metadata: authors: 0: author: Karp, Larry
1: author: Siddiqui, Sauleh
2: author: Strand, Jon
count: World
docna: 0: docna: Dynamic climate policy with both strategic
            and non-strategic agents : taxes versus quantities
docty: Policy Research Working Paper
subtopic: Debt Markets,Economic Theory & Research,Emerging Markets,Markets and Market Access,Climate Change Economics
teratopic: Finance and Financial Sector Development,Private Sector Development,Macroeconomics and Economic Growth
abstracts: cdata!: This paper studies a dynamic game where
            each of two large blocs, of fossil fuel importers and
            exporters respectively, sets either taxes or quotas to
            exercise power in fossil-fuel markets. The main novel
            feature is the inclusion of a "fringe" of non-
            strategic (emerging and developing) countries which both
            consume and produce fossil fuels. Cumulated emissions over
            time from global fossil fuel consumption create climate
            damages which are considered by both the strategic importer
            and the non-strategic countries. Markov perfect equilibria
            are examined under the four combinations of trade policies
            and compared with the corresponding static games where
            climate damages are given (not stock-related). The main
            results are that taxes always dominate quota policies for
            both the strategic importer and exporter and that
            "fringe"countries bene?t from a tax policy as
            compared with a quota policy for the strategic importer, as
            the import fuel price then is lower, and the strategic
            importer's fuel consumption is also lower, thus causing
            fewer climate damages.
repnme: repnme: Dynamic climate policy with both strategic
            and non-strategic agents : taxes versus quantities
display_title: Dynamic climate policy with both
            strategic and non-strategic agents : taxes versus quantities
listing_relative_url: /research/2013/10/18447749/dynamic-climate-policy-both-strategic-non-strategic-agents-taxes-versus-quantities
publishtoextweb_dt: 2013-10-01T00:00:00Z
historic_topic: Finance and Financial Sector Development,Private Sector Development,Macroeconomics and Economic Growth
no_of_pages: 34
model_extractions: WPS6679 Policy Research Working Paper 6679 Dynamic Climate Policy with Both Strategic and Non-Strategic Agents Taxes Versus Quantities _Larry Karp Sauleh Siddiqui Jon Strand_ The World Bank Development Research Group Environment and Energy Team October 2013
Policy Research Working Paper 6679

# **Abstract**

This paper studies a dynamic game where each of two large blocs, of fossil fuel importers and exporters respectively, sets either taxes or quotas to exercise power in fossil-fuel markets. The main novel feature is the inclusion of a “fringe” of non- strategic (emerging and developing) countries which both consume and produce fossil fuels. Cumulated emissions over time from global fossil fuel consumption create climate damages which are considered by both the strategic importer and the non-strategic countries. Markov perfect equilibria are examined under the four combinations of trade policies and compared with the corresponding static games where climate damages are given (not stock-related). The main results are that taxes always dominate quota policies for both the strategic importer and exporter and that “fringe”countries bene t from a tax policy as compared with a quota policy for the strategic importer, as the import fuel price then is lower, and the strategic importer’s fuel consumption is also lower, thus causing fewer climate damages.

This paper is a product of the Environment and Energy Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at jstrand1@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Dynamic climate policy with both strategic and non-strategic agents: Taxes versus quantities Larry Karp Sauleh Siddiqui Jon Strand<sup>∗</sup> October 28, 2013 Abstract This paper studies a dynamic game where each of two large blocs, of fossil fuel importers and exporters respectively, sets either taxes or quotas to exercise power in fossil-fuel markets. The main novel feature is the inclusion of a “fringe” of nonstrategic (emerging and developing) countries which both consume and produce fossil fuels. Cumulated emissions over time from global fossil fuel consumption create climate damages which are considered by both the strategic importer and the non-strategic countries. Markov perfect equilibria are examined under the four combinations of trade policies and compared with the corresponding static games where climate damages are given (not stock-related). The main results are that taxes always dominate quota policies for both the strategic importer and exporter and that “fringe”countries benefit from a tax policy as compared with a quota policy for the strategic importer, as the import fuel price then is lower, and the strategic importer’s fuel consumption is also lower, thus causing fewer climate damages.

