A Simple Model of International Environmental Agreements with a Stock Pollutant

A SIMPLE DYNAMIC MODEL OF INTERNATIONAL ENVIRONMENTAL AGREEMENTS WITH A STOCK POLLUTANT

SANTIAGO RUBIO

(University of Valencia)

AND

ALISTAIR ULPH

(University of Southampton)

Very Preliminary Version

November 2001

ABSTRACT

Much of the literature on international environmental agreements uses purely static models of transboundary or global pollution, despite the fact that the important problems to which this literature is addressed involve stock pollutants. The few papers that study IEAs in the context of dynamic models of stock pollutants do not allow for the possibility that membership of the IEA may change endogenously over time in response to the dynamics of the stock pollutant. In this paper we set up a very simple two-period model of a stock pollutant with symmetric countries, but where countries are limited to two actions (pollute or abate). We study an open-loop equilibrium, where IEA membership is determined at the outset and then countries choose their emissions over the two periods, and a feedback equilibrium, where at the start of each period, countries have to decide both whether to join an IEA for that period and whether to abate or pollute. In this simple context we show that: (i) the open-loop model may have two stable IEAs, the most common of which IEA members abating only in the first period; (ii) the feedback equilibrium involves membership rising sharply over the two periods; (iii) one needs to model carefully how membership decisions get linked over time; (iv) the feedback equilibrium may have no stable IEA in period 1 but significant membership in period 2; generally welfare is higher in the feedback equilibrium than in the open-loop. We suggest that some of these results may not be robust to extension beyond two periods, in particular that the feedback equilibrium may involve IEA membership falling over time. Nevertheless the important point is that the dynamics of IEA membership may be closely linked to the dynamics of the stock pollutant.

Key words: international environmental agreement, stock pollutant, stable

agreements, open-loop equilibrium, feedback equilibrium

JEL Classification: F02, F18, Q20

1. Introduction.

There is now an extensive literature on international environmental agreements (see Barrett (2001) and Finus (2001) for excellent recent books summarising this literature). Yet for the most part this literature works with simple static models of pollution despite the fact that the important problems (climate change, ozone depletion, acid rain) which this literature seeks to shed light on involve stock pollutants. Does this matter? The only way to find out is to see what happens when one works with models of dynamic pollutants, and there are now some papers which do this.

Rubio and Casino (2001) extend the simple Barrett (1994) model of IEA formation to allow for a stock pollutant. The model is analysed in two stages: countries first of all decide whether or not to sign an IEA; then countries choose their paths of emissions. These emission paths are calculated by solving a differential game in either open-loop or feedback strategies assuming that the IEA signatories act to maximise their joint welfare while non-signatories just maximise individual welfare. Unlike the static model they argue that there is no difference between the case where the IEA signatories act in Cournot or Stackelberg fashion with respect to the non-signatories. Having solved for the emission paths and hence evaluated payoffs to signatories and non-signatories, Rubio and Casino then ask how many countries will want to sign the IEA, using the stability analysis familiar from the work of Barrett (1994), Carraro and Siniscalco (1993) etc. Rubio and Casino derive a rather more pessimistic result than the static framework. In the static framework Barrett (1994) showed that when gains to cooperation are high few countries will join an IEA. Rubio and Casino show that the number of signatories will be 2 no matter what the gains to cooperation are. However, the reason for this more pessimistic result is that they impose a restriction on parameter values to ensure that the signatories will produce strictly positive emissions in steady state for any size of IEA, and these parameter values drive down the size of the stable IEA. But it may be appropriate that for some numbers of signatories, the optimal policy would involve signatories producing zero emissions in steady-state, and it would be interesting to check whether relaxing this constraint would allow a bigger range of stable IEAs.

Germain, Toint, Tulkens and de Zeeuw [GTTZ] (2000) extend the framework of Chander and Tulkens (1995) to a dynamic model of a stock pollutant. As is now well understood the Chander and Tulkens approach, based on core concepts, is able to get much more optimistic conclusions about IEA formation, with the grand coalition being formed, because they assume that if one country defects from the grand coalition all countries will revert to non-cooperative behaviour. This punishment is sufficiently severe to deter defections. By contrast the stability analysis of Barrett and others assumes that if one country defects, the remaining countries will act to optimise their joint interests; this is less damaging to a defect and so encourages fewer countries to sign. With asymmetric countries it will also be important to use income transfers to ensure the stability of the grand coalition and Chander and Tulkens calculated the transfers needed to ensure this stability. GTTZ(2000) extend this analysis by calculating the transfers that will be needed to ensure stability of the grand coalition when there is a stock pollutant.

In both these cases, however, it would seem that the extension of the analysis from a static model to a dynamic model of pollution is not really generating any fundamental new insights. So maybe it does not matter very much if our analysis of IEAs draws heavily on an essentially static model. In this note we argue that this conclusion may be premature, because the papers cited above do not really allow the dynamics of the pollution problem to affect the formation of the IEA. This is seen most clearly in the Rubio and Casino model where countries essentially have to commit at the outset to being a signatory or a non-signatory. The dynamics of the pollution problem affects the calculation of the payoffs to signatories and non-signatories but the game of whether or not to join the IEA is just as static as in the Barrett framework. Moreover, while they analyse both open-loop and feedback strategies, with the latter allowing countries to set their emissions as a function of the stock, countries are not allowed to condition their decision on whether to be a signatory or a non-signatory on the stock. So one cannot ask questions such as would the number of signatories increase or decrease as the pollution stock increases. Equally in the GTTZ model, transfers are designed to maintain the grand coalition, so again the question of whether the number of signatories may change over time due to the pollution dynamics does not arise. At least in this case the fact that the number of signatories does not change over time is a conclusion from the analysis rather than an a priori assumption.

