Dynamic Games in the Economics and Management of Pollution

Transcription

1 Dynamic Games in the Economics and Management of Pollution Ste en Jørgensen Department of Business and Economics University of Southern Denmark, Denmark Guiomar Martín-Herrán Departamento de Economía Aplicada (Matemáticas) Universidad de Valladolid, Spain Georges Zaccour Chair in Game Theory and Management, GERAD, HEC Montréal, Canada Abstract The paper provides a survey of the literature which utilizes dynamic state-space games to formulate and analyze intertemporal, many-decision maker problems in the economics and management of pollution. Key Words: Environment; Economics; Pollution; State space dynamic games; Cooperative and noncooperative games. 1 Introduction Environmental and resource economics are concerned with the economic aspects of the utilization of natural renewable resources (forests, sheries), natural exhaustible resources (oil, coal, minerals) and environmental resources (soil, water, air). The focus of this survey is pollution, a major environmental issue. The extraction and use of natural resources, in particular nonrenewable ones, for production, heating, transportation, etc. causes pollution while the abatement of pollution requires equipment and the expenditure of resources. The concept of an externality is central in environmental economics. An externality is an important instance of market failure which produces a deviation from the rst-best (Pareto optimal) solution. The problem is that market prices do not necessarily re ect the true social costs or bene ts. In such cases, regulatory institutions and instruments are needed. For the global environment, however, there is no supranational regulating agency which can dictate nations their environmental behavior; improvements can only be achieved by voluntary cooperation involving many countries with very diverse interests. 1

2 Externalities can be positive or negative. A negative externality is pollution generated by manufacturing rms and power plants and a ecting negatively the welfare of individuals and others; rms and power plants do not pay the full costs of their decisions. An important notion in environmental economics is ownership. As concerns natural resources, ownership can be clearly de ned (oil, minerals) or it can be common-property (open-access) in which case a resource is owned by nobody (high-seas shery). A great challenge lies in the fact that very serious problems of pollution do not have a speci c ownership. It is clearly a matter of speech if we say that the atmosphere and the ozone layer are open-access resources, owned by nobody, or they are owned by the world as a whole (Cesar (1994)). The absence of clearly de ned property rights (ownership) typically leads to externalities. The environmental resources (air, oceans) are common property resources: Everybody can, in principle, use the environment to deposit pollutants. Most environmental problems have three characteristics that should be taken into consideration in the modeling and analysis of these problems. 1. Interdependence: Strategic interdependence is present when actions of an individual economic agent a ect the welfare (payo, utility) of the agent, but in uence also the welfare of other agents. This is certainly true in international transboundary pollution problems (e.g., acid rain), but it also applies in downstream pollution and between buyers and sellers in markets for tradeable pollution permits. Environmental interdependence is related to environmental externalities. 2. Time. Environmental problems are intrinsically dynamic and only as a rst approach they should be studied as one-shot phenomena. For example, in the regulation of polluting behavior time is a key element because the interactions between regulator and rms take place over time, because externalities are intertemporal, and because the credibility and e ciency of regulating policies can only be examined over time. 3. Strategic and forward-looking behavior on the part of the agents (business rms, communities, regions, nations) who take actions that a ect the environment. Con icts arise because agents have their own objectives and their own preferred courses of action. Agents act strategically and are aware that other strategically acting agents are part of the problem and that their decisions in uence the outcome. Agents take into account the present and future consequences of their own actions and those of other agents. To represent time, strategic behavior and interdependencies in mathematical models of environmental problems, dynamic games have proven to be of considerable value from a prescriptive point of view. By a dynamic game we understand in this survey the so-called state-space games. A state-space game is one which contains a set of state variables that describe the main features 2

3 of a dynamic system at any instant of time during the game. The idea is that the state variables adequately summarize all relevant consequences of the past history of the game. The modeler decides if time should be continuous or discrete and the dynamics of an environmental dynamic game model is a set of di erential or di erence equations called the state equations. An advantage of state space games in applications to problems of pollution is the opportunity to model not only ow pollution damage e ects (which can be accounted for in static models), but also damage caused by stocks of accumulated pollution (e.g., concentrations of greenhouse gases in the atmosphere or acidi cation of soils). For an introduction to di erential and dynamic games, see Basar and Olsder (1999), Dockner et al. (2000). Apart from state variables, a dynamic game model also contains decision (or control) variables of the players. In a pollution problem, control variables of rms may be their rates of pollution emission and investments in abatement equipment. To see how the dynamics of a simple di erential game of a pollution problem might look, consider the following example. Example 1 Suppose that n rms are located in a geographical area. Each rm emits quantities of a pollutant, say, XYZ. Emissions stay within the area and accumulate in a stock. Represent by the state variable S(t) 0 the stock of XYZ by time t 2 [0; T ] where T is the rms planning horizon. The emission rate of the pollutant at time t of rm i 2 f1; 2; :::; ng is its control variable and is denoted by e i (t) 0: Let 0 be a parameter. The evolution of the stock S is prescribed by the ordinary di erential equation ds dt (t) = n X i=1 e i (t) S(t); where the rst term on the right-hand side is the aggregate emission rate and the second term represents natural decay of the stock of pollution. The assumption is that the natural decay rate is proportional to the stock level. Dynamic game analyses of multi-agent pollution problems are typically concerned with one or both of the following scenarios. The rst-best, Pareto optimal or e cient, solution occurs if a social planner has been given the authority to select actions that maximize the welfare of all agents involved in the environmental problem. The rst-best solution can also be achieved if all agents agree to cooperate and decide and implement a rst-best solution. Clearly this is an ideal case. If there is no social planner or agents cannot agree to cooperate, a noncooperative game is played. The standard assumption in the literature is that the game is played with open-loop or with feedback strategies. An open-loop strategy is one where an agent precommits to the use of a xed time-function as her strategy. If the game is played with feedback (Markovian) strategies, actions depend on time as well as the current 3

