Dynamic Games in the Economics and Management

Environ Model Assess DOI 10.1007/s10666-010-9221-7

Dynamic Games in the Economics and Management of Pollution Steffen Jørgensen · Guiomar Martín-Herrán · Georges Zaccour

Received: 30 March 2009 / Accepted: 6 January 2010 © Springer Science+Business Media B.V. 2010

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. Keywords 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, fisheries), 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, trans-

S. Jørgensen Department of Business and Economics, University of Southern Denmark, Odense, Denmark G. Martín-Herrán Departamento de Economía Aplicada (Matemáticas), Universidad de Valladolid, Valladolid, Spain G. Zaccour (B) Chair in Game Theory and Management, GERAD, HEC Montréal, Montréal, QC, Canada e-mail: [email protected]

portation, 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 first-best (Pareto optimal) solution. The problem is that market prices do not necessarily reflect the true social costs or benefits. 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. Externalities can be positive or negative. A negative externality is pollution generated by manufacturing firms and power plants and affecting negatively the welfare of individuals and others; firms 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 defined (oil, minerals) or it can be common property (open access) in which case a resource is owned by nobody (high-seas fishery). A great challenge lies in the fact that very serious problems of pollution do not have a specific 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 [45]. The absence of clearly defined 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.

S. Jørgensen et al.

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 affect the welfare (payoff, utility) of the agent but influence 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 first 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 firms take place over time, because externalities are intertemporal, and because the credibility and efficiency of regulating policies can only be examined over time. 3. Strategic and forward-looking behavior on the part of the agents (business firms, communities, regions, nations) who take actions that affect the environment. Conflicts 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 influence 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 socalled state-space games. A state-space game is one which contains a set of state variables that describe the main features 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 differential or difference equations called the state equations. An advantage of state-space games in applications to problems of pollution is the opportunity to model not only flow pollution damage effects (which can be accounted for in static models) but also damage caused by stocks of accumulated pollution (e.g., con-

centrations of greenhouse gases in the atmosphere or acidification of soils). For an introduction to differential and dynamic games, see Basar and Olsder [15] and Dockner et al. [57]. Apart from state variables, a dynamic game model also contains decision (or control) variables of the players. In a pollution problem, control variables of firms may be their rates of pollution emission and investments in abatement equipment. To see how the dynamics of a simple differential game of a pollution problem might look, consider the example below. Example 1 Suppose that n firms are located in a geographical area. Each firm 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 ∈ [0, T] where T is the firms’ planning horizon. The emission rate of the pollutant at time t of firm i ∈ {1, 2, ..., n} is its control variable and is denoted by ei (t) ≥ 0. Let δ ≥ 0 be a parameter. The evolution of the stock S is prescribed by the ordinary differential equation dS ei (t) − δS(t), (t) = dt i=1 n

where the first 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 first-best, Pareto optimal or ef f icient, 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 first-best solution can also be achieved if all agents agree to cooperate and decide and implement a first-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 fixed time function as her strategy. If the game is played with feedback (Markovian) strategies, actions depend on time as well as the current state. In deterministic games, the use of feedback strategies requires an assumption that players can observe the state without error.

Dynamic games in the economics and management of pollution

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 dif ferential/dif ference 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 effects. Effects 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 effects of pollution. Global pollution is caused by emissions of pollutants from countries around the world and may affect a few, many, or all countries. Examples of the latter are the greenhouse effect, the depletion of the ozone layer, and acid rain. In a differential game model, List and Mason [142] (see also [141]) 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 firms’ payoffs is local or central. List and Mason consider two scenarios. In the first, a central regulatory agency fixes uniform standards for the local jurisdictions. In the second, pollution control is decentralized such that local authorities can choose different instruments in different jurisdictions. The paper shows that decentralized control increases the joint payoff of players if there are significant asymmetries in payoffs and initial pollution levels are sufficiently 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 different shadow prices for different stocks. Environmental policy instruments can be divided into two broad groups: •

•

Command-and-control instruments. This group includes the prohibition of specific inputs, processes, and technologies, discharge permits, emission standards and quotas, as well as technological specifications 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-and-control 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 effective when the machinery to enforce regulations is weak or nonexisting at local levels. Segerson and Miceli [178] study voluntary agreements as a two-stage game. The results suggest that although voluntary agreements may offer cost savings for firms and regulators, concerns about reductions in environmental quality may be justified if, for example,

1 Command-and-control instruments require in general some monitoring effort to ensure compliance with the regulation. Before the issue of environmental regulation became popular, Filar [74] and Filar and Schultz [76] 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 single-controller, zerosum, undiscounted stochastic game. It is well known that solving such problems is hard. However, in Filar and Schultz [76], 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 [56] ask the question why market-based instruments have been less exploited and provide an explanation using a positive approach to instrument choice, based on identification of interests, conflict resolution, and pressure from politicians and bureaucrats.


subsidies are expensive and firms have considerable bargaining power. Alberini and Segerson [2] 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 identifies some main factors which could increase the efficiency of voluntary agreements.3 2.1 Taxes We start with an example of how a differential game of pollution taxation can be modeled [21]. Example 2 An oligopolistic industry consists of n identical firms producing a homogeneous good. Let the unit cost of production be constant, equal to c, and let qi (t) be the production rate of firm i. Industry output equals n qi (t) and the market price P(t) is given Q(t) = i=1 by the inverse demand function P = P(Q) such that P (Q) < 0, P(0) > c. Production generates pollution which is measured by the size of the stock S(t). A firm’s pollution emission rate ei (t) equals its production rate qi (t). The pollution dynamics are simple: dS (t) = Q(t) − δS(t), dt where δ ≥ 0 is the natural decay rate of the stock S. Firms are free to choose their output rates, but they must pay taxes. A firm’s tax bill (more precisely: tax payment rate) at time t is a function of its own emission rate ei (t) = qi (t) and the size of the overall stock S(t). The tax bill thus is Ti = T(qi (t), S(t)). Function T is the same for all firms: Equal firms are treated equally. The function T(·, ·) could, for instance, be linear in the production rate: Ti = f (S)qi . The profit function of firm i is its long-run, discounted profit: ∞ πi = e−rt {P(Q(t)) − c − f (S(t))} qi (t)dt. 0

Each firm knows that its current production will add to the future stock of pollution and thus affect its future tax payments. At time t = 0, the government announces the per unit tax rule f (S (·)) that applies to all firms at all future instants of time. The problem of the government is to find an efficient tax rule which achieves a social optimum which is one in which the

3 The

volume edited by Ten Brink [188] surveys the area of voluntary environmental agreements.

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 firms to choose production paths that are socially optimal. Dawid et al. [51] study an infinite horizon Stackelberg differential game played by a government and a number of firms.4 Firms come in two versions: Believers who take tax announcements of the government for face value and Nonbelievers who perfectly anticipate, albeit at a cost, the government’s decisions. The proportion of firms in a specific category changes over time according to the difference in the profits made of the two types of firms. A main issue of the paper is the time consistency of government policy (see the remark below). Dawid et al. [51] suggest a new solution to the problem of time inconsistency in environmental policy making: The regulator should not precommit to a tax policy, but rather make nonbinding announcements and implement taxes such that “Believers”make good profits. Remark 1 The issue of time consistency of macroeconomic policies was first raised by Kydland and Prescott [135]. 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 firms (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 defined 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 International Environmental Agreements (IEA). Time consistency of environmental policies is discussed in Batabyal [16, 17] who propose a differential game between a government agency (Stackelberg leader) and a number of polluting firms (followers). A specific issue is the possible time inconsistency of 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 twoplayer game, the first 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 honor the commitment.


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 different effects 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 [122] is used. It is shown that open-loop and consistent policies are equivalent if firms’ production costs are unrelated to the stock of pollution (the state variable). In such a case, it makes no difference whether the agency announces a tax policy at the start of the game or revises continually its policy. 2.1.1 Optimal Intertemporal Taxation Schemes Pigou [164] showed that in a problem of a single-period externality, the Pareto ef f icient pollution tax is equal to the marginal environmental damage from pollution, evaluated at the socially optimal level of emissions. This is a first-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 affect the effectiveness of environmental policy instruments, including taxes (see also [39]). A considerable part of the dynamic game literature on pollution taxes has its focus on optimal taxation schemes. The idea is to find intertemporal pollution taxation schemes that sustain Pareto efficient outcomes. Hoel [101, 102] considers a finite horizon difference game (where the trend in pollution economics seems to be an infinite 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. 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 openloop 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 conflict with Benchekroun and Long [21] and Xepapadeas [213]. Benchekroun and Long [21] (see also the above example) ask the question whether there exists a time-independent output tax rule such that polluting, oligopolistic firms will choose socially optimal produc-

tion and pollution paths. Firms play a noncooperative differential game with open-loop or feedback strategies. Which game is played affects 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 figure out which game (open-loop or feedback) the firms actually will play. Karp and Livernois [124] 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 firms’ 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 firms, the adjustment mechanism is well known. In a dynamic setting with strategically behaving firms, 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 firms will try to affect the tax rate to their own advantage. The adjustment rule is exogenous and depends on the actions of the firms. A differential game is analyzed with openloop and feedback strategies. The approach in Karp and Livernois is related to that in Conrad and Wang [46] who examine the steady-state properties of a tax adjustment mechanism in situations where government has no information about firms’ 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 CO2 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 Organization of the Petroleum Exporting Countries) and a coalition of resource-importing (and resource-consuming) countries. A primary issue here is the strategic aspects of a taxation of CO2 emissions in resource-importing countries. The following papers belong to this area: [140, 148, 171, 186, 202–205, 209, 211]. Martin et al. [148] use a dynamic game of CO2 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 benefits 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 different 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 [102], but Martin et al. assume that players’ objective functions are not the same. Wirl [204] 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 [211] extend the cartel vs. resource-importing countries model by introducing an additional objective of the government in the consuming country: Apart from the conventional externalitycorrecting 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 profits while disregarding the pollution state equation. This means that the cartel does not care about the stock externality. Confining 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 [206] combines the approach in Wirl and Dockner [211] with the internationally coordinated carbon taxes in Hoel [102]. The model is a linear-quadratic, infinite-horizon differential 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 [203] considers the dynamic interactions between resource suppliers and consumers in a model with a flow externality (e.g., acid rain) and stock externality (e.g., global warming). Wirl [209] 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 [186] extends the model of Wirl [203] 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 differential game 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 [205] considers a dynamic game between a cartel of resource-producing and resource-exporting countries and a coalition of resource-importing and resource-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 differential game with two state variables. An analytical solution is possible only in a simplified case in which the state variables are identical. The paper extends the setup in [203] by allowing for depreciation of the stock of pollution and resource extraction costs that increase by the quantity already extracted. Rubio and Escriche [171] use the model of Wirl [205] 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 profits? 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 profits in a feedback Stackelberg equilibrium. Liski and Tahvonen [140] also study a differential game being close to the one in [205] and [171] (see also [186]). The objective of Liski and Tahvonen is to design an optimal carbon tax in the cartel vs. coalition differential 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-tariff component which shifts rents from the cartel to the coalition. Because of the trade policy component, the optimal carbon tax is in general different for the neutral Pigouvian tax (cf. [171]) which internalizes the damage cost of pollution (i.e., the stock externality) only.


Yanase [219] 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 differential 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 [221] and Yeung and Cheung [224] study the capital accumulation problem of a firm (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 differential game with state equations for capital and pollution accumulation. Open-loop and feedback equilibria are considered, and it is shown that the steadystate levels of capital and pollution are the highest in open-loop equilibrium. Extending the setup in [224], Yeung [222] considers a differential game in which the depreciation of the productive capital stock is not constant but increases with the stock of pollution. 2.1.2 Nonpoint Source Pollution In point source pollution (e.g., industrial or municipal emissions), the source, volume, and characteristics of discharges can be identified by regulators with reasonable precision and cost. In nonpoint source pollution (e.g., agriculture, vehicles), neither the source nor the volume of individual emissions is observable by regulators. The regulator can measure the ambient pollution at specific points but is unable to assign a particular proportion of the pollutant level to a specific 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 input-based schemes that tax observable, polluting inputs. Xepapadeas [216] provides a brief survey of nonpoint source pollution problems; Shortle et al. [180] 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 [213] 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 differential equation and a stochastic diffusion 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 profit-maximizing firms 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 firms. If they use feedback strategies, the tax needs to be higher than if firms 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 firms, the damages caused by accumulation of pollutants, and the type of strategies that firms will employ (cf. [21] 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 [123] notes that an ambient tax can sustain a social optimum, given that firms recognize that their decisions affect the aggregate emission level. This means that an ambient tax makes sense only in situations where firms behave strategically and recognize that they may affect the tax rate. The incentive to do so is 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. [213]). A surprising result is that even if the regulator has perfect information about firms’ emissions, they might get a higher payoff if the regulator acted as if it were unable to observe individual emissions! Note that an ambient tax has two limitations: Polluting firms must be large enough to realize that their emissions affect the aggregate pollution level and the tax may result in large transfers to/from a firm. 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


efficient income generator for the government. This effect has been called the double-dividend hypothesis and is discussed in [52]. 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 define 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 ([100], p. 231) provides a summary of factors affecting the choice of a tax versus a standard. To illustrate, efficiency factors that are in favor of taxes are differences in abatement costs among polluting firms, entry-exit conditions, and effects on technological change. Among the factors in favor of standards are correlated marginal benefits and costs as well as distributional effects. Helfand [100] 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 firm 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 firms 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 qi of firm i is a function of its stock of capital Ki (t) and the rate of utilization ei (t) of the polluting input. Thus qi = qi (ei (t), Ki (t)). The firm’s revenue is a function of the outputs of both firms: Ri = Ri qi (ei , Ki ), q j(e j, K j) . Firm i can invest at rate Ii (t) in its capital stock for which the accumulation dynamics are dKi (t) = Ii (t) − ai Ki (t), dt where ai is a constant and positive rate of depreciation. Investment costs are given by a convex and increasing function C(Ii ), being the same for both firms. The unit

price of the polluting input is p(t) which is exogenously given. The profit function of firm i is πi =

∞

e−rt {Ri qi (ei (t), Ki (t)), q j(e j(t), K j(t))

0

− p(t)ei (t) − C(Ii (t)}dt which is to be maximized, subject to the capital accumulation dynamics and—if applicable—an emission standard constraint: ei (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, firm i pays p(t) + τ i (t) per unit of input, where τ i (t) is the per unit tax. Feenstra et al. [68] study the production capital accumulation game in the example. The governments in the countries of the two firms are not players in the game: They fix tax rates or standards which act as timedependent parameters in the optimization problems of the firms. 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 specification of the functional forms of demand, production, and cost functions (and symmetric firms). The interesting question is whether taxes will lead to more investment than standards? Such a result is valid in a multistage setup in [194] and the open-loop differential game in [70]. In [68], it cannot be concluded that taxes lead to more investment than standards when firms use feedback investment strategies. Stimming [183, 184] also studies a duopoly capital accumulation differential game in which each firm 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 firm’s emissions with a constant tax rate. Alternatively, firms 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 firms, given an exogenous tax scheme or permit system. In the case of open-loop strategies and if one firm 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 firm being taxed (the impact on the clean technology is ambiguous). For the other firm, a higher tax with stimulate long-run investment in both technologies. Feedback equilibria are not analytically tractable.


Feenstra [67] 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 firms 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 differential 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 conflict of interest and the government can implement a first-best solution. Feenstra et al. [69] consider a duopoly abatement capital differential game. Players are two firms in different countries and each country has a government which may potentially interfere with the firms’ decisions. The benchmark is the case in which a government directly controls the decisions (output and investment in abatement capital) of its respective firm. 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 behavior of its firm. 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 offered 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 firms 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 firms do not need to reduce their output or engage in abatement (end-of-pipe

emissions reduction). Carraro and Topa [43] 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 question is the timing of innovation where it turns out that firms have an incentive to postpone innovations, compared to socially optimally adoption dates. A tax can induce firms 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 firms, 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 sufficient. Katsoulacos and Xepapadeas [126] study optimal environmental policy rules in a duopoly with pollutiongenerating firms (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 firms that are simultaneously investing in R&D. The tax, on the other hand, corrects the pollution externality. Krawczyk and Zaccour [133, 134] 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 [133] 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 [134] allow for time-varying tax and subsidy rates and introduce a third instrument for the government, its pollution cleaning effort. 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’ payoffs of politically “acceptable” subsidy and cleaning effort 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 [80], the deforestation problem is cast as a twoplayer differential 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-off 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 differential game in which North takes the role of a leader who offers 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 [80], 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 sufficiently large, it can design a subsidy scheme which slows down deforestation. Fredj et al. [81] study the same problem and characterize both short-term (finite-horizon) and infinite-horizon sustainable deforestation policies. North 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. [149] also address the problem of deforestation, using Stackelberg differential 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, fixedcoefficient 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 [150] consider different transfer mechanisms through which the donor country can subsidize the recipient country and compared those mechanisms with respect to the effect 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 [48]). 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 [129] gives a brief introduction to tradeable permits in economic theory, and Tietenberg [191] discusses the practical experiences of using transferable permits for the particular case of air pollution in the USA.

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-effective way, independently of the initial allocation of permits. If the perfect competition assumption for the permits market is abandoned, firms can (and will) influence permit prices and there will be a loss of efficiency. Carraro [39] studies three, although static, scenarios which lead to different results concerning efficiency 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 permit 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 [1]. The permit system involves multiple types of permits, not just the conventional single 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 benefit 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 first-best outcomes. The permit system is a financial arrangement that leads to first-best steady-state outcomes, without preplay bargaining or commitments to emission strategies. Germain and Van Steenberghe [84] and AltamiranoCabrera and Finus [5] investigate various rules for allocating tradeable permits (quotas) for CO2 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 efficient, “fair”, individually rational, and selfenforceable.


Germain and Van Steenberghe [84] observe that any allocation of permits is likely to reflect some notion of equity. Since participation in an agreement is voluntary, an equitable rule must at least be individually rational. If there is an equitable allocation rule, not being individually 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 difference game of Germain et al. [83] and so is the financial compensation system between countries. The purpose of the tradeable quotas is to ensure core-theoretic cooperation. Two types of rules are investigated (cf. [166]): allocation-based rules which apply directly to the assignment of quotas and outcomebased rules which apply to the surplus of cooperation. Allocation-based rules are, for example, the egalitarian rule (same amount of quotas per capita), the gross domestic product (GDP) rule, and 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 USA 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 [5] consider a selection of allocation-based rules. The setup is a two-stage game. In the first 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 includes, for instance, the egalitarian rule. The schemes are assessed by making numerical calculations in the empirical module. As in [84], the grand coalition is not dynamically stable because individual rationality cannot be satisfied. It seems that pragmatic schemes are superior to the equitable ones. The papers by Haurie and Viguier [95] and Bernard et al. [22] are concerned with the trading of carbon emission permits. Russia and China are players in a Cournot-type duopoly difference game in which there

is one state variable for each player, namely the stock of emission permits banked. Stocks can be replenished through abatement activities. Haurie and Viguier [95] employ a stochastic setup and use the concept of an S-adapted equilibrium.5 Uncertainty about the timing of China’s entry on the permit market affects the behavior of Russia which banks less permits in the short run and more in the long run. The supply of permits from Russia increases if it is highly probable that China will not enter. Bernard et al. [22] include Annex B countries in the game and look for an open-loop Nash equilibrium in a deterministic setup. Despite an open-loop equilibrium may be conceptually unappealing, it has a number of advantages to the modeler, e.g., results about the existence and uniqueness of equilibrium. A numerical approximation of the equilibrium can be found by using methods from mathematical programming. Simulations suggest that competition between Russia and China significantly lowers permit prices compared to the situation in which Russia acts as a monopolist. The next two papers deal with joint implementation of environmental projects, one of the flexible mechanisms of the Kyoto Protocol. Breton et al. [32] consider a two-player finite-horizon differential game of joint implementation where three scenarios are contrasted, namely (a) a noncooperative, business-as-usual game where players do not face constraints on emissions, (b) a noncooperative game where each player has an emission target at horizon date, and (c) a joint implementation game in which players can invest abroad in environmental projects and get emissions credit units that apply to their target constraint. Assuming linear damage costs, it is shown that only one country invests abroad at a given instant of time, the investor increases its emissions at home, the mechanism leads to higher total payoffs of the countries, and in some instances is Pareto-improving. In the follow-up paper [29], each player faces a nonlinear damage cost which may be more in line with empirical studies. The authors conclude that the impact of the availability of the joint implementation option on local pollution and investments cannot be described in general; it depends on the specific values of the model parameters.

5 In

such an equilibrium, the players use the S-adapted openloop information, i.e., at any time, each player’s information set includes the current calendar time, the current state, and the distribution of future demand (or other variables of interest). This concept has been introduced in [99] and further developed in [98].


2.5 Assessment of Policy Instrument Effects A major concern in dynamic pollution control is how policy instruments “work”. The issue has a theoretical and an empirical side, but in any case, one needs criteria for judging and comparing various policy instruments. Traditionally, assessments of instruments were made in static settings where efficiency criteria seem to work nicely. To illustrate, in a static setup, Jung et al. [114] provide the following unambiguous ranking of instruments: auctioned permits, emission taxes and subsidies, marketable permits, and performance standards. The ranking of instruments is invariant with respect to firm size, industry size, and industry abatement cost structure. A discussion of the assessment literature is provided by Russell and Powell [174] who note that none of the work which they survey is really dynamic: Time may be explicit, but decision makers do not take into account that current decisions affect future decision situations. Russell and Powell discuss the problems that need to be addressed when assessments are done in dynamic settings. The authors note the many complications that arise and need to be addressed. For example, the set of pollution sources changes over time and production technology as well as pollution control technologies change. The authors conclude, in our opinion not rightly, that economists faced with such complications have not built the dynamic models and made the analysis but have resorted to ‘qualitative’ statements. Our “proof” lies in the survey at hand. An important question in policy assessment is, as already mentioned, whether a single instrument might suffice to correct an externality or whether the situation requires that two or more instruments are simultaneously employed. To answer this question, one should be aware of the significance of the market structure. Carraro [39] writes: “If the market structure is oligopolistic, no general conclusion about the effects of environmental policy can be derived, because the presence of multiple market externalities, both positive and negative, makes the use of a single policy instrument designed to correct for the environmental externality largely suboptimal” ([39], p. 245; see also [33]). Xepapadeas [215] is concerned with global warming and notes, given the uncertainties involved, that two questions need to be answered. First, what is an optimal policy in terms of reductions of emissions and, second, when should the policy be implemented? The author uses the theory of optimal stopping time (from stochastic optimal control theory) to analyze optimal emission reduction policies under stochastic marginal damages. Xepapadeas’ method designs a policy func-

tion and using this function, the decision maker should implement the reduction policy once the marginal damages reach a critical level. A cooperative problem and a noncooperative game with linear Markovian strategies are analyzed. The general result from deterministic dynamic games that each country emits more in the noncooperative solution than in the cooperative one is valid also here, for any realization of the random variable.

