Voluntary Prices vs. Voluntary Quantities

Transcription

1 Voluntary Pries vs. Voluntary Quantities Torben K. Mideksa Martin L. Weitzman April 12, 2018 Abstrat We extend the standard Pries vs. Quantities framework to over two independent and idential jurisditions, A and B. Both jurisditions set a prie or quantity to maximize their own expeted welfare onditional on the instrument type and amount hosen by the other jurisdition. With iid unertainty, a dominant strategy of both jurisditions is to hoose a prie instrument when the slope of marginal benefit is less than the slope of marginal ost and a quantity instrument when the ondition is reversed. With n ountries, if the slope of marginal benefit is equal to the slope of marginal ost, the welfare ost at the equilibrium in whih ountries oordinate on pries is higher, by a fator of n, than the welfare ost at the equilibrium in whih ountries oordinate on quantity. By extending the standard Pries vs. Quantities riterion from the basi hoie framework to a strategi setting, we allow the hoie of poliy type and amount to take into aount the free-riding by other jurisditions and disover the welfare benefit of oordination on quantities. JEL Codes: C7, D8, F5, H21, Q28, Q58 Keywords: pries versus quantities, regulatory instruments, pollution, limate hange The draft has benefited from the reations of audienes at Harvard Environmental Eonomis Lunh Seminar. Please ontat us for omments. Harvard John A. Paulson Shool of Engineering and Applied Sienes, Harvard Environmental Eonomis Program, and the Weatherhead Center for International Affairs at Harvard University (torben mideksa@post.harvard.edu) Department of Eonomis, Harvard University (mweitzman@harvard.edu) 1

2 1 Introdution Choosing the best instrument for ontrolling negative externality from pollution has been a long-standing entral issue in environmental eonomis. Pigou (1920) introdued and subsequently popularized the entral onept of plaing a prie-harge on pollution (sine alled a Pigouvian tax ) as an effiient way to orret a pollution externality. This Pigouvian-tax approah dominated eonomi thinking about the pollution externality problem for about the next half-entury. Dales (1968) introdued the idea of reating property rights in the form of tradeable pollution permits (aka allowanes ) as an effiient alternative to a Pigouvian tax. Montgomery (1972) rigorously proved the formal equivalene between a prie on pollution and a dual quantity representing the total allotment of tradeable permits. Heneforth, it beame widely aepted that there is a fundamental isomorphism between a Pigouvian tax on pollution and the total quantity of aps allotted in a ap-and-trade system, with all permits trading at the same ompetitiveequilibrium market prie as the Pigouvian tax. For every Pigouvian tax, there exists a quantity of tradeable permits allotted whose ompetitive-equilibrium market prie equals the Pigouvian tax. And for every total quantity of tradeable permits, there exists a ompetitive-equilibrium market prie that would yield the same result if imposed as a Pigouvian tax. So far all the analysis has taken plae in a deterministi ontext with full ertainty. Weitzman (1974) shows that there is no longer an isomorphism between prie and quantity instruments when there is unertainty in ost and benefit funtions. When unertainty is introdued, setting a fixed prie stabilizes marginal ost while leaving the total quantity of pollution variable, whereas setting a fixed total quantity of tradeable permits stabilizes total quantity while leaving prie (or marginal ost) variable. The question then beomes: whih instrument is better under whih irumstanes? Weitzman (1974) derives a relatively simple formula for the omparative advantage of pries over quantities, denoted in the paper as. The sign of depends on the relative slopes of the marginal abatement-ost urve and the marginal abatement-benefit urve. When the marginal benefit urve is flatter than the marginal ost urve, the sign of is positive (pries are favored over quantities). Conversely,when the marginal benefit urve is steeper than the marginal ost 2

