Global warming: Prices versus quantities from a strategic point of view

Journal of Environmental Economics and Management 64 (2012) 217–229

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Global warming: Prices versus quantities from a strategic point of view Franz Wirl n University of Vienna, Chair Industry, Energy and Environment, Br¨ unnerstr. 72, A-1210 Vienna, Austria

a r t i c l e i n f o

abstract

Article history: Received 15 June 2011 Available online 7 December 2011

This paper investigates how the choices of the instruments affect the interactions in a stock externality game (global warming) between cartelized fossil fuel suppliers and consumers. More precisely, the paper studies the equilibria in Markov strategies in a dynamic game with each player choosing either the quantity or the price strategy including short-run first mover advantages. Indeed OPEC and its opponent IEA have tried both instruments in the past and play currently in quantities. Given such a noncompetitive setting, both players should prefer the price instrument. Therefore, both players are expected to switch back to price and tax policies if global warming will be treated effectively. & 2011 Elsevier Inc. All rights reserved.

Keywords: Prices versus quantity Carbon taxes versus permits Differential games Nash and Stackelberg Global warming

1. Introduction This paper investigates the strategic choices of quantity or price instruments in the dynamic context of global warming. It is thus complementary to the existing literature following the seminal paper of Weitzman [28] that uses uncertainty to draw a wedge between instruments that are equivalent in a deterministic and static setting. Hoel and Karp (2001) extend Weitzman’s question into a dynamic setting (global warming) and conclude that price dominates quantity at least in the case of multiplicative uncertainty; D’Amato and Dijkstra [5] investigate in the standard Weitzman model the consequences of multiplicative uncertainty. Similar comparisons applied to global warming are made in Pizer [20] and Newell and Pizer [19], while Pizer [21] recommends combining both instruments over time. Karp and Zhang [12] stress the (potential) time inconsistency of taxes but not of quotas under cost shocks. Recently, Strand [25,26] considers carbon taxes versus cap and trade policies, focussing on a bloc of importers that enact a climate policy with price setting oil exporters. Earlier static investigations in a strategic context are Karp [10] and Wirl [29]. Montero [17] stresses that quantity instruments (plus subsidies) dominate prices in fostering R&D if governments can commit to an instrument but not to the corresponding level. The following analysis uses a dynamic stock externality game between energy consumers and fossil fuel producers in order to compare the strategic implications of price or quantity instruments applied on the demand as well as the supply side. Although the following investigation is theoretical, it is motivated by real world interactions between noncompetitive fossil fuel producers (the Organization of Petroleum Exporting Countries, OPEC) and cartelized energy consumers. In fact, the International Energy Agency (IEA) was founded in 1974 as a response to OPEC (itself founded in 1960) and the first oil price shock in November 1973 with the objective of protecting consumers’ interests; how much it has achieved so far is an open question. The consumer cartel cares, in contrast to the suppliers, about the damages from

n

Fax: þ 43 1427738104. E-mail address: [email protected]

0095-0696/$ - see front matter & 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jeem.2011.11.002

