Optimal Tariffs on Exhaustible Resources: A Leader-Follower Model

Transcription

1 Optimal Tariffs on Exhaustible Resources: A Leader-Follower Model Kenji Fujiwara Department of Economics, McGill University & School of Economics, Kwansei Gakuin University Ngo Van Long Department of Economics, McGill University January 8, 010 Abstract This paper derives the time consistent feedback Nash equilibrium and two feedback Stackelberg equilibria for a dynamic game where an importing country and an exporting country choose their decision rules for setting tariffs and producer prices, respectively We obtain the equilibrium strategies, and numerically compare across scenarios) the welfare levels of each country and the world Using plausible parameter values, we find that the world welfare is highest in the Nash equilibrium while it is lowest in the Stackelberg equilibrium in which the importing country is a leader Keywords: dynamic game, exhaustible resource, feedback Nash equilibrium, feedback Stackelberg equilibrium JEL Classification: C73, Q34, F18 Correspondence Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec, H3A T7, Canada Tel: Fax: ngolong@mcgillca 1

2 1 Introduction There is a large theoretical literature on the exercise of market power in the trading relationship between a resource poor economy and a resource rich economy Broadly, this literature consists of three types of models The first group of model is characterized by the assumption that the seller exercises its market power while the importing country is passive 1 The second group of models considers the opposite scenario: the importing country imposes a tariff to shift resource rents from competitive, price-taking producers to the consumers The third group of models deals with the case where both seller and buyer realize they have market powers and behave strategically This group includes Karp 1984), Wirl 1994, 1995), Wirl and Docker 1995), Tahvonen 1995,1996), Rubio and Escriche 001), Liski and Tahvonen 004), Rubio 005), and Chou and Long 009) It consists of two subsets of models The first subset deals with Nash equilibrium only The second subset considers also Stackelberg equilibrium: both seller and buyer exercise their market power, but one of the player acts as a Stackelberg leader 3 Our paper is a contribution to the analysis of Stackelberg equilibrium when both buyer and seller have market power We shed new lights by clarifying the precise nature of the choice of strategies by the leader, and the response of the follower In Karp 1984), a section is devoted to a model where the feedback Stackelberg leader is the government of the importing country and the follower is the resource-extracting monopolist In that model the monopolist knows that at any point of time t, given the tariff, the greater is the quantity qt) he offers, the lower will be the current price pt); however the monopolist does not take into account the fact that he is facing a leader that uses a feedback policy where the tariff is dependent on the state vari- 1 See Kemp and Long 1979, Section 4) for a dynamic two-country general equilibrium formulation There is also a large literature on monopolistic supply in an exhaustible resource market in a partial equilibrium framework, see Long 009) for a survey See Newbery 1976), Kemp and Long 1980), Bergstrom 198), Brander and Djajic 1983), Karp 1984), Maskin and Newbery 1990), Karp and Newbery 1991, 199) 3 The second subset includes Karp 1984), Tahvonen 1996), Rubio and Escriche 001), and Rubio 005)

3 able) If he did take this into account, his behavior would be different from what the model describes Tahvonen 1996), Rubio and Escriche 001), and Rubio 005) use a Stackelberg formulation that has been called stage-wise Stackelberg leadership 4 model 5 This conception is best explained in a discrete time, finite horizon Assume the leader is the importing country In each period, the leader chooses the level of her control variable say the unit tariff τ) first, and the follower observes before he chooses the level of his control variable say the wellhead price of oil) Consider the last period, T, when players face the opening stock level, s T 1 The game in that period is then a simple static Stackerberg leadership game The equilibrium payoff of each player can be obtained, and expressed as a function of s T 1 Working backward, the solution for the second-last period can be computed, and so on Stage-wise Stackelberg leadership is thus conceptually very simple 6 Tahvonen 1996), Rubio and Escriche 001) and Rubio 005) find that when the seller is the stage-wise) Stackelberg leader, the equilibrium strategies of the leader and the followers are identical to the feedback Nash equilibrium In our paper, we use a different conception of feedback Stackelberg leadership What we are interested in may be called the hierarchical feedback Stackelberg leadership, as discussed in Dockner et al 000) 7 According to this conception, there is a hierarchical structure: the leader lets the follower know her decision rule, to which the latter responds by choosing the level of his control variable say his wellhead price) optimally His best reply can be expressed as a decision rule where the control level is a function of both the state variable and the decision rule of the leader Knowing this, the leader then chooses among all her possible decision rules the one that maximizes 4 These authors did not use the expression stage-wise This terminology appeared in Mehlmann 1988) 5 Tahvonen 1996), Rubio and Escriche 001), and Rubio 005) all use a continuous time, infinite horizon formulation 6 See Section 4 of Simaan and Cruz 1973), also Kydland 1975) and de Zeeuw and van der Ploeg 1991), where the formulations are in discrete time with a finite horizon For an application, see Turnovsky et al 1988) 7 See Shimomura and Xie 008), Long and Sorger 009) for further discussion 3