3 urve, the sign of is negative (quantities are preferred over pries). This led to the development of a sizable literature on the optimal hoie of prie vs. quantity poliy instruments under unertainty. 1 Adar and Griffin (1976), Fishelson (1976), and Roberts and Spene (1976) analyzed seemingly alternative (but ultimately similar) forms of unertainty. Weitzman (1978), Yohe (1978), Kaplow and Shavell (2002), and Kelly (2005) extended the basi model to over various aspets of nonlinear marginal benefits and nonlinear marginal osts. Yohe (1978) and Stavins (1995) analyzed a situation where unertain marginal osts are orrelated with unertain marginal benefits. Chao and Wilson (1993), and Zhao (2003) inorporated investment behavior into the basi framework of instrument hoie under unertainty. In these extensions, the results generally preserve the earlier insight that, all else held equal, flatter marginal benefits or steeper marginal osts tend to favor pries while steeper marginal benefits or flatter marginal osts tend to favor quantities. 2 Extensions to over stok externalities in a dynami multi-period ontext were made by Hoel and Karp (2002), Pizer (2002), Newell and Pizer (2003), and Fell, MaKenzie and Pizer (2012), among others. The extensions to dynami stok externality kept muh of the flavor of the original story, whih was phrased in terms of emission flows throughout a regulatory period (followed by a new regulatory period with new deision-relevant parameters). In partiular, for the ase of limate hange from aumulated stoks of atmospheri arbon dioxide (CO 2 ), this stok-based literature onludes that Pigouvian taxes are strongly favored over ap-andtrade throughout the relevant regulatory period. This is beause the flow of CO 2 emissions throughout a realisti regulatory period is only a tiny fration of the total stok of atmospheri CO 2 (whih atually does the damage), and therefore the orresponding marginal flow benefits of CO 2 abatement within, say, a five to ten year regulatory period are very flat, implying that pries have a strong omparative advantage over quantities. In addition to the stok dimension, free-riding at an international level lies at the heart of the CO 2 problem. This problem manifests itself in weak mitigation efforts among ountries, whih do not apture the full benefits of their abatement. At worst, mitigation efforts in 1 Given the number of published papers on Pries vs. Quantities, we have only inluded a subset that we subjetively judge to be most relevant to this paper. 2 Note that ombinations of instruments, suh as a fixed prie with a floor and eiling on quantities, must supersede in expeted welfare both a pure prie and a pure quantity, beause both of these pure instruments are speial ases of suh ombinations of instruments. This insight traes bak to Roberts and Spene (1976). 3

4 other ountries may derease due to arbon leakage and the migration of fossil-fuel intensive prodution aross national jurisditions. The Paris Climate Agreement, onluded in 2015, attempts to take a bottom-up approah to the international free-rider problem using Intended Nationally Determined Contributions (INDCs), hosen voluntarily and non-ooperatively by eah ountry. As the name indiates, INDCs are both intended and voluntary promises that are meant to be enfored by blame and shame mehanism. However, it is not lear if the soial fores of blame and shame that aim at steering individuals towards soially aepted behaviors an similarly steer ountries and enfore the INDCs. To be redible, these promises must be equilibrium hoies, i.e., onsistent with maximization of a national welfare. To study the role of the international free-rider problem, we extend the standard Pries vs. Quantities riterion into a game-theoreti equilibrium framework. We fous on two independent idential twin jurisditions A and B, whih eah set a prie or quantity independently to maximize their expeted welfare onditional on the instrument type and value hosen by the other jurisdition. Let b be the ommon slope of both marginal benefit funtions and be the ommon slope of both marginal ost funtions. With iid unertainty, we find that it is a dominant strategy for both jurisditions to hoose a prie instrument when b < and a quantity instrument when b >. For n ountries, if b =, then the welfare ost at the equilibrium in whih ountries oordinate on pries is n times higher than the welfare ost at the equilibrium in whih ountries oordinate on quantity. Thus, the result here extends the standard Pries vs. Quantities riterion from the basi hoie framework to a setting with multiple ountries using the dominant strategy equilibrium framework. In doing so, we not only formalize the voluntary promises of the INDCs, but also identify the onditions under whih for voluntary pries and voluntary quantities regulation an fulfill the INDCs. In addition, we demonstrate the olletive benefits of oordinating on quantities despite eah ountry being indifferent between prie and quantity when b =. 4