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global warming. Furthermore, the strategic instruments – price or quantity – have changed during the recent decades on both sides of the energy market. OPEC used price strategies up to 1985 but has pursued a quantity policy since 1986. More precisely OPEC’s (biannual) Conference announced and fixed the ‘official selling price’ for its reference crude ‘Arab Light’ until 1985, but since 1986 has announced production quotas and let the market determine the price. On the demand side, oil products are heavily taxed in most parts of the world. For example, revenues from taxing petroleum exceed 50% in Europe on refined products (www.opec.org) and amount to around 10% of all tax revenues and to a quarter of (all) income taxes in Germany [23]. In spite of this history of taxation, industrialized country governments prefer quantity instruments for global warming mitigation. This switch to the quantity instrument is documented by the case of the European Union running its Emission Trading System (ETS) for carbon permits since 2005, and in the US by the long discussed Waxman– Markey cap-and-trade bill. Finally, the alternative of regulating producers is infeasible if the crucial fossil fuel producers (OPEC, Russia) are governments of foreign countries. Related are the questions about taxing non-renewable resources (e.g., [9,24]) and how a resource importing monopsony can lower the competitive price by committing (or not) to a consumption path; first Maskin and Newbery [16] and then ¨ Horner and Kamien [8] pointing out the formal equivalence with the durable good monopoly. Liski and Montero [13] ¨ depart from Horner and Kamien [8] but require subgame perfection since commitment strategies are implausible (a similar issue arises with storable permits [14]); Bosetti and Victor [2] stress the importance of credible policies. Therefore the analysis in this paper is restricted to subgame, more precisely Markov, perfect equilibria and thus the analysis of open loop solutions is eschewed. Formally, the paper departs from the asymmetric differential game in Wirl [30], Tahvonen [27], Rubio and Esriche [22], Liski and Tahvonen [15], Wirl [31] and recently in Dullieux et al. [7] for constraining global warming ex ante (e.g., to the Kopenhagen target of % þ2 1C). All these papers derive equilibria in Markov strategies when a benevolent government of consumers and a cartel of fossil fuel suppliers play in prices (i.e., the government charges emission taxes). However, strategic players can choose quantities instead of prices if it advances their position. Therefore, the framework in Wirl [30] is extended to investigate how quantities – permits issued by the government and a Cournot playing supply cartel – affect the intertemporal outcomes and to whose advantage. Another sensible extension – assuming that one party is able to commit in the short run and thus announces its action just prior to the opponent’s action – is also explored, in particular, if the simultaneous move game does not allow for an interior outcome. What are the outcomes of these different combinations of instruments? It turns out that quantities are poor instruments because an equilibrium with positive emissions need not exist. These differences are a consequence of dynamics due to the stock externality, because the choice of a price or quantity instrument does not matter in the static Nash equilibrium. Therefore the current choice of quantities as strategies should not persist if global warming becomes a crucial policy issue such that the antagonism between consumer governments and fossil fuel producers intensifies. In particular the strategy of permits is a bad choice since it eliminates rents from a first mover advantage, a point that is emphasized in Tahvonen [27] and Liski and Tahvonen [15]; similarly for the supply side, choosing a quantity policy surrenders profits if carbon emissions are taxed. In short, the price policies are the undominated strategies on both sides of the market. 2. Model The following framework is an adaptation of Wirl [30]. It considers three players: consumers, a fossil fuel supplier and a consumer government. Consumers are passive and the other two act strategically. The payoffs are affected by a stock externality, which demands a dynamic analysis. Thus a differential game framework is applied. Competitive consumers enjoy the surplus,   q2 UðqÞ ¼ q Pq, ð1Þ 2 from consuming fossil fuels of the amount (q). This surplus consists of the normalized linear-quadratic utility between the brackets minus the expenses where P is the final price paid by consumers. As a consequence, inverse demand is U 0 ¼ 1q ¼ P:

ð2Þ

Consumption of fossil fuels leads to CO2 emissions that accumulate in the atmosphere according to X_ ¼ q,

Xð0Þ ¼ 0:

ð3Þ

This accounting ignores depreciation (small given the large time constant that a ton of CO2 remains in the atmosphere) and resource constraints (i.e., global warming determines the ultimate use of fossil fuels, which rules out the green paradox). Since there is no depreciation, a pristine environment is assumed at the start to exclude trivial outcomes with initial pollution stocks above the steady state. Measuring the stock externality as well as consumption¼emission in terms of their net contribution to global warming, this stock induces convex (for simplicity quadratic) external costs, CðXÞ ¼

c 2 X : 2

ð4Þ

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219

If p denotes the price received by the producers, then any difference Pp ¼ t or p

ð5Þ

corresponds either to the tax ðtÞ or the price for a permit ðpÞ. Assuming that the producers do not care about the external costs, their objective is to maximize their net present value of profits using the discount rate r 40, and VðXÞ denotes the corresponding value function, Z 1 VðXÞ :¼ max ert pðtÞqðtÞ dt: ð6Þ 0

The government maximizes the net present value (using the same discount rate r 4 0) of their consumers’ surplus minus the external costs from global warming, i.e.,   Z 1 q2 ðtÞ pðtÞqðtÞCðXðtÞÞ dt, ð7Þ ert qðtÞ WðXÞ :¼ max 2 0 subject to the dynamic constraint (3), the suppliers decisions, and market clearing (2). Objective (7) assumes that tax revenues and similarly the proceeds from permit sales are transfers that do not count. WðXÞ denotes the government’s value function that depends on the choice of instruments and equilibrium, and thus differs across the different cases. This is also the case for the supplier’s value functions V. Adding both players’ objectives and solving for the optimal intertemporal emission policy determines the (socially) efficient outcome with stationary pollution denoted X n ¼ r=c, see Wirl [30]. This setup makes a number of simplifying assumptions – no extraction costs, no resource constraint since the atmosphere is the ultimately binding constraint, linear demand – and assumes (optimistically) that most countries join a global warming compact and (pessimistically) increased cartelization: a GASPEC (e.g., Russia, Iran and Qatar, which hold more than 50% of natural gas reserves) following OPEC and both joining hands. The robustness and potential extensions are addressed in the last section.

3. Markov perfect equilibria The analysis is restricted to equilibria in Markov strategies, considering first simultaneous move Nash equilibria and then the consequences of first mover advantages. However, all first mover advantages are short run, i.e., one player may announce the strategy in period t before the opponent, of course taking the opponent’s reaction into account (compare Basar and Olsder [1] and Dockner et al. [6]).

3.1. Prices versus taxes The Hamilton–Jacobi–Bellman equations (short HJBs) implied for the price–tax game are rV ¼ maxfð1ptÞðp þ V 0 Þg,

ð8Þ

p

( ) ð1pÞ2 t2 c 2   X þ W 0 ð1ptÞ : t 2 2 2

rW ¼ max

ð9Þ

Solutions of these two functional equations (and all HJBs considered below) must satisfy the conditions of no arbitrage (also known as value matching) and of smooth pasting when emissions stop. Due to no depreciation, the stopping level is identical to the steady state, denoted by X 1 , i.e., q¼0 8X 4 X 1 . Hence the corresponding value matching conditions are VðX 1 Þ ¼ 0,

WðX 1 Þ ¼ 

c 2 X , 2r 1

ð10Þ

since the cartel’s value function must be zero in this domain of no emissions ¼no sales and the government’s value function equals the aggregate net present value of damages. Smooth pasting requires in addition continuity of the derivatives across the stopping¼steady state level, V 0 ðX 1 Þ ¼ 0,

W 0 ðX 1 Þ ¼ 

cX 1 : r

ð11Þ

Simultaneous maximization of both right hand sides in (8) and (9) implies 1V 0 þ W 0 2 t ¼ W 0

p¼

¼)

P ¼ pþt ¼

1ðV 0 þ W 0 Þ , 2

ð12Þ

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F. Wirl / Journal of Environmental Economics and Management 64 (2012) 217–229

and substitution of (12) into (8) and (9) yields rV ¼

rW ¼

ð1 þ ðV 0 þ W 0 ÞÞ2 , 4

ð13Þ

ð1 þ ðV 0 þW 0 ÞÞ2 c 2  X : 2 8

ð14Þ

Solving these two interdependent functional equations is sketched in another case below, because Wirl [30] reports already. Proposition 1. The explicit solution of the strategies pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð r 2 þ 3crÞ  r X , P ¼ 1þ 3 c p¼

ð15Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r 2 þ 3cðr 2 þ6cÞ  r X , 9r c

t ¼ 1þ

ð16Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r r 2 þ 3c2r 2 þ6c  r X 9r c

ð17Þ

implies that consumer price and tax increase up to the choke price, P and t-1, which is asymptotically reached at the steady state, X1 ¼ Xn ¼

r : c

ð18Þ

Therefore, the steady state of this non-cooperative dynamic game is the socially efficient one ðX n Þ. However, the intertemporal allocation is not efficient, more precisely, emissions are too conservationist, because the socially optimal consumer price is lower for all X oX 1 . The producer price declines monotonically to zero (again reached at the steady state) yet exceeds the usual monopoly price ðp ¼ 1=2Þ for 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ c r 4 þ3r 2 c 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X : X o X~ :¼ 1 r 2 þ6c r 4 þ3r 2 c

ð19Þ

In fact, larger external costs increase the initial price and the relative preemption domain ðX~ =X 1 Þ with lim ðX~ =X 1 Þ ¼ 1=4. c-1

The conditions of value matching and smooth pasting are satisfied without being used in the derivation of the linear strategies. However, these conditions turn out to be useful if only nonlinear strategies allow for emissions. Furthermore note that the steady state as well as the split of P into wellhead price and tax at this steady state, pðX 1 Þ ¼ 0, PðX 1 Þ ¼ tðX 1 Þ ¼ 1, follows immediately from the value matching and smooth pasting conditions and (13) and (14). The stock externality induces the monopoly to preempt the tax. More precisely, the monopoly charges at any point in time the price above the static Nash response, p 4 ð1tÞ=2 (weak preemption), and this price even exceeds the (static) monopoly price, p 4 1=2 for some time (strong preemption) with the limit (for c-1) that a quarter of the total will be sold above the usual monopoly price. The economic intuition is that the monopoly recognizes that selling a unit today will increase carbon taxes in the future. Accounting for this reduction in future demand leads to a higher price today in order to deter and to delay taxation; this incentive increases with higher marginal costs. Furthermore, the volume that the monopoly can sell is bounded by the sink. Hence its aim is to sell this limited volume ðr=cÞ at the highest price possible. There are real world precedents of preemptive moves. For example, The Economist (1992) explains the oil price increase of almost $4 per barrel prior to the Earth Summit in Rio de Janeiro between March and June 1992 was a consequence of the Saudi Arabian policy ‘to see the price rise by $3 a barrel to match the effect of the first step in the EC’s new carbon-tax plan’. The high oil prices in mid-2008 proved the power of preemption as these high prices deterred the introduction of planned CO2 taxes (e.g., in New Zealand) and urged many politicians to ask for a reduction in petrol taxes. Taking the view of the government, a positive by-product of taxing carbon is that the importing countries get lower prices, p o1=2, for at least 34 of the total volume imported. 3.1.1. Short run commitments The above analysis is now extended by giving one of the players the ability of short run commitment, i.e., to announce his policy in each period t just ahead of his rival. Allowing for such intraperiod first mover advantages modifies the results in two ways. First, the short run commitment by the supplier reproduces the above Nash outcome, because the government’s action is still to set the tax equal to the marginal damage, t ¼ W 0 according to (12). Second, letting the government choose the tax tðtÞ at any point in time ahead of the producers yields a different outcome. Substituting the