4 her life-time payoff Interestingly, we find that, in sharp contrast to Tahvonen 1996), Rubio and Escriche 001) and Rubio 005), the Stackelberg equilibrium when the seller is the leader is different from the Nash equilibrium This is also true when the buyer is the leader Developing a two-country model of exhaustible resource extraction where an importing country levies an import tariff on imports from the foreign country which monopolistically chooses a price, we derive three equilibria; i) feedback Nash equilibrium, ii) feedback Stackelberg equilibrium in which the importer leads, and iii) feedback Stackelberg equilibrium in which the exporter leads By making use of the time consistency condition, we seek to characterize their analytical solutions Then, we numerically compare welfare levels of each individual country and the world A large set of parameter values allows us to establish that the world welfare is highest in the Nash equilibrium whereas it is lowest in the Stackelberg equilibrium in which the importing country is a leader This implies that eliminating leadership of either the importing country, eg, Japan, or the exporting country, eg, OPEC, is welfare-improving from the world perspective 8 This paper is organized as follows Section builds a basic model Sections 3 solves for the feedback Stackelberg equilibrium in which the importing country is a leader and Section 4 turns to the other Stackelberg equilibrium with a leadership of the exporting country Based on the related paper, Chou and Long 009), Section 5 briefly addresses the Nash equilibrium Section 6 makes welfare comparisons among these three cases Section 7 discusses the difference in implications between our solution and another feedback Stackelberg solution considered in the literature Section 8 concludes the paper 8 This result may, however, be no longer valid if the exporting country chooses a quantity decision rule instead of a price decision rule, as shown in a companion paper, Fujiwara and Long 009) 4

5 The Model There are two countries, an importing country Home) and an exporting country Foreign) Foreign exclusively exports the good to Home Home imposes a specific tariff on imports Foreign has a stock of resource X The surface area is unity, so the depth at which the last unit of resource can be found is X The marginal cost of extraction increases with the depth of the mine Let S be the depth reached and q be the rate of extraction, which is the same as the rate at which the depth increases as extraction proceeds: Ṡ = q 9 At any time, the cost of extracting q is csq, ie, the marginal cost of extraction is cs Thus, the deeper one has to go down, the higher is the marginal cost Home s inverse demand function of the resource good is p c = a q, a > c, 1) where p c is the price the consumers have to pay per unit The parameter a is the choke price It is the marginal utility of consuming the first unit Let S denote the depth at which the marginal extraction cost equals the choke price, ie, cs = a We assume that X is larger than S Then, efficiency implies that the resource stock be abandoned at S = a/c, ie, before physical exhaustion of the stock Let τ be a specific tariff rate levied on imported resources Then, the consumer price is the sum of the producer price p and the tariff rate: p c = p + τ ) From 1) and ), the quantity demanded can be expressed as a function of p and τ as q = a p τ, from which the resource dynamics is described by Ṡ = a p τ 3) Chou and Long 009) solve for a feedback Markov perfect) Nash equilibrium of a game between Home and Foreign They assume that Foreign, 9 In what follows, we suppress the time argument t unless confusion arises 5