5 2 A Model of Regulatory Game Suppose that there are two idential ountries A and B. Following standard onvention, goods are good, and instead of fousing on pollution, we fous on abatement. A ountry i {A, B} benefits from the total global abatement of q A + q B and inurs a private abatement ost. A ountry s abatement is hosen by firms loated in the ountry. From the viewpoint of a regulator in ountry i, the net soial benefit of abatement is desribed by W i = B (q A + q B, η i ) C (q i, θ i ). (1) To failitate omparability to the literature that follows Weitzman (1974), we assume: B (q A + q B, η i ) [β + η i ] [q A + q B ] b 2 [q A + q B ] 2, and C (q i, θ i ) [γ + θ i ] q i + 2 q2 i. In this setting, eah regulator hooses the type of regulatory poliy instrument P i { p i, q i }, where abatement is regulated through prie p i or quantity q i under informational onstraint. Commitment Stage P i Choosen by Regulators η i, θ i Drawn by Nature Abatement Stage q i Choosen by Firms Figure 1: Timing of the Game. After ountries simultaneously ommit to P i, nature reveals the independent values of θ i and η i for firms in eah ountry. The shoks are iid, with E [θ i ] = E [η i ] = 0 and varianes of σθ 2 and ση. 2 Then, a representative firm in ountry i hooses abatement q i given the realization of the shoks and the poliy instrument the regulator has hosen. Finally, payoffs are realized for both ountries. 5

6 3 Analysis We use bakward indution to solve for the equilibrium under the four potential regulatory regimes. In the final stage, firms omply with regulation given the realization of shoks and the regulatory poliy P i hosen by the regulator in ountry i. For a ountry, the ex-post welfare is maximized when q i is hosen to maximize W i in (1) given realized η i and θ i and the abatement in the other ountry q i. The ex-post optimal abatement qi is given by q A = β γ b + b b + q B + η A θ A, and (2) b + qb = β γ b + b b + q A + η B θ B b +. (3) Note that the reation funtions (2) and (3) apture the weak and strong forms of the international free-rider problem. A ountry only aounts for its own benefit, so the interepts of the reation funtions are lower than that of a global planner. Moreover, the reation funtions have negative slopes beause of the strong form of the international free-rider problem. When a ountry ommits to more abatement, the other ountry redues its abatement. As b inreases, so do redutions in abatement due to the rowding-out. The regulators annot implement their ideal hoie (2) and (3) due to an informational gap. Instead, eah regulator must set a minimum quantitative quota q i or harge a prie per unit of abatement p i to inentivize firms to inorporate the extra information about the shoks. Given the regulator s hoie of poliy, a firm s optimal reation funtion under the regulatory onstraint, is given by q i ( q i, θ i ) arg min q i [γ + θ i ] q i + 2 q2 i suh that q i q i (4) when abatement is regulated through quantity or q i ( p i, θ i ) arg max { p i q i [γ + θ i ] q i } q i 2 q2 i (5) when abatement is regulated through prie. While firms must omply with the quantitative restrition, their ompliane balanes the marginal benefit of ompliane p i and its marginal 6

7 ost, γ + θ i + q i under the prie regulation. The marginal ompliane ost depends on the realization of θ i, observed by firms prior to ompliane. When the regulator hooses the form of regulation, it knows that firms will reat optimally to their extra information given the regulatory onstraint. Thus, if the regulator in ountry A ommits to a quantitative quota q A and the regulator in ountry B ommits to a quantitative quota q B, ountries fae an ex-post welfare loss DW L i due to the disrepany between q i and q i. The welfare loss is given by DW L A ( q A, q B ) = b + 2 DW L B ( q A, q B ) = b + 2 [ q A β γ b q ] 2 B + η A θ A, and (6) b + [ q B β γ b q ] 2 A + η B θ B. (7) b + The welfare ost that eah ountry faes depends on its own regulation, but also on the regulatory outome in the other jurisdition. The value of q A that minimizes the expeted welfare ost E[DW L A ( q A, q B )] is q A = β γ b + b b + q B, (8) and the value of q B that minimizes the expeted welfare ost E[DW L B ( q A, q B )] is q B = β γ b + b b + q A. (9) The best reation funtions (8) and (9) exhibit the regulatory free-rider problem wherein a regulator redues its quantitative quota when the other regulator raises its quantitative quota of abatement. Figure 2 plots the two regulators quantitative best reation funtions. As shown in the figure, there is a unique fixed point on the best response funtions from whih neither regulator has an inentive to unilaterally deviate. The unique fixed point ( q A, q B ) at whih the best response funtions interset is: q A = β γ 2b + and q B = β γ 2b +. (10) 7

