# American Institute of Mathematical Sciences

April  2017, 4(2): 149-173. doi: 10.3934/jdg.2017009

## Nash and social welfare impact in an international trade model

 1 Department of Mathematics and LIAAD-INESC, Faculty of Sciences, University of Porto, Rua do Campo Alegre, 687,4169-007 Porto, Portugal 2 IMPA, Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, RJ 22460-320, Rio de Janeiro, Brasil

Received  December 2015 Revised  January 2017 Published  March 2017

We study a classic international trade model consisting of a strategic game in the tariffs of the governments. The model is a two-stage game where, at the first stage, governments of each country use their welfare functions to choose their tariffs either (ⅰ) competitively (Nash equilibrium) or (ⅱ) cooperatively (social optimum). In the second stage, firms choose competitively (Nash) their home and export quantities. We compare the competitive (Nash) tariffs with the cooperative (social) tariffs and we classify the game type according to the coincidence or not of these equilibria as a social equilibrium, a prisoner's dilemma or a lose-win dilemma.

Citation: Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics and Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009
##### References:
 [1] K. Bagwell and R. Staiger, A theory of managed trade, American Economic Review, 80 (1990), 779-795. [2] K. Bagwell and R. Staiger, Enforcement, private political pressure, and the GAT/WTO escape clause, Journal of Legal Studies,, 34 (2005), 471-513. [3] N. Banik, F. A. Ferreira, J. Martins and A. A. Pinto, An Economical Model for Dumping by Dumping in a Cournot Model, chapter in "Dynamics, Games and Science Ⅱ: DYNA 2008, in Honour of Maurício Peixoto and David Rand", editors: M. M. Peixoto, A. A. Pinto and D. A. Rand, pp. 141-154, Springer, 2011. doi: 10.1007/978-3-642-14788-3_11. [4] J. A. Brander, Intra-industry trade in identical commodities, Journal of International Economics, 11 (1981), 1-14. [5] J. A. Brander and B. J. Spencer, Export subsidies and international market share rivalry, Journal of International Economics, 18 (1985), 83-100. [6] J. I. Bulow, J. D. Geanakoplos and P. D. Klemperer, Multi-market oligopoly: Strategic substitutes and complements, Journal of Political Economy, 93 (1985), 488-511. [7] M. Choubdar, J. P. Zubelli and A. A. Pinto, Nash and Social Welfare Impact in International Trade, Recent Advances in Applied Economics, Proceedings of the 6th International Conference on Applied Economics, Business and Development (AEBD'14) Lisbon, Portugal, (2014), 23-26. [8] M. Choubdar, E. Farias, F. A. Ferreira and A. A. Pinto, Uncertainty costs on an international duopoly with tariffs, Proceedings of the 6th International Conference on Applied Economics, Business and Development (AEBD'14) Lisbon, Portugal, (2014), 13-16. [9] A. Dixit, International trade policy for oligopolistic industries, Economic Journal, 94 (1984), 1-16. [10] A. Dixit, Strategic aspects of trade policy, in Bewley, T. (Editor), Advances in Economic Theory, Cambridge University Press, 329–362. [11] A. Dixit and G. Grossman, Targed export promotion with several oligopolistic industries, Journal of International Economics, 21 (1986), 233-249. [12] G. Eaton and G. Grossman, Optimal trade and industrial policy under oligopoly, Quarterly Journal of Economics, 101 (1984), 383-406.  doi: 10.2307/1891121. [13] F. A. Ferreira, Applications of Mathematics to Industrial Organization, Ph. D. Thesis, Universidade do Porto, Portugal, 2007. [14] F. A. Ferreira, F. Ferreira, M. Ferreira and A. A. Pinto, Quantity competition in a differentiated duopoly, Chapter in J. A. Tenreiro Machado, Bela Patkai and Imre J. Rudas (Eds.): Intelligent Engineering Systems and Computational Cybernetics. Springer Science+Business Media B. V. , (2008), 365–374. [15] F. A. Ferreira, F. Ferreira and A. A. Pinto, Flexibility in Stackelberg leadership, Chapter in J. A. Tenreiro Machado, Bela Patkai and Imre J. Rudas (Eds.): Intelligent Engineering Systems and Computational Cybernetics, Springer Science+Business Media B. V. , (2008), 399–405. [16] F. A. Ferreira, F. Ferreira and A. A. Pinto, Bayesian price leadership, Chapter in Tas, K. et al. (eds.): Mathematical Methods in Engineering, Springer, Dordrecht, (2007), 371–379. [17] F. A. Ferreira, F. Ferreira and A. A. Pinto, Unknown costs in a duopoly with differentiated products, Chapter in Tas, K. et al. (eds.): Mathematical Methods in Engineering, Springer, Dordrecht, (2007), 359–369. [18] F. A. Ferreira, F. Ferreira and A. A. Pinto, Uncertainty on an asymmetric duopoly, Progress in Industrial Mathematics at ECMI 2006, Proceedings 14th European Conference on Mathematics for Industry, Madrid, Spain, July 10-14. Springer, Berlin (2006) to appear. [19] E. O'N. Fisher and C. A. Wilson, Price competition between two international firms facing tariffs, International Journal of Industrial Organization, 3 (1995), 67-87. [20] R. Gibbons, A Primer in Game Theory, Pearson Prentice Hall, Harlow, 1992. [21] G. M. Grossman, Strategic export promotion: A critique, Chapter 3 in: Strategic Trade Policy and the New International Economics, P. R. Krugman (editor), MIT Press, Cambridge MA, (1986), 47–68. [22] G. V. Haberler, The Theory of International Trade with its Application to Commercial Policy, Macmillan, New York, 1937. [23] E. Helpman, Increasing Returns, Imperfect Markets, and Trade Theory, Jones, R. W. , Kenen, P. B. (eds.): Handbook of International Economics, 1 North Holland Press, Amesterdam, 1984, Chapter 7. [24] J. -M. M. Kilolo, Country Size, Trade Liberalization and Transfers, MPRA Paper, University Library of Munich, Germany, 2013. [25] K. Krishna, Trade restrictions as facilitating practices, Journal of International Economics, 26 (1989), 251-270. [26] P.-C. Liao, Rivalry between exporting countries and an importing country under incomplete information, Academia Economic Papers, 32 (1990), 605-630. [27] N. Limao and K. Saggi, Tariff retaliation versus financial compensation in the enforcement of international trade agreements, Journal of International Economics, 76 (2008), 48-60. [28] N. Lim'ao and K. Saggi, Size inequality, coordination externalities and international trade agreements, y, coordination externalities and international trade agreements,, 63 (2013), 10-27. [29] J. Martins, N. Banik and A. A. Pinto, A Repeated Strategy for Dumping, to appear in Discrete Dynamical Systems and Applications, ICDEA 2012, editors: Lluís Alseda, Jim M. Cushing, Saber Elaydi and A. A. Pinto, Springer. doi: 10.1007/978-3-662-52927-0_11. [30] J. Martins and A. A. Pinto, Deviation from collusion with and without dumping, to appear in Modeling, Dynamics, Optimization and Bioeconomics Ⅱ, editors: A. A. Pinto and D. Zilberman, Springer. [31] J. McMillan, Game Theory in International Economics, Harwood Academic Publishers, Chur, Switzerland, 1986. doi: 10.1080/00036846900000024. [32] A. A. Pinto, B. M. Oliveira, F. A. Ferreira and F. Ferreira, Stochasticity favoring the effects of the R & D strategies of the firms, Chapter in J. A. Tenreiro Machado et al. (Eds.): Intelligent Engineering Systems and Computational Cybernetics, Springer Science+Business Media B. V. , (2009), 415–423. [33] A. A. Pinto, B. M. Oliveira, F. A. Ferreira and F. Ferreira, Investing to survive in a duopoly model, Chapter in J. A. Tenreiro Machado et al. (Eds.): Intelligent Engineering Systems and Computational Cybernetics, Springer Science+Business Media B. V. , (2009), 407–414. [34] B. J. Spencer and J. A. Brander, International R & D rivalry and industrial strategy, Review of Economic Studies, 50 (1983), 707-722.  doi: 10.3386/w1192. [35] R. Staiger, International rules and institutions for trade policy, in: Grossman, Gene, Rogoff, Kenneth (Eds.): Handbook of international economics, vol. 3, Elsevier, North-Holland, 1495–1551.

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##### References:
 [1] K. Bagwell and R. Staiger, A theory of managed trade, American Economic Review, 80 (1990), 779-795. [2] K. Bagwell and R. Staiger, Enforcement, private political pressure, and the GAT/WTO escape clause, Journal of Legal Studies,, 34 (2005), 471-513. [3] N. Banik, F. A. Ferreira, J. Martins and A. A. Pinto, An Economical Model for Dumping by Dumping in a Cournot Model, chapter in "Dynamics, Games and Science Ⅱ: DYNA 2008, in Honour of Maurício Peixoto and David Rand", editors: M. M. Peixoto, A. A. Pinto and D. A. Rand, pp. 141-154, Springer, 2011. doi: 10.1007/978-3-642-14788-3_11. [4] J. A. Brander, Intra-industry trade in identical commodities, Journal of International Economics, 11 (1981), 1-14. [5] J. A. Brander and B. J. Spencer, Export subsidies and international market share rivalry, Journal of International Economics, 18 (1985), 83-100. [6] J. I. Bulow, J. D. Geanakoplos and P. D. Klemperer, Multi-market oligopoly: Strategic substitutes and complements, Journal of Political Economy, 93 (1985), 488-511. [7] M. Choubdar, J. P. Zubelli and A. A. Pinto, Nash and Social Welfare Impact in International Trade, Recent Advances in Applied Economics, Proceedings of the 6th International Conference on Applied Economics, Business and Development (AEBD'14) Lisbon, Portugal, (2014), 23-26. [8] M. Choubdar, E. Farias, F. A. Ferreira and A. A. Pinto, Uncertainty costs on an international duopoly with tariffs, Proceedings of the 6th International Conference on Applied Economics, Business and Development (AEBD'14) Lisbon, Portugal, (2014), 13-16. [9] A. Dixit, International trade policy for oligopolistic industries, Economic Journal, 94 (1984), 1-16. [10] A. Dixit, Strategic aspects of trade policy, in Bewley, T. (Editor), Advances in Economic Theory, Cambridge University Press, 329–362. [11] A. Dixit and G. Grossman, Targed export promotion with several oligopolistic industries, Journal of International Economics, 21 (1986), 233-249. [12] G. Eaton and G. Grossman, Optimal trade and industrial policy under oligopoly, Quarterly Journal of Economics, 101 (1984), 383-406.  doi: 10.2307/1891121. [13] F. A. Ferreira, Applications of Mathematics to Industrial Organization, Ph. D. Thesis, Universidade do Porto, Portugal, 2007. [14] F. A. Ferreira, F. Ferreira, M. Ferreira and A. A. Pinto, Quantity competition in a differentiated duopoly, Chapter in J. A. Tenreiro Machado, Bela Patkai and Imre J. Rudas (Eds.): Intelligent Engineering Systems and Computational Cybernetics. Springer Science+Business Media B. V. , (2008), 365–374. [15] F. A. Ferreira, F. Ferreira and A. A. Pinto, Flexibility in Stackelberg leadership, Chapter in J. A. Tenreiro Machado, Bela Patkai and Imre J. Rudas (Eds.): Intelligent Engineering Systems and Computational Cybernetics, Springer Science+Business Media B. V. , (2008), 399–405. [16] F. A. Ferreira, F. Ferreira and A. A. Pinto, Bayesian price leadership, Chapter in Tas, K. et al. (eds.): Mathematical Methods in Engineering, Springer, Dordrecht, (2007), 371–379. [17] F. A. Ferreira, F. Ferreira and A. A. Pinto, Unknown costs in a duopoly with differentiated products, Chapter in Tas, K. et al. (eds.): Mathematical Methods in Engineering, Springer, Dordrecht, (2007), 359–369. [18] F. A. Ferreira, F. Ferreira and A. A. Pinto, Uncertainty on an asymmetric duopoly, Progress in Industrial Mathematics at ECMI 2006, Proceedings 14th European Conference on Mathematics for Industry, Madrid, Spain, July 10-14. Springer, Berlin (2006) to appear. [19] E. O'N. Fisher and C. A. Wilson, Price competition between two international firms facing tariffs, International Journal of Industrial Organization, 3 (1995), 67-87. [20] R. Gibbons, A Primer in Game Theory, Pearson Prentice Hall, Harlow, 1992. [21] G. M. Grossman, Strategic export promotion: A critique, Chapter 3 in: Strategic Trade Policy and the New International Economics, P. R. Krugman (editor), MIT Press, Cambridge MA, (1986), 47–68. [22] G. V. Haberler, The Theory of International Trade with its Application to Commercial Policy, Macmillan, New York, 1937. [23] E. Helpman, Increasing Returns, Imperfect Markets, and Trade Theory, Jones, R. W. , Kenen, P. B. (eds.): Handbook of International Economics, 1 North Holland Press, Amesterdam, 1984, Chapter 7. [24] J. -M. M. Kilolo, Country Size, Trade Liberalization and Transfers, MPRA Paper, University Library of Munich, Germany, 2013. [25] K. Krishna, Trade restrictions as facilitating practices, Journal of International Economics, 26 (1989), 251-270. [26] P.-C. Liao, Rivalry between exporting countries and an importing country under incomplete information, Academia Economic Papers, 32 (1990), 605-630. [27] N. Limao and K. Saggi, Tariff retaliation versus financial compensation in the enforcement of international trade agreements, Journal of International Economics, 76 (2008), 48-60. [28] N. Lim'ao and K. Saggi, Size inequality, coordination externalities and international trade agreements, y, coordination externalities and international trade agreements,, 63 (2013), 10-27. [29] J. Martins, N. Banik and A. A. Pinto, A Repeated Strategy for Dumping, to appear in Discrete Dynamical Systems and Applications, ICDEA 2012, editors: Lluís Alseda, Jim M. Cushing, Saber Elaydi and A. A. Pinto, Springer. doi: 10.1007/978-3-662-52927-0_11. [30] J. Martins and A. A. Pinto, Deviation from collusion with and without dumping, to appear in Modeling, Dynamics, Optimization and Bioeconomics Ⅱ, editors: A. A. Pinto and D. Zilberman, Springer. [31] J. McMillan, Game Theory in International Economics, Harwood Academic Publishers, Chur, Switzerland, 1986. doi: 10.1080/00036846900000024. [32] A. A. Pinto, B. M. Oliveira, F. A. Ferreira and F. Ferreira, Stochasticity favoring the effects of the R & D strategies of the firms, Chapter in J. A. Tenreiro Machado et al. (Eds.): Intelligent Engineering Systems and Computational Cybernetics, Springer Science+Business Media B. V. , (2009), 415–423. [33] A. A. Pinto, B. M. Oliveira, F. A. Ferreira and F. Ferreira, Investing to survive in a duopoly model, Chapter in J. A. Tenreiro Machado et al. (Eds.): Intelligent Engineering Systems and Computational Cybernetics, Springer Science+Business Media B. V. , (2009), 407–414. [34] B. J. Spencer and J. A. Brander, International R & D rivalry and industrial strategy, Review of Economic Studies, 50 (1983), 707-722.  doi: 10.3386/w1192. [35] R. Staiger, International rules and institutions for trade policy, in: Grossman, Gene, Rogoff, Kenneth (Eds.): Handbook of international economics, vol. 3, Elsevier, North-Holland, 1495–1551.
The Welfare Game Type: Green -$\textbf{L}_j\textbf{W}_i$; Red -PD; Yellow -$\textbf{L}_i\textbf{W}_j$
The Nash (Social) tariffs for the home quantities, total quantity in the market, inverse demand, custom revenue and custom surplus, resulting in a social equilibrium. $h$ -Home quantities; $Q$ -Aggregate quantity in each country; $p$ -Inverse demand; $CR$ -Custom revenue; $CS$ -Consumer surplus
 SE game Economic quantity $h$ $e$ $Q$ $p$ $CR$ $CS$ Nash (Social) tariff of country i $T_i$ 0 0 $T_i$ $T_i / 2$ 0 Nash (Social) tariff of country j $T_j$ 0 0 $T_j$ $T_j/2$ 0
 SE game Economic quantity $h$ $e$ $Q$ $p$ $CR$ $CS$ Nash (Social) tariff of country i $T_i$ 0 0 $T_i$ $T_i / 2$ 0 Nash (Social) tariff of country j $T_j$ 0 0 $T_j$ $T_j/2$ 0
Comparing total quantities of the two countries with Nash tariffs and social tariffs with different cost similarities and concluding the game type
 Total quantities $(q_i,q_j)$ produced by the firms Condition Nash tariffs Social tariffs Game type If $2T_j < T_i$ $(T_i,T_j)$ $(0,0)$ $\textbf{L}_i\textbf{W}_j$ If $T_i/2 \leq T_j\leq 2T_i$ $(T_i,T_j)$ $(0,0)$ PD If $2T_i  Total quantities$(q_i,q_j)$produced by the firms Condition Nash tariffs Social tariffs Game type If$2T_j < T_i(T_i,T_j)(0,0)\textbf{L}_i\textbf{W}_j$If$T_i/2 \leq T_j\leq 2T_i(T_i,T_j)(0,0)$PD If$2T_i
Comparing profits of the firms of the two countries with Nash tariffs and social tariffs, where $H_i$ and $H_j$ are the tax-free home production indexes
 Profits $(\pi_i,\pi_j)$ of the firms Condition Nash tariffs Social tariffs Game type If $H_i < 3/5$ $(T_i,T_j)$ $(0,T_j)$ $\textbf{L}_i\textbf{W}_j$ If $H_i > 3/5$ and $H_j > 3/5$ $(T_i,T_j)$ $(T_i,T_j)$ SE If $H_j < 3/5$ $(T_i,T_j)$ $(T_i,0)$ $\textbf{L}_j\textbf{W}_i$
 Profits $(\pi_i,\pi_j)$ of the firms Condition Nash tariffs Social tariffs Game type If $H_i < 3/5$ $(T_i,T_j)$ $(0,T_j)$ $\textbf{L}_i\textbf{W}_j$ If $H_i > 3/5$ and $H_j > 3/5$ $(T_i,T_j)$ $(T_i,T_j)$ SE If $H_j < 3/5$ $(T_i,T_j)$ $(T_i,0)$ $\textbf{L}_j\textbf{W}_i$
Comparing welfares of the two countries with Nash tariffs and social tariffs where $H_i$ and $H_j$ are the tax-free home production indexes satisfying $0 < H_i < 2/3 < H_j < 1$
 Welfares $(W_i,W_j)$ of the countries Condition Nash tariffs Social tariffs Game type $H_j\geq 5/6$ $( A_{W,i}, T_j)$ $(0,T_j)$ $\textbf{L}_i\textbf{W}_j$ $4/5< H_j< 5/6$ $(A_{W,i}, A_{W,j})$ $(0, B_{W_S,j})$ LW or PD $H_j \leq 4/5$ $(A_{W,i}, A_{W,j})$ $(0,0)$ LW or PD
 Welfares $(W_i,W_j)$ of the countries Condition Nash tariffs Social tariffs Game type $H_j\geq 5/6$ $( A_{W,i}, T_j)$ $(0,T_j)$ $\textbf{L}_i\textbf{W}_j$ $4/5< H_j< 5/6$ $(A_{W,i}, A_{W,j})$ $(0, B_{W_S,j})$ LW or PD $H_j \leq 4/5$ $(A_{W,i}, A_{W,j})$ $(0,0)$ LW or PD
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