L L 0 k(cid:48)  k (cid:96) (1+β·k(cid:48)V(cid:48) )≤V +τ , f +Ψ(i)1− 1 αE k(cid:48) w  −β (cid:2) ϕ V(cid:48) +(1−ϕ )ϕ V(cid:48) (cid:3)  L Γp(cid:48) L 0 Γd(cid:48) 34

with equality when f > 0. The left-hand side of the above inequality gives the marginal benefit from consumption and from future capital accumulation, respectively. The first term of the right-hand side, V = βV(cid:48) , is the price w w−f of fossil fuel. The second term is the tax on consumption of fossil fuel. Note that τ ∈ [0,1], so this tax could take any value from zero to the total value of the present and future damages resulting from GHG emissions. We will find it convenient to rewrite the above inequality as follows: (1−α)(1−θ −θ ) f +Ψ(i)1− 1 αE ≥ (cid:110) (cid:111) k (cid:96) , (68) τπ ϕL + (1−ϕL)ϕ0 +(1−βΘ)V 1−β 1−β(1−ϕ) w with equality for f >0. For f,f(cid:48) >0, using V =βV(cid:48) and (68), we obtain w w(cid:48) (cid:110) (cid:111) (cid:16) f +Ψ(i)1− 1 αE (cid:17)−1 = β (cid:16) f(cid:48)+Ψ(i(cid:48))1− 1 αE(cid:48) (cid:17)−1 +(1−β) τπ 1 ϕ − L β + 1 (1 − − β ϕ (1 L − )ϕ ϕ 0 ) . (1−α)(1−θ −θ ) k (cid:96) (69) We use equation (69) to find the equilibrium path of fossil fuel consumption by solvingthispathbackward. Todoso,wealsoneedtodeterminetheequilibrium path of the renewable energy productivity. The optimal choice for i, when the representative agent takes into account onlyξfractionofthebenefitofhigherionfuturerenewableproductivity,implies 1 Ψ(cid:48)(i)Ψ(i)1− 1 α −1E (cid:26) Θ (cid:27) 0=(1−α)(1−θ −θ )1−α 1+β +β·ξeiEV(cid:48) (70) k (cid:96) f +Ψ(i)1− 1 αE 1−βΘ (cid:124)(cid:123)(cid:122)(cid:125) E(cid:48) E(cid:48) or −Ψ(cid:48)(i)(1−θ k −θ (cid:96) ) Ψ(i)1− 1 αE =β·ξE(cid:48)V(cid:48) . (71) Ψ(i) 1−βΘ f +Ψ(i)1− 1 αE E(cid:48) Utilizing the envelope theorem, we have EV = (1−α)(1−θ k −θ (cid:96) ) Ψ(i)1− 1 αE +βE(cid:48)V(cid:48) . (72) E 1−βΘ f +Ψ(i)1− 1 αE E(cid:48) Combining the above equation with (71) , we obtain 1 Ψ(i(cid:48))1−αE(cid:48) −Ψ(cid:48)(i) 1 (cid:18) −Ψ(cid:48)(i(cid:48)) (cid:19) f(cid:48)+Ψ(i(cid:48))1−αE(cid:48) =β· ξ(1−α)+ . (73) Ψ(i) 1 Ψ(i(cid:48)) Ψ(i)1−αE 1 f+Ψ(i)1−αE Equation (73) shows how the evolution of the elasticity of the technology adoption cost with respect to the adoption rate, −Ψ(cid:48), between two consecutive pe- Ψ riods depends on the corresponding ratio of the share of renewable energy, 1 Ψ(i)1−αE . To determine the path of i and f, we begin by determining (cid:98)i, 1 f+Ψ(i)1−αE 35

the long-run i. On a long-run balanced growth path, we have f = f(cid:48) = 0, and i=i(cid:48) =(cid:98)i, where(cid:98)i is determined by −Ψ(cid:48)((cid:98)i) β = ·ξ(1−α). (74) Ψ((cid:98)i) 1−β The minimum E for which f is zero follows from (68): (1−α)(1−θ −θ ) Ψ((cid:98)i)1− 1 αE = (cid:110) k (cid:96) (cid:111) . (75) τπ ϕL + (1−ϕL)ϕ0 1−β 1−β(1−ϕ) Note that E < ∞ only if τ > 0. The representative agent could exhaust the stock of fossil fuel before the productivity of the renewable energy reaches E, in which case the price of fossil fuel will be positive. However, once E ≥E, the stock of fossil fuel is not