4 state. In deterministic games the use of feedback strategies requires an assumption that players can observe the state without error. If the assumption of certainty is abandoned, one can turn to the theory of stochastic dynamic games. This theory includes games in which the state dynamics are stochastic di erential/di erence equations as well as games called piecewise deterministic games. In the latter, the dynamics are deterministic but change randomly at random instants of time. We proceed as follows. Section 2 surveys pollution control instruments, including standards and quotas, tradeable permits, various tax schemes, subsidies, and combinations of instruments. Section 3 deals with transboundary pollution problems including downstream and global pollution problems. Noncooperative and cooperative approaches are discussed as well as international environment agreements and empirical studies. Section 4 is devoted to macroeconomic issues and includes economic and population growth, climate change, income and technology transfers, and sustainable development. Section 5 contains our conclusions and points to some avenues for future research. 2 Pollution Control Instruments There are many sources of pollution and a single source of pollution can have multiple e ects. E ects may, roughly, be characterized as local or global. Locally, soil, lakes, rivers and ground water may be polluted by fertilizers and pesticides used in agricultural production, by emissions of chemical residues from industrial production, or by household waste. Local authorities can introduce environmental policies within their jurisdiction to try to manage the sources and e ects of pollution. Global pollution is caused by emissions of pollutants from countries around the world and may a ect a few, many or all countries. Examples of the latter are the greenhouse e ect, the depletion of the ozone layer, and acid rain. In a di erential game model, List and Mason (2001) (see also List and Mason (1999)) ask the question: Given a second-best world only, should environmental regulation be done locally or centrally? Usually, dynamic game models make no distinction between local and central environmental regulation or whether information on rms payo s is local or central. List and Mason consider two scenarios. In the rst, a central regulatory agency xes uniform standards for the local jurisdictions. In the second, pollution control is decentralized such that local authorities can choose di erent instruments in di erent jurisdictions. The paper shows that decentralized control increases the joint payo of players if there are signi cant asymmetries in payo s and initial pollution levels are su ciently small. The result is driven by a well-known fact that a central regulator has a single shadow price for all pollution stocks, while local regulators use di erent shadow prices for di erent stocks. Environmental policy instruments can be divided into two broad groups. Command-and-control instruments. This group includes the prohibition 4

5 of speci c inputs, processes, and technologies, discharge permits, emission standards and quotas, as well as technological speci cations for the handling of pollutants. 1 Market-based instruments are economic incentives. This group contains tradeable emission permits, emission charges and other tax schemes, subsidies, and liability law provisions. In the past, environmental policies have been dominated by command-andcontrol instruments. 2 In contrast, internationally applied policies, typically concerning the reduction of emissions, must be based on voluntary and multilateral agreements. The reason is clear: At the international level there are no authorities to enforce environmental policies. Voluntary agreements on environmental issues are also used at the national level, between an industry (or industry association) and a government. Such a practice is known from industrialized countries and has gained some popularity in developing and transition countries (particularly in Latin America). However, in developing and transition countries it is not clear whether voluntary agreements will be e ective when the machinery to enforce regulations is weak or non-existing at local levels. Segerson and Miceli (1998) study voluntary agreements as a two-stage game. The results suggest that although voluntary agreements may o er cost savings for rms and regulators, concerns about reductions in environmental quality may be justi ed if, for example, subsidies are expensive and rms have considerable bargaining power. Alberini and Segerson (2002) discuss the relative merits of voluntary and mandatory approaches as well as theoretical and empirical issues that arise in the assessment of a voluntary agreement. The paper identi es some main factors which could increase the e ciency of voluntary agreements Taxes We start with an example of how a di erential game of pollution taxation can be modeled (Benchekroun and Long (1998)). Example 2 An oligopolistic industry consists of n identical rms producing a homogeneous good. Let the unit cost of production be constant, equal to c; and let 1 Command-and-control instruments require in general some monitoring e ort to ensure compliance with the regulation. Before the issue of environmental regulation became popular, Filar (1985) and Filar and Schultz (1986) proposed a dynamic inspection process model, termed The Traveling Inspector Model, where one player, the inspector, devises a monitoring strategy to minimize deviations (or its cost) from a norm (e.g., level of pollution). The problem is formulated as a singlecontroller, zero-sum, undiscounted stochastic game. It is well known that solving such problems is hard. However, in Filar and Schultz (1986), the authors show that the special structure of the game allows for a solution by using a relatively simple linear program. 2 Dietz and Vollebergh (2002) ask the question why market-based instruments have been less exploited and provide an explanation using a positive approach to instrument choice, based on identi cation of interests, con ict resolution, and pressure from politicians and bureaucrats. 3 The volume edited by Ten Brink (2002) surveys the area of voluntary environmental agreements. 5

