Conflict and agreement in the collective choice of trade policies: implications for interstate disputes

Abstract

From the collective choice perspective, this paper examines how different trade regimes have differing implications for two enemy countries' arming decisions in a three-country world with a neutral third-party state. We compare the two adversaries' aggregate arming (i.e., overall conflict intensity) and show that it is in ascending order for the following regimes: (i) a free trade agreement (FTA) between the adversaries, leaving the third-party state as a non-member, (ii) worldwide free trade in the presence of the interstate conflict, (iii) trade wars with Nash tariffs, and (iv) an FTA between the third country and one adversary, excluding the other adversary from the trade bloc. These results have policy implications for interstate conflicts. First, “dancing between two enemies” with an FTA results in lower aggregate arming than under worldwide free trade. Second, the world is “more dangerous” in tariff wars than under free trade. Third, an FTA between one adversary and the third party while keeping the other adversary as an outsider is conflict-aggravating since aggregate arming is the highest compared to all other trade regimes. We also analyze aggregate arming under a customs union (CU) and discuss differences/similarities in implications between a CU and an FTA for interstate conflicts.

Appendices

Appendices

1.1 A-1: Market equilibrium condition for good a in country A

Alternatively, we have the following equilibrium condition:

$$(\alpha - \beta P_{a}^{A} ) + \left[ {(\alpha - \beta P_{a}^{B} ) - \left( {\frac{{G^{B} }}{{G^{A} + G^{B} }}} \right)3} \right] + (\alpha - \beta P_{a}^{C} ) = \left( {\frac{{G^{A} }}{{G^{A} + G^{B} }}} \right)3 - G^{A} - K^{A} .$$

(a.1)

The second bracket term on the LHS of Equation (a.1) is the consumption of good a by country B, $(\alpha - \beta P_{a}^{B} ),$ minus the quantity of the good that B appropriates from A, ${{[G^{B} } \mathord{\left/ {\vphantom {{[G^{B} } (}} \right. \kern-0pt} (}G^{A} + G^{B} )]3.$ This difference gives the amount of good $a$ that country B imports from country A. The term on the RHS of Equation (a.1) is the quantity of good a that country A supplies, which is given by $Z_{a}^{A}$ in (2). It is easy to verify that Equation (a.1) is identical to Eq. (8).

1.2 A-2: Market equilibrium condition for good b in country B

Alternatively, we have the following equilibrium condition:

$$\left[ {(\alpha - \beta P_{b}^{A} ) - \left( {\frac{{G^{A} }}{{G^{A} + G^{B} }}} \right)3} \right] + (\alpha - \beta P_{b}^{B} ) + (\alpha - \beta P_{b}^{C} ) = \left( {\frac{{G^{B} }}{{G^{A} + G^{B} }}} \right)3 - G^{B} - K^{B} .$$

(a.2)

The first bracket term on the LHS of Equation (a.2) is the consumption of good b by country A, $(\alpha - \beta P_{b}^{A} ),$ minus the amount of the good that A appropriates from B, ${{[G^{A} } \mathord{\left/ {\vphantom {{[G^{A} } (}} \right. \kern-0pt} (}G^{A} + G^{B} )]3.$ This difference gives the quantity of good b that country A imports from country B. The term on the RHS of Equation (a.2) is the quantity of good b that country B supplies, as given by $Z_{b}^{B}$ in (2). It is easy to verify that Equation (a.2) is identical to Eq. (11).

1.3 A-3: Comparative static results for the protectionist regime

Based on the optimal tariffs under the protectionist regime, as shown in (22), we have the following results:

$$\begin{aligned} & \frac{{\partial \tau_{b}^{A,PR} }}{{\partial G^{A} }} = - \frac{{8G^{B} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0,\frac{{\tau_{b}^{A,PR} }}{{\partial G^{B} }} = - \frac{{(G^{A} + G^{B} )^{2} - 8G^{A} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0, \\ & \frac{{\partial \tau_{a}^{B,PR} }}{{\partial G^{A} }} = - \frac{{(G^{A} + G^{B} )^{2} - 8G^{B} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0,\frac{{\partial \tau_{a}^{B,PR} }}{{\partial G^{B} }} = - \frac{{8G^{A} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0, \\ & \frac{{\partial \tau_{a}^{C,PR} }}{{\partial G^{A} }} = - \frac{{(G^{A} + G^{B} )^{2} - G^{B} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0,\frac{{\partial \tau_{a}^{C,PR} }}{{\partial G^{B} }} = - \frac{{G^{A} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0, \\ & \frac{{\partial \tau_{b}^{C,PR} }}{{\partial G^{A} }} = - \frac{{G^{B} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0,\frac{{\partial \tau_{b}^{C,PR} }}{{\partial G^{B} }} = - \frac{{(G^{A} + G^{B} )^{2} - G^{A} }}{{7\beta (G^{A} + G^{B} )^{2} }} < 0, \\ & \tau_{c}^{A,PR} - \tau_{a}^{C,PR} = \frac{{G^{A} + K^{A} }}{7\beta } + \frac{{G^{B} }}{{7\beta (G^{A} + G^{B} )}} > 0, \\ & \tau_{c}^{B,PR} - \tau_{b}^{C,PR} = \frac{{G^{B} + K^{B} }}{7\beta } + \frac{{G^{A} }}{{7\beta (G^{A} + G^{B} )}} > 0. \\ \end{aligned}$$