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221

cartel’s price reaction from (12) into the right hand side of (9) and then maximizing with respect to t yields

t¼

1 þ V 0 2W 0 1 þW 0 2V 0 ¼)p ¼ , 3 3

P¼

2ðV 0 þ W 0 Þ : 3

ð20Þ

Substitution of these strategies (20) into (8) and (9) yields rV ¼

ð1 þðV 0 þ W 0 ÞÞ2 , 9

rW ¼

ð1 þ ðV 0 þ W 0 ÞÞ2 c 2  X : 2 6

ð21Þ

ð22Þ

This pair of functional equations is identical to those of a taxing government (including its short run commitment ability) and a quantity setting producer. The corresponding result is shown formally in the following section. 3.2. Taxes and production quotas Assuming Cournot behavior of the fossil fuel cartel, the objectives of the two players are   Z 1 q2 ert q ð1qtÞqC dt, max t 2 0 max q

Z

ð23Þ

1

ert ð1qtÞq dt:

ð24Þ

0

Hence, the Hamilton–Jacobi–Bellman equations are

q2 rW ¼ max qt þ C þW 0 q , t 2 rV ¼ maxfð1qtÞqþ V 0 qg: q

ð25Þ ð26Þ

The linearity of the right hand side of (25) with respect to the tax instrument arises because taxes allow to accrue the cartel’s total (rectangular) rent at only a triangle loss. As a consequence, the only Nash equilibrium is highly inefficient at the emission free corner, ðt ¼ 1, q ¼ 0Þ. 3.2.1. Short run commitments Allowing the cartel to move first at time t does not alter this conclusion. However, providing a short run commitment to the government allows for an interior policy because substituting the supply cartel’s Nash–Cournot reaction obtained from (26), q¼

1t þ V 0 , 2

into the RHS of (25) yields ( ) W 0 ð1þ V 0 tÞ ð1 þ V 0 þ tÞ2 t2 q2 þ  þ C : rW ¼ max t 2 8 2 2

ð27Þ

ð28Þ

Maximization of the right hand side of (28) yields the tax,

t¼

1 þ V 0 2W 0 , 3

ð29Þ

which is identical to the tax derived for a government having the first move (at time t) in the price–tax game as mentioned above. Substituting this tax (29) into (27) implies the cartel’s output is q¼

1 þV 0 þ W 0 : 3

ð30Þ

As a consequence, emissions and thus consumer prices depend only on the sum of the value functions’ derivatives as in the price–tax game, P¼

2ðV 0 þ W 0 Þ : 3

ð31Þ

Substitution of the strategies (29) and (30) into the right hand sides of (26) and (28) yields (21) and (22). Therefore, not only the final consumer price but also the right hand sides in the HJBs depend only on this aggregate Z :¼ V þ W. Adding up leads to a single functional equation, rZ ¼

5ð1 þZ 0 Þ2 c 2  X : 2 18

ð32Þ

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F. Wirl / Journal of Environmental Economics and Management 64 (2012) 217–229

Guessing a quadratic solution, Z ¼ z0 þ z1 X þ ðz2 =2ÞX 2 , comparing coefficients in (32), and choosing the root that implies a stable strategy, yields the coefficients, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 5z2 ð3r 9r 2 þ20cÞ, z1 ¼ z2 ¼ , 10 9r5z2 which allows calculation of V and W and the players’ strategies. Proposition 2. An interior equilibrium in Markov strategies for a quantity setting supply cartel and a taxing government requires short run commitment of the government when setting the intraperiod tax first. The equilibrium is characterized by the efficient steady state (18) and qualitative properties similar to Proposition 1. Specifically, consumer prices, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2z1 z2 X r 9r 2 þ20c3r  ¼ 1þ X , ð33Þ P¼ 3 10 c and taxes, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10c=r þ 2ð 9r 2 þ 20c3rÞ  r X , t ¼ 1þ 25 c

ð34Þ

are increasing to the choke price, and the producer price, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 1 9r 2 þ 20cð20c=r þ 3rÞ  p¼ X o , 50 c 2

ð35Þ

is declining to zero. However, the tax is higher (conditional on X) and the producer price is lower, such that (strong) preemption is eliminated since p o 1=2. The final consumer price is higher, which lowers emissions and delays global warming. Moreover, this outcome is identical to the case if a taxing government has the first move (at any period t) in the price versus tax game as mentioned above. Proof. Above derivation; for the remaining arithmetical claims see Appendix. Fig. 1 compares the two different scenarios addressed in Propositions 1 and 2 using a numerical example. The nearly identical consumer price but the substantial difference in wellhead prices and thus also in taxes highlights the rent that is accrued by a taxing and first moving government, independent of whether the suppliers choose quantity or price. However, this increase in surplus from emissions due to lower import prices does not lead to an increase in global warming. 3.3. Permits and price setting cartel Assuming that the government issues q permits implies that either the volume of permits or the price charged by the cartel determines final demand and thus emission, q ¼ minfq,1pg, because not all permits will be used if the cartel’s price is too high. Therefore, it is clearly ineffective for the government to issue more permits than ð1pÞ and maximizing the RHS of the corresponding HJB,

q2 q pqC þ W 0 q , ð36Þ rW ¼ max 2 0 r q r 1p

P 1.0 0.8

tax 0.6

static monopoly price

0.4

p

0.2

0.2

0.4

0.6

0.8

1.0

X

Fig. 1. Comparing different mix of strategies for ðr ¼ 0:05 ¼ c ¼ 4 X 1 ¼ 1Þ: 1. Tax versus price (dashing) and Nash equilibrium. 2. Taxes versus output quotas with the government having the first move (¼ outcome of the tax–price game too given this first mover advantage).

F. Wirl / Journal of Environmental Economics and Management 64 (2012) 217–229

223

yields q¼

1p þ W 0 0

if 1p þ W 0

Z o

0,

ð37Þ

because the upper bound in (36) cannot bind due to W 0 r 0. The HJB of the price setting cartel facing q permits is, rV ¼ max fpð1pÞ þ Vð1pÞg ¼ maxfpq þ V 0 qg:

ð38Þ

p

p Z 1q

No matter how many permits are issued, charging a price below the market clearing level ðp o1qÞ leaves money on the table. Therefore, suppliers will choose at least the market clearing price that drives the permit price to zero, p Z 1q. Maximizing the right hand side in (38) implies, p¼

1V 0 Z 1 þV 0 , 2 if q o 2 1q

ð39Þ

which is shown in an inverted form in Fig. 2. In words, the cartel will charge at least p ¼ ð1V 0 Þ=2 and the higher market clearing price, p ¼ 1q, for less permits ðq o 1ðð1V 0 Þ=2ÞÞ. A consequence of these Nash reactions (37) and (39) is that no interior simultaneous move Nash equilibrium exists, see Fig. 2, since q¼0 is the only point of intersection. Therefore, the alternatives of short run commitment by either player are considered in two subsections. Proposition 3. If the consumers’ government issues permits and the supply cartel sets prices, then only the boundary solution, p¼1

and

q ¼ 0,

ð40Þ

is a candidate for a Nash equilibrium in Markov strategies. And it is an equilibrium because (40) is compatible with the corresponding HJBs (36) and (38) and their boundary conditions (10) and (11). 3.3.1. Cartel moves first Assume that the price setting cartel moves first at each point in time. The government’s Nash reaction is to issue permits below the market clearing level according to (37). Therefore the HJB equation of the first moving cartel must satisfy rV ¼ maxfð1p þ W 0 Þðp þV 0 Þg

¼)

p

p¼

1 þ W 0 V 0 : 2

ð41Þ

Substitution of this price and the government’s Nash reaction (37) into the HJBs (36) and (38) yields rV ¼

ð1 þðV 0 þ W 0 ÞÞ2 , 4

rW ¼

ð42Þ

ð1 þ ðV 0 þ W 0 ÞÞ2 c 2  X : 2 8

ð43Þ

These functional equations are identical to those obtained under the assumptions that price and tax are the instruments and that both parties move simultaneously. As a consequence, the same allocation results as in Proposition 1. Since the

1−V ' 2

q 1− q

1+V ' 2

many permits government’s Nash reaction

1− p +W '

cartel’s Nash reaction few permits, clearing demand 1− p no permits

p

Fig. 2. Intraperiod Nash reaction functions for a permit issuing government and a price setting cartel.

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F. Wirl / Journal of Environmental Economics and Management 64 (2012) 217–229

cartel moves first, its price will not clear the market, p o1q, because the government allocates permits below the market clearing price. This implies a positive permit price, p ¼ 1qp, that remains within consuming countries as the tax revenues in Section 3.1, either as proceeds from auctioning permits or as consumer rent if permits are grandfathered. Proposition 4. A government issuing permits facing a price setting cartel, which has the first move, reproduces the outcome in Proposition 1. The government will issue permits below the market clearing level given the cartel’s supply price (determined as outlined in Proposition 1). As a consequence, the permit price covers the wedge between producer’s and consumer’s price and this difference is equal to the tax given in Proposition 1, p ¼ t. 3.3.2. Government moves first Assume that the government moves first and distributes q permits. Since the cartel will charge at least p ¼ ð1V 0 Þ=2, issuing permits above the corresponding market clearing level, q 4 1ðð1V 0 Þ=2Þ ¼ ð1þ V 0 Þ=2, is without any consequence. Thus

1þ V 0 ¼ q, ð44Þ q ¼ min q, 2 if the government’s choice of permits is binding. Accounting for this constraint (44) and substituting the cartel’s choice of the market clearing price, pðqÞ ¼ 1q, into (36) yields ( ) q2 0 C þW q : ð45Þ rW ¼ max 2 0 r q r ð1 þ V 0 Þ=2 The resulting consumer surplus is convex in the government’s instrument ðqÞ if the cartel maximizes its rents by driving the permit price to zero ðp ¼ 0Þ. Therefore, the maximum on the right hand side of (45) must be at a boundary; either no permits or permits equal to the cartel’s maximum supply, 1 þV 0 1 þV 0 4 q¼ 2 if 2W 0 : 2 r 0

ð46Þ

The choice between these alternatives depends on the maximum volume supplied by the cartel (and thus on V 0 ) and on the shadow costs from global warming ðW 0 o 0Þ. The first inequality in (46) holds if the social surplus from emissions (valuing the damage at W 0 per unit emission) is still positive although the permits have no market value. If this is the case, the number of permits equals the market clearing demand at the cartel’s price, the first line in (46). Therefore, the hypothesis of positive emissions implies the following pair of functional equations: rW ¼

rV ¼

1 þV 0 ð1þ V 0 þ 4W 0 ÞC, 8



1þ V 0 2

ð47Þ

2 :

ð48Þ

Proposition 5. There exists no equilibrium in Markov strategies supporting positive quantities. Hence, q ¼ 0, p¼ 1 results as the unique equilibrium as in Proposition 3. Proof. No pair of solution curves of (47) and (48) qualifies for a value function since they cannot satisfy the boundary conditions (10) and (11), more precisely, V ¼ V 0 ¼ 0 cannot be met. In contrast, substituting the corner solution ðq ¼ 0, p ¼ 1Þ into (38) and (45) satisfies the corresponding functional functions (38) and (45) and the boundary conditions. & 3.4. Permits and quotas1 The currently observed regime pairs permits and quotas. EU-countries have issued permits since 2005 and OPEC determines output quotas at its biannual conference. At first sight it seems impossible to let both parties fix the quantity at the same time. However, each party may overwrite the other’s choice. For example, the suppliers plan to sell Q yet the government deems this too high and issues fewer permits, q o Q . Conversely, the government may issue a lot of permits yet the suppliers find it in their interest to supply less, Q oq. Therefore q ¼ minfq,Q g

ð49Þ

with the following implications on consumer, wellhead and permit prices: P ¼ 1q,

1

p ¼ 1Q ,

p ¼ maxf0,Q qg:

The following argument owes a lot to comments by Larry Karp.

ð50Þ

F. Wirl / Journal of Environmental Economics and Management 64 (2012) 217–229

The HJBs of a government grandfathering permits and of a quota setting supplier are

q2 rW ¼ max q ð1Q ÞqC þW 0 q , 2 q

225

ð51Þ

rV ¼ maxfð1Q Þq þ V 0 qg,

ð52Þ

Q

both accounting for (49). The government’s Nash reaction if facing supply Q allows for two cases: q 4 Q in which case the government’s choice is irrelevant and thus it may choose q ¼ Q without any loss; or it is determinant, q oQ . Therefore accounting for this constraint, the government’s optimal reaction is 0 r q ¼ Q þW 0 r Q ,

ð53Þ

and the constraint on the right hand side is not binding due to W 0 r0. Facing the permit constraint, any choice of Q 4 q is clearly suboptimal since it only lowers the wellhead price without increasing sales. Therefore, if the choice of Q is determinant, then q ¼ Q rq, which implies the supplier’s Nash reaction, 1þ V 0 Z 1þV0 : Q¼ 2 if q o 2 q

ð54Þ

Fig. 3 plots the corresponding reaction functions holding V 0 and W 0 fixed. This figure highlights that these reactions are a mirror image from those in Fig. 2 and that the corner, Q ¼ q ¼ 0:

ð55Þ

is the only point of intersection. Proposition 6. There exists no Nash equilibrium in Markov strategies with positive emissions if both players choose quantities. Therefore (55) is the unique equilibrium. Allowing the government the first move of issuing its permits does not change this conclusion. Assume that the government moves first, then substituting the cartel’s reaction (54) into the government’s HJB (51) yields ( ) q2 0 rW ¼ max0 q ð1qÞqC þ W q , ð56Þ 2 q r ð1 þ V Þ=2 because any commitment to q 4 ð1þ V 0 Þ=2 is undercut by the cartel’s supply ð1þ V 0 Þ=2. The right hand side in (56) is convex in the government’s instrument and the same as in (45). Therefore, the optimal number of permits is at the boundaries: either zero or equal to the maximum that the cartel is willing to supply, q¼

1þ V 0 1þV0 Z if 2W 0 , 2 2 o 0

ð57Þ

1+V ' 2

q

cartel’s Nash reaction

government’s Nash reaction

1+V ' 2

-W´ 45°

Q

Fig. 3. Intraperiod Nash reactions between a permit issuing government and a quantity setting cartel depending on the shadow prices (V 0 and W 0 ).

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and the latter only for sufficiently large cartel supply. Substitution of positive emissions, q ¼ ð1 þV 0 Þ=2 4 2W 0 yields the already known pair of HJBs (47) and (48) with the above mentioned consequence. 3.4.1. Cartel moves first Substituting the government’s Nash reaction (53) into the cartel’s optimization problem, rV ¼ maxfð1Q ÞðQ þ W 0 Þ þ V 0 ðQ þW 0 Þg,

ð58Þ

Q

and then maximizing the right hand side yields (in the interior, Q 40) Q¼

1 þV 0 W 0 : 2

ð59Þ

Substitution of (59) and q ¼ ð1 þV 0 þ W 0 Þ=2 due to (53) into (56) and (58) yields the functional equations (13) and (14). Therefore: Proposition 7. If the cartel has the first move of setting its quota facing a permit issuing government, the same outcome as already characterized in Proposition 1 results including the market clearing prices, except that the tax is replaced by the identical price for grandfathered permits. 4. Comparison and summary This paper considered the strategic implications of price and quantity instruments in a game between a consumer government concerned about global warming and profit maximizing fossil fuel suppliers, with both sides being perfect cartels. The results are summarized in Table 1. If the government sets a tax while the monopolist simultaneously sets the price, there is a unique Markov perfect equilibrium (MPE) in linear strategies. In this equilibrium, the tax increases over time, the monopolist’s price declines, and the stock of pollution, X, increases and converges to the efficient level, X n (but at less than the socially optimal speed) written as XmX n . If, at each moment in time, the monopolist sets the price before the country sets its tax, the outcome is given in the second sub-column of the first column and it is identical to the simultaneous move case. If the government acts first, there also exists a unique MPE in linear strategies, but the tax is now larger in order to force the monopolist to reduce its price. Although X converges to X n as before, it increases more slowly than in the first case since the final consumer price is higher; this difference is indicated by writing XsX n . In the case of permit and quota policies, only short run commitment by the cartel produces an equilibrium with positive emissions, and this outcome is identical to the simultaneous move price–tax game. The interpretation of all outcomes is then straightforward with the help of Table 1. Therefore given the different combinations analyzed above, what are the likely choices of these instruments? Starting with the supply side, price and quantity are equivalent instruments without externalities (and thus no carbon policy), at least in the Nash setting. Using the price–tax Nash equilibrium as a benchmark, choosing quantity as the strategic variable renders either a strategic advantage to an externality taxing government, or eliminates the scope for sales entirely. Therefore, a rational monopoly will prefer the price strategy in the case of an emerging rent contest between OPEC and IEA over carbon taxes. As a consequence, OPEC should be expected to switch back to pricing if consumer governments tax carbon. Assuming that the supply cartel sets prices, the government is not indifferent between the price and quantity instrument. Choosing a permit strategy, the government surrenders its strategic advantage such that either the corner equilibrium or the outcome of (partially) preemptive cartel behavior under the price–tax regime emerges. Moreover, the government’s ability of short run commitment is only beneficial if it taxes carbon in which case it eliminates (strong) preemption by the cartel. Summarizing, quantities are bad choices for both parties in this strategic and non-competitive setup such that prices and taxes are the natural choices. More precisely, considering the payoffs ðVð0Þ,Wð0ÞÞ in Table 1 for the three meta-games of simultaneous moves or either party leading with prices and quantities as the two players’ strategies, then (price, tax) is the undominated Nash equilibrium in all three meta-games. The inefficiency of quantity strategies is due to the lack of interior equilibria. The reason for this is that once a party chooses the quantity strategy it tempts the other party to grab all rents. Yet with all rents taken away the affected party has no incentive to supply (either permits or fuel). Table 1 Comparison of the results. Government (G)/Monopoly (M) strategies

G-tax G-permit

M-price

M-quota

Sim

M 1st

G 1st

Sim

M 1st

G 1st

XmX n X¼0

XmX n XmX n

XsX n X¼0

X¼0 X¼0

X¼0 XmX n

XsX n X¼0

M, monopoly; G, consumer government; sim, simultaneous moves; m, convergence; s, slower convergence due to higher consumer price P; X n ¼ 1st best steady state.