6 q i (t) be the production rate of rm i: Industry output equals Q(t) = P n i=1 q i(t) and the market price P (t) is given by the inverse demand function P = P (Q) such that P 0 (Q) < 0; P (0) > c: Production generates pollution which is measured by the size of the stock S(t): A rm s pollution emission rate e i (t) equals its production rate q i (t): The pollution dynamics are simple: ds (t) = Q(t) dt S(t); where 0 is the natural decay rate of the stock S: Firms are free to choose their output rates, but they must pay taxes. A rm s tax bill (more precisely: tax payment rate) at time t is a function of its own emission rate e i (t) = q i (t) and the size of the overall stock S(t): The tax bill thus is T i = T (q i (t); S(t)): Function T is the same for all rms: Equal rms are treated equally. The function T (; ) could, for instance, be linear in the production rate: T i = f(s)q i : The pro t function of rm i is its long-run, discounted pro t: i = Z 1 0 e rt fp (Q(t)) c f(s(t))g q i (t)dt: Each rm knows that its current production will add to the future stock of pollution and thus a ect its future tax payments. At time t = 0 the government announces the per unit tax rule f(s ()) that applies to all rms at all future instants of time. The problem of the government is to nd an e cient tax rule which achieves a social optimum which is one in which the government makes the production decisions such that aggregate welfare of the society is maximized. For this purpose the government can design a tax rule (an incentive) which induces rms to choose production paths that are socially optimal. Dawid et al. (2005) study an in nite horizon Stackelberg di erential game played by a government and a number of rms. 4 Firms come in two versions: Believers who take tax announcements of the government for face value and Non-Believers who perfectly anticipate, albeit at a cost, the government s decisions. The proportion of rms in a speci c category changes over time according to the di erence in the pro ts made of the two types of rms. A main issue of the paper is the time consistency of government policy (see the remark below). Dawid et al. (2005) suggest a new solution to the problem of time inconsistency in environmental policy making: The regulator should not precommit to a tax 4 As opposed to situations in which players make their decisions simultaneously, so-called Stackelberg games assume that decisions are made (or actions are taken) sequentially. In a two-player game, the rst mover is a leader and the second mover is a follower. One interpretation of the sequential setup is that the follower observes the decision (action) of the leader. Another interpretation is that the leader commits to a decision and announces it to the follower who then makes her decision, believing that the leader will honour the commitment. 6

7 policy, but rather make non-binding announcements and implement taxes such that Believers make good pro ts. Remark 1 The issue of time consistency of macroeconomic policies was rst raised by Kydland and Prescott (1977). Applied to the dynamic Stackelberg game considered in Dawid et al. time consistency means the following. The government acts as the Stackelberg leader but if it announces that it will implement the Stackelberg solution, such an announcement is not credible because rms (the followers) know that government has an incentive to re-optimize once followers have acted. The reason for this incentive is that Stackelberg equilibrium is not de ned by mutual best replies: It only requires that the follower uses a best reply to the leader s announced action. About time consistency, see also the section Cooperative Game Approach to IEA. Time consistency of environmental policies is discussed in Batabyal (1996a, 1996b) who propose a di erential game between a government agency (Stackelberg leader) and a number of polluting rms (followers). A speci c issue is the possible time inconsistency of a tax policy announced by the agency which taxes the production of a polluting good by using a unit tax or an ad valorem tax. These two instruments can have quite di erent e ects on tax revenue as well as welfare. As expected, open-loop tax policies are time inconsistent. To obtain time consistent tax policies, a method employed by Karp (1984) is used. It is shown that open-loop and consistent policies are equivalent if rms production costs are unrelated to the stock of pollution (the state variable). In such a case it makes no di erence whether the agency announces a tax policy at the start of the game or revises continually its policy Optimal Intertemporal Taxation Schemes Pigou (1947) showed that in a problem of a single-period externality, the Pareto e cient pollution tax is equal to the marginal environmental damage from pollution, evaluated at the socially optimal level of emissions. This is a rst-best solution and the internalization of the external damages is complete. Later it has been shown that the degree of internalization depends on the market structure. Markets targeted for environmental regulation are seldom monopolies or perfectly competitive. In oligopolistic markets there are other externalities, apart from the environmental ones, and these externalities a ect the e ectiveness of environmental policy instruments, including taxes. See also Carraro (2002). A considerable part of the dynamic game literature on pollution taxes has its focus on optimal taxation schemes. The idea is to nd intertemporal pollution taxation schemes that sustain Pareto e cient outcomes. Hoel (1992, 1993) considers a nite horizon di erence game (where the trend in pollution economics seems to be an in nite horizon). Neglecting taxes, the social optimum is determined and a noncooperative game is played with open-loop and feedback strategies. Total cumulative emissions are, as expected, higher in the feedback equilibrium than in the open-loop equilibrium, and total cumulative emissions are higher in the feedback equilibrium than in the social optimum. 7