1.4 A-4: Decomposing the aggregate payoff effect of arming for a contending country under the protectionist regime

Under symmetry, we can look at country A. The country's aggregate payoff function is

$\begin{aligned} \Pi^{A} & = CS^{A} + PS^{A} + TR^{A} \\ & = \frac{1}{2\beta }[(\alpha - \beta P_{a}^{A} )^{2} + (\alpha - \beta P_{b}^{A} )^{2} + (\alpha - \beta P_{c}^{A} )^{2} ] + [P_{a}^{A} (Z_{a}^{A} ) + P_{b}^{A} (APP_{b}^{A} )] + (\tau_{b}^{A} M_{b}^{A} + \tau_{c}^{A} M_{c}^{A} ), \\ \end{aligned}$ where $APP_{b}^{A} = {{[G^{A} } \mathord{\left/ {\vphantom {{[G^{A} } (}} \right. \kern-0pt} (}G^{A} + G^{B} )]3$ is the amount of good b appropriated by country A. Taking the derivative of $SW^{A}$ with respect to $G^{A}$ yields

$$\begin{aligned} \frac{{\partial \Pi^{A} }}{{\partial G^{A} }} & = \left[ { - (\alpha - \beta P_{a}^{A} )\frac{{\partial P_{a}^{A} }}{{\partial G^{A} }} - (\alpha - \beta P_{b}^{A} )\frac{{\partial P_{b}^{A} }}{{\partial G^{A} }}} \right] \\ & \quad + \left[ {\frac{{\partial P_{a}^{A} }}{{\partial G^{A} }}Z_{a}^{A} + \frac{{\partial Z_{{}}^{A} }}{{\partial G^{A} }}P_{a}^{A} + \frac{{\partial P_{b}^{A} }}{{\partial G^{A} }}(APP_{b}^{A} ) + \frac{{\partial (APP_{b}^{A} )}}{{\partial G^{A} }}P_{b}^{A} } \right] \\ & \quad + \left( {\tau_{b}^{A} \frac{{\partial M_{b}^{A} }}{{\partial G^{A} }} + \tau_{c}^{A} \frac{{\partial M_{c}^{A} }}{{\partial G^{A} }} + M_{b}^{A} \frac{{\partial \tau_{b}^{A} }}{{\partial G^{A} }} + M_{c}^{A} \frac{{\partial \tau_{c}^{A} }}{{\partial G^{A} }}} \right). \\ \end{aligned}$$

Note that changes in country A's arming do not affect $M_{c}^{A}$ and $\tau_{c}^{A} .$ That is, ${{\partial M_{c}^{A} } \mathord{\left/ {\vphantom {{\partial M_{c}^{A} } {\partial G^{A} }}} \right. \kern-0pt} {\partial G^{A} }} = 0$ and ${{\partial \tau_{c}^{A} } \mathord{\left/ {\vphantom {{\partial \tau_{c}^{A} } {\partial G^{A} = 0}}} \right. \kern-0pt} {\partial G^{A} = 0}}.$ Note also that country A's import demand for good b is given by its total consumption of good b minus the amount of the good appropriated, i.e., $M_{b}^{A} = (\alpha - \beta P_{b}^{A} ) - A_{b} .$ We incorporate the zero derivatives and this definition into the derivative, after re-arranging terms. This exercise yields

$$\frac{{\partial \Pi^{A} }}{{\partial G^{A} }} = \left[ {Z_{a}^{A} - (\alpha - \beta P_{a}^{A} )} \right]\frac{{\partial P_{a}^{A} }}{{\partial G^{A} }} + \left[ {\left( {\tau_{b}^{A} \frac{{\partial M_{b}^{A} }}{{\partial G^{A} }} + M_{b}^{A} \frac{{\partial \tau_{b}^{A} }}{{\partial G^{A} }}} \right) - M_{b}^{A} \frac{{\partial P_{b}^{A} }}{{\partial G^{A} }}} \right] + \frac{{\partial Z^{A} }}{{\partial G^{A} }}P_{a}^{A} + \frac{{\partial (APP_{b}^{A} )}}{{\partial G^{A} }}P_{b}^{A} .$$

(a.3)

This derivative contains four different terms:

(i) The first term $[Z_{a}^{A} - (\alpha - \beta P_{a}^{A} )]\frac{{\partial P_{a}^{A} }}{{\partial G^{A} }}$ reflects a terms-of-trade effect of arming, which is payoff-increasing since $[Z_{a}^{A} - (\alpha - \beta P_{a}^{A} )] > 0$ and $\frac{{\partial P_{a}^{A} }}{{\partial G^{A} }} > 0.$

(ii) The second bracket term $[(\tau_{b}^{A} \frac{{\partial M_{b}^{A} }}{{\partial G^{A} }} + M_{b}^{A} \frac{{\partial \tau_{b}^{A} }}{{\partial G^{A} }}) - M_{b}^{A} \frac{{\partial P_{b}^{A} }}{{\partial G^{A} }}]$ reflects the (net) effect of country A's arming on tariff revenue from the import of good b minus import spending. Note that

$$\left( {\tau_{b}^{A} \frac{{\partial M_{b}^{A} }}{{\partial G^{A} }} + M_{b}^{A} \frac{{\partial \tau_{b}^{A} }}{{\partial G^{A} }}} \right) = \tau_{b}^{A} \left[ { - 3\frac{{G^{B} }}{{(G^{A} + G^{B} )^{2} }}} \right] + M_{b}^{A} \left[ { - \frac{{8G^{B} }}{{7\beta (G^{A} + G^{B} )^{2} }}} \right] < 0.$$

We also consider how arming affects the price of good b in country A, which is ${{\partial P_{b}^{A} } \mathord{\left/ {\vphantom {{\partial P_{b}^{A} } {\partial G^{A} }}} \right. \kern-0pt} {\partial G^{A} }}.$ This derivative is positive since country A's arming causes country B to raise its price for good b. The second bracket term $[(\tau_{b}^{A} \frac{{\partial M_{b}^{A} }}{{\partial G^{A} }} + M_{b}^{A} \frac{{\partial \tau_{b}^{A} }}{{\partial G^{A} }}) - M_{b}^{A} \frac{{\partial P_{b}^{A} }}{{\partial G^{A} }}]$ is thus unambiguously negative.

(iii) The third term $\frac{{\partial Z_{a}^{A} }}{{\partial G^{A} }}P_{a}^{A}$ reflects an output distortion effect since allocating more resources to arming lowers the amount of resources for final good production and consumption, which is payoff-reducing.

(iv) The fourth term $\frac{{\partial (APP_{b}^{A} )}}{{\partial G^{A} }}P_{b}^{A}$ is a resource appropriation effect, which is payoff-increasing.

It follows from (a.3) that we can decompose the effect of country A's arming on its aggregate payoff into four different effects as follows:

$$\begin{aligned} \frac{{\partial \Pi^{A} }}{{\partial G^{A} }} & = \underbrace {{\left[ {Z_{a}^{A} - (\alpha - \beta P_{a}^{A} )} \right]\frac{{\partial P_{a}^{A} }}{{\partial G^{A} }}}}_{{{\mathrm{Export-revenue}}\,{\mathrm{effect}}\,{\mathrm{of}}\,{\mathrm{arming(+)}}}} \, + \underbrace {{\frac{{\partial (APP_{b}^{A} )}}{{\partial G^{A} }}P_{b}^{A} }}_{{{\mathrm{Resource-appropriation}}\,{\mathrm{effect}}\,{\mathrm{of}}\,{\mathrm{arming}}\,{(+)}}} \\ & \quad + \underbrace {{\left[ {\left( {\tau_{b}^{A} \frac{{\partial M_{b}^{A} }}{{\partial G^{A} }} + M_{b}^{A} \frac{{\partial \tau_{b}^{A} }}{{\partial G^{A} }}} \right) - M_{b}^{A} \frac{{\partial P_{b}^{A} }}{{\partial G^{A} }}} \right]}}_{{{\mathrm{Tariff-revenue}}\,\&\,{\mathrm{import-spending}}\,{\mathrm{effect}}\,{\mathrm{of}}\,{\mathrm{arming}}\,{(-)}}}{ + }\underbrace {{\frac{{\partial Z_{a}^{A} }}{{\partial G^{A} }}P_{a}^{A} }}_{{{\mathrm{Output-distortion}}\,{\mathrm{effect}}\,{\mathrm{of}}\,{\mathrm{arming}}\,{(-)}}} = 0. \\ \end{aligned}$$

1.5 A-5: Optimal arming is lower under the FTA (A&B) regime than under worldwide free trade

We evaluate the slopes of $SW_{i}$ (for $i = A,B)$ under the WFT regime at the equilibrium arming allocations under the FTA(A&B) regime, $\{ G_{{}}^{A,FTA(A\& B)} ,G_{{}}^{B,FTA(A\& B)} \} .$ With symmetry that $G_{{}}^{A,FTA(A\& B)} = G_{{}}^{B,FTA(A\& B)} = G_{{}}^{FTA(A\& B)} ,$ we look at country A. Since $\tau_{b}^{A} = \tau_{a}^{B} = 0$ under the FTA(A&B) regime, we have from the aggregate payoff decomposition in (24) that the FOC for country A is

$$\begin{aligned} \frac{{\partial \Pi^{FTA(A\& B)} }}{{\partial G^{A} }} & = \left[ {Z_{a}^{A} - (\alpha - \beta P_{a}^{A,\,FTA(A\& \,B)} )} \right]\frac{{\partial P_{a}^{A,\,FTA(A\& \,B)} }}{{\partial G^{A} }} + \frac{{\partial APP_{b}^{A} }}{{\partial G^{A} }}P_{b}^{A,\,FTA(A\& B)} \\ & \quad + \frac{{\partial Z_{a}^{A} }}{{\partial G^{A} }}P_{a}^{A,\,FTA(A\& \,B)} = 0, \\ \end{aligned}$$

where $APP_{b}^{A} = {{[3G^{A} } \mathord{\left/ {\vphantom {{[3G^{A} } (}} \right. \kern-0pt} (}G^{A} + G^{B} )]$ is the amount of good b appropriated by country A. Next, we derive results for each of the terms as shown in country A's FOC. Substituting $\tau_{a}^{A,\,FTA(A\& B)} = {{(3 - G^{A} - K_{A} )} \mathord{\left/ {\vphantom {{(3 - G^{A} - K_{A} )} {8\beta }}} \right. \kern-0pt} {8\beta }}$ from (26a) into $P_{a}^{A,\,FTA(A\& B)}$ in (25) yields

$$P_{a}^{A,FTA(A\& B)} = \frac{{3\alpha + G^{A} + K^{A} - \beta \tau_{a}^{C} - 3}}{3\beta } = \frac{{8\alpha + 3G^{A} + 3K^{A} - 9}}{8\beta },$$

(a.4)

which implies that

$$\frac{{\partial P_{a}^{A,\,FTA(A\& \,B)} }}{{\partial G^{A} }} = \frac{3}{8\beta }.$$

(a.5)

The appropriation of good b by country A, $APP_{b}^{A} = {{[3G^{A} } \mathord{\left/ {\vphantom {{[3G^{A} } (}} \right. \kern-0pt} (}G^{A} + G^{B} )],$ implies that

$$\frac{{\partial APP_{b} }}{{\partial G^{A} }} = \frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}.$$

(a.6)

Country A's production of good a, $Z_{a}^{A} = {{[3G^{A} } \mathord{\left/ {\vphantom {{[3G^{A} } (}} \right. \kern-0pt} (}G^{A} + G^{B} )] - G^{A} - K^{A}$, implies that

$$\frac{{\partial Z_{a}^{A} }}{{\partial G^{A} }} = \frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }} - 1.$$

(a.7)

Substituting $\tau_{b}^{C} = {{(3 - G^{B} - K^{B} )} \mathord{\left/ {\vphantom {{(3 - G^{B} - K^{B} )} {(8\beta )}}} \right. \kern-0pt} {(8\beta )}}$ from (26c) into $P_{b}^{A,\,FTA(A\& B)}$ in (25) yields

$$P_{b}^{A,FTA(A\& B)} = \frac{{3\alpha + G^{B} + K^{B} - \beta \tau_{b}^{C} - 3}}{3\beta } = \frac{{8\alpha + 3G^{B} + 3K^{B} - 9}}{8\beta }.$$

(a.8)

The substitution of the results from (a.4)–(a.8) back into country A's first-order condition implies that

$$\begin{aligned} & \frac{{\partial \Pi^{FTA(A\& B)} }}{{\partial G^{A} }} = \underbrace {{\frac{3}{8\beta }\,\left[ {\left( {\frac{{3G^{A} }}{{G^{A} + G^{B} }} - G^{A} - K^{A} } \right) - \left[ {\alpha - \beta \left( {\frac{{8\alpha + 3G^{A} + 3K^{A} - 9}}{8\beta }} \right)} \right]} \right]}}_{{{\text{Export - revenue}}\,{\text{effect}}\,{\text{of}}\,{\text{arming}}\,{(+)}}} \\ & \quad + \underbrace {{\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}(\frac{{8\alpha + 3G^{B} + 3K^{B} - 9}}{8\beta })}}_{{{\text{Resource - appropriation}}\,{\text{effect}}\,{\text{of}}\,{\text{arming}}\,{(+)}}} + \underbrace {{\left[ {\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }} - 1} \right]\left( {\frac{{8\alpha + 3G^{A} + 3K^{A} - 9}}{8\beta }} \right)}}_{{{\text{Output - distortion}}\,{\text{effect}}\,{\text{of}}\,{\text{arming (}} - {)}}}\, = 0 \\ \end{aligned}$$

(a.9)

where $G^{A} = G^{B} = G_{{}}^{FTA(A\& B)} .$

Under the WFT regime, the slope of country A's aggregate payoff function with respect to its arming is

$$\frac{{\partial \Pi^{WFT} }}{{\partial G^{A} }} = \frac{{\partial P_{a}^{A,WFT} }}{{\partial G^{A} }}\left[ {Z_{a}^{A} - (\alpha - \beta P_{a}^{A,WFT} )} \right] + \frac{{\partial (APP_{b}^{A} )}}{{\partial G^{A} }}P_{b}^{A,WFT} + \frac{{\partial Z_{a}^{A} }}{{\partial G^{A} }}P_{a}^{A,WFT} ,$$

where

$$\begin{aligned} & P_{a}^{A,\,WFT} = \frac{{3\alpha + G^{A} + K^{A} - 3}}{3\beta },\frac{{\partial P_{a}^{A,\,WFT} }}{{\partial G^{A} }} = \frac{1}{3\beta },P_{b}^{A,\,WFT} = \frac{{3\alpha + G^{B} + K^{B} - 3}}{3\beta },\frac{{\partial P_{b}^{A,\,WFT} }}{{\partial G^{A} }} = 0, \\ & APP_{b}^{A} = \frac{{3G^{A} }}{{G^{A} + G^{B} }},\frac{{\partial (APP_{b}^{A} )}}{{\partial G^{A} }} = \frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}, \\ & Z_{a}^{A} = \frac{{3G^{A} }}{{G^{A} + G^{B} }} - G^{A} - K^{A} ,\frac{{\partial Z_{a}^{A} }}{{\partial G^{A} }} = \frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }} - 1. \\ \end{aligned}$$

After substituting, we have

$$\begin{aligned} & \frac{{\partial \Pi^{WFT} }}{{\partial G^{A} }} = \underbrace {{\frac{1}{3\beta }\left[ {\left( {\frac{{3G^{A} }}{{G^{A} + G^{B} }} - G^{A} - K^{A} } \right) - \left[ {\alpha - \beta \left( {\frac{{3\alpha + G^{A} + K^{A} - 3}}{3\beta }} \right)} \right]} \right]}}_{{{\text{Export - revenue}}\,{\text{effect}}\,{\text{of}}\,{\text{arming}}\,{(+)}}} \\ & \quad + \underbrace {{\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}\left( {\frac{{3\alpha + G^{B} + K^{B} - 3}}{3\beta }} \right)}}_{{{\text{Resource - appropriation}}\,{\text{effect}}\,{\text{of}}\,{\text{arming}}\,{(+)}}} + \underbrace {{\left[ {\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }} - 1} \right]\left( {\frac{{3\alpha + G^{A} + K^{A} - 3}}{3\beta }} \right)}}_{{{\text{Output - distortion}}\,{\text{effect}}\,{\text{of}}\,{\text{arming}}\,{(} - {)}}} \\ \end{aligned}$$

(a.10)

where $G^{A} = G^{B} = G_{{}}^{WFT} .$ We evaluate ${{\partial \Pi^{WFT} } \mathord{\left/ {\vphantom {{\partial \Pi^{WFT} } {\partial G^{A} }}} \right. \kern-0pt} {\partial G^{A} }}$ in (a.10) at the FTA(A&B) equilibrium arming allocations where $G_{{}}^{A} = G_{{}}^{B} = G_{{}}^{FTA(A\& B)} ,$ taking into account the FOC as shown in (a.9). We have the following:

(i) Comparing the export-revenue effect

$$\begin{aligned} & \frac{1}{3\beta }\left[ {\left( {\frac{{3G^{A} }}{{G^{A} + G^{B} }} - G^{A} - K^{A} } \right) - \left[ {\alpha - \beta \left( {\frac{{3\alpha + G^{A} + K^{A} - 3}}{3\beta }} \right)} \right]} \right] \\ & \quad - \frac{3}{8\beta }\,\left[ {\left( {\frac{{3G^{A} }}{{G^{A} + G^{B} }} - G^{A} - K^{A} } \right) - \left[ {\alpha - \beta \left( {\frac{{8\alpha + 3G^{A} + 3K^{A} - 9}}{8\beta }} \right)} \right]} \right] \\ & \quad \quad = \frac{{[7(G^{A} )^{2} + 7G^{A} G^{B} - 21G^{A} + 51G^{B} + 7(G^{A} + 7G^{B} )K^{A} ]}}{{576\beta (G^{A} + G^{B} )}} > 0. \\ \end{aligned}$$

(ii) Comparing the resource-appropriation effect

$$\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}\left( {\frac{{3\alpha + G^{B} + K^{B} - 3}}{3\beta }} \right)\, - \frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}\left( {\frac{{8\alpha + 3G^{B} + 3K^{B} - 9}}{8\beta }} \right) = \frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }}\,\left( {\frac{{3 - G^{B} - K^{B} }}{24\beta }} \right) > 0.$$

(iii) Comparing the output-distortion effect

$$\begin{aligned} & \left[ {\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }} - 1} \right]\left( {\frac{{3\alpha + G^{A} + K^{A} - 3}}{3\beta }} \right)\, - \,\left[ {\frac{{3G^{B} }}{{(G^{A} + G^{B} )^{2} }} - 1} \right]\left( {\frac{{8\alpha + 3G^{A} + 3K^{A} - 9}}{8\beta }} \right) \\ & \quad = - \frac{{[(G^{A} )^{2} + (G^{B} )^{2} + 2G^{A} G^{B} - 3G^{B} ](3 - G^{A} - K^{A} )}}{{24\beta (G^{A} + G^{B} )^{2} }} < 0. \\ \end{aligned}$$

Putting together the three effects, (i)–(iii), we have under symmetry ($G_{{}}^{A} = G_{{}}^{B} = G_{{}}^{FTA(A\& B)}$) that

$$\frac{{\partial \Pi^{WFT} }}{{\partial G^{A} }}\left| {_{{_{{G_{{}}^{A} = G_{{}}^{B} = G_{{}}^{FTA(A\& B)} }} }} } \right. = \frac{{[31(G_{{}}^{FTA(A\& B)} )^{2} - 93G_{{}}^{FTA(A\& B)} + 31G_{{}}^{FTA(A\& B)} K - 36K + 108]}}{{576\beta G_{{}}^{FTA(A\& B)} }} > 0.$$

The strict concavity of the aggregate payoff function implies that the optimal arming under the FTA(A&B) regime is lower than that of the global free trade regime. That is, $G^{FTA(A\& B)} < G^{WFT} .$ Starting from the FTA(A&B) regime, a move to the WFT regime will encourage each contending country to increase arming, since the export-revenue effect plus the resource-appropriation effect (i.e., the marginal revenue of arming) exceeds the output-distortion effect (i.e., the marginal cost of arming).

A-6: Optimal arming allocations of two adversary countries that form a CU

For a CU between countries A and B, denoted as the CU(A&B) regime, we have $\tau_{b}^{A,CU(A\& B)} = \;\tau_{a}^{B,CU(A\& B)} = 0.$ At the trade policy stage, A and B jointly determine a common external optimal tariff, denoted as $\tau_{c}^{m,CU(A\& B)}$, on their imports of good c. Country C sets an optimal tariff structure, $\left\{ {\tau_{a}^{C} ,\;\tau_{b}^{C} } \right\}$, on its imports of good a and b. Making use of the price equations in (9), (12), and (15), and considering that $\tau_{b}^{A,CU(A\& B)} = \;\tau_{a}^{B,CU(A\& B)} = 0,$ the equilibrium prices under the CU(A&B) regime are

$$\begin{aligned} & P_{a}^{A,CU(A\& B)} = P_{a}^{B,CU(A\& B)} = \frac{{3\alpha - \beta \tau_{a}^{C} - (3 - G^{A} - K^{A} )}}{3\beta }, \\ & P_{b}^{A,CU(A\& B)} = P_{b}^{B,CU(A\& B)} = \frac{{3\alpha - \beta \tau_{b}^{C} - (3 - G^{B} - K^{B} )}}{3\beta }, \\ & P_{a}^{C,CU(A\& B)} = \frac{{3\alpha + 2\beta \tau_{a}^{C} - (3 - G^{A} - K^{A} )}}{3\beta },P_{b}^{C,CU(A\& B)} = \frac{{3\alpha + 2\beta \tau_{b}^{C} - (3 - G^{B} - K^{B} )}}{3\beta }, \\ & P_{c}^{A,CU(A\& B)} = \frac{{3\alpha + \beta \tau_{c}^{m,CU(A\& B)} - 3}}{3\beta },P_{c}^{B,CU(A\& B)} = \frac{{3\alpha + \beta \tau_{c}^{m,CU(A\& B)} - 3}}{3\beta }, \\ & P_{c}^{C,CU(A\& B)} = \frac{{3\alpha - 2\beta \tau_{c}^{m,CU(A\& B)} - 3}}{3\beta }. \\ \end{aligned}$$

In determining their common external tariff on the import of good c, countries A and B jointly maximize their aggregate payoffs: $\Pi^{A\& B,CU(A\& B)} = \Pi^{A,CU(A\& B)} + \Pi^{B,CU(A\& B)} ,$ where

$$\Pi^{A,CU(A\& B)} = CS^{A,CU(A\& B)} + PS^{A,CU(A\& B)} + \tau_{c}^{m,CU(A\& B)} M_{b}^{A,CU(A\& B)} ,$$

(a.11)

$$\Pi^{B,CU(A\& B)} = CS^{B,CU(A\& B)} + PS^{B,CU(A\& B)} + \tau_{c}^{m,CU(A\& B)} M_{b}^{B,CU(A\& B)} .$$

(a.12)

The FOC for aggregate payoff maximization implies that the common external tariff on good c is

$$\tau_{c}^{m,CU(A\& B)} = \frac{6}{5\beta }.$$

(a.13)

Country C determines an optimal tariff structure,$\{ \tau_{a}^{C} ,\tau_{b}^{C} \} ,$ to maximize its domestic aggregate payoff:

$$\Pi^{C,CU(A\& B)} = CS^{C,CU(A\& B)} + PS^{C,CU(A\& B)} + \tau_{a}^{C,CU(A\& B)} M_{a}^{C,CU(A\& B)} + \tau_{b}^{C,CU(A\& B)} M_{b}^{C,CU(A\& B)} .$$

The FOCs for country C imply that the optimal tariffs are

$$\tau_{a}^{C,CU(A\& B)} = \frac{{(3 - G^{A} - K^{A} )}}{8\beta }\,\,{\text{and}}\,\,\tau_{b}^{C,CU(A\& B)} = \frac{{(3 - G^{B} - K^{B} )}}{8\beta }.$$

(a.14)

We proceed to the security stage at which A and B independently and simultaneously determine their optimal arming decisions. Substituting the optimal tariffs from (a.13) and (a.14) into the aggregate payoff functions in (a.11) and (a.12), we have the FOCs for A and B:

$$\frac{{\partial \Pi^{A,CU(A\& B)} }}{{\partial G^{A} }} = 0\,\,{\text{and}}\,\,\frac{{\partial \Pi^{B,CU(A\& B)} }}{{\partial G^{B} }} = 0.$$

Denote the Nash equilibrium levels of arming as $\{ G^{A,CU(A\& B)} ,G^{B,CU(A\& B)} \} .$ Under symmetry in all dimensions, we have $G^{A,CU(A\& B)} = G^{B,CU(A\& B)} = G^{CU(A\& B)} .$ Calculating the optimal arming yields

$$G^{CU(A\& B)} = \frac{{\sqrt {4096\alpha^{2} - 3159 + K(1521K + 4992\alpha - 3510)} }}{78} - \frac{32\alpha }{{39}} - \frac{K}{2} + \frac{3}{2}.$$

It is easy to verify that $G^{FTA(A\& B)} = G^{CU(A\& B)} .$ Evaluating the slope ${{\partial \Pi^{A,CU(A\& B)} } \mathord{\left/ {\vphantom {{\partial \Pi^{A,CU(A\& B)} } {\partial G^{A} }}} \right. \kern-0pt} {\partial G^{A} }}$ at the point where $G^{A} = G^{A,PR}$, we have

$$\frac{{\partial \Pi^{A,CU(A\& B)} }}{{\partial G^{A} }}\left| {_{{G^{A} = G^{A,PR} }} } \right. < 0,$$

which implies that $G^{A,CU(A\& B)} = G^{B,CU(A\& B)} = G^{CU(A\& B)} < G^{A,PR} .$

2.1 A-7: CU between one contending country and a neutral third country

For the scenario where there is a CU between countries A and C, denoted as the CU(A&C) regime, we have $\tau_{c}^{A,CU(A\& C)} = \;\tau_{a}^{C,CU(A\& C)} = 0.$ At the trade policy stage, countries A and C jointly determine a common external tariff, denoted as $\tau_{b}^{m,CU(A\& C)} ,$ on their imports of good b. Simultaneously, country B sets an optimal tariff structure, $\{ \tau_{a}^{B} ,\tau_{c}^{B} \} ,$ on its imports of goods a and c. Making use of the price equations in (9), (12), and (15), and considering that $\tau_{c}^{A,CU(A\& C)} = \;\tau_{a}^{C,CU(A\& C)} = 0,$ the equilibrium prices under the CU(A&C) regime are

$$\begin{aligned} & P_{a}^{A,CU(A\& C)} = P_{a}^{C,CU(A\& C)} = \frac{{3\alpha - \beta \tau_{a}^{B} - (3 - G^{A} - K^{A} )}}{3\beta }, \\ & P_{c}^{A,CU(A\& C)} = P_{c}^{C,CU(A\& C)} = \frac{{3\alpha - \beta \tau_{c}^{b,CU(A\& C)} - 3}}{3\beta }, \\ & P_{a}^{B,CU(A\& C)} = \frac{{3\alpha + 2\beta \tau_{a}^{B} - (3 - G^{A} - K^{A} )}}{3\beta },P_{b}^{A,CU(A\& C)} = \frac{{3\alpha + \beta \tau_{b}^{m,CU(A\& C)} - (3 - G^{B} - K^{B} )}}{3\beta }, \\ & P_{b}^{B,CU(A\& C)} = \frac{{3\alpha - 2\beta \tau_{b}^{m,CU(A\& C)} - (3 - G^{B} - K^{B} )}}{3\beta }, \\ & P_{b}^{C,CU(A\& C)} = \frac{{3\alpha + \beta \tau_{b}^{m,CU(A\& C)} - (3 - G^{B} - K^{B} )}}{3\beta },P_{c}^{B,CU(A\& C)} = \frac{{3\alpha + 2\beta \tau_{c}^{B} - 3}}{3\beta }. \\ \end{aligned}$$

In determining their tariff on the import of good b, countries A and C set a common external tariff that maximizes their aggregate payoff: $\Pi^{AC,CU(A\& C)} = \Pi^{A,CU(A\& C)} + \Pi^{C,CU(A\& C)}$, where

$$\begin{aligned} & \Pi^{A,CU(A\& C)} = CS^{A,CU(A\& C)} + PS^{A,CU(A\& C)} + \tau_{b}^{m,CU(A\& C)} M_{b}^{A,CU(A\& C)} , \\ & \Pi^{C,CU(A\& C)} = CS^{C,CU(A\& C)} + PS^{C,CU(A\& C)} + \tau_{b}^{m,CU(A\& C)} M_{b}^{C,CU(A\& C)} . \\ \end{aligned}$$

The FOC for the joint payoff maximization problem is ${{\partial \Pi^{AC,CU(A\& C)} } \mathord{\left/ {\vphantom {{\partial \Pi^{AC,CU(A\& C)} } {\partial \tau^{m,CU(A\& C)} }}} \right. \kern-0pt} {\partial \tau^{m,CU(A\& C)} }} = 0.$ Solving for the optimal common external tariff yields

$$\tau_{b}^{m,CU(A\& C)} = \frac{{2G^{B} (3 - G^{A} - G^{B} ) - 3G^{A} - 2K^{B} (G^{A} + G^{B} )}}{{5\beta (G^{A} + G^{B} )}}.$$

(a.15)

Similarly, country B determines an optimal tariff structure,$\{ \tau_{a}^{B} ,\tau_{c}^{B} \} ,$ to maximize its domestic aggregate payoff:

$$\Pi^{B,CU(A\& C)} = CS^{B,CU(A\& C)} + PS^{B,CU(A\& C)} + \tau_{a}^{B,CU(A\& C)} M_{a}^{B,CU(A\& C)} + \tau_{c}^{B,CU(A\& C)} M_{c}^{B,CU(A\& C)}$$

Making use of the FOCs for country B, we solve for its optimal tariffs:

$$\tau_{a}^{B,CU(A\& C)} = \frac{{G^{A} (3 - G^{A} - G^{B} ) - 6G^{B} - K^{A} (G^{A} + G^{B} )}}{{8\beta (G^{A} + G^{B} )}}\,\,{\text{and}}\,\,\tau_{c}^{B,CU(A\& C)} = \frac{3}{8\beta }.$$

(a.16)

We proceed to the security stage at which countries A and B independently and simultaneously make their arming decisions. Country A determines an optimal arming, denoted as $G^{A,CU(A\& C)} ,$ that maximizes its aggregate payoff:

$$\Pi^{A,CU(A\& C)} = CS^{A,CU(A\& C)} + PS^{A,CU(A\& C)} + \tau_{b}^{m,CU(A\& C)} M_{b}^{A,CU(A\& C)} .$$

Evaluating the slope ${{\partial \Pi^{A,CU(A\& C)} } \mathord{\left/ {\vphantom {{\partial \Pi^{A,CU(A\& C)} } {\partial G^{A} }}} \right. \kern-0pt} {\partial G^{A} }}$ at the point where $G^{A} = G^{A,PR}$, we have

$$\frac{{\partial \Pi^{A,CU(A\& C)} }}{{\partial G^{A} }}\left| {_{{G^{A} = G^{A,PR} }} } \right. > 0.$$

The strict concavity of the aggregate payoff function implies that $G^{A,CU(A\& C)} > G^{A,PR} .$

Country B determines an optimal arming, denoted as $G^{B,CU(A\& C)} ,$ that maximizes its aggregate payoff: $\Pi^{B,CU(A\& C)} = CS^{B,CU(A\& C)} + PS^{B,CU(A\& C)} + \tau_{c}^{B} M_{c}^{B,CU(A\& C)} + \tau_{a}^{B} M_{a}^{B,CU(A\& C)} .$

Evaluating the slope ${{\partial \Pi^{B,CU(A\& C)} } \mathord{\left/ {\vphantom {{\partial \Pi^{B,CU(A\& C)} } {\partial G^{A} }}} \right. \kern-0pt} {\partial G^{A} }}$ at the point where $G^{B} = G^{B,PR}$, we have

$$\frac{{\partial \Pi^{B,CU(A\& C)} }}{{\partial G^{B} }}\left| {_{{G^{B} = G^{B,PR} }} } \right. < 0,$$

which implies that $G^{B,CU(A\& C)} < G^{B,PR} .$