3 Transboundary Pollution Transboundary pollution problems (TPP) are environmental problems involving more than one independent jurisdiction.6 TPP includes (a) unidirectional or downstream pollution problems and (b) international and global pollution problems. In the former, a firm/region/country emits pollutants that damage the environment of one or more neighbors. Examples of unidirectional pollution include (1) an upstream plant or a farmer using chemicals or fertilizers that pollute a downstream country/region and (2) acid rain generated by the industrial activities in one country that pollute the soil of a neighboring country. In international or global pollution problems, many or all nations generate emissions, and many or all of them are suffering from them. A celebrated example is the emission of greenhouse gases (GHG) that cause global warming. One basic element of TPP is the absence of a transnational institution that can impose an environmental policy. Given this, an environmental solution can be implemented only if the parties involved in the TPP agree to the solution. The papers dealing with TPP are discussed under three headings: 1. Noncooperative and cooperative solutions. The main focus here is on the determination and the comparison of the two types of solutions. We also deal with noncooperative dynamic games that involve common environmental constraints. 2. International Environmental Agreements. We consider noncooperative and cooperative game approaches to the issue of determining stable environmental agreements among nations. 3. Empirical dynamic games of TPP. We report on papers that have estimated, with real (or realistic) data, the parameters of the models and the outcomes of games.

6 Missfeldt

[153] reviews game-theoretic modeling of TPP.


The categorization is not unambiguous and many of the surveyed papers fall in more than one category. Alternative classifications could have been based on the type of situations (unidirectional vs. global pollution problems), the mode of play (cooperative or noncooperative), etc. Any categorization has some arbitrariness, but we feel that the above is useful in giving an exposition of the contributions. 3.1 Noncooperative and Cooperative Solutions Van der Ploeg and de Zeeuw [197] were among the first to characterize and contrast noncooperative and cooperative pollution strategies and outcomes in a transboundary pollution context (see also [196]). They consider a world made of N identical countries engaged in production activities (which are control variables) that generate pollution which accumulates over time according to the simple dynamics: N dS α Yi (t) − δS (t) , (t) = dt N i=1

S (0) = S0 ,

where Yi (t) is the production rate of country i, i = 1, . . . , N, at time t, S(t) is the stock of pollution, and δ ≥ 0 is the constant decay rate of the stock of pollution. Country i derives profits from production, measured by a concave function B(Yi ), and incurs a damage cost D(S) due to pollution; D (S) > 0, D (S) ≥ 0. Player i (i.e., the government of country i) maximizes the welfare function ∞ e−rt (B (Yi (t)) − D (S (t))) dt, Wi = 0

subject to the pollution stock dynamics; r > 0 denotes the social rate of discount. The authors determine environmental charges in two scenarios. In the first, there is international coordination of emissions and governments choose their production levels {Y1 (t), . . . , Y N (t), t ≥ 0} in order to N maximize their joint payoff i=1 Wi . In the second, each government chooses its emissions without any consideration of the adverse effects on the other countries, and open-loop and feedback Nash equilibrium strategies are characterized. The major conclusions are that production and emission levels under international coordination are lower than in the noncooperative case, and open-loop equilibrium leads to lower production and emissions levels than in the feedback equilibrium. Long [143] analyzes a two-player transboundary differential game. Each player aims at maximizing a stream of welfare which depends positively on consumption and negatively on the stock of pollution. The

author considers three solutions: joint maximization of welfare, open-loop Nash equilibrium, and openloop Stackelberg equilibrium. As expected, the joint optimization solution gives lower pollution than the two noncooperative ones. The interesting results of the paper are that the steady state of the pollution stock under Stackelberg leadership is higher than in the Nash equilibrium and that the leader emits more and the follower less than in Nash equilibrium. An open-loop Stackelberg equilibrium is in general time inconsistent, and it makes sense to use it only if the leader makes a binding commitment. Motivated somehow by the above observations on open-loop equilibria, Dockner and Long [58] reconsider the model of Long [143] and analyze feedback Nash equilibria along with the cooperative solution. They assume a linear-quadratic structure of the differential game because then an informed guess is that feedback strategies are linear in the state. As expected, the cooperative joint maximization solution outperforms, in terms of total payoff and pollution stock, the feedback Nash equilibrium. However, the most interesting results in the paper lie elsewhere. The authors prove the existence of other feedback equilibrium strategies that are nonlinear in the pollution stock and, remarkably, show that if the players have a low discount rate, then the cooperative solution can be sustained as an equilibrium outcome with nonlinear strategies. Rubio and Casino [169] show that for the Dockner and Long result to hold, it is necessary that the initial value of the stock of pollution be higher than the cooperative solution stock. On the same topic, Wirl [208] shows that the existence of multiple equilibria does not necessarily lead to more conservation. Wirl [210] extends the analysis of this issue to a stochastic differential game setting (see also [207]) and characterizes equilibrium strategies when pollution is reversible (through, e.g., cleaning-up activities) and when it is not. The author shows that increasing uncertainty reduces pollution and irreversibility of pollution accentuates this reduction. Xepapadeas [214] analyzes a differential game of pollution control where emissions per unit of output can be reduced through technical progress in production processes. The author determines a global international optimum where each country contributes to a common (or global) R&D effort. Next, open-loop and feedback Nash equilibria are characterized under two different assumptions on investments in R&D at the country level. In the first, each country optimizes its individual welfare while contributing to a global R&D effort, whereas in the second, each country has its own


R&D stock. Noncooperation leads to lower provision of R&D and higher emissions. It is also discussed how R&D subsidies and emissions taxes can lead to the implementation of a first-best outcome. Dockner and Nishimura [59] analyze a discrete-time, infinite-horizon dynamic game of transboundary pollution and introduce three scenarios of how consumers in a country are affected by the pollution of other countries. In the first, countries are located on a circle and consumers in each country suffer from their own pollution and from that of one neighboring country (downstream/unidirectional pollution). In the second, consumers incur costs from all stocks of pollution in the world economy. Finally, there is a scenario where the individual cost depends on the totality of pollution stocks. The paper characterizes Markov-perfect equilibria for the three scenarios. An often made assumption in dynamic pollution games is that a player’s objective function depends directly on his own production activities and emissions only. Yanase [218] deviates from this assumption by supposing that the objective function of a player depends on all players’ emissions. Moreover, there is a within-period economic interaction between the players (due to a change of the terms of trade) in addition to the long-term environmental interaction. In the absence of within-period interaction, the author obtains the standard result that cooperation leads to a lower steady-state pollution stock. When the interaction is considered, Yanase shows the surprising result that cooperation may actually lead to a higher steady-state stock of pollution than its noncooperative counterpart. Further, the author designs trigger strategies that can support the cooperative outcome as an equilibrium of a noncooperative game. Differential games in environmental economics literature have often been of the linear-quadratic variety, mainly for tractability reasons. Kossioris et al. [128] propose a numerical method to derive nonlinear feedback Nash equilibria for differential games that are not linear-quadratic. An application to a shallow lake pollution game is provided. 3.1.1 Noncooperative Games with Coupling Constraints Haurie [87] and Haurie and Zaccour [97] consider the problem of n players who face a common constraint, e.g., that their total emissions must not exceed a prescribed value. An international agency is in charge of coordinating the policies of the countries by means of taxes or emissions permits, with the aim to achieve a long-run sustainable and efficient development. Each player reacts dynamically and individually to the co-

ordinating actions of the agency. The presence of the common, or coupling, constraint implies that an equilibrium cannot be considered as a fully noncooperative solution; in any event, it needs a special treatment. Rosen [167] was the first to deal with the problem of determining coupled-constraint Nash equilibria. To see how it works in a static setting, denote by ei the emission level (control variable) of player i and denote by Ei the set of admissible actions. The n players must satisfy the common constraint (e1 , . . . , en ) ∈ E ⊂ E1 × . . . × En .

(1)

An example of such a constraint is n

ei ≤ e¯,

(2)

i=1

where e¯ is a prescribed total emission level. Let πi (e1 , . . . , en ) be the payoff of player i. A coupled equilibrium is defined as an n-tuple (e∗1 , . . . , e∗n ) ∈ E such that πi e∗1 , . . . , e∗n ≥ πi e∗1 , . . . , ei , . . . , e∗n ∀ei ∈ Ei ∗ e1 , . . . ei , . . . e∗n ∈ E, i ∈ {1, . . . , n} . Each player may consider unilateral actions that keep the common constraint satisfied. If E is defined by a set of inequality constraints and under some regularity conditions, there is for each player i a Karush–Kuhn– Tucker multiplier λi associated with the constraint (1). Rosen used the term “normalized equilibrium” for a coupled equilibrium in which multipliers satisfy λi =

λ0 , i = 1, . . . , n, ri

where λ0 ≥ 0 is a given vector and ri > 0, i = 1, . . . , n, are given weights. Under a concavity condition, Rosen showed that there exists a unique normalized equilibrium associated with each positive weighting vector r = (r1 , ..., rn ). In the simple context of a coupling constraint as the one in (2), the vector r is an indication of how the regulator has distributed the burden of satisfying the constraints (e.g., the taxation of emissions) among the players. Haurie and Krawczyk [92] apply Rosen’s approach to design a Pigouvian effluent tax by a regional authority of economic agents located in a river basin. The paper offers interesting clues to how to model lumped and distributed pollution and how to tax players in order to induce cooperation to satisfy a common constraint. Krawczyk [130] deals with static as well as dynamic games of a river basin pollution problem and provides a nice economic interpretation of the Rosen formalism.