Keywords: bilateral market power, optimal tax and quota, carbon-based fuel trade, dynamic games, commodity markets JEL classification numbers : C63; C73; Q41; Q54. Sector boards: Environment; Energy

> ∗Karp: Department of Agricultural and Resource Economics, University of California, Berkeley, and the Ragnar Frisch Center for Economic Research; email: karp@berkeley.edu. Siddiqui: Department of Civil Engineering and the Johns Hopkins Systems Institute, Johns Hopkins University, Baltimore, MD 21218; email: siddiqui@jhu.edu. Strand: Development Research Group, Environment and Energy Team, The World Bank, Washington DC 20433; email:Jstrand1@worldbank.org. We benefitted from comments by Mike Toman, Franz Wirl, and seminar participants at the World Bank and the International Energy Workshop in Paris, June 2013. This research was supported by a grant from the World Bank’s Research Support Budget. The conclusions and viewpoints presented here are those of the authors and not necessarily of the World Bank, its management, directors or staff.
# 1 Introduction

A small group of countries account for most exports of fossil fuels, in particular oil on which this paper focuses. Another main group of countries, which includes most of the OECD, are fuel importers with only limited own oil production. The latter group of countries has already established, or may soon be expected to establish, policies for limiting the carbon emissions resulting from their fossil fuel consumption. The global consumption of fossil fuels increases stocks of greenhouse gasses (GHGs), likely altering the climate. Climate policy might serve as a coordination device, enabling a stragetic bloc of importers to affect the price of fossil fuels, while also controlling carbon emissions. In a game involving strategic importers and exporters, the equilibrium policy levels depend on the choice of instrument, e.g. a trade tax or quota. The equilibrium may also be sensitive to the presence of nonstrategic (passive) countries with fixed trade policies. These nonstrategic countries are “innocent bystanders”in the game among the strategic countries, but their presence alters the equilibrium to the game. The strategic exporter in our setting represents OPEC, and the strategic importer represents a subset of developed countries that might at some point in the future agree on a unified climate policy. That policy would likely involve trade measures, and would therefore affect both terms of trade and climate-related outcomes. The nonstrategic “innocent bystander” represents the poorer countries. We have two research questions: How does the presence of these countries affect the equilibrium outcomes in the games involving the strategic importer and exporter, under various combinations of policy instruments? How do the equilibria in the different games affect the welfare of the poorer countries?

In order to address these questions, we study a model in which a monopsonistic importer and a monopolistic exporter exercise market power, using either a trade tax or quota. There are four policy combinations, leading to four different games. We call the nonstrategic agent “R”, for Rest-of-World. R’s trade policy is fixed and exogenous: free trade in our setting. GHGs related to fossil fuel consumption accumulate in a stock variable that causes differing levels of damages to both the strategic importer and R; the strategic exporter incurs no damages. This assumption captures the idea that climate-related damages differ across countries, both in their actual effects, and in the way in which countries take these effects into consideration when determining their climate-related policies. In particular, we assume that, while OPEC countries may suffer damages from adverse climate developments, such damages are not reflected in the fuel-related policies of these countries.

In order to understand the forces at work, it helps to begin with a static framework in which there are no climate damages. In general, a country’s optimal trade tax equals the 1
inverse of their trading partner’s (tax or quota inclusive) elasticity of import demand or export supply. If a large importer and exporter are the sole agents in this market, and both use a trade tax, the Nash equilibrium taxes are positive. These taxes lower aggregate world welfare, but do not eliminate trade (Johnson 1953). In contrast, if both of these agents use quotas, there is zero trade in the Nash equilibrium (Tower 1975). For example, suppose that the importer takes the exporter’s quota as given, as in a Nash equilibrium. For any export quota, the importer has an incentive to set a still lower import quota, in order to render the export quota non-binding and thereby capture all of the quota rents from the exporter. The exporter who takes the import quota as given has the same incentive. Because this incentive holds for any positive quota, the only Nash equilibrium in the quota-setting game involves zero quotas for both agents, and thus zero trade.

In a two stage game where the countries choose their policy instrument (either a tax or a quota) in the first stage and the level of that policy in the second stage, countries’ first-stage dominant strategy is a tax. A country does not want to select a quota in the first stage, because it understands that if it does so, the second stage equilibrium involves zero trade if the rival also chooses a quota; if the rival chooses a tax, it extracts all of the quota rent by setting its tax at a level that makes the quota non-binding. This qualitative comparison survives in a dynamic setting where the level of a policy instrument (tax or quota) changes over time, as the stock of GHGs or some other state variable changes endogenously (Wirl 2012), (Rubio 2005).

The introduction of a nonstrategic third agent, R, into the static game qualitatively alters incentives, and therefore potentially alters the equilibrium (Karp 1988). Suppose for example that the exporter sets a quota and R has a downwardly sloping excess demand for the commodity. The combination of the export quota and R’s excess demand function causes the strategic importer to face a kinked, but not perfectly inelastic excess supply function. In this situation, the strategic importer does not, in general, want to set its trade policy to render the exporter’s quota non-binding. Such a policy still eliminates the exporter’s quota rents, but it (typically) is too costly for the importer, because most or all of the rents might be transferred to R. The exporter faces an analogous situation. It does not matter whether, in equilibrium, R is an importer or an exporter; its mere presence causes a country, whose rival uses a trade quota, to face a downwardly sloping excess demand function for some range of prices. Thus, the introduction of R eliminates both of the earlier results: the quota equilibrium does not drive trade to zero, and it may not be the case that choosing a tax is a dominant strategy in the first-stage game where countries choose their policy instrument. Even with R, the use of a quota causes the partner to face a less elastic excess supply or 2
demand curve (relative to the curves under a tax). Thus, using a quota encourages the trading partner to use an aggressive trade policy. For that reason, the forces that promote the adoption of taxes rather than quotas operate even with the presence of R, but they might no longer be determinative.