8 Next, a time-dependent emission tax is introduced, being the same for all countries. It turns out that the tax providing the Pareto optimum is the same for feedback and open-loop equilibria. It is surprising that the optimal tax does not depend on the kind of strategic behavior that the countries adopt and the result is in con ict with Benchekroun and Long (1998), Xepapadeas (1992). Benchekroun and Long (1998) (see also the above example) ask the question whether there exists a time-independent output tax rule such that polluting, oligopolistic rms will choose socially optimal production and pollution paths. Firms play a noncooperative di erential game with open-loop or feedback strategies. Which game is played a ects the choice of parameters in the tax rule. In any case there exists a time-independent tax (per unit of output) rate that depends on the current level of the pollution stock. If the stock level is low, the tax may be negative (i.e., a subsidy). Apart from knowledge of the damage function, costs and other parameters, the government must try to gure out which game (open-loop or feedback) the rms actually will play. Karp and Livernois (1994) are concerned with the question of the level of an emissions tax that is needed to achieve a target level of pollution. If government does not have full information about rms abatement costs, the tax level is unknown. The paper studies a linear mechanism that adjusts the tax when aggregate emissions deviate from the target level. In a static setting with nonstrategic rms, the adjustment mechanism is well known. In a dynamic setting with strategically behaving rms, it is not clear if an adjustment rule will solve the information problem. The aim of the paper is to study the performance of a linear emissions tax adjustment mechanism, given that rms will try to a ect the tax rate to their own advantage. The adjustment rule is exogenous and depends on the actions of the rms. A di erential game is analyzed with openloop and feedback strategies. The approach in Karp and Livernois is related to that in Conrad and Wang (1993) who examine the steady-state properties of a tax adjustment mechanism in situations where government has no information about rms abatement costs. A fair part of the literature on the taxation of pollution is concerned with carbon taxes. A carbon tax is an energy pricing instrument to control CO 2 emissions. The tax can be levied on the carbon content in fuels, as a proxy for the emission that is produced when fuels are combusted. Some of the literature is concerned with optimal taxation (as the above papers). Another topic is dynamic bilateral interaction between a resource-exporting cartel (typically: OPEC) and a coalition of resource-importing (and resource-consuming) countries. A primary issue here is the strategic aspects of a taxation of CO 2 emissions in resource-importing countries. The following papers belong to this area: Martin et al. (1993), Wirl (1994a, 1994b, 1994c, 1995, 2007b), Wirl and Dockner (1995), Tahvonen (1996), Rubio and Escriche (2001), Liski and Tahvonen (2004). Martin et al. (1993) use a dynamic game of CO 2 emissions with asymmetric players to study the impact of imposing a carbon tax. Asymmetry of players with respect to, typically, cost and damage functions, is usual in the literature. However, the asymmetry in the Martin et al. paper is such that one player 8

9 bene ts from a climate change, the others are losing. (The overall impact of climate change is assumed to be negative). The implication is that players are likely to have di erent attitudes concerning global climate changes. The paper assesses the impact of a tax scheme on players strategies in feedback Nash equilibrium. The tax policy is a Pigouvian emissions tax where some of the total revenues are returned to the players. The approach is similar to Hoel (1993), but Martin et al. assume that players objective functions are not the same. Wirl (1994c) addresses the question whether a Pigouvian tax on pollution should be introduced gradually, starting with low tax rates. A government acts as a kind of Stackelberg leader while consumers and producers are followers who determine their optimal responses to the government pollution tax policy by solving one-sided optimal control problems. An optimal tax strategy of the government is determined, taking into account the rational reactions of producers and consumers. It turns out that it is optimal to use the static Pigouvian tax which equals the marginal pollution damage at each instant of time. Moreover, the tax should decline from an initial high rate, toward a steady-state level. Wirl and Dockner (1995) extend the cartel vs. resource-importing countries model by introducing an additional objective of the government in the consuming country: Apart from the conventional externality-correcting purpose of a carbon tax, the government likes tax revenues as such. (This seems to be a realistic assumption in many countries). The government faces a competitive or a cartelized supply. In the latter case, which perhaps is the most realistic, cartelized producers employ feedback pricing strategies to maximize their pro ts while disregarding the pollution state equation. This means that the cartel does not care about the stock externality. Con ning their interest to linear feedback strategies, the authors show that the government raises initial taxes considerably. This lowers initial emissions, but increases the long-run stock compared to a neoclassical benevolent government. Wirl (1996) combines the approach in Wirl and Dockner (1995) with the internationally coordinated carbon taxes in Hoel (1993). The model is a linear-quadratic, in nite-horizon di erential game and the author characterizes linear feedback strategies. In the long run (i.e., in steady state), governments that like tax revenues may worsen environmental conditions Wirl (1994b) considers the dynamic interactions between resource suppliers and consumers in a model with a ow externality (e.g., acid rain) and stock externality (e.g., global warming). Wirl (2007b) extends this problem by introducing uncertainty in the sense that global warming follows a stochastic process. Under uncertainty one needs to distinguish reversible and irreversible emissions. In the case of reversible emissions, linear tax strategies can be analytically derived. Otherwise, one has to resort to numerical analysis. Tahvonen (1996) extends the model of Wirl (1994b) by introducing, although not initially, decay of pollution and extraction costs that depend on the stock of resource. It turns out that this extension changes the main results obtained in the simpler model. Formally, the setup is a linear-quadratic di erential game 9

10 with two state variables. Tahvonen assumes that the suppliers cartel has an advantage and can act as a Stackelberg leader. A feedback Stackelberg equilibrium is determined numerically. Wirl (1995) considers a dynamic game between a cartel of resource-producing and exporting countries and a coalition of resource-importing and consuming countries (represented as one decision maker, a government). The analysis focuses on linear feedback strategies (cartel price and government tax, respectively). The problem is a mixed one of exhaustible resource extraction and pollution (e.g., carbon dioxide); the model is a di erential game with two state variables. An analytical solution is possible only in a simpli ed case in which the state variables are identical. The paper extends the setup in Wirl (1994b) by allowing for depreciation of the stock of pollution and resource extraction costs that increase by the quantity already extracted. Rubio and Escriche (2001) use the model of Wirl (1995) and raise the question (also noted by Wirl) whether a carbon tax could enable the coalition to correct the pollution externality and appropriate part of the cartel pro ts? In a feedback Nash equilibrium, the carbon tax corrects the pollution externality only. However, if the coalition can act as a Stackelberg leader, taxation allows the coalition to seize part of the cartel s pro ts in a feedback Stackelberg equilibrium. Liski and Tahvonen (2004) also study a di erential game being close to the one in Wirl (1995) and Rubio and Escriche (2001); see also Tahvonen (1996). The objective of Liski and Tahvonen is to design an optimal carbon tax in the cartel vs. coalition di erential game. A main result is the explicit isolation of a Pigouvian and a strategic trade policy component of the optimal tax for a stock pollutant. When resource importing countries coordinate the taxation of emissions, the optimal tax includes an import-tari component which shifts rents from the cartel to the coalition. Because of the trade policy component, the optimal carbon tax is in general di erent for the neutral Pigouvian tax (cf. Rubio and Escriche (2001)) which internalizes the damage cost of pollution (i.e., the stock externality) only. Yanase (2007) studies emission taxes vs. quotas in a model of international trade where the production of a traded commodity causes transnational pollution. The problem is formulated as a di erential game between governments that determine national environmental policies in the form of Markov-perfect strategies. In a linear-quadratic, duopolistic version it is demonstrated that an emission tax causes greater pollution and lower welfare than an emission quota system. The author also studies an asymmetric case where one country uses an emission tax while the other country uses an emission quota system. Closed-form results cannot be derived for this problem. Yeung (1992), Yeung and Cheung (1994) study the capital accumulation problem of a rm (or industrial sector), subject to taxation. A time-dependent output tax is imposed by a regulator in order to collect money to be spent on abatement. The model is a two-player di erential game with state equations for capital and pollution accumulation. Open-loop and feedback equilibria are considered and it is shown that the steady-state levels of capital and pollution are highest in open-loop equilibrium. Extending the setup in Yeung and Cheung 10

11 (1994), Yeung (1995) considers a di erential game in which the depreciation of the productive capital stock is not constant, but increases with the stock of pollution Nonpoint Source Pollution In point source pollution (e.g., industrial or municipal emissions), the source, volume and characteristics of discharges can be identi ed by regulators with reasonable precision and cost. In nonpoint source pollution (e.g., agriculture, vehicles), neither the source nor the volume of individual emissions are observable by regulators. The regulator can measure the ambient pollution at speci c points, but is unable to assign a particular proportion of the pollutant level to a speci c discharger. Nonpoint source pollution creates problems of monitoring and measurement, caused by informational asymmetries between dischargers and regulators, and standard regulatory instruments (Pigouvian taxes or emission standards) are useless as incentives to make dischargers adopt socially preferable policies. There has been increasing focus on policy instruments that may be appropriate for nonpoint source pollution problems. There are two broad categories of instruments: Tax schemes, based on observed ambient pollution, and inputbased schemes that tax observable, polluting inputs. Xepapadeas (2002) provides a brief survey of nonpoint source pollution problems; Shortle et al. (1998) outline some research issues that arise in the analysis of input taxes and ambient taxes. The authors identify, although in a static setting only, theoretical and empirical issues in the choice between the two tax approaches. The aim in Xepapadeas (1992) is to design an incentive scheme that induces dischargers to select policies that lead to a socially desirable steady-state pollution level. Two sets of pollution dynamics are investigated: An ordinary di erential equation and a stochastic di usion process. The incentive scheme is designed in such a way that it charges, at any instant of time, a tax per unit of deviation between the socially desirable and the observed ambient pollution levels. The scheme can be seen as a Pigouvian tax on deviations from the socially optimal path. If pro t-maximizing rms are subjected to this scheme, the pollution level converges to the socially desirable steady state. The size of the tax depends on the strategic behavior of the rms. If they use feedback strategies, the tax needs to be higher than if rms use open-loop strategies. With a view to implementation, the data requirements could be formidable. A regulator needs to know about production and abatement technologies of rms, the damages caused by accumulation of pollutants and the type of strategies that rms will employ (cf. Benchekroun and Long (1998) above). An ambient tax is one in which the regulator charges a unit tax based on the aggregate level of pollution (e.g., in an industry or region). Karp (2005) notes that an ambient tax can sustain a social optimum, given that rms recognize that their decisions a ect the aggregate emission level. This means that an ambient tax makes sense only in situations where rms behave strategically and recognize that they may a ect the tax rate. The incentive to do so is 11