Both Rubio and Casino and GTTZ analyse models in which an IEA operates over the whole life of a stock pollutant. A paper which is closer to the spirit of the issues we seek to address is the paper by Karp and Sacheti (1997). They consider a two-period model of a stock pollutant but assume that an IEA will only form in one of the periods – either just in the first, because the IEA will fall apart at the end of one period, or in the second, because there may be substantial delays in forming an IEA – and ask how the dynamics of the pollution problem affect the incentives to join an IEA[1] as well as by differences in the extent to which the pollutant is local or global and the extent to which planners discount the future. They argue that static models of pollution may overstate or understate the difficulties of forming an IEA and that it is important to consider the dynamics of the pollution problem itself. This is in the same spirit as our approach, except that we do not restrict an IEA to only last one period but rather ask how the number of signatories may vary over time. However, we shall show that in one formulation of the feedback equilibrium the most likely outcome is that there will be no signatories to an IEA in the first period but significant numbers in the second period, but this is determined endogenously. We shall also see that for the most likely equilibrium of the open-loop model, although, by construction, there will be signatories to the IEA in both periods, in the second period the IEA is ineffective – signatories do not abate pollution.

In this note we shall use the Rubio and Casino framework but now allow for the possibility that in the feedback solution the number of signatories each period is determined as a function of the stock of the pollutant in existence at the start of the period. Hence the number of signatories may vary over time. This reflects the key point emphasised by Barrett (2001) that the fundamental problem of IEAs is the essentially anarchic nature of international relations, since national sovereignty allows countries the freedom to choose their actions, and if it is their interest to join or leave an IEA they will do so. As Barrett shows, most IEAs do allow for countries to join at different dates, and any punishment for leaving has to be built into the IEA itself, i.e. IEAs must be self-enforcing. However the model we use is a drastically simplified version of the Rubio and Casino model in which there are just two periods and we restrict countries to just two actions. The reason for simplification is to make the solution of the feedback equilibrium tractable. One problem is that the stable IEA is defined as an integer number of signatories, which makes the value function in the feedback equilibrium non-differentiable. Even if one ignores this integer problem, the value function with continuous actions and endogenously determined IEA membership becomes very non-linear. In the earlier literature on static models of IEAs determination of the stable IEA often resorted to numerical solutions because payoff functions were non-monotonic. To allow us to identify the different factors at work and to minimize the reliance on numerical solutions we have chosen to work with a model of discrete actions. However, we view this as a first step towards a more general analysis which will provide some guidance on modeling issues in a more general model. We hope to report results on a model with continuous actions in a later paper.

Despite the simplicity of the model, we find some interesting results: (i) there can be multiple stable IEAs in the open-loop model; (ii) the open-loop equilibrium usually involves only partial cooperation by IEA signatories, in the sense that they cut emissions only in the first period and set high emissions in the second period; (iii) in the feedback equilibrium the number of signatories rises over time; (iv) for a wide range of parameter values, the number of signatories in the feedback equilibrium and the level of abatement is at least as high in each period as in the open-loop equilibrium; (v) there are some equilibria of the feedback model where there is no stable IEA in period 1; this depends in part on assumptions about how countries are assigned to membership of the IEA in period 2; (vi) for most (but not all) parameter values, the feedback equilibrium generates higher welfare than in the open-loop equilibrium, contrary to the result in standard non-cooperative differential game models of stock pollutants (see e.g. Hoel (1992), van der Ploeg and de Zeeuw (1992)).

In the next section we set out the static version of the model. In section 3 we analyse the simplest possible extension of the static model to capture a stock externality by assuming that second period damage costs depend on the total stock of pollution at the end of that period, which includes the stock of pollution inherited from period 1. We assume that unit damage costs are not dependent on the pollution stock and solve for the open-loop and feedback equilibria of this model. While the feedback equilibrium has a higher number of signatories in period 2 than period 1, the number of signatories in period 2 is independent of the stock of pollution carried over from period 1. While this is unsatisfactory the simplicity of the model allows us to get sharp results. In section 4 we make a further extension to allow the unit damage costs of pollution in period 2 to depend on the stock of pollution inherited from period 1, and solve for the open-loop and feedback equilibria. Again the number of signatories in period 2 of the feedback model will be higher than in period 1, but now the number of signatories depends (negatively) on the stock of pollution carried over from period 1. In section 5 we speculate on how our results might be affected by weakening two key restrictions: countries have only two actions, and there are only two periods. We sketch an argument to show that in an infinite horizon model the feedback equilibrium could have the number of IEA members declining over time as the stock of pollution increases. However this does not contradict a basic message of this paper that there could be an important link between the dynamics of accumulation of a stock pollutant and dynamics of IEA membership. Section 6 concludes.

2. The Simple Static Model.

There are N identical countries, indexed i, each of whom can choose two possible emission levels, qi, low or high, which we normalise to take values 0,1 respectively. We denote aggregate emissions by all countries by Q, and aggregate emissions by all countries other than i by Q-i. The net payoff benefit function of country i is: