Pushing and Pulling Environmental Innovation: R&D Subsidies and Carbon Taxes

Transcription

1 Pushig ad Pullig Evirometal Iovatio: R&D Subsidies ad Carbo Taxes Matthew S. Clacy ad GiaCarlo Moschii * Abstract We use a ovel modelig framework that icorporates free etry ito the R&D sector ad ucertaity about techological opportuity to evaluate three policy regimes (relative to laissez faire) desiged to address a market with egative evirometal exteralities: a carbo tax, a R&D subsidy, ad a mix of the two istrumets. We show carbo taxes o their ow are sufficiet to obtai most of the welfare gais achieved by a optimal policy mix, ad that the optimal carbo tax level is relatively robust to chagig modelig assumptios, i cotrast to optimal R&D subsidies. We also show R&D subsidies ted to produce more disperse outcomes tha a carbo tax: either more R&D etrats (whe techological opportuity is favorable) or oe at all (whe techological opportuity is ufavorable). Key Words: Carbo tax, Icetive, Iovatio, Reewable eergy, R&D subsidy, Welfare. JEL codes: H3, O3, Q4, Q55, Q58 Selected Paper prepared for presetatio at the Agricultural & Applied Ecoomics Associatio aual meetig, Bosto, MA, July 3-August, 06. This versio: May 4, 06 * Matthew S. Clacy (mattsclacy@gmail.com) is a ecoomist with the Ecoomic Research Service, U.S. Departmet of Agriculture, ad GiaCarlo Moschii (moschii@iastate.edu) is professor ad Pioeer Chair i Sciece ad Techology Policy, Departmet of Ecoomics ad Ceter for Agricultural ad Rural Developmet, Iowa State Uiversity. The views expressed i this article are those of the authors ad may ot be attributed to the Ecoomic Research Service or the USDA. 0

2 . Itroductio Promotig the developmet ad use of alterative forms of eergy is a stadard compoet of policies aimed at adaptig to or mitigatig climate chage. At preset, most alterative eergy caot be produced at competitive-eough costs to capture a large share of the eergy market, which remais domiated by covetioal fossil fuels. The eed for policies to promote evirometal research ad developmet (R&D) activities stems from the existece of two major sets of market failures. As for ay iovatio udertake by the private sector, idivisibilities, risk, ad imperfect appropriability ca severely reduce the icetive to iovate to below what is socially desirable (Arrow 96). But for evirometal iovatios, this problem is compouded by the fact that the presece of ucompesated exteralities reduce the private value of ew techology (Jaffe, Newell ad Stavis 005, Newell 00). What are the most appropriate policies to promote iovatio i this settig? Broadly speakig, iovatio policies may be grouped ito two categories: push ad pull policies (Nemet 009). Push policies operate at the level of the R&D decisio, for example, by directly subsidizig R&D. Pull policies operate at the level of the market for the R&D output, for example, by subsidizig use of ew techologies. For evirometal exteralities, policies that lead agets to iteralize the exteral cost of pollutio such as a carbo tax also work as a pull policy vis-à-vis the promotio of alterative (clea) techologies. The relative effectiveess of carbo taxes ad R&D subsidies to promote evirometal iovatio has bee ivestigated i a few studies. A key fidig relates to their complemetarity: because of the eed to address multiple market failures, ad their differet mode of actio, both istrumets are desirable compoets i a optimal policy portfolio (Arrow et al. 009; Popp, Newell ad Jaffe 00). I a two-sector growth model with edogeous techical chage, Acemoglu et al. (0) show that the joit use of a R&D subsidy ad a carbo tax is crucial for directig the ecoomy away from usig iputs from the dirty sector. There is also some cosesus that a carbo tax is the most importat of the two policies. By simulatig a climate policy model, Popp (006) fids that the carbo tax aloe ca achieve as much as 95% of the welfare gais from the combied policies, whereas the R&D subsidy aloe achieves oly % of such beefits. Fischer ad Newell (008) cosider a broader set of six policies ad, i a umerical calibratio of the US electricity sector, fid the carbo tax to be the most efficiet istrumet, ad the R&D subsidy to be the least effective oe. Such coclusios appear supportive of the results of Parry, Pizer, ad Fischer (003), whose umerical aalysis fids that the gais from correctig the R&D market failure are smaller tha

3 those that arise from correctig the exterality. By cotrast, Acemoglu et al. (0) war agaist the dager of relyig solely o the carbo tax ad emphasize the critical role of the R&D subsidy for directig techological chage. I this paper we revisit the questio of the relative effectiveess of carbo taxes ad R&D subsidies to promote evirometal iovatio i the cotext of a ew model that maitais some critical features of iovatio i reewable eergy. The aalysis relies o the modelig framework of Clacy ad Moschii (05), who build a origial stochastic iovatio model to study the effectiveess of quatity madates as iovatio icetives. The model s structure is meat to capture some essetial log-ru features of the iovatio process ad evisios three distict stages: the choice of policy istrumet ad its level; the forward-lookig decisio of iovators to ivest i R&D, give the policy cotext ad their iformatio about techological opportuity; ad, ex post licesig of successful iovatios to adopters, followed by productio ad cosumptio decisios. Two sources of ucertaity are explicitly represeted: R&D ucertaity (iovatio is stochastic), ad policy ucertaity about the outlook for techological iovatio. More specifically, the model cosiders two sources of eergy, reewable eergy ad fossil fuels. The latter impose a egative exterality o society. Reewable eergy has o such exterality, ad the cost of producig it ca be lowered through R&D. As i Parry (995), Laffot ad Tirole (996) ad Deicolo (999), the R&D sector is distict from the productio sector adoptig the ew techology. The profit opportuity that motivates iovators is directly iflueced by policies that pealize dirty eergy use (e.g., the carbo tax) or directly promote iovatio (e.g., the R&D subsidy). The presumptio is that iovatios are pateted, ad successful iovators profit from licesig their techology to the productio sector. The licesig framework implemeted i the model permits a ovel free-etry represetatio of the R&D eterprise where the umber of iovators is edogeously determied, followig the approach of Spulber (03) ad Clacy ad Moschii (05). Multiple iovators ca raise welfare through two chaels: a icrease i the umber of iovatig firms icreases the expected quality of the best iovatio that will be discovered, ad, the ex post royalty rate for the best iovatio is reduced by the presece of competitors. This licesig formulatio also effectively captures the welfare spillover effect of iovatios ad the associated appropriability problem that is oe of the roots of R&D uder-provisio. The model also maitais a plausible presumptio about the iovatio process: by the time they choose R&D ivestmets, firms have better iformatio tha