Our chief policy question concerns the equilibrium welfare effect, of different policy choices, on the poorer countries, R. In our setting, R is a net importer of fossil fuels. The strategic importer has two targets, its terms of trade and climate-related damages, and a single (state-dependent) instrument, the tax or quota. Both of its objectives encourage the importer to restrict trade. Its trade restriction lowers the equilibrium world fossil fuel price and slows the growth of the GHG stocks. R is a free rider; it benefits from both of these changes, because it is an importer and it suffers climate-related damages. R therefore prefers the strategic importer to use an aggressive trade restriction. The strategic exporter has a single target, improving its terms of trade, and a single instrument. A more aggressive export restriction raises the world price, harming R, and slows the accumulation of GHGs, benefiting R. Therefore, the welfare effect, on R, of the export policy is ambiguous. The presence of R causes carbon leakage. As the importer lowers the market price by restricting its own demand for fossil fuel imports, R increases its demand for those imports. As the exporter increases the market price by restricting its exports, R shifts from imports to domestic production.

Our calibration assumes that climate-related damages are small to moderate relative to the benefit of consuming fossil fuel. As a consequence, terms of trade considerations are more important than climate-related damages for both the strategic importer and R. Higher pollution stocks cause the strategic importer to use more aggressive equilibrium trade restrictions, in order to reduce future damages. As this importer’s demand for fossil fuels diminishes, with higher pollution stocks, the strategic exporter also lowers its export quota. The higher stocks directly harm the strategic importer, but affect the strategic exporter only indirectly, via reduced importer demand. Consequently, the importer’s policy is much more sensitive to the pollution stock, compared to the exporter’s policy: higher stocks reduce both equilibrium (strategic) imports and exports, but the effect on the former is greater. Therefore, higher pollution stocks increase the supply of imports available for R. At least for low stock levels (and in some policy scenarios for all stock levels), climate-related damages actually increase R’s welfare, simply because these damages cause I to reduce its demand for imports.

We also find that R’s payoff is highest when the strategic importer uses a tax, and the strategic exporter uses a quota. The exporter’s use of a quota encourages the strategic 3
importer to use a high tariff, in order to capture quota rents. The high tariff reduces the world fossil fuel price, benefitting R. If the strategic countries can choose the policy instrument (in addition to choosing the level of the policy), the unique Nash equilibrium in the policy selection game is for each to use a tax, just as in the simple static model without R. The tax is a dominant strategy for both players at every level of stock, so this equilibrium is subgame perfect, as is the level of every policy, conditional on the stock. Given our calibrations, the first-best stock trajectory under the social planner who uses a Pigouvian tax lies below the equilibrium trajectories in the four games corresponding to the four combinations of trade policy. The emissions reductions arising from strategic countries’ desire to improve their terms of trade, exceed the reductions due to the Pigouvian tax. Under the Pigouvian tax, there are no terms of trade incentives.

To check robustness, we consider other calibration assumptions. Plausibly, the climate externality could dominate the consumption benefits for some countries, notably countries in the fringe R; many of these could suffer substantial climate damages. In such cases, welfare to R would be higher in the long run, when bloc I uses quotas. I’s use of the quota lowers accumulation of atmospheric carbon, benefitting R in the long run.

Recent papers compare taxes and quotas in static models of the fossil fuel markets, with two strategic blocs and two fuels (Strand 2011), and with one fuel and a non-strategic bloc as in our paper (Strand 2013). Both papers find (as do we) that the strategic fuel importer prefers a tax policy over a quota policy. Earlier papers focus on using an import tax to capture a seller’s resource rent, (Bergstrom 1982), (Brander and Djajic 1983), (Karp 1984), (Karp and Newbery 1991). Climate policy may also be a means of capturing resource rents (Wirl and Dockner 1995), (Wirl 1995), (Amundsen and Schöb 1999), (Liski and Tahvonen 2004), (Rubio 2005), (Kalkuhl and Edenhofer 2010), (Njopmouo 2010). Wirl (2012), the paper closest to ours, studies a dynamic model with only two (strategic) blocs and no third (passive) bloc. This simpler model can be solved analytically. In this setting also, tax policies are dominant for both importer and exporter. Dong and Whalley (2009)’s computable general equilibrium model suggests that a 20% ad valorem carbon tax could increase real income in the U.S., E.U. and China by 0.4 —0.8%, while reducing OPEC real income by 5%. Jørgensen, Martín-Herrán, and Zaccour (2010) and Long (2010) survey applications of the type of dynamic game that we, and many of the other cited papers, use.

