Trade Openness, International Conflictand the "Paradox of Power"*

by

Constantinos SyropoulosDepartment of Economics and International Business

LeBow College of BusinessDrexel University

503-N Matheson HallPhiladelphia, PA 19104

E-mail: c.syropoulos@drexel.edu

Current Version: December 2006

Abstract

This paper explores the possible connections between trade openness, interstate disputes andinternational conflict over contested resources in the context of a neoclassical trade model in which two“small” countries with ownership claims over a productive resource (e.g., land or oil) resolve their disputenon-violently in a military contest. We first clarify how the initial distribution of secure endowments andtechnology condition arming incentives, the division of the contested resource, and the contending countries’welfare under autarky and free trade. We then identify the conditions under which free trade, through itsimpact on product and factor prices, “levels the playing field” in arms competition and intensifies conflict.In addition, we (i) unveil how the resource cost of conflict may overwhelm the traditional gains from tradefor one or both contending countries; (ii) clarify how relative power, the volumes and patterns of thecontesting countries’ trade flows are determined by the interplay between technology, the distribution ofendowments, and the world price; (iii) demonstrate how terms of trade improvements can be “immiserizing”in the presence of international conflict, and (iv) elaborate on the linkages between trade and securitypolicies.

JEL Classification: D30, D70, D72, D74, F2, F10.Keywords: trade openness, property rights, interstate disputes, conflict, security policies.

_______________________

* For their thoughtful comments and helpful discussions on earlier versions of the paper, I wish to thank RickBond, Spiros Bougheas, Ngo Van Long, Bill Polley, Stergios Skaperdas, and participants at various seminarsand conferences.

1. INTRODUCTION

In the decades that preceded World War I, the proportion of world trade to world GDP had

reached unprecedented heights (O�Rourke and Williamson, 2000). Yet, international con�ict ensued

with much ferocity and despite expectations to the contrary.1 Similarly, the expansion of trade in

the post World War II era has been spectacular. Still, insecurity and contention have not only

subsided but are �aring in many parts of the world.2 The European Economic Community (now

European Union) has been a notable exception� the integration of markets there appears to have

coexisted with diminished tensions and increased security for member states. Nevertheless, the

perceived need to maintain armies within that family of nations persists.

Is there a relationship between trade openness and international con�ict? This question has

been of enormous interest to historians, political scientists, and scholars of international relations

who have debated it extensively and have come to share a rich literature on it.3 Disturbingly,

economists have explored this topic sparingly.4 This paper aims to kindle interest in this issue and

help bridge the existing gap in scholarship.

The causes for international con�ict are, of course, numerous. There is strong evidence though

1The prediction before World War I, for example, that war was impossible or unthinkable� because Britain andGermany had become so economically interdependent that con�ict would be �commercial suicide� (Angell, 1933)�was �atly contradicted by experience.

2From the Mediterranean and Africa to the Middle East, Caucasus, Central Asia, Kashmir, and the Spratly islandsin the South China sea, violent con�ict devastates societies with destruction, poverty, and despair. But, by divertingresources away from productive activites, non-violent con�ict ("cold wars") too can be very costly to societies. Hess(2003) estimates the welfare costs of con�ict coming from its e¤ects on consumption alone for 147 countries spanningthe period 1960-1992 to be on average 8 percent of steady-state consumption.

3Two prominent schools of thought stand out: the classical liberal school, which posits that freer trade alleviatescon�ict because of its high opoortunity costs� con�ict preempts the possibility of realizing the classical gains fromexchange and specialization in production (Polachek, 1980); and the realist/neorealist schools, which view relationsbetween states primarily as adversarial with trade being an instrument that may in fact hurt one or all parties.Realists argue that the bene�ts from freer trade may fuel frictions and con�ict because some states might perceivethat they (or their rivals) will develop a military edge in security competition (Waltz, 1979; Grieco, 1988). Neorealistspoint out that freer trade between adversaries creates an adverse �security externality� that may in fact outweighthe gains from trade (Gowa and Mans�eld, 1993; Gowa, 1994).The classical liberal view also emphasizes the possibility that free trade may create constituencies that favor peace

by expanding global contacts and increasing commercial interests that oppose con�ict (Schneider et al., 2003). Relatedperspectives also propound the notion that, by fostering understanding of cultures, mutually advantageous exchangediscourages violence and diminishes the likelihood of con�ict.

4Recent research that has considered aspects of the relationship between security and trade include Anderson andMarcouiller (2005) and Anderton et. al. (1999) who have analyzed Ricardian models in which traded commoditiesare insecure either because of the presence of pirates and bandits or because the two sides in�uence the terms oftrade through arming. Both approaches emphasize the important point that international trade can be hampered bythe anarchic nature of international relations. Skaperdas and Syropoulos (2001, 2002) have addressed some of theimplications of insecure property in simple models of exchange. Barbieri and Schneider (1999) have surveyed muchof the theoretical and empirical literatures on the subject.

1

that territorial disputes or, more generally, competing claims over resources (most notably land, oil,

natural gas and water) have sparked numerous con�icts throughout history and are also the culprit

of several (actual and/or potential) con�icts today.5 Typically this is so because international peace-

promoting institutions are weak, supra-national authorities in charge of enforcing international

treaties lack the requisite power, and/or the costs of contract enforcement are extremely large. It

is thus interesting that, even though property rights over certain resources cannot be meaningfully

viewed as "complete" or "secure" and the barrel of the gun often serves as the ultimate arbiter of

inter-state disputes, the extant trade theory has ruled these possibilities out by assumption.6 This

paper seeks to broaden the scope of this theory by recognizing that sovereign nations contest the

ownership of productive resources, and considering a model of appropriative con�ict/trade that

casts new analytical light on the link between trade openness and intestate con�ict.

In particular, we develop a general equilibrium model in which con�ict takes the form of a contest

over a disputed resource (land) by two contending nations: the larger the quantity of arms (i.e.,

military capability, or "guns" for short) of a contender, the bigger its share of the contested resource

and therefore its national income. But production of guns is socially costly� to produce guns a

country must divert resources away from income-generating activities, notably the production of

consumption goods.7 Thus, adversaries inevitably face the fundamental tradeo¤ between power

and income or, more starkly, between "guns" and "butter."

A novel feature of our approach to the relationship between the international economy and

national security policies is its emphasis on domestic distributional and factor-price e¤ects.8 This

5Klare (2001) provides a fascinating account of past and ongoing con�icts for oil in the Persian Gulf, the �ghtsover undersea oil and natural gas reserves in the Caspian Sea Basin (that directly involve Azerbaijan, Iran, Armenia,Georgia and Uzbekistan), and the South China Sea (with China, Malaysia, the Philipinnes, Taiwan and Vietnambeing key actors). He also details the expanding involvement of major powers (including the United States andRussia) in local power dynamics, as indicated by their military presence and show of force. Klare also discusses thecontinuing con�icts over water and river systems, such as the Nile, Jordan, Euphrates, and Indus river basins, citingthe numerous disputes and clashes for usage rights.

6Notable exceptions are Chichilnisky (1994), who argued that trade may reduce welfare in the South by accen-tuating the over-exploitation of an open-access resource it has a comparative advantage in, and Brander and Taylor(1997a) who formally proved this idea. See also Hotte et. al. (2000) who studied the e¤ects of trade in an open-accessresource and extended the analysis to consider the evolution of private enforcement in dynamic environments.

7Of course, the social costs of violent con�ict extend far beyond the "socially unproductive" use of resources weconsider in this paper� they surely include the loss of lives in combat and the destruction of productive assets thatwar normally brings along. At the same time, however, con�ict may bene�t one or more parties by enabling themto appropriate a larger share of a disputed prize, or by preempting costly frictions in the future. As we will see,to explore how trade openness matters in such settings, it is imperative to use general equilibrium models in whichtrade openness a¤ects factor prices and, through these prices, arming incentives.

8Rowe (1999) embraced this approach and argued how it could be fruitfully used to demonstrate that many of theorigins of World War I can be traced to the expansion of the world economy in the several decades that preceded it.

2

emphasis is appropriate because it captures compellingly the nuanced in�uence of trade openness

on the ability of sovereign nations to mobilize resources for security purposes. Borrowing several key

elements from the Heckscher-Ohlin-Samuelson (HOS) or factor-proportions theory, in the model

there are two intersectorally mobile (but internationally immobile) factors of production and two

consumption goods that can be traded internationally at �xed prices (i.e., countries are "small"

in world markets). Viewing "guns" as a composite commodity, we suppose they can be produced

domestically in a third sector that relies on the economy�s available resources. Abstracting from

domestic politics, here we focus on the actions of "benevolent" policymakers that use arming to

maximize national welfare and capture trade openness by way of reference to two extreme regimes:

autarky (no trade) and free trade.

Under either trade regime, we disentangle the dependence of the contending countries�military

capability and economic prowess on fundamentals (i.e., technology, factor endowments, consumer

preferences, the degree of land insecurity). This enables us to address the following speci�c ques-

tions. How does trade openness a¤ect arming incentives and the balance of power? Is it possible

for an economically disadvantaged side to obtain a bigger share of the contested resource (i.e.,

be militarily more powerful)? Or, as Jack Hirshleifer�s (1991) put it, when does the "Paradox of

Power" arise in international relations? Should international con�ict over resources be viewed as

an activity that raises "trade costs?" Or is it more fruitful to think of it as an activity capable

of altering the structure of a country�s comparative advantage? In the presence of con�ict, is free

trade superior to commercial isolation?

Starting with autarky, we �rst identify secure endowment distributions between the contending

countries which ensure their production identical quantities of arms and therefore equally balanced

power. We then show how technology and asymmetries in distribution of factor ownership a¤ect

equilibrium investments in arms and the prices of consumption goods. Our analysis reveals that,

depending on technology, the relatively less a­ uent (in resources) contender may arm more heavily

than its adversary, but that delicately depends on the composition of its initial endowments.

Turning to free trade in consumption goods we demonstrate that there exist circumstances

under which such trade leads to international factor price equalization. We then argue that this

could result in the equalization of marginal bene�ts and marginal costs of arming across countries

and, ultimately, arming incentives. This is an intriguing �nding as it suggests that, under certain

3

circumstances (which we make precise), free trade in consumption goods will "level the playing

�eld" in arms competition and, strikingly, equalize power. Of course, this is rarely the case in the

real world. Nonetheless, the result is noteworthy because its unveils a number of factors (e.g., those

that ensure international factor price equalization) that play prominent roles in the determination

of the balance of power.

But while opening up borders to free trade may equalize power, it does not necessarily imply

that the contending countries will devote less resources to arming, as compared to autarky. Indeed,

depending on world prices and their e¤ect on domestic factor prices, it is possible for trade to reduce

the opportunity cost of arming and thereby amplify national incentives to build guns. Since, from

society�s perspective, arming is purely an internationally redistributive activity, it is possible that

the resource cost it generates may overshadow the traditional gains from free trade, and thereby

cause free trade to become Pareto-dominated by autarky (or, depending on fundamentals, welfare

inferior to autarky for one side). In other words, we formally establish the sense in which, and some

of the circumstances under which, autarky may dominate free trade in the presence of international

con�ict.

Since the implementation of security policies entails the absorption of real resources and trade

regimes shape arming incentives, a contending country�s residual factor supplies� the supplies left

for "useful" production� will be trade-regime dependent. This has important implications for a

contender�s apparent comparative advantage, which may di¤er from the country�s natural compar-

ative advantage. Our analysis shows that resource disputes and international con�ict can interfere

with the determination of trade patterns and, moreover, that con�ict may spur (rather than sup-

press) a country�s trade �ows. In addition, we �nd that, in the presence of international con�ict,

neither autarky prices nor intercountry di¤erences in relative endowments are necessarily accurate

predictors of trade patterns.

Our analysis also sheds light on the possible connection that might exist between international

con�ict and the "resource curse" problem� that resource-abundant countries may su¤er in terms

of growth. Existing research on the subject emphasized the importance of domestic rent-seeking

(e.g., Torvik, 2002; Mehlum et. al. 2005), redistributive politics (e.g., Robinson et. al., 2005),

and domestic con�ict (Gar�nkel et. al., 2005) for such economies. We show that, through their

stimulant e¤ect on arming, terms of trade improvements can hurt countries that export contested-

4

resource-intensive commodities and are immersed in international con�ict.