4 policymakers did whe they set the policy. We ote at this jucture that the uderlyig assumptio is that the regulator ca commit to their policy choice. Of course, this is without loss of geerality for the R&D subsidy, a iheretly ex ate tool. But the carbo tax, i priciple, could be chaged ex post, that is after the realizatio of the iovatio. To keep the compariso meaigful, i this paper we assume that the govermet ca commit to the carbo tax. I the cotext of this model, we cosider three policy regimes (i additio to laissez faire) to address the sub-optimal provisio of R&D discussed earlier: a carbo tax, a R&D subsidy, ad a mixed policy that uses both subsidies ad taxes. The carbo tax is a pull policy that iduces iovatio by raisig the price of covetioal eergy ad therefore the price of clea eergy (which is priced to compete with it). The R&D subsidy is a push policy that iduces iovatio by reducig the cost of R&D. We use two methods to compare ad cotrast the impact of these policies. Uder some simplifyig assumptios, aalytical results are possible. We complemet the aalytical approach with a umerical simulatio that relaxes a umber of assumptios ad that allows us to characterize optimal (welfare-maximizig) policies. Our aalytical results show the optimal R&D subsidy does ot deped o the shape of demad or the outlook for techological opportuity whe there is a sigle iovator. The optimal carbo tax does deped o these parameters, ad we show that the optimal tax will be higher whe iovatio is take ito accout. We also discuss how the choice of policy impacts the distributio of outcomes; i geeral, a R&D subsidy will iduce a greater variace i R&D. Ideed, eve though a R&D subsidy operates directly o the R&D decisio, we fid a carbo tax does a better job of esurig iovatio occurs whe techological opportuity is low. We supplemet these results with a umerical simulatio that allows us to compare the welfare implicatios of alterative policies. Our umerical simulatios agree with much of the earlier literature. R&D subsidies o their ow achieve oly a fractio of the welfare gais attaied by a carbo tax o its ow, ad addig R&D subsidies to a carbo tax leads to mior additioal welfare gais. We also use our umerical simulatio to assess the sesitivity of optimal policies to differet parameter assumptios. We fid the optimal carbo tax is relatively robust to chagig assumptios, i cotrast to the optimal R&D subsidy. The well-kow time cosistecy problem of evirometal policies is aalyzed, amog others, by Laffot ad Tirole 996, Deicolo 999, Keedy ad Laplate 999, ad Motero, 0. 3

5 . The Model The stochastic iovatio model developed i Clacy ad Moschii (05) presumes that R&D is carried out by specialized firms which, if successful, ca licese their iovatios to a competitive sector. The purpose of R&D is to devise more efficiet techologies for the productio of reewable eergy, which represets a cleaer alterative for covetioal eergy. Without much loss of geerality, reewable eergy is assumed to have zero emissios. The amout of covetioal eergy is deoted Q, ad that of reewable eergy is deoted Q. These two sources of eergy are perfect substitutes from the cosumer s perspective: cosumers (iverse) demad fuctio is PQ ( ), where Q Q Q represet total eergy used. Total damage from emissios is X xq, where x is the (costat) margial evirometal damage rate. O the supply side, both forms of eergy are produced by competitive idustries, with idustry cost fuctios C( Q ) ad C( Q, ), where 0 deotes the quality of the iovatio. We maitai the presumptio that reewable eergy, eve after iovatio, is ulikely to completely supplat covetioal eergy. To capture this asymmetry, covetioal eergy is assumed to be produced with costat margial cost, whereas the ew clea techology displays a upward-slopig (idustry) margial cost fuctio, i.e., C( Q) c Q () C( Q, ) c Q Q () where c ad c are fixed parameters, with c c. These margial costs are illustrated i Figure. The quality of iovatio, deoted by, represets the realizatio of a radom variable. Specifically, a research firms that coducts a R&D projects, upo icurrig a (fixed) cost k 0, gets a draw 0 from the coditioal distributio fuctio F(. ) Here, the parameter 0, captures the outlook for iovatio, ad permits the model to represet a asymmetry betwee what researchers kow whe they make the R&D ivestmet, ad what the policy maker This commoly ivoked coditio that the margial evirometal damage of the exterality is costat, together with the assumptio that the coditioal distributio of firms iovatio outcomes is uiform (see below), simplifies the aalysis ad permits the derivatio of explicit results. 4

6 kows whe policies are chose. I particular, the timelie we cosider is as follows. First, the policymaker chooses the policy: either a R&D subsidy rate s [0,] that reduces the cost of R&D from k to ( sk ), a per-uit carbo tax t 0, or both. Whe makig this choice the policymaker is ucertai about the outlook for iovatio ad oly kows its distributio fuctio G( ). Coversely, research firms observe the actual realizatio of the parameter, ad of course kow the chose policy parameter prior to makig their R&D ivestmet. Whereas the distributio fuctio G( ) is urestricted, apart from the stadard mootoicity ad cotiuity properties, the aalytical results that we preset rely o postulatig that F( ) is a uiform distributio. The desity fuctio of this distributio is: if 0, f ( ) 0 otherwise (3) This represetatio provides a straightforward iterpretatio of the techological opportuity parameter: both the expected value ad the upper boud of the iovatio draw are icreasig i. But because eve the most promisig iovatio ca fail, the lower boud o iovatio quality is always zero, regardless of the prospect for iovatios. Figure. Covetioal ad reewable eergy: Iovatio, supply ad demad p C ( Q,0) Q C ( Q, ) Q c ˆ c C ( Q ) Q c PQ ( ) Q 5

