A game-theoretic analysis of international trade and political conflict over external territories

Abstract

For analyzing conflict between two large open countries over external territories rich in natural resources, we develop a game-theoretic model of trade under resource appropriation possibilities. We show that greater trade openness (by lowering trade costs) reduces the overall intensity of arming when the adversary countries are symmetric in all dimensions. This finding is consistent with the liberal peace hypothesis that trade reduces conflict. We further analyze how equilibrium is affected by differences in national resource endowments. The resulting asymmetric equilibrium reveals that arming by the more endowed country exceeds that of the less endowed country and the two adversaries respond to lower trade costs differently: the more endowed country cuts back on its arming, whereas the less endowed country may increase it. Under resource endowment asymmetry, the aggregate arming allocations of the adversaries could increase despite greater trade openness.

Appendix

1.1 Appendix 1: Equilibrium prices, consumer surplus, and producer surplus

After substituting $X_{A}$ from (2a) and $Y_{B}$ from (2b) into the final good prices in (11), we have:

$$P_{X} = \frac{{G_{A} (G_{A} + G_{B} - Z) + (2\alpha - R_{A} - t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}},$$

(24)

$$P_{Y} = \frac{{G_{B} (G_{A} + G_{B} - Z) + (2\alpha - R_{B} + t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}.$$

(25)

$$H_{X} = \frac{{G_{A} (G_{A} + G_{B} - Z) + (2\alpha - R_{A} + t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}},$$

(26)

$$H_{Y} = \frac{{G_{B} (G_{A} + G_{B} - Z) + (2\alpha - R_{B} - t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}},$$

(27)

Substituting these equilibrium prices into the demand equations in (3), we then use Eqs. (4) and (5) to calculate consumer and producer surplus. For country A, we have

$$\begin{aligned} CS_{A} = \frac{1}{2}\left[ {\frac{{G_{A} (Z - G_{A} - G_{B} ) + (R_{A} + t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right]^{2} + \frac{1}{2}\left[ {\frac{{G_{B} (Z - G_{B} - G_{A} ) + (R_{A} - t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right]^{2} , \hfill \\ PS_{A} = \left( {\frac{{(2\alpha - R_{A} - t)(G_{A} + G_{B} ) - G_{A} (Z - G_{A} - G_{B} )}}{{2(G_{A} + G_{B} )}}} \right)\quad\left[ {R_{A} - G_{A} + \left( {\frac{{G_{A} }}{{G_{A} + G_{B} }}} \right)Z} \right]. \hfill \\ \end{aligned}$$

(28)

For country B, we have

$$\begin{aligned} CS_{B} = \frac{1}{2}\left[ {\frac{{G_{B} (Z - G_{B} - G_{A} ) + (R_{B} + t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right]^{2} + \frac{1}{2}\left[ {\frac{{G_{A} (Z - G_{A} - G_{B} ) + (R_{A} - t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right]^{2} , \hfill \\ PS_{B} = \left( {\frac{{(2\alpha - R_{B} + t)(G_{A} + G_{B} ) - G_{B} (Z - G_{A} - G_{B} )}}{{2(G_{A} + G_{B} )}}} \right)\left[ {R_{B} - G_{B} + \left( {\frac{{G_{B} }}{{G_{A} + G_{B} }}} \right)Z} \right]. \hfill \\ \end{aligned}$$

(29)

1.2 Appendix 2: Decomposing the effect of country A’s arming

Making use of ${{\partial CS_{A} } \mathord{\left/ {\vphantom {{\partial CS_{A} } {\partial G_{A} }}} \right. \kern-0pt} {\partial G_{A} }}$ in (14) and ${{\partial PS_{A} } \mathord{\left/ {\vphantom {{\partial PS_{A} } {\partial G_{A} }}} \right. \kern-0pt} {\partial G_{A} }}$ in (15), the effect of country A’s arming on its aggregate payoff is calculated as follows:

$$\begin{aligned} \frac{{\partial \varPi_{A} }}{{\partial G_{A} }} & = \frac{{\partial CS_{A} }}{{\partial G_{A} }} + \frac{{\partial PS_{A} }}{{\partial G_{A} }} \\ & = \left( {D_{X} \frac{{\partial D_{X} }}{{\partial G_{A} }} + M_{Y} \frac{{\partial M_{Y} }}{{\partial G_{A} }}} \right) + \left( {P_{X} \frac{{\partial X_{A} }}{{\partial G_{A} }} + X_{A} \frac{{\partial P_{X} }}{{\partial G_{A} }}} \right) \\ & = D_{X} \frac{{\partial D_{X} }}{{\partial G_{A} }} + M_{Y} \frac{{\partial M_{Y} }}{{\partial G_{A} }} + P_{X} \frac{{\partial X_{A} }}{{\partial G_{A} }} + X_{A} \left( { - \frac{{\partial D_{X} }}{{\partial G_{A} }}} \right) \\ & = \left[ { - (X_{A} - D_{X} )\frac{{\partial C_{X} }}{{\partial G_{A} }}} \right] + M_{Y} \left( {\frac{{\partial M_{Y} }}{{\partial G_{A} }}} \right) + P_{X} \frac{{\partial X_{A} }}{{\partial G_{A} }}\quad \left( {{\text{noting}}\;{\text{ that}}\ \frac{{\partial M_{Y} }}{{\partial G_{A} }} = - \frac{{\partial P_{Y} }}{{\partial G_{A} }}} \right) \\ & = (X_{A} - D_{X} )\frac{{\partial P_{X} }}{{\partial G_{A} }} + \left[ {M_{Y} \left( { - \frac{{\partial P_{Y} }}{{\partial G_{A} }}} \right)} \right] + P_{X} \frac{{\partial X_{A} }}{{\partial G_{A} }} \\ \end{aligned}$$

The above derivative can further be re-written as $\frac{{\partial \varPi_{A} }}{{\partial G_{A} }} = E_{X} \frac{{\partial P_{X} }}{{\partial G_{A} }} + M_{Y} \left( { - \frac{{\partial P_{Y} }}{{\partial G_{A} }}} \right) + P_{X} \frac{{\partial X_{A} }}{{\partial G_{A} }}.$ Similarly, country B determines an arming allocation $G_{B}$ that solves the following maximization problem: $\mathop {\text{Max}}\limits_{{\{ G_{B} \} }} \varPi_{B} = CS_{B} + PS_{B} ,$ where $CS_{B}$ and $PS_{B}$ are consumer and producer surplus as given in Appendix 1. We decompose the effect of country B’s arming into three separate terms:

$$\frac{{\partial \varPi_{B} }}{{\partial G_{B} }} = \underbrace {{E_{Y} \frac{{\partial P_{Y} }}{{\partial G_{B} }}}}_{\begin{subarray}{l} {\text{ Export - revenue effect }} \\ {\text{ of arming }} \\ { ( + )} \end{subarray} } + \underbrace {{M_{X} \left( { - \frac{{\partial P_{X} }}{{\partial G_{B} }}} \right)}}_{\begin{subarray}{l} {\text{Import - expenditure effect }} \\ {\text{ of arming }} \\ { (} - )\end{subarray} } + \underbrace {{P_{Y} \frac{{\partial Y_{B} }}{{\partial G_{B} }}}}_{\begin{subarray}{l} {\text{Output - distortion effect}} \\ {\text{ of\, arming}} \\ { (} - )\end{subarray} }$$

where $E_{Y} \equiv (Y_{B} - D_{Y} )$ is the amount of the final good Y exported from B to A.

Alternatively, we can use (14)–(16) to decompose the effect of country A’s arming into three separate terms explicitly in terms of $G_{A}$ and $G_{B}$ as follows:

$$\begin{aligned} \frac{{\partial \varPi_{A} }}{{\partial G_{A} }} & = \frac{{\partial CS_{A} }}{{\partial G_{A} }} + \frac{{\partial PS_{A} }}{{\partial G_{A} }} \\ & = \left( {D_{X} \frac{{\partial D_{X} }}{{\partial G_{A} }} + M_{Y} \frac{{\partial M_{Y} }}{{\partial G_{A} }}} \right) + \left( {P_{X} \frac{{\partial X_{A} }}{{\partial G_{A} }} + X_{A} \frac{{\partial P_{X} }}{{\partial G_{A} }}} \right) \\ & = D_{X} \left( { - \frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right) + M_{Y} \left( { - \frac{{G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right) \\ & \quad + P_{X} \left( { - \frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{\left( {G_{A} + G_{B} } \right)^{2} }}} \right) + X_{A} \left( {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right) \\ & = \underbrace {{\left( {X_{A} - D_{X} } \right)\left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{\begin{subarray}{l} {\text{ Export - revenue\, effect }} \\ {\text{ of arming }} \\ { ( + )} \end{subarray} }\underbrace {{ - M_{Y} \frac{{G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}}}_{\begin{subarray}{l} {\text{ Import - expenditure\, effect }} \\ {\text{ of arming }} \\ { (} - )\end{subarray} }\underbrace {{ - P_{X} \left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{\begin{subarray}{l} {\text{Output - distortion\, effect}} \\ {\text{ of arming}} \\ { (} - )\end{subarray} }, \\ \end{aligned}$$

where $X_{A} ,$$D_{X} ,$$M_{Y} ,$ and $P_{X}$ are functions of $G_{A}$ and $G_{B} .$ That expression implies that country A increases its arming up to where marginal benefit equals marginal cost, that is,

$$\underbrace {{\left( {X_{A} - D_{X} } \right)\left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{\text{ A's\ marginal\ revenue\ of\ arming}} = \underbrace {{M_{Y} \frac{{G_{B} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }} + P_{X} \left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{B} Z}}{{\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{\text{A's\ maerginal\ cost\ of\ arming}}.$$

As for country B, we have the following FOC:

$$\frac{{\partial \varPi_{B} }}{{\partial G_{B} }} = \underbrace {{\left( {Y_{B} - D_{Y} } \right)\left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{A} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{{ ( { + )}}}\underbrace {{ - M_{X} \frac{{G_{A} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}}}_{ (- )}\underbrace {{ - P_{Y} \left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{A} Z}}{{\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{ (- )} = 0.$$

Country B’s arming likewise is chosen where marginal benefit equals marginal cost, namely,

$$\underbrace {{\left( {Y_{B} - D_{Y} } \right)\left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{A} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{\text{B's\ marginal\ revenue\ of\ arming}} = \underbrace {{M_{X} \frac{{G_{A} Z}}{{2\left( {G_{A} + G_{B} } \right)^{2} }} + P_{Y} \left[ {\frac{{\left( {G_{A} + G_{B} } \right)^{2} - G_{A} Z}}{{\left( {G_{A} + G_{B} } \right)^{2} }}} \right]}}_{\text{B's\ marginal\ cost\ of\ arming}}.$$

1.3 Appendix 3: Proof of Lemma 1

For country A, ${{\partial \varPi_{A} } \mathord{\left/ {\vphantom {{\partial \varPi_{A} } {\partial G_{A} }}} \right. \kern-0pt} {\partial G_{A} }} < 0$ when $E_{X} ({{\partial P_{X} } \mathord{\left/ {\vphantom {{\partial P_{X} } {\partial G_{A} }}} \right. \kern-0pt} {\partial G_{A} }})$ is less than the sum of $M_{Y} ( - {{\partial P_{Y} } \mathord{\left/ {\vphantom {{\partial P_{Y} } {\partial G_{A} }}} \right. \kern-0pt} {\partial G_{A} }})$ and $P_{X} ({{\partial X_{A} } \mathord{\left/ {\vphantom {{\partial X_{A} } {\partial G_{A} }}} \right. \kern-0pt} {\partial G_{A} }})$ in absolute value. That is, $MR_{A}^{Arms} < MC_{A}^{Arms} .$ Similarly, ${{\partial \varPi_{B} } \mathord{\left/ {\vphantom {{\partial \varPi_{B} } {\partial G_{B} }}} \right. \kern-0pt} {\partial G_{B} }} < 0$ when $MR_{B}^{Arms} < MC_{B}^{Arms} .$ As a result, we have $G_{A} = G_{B} = 0.$□

1.4 Appendix 4: Comparative statics of the equilibrium arming under symmetry

Taking the derivative of $G^{ * }$ (18) with respect to $Z$, R, and t, respectively, we have the following derivatives:

$$\frac{{\partial G^{ * } }}{\partial Z} = \frac{6R + Z - 16\alpha + 10t + 5\sqrt K }{12\sqrt K } > 0,\frac{{\partial G^{ * } }}{\partial R} = \frac{6R + Z - 8\alpha + 2t + \sqrt K }{2\sqrt K } > 0,\frac{{\partial G^{ * } }}{\partial t} = \frac{(6R + 5Z - 8\alpha + 2t + \sqrt K )}{6\sqrt K } > 0.$$

1.5 Appendix 5: Aggregate payoff functions under asymmetry in national endowments

Substituting $R_{A} = (R_{o} + \delta )$ and $R_{B} = (R_{o} - \delta )$ into (28) and (29) in Appendix 1, we have the following aggregate payoff functions for countries A and B:

$$\begin{aligned} & \varPi_{A} (G_{A} ,G_{B} ;\delta ) \\ & = CS_{A} (G_{A} ,G_{B} ;\delta ) + PS_{A} (G_{A} ,G_{B} ;\delta ) \\ & = \frac{1}{2}\left( {\frac{{G_{A} (Z - G_{A} - G_{B} ) + (R_{o} + \delta + t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right)^{2} + \frac{1}{2}\left( {\frac{{G_{B} (Z - G_{B} - G_{A} ) + (R_{o} - \delta - t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right)^{2} \\ & \quad + \left( {\frac{{(2\alpha - R_{o} - \delta - t)(G_{A} + G_{B} ) - G_{A} (Z - G_{A} - G_{B} )}}{{2\beta (G_{A} + G_{B} )}}} \right)\left[ {(R_{o} + \delta ) - G_{A} + \left( {\frac{{G_{A} }}{{G_{A} + G_{B} }}} \right)Z} \right] \\ \end{aligned}$$

and

$$\begin{aligned} & \varPi_{B} (G_{A} ,G_{B} ;\delta ) \\ & = CS_{B} (G_{A} ,G_{B} ;\delta ) + PS_{B} (G_{A} ,G_{B} ;\delta ) \\ & = \frac{1}{2}\left( {\frac{{G_{B} (Z - G_{A} - G_{B} ) + (R_{o} - \delta + t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right)^{2} + \frac{1}{2}\left( {\frac{{G_{A} (Z - G_{A} - G_{B} ) + (R_{o} - t)(G_{A} + G_{B} )}}{{2(G_{A} + G_{B} )}}} \right)^{2} \\ & \quad + \left( {\frac{{(2\alpha - R_{o} + \delta + t)(G_{A} + G_{B} ) - G_{B} (Z - G_{A} - G_{B} )}}{{2\beta (G_{A} + G_{B} )}}} \right)\left[ {(R_{o} - \delta ) - G_{B} + \left( {\frac{{G_{B} }}{{G_{A} + G_{B} }}} \right)Z} \right]. \\ \end{aligned}$$