4
# 2 The Dynamic Game

There are three agents in the game, representing three regions: the strategic importer bloc (I), the strategic exporter bloc (E), and the nonstrategic rest of the world (R). The importer and exporter blocs, I and E, exercise market power, using either a trade tax or a quota. We take as given the combination of policy instruments and calculate the equilibria under the four policy combinations. By comparing payoffs, we determine the equilibrium policy choice. Region R is a price taker and can be either a net importer or exporter of fossil fuels, depending on the world price.

In period t, the strategic importer (I) incurs damages resulting from the stock of GHGs, xt. In order to emphasize the situation where agent I is more concerned than agent E about GHG accumulations, we suppose that only I and R suffer stock-related damages. These stocks are the only source of dynamics. In particular, we assume that extraction costs are independent of cumulative extraction, and we ignore the fact that resource stocks are finite. These assumptions produce a model with a single state variable, xt. In view of our functional assumptions and reliance on numerical methods, we could extend the model to include a second state variable, cumulative extraction, and thereby take into account the non-renewable resource aspect of the problem. However, in the one-state variable model we can present all important results graphically; those graphs would be less useful in a two-state model, and the results would be harder to interpret. Given the complexity of results in even the one state variable model, it is worth beginning there, despite the fact that such a model does not capture the real-world property that fossil resources are exhaustible.

The trajectory of the stock of GHGs is endogenous to the model. Our solution concept is a Markov Perfect equilibrium. In any period t, the current stock is predetermined, a function of past stocks and emissions. Both strategic players condition their period-t policy (level) on the period t stock level, the only “directly payoff relevant”state variable in this model. The equilibrium level of I’s policy in period t is a function of xt, which makes the importer’s problem dynamic. The GHG stock does not directly affect the exporter’s payoff, because by assumption E does not incur climate-related costs. However, I’s equilibrium policy (level) is conditioned on the stock, and I’s policies directly enter E’s payoff. Therefore, E also has a dynamic problem, and its equilibrium policy level also depends on the stock of GHGs. Because E and I solve mutually related dynamic problems, they play a dynamic game.

The rest of the world, R, responds passively, taking the world price as given. R’s presence in the model is essential for two reasons. First, we want to know how the strategic interaction of large buyers and sellers affects nonstrategic agents, in particular, developing countries.

5
Second, the presence of R’s net demand means that when either I or E use a quota, the other strategic agent does not face a perfectly inelastic demand or supply function. In the absence of R, there is 0 trade in the equilibrium when both strategic agents use a quota; if only one strategic agent uses a quota, the other strategic agent can capture all of the gains from trade by using a price policy, absent R. Matters are more complex in the more realistic situation where R is present in the market.

# 2.1 Flow payoffs

We assume that supply and demand curves are linear, the stock-related damage function quadratic, and that E and R’s average production costs increase in the rate of output (but are independent of the stock). The world fuel price, defined as the price that E receives and R pays is p. Consumers in I pay the price P . P − p equals the (possibly implicit) unit tax or quota rent in I. We first state the single period payoffs of the three agents, and then use these to define the dynamic game.

Country I has no domestic production; its demand for imports equals A − BP . The tariff revenue or the quota rents equal (P − p) (A − BP ). The climate-related damages, conditional on x, is<sup><u>d</u></sup> 2<sup>x2wheredisaconstant.Thestockxisanamalgramofallclimate-</sup> related variables, e.g. carbon stocks and temperature changes, and thus does not have a simple physical interpretation. Merely for purpose of exposition, we refer to it as the pollution stock. I’s single period payoff equals consumer surplus plus tariff revenue (or quota rents) minus environmental damages: At price p, R’s domestic demand is a − b0p and its domestic supply is b1p, so its net imports equal a − bp, with b0 + b1 ≡ b. R’s gains from trade minus its climate related damages<sup><u>κ</u></sup> 2<sup>x2equalitsflowpayoff:</sup> This payoff is not relevant to the solution to the game, because R is passive. However, the solution to the game determines the equilibrium trajectories of p and x. That information, together with R’s single period payoff, enables us to calculate the present value of the stream 6
of R’s payoff, and thereby enables us to see how different policies affect R’s welfare.

The exporter, E, has no domestic consumption and faces no stock-dependent costs. If the fuel export price is p and the (possibly implicit) export tax in region E is τ , E’s producers receive the price p−τ . These producers’marginal cost function, equal to E’s supply function, is g + f (p − τ ), where g and f are constants. The exporter’s single period payoff equals its domestic profits plus the tax revenue or quota rents Each agent has the same constant discount factor, β. Welfare for each agent equals the discounted stream of their single period payoff.

# 2.2 Single period equilibrium

We can express single period payoffs as functions of the state variable, x, and the control variables. The identity of the control variables depends on the policy scenario. A strategic player can either use a quota, Q for I and q for E, or they can use a unit tax, T for I or τ for E. It is also possible that one agent uses a quota and the other a tax, resulting in four scenarios.

If the agents both uses quotas (Q and q), the equilibrium conditions in E and in the world at large are If they both use taxes (T and τ ) equilibrium requires If I uses the tax T and E uses the quota q, equilibrium requires 7
If I uses the quota Q and E uses the tax τ , equilibrium requires The first two equations on the second lines of each of (4) —(7) give the equilibrium values of p and P as linear functions of the control variables (a combination of Q, q, T and τ ) for the four scenarios. If E chooses the tax, τ , as in the scenarios that correspond to equations (5) and (7), the optimality condition for E’s problem determines τ . If E chooses a quota, q, the equilibrium condition q = g + f (p − τ ), together with the requirement that aggregate demand equal aggregate supply, leads to an implicit tax, τ . This implicit tax is a linear function of the control variables, as shown by the third equation in the second lines of (4) and (6). The payoffs, presented in Section 2.1, are quadratic functions of the prices, controls and stock, T , τ , and x, and thus are quadratic functions of the control variables and x in each of the four scenarios. Given its rival’s level of trade policy, a country is indifferent whether it supports its own trade restriction using a quota or a tax. However, the level of its rival’s equilibrium trade restriction depends on both the level and form (a tax or quota) of its own policy instrument.

# 2.3 Dynamics

In a period, the current level of GHG stocks is predetermined, at level x. Region R’s domestic supply is b1p and E supplies q, so total emissions are b1p + q. The constant decay rate is δ so next period stock, denoted x<sup>′</sup> , is We use the same procedure as above to write the right side of this equation in terms of the control variables. For example, if both countries use quotas, we replace p with<sup>−q+</sup> b<sup><u>Q+a</u></sup> .

# 2.4 Calibration

This game is too complicated to easily produce analytic results, but too simple to provide an accurate empirical description of fossil fuel markets. We solve the model numerically and select parameter values to provide an economically meaningful context, so that the results are informative about world markets. We assume that, if I uses no trade restrictions, I 8
imports the fraction Λ < 1 of E’s exports, and R imports the remainder: for any price, I’s elasticity of demand equals R’s elasticity of net demand. We define Γ =<sup><u>b</u></sup> b<sup><u>0</u>,theslopeof</sup> R’s demand relative to the slope of its import demand. By varying Γ, we can change R’s fraction of world production in a competitive equilibrium. By choice of units, we set I’s demand intercept A = 8, and E’s supply slope f = 1. We assume that E’s production equals 0 at p = 0, implying g = 0. This assumption and the linearity of E’s supply implies that E’s elasticity of supply everywhere equals 1. Our second calibration assumption is that I’s elasticity of demand, evaluated at free trade, also equals 1. With these two calibration assumptions (and g = 0) and the normalizations A = 8 and f = 1, the choice of Γ and Λ determine the remaining supply and demand parameters.

|Variable|A= 8Λ<br>B =Λ|a= 8(1|−Λ)|b0=Γ(1−Λ)|b1=|(1−|Γ) (1−Λ)|
|---|---|---|---|---|---|---|---|
|Value|5.6<br>0.7|2.4||0.05||0|.25|
|b=b1+b2= 1−|Λ<br>g<br>Λ<br>Γ=|0.1667|f|d|δ|β|κ= 0.0000991|
|0.3|0<br>0.7<br>0.25|<sup>−3.0+4.0Λ</sup><br>−1.0+Λ|1<br>|2.31331×10<sup>−4</sup>|.99|.95|d(1−Λ)<br>Λ|

Table 1: Benchmark parameter values For our baseline, we set Λ = 0.7, so (absent import restrictions) I accounts for 70% of E’s exports, and Γ = 0.1667, so that in a competitive equilibrium E accounts for 80% of world supply Table 1 summarizes our parameter choices and the relation between Λ and Γ and the supply and demand parameters. Table 2 shows the formulae relating model parameters to the elasticities and Γ, Λ.

To assess the sensitivity of our results to parameters, we also considered the alternative Λ = 0.3. The choice Λ = 0.3 corresponds, roughly, to the situation where I represents Annex B countries under the Kyoto Protocol; Λ = 0.7 corresponds to a more aggressive policy scenario, where I includes the US and China; including all BRIC countries in I increases Λ above 0.8. The cumulative supply is about 10% higher with Λ = 0.3 compared to Λ = 0.7, because with the lower Λ, I has less market power and restricts its fuel demand less. But results are qualitatively unchanged. The results for this alternative calibration can be made available upon request.

|I’s demand elasticity =|E’s supply|E’s production|I’s consumption|
|---|---|---|---|
|R’s net demand elasticity|elasticity|share|share|
|8−g|<sup>8−g</sup>|−8f−g|Λ|
|8f+g|f <br>8f+g|−8f−8+8Λ−Λg+8Γ−Γg−8ΓΛ+ΓΛg||
|1|1|0.8|0.7|