8 q A β γ b R A (q B ) β γ b+ q A R B (q A ) 0 q B β γ b+ β γ b q B Figure 2: Best Response Abatement Funtions. Sine these reation funtions are entirely non-ooperative, they orrespond to the INDCs enshrined in the Paris Climate Agreement. The minimized expeted welfare loss at the equilibrium value of the INDCs profile ( q A, q B ) is E[DW L i ( q A, q B )] = σ2 η + σθ 2, for i {A, B}. (11) 2[b + ] Equation (11) suggests that, all else held equal, the higher the variability of the shoks or the lower the sum of the slopes of marginal benefit and ost of abatement, the greater the expeted welfare ost of ommitting to a given quantitative regulation. For ease of presentation, we work with normalized the payoffs, first multiplying the value of E[DW L i (.,.)] with 2[b + ], subtrating ση, 2 and finally dividing by σθ 2. Thus, the normalized welfare ost when both ountries adopt quantity regulation is [2[b + ]E[DW L i ( q A, q B )] σ 2 η]/σ 2 θ = 1, for i {A, B}. (12) A regulatory deviation by ountry A to prie p A, given that the regulator in ountry B ommits 8

9 to a quantitative quota q B, results in an ex-post welfare loss due to the disrepanies between q A ( p A ) and qa, and q B and qb. The regulatory reation funtions minimize the expeted welfare ost of regulation. The value of p A that minimizes the expeted welfare ost E[DW L A ( p A, q B )] and the value of q B that minimizes the expeted welfare ost E[DW L B ( p A, q B )] are: p A = β + bγ b + b b + q B, and (13) q B = β + [b ]γ [b + ] b [b + ] p A. (14) Like reation funtions (8) and (9), these reation funtions are downward sloping due to the international free-rider problem. 3 The unique fixed point ( p A, q B ) at whih the best response reation funtions (13) and (14) interset is: p A = β + 2bγ 2b + and q B = β γ 2b +. (15) Given ( p A, q B ) in (15), the normalized value of the minimized expeted welfare loss is given by [2[b + ]E[DW L A ( p A, q B )] σ 2 η]/σ 2 θ = [ ] 2 b, and (16) [2[b + ]E[DW L B ( p A, q B )] σ 2 η]/σ 2 θ = 1 + [ ] 2 b. (17) With (11), (16), and (17), we have all the neessary piees to establish the ondition under whih ommitting to one regulatory form would be more benefiial than ommitting to another. The differene in expeted welfare from prie regulation versus quantity regulation given the other ountry uses quantity regulation is given by A ( q B ) = b 2 2 σ2 θ. (18) 3 Counterintuitively, it may appear that when regulation aross ountries takes different forms, the free-rider response in (13) depends on the slope of the marginal ompliane ost in addition to the slope of marginal benefit, b. This ontrasts with the ase in whih regulation is idential in type, (e.g. (8) and (9)). However, this observation is deeptive sine the reation funtion (14) is undefined if = 0. 9

