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