7 I this settig, we wat to evaluate the effectiveess of a R&D subsidy ad/or a carbo tax as policy tools to promote iovatio. For a meaigful bechmark, we compare these policy scearios to the laissez faire (o policies) situatio.. Iovatio with a sigle iovator For all cases cosidered laissez faire situatio (absece of govermet policy), R&D subsidy or carbo tax the residual iverse demad curve facig producers of reewable eergy is: P Q c t if Q P ( c t) PQ ( ) otherwise (4) where t deotes the carbo tax (per uit of dirty eergy). For the laissez faire ad R&D subsidy cases, t 0. If clea eergy is priced below the cost of dirty eergy ( c t ), the it captures the etire market; if it is priced above the cost of dirty eergy, demad for clea eergy falls to zero; ad, ay quatity Q 0, P ( c t) eergy. ca be sold whe clea eergy is priced at the cost of dirty As oted earlier, the realistic sceario is that the ew reewable eergy source does ot completely replace the pre-existig covetioal source. That is, the iovatio is odrastic i Arrow s (96) termiology. The followig coditio, which we maitai throughout (but which is relaxed i the umerical aalysis), will guaratee this outcome. Coditio. The upper boud o techological iovatio satisfies c c P ( c ). I this sectio we cosider the case whe there is oly oe firm capable of iovatig (this assumptio is relaxed i later sectios). This settig is commo i the literature, ad relevat for applied policy aalysis whe there are substatial barriers to etry ito the iovatio market or whe the huma capital required to coduct R&D is scarce ad cocetrated. To characterize the iovator s decisio problem, cosider first the licesig stage for a arbitrary iovatio of quality. The iovator essetially acts as a moopolist with a competitive frige, ad sets the per-uit royalty r to maximizes profits coditioal o the adoptio costrait by the competitive producers of reewable eergy (which, give the foregoig cosideratios, face a perfectly elastic demad at price equal to c t). Thus, the iovator s optimal royalty maximizes rq, where the demad from 6

8 the competitive adoptig clea eergy sector, for Q 0, satisfies c Q r c t. Whe c c t there is o strictly positive licese fee that ca result i ay adoptio: the iovatio is isufficiet to be cost-competitive with the dirty techology. Thus, profitable licesig oly occurs if the iovative step is sufficietly large. More specifically, let ˆ c c t defie the miimum iovative step beyod which the iovatio becomes profitable (see Figure for the laissez-faire case for which t 0 ). For ˆ, the optimal royalty is r* ( ˆ ), ad at this price the quatity licesed is Q ˆ ( ). The maximum profit a iovator with techology ca obtai, whe ˆ, is ( ˆ ) 4 (ad, of course, 0 whe ˆ ). We will restrict attetio to situatios whe iovatio is required to make reewable eergy cost competitive with covetioal eergy: Coditio. The miimum ivetive step is o-egative, i.e., ˆ 0. Clearly, a researcher with techological opportuity ˆ expects zero profit. For ˆ the iovatio ca still yield zero profit wheever ˆ, which happes with probability ˆ, ad thus the researcher expects to make positive profit with probability ˆ. Expected licesig profit coditioal o, deoted, ca therefore be writte as: ˆ ( ˆ ) ( ) 4( ˆ ) 3 ˆ d ˆ (5) A risk eutral iovator will choose to coduct research if this expected licesig profit exceeds the subsidized costs of R&D, i.e., whe ( ) sk where 0, 7 s deotes the R&D subsidy rate. Note that s 0 for the laissez-faire case ad for the carbo tax oly regime. This implies the existece of a threshold ˆ, which satisfies ˆ ad oly if ˆ ˆ. ( ) s k, such that iovatio is udertake if Expected welfare ca be expressed i terms of the pre-iovatio static allocatio ad chages to this allocatio that are brought about by iovatio. Note that, give the presumptio that iovatio is o-drastic, eergy is always priced at c t. Accordigly, the total quatity of eergy Q, ad cosumer surplus, are ot affected by iovatio. Istead, iovatio affects the share of

9 eergy produced by reewable sources, ad reduces the damage from exteralities relative to the status quo ate by xq. Accoutig for the miimum iovatio step, ad proceedig aalogously to (5), expected clea eergy is EQ ( ˆ ) 4. Licese profits are give i equatio (5). Clea 3 producer profits ca be show to be ( ˆ ) 4 i expectatio. We assume subsidies ad taxes are possible as frictioless lump-sum trasfers. All told, therefore, expected welfare is Q ˆ 3 ˆ 3 ˆ ( ) ( ) ( ) E W P q c 0 x dq x k dg ˆ 4 4 (6) where the first itegral deotes pre-iovatio welfare ad the secod itegral i (6) is the expected cotributio of iovatio to welfare. Equatio (6) illustrates three potetial market failures: the evirometal exterality, the iovatio market failure, ad the iteractio of the two. First, abset govermet itervetio, cosumers cosume too much dirty eergy, so that P Q c c x. This misallocatio is captured by the first itegral. Secod, firms coduct R&D if private licesig profits (the first term uder the secod itegral) exceed k, but do ot take ito accout the spillovers of their discovery to the private sector. This private spillover is reflected i the gais to clea producers, give by the secod term i the secod itegral. Third, firms do ot take ito accout the reductio of evirometal damages that their discoveries eable. This is captured by the third term uder the secod itegral. These terms illustrate the reasos why iovatio is doubly uder-provisioed, as discussed by Jaffe, Newell ad Stavis (005). Iovatio here is socially desirable for some ˆ, but uder laissez faire is ot coducted. We ow cosider two policies to address these market failures.. R&D Subsidies A social plaer would like R&D to occur wheever: ˆ 3 ˆ 3 ˆ ( ) ( ) ( ) x k (7) 4 4 But as discussed above, uder laissez faire the lower boud ˆ is computed to take ito accout oly the first term i equatio (7). We model the R&D subsidy as the fractio s (0,) of the R&D fixed 8

10 cost that is paid by the govermet, i.e., the iovator receives a lump-sum subsidy sk. Note that, because techological opportuity is ot kow to policymakers whe the policy choice is made, the subsidy caot be tailored to. If ˆs deotes the threshold such that equatio (7) holds as a equality, the ˆ ˆ ad the optimal subsidy to R&D is such that the iovator s expected s licesig profit is equal to the usubsidized portio of R&D cost, that is s solves: ˆ ˆ 3 ( s ) ( sk ) (8) ˆ s Note that the optimal R&D subsidy does ot deped o kowledge of P Q or G. Because G describes the shape of beliefs about techological opportuity, its form may be highly ucertai ad disputed. A virtue of the optimal R&D subsidy is that it does ot require a cosesus for G. Remark : The optimal R&D subsidy does ot deped o demad parameters P Q or the policy-maker s beliefs about techological opportuity G. A R&D subsidy improves welfare by iducig iovators to take bigger risks ad choose to coduct R&D eve whe ˆ, ˆ (i.e., techological opportuity is relatively low). However, it s is importat to ote that, similar to the laissez faire, oly iovatios with ˆ will be profitably licesed. These iovatios are less likely to occur whe ˆ, ˆ s. Hece, usig a R&D subsidy icreases the frequecy of failures (ex post, the public might perceive such subsidies to be bad ivestmets). Remark : With a sigle iovator, R&D subsidies iduce more iovatio that is ulikely to be useful (i.e., with ˆ ). Ufortuately, a R&D subsidy does ot address the evirometal exterality except idirectly through the third term i equatio (7). A carbo tax, i cotrast, directly addresses this issue, but oly idirectly impacts the icetives to iovate. 9

11 .3 The aïve carbo tax For both the laissez faire ad the R&D subsidy cases, welfare is suboptimal because, iter alia, the ucompesated egative exterality meas there is excess productio of dirty fuel. The caoical solutio to a exterality of this type is a Pigouvia tax o the dirty fuel, e.g., a carbo tax. Because use of fossil fuels icurs a social cost x per uit, if oe igores the prospect of iovatio the tax should be set at t x. This tax iduces the optimal mix of dirty ad reewable eergy i the absece of iovatio, but iduces isufficiet iovatio. This is because, as oted earlier, iovatig firms oly take ito accout the impact of R&D o their ow licesig profits, ot the positive spillovers ejoyed by producers ad cosumers (i the form of reduced depedece o dirty fuel). Ideed, it ca be show: Remark 3: The optimal carbo tax with a sigle iovator is greater tha the aïve tax t x. To see why, differetiate the expected welfare i (6) with respect to the tax (recall that ˆ c c t ). Via Leibitz rule, recallig that ˆ k, ad simplifyig, we obtai: EW S ˆ ˆ ˆ 3 ˆ 0 9( ) ( ) ( ˆ ) ( ˆ ) ˆ x dg x t t ˆ t (9) Q where S0 P q c x dq. The optimal tax satisfies EW / t 0 0. At t x, however, S 0 / t 0. Moreover, we ote that profit is icreasig i t ad so ˆ / t 0. Thus; E W 3 9( ˆ ) ( ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) ˆ x dg x 0 t ˆ t (0) tx where the sigs of each term are displayed below them. Thus, expected welfare ca be icreased by raisig the carbo tax above its aïve level. Ituitively, it is worth reducig the total cosumptio of eergy by settig P Q c x because doig so iduces more iovatio, ad also iduces more productio of clea eergy (which is otherwise uderprovisioed because the iovator has market power). 0

12 3. Multiple iovators Whereas the precedig discussio pertais to the case of a sigle research firm, a more realistic descriptio of the research idustry should cosider may (competig) iovators egaged i R&D projects. Clacy ad Moschii (05) model this case by postulatig the existece of a large umber of potetial iovators ad free etry ito the reewable eergy iovatio sector. Iovators are ex ate idetical ad observe a commo techological opportuity sigal. If they choose to coduct R&D, they obtai idepedet draws from f. The iovator who draws the highest, deoted, has the best techology ad becomes the exclusive licesor to the reewable eergy productio sector. However, as i Spulber (03), the choice of royalty by the iovator who draws is ow costraied by the presece of competig iovators. Uder Bertrad competitio, the secod-highest draw, deoted, is the bidig costrait. Essetially, as compared with the foregoig aalysis, plays the same role as the pre-iovatio productio techique 0 for the sigle iovator case. But, of course, i the multiple iovator settig is edogeous. The pricig of iovatio i this multiple-iovators settig is characterized by Clacy ad Moschii (05). Cosider first the laissez faire settig. For low realizatios of, the costrait imposed by the secod-best techology does ot bid, the sigle-iovator results cotiue to hold, ad the solutio is r* Q ˆ ( ). But wheever ˆ ( ), the optimal royalty is r*, ad Q ˆ. The best iovator s maximum profit, deoted, is therefore give by: ( ˆ ) if ˆ ( ) () 4 ˆ if ( ˆ ) () The expected profit of a potetial etrat ow depeds o the distributio of ad, which are best described by the cocepts of order statistics widely used i auctio theory (Krisha 00). Specifically, give iovators, the probability that a iovator s draw of is the maximum draw is equal to the probability that the other draws are smaller tha. Because we have assumed a uiform distributio for the iovatio projects, this probability equals ( ).

13 Moreover, coditioal o a give beig the maximum draw, the secod highest realizatio is the maximum of idepedet draws from the uiform distributio o the support of 0,. Hece, the secod highest realizatio has cumulative distributio fuctio ( ) ad desity fuctio ( ). Usig these results o the distributio of the first ad secod best iovatios, we ca determie the expected profitability of participatig i the R&D cotest. Specifically, with etrats, the expected licesig profit of each iovator, give techological opportuity, ca be writte as: ˆ ˆ ( ) ˆ (, ) ( )( ) d d 4 ˆ ( ˆ ) (3) This term itegrates over the rage of values for that are both feasible ad ear positive profit. Withi the itegral, profits are divided ito two terms. Whe ˆ ( ), which occurs with probability ( ˆ ), profit is give by equatio (). This is the first term uder the itegral. Coversely, wheever ˆ ( ), profit is give by equatio (). This is captured by the secod term, itself a itegral over possible values of. The equilibrium umber of iovators is determied by the free etry coditio. I equilibrium, otig that is a iteger, the umber of iovators * satisfies:, s k, (4) Uder free etry, the choice of policy may have a sigificat impact o the distributio of outcomes. Remark 4: A carbo tax o its ow iduces (weakly) more etry of iovators for low values of techological opportuity ( c c) tha a R&D subsidy o its ow. To see why, recall that the miimum iovative step is defied to be ˆ c c t. Uder a pure R&D subsidy policy, t 0 ad therefore licesig profit is zero for c c. A carbo tax lowers this threshold, allowig for positive licesig profit for lower levels of techological opportuity. Provided k is ot too large relative to these licesig profits, there will be additioal etrats uder

14 a carbo tax relative to a R&D subsidy for c c. The opposite effect takes place for high values of techological opportuity. Remark 5: For a give tax t ad subsidy s, for sufficietly high techological opportuity, a R&D subsidy iduces more iovatio etrats tha a carbo tax. A proof is preseted i the appedix, but the ituitio here is that, for large iovatios, the expected profits of a potetial etrat uder a R&D subsidy are close to those uder a carbo tax (the additioal price premium provided by the tax is less importat). Although the expected licesig profits are of similar magitude uder either policy tools, however, etrats i a R&D subsidy regime compare profits to sk while etrats i a carbo tax regime compare them to the full cost k. Ceteris paribus, the former supports more iovators Numerical Aalysis The foregoig aalysis is uable to assess the relative magitude of the welfare effects, or the gais that come from a optimal mix of both policies. This is because welfare coclusios are boud to deped o the particular shape of the demad fuctio PQ ( ) ad o the distributio of techological opportuities G( ). I additio, our aalytical results have bee cotiget o the assumptios that clea eergy caot capture the etire market (Coditio ). I this sectio we relax this coditio ad specify explicit fuctioal forms for PQ ( ) ad G( ) so that we may cosider the impacts of the policy istrumets of iterests i a more geeral cotext by meas of a umerical aalysis. 4. Parameterizatio We employ the same parameterizatio as Clacy ad Moschii (05). The margial cost of covetioal eergy is ormalized to c 00, so that a tax o dirty eergy ca be iterpreted as a percet of the laissez-faire price level. I the baselie parameterizatio the exterality is calibrated to 3 Note that Remark 5 oly applies whe is greater tha some 0, as discussed i the appedix. If 0, that satisfies Remark 5. 0 the there may be o 3

15 x 0, so that it amouts to 0% of the private cost of dirty eergy, 4 ad we put c 0 so that reewable eergy is o the cusp of beig socially desirable, but still requires iovatio. Next, we postulate the iverse demad fuctio p( Q) ( a l Q) b or, equivaletly, that the direct demad fuctio for eergy takes the semi-log form: l Q a bp (5) This is a coveiet parameterizatio which, amog other desirable features, ca accommodate various hypotheses cocerig demad elasticity lq l p. For this fuctio bp, hece the parameter b ca be varied to implemet alterative elasticity values. The parameter a is calibrated so that total demad for eergy at price p c (ad at the baselie elasticity value) is equal to Q 00, that is we put a bc l00. As for G( ), we assume that is distributed o 0, by a appropriately scaled beta distributio. The probability desity fuctio g( ) is therefore give by: g ;, / / (6) where the parameters ad determie the momets of this distributio ad gover its shape. This distributio is very flexible, ad alterative choices of ad ca yield both symmetric ad skewed desity fuctios. We ormalize 0 so that, uder all possible iovatio, the margial cost of clea eergy remais o-egative everywhere. Give the foregoig fuctioal form assumptios ad parametric ormalizatios, there are four free parameters that ca be varied to gai some isights i the ature of the results. The first of these is the elasticity of demad. Because this value depeds o the evaluatio price, for clarity we will 4 This value for the exterality cost is meat to be somewhat represetative of estimates for the social cost of carbo relative to the cost of trasportatio fuel. The US govermet s estimate for the 05 social cost of carbo, i 007 dollars, is $37/to of CO if a 3% discout rate is used, ad $57/to of CO if a.5% discout rate is used (US Govermet 03, p. 3). These discout rates have bee criticized for beig too high (Johso ad Hope 0), ad so we use the figure associated with the lower.5% discout rate as our baselie. Covertig this estimate to 05 dollars yields a social cost of $65/to of CO. The carbo emissio coefficiet is 8.9 kg CO/gallo of gasolie (EPA 04), which implies a social cost of carbo is $0.58 per gallo. Takig the bechmark price of gasolie to be $3.00/gallo, the the damage imposed by the carbo exterality is approximately 0% of the cost of fuel, which is reflected i our baselie value of x 0. 4

16 always measure elasticity with referece to the laissez-faire price of eergy, where p c. For our baselie, we set b so that 0.5. We also cosider the cases where 0.5 ad (these values reflect the widely-held belief that eergy demad is ielastic; see Toma, Griffi ad Lempert 008, p. 8). Secod, we vary the cost of the exterality x. As oted, for the baselie we set x 0, but we also cosider the cases of x 0 ad x 40. Third, we vary the R&D cost k. To calibrate this parameter we relate it to the magitude of profits that iovatio ca produce i the laissez-faire baselie. Uder the highest level of techological opportuity, the expected profit for a sigle iovator, i view of (5) ad the chose ormalizatios, is equal to t( ) 6,50 9. We cosider values of k equal to 3%, 6%, ad % of this profit level, with 6% correspodig to the baselie. Fourth, we vary the shape of the distributio of techological opportuity G( ). The first momet of the assumed beta distributio is E ( ). We set ad, by varyig the parameters ad, we obtai both differet values for E ad differet shapes. The baselie parameters are 0.5 ad.5, which yield E This is a positively skewed distributio (low draws of are more likely tha high oes), which reflects the belief that techological opportuity is more likely to be cosistet with icremetal iovatio tha major breakthroughs. The other two cases we cosider are 0.5 ad.75, which yield E 5 (ad correspod to a eve more positively skewed distributio), ad ad, which yield E 60 (ad correspod to a uiform distributio where high draws of are equally likely as low oes). As for the policies t ad s, for each set of parameters that we cosider, we umerically solve for the value of the policy istrumet that maximizes welfare (expected Marshallia surplus) for a R&D subsidy o its ow, a carbo tax o its ow, ad a combiatio of R&D subsidy ad carbo tax. All calculatio are coded i Matlab. 4.. Results Some basic descriptive results for the baselie parameters are reported i Table. For the sigle iovator case the expected umber of iovators E ca be iterpreted as the probability that R&D will be coducted. I the baselie settig, uder a laissez-faire policy, R&D is coducted with probability 0.5 for the sigle iovator case. The expected quality of iovatio E is 9.6, which improves to 5.9 with multiple iovators. Hece, i either case the average techology

17 uder laissez faire is isufficiet to compete with fossil fuels (the miimum ivetive step here is ˆ 0 ). Still, some iovatio does take place uder laissez faire, because some better-tha-average draws will be profitable. The expected quatity of clea eergy cosumed is small but ot egligible, at.6 ad 8.7 uder the sigle iovator ad free etry coditios respectively (recall that the laissezfaire quatity of total eergy cosumed was ormalized to 00). Table. Numerical Results for Baselie Laissez Faire R&D Subsidy Carbo Tax Mixed Policy Etrats Sigle Multiple Sigle Multiple Sigle Multiple Sigle Multiple Subsidy rate Carbo Tax E Var( ) E Var( ) EQ Var( Q ) E W 8,3 8,40 8,9 8,455 8,30 8,676 8,30 8,679 Var( W ) Note: the baselie parameters are 0.5, x 0. c, k 0.06 ( ), ad 0.5 ad.5 (i.e., E[ ] 30 ). A optimal policy (tax or R&D subsidy) raises all these quatities, ad also improves welfare. The expected quality of iovatio E, as well as the expected quatity of clea eergy produced EQ, is sigificatly icreased by the carbo tax, but less so by the R&D subsidy, despite the fact that the optimal level of the subsidy is quite high (83% for the sigle iovator case ad 65% uder free etry of iovators). R&D subsidies are particularly poor at icreasig the use of clea eergy. These features reflect the fact that R&D subsidies iduce lower-quality R&D projects that would ot otherwise be pursued, ad which are less likely to exceed the miimum ivetive step, as oted i Remark. Note also that, i the sigle iovator settig, eve though the subsidy is quite large, 6

18 R&D is more likely to occur uder a carbo tax ( E is 0.56 istead of 0.36). This reflects the superior ability of a carbo tax to iduce iovatio for ˆ. I both sigle ad multiple iovators cases, welfare is also higher uder a optimal carbo tax, but of course highest uder the optimal mixed policy. To gai further isights ito the performace of each policy, Table illustrates the sesitivity of optimal policies to chages i the calibrated parameters. Table. Optimal Policy Istrumets Uder Alterative Assumptios R&D Subsidy Carbo Tax Etrats Sigle Multiple Sigle Multiple Policy Mix S S+T S S+T T S+T T S+T Baselie x 0 x 40 k 0.03 k E E Note: Each row chages oe parameter, all other parameters as i the baselie The first row of Table reiterates the optimal policies for the baselie parameterizatio reported i Table. Each subsequet row presumes the same parameters as the baselie, except alog oe dimesio. For example, i the secod row the elasticity of demad, evaluated at the laissez-faire price, is chaged to 0.5. Each colum gives the optimal policy value for a R&D subsidy or carbo tax i the presece of a sigle or multiple etrats, ad whe the policy is cosidered aloe ( S for subsidy or T for carbo tax) or as part of a pair (deoted S+T ). O its ow, the optimal R&D subsidy is substatial ad everywhere greater tha 50%. The subsidy 7

