Gunboat DiplomacyPolitical Bias, Trade Policy, and War∗

Brendan Cooley†

13 November 2019

Abstract

Countries with deep trading relationships rarely �ght wars with one another. Here,I develop a theory of trade, war, and political bias, in which both trade and war areendogenous objects. Governments can rectify poor market access conditions abroadthrough war and subsequent regime change, in which the victorious country installsa liberal “puppet” government abroad. Trade policy bargaining is therefore conducted“in the shadow of power,” with counterfactual wars shaping the policy choices thatprevail in times of peace. When peace prevails, militarily weak countries are moreopen to trade than powerful ones, all else equal. Equilibrium trade policies balancedomestic interests against military threats from abroad. War is less likely betweenliberal governments because they prefer less protectionist trade policies. As a result,trade �ows and the probability of peace are positively correlated in equilibrium, eventhough trade does not cause peace.

JEL Classi�cation Codes: D72, D74, F13, F51, F52, F54

∗Ph.D. candidate, Department of Politics, Princeton University. Previous versions of this paper werecirculated under the titles “Trade Wars, Hot Wars, and the Commercial Peace” and “Trade Policy in theShadow of Power.” For helpful comments and discussions on earlier drafts of this paper, I thank Timm Betz,Adrien Bilal, Tyson Chatagnier, Rob Carroll, Andrew Coe, Noel Foster, Erik Gartzke, Kishore Gawande,Dan Gibbs, Joanne Gowa, Gene Grossman, Bobby Gulotty, Matias Iaryczower, Amanda Kennard, ColinKrainin, Melissa Lee, James Mao, Helen Milner, John Oneal, Kris Ramsay, Bryan Schonfeld, Brad Smith,Sondre Solstad, and Jack Zhang as well as audiences at Princeton’s Political Economy Graduate Colloquium,Princeton’s International Relations Graduate Seminar, Princeton’s Fellowship of Woodrow Wilson Scholars,the Midwest Political Science Association’s 2018 Annual Meeting, Southern Political Science Association’s2019 Annual Meeting, and the International Studies Association’s 2019 Annual Meeting.

†Ph.D. Candidate, Department of Politics, Princeton University

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IntroductionCountries with deep trading relationships rarely �ght wars with one another. Some arguethis “commercial peace” is due to the pacifying e�ect of trade — trade causes peace.1 Otherssay amicable political relations cause trade.2 Trade is usually considered to be exogenousto con�icts of interest in international relations. Governments are posited to �ght warsover territorial, ideological, or other non-economic con�icts of interest. Preexisting traderelations make these con�icts more costly, and less likely as a result.

Yet, trade policy is itself a central object of contention in international relations. Gov-ernments are mercantilist to some extent (Gawande, Krishna, and Olarreaga 2009). Theydesire some degree of protection at home and “open doors” abroad and have historicallybeen willing to �ght wars to compel market openness abroad.3 In the 1850s, the U.S.gunboats compelled an autarkic Japanese government to open its markets. Britain andFrance prosecuted the Opium Wars (1839-1842; 1856-1860) to compel a recalcitrant Chinesegovernment to reform its trade policies. Recently, a proposal to integrate the economies ofUkraine and the European Union led to war between Ukraine and Russia.4 These episodeshighlight linkages between trade policy and war. But such linkages may be more common,as the absence of war need not imply the absence of coercive bargaining (Fearon 1995).

Here, I develop a theory of trade policy bargaining in the shadow of military power, or“gunboat diplomacy.” The value of international trade �ows and the probability of peaceare positively correlated in equilibrium. This correlation emerges not because trade causespeace. Rather, liberal trade policy preferences generate incentives for both trade andpeace. When peace prevails, latent military threats in�uence equilibrium trade policies,which balance domestic political-economic interests against military threats from abroad.Militarily weak countries are more open to trade than powerful ones, all else equal.

The model considers the interaction between two governments (home and foreign), whicheach value a weighted average of consumer welfare and �rm pro�ts. These componentsof government utility themselves depend on the workings of an underlying “new trade”international economy (Krugman 1980; Venables 1987). Governments di�er in how muchin�uence consumers have over policymaking.5 I refer to this variation as the governments’degree of political bias.6 Tari�s help �rms by shielding them from competition, but raise

1This literature is vast. See Gartzke and Zhang (2015) for a complete survey. Angell (1911), Polachek(1980), and Philippe, Mayer, and Thoenig (2008) are representative of this view.

2See, for example, Pollins (1989), Barbieri and Levy (1999), Benson and Niou (2007)3See Findlay and O’Rourke (2007) for a chronicle of trade con�icts over the past millennium.4In this case, Moscow conditioned its coercion on the trade policy choices of Kiev, see James Marson

and Naftali Bendavid, “Ukraine to Delay Part of EU Pact Opposed by Russia,” The Wall Street Journal, 12September 2014.

5This setup mirrors Grossman and Helpman (1994).6This phraseology borrows from Jackson and Morelli (2007). In their model, political bias determines the

extent to which the pivotal decision maker internalizes the costs of war. Conceptually, bias is similar to thesize of the selectorate in the model of Bueno De Mesquita et al. (2003).

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prices for consumers. Low bias or liberal governments prefer lower tari�s.

Governments care about trade policy choices abroad because of market access externalities(Ossa 2011, 2012). Firms’ pro�ts depend on their ability to access foreign markets. Hightari�s shift pro�ts from foreign to home �rms. Therefore, �rms on opposite sides of aborder experience a con�ict of interest over trade policy. Firms desire protection at homeand liberalization abroad. The greater their bias, the more the governments internalizethese interests.

As a consequence, governments themselves experience con�icts of interest over tradepolicy, which vary as a function of the governments’ bias. When governments valueconsumer welfare, they prefer to adopt low barriers to trade. In doing so, they imposesmall market access externalities on their trading partners. When both governments holdliberal policy preferences, their relations are harmonious – there is no incentive for con�ict,militarized or otherwise. As governments become less liberal, con�icts of interest becomemore severe.

If a government wins a war, it earns the right to impose regime change and install a “puppet”government abroad.7 Puppets open their markets to foreign �rms, allowing victoriousgovernments to impose their trade policy preferences by force. This is the threat pointgovernments leverage in bilateral trade policy negotiations. War sometimes occurs due toinformation frictions.

Two primary insights emerge from this environment. First, governments’ degree of biasa�ects their propensity to trade and �ght wars. When both governments are liberal, thecosts of regime change never exceed its policy bene�ts. As a result, highly liberal govern-ments never �ght wars with one another. Their liberal preferences also result in liberalequilibrium trade policies. Naturally, lowering barriers to trade increases trade itself. Thegovernments’ preference compatibility produces a relationship between trade and peace,but this relationship is spurious — trade itself has no pacifying e�ect. Second, even whengovernments avoid con�ict, trade policies re�ect the balance of military power betweenthe governments. Powerful countries can credibly threaten to impose regime change. Theyleverage this power to extract trade policy concessions and resist liberalization. Afterbargaining, powerful countries are more protectionist than weaker ones.

The model rationalizes several well-established empirical facts in international relations.Bilateral trade tends to decrease before wars and rebound thereafter.8 In the model,protectionist preference shocks decrease trade, but also increase the likelihood of war. Asa consequence, periods of depressed trade correlate with war onset. Conversely, regimechange following war causes a liberal preference shock to the losing country’s government.Trade increases after wars, as in the data.

Some argue democracies have more liberal trade policy preferences than autocracies7See Owen IV (2002) for an empirical study of regime change.8This relationship is shown in Figure 5 in the Appendix.

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(Milner and Kubota 2005). Because consumers (voters) prefer free trade, they punishprotectionist politicians (Mayer 1984; Grossman and Helpman 1996).9 This provides a checkon the protectionist in�uence of special interest groups. Translated into this framework,these theories deem democracies less biased than autocracies. If this “liberal democracy”hypothesis holds, the analysis of low bias governments extends to democratic dyads. Themodel then jointly rationalizes the democratic peace and democracies’ propensity for tradeopenness.10

The theory’s implications about power and protectionism are, as far as I know, novel. Iconsider these empirical implications in more detail in the Discussion section. There, Ialso relate the theory to militarism, imperialism, and territorial con�ict.

Antràs and Padró i Miquel (2011) and Carroll (2018) are two closely related papers thatmerit some discussion. Antràs and Padró i Miquel (2011) consider a similar model, inwhich foreign governments can interfere in the domestic political economy of trade. Asin this model, foreign in�uence has a liberalizing e�ect. In the anarchy of world politics,however such in�uence can always -– in principle -– take the form of threats, displays,or uses of military force (Fearon 1997). Analyzing this form of in�uence allows me torelate the domestic political economy of trade to military power and con�ict propensity.Carroll (2018) uni�es militarized competition and economic exchange in a more generalsetting. There, countries’ convert commodities into military power, which can in turnbe employed to seize others’ commodities. Military power itself is endogenous to thegeneral equilibrium of the international economy. I take power as exogenous and focus oncompetition over trade policy. This more narrow focus allows me to incorporate domesticpolitical economy considerations and make empirical predictions about power, trade policy,and war.

Others have considered policy competition in the shadow of power more generally. Bilsand Spaniel (2017) study a model of coercive bargaining over the location of a spatialpolicy. Like the model studied here, governments vary in the location of their ideal points.They study how uncertainty over the states’ ideal points a�ects con�ict propensity. Here, Imicrofound governments’ preferences with a political economy of trade policy and explorehow domestic political economic shocks a�ect the international bargaining environment.This generates unique comparative statics and empirical implications.

EnvironmentHere and in the Analysis section, I relegate proofs and derivations of key quantities to theAppendix, in order to ease exposition. I �rst present the context in which governments

9For a skeptical take on this mechanism, see Guisinger (2009) and Betz and Pond (2019).10These facts are depicted in Figure 6 in the Appendix. For a recent review on the relationship between

democracy and peace, see Reiter (2017). Milner and Kubota (2005) show democratization tends to lead todecreased protectionism.

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bargain, followed by the international economy. The general equilibrium of the economydetermines how trade policies a�ect prices, wages, and trade �ows. These in turn determinethe welfare of consumers, �rms, and the governments that represent them. Proposition1, presented in this section, states that given our assumptions, an economic equilibrium(De�nition 2) will exist for any feasible trade policy choices. This economy allows usto write government preferences as an indirect function of trade policy choices. Asgovernments become more liberal, they prefer to adopt low barriers to trade. This variationa�ects how governments behave in the coercive bargaining game that follows.

International BargainingTwo governments, home (i) and foreign (j) bargain over their joint trade policies τ =(τi, τj) ∈ [1, τ ]2.11 By controlling the degree of market access a�orded to foreign �rms,governments’ trade policies impose externalities on one another. Government preferencesdepend on an exogenous parameter ai ∈ [

¯a, a], which controls the value these governments

place on consumer welfare, relative to �rm pro�ts.12 Government utility is denotedGi(τ |ai).13 Higher tari�s increase �rm pro�ts by shifting market share to local �rms. Thiscomes at the expense of consumers, however, who bene�t from having access to a varietyof products, home and foreign. Higher tari�s also harm foreign �rms and the foreigngovernment. This is the model’s core con�ict of interest. Each government would like toimplement some degree of protectionism at home, while maintaining access to marketsabroad.

Bargaining occurs in the shadow of power. Government i makes a take-it-or-leave-it(TIOLI) o�er τ = (τi, τj) to Government j. Government j can either accept the o�eror declare war, a choice denoted with ω ∈ {accept,war}. This is a simple coercivebargaining framework following Fearon (1995).14 Here, however, war results in regimechange, rather than a simple costly division of a �xed surplus. Regime change is modeled asa change in a vanquished government’s preferences. If government i wins a war, it replacesthe government of its counterpart, �xing its preference parameter at a?. ρ denotes theprobability that Government i is successful in a war for regime change.15 ci is the cost thatgovernment i must pay if a war occurs. cj > 0 is held as private information. Governmenti believes cj is distributed according to F where F is the uniform c.d.f. on [

¯cj, cj].

As is standard in bargaining models of war, the costs of war must be bounded, or the11Here, τ is an arbitrary prohibitively high tari� that shuts down bilateral trade.12a is de�ned below.13I develop the international economy from the home country’s perspective, but analogous primitives

exist for the foreign country.14While this bargaining protocol is restrictive, Fey and Kenkel (2019) show more complex bargaining

processes are equivalent to a TIOLI o�er in terms of their associated payo�s and probability of war.15With complementary probability, the initiating government is overthrown. A more realistic formulation

might allow for the possibility that no regime change occurs, with ρi + ρj ≤ 1. While this “all or nothing”conception of war is stark, it simpli�es the analysis and highlights the forces at play.

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proposing country will never risk con�ict. Assumption 1 formalizes this intuition.

Assumption 1: cj ≤ κj and ci < κi(cj) where κj and κi(cj) are positive constants de�nedin the Appendix.

A strategy for Government i is an o�er, τ (ai, ci, ρ). A strategy for Government j, denotedω(τ ; aj, cj, ρ) is a mapping between this o�er and a choice of whether or not to attemptregime change

ω : τ → {accept,war} .Let Gk(τ , ω|ak, ck, ρ) denote government k’s utility as a function of these choices. Fromthese objects we can de�ne a subgame perfect bargaining equilibrium.

De�nition 1: A subgame perfect bargaining equilibrium is pair of strategies, τ ?(ai, ci)and ω?(τ ; aj, cj) such that

ω?(τ ; aj, cj, ρ) = arg maxω∈{accept,war}

Gj (τ , ω; aj, cj, ρ)

andτ ?(ai, ci, ρ) ∈ arg max

τ∈[1,τ ]2Ef(cj)

[Gi (τ , ω

?(τ ; aj, cj, ρ); ai, ci, ρ)].

International EconomyGovernment preferences in the game described above depend on the mechanics of theinternational economy. To simplify the presentation and focus on the dynamics of coercivebargaining in this political economy, I consider the special case in which countries aremirror images of one another in terms of their economic primitives. Each country isinhabited by a representative consumer with labor endowment Li = Lj = L. Consumersvalue varieties of manufactured goods and goods from an undi�erentiated outside sector,which I’ll call agriculture. By providing their labor to local producers of these goods, theyearn an endogenous wage wi. Consumers’ income inclusive of tari� revenues ri(τi) isIi(τi) = wiLi + ri(τi). A unit of labor can produce one unit of both di�erentiated goodsand agricultural goods. There is a mass of �rms of measure 1 in each economy whichproduce di�erentiated manufactured goods, indexed νi.16 Agricultural goods are producedcompetitively. The setup borrows from Venables (1987) and Ossa (2012).

Tari�s and Prices

Firms engage in monopolistic competition, setting prices in each market to maximizepro�ts, given consumer demand. Governments can shift the prices that consumers pay

16In a completely general equilibrium, this quantity would also be an endogenous object. Fixing thenumber of �rms allows each �rm to derive positive pro�ts, providing biased governments with an incentiveto implement a positive tari�. In this sense, the model is in a “short run” equilibrium in which pro�ts havenot yet been competed away.

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for foreign goods by charging a uniform import tari� on manufactured goods, τi − 1.This drives a wedge between the price set by foreign �rms, pj(νj), and the price paid byconsumers for foreign goods. The price of foreign manufactured goods in the home marketis pij(νj) = τipj(νj). The price in the agricultural sector serves as the numeraire, pyi = 1.The government collects the revenue raised from tari�s.

Consumption

Consumer preferences over agricultural goods Yi and aggregated di�erentiated varietiesXi are Cobb-Douglas, where an exogenous parameter α ∈ [0, 1] controls the consumers’relative preference for di�erentiated varieties. Consumers therefore solve the followingproblem

maxXi,Yi

Xαi Y

1−αi

subject to PiXi + Yi ≤ wiL(1)

where Xi is a CES aggregate of manufactured goods x, à la Dixit and Stiglitz (1977). Con-sumers value each di�erentiated good equally. Home and foreign goods are distinguishedonly by their price. Let xij(νj) denote the quantity of di�erentiated goods produced incountry j that are consumed in country i. Consumer’s utility over di�erentiated goods is

Xi =

(∫νi

xii(νi)σ−1σ dνi +

∫νj

xij(νj)σ−1σ dνj

) σσ−1

(2)

where σ > 1 is the elasticity of substitution between varieties. The real price level ofdi�erentiated goods in each country is described by the CES exact price index

Pi(τi) =

(∫νi

pii(νi)1−σdνi +

∫νj

pij(νj)1−σdνj

) 11−σ

. (3)

Equilibrium demand for manufactured goods from j in i is

x?ij(νj) = pij(νj)−σPi(τi)

σ−1αIi(τi). (4)

With prices of agricultural goods serving as numeraire, Yi = (1− α)Ii(τi) and consumerindirect utility is

Vi(τi) = αα(1− α)1−α Ii(τi)

Pi(τi)α. (5)

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Production

Firms set prices to maximize pro�ts across home and foreign markets, given consumerdemand. Because all �rms in country i face the same demand curves at home and abroad,they all set the same factory-gate price. The quantity that each �rm i produces for marketj is equal to the demand for i’s goods in j, x?ij(νj). It takes one unit of labor to produceeach unit of a manufactured good. The �rms’ problem is given by

maxpi(νi)

Πi (pi(νi)) = (pi(νi)− wi)(x?ii(νi) + x?ji(νi)

)subject to x?ii(νi) = pi(νi)

−σPi(τi)σ−1αIi(τi)

x?ji(νi) = (τjpi(νi))−σPj(τj)

σ−1αIi(τj).

(6)

This problem yields equilibrium prices

p?i (νi) =σ

σ − 1wi. (7)

Since prices are constant across �rms, I suppress the variety indices and write p?i (νi) = p?i .Total consumption of manufactured goods from i in j is

x?ji(τj) =

∫νi

x?ji(νi)dνi.

Total pro�ts for all �rms in country i can then be computed as

Πi(τi, τj) =

∫vi

Πi(pi(νi)) = (p?i − wi)(x?ii(τi) + x?ji(τj)

). (8)

By raising the price of foreign varieties, tari�s shift pro�ts from foreign to home producers.As tari�s get large (τj → τ ), demand for imported manufactured goods contracts (x?ji(τj)→0). Consumers substitute toward home varieties (p?jx?jj(τj) → αIj(τj)), increasing localpro�ts. While consumers are harmed by the imposition of tari�s (∂Vj

∂τj< 0), local producers

bene�t. The preferences of home consumers are aligned with those of foreign �rms, bothof which desire liberal trade policies from the home government.

I assume that �rms’ welfare is dependent only on their pro�ts, and not in�uenced bythe aggregate price level (Pi(τi)) within the economy. This is consistent with the specialcase of Grossman and Helpman (1994) in which �rm owners are “small” in the broaderpopulation. A more complex preference structure would emerge if this assumption wereviolated, or if �rms employed intermediate goods in production.

Tari� Revenue

For every unit of manufactured goods imported, the government collects (τi − 1)p?j intari� revenue. Total tari� revenue can be written

ri(τi) = (τi − 1)p?jx?ij(τip

?j). (9)

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Economic Equilibrium

Consumers lend their labor endowment to the manufacturing and agricultural sectorsin order to maximize their income. If both sectors are active, then wi = 1 because theagricultural sector is competitive and serves as numeraire. Let Lxi denote the amount oflabor i allocates toward manufacturing and Lyi the amount of labor i allocates towardagriculture. Let w = (wi, wj) and L = (Lxi , L

yi )i∈{i,j}.

De�nition 2: An economic equilibrium is a function h : τ → (w,L) mapping trade policychoices to endogenous wages and labor allocations such that goods and factor marketsclear given equilibrium prices and corresponding demands.

If the agricultural sector is active, it pins down wages and nulli�es incentives for govern-ments to employ tari�s for purposes of manipulating the terms of trade. Assumption 2guarantees that the agricultural sector will remain active regardless of the governments’choices of trade policies. Substantively, it requires that consumers spend a large enoughproportion of their income on agricultural goods to prevent the specialization of eithercountry in the production of manufactured goods. This allows me to focus analysis onpro�t shifting incentives for trade policy, as in Ossa (2012).

Assumption 2:

α <2

3

σ

σ − 1.

Proposition 1: If Assumption 2 is satis�ed, then a unique economic equilibrium existswith Lxi , L

yi , L

xj , L

yj > 0 and wi = wj = 1 for all τ ∈ [1, τ ]2.

Government Preferences

Governments value a combination of consumer welfare and producer pro�ts. With thesequantities derived, we can write

Gi(τ |ai) = aiVi(τi) + Πi(τi, τj). (10)

The exogenous parameter ai controls the relative weight the government places on con-sumer welfare, relative to pro�ts and revenue. This conception of government preferencesfollows Grossman and Helpman (1994), in which ai represents the value the governmentplaces on campaign contributions relative to consumer welfare.17

I take ai as a measure of the representativeness of i’s government. When ai is small (highbias), the government privileges the narrow interests of �rms and its own revenue. As

17In their model, �rms lobby for protective tari�s (or export subsidies), promising campaign contributionsin exchange for policy deviations from the consumer welfare-maximizing ideal. Grossman and Helpman(1996) provide additional microfoundations for this objective function in a model of electoral competition,in which the government can employ campaign contributions to in�uence the vote choice of “uneducated”voters.

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ai gets larger (low bias), the welfare of consumers plays a larger role in determining thegovernments’ preferences. If democracies are more sensitive to the interests of consumers,then we would expect them to have higher values of ai than autocracies.

Assumption 3 guarantees that the government prefers an interior tari� (τi < τ ). Thisrequires that the government’s weight on consumer welfare be su�ciently large.

Assumption 3: ai ∈ (¯a, a] for all i where

¯a is a positive constant de�ned in Appendix C

and a is an arbitrarily large but �nite number.

Notably,¯a depends positively on the consumers’ elasticity of substitution, σ. As σ increases,

manufactured varieties become more substitutable, and foreign varieties become lessvaluable to consumers. Governments therefore prefer higher tari�s, all else equal.

¯a

increases with σ in order to ensure that no government prefers prohibitive tari�s.

Lemma 1 formalizes the claims that ideal points, denoted τ ?i (ai), are interior to the policyspace.

Lemma 1: τ ?i (ai) ∈ (1, τ) .

AnalysisRecall from De�nition 1 that a bargaining equilibrium is a trade policy o�er from thehome country, and a decision of whether or not to declare war, given this o�er, from theforeign country. This section analyzes how these equilibrium choices vary as a function ofthe governments’ bias types.

The results can be summarized as follows. Because they internalize the welfare of con-sumers, liberal governments prefer to adopt lower tari�s (Lemma 2). If governments wereunable to bargain, a non-cooperative equilibrium (De�nition 3) would emerge in whichgovernments simply implemented their ideal tari�s. This non-cooperative equilibriumserves as a baseline from which governments compare o�ers in a bargaining equilibrium(De�nition 1). As governments’ degree of bias increases, they impose larger and largerexternalities on one another in a non-cooperative equilibrium. This increases the degreeof con�ict of interest between the governments (De�nition 4), and makes regime changerelatively more appealing. Liberal governments experience smaller con�icts of interestwith one another (Proposition 3) which makes them unwilling to initiate wars (Proposition4). Because they prefer lower trade barriers, trade also increases as governments becomemore liberal so long as the shadow of war is absent (Proposition 5). Finally, militarilypowerful governments adopt higher barriers to trade in equilibrium (Proposition 6)

PreferencesFigure 1 depicts the governments’ objective functions as a function of their own tari�choice, τi. As the government becomes more representative, the peak of the curve shifts to

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G(τi|a)

G(τi|a‘)

1 τi∗(a) τi

∗(a‘)τi

G(τ

i)

Government Objective Function

Figure 1: Government preferences over own tari� rates with a > a′

the left, indicating that the government prefers a lower tari�. This is a natural result. Asthe government becomes more representative, it values the welfare of the consumer moreand more. This pushes the government to adopt a policy that is closer to the consumer’sideal.

Figure 2 depicts the government’s welfare in (τi, τj) space. By decreasing the market accessa�orded to �rms in i, non-zero tari�s in j strictly decrease the government’s welfare. Forany given τi, the government’s welfare is increasing as τj decreases.

Lemma 2: Gi(τj) is strictly decreasing in τj .

If the governments were prohibited from bargaining, they would each simply choose thepolicy that maximized their utility, taking the other country’s policy choice as given.

De�nition 3: A noncooperative equilibrium is a pair of policies(τ ?i (ai), τ

?j (aj)

)such that

τ ?i (ai) = arg maxτi∈[1,τ ]

Gi(τi; ai)

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Increasing iso−welfare

1

τj∗(aj)

1 τi∗(ai)

τi

τ j

Government Iso−Welfare Curves

Figure 2: Government iso-welfare curve over home and foreign tari� rates

andτ ?j (aj) = arg max

τj∈[1,τ ]

Gj(τj; aj).

Our next result shows that as governments become more liberal, their optimal tari�s fall.

Lemma 3: τ ?i (ai) is strictly decreasing in ai.

Figure 3 depicts each governments’ best response curves through the policy space. Becausethe governments’ optimal choices do not depend on one another’s policy choice, theirbest response curves are straight lines. Their intersection constitutes the noncooperativeequilibrium. As the governments’ preferences become more biased, these curves shiftoutward, resulting in a more autarkic noncooperative equilibrium.

Regime ChangeIt is now clear that each government cares indirectly about the preferences of its bargainingpartner. More welfare-conscious governments adopt lower barriers to trade (Lemma 3) in

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τi∗(ai),τj

∗(aj)

1

τj∗(aj)

1 τi∗(ai)τi

τ j

Government Best Responses

Figure 3: Government best response functions

a non-cooperative equilibrium, which bene�ts governments abroad by providing greatermarket access to their �rms. If each government were able to choose the preferences oftheir negotiating partner, they would do so in order to minimize trade barriers. This is thepurpose of regime change in this model. If a government wins a war, it earns the right toreplace the government with a puppet with more “dovish” preferences. Regime change istherefore used instrumentally to pry open international markets. Let a? ∈ (

¯a, a] denote

the type of the optimal puppet government.

The optimal puppet’s type solves

maxaj∈(

¯a,a]

Gi(τ?i (ai), τ

?j (aj); ai).

Proposition 2: a? = a .

If a government wins a war, it will replace the vanquished government with a maximally-responsive puppet. This government will adopt no trade barriers, providing maximalmarket access for the victorious country’s �rms. This is the threat point that governmentsleverage in international coercive bargaining.

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Con�icts of InterestIf a government wins a war it adopts its optimal policy and enjoys complete access to themarkets of its trading partner. This best case scenario yields the government utility

Gi(ai) = Gi(τ?i (ai), τ

?j (a); ai).

If a government loses a war, it is replaced by a puppet and must su�er under the policiesimplemented by the puppet regime. This is consistent with a notion of the government asa particular amalgamation of social actors that continues to exist at the conclusion of awar. The vanquished government yields utility

¯Gi(ai, aj) = Gi(τ

?i (a), τ ?j (aj); ai).

These outcomes represent upper and lower utility bounds on the outcome of any coercivenegotiation. Each government can be made no worse o� than if it were to lose a war. Andeach government can secure no better bargaining outcome than if they were to (costlessly)win a war for regime change. The welfare di�erence between these two scenarios is takento be i’s con�ict of interest with j. Note that this con�ict of interest, unlike standard modelsof bargaining and war, need not be symmetric. The “pie” at stake in the negotiation overtrade policies may be valued di�erently by each government — i’s preference intensitymay be stronger than j’s or vice versa. This variation in preference intensity, combinedwith variable military power, determines bargaining outcomes.

De�nition 4: The magnitude of government i’s con�ict of interest with government j is

Γi(ai, aj) = Gi(ai)−¯Gi(ai, aj). (11)

Proposition 3: Γi(ai, aj) is decreasing in ai and aj .

Proposition 3 states that as government i becomes more welfare-conscious, the magnitudeof its con�ict of interest decreases. Likewise, as government j becomes more welfare-conscious, i’s con�ict of interest with it decreases. As government i becomes more welfare-conscious, it prefers less protectionism. This decreases the di�erence between i’s idealpolicy and the (free trading) policy that will be imposed upon it if j is victorious in a war.As j becomes more welfare-conscious, it imposes smaller market access externalities oni. Regime change becomes relatively less appealing, because the distance between j’spreferred policy and the policy that a puppet would impose shrinks. In the corner casewhere ai = aj = a, the con�ict of interest evaporates – puppets would implement theexact same policies as the sitting governments.

BargainingThese con�icts of interest structure what sets of policies i will o�er and what o�ers j willprefer to war. Working backward, recall from De�nition 1 that ω?(τ ; aj, cj, ρ) is a function

14

that takes an o�er from i and returns a choice of whether or not to declare war. j’s utilityfor war is given by

Gj(aj, ai) = (1− ρ)Gj(aj) + ρ¯Gj(aj, ai)︸ ︷︷ ︸

Wj(aj ,ai)

−cj = (1− ρ)Γj(aj, ai) +¯Gj(aj, ai)− cj.

Note that j’s utility can be written in terms of its con�ict of interest with i. j will preferwar to i’s o�er whenever

Gj(aj, ai) ≥ Gj(τ ; aj).

This condition allows us to characterize ω?(τ ; aj, cj, ρ).

Lemma 4:

ω?(τ ; aj, cj, ρ) =

{war if Gj(aj, ai) ≥ Gj(τ ; aj)

accept otherwise

If j’s con�ict of interest with i is small enough, then i can simply o�er its ideal point andall cost types of j will accept.

Lemma 5: IfWj(aj, ai)−Gj(τ

?j (a), τ ?i (ai); aj) = Γj(aj, ai) ≤

¯cj

thenτ ? = (τ ?i (ai), τ

?i (a))

andω?(τ ?i (ai), τ

?j (a); aj, cj, ρ) = accept

for all cj ∈ [¯cj, cj].

Given our assumptions on the costs of war, we can always �nd a cutpoint bias type for theforeign country such that all types more liberal than the cutpoint accept i’s ideal point.

Lemma 6 (Zone of Peace): For every¯cj ∈ [0, cj) there exists a aj(

¯cj, ai) such that for all

aj ∈ [aj(¯cj, ai), a) the probability of war is 0.

Lemma 6 proves the existence of a “Zone of Peace” – a set of foreign bias types that never�ght in equilibrium. Combining this observation with the fact that j’s con�ict of interestwith i is decreasing in i’s bias type yields our �rst core result. Namely, that the size of thiszone of peace is increasing as i becomes more liberal in its policy preferences. Figure 4depicts this result.

Proposition 4 (Liberal Peace): aj(¯cj, ai) is weakly decreasing in ai.

Whenever aj ≥ aj(¯cj, ai), i o�ers its ideal point which is accepted by j. This guarantees

peace. j is more willing to accept i’s ideal point as it becomes more liberal, because i’sideal policy imposes smaller and smaller externalities on j. This generates a “Liberal Peace.”

15

¯a ai ↑ =⇒← aj(cj , ai) a

Shadow of War Zone of Peace

Figure 4: As the home government becomes more liberal, the set of foreign types that accept itsideal point expands.

When peace prevails, liberal governments also settle on more open trade policy regimesoverall. Naturally, reducing trade costs increases trade between the governments.

Proposition 5 (Liberal Trade): If aj ≥ aj(¯cj, ai) then trade in manufactured goods is

increasing in ai.

If aj < aj(¯cj, ai), however, then i faces a risk-return tradeo� (Powell 1999). O�ers closer

to i’s ideal point yield higher utility conditional on acceptance, but also generate a higherrisk of war. Here, the shadow of power a�ects equilibrium policies.

For any o�er, the probability that j will declare war is given by

Pr (cj ≤ Wj(aj, ai)−Gj(τ ; aj)|τ , ai, aj) = F (Wj(aj, ai)−Gj(τ ; aj)) . (12)

With this quantity known, we can work to characterize i’s o�er function, τ ?(ai). If waroccurs, i receives utility

Gi(ai, aj) = ρΓi(ai, aj) +¯Gi(ai, aj)− ci.

With the probability of war given in Equation 12, we can write i’s utility for any o�er as

Gi (τ , ω?(τ ; aj, cj, ρ); ai, ci, ρ) = (1− F (Wj(aj, ai)−Gj(τ ; aj))) (Gi(τ ; ai))︸ ︷︷ ︸

¬war

+

F (Wj(aj, ai)−Gj(τ ; aj))(Gi(ai, aj)

)︸ ︷︷ ︸

war

.(13)

By De�nition 1, i’s equilibrium o�er will maximize this objective function. Lemmas 7 and8 state that an o�er will lie inside the pareto set and that proposed trade policies will beweakly more liberal than those in a noncooperative equilibrium (De�nition 3).

De�nition 5: The pareto set is given by

P =

{τ ∈ [1, τ ]2|τ ∈ arg max

τ∈[1,τ ]2λGi(τ ; ai) + (1− λ)Gj(τ ; aj)

}

16

for some λ ∈ [0, 1].

Lemma 7: τ ?(ai, ci, ρ) ∈ P .

Lemma 8: τ =(τ ?i , τ

?j

)≤(τ ?i (ai, ci, ρ), τ ?j (ai, ci, ρ)

)with ≤ the natural vector order.

These Lemmas establish that any policy proposal is e�cient and that tari�s are weaklylower than those in the noncooperative equilibrium. How i chooses to resolve the risk-return tradeo� depends on its military power. Relatively powerful governments canimplement their ideal point with high probability through war. They run little risk thatthe foreign government would reject an o�er close to their ideal point. Conversely, weakgovernments are likely to lose a war over market access, and therefore must concedemore to their counterpart. Military power therefore a�ects trade policy. Because i’s idealpoint features more protection of its own market than j’s ideal point, as i becomes morepowerful, it proposes higher levels of protection for itself. If this o�er is accepted andpeace prevails, powerful countries will be more closed to international trade.

Proposition 6 (Power and Protection): If aj < aj(¯cj, ai) and peace prevails, govern-

ment i’s trade barriers are increasing in its military strength, i.e. τ ?i (ai, ci, ρ) is increasingin ρ.

DiscussionJointly, Propositions 4 and 5 establish that the most liberal governments never �ght andalso trade more than illiberal governments. Contra the commercial peace framework, tradeis endogenously determined by governments trade policy choices. These endogenous tradepolicies determine trade �ows and generate con�icts between governments. Economicintegration is not exogenously given. Rather, participation in the international economy isa choice. Even in today’s globalized era, such policy frictions persist (Anderson and VanWincoop 2004; Cooley 2019).

These policy choices are the object of contention between governments in the model.Protectionist barriers cause con�icts of interest between market access-motivated govern-ments. McDonald (2004) shows that measures of protection are better predictors of con�ictthan trade.18 Trade can persist in the presence of trade barriers. For example, World War Ibroke out during an era of rapid globalization. McDonald and Sweeney (2007) show that thegreat powers maintained high protective tari�s during this era, which provided a rationalefor con�ict over market access conditions, despite high trade volumes. Chatagnier andKavakli (2015) examine governments whose �rms compete in the same export markets.They show these governments are more likely to experience international con�icts.

Firms are the source of belligerent foreign policies in the theory. In Imperial Germany,“iron and rye” advocated for protectionism and expansionist foreign policies (Gerschenkron

18His analysis covers the years 1960-2000.

17

1943). Similar domestic political coalitions emerged in the United States. Fordham (2019)traces the development of the United States as a naval power in the 19th century. He �ndsprotectionist interests were the strongest advocates for the nascent U.S. �eet. At the time,U.S. trade policy was protectionist. Washington also sought preferential market accessin developing countries, particularly in Latin America. The �eet served to protect theseobjectives against military interference from Europe. Commercial objectives also motivatedWashington at the dawn of the Cold War (Fordham 1998). Under Soviet in�uence, EasternEurope became closed to trade with the United States. Congressmen representing export-oriented districts tended to support an aggressive military posture toward the Soviet Union.The goal of export-oriented �rms, argues Fordham, was to secure market access in Europeand Japan. In the post-Cold War era, Congressmen representing import-exposed districtshave tended to support hostile foreign policies toward China, whose exports (plausibly)harm their constituent �rms (Kleinberg and Fordham 2013).

Domestic political institutions connect these underlying economic interests to govern-ment preferences. I treat these institutions in reduced form, focusing on variation inconsumers’ ability to in�uence policy. Consumers pacify foreign policy preferences. Ifdemocratic political institutions privilege the interests of consumers, then Propositions 4and 5 support a commercial-democratic peace. Observational analyses uncover positivecorrelations between democracy, trade, and peace because of the trade policy preferencesof democracies.19 Liberal preferences increase trade and reduce con�ict.

Observed trade policies are not a su�cient statistic for government preferences, how-ever. Proposition 6 states that relative military power e�ects trade policy in peacetime.Governments’ trade policies re�ect their preferences only up to a war constraint. Liberalpolicies do not imply liberal preferences. Several studies have employed Grossman andHelpman (1994) and Grossman and Helpman (1995) to structurally estimate governments’welfare-consciousness (Goldberg and Maggi 1999; Mitra, Thomakos, and Ulubasoglu 2006;Gawande, Krishna, and Olarreaga 2009, 2012, 2015; Ossa 2014). International strategicconsiderations are e�ectively absent from these models. The domestic political econ-omy determines outcomes. Therefore, a simple inversion on the policy function recoverspreferences. The model developed here highlights the importance of the war constraint.Militarily weak, illiberal governments adopt the same policies as liberal governments.

Territory plays a central role in theories and empirical studies of interstate con�ict.20

Wars often redraw international borders. In doing so, they also relocate customs barriersand modify the trade policies of captured regions. Gunboat diplomacy and territorialcon�ict are plausibly substitutes for one another. Governments can acquire foreign marketaccess through territorial annexation or regime change. Territory and trade policy are not

19See Oneal and Russett (1999) for a representative study.20The “pie” at stake in bargaining models of war is often motivated as the distribution of territory between

the countries. Empirical studies of territorial con�ict often conceptualize territorial control as a consumptiongood, rather than a means to implement policy. See, for example, Caselli, Morelli, and Rohner (2015). For areview of this literature, see Schultz (2015).

18

exclusive realms of international con�ict.

ConclusionThe model envisions a stylized world. Two governments preside over identical economiesexogenous military capacities and political bias types. If the countries were heterogeneousin market size (Li), the economically larger country would possess an additional source ofbargaining power. Whether this outside option a�ects bargaining outcomes would dependon the distribution of economic and military strength. This analysis might shed light thesubstitutability of economic and military coercion (Hirschman 1945).

Military power (ρ) is also treated as exogenous here. If military investment was possible, ρwould depend on economic and political primitives.21 Because they have more at stake inbargaining, illiberal governments might invest more in their militaries. This result wouldhold especially if the costs of militarization are borne by consumers (Jackson and Morelli2007; Chapman, McDonald, and Moser 2015). This variant might explain why democraciesspend less on their militaries (Fordham and Walker 2005).

If the game developed here was dynamic, regime type itself would be an endogenous object.Wars impose puppet governments more liberal than those that preceded them. The worldwould democratize over time, as conquering powers installed liberal governments abroad.(McDonald 2015).

While imperial powers sometimes seek to democratize defeated countries, they ofteninstead install allied strongmen or colonial administrations. If the liberal democracyhypothesis holds, these actions are puzzling. Controlling these governments presentsagency problems absent in relations with liberal regimes. Liberal regimes adopt liberaltrade policies of their own accord. Conquerors must incentivize their agents to adopt thesepolicies.

A multi-country variant of this framework might provide a rationale for this behavior.Consider a world populated with three countries — A, B, and C. For �rms in A, an idealpolicy for B is one that is open to trade with A but closed to trade from C. This allowsA’s �rms to maximize their share of B’s market.22 In other words, �rms seek preferentialaccess to foreign markets. It is immediate that a low bias government would not providesuch preferential access. Coercing governments might be willing to su�er the agency costsof obtaining preferential access to B if B’s market was valuable enough. This frameworkmight be pro�tably applied to the study of imperialism (Gallagher and Robinson 1953).

Finally, the theory highlights an underappreciated prerequisite for international con�ict.For governments to war with one another, they must both 1) possess con�icts of interestlarge enough to justify the costs of con�ict and 2) be unable to resolve these con�icts

21See, for example, Jackson and Morelli (2009a).22This assumes, of course, that B cannot tax its own �rms.

19

peacefully (Jackson and Morelli 2009b). Theoretical research on international con�ict hasfocused on the latter. In abstracting away from the exact nature of the dispute at hand,these models direct our attention away from the question of why international disputesemerge in the �rst place. What do governments want, and why do their objectives bringthem into con�ict with one another (Moravcsik 1997; Coe, n.d.)? While war is rare ininternational politics, antagonistic and militarized political relations are common. Focusingattention on the con�icts of interest underlying these antagonisms might help explaintheir emergence and termination.

20

Appendix

A: Trade, War, and Democracy: Empirical Facts

−10 −5 0 5 10Years After Beginning of War

Dya

dic

Eco

nom

ic In

tegr

atio

nTrade and War

Figure 5: Plot depicts trade relations between dyads that experienced wars, 10 years prior to and 10years following the outbreak of hostilities. Economic integration is measured as the average of thecountries’ directed imports to gdp ratio. An inverse hyperbolic sine transformation was applied tonormalize this measure. Data from Barbieri, Keshk, and Pollins (2008), Barbieri, Keshk, and Pollins(2009), Sarkees and Wayman (2010), Bolt et al. (2018).

21

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Austria−Hungary − Russia, 19142.9 million battle deaths

Armenia − Azerbaijan, 199314.0 thousand battle deaths

France − Germany, 2007

●

Notes: Each point is one dyad−year. Red−orange points are dyad−years in which a warbegan, sized by the number of battle deaths each side incurred during that war. Light bluepoints are a sample of 10,000 dyad−years in which no wars occured. Economic integration ismeasured as the average of the countries' directed imports to gdp ratio. An inversehyperbolic sine transformation was applied to normalize the economic integration measure.Economic Integration score lagged by one year. Margin plots show the density of thetransformed economic integration measure and the average POLITY score. A trend lineshows the correlation between economic integration and average POLITY score in thesample of peaceful dyads. The distribution of average POLITY scores for war dyads forwhich trade or gdp data were not available is plotted in a histogram below the scatterplot.

●

−10 −5 0 5 10

−10 −5 0 5 10Dyad−Year Average POLITY Score

Dya

d−Ye

ar E

cono

mic

Inte

grat

ion

War

s w

ithM

issi

ng T

rade

The Commercial−Democratic Peace, 1870−2014

Figure 6: Data from Marshall, Jaggers, and Gurr (2002), Barbieri, Keshk, and Pollins (2008), Barbieri,Keshk, and Pollins (2009), Sarkees and Wayman (2010), Bolt et al. (2018).

22

B: International EconomyDemand for Manufactured Goods: Total expenditure on manufactured goods is αIi =PiXi. Cobb Douglas preferences ensure that consumers will spend an α-share of theirincome on manufactured goods. We can derive Equation 4 by solving Equation 2 subjectto the constraint ∫

νi

pii(νi)xii(νi)dνi +

∫νj

pij(νj)xij(νj) ≤ αwiL. (14)

For any two domestic varieties, νi and ν ′i, we must have

x?ii(νi)pii(νi)σpii(ν

′i)

1−σ = pii(ν′i)x

?ii(ν

′i)

x?ii(νi)pii(νi)σ

∫ν′i

pii(ν′i)

1−σdν ′i =

∫ν′i

pii(ν′i)x

?ii(ν

′i)dν

′i.

The same must hold for foreign varieties:

x?ij(νj)pij(νj)σ

∫ν′j

pij(ν′j)

1−σdν ′j =

∫ν′j

pij(ν′j)x

?ij(ν

′j)dν

′j.

Summing these conditions and noting x?ij(νj)pij(νj)σ = x?ii(νi)pii(νi)σ at equilibrium

consumption gives

x?ii(νi)pii(νi)σ

(∫ν′i

pii(ν′i)

1−σdν ′i +

∫ν′j

pij(ν′j)

1−σdν ′j

)=

∫ν′i

pii(ν′i)x

?ii(ν

′i)dν

′i +

∫ν′j

pij(ν′j)x

?ij(ν

′j)dν

′j

x?ii(νi)pii(νi)σPi(τi)

1−σ = αIi(τi)

x?ii(νi) = pii(νi)−σPi(τi)

σ−1αIi(τi).

Indirect Utility: Indirect utility is Xαi Y

αi evaluated at equilibrium consumption. Substi-

tuting our demand equations 4 into Equation 2 gives

X?i =

(∫νi

x?ii(νi)σ−1σ dνi +

∫νj

x?ij(νj)σ−1σ dνj

) σσ−1

= Pi(τi)σ−1Ii(τi)

(∫νi

pii(νi)1−σ + pij(νj)

1−σ) σ

σ−1

= αIi(τi)

Pi(τi).

Because they serve as numeraire, equilibrium consumption of agricultural goods is equiv-alent to expenditure: Y ?

i = (1− α)Ii(τi). Substituting these into the consumer’s utilityfunction yields Equation 5.

23

Prices: The �rms’ �rst order condition is∂Π (p?i (νi))

∂p?i (νi)= (p?i (νi)− wi)

(∂x?ii(νi)

∂p?i (νi)+∂xji(νi)

∂p?i (νi)

)+ x?ii(νi) + x?ji(νi) = 0

where∂xii(νi)

∂pi(νi)= −σp(

iνi)−σ−1P σ−1

i αIi

and∂xji(νi)

∂pi(νi)= −στ−σj p

(iνi)

−σ−1P σ−1j αIi.

Note that− σ

p?i (νi)

(x?ii(νi) + x?ji(νi)

)=∂xii(νi)

∂p?i (νi)+∂xji(νi)

∂p?i (νi).

The �rst order condition then becomes

σwi

p?i (νi)

(x?ii(νi) + x?ji(νi)

)− σ

(x?ii(νi) + x?ji(νi)

)+(x?ii(νi) + x?ji(νi)

)= 0

σwi

p?i (νi)− σ + 1 = 0

σ

σ − 1wi = p?i (νi).

Economic Equilibrium:

With unit costs of production in manufacturing and agriculture, goods market clearingrequires

Lyi + Lyj = (1− α) (Ii(τi) + Ij(τj))

Lxi =(x?ii(τi) + x?ji(τj)

)Lxj =

(x?ij(τi) + x?jj(τj)

)Lxi + Lxj + Πi(p

?i ) + Πj(p

?j) = α (Ii(τi) + Ij(τj)) .

Domestic factor market clearing requires

Lxi + Lyi = L

Lxj + Lyj = L.

C: Constant De�nitionsAssumption 1

I restrict j’s costs of war such that there exists some government j with aj ∈ (¯a, a] that

would be willing to �ght if victory were certain and i o�ered its ideal point. Formally, thisrequires

κj = Γj(¯a, ai).

24

I then restrict i’s costs of war to ensure that it never o�ers j’s ideal point – an interiorsolution exists outside of the zone of peace. Formally, this requires

κi(cj, ai) = minτ∈P−cj (1− F (Wj(

¯a, ai)−Gj(·, τi; aj)))

∂Gi(τi,·;ai)∂τi

∂Gj(·,τi;ai)∂τi

.

Because ∂Gj(·,τi;ai)∂τi

< 0, this quantity is guaranteed to be positive.

Assumption 3

Let

¯a = (σ − 1)k(α)−1(p?)α(1− (p?)−1)

wherek(α) = αα(1− α)1−α.

This quantity is derived by letting

¯a =

{ai

∣∣∣∣ limτi→∞

aiV′i (τi) + Π′i(τi) = 0

}.

Note that it is positive.

D: ProofsRestatements of results from the main text proceed all proofs. Lemmas 2, and 3 exploit thefollowing de�nitions:

A(τi) = p?x?ij(τi)(αI(τi))−1 =

(1 + τ 1−σ

i

)−1τ−σi

B(τi) = (σ − 1)A(τi)− στ−1i .

Proposition 1: If Assumption 2 is satis�ed, then a unique economic equilibrium existswith Lxi , L

yi , L

xj , L

yj > 0 and wi = wj = 1 for all τ ∈ [1, τ ]2.

Proof: A competitive agricultural sector guarantees that agricultural producers make zeropro�ts. This zero pro�t condition implies

(1− wi)Yi = 0

which implies wi = 1 whenever the agricultural sector is active, Yi > 0. From Equation 7,this implies p?i = p?j = p? = σ

σ−1. Suppose for now that the agricultural sector is active

in both countries, implying wages are equalized across countries and sectors. Below, weverify that this is the case if Assumption 2 is satis�ed.

25

Labor allocations to each sector depend on tari� levels. The labor allocation in country i tosector k ∈ {x, y} can then be writtenLki (τ ). The total labor allocation to the manufacturingsector in country i is

Lxi (τ ) = x?ii(τi) + x?ji(τj).

Because x?ii(τi) is increasing in τi and x?ji(τj) is decreasing in τj (see Lemma 2.), Lxi (τ ) ismonotone increasing in τi and monotone decreasing in τj . This implies Lxi (τ ) attains itsmaximum at (τ, 1):23

Lxi (τ, 1) = p−σPi(τ)σ−1αL+ (1p)−σPj(1)σ−1αL

=p−σαL

p1−σ +p−σαL

2p1−σ

=αL

p+

1

2

αL

p

=3

2

σ − 1

σαL.

Allocation to the agricultural sector is then, by the labor market clearing condition,

Lyi (τ, 1) = L− Lxi (τ, 1).

If α < 23

σσ−1

, then Lyi (τ, 1) > 0. Because total labor allocation to the manufacturing sectorachieves its maximum at (τ, 1), Lyi (τ ) > 0 for all τ ∈ [1, τ ]2. Moreover, Lxi (τ ) > 0 for allτ ∈ [1, τ ]2.24 This demonstrates the proposition. �

Lemma 1: τ ?i (ai) ∈ (1, τ) .

Proof: The �rst order condition is

0 = −αaiVi(τi)A(τi) + air′i(τi)Pi(τi)

−α + Π′i(τi). (15)

By construction,limτi→∞

∂Gi

∂τi< 0

for all ai >¯a. Additionally,

∂Gi

∂τi

∣∣∣τi=1

=∂Π(τi)

∂τi

∣∣∣τi=1

> 0.

This precludes τi ∈ {1, τ} from being optimal. From each corner point, the directionalderivative toward the interior of the policy space is positive. �

Lemma 2: Gi(τj) is strictly decreasing in τj .23Here we note the dependence of the price index on the home tari� Pi(τi).24This follows from the fact that Lx

i (1, τ) > 0 and the monotonicities established above.

26

Proof: It is su�cient to show that

∂Gi(τj)

∂τj< 0.

Note that∂Gi(τj)

∂τj=∂Πi(τi, τj)

∂τj= (p? − 1)x?′ji(τj).

The derivative of home exports with respect to the foreign tari� is

x?′ji(τj) = B(τj)x?ji(τj) +

x?ji(τj)

I(τj)r′j(τj)

= B(τj)x?ji(τj) +

x?ji(τj)

I(τj)

((τj − 1)p?x?′ji(τj) + p?x?ji(τj)

)= B(τj)x

?ji(τj) +

x?ji(τj)

I(τj)

(rj(τj)

x?ji(τj)x?′ji(τj) + p?x?ji(τj)

)= B(τj)x

?ji(τj) + λ(τj)x

?′ji(τj) + p?x?ji(τj) +

x?ji(τj)

I(τj)

(1− λ(τj))x?′ji(τj) = x?ji(τj)

(B(τj) + p?x?ji(τj)I(τj)

−1)

= x?ji(τj) (B(τj) + αA(τj))

< x?ji(τj) (B(τj) + A(τj))

= στ−1((

1 + τ 1−σj

)−1τ−σ−1j − 1

)︸ ︷︷ ︸

<0

.

�

Lemma 3: τ ?i (ai) is strictly decreasing in ai.

Proof: Government i’s optimal policy does not depend on the policy choice of j. As such,it is su�cient to show that the government’s objective function has a negative cross partialwith respect to τi, ai, or

∂2Gi

∂ai∂τi< 0.

27

This can be written

∂2Gi

∂ai∂τij= V ′i (τi)

= r′(τi)Pi(τi)α − αPi(τi)αI(τi)A(τi)

= αPi(τi)αI(τi)

((αI(τi))

−1r′(τi)− A(τi))

= αPi(τi)αI(τi)

((αI(τi))

−1((τi − 1)p?x?′ij(τi) + p?x?ij(τi)

)− A(τi)

)= Pi(τi)

αI(τi)((αI(τi))

−1(τi − 1)p?x?′ij(τi) + A(τ)i − A(τi))

= Pi(τi)αα−1(τi − 1)p? x?′ij(τi)︸ ︷︷ ︸

<0

where the �nal inequality follows from Lemma 2. �

Proposition 2: a? = a .

Proof: Follows immediately from Lemma 2 and Lemma 3. �

Proposition 3: Γi(ai, aj) is decreasing in ai and aj .