10 The following lemma summarizes the substantive impliation of A ( q B ) formula. Lemma 1. If the regulator in ountry B ommits to quantity regulation, then the regulator in ountry A is stritly better off by ommitting to prie regulation if > b and quantity regulation if b >. Lemma 1 suggests that as long as b 0, the payoff from adopting one regulatory form dominates the payoff from adopting another, provided that the other ountry uses quantity regulation. What happens if ounty B, instead, ommits to prie regulation? If the regulator in ountry A ommits to a prie p A and the regulator in ountry B ommits to a prie p B, firms hoose abatement suh that max q i p i q i The optimal abatement in eah ountry is: [ [γ + θ i ] q i + ] 2 q2 i. q A ( p A ) = p A γ θ A, and (19) q B ( p B ) = p B γ θ B. (20) The ex-post welfare loss ours due to the disrepanies between q A ( t A ) in (19) and qa in (2) and q B ( p B ) in (20) and q B in eah ountry are: in (3). The pries that minimize the expeted welfare ost of regulation p A = p B = β + bγ b + β + bγ b + b b + p B, and (21) b b + p A. (22) Interepts and slopes of the reation funtions (21) and (22) are onsistent with those of a publi good. The unique fixed point ( p A, p B ) at whih the best response reation funtions (21) and (22) interset is p A = p B = and the minimized expeted welfare loss is β + 2bγ 2b +, (23) 10

11 [ ] 2 b [2[b + ]E[DW L i ( p A, p B )] ση]/σ 2 θ 2 = 2, for i {A, B}. (24) If ountry A instead deviates to regulation using a quantity q A, given that the regulator in ountry B ommits to a prie p B, there is an ex-post welfare loss due to the disrepany between q A and q A, and q B ( p B ) and q B. The value q A that minimizes the expeted welfare ost E[DW L A ( q A, p B )] is q A = β + [b ]γ [b + ] b [b + ] p B, (25) whereas the value of p B that minimizes the expeted welfare ost E[DW L B ( q A, p B )] is p B = β + bγ b + b b + q A. (26) The unique point ( q A, p B ) at whih the reation funtions (25) and (26) interset is q A = β γ 2b + and p B = and the resulting minimized expeted welfare loss is [2[b + ]E[DW L A ( q A, p B )] σ 2 η]/σ 2 θ = 1 + β + 2bγ 2b +, (27) [ ] 2 b, (28) [2[b + ]E[DW L B ( q A, p B )] σ 2 η]/σ 2 θ = [ ] 2 b. (29) Thus, the differene in expeted welfare loss from prie regulation versus quantity regulation given the other ountry uses prie is A ( p B ) = b 2 2 σ2 θ. (30) Lemma 2. If the regulator in ountry B ommits to prie regulation, then the regulator in ountry A is stritly better off by ommitting to prie regulation if > b and quantity regulation if b >. 11

12 Unexpetedly, the expressions in (30) and (18) are idential, suggesting that a ountry s welfare from ommitting to a given instrument rests entirely on the parameters b and. Speifially, it depends on the differene between the parameter representing international free-riding and the parameter representing the slope of the marginal ompliane ost. Beause the two ountries are idential, the results in Lemma 1 and Lemma 2 also apply to ountry B. Lemmas 1 and 2, together with the symmetry between the two ountries, imply the main result of this paper: Proposition. If b =, then there are multiple Pareto ranked pure strategy equilibria. For eah ountry, hoosing quantity regulations is a Pareto dominant Nash-Equilibrium. If b <, then using prie is a dominant strategy equilibrium. Similarly, if b >, then using quantity is a dominant strategy equilibrium. The logi an be easily understood with the help of the payoff matrix in Table 1. Comparing the payoffs in Table 1, one finds that a dominant strategy for both jurisditions is to hoose a prie instrument when b < and to hoose a quantity instrument when b >. p B Country B Country A p A 2 [ b ] 2, 2 [ b ] 2 [ b ] 2, [ b ] q A [ b ] 2 + 1, [ b ] 2 1, 1 q B Table 1: The Expeted Welfare Cost of Committing to a Prie or Quantity. An interesting result shows up when b =. When this is the ase, there are multiple pure strategy equilibria Nash Equilibria. This is beause a regulator is indifferent between prie and quantity given the strategy of the other regulator. However, sine the payoffs assoiated with the different equilibria an be Pareto ranked, the equilibrium in whih eah ountry hooses quantities is ompelling beause it is Pareto dominant. With two ountries, the welfare ost at the equilibrium in whih ountries oordinate on pries is two fold higher than the welfare ost at the equilibrium in whih ountries oordinate on quantity. This is beause the equilibrium in whih ountries oordinate on pries is produes too muh volatility in abatement. The 12

