Nash and social welfare impact in an international trade model

Filipe Martins; Alberto A. Pinto; Jorge Passamani Zubelli

doi:10.3934/jdg.2017009

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2017, Volume 4, Issue 2: 149-173. Doi: 10.3934/jdg.2017009

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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: November 30, 2015

Revised: December 31, 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 91B14, 91B15, 91B60; Secondary: 91B64.

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Figure 1. The Welfare Game Type: Green -$\textbf{L}_j\textbf{W}_i$; Red -PD; Yellow -$\textbf{L}_i\textbf{W}_j$

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Table 1. 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

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Table 2. 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 $2T_i <T_j$	$(T_i,T_j)$	$(0,0)$	$\textbf{L}_j\textbf{W}_i$

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Table 3. 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 $H_j < 3/5$	$(T_i,T_j)$	$(T_i,0)$	$\textbf{L}_j\textbf{W}_i$

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Table 4. 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

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