Drouet et al. [60] show how to formulate and solve numerically a noncooperative game which intends to model climate negotiations to reach a post-Kyoto agreement. The starting point is that countries agree on a coupled constraint in terms of carbon concentrations not exceeding a certain maximum level in 2050. The time interval is divided into two commitment periods, namely 2000–2025 and 2025–2050, with the assumption that by the end of the first period, the uncertainty of the climate sensitivity to emissions would be revealed. The paper provides a series of numerical simulations to help decision makers in negotiations. Bahn and Haurie [9] deal with the design of equilibrium solutions with coupled constraints in dynamic games of greenhouse gas emissions abatement. An international agency can impose on countries a coupled constraint on the total emissions allowed over the twenty-first century, or on the concentration of carbon reached at the end of the century. Interestingly, it is shown that the normalized equilibrium is close to a Pareto-optimal solution. The relationship between Pareto optimality and the weights in a Rosen equilibrium has been investigated in a static context in Tidball and Zaccour [189] who show that a Pareto solution can indeed be attained by a suitable choice of these weights. Tidball and Zaccour [190] show that the result does not carry over to a dynamic game. For a survey of numerical solutions to normalized equilibrium problems, see [131], and for their computations, see [132].

3.2 International Environmental Agreements During the last two decades, a significant literature has focused on the design of IEA. The literature has adopted one of two approaches: 1. A noncooperative game, justified by the argument that participation in an IEA is inherently voluntary, in view of the fact that there is no transnational institution that can enforce such agreements. The main idea is to search for mechanisms that, when implemented, will lead to the largest possible and stable coalition of countries that adhere to the IEA because it is in their best interest to do so. 2. A cooperative game, based on the premise that the coordination of emissions of all countries leads to the best environmental and economical outcomes. Once such an outcome is determined, the next step is to find allocations of the joint burden that guarantee satisfaction of properties like intertemporal stability of the agreement and fairness.

3.2.1 Noncooperative Game Approach to IEA Following d’Aspremont et al. [50] who was interested in the stability of a cartel, this branch of literature determines an equilibrium membership of an IEA by invoking two conditions. Internal stability means that no member has an interest in leaving the agreement; external stability means that no nonmember wishes to join the agreement. A typical contribution in this area adopts a twostage game formalism where countries decide in the first stage to join the IEA or not. This is called the membership game. In the next stage, they decide their emissions (called the emissions game).7 Using these two conditions has led to the conclusion that successful cooperation, i.e., creating an IEA with many members, is in general hard to achieve. Note that this theoretical result contrasts with the high level of participation observed in some real-life agreements, e.g., the Montreal Protocol.8 In an attempt to explain this discrepancy, a series of different elements have been incorporated in models: • • • • • •

Stackelberg leadership [11, 55, 172] Transfers [40, 103] Reputation effects [36, 103, 106] Issue linkages [12, 26, 41, 125, 127, 139, 154] Punishments [12, 13] Regional rather than global agreements [7]

Other references on the use of noncooperative game theory for creating an IEA are [42, 65, 77, 78, 201]. In all the above references, the game is static. Breton et al. [31] point out that static games applied to pollution emissions and IEAs may be criticized on two important grounds. First, the stock externality “transboundary environmental damage” is mainly related to the accumulation of pollution rather than the flow of emissions. Second, in a static setup, there is no room for countries to reconsider their participation in an IEA which is particularly relevant in circumstances where environmental damage changes over time. Noncooperative dynamic games applied to IEAs are still in their infancy, with only few published papers that include the participation decision as an element of the game. Rubio and Casino [170] analyze a game where the decision whether to join the agreement is

7 The papers cited in this section do not deal with the negotiation process itself. For an example of the modeling of this process, the interested reader may consult Caparrós et al. [37]. 8 The

Montreal Protocol on Substances that Deplete the Ozone Layer was signed in 1987 and at present it counts 191 nations.


taken once and for all, and signatories and nonsignatories select their emission strategies in an infinitehorizon differential game. Numerical results show that the stable size of an agreement is two. When requiring a minimum number of signatories for the agreement to be in force, a stable agreement has precisely the same size as in the minimum clause. In the Rubio and Casino [170] paper, the dynamics of the pollution stock affect the choice of emission strategies, but not the IEA membership. Rubio and Ulph [173] correct for this. In a discrete-time dynamic game setting, they assume that n symmetric players in each period solve an emission game and a membership game such that the number of signatories changes over time with the stock of pollution. Under the symmetry assumption, the authors can only determine the number of signatories. To remedy this, it is assumed that signatories are randomly selected such that the stability concept is equality of welfare of signatories and nonsignatories. Rubio and Ulph show the existence of a unique steady state of the pollution stock and a corresponding steady state size of a stable IEA. As in static games, the number of signatories in a stable agreement is a nonincreasing function of the pollution stock. De Zeeuw [54] extends the idea of a farsighted stable agreement that has been used in a static context to a dynamic setup. Farsightedness means that when a player computes her gains from joining or leaving an agreement, she takes into account that her decision may incite other players to change their membership status. Static game literature has demonstrated that under farsightedness one may obtain both small and large stable coalitions. De Zeeuw shows that this result extends to a dynamic game only if the costs of emissions are sufficiently small compared to abatement costs.9 Breton et al. [30] consider a symmetric discrete-time game and adopt a replicator dynamics. This produces evolutionary pressures in favor of the group which obtains the highest payoff. In the spirit of evolutionary games, the group that achieves a better result is joined by a fraction of new players. The adjustment speed at which countries switch to the superior strategy reflects the “psychological” or “physical” cost of changing behavior. The evolutionary process ends in an IEA which is stable over time and at the steadystate pollution stock. In this process, the evolution of players’ welfare over time depends not only on the dynamics of emissions and pollution but also on the

9 Note

that the model of de Zeeuw, somewhat unconventionally, uses emissions as the state variables.

evolution of the composition of the different groups. In [8], the authors calibrate the model’s parameters of [30] using the MERGE climate policy assessment model and provide numerical illustrations. They also review briefly the cooperative and noncooperative approaches to define stability of IEAs. 3.2.2 Cooperative Game Approach to IEA In a cooperative dynamic game approach to IEA, the starting point is the grand coalition (that is, all n countries forming one coalition). A main objective is to identify mechanisms that guarantee the intertemporal stability/sustainability of the grand coalition. Haurie [86] shows that an agreement which is individually rational at the start of the game, t0 ,, may fail to be so till its maturity date T. He distinguished between two possible reasons for an instability: •

•

If players agree to renegotiate the original agreement at some intermediate instant of time τ ∈ (t0 , T], it is not sure that they would wish to continue with the agreement. In fact, they will not if the original agreement fails to be a solution of the cooperative game that starts out at time τ . Suppose that a player considers to deviate from the agreement, that is, as of time τ ∈ (t0 , T] she will use a strategy different from the one prescribed for her by the cooperative solution. A player should deviate if it gives her a payoff in the continuation game that is greater than the one she stands to receive through continued cooperative play.

Folmer et al. [79] and Missfeldt [153] discuss the problem of free-riding on agreements on pollution reduction. By deviating from its part of the cooperative agreement, an agent (group of agents) can do better. For example, free-riders could save on abatement expenditures, relying on other agents will pay their abatement expenditures according to the agreement. An agreement is sustainable if no player or subset of players would wish to deviate from their prescribed parts of the cooperative, agreed strategy. A sustainable solution is also called a dynamically stable solution. A key question is: Which approaches and instruments could make an agreement dynamically stable? The literature seems to have followed two paths: 1. Cooperative equilibrium approach. The idea is make a cooperative solution an equilibrium of an associated noncooperative game. Once the solution has been agreed, there is no need for further communication or cooperation because, due to the equilibrium property, no player has incentive to de-


viate from the agreement. The agreement is said to be self-enforcing. To achieve this, various methods are available. One method applies trigger strategies. A trigger strategy conditions a player’s action on the history of actions and letting players use such strategies, a cooperative solution can be made an equilibrium. The natural setup for trigger strategies is a repeated game where the “Folk Theorem” shows that cooperation can be sustained if players have a sufficiently low discount rate. In differential games, the application of trigger strategies causes some technicalities (see, e.g., [57]). Cesar [45] gives an application of trigger strategies in a differential game of the greenhouse effect (see also [91, 94, 116, 192]). A second method is to design incentive strategies. This strategy concept originates in Stackelberg games where the leader wishes to design an incentive for the follower such that the latter behaves in a way which is desired by the leader (or by both players). An incentive strategy of a player conditions the player’s action on the action of the other player. An equilibrium has the property that when one player implements his strategy, the other player can do no better than to act in accordance with the agreement (see, e.g., [31, 62–64, 111, 151, 152]). Yet another idea is unilateral actions where an agent (or group of agents) takes a certain action to show a “good example” to other agents and lead them to take similar actions. Zagonari [229] uses a two-player, linear-quadratic differential game to study the question whether unilateral actions to reduce pollution, advocated by environmentalists in one country, may result in higher total emissions. When both countries use linear feedback strategies in a noncooperative game and one country does not care very much about the environment, the equilibrium involves less pollution than the cooperative solution. To establish these results, one needs to assume that the other country has great concern for the welfare of future generations and has low bargaining power. 2. Dynamic rationality. The idea is to try to sustain a cooperative agreement over time by appropriate side payments. As already said, a cooperative agreement (e.g., core, Shapley value, Nash bargaining solution) is time consistent if, at no instant of time, no player or group of players wishes to defect on the agreement. Time consistency is verified along the cooperative state trajectory; the assumption is that players have cooperated so far. A stronger condition is used in the concept of agreeability where the verification is made along

any state trajectory [117]. Time consistency and agreeability in pollution control differential games has been dealt with in [108, 109, 112, 118, 163, 223, 225, 226, 228]. Time consistency or agreeability can be achieved by using side payments. In a differential game, such payments are transferred continually among the parties of the agreement and can be based on the total side payment as determined by one of the classical solutions of a cooperative game (core, Shapley value, Nash bargaining solution). Jørgensen and Zaccour [110] propose a decomposition over time of the total side payment as determined by the Nash bargaining solution to sustain cooperation in a downstream pollution game. Germain et al. [83] and Jørgensen [107] use the core solution and apply a particular side payment scheme to ensure individual and coalitional rationality throughout the game. The use of side payments in environmental agreements has been criticized on various grounds. Side payments may induce countries to use “minimal” environmental policies in anticipation of large side payments in the future and countries offering side payments may weaken their negotiation power. Moreover, side payments imply the “victim pays” principle, not the “polluter pays” principle. In what follows we report on papers which have adopted a cooperative game approach to design bilateral or multilateral environmental agreements. Bilateral Environmental Agreements Jørgensen and Zaccour [110] study a differential game where industrial activities of an upstream country pollute a country located downstream. The aim of the paper is to suggest a side payment rule that allocates over time the surplus gained if players coordinate their downstream pollution control policies. The rule allocates to each country at each instant of time its share of instantaneous dividend of cooperation, based on the egalitarian principle, plus a term that depends on the imputed value of the deviation between the cooperative and noncooperative trajectories of the stocks of pollution and abatement capital. It is shown that side payments always flow from the polluted (vulnerable) country to the nonvulnerable one. Thus, it is more efficient for the vulnerable country to “buy” a reduction of emissions upstream rather than to suffer from its damage. The conditions under which cooperative outcomes are time consistent are discussed. Jørgensen and Zaccour [111] consider two neighboring countries that implement a cooperative solution by using linear incentive strategies . As mentioned, the idea of incentive strategies is that player i designs a strategy which depends on player j’s control such that