6 taking as given Home s feedback tariff rule τ = τs), chooses a feedback producer price rule p = ps) to maximize the discounted stream of profit: 0 e rt πdt = 0 e rt p cs) [a p τs)] dt 4) Taking Foreign s feedback producer price rule p = ps), Home chooses a feedback tariff rule τ = τs) to maximize the discounted stream of the sum of consumer surplus and tariff revenue: 0 e rt W dt = 0 rt [a ps) + τ] [a ps) τ] e dt 5) Assuming that both players simultaneously move, Chou and Long 009) show that the feedback Nash equilibrium has the following properties: i) ps) is linear affine in S with p S) > 0 and p S ) = a ii) τs) is linear affine in S with τ S) < 0 and τ S ) = 0, and iii) the depth St) approaches S as t The feedback Nash equilibrium has an appealing property: as long as marginal cost cs is below the choke price a, extraction should proceed And extraction should never be at a depth where marginal cost cs exceeds the choke price a This paper intends to find feedback Stackelberg equilibria of this game We consider both the case in which Home leads and the case in which Foreign leads We begin with the former case, in the sense that Home announces in advance to Foreign that it is committed to a feedback tariff rule τs) Clearly, the leader can choose the Nash equilibrium tariff rule that Chou and Long 009) find, and thus achieve exactly the same outcome in terms of price, quantity, and welfare as in the Nash equilibrium But, the leader may be able to do better 6

7 3 Feedback Stackelberg Equilibrium with Importer s Leadership To find a Stackelberg equilibrium in which Home is a leader, we suppose that it announces right at the beginning of the game a linear feedback tariff rule τs) = αs + β Since the game is solved backward, let us consider the follower s problem Foreign s Hamilton-Jacobi-Bellman HJB) equation is rv S) = max {p cs)a p αs β) + V p S S)a p αs β)}, where V S) is Foreign s value function and V S S) dv S)/dS The firsorder condition for maximizing the right-hand side yields Foreign s strategy: ps) = V S S) + α + c)s + a β Given the linear-quadratic structure of the game, it is plausible to guess that V S) is quadratic in S: V S) = A S + B S + C for S [ 0, S ], 6) where A, B and C are undetermined coefficients which are endogenously derived below This immediately leads to V S S) = A S + B and the above strategy is rewritten as p = A α + c)s B + a β 7) Substituting these results into 6), we have an identity in S: A ) [ A r S + B S + C α c)s + B ] + a β = Equating the coefficients multiplied by S and S, and constant terms, we have ra = A α c ) rb = A α c)b + a β) B rc ) + a β = 7

8 Solving the first equation for A yields A = α + c + r ± rα + c + r) > 0 Substituting 7) into the resource dynamics yields Ṡ = a αs β A α + c)s B + a β = r ± S + B + a β Therefore, in order to guarantee the asymptotic stability, we need to require r ± ) / < 0 As a result, A is determined as A = α + c + r 8) Substituting 8) into the second equation above and solving for B, we obtain ) α c r + a β) B = 9) α + c Substituting 9) into the third equation, C becomes ) C = 1 r + a β) 10) 4r α + c Finally, substituting 8) and 9) into 7), the exporting firm s strategy is explicitly derived as ps) = α S + β = α r + ) α + c + r a β) S + 11) α + c) Having described the follower s behavior, let us turn to the leader s problem To this end, substituting 11) into 3), the resource dynamics under linear strategies is Ṡ = α + α )S + a β β, 8

9 the solution of which is S = e α+α )t S 0 a β β α + α ) + a β β α + α The instantaneous objective of Home under linear strategies τs) = αs + β and ps) = α S + β is W = α α ) S [αβ + α a β )] S + a β ) β Substituting the above solution of S, and α and β into this and rearranging terms, we obtain W = r 3α + c + r) α + r) e r )t S 0 a β α + c ) r αa + βc) + e r t S 0 a β ) α + c α + c Taking the integral of this function, Home s objective function is finally obtained as 0 e rt W dt = r3α + c + r) α + r) S 0 a β α + c + r ) αa + βc) ) S 0 a β ) 1) r + α + c) α + c Home chooses α and β to maximize 1) Therefore, they are obtained by solving the first-order conditions by differentiating 1) with respect to α and β However, such solutions for α and β would depend on S 0, which implies that any replanning at a future date t 1 where St 1 ) > S 0 will give a new solution, ie, the feedback rule would be time inconsistent To ensure time-consistency, we impose the restriction that αa + βc = 0 Under this restriction, the above maximization problem amounts to max α r3α + c + r) α + r) S 0 a c ) The first-order condition is [ r 1 3 ] α + c + r) + c + r = α + c + r) 3 9 ) )