Proof: To establish that Γi(ai, aj) is decreasing in ai, note that derivative of Γi(ai, aj)taken with respect to ai is

∂Γi(ai, aj)

∂ai=∂Gi(τ

?i (ai), τ

?i (a); ai)

∂ai

∣∣∣∣(τ?i (ai),τ?j (a))

+∂Gi(τ

?i (ai), τ

?j (a); ai)

∂τ ?i (ai)︸ ︷︷ ︸=0

∂τ ?i (ai)

∂ai−

∂Gi(τ?i (a), τ ?j (aj); ai)

∂ai

∣∣∣∣(τ?i (a),τ?j (aj))

=Vi(τ?i (ai))− Vi(τ ?i (a))︸ ︷︷ ︸

<0

where the �nal inequality holds because τ ?i (ai) > τ ?i (a) for all ai < a. To see that Γi(ai, aj)is decreasing in aj , note

∂Γi(ai, aj)

∂aj= − ∂Gi(τ

?i (a), τ ?j (aj); ai)

∂τ ?j (aj)︸ ︷︷ ︸<0

∂τ ?j (aj)

∂aj︸ ︷︷ ︸<0

where the inequalities follow from Lemmas 2 and 3. �

Lemma 5: IfWj(aj, ai)−Gj(τ

?j (a), τ ?i (ai); aj) = Γj(aj, ai) ≤

¯cj

thenτ ? = (τ ?i (ai), τ

?i (a))

28

andω?(τ ?i (ai), τ

?j (a); aj, cj, ρ) = accept

for all cj ∈ [¯cj, cj].

Proof: By Lemma 4, τ = (τ ?i (ai), 1) will be accepted for all cost types cj ∈ [¯cj, cj]. Since

this o�er maximizes i’s utility conditional on peace,

τ ? =(τ ?i (ai), τ

?j (a)

).

�

Lemma 6: For every¯cj ∈ [0, cj) there exists a aj(

¯cj, ai) such that for all aj ∈ [aj(

¯cj, ai), a)

the probability of war is 0.

Proof: Government j will accept i’s ideal point so long as

Wj(aj, ai)−Gj(τ?j (a), τ ?i (ai); aj) ≤

¯cj.

Note that this condition can be rewritten as

Γj(aj, ai) ≤¯cj.

If Γj(aj, ai) ≤¯cj for all aj ∈ [0, a), set aj(

¯cj, ai) = 0. Otherwise, let aj(

¯cj, ai) solve

Γj(aj(¯cj, ai), ai) =

¯cj.

Recall from Proposition 3 that Γj(aj, ai) is decreasing in both arguments. By Assumption1,

¯cj < cj ≤ κj = Γj(

¯a, ai).

Since Γj(aj, ai) is continuous and decreasing in aj , a solution exists for¯cj large enough.

Then, by construction,Γj(aj, ai) ≤

¯cj

for all aj ≥ aj(¯cj, ai) and j accepts i’s ideal point. By Lemma 5, this guarantees peace. �

Proposition 4: aj(¯cj, ai) is weakly decreasing in ai.

Proof: We have two cases, either aj(¯c, ai) =

¯a or

aj(¯c, ai) = Γ−1

j (¯c; ai).

Since Γj(aj, ai) is decreasing in ai (Proposition 3), so is its inverse. This is su�cient toguarantee that aj(

¯cj, ai) is decreasing in ai. �

Lemma 7: τ ?(ai, ci, ρ) ∈ P .

29

Proof: Suppose τ ?(ai) /∈ P . It is straightforward to show that this produces a contradic-tion, namely

τ ?(ai, ci, ρ) /∈ arg maxτ∈[1,τ2]

Gi (τ , ω?(τ ; aj, cj, ρ); ai, ci, ρ) .

First, if τ ? /∈ P , then there exists τ ′ ∈ [1, τ ]2 such that either 1) Gi(τ′) > Gi(τ

?) andGj(τ

′) ≥ Gj(τ?) or 2) Gj(τ

′) > Gj(τ?) and Gi(τ

′) ≥ Gi(τ?).

Take the �rst case and recall

Gi (τ , ω?(τ ; aj, cj, ρ); ai, ci, ρ) = (1− F (Wj(aj, ai)−Gj(τ ; aj))) (Gi(τ ; ai)) +

F (Wj(aj, ai)−Gj(τ ; aj))(Gi(ai, aj)

).

In the proof of Proposition 6 (below), I show Gi(τ?) > Wi(ai, aj) ≥ Gi(ai, aj).

Also, F (τ ?) ≤ F (τ ′). Then, if 1) or 2), then Gi (τ′, ω?(τ ; aj, cj, ρ)|ai, ci, ρ) >

Gi (τ?, ω?(τ ; aj, cj, ρ); ai, ci, ρ), producing the desired contradiction. �

Lemma 8: τ =(τ ?i , τ

?j

)≤(τ ?i (ai, ci, ρ), τ ?j (ai, ci, ρ)

)with ≤ the natural vector order.

Proof: Suppose, for sake of contradiction, that for some τ ?, τ ?i > τ ?i (ai). By Lemma 7, τ ?must lie in the pareto set. By the de�nition of τ ?i (ai), Gi(τ

?i , ·; ai) < Gi(τ

?i (ai), ·; ai). By

Lemma 2, Gj(·, τ ?i ; aj) < Gj(·, τ ?i (ai); ai). Thus, a pareto improvement exists, contradict-ing the hypothesis that τ ? is an equilibrium o�er. �

Proposition 6: If aj < aj(¯cj, ai) and peace prevails, government i’s trade barriers are

increasing in its military strength, i.e. τ ?i (ai, ci, ρ) is increasing in ρ.

Proof: By Assumption 1, i’s �rst order condition must characterize τ ?i when aj < aj(¯cj, ai).

Here, we have

∂Gi(τi)

∂τi= (1− F (Wj(aj, ai)−Gj(·, τi|aj)))

∂Gi(τi)

∂τi+

1

cj −¯cj

∂Gj(τi)

∂τi

(Gi(τi)− Gi(ai, aj)

)= 0.

Rearranging,

(1− F (Wj(aj, ai)−Gj(·, τi; aj)))∂Gi(τi)

∂τi=

1

cj −¯cj

∂Gj(τi)

∂τi

(Gi(ai, aj)−Gi(τi)

)(1− F (Wj(aj, ai)−Gj(·, τi; aj)))

∂Gi(τi)

∂τi=

1

cj −¯cj

∂Gj(τi)

∂τi(Wi(ai, aj)− ci −Gi(τi))

(1− F (Wj(aj, ai)−Gj(·, τi; aj)))∂Gi(τi)

∂τi=

1

cj −¯cj

∂Gj(τi)

∂τi(Wi(ai, aj)− ci −Gi(τi))

(cj −¯cj) (1− F (Wj(0, ai)−Gj(·, τi; aj)))

∂Gi(τi,·;ai)∂τi

∂Gj(·,τi;ai)∂τi

+ ci = Wi(ai, aj)−Gi(τi).

30

By Assumption 1, the LHS of this expression must be negative, this ensures that i’s payo�at the solution is higher than its war value,25

Wi(ai, aj) < Gi(τi).

Now note that∂Wj

∂ρ= Gj − Gj < 0

and∂Gi

∂ρ= Γi > 0.

We have

∂2Gi(τi)

∂τi∂ρ= − 1

cj −¯cj

∂Gi(τi)

∂τi︸ ︷︷ ︸>0

∂Wj

∂ρ︸︷︷︸<0

− 1

cj −¯cj

∂Gj(τi)

∂τi︸ ︷︷ ︸<0

∂Gi

∂ρ︸︷︷︸>0

> 0

which implies∂τ ?i (ai)

∂ρ> 0

as desired. �

Proposition 5: If aj ≥ aj(¯cj, ai) then trade in manufactured goods is increasing in ai.

Proof: If aj ≥ aj(¯cj, ai) then τ ? =

(τ ?i (ai), τ

?j (a)

)by Lemma 5. Equilibrium trade in

manufactured goods is then

x?ij(τ?i (ai)) + x?ji(τ

?j (a)).

By Lemma 3, τ ?i (ai) is decreasing in ai and x?ij(τi) is decreasing in τi. Then, within thezone of peace, equilibrium trade in manufactured goods is increasing in ai. �

25Note also that by applying the de�nition of Wi, wwe see this condition implies Gi(τ ) is concave aboutthe pareto set.

31

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