if player j implements her cooperative decision, then player i selects her cooperative decision. If not, player i modifies her decision by an amount which is linear in the deviation of player j’s decision from her cooperative decision. The paper provides a decomposition over time of cooperative payoffs such that at each instant of time, the individual cooperative payoffs are not lower than their noncooperative counterparts. This property, which could be termed instantaneous individual rationality, is much stronger than “standard” individual rationality which is defined in terms of payoffs-to-go. Cabo et al. [34] achieve the same result as Jørgensen and Zaccour [111] in a context of two countries that are trading an intermediate good and share an environmental concern. Cabo et al. characterize and contrast two equilibria, feedback and open-loop, and a cooperative solution. In the latter, players reduce their production and trade but still enjoy a higher total cooperative payoff thanks to an environmental gain. Multilateral Environmental Agreements The standard approach of coalitional cooperative games is to determine the characteristic function values for all coalitions and then allocate the grand coalition’s total payoff among the players. The emissions that are allowed under the IEA are those obtained by minimizing the total payoff of the grand coalition. Although these emissions are collectively optimal, they are not necessarily dynamically individually rational. Thus, it may happen that some players obtain a lower payoff by joining the IEA than by free riding, either at the initial date or at a future date. One of the first papers to address the formation of an IEA using a cooperative game approach is Filar and Gaertner [75] who use the Shapley value to allocate the total emissions in each period among the four players involved in the game (Organization for Economic Cooperation and Development (OECD), Former Union of Soviet Socialist Republics and East Europe, China and C.P. Asia, Rest of the World). Emissions are obtained from the coupled economic/greenhouse environmental optimization model Optimization Model for Economic and Greenhouse Assessment (OMEGA). OMEGA couples the Dynamic Integrated Model of Climate and Economy (DICE) model [158]) and the global climate model Integrated Model to Assess the Greenhouse Effect, due to Rotmans [168]. Trade flows between regions are used to measure the strength of all possible coalitions and to define their characteristic function values. Thus, characteristic function values are determined by exogenous data. The aim of the paper is to provide some preliminary hints to what could be a fair sharing of total emissions, possibly to be entrenched in a future IEA (see also [82]).

Germain et al. [83] and Petrosjan and Zaccour [162] consider the problem of n countries wishing to coordinate their emissions strategies to minimize their total cost. Germain et al. [83] propose a transfer mechanism to sustain cooperation over time having the properties of being individually and coalitionally rational in the core-theoretic sense. Petrosjan and Zaccour [162] allocate the total cost using a timeconsistent decomposition over time of each player’s cost as determined by the Shapley value. The latter has the property of being fair. In an example, it is shown that the Shapley value belongs to the core. Apart from the difference in the solution concept, the two papers differ in their definition of the characteristic function. Germain et al. use the γ -characteristic function which assigns a value to a coalition being the coalition’s Nash equilibrium outcome in a noncooperative game with the left-out players. Petrosjan and Zaccour assume that left-out players maintain their emissions at their equilibrium levels in a fully noncooperative game. This approach has a drawback since that one expects that left-out players will react strategically to the formation of a coalition. However, the Petrosjan–Zaccour approach requires solving only one equilibrium problem; all other problems are optimization problems. Using the concept of a γ -characteristic function, one needs to solve one optimization problem (for the grand coalition) and 2n − 2 equilibrium problems.10 This could be very demanding in large-scale economics– environmental models including thousands of variables (MARKAL, TIMES, etc.). Eyckmans and Tulkens [66] and Labriet and Loulou [137] use large-scale models.11 The former uses the CLIMNEG world simulation model, derived from the RICE model, to determine collectively optimal emissions and transfers among the six players (regions) to obtain an agreement which is stable in the sense of the core of a cooperative game with the γ -characteristic function. Contrary to what is usually advocated in the noncooperative approach to IEAs, a stable agreement including all players turns out to be feasible. Labriet and Loulou [137] consider a 15-player game and compute and contrast the numerical results of using various allocation mechanisms. The paper includes a discussion of the internal stability of the agreement and the impact of possible deviations of the players. The sharing and transfer problem is dealt with once for all, and hence, γ -characteristic function and the characteristic function used in Petrosjan and Zaccour coincide for linear-state dynamic games (see [227]).

10 The

11 See

also [136].


the issue of intertemporal consistency is evacuated. Numerical simulations provided in both papers are an interesting input to policy makers involved in negotiating IEAs. Given the computational difficulties involved in the determination of feedback equilibria, both papers adopt the more tractable open-loop information structure. 3.3 Empirical Transboundary Games What mainly distinguishes this section from the previous one is that the parameters of the economics– environmental models are estimated using real-life data. Consequently, payoffs and side payments are empirically determined. A series of papers analyze the acid rain dif ferential game that extends the static one in [145, 156, 187]. The game involves countries emitting air pollutants (sulfur and nitrogen) that affect their neighbors, leading to soil acidification and negative impacts on forest growth. Polluted countries wish to negotiate with polluters to find an agreement that controls the emissions (De Zeeuw [53] surveys the acid rain game literature). The first analysis of the acid rain differential game appeared in [119–121]. The model specifies, among other things, sulfur emissions and their depositions, acidification dynamics, the emissions abatement costs as well as the benefits from the quality of the environment. The authors compute a cooperative solution, the fully noncooperative solution, and cheating outcomes where one player deviates from the cooperative solution. An interesting feature of the 1992 papers is the assessment of the benefits of the agreement on sulfur emissions reduction signed by Finland and the Union of Soviet Socialist Republics. One conclusion is that Finland benefits from any form of cooperation. However, Finland needs to compensate financially the Union of Soviet Socialist Republics if the four regions of the Union of Soviet Socialist Republics that are closest to Finland should find an interest in cooperating. Kaitala et al. [115] reconsider the acid game with three players (Finland, Russia, and Estonia) and relax the assumption that the players have full information when they negotiate to establish a cooperative emission reduction program. The paper follows the approach of Tulkens [193] where players have access to local information only, i.e., the current values of their marginal abatement and damage costs. A main result of the paper is that local information is sufficient for determining a succession of cooperative emissions abatement programs that converge to the cooperative solution. Mäler and de Zeeuw [146] consider a similar game but with finite horizon and different acidification dy-

namics. A deposition of sulfur and nitrogen oxides that exceeds the critical level causes a depletion of the acid buffer stock and implies a damage cost. The authors compute open-loop and feedback Nash equilibria as well as the cooperative solution and show that the depositions converge to critical loads, but the steadystate levels of the buffer stocks differ. The model is applied to an acid rain game between Great Britain and Ireland. Mäler and de Zeeuw obtain that cooperation, compared to Nash equilibria, is beneficial to Ireland and does not lead to a significant gain for Great Britain. An early empirical differential game on the global warming issue is [185] where a five-player framework is assumed (USA, other OECD countries, Russia, China, Rest of the World). The two state variables are the difference between present and preindustrial global mean temperature and CO2 concentration; control variables are the levels of emissions. Noncooperative and Pareto solutions are compared, with the conclusion that the Pareto optimal agreement is beneficial to developing countries but is more costly for the industrial world. Yang [220] proposes an algorithm to obtain a closed-loop Nash equilibrium to a TPP as a sequence of open-loop Nash equilibria. The approach is illustrated with the RICE model and the policy and methodological implications of the closed-loop strategies are discussed. It is worth noting that this approach is especially useful when the differential game model is intractable and the negotiation rules have a clause allowing for periodical reevaluation of strategies. Fernandez [71, 72] examines water pollution problems, due to wastewater emissions from countries in a shared waterway, along the US–Mexico border. To the best of our knowledge, these studies are the only ones that use a differential game format to assess empirically the costs and benefits of trade and water pollution. The main questions are: • •

What are the optimal strategies for managing water quality under two scenarios related to trade liberalization? What are the effects on water quality, trade, and welfare when both countries coordinate their pollution control and when they acting selfishly?

In noncooperative scenarios, the author computes a feedback Nash equilibrium. Interestingly, the formulation of the scenarios allows one to distinguish between the added value of trade liberalization and the benefits of coordinating emissions strategies. Results show that if trade liberalization is implemented, Mexico treats up to 60% of its wastewater which reduces considerably the transboundary pollution. The steady-state pollution stock is the lowest in a scenario of trade liberalization


and cooperation. Trade and employment are at their highest levels. Fernandez [72] offers a statistical analysis of the determinants of environmental project acceptance by the Border Environmental Cooperation Commission and shows that there is a higher level of US projects approval. Equity between the two countries does not seem to be a goal in this context. Bayramoglu [18] deals with a water pollution problem, the eutrophication of the Black Sea. She limits her analysis to the two most affected coastal countries, Romania and Ukraine, and specifies a utility function for each country that depends on agriculture and fish consumption. Agricultural activities generate pollution such that the resulting stock has a negative impact on the stock of fish. The following solutions are assessed: •

• • •

Joint optimization which assumes that a regulator either chooses an emission rate for both countries (if the sources of pollution cannot be distinguished) or two different pollution rates A feedback Nash equilibrium The common policy proposed by The Black Sea Commission and the International Commission on the Protection of the Danube River A solution based on per capita standards.

One conclusion of the paper is that the noncooperative solution dominates the policy in which the regulator optimizes the joint payoff subject to having the same rate of pollution for both countries. Another conclusion is that the noncooperative solution is better, welfarewise, than the common policy proposed by the two commissions. Fernandez [73] contrasts the feedback-Nash equilibrium with the joint-optimization solution of a transboundary differential game where pollution takes the form of invasive species due (mainly) to marine shipping. The empirical part of the paper focuses on two players (Canada and USA) and two species. The author obtains that preventive abatement lowers damages for the USA by 40% and lowers the steady-state stock of invasive species by 21%. Canada gains from cooperation with 38% less stock and a 61% decrease in damages.

4 Macroeconomic Problems The introduction of the Handbook of Environmental and Resource Economics (Edward Elgar, 1999, p. 12) states that “Environmental macroeconomics may be regarded as covering theoretical and empirical issues related to growth, sustainable development, and the physical scale of the economy”. Daly [49] pointed out

that macroeconomic textbooks have neglected environmental issues, such as the use of natural resources, emissions of pollutants, and their environmental impacts. About a decade later, however, Munasinghe [155] could write: “Many key ideas on macroeconomics and the environment have been developed within the last two decades, although some historical roots are discernible in several classical papers”. The increasing attention that sustainable development has received from world decision makers seems to have promoted the study of macroeconomic policies and their environmental impacts. In a review of macroeconomic theories and models for the environment, Van Ierland [198] writes: “New models have been developed to analyze the macroeconomic impacts of environmental policy measures, to study the characteristics of sustainable development in macroeconomic growth models and to analyze the economic aspects of global warming in integrated dynamic climate and economy models”. From a dynamic game perspective, the macroeconomic area of environmental economics has not yet been broadly developed. One exception are regionalized macroeconomic world models, cast as gametheoretical models, which have been widely applied in the analysis of climate change. In this section, we survey the literature devoted to the analysis of various macroeconomic problems using a dynamic game framework. First, we look at economic growth where we consider exogenous and endogenous growth models as well as closed and open economies. Second, we discuss economic–environmental problems of climate change, modeled as dynamic games. The effect of population growth and migration on macroeconomic policies is the topic of the papers surveyed in the third subsection. We also look at the use of income transfers as a mechanism to improve environmental quality and review a number of papers dealing with sustainable development.

4.1 Economic Growth Although economic growth and changes in the environment have run parallel in history, the study of the interaction between growth and environmental problems was not initiated until the 1970s when exhaustible natural resources and pollution were incorporated in neoclassical growth models. The neoclassical growth model of Solow [182] has been widely used to analyze the relationship between economic growth, technological progress, the use of natural resources, and the emission and accumulation of pollutants. In the early neoclassical growth models, long-run economic growth


was either considered to be driven by exogenous factors, or it was absent. Neoclassical growth models have been used to analyze sustainability issues, for instance, by using the environmental Kuznets curve that hypothesizes an inverted U-shaped relationship between per capita income and the use of natural resources and/or the level of emissions of pollutants. According to Kuznets, at low income levels, the use of natural resources and/or the level of emissions of pollutants increases with income more than proportionally and after reaching a maximum, the use of the natural resources and/or the emissions tend to decline with income. Empirical analyses show that this relationship may hold in some cases although for some resources and/or pollutants the relationship does not hold. Neoclassical growth models have also been extended to an integrated economic analysis of energy issues and global warming. As we shall see, these models provide a quite strong analytical framework for studying important questions in the area of environmental economics. Starting in the 1990s, the new endogenous growth literature has introduced environmental variables in general equilibrium growth models, allowing the endogenous determination of aggregate economic growth rates. Contrary to neoclassical or exogenous growth models, endogenous growth models assume that longrun economic growth is endogenously determined by factors such as technological progress, public investment, human capital, or other forms of knowledge. An excellent survey of endogenous growth models and their applications in environmental economics is [181]. Endogenous growth models have been used to analyze environmental issues. Among the most important are: •

nomic activities and the environment in a dynamic perspective. However, the more interactions, the more complex the model, and the smaller the prospect of obtaining closed-form results. Compared to the substantial amount of literature assuming a single optimizing agent, this may be a reason for the limited number of works on economic growth and environmental problems involving two or more agents who act strategically over time. Before surveying the papers in the area, we present a typical differential game model of economic growth and transboundary pollution. Example 4 Consider two countries indexed by i = 1, 2. Social welfare, expressed by a utility function U i (Ci , S), depends positively on consumption Ci and negatively on the damage caused by the stock of pollution, S. The gross benefit of consumption is an increasing and concave function Bi (Ci ) such that as consumption tends to zero, marginal benefits tend to infinity. The pollution damage function Di (S) satisfies

Di (S) > 0, Di (S) ≥ 0 ∀S ≥ 0, Di (0) = 0, and the utility function is U i (Ci , S) = Bi (Ci ) − Di (S). Each country produces a single homogenous good that can be consumed or invested in productive capital. The production function Fi (·, ·) uses the stock of capital, Ki , and a polluting input, ei , to produce consumption goods in amount of Ci . The pollutants emitted, ei , are accumulated in the stock S which evolves over time according to the differential equation dS (t) = e1 (t) + e2 (t) − δS(t), dt

Under which conditions are economic growth and environmental preservation compatible? What is the relationship between optimal growth and sustainability? Which are the consequences of environmental policy for growth?

where δ is a nonnegative parameter that represents the constant natural absorption of the stock of pollution. The accumulation dynamics of the capital stocks are

Most of the studies of exogenous or endogenous economic growth models have focused on research questions cast in the framework of a closed economy and have involved the solution of a social planner’s optimal control problem. When open economies are considered, in most cases, only one agent is involved, solving a dynamic optimization problem. Thus, the assumption is nonstrategic behavior on the part of the remaining economic agents. Models of economic growth and the environment need to represent the interactions between the eco-

where γi is the depreciation rate of capital in country i. The government in each country has a positive and constant rate of time preference ρ and chooses its time paths for consumption and emissions in order to maximize the flow of discounted utility over an infinite horizon: ∞ Wi = e−ρt U i (Ci (t), S(t))dt,

• •

dKi (t) = Fi (Ki (t), ei (t)) − γi Ki (t) − Ci (t), dt

0

subject to the capital accumulation and pollution dynamics.


Casino [44] analyzes in an exogenous economic growth framework the relationship between transboundary pollution abatement and national income. A two-country noncooperative differential game with open-loop strategies replicates the J-curve for pollution abatement, derived by Selden and Song [179] in a one-country framework. For low levels of current expenditure and capital, it is optimal to spend no resources on abatement and to postpone cleaning activities. However, there is a critical value of current expenditure (consumption plus abatement expenditures) from which welfare improves by allocating resources to abatement. The optimization problem is solved in two stages. The first stage is a static problem that establishes the optimal allocation between consumption or abatement. The second stage is a dynamic capital accumulation problem that determines the optimal level of current expenditure. The two countries differ in the emissions per unit of output (more or less polluting technology), the discount rate, and the depreciation rate of capital. The country with the smallest rate of discount and/or the less polluting technology attains the highest steady-state levels of per capita capital, current expenditure, and abatement effort. Verdier [199] is one of the pioneering investigations of the question whether economic growth and environmental preservation are compatible. The author studies an endogenous growth model which includes environmental pollution. Firms produce existing products and use R&D to develop new ones. New products can be developed to be “cleaner” than existing ones, at an extra cost. The paper studies the effects of emission taxes and technological standards on the long-run equilibrium growth rate of the economy. A technological standard is a requirement imposed on production that it must have a certain output-emission ratio. The author concludes that the ranking of taxes and standards is not clear-cut when the regulator has a limited set of instruments to obtain a first-best solution. A rather unusual feature of the model is that there are no transient path. The economy jumps instantaneously to the steady state. Asada [6] uses the “AK model” , one of the simplest models in endogenous growth theory, to study two dynamic models of capital accumulation involving an environmental factor (specifically, a stock of pollution). The analytical framework is the Lancaster [138] differential game of capitalism between workers and capitalists. The effect of pollution on the welfare of the classes is asymmetric. Pollution affects negatively workers’ welfare while capitalists’ welfare depends on their consumption and capital stock. The stock of pollution simply is assumed proportional to the capital stock; the proportionality factor depends positively on the

economic growth rate. The model is of the linear-state variety, and therefore, the open-loop Nash equilibrium coincides with the feedback one. In a case where there are two equilibria, one has a high rate of growth and a high pollution level; the other one has a low growth rate and pollution level. The latter equilibrium leads to some counterintuitive results. For example, an increase of workers’ disutility of pollution, or an increase of the adverse pollution effect of capital accumulation, does not necessarily decrease the economic growth rate. A growth model with an environmental asset is studied in [20] who extend the Golden Rule of economic growth to the environmental area. The Green Golden Rule is a path that maximizes long-run sustainable utility of consumption and environment. Palokangas [161] searches for the Green Golden Rule in a union of symmetric countries that wish to coordinate their emission policies. In each country, a social planner maximizes welfare that depends positively on the level of consumption and negatively on total emissions in the union. Emissions can be reduced by improving the production process through R&D which generates technological change. The main objective of the paper is to characterize a Pareto optimal policy for the union, designed by a central planner who maximizes the union’s welfare and influences local planners’ decisions through emission taxes. It is shown that the higher the absorption rate of pollution, the more emissions or the more pollution are disliked, the higher the Pareto optimal growth rate. The optimal emissions tax leading to the Green Golden Rule increases with the number of countries. An important stream of literature which emerged in the early 1990s explores the linkages between international trade, environmental degradation, and growth. Dynamic games have only been adopted by a smaller number of papers, focusing on the so-called North– South dynamic trade game. Alemdar and Özyildirim [3, 4] contribute to this literature. Both papers assume a global economy comprised of two regions, North and South. The former produces manufactured goods which are consumed, invested in the region, or exported to the South at a fixed world market price. Inputs in North’s production function are North’s technological knowledge and raw materials imported from South at a monopoly price determined by South. North’s stock of knowledge accumulates over time, indirectly affected by South via the terms of raw materials trade. North chooses a consumption plan and the amount of raw materials imported from South. South extracts raw materials using labor; resource extraction causes pollution. North’s knowledge diffuses and impacts the damage done by resource extraction on South’s environment.


The main difference among the Alemdar and Özyildirim models is that in the first paper, pollution does not accumulate over time. Pollution is positively related to resource extraction and negatively related to the stock of North’s knowledge. In the second paper, pollution accumulates over time and damages South’s environment. The second paper includes two additional features. First, some waste material (a by-product of North’s production) is dumped in the South. Second, there are multiple resource producers and resource extraction activities only damage the local environment. The diffusion of knowledge from the North affects the local environmental damage. Southern producers choose their production plans, given the Northern demand for resources, taking into account the evolution of their stocks of pollution. Pollution is only locally internalized in the South. In both papers, cooperative and open-loop noncooperative Nash equilibria of the dynamic trade game are simulated. Sensitivity analyses demonstrate the role of the rate of diffusion of knowledge and the market structure in South. As expected, uncoordinated resource extraction can cause a significant deterioration in South’s welfare. Cooperation between South’s resource producers improves the global welfare, though South has more to gain. Furthermore, even in the absence of global cooperation, both regions experience substantial welfare gains when productive activities pollute less and create more knowledge spillovers. A North–South trade differential game is analyzed in [35] to study how transfers from North to South affect capital growth and biodiversity conservation of the South ecosystem. Biodiversity losses affect both economies. North agrees to transfer income to South to diminish biodiversity losses. Such transfers are used by a R&D sector that improves technology and enhances South’s labor productivity. This leads to a reduction of the price of the traded good and improved biodiversity conservation in South. South chooses a range of natural species used in the production of an intermediate “natural” good required in North’s productive process.12 North may select either its saving rate, the transfer rate, or both. Two types of open-loop equilibrium strategies are analyzed: first, strategies leading to an initial period of economic growth, followed by one of constant capital stock in North, and second, strategies implying that the capital stock grows throughout the game. It turns out that whether North only chooses the saving rate or both rates, as income in the North increases, the burden of the transfers for North decreases. Increasing the trans-

12 The

range is a proxy for losses of biodiversity.

fer does not have an unambiguous effect on biodiversity conservation. When North controls the transfer rate and the savings rate is constant, an increase in the latter leads to better biodiversity conservation, despite an indeterminate effect of a higher saving rate on the transfer rate. 4.2 Economic–Environmental Models of Climate Change Aiming at a better understanding of the problems of climate change, an important literature on economics– environmental dynamic games has emerged during the last decade. Models are complex and involve the dynamics of economic and environmental variables as well as relationships between main economic and policy variables. In most cases, there are many players. These features necessitate the use of numerical methods if one wants to characterize equilibrium strategies. Models are calibrated using historical data and involve economic growth and transboundary pollution; stocks of capital (physical, knowledge, abatement) are accumulated and the dynamics of environmental variables are taken into account (mass of GHG in the atmosphere, atmospheric temperature relative to a base period). The aim is to analyze the impacts of climate policies and identify cooperative solutions to climate change problems. Scheffran and Pickl [177] and Scheffran [175, 176] are among the first to analyze the interaction between economic output and emission reductions using a dynamic game framework. The first paper analyzes the design of a Joint Implementation program, to evaluate the impact of cooperation between industrialized and developing countries. The paper focuses on Joint Implementation abatement projects at the government level and studies the conditions that allow an industrialized and a developing country to cooperatively achieve their goals at lower costs than those which must be paid if actions were uncoordinated. The industrialized country’s goal is to reduce emissions to a certain level while the developing country’s goal is to attain a given growth in GNP. Both countries have technologies which are more and less efficient in terms of emissions and costs. The degree of cooperation is measured through technology transfers and capital flow from the industrialized to the developing country. The model is extended to more than two agents and a coalition forming process is studied by means of a dynamic game. In two papers, Scheffran [175, 176] studies cooperation between an industrialized and a developing country, through the introduction of reduced-emission technology in the developing country with financial support


from the industrialized country. Simulations of energy consumption, economic output, emissions, investment, value, and technology priority are carried out. Dynamic integrated growth and climate models have been used to evaluate the incentives of countries (regions) to sign an international treaty on climate change control, such as the Kyoto Protocol. Bosello et al. [23] study how the equity criterion influences the developing countries’ participation decision (to sign the Kyoto Protocol). The first criterion is equal average abatement costs, the second is equal per capita abatement costs, and the third is equal abatement costs per unit of GDP. The analysis utilizes the RICE optimization model [160]. This model, see also “Multilateral Environmental Agreements” in Section 3.2.2, is frequently used as the basis of empirical dynamic games to evaluate the effectiveness of climate policy. The results of Bosello et al. show that the adoption of any of the three equity rules increases the profitability, but not the stability, of a climate agreement. With the aim of offsetting incentives to free ride, the authors propose a Pareto optimal transfer mechanism which, however, does not provide a global agreement on climate change (i.e., the grand coalition does not form). Scientists and (maybe) politicians agree that technological innovation plays an important role in reducing the cost of stabilizing GHG concentrations in the atmosphere. A model called World Induced Technical Change Hybrid (WITCH) has been introduced in [24, 25], with the aim of evaluating the effects of international knowledge flows on the dynamics of domestic R&D sectors as well as major economic and environmental variables. A main characteristic of WITCH is that technological change is endogenous and depends on climate policy, international spillovers, and other economic effects. The model takes into account learning-by-doing effects and the accumulation of knowledge through R&D. A Ramsey-type neoclassical optimal growth model is defined for 12 macroregions that play an open-loop noncooperative intertemporal game. The equilibrium characterizes the optimal investment in all energy technologies, in physical capital and R&D, and the direct consumption of natural resources (fossil fuels). The model is calibrated by using data from the empirical literature. It turns out that there are incentives to free ride on carbon-free investments, implying a delay in the adoption of climate friendly technologies in the first periods of the game. Later on, a faster introduction of the climate friendly technology is caused by a decrease in the investment cost, driven by learning by doing. The results show that emissions in the cooperative case would be around half the level of the noncooperative case.

Bosetti et al. [25] use WITCH to analyze the role of international knowledge f lows in advancing the development of more efficient energy technologies. The setup includes international knowledge spillovers and allows an analysis of the cost reductions that may be achieved by increased knowledge diffusion. The analysis focuses on disembodied knowledge spillovers and compares three scenarios: • • •

International knowledge pool available Absorption capacity, i.e., the ability of each country to benefit from the international knowledge pool Interaction between international spillovers and domestic R&D sectors

The results show that the endogenization of international energy R&D spillovers increases the incentives to free ride and decreases investments in energy R&D. The implications are that neither the overall domestic pool of knowledge nor the cost of stabilizing world CO2 concentrations in the atmosphere are greatly affected. However, the cost of stabilizing the GHG emissions can be significantly reduced by implementing a stabilization policy. Such a policy should be based on a global permit market, combined with a technology policy that is designed to help knowledge dissemination and, in particular, to enhance absorption capacity in low income countries. A main objective of the paper by Viguier et al. [200] is an evaluation of the effect of endogenous technological learning in atmospheric CO2 stabilization scenarios. The evaluation is done by using two models: • •

MERGE, an integrated assessment model [147] GMM, a multiregional energy system model [10, 165]

Results show how R&D policies and demonstration and deployment programs can be used to attain a cost reduction through learning effects. Various scenarios related to CO2 emission control are introduced to illustrate the results that can be generated by MERGE and GMM. 4.3 Population Growth Dutta and Radner [61] allow population growth in a discrete-time dynamic game used to study the global warming problem. The focus is on the economic costs of warming and the future benefits derived from controlling the greenhouse effect. The global stock of GHG emissions evolves endogenously in the model, while the population in each of the I countries changes exogenously over time. The Markovian Nash equilibria of the game show that each country’s emission is independent


of the global stock of GHG and depends on its own population level only. Global Pareto optimal emission rates are also independent of the stock of GHG but depends on all countries’ population. As expected, cooperative emission rates are lower than noncooperative emission rates. The population effect on the emissions is unclear, both in the noncooperative and cooperative framework: Depending on the specification of utility functions, an increase of the population can lead to an increase or to a decrease in the level of emissions. The model also considers R&D and development efforts. In [85], the total population of an economy is fixed, but household mobility is introduced in a transboundary pollution problem. In a static framework, it has been proved that national environmental policies are socially optimal under perfect household mobility (i.e., moving from one country to another is costless). This result is due to the fact that household mobility levels off regional welfare differences. In a differential game, Haavio shows that the static result is not necessarily robust to the introduction of migration costs. Under perfect household mobility, the static result is replicated, but when migration costs are included and policymakers use feedback strategies, household mobility can be viewed as equivalent to transboundary pollution, in the sense that both generate too large emissions by individual countries. Furthermore, the steady-state pollution stock is larger than under no household mobility. The author demonstrates that environmental problems tend to worsen as household mobility increases. 4.4 Income Transfers and Environmental Quality It has been proposed in the literature that income transfers between countries should be used to improve the noncooperative Nash equilibrium outcome which only in rare instances is Pareto efficient. In a two-country dynamic game of environmental management, Niho [157] examines the long-run and short-run effects of international income transfers on environmental quality and welfare. Producing a consumption good in a country generates pollution. A country allocates its resources between the production of consumption goods and the cleanup of pollution. Utility of a country depends on the amount of consumption goods it produces and the environmental quality. Assuming open-loop strategies, Niho shows that the neutrality theorem applies if the cleanup technologies are the same in both countries. Neutrality means that international income transfers do not affect neither the level of environmental quality nor the utility level of the donor and the recipient country. However, if the cleanup technologies are different, neutrality no longer applies. Unexpectedly, an income

transfer from a country with a more efficient cleanup technology to a country with a less efficient technology causes a deterioration in the environmental quality and the utility of the donor as well as the recipient. The result is obtained assuming an interior solution and ignoring the possibility that one country does not contribute to cleanup activities. These assumptions are relaxed in [217] where corner solutions are allowed. The author focuses on long-run effects of income transfers and a steady-state equilibrium. Yanase shows that the effect of an international income transfer depends on whether both countries undertake cleanup activities (Niho’s assumption) or only one country contributes to the cleanup activity. The income transfer may deteriorate or improve the environment depending on the initial distribution of resources in each country. Long [144] also incorporates income transfers when studying the optimal use of productive capacity and optimal investments in environmental quality when the latter positively affects the production process. An initial analysis of a single country is extended to two countries, having different preferences and technological efficiency. Each economy produces a final output using as inputs country-specific physical capital and the environmental quality, shared by the two countries. Countries play a noncooperative differential game with open-loop strategies. It is possible that one country uses its capacity fully, while the other country underutilizes its capacity. To improve the Nash equilibrium outcome, which it is not Pareto efficient, a constant fraction of one country’s income can be paid to induce the other country to reduce its capacity utilization (production) and still be better off. If countries have the same discount rate, the optimal fraction of one country’s income to be transferred to the other country is time invariant and the dynamic inconsistency problem is avoided.

4.5 Sustainable Development Since the report of the World Commission on Environment and Development [212], known as the Brundtland Report, sustainability has increasingly been incorporated in governmental policy. The Brundtland report has frequently been quoted for its definition of sustainable development as “The development that meets the needs of the present without compromising the ability of the future to meet their own needs”. Sustainable development became a cornerstone of the United Nations’ Agenda 21 which has three dimensions: economy, environment, and society. Agenda 21 called for action on sustainability issues at global, national, and


regional levels, and one result is the Kyoto Protocol on climate change. Beltratti [19] is, to our best knowledge, the first to study sustainability and international coordination in a dynamic game-theoretic framework. Beltratti points out that the definition of sustainable development essentially is based on the role of stock variables, in particular environmental resources, as dynamic elements connecting the present and the future. A two-country, linear-quadratic model of a resource being used for providing consumption goods is studied both in a static and in a dynamic setup. The author develops a simplified version which ignores physical capital and pollution. The analysis is mainly in terms of the steady state. As expected, the stock of the environmental asset in the long-run Markovian Nash equilibrium is lower than the first-best solution achieved by an international planner. Numerical simulations suggest that the lower the time preference rate, the lower the ratio between the steady-state stocks of the environmental asset in the cooperative and noncooperative solutions. More recently, Haurie [88, 90] focuses on sustainable development through the study of a multigeneration game. Such a game is an alternative way to represent time preferences in cost-benefit analyses of very longterm projects that overlap several generations.13 The intergenerational game model introduces a form of altruism in long-term cost-benefit analysis. The model is based on a stochastic interpretation of the discounting process and combines a piecewise deterministic control structure with a multigeneration stochastic game approach. The Markov-perfect equilibrium in the game with players belonging to successive generations escapes the time-inconsistency problem often encountered in sustainability studies. Haurie and Moresino [93] use the same formalism proposed in [88, 90] and re-expose some of the concepts and results of these papers. In [90], each generation is concerned with its own expected consumption and that of the immediately following generation.14 The setup is a stationary control system where each generation has an exponentially distributed life duration. The steady-state attractors (turnpikes) for the equilibrium trajectory are compared under different assumptions concerning the time preference or “altruism parameters”. The latter defines the weight that one generation places on the rewards of later generations when it computes its payoff. A stationary intergenerational

13 A 14 In

equilibrium is characterized and the optimal strategy of each generation is obtained as the solution of a family of infinite horizon optimal control problems. The equilibrium is also Pareto optimal. Finally, the equilibrium concepts are used in the context of an integrated assessment model of global climate change, in particular a simplified continuous-time version of the DICE model [158, 159]. Haurie [89] develops integrated assessment models for the evaluation of climate change policies. The starting point is the simple but important observation that climate and economic subsystems interact and they evolve on very different time scales (on the interaction of climate and economic systems, see [96]). International negotiations and agreements on climate control are supposed to happen on a “slow” time scale while economic adjustments take place on a “fast” time scale. The fast time index is t and the slow time index is t/ε, where ε > 0 (and small) is the ratio between the time scales.15 At any instant of time, the economy and the climate, respectively, can be in a specific “mode” ..... Modes change according to a controlled jump process. The combined economic and climate system has two time scales and is a piecewise deterministic system [38]. Haurie defines the dynamic games played by nations in international negotiations on greenhouse gas emission reductions. Solving the games is technically demanding, but one can exploit the hierarchical structure induced by the two time scales. This leads to an “approximate” dynamic game which is considerably simpler to analyze. An instructive paper on the use of fast and slow processes in the modeling of natural resources is [47].