exhausted. In the period right before the use of fossil fuel ends–i.e., when f > f(cid:48) = 0, following (73) and given that i(cid:48) =(cid:98)i–we have: (cid:32) (cid:33) −Ψ(cid:48)(i) f +Ψ(i)1− 1 αE −Ψ(cid:48)((cid:98)i) = β· ξ(1−α)+ Ψ(i) Ψ(i)1− 1 αE Ψ((cid:98)i) f +Ψ(i)1− 1 αE ξ(1−α) = β· · . (76) Ψ(i)1− 1 αE 1−β Substitutingf(cid:48) =0in(69)tosolveforf+Ψ(i)1− 1 αE,andnotingthatE =e−iE(cid:48), the above equation gives −Ψ(cid:48)(i) 1 ξ(1−α) ·Ψ(i)1− 1 αe−iE(cid:48) =β· · , (77) Ψ(i) (cid:124)(cid:123)(cid:122)(cid:125) β + (1−β) 1−β E Ψ((cid:98)i)1− 1 αE(cid:48) τπ (cid:26) (1 1 ϕ − − L α β ) + (1 ( − 1 1 − θ − k β ϕ − (1 L θ − ) (cid:96) ϕ ϕ ) 0 ) (cid:27) which uniquely determines i, given the next period’s productivity, E(cid:48). Given i and utilizing (69), we have 1 f˜= −Ψ(i)1− 1 αe−iE(cid:48). (78) β + (1−β) (cid:124)(cid:123)(cid:122)(cid:125) Ψ((cid:98)i)1− 1 αE(cid:48) τπ (cid:26) (1 1 ϕ − − L α β ) + (1 ( − 1 1 − θ − k β ϕ − (1 L θ − ) (cid:96) ϕ ϕ ) 0 ) (cid:27) E Note that f˜ is the maximum level of fossil fuel consumption prior to ending its use. That is, if the stock of remaining fossil fuel was larger than f˜, some of the fossil fuel would be left for consumption in the next period.22 Thus, it is possible that the stock of remaining fossil fuel is in fact lower than f˜. In such cases, equation (69) does not hold, since f(cid:48) = 0 and (68) is an inequality. 22Equation(69)holdswhenbothf andf(cid:48) arepositive,butitalsoholdsiff =f˜and(68) holdswithequalityforf(cid:48)=0. 36

Nevertheless, for any value of f <f˜, we can determine i simply by noting that E =e−iE(cid:48) and using −Ψ(cid:48)(i) Ψ(i)1− 1 αe−iE(cid:48) ξ(1−α) · =β· . (79) Ψ(i) f +Ψ(i)1− 1 αe−iE(cid:48) 1−β Whenf,f(cid:48) >0,using(69)tosolveforf+Ψ(i)1− 1 αE in(73)andnotingthat E =e−iE(cid:48), we obtain −Ψ(cid:48)(i) Ψ(i)1− 1 αe−iE(cid:48) = β· 1 · Ψ(i(cid:48))1− 1 αE(cid:48) Ψ(i) (cid:124)(cid:123)(cid:122)(cid:125) β + (1−β) f(cid:48)+Ψ(i(cid:48))1− 1 αE(cid:48) E f(cid:48)+Ψ(i(cid:48))1− 1 αE(cid:48) τπ (cid:26) (1 1 ϕ − − L α β ) + (1 ( − 1 1 − θ − k β ϕ − (1 L θ − ) (cid:96) ϕ ϕ ) 0 ) (cid:27) (cid:18) −Ψ(cid:48)(i(cid:48)) (cid:19) × ξ(1−α)+ , (80) Ψ(i(cid:48)) which allows us to uniquely determine i for a given E(cid:48), i(cid:48), and f(cid:48). Then, using i and (69), we obtain the equilibrium path f: 1 f = −Ψ(i)1− 1 αe−iE(cid:48). (81) β + (1−β) (cid:124)(cid:123)(cid:122)(cid:125) f(cid:48)+Ψ(i(cid:48))1− 1 αE(cid:48) τπ (cid:26) (1 1 ϕ − − L α β ) + (1 ( − 1 1 − θ − k β ϕ − (1 L θ − ) (cid:96) ϕ ϕ ) 0 ) (cid:27) E Byusingthisbackwardcalculationwecandeterminetheentireequilibriumpath for all possible initial stock of fossil fuel and renewable productivity levels.23 Finally, if f =0, we have (1−α)(1−θ −θ )1 V = k (cid:96) . (82) E (1−β)(1−βΘ) E Hence, V (cid:0) k;A,L,E;0,Γp,Γd(cid:1) = C+ Θ lnk 1−βΘ 1 + {lnA+θ lnL+(1−α)(1−θ −θ )lnE} (1−β)(1−βΘ) l k (cid:96) π π(1−ϕ) − Γp− Γd, (83) (1−β)(1−βΘ) (1−(1−ϕ)β)(1−βΘ) whereC isaconstant. Notethatlog utility,fulldepreciation,andthestructure of the damage function imply that the above expression is linear in Γp and Γd. 23Wecanshowthatgoingbackward,iconvergesto(cid:98)if determinedby −Ψ(cid:48)((cid:98)if) βe(cid:98)if = (1−α), Ψ((cid:98)if) 1−βe(cid:98)if where(cid:98)if >(cid:98)i. 37