9
Table 2: Economic interpretation of demand and supply parameters; all formulae evaluated at competitive equilibrium We choose the unit of time equal to a year and set the discount factor β = 0.95, for an annual discount rate of about 5.3%. The persistence parameter δ = 0.99 implies a half-life of the pollution stock of approximately 90 years. Despite the lack of a physical interpretation of the stock x (see above), it is important that there be an economic and physical interpretation of the parameter d, in order to give context to the model results. We obtain the parameter d as a function of previously chosen parameters and the level of a threshold stock above which it is optimal for I to consume nothing. We can choose the value of this threshold, and thereby choose the value of d, by answering the following question: How many years of consumption at the competitive level would it take to reach the threshold stock? Our choice of d is consistent with the answer “105 years”, implying a threshold value of x = 900, with an initial value x0 = 0. Appendix A explains this calibration procedure, which we intend only as a means of providing context for a numerical value that would otherwise be hard to interpret. Our results imply that for this value of d the environmental objectives are low relative to the terms of trade objectives; in that respect, our calibration represents low to moderate levels of damages.

We set R’s damage parameter κ = d(1Λ−Λ) . With this choice, the ratio of I and R’s damage, for any stock, equals the ratio of their import demand absent trade restrictions: I and R have the same relative benefit of consumption to cost of stock-related damage; they merely differ in size. As a second sensitivity experiment, we hold fixed other parameter values and double the value of κ, to represent a situation where R has much higher damages than I, taking into account their size difference.

# 3 Results

We study four scenarios, in which I chooses a sequence of either taxes or quantities, represented by T or Q, and E chooses a sequence of either taxes or quantities, τ or q. In each case, a player’s equilibrium control rule is a linear function of x, equal to ρ + σx for I and λ + μx for E. For example, if I chooses T and E chooses q, we have T = ρ + σx and q = λ + μx. The values of the four endogenous parameters, ρ, σ, λ, μ are different in the different scenarios. R does not use a policy, so it has no control rule. The equilibrium payoff — of each of the agents —the present discounted value of that agent’s future payoff stream is a quadratic function of the current stock. The payoff for I is V (x) = χ + ψx +<sup><u>ω</u></sup> 2<sup>x2,for</sup> 10
E is W (x) = ε + νx +<sup><u>φ</u></sup> 2<sup>x2,andforRisY (X)=ς+ ηx +</sup><sup><u>γ</u></sup> 2<sup>x2.</sup> Appendix B explains how we obtain the value of these parameters in the four scenarios. Table 3 lists the parameter names.

|Parameter|Importer|Exporter|ROW|
|---|---|---|---|
|Coef cient of x<sup>2</sup>|ω|φ|γ|
|Coef cient of x|ψ|ν|η|
|Constant in Value Function|χ|ε|ζ|
|Coef cient of x in Constant Rule|σ|μ|-|
|Constant in Control Rule|ρ|λ|-|

Table 3: Definition of endogenous parameters We use the model parameters from Section 2.4. We first discuss the parameter values for the endogenous value functions and control rules for the case where both strategic agents use quotas. We then compare the equilibrium stock trajectories, payoffs and prices in the four scenarios. We use information on the payoffs to determine the equilibrium to the game in which agents choose their policy instrument (a tax or quota).

With one exception, we compare results across different policy scenarios by comparing them to the corresponding result in the scenario where both strategic agents uses quotas, which we call the “reference scenario”. The graphs labeled ImpTExpT, ImpQExpT, and ImpTExpQ refer, respectively, to the graph of an outcome when both agents use tax policies, when I chooses a quota and E chooses a tax policy, and when I chooses a tax and E chooses a quota. In all cases but one, the outcome (e.g. a payoff, price, or quantity) is relative to the corresponding outcome in the reference scenario.

The exception is for I’s payoff, V (x), where the reference scenario trajectory passes through zero. For each of the four scenarios, these payoffs are initially positive, because the initial value of the stock is x = 0. However, as x increases, the payoffs become negative. The switch in sign occurs at a different time in each of the scenarios. Normalizing I’s payoff in scenario ImpTExpT, for example, by dividing by the payoff when both agents use quotas (ImpQExpQ) would involve dividing by 0. To avoid this problem, we show the payoffs for I in the four scenarios as levels, rather than ratios.