12 greater when taxes are paid on aggregate rather than individual emissions. Karp compares, in steady state, the tax burden under the ambient tax and under a tax charged on individual emissions. (The aggregate emissions in steady state are the same for both taxes). The tax rule charges at each instant of time a tax per unit of deviation between industry emissions and an exogenous target emission level (cf. Xepapadeas (1992)). A surprising result is that even if the regulator has perfect information about rms emissions, they might get a higher payo if the regulator acted as if it were unable to observe individual emissions! Note that an ambient tax has two limitations: Polluting rms must be large enough to realize that their emissions a ect the aggregate pollution level and the tax may result in large transfers to/from a rm. Finally, let us note that economists have discussed the possibility of an environmental tax reform. The idea is to shift at least some of the tax burden from income to pollution, but it remains to be seen if this would improve environmental quality and make taxation a more e cient income generator for the government. This e ect has been called the double-dividend hypothesis and is discussed in de Mooij (2002). The name comes from what the hypothesis claims: Changing the taxation system yields two dividends, a green (environmental) one and a blue (non-environmental) one. There has, however, been a debate on how one should de ne and interpret these dividends. 2.2 Standards and Taxes Pollution taxes and standards have been implemented in practice, predominantly taxes in European countries, standards in the United States. Helfand (2002, p. 231)) provides a summary of factors a ecting the choice of a tax versus a standard. To illustrate, e ciency factors that are in favour of taxes are di erences in abatement costs among polluting rms, entry-exit conditions, and e ects on technological change. Among the factors in favour of standards are correlated marginal bene ts and costs as well as distributional e ects. Helfand (2002) also provides a discussion of standards versus taxes in pollution control. Pollution taxes essentially are incentives for reducing emissions while standards are mandatory requirements, e.g., an upper limit on the emissions of a polluting rm per time unit. Standards can also be applied to pollutive inputs. Papers which have studied the standards versus taxes problem in a dynamic game setup have focused on capital accumulation. Capital could be productive capital, abatement capital, or both. We give an example of the modeling of such a game. Example 3 Two rms decide on their investments and their use of a polluting input. Firms operate on the same output market. Environmental policy is exogenously determined by a regulator and is either a tax or an emission standard. The output q i of rm i is a function of its stock of capital K i (t) and the rate of utilization e i (t) of the polluting input. Thus q i = q i (e i (t); K i (t)): The rm s 12

13 revenue is a function of the outputs of both rms: R i = R i [q i (e i ; K i ); q j (e j ; K j )] : Firm i can invest at rate I i (t) in its capital stock for which the accumulation dynamics are dk i dt (t) = I i(t) a i K i (t); where a i is a constant and positive rate of depreciation. Investment costs are given by a convex and increasing function C(I i ); being the same for both rms. The unit price of the polluting input is p(t) which is exogenously given: The pro t function of rm i is i = Z 1 0 e rt fr i [q i (e i (t); K i (t)); q j (e j (t); K j (t))] p(t)e i (t) C(I i (t)g dt which is to be maximized, subject to the capital accumulation dynamics and - if applicable - an emission standard constraint: e i (t) e i (t) for i = 1; 2 and t 0; where e i (t) is an exogenously given emission standard. In case of a tax, rm i pays p(t) + i (t) per unit of input, where i (t) is the per unit tax. Feenstra et al. (2001) study the production capital accumulation game in the example. The governments in the countries of the two rms are not players in the game: They x tax rates or standards which act as time-dependent parameters in the optimization problems of the rms. The authors determine equilibrium strategies for input use as well as investments in production capital. For the latter, feedback strategies are found for a particular speci cation of the functional forms of demand, production and cost functions (and symmetric rms). The interesting question is whether taxes will lead to more investment than standards? Such a result is valid in a multistage setup in Ulph (1992) and the open-loop di erential game in Feenstra et al. (1996). In Feenstra et al. (2001) it cannot be concluded that taxes lead to more investment than standards when rms use feedback investment strategies. Stimming (1999a,b) also studies a duopoly capital accumulation di erential game in which each rm has two linear production activities (technologies): A clean one and a dirty one. The clean activity is less productive than the dirty. A government has the option of taxing a rm s emissions with a constant tax rate. Alternatively, rms may have tradeable emission permits the price of which is exogenously given. The author looks for open-loop as well as feedback equilibria in the noncooperative game between the two rms, given an exogenous tax scheme or permit system. In the case of open-loop strategies, and if one rm faces an emission tax and the other a tradeable permit system, a higher tax will reduce steady-state investment in the dirty technology in the rm being taxed. (The impact on the clean technology is ambiguous). For the other rm, 13

14 a higher tax with stimulate long-run investment in both technologies. Feedback equilibria are not analytically tractable. Feenstra (1998) addresses the practice of environmental dumping where, due to domestic and foreign competition and transboundary pollution, governments employ environmental policies that are laxer than those they would otherwise use. The implication is a worsening of the environment. The paper assumes that a government wishes to improve the conditions of domestic rms but also wants to have a good environmental quality. The aim of the analysis is to compare emission taxes and standards in the framework of Stackelberg di erential games, taking a cooperative joint-maximization solution as the benchmark. A government plays against an industrial sector and can employ a tax or a standard to induce the sector to internalize environmental damage. Apart from environmental damage, the government and the sector have no con ict of interest and the government can implement a rst-best solution. Feenstra et al. (2002) consider a duopoly abatement capital di erential game. Players are two rms in di erent countries and each country has a government which may potentially interfere with the rms decisions. The benchmark is the case in which a government directly controls the decisions (output and investment in abatement capital) of its respective rm. The paper compares the steady-state level of abatement capital in the benchmark case with those in cases where a government uses taxes or standards to regulate the behaviour of its rm. The model does not allow an analytical solution but numerical experiments suggest that taxes can lead to larger deviations in steady-state abatement capital levels than standards. 2.3 Subsidies Subsidies sometimes occur as negative taxes in problems of optimal taxation. One can see this as a technicality when the focus is on taxation, not subsidies. We note that in the dynamic game literature, only a few papers are concerned with subsidies as the primary environmental policy instrument. Subsidies are o ered as incentives for encouraging decision makers to take environmentally favorable actions, for instance, Firms should engage in R&D activities with the aim of developing cleaner production technologies Polluting rms or power plants should adopt existing, cleaner technologies Countries or regions should reduce the rate of deforestation of their land. The successful development of new and cleaner technologies is desirable because rms do not need to reduce their output or engage in abatement (endof-pipe emissions reduction). Carraro and Topa (1995) study an oligopolistic industry where a government introduces a tax on pollution. Firms react by decreasing output and investing in R&D to develop cleaner technologies. Investment takes place only if the tax scheme is properly designed. An interesting 14

15 question is the timing of innovation where it turns out that rms have an incentive to postpone innovations, compared to socially optimally adoption dates. A tax can induce rms to adopt cleaner technologies, but in the absence of appropriate incentives, the timing of adoption is socially suboptimal. The recommendation to the government is to subsidize the R&D costs of rms, in order not to delay innovations, and the optimal policy of the government thus is a pair of instruments (a stick and a carrot), not a single one. The paper contributes to the discussion whether a single environmental policy instrument is su cient. Katsoulacos and Xepapadeas (1996) study optimal environmental policy rules in a duopoly with pollution-generating rms (a negative externality), but such that there are R&D spillovers of environmental innovations (a positive externality). A scheme is designed that incorporates emission taxes as well as subsidies on environmental R&D. Taxes provide funding for the subsidies. Since there are R&D spillovers, the subsidy will correct the appropriability problem faced by rms that are simultaneously investing in R&D. The tax, on the other hand, corrects the pollution externality. Krawczyk and Zaccour (1996, 1999) study the problem of a local government who can subsidize economic agents located in a particular area to build their abatement capacities. Krawczyk and Zaccour (1996) analyze the environmental impact and budget implications in a context where the local government (Stackelberg leader) implements constant subsidy and tax rates. Krawczyk and Zaccour (1999) allow for time-varying tax and subsidy rates and introduce a third instrument for the government, its pollution cleaning e ort. The model does not allow for a closed-form characterization of Stackelberg equilibrium strategies and the authors design a decision-support system for the local government. This system is helpful in the assessment of the impacts on agents payo s of politically acceptable subsidy and cleaning e ort policies. The increasing deforestation of tropical forests is an environmental problem of considerable importance. Deforestation occurs due to excessive harvesting of trees for commercial use and/or because forests are cut down/burned to give space for agricultural activities. In Fredj et al. (2004) the deforestation problem is cast as a two-player di erential game where North is a set of nations who wish to have as much tropical forest as possible. The other player, South, faces a trade-o between the exploitation of its forests (timber production) and agricultural activities. The benchmark case is one in which South ignores North and solves a dynamic optimization problem. The strategic scenario is a Stackelberg di erential game in which North takes the role of a leader who o ers South a subsidy if it reduces the deforestation rate. The subsidy must depend on the deforestation rate, not be a lump-sum payment which sometimes has been suggested. In Fredj et al. (2004), the subsidy rate also depends on the size of the forest. The game is played with open-loop strategies which, to avoid time inconsistency, requires precommitment on the part of the leader. If North s budget for subsidies is su ciently large, it can design a subsidy scheme which slows down deforestation. Fredj et al. (2006) study the same problem and characterize both short-term ( nite-horizon) and in nite-horizon sustainable deforestation policies. North 15