13 relative variability would be the same with one regulator (Weitzman, 1974). In fat extending the number of ountries to a finite n so that Q n q j and (1) beomes W i = [β +η i ]Q b 2 Q2 [γ + θ i ]q i 2 q2 i gives the following general theorem, whose proof is in the appendix. Theorem: For a finite n number of ountries, if b =, then the welfare ost at the equilibrium in whih ountries oordinate on pries is n times higher than the welfare ost at the equilibrium in whih ountries oordinate on quantity. To emphasize the signifiane of our results, we remind the reader that this result is based on a dominant strategy equilibrium onept, whih imposes the weakest possible ondition of rationality, and alls upon rational deision makers to exlude strategies with payoffs that are stritly dominated by others. Sine every dominant strategy is a Nash equilibrium, the result is also a Nash Equilibrium, and holds even when we weaken the assumptions of rational expetations or the Nash-onjeture. Moreover, the equilibria in (10), (15), (23), and (27) are non-ooperative, and thus they orrespond to the INDCs enshrined in the Paris Climate Agreement. Our model not only formalizes the voluntary ontributions but also identifies the onditions under whih the optimal Prie vs. Quantity rule an be implemented under the Paris agreement. j=1 4 Conluding Disussion The model presented in this paper makes a number of simplifying assumptions. We assume a ompletely symmetri idential twin jurisditions with the same linear marginal ost and marginal benefit funtions, differing only by iid marginal ost shoks and iid marginal benefit shoks. 4 Nonetheless, this simplifiation allows us to extend the standard Pries vs. Quantities framework to over two independent idential twin jurisditions, whih eah set a prie or quantity to maximize their own expeted welfare onditional on the instrument hoie and value hosen by the other jurisdition. 4 Hoel and Karp (2002), after disussing about the onsequenes of departing from iid assumption, settle for the iid shoks. Karp and Zhang (2005) onsider orrelated abatement ost in the ontext of limate hange and onlude that for a range of parameter values ommonly used in global warming studies, taxes dominate quotas... regardless of the extent of ost orrelation. In our ase, allowing perfet orrelation of shoks does not affet the onlusion when b >. Using Pries ontinue to be a dominant strategy if > f(b) instead of > b. 13

14 While we only present the symmetri ase, our results are robust to onsiderations of alternative sequenes of moves by regulators, multiple heterogeneous ountries, oalitional deision making by groups of ountries, and asymmetri poliy environments. Beause of the dominane property, the equilibrium outome of the regulatory game is robust to alternative sequenes of ommitments. This allows us to desribe equilibrium outomes in ases where regulators in different ountries ommit after others and that information is publily known. Similarly, the assumption of the two idential ountries might appear limiting. After all, with multiple and heterogeneous ountries, deviation needs to be onditioned on any prie-quantity ommitment onfigurations by the other n 1 ountries. Although, for ease of presentation, we have presented a model of two idential ountries, our results generalize to any finite number of heterogeneous ountries having own b i and i. 5 Asymmetri poliy environments in whih some ountries hoose quantity and others hoose prie (e.g., Chile and South Afria) are also aptured by the model, whih generalizes to an asymmetri poliy environment. Taken together, the key result demonstrates the relative advantage of prie over quantity for a given ountry, independent of the type of poliy instrument other ountries hoose; and it is the same riterion as the original Pries vs. Quantities formula. Despite the simplifiations, our results suggest that the original riterion may still have wider appliability for determining instrument hoie in a non-ooperative strategi environment. 5 Appendix: Proof The minimized expeted welfare loss from a quantity regulation, given a subset ˆM = m < n ountries adopt prie regulation, is ˆM of with [2 [b + ] EDW L i ( q i, n [ ] 2 b q j q i ) ση]/σ 2 θ 2 = 1 + m. (31) j=1 5 In showing the result is robust in any finite n number of heterogeneous ountries, are needs to be taken in verifying that a ountry has no inentive to deviate from the equilibrium, in whih deviation needs to be onditioned on any prie-quantity ommitment onfigurations by all the possible subsets of the n 1 ountries. In the proess, one would observe that the results generalize also to a setting in whih a oalition of group of ountries like the European Union is a player. 14