19 is largest whe there is a sigle iovator. Note the optimal R&D subsidy for a sigle iovator is uaffected by variatios of parameters pertaiig to P Q ad G, which are give by the first three ad last two rows, as oted i Remark. The optimal policy is substatially impacted by the presece or absece of a complemetary carbo tax. Whe a carbo tax is also i place, the optimal R&D subsidy rate drops to 30% or less i all cases cosidered. I cotrast, the optimal carbo tax is oly sigificatly impacted oly by chages i the level of the exterality, but it is very robust otherwise. I particular, it is isesitive to factors that perturb the optimal subsidy, icludig the umber of iovators ad the presece or absece of a complemetary R&D subsidy. The optimal carbo tax is everywhere greater tha the aïve carbo tax of t x, as oted i Remark 3. I geeral, the optimal carbo tax is reduced by a small amout whe paired with a complemetary R&D subsidy, while the subsidy is substatially reduced. This suggests the carbo tax is relatively more importat for the welfare improvemet due to the optimal policy. To examie this cojecture explicitly, Table 3 computes welfare gais relative to laissez faire uder each of the policies. Table 3. Welfare Gais Uder Alterative Assumptios R&D Subsidy Carbo Tax Both Policies Etrats Sigle Multiple Sigle Multiple Sigle Multiple Baselie x 0 x 40 k 0.03 k E E

20 Table 3 strogly supports the idea that carbo taxes do the majority of the work i improvig welfare. I all cases cosidered, a carbo tax o its ow outperforms a R&D subsidy o its ow by a substatial margi. Moreover, addig a carbo tax to a pure subsidy program leads to substatial welfare gais typically multiple times as large as the gais from a R&D subsidy program o its ow. I cotrast, addig a R&D subsidy to a pure carbo tax program leads to oly mior improvemets. The largest gais (proportioally) are whe the exterality is smallest. Fially, as oted i sectio 3, the choice of policy may have sigificat effects o the distributio of outcomes. To more closely examie these distributioal cocers, table 4 displays the stadard deviatio of the umber of R&D etrats, assumig free etry ad uder various parameter combiatios. The last colum also displays the relative size of the R&D subsidy stadard deviatio compared to the carbo tax stadard deviatio. Table 4: Stadard Deviatio of Etrats Uder Alterative Assumptios ad Free Etry R&D Subsidy Carbo Tax Both Policies R&D Subsidy / Carbo Tax Baselie x x k k E E As oted i sectio 3, carbo taxes ca iduce more R&D etrats whe techological opportuity is low (Remark 4), but fewer etrats whe techological opportuity is high (Remark 5). This suggests the variace of etrats will be larger uder a R&D subsidy tha uder a carbo tax, which is bore out i Table 4. Moreover, we would aticipate the differece betwee the two will be 9

21 largest whe techological opportuity itself has a wider dispersio, a fidig cosistet with the last two rows of Table Coclusio Give the perceived eed to promote evirometal iovatios, both pull ad push policy tools ca i priciple help. I this paper we examie the efficacy of two such policies i the cotext of a model icorporatig free etry ad ucertaity about techological opportuity at the time of the policy choice. Our umerical results show a carbo tax o its ow suffices to obtai most of the welfare gais that a optimal mix of carbo taxes ad R&D subsidies achieves. Our model also allows us to make some claims about the robustess of differet policies to chagig parameters. While the optimal R&D subsidy for a sigle iovator does ot deped o the shape of demad or forecast techological opportuity, this turs out to be a special case. I geeral, the optimal R&D subsidy is highly cotiget o whether free etry is modeled ad whether the subsidy is paired with a carbo tax. I cotrast, while the optimal carbo tax does deped o the shape of demad ad the outlook for techological opportuity, the magitude of these effects is small. Moreover, the optimal carbo tax level is fairly robust to chagig assumptios about free etry or whether or ot a R&D subsidy is also beig implemeted. I geeral, the optimal carbo tax is slightly larger tha a aïve tax that exactly offsets the evirometal exterality. Fially, our model also allows us to study the impact of differet policy choices o the distributio of outcomes, i additio to their expected values. Compared to a carbo tax, subsidies are more likely to yield R&D that produces uused iovatios. We further show umerically ad aalytically that R&D subsidies are associated with more disperse outcomes whe iovatio ivolves a miimum ivetive step. This is because R&D subsidies are ieffective for low levels of techological opportuity, but ted to iduce more iovators tha carbo taxes whe techological opportuity is so high that taxes become of secod-order importace. 0

22 Refereces Acemoglu, D., P. Aghio, L. Burszty, ad E. Hemous. The Eviromet ad Directed Techical Chage. America Ecoomic Review 0(), 0: Arrow, K. J., Cohe, L., David, P.A., Hah, R.W., Kolstad, C.D., Lae, L., Motgomery, W.D., Nelso, R.R., Noll, R.G. ad Smith, A.E. A statemet o the appropriate role for Research ad Developmet i climate policy. The Ecoomists' Voice, 6(), February 009. Clacy, M.S. ad Moschii, G. Madates ad the Icetive for Evirometal Iovatio. CARD Workig Paper No. 557, Iowa State Uiversity, Jue 05. Deicolo, V. Pollutio-reducig iovatios uder taxes or permits. Oxford Ecoomic Papers 5(), 999, EPA. Greehouse Gas Emissios from a Typical Passeger Vehicle. Questios ad Aswers. U.S. Evirometal Protectio Agecy, EPA-40-F-4-040a, May 04. Fischer, C. ad R.G. Newell. Evirometal ad techology policies for climate mitigatio. Joural of Evirometal Ecoomics ad Maagemet 55()(008): 4-6. Jaffe, A.B., R.G. Newell ad R.N. Stavis. A Tale of Two Market Failures: Techology ad Evirometal Policy. Ecological Ecoomics 54, 005: Johso, Laurie T., ad Chris Hope. The social cost of carbo i U.S. regulatory impact aalyses: a itroductio ad critique. Joural of Evirometal Studies ad Sciece (3) (0): 05-. Khisha, V. Auctio Theory, d editio. Amsterdam: Elsevier, 00. Keedy, P.W. ad B. Laplate. Evirometal policy ad time cosistecy: emissios taxes ad emissios tradig. I: E. Petrakis, E.S. Sartzetakis ad A. Xepapadeas, eds. Evirometal Regulatio ad Market Power: Competitio, Time Cosistecy ad Iteratioal Trade. Edward Elgar Pub, 999. Laffot, J-J. ad J. Tirole. Pollutio permits ad evirometal iovatio. Joural of Public Ecoomics 6(), 996, Motero, J. P. A ote o evirometal policy ad iovatio whe govermets caot commit. Eergy Ecoomics 33 (0): S3-S9.

23 Nemet, G.F., 009. Demad-pull, techology-push, ad govermet-led icetives for oicremetal techical chage. Research Policy, 38(5), pp Newell, R.G., 00. The role of markets ad policies i deliverig iovatio for climate chage mitigatio. Oxford Review of Ecoomic Policy, 6(), pp Parry, I.W.H. Optimal Pollutio Taxes ad Edogeous Techical Progress. Resource ad Eergy Ecoomics 7, 995: Parry, I.W., Pizer, W.A. ad Fischer, C., 003. How large are the welfare gais from techological iovatio iduced by evirometal policies?. Joural of Regulatory Ecoomics, 3(3), pp Popp, David. "R&D subsidies ad climate policy: is there a free luch?." Climatic Chage 77, o. 3-4 (006): Popp, D., R.G. Newell ad A. Jaffe. Eergy, the eviromet, ad techological chage. Chapter i: Hall, B. ad N. Roseberg (Eds.), Hadbook of the Ecoomics of Iovatio, vol. (pp ). Elsevier, Amsterdam, 00. Toma, M., J. Griffi, ad R. J. Lempert. Impacts o U.S. Eergy Expeditures ad Greehouse-Gas Emissios of Icreasig Reewable-Eergy Use. Techical Report, RAND Corporatio, 008. US Govermet. Techical Support Documet: - Techical Update of the Social Cost of Carbo for Regulatory Impact Aalysis. Executive Order 866, Iteragecy Workig Group o Social Cost of Carbo, May 03, Revised November 03.

24 Appedix: Proof of Remark 5 The proof draws o the followig two lemmas. Lemma : For ay m ad polyomial i bx i where 0 i0 b, there is some x 0 such that for all x x0 : mb x i bix (7) i0 Proof: Rewrite equatio (7) as: m i i i0 i0 (8) b x b x b x Re-order equatio (8) as follows: i m i x bx bi 0 (9) i0 The above is satisfied for: / i bi x0 max b m (0) Lemma : For ay m ad polyomial i bx i where 0 i0 b, there is some x 0 such that for all x x0 : i bix bx / m () i0 Proof: Let bi true. mbi. By Lemma, there exists some 0 x such that for all x x0 the followig is 3

25 i i i0 mb x b x b x () Rewrite equatio () as: i bx bx / m bix i0 i i i0 b x b x b x / m (3) This completes the proof. Remark 5: For a give tax t ad subsidy s, for sufficietly high techological opportuity, a R&D subsidy iduces more iovatio etrats tha a carbo tax. Proof: For expositioal clarity, we iclude the term ˆ i the expected licesig equatio (3) as follows: ˆ ˆ ˆ ( ) ˆ (,, ) ( ˆ ( ˆ )( ) d d ) 4 (4) Where ˆ c c t. To reiterate sectio 3, the th iovator will eter if:,, ˆ For the case of pure carbo tax, ˆ c c t ad s 0 ˆ c c ad s 0 s k (5), while for a pure R&D subsidy,. Equatio (5) ca be re-expressed for a carbo tax as:,, c c t k (6) 4

26 Ad for a R&D subsidy as:,, / c c s k (7) We will establish that for ay s,, t, there is a 0 such that:,, c c / s,, c c t for all (8) 0 Whe this coditio is satisfied, there are situatios where equatio (7) holds but equatio (6) does ot (but ever the reverse). I these situatios, R&D subsidies iduce the etry of more iovators tha a carbo tax. We expad equatio (4) i order to complete the proof. To simplify otatio ad keep thigs compact, defie: x ˆ (9) The ier itegral of equatio (4) ca be writte as: ( ˆ ) ˆ x x ˆ x ˆ x d ( )( ) (30) With some substitutio ad simplificatio, we ca rewrite equatio (4) as: ˆ (,, ) Where we defie the followig for compactess: ˆ 0 d (3) 4 x x x (3) ˆ ˆ ˆ x x (33) 5

27 0 ˆ ˆ ˆ x 4 (34) To expad equatio (3), we eed to solve may itegrals of the form ˆ x m d. Substitutig i equatio (9) ad usig the biomial theorem, the expaded solutio to these terms takes the followig form: m m i ˆ i m i ˆi ˆ x d (35) i0m i Equatio (35) expresses ˆ x m d as a polyomial. As show i Lemmas ad, as is icreased, the raised to the highest expoet comes to domiate all other terms. Let L f x deote the largest expoet term of polyomial f x. That is: L b x b x i i (36) i0 Equatio (35) implies: L m m ˆ x d (37) Sice equatio (3) is the sum of polyomials, it too is a polyomial. By equatio (37), the largest expoet of (3) is: (,, ˆ ) L x x x d ˆ 4 This ca be simplified to be: (38) / L (,, ˆ ) (39) 6

28 By Lemmas ad, equatio (39) meas for ay m there exists 0 such that for all 0 :,, / / ˆ m m (40) For ay m, there is some 0 such that for 0 :,, / c c t m (4) Ad / m s,, c c / s (4) Set m s, which implies: / / m m s (43) The cojuctio of equatios (4), (4), ad (43) implies: Completig the proof.,, c c / s,, c c t for all (44) 0 7