16 can design a subsidy mechanism such that it is in the best interest of the South to follow in the short run the sustainable exploitation path of its forest. Martín-Herrán et al. (2006) also address the problem of deforestation, using Stackelberg di erential games, and introduce alternative assumptions for the information structure. The paper considers a feedback Stackelberg equilibrium and a case where the follower (South) uses a feedback strategy while the leader (North) designs the subsidy scheme as a linear, xed-coe cient incentive in the state variable (the volume of forest). The paper demonstrates that there are conditions under which the size of the forest can be increased when both players use feedback strategies. It also shows that it is not always optimal to employ a subsidy rule that depends on both the deforestation rate and the volume of forest. Martín-Herrán and Tidball (2005) consider di erent transfer mechanisms through which the donor country can subsidize the recipient country and compared those mechanisms with respect to the e ect of forest exploitation in the short as well as and the long run. 2.4 Tradeable Emission Permits Tradeable emission permits (quotas, rights) were discussed in economic theory in the late 1960s (see Dales (1968)). Tradeable permits is a market-based policy instrument that after the Kyoto Protocol of 1997 has gained increased attention as one of the most promising approaches. Koutstaal (2002) gives a brief introduction to tradeable permits in economic theory and Tietenberg (2002) discusses the practical experiences of using transferable permits for the particular case of air pollution in the United States. If one makes the strong assumption of perfect competition in product and pollution permit markets, it can be shown that tradeable permits reduce pollution in a cost-e ective way, independently of the initial allocation of permits. If the perfect competition assumption for the permits market is abandoned, rms can (and will) in uence permit prices and there will be a loss of e ciency. Carraro (2002) studies three, although static, scenarios which lead to di erent results concerning e ciency of regulatory instruments: 1. Competitive product market and imperfect permit market 2. Oligopolistic product market and competitive permit market 3. Imperfect competition in both markets. Unless product and permits markets are perfectly competitive, the initial allocation of permits is important when designing and implementing a tradeable permit system. The allocation is a matter of negotiation among the parties involved in the agreement. Two main issues need to be addressed: How should permits be designed and how should permits be allocated among the participants in a cooperative agreement? The design of a permit system is the main topic of Ahn and Kim (2001). The permit system involves multiple types of permits, not just the conventional single 16

17 type of permit. There is one type of permit for each country in the agreement and each country is allowed to issue permits of its own type. The purpose of having individual permits is to give countries a way of revealing their damage and bene t functions, through the permit prices. If a country wishes to emit one unit of the pollutant, it must acquire a permit bundle, consisting of one unit of each of the permit types. All participating countries must agree unanimously on the intended pollution. Under the multiple-type permit system, the steady-state stock and emission levels in stationary Markov-perfect equilibria coincide with rst-best outcomes. The permit system is a nancial arrangement that leads to rst-best steady-state outcomes, without preplay bargaining or commitments to emission strategies. Germain and Van Steenberghe (2003), Altamirano-Cabrera and Finus (2006) investigate various rules for allocating tradeable permits (quotas) for CO 2 emissions. Both papers combine a dynamic game model with an empirical module and argue that a cooperative agreement, based on tradeable emission permits, must be e cient, fair, individually rational, and self-enforceable. Germain and Van Steenberghe (2003) observe that any allocation of permits is likely to re ect some notion of equity. Since participation in an agreement is voluntary, an equitable rule must at least be individually rationality. If there is an equitable allocation rule, not being individual rational, the authors suggest a method that generates a new allocation which is individually rational and is as close as possible to the original, equitable rule. The model under consideration is the di erence game of Germain et al. (2003) and so is the nancial compensation system between countries. The purpose of the tradeable quotas is to ensure core-theoretic cooperation. Two types of rules are investigated (cf. Rose et al. (1998)): Allocation-based rules which apply directly to the assignment of quotas and outcome-based rules which apply to the surplus of cooperation. Allocation-based rules are, for example, the egalitarian rule (same amount of quotas per capita), the GDP (gross domestic product) rule, the Grandfathering rule (quotas allocated proportionally to historical emissions). Computations show that none of these rules satisfy individual rationality so this is where the authors method comes in. As an illustration of the results, to make the egalitarian rule individually rational requires that the United States gets three times the amount of quotas that it would receive otherwise. Under the Grandfathering rule, the result is the opposite. Altamirano-Cabrera and Finus (2006) consider a selection of allocationbased rules. The setup is a two-stage game. In the rst stage, countries decide if they will be member or not of the agreement; in the second stage, countries decide their abatement strategies. Of particular interest is the coalition structure where the notion of a partial Nash equilibrium with respect to a coalition is employed. In such an equilibrium the coalition plays a game with the remaining players who use their individual best responses. Two types of allocation schemes are studied. Pragmatic schemes in which all members in the agreement get permits that are the same percentage of some base-line emission level (business-as-usual or noncooperative Nash equilibrium). An equitable scheme allocates permits based on some normative criterion. This class of rules in- 17