# 3.1 Equilibrium parameters when both agents use quotas

Table 4 shows the equilibrium values of the endogenous parameters under our baseline calibration, when both E and I use quotas. For all x, the importer’s payoff decreases with x, 11
so ω < 0 and ψ < 0. I’s equilibrium imports decrease as the stock rises, so σ < 0.

|Parameter|Importer|Exporter|ROW|
|---|---|---|---|
|Coef cient of x<sup>2 </sup>in value function|ω =−3.3×10<sup>−3</sup>|φ= 3.2156×10<sup>−6</sup>|γ = 2.3663×10<sup>−6</sup>|
|Coef cient of x in value function|ψ =−0.1435|ν =−2.75×10<sup>−2</sup>|η = 9.5×10<sup>−3</sup>|
|Constant in value Function|χ= 14.4634|ε= 120.9244|ζ = 19.6012|
|Coef cient of x in control rule|σ =−3.9290×10<sup>−4</sup>|μ=−1.7066×10<sup>−4</sup>|-|
|Constant in Control Rule|ρ= 0.5049|λ= 1.2624|-|

Table 4: Equilibrium values of endogenous parameters when E and I both use quotas.

Over relevant state space, E’s value function also decreases in the stock: ν < 0 and |ν| is large relative to φ. However, φ > 0, so E’s value function is convex in x. The stock has no direct effect on E, but as x increases, E faces decreasing demand from I (because σ < 0). I eventually becomes a negligible part of the market, so further decreases in its demand have a negligible effect on E’s payoff; hence, the convexity of E’s payoff in x. As I’s demand falls with the increase in x, E’s exports also fall: μ < 0. The fact that E suffers no direct loss in utility due to higher pollution stock means that its payoff is much less sensitive to x, compared to I’s payoff. (Compare the magnitudes of ω and φ and of ψ and ν.) I’s equilibrium quota is about twice as sensitive to the stock, compared to E’s equilibrium quota:<sup><u>σ</u></sup> μ<sup>≈2.</sup>

R’s payoff is a convex increasing function of the pollution stock (both γ and η are positive); over the relevant range of stocks, the relation is approximately linear (<sup><u>γ</u></sup> η<sup>≈0).</sup> A higher stock has offsetting effects on R’s payoff. The higher stock increases R’s damages, lowering its net payoff. The higher stock also decreases E’s supply and I’s imports, but the second effect is approximately twice as large as the first (<sup><u>σ</u></sup> μ<sup>≈2), so on balance a higher stock</sup> increases the supply absorbed by R, increasing its gains from trade. With our calibration, the higher gains from trade dominate the higher damages, so on balance higher pollution stocks R.

# 3.2 Equilibrium stock trajectories

Figure 1 (a) shows the pollution stock trajectories as functions of time in the quota-setting game, and in the first-best scenario where the social planner uses Pigouvian taxes. We defer discussion of the outcome under the social planner until Section 3.3 and here discuss the stock trajectories under the games corresponding to different combinations of trade policies.

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When an agent uses a quota rather than a tax, its trading partner faces a less elastic excess supply or demand function, and therefore has an incentive to use a more aggressive trade restriction. Figure 1 (b) shows that this comparison also holds in our dynamic setting.

The steady state when both countries use taxes is x = 329, much lower than the assumed threshold of x = 900; after 150 years the stock reaches 80% of its steady state level. The steady state stocks in the other policy scenarios range from 254 to 280; by year 150 the stocks in these scenarios also equal about 80% of their respective steady states.

# 3.3 Payoffs and instrument selection

Figure 2 shows the importer and exporter continuation payoffs (value functions V and W ) as functions of the stock. (Recall that the former is in levels, and the latter shows graphs relative to the ImpQExpQ levels, accounting for the difference in scale of the two figures.) The principal information from these figures is that the tax is a dominant strategy for both countries, at every stock level. If both countries believe that they can choose their policy instrument in perpetuity in the initial period, the unique Nash equilibrium is for both to choose a tax. If they have the opportunity to revisit this decision at any time in the future (i.e. at any stock level), the equilibrium policy choice does not change. Consider the more complex game in which, at each period, agents choose both their policy instrument and the level of the instrument. In the MPE to this game, both countries choose the tax, and the tax equals that of ImpTExpT.

Both countries’ payoffs decrease with the stock. The importer suffers stock-related damages. As the stock increases, I tightens its trade restriction, reducing the aggregate demand that E faces, and reducing E’s flow payoff and its continuation payoff. Because the importer suffers direct damages, its payoff is more sensitive to the stock, relative to the exporter’s payoff. The dots on the vertical axis identify the payoffs in the static game, obtained by setting the damage parameter to 0. Comparison of the dots corresponding to the static games and the intercepts corresponding to the dynamic games shows two facts. First, the payoff ranking is the same in the static and in the dynamic settings. Second, the payoffs in the dynamic setting lie only slightly below the corresponding values in the static setting. The differences reflect the fact that I suffers from damages in the dynamic game (harming I) and therefore uses more aggressive trade restrictions (harming E). Our calibration is consistent with relatively small damages. The static terms of trade considerations are much more important to I’s payoff, compared to environmental damages. The largest difference 14
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The non-monotonicity is easiest to see in Figure 3 (b), where we double R’s damages by doubling κ. Higher damages do not alter the comparison of static and dynamic payoffs; in this respect, damages remain small relative to the gains from trade, even when κ doubles. However, for higher damages R’s payoff is non-monotonic in time. Because the stock is monotonically increasing over time, we conclude that R’s payoff is non-monotonic in the stock. As above, a higher stock decreases I’s demand more than it lowers E’s supply, thereby increasing the supply available to R and increasing its gains from trade; and the higher stock increases R’s damages. At low stocks, early in the program, the first effect dominates; at high stocks, later in the program, the second effect dominates when we double the damage parameter κ. In this case, the relation between R’s payoff is first increasing and then decreasing over both time and over stock levels. When the climate damages are suffi ciently important for R, relative to fossil-fuel consumption, R’s payoff in the ImpTExpT game must eventually fall below its payoff in the ImpQExpT game, simply because the rate of accumulation of the carbon stock is greater in the former case.

# 3.5 Price and policy trajectories

Figure 4 (a) shows the equilibrium world price, p (the price that R pays and E receives) and Figure 4 (b) shows the importer’s domestic price P . As the stock increases and I tightens its trade restriction, P rises. As the stock increases and I’s import demand falls, the world price falls. E is in the strongest position to exercise market power when it uses a tax and I uses a quota; therefore, this scenario leads to the highest market fuel world price. I is in the strongest position to exercise market power when it uses a tax and E uses a quota; therefore, this scenario leads to the lowest world price. The other scenarios, where both agents use taxes or both use quotas, result in intermediate levels of the world price.

Recall that absent R, the equilibrium when both agents use quotas implies that no fuel is traded. As discussed in the Introduction, the presence of R moderates this extreme result. With R, it is too costly for the strategic agents to try to capture all of their rival’s quota rent. Nevertheless, trade between I and E is lowest in the quota setting game, so that scenario results in the highest domestic price for I. For similar reasons, trade between I and E is highest when both agents use tariffs, so that scenario leads to the lowest domestic price for I. The dots on the vertical axes show the equilibrium prices in the static games, where damage equal 0.

Figure 5 graphs the explicit or (in the case where an agent uses a quota) implicit trade tax. Consistent with our previous discussion, these figures show that an agent has the 17
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policy. The presence of this nonstrategic bloc means that even if a strategic country uses a trade quota, the excess supply or demand function facing its trading partner is not perfectly inelastic. We find, under our preferred calibration assumptions, that a tax policy by both the strategic importer and exporter constitutes the Markov (or subgame) equilibrium to this game, at any value of the state variable. This result echos results from related models, especially the static three-bloc model in Strand (2013), and the dynamic two-bloc model (without the fringe) in Wirl (2012).

The strategic importer and exporter both use trade policies to improve their terms of trade. The strategic importer also uses trade policy to control the future stock-related damages, but does not internalize the damages facing R. The fact that the stock changes over time renders the importer’s problem dynamic. Although the exporter has no intrinsic concern about the stock, its equilibrium trade policy depends on the importer’s policy and therefore is also stock dependent. For our calibration, the terms of trade objectives dominate the environmental objective in explaining policy levels. OPEC countries appear to be concerned that a unified climate policy among OECD countries might provide both “green cover”and a coordinating device that would enable the OECD countries to exercise greater market power in the fuel markets. Our results indicate that OECD countries might indeed have an incentive to behave strategically in this way; although our model has little to say about whether unified climate policy would actually induce such behavior. In our calibration, the strategic countries’terms of trade objectives and concern for country-specific damages, lead to smaller equilibrium pollution stocks than under the social planner who can use a Pigouvian tax.

The nonstrategic agent, R, also suffers stock-related damages. This country, a net fossil fuel importer, is a free rider, benefiting from the importer’s trade restriction; that restriction lowers the equilibrium price of fossil fuels and also reduces the equilibrium stock trajectory, lowering damages to R. R’s equilibrium payoff is higher in the dynamic setting, where it incurs stock related damages, compared to the static setting where it incurs no damages. The explanation is that stock-related damages cause the strategic importer to use more aggressive trade restrictions, benefiting R. The reduced competition for fossil fuel imports more than offsets the stock-related damages. The social planner’s optimal solution is to set a Pigouvian tax applied to all fossil fuel consumption, including by the fringe. In our model, by contrast, the fringe faces lower fossil fuel prices than the strategic importer.

Our calibration assumes that, under free trade, the strategic importer accounts for 70% of fossil fuel imports. This scenario corresponds to a situation where most large countries cooperate on trade and environmental policy; those two policies are indistinguishable in our 20
setting, where the strategic importer consumes but does not produce fossil fuels. We have also considered an alternative calibration, where strategic importers account for only 30% of imports under free trade. The qualitative results in the two cases are similar, although the smaller importer obviously has less market power and therefore uses less aggressive trade restrictions.

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