8.5 Productivity Growth in the Fossil Fuel Sector Here we characterize the equilibrium allocation across the transition in an extension of the model where we allow for growth in the fossil fuel sector. We let V(k;A,A ,L,E,w;Γp,Γd) denote the value given that the productivity of the f fossil fuel sector is A and the growth in productivity of the fossil fuel sector is f gf. The recursive formulation for V(·) is given by V (cid:0) k;A,A ,L,E,w;Γp,Γd(cid:1) = max{ln((1−βΘ)y) f i,f +βV(βΘy;gA,gfA ,gll,eiE,w−f;Γp(cid:48),Γd(cid:48)) (cid:9) f where y =e−π(Γp(cid:48)+Γd(cid:48)−Γ¯)ALθ(cid:96) (cid:16) A f f +Ψ(i)1− 1 αE (cid:17)(1−α)(1−θk−θ(cid:96)) kΘ, Γp(cid:48) =Γp+ϕ f, L Γd(cid:48) =(1−ϕ)Γd+(1−ϕ )ϕ f. (84) L 0 Proceeding as before, we obtain E (1−α)(1−θ −θ ) f +Ψ(i)1− 1 α ≥ (cid:110) (cid:111) k (cid:96) , (85) A f τπ ϕL + (1−ϕL)ϕ0 +(1−βΘ)V 1−β 1−β(1−ϕ) w with equality for f >0. For f,f(cid:48) >0, using V =βV(cid:48) and (85), we obtain w w(cid:48) 1 β 1−β = + . f +Ψ(i)1− 1 α A E f f(cid:48)+Ψ(i(cid:48))1− 1 α A E (cid:48) f (cid:48) τπ ( (cid:110) 1− 1 ϕ − α L β )( + 1− ( 1 1 − θ − β k ϕ ( − 1 L − θ )ϕ (cid:96) ϕ ) 0 ) (cid:111) (86) Like before, we use equation (69) to find the equilibrium path of fossil fuel consumption by solving this path backward. The equation determining the optimal choice for i together with the envelope theorem implies 1 Ψ(i(cid:48))1−αE(cid:48) −Ψ(cid:48)(i) A(cid:48)f(cid:48)+Ψ(i(cid:48))1− 1 αE(cid:48) (cid:18) −Ψ(cid:48)(i(cid:48)) (cid:19) =β· f ξ(1−α)+ . (87) Ψ(i) 1 Ψ(i(cid:48)) Ψ(i)1−αE 1 Aff+Ψ(i)1−αE Equation (73) shows how the evolution of the elasticity of the technology adoption cost with respect to the adoption rate, −Ψ(cid:48), in two consecutive periods de- Ψ 1 pendsonthecorrespondingratiooftheshareofrenewableenergy, Ψ(i)1−αE . 1 Aff+Ψ(i)1−αE The minimum E for which f is zero follows from (85): E (1−α)(1−θ −θ ) Ψ((cid:98)i)1− 1 α = (cid:110) k (cid:96) (cid:111) . (88) A f τπ ϕL + (1−ϕL)ϕ0 1−β 1−β(1−ϕ) 38

Notethatwiththegrowthintheproductivityoffossilfuel,thelevelofE,for which fossil fuel is no longer used, depends on the productivity of the fossil fuel sectorA . Therefore,thislevelexistsonlyifthelong-rungrowthinproductivity f of the renewable sector, e(cid:98)i, which is given by (74), is larger than the growth in productivity of the fossil fuel sector, gf.24 Without loss of generality, we assume that when the productivity in the renewable sector is at E, the productivity in the fossil fuel sector is A = 1. f Therefore, for the levels of the renewable productivity E larger than E, when the renewable productivity grows at rate e(cid:98)i while fossil fuel productivity grows at rate gf, we should have (cid:16) (cid:16) (cid:17)(cid:17) A f =exp log(gf)× log(E/E)/(cid:98)i . The rest of the backward solution of the model works similar to the case without productivity growth in the fossil fuel sector. For example, when f > f(cid:48) =0wecandetermineiuniquelyfromthefollowingexpression, whichisakin to expression (77): −Ψ(cid:48)(i) 1 ξ(1−α) ·Ψ(i)1− 1 αe−iE(cid:48) =β· · . Ψ(i) (cid:124)(cid:123)(cid:122)(cid:125) β + (1−β) 1−β E Ψ((cid:98)i)1− 1 α A E (cid:48) f (cid:48) τπ (cid:26) (1 1 ϕ − − L α β ) + (1 ( − 1 1 − θ − k β ϕ − (1 L θ − ) (cid:96) ϕ ϕ ) 0 ) (cid:27) And then use i to determine f˜, which is the maximum level of fossil fuel consumption prior to ending its use, from the following expression akin to (78): 1 (gf)−1A(cid:48) f ·f˜= β + (1−β) −Ψ(i)1− 1 αe (cid:124) − (cid:123) i (cid:122) E (cid:125) (cid:48). (cid:124) A (cid:123)(cid:122) f (cid:125) Ψ((cid:98)i)1− 1 α A E (cid:48) f (cid:48) τπ (cid:26) (1 1 ϕ − − L α β ) + (1 ( − 1 1 − θ − k β ϕ − (1 L θ − ) (cid:96) ϕ ϕ ) 0 ) (cid:27) E 8.6 Calibrating the Spillover Externality Herewediscussanalternativewayofcalibratingξ,theparametercapturingthe importanceofspilloversininnovationintherenewableenergysector. Although coming from a different viewpoint, this method results in a similar value to the one we use in the paper. In our model, the spillover externality depends on the aggregate amounts of investment and capital stock in the renewable energy sector and affects the firms’ productivity. While measuring the direct effect of spillovers on productivity is challenging, one approach is to concentrate on knowledge spillovers. If knowledge, innovation, and productivity improvements are proportional to each other, then quantifying knowledge spillovers can be informative about the magnitude of spillovers in actual productivity improvements. It is common to use patents as a measure of acquired knowledge and 24If gf ≥e(cid:98)i, the use of fossil fuel never stops, as it is optimal to leave some fossil fuel for useinthefuturewhentheproductivityofproducingenergywithitstillexceedsthedamage. 39

patent citations as a measure of the reliance of new discoveries on existing knowledge. Cross-citations can then serve as a measure of connectivity across different technologies, sectors, or geographic locations. To get a handle on knowledge spillovers related to renewable energy, we investigate the interconnections between new and existing patents in renewable technologies. While we did not have access to cross citations across individual firms, Conti et al (2018) document aggregate cross-citations in renewable technology between the US, the EU, and Japan from 2000 to 2010. These are summarized below: Citing country Domestic Foreign EU .76 .24 Japan .61 .39 US .42 .58 Forexample,assumingergodicity,thiswouldimplythattheaveragenewEU patent citations constitute 76 percent of existing EU patents and 24 percent of existing Japanese and US patents. Simple averaging would lead to a ξ of 0.59, which is quite close to the calibrated value of 0.54 used in the paper. 8.7 Lack of Commitment In order to explore the implications of commitment, here we briefly consider an extension of the model in the main text. Suppose that the government experiences“electoraldeath”withprobabilityω ineachperiod. Moreprecisely, for any period t, with probability ω ∈ (0,1], the government learns at the end of t that it will not be around at the beginning of period t+2. The implied discountingsequencesforthegovernment,βG,andfortherepresentativeagent, βA, are given by βG = {1,β,β2(1−ω),β3(1−ω)2,...} βA = {1,β,β2,β3,...}. (89) We assume lack of commitment.25 Thus, at the beginning of each period, the taxes and subsidies are set for the current period. We examine the case where ξ = 1, so there are no technology spillovers and the only externality is the one associated with GHG emissions. In each period, the government chooses the Pigouvian tax rate on fossil fuel consumption, τf. Under no commitment, this t tax needs to be set in a time-consistent fashion. The optimal allocation can again be supported by a Pigouvian tax on emissions. This tax is lower than in the case studied in the earlier model. The government’s objective is a modified version of the planner’s objective in the previous section and it is given by 25SeeHarstad(2019)foradiscussionoftimeinconsistencyinarelatedmodel. 40

∞ (cid:88) u(c )+ (1−ω)t−1βtu(c ). (90) 0 t t=1 All feasibility constraints remain the same. As a result, the optimal allocation is characterized by similar FOCs as in the previous section, with the only modificationbeinginthediscountsequence. Wecandemonstratethefollowing. Proposition 4 (1) The government’s optimal allocation can be supported by a Pigouviantaxgivenbyτf =π y∗(1−d )+ (cid:80)∞ (1−ω)j−1βju(cid:48)(c∗ t+j ) π y∗ (1− t t t 0 j=1 u(cid:48)(c∗) t+j t+j d ), where {c∗,y∗,i∗}∞ is the solution to the government’s pro t blem, and 1− j t t t t=0 d =ϕ +(1−ϕ )ϕ (1−ϕ)j. (2) If u(c)=log(c), α =α =α, π =π, all t, j L L 0 r f t (cid:104) (cid:105) and δ =1, then τf =y π ϕ +(1−ϕ )ϕ + βϕL + β(1−ϕ)(1−ϕL)ϕ0 does t t L L 0 1−(1−ω)β 1−(1−ω)(1−ϕ)β not depend on the growth rate of the economy. (3) The tax is strictly lower than the optimal Pigouvian tax of the previous section for all t. 8.8 A Continuous Time Interpretation Here we provide a continuous time foundation of how our modeling of the technologyadoptioncostinthemaintextcanbeinterpretedasthecostsassociated with the scraping process. The instantaneous production function in this case is given by er(t)=[E (t)]1−αr (cid:2) kr(t) (cid:3)αr. (91) j j j Asthenewtechnologyadoptionrate,i (t),requiresscrappingofcurrentcapital, j capital evolves according to dkr(t) j =−f(i (t))dt, (92) kr(t) j j where f(cid:48) >0. Adoption of a newer technology, i (t), improves the productivity j of firm j according to dE (t) j =g(i (t))dt, (93) E (t) j j where g(cid:48) >0. The resulting change in production is der(t)=α [E (t)]1−αr (cid:2) kr(t) (cid:3)αr−1 dkr(t)+(1−α )[E (t)]−αr (cid:2) kr(t) (cid:3)αrdE (t). j r j j j r j j j (94) After replacing for the E (t) and kr(t) processes, we obtain j j der(t)=Ψ[i (t)][E (t)]1−αr (cid:2) kr(t) (cid:3)αrdt, (95) j j j j where Ψ[i (t)]≡(1−α )g(i (t))−α f(i (t)). We assume that Ψ(cid:48) <0. j r j r j 41

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