In the presence of resource disputes, activist commercial policies can have important e¤ects on

appropriative competition and welfare even when countries are unable to in�uence world prices.

Our analysis reveals that this possibility arises when policymakers design trade policies before

they implement security policies (as when they enter multilateral or preferential trade agreements).

As in perfectly competitive environments, for any given arms, trade policies generate the usual

deadweight losses. However, in the presence of international con�ict, trade policies, through their

impact on domestic distribution and factor prices, can have important e¤ects on strategic arming

incentives and, ultimately, welfare.9

In the next section, we present the formal model and a preliminary analysis that proves useful

in subsequent sections. Then, in Section 3, we investigate optimal security policies, con�ict, and

the balance of power, under autarky and free trade. In Section 4, we explore the implications of

international con�ict for trade patterns and trade volumes. In Section 5, we compare autarky and

free trade in terms of their impact on arming and welfare. In Section 6, we brie�y examine some

additional issues, including the choice of trade regime, and the implications of commercial policies.

Lastly, in Section 7, we o¤er several concluding comments. All technical proofs have been placed

in the Appendix.

2. FRAMEWORK AND PRELIMINARY ANALYSIS

We consider a world that consists of two countries, labeled 1 and 2, and the rest of the world

(ROW) which for simplicity we treat as a single entity. Every country may produce two consumption

goods, also labeled 1 and 2, using two resources, "labor" and "land," under conditions of constant

returns to scale. For speci�city, we suppose good 2 is land-intensive, but we indicate how the

ranking of factor intensities across industries matters. In the spirit of the HOS trade model, we

rule out factor intensity reversals and allow all countries to have access to the same technology.

Moreover, we suppose consumers have identical and homothetic preferences, goods 1 and 2 are

essential in consumption, and all markets are perfectly competitive.9That trade and security policies should be related appears obvious. In fact, many embargoes, sanctions, and

various other forms of trade restrictions that have been used throughout history and continue to be used today canbe considered extensions of security policies (Hirschman, 1945). The British policy of appeasement towards Germanyduring the 1930s was based on classical liberal arguments about the use of economic carrots to avoid war (Kagan,1995).

5

Each country i = 1; 2 possesses Li units of secure labor andKi units of secure land. (Land could

also be interpreted as a natural resource� e.g., oil, water� or just physical capital.) In addition,

there are K0 units of insecure land which the two countries contest. The stage is anarchic�

enforceable binding contracts on the proliferation of arms and the partition of K0 are impossible�

and policymakers use arming to maximize national welfare.10

Denote with Gi country i�s guns� but keep in mind that it is more accurate to think of this

variable as a producible composite commodity that encapsulates country i�s military capability.

Now �i be country i�s share of K0, and identify it with the contest success function (CSF)

�i(Gi; Gj) =

8><>:f(Gi)

f(Gi)+f(Gj)if Gi +Gj > 0 for i 6= j = 1; 2

12 if G1 +G2 = 0

(1)

where f(�) � 0, f(0) = 0, f 0(�) > 0, limGi!0 f0(Gi) =1, and f 00(�) � 0. In words, a country�s share

of the contested resource is increasing in its own guns (i.e., �iGi = @�i=@Gi > 0) but decreasing

in the guns of its adversary (�iGj = @�i=@Gj < 0 for i 6= j).11 This dependence of a country�s

contested share on guns can be taken literally or could be interpreted as the reduced form of a

bargaining process in which relative arming �gures prominently in the division of the contested

resource.12

A country builds its own military constellation but often it imports some weapons and hires

mercenaries or foreign security experts. To stay on course, here we abstract from the possibility of

trade in arms and focus on the case in which guns can only be produced domestically. A contending

country has an interest in producing guns because more guns translate into a bigger share of the

contested land and, as we will see, more income. But there will also exist an opportunity cost to

producing guns� the loss in income due to the diversion of resources from consumption goods.

The sequencing of decisions will be as follows. (i) Each contender i determines its (irreversible)

10Of course, many of our assumptions can (and should) be relaxed. But, since our main objective here is to providea uni�ed framework for the analysis of arming incentives and trade openness, such modi�cations must necessarily beleft to future research.11Condition limGi!0 f

0(Gi) = 1 could be relaxed without much loss of generality. In the Appendix we describeseveral additional properties of �i that prove useful in our analysis. A functional form for f(�) that has been widelyused in the rent-seeking literature, as well as the literatures on tournaments and con�ict, is f(G) = G where 2 (0; 1](Tullock, 1980). See Skaperdas (1996) for an axiomatization of this function and Hirshleifer (1989) for a comparisonof the properties of this form with f(G) = e G.12See Anbarci et. al. (2002) for an analysis of this issue and how, in particular, di¤erent bargaining solution

concepts lead to division rules that vary in their sensitivity to guns.

6

production of guns, Gi, for a any given choice of guns, Gj (j 6= i), by its adversary and given its

initial secure factor endowments. (ii) Once guns choices are made, each contender i seizes �iK0

unit of the contested resource. (iii) With countries�residual factor endowments now having been

determined, private production and consumption decisions take place. Under autarky, prices adjust

until markets clear. Under free trade, the prices of consumption goods are �xed in the world market.

A con�ictual equilibrium will be the Nash equilibrium in guns conditional on the prevailing trade

regime. To complete the model, we now detail the supply and demand sides in each economy.

Let i � (wi; ri) and cij � cj(wi; ri) respectively represent the unit cost functions of guns

and good j = 1; 2, where wi and ri are the competitive wage and rental rates paid to workers and

landowners, respectively. These functions have the usual properties, including linear homogeneity

in factor prices. By Shephard�s lemma, iw � @ i=@wi and ir � @ i=@ri capture the unit labor and

land requirements in arms, respectively. Similarly, the unit land and labor requirements in good j

are aiKj � @cij=@ri and aiLj � @cij=@w

i; therefore, the land/labor ratio in guns is kiG � ir= iw and

the corresponding ratio in industry j is kij � aiKj=aiLj . We say that industry 2 is land-intensive if

ki2 > ki1 (or labor-intensive if ki2 < ki1) at all relevant factor prices.

Identifying good 1 as the numeraire, denote with pi the relative price of good 2 in country i.

With diversi�cation in production, perfect competition in product markets requires

c1(wi; ri) = aiL1w

i + aiK1ri = 1 (2)

c2(wi; ri) = aiL2w

i + aiK2ri = pi: (3)

These equations and the properties of unit cost functions imply that the relative price of labor,

!i � wi=ri, can be written as a function of pi (see Lemma 1 below). More speci�cally, !i = !(pi)

because every country has access to the same technology by assumption.

Let Xij be the output of consumption good j in country i. For given guns, the vector (K

iX ; L

iX)

captures the residual quantities of resources left for the production of consumption goods in country

i. When both consumption goods are produced, factor market clearing requires

aiK1Xi1 + a

iK2X

i2 = Ki

X ( � Ki + �iK0 � irGi) (4)

aiL1Xi1 + a

iL2X

i2 = LiX ( � Li � iwGi). (5)

7

These equations together with the linear homogeneity of unit cost functions and the fact that

!i = !(pi) imply that the relative supply of the non-numeraire, RSi � Xi2=X

i1, can be written as

a function of pi and the residual land/labor ratio,

kiX �KiX

LiX=Ki + �iK0 � irGi

Li � iwGi: (6)

In other words, RSi = RS(pi; kiX). Lemma 1 below unveils the dependence of !i and RSi on their

arguments.

Lemma 1. If production in a country is diversi�ed (i.e., Xij > 0, 8i 6= j), then

(a) @!i=@pi

!i=pi7 0 if ki2 ? ki1;

(b) @RSi=@kiXRSi=kiX

? 0 if ki2 ? ki1;

(c) @RSi=@pi

RSi=pi> 0.

Part (a) of Lemma 1 is a version of the Stolper-Samuelson (1941) theorem� an increase in the

relative price of the land- (labor-) intensive good reduces (raises) the relative price of labor. Part

(b) is a variant of the Rybczynski (1955) theorem and clari�es that an increase in the residual

relative supply of land causes the relative supply of the land-intensive good to rise. Part (c) is a

re�ection of the fact that production exhibits increasing opportunity costs.

In Lemma 1 the residual land/labor ratio, kiX , is treated as exogenous. However, from (6) it

is clear that kiX depends on price, guns, and factor supplies. Lemma 2 below unveils the exact

relationship.

Lemma 2. Let ki � Ki+�iK0

Liand suppose the production of consumption goods is diversi�ed.

Then kiX = kiX(pi; Gi; Gj ;K0;K

i; Li) and

(a) @kiX@pi

? 0 if ki2 ? ki1;

(b) @kiX@Gi

> 0; 8Gi that satisfy Ki0�

iGi �

i=ri + " > 0, for some " > 0;

(c) @kiX@Gj

< 0; 8i 6= j;

(d) @kiX@Gi

+@kiX@Gj

R 0 if ki R kiG whenever Gi = Gj ; 8i 6= j;

8

(e) @kiX@Li

< 0 and @kiX@K0

> 0;@kiX@Ki > 0.

Suppose ki2 > ki1. Part (a) of Lemma 2 establishes that, when both consumption goods are

produced, the residual land/labor ratio, kiX , is increasing in the relative price, pi. By Lemma 1(a),

for given guns, an increase in pi forces the wage/rental ratio, !i, to fall. In turn, this reduces

(raises) the quantity of land (labor) demanded in the arms sector, and causes the residual amount

of land (labor) to rise (fall).

As we will see shortly, part (b) provides a condition that will necessarily be satis�ed if the

marginal bene�t to country i of producing guns is not overwhelmed by its corresponding opportunity

cost. Part (c) follows readily upon recognition of the fact that a country�s share of the contested

land is decreasing in the adversary�s guns. Part (d) notes that a simultaneous and equal increase in

both countries�arms raises or reduces kiX depending on how the land/labor intensity in the arms

sector, kiG, compares with the country�s overall land/labor ratio, ki. Part (e) is fairly obvious.

Turning to preferences, for simplicity we suppose consumers are risk neutral. Thus, we may

write country i�s indirect utility (aggregate welfare) function as

V i � V i(pi; Gi; Gj ;K0;Ki; Li) = �(pi)Ri(pi;Ki

X ; LiX) (7)

where Ri and �i � �(pi) respectively denote net national income and the marginal utility of

income. We are assuming that a country �nances its cost of producing arms with nondistortionary

income taxes. This assumption together with the prevalence of perfect competition imply that Ri

is the country�s maximized value of domestic production of consumption goods or, equivalently,

the minimized value of rental payments paid to owners of KiX and LiX . Thus, R

i = Xi1 + piXi

2 =

riKiX + w

iLiX , which explains the arguments of Ri (and V i) in (7).13

Country i�s demand function for the non-numeraire is Di2 = �iDR

i=pi, where �iD � �D(pi) =

�pi�ip=�i is the associated expenditure share. Total di¤erentiation of (7) and utilization of these

ideas gives

dV i = �(pi)��M idpi + ri

�K0�

iGi �

i=ri�dGi + riK0�

iGjdG

j�

(8)

13Ri should be identi�ed with the familiar gross domestic product (GDP) or revenue function (Dixit and Norman,1980), excluding arms expenditures. Ri is increasing and convex in pi, and increasing and concave in factor inputs(Ki

X ; LiX). It can be veri�ed that X

i2 = Rip (� @Ri=@pi) and @Xi

2=@pi = Ripp � 0. Furthermore, factor prices satisfy

wi = RiL (� @Ri=@LiX) and ri = RiK (� @Ri=@Ki

X).

9

for j 6= i, where M i � Di2 �Xi

2 is the excess demand of the non-numeraire. (To avoid cluttering

later, we omit subscript "2" from M i2.) The �rst term inside the square brackets (weighted by the

marginal utility of income) captures the familiar terms of trade (TOT) e¤ect on a country�s welfare.

In words, an increase in pi is welfare-reducing for an importer of good 2, but welfare-improving for

an exporter of this product.

The second term in the brackets captures the income e¤ect of a change in country i�s guns.

Ceteris paribus, an increase in Gi expands country i�s share of the contested land (the �rst term in

the parentheses) and thus its national income. At the same time, however, the Gi increase draws

resources away from the production of consumption goods and thus reduces income (the second

term in the parentheses). Consequently, a country will be able to improve its welfare by expanding

its arms production only if the expression in the parentheses is positive.

The third term in the brackets captures the e¤ect of a change in arms by country i�s adversary�

an increase in Gj a¤ects country i adversely because it reduces country i�s appropriated share of the

contested resource and thus its income. If Gi = Gj initially and both countries expand their gun

production proportionately, both countries will necessarily become worse o¤ because the division

of the contested resource remains una¤ected but their expenditure on guns rise.

We will use the above ideas to explore optimal security policies (arming), and the dependence

of international con�ict on alternative trade regimes. A distinguishing feature of this problem is

that, at interior solutions, country i�s �rst-order necessary condition (FOC) satis�es

V iGi �

@V i

@Gi= �(pi)r(pi)

�K0�

iGi �

i=ri�= 0 (9)

regardless of the trade regime we consider.14 Under autarky, product prices adjust until M i = 0.

This causes the �rst term inside the brackets in (8) to vanish and yields (9) as the relevant FOC.

But this term also vanishes under free trade in consumption goods because product prices are

invariant to security policies when countries are "small" in world markets. It should thus be clear

from (9) that a contending country�s marginal bene�t of producing guns (measured in land units)

is MBi = K0�iGi , which is independent of p

i and, consequently, of the trade regime considered. On

14Our discussion here and later in the paper is based on the assumtion that under autarky the distribution of factorendowments between the adversaries is such that their production of arms is not constrained by their secure landholdings. Under free trade, we assume that technology, the distribution of factor endowments, the quantity of thecontested resource and the world price are such that production of consumption goods is diversi�ed.

10

the other hand, the country�s marginal cost of producing guns (again measured in land units) is

MCi = i=ri, which is trade-regime dependent. We thus arrive at

Lemma 3. A country�s indirect utility function, V i, has the following properties:

(a) V iGiGi

< 0;

(b) V iGiGj

R 0 if Gi R Gj, 8i 6= j;

(c) V iGipi

? 0 if ki2 ? ki1 at the Gi that solves V i

Gi= 0;

(d) V i is strictly quasi-convex in pi and minimized at the pi that solves M i = 0:

Part (a) establishes that V i is strictly concave in country i�s guns, and thus explains the down-

ward sloping shape ofMBi, portrayed in Fig. 1 by the solid-line curve.15 Part (b) clari�es the idea

that a country�s net marginal payo¤ to producing arms rises or falls with its rival�s arms (i.e., MBi

shifts up or down) depending on which contender produces more guns to start with. As shown in

the Appendix, both parts (a) and (b) follow from the properties of the CSF.

Part (c) captures the idea that at the optimum the net marginal welfare e¤ect of country i�s

arming rises or falls with pi depending on the ranking of factor intensities between industries 1 and

2. This e¤ect is a consequence of the Stolper-Samuelson theorem and is driven by the impact of a

price change on the country�s marginal cost, MCi. The linear homogeneity of i in factor prices

implies that MCi = i=ri = (!i; 1) is increasing in the wage/rental ratio, !i. But, by Lemma

1(a), !i is decreasing (increasing) in pi if ki2 > ki1 (ki2 < ki1). Since, under global free trade, MCi is

pinned down by the world price� and is, therefore, invariant to changes in Gi� it must be the case

that MCi is a constant function under this regime (the dotted-line curve in Fig. 1). In the next

section, we show that under autarky MCi is generally increasing in Gi (the dashed-line curve in

Fig. 1).

Part (d) is a well-known property of indirect utility functions (Dixit and Norman, 1980) that

highlights the important idea that, for given guns, a country�s welfare is increasing with the distance

of the price from its autarky level.

15From the de�nition of MBi and the properties of the CSF, we know that MBi depends on K0 and guns, andlimGi!0MBi =1.

11

Wemay now state a noteworthy result that arises when security policies are individually optimal.

Let a star "*" over variables capture these policies. Lemma 4 below unveils how the technology in

consumption goods and product prices are related to optimal security policies.

Lemma 4. Suppose the production of consumption goods is diversi�ed when the contendingstates design their security policies optimally. Then�

ki2 � ki1� �p1 � p2

�R 0, G1� R G2�:

Note that Lemma 4 would remain valid even if product prices were endogenous (as will be the

case under autarky). This proves quite useful in our characterization of con�ictual equilibria and

comparative advantage. Suppose good 2 is land-intensive. Lemma 4 suggests that the country with

the largest relative price of good 2 is also the largest producer of guns and conversely.

3. TRADE REGIMES AND CONFLICT

In this section, we explore the implications of autarky and free trade for arming and welfare.

Letting a star "*" now identify Nash equilibrium values, we di¤erentiate equilibrium quantities

between trade regimes with subscripts "A" for autarky and "F" for free trade.

3.1 Con�ict under autarky

Equations (2)-(5) de�ne factor prices and outputs as functions of product prices and guns.

Moreover, by (9), optimal security policies depend on product prices. Thus, to close the model we

need to determine autarky prices, p1A and p2A, as functions of guns. We can do this by searching

for those prices that ensure the relative demand of the non-numeraire equals its relative supply. A

key advantage of this approach is transparency� the channels through which the e¤ects of security

policies travel become fairly clear. (An equivalent procedure is to �nd the prices that ensure

M i = 0.)

Homotheticity of preferences implies RDi is uniquely determined by price pi.16 By Lemma

1, we know the dependence of RSi on its arguments pi and kiX ; and, by Lemma 2, we know the

dependence of kiX on pi, Gi, Gj , and factor endowments. But the autarky price, piA, solves

RD�pi�= RS

�pi; kiX

�pi; Gi; Gj ; �

��: (10)

16Since Di2 = �iDR

i=pi, we will have Di1 = (1� �iD)R

i; therefore, RDi � Di2=D

i1 =

1pi

�D(pi)

1��D(pi)is a function of pi:

12

We thus establish

Lemma 5. A country�s autarky price is decreasing (increasing) in its residual land/labor supply

ratio if the non-numeraire good is land- (labor-) intensive; that is, @piA@kiX

7 0 if ki2 ? ki1. Thus

(a) @piA@Gi

7 0 if ki2 ? ki1, 8Gi that satisfy V iGi+ " > 0, for some " > 0;

(b) @piA@Gj

? 0 if ki2 ? ki1, 8i 6= j;

(c)�ki2 � ki1

� �@piA@Gi

+@piA@Gj

�Q 0 for Gi = Gj if ki R kiG; 8i 6= j;

(d) @piA@Li

? 0 and @piA@K0

7 0; @piA

@Ki 7 0 if ki2 ? ki1.

By Lemma 1(b), an increase in country i�s residual relative supply of land, kiX , raises the

relative supply of its land-intensive product. Market stability then requires the equilibrium price of

the land-intensive good to fall, as stated in Lemma 5. This observation together with parts (b)-(e)

of Lemma 2 (which explain the dependence of kiX on guns and endowments) readily imply parts

(a)-(d) of Lemma 5.

To see the signi�cance of Lemma 5, recall that MCiA = (!(piA); 1). From Lemmas 1(a)

and 5(a) it can now be seen that MCiA is increasing in country i�s policy instrument beyond its

intersection withMBi (see Fig. 1) regardless of the ranking of factor intensities in the consumption

goods industries.17 The intersection of MCiA with MBi at point A in Fig. 1 gives country i�s best-

response function, BiA(G

j). By Lemmas 1 and 5, we may also verify that an increase in the quantity

of guns produced by country i�s adversary will shift country i�s marginal cost schedule downward

which typically elicits a more aggressive response by country i (see below); thus, the endogeneity of

product prices under autarky tilts the balance toward the emergence of strategic complementarities

in security policies.18

At this point it should be clear that technology, institutions (the degree of resource insecurity),

and the size of factor endowments ought to have important implications for optimal security policies

because they condition autarky prices and, through them, marginal costs. To unveil the signi�cance

17The intuition for this is simple. If ki2 > ki1 (ki2 < ki1), then, by Lemma 5(a), an increase in country i�s guns will

reduce (raise) its autarky price (for the range of Gis considered). In turn, by the Stolper-Samuelson theorem (Lemma1(a)), this price adjustment will force the wage/rental ratio to rise; therefore, MCi

A must be increasing in Gi.

18Of coures, there is a second determinant of the shape of a country�s best-response function� the dependence ofMBi on the country rival�s security policies (Lemma 3(b)).

13

of these ideas, we �rst establish existence of a pure-strategy equilibrium in security policies. Since

the dependence of con�ictual equilibria on trade regimes is central to our analysis, we also identify

(su¢ cient) conditions for uniqueness of equilibrium, which we then proceed to characterize.

Theorem 1. (Autarky) An interior Nash equilibrium in pure strategies (security policies) ex-ists. Moreover, the equilibrium is unique if the technology for arms is su¢ ciently labor-intensiveor the inputs to arms are not very close complements.

The proof of Theorem 1 points out that uniqueness of equilibrium can be assured under fairly

general circumstances.19 The proof also clari�es how the endogeneity of prices matters for the

shapes of best-response functions, B1A and B2A, which are illustrated in Fig. 2 by the solid-line

curves� ignore the other curves for now.

First, note that every best-response function is upward-sloping (strategic complementarity)

beyond its point of intersection with the 45� line� at some point beyond this intersection, best-

response functions may become negatively sloped (strategic substitutability). This suggests that

policy instruments are necessarily strategic complements around the 45� line. (As mentioned earlier,

this is due to the endogeneity of product prices.) Second, point (0; 0) is not a Nash equilibrium

because a contestant always has an incentive to produce some arms when its rival produces none.

Third, the intersection of B1A and B2A at point A is the con�ictual (Nash) equilibrium. Curves B

10A

and B20A depict two alternative best-response functions and point A0 is the associated equilibrium.

In this latter case, country 1 produces more guns (and therefore obtains a larger share of the

contested resource) than its adversary country 2. Furthermore, in the neighborhood of A0 country

1�s guns remain a strategic complement for its adversary�s guns, but country 2�s instrument may

be a strategic substitute. Of course, at this level of abstraction, these possibilities appear arbitrary.

Nonetheless, they raise the following important questions.

What are the determinants of arming incentives and military superiority (power) under autarky?

Does the distribution of secure factor endowments between the two adversaries matter? How?

Is economic might equivalent to military might? Or, in the words of Jack Hirshleifer (1991, p.

177), "Can initially weaker or poorer contenders end up gaining on initially stronger or wealthier

19The proof is based on the assumption that secure land supplies are not absorbed into the production of guns butcan be ammended to entertain this possibility. One su¢ cient condition that precludes this complication is that thedegree of land insecurity (i.e., the fraction of land that is contested) is not very big. Another condition is that laboris the only input in the production of guns.

14

opponents?" And what about autarky prices? Will the (ex-ante) land-abundant country necessarily

have a lower autarky price for the land-intensive good?

We begin to address these questions with the help of Fig. 3 where for speci�city we assume

ki2 > ki1 > kiG. By construction, the sides of the outer rectangle depict the aggregate supplies

of labor and land that are available to the contenders, including K0. The sides of the inner box

A1B1A2B2 identify the aggregate quantities of secure land and labor. Allocations in this inner

box identify the distribution of secure resources across countries with point Ai being country i�s

origin. The vertical distances of A1B1A2B2 from the horizontal sides of the outer rectangle depict

the quantity of K0 each country seizes in the contest.

Let us now focus on the benchmark case in which countries 1 and 2 have identical secure

endowments, as indicated by midpoint by D on diagonal A1A2. To di¤erentiate the resulting

equilibrium from others, we place a tilde "~" over variables. Due to symmetry, both countries will

face identical arming incentives; therefore, Gi�A = eG�A with each country thus receiving one half ofthe contested resource. Thinking of point Oi as country i�s origin of overall endowments, we will

have O1A1 = O2A2 = 12K0, with vectors OCi and CiD depicting the resources country i devotes

to the production of arms and consumption goods, respectively.20 By symmetry, O1C1 = O2C2

and C1D = C2D. Moreover, by Lemma 4, pi�A = ep�A and thus !i�A = e!�A and ki�X = ek�X for each i.

Consistent with our factor intensity assumptions, the sides of parallelogram C1J1C2J2 capture the

aggregate sectoral distribution of resources in the production of good 1 (sides C1J1 = C2J2) and

the production of good 2 (sides C1J2 = C2J1) in each country.

The heart of the "Paradox of Power" lies in the answer to this question: Is the symmetric

distribution of secure endowments the only distribution that yields a symmetric equilibrium? To

see the answer, let us proceed as follows. In the context of Fig. 3, de�ne these sets: S0 � F 1F 2,

S1 � F 1F 2A2B1, and S2 � F 2F 1A1B2.21 We can now establish

Proposition 1. (Arming and prices) Under autarky, there exists a non-empty set of asymmetricfactor distributions under which contending states face identical market-clearing prices, produceidentical quantities of arms, and are equally powerful. All other distributions generate di¤erentprices and unequal arming and power. Speci�cally, for each country i 6= j = 1; 2,

20 It is easy to show that the quantity of capital, KiG = irG

i, employed in the production of arms in country i doesnot exceed 1

2K0.

21We exclude endowment distributions in the shaded regions of Fig. 3 because they imply that secure land holdingsconstrain countries�arms choices.

15

(a) Gi�A =eG�A and pi�A = ep�A for factor distributions in S0;

(b) if ki2 ? ki1 then Gi�A > Gj�A and pi�A ? pj�A for factor distributions in Si.

Proposition 1 emphasizes that the symmetric equilibrium ( eG�A; ep�A) arises not only when thecontending sides are symmetric but also when they face certain well-de�ned asymmetries in factor

ownership. Indeed, the symmetric equilibrium arises for all secure factor allocations along the thick

solid-line set, S0, that goes through point D in Fig. 3. An important property of these allocations

is that they ensure k1X = k2X = ek�X . To see the logic of part (a), let us consider the followingexperiment. Starting at point D, transfer resources from country 2 to country 1 so that we move

along S0 in the direction of point E. Since, at constant guns and prices these redistributions do not

a¤ect the residual land/labor ratios, k1X and k2X , both countries�relative supply and relative demand

functions will not shift; consequently, there will be no reason for autarky prices to change. But,

if product prices do not change, arming incentives will not change either and thus parallelogram

C1J1C2J2 will continue to capture the overall allocation of resources into goods 1 and 2.22

Part (b) unveils the circumstances under which a country is likely to arm more heavily than its

adversary. It also clari�es how autarky prices compare across two countries immersed in con�ict as

functions of technology and factor ownership. For some intuition, recall that ki2 > ki1 in Fig. 3, and

consider an initial allocation on S0, say point E (in Fig. 3). By Proposition 1(a), the con�ictual

equilibrium will initially be on the 45� line of Fig. 2, as portrayed by point A, where functions

B1A and B2A intersect. Now arbitrarily transfer labor from country 2 to country 1, as indicated

by the move from point E to point H. By Lemma 5(d), this transfer will raise (reduce) country

1�s (2�s) autarky price which, by Lemma 3(c), will reduce (raise) country 1�s (2�s) opportunity

cost of producing guns. As a consequence, country 1 (2) will behave more (less) aggressively, as

shown by the clockwise rotation of B1A and B2A to B

10A and B2

0A , respectively. The properties of

the best-response functions described earlier (and in the proof to Theorem 1) ensure that the new

equilibrium point will be below the 45� line, as illustrated by point A0 in Fig. 2, where clearly

country 1 arms more heavily than its adversary. By Lemma 4, we must have p1�A > p2�A .

Two important ideas deserve some emphasis here. First, under autarky the relationship between

factor abundance, arming, and military superiority is complex. Consistent with the "Paradox of22The dotted-line parallelograms inside C1J1C2J2 with origins at C1 and C2 capture the decomposition of the

endowment vector at E into vectors of sectoral input demands in each country.

16

Power" (Hirshleifer, 1991), the initially resource disadvantaged side does not necessarily obtain a

smaller share of the contested pie. For example, even though country 1�s secure factor endowments

(and income) are smaller than its opponent�s at, say, point Q in set S1 of Fig. 3, country 1

builds more arms and commands a bigger share of the contested resource than its rival. It may be

tempting to conclude that country 1 arms more heavily in this case because it is relatively less well

endowed in the contested resource (land). However, this reasoning is incorrect because country 1

also produces more arms at point H, where it is relatively more abundant in land. What part (b)

makes clear, then, is that a necessary condition for military superiority is that a country�s secure

endowments exceed certain threshold levels.

Second, Proposition 1 establishes that, in the presence of insecure property, the relation-

ship between a country�s autarky price and secure land/labor endowment ratio is not necessarily

monotonic. Alternatively, and in contrast to the corresponding relationship obtained in the world

of the HOS trade model where property is secure, the country with the largest land/labor ratio does

not necessarily enjoy the lowest relative price of the good that employs land intensively� compare,

for example, relative endowments at di¤erent points along S0 where pi�A = ep�A.23 What matters

for autarky prices is the residual ex-post endowment ratios left for the production of consumption

goods. But, in the presence of con�ict, these ratios are endogenous. As we will see later, this

matters for the direction of international trade �ows.

We have seen that the international distribution of secure resources has important implications

for arming and power. The next proposition, which outlines the implications of secure endowment

reassignments, proves useful in our upcoming comparison of trade regimes.

Proposition 2. For initial factor allocations in S0, a small transfer of a secure resource fromcountry j to its adversary i (6= j) has the following implications for arming and welfare:

(a) dGi�AdLi

= �dGj�AdLi

> 0 but dGi�AdKi = �

dGj�AdKi < 0;

(b) dV i�AdLi

= �dV j�AdLi

> �(ep�A)w(ep�A) but dV i�AdKi = �

dV j�AdKi < �(ep�A)r(ep�A):

Part (a) of Proposition 2 unveils how the willingness of countries to contest the disputed territory

varies with the type of resource transfer considered. Interestingly, while the country with the23This point can also be seen when we consider endowment con�gurations along A1A2 where both countries have

identical secure land/labor ratios. In this case, market clearing prices would be identical across countries if propertywere secure but they are not when some property is insecure.

17

increased (reduced) labor endowment produces more (less) guns in the new equilibrium, exactly

the opposite is true for redistributions of secure land. The driving force behind these e¤ects is the

impact of endowment changes on autarky prices, which in�uence factor prices and ultimately the

contestants�marginal costs. Thus, under autarky, the composition of secure assets in the contending

states has important implications for relative arming and the emergence of the Paradox of Power

we referred to earlier.24

To understand part (b) we must extend the welfare decomposition in (8) to include the e¤ect of

factor supply changes. Focusing on labor redistributions, utilization of the envelope theorem along

with the fact that M i = 0 under autarky yields

dV i�A

dLi= �(pi�A)

"w(pi�A) + r(p

i�A)K0�

iGjdGj�AdLi

#(11)

for j 6= i. (Of course, pi�A = ep�A for initial distributions in S0.) Thus, a labor transfer generatestwo e¤ects: A wage-income adjustment to the country�s labor endowment (the �rst term in the

brackets); and a strategic e¤ect (the second term in the brackets) which by Proposition 2(a) is

positive for the recipient and negative for the donor country. Similar logic can be used to sort out

the welfare e¤ects of land redistributions.25

3.2 Con�ict under free trade

Turning to trade, we suppose the contending countries are "small" in world markets and that

there are no trade costs. Letting � denote the international price of the non-numeraire, free trade

is consumption goods requires pi = �, 8i. Since � is independent of national security policies, a

country�s payo¤ function can be identi�ed with its indirect utility function, V i.

Depending on fundamentals, the degree of land insecurity, and the international price level, it is

now possible for one or both contestants to specialize completely in the production of a consumption

good. It is also possible for arms production to be constrained by countries�secure endowments.24 In part (a), the o¤setting changes in the contestants� guns imply that the resource redistributions considered

leave the aggregate production of guns unchanged. It can be shown that, if the initial resource allocation is inSi, redistributions of the type considered in part (a) typically generate a reduction in the aggregate production ofguns. Even more interestingly, when the initial allocation is su¢ ciently asymmetric, resource redistributions mayinduce both countries to produce less guns. Thus, the extent of the asymmetry in secure endowments has importantimplications for the intensity of con�ict.25The analysis could also be extended to consider the e¤ects of changing K0. It can be shown that, for initial

resource allocations in S0, an increase in K0 normally induces both adversaries to produce more arms and thusgenerates an adverse strategic welfare e¤ect.

18

To place at center stage the factor price e¤ects of opening borders up to free trade and to highlight

the striking implications this can have on arming incentives, we focus on incomplete specialization.

With the help of parts (a) and (b) of Lemma 3, we may �rst obtain

Theorem 2. (Free Trade) If the world price, technology, the distribution of secure endowmentsand the degree of land insecurity are such that (i) free trade in consumption goods leads to inter-national factor price equalization, and (ii) the contending states�arms production is unconstrainedby their secure endowments, an interior Nash equilibrium in security policies will exist, and willbe unique and symmetric.

To understand Theorem 2 and the conditions that ensure its validity, let us suppose � = ep�Ainitially and let us imagine that the conditions of the theorem are ful�lled. From Fig. 1 it becomes

apparent that the intersection of MBi and MCiF at point A determines country i�s best-response,

BiF (G

j). Since @BiF =@G

j = �V iGiGj

=V iGiGi

= ��iGiGj=�iGiGi , the shapes of best-response functions

are determined solely by the properties of the CSF. As indicated by the dashed-line curves, B1F

and B2F , in Fig. 2, the following ideas now spring to the forefront. First, best-response functions

exhibit strategic complementarity up to their point of intersection with the 45� line, and strategic

substitutability thereafter. Thus, in contrast to autarky, here strategic complementarity does not

arise on or beyond the 45� line� this is so because product prices are �xed. Second, because free

trade in consumption goods brings about international factor price equalization and secure resource

endowments are not exhausted in the production of guns, the contending states will have identical

marginal bene�t and marginal cost functions; hence the symmetric Nash equilibrium at point A in

Fig. 2, where B1F and B2F intersect.

26 (As before, point (0; 0) is not an equilibrium.)27 We may

now re�ect on the conditions that generate this equilibrium.

In addition to assuming � = ep�A, let us suppose the degree of land insecurity is su¢ cientlysmall so that S0 is non-empty and parallelogram C1J1C2J2 in Fig. 4 captures the joint sectoral

decomposition of inputs in the consumption goods sectors for the two contestants. (Fig. 4 is

26The conditions for international factor price equalization include: constant returns to scale in production, theabsence of factor intensity reversals, identical technologies across countries, diversi�cation in production, absenceof market failures or distortions, no trade barriers, and the existence of at least as many productive factors in thetradable goods sectors as there are traded goods (Samuelson, 1949).27Existence and uniqueness of equilibrium arise under less restrictive conditions than the ones stated in Theorem 2.

The analysis could be extended to entertain the possibilities that factor prices may not be equalized internationallyand the production of guns may be constrained by resource constraints. Since these possibilities complicate theanalysis without altering the key insights from our comparison of con�ict under autarky and free trade we choose notto treat them formally here.

19

largely a reproduction of Fig. 3.) Since Gi�F = eG�A 8i, we may temporarily abstract from resource

constraints in the production of guns and view C1C2 as the vector of residual factor supplies of an

integrated economy (i.e., an economy in which both goods and factors are tradable) that replicates

the free trade in goods equilibrium that arises when countries 1 and 2 are considered separately

and their residual factor endowments are in C1J1C2J2 (Dixit and Norman, 1980; Helpman and

Krugman, 1985). Under these circumstances, trade in consumption goods causes factor prices to

be equalized internationally for all residual resource distributions in C1J1C2J2. But every country

i�s secure factor endowments must cover the AiN i = OiCi vector of resources that are absorbed

in its arms sector; therefore, the set of secure endowment distributions that are consistent with

international factor price equalization, and secure resources that do not constrain the production

of guns is the shaded hexagon in Fig. 4, which we label the "arms equalization set" (AES).28 An

immediate consequence of Theorem 2 is

Corollary 1. Endowment redistributions within the AES have no e¤ect on arming. Thus, thewelfare e¤ects of such redistributions within each contending state coincide with the factor pricerelated changes in national income weighted by the marginal utility of income.

As before, the welfare e¤ects of endowment redistributions can be visualized by considering,

say, a labor transfer from country j to country i(6= j). Using a welfare decomposition similar to

that in (11), and noting that the strategic e¤ect of such a transfer vanishes, yields dV i�F =dL

i =

�dV i�F =dL

i = �(�)w(�).29

The size of the AES relative to the set of possible secure endowment distributions provides a

measure of the likelihood that arms will be equalized internationally. This measure depends on

the nature of technology, the degree of land insecurity, and the world price. Ceteris paribus, the

higher the degree of similarity between the consumption goods technologies, and the larger the

deviation of the international price from ep�A, the smaller the factor price equalization (FPE) setand thus the smaller AES. Similarly, the larger the degree of land insecurity, the larger the set of

factor distributions over which secure land constrains arms production, and thus the smaller AES.

28For factor distributions outside the AES, at least one country will specialize in the production of a consumptiongood. Such specialization precludes the possibility of international factor price equalization and renders a country�smarginal cost of producing arms independent of the world price, but increasing in its arms.29The e¤ects of K0 on arming di¤er from the e¤ects of redistributing secure resources. From (9) it can be seen

that the larger K0, the larger a country�s marginal bene�t of arming and thus the larger its incentive to arm. (Unlikeautarky, a contestant�s marginal cost remains intact here because product prices pin down factor prices.)

20

Clearly, it is possible for AES to be empty. Yet, there always exist combinations of technologies,

secure factor distributions, degrees of land insecurity, and world prices that preclude this possibility.

Recapitulating, for factor distributions in AES, free trade in consumption goods levels the play-

ing �eld in arms competition and equalizes power internationally. We thus have another instance

of (the "strong version" of) the Paradox of Power (Hirshleifer, 1991)� where the relatively more

a­ uent adversary devotes a smaller fraction of its income on arms� for a potentially large set of

factor distributions.

But what are the implications of international prices for arming, trade �ows and welfare? We

address these questions in Propositions 3, 4 and 5, respectively.

Proposition 3. (Guns) If free trade in consumption goods leads to international arms equal-ization for a certain range of international prices, then

(a) an increase in the world price of the contested-resource-intensive good within this rangewill induce all adversaries to produce more guns (i.e., dGi�F =d� ? 0 if ki2 ? ki1, 8i);

(b) there will exist threshold prices, � < �, such that for all prices outside (�; �) equilibriumguns will be insensitive to price changes (i.e., dGi�F =d� = 0, 8� =2 (�; �)).

We illustrate Proposition 3 with the help of Fig. 5 which deals with the (less general) case of

identical adversaries and is based on the assumption that the non-numeraire is land-intensive. (For

simplicity, in the remainder of this section, we assume ki2 > ki1.) The shaded regions in this �gure

capture the idea that both adversaries will specialize in the production of a consumption good if

international prices are su¢ ciently high or su¢ ciently low, as pointed out in part (b).

Starting at any world price that ensures diversi�cation in production, let that price rise. By the

Stolper-Samuelson theorem (Lemma 1(a)), the wage/rental ratio in every contending country will

fall. In turn, by parts (a)-(c) of Lemma 3, this factor price adjustment will reduce the opportunity

cost to arming in every contending country and will thus induce all countries to arm more heavily,

as indicated by the positively sloped solid-line curve G�(�) in Fig. 5. (If ki2 < ki1, curve G�(�)

would become negatively sloped.) This explains part (a).

With the help of (3), (4) and (5), it can be veri�ed that price increases beyond � force all

factor prices to rise proportionately and thus have no e¤ect on the marginal costs of arming. (A

similar e¤ect arises for price decreases below �.) The dotted-line extensions of G�(�) in the regions

21

of specialization portray the independence of optimal guns from the world price, as noted in part

(b).30

Proposition 4. (Trade patterns) For each contending country i there is a world price, �iA, suchthat M i�

F (�) S 0 if � T �iA. For factor distributions in the AES of S0, we have �iA = ep�A; but for

factor distributions in the AES of Si, we have �iA > pi�A and �jA < pj�A (j 6= i).

Obviously, the direction of a country�s trade �ows depends on the world price. The interesting

part of Proposition 4 is that, in the presence of asymmetries in secure factor ownership, the inter-

national price that eliminates a contending country�s trade volume di¤ers from the price that arises

under autarky. Intuitively, this is so because the introduction of free trade, through its impact on

arming incentives, alters countries�residual factor endowments and therefore their excess demand

functions. To see this point more clearly, assume � = p2�A and consider a distribution of factors

in the AES of S1. By Proposition 1(b), the autarkic con�ictual equilibrium can be identi�ed with

point A0 in Fig. 2. Since free trade levels the playing �eld in arms competition and, as a result,

induces country 2 to become relatively more aggressive (compare the free trade equilibrium at point

A to the autarkic equilibrium at point A0), country 2�s residual land/labor ratio, k2X , will increase

(Lemma 2(a,b)). By Lemma 1(b), country 2 will experience an excess supply of the non-numeraire

good, so the world price will have to fall below p2�A for trade to be eliminated. (We can use similar

logic to show that �1A > p1�A .) We revisit this issue in the next section where we discuss the broader

implications of con�ict for trade patterns.

Turning to welfare, the decomposition in (8) together with the envelope theorem give

dV i�F

d�= �(�)

"�M i�

F + r(�)K0�iGjdGj�Fd�

#, 8 i 6= j. (12)

The �rst term inside the brackets captures the TOT e¤ect of a price change and its sign is de-

termined by the country�s trade pattern. The second term captures the strategic welfare e¤ect

of a price change. Since ki2 > ki1, this e¤ect is negative when arms are equalized internationally

(Proposition 3(a)), but vanishes when the world price rises above � or falls below � (Proposition

3(b)). Proposition 5 summarizes several noteworthy implications of (12).

30 In the presence of asymmetries in factor ownership, there also exist prices ranges under which only one countryspecializes completely.

22

Proposition 5 (Welfare) If dGj�F (�iA)=d� > 0, then a contending country i�s welfare will be

(a) decreasing in its terms of trade (the relative price of its exportable) for internationalprices in the neighborhood of �iA;

(b) minimized at an international price, �imin, that exceeds �iA (i.e., �

iA < �imin).

Part (a) points out that an improvement in a contending country�s TOT will necessarily be

"immiserizing" if the world price does not di¤er considerably from �iA and the country exports the

disputed-resource-intensive product. The intuition is simple. The positive volume-of-trade e¤ect

of a TOT improvement is overwhelmed by the loss in income due to an increase in the intensity in

arms competition.

Part (b) clari�es that a contestant�s welfare is minimized at some price, �imin, where at the

margin the bene�cial e¤ect of a TOT improvement is not less than the adverse e¤ect of increased

arming. By its unveiling of the existence of a range of prices under which a TOT is welfare-

reducing, in a sense this �nding strengthens the result in part (a). The important point is that

booms in contested-resource intensive products may force contending countries with an apparent

comparative advantage in such products to experience the "resource curse" problem. Interestingly,

this possibility does not require policymaking to be driven by domestic rent-seeking (e.g., Torvik,

2002; Mehlum et. al. 2005), redistributive politics (e.g., Robinson et. al., 2005), or domestic con�ict

(Gar�nkel et. al., 2005). In short, the absence and/or ine¤ectiveness of international institutions

aimed at managing international con�ict may also explain the resource curse problem.

We have seen that, for endowment distributions in the AES, country size is inconsequential

with regards to the quantity of guns adversaries produce under free trade. However, country size

plays an important role in the determination of the range of international prices over which the

resource curse problem arises. Consider, for example, an asymmetric distribution in the AES of S0

in Fig. 4 (where � = ep�A), such as point E. From (12), it can be ascertained that the marginal cost

of a TOT improvement (the second term in (12)) does not di¤er across adversaries. However, the

marginal bene�t does di¤er. Since by assumption country 1 is larger than country 2, we will have

�M1�F > �M2�

F > 0 for � > ep�A; therefore, �1min < �2min. We may thus state

Corollary 2. For asymmetric factor distributions in the AES of S0, the relatively smalleradversary will experience the resource curse problem over a larger range of international prices.

23

4. TRADE PATTERNS AND TRADE VOLUMES

By inducing countries to allocate resources in the production of arms for appropriative purposes,

international con�ict alters a country�s residual factor endowments and may thus a¤ect its ex post

comparative advantage. Thus, con�ict over resources may a¤ect not just the volume of trade

�ows but also their direction. In this section we contribute two ideas to the literature. First, we

illustrate how con�ict may distort a contending country�s natural comparative advantage. Second,

we show that the trade patterns of contending countries may not be accurately predicted simply

by comparing international and autarky prices.

To illustrate the idea that con�ict may distort a contending country�s natural comparative

advantage, it is instructive to focus on identical adversaries and show how their trade pattern (with

con�ict) compares to the (hypothetical) trade pattern we would observe if property were secure

and there were no arming. We may explain the key ideas with the help of Fig. 5.

As noted earlier, curve G�(�) depicts the equilibrium choice of guns as a function of price. The

downward sloping solid-line curve, pA(G), describes the dependence of the representative country�s

autarky price on the common quantity of guns G. This relationship is a consequence of Lemma

5(c) and the assumption that ki > kiG. (Curve pA(G) would be increasing in G if ki < kiG.)

Point A, where curves G�(�) and pA(G) intersect, identi�es the con�ictual equilibrium under

autarky. Clearly, if � = ep�A point A would also describe the con�ictual equilibrium under free

trade. Owing to symmetry, the international price that eliminates a contending country�s trade

�ows equals the equilibrium price that obtains under autarky (i.e., �iA = ep�A); therefore, undercon�ict and trade, both adversaries will export the land-intensive good 2 if � > ep�A and will importit if � < ep�A.

Let us now consider the possibility of no arming. As can be inferred from the de�nition of CSF

in (1), in this case each adversary will receive 12K0 of the contested resource, and autarky prices will

coincide with price pnA = pA(0) (superscript "n" stands for "no con�ict"). Thus, in the absence of

con�ict, the representative country will export the land-intensive good 2 if � > pnA and will import

it if � < pnA. Putting these ideas together yields

Proposition 6. (Identical adversaries) Suppose ki2 > ki1 and let free trade in goods lead to armsequalization. Then, competition for land will reverse the contending countries�natural comparativeadvantage for all � 2 (ep�A; pnA) if ki > kiG(!), or for all � 2 (pnA; ep�A) if ki < kiG(!).

24

Proposition 6 clari�es how con�ict and the nature of technology in arms interact to determine

trade patterns. For some intuition, let us focus on ki > kiG(!) and let us imagine � 2 (ep�A; pnA)initially, as indicated by �0 in Fig. 5. Obviously, under no con�ict, both contestants will import

good 2. Now observe that, if every adversary produced G0 guns, neither country would engage in

trade because �0 = pA(G0), as indicated by point A0. But, under free trade, both adversaries�gun

production will exceed G0� compare, for example, the quantities of guns associated with points A0

and B. Consequently, by Lemma 2(d), the contestants residual land/labor ratios under free trade

would be larger than the ones at point A0. By the Rybczynski theorem (Lemma 1(b)), this would

create an excess supply for good 2 in each country i (because RSi would exceed RDi); therefore,

in stark contrast to the "no con�ict" case, both countries would have to export this product under

con�ict. (We can use similar logic to explain the reversal in trade patterns when ki < kiG(!).)31

Several observations are now in order. First, resource insecurity and the con�ict it engenders

may cause a country�s apparent comparative advantage to diverge from its natural comparative

advantage. As we have just seen, the nature of technology and world price levels play key roles in

this regard. Second, it may inappropriate to view international con�ict over productive resources as

a type of trade cost that necessarily reduces the size of a contending country�s trade �ows. Suppose,

for example, � = pnA which implies the contestants will not engage in trade in the absence of con�ict.

But, as we have just seen, in the presence of con�ict, the contending states�will be exporters of

the land-intensive good if ki > kiG(!); consequently, depending on technology, international con�ict

may spur, rather than sti�e, trade.32

In the world of neoclassical trade theory, a country�s trade pattern is determined by comparing

the world price to its autarky price. In the world of insecure property and international con�ict

though unquali�ed application of this logic may lead to erroneous inferences about trade patterns.

In other words, insecure property and con�ict over resources may not only alter trade patterns but

also the standard logic we normally use in trade theory to predict trade patterns.

31An additional possibility (not explored in Proposition 6) is that the ranking of between ki and kiG(!) may getreversed as gun production rises. In this case, pA(G) in Fig. 5 would not necessarily be decreasing everywhere.(This possibility does not arise if production functions are Cobb-Douglas.) Still, the main message survives: as longas ep�A 6= pnA, there will exist price ranges under which it will be possible for con�ict to reverse a country�s naturalcomparative advantage.32The relationship between the volume of trade and con�ict has been addressed empirically in the political science

literature (e.g., Barbieri, 2002) which appears to �nd support to the idea that con�ict may stimulate trade.

25

Proposition 7. (Asymmetric adversaries) If there are asymmetries in factor ownership, then,in the presence of international con�ict,

(a) it may be impossible to determine a contending country�s trade pattern by comparingthe international price to its autarky price;

(b) a contending country may not necessarily export the good that employs intensively itsrelatively abundant factor.

We establish the validity of Proposition 7 informally and with the help of Proposition 4 which

showed that the price that eliminates a country�s trade di¤ers from its autarky price (i.e., �iA 6= pi�A)

for factor distributions in Si that ensure trade brings about arms equalization when � = pi�A . Let

us consider such a distribution of resources under the assumption that ki2 > ki1. Clearly, if arming

did not depend on the trade regime considered, country i would not engage in trade if � = pi�A .

But, as was shown in Proposition 4, arming does depend on the trade regime considered. In other

words, the direction of trade �ows is not determined by how the international price di¤ers from

the country�s autarky price, pi�A , but rather how it di¤ers from �iA.33

To see the logic behind part (b), suppose � = ep�A and consider the allocation H in Fig. 4

which implies that country 1 will export the labor-intensive good 1 and country 2 will export the

land-intensive good 2. Now observe that at point H country 1 is land-abundant and country 2 is

labor-abundant, both in terms of their secure endowments and in terms of their �nal endowments

under free trade. This con�rms part (b).34

5. COMPARISON OF TRADE REGIMES

In this section, we illustrate two ideas: (1) how resource insecurity and competition to establish

property rights a¤ects arming incentives under free trade relative to autarky; and (2) how the costs

of arming may overwhelm a country�s traditional gains from trade. The analysis is noteworthy not

only because it clari�es how international con�ict generates a trade regime-dependent distortion

(Bhagwati, 1971), but also because it sheds light on the conditions under which free trade intensi�es

33The idea that a country�s apparent comparative advantage depends on the nature of property rights is reminiscentof a related point in Brander and Taylor (1997b) who showed that over time the depletion of a common-pool resourcein a country with ill-de�ned propery rights may reverse its comparative advantage. In our setting, residual factorendowments (and thus comparative advantage) may change because the dissipation of resources in con�ict is traderegime-dependent.34 It is also worth noting that, since p1�A > p2�A at pointH, the land- (labor-) intensive commodity does not necessarily

command the lowest (largest) autarky price in the land- (labor-) abundant country.

26

this distortion. To substantiate these points we consider two possibilities: (i) when adversaries

are identical (Proposition 8)� this case unveils the gist of the argument and the circumstances

under which autarky dominates trade; (ii) when adversaries have asymmetric endowment pro�les

(Proposition 9), which sheds some light on the conditions under which national preferences over

trade regimes may diverge.

Proposition 8. (Symmetric adversaries) If free trade in consumption goods induces adversarieswith identical endowment pro�les to

(a) export the disputed-resource-intensive commodity, then the adversaries will arm moreheavily under free trade than under autarky, and free trade will be Pareto dominated byautarky for a certain range of international prices;

(b) import the disputed-resource-intensive product, there will be less arming under free thanunder autarky, and free trade will Pareto dominate autarky.

Proposition 8 can be illustrated with the help of Fig. 5. Since point A describes the con�ictual

equilibrium under autarky and the corresponding equilibrium under free trade is on G�(�), the

comparison of equilibrium arming under the two trade regimes should be fairly clear.35

Turning to payo¤s, we begin by unveiling the shape of welfare contours in the (G; �) space.

First note that the representative country�s welfare is decreasing in guns, G, regardless of the trade

regime considered. This is so because equi-proportional arms increases do not alter the division of

the contested land but raise resource costs. Now observe that, for given guns, a country�s welfare

increases with the deviation of the world price from its autarky level pA(G) (Lemma 3(d)). It thus

follows that a contestant�s welfare contours will have the shape indicated in Fig. 5, with contours

further to the right capturing lower welfare levels for the two contestants.

Noting that every contestant exports the contested-resource-intensive product at prices � > ep�A,Fig. 5 illustrates that autarky will Pareto dominate free trade for all international prices within

the (ep�A; �0) interval but not for prices beyond �0. The �gure also shows that free trade is Paretosuperior to autarky when countries import the land-intensive product.

35Hirshleifer (1991) observed that the diversion of resources into arms falls with the degree of market integration butthe size of this e¤ect is small. However, Hirshleifer considered con�ict over output and identi�ed market integrationwith the degree of complementarity between the inputs in useful production. Our approach suggests that, whencon�ict is over resources and market integration takes the form of a move from more protected (autarky) to lessprotected (free trade) trade regimes, the intensity of arming may rise or fall depending, among other things, ontechnology, the degree of land insecurity and international prices.

27

Let us now consider adversaries with asymmetric endowment pro�les. Arbitrary factor distrib-

utions in Si complicate the welfare ranking of trade regimes for at least two reasons: (1) because

adversaries begin to specialize in production at di¤erent international prices, it becomes necessary

to investigate arming incentives outside the AES for one country initially and eventually for both;

(2) because the endogeneity of trade patterns together with the fact that V i�F 6= V i�

A at � = pi�A

for arbitrary distributions in Si make it di¢ cult to identify workable benchmarks for comparison

purposes. Still, as Proposition 9 illustrates, there exists two noteworthy asymmetries that yield

tractable comparisons.

Proposition 9. (Welfare comparison of trade regimes for asymmetric adversaries)

(a) For any asymmetric factor distributions in the AES subset of S0, there will exist a rangeof international prices that render autarky Pareto superior to free trade.

(b) If � = ep�A, there will exist subsets Di � Si of factor distributions adjacent to the AESof S0 such that one country prefers autarky over free trade while its adversary does not.

Part (a) is an extension of Proposition 8 to asymmetric distributions in the AES subset of

S0. Part (b) clari�es how countries�preferences over trade regimes may di¤er when more general

factor endowment asymmetries are considered. As shown in the proof, the key to the-just noted

divergence in preferences is the presence of a strategic e¤ect when redistributing resources under

autarky (Proposition 2), and the absence of such a strategic e¤ect under free trade (Corollary 1).

6. CHOICE OF TRADE REGIMES AND TRADE POLICIES

In previous sections, we studied the consequences of trade regimes but we did not allow poli-

cymakers to make choices over trade regimes or over trade policies. In this section we extend the

analysis in two directions. In the �rst subsection, we allow policymakers to noncooperatively choose

the trade regime that ful�lls their objective of national welfare maximization. Then, in the second

subsection, we illustrate how a country could use its own trade policy to tilt the balance of power

to its favor.

6.1 Choice of trade regimes

If the contending countries implement their security policies before they commit to a particular

trade regime, it will always be the case that they will choose free trade over autarky. This is so

28

because, for any given arms con�guration, free trade maximizes their welfare by virtue of the fact

that they are price takers in world markets.

However, if countries can commit to choosing a trade regime before they implement their se-

curity policies then, depending on the international distribution of resources, two possible types of

equilibria exist. One in which free trade is the chosen trade regime, and another in which both

autarky and free trade are possible equilibria. One interesting aspect of this setup is that the free

trade equilibrium may have the features of a prisoner�s dilemma...

6.2 Trade policies

So far, our analysis focused on the extreme regimes of complete autarky and global free trade,

so it would be of interest to examine the implications of less restrictive policies. Suppose country

i intervenes in trade with an ad valorem trade tax, � i, on good 2. (This requires � i > 0 when

M i > 0, and � i < 0 when M i < 0.) Under these circumstances, pi = (1 + � i)� > 0 for i = 1; 2.

Assuming that in each country i tari¤ revenues, � i�M i = (pi��)M i are redistributed to consumers

in lump-sum fashion, country i�s welfare changes may be decomposed as follows:

dV i = �(pi)��M id� + � i�dM i +

�riK0�

iGi �

i�dGi + riK0�

iGjdG

j�

(13)

for i 6= j. The above equation clari�es the channels through which the e¤ects of security and trade

policies will travel now. The �rst and second terms inside the brackets capture the familiar terms of

trade and volume of trade e¤ects of trade policies, respectively. By our assumption that countries 1

and 2 are "small" in world markets, the �rst term will vanish when we consider the e¤ects of trade

and security policies. If country i participates in world trade, the second term will not vanish; it

will depend on country i�s trade and security policies, and on its adversary j�s security (but not

trade) policy. The third and fourth terms in (13) capture the direct e¤ects of security policies that

we discussed earlier.

Now suppose country i�s trade and security policies are simultaneously determined. It can be

easily inferred from (13) that its optimal trade policy will be free trade (i.e., � i� = 0). Since this

implies that the second term will also vanish, country i�s optimal security policy will coincide with

the one we described earlier in the context of global free trade. It is straightforward to verify that

if security policies are determined prior to trade policies the analysis is similar.

29

The above reasoning raises the question of whether trade policy commitments before the im-

plementation of security policies alter the analysis in a substantive way. To explore this possibility,

consider a two-stage game in which countries determine their trade policies in stage 1 and their

security policies in stage 2. In the presence of trade taxes, country i�s optimal security policy would

have to include its possible e¤ect on the volume of trade. Starting with the last stage, at an interior

solution, country i�s (= 1; 2) FOC for welfare maximization will be

@V i

@Gi= �(pi)

�� i�

@M i

@Gi+ riK0�

iGi �

i

�= 0: (14)

The e¤ects of trade policy on arming and power can be identi�ed with standard comparative statics

exercises performed on (14). The important point for our purposes is that precommitments on trade

policy can strategically a¤ect the security policies of even small countries.

Now, identify with a star (�) the solution to the above system of equations and let subscript

T re�ect the presence of tari¤s. For simplicity, suppose country i imports good 2. Going back to

stage 1, we may summarize the welfare e¤ect of a change in country i�s trade policy as follows:

@V iT

@� i= �(pi)

"� i�

@M i

@� i+

�� i�

@M i

@Gj+ riK0�

iGj

�@Gj�T@� i

#; i 6= j: (15)

In (15), the direct e¤ect of trade policy on country i�s optimal security strategy vanished, by

the envelope theorem. An increase in country i�s tari¤ will have a negative welfare e¤ect due to its

distortionary impact on the country�s volume of trade (i.e., @M i=@� i < 0), as indicated by the �rst

term in (15). But there also exists a strategic e¤ect� one associated with the possible impact of the

country�s trade policy on the rival country�s security policy, captured by the second term inside the

brackets. This latter e¤ect has two components. Suppose a restrictive trade policy induces country

i�s adversary to behave less aggressively in security competition (i.e., @Gj�T =@�i < 0). Then, if

ki2 > ki1, the �rst component of the strategic e¤ect will be negative because @Mi=@Gj > 0, but

the second will be negative because �iGj < 0. The former e¤ect is due to the volume e¤ect of

the rival country�s security policy and the latter e¤ect is due to the income e¤ect of this policy.

Numerical analysis of a model with Cobb-Douglas production and utility functions, and a Tullock

CSF indicates that restrictive unilateral trade policies do not raise welfare.

30

7. CONCLUDING REMARKS

In this paper, we have constructed a simple but su¢ ciently general model of trade and inter-

national con�ict over a valuable but disputed resource that enabled us to provide a systematic

assessment of the implications of di¤erent trade regimes for arming and welfare. By design, our

focus was on "small" countries, so that we could abstract from the possible terms of trade e¤ects

of security and trade policies. This is appropriate for some countries but not necessarily for others

that have monopoly/monopsony power in world trade, so it would be interesting and worthwhile

to extend the formal analysis in that direction. One important di¤erence from the current setting

is that countries� incentives to produce guns and intervene in trade become more complex, with

trade no longer inducing the equalization of arming incentives under conditions of factor price

equalization. Still, the qualitative welfare e¤ects can be shown to be broadly consistent with the

ones obtained here.36 Another di¤erence is that trade and security policies can be used simulta-

neously, the former to balance terms of trade with volume of trade e¤ects, and the latter security

considerations. This is a rich and promising environment within which the implications of policy

interactions could be explored, including the economics of incentive-compatible free trade agree-

ments and their possible spillover e¤ects on national security. An additional extension in a di¤erent

direction is the possible examination of overt con�ict� as that may be captured, for example, in

a "winner-take-all" contest� and the identi�cation of circumstances under which con�ict with no

trade may arise as an equilibrium outcome. Pursuing this alternative in the environment of this

paper and exploring in �ner detail the interactions of trade and security policies are clearly goals

worthy of further exploration.

Ultimately, solving the problem of insecurity entails the development of commitment devices

that aim to reduce, and possibly eliminate, the need to arm. Such commitment devices, however,

are not easy to come by and, judging from particular historical instances, they take a long time

to develop. Europe is a good example of this. After the experience of the two world wars, the

original six members of the European Community slowly began to develop mechanisms of economic

integration that were in large part institutions of con�ict management. That twin process of

36Skaperdas and Syropoulos (2002) explored a framework similar to the one considered here but with the terms oftrade determined by bargaining. Welfare under unrestricted trade in that article is shown to be higher than welfareunder autarky only when the countries are su¢ ciently dissimilar in the secure endowments� so that their gains fromtrade are large enough.

31

economic integration and con�ict resolution through bureaucratic and political struggle, instead of

con�ict in the battle�eld, is ongoing and far from complete after a century of tribulations. Trade

openness and, more generally, economic interdependence may ameliorate con�ict, but it would be

naive to think that they could achieve this by themselves.

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Waltz, K. (1979), Theory of International Politics, New York: Random House.

9. APPENDIX

We �rst unveil several useful properties of the CSF in (1): For simplicity, de�ne fi � f(Gi).

Recalling that f 0i > 0 and f00i � 0, di¤erentiate �i(Gi; Gj) with respect to its arguments to obtain

�iGi =f 0ifj

(f1 + f2)2> 0 (A.1)

�iGj = �f 0jfi

(f1 + f2)2< 0 (A.2)

�iGiGi =fj

(f1 + f2)3[f 00i (f1 + f2)� 2(f 0i)2] < 0 (A.3)

�iGiGj =(fi � fj)f 0if 0j(f1 + f2)3

R 0 if Gi R Gj for i 6= j: (A.4)

34

Proof of Lemma 1: Following Jones (1965), denote with �ihj � riaihj=cij the shares of factor

h = K;L in the cost of producing good j = 1; 2, and with �iKG � ri ir= i and �iLG � wi iw=

i the

corresponding cost shares in guns. Furthermore, let �iKj � Kij=L

iX and �iLj � Lij=L

iX respectively

capture the proportion of capital and labor employed in industry j = 1; 2 and let a hat (^) over a

variable denote percentage change (e.g., bx = dxx ).

Part (a): Di¤erentiating (2) and (3) totally, using the above de�nitions and solving the resulting

system of equations for the pi-induced changes in factor prices yields

piwipwi

= ��iK1���i�� and

piripri

=�iL1���i�� (A.5)

where���i�� � �iK2 � �iK1 = �iL1 � �iL2 =

!i(ki2�ki1)(!i+ki1)(!i+ki2)

? 0 if ki2 ? ki1. It can be con�rmed that j�ij

is the determinant of the coe¢ cient matrix obtained from di¤erentiating (2) and (3). From (A.5)

it can now be seen thatpi!ip!i

=piwipwi

� piripri= � 1

j�ij , which completes the proof to part (a).

Parts (b) and (c): First note that we can combine (4) and (5) to obtain �iL1ki1 + �iL2k

i2 = kiX .

Di¤erentiating (4) and (5) totally and solving the resulting system of equations gives

bXi1 =

1���i�����iL2 bKi

X + �iK2bLiX�� 1���i�� ���i�� ��iL2�iK + �iK2�iL� bpibXi

2 =1���i���+�iL1 bKi

X � �iK1bLiX�+ 1���i�� ���i�� ��iL1�iK + �iK1�iL� bpiwhere �iK � �iK1�

iL1�

i1 + �iK2�

iL2�

i2, �

iL � �iL1�

iK1�

i1 + �iL2�

iK2�

i2, with �ij being the elasticity

of substitution between land and labor in industry j, and���i�� � �iK2 � �iL2 = �iL1 � �iK1 =

(ki2�kiX)(kiX�ki1)kiX(k

i2�ki1)

? 0 if ki2 ? ki1. Noting that bkiX = bKiX � bLiX , it follows that

cRSi = bXi2 � bXi

1 =1���i��bkiX + �iK + �

iL���i�� ���i�� bpi (A.6)

thus completing the proof to this part and, therefore, Lemma 1. k

Proof of Lemma 2: Denote with siK �riKi

X

Riand siL �

wiLiXRi

country i�s land and labor shares

in total net income Ri, and let �iG = i iwr iw

irbe the elasticity of substitution between land and labor

35

in the military sector. Total di¤erentiation of (6) yields

bkiX =

i�iLG�

iKG�

iGG

i���i��RisiKsiL! bpi +

i

RisiKsiL

�risiL i

�K0�

iGi �

i

ri

�+ �iLG

�dGi +

riK0�iGj

RisiKdGj +

�idK0

KiX

+dKi

KiX

� dLi

LiX: (A.7)

Parts (a)-(c) & (e): The proofs follow from (A.7) and the linear homogeneity of i.

Part (d): Suppose Gi = Gj so that �iGi = ��iGj . Now use dGi = dGj in (A.7) to obtain

@kiX=@Gi

kiX= � ir

KiX+ iw

LiX. Utilizing the de�nitions of Ki

X and LiX in (4) and (5) we may transform

this relationship into

@kiX=@Gi

kiX= iw�kiX � kiG

�KiX

= iwL

i�ki � kiG

�KiXL

iX

= i��iLG � siL

�RisiKs

iL

(A.8)

which proves part (d). jj

Proof of Lemma 3. Part (a): Di¤erentiate (9) with respect to Gi and utilize (A.3) to obtain

V iGiGi = �iriK0�

iGiGi < 0. (A.9)

Part (b): Di¤erentiation of (8) with respect to Gj and utilization of (A.4) gives

V iGiGj = �iriK0�

iGiGj R 0 if Gi R Gj . (A.10)

Part (c): Di¤erentiating (8) with respect to price and evaluating the resulting expression at the

optimum gives (by Lemma 1(a))

V iGipi = ��iri@(

i=ri)

@pi= ��i

� i

pi

��wi iw i

� pi!ip!i

!

= �iri� i=ri

pi

��iLG���i�� ? 0 if ki2 ? ki1. (A.11)

Part (d): This is a standard property of indirect (trade) utility functions. k

36

Proof of Lemma 4. In (9) we can use (A.1) and (A.2) and the fact that MCi = i=ri =

(!i; 1) to obtainMB1

MB2=

f 0(G1)=f(G1)

f 0(G2)=f(G2)= (!(p1); 1)

(!(p2); 1)=MC1

MC2

under interior solutions where, for simplicity, we omitted stars. Now suppose ki2 � ki1 > 0. By

Lemma 1(a), (!(pi); 1) is decreasing in pi; therefore, MC1=MC2 S 1 if p1 � p2 R 0 which by the

above equation requiresMB1=MB2 S 1. By the concavity of f(�) in the CSF (and thus in theMBi

function), we must have G1 R G2. On the other hand, if ki2 � ki1 < 0, by Lemma 1(a), (!(pi); 1)

would be increasing in pi and would imply MC1=MC2 R 1 if p1 R p2. But then MB1=MB2 R 1

which would require G1 S G2. We can prove the only if part using similar reasoning. k

Proof of Lemma 5. Let �iD be the elasticity of substitution in consumption. Focusing on

percentage changes, note that dRDi= ��iDbpi and that the expression for cRSi is given in (A.6).

Totally di¤erentiating (10) and rearranging terms gives

dRDi= cRSi =)

�iD +

�iK + �iL���i�� ���i��! bpi + 1���i��bkiX = 0:

The above relation and the de�nition of���i�� reveal that pi is decreasing (increasing) in kiX if ki2 > ki1

(ki2 < ki1). Utilizing (A.7) in the above expression gives

bpiA = � 1

�i���i��

�@kiX=@G

i

kiXdGi +

@kiX=@Gj

kiXdGj +

�idK0

KiX

+dKi

KiX

� dLi

LiX

�(A.12)

where �i � �iD +�iK+�

iL

j�ijj�ij + i�iLG�

iKG�

iG

j�ijj�ijRisiKsiLGi > 0: The proofs to parts (a)-(d) now follow from (A.12)

and (A.7). k

Proof of Theorem 1. (Existence) We establish existence of equilibrium in pure strategies,

by showing that every country i�s payo¤ function V iA is strictly quasi-concave in its strategy G

i.

To establish strict quasi-concavity of V iA in Gi it is su¢ cient to show either that V i

A is strictly

monotonic in Gi or that V iA is �rst strictly increasing and then strictly decreasing over the agent�s

strategy space.

Let F (KiG; L

iG) be the production function for guns that is dual to the unit cost function

(wi; ri) and de�ne Gi � F (Ki; Li). Country i�s strategy space is [0; G

i]. For any Gj 2 [0; Gj ],

37

country i (6= j) will be unable to produce any of the consumption goods if Gi = Gi; therefore,

V iA(G

i; Gj) < V i

A(Gi; Gj) for any Gi 2 [0; Gi) which implies that, under autarky, no country will

use all its resources to produce arms. Furthermore, since limGi!0 f0(Gi) = 1, we must have

@V iA=@G

i > 0 as Gi ! 0. By the continuity of V iA in G

i, there will exist a BiA(G

j) � minfGi 2

(0; Gi) j @V i

A=@Gi = 0g with the property that @V i

A=@Gi > 0 8Gi < Bi

A(Gj). Thus, to establish

strict quasi-concavity of V iA in G

i it remains to prove that @V iA=@G

i < 0 8Gi > BiA(G

j).

Suppose @V iA=@G

i � 0. Since V iA must eventually fall to V

iA(G

i; Gj), the function must attain

a local minimum at some Gi > Bi(Gj) which would imply that @2V iA=@(G

i)2 > 0. We now prove

that this is impossible. Recalling that piA = piA(Gi; Gj) under autarky and that !i = !(pi), we may

di¤erentiate (9) with respect to Gi and apply (9) on the resulting expression to obtain

@2V iA

@(Gi)2=

(�)�V iGiGi

�pi=piA

+

(�)hV iGipi

ipi=piA

(�)�@piA@Gi

�< 0. (A.13)

By Lemma 3(a), the �rst term in the right-hand side (RHS) of the above expression is negative

regardless of the ranking of factor intensities. Furthermore, by Lemmas 3(c) and 4, the product of

the expressions in the second term will also be negative.37 It follows that @2V iA=@(G

i)2 < 0 at any

Gi point at which @V iA=@G

i = 0 regardless of the ranking of factor intensities. This proves BiA(G

j)

is unique. In addition, it establishes the existence of a pure-strategy equilibrium.

(Uniqueness) To establish uniqueness of equilibrium we prove that at any equilibrium point the

determinant of the Jacobian of the marginal payo¤s in (9) is positive (i.e., jJ j = @2V 1A@(G1)2

@2V 2A@(G2)2

�@2V 1A

@G1@G2@2V 2A

@G2@G1> 0) and some boundary conditions are satis�ed (Kolstad and Mathiesen, 1987).

First, note that

@2V iA

@Gi@Gj=�V iGiGj

�pi=piA

+

(�)hV iGipi

ipi=piA

(�)�@piA@Gj

�. (A.14)

By Lemma 3(b) (see also (A.10)), the �rst term in the RHS of the above expression is positive or

negative depending on whether BiA(G

j) > Gj or BiA(G

j) < Gj , respectively. By Lemmas 3(c) and

4, the second term is always positive.

37 In (A.15) and elsewhere, the top signs in \� " and \� " apply when ki2 > ki1 and the bottom signs apply whenki2 < ki1.

38

Utilizing (A.13) and (A.14), we may write the slope of country i�s best-response function as

@BiA

@Gj= �@

2V iA=@G

i@Gj

@2V iA=@(G

i)2= �

�V iGiGj

�pi=piA

+hV iGipi

ipi=piA

�@piA@Gj

��V iGiGi

�pi=piA

+hV iGipi

ipi=piA

�@piA@Gi

� . (A.15)

From (A.13), (A.14) and (A.15) it can be seen that jJ j > 0 if�@B1A=@G

2� �

@B2A=@G1�< 1 at

an equilibrium point. Since @2V iA=@(G

i)2 < 0, the sign of @BiA=@G

j is determined by the sign of

@2V iA=@G

i@Gj . As can be a¢ rmed from the above, @2V iA=@G

i@Gj > 0 when BiA(G

j) � Gj , so Gi

is a strategic complement for Gj in this case. Inspection of (A.14) reveals that Gi can become a

strategic substitute for Gj when BiA(G

j) is su¢ ciently smaller than Gj . Furthermore, from (A.10)

it follows that sign�V 1G1G2

�= �sign

�V 2G2G1

�since �1G1G2 = ��2G2G1 ; therefore, there are two

possibilities with regards to the signs of best-response functions at an equilibrium point. Either (i)

@BiA=@G

j > 0 and @BjA=@G

i � 0 for i 6= j = 1; 2, or (ii) @BiA=@G

j > 0 8i 6= j = 1; 2. It is easy to

check that, in case (i), jJ j > 0. Turning to case (ii), we now establish the existence of (su¢ cient)

conditions that ensure�@B1A=@G

2� �@B2A=@G

1�< 1 and therefore jJ j > 0.38

First, apply (9) onto (A.7�) and then onto (A.12) to obtain

@piA@Gi

= � piA i

�i���i��RisiKsiL �iLG

@piA@Gj

=piA

i

�i���i��RisiKsiL

��iGj

�iGi

!siL:

The above expressions together with (A.9), (A.10), and (A.11) can be substituted into (A.15) to

obtain @BiA=@G

j = �(�iGj=�iGi)�

iA, where

�iA =��i

GiGj

�iGj�i + i

j�ijj�ijRisiKsiL(�iLGs

iL)

��iGiGi

�iGi�i + i

j�ijj�ijRisiKsiL(�iLG)

2

. (A.16)

From (A.1) and (A.2), we have (�1G2=�1G1)(�

2G1=�

2G2) = 1 which implies

�@B1A=@G

2� �@B2A=@G

1�=

�1A�2A; therefore, if �

iA 2 (0; 1) 8i = 1; 2, then jJ j > 0. But, in case (ii), both the numerator

and the denominator of �iA are positive, so �iA > 0. Now subtract the numerator of �iA from its

38Note that, in case (ii), jJ j > 0 is also the condition for local stability of equilibrium.

39

denominator to obtain

�i

Gi

�iD +

�iK + �iL���i�� ���i��!+

i�iLG���i�� ���i��RisiKsiL��iLG + �

iKG�

iG�

i � siL�

(A.17)

where �i � Gif 0i=fi � Gif 00i =f0i (> 0). (To derive (A.17) we used the de�nition of �i in (A.13)

and (A.1)-(A.4) which imply ��iGiGi=�iGi + �iGiGj=�

iGj = �i=Gi.) Clearly, a su¢ cient condition

for �iA < 1 is that (A.17) is positive. Inspection of (A.17) reveals that this is almost always true.

A somewhat restrictive (but hardly necessary) condition for �iA < 1 is �iLG + �iKG�iG�

i � siL � 0

which is satis�ed under a wide range of circumstances including: (i) �iG�i � siL which requires arms

inputs not to be close complements;39 (ii) �iLG � siL (or, by (A.8), ki > kiG) which requires the

guns sector to be su¢ ciently labor-intensive, regardless of the degree of substitutability between

inputs in arms. The above conditions and BiA(G

j) 2 (0; Gi) 8i 6= j (boundary conditions) establish

uniqueness of equilibrium. k

Proof of Proposition 1. Since the logic behind part (a) was outlined in the main text, here

we prove part (b). A redistribution of a secure resource from country j to country i(6= j) expands

(contracts) the "recipient" ("donor") country�s resource endowment. Di¤erentiating country i�s

FOC condition in (9) appropriately gives

@2V iA

@(Gi)2dBi

A +@2V i

A

@Gi@H idH i = 0 =) dBi

A

dH i= �@

2V iA=@G

i@H i

@2V iA=@(G

i)2

for H = L;K, with @2V iA=@(G

i)2 < 0 from (A.13). Thus, sign[dBiA=dH

i] = sign[@2V iA=@G

i@H i].

Di¤erentiation of (9) yields@2V i

A

@Gi@H i=hV iGipi

ipi=piA

dpiAdH i

: (A.18)

From Lemma 3(c) and Lemma 5(d), it follows that dBiA=dL

i > 0 whereas dBiA=dK

i < 0; therefore,

for any arms choice by its rival, the recipient (donor) country�s best-response entails producing

more (less) arms than before. Thus, if we start with an arbitrary endowment con�guration in S0

and transfer labor from country j to country i, so that we end up somewhere in Si, the properties

of best-response functions ensure that at the new equilibrium we will necessarily have Gi�A > Gj�A .39This condition is always satis�ed when the production function for guns is Cobb-Douglas and the CSF assumes

the Tullock form (i.e., f(Gi) = (Gi) ; 8 2 (0; 1]). This is so because �iG = 1 and �i = 1, hence, �iLG+�iKG�iG�i�siL =1� siL � 0

40

By Lemma 4, we must also have pi�A ? pj�A if ki2 ? ki1. k

Proof of Proposition 2. To identify the e¤ects of endowment changes on equilibrium security

policies we di¤erentiate the FOCs in (9) and solve the resulting system of equations to obtain

0B@ dG1�A

dG2�A

1CA =1

jJ j

0B@ @2V 2A@(G2)2

� @2V 1A@G1@G2

� @2V 1A@G2@G1

@2V 1A@(G1)2

1CA0B@ � @2V 1A

@G1@H1dH1

� @2V 2A@G2@H2dH

2

1CA (A.19)

for H i = Li;Ki where jJ j > 0 and all expressions are evaluated at the equilibrium. Start with an

endowment allocation in S0, so that Gi�A = eG�A and pi�A = ep�A for i = 1; 2. Evaluation of expressionsat such allocations implies: (i) @2V 1A

@G1@G2=

@2V 2A@G2@G1

> 0 by (A.14) and because V 1G1G2 = V 2G2G1 = 0

(Lemma 3(b)); (ii) @2V 1A@(G1)2

=@2V 2A@(G2)2

< 0 by (A.13); (iii) @B1A@G2

= �@2V 1A=@G1@G2

@2V 1A=@(G1)2

= �(�1G2=�1G1)�

1A =

�1A 2 (0; 1) by (A.15), (A.16) and the related discussion in the proof of Theorem 1; and (iv)

@2V 1A@G1@L1

=@2V 2A@G2@L2

> 0 whereas @2V 1A@G1@K1 =

@2V 2A@G2@K2 < 0 by (A.18) and the related discussion.

Part (a): Let us now focus on a small transfer of labor from country 2 to country 1, so that

�dL2 = dL1 > 0. Using the above observations in (A.19) yields

dG1�AdL1

= �dG2�A

dL1=

(+)

1

jJ j

(+)�� @2V 1A@(G1)2

� (+)�1� @B1A

@G2

� (+)�@2V 1A@G1@L1

�> 0:

For land redistributions, we can use similar logic to show that dG1�A =dK1 = �dG2�A =dK2 < 0.

Part (b): The proof is established by invoking symmetry and applying part (a) to (11). k

Proof of Theorem 2. The proofs of existence and uniqueness of equilibrium are similar to

the ones in Theorem 1 and are omitted. The proof of symmetry is outlined in the text. k

Lemma A1. For given � and a factor distribution in the intersection of Si and the AES, acountry�s residual land/labor ratio, kiX = kiX

��;Bi

F (Gj); Gj ; �

�, will change as follows along its free

trade best-response function, BiF (G

j), for i 6= j:

bkiX = i

RisiKsiL

f 0jfi

fjf 0i

240@ fi�fjfi+fj

2�i � Gif 00if 0i

fiGif 0i

1A �iLG � siL

35 dGj : (A.20)

(a) If Gi � Gj , then dkiX=dGj jGi=BiF (Gj) < 0;

(b) If Gi > Gj , then dkiX=dGj jGi=BiF (Gj) < 0 almost always. A su¢ cient (but hardly neces-

41

sary) condition is �iLG < 2siL.

Proof. Equation (A.20) is obtained from country i�s FOC in (A.7), and the expressions for

�iGiGi and �iGiGj from (A.3) and (A.4), respectively, used in dBi

F =dGj = ��iGiGj=�

iGiGi . Parts (a)

and (b) follow from inspection of (A.20). k

Proof of Proposition 4. Recalling that ki2 > ki1, consider an asymmetric distribution of

resources in the AES portion of S1 associated with � = p2�A . By Proposition 1(b), G1�A > G2�A and

p1�A > p2�A under autarky. Keeping � at p2�A , let the contestants move toward free trade, so that

we travel along B2F (G1) in Fig. 2 from equilibrium point A0 under autarky to equilibrium point A

under free trade. By Lemma A1, the adjustment in guns will cause k2X to rise which will generate

an excess supply for the non-numeraire (by forcing RS2 to shift rightward (Lemma 5) relative to

RD2) and will thus require � to fall below p2�A to eliminate country 2�s volume of trade; therefore,

�2A < p2�A . We can use similar logic (together with Lemma A1) to establish that �1A > p1�A . k

Proof of Proposition 5. Part (a): This part follows readily by noting that M i�F (�

iA) = 0 in

(12) and recognizing that the strategic welfare e¤ect is negative when ki2 > ki1.

Part (b): By part (a), we will have dV i�F (�

iA)=d� < 0. Moreover, by Proposition 3, there will

exist a su¢ ciently high price, �, such that dGj�F =d� = 0 for all � > �. But then by part (a) and

(12), we will have dV i�F =d� > 0 for all � > �. Thus, there must exist a price �imin (> �iA), that

minimizes country i�s welfare and is such that the country exports the land-intensive product. k

Proof of Proposition 8. The focus on symmetric adversaries (point D in Figs. 3 and 4)

implies: (i) Gi�F = G�F , Vi�F = V �F , �

iA = ep�A 2 (�; �) and �imin = �min 8i; (ii) if � = ep�A, then

G�F =eG�A and V �F = eV �A; (iii) all adversaries export the land-intensive good if � ? ep�A when ki2 ? ki1.

It can now be seen that arming comparisons are a direct consequence of Proposition 3. To establish

the welfare result, �rst note that the requirement dGi�F (ep�A)=d� > 0, 8i in Proposition 5 is satis�edbecause ep�A 2 (�; �). Then, by part (b) of Proposition 5, there will exist a �0 (? ep�A as ki2 ? ki1)

that solves V i�F (�) =

eV �A. We may now complete the proof by noting that all contestants exportthe land-intensive good and V �F (�) < eV �A, 8� 2 (ep�A; �0) if ki2 > ki1, or V

�F (�) <

eV �A, 8� 2 (�0; ep�A) ifki2 < ki1). k

42

Proof of Proposition 9. For concreteness, we suppose ki2 > ki1. Since we consider allocations

in the AES of S0, it will necessarily be the case that pi�A = ep�A, 8i and thus �iA = ep�A.Part (a): Since �iA = ep�A, 8i, dGi�F (ep�A)=d� > 0 as in Proposition 8. By Proposition 5(b),

there will thus exist a price �i0, 8i = 1; 2 such that V i�F (�) <

eV i�A , 8� 2 (ep�A; �i0). Now de�ne

�00 = minf�10; �20g. It follows that V i�F (�) <

eV i�A , 8� 2 (ep�A; �00).

Part (b): Starting at an arbitrary allocation in the AES of S0, transfer a small quantity of

labor from country 2 to country 1 (i.e., �dL2 = dL1) so that the �nal allocation is in S1, as

indicated by the move from point E to point H in Figs 3 and 4. Proposition 2(b) and Corollary

1 imply that dV 1�F =dL1 < dV 1�A =dL1 and dV 2�F =dL1 > dV 2�A =dL1. Since V i�F = V i�

A initially, we will

have V 1�F < V 1�A and V 2�F > V 2�A after the transfer. By continuity, there will exist additional labor

reallocations with the just described preferences over trade regimes. k

43

Figure 1

Individually Optimal Security Policies

Figure 2

Best-Response Functions in Security Policies

Figure 3

The Distribution of Factor Endowments, Sectoral Decomposition of Production,and Arming Incentives under Autarky

Figure 4

Free Trade in Consumption Goods and the Arms Equalization Set

Figure 5

Trade Patterns, Welfare, and Equilibrium Security Policiesof Symmetric Adversaries