10 While it is extremely difficult to obtain an explicit solution of α in this equation, we can guarantee the existence of the solution Since we want rα + c + r) > 0, let us define λ α + c + r and rewrite the above equation as 3r λ + r Squaring both sides, we have 9r 16 λ + 3r 4 c + r c + r ) = λ 3 ) λ + r c + r = λ 4 ) 3 Let us define fλ) = λ 3 9r 16 λ 3r c + r ) λ 4 ) The rest of our task is to find λ > 0 that satisfies fλ) = r 4 c + r The ) function fλ) has the properties that f0) = 0 and f 0) = 3r 4 c + r < 0 Noting that f ) = and f ) =, we conclude that fλ) = 0 at three values, λ = 0, λ 1 < 0, and λ > 0, and that there exists a unique λ ) which satisfies fλ ) = r 4 c + r This implies that there exists a unique value of α which maximizes Home s objective function Finally, β is derived as β = αa/c This result is summarized as follows Proposition 1 There exists a unique feedback Stackelberg equilibrium in linear strategies where Home the importing country) is a leader 4 Feedback Stackelberg Equilibrium with Exporter s Leadership This section turns to the other stage-wise game in which Foreign is a leader Supposing that Foreign chooses a feedback rule ps) = α S + β, Home s problem is max e rt a α S β + τ)a α S β τ) dt τ 0 st Ṡ = a α S β τ 10

11 The HJB equation associated with this problem is rv S) = { a α S β + τ)a α S β τ) max τ +V S S) a α S β τ)}, where V S) is Home s value function and V S S) is its derivative with respect to S The first-order condition for maximizing the right-hand side yields by assuming V S) = AS / + BS + C equation, we have an identity in S: τs) = V S S) = AS B, 13) Substituting this into the HJB ) A r S + BS + C = [A α )S + B + a β ] By applying the procedure developed in the last section, the three parameters are A = α + r Γ 14) B = α + r Γ ) a β ) 15) α ) C = 1 r + Γ a β ) 16) r α Γ r 4α + r) > 0 Substituting these into 13), the follower s strategy is τs) = α r + Γ S + Since the dynamics of S is Ṡ = a α S β + AS + B, the solution is obtained as S = e A α )t S 0 a ) β + B α A = e r Γ t S 0 a ) β α 11 α + r Γ ) a β ) α 17) + a β α + a β + B α A

12 Let us now consider the exporting firm s problem profit is The instantaneous π = p cs)a p τ) = α S + β cs)a α S β + AS + B) Substituting A, B and the explicit solution of S into this and making some arrangements yield π = α c) r Γ ) Taking the integral, we have e r Γ)t + [aα ca β )] r Γ ) α S 0 a β α e r Γ t ) S 0 a β α e rt πdt = α c) 4α r + Γ ) S 0 4α 0 a ) β + r α [aα ca β )] α + r Γ ) S α 0 a ) β α ) 18) As was in the last section, let us make the time consistency condition: aα ca β ) = 0 under which our problem reduces to α c) 4α r + Γ ) max S α 4α 0 a ) + r c The first-order condition is α + c + r) [r4α + r)] 1 = 4α + r), which is again almost impossible to explicitly solve Thus, we prove the unique existence of solution by transforming a variable and resort to a numerical analysis Let us define γ = 4α + r and square the above equation Then, we have γ r + c + r ) γ = γ 4, 1

13 which is equivalent to a cubic equation of γ: r [ γ + 4c + r) + 4c + r)γ ] = 4γ 3 We must find γ > 0 that satisfies this condition Defining gγ) = 4γ 3 rγ r4c + r)γ, the rest of our task to find a positive γ which satisfies gγ ) = rr + 4c) Since g0) = 0, g 0) < 0, lim γ gγ) = and lim γ gγ) =, we find three solutions to gγ) = 0: γ = 0, γ 1 < 0 and γ > 0 Therefore, we have arrived at: Proposition There exists a unique feedback Stackelberg equilibrium in linear strategies where Foreign the exporting country) is a leader 5 Feedback Nash Equilibrium Having derived two types of Stackelberg equilibria, we briefly consider the Nash equilibrium 10 Chou and Long 009) show that the feedback Nash equilibrium involves the following pair of the HJB equations rv S) = 1 8 [a cs + V SS) + V S S)] rv S) = 1 4 [a cs + V SS) + V S S)], ie, V S) = V S) and V S S) = V S S) Substituting these into Home s HJB equation yields rv S) = 1 8 [a cs + 3V SS)] Let us suppose a quadratic value function V S) = AS / + BS + C Then, we have V S) = A S + BS + C = µ 8r S aµ 4rc S + 1 r 10 For the details in this section, see Chou and Long 009) ) aµ, c 13

14 [ where µ = 3 r + 3cr) 1/ r ] Since V a/c) = 0, the boundary condition is satisfied Starting at time zero where S = 0, the equilibrium welfare of each country is V 0) = 1 ) aµ r c V 0) = 1 ) aµ r c Then, the Nash equilibrium strategy pair is τs) = a c S ) µ 4r ps) = c + µ S + 6 ar + µ)µ rc Note that τa/c) = 0 and that pa/c) = a The equilibrium path of accumulated extraction is S = a c ) 1 e µt The equilibrium producer s price path is [ ) c + µ p = a 1 6c e µt which allows us to know that pt) > 0 for all t The equilibrium tariff is Thus, we can establish: ) τ = µ a e µt 4r c ], Proposition 3 There exists a unique feedback Nash equilibrium in linear strategies 6 Welfare Comparison We now have three feedback solutions and the corresponding levels of welfare of each country and the world Our finding is summarized in Tables These tables provide us with several interesting observations 11 Detailed calculations are available from the authors upon request 14

15 First, the world welfare which is the sum of the two countries welfare is largest in the Nash equilibrium and lowest in the Stackelberg equilibrium where the importing country is a leader This result relies crucially on the assumption that the exporting country uses a price decision rule Might it be reversed once the exporting country s strategy on price a price decision rule) is replaced by a quantity decision rule? In a companion paper, assuming that the exporting firm uses a quantity decision rule, Fujiwara and Long 009) show that the Nash equilibrium indeed yields the smallest world welfare compared to either Stackelberg equilibrium; in particular the Stackelberg equilibrium with the importer s leadership yields the largest world welfare 1 The second point, which is related to the first one, is that under price strategies the Stackelberg equilibria are not Pareto superior to the Nash equilibrium In other words, both players welfare can not improve simultaneously by moving from the Nash equilibrium to either Stackelberg equilibrium This is again reversed in the quantity setting model of Fujiwara and Long 009), where in any one of the two Stackelberg equilibria, both players payoffs are higher than in the Nash equilibrium That is, when the exporting country determines the output, there is no conflict of interest between countries concerning the movement from Nash to Stackelberg Finally, we note that our results in Table 3 are different from those of Karp 1984, p 90) who analyzes a case in which the importing country Home in our context) leads Karp provides a numerical example, in which the importing country s welfare is 004 and the exporting country s welfare is 0103, implying that the exporting country achieves a higher welfare level than the importing country In contrast, under the same parameter values, our example indicates the reverse: the welfare of the importing country is about 03a, which is much higher than that of the exporting country, about 005a The reason for this difference in results can be explained as follows Karp 1984) mentions two methods that can ensure time-consistency of the leader s 1 This contrast is analogous to the sharp contrast between the implications of the Brander-Spencer model and those of the Eaton-Grossman model of strategic trade policies 15

16 tariff policy The first method, originating from Simaan and Cruz 1973), assumes that the leader treats the follower s dynamic programming equation as a constraint and solves her problem using the dynamic programming method The second method, proposed and used by Karp 1984) to compute his numerical example, is based on two assumptions: first, the leader acts as if she ignored the effect of her decision on the rent of the exporting country, and second, when the follower calculates the shadow price of his stock, he ignores the fact that the leader will follow a feedback rule 13 These assumptions make Karp s result different from ours 7 Discussion The feedback Stackelberg equilibria we have derived can be called a hierarchical feedback Stackelberg equilibria since the leader determines its strategy over the entire horizon, prior to the strategy choice of the follower On the other hand, a different concept of feedback Stackelberg equilibrium has been used in literature, which has been called a stagewise feedback Stackelberg equilibrium by Basar and Olsder 1995) and Mehlmann 1988) In a framework similar to ours, Tahvonen 1996) and Rubio and Escriche 001) characterize the stagewise feedback Stackelberg equilibria and conclude that, when the exporting country leads, the Stackelberg equilibrium is identical to the Nash equilibrium They also find that this is not the case when the importing country leads This subsection shows how stagewise leadership leads to numerical values that are totally different from hierarchical leadership Assuming quadratic value functions, V S) = AS / + BS + C and V S) = A S / + B S + C, the stagewise feedback Stackelberg equilibrium with Home s leadership is obtained as follows 14 A = 7r + 30c 9 Ψ There is a third assumption, which is less important: the control is not at a boundary of the feasible set 14 Derivations are found in Appendix A in Rubio and Escriche 001) 16

17 B = 3 r Ψ ) 5 3r + Ψ ) ) C = 3 3r Ψ a r 10c A = 3 A, B = 3 B, C = 3 C Ψ = r9r + 0c) > 0 Tables 4-6 show each country s payoff in stagewise feedback Stackelberg equilibria under the same parameter specifications as the previous sections In comparison with Tables 1-3, the differences are striking 8 Concluding Remarks We have considered some welfare implications of a dynamic game model of international trade involving exhaustible resource extraction This paper possibly has contributed to both international economics and dynamic game theory While we have characterized three feedback solutions in a dynamic trade model with a linear-quadratic structure where both countries exercise market power, our technique of deriving the feedback solutions through using the time consistency condition is applicable to a wide variety of dynamic games 15 However, our attempt is preliminary, leaving much unexplored Among others, we have exclusively focused on the linear feedback solutions As is recently documented by Shimomura and Xie 008), dynamic Stackelberg games can admit nonlinear strategies which may lead to a superior outcome than linear strategies 16 We plan to tackle non-linear strategies in our future research 15 Kemp, Long and Shimomura 001) derive the feedback Nash equilibrium of a twocountry model of tariff war in a model with the linear-state structure In their model, the feedback strategies are independent of state variables 16 Shimomura and Xie 008) derive a useful Implementation Lemma, but their argument assumes the existence of the equilibrium Further exploration is found in Long and Sorger 009) 17

18 References [1] Basar, T and G Olsder 1995), Dynamic Noncooperative Game Theory, San Diego, Academic Press [] Bergstrom, T C 198), On Capturing Oil Rents with A National Excise Tax, American Economic Review 7: [3] Brander, J and S Djajic 1983), Rent Extracting Tariffs and the Management of Exhaustible Resources, Canadian Journal of Economics, 16, [4] Chou, S and N V Long 009), Optimal Tariffs on Exhaustible Resources in the Presence of Cartel Behavior, Asia-Pacific Journal of Accounting and Economics, forthcoming [5] de Zeeuw, A J and F van der Ploeg 1991), Difference Games and Policy Evaluation: A Conceptual Framework, Oxford Economic Papers 43: [6] Fujiwara, K and N V Long 009), Optimal Tariffs on Exhaustible Resources: the Case of Quantity-Setting, Working Paper, Department of Economics, McGill University [7] Karp, L 1984), Optimality and Consistency in a Differential Game with Non-Renewable Resources, Journal of Economic Dynamics and Control 8: [8] Karp, L and D M Newbery 1991), Optimal Tariffs on Exhaustible Resources, Journal of International Economics, Vol 303-4): [9] Karp, L, and D M Newbery 199), Dynamically Consistent Oil Import Tariffs, Canadian Journal of Economics, 5, 1-1 [10] Kemp, MC and N V Long 1979), The Interaction between Resource- Poor and Resource-Rich Economies, Australian Economic Papers 18: 18

19 58-67, reprinted as Essay 17 in M C Kemp and N V Long Eds), Exhaustible Resources, Optimality and Trade, 1980, pp197-09, North Holland, Amsterdam [11] Kemp, M C, N V Long and K Shimomura 001), A Differential Game Model of Tariff War, Japan and the World Economy, 133): [1] Kydland, F 1975) Non-cooperative and Dominant Player Solutions in Discrete Dynamic Games, International Economic Review 16: [13] Liski, M and O Tahvonen 004), Can Carbon Tax Eat OPEC s Rents? Journal of Environmental Economics and Management, 47: 1 1 [14] Long, N V 009), A Survey of Dynamic Games in Economics, World Scientific to appear) [15] Long, N V and G Sorger 009), A Dynamic Principal-Agent Problem as a Feedback Stackelberg Differential Game, Working Paper No 0905, Department of Economics, University of Vienna [16] Maskin and Newbery 1990) Disadvantageous Oil Tariffs and Dynamic Consistency, American Economic Review 801): [17] Mehlmann, A 1988), Applied Differential Games, New York, Kluwer Academic Publishers [18] Newbery, D 1976), A Paradox in Tax Theory: Optimal Tariffs on Exhaustible Resources, Unpublished manuscript [19] Newbery, D 1981), Oil Prices, Cartels, and the Problem of Dynamic Inconsistency, Economic Journal 91: [0] Rubio, S 005), Tariff Agreements and Non-Renewable Resource International Monopolies: Prices versus Quantity, Working Paper WP-AD , University of Valencia 19

20 [1] Rubio, Santiago and Luisa Escriche 001), Strategic Pigouvian Taxation, Stock Externalities and Polluting Non-renewable Resources, Journal of Public Economics 79: [] Shimomura, K and D Xie 008), Advances on Stackelberg Open-Loop and Feedback Strategies, International Journal of Economic Theory, Vol 41): [3] Simaan, M and J B Cruz 1973), Additional Aspects of the Stackelberg Strategy in Non-Zero Sum Games, Journal of Optimization Theory and Applications, 11: [4] Smith, J L, 009), World Oil: Market or Mayhem?, Journal of Economic Perspective, Vol 33): [5] Tahvonen, O 1996), Trade with Polluting Non-Renewable Resources, Journal of Environmental Econ and Management 30: 1-17Wirl, F 1994), Pigouvian Taxation of Energy for Flow and Stock Externalities and Strategic, Non-competitive Energy Pricing, Journal of Environmental Economics and Management 6: 1-18 [6] Wirl, F 1995), The Exploitation of Fossil Fuels under the Threat of Global Warming and Carbon Taxes: A Dynamic Game Approach, Environmental and Resource Economics 5: [7] Wirl, F and Dockner, E 1995) Leviathan Governments and Carbon taxes: Costs and Potential Benefits, European Economic Review 39:

21 Home Foreign Total Nash a a a Stackelberg Home is leader) 075a 005a 035a Stackelberg Foreign is leader) a a a Table 1: Example 1: payoffs under S 0 = 0, r = 005 and c = 1 Home Foreign Total Nash 04a 08a 1a Stackelberg Home is leader) a a a Stackelberg Foreign is leader) a a a Table : Example : payoffs under S 0 = 0, r = 005 and c = 05 Home Foreign Total Nash & Foreign is leader a a a Stackelberg Home is leader) a a a Stackelberg Foreign is leader) a a a Table 3: Example 3: payoffs under S 0 = 0, r = 01 and c = 1 1

22 Home Foreign Total Nash & Foreign is leader a a a Stackelberg Home is leader) a a a Table 4: Example 1 stagewise solution): payoffs under S 0 = 0, r = 005 and c = 1 Home Foreign Total Nash & Foreign is leader 04a 08a 1a Stackelberg Home is leader) a a a Table 5: Example stagewise solution): payoffs under S 0 = 0, r = 005 and c = 05 Home Foreign Total Nash a a a Stackelberg Home is leader) a a a Table 6: Example 3 stagewise solution): payoffs under S 0 = 0, r = 01 and c = 1