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This strong finding contradicts both players’ preference for quantity strategies with consumer governments eschewing carbon taxes and issuing permits and with OPEC announcing quantities instead of prices. One explanation is that markets are (more) competitive and that the real world players have less leverage than what is assumed. In particular, tax and cap instruments are equivalent if both parties cooperate or if supply is competitive. Although there is much talk about a producer and consumer dialogue (initiated by Dr. Birol at the IEA who joined IEA after working for OPEC), competitive supply seems unlikely considering the large benefits for the few players holding the bulk of the reserves as well as the high prices. However, this suggests extensions that relax this assumption, as Karp et al. [11] do by adding a third group of passive consumers ignorant of global warming (thus avoiding the additional layer of a tragedy of the commons). This extension can, surprisingly, modify the conclusions. However, most of the other simplifying assumptions used here should leave the conclusions intact. The assumption that global warming limits fossil fuel use allows to eliminate scarcity rents and thereby to restrict the analysis to stationary Markov strategies. Neither extraction costs nor depreciation seem crucial, considering Tahvonen [27] and Liski and Tahvonen [15]. The choice of pure objectives, in which only the importing government cares about global warming is made to stress the antagonism between the players, even if exporters may also care. This assumption is not far off the mark, because the population of major fossil fuel exporters is only a small share of the world total. A correspondingly small damage parameter (say at 5% or less from c) has little impact on the results, at least as long as exporters’ own consumption is ignored (details are available upon request). The results in Karp et al. [11] suggest that one should include further aspects, but also warn that any substantial extension may render closed form solutions impossible or intractable. This difficulty applies even to extending the analysis for the global Stackelberg equilibrium—the leader commits not only to the current strategy but to his entire strategy profile without the need to change it later; further details are available upon request. And there are of course explanations outside of the model. Many of them are presumably political in nature, in particular with respect to the preference for carbon permits over taxes, because increasing fuel taxes harms consumers. This aspect is of obvious concern for politicians in consuming countries, but also in producing countries. Neither the King of Saudi Arabia and more surprisingly not even Hugo Chavez want to appear as villain in the eyes of US consumers and OPEC’s quota policy allows them to condemn speculators as being responsible for high prices in particular during mid-2008 when prices reached $140=barrel. Therefore, accounting for political objectives of suppliers (e.g., [32]) and governments (e.g., Leviathan motives as in [33]) seems another useful agenda for future research.

Acknowledgments This paper has been presented at Institut de Recherches Economiques et Sociales, UCLouvain and at the last EAERE meeting in Rome. I thank my discussants Katheline Schubert and Anne-Sophie Cre´pin for their helpful comments. I have also received very helpful comments from Larry Karp, Jon Strand, Engelbert Dockner, and two anonymous referees. Last but not least, the editor (Dan Phaneuf) considerably improved the presentation. Appendix A A.1. Proof of Proposition 2 A.1.1. Consumer price Given the same steady state and the same consumer price at the steady state it is sufficient to compare the intercepts. The game in prices and taxes yields, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ3c r 4 þ3r 2 c Pa ð0Þ ¼ 3c and for a quantity setting cartel confronted with a first moving and taxing government, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3r 2 þ 10c 9r 4 þ20r 2 c : Pb ð0Þ ¼ 10c The claim is equivalent to P b ð0ÞPa ð0Þ 4 0. Cancelling the term c from the denominator, i.e., pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 r 4 þ3r 2 c3 9r 4 þ20r 2 cr 2 b a : DðcÞ :¼ cðP ð0ÞP ð0ÞÞ ¼ 30 Note that Dð0Þ ¼ 0 and that   r 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D0 ¼ 2 r 2 þ3c 9r 2 þ 20c is definitely positive, because the bracket is positive iff pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9r 2 þ 20c 42 r 2 þ 3c () 9r 2 þ20c 4 4r 2 þ 12c:

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A.1.2. Producer price Similar for the producer prices, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ 6c r 4 þ 3r 2 c , pa ð0Þ ¼ 9c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3r 2 þ 20c 9r 4 þ 20r 2 c : pb ð0Þ ¼ 50c The claim is equivalent to pa ð0Þpb ð0Þ 4 0. Cancelling the term c from the denominator, i.e., pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 120c þ23r 2 þ 9 9r 4 þ20r 2 c50 r 4 þ3r 2 c , DðcÞ :¼ cðpa ð0Þpb ð0ÞÞ ¼ 450 which implies again that Dð0Þ ¼ 0 and that the derivative   1 6r 5r 8 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D0 ¼ 30 9r 2 þ20c r 2 þ 3c is positive, because the term between the brackets is positive iff, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 8 9r 2 þ 20c 9r 2 þ 20c þ 4 : 5 5 r2 r 2 þ 3c A.1.3. No preemption The claim of no preemption follows also immediately from the explicit solution. Preemption, if taking place at all must apply at X¼0, therefore the claim implies, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20c þ 3r 2  9r 4 þ20cr 2 1 o , pð0Þ ¼ 2 50c where the claimed inequality holds iff, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3r 2 5c o 9r 4 þ 20cr2 :

ð60Þ

This inequality holds trivially if the LHS is negative. Hence, violation of the claim requires at the minimum, 0 o 3r 2 5c

¼)

co

3 2 r : 5

ð61Þ

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