5 Conclusions This survey should have demonstrated to the reader that dynamic cooperative and noncooperative games are promising tools in the study of a large variety of environmental pollution problems. Taking stock on what has been achieved during the last two decades, we discuss in the sequel some areas where more research efforts are particularly needed. 5.1 Representation of Interactions Among Players The dynamic games literature in pollution control and environmental economics in general have often

good example is the study of global climate changes.

[88], each generation considers in its payoff the whole sequence of expected rewards to all future generations.

15 The

study of control problems with two time scales is known as singular perturbation theory.


adopted a modeling strategy of downscaling to few links (if not just one) a reality that is of formidable complexity in terms of interactions among players. Nevertheless, working with simplified interactions has allowed researchers to construct a body of knowledge that should improve our understanding of the economics of pollution control in a strategic and dynamic setting. The agenda for the future, which clearly is an ambitious research program, should include the relaxation of some of the simplifying assumptions on interactions among players. The items to consider are discussed below. Trade Most of the literature focuses on closed economies and disregards that countries and regions are not isolated but trade with each other. Explicitly including in the models variables and equations that describe trade relationships between players would undoubtedly enrich the conclusions and recommendations derived from models. Trade relationships may include the exchange of final consumption goods, intermediate goods used in the production of final goods, natural resources, technology, know-how, etc. Taking trade relationships into account is necessary in a meaningful analysis of questions such as the economic and environmental sustainability of growth in an increasingly globalized world. Till now, however, most of the studies in the environmental branch of exogenous and endogenous economic growth theories have focused on closed economies with a single agent. The endogenous determination of the prices of the goods exchanged in the market is an important step to enhance the applicability of models. There is also a need to extend the links between trade and technological transfers and international agreements on pollution control to a dynamic setting. Such links have been explored in static games, and it is of interest to see if trade and technological support to developing countries can help in building sustainable cooperation in the environmental area. Strategic Interactions In noncooperative differential/difference games of pollution, it has often been assumed that players use open-loop or feedback strategies. Quite many papers study both cases. By choosing a particular type of strategy, the modeler makes an assumption about the information players employ when designing their strategies. This choice is often left unmotivated but should be part of the institutional setting of the game. Some researchers argue that the strategy choice of players should be an endogenous part of the game. Open-loop strategies have been considered “less satisfying” as representations of real-life strategic behavior, but there may be no reason for such concern: Open-loop strategies represent behavior of agents who can and will precommit to their future actions, or agents who are unable to observe the evolution of pollution

stocks in real time.16 Arguments in favor of openloop strategies include that in many cases, they lead to lower stocks of pollution than when agents use feedback strategies and they are computationally feasible in large-scale climate/econometrics models, providing information that should be valuable for decision makers. Intergenerational Interactions The issue of sustainable development is intimately related to interactions between different generations of players. The larger part of environmental economics literature has ignored, or greatly simplified, the interactions between generations. The definition of criteria that take into account strategic behavior of the agents and allow to achieve intragenerational and intergenerational equity has been the focus of some recent works, but more research is needed to refine concepts, for example, the strategic behavior of future generations, time preference rates, valuation of biodiversity, etc., and the computation of sustainable development paths. 5.2 Extension of Models Writing this survey has strengthened our belief that utilizing dynamic games provides an understanding of the economics and management of pollution which goes significantly further than what static game theory has provided. However, pollution control models still need to be and can be improved. We provide a list of areas in which extensions and modifications could be done. The list relates to the papers that we have surveyed, and our suggestions may already have been noted elsewhere in the literature. Making out the list, the practical implementability of our suggestions has been kept in mind. Sustainable Agreements The cooperative game approach to the formation and sustainability over time of an IEA has been successfully extended to a dynamic setting, and a number of empirical contributions have been published. While static noncooperative game literature on IEA started in the early 1990s, the dynamic games literature is taking its first steps. This may be surprising in view of the inherently dynamic aspects of damages caused by pollution stocks, the possibility of entry and exit of IEA members over time, and so forth. Research is needed to better understand the formation of large and dynamically stable IEAs, taking into consideration factors such as heterogeneity of players,

16 Certainly, this does not exclude the need for studies that assume state-dependent strategies, reflecting quite another way of decision making.


uncertainty in climate systems, linkage of multiple negotiation themes (environment, trade, R&D transfers) and the presence of regional agreements. Population Growth The two main components of population growth, net natural variations and migration, are ignored in most of the economicenvironmental models formulated as dynamic games. Empirical works tend to underestimate the effects of these two components of population growth. The operationalization of the economic–environmental models of climate change for practical purposes requires the introduction of population growth. The high population growth rates in some countries and the migration flows from developing to developed countries cannot be ignored in the characterization of the conditions and policies for long-run sustainable development. R&D and Technological Progress There seems to be general consensus among scientists and politicians that technological improvement is a key factor in reducing the cost of stabilizing greenhouse gas concentrations in the atmosphere. Technological improvement requires either the investments in a country’s own R&D sector or flows of knowledge through international or interregional trade. Macrogrowth models that intend to analyze the impacts of climate policies have defined the technology part of a model in a general way. A more detailed description of the effects of investments, knowledge flows, and learning-by-doing effects on the dynamics of the knowledge stock should provide better insights into the effects of climate policy and the conditions for the stability of IEAs on climate change. Model components The assumptions made in dynamic game models of pollution are often remarkably simple. They reflect the modeler’s choice which could be based on rather diverse considerations: •

Descriptions of pollution flows and stocks and their dynamics are often highly stylized, e.g., a single stock of pollutant, pollution dynamics are time invariant, impacts of emissions are homogenous over time, pollution dynamics are deterministic, and decay of pollution stocks is modeled in a simplistic way.17 A more detailed description of the economic and environmental variables is needed if we wish to present dynamic games as a relevant decisionsupport tool of the stakeholders in the pollution control arena. Some attempts in this direction have recently been made, particularly in the literature of climate change. Another possible improvement

•

•

•

•

comes from a better description of the interactions between different types of pollutants on one hand and abatement investments or green technology on the other. The description of decision makers has been simplified. For example, North vs. South, government vs. industrial sector, upstream country vs. downstream country, one social planner, decision makers are homogenous with respect to discount rates, cost functions, damage functions, and so forth. There could be a need of introducing, e.g., local regulators in addition to a central regulator and more detailed descriptions of the agents in polluting industries. Also more sophisticated informational assumptions can be entertained: Some decision makers use state information, others do not, and strategies could be history dependent. The benchmark against which outcomes are assessed often is an optimum determined by a social planner. This may be far from a reality in which the socially desirable is not well defined, reflecting diverse interests of politicians, environmentalists, polluting industries, abatement equipment producers, and so forth. The joint optimization benchmark solution involves weights that may not be easily assessed. Solution concepts originating in cooperative game theory, e.g., the core or Shapley value, have been used only in a few cases. It may be worthwhile to utilize cooperative game theory more in dynamic games of pollution and the environment. The time horizon has an important, but somewhat neglected role in applied dynamic games. Often it is routinely chosen as infinity, without a discussion of the choice and its implications.18 In pollution problems modeled as dynamic games, decision makers may very well have different time horizons, as they may have different time preference rates. Intergenerational effects are of paramount importance in the discussion of global warming and climate change and could be addressed in the setup of overlapping generations models. In noncooperative dynamic games of pollution, strategies are routinely chosen to be of the openloop or feedback variety, but a theory of historydependent strategies exists. Using such strategies, decision makers can condition their current actions upon the whole history of the game, or a part of the history.

17 Exceptions are acid rain games and large scale IEA and climate

change models.

18 Analytical

tractability may be one motivation.


5.3 Larger Variety of Tools Differential games literature in environmental economics has predominantly used an analytical approach. The strength of analytical methods lies in the fact that model parameters and functional forms may be left unspecified. The drawback is that only models having a simple structure are tractable. The strength of numerical methods is the possibility to analyze games of considerable complexity, but this comes at the cost of a limited generality of conclusions. To generate results, the model structure and parameter values must be specified. The identification of an equilibrium of a differential game often relies on the Maximum Principle or the Hamilton–Jacobi–Bellman equations. In both approaches, the required work is considerably simplified when the game has some particular feature. The purpose of this subsection is to call attention to a series of theoretical developments in differential/difference games that only rarely have been applied to environmental problems.19 Piecewise Deterministic Systems The basic idea is that the state dynamics switch randomly, at random instants of time, between different types of dynamics. Piecewise deterministic systems are useful to model, for example, situations in which sudden changes in parameter values affect the system but have been used much less than the white noise approach where uncertainty influences the dynamical system continuously. Almost any system in practice is influenced by exogenous disturbances that in many cases tend to degrade the performance of the system (see [27] for an introduction to the theory and [88, 90] and [93] for applications in environmental economics). H ∞ -Optimal Control Theory The aim is to find worst-case designs which means that a decision maker chooses her control in the expectation that the opponent will choose an outcome that is maximally harmful to the decision maker. It has been discovered that H ∞ -optimization has important links to zero-sum differential games. More precisely, many H ∞ -problems are min–max optimization problems and in a gametheoretic setting they can be seen as two-person zerosum games (see [14] for an introduction to the theory). Impulsively Controlled Systems Consider a twoplayer differential game with separated dynamics. The first player controls a system consisting of an ordinary differential equation for a state variable x(t) to describe the continuous motion and a discrete equation describing jumps in the state. The first player chooses

19 Our

exposition is based on [113].

a continuous control u(t) to influence the evolution of x(t) and discontinuous control μ to produce a jump in the state. The second player controls a similar system. Basically, the game has a qualitative payoff, i.e., it is a “game of kind” in the terminology of Isaacs [105]. Stochastic Hybrid Models These models combine versatility and generality as they involve both discrete (logic-based), continuous, and probabilistic elements. Models contain deterministic dynamics, stochastic differential equations, impulsive jumps in state variables, and random switches between different dynamics at random instants of time. Extensions to differential games are still in their infancy (see [28] for an exposition of stochastic hybrid models). Nonlinear Dynamics Most economic time series are characterized by regular (stable) and irregular (erratic) behavior. The classical view is that erratic fluctuations should be seen as products of exogenous random shocks, but such that the economic system basically is stable. A more recent explanation is that erratic behavior could be caused by inherent nonlinearities in the “laws” that govern an economic system. For the last 20 to 30 years, there has been a growing interest in economic systems described by nonlinear dynamics since it was discovered that deterministic nonlinear systems can exhibit complicated and erratic dynamical behavior, also known as deterministic chaos. Thus, it does not take a stochastic system to generate unpredictable behavior (see, for instance, [104]). Acknowledgements The authors wish to thank Denis Claude, Sophie Legras, Henry Tulkens, and Franz Wirl for their valuable comments. We also thank the Associate Editor and the Reviewer for their helpful suggestions. Any errors are our responsibility.

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