15 The minimized expeted welfare loss from a prie regulation, given a subset ˆM = m < n ountries adopt prie regulation, is ˆM of with [2 [b i + i ] EDW L i ( p i, n q j q i ) ση]/σ 2 θ 2 = j=1 [ ] 2 b + m [ ] 2 b. Thus, if b =, then the normalized welfare ost when all ountries ommit to quantity takes the value of 1 whereas the normalized welfare ost when all ountries ommit to prie takes the value of n. Thus, setting m = 0 vs. m = n 1 gives the laimed result. QED Referenes [1] Adar, Zvi, and James M. Griffin (1976). Unertainty and the Choie of Pollution Control Instruments. Journal of Environmental Eonomis and Management, 3: [2] Chao, Hung-Po, and Robert Wilson (1993). Option Value of Emission Allowanes. Journal of Regulatory Eonomis, 5: [3] Dales, John H. (1968). Pollution, Property, and Pries: An Essay on Poliy-making and Eonomis. Toronto: University of Toronto Press. [4] Fell, Harrison, Ian A. MaKenzie, and William A. Pizer (2012). Pries versus Quantities versus Bankable Quantities. Resoure and Energy Eonomis, 34(4): [5] Feng, Hongli, and Jinhua Zhao (2006). Alternative Intertemporal Trading Regimes with Stohasti Abatement Costs. Resoure and Energy Eonomis, 28(1): [6] Fishelson, Gideon (1976). Emission Control Poliies under Unertainty. Journal of Environmental Eonomis and Management, 3: [7] Goulder, Lawrene H., and Andrew R. Shein (2013). Carbon Taxes vs. Cap and Trade: A Critial Review. Climate Change Eonomis, 4(3): [8] Hoel, Mihael, and Larry Karp (2002). Taxes versus Quotas for a Stok Pollutant. Resoure and Energy Eonomis, 24(4): [9] Kaplow, Louis, and Steven Shavell (2002). On the Superiority of Corretive Taxes to Quantity Regulation. Amerian Law and Eonomis Review, 4(1):

16 [10] Karp, Larry, and Jiangfeng Zhang (2005). Regulation of Stok Externalities with Correlated Abatement Costs. Environmental and Resoure Eonomis, 32 (2): [11] Kelly, David L. (2005). Prie and Quantity Regulation in General Equilibrium. Journal of Eonomi Theory, 125: [12] Montgomery, W. David (1972). Markets in Lienses and Effiient Pollution Control Programs. Journal of Eonomi Theory, 5(3): [13] Newell, Rihard, and William Pizer (2003). Regulating Stok Externalities under Unertainty. Journal of Environmental Eonomis and Management, 45: [14] Pigou, Arthur C. (1920). The Eonomis of Welfare. London: Mamillan. [15] Pizer, William A. (2002). Combining Prie and Quantity Controls to Mitigate Global Climate Change. Journal of Publi Eonomis, 85: [16] Roberts, Mark J., and A Mihael Spene (1976). Effluent Charges and Lienses under Unertainty. Journal of Publi Eonomis, 5: Rubin, Jonathan D. (1995). A Model of Intertemporal Emission Trading, Banking, and Borrowing. Journal of Environmental Eonomis and Management, 31: [17] Weitzman, Martin L. (1974). Pries vs. Quantities. Review of Eonomi Studies, 41(4): [18] Weitzman, Martin L. (1978). Optimal Rewards for Eonomi Regulation. Amerian Eonomi Review, 68: [19] Yohe, Gary W. (1978). Towards a General Comparison of Prie Controls and Quantity Controls under Unertainty. Review of Eonomi Studies, 45: