Growth, Import Dependence and War

Existing theories of pre-emptive war typically predict that the leading country may choose to launch a war on a follower who is catching up, since the follower cannot credibly commit to not use their increased power in the future. But it was Japan who launched a war against the West in 1941, not the West that pre-emptively attacked Japan. Similarly, many have argued that trade makes war less likely, yet World War I erupted at a time of unprecedented globalization. This paper develops a theoretical model of the relationship between trade and war which can help to explain both these observations. Dependence on strategic imports can lead follower nations to launch pre-emptive wars when they are potentially subject to blockade.


Introduction
This paper develops a model of trade and war that speaks to two distinct literatures. The first is the literature on whether or not trade helps reduce the likelihood of warfare. The argument that it does so sits uneasily with the observation that World War I erupted at a time of unprecedented globalization. The second is the literature on war between established and rising powers. A typical prediction is that the established power (or leader) may launch a pre-emptive war against the rising power (or follower), since the latter cannot credibly commit to not use their increased power in the future. And yet it was Japan who attacked the West in 1941, not vice versa.
Our model can help to resolve both apparent paradoxes. We show that import dependence can lead a follower country to lauch pre-emptive wars against the leader if two conditions hold.
First, import dependence must increase over time. Second, the country must be vulnerable to blockade in the event of war. The model can be regarded as a formalization of arguments about trade and war made by some realist scholars in the international relations literature.
Ours is a model of hegemonic war, and hegemonic wars are too infrequent for our arguments to be testable econometrically. We therefore provide a brief historical narrative in which we show how our model can help to make sense of three historical episodes: Anglo-German rivalry prior to World War I; Hitler's expansionist ambitions, and his decision to attack the Soviet Union in 1941; and Japan's decision to attack the West later in the same year. Our model formalizes some of the arguments made about these three episodes by prominent historians: Avner O↵er's book on Anglo-German rivalry (O↵er 1989), Adam Tooze's book on the Nazi German war economy (Tooze 2006), and Michael Barnhart's book on Japan's "preparation for total war" (Barnhart 1987).
We are sure that none of these historians would argue that the mechanism that we describe here "explains"any of these three conflicts in some monocausal way. Lest there be any misunderstanding on the subject, we do not make such a claim either: the origins of the first and second world wars were much too complicated to be "explained"by this or any other formal model. Our model has just two players, but there were many players involved in these conflicts (and so a country like Germany could be a follower relative to the UK, but a leader relative to Russia).
It assumes that conflict is motivated by just one cause (a "pie"which both players are struggling to obtain), but international rivalries in the 1910s and 1930s were multi-dimensional. It assumes that countries can be modelled as unitary actors, but internal divisions were important in Wilhelmine Germany, Imperial Japan and elsewhere. And it assumes rationality, even though many important actors in these three episodes were motivated by sentiments such as honour and dignity, or by racial or religious prejudice, or were over-optimistic about their chances in a war, or under-estimated their opponents.
Nevertheless, we hope to convince the reader that the mechanism described by our model was one factor among many at work during these three episodes, and that trade dependence can sometimes make war more rather than less likely. We should not expect an economic model to be able to explain on its own something as complicated as the outbreak of a world war, but this does not mean that it has nothing to tell us about the past, or that it cannot provide us with lessons that may be useful in the future.

Trade and war
The optimistic, liberal argument that international trade promotes peace is ancient but controversial (see e.g. Barbieri 1996, Rowe 2005 . One objection is that trade can make countries dependent on others, and therefore vulnerable, in the context of an anarchic world in which countries have fundamentally di↵erent interests. In the words of John Mearsheimer, "states will struggle to escape the vulnerability that interdependence creates, in order to bolster their national security. States that depend on others for critical economic supplies will fear cuto↵ or blackmail in time of crisis or war; they may try to extend political control to the source of supply, giving rise to conflict with the source or with its other customers" (Mearsheimer 1990, p. 45).
There is a critical di↵erence between international and domestic trade, argues Kenneth Waltz: regions within a country "are free to specialize because they have no reason to fear the increased interdependence that goes with specialization", whereas in an anarchic world, states may fear specialization on the grounds that their potential competitors may gain more than they do, or because trade makes them "dependent on others through cooperative endeavors and exchanges of goods and services" (Waltz 2006, pp. 104, 106; see also Gilpin 1981, p. 220).
There is also a large literature on hegemonic wars between rising challengers and dominant powers (Gilpin 1981). Our paper develops a model of trade and hegemonic warfare, in the tradition of recent papers on "rationalist explanations for war" (Fearon 1995, Powell 2006).
These start from the premise that wars are costly, and that rational unitary states in dispute with each other should be able to bargain their way to compromises that leave both better o↵ (in probabilistic terms) than they would be in the event that war breaks out. Powell (2006) argues that wars can nevertheless arise as a result of commitment problems. He does so in the context of models in which a pie has to be divided between countries in a setting where (1) countries cannot pre-commit to particular divisions of the pie in the future; (2) countries have the option to launch a war to "lock in" an expected share of future flows; (3) wars are costly, in that they reduce the overall size of the pie; and (4) the distribution of power, which a↵ects how much of the pie countries can lock in, exogenously changes over time (p. 181). For example, consider the case in which a follower exogenously catches up on a leader (Fearon 1995). The follower has an incentive to forestall a pre-emptive war by the leader, by promising the leader a su ciently big slice of the pie in the future. Since it cannot pre-commit to this, and has an incentive to use its greater power in the future to secure a greater share of the pie, the leader may chose to launch a pre-emptive war in order to lock in a higher share of the spoils while it still has the chance.
In our model, we find that it is the follower who may declare war on the leader. International trade, and the opportunities and vulnerabilities which it implies, are central to establishing this otherwise counter-intuitive result. Central to our analysis is the assumption that the follower needs to import raw materials from the rest of the world.
We model the link between growth and changes in the distribution of power in a context in which the follower becomes increasingly dependent on imported raw materials. We assume that the leader, as befits the hegemon, can control the follower's access to imported raw materials, either because it controls the sources of supply (via formal or informal empire), or because it controls world shipping lanes and can mount a blockade of the follower. We show that if dependence on imported raw materials increases over time, the follower can become militarily weaker and not stronger, even if it is growing more rapidly, and can therefore have an incentive to start a pre-emptive war. International trade can thus be crucial in determining the likelihood of war.
While we borrow our basic theoretical mechanism from the existing literature (Powell 2006), our application of these ideas is novel. Furthermore, in our setup both the leader and the follower also care about consumption, allowing us to endogenise the share of their GDP that countries wish to devote to their armed forces. The paper closest in spirit to ours is Copeland (1996), who constructs a similar argument in which pessimistic expectations of future trade levels can lead trade-dependent countries to declare war. Our contribution is di↵erent from his, in that we provide a formal theoretical analysis, which he does not. This means among other things that we can endogenously figure out where these trade expectations come from. We also tell a story in which the processes of catch-up and structural change, and the strategic nature of trade, play central roles. 1

Model description
We consider a world with two industrial countries, L and F (for "Leader" and "Follower"), and a third resource-rich country C. In each country, there is a large number of agents, allowing markets to be competitive. There is an infinite number of periods, indexed by t = 1, ..., 1.
Agents everywhere care about consumption of a final good, z. In addition, in L and F , agents also care about consumption of a "pie", p, which we may interpret as a range of contested issues 1 There is a growing literature on the relationship between trade and war. Glick and Taylor (2010) estimate the impact of war on trade flows, and find that it is large. Acemoglu et al. (2012) present a dynamic model of resource trade and war, focussing on how, in the presence of an inelastic demand for resources, progressive depletion may increase the value of a resource-rich region, thus increasing the incentives for a resource-scarce country to invade the country the region belongs to (and thus appropriate the resource). They study how di↵erent market structures in the natural resource industry -perfectly competitive, or monopolistically controlled by the government of the resource-rich country -may be associated with di↵erent probabilities of war. While the main focus of their paper is on wars between resource-rich and resource-scarce countries, ours is on wars between resource-scarce industrialized countries. Caselli et al. (2015) find that war between pairs of countries is more likely when at least one country has natural resources, and when these are located near borders. Finally, a series of papers by Stergios Skaperdas and co-authors (see Garfinkel et al. 2012 for a good overview) study the pattern and welfare implications of trade in a context in which two countries may fight over a contested region. The focus of these papers is di↵erent from our own: they present static models of the impact of trade (between the two countries and the rest of the world) on the incentives for the two countries to arm and go to war over the contested region.
Ours is a dynamic model of trade between the two countries and the rest of the world, where the dynamics of relative power and trade dependence determine the likelihood of war. that the two countries must settle. Preferences in L and F are described by period t utility where z J t and p J t denote, respectively, consumption of the final good and the pie by the representative agent in country J 2 {L, F }. The present discounted value of utility in J is where s=t s t p J s , and < 1 is the discount factor. Period t and present discounted value utility in C are similarly equal to z C t and Z C t . However, C does not make any strategic decisions in our model: it is L and F who compete for the pie, and whose decisions determine whether or not there will be a war. 2 In both L and F , social planners maximise eq. (1). The essential tradeo↵ they face is that resources can be allocated either to the production of the final consumption good, z, or to the production of an army. Armies are not valuable per se, but are useful in securing a greater share of the pie. The planners thus face a trade o↵ between the consumption of z and of p. In this paper, we develop a model of the strategic interaction of the two planners over an infinite number of discrete periods, as they attempt to maximise eq. (1). In each period, the planners first simultaneously set the size of their armies. Next, they decide how to share the pie (by going to war, or through peaceful negotiations). Finally, given the planners' arming and war decisions, production of the final good takes place, the pie is allocated, and consumption is realised.
We begin by describing how the final good and army are produced, as well as the two economies' endowments (Section 2.1); the way in which the pie is divided (2.2); the exact timing of actions (2.3); and the equilibrium concept used in the paper (2.4). The equilibrium of the game is then characterised in Sections 3 and 4. For brevity, we use "L" and "F " as a short-hand for "L's social planner" and "F 's social planner". We use the following notation. As above, a lower case latin letter, e.g. x J t , denotes the value ⇤ .
The industrial input is not produced but something with which economies are endowed.
Raw material supplies are also given by endowments. Endowments evolve over time, following an exogenous growth process described below. We choose z as the numéraire. All owners of endowments are small enough to be price takers.
We interpret country L as an industrial leader that, by the beginning of period 1, has completed its process of structural transformation, and whose economy grows at a constant, steadystate rate in all sectors. In contrast, F is a follower that is still undergoing structural transformation in period 1, and only reaches steady state in period 2. By "structural transformation" we mean that F is undergoing catch-up growth, and reallocating resources from the primary sector to the industrial sector. Its industrial inputs, then, initially grow faster than in steady state, while its raw materials sector (here modelled simply as an endowment of raw materials available in every period) grows more slowly (and possibly at a negative rate). Finally, we assume that, in all periods, F is scarce in raw materials. To capture all this, we assume the following endowments of the two inputs in L and F : where 1 is steady state growth, ⇢ 0 captures the availability of raw materials in L, 0 < ↵  ↵ and 0 capture structural transformation in F , and we assume ↵ to make sure that F is scarce in raw materials. 4 After period 2, all endowments grow at a constant rate . Note that we have normalised the initial size of L's economy to 1, while F 's economy can have any initial size (↵ unconstrained). As for C, we interpret it as a peripheral country that is abundant in raw materials. It is a large economy, relative to both L and F . It produces both the final consumption good and raw materials, and the world relative price of raw materials in terms of the consumption good, ⌘ 2 [0, 1), is determined there. There are no transportation costs, and L and F can exchange unlimited quantities of raw materials for the final consumption good, or vice versa, in C's markets at this fixed relative price.
All goods are tradable internationally, except for y which is non-tradable. This implies that, given its scarce domestic supply, F will import raw materials, and export the final good in return.
If ⇢ 2 [0, 1), L will have similar trade patterns, and both industrial countries will import raw materials from C. If ⇢ > 1, L is abundant in raw materials, and exports these in exchange for imports of z. F will import from either C or L (or both). Given that raw materials cost ⌘, perfect competition in production of the final good and the fact that z is the numéraire together imply that the equilibrium price of y will be 1 ⌘. 5 In L and F , the planner can divert resources from production of the final good to production of an army, a. While the army does not increase utility directly, it may do so indirectly by increasing the portion of the pie that a country is able to obtain. If y J t a and x J t a units of the two inputs are allocated to the production of a, then an army of size is produced. 6 We assume a more advanced country has a lower cost of producing an army, relative to the final good, perhaps because of a superior technology. For example, a more advanced country could have a technology that yields a more powerful army for given military expenditure. We denote military expenditure by m J t = c J t a J t .

Political environment
Our model follows closely the model of pre-emptive war in Powell (2006). In every period, there is a pie that the two countries must partition. The pie has size ⇡ 1 > 0 in period 1, and grows at a constant rate in all periods after that. 7 The partition of the pie can be done in two ways. On the one hand, in every period t in which there has been no previous war (thus, at least in period 1), the two countries may try to negotiate a peaceful partition of the pie involving J getting a share s J t . Alternatively, they may go to war. This is won by J with probability 6 The planner could either appropriate the inputs directly, and produce the army by itself, or impose a lump sum tax on final good consumption, and use it to purchase the army from the private sector. Both interpretations, as well as a mixture of the two, are consistent with our model. 7 The pie may represent a range of contested issues that L and F must settle. These could be non-economic issues, such as the division of territory that matters purely for matters of prestige, or issues that arise because of ideological concerns. Or they could be economic issues, such as the division of territories with an economic value. and by J with reciprocal probability 1 q J t . The war gives the winner the entire current and all future pies. However, war also costs a share  2 [0, 1) of the present discounted value of all pies. In summary, war implies that the present discounted value of consumption of the pie, P J t , will be equal to q J t ⇧ t (1 ), while peaceful partition implies that p J t will be given by s J t ⇡ t , with P J t+1 remaining to be determined in subsequent periods. Negotiations to reach a peaceful partition work as follows. First, L decides whether to enter negotiations, or to immediately start a war. In the former case, it o↵ers F a share s F t of the current pie (so that a share s L t = 1 s F t would remain for itself). Given this o↵er, F decides whether to accept, or to reject and start a war. If it accepts, the pie is peacefully partitioned, and the two countries move on to the next period. 8 Note that, while war allocates the entire future stream of pies, negotiatiors cannot commit to the sharing of future pies. This lack of commitment is the key friction in the model, which may lead to a welfare-reducing war occurring in equilibrium. 9 To see why, suppose that a country expects to become weaker over time. Then, it knows that, unless it secures the future pies by winning a war today, it will get little of them as a result of future negotiations or conflict. Since lack of commitment prevents today's negotiators from alleviating this country's concerns, it may start a war even if, in principle, there is an overall surplus that the two parties could share. It turns out that, in equilibrium, such an ine cient war can only occur in period 1, the period in which structural transformation leads to a shift in relative power between L and F .
As with the production of the final good, producing an army relies on raw materials, which may have to be imported. This potential dependence of the army on international trade makes it important to specify the e↵ect of war on the two countries' capacity to trade. In this paper, we consider two alternative cases. The first is a symmetric case in which war does not a↵ect the capacity of either country to trade. In this case, dependence on imported raw materials does not matter for relative military power. The second case is an asymmetric one, in which L 8 Note that this structure of negotiations allocates all of the bargaining power to L. We have assumed this extreme distribution of bargaining power just for simplicity: to relax this assumption would not qualitatively change our results. 9 As we will see, war can also be "welfare-increasing" in this model since it implies that no future military expenditures will be undertaken. Since our interest is in welfare-reducing war, we will rule this possibility out in Section 3.5 and subsequently. may blockade F in times of war, but not the other way around. This involves both F 's trade with L (if any), and F 's trade with C. We refer to this second case as "L having the capacity to blockade". It is easy to show that, if L has the capacity to blockade, it always uses it in times of war. Intuitively, the disruption of F 's trade does not carry a direct economic cost for L, since it can still trade with C. On the other hand, as clarified below, to blockade F can reduce the latter country's probability of winning the war. Thus, L having the capacity to blockade is synonymous with L blockading F in times of war. 10 We believe this second case is an important one, since hegemonic countries may develop a naval superiority that allows them to control trade routes in case of conflict. In this second case, F 's dependence on imported raw materials may have important consequences for relative military power. 11 We define a state variable B, which is one if and only if L has the capacity to blockade.
An indicator variable w J t is one if and only if country J starts a war in period t. Then, w t ⌘ w L t + w F t is one if and only if a war occurs in period t. We also define a state variable W J t , which is one if and only if J has won a war in some previous period T < t. Then, W t ⌘ W L t + W F t is a state variable indicating whether or not war has already occurred in period t.

Timing
Each period t can be divided into three sub-periods, during which the following events take place: t.1 L and F simultaneously set a L t and a F t . 10 We do not consider the possibility that L uses the capacity to blockade in times of peace. An obvious justification for this assumption is that a blockade could, in itself, be regarded as an act of war. 11 The capacity to blockade could be thought of as arising in two ways. It could arise in the context of a world in which C remains independent, but in which L gains control over the trade routes linking C to its industrial rival. In this interpretation, the key determinant of the capacity to blockade is the relative size of the countries' navies: L will have the largest navy, and will then have the ability to blockade F (but not vice-versa). The capacity to blockade could also arise in a world in which L gained colonial control over C. Colonial control would give L the power to deprive its rival of the ability to import raw materials, which is what a blockade means in the context of our model. We think that the first interpretation is more consistent with the structure of our model. Our assumption is that the capacity to blockade is indivisible, and is therefore a↵ected by war in a way that it cannot be by peaceful negotiations between the two countries. If L's capacity to blockade originated from the control of colonial empires, it would be quite hard to argue for its indivisibility, since colonial empires can be divided in many di↵erent ways. In contrast, negotiations over naval power are much more discontinuous in nature -a navy is either dominant, or it is not -and so it is possible that the expected impact of war on naval power cannot be obtained through peaceful negotiations. t. 2 If there has been a war at some T < t, the winner gets the entire period t pie. If there hasn't been a war, L can either make an o↵er s F t on how to share the period t pie, or start a war (w L t = 1). If it makes an o↵er, F may either accept, in which case the pie is peacefully partitioned, or reject the o↵er and start a war (w F t = 1).
t.3 Trade and production take place (if someone has started a war, i.e. w t = 1, and L has the capacity to blockade, i.e. B = 1, then F cannot trade). After production has taken place, if someone has started a war, it now occurs. Finally, consumption takes place.

Definition of equilibrium
We focus on Markov-perfect Subgame Perfect Nash Equilibra (SPNE). Then, in each period, all relevant information about the previous history is summarised by the state variable W t , which specifies whether or not war has already occurred. That is to say, given W t , equilibrium strategies must prescribe optimal actions for each possible action played in all previous sub-periods only.
In period t, in any node such that W t = 0, the structure of the game is as follows where the expressions at the end of each branch denote payo↵s, and Z J t (w t = 1) and Z J t (w t = 0) are the present discounted values of consumption with and without war in period t. If there is war, each player gets its expected share of the pie arising from war, q J t ⇧ t (1 ). If there is no war, each player gets its peacefully negotiated share of the pie this period, plus the present discounted value of its share of the pie in the following period (which will depend among other things on whether there is a war in the following or subsequent periods). Equilibrium strategies must prescribe actions The di↵erence between these two expressions reflects the fact that only L can o↵er a peaceful partition of the pie, s F t ; F has to take this as given. In any node such that W t = 1, equilibrium strategies must prescribe actions It is easy to anticipate that, in this second case, in which the allocation of all the pies has already been determined (it is given by the second expression on the right hand side of (6)), and the sole concern is to maximise Z J t , arming will always be set equal to zero in equilibrium. To simplify the identification of a unique SPNE, we focus on a subset of equilibria which we call "balanced growth path SPNEs". These are defined by Definition 1. A balanced growth path SPNE is a SPNE in which, in t 2 where war has not yet occurred, if war does not occur, then m J t+1 = m J t for J 2 {L, F }.
By focusing on balanced-growth path SPNEs, we impose the requirement that, from period 2 onwards and until there is a war (if ever), military expenditures grow at rate . This is a reasonable restriction given that, from period 2 onwards, all relevant parameters for the arming decisions are scaled up by a factor in every period. Our strategy is consistent with the standard approach in the growth literature, which is to focus on balanced growth paths only.

Preliminary results
In every period in which war has not yet occurred, the planners first allocate resources between producing the consumption good and the army (these decisions we will henceforth refer to as arming decisions). These decisions determine their bargaining power in negotiations, and whether or not they decide to go to war. Given arming decisions (taken in sub-period t.1) and given the occurrence or non-occurrence of war (decided in sub-period t.2), agents optimally trade, produce, and consume (in sub-period t.3). To solve the game, we need to derive optimal arming and war decisions in every period. This is complicated, since it depends on dynamic calculations about future behaviour in both countries. We therefore proceed in steps, as follows.
In Section 3.1, we begin by deriving equilibrium consumption, determined in sub-period t.3, given arming decisions and given the occurrence or non-occurrence of war (determined in the previous two sub-periods). Next, we turn to optimal war decisions, given arming decisions in the previous sub-period. Finally, we consider optimal arming decisions.
In order to decide whether to go to war or not, planners need to compare payo↵s with and without war. These payo↵s depend both on consumption and on the share of the pie. Payo↵s with war are relatively easy to find, since: the war determines who will get the pie in all subsequent periods; the war means that there will be no future wars, and that optimal arming will be zero in subsequent periods; and we have already determined optimal consumption given arming and war decisions in Section 3.1. We present these payo↵s, for given arming decisions in the previous sub-period, in Section 3.2. Payo↵s without war are harder to calculate, as they depend on future arming and war decisions, which are themselves part of the equilibrium to be determined.
We proceed in several steps: the remainder of Section 3 establishes some essential preliminary findings, before our main results are established in Section 4. 12 In a first step, in Section 3.3, we derive optimal arming in a period when there is war. This arming decision is important for two reasons. First, it allows us to completely specify payo↵s if there is a war, given the results in Section 3.2. Second, it turns out that arming decisions with peace from period 2 onwards are the same as if there were war. 13 Having derived this optimal arming decision, we can now track the evolution of relative power over time (Section 3.4). In a third step, in Section 3.5, we impose restrictions on the parameter space to rule out an uninteresting case in which war must always occur. This case arises because of the channel identified by Garfinkel and Skaperdas (2000): by going to war now, countries can save on future military expenditure. This channel implies that, if the cost of war  is low enough, the e↵ective cost of war may be negative (that is, war may be welfare increasing), in which case war must occur. In order to focus on our own channel, this possibility will be ruled out by assuming  , where is a threshold cost of war between zero and one. In a fourth step, in Section 3.6, we present a lemma (Lemma 3) showing that, in the remaining parameter space, if there is no war in period 1, war will never occur, but that (as previously mentioned) countries will continue to arm in every period as if there were war.
Lemma 3 gives us everything we need to calculate payo↵s without war in period 1, given period 1 arming decisions, while Sections 3.1-3.3 give us everything we need to calculate payo↵s with war in period 1, again taking that period's arming decisions as given. In Section 4, therefore, we can finally turn to optimal war decisions in period 1, by comparing the payo↵s with and without war (taking arming decisions in period 1 as given). Finally, we complete the derivation of the equilibrium by calculating optimal arming decisions in period 1. Because our goal is to show that war may occur, to simplify the analysis we focus on the case in which the e↵ective cost of war is close to zero ( is close to).

Equilibrium consumption given arming and war decisions
The payo↵s of both L and F depend in part on their consumption of the final good. It is therefore useful to begin by showing how equilibrium consumption in t.3 depends on war and arming decisions, taken in t.2 and t.1 respectively.
Suppose that, at time t, war has not yet occurred (W t = 0). If no war is started in t.2 (w t = 0), international trade is not disrupted. Given abundance of x in the world as a whole, endowments of y must be fully utilised in equilibrium. Then, in F , y F t x F t units of raw materials must be imported at a price of ⌘. Balanced trade requires that F export i F t units of the final good, where i F t , the net consumption cost of imports, is given by At least i F t of the final good must be produced, and arming decisions, a F t , must have taken this into account: it must be the case that c F t a F t  y F t i F t . 14 Given arming decisions, resource utilisation by the army is and resource utilisation in the production of the final good, It follows that consumption of the final good is Turning now to L, there are two cases. If y L t > x L t , then y L t x L t units of raw materials must be imported, at a cost ⌘ y L t x L t . If y L t  x L t , instead, L is abundant in raw materials, and will export x L t y L t units to either F or C. This raises L's consumption of the final good by It follows that the net consumption cost to L of its trade in raw materials is and that L's consumption of the final good is Suppose next that a war has been started in t.2 (w t = 1). If B = 0 (L does not have the capacity to blockade), the war does not lead to any trade disruption. Then, consumption by both countries is still as above. If B = 1, F 's imports are constrained to be zero. Given arming decisions, resource utilisation by the army is as above, while in the production of the final good we have Note that, if B = 1, the maximum amount of raw materials available to F 's army in times of war is x F t . But since the army is only actually used in times of war, arming decisions must have taken this constraint into account: it must be the case that c F Putting together the various cases considered so far, F 's consumption of the final good can be written as Turning now to L, even if there is a blockade it can still trade with C. It follows that the net consumption cost of L's trade in raw materials is still i L t , and L's consumption of z can still be written as in (8).
Finally, consider the case in which war has already occurred (W t = 1). Because war and blockades cannot occur anymore, endowments of y must be fully utilised in period t. Consumption of the final good can still be written as in eq. (8) and (10), with w t = 0. Thus, these two expressions denote equilibrium consumption in all cases.

Payo↵s with war
Suppose again that war has not yet occurred in period t. We now derive payo↵s if a country starts a war, taking arming decisions in the previous sub-period as given. To begin with, note that war may occur at most once. Then, if a war occurs in one period, we would expect that arming will be set to zero in all subsequent periods. This intuition is confirmed by Proof. Take any t > 1, and suppose W t = 1. Substituting (8) and (10) into (6), we see that, in a SPNE, arming decisions must satisfy Since a J t only enters the maximands negatively, the solution is clearly We are now ready to derive the payo↵s if a country starts a war. Let these be denoted by (8) and (10), together with Lemma 1, we can find Z J t (w t = 1). Then, As discussed above, war implies destruction (the size of ⇧ t is decreased by ) and possibly trade disruption (if B = 1, F 's consumption is decreased by g F t ). However, it also gives the winner control of the contested resource for the rest of time (⇧ t ), and creates a peaceful world in which no further consumption is sacrificed to wasteful arming (no c J s a J s is subtracted from payo↵s in any period s > t).
Note that eq. (11) and (12) give payo↵s from war, taking arming decisions in the previous sub-period as given. We still have to derive payo↵s from peace, taking arming decisions as given; and derive optimal arming decisions. It is however convenient to first derive optimal arming decisions in a period when there is war.

Optimal arming in a period when there is war
Consider any period t such that, in sub-period t.2, one country starts a war. If the occurrence of war was exogenously given, then arming decisions in t.1 would be extremely simple: anticipating the exogenous coming of war, countries would, in equilibrium, choose a L t and a F t that simultaneously maximise (11) and (12), subject to the relevant constraints. In reality, of course, the occurrence of war in t.2 is endogenous to arming decisions in t.1. However, we show in Online Appendix B that if, in a SPNE, war occurs in period t, then optimal arming levels must be precisely those that simultaneously maximise (11) and (12). Indeed, those arming levels turn out to be selected even in periods when there is peace. 17 In what follows we therefore derive those arming levels.
When selecting a L t and a F t that simultaneously maximise (11) and (12), countries face the two types of constraints discussed in Section 3.1. First, armies cannot use more than the available To simplify, we assume that the former constraint is not binding. 18 That allows us to consider the impact of the second constraint.
Suppose first that not even the second constraint is binding. We are then looking for a L t and a F t that simultaneously maximise (11) and (12), subject to no constraint. Set @V J t w t = 1|a L t , a F t /@a J t = 0 for J 2 {L, F }, then solve for a J t as a function of a J t . This yields the best response functions which are plotted in Figure 1 (drawn for the case c L t < c F t ). Solving them together yields the unconstrained optimum, In words, F is relatively more powerful, the larger is its economy relative to L's.
Next, suppose that the constraint c F t a F t  x F t is binding: in other words, B = 1, and and is represented by point B in the figure. 19 At the constrained optimum, relative military power can be written as In words, F 's relative power is now constrained by domestic availability of raw materials, x F t . It is still increasing in the relative development of F 's economy, however, as this determines how e cient F 's army is in using available raw materials.
In what follows, we assume that, if L has the capacity to blockade (B = 1), F is always Then, optimal arming levels are with q J t w and m J t w similarly defined. 20 Let V J t (w t = 1) denote payo↵s in a period where countries go to war. Putting together the results of this and the previous section, we can write  structural transformation, it becomes more dependent on imported raw materials. If F catches up faster than it becomes more import-dependent (↵/↵ > / ), then its relative power increases between periods 1 and 2. However, at lower speeds of catching up (↵/↵ < / ), F 's relative power decreases. Intuitively, even though F becomes more e cient at arming, its increased dependence on imported raw materials that are subject to blockade has a stronger, negative e↵ect on its capacity to arm. In the knife edge case in which ↵/↵ = / F 's relative power remains constant between periods 1 and 2. 21

Ruling out welfare-increasing war
There are two distinct reasons why war may occur in this model. The first is the one highlighted by Garfinkel and Skaperdas (2000). As these authors pointed out, a desirable feature of war is that, by permanently allocating the pie to the winner, it removes the need to arm in future periods. In contrast, so long as there is peace, there is a pie that needs to be allocated in every period, and this forces countries to arm so as to strengthen their position in negotiations.
Because arming is costly, this e↵ect makes war more attractive for both countries. Indeed, if it is strong enough, war becomes welfare increasing, as the joint payo↵ of the two countries is higher with war than without. When this is the case, negotiations can never succeed, since the maximum that one country is willing to o↵er is less than the minimum that the other is willing to accept. War must then always occur.
In this paper, we want to focus on a second channel, in which war may occur as the result of trade-related shifts in relative power. We therefore want to rule out the case in which, because of a high future cost of arming, war must always occur. As it turns out, this can be done by ruling out very low values of , the exogenous cost of war. When  is low, war is likely to be welfare increasing for two reasons. On the one hand, a low  means that the war has limited destructive e↵ects. On the other hand, this implies that, in negotiations, the outside option of going to war is valuable: in turn, this induces countries to invest a lot in arming until there is war, in order to strengthen their position in negotiations. Indeed, given Assumption 1.
(an assumption that we further comment on in footnote 22) the following lemma establishes that war must occur immediately if  is su ciently low, for any value of the parameters: war is welfare increasing, and always occurs in period 1.
Proof. In Online Appendix B.
The threshold is such that, in period 1, the "e↵ective" cost of war -that is, the cost of war net of savings related to the future cost of arming -is zero if  =. Intuitively, the fact that there is now a benefit from going to war pushes up the zero cost-of-war threshold, relative to a model with no costly arming where the threshold would be at  = 0. 22 To focus on our own channel, we impose Given Assumption 2, the rest of the paper focuses on the case in which, in equilibrium, war is welfare decreasing, or its e↵ective cost is positive. As we show below, war will then only occur in the presence of shifts in relative power. Since such shifts will only occur between periods 1 and 2, this will imply that war can only occur in period 1. 23 22 If B = 1, war has an additional cost, due to the fact that trade disruption occurs immediately, as opposed to at some future date. If this cost is large, the e↵ective cost of war is negative for  = 0, and the threshold is negative. The role of Assumption 1 is to rule out this case, by requiring that the discount rate be high (and that therefore the cost from anticipating trade disruption be low). It is desirable for the model to feature > 0, since, as further explained below, this allows us to consider the case in which the e↵ective cost of war is close to zero ( ! from above). 23 Assumption 2 implies that the future cost of arming will not on its own eliminate the bargaining range. On the other hand, in period 1, when war remains possible because of shifts in relative power, the future cost of arming will still be a determinant of the size of the bargaining range, as we will see below (see eq. 28).

Subgame starting in period 2
Before proceeding further, we introduce Assumption 3 only poses a restriction on parameters if B = 1. It requires F 's economy to be large, relative to its gains from trade. The assumption is needed in order to ensure the existence of a balanced growth path SPNE of the game. 24 The following lemma describes the SPNE of the subgame starting in period 2. For conciseness, we only present the equilibrium path. The full description of the equilibrium is presented in Online Appendix B.
Lemma 3. Suppose war does not occur in period 1. In the unique balanced growth path SPNE of the subgame that starts in period 2, war never occurs. For all t 2, • Negotiators agree on an allocation of the pie that leaves F exactly as well o↵ as with war.
Proof. In Online Appendix B.
If war is avoided in period 1, it does not occur anymore. Intuitively, negotiations can only fail if one country expects to become relatively weaker over time: the impossibility to commit to a future sharing of the pie can lead to a situation in which any o↵er is not good enough for this country. However, from period 2 onwards, the economies of L and F grow at the same rate.
Then, their arming technology gets better at the same rate, and no shift in the balance of power is expected (see Figure 2). This is enough to ensure that the minimum share F must be o↵ered is less than the entire current pie, and that the maximum share L is willing to o↵er is greater than zero. Negotiations must then succeed in every period. 24 As stated in Lemma 3 below, in the unique equilibrium, F 's payo↵ is equal to its payo↵ from going to war, and F arms so as to maximise this payo↵. By requiring that the trade cost of war be moderate, Assumption 3 ensures that this payo↵ be positive: if it were negative, F would choose not to arm, and the equilibrium would collapse. Full details are provided in the proof to Lemma 7 in Online Appendix B (which is referred to by the proof to Lemma 3) Note that arming decisions with peace are the same as if there were war. This is because, in equilibrium, countries receive a payo↵ which is equal to their outside options, plus a share (which is one for L, and zero for F ) of the surplus from not going to war. Since the outside options are payo↵s with war, and neither the surplus nor the way it is shared depend on current arming levels, 25 countries arm so as to maximise their payo↵s with war. Given that equilibrium arming is a J t w , military expenditure is in both countries a constant share of GDP.
Also, note that F is o↵ered (and accepts) a share of the pie such that the entire surplus from not going to war is captured by L in every period. Intuitively, by moving first, L can o↵er F the minimum it requires for not starting a war, and keep the rest of the surplus for itself.
2 onwards. Payo↵s without war can then be written as The surplus from avoiding war from period 2 onwards is made up of the last three terms in eq. (22). First, there is a positive term, ⇧ 2 , which captures the fact that the destruction associated with war is permanently avoided. Second, a negative term captures the fact that, contrary to what would happen if there was a war in period 2, countries must pay for military expenditures not only in period 2 (which cost is included in V F 2 (w 2 = 1)), but also in all subsequent periods. Considering military expenditure in both countries, this carries a combined cost (m 3 ) w in the next period (3), which then grows (in discounted value terms) at a constant rate in subsequent periods. Finally, a further benefit of never going to war is that the trade disruption implied by war is avoided. This is captured by the term Bg F 2 . Although such trade costs are born by F , a higher trade cost of war actually increases L's payo↵ in equilibrium, since it makes F 's outside option less attractive and thus weakens its position in negotiations.
We can now proceed using backward induction. Suppose L's negotiators have o↵ered F a share s F 1 of the pie. When does F accept? To answer this question, let s F 1 a L 1 , a F 1 be the share that leaves F indi↵erent between accepting or not. Clearly, then, s F 1 a L 1 , a F 1 is also the minimum share that F is willing to accept. It is given by Using (12) and re-arranging, the threshold can be re-written as Eq. (24) is an important equation that we comment on in detail below. Before doing that, however, we also derive the share that leaves L indi↵erent between making an o↵er that gets accepted and starting a war. This is given by arg which, using (11) and re-arranging, can be written as This is the maximum share that L is willing to o↵er (provided it expects its o↵er to be accepted).
The di↵erence between the minimum share that F is willing to accept and the maximum share that L is willing to o↵er must be equal to the surplus from striking an agreement. Given that, if countries strike an agreement in period 1, war will never occur (Lemma 3), this surplus must be equal to the surplus from permanently avoiding war from period 1 onwards. Indeed, this is what we see in equation (25) (note that, compared to the expression for the previously discussed surplus from avoiding war from period 2 onwards, all subscripts are one period earlier). Again, the surplus from permanently avoiding war is made up of the gain from avoiding destruction and trade disruption in perpetuity, but there is also a cost due to the fact that both countries must continue to arm in all subsequent periods. We denote this gain by K 1 in what follows. As explained in Section 3.5, the threshold is such that K 1 = 0 if and only if  =. Because K 1 0 in our range of parameters, war is welfare reducing. Then, one might expect that negotiators should be able to avoid war. This however does not need to be the case. Lack of commitment explains this ine ciency: since negotiators cannot commit to future agreements, they may be unable to o↵er enough to a country whose war prospects are better today than tomorrow. To gain some intuition, consider equation (24). If F expects to become weaker over , or if the cost of arming for one more period is very high ( m F 2 w is high), or if the trade cost of war increases over time (g F 1 < g F 2 ), then F may require more than the entire current pie to be induced not to start a war (s F 1 a L 1 , a F 1 > 1). Since negotiators cannot allocate future pies, they cannot avoid war. Similarly, from (25), if L expects to become weaker over time, or expects that the future cost of arming will be high, not even the possibility of keeping the entire pie for itself will be enough to prevent it from going to war. 26   < 0? In the first case, the best L can do is to o↵er 1. This, however, is not su cient to avoid war, which must then occur.
In the second case, the best L can do is to o↵er 0. This is more than what L would ideally like to o↵er, but it is su cient to avoid war, and the cheapest feasible way to do so. But does L want to make such an o↵er? Clearly, it does so if s F 1 a L 1 , a F 1 K 1 , since then the maximum that L is willing to o↵er (eq. 25) is more than 0. Otherwise, this country prefers to start a war than to o↵er anything.
We next introduce the following: 26 As anticipated in Section 3.5, the future cost of arming is a determinant of the size of the bargaining range: a higher m F 2 w increases the minimum that F must be o↵ered, while a higher m L 2 w decreases the maximum that L is willing to o↵er.

Definition 2.
A J-led war is a war that takes place when there exists a peaceful partition that would induce J to prefer peace to war, but J prefers war to such a partition.
Applying this definition to the case just discussed, it is evident that there is a F -led war in period 1 if and only if s F 1 a L 1 , a F 1 > 1, and there is an L-led war if and only if s F 1 a L 1 , a F 1 < K 1 . 27 Having found how war decisions depend on s F 1 a L 1 , a F 1 , we move back one sub-period and examine the arming decisions that determine this threshold. These must simultaneously satisfy where The above maximands have an intuitive interpretation. If arming decisions are such that is either greater than 1 or smaller than K 1 , then a war occurs, and both countries obtain their payo↵s with war. If this minimum share lies between zero and one, then war is 27 We prefer to characterise a war based on Definition 2, and not based on who starts the war, because the latter approach depends on the specific tie-breaking rule used, while the former approach does not. When deriving the full equilibrium in Online Appendix B, we show that, under reasonable tie-breaking rules, if there is a J-led war, this is always started by J. These rules can be summarised as follows: given equal wartime and peacetime payo↵s, a country prefers not to start a war. Note that, had we assumed the opposite, both an L-led war and an F -led war would be started by L, the country who moves first. More details are provided in the proofs to Propositions 1 and 2.
avoided, and L can o↵er the minimum that F is willing to accept. It follows that F is driven down to its war payo↵, while L reaps the entire gain from not going to war, K 1 . Finally, if the minimum share is just below zero, then war is avoided, but L must o↵er more than the minimum.
Relative to the previous case, F 's payo↵ must be higher, and L's payo↵ lower (note that the term s F 1 a L 1 , a F 1 ⇡ 1 is negative in this range). What armies will countries have in equilibrium, and will this lead to war? To provide a general answer to this question would require maximising the expressions in (26) and (27), two complicated functions of a L 1 and a F 1 . Here, we adopt a simpler approach: instead of looking at the entire range  2 [, 1], we focus on the case in which  is close enough to. This is enough for our purposes: to show that a welfare reducing war can occur, it is enough to show that it can occur if its e↵ective cost is small enough (but still positive). 28 This approach simplifies the problem in (26) and (27) From the previous discussion, we know that an L-led war occurs if and only if the above share is smaller than K 1 , and an F -led war occurs if and only if it is greater than one. However, for  close enough to, K 1 is close to zero. So, what we have to check is whether or not the above share is smaller than 0 or greater than 1. All have we to do, then, is to write q F Proof. In Online Appendix B.
If L does not have the capacity to blockade, F 's military power increases as it catches up to the leader (see Figure 2). Then, the term q F 1 w q F 2 w in eq. (28) can be negative.
If this shift in relative power is large enough, this may make s F t a L 1 , a F 1 negative, leading to an L-led war. The condition for this to happen is presented in condition (29). The expression is true if F catches up fast enough, that is if ↵ is large relative to ↵. Proposition 1 is simply the well-known result that an industrial leader may find it optimal to start a pre-emptive war against a catching-up follower. Intuitively, catching up will make the follower more powerful in the future (q F 2 > q F 1 ), and the follower cannot commit not to use this augmented power against the leader. In these circumstances, L may want to start a pre-emptive war so as to defeat the follower before it is too late. In Online Appendix C, we present two vectors of parameters which satisfy all assumptions of the paper, and such that, if B = 0, there is, respectively, no war and a L-led war in period 1.
If L does have the capacity to blockade, we can obtain the results reported in while there is an L-led war if and only if the LHS of the above inequality is less than zero.
Proof. In Online Appendix B.
If L has the capacity to blockade, an F -led war is now possible. Two channels make it potentially attractive for F to start a war. First, there is the shift in relative power channel, which is captured by the first term in (28), or on the LHS of (30). From Figure 2, we know that F may now become weaker over time ( q F 1 w > q F 2 w ): that happens when F 's catching up is not fast enough to make up for its increased dependence on imported raw materials, ↵/↵ < / .
In this case, the first term in (28), or on the LHS of (30), is large and positive: this may imply that the minimum share that F is willing to accept is greater than one, making a F -led war unavoidable. Intuitively, if faced with a large enough decline in its relative power, F may find it optimal to go to war immediately. The second channel is the increase in the trade cost of war channel, which is captured by the third term in (28), or on the LHS of (30). Intuitively, if dependence on imported raw materials grows fast (so that ↵ is much greater than ↵ ), F faces a much higher trade cost of war in period 2 than in period 1. Anticipating that this will make it weaker over time, by a logic similar to that of a decline in relative power, F may then decide to start a war immediately. It turns out that, even if F 's relative power is constant or increasing (which, from Figure 2, is the case if ↵/↵ / ), this second channel may still make an F -led war unavoidable. 29 On the other hand, an L-led war may still occur: this may happen when ↵/↵ is much larger than / . Intuitively, F 's prodigious economic growth makes its military technology increasingly sophisticated, and increasingly good at using scarce domestic raw materials. Then, F 's relative power increases fast between period 1 and period 2, making the first term in (28), or on the LHS of (30), negative. If this e↵ect is large enough, (28), or the LHS of (30), may then be negative, making a L-led war unavoidable.
These results are summarised by the following: there is either peace or a F -led war. If ↵/↵ > / , there can be a 29 As explained in footnote 23, the third channel contained in eq. (28), the additional military expenditure channel, cannot, in itself, be a cause of war. This can be seen by setting ↵/↵ = / = 1, which "shuts down" both the shift in relative power channel and the increase in the trade cost of war channel: as shown in the proof to Corollary 1, there can never be a war in this case. However, the additional military expenditure channel still contributes to determining the size of the bargaining range, and may therefore matter in conjunction with the other two channels.
L-led war, peace or a F -led war.
Proof. In Online Appendix B.
In Online Appendix C, we present three vectors of parameters which satisfy all assumptions of the paper, and such that, if B = 1, there is, respectively, no war, an F -led war, and an L-led war in period 1.

Extensions
So far, we have assumed that, if L has the capacity to blockade, countries must take this initial condition as given. In this section, we briefly consider two ways in which this assumption can be relaxed. We begin by looking at a case in which F can attack C, and thus conquer enough raw materials to become immune to a blockade. We then consider the possibility that L may surrender the capacity to blockade.

Conquest of C
Suppose that, before any other event takes place, F may attack C. We assume that, if F is indi↵erent between attacking or not, it does not attack. For simplicity, we also assume that an attack costs nothing (or very little), and is always successful. As a result of a successful attack, F annexes a portion of C's territory producing no less than ↵ in raw materials in period 1, and t 1 times that amount in period t > 1. In order to focus on our security of supply channel, we assume that even if F conquers a portion of C, it must still pay for any raw materials it imports from there, at the original price ⌘. Thus, if B = 0, F will never attack C, since it gains nothing by doing so. However, as soon as conquered resources become part of F 's endowment, they become non-blockadable by L. This can give the follower a strategic incentive to attack.
Suppose then that B = 1. The choice to attack or not is equivalent to a choice between playing the baseline game when B = 0, or playing it when B = 1. Because F receives its wartime payo↵ in both cases, it will attack if and only if its wartime payo↵ is higher in the former case, than in the latter. In terms of Figure 1, it will attack if and only if its wartime payo↵ is higher at point A than at point B. It is possible to show that, if F is equal in size (in terms of its endowment of y) or larger than L, or if it is smaller, but is severely constrained in its arming decisions, then its wartime payo↵ is higher at point A than at point B. In either case, F attacks C. Intuitively, by attacking, F can increase its chances of winning a war against L, as well as reduce its trade costs from such a war. This benefits F both if the war actually occurs, and if it does not, since it increases its bargaining power in negotiations. 30 What does F 's attack on C actually imply for bilateral relations between L and F ? Comparing Propositions 1 and 2, we see that the model allows for a rich set of cases. If Finally, because we have assumed that an attack makes F fully self-su cient, it can never be the case that, after an attack, there is an F -led war. However, it is easy to imagine a more general case in which F would first attack C, and then attack L. All is required for this is for F to be capable of conquering only a portion of C, so that the raw materials that it can grab from C are not enough to make it fully self-su cient. In this case its dependence on imported raw materials may still grow fast enough during structural transformation for it to attack the leader.
However, it may well make sense for F to first attack C, so as to increase the probability that it 30 Perhaps surprisingly, F 's wartime payo↵ is not always higher at point A than at point B. F 's arming decision at point A is optimal given L's arming decision. This does not rule out the possibility that if both countries choose di↵erent arming levels (e.g. those at point B), F 's payo↵ might be higher. It turns out that, if F is smaller than L, and B is not too far from A, then F 's wartime payo↵ can be higher at point B. This is because, in this case, at point A F arms "too much" against a powerful opponent. Interestingly, then, the model admits a case in which F chooses not to attack C in order to avoid an escalation in arming levels.
will defeat the leader. 31

Surrendering the capacity to blockade
We again focus on the case B = 1. Suppose that, before any other event takes place, L may decide to surrender the capacity to blockade, with immediate e↵ect. This is again a choice between playing the baseline game when B = 1, or playing it when B = 0. The e↵ect on bilateral relations between L and F will depend on economic fundamentals, as in the previous section.
When does L surrender to capacity to blockade? To answer this question, note that L's payo↵ is equal to its wartime payo↵ if there is a war, and to its wartime payo↵ plus the surplus K 1 if there is no war. It is possible to show that L's wartime payo↵ is always higher for B = 1 than for B = 0, because its chances of winning the war are higher in the former case. 32 Then, of the three examples considered in the previous section, L will only consider surrendering the capacity to blockade in the first, when F 's import dependence is growing fast but its economy is not growing too rapidly. In this case surrendering the capacity to blockade can stave o↵ an F -led war: L would then have to trade o↵ the fall in its wartime payo↵ against its ability to reap the surplus K 1 from not going to war. In both other cases, the fact that F 's economy is growing fast implies that, if L surrenders the capacity to blockade, it will then have to start a war against F : this cannot possibly be an optimal choice for L.

A brief historical discussion
Our model predicts that war can arise when a rising power finds its import dependence growing more rapidly than it is converging on the industrial leader. In this case, its relative military strength will be declining over time, and the follower may have an incentive to strike before it is too late. Indeed, the rising cost to consumers of wartime blockades may give the follower an 31 Conquering a portion of C will move the equilibrium along the leader's best response function in Figure 1 from B in the direction of A. As in the case when the follower is capable of conquering all of C, and the equilibrium moves all the way to A, for many parameter values shifting the equilibrium in this way will increase the follower's payo↵ from war. 32 That is to say, contrary to what we saw for F , points A and B in Figure 1 are unambiguously ranked in L's preferences: B always dominates A.
incentive to go to war, even in circumstances when its relative power is not declining. The model also predicts that the follower may attack resource-rich peripheral areas, in an attempt to become more self-su cient, or entirely self-su cient, in raw materials. It may do so prior to launching an attack on the leader. It may even do so in circumstances when it knows that this will provoke an attack upon it by the leader, when otherwise the two countries would not have gone to war. If the follower is not only becoming rapidly more import-dependent, but is also converging rapidly on the leader, then conquering the country supplying raw materials transforms what would have been a follower-led war into a leader-led war. In this case, while it is the leader who decides to go to war against the follower, the root cause of the war remains the follower's incentive to fight, arising from its import-dependence.
There is a substantial body of historical literature which suggests that this trade-dependence mechanism was at work in the first half of the twentieth century, and that concerns over the supply of imported raw materials was an important motivating factor at various points in time for both German and Japanese military planners. In the words of Azar Gat, "the quest for self-su ciency in strategic war materials became a cause as well as an e↵ect of the drive for empire, most notably in the German and Japanese cases towards and during the Second World War" (Gat 2006, p. 556). This seems especially obvious in the Japanese case.
To repeat: the world is much more complicated than the simple structure envisaged in our model, or any other, and we do not argue that our mechanism can "explain" the Second World War in some monocausal way. However, our model provides useful insights into the origins of this war, especially in the Pacific. It is much less useful in understanding the origins of the First World War, which lie elsewhere, but does provide insights into the Anglo-German naval rivalry which preceded it, and which helps explain Britain's decision to join the war once it had started.
We therefore provide a very brief account of the build-ups to the Second World War in Asia, the Second World War in Europe, and the First World War. In each case, we indicate how the mechanisms identified by our model are relevant in understanding the episode in question, as well as some of the ways (certainly not all) in which reality was more complex than allowed for in the theoretical discussion above.

World War II in Asia
Japan's industrial output had been growing more rapidly than American output since 1890 (Bénétrix et al. 2015). Between 1920 and 1938, Japan's industrial output grew at an average of 6.7 per cent per annum, much higher than the growth rates recorded in the USA (1.2 per cent, although that reflected the severity of the Great Depression) and UK (3 per cent) over the same period. Rapid growth meant an increase in Japan's relative military power, already dramatically displayed during the Russo-Japanese war of 1904-5. This was a case where ↵/↵ was unambiguously high.
However, / was also very high in interwar Japan. Japan was endowed with very few natural resources, and rapid growth meant greater dependence on trade: by the eve of the war Japan was producing "only 16.7 per cent of her total iron ore consumption, 62. Manchuria and China, but Southeast Asia as well, then planners estimated that she would be self-su cient in the major strategic commodities, aside from nickel (ibid ).
A group of "total war" military o cers became convinced that Japan would only be secure if it was self-su cient. "War hereafter would be protracted...and nations had to be able to supply themselves during wartime with adequate quantities of raw materials and manufactured goods.
Reliance on other countries for the materiel of war was a sure path to defeat... The need for security became, slowly, an impulse for empire, and it led directly to the Pacific War" (Barnhart 1987 Unlike what we assume in the extension to our simple model (Section 5.1), conquering China was far from costless, and increased the need for imported raw materials from the West (Yasuba 1996). It also increased Western suspicion of Japan and aid to China. The US response confirmed in the minds of Japanese planners that their basic assumption, that a reliance on trade was dangerous for national security, was correct. In July 1940 the President was empowered to ban the export of strategic commodities, and soon the US had banned the export of scrap iron and steel, aviation fuel and other commodities. While in the short run Japan could live with this, having stockpiled American raw materials since 1937, the ban on oil exports which came in July 1941 was a di↵erent matter, and was seen as a de facto declaration of war. 33 The fact that critical raw materials were now in short supply became an argument, not for restraint, but for an immediate all-out war (Ferguson 2007), since it implied that q F t+1 < q F t . This case seems the one that best fits our model. Japan was growing relatively rapidly, and becoming more dependent on imported raw materials, just as is true of the follower country in our model. The European imperial powers and the United States possessed colonies which produced vital raw materials, or (as in the case of US oil) produced those raw materials domestically. This gave them an ability to blockade which was used by the United States in the run-up to war.

Japan's invasions of Manchuria, China and Southeast Asia corresponded to invasions first of C,
and then of L, in our model, a possibility discussed in Section 5.1. They were motivated by a desire for economic and strategic self-su ciency, to be formalised via the creation of a Greater East Asia Co-Prosperity Sphere. This would have deprived the Western powers of the ability to blockade Japan. But trying to achieve such self-su ciency was a high risk gamble, since attacking Southeast Asia required launching an attack on the Western powers, despite Japan's economic and military inferiority relative to America.

World War II in Europe
Again and again, Hitler returned in his speeches and writings to the need for secure supplies of both food and raw materials. The key was the Soviet Union. As early as 1931 he told a Party member that "Europe needs the grain, meat, the wood, the coal, the iron, and the oil from Russia in order to be able to survive" (Overy 2009, p. 51), and shortly before the war began he 33 As is well known, Roosevelt had not envisaged the oil embargo as being a complete one, but the State Department o cials who implemented the embargo ensured that it became one (Iriye 1987, p. 150). told a Swiss diplomat that "I need the Ukraine, so that no one will starve us out as they did in the last war" (Hildebrand 1973, p. 88).
In a speech to the heads of the armed forces of November 1937, Hitler stated that: "There was a pronounced military weakness in those States which depended for their existence on foreign trade. As our foreign trade was carried on over the sea routes dominated by Britain, it was more a question of security of transport than one of foreign exchange, which revealed in time of war the full weakness of our food situation. The only remedy...lay in the acquisition of greater living space...areas producing raw materials can be more usefully sought in Europe, in immediate proximity to the Reich, than overseas..." 34 Germany was extremely or entirely dependent on imports for its supplies of such strategically vital raw materials as bauxite, chromium, copper, iron, lead, nickel, oil, rubber, and zink (Volkmann 1990, p. 246). The 1934 New Plan and 1936 Four Year Plan therefore tried to promote import substitution: in terms of our model, increasing . The annexations of Austria and Czechoslovakia in 1938 and 1939 provided the Reich with lignite, coal, and iron ore, as well as heavy industry (Overy 2002, pp. 197, 227), and Germany also tried to increase its economic hold over the resources of Hungary, Bulgaria and Romania via a series of bilateral deals. Dominating Poland was "necessary, in order to guarantee the supply of agricultural products and coal for Germany" (Overy 2002, p. 222). However, the ultimate prize, Russian resources, were still essential in order to make the Nazi empire blockade-proof (Kaiser 1980, pp. 277-9;Volkmann 1990, p. 258; Hildebrand 1973, p. 92). The conclusion of the Nazi-Soviet pact was thus crucial for Hitler, who could now invade Poland confident that even if Britain and France intervened, "We need not be afraid of a blockade. The East will supply us with grain, cattle, coal, lead and zinc."And indeed, in 1940 the USSR supplied Germany with 74 per cent of its phosphates imports, 67 per cent of its imported asbestos, and 34 per cent of its oil (Tooze 2006, p. 321). Ultimately, however, Hitler's aim was to grab these resources, so as to be able to rival the Anglo-American powers, rather than to buy them from the Communist enemy. It is in that light that his decision to invade the Soviet Union in 1941 needs to be understood.
There was nothing rational about Hitler's racial theories and rabid nationalism. However, his desire for Lebensraum is quite consistent with our model. Visà vis the Western nations, the Nazi state was a rising power. However, its dependence on trade left it vulnerable to blockade by sea. One obvious solution was to attack Eastern Europe, which corresponded to C in our model. Indeed, as Section 5.1 argues, it might even have made sense to attack Poland in 1939, despite the fact that this risked war with France and Britain. And in the long run, conquering Russia was the only way to achieve complete self-su ciency in raw materials. per cent of these imports were arriving by sea, either directly or indirectly (O↵er 1989, p. 335), implying that they were potentially vulnerable to blockade by the British.

Anglo-German naval rivalry and World War I
According to Avner O↵er (1989), a key factor underlying Anglo-German naval rivalry was the fact that both Germany and Britain were increasingly dependent on overseas imports of food and raw materials. "The economies of both Britain and Germany came to depend on hundreds of merchant ships that entered their ports every month. Overseas resources, the security of the sea lanes and the economics of blockade a↵ected the war plans of the great powers and influenced their decision to embark on war" (O↵er 1989, p. 1).
In 1898 Germany embarked on a naval buildup whose aim was to achieve naval parity with Britain, not globally, but locally (that is to say, in the waters between the two countries). But this strategy completely underestimated the importance of preserving naval hegemony in British eyes: it was essential both for the security of the Empire, and of Britain herself. The result was a naval arms race which Britain eventually won, but which in the process helped to shift British strategic thinking in an anti-German, rather than a pro-German, direction. As Sir Edward Grey, As Section 5.2 suggests, abandoning the capacity to blockade a rival that was growing as rapidly as Germany was unthinkable to the British.
The failure to make any headway in challenging Britain's naval superiority prompted some in Germany to argue for a strategy of German continental dominance, based on a European economic bloc with Germany at its centre (Strachan 2001, pp. 46-7). This was also unacceptable to Britain, since it would have granted Germany access to Atlantic ports, weakening or eliminating Britain's capacity to blockade her. As Grey said in 1911, if a European power achieved continental hegemony Britain would permanently lose its control of the sea, which would in turn mean its separation from the Dominions and the end of the Empire (Howard 1972, pp. 51-52).
Paradoxically, Britain's traditional maritime orientation meant that it was more likely that she would intervene in a war in which France risked being destroyed by Germany.

Conclusions
This paper has developed a model of the links between growth, trade and military power in which a follower country may choose to launch a pre-emptive attack on a leader, despite the fact that it is growing more rapidly. Faster growth may not translate into greater future military strength if it is accompanied by increased dependence on imported raw materials, and the leader has the capacity to blockade; since the leader cannot pre-commit to not use this capacity in the future, the follower may choose to launch a pre-emptive war.
In our view, this mechanism is most likely to have been at work during the post-Industrial Revolution period. Rapid industrial growth involved profound structural change, and it is this structural change which made rising powers potentially vulnerable to blockade. We would not therefore expect to see our mechanism at work during the 18th century or earlier, even though there were of course many wars involving the English/British, Dutch and French trying to establish naval superiority over each other.
But neither was it the case that rapid industrial growth and structural change necessarily led to war from the 19th century onwards. There are several historical examples of countries rapidly catching up on established leaders, without their attacking either the leader or adjacent sources of raw materials. Sometimes this was because these rising powers were impossible to blockade. Thus, a classic example of a follower country catching up on and overtaking a leader, without provoking a war, is the United States' ascent relative to Britain. Our mechanism could not have been at work in this instance, since the United States was a vast continental economy abundant in raw materials, and impossible to blockade. Nor did Russia, or the Soviet Union, launch preemptive wars against Germany in 1914, 1939 or 1941. Again, our mechanism would not have been expected to work in this instance, since Russia was another vast, resource-abundant country that was impossible to blockade.
On the other hand, neither did the USSR attack the West after 1945 (or vice versa), despite the fact that the former was growing more rapidly than the latter until the 1970s, and that Russia was importing food by the end of this period. Nuclear weapons are one obvious reason why the peace was kept on this occasion. Nor has China's rise over the past three decades provoked an attack on its trading partners, despite the fact that it is becoming increasingly import-dependent.
Several historians have noted that there was a circularity to some of the strategic and military logics driving nations to war in the 1930s. In the case of Germany, David Kaiser (1980, p. 282) wrote that "Having insisted upon rearmament for the sake of conquest, he (Hitler) found himself in a situation where conquest was the only means of continuing rearmament. His belief that Germany must conquer a self-su cient economic empire, rather than rely upon world trade, had become a self-fulfilling prophecy." In the case of Japan, Hatano and Asada (1989, pp. 399-400) comment that Japanese military thinking during this period "was characterised by peculiarly circular reasoning: to prepare for hostilities with the Anglo-American powers, Japan would have to march into Indochina to obtain raw materials; the United States would counter by imposing an economic embargo; this in turn would compel Japan to seize the Dutch East Indies to secure essential oil, a step that would lead to hostilities with the United States." Ralph Hawtrey (1952, p. 72) wrote that "the principal cause of war is war itself", in that "the aim for which war is judged worth while is most often something which itself a↵ects military power." As Kaiser noted, the danger with circular logics is that they can become self-fulfilling. Standard political economy considerations imply that it would be di cult if not impossible to unwind today's globalization, on which the Chinese economy depends: production is so fragmented, and the Chinese and Western economies so inter-dependent, that a move away from free trade would be impossibly costly, not just in the aggregate, but for large corporations that wield considerable political as well as economic power. This paper sounds a cautionary note (although one hopes that the costs of war have now become so enormous as to make it unthinkable): if strategic considerations were ever allowed to gain an upper hand, globalization would become more fragile, and the world would become a much more dangerous place.
Appendix A: key variables as functions of parameters Lemma 5. Let k t be the joint welfare gain from delaying war by one period in period t (expressed as a share of period t pie). Such a gain can be written as If B = 0, such a gain is constant from period 1 onwards, while if B = 1 it is lower in period 1 than in period 2, and constant from period 2 onwards. In either case, let k be the expression denoting this constant value of k t . There exists  2 (0, 1) such that Proof. By Lemma 4, if war occurs in period t + 1, it must be that a L t+1 = a L t+1 w and a F t+1 = a F t+1 w . By definition, k t can then be written as 1/⇡ t times which, using (11)- (12) and (20)- (21) and re-arranging, can be written as in (49). If B = 0, k t is constant over time, since, as evident from (44) and (46), (m t+1 ) w grows at a rate like ⇡ t . If B = 1, the term k t is constant from period 2 onwards, since g F t g F t+1 can then be written as (1 ) g F t , which also grows at a rate . However, k 1 < k 2 , since g F t grows faster than between period 1 and 2. From (44) and (46), we see that (m t+1 ) w = m L t+1 w + m F t+1 w is continuously decreasing in . It follows that k is continuously increasing in . All we need to show is that there exists  2 (0, 1) such that k = 0 if  = . If B = 0,  is easy to derive explicitly. Using (35) and (36), If B = 1, using (43)-(46), we can write: Because k is continuously increasing in , all we need to show is k < 0 for  = 0, and k > 0 for  ! 1. If  = 0, condition k < 0 can be written as By assumption, it is x F t  m F t w,u . Using (36) and (2), it is easy to see that this implies Then, a su cient condition for (81) to hold is which is the same as Assumption 1. If  ! 1, condition k > 0 converges to 1 + (↵ 1) Again, using (36) and (2), it is easy to see that the assumption Then,  ! 1 implies /⇡ 1 ! 0 (or, in other terms, (m t+1 ) w /⇡ t ! 0). It follows that (53) must hold true.
which, givenm s+1 = m s in a balanced growth path SPNE, and given ⇡ s = ⇡ s 1 and g F s = g F s 1 , does not depend on s. Then, the joint surplus from not going to war in period s can be written as P 1 v=s v sk ⇡ v =k⇧ s . Letê J s+1 2 [0, 1] be the share of such surplus appropriated by J in the subgame starting in period s + 1, and let be the share that leaves F indi↵erent between starting a war in period s, or not. Then, backward induction and tie-breaking rule 1 requirê Next, note that the share that leaves L indi↵erent between o↵ering that share (and being accepted) and starting a war, can, substituting (11) ⇤ , would follow, respectively, from backward induction, and from backward induction and tie-breaking rule 2. But givenŵ F s a L s , a F s , s F s ,ŝ F s (a L s , a F s ) andŵ L t (a L t , a F t ), it would be the case that w s = 1 for any a L s and a F s , a contradiction.
Step 2.4. It cannot be the case that, in the SPNE under consideration, war never happens, and a J s 6 = a J s w for at least one J 2 {L, F }. Suppose this was the case. It cannot be thatk < 0, or, by the logic of Step 2.3, we would have w s = 1, a contradiction. Givenk 0, the share that leaves L indi↵erent,ŝ F s (a L s , a F s ,ê F s+1 ) +k⇧ s , is at least as high asŝ F s (a L s , a F s ,ê F s+1 ). Then, follow, respectively, from backward induction and tie-breaking rule 1, and from backward induction and tie-breaking rule 2. Also, Then, substituting in (11)- (12) and (64), and re-arranging, payo↵s in the case under consideration can be written as where the d J s ê F s+1 are expressions which do not depend on a J t . It follows that at least one country could obtain a higher payo↵ by decreasing arming by ✏. 36 Next, it cannot be the case Since it cannot be the case thatŝ F s (â L s ,â F s ,ê F s+1 ) 2 h k ⇧s ⇡s , 1 i , a war must occur in s, a contradiction.
Step 2.5. In summary, we have shown that, in any balanced growth path of the subgame starting in period t, war cannot happen in T > t (Step 2.1), nor can it be the case that war never occurs (Step 2.2). It follows that war must occur immediately.
Step 3. The proposed SPNE is the unique balanced growth path SPNE of the subgame starting in period 2. In t > 2, if war has already occurred, by Lemma 1, a L t ⇤ and a F t ⇤ are the unique optimal actions. Next, take any t 2, and suppose that war has not yet occurred. If w t = 0, by Step 2, it is the case that w t+1 = 1. By the logic of Step 1.2, then, Proof. Preliminaries. First, note that the term k, defined in Lemma 5, is the joint welfare gain (expressed as a share of the current pie) from delaying war by one period. As shown in that lemma, given  , we have k 0. Next, note that in t 2, if war has not yet occurred, the actions in (67)-(71) imply w substituting (12) where we have used the fact that q F t w = q F t+1 w . On the one hand, s F t a L t w , a F t w < 1, since (73) can be re-written as On the other hand, under Assumption 3, s F t a L t w , a F t w > 0. To see this, note that, using (73) and the fact that m F t+1 w = m F t w , such inequality can be re-written as A su cient condition for (74) to hold can be obtained by subtracting m F t w > 0 from the LHS and re-arranging, The LHS of (75) is F 's gain from war (gross of any trade cost) if a L t = a L t w and a F t = a F t w .
Because a F t = a F t w > 0 maximises such gain, and not a F t = 0 (the latter being a level of arming at which the LHS is zero), the LHS must be positive. 37 If B = 0, then, (75) always holds. If B = 1, substituting in (48) and (46), (75) can be written as can then be written as (1 ) > 0. If B = 1, as shown below, the LHS can be written as By assumption, it is x F t  m F t w,u . Using (36) and (2), it is easy to see that this implies Then, a su cient condition for (76) to hold is (1 + ↵) = ↵ (1 ⌘) ↵ , that is Assumption 3. We have shown that, in t 2, if war has not yet occurred, w t = 0. This implies that it is also true that w s = 0 for s > t (war is forever delayed). Given that, as shown, F receives payo↵ V F t (w t = 1) in period t, L must then receive payo↵ The rest of the proof proceed in four steps, and in various sub-steps.
Step 1. The proposed outcome is a balanced growth path SPNE of the subgame starting in period 2. Step 1.1. In t > 2, if war has already occurred, by Lemma 1, (a L t ) ⇤ and (a F t ) ⇤ are optimal actions. Step 1.2. Take t 2, and suppose war has not yet occurred. If w t = 0, by the Preliminaries, in period t+1, F and L receive payo↵s , F 's payo↵ in the latter case is at most y F t i F t + V F t+1 (w t+1 = 1). Then, a su cient condition for a F t w to solve (79) given a L t = a L t w is Using (12), (80) can be re-written as Condition (81) that is the share that leaves F indi↵erent between starting a war in period s, or not. Next, note that the share that leaves L indi↵erent between o↵ering that share (and being accepted) and starting a war, can, substituting (11) into (82) and re-arranging, be written as s F s a L s , a F s , e F s+1 + k⇧ s . Then, backward induction and tie-breaking rules 1 and 2 require actions to be as in (59) = 1), which in turn implies that L receives payo↵ V F s (w s = 1) + k⇧ s . 38 Step 3 (intermediate result). In t 2, if war has not yet occurred, in any balanced growth path SPNE of the subgame starting in period t, war never occurs. Take any balanced growth path SPNE of the game starting in period t. We now show that, in such a SPNE, in any s t, war does not occur. In order to do that, we derive optimal actions in period s for any possible equilibrium path of the subgame starting in s + 1, and show that those actions do not lead to war. There are two possible cases. Step 3.1. First, in the equilibrium path of the subgame that starts in period s +1, war never occurs. By Step 2, in period s +1, F and L receive payo↵s V F s+1 (w s+1 = 1) and V L s+1 (w s+1 = 1) + k⇧ s+1 . By the logic of Step 1.2 (replacing t with s), ⇤ and a F s ⇤ follow from backward induction and from tie-breaking rules 1 and 2. By the Preliminaries, it is then the case that w s = 0.
Step 3.2. Second, in the equilibrium path of the subgame that starts in period s + 1, war occurs in T s + 1. By Lemma 4, a J T = a J T w . It follows that, if w T 1 = 0, then w T = 1, and, in period T , F and L receive payo↵s V F T (w T = 1) and V L T (w T = 1). Consider optimal actions in period T 1. By the logic of Step 1.2 (replacing t with T 1), follows from backward induction, and from tie-breaking rule 1. Next, note that, the share that leaves L indi↵erent between o↵ering that share (and being accepted) and starting a war, can, substituting (11) into (85) and re-arranging, be written as s F T 1 a L T 1 , a F T 1 + k. Again by the logic of Step 1.2 (additionally replacing k⇧ T 1 with k⇡ T 1 ), ⇤ follow from backward induction, and from tie-breaking rule 2. By the logic of the Preliminaries (replacing t with T 1, and k⇧ T 1 with k⇡ T 1 ), these actions imply w T 1 = 0, and that F and L receive payo↵s in period T 1, F and L receive payo↵s V F T 1 (w T 1 = 0) and V L T 1 (w T 1 = 0) + k. Consider optimal actions in period T 2. By the logic of the analysis just conducted for period T 1 (now replacing t with T 2, and k⇧ T 2 with k⇡ T 2 + k⇡ T 1 ) it follows that w T 2 = 0, and F and L receive payo↵s V F T 2 (w T 2 = 0) and V L T 2 (w L 2 = 0) + k⇡ T 2 + k⇡ T 1 . If T = s + 2, it is established that w s = 0. If T > s + 2, w s = 0 can be established using the logic of the analysis just conducted recursively.
Step 4. The proposed SPNE is the unique balanced growth path SPNE of the subgame starting in period 2. In t > 2, if war has already occurred, by Lemma 1, a L t ⇤ and a F t ⇤ are the unique optimal actions. Next, take any t 2, and suppose that war has not yet occurred. If w t = 0, by Step 2 and 3, in period t + 1, F and L receive payo↵s V F t+1 (w t+1 = 1) and V L t+1 (w t+1 = 1) + k⇧ t+1 . By the logic of Step 1.2, then, that are used in the proof, and we then proceed to the proof itself.
Summary of earlier results. In Lemma 5, we derived the joint welfare gain from delaying war by one period (expressed as a share of the current pie); we showed that, if B = 0, this is constant from period 1 onwards, while if B = 1, it increases between period 1 and period 2, and is then constant from period 2 onwards; and, letting k denote this constant value of the joint welfare gain, we showed that there exists  2 (0, 1) such that k < 0 if and only if  < . In Lemmas 6 and 7, we showed that, in the unique balanced growth path SPNE of the subgame starting in period 2, if  < , war is welfare increasing (since the gain from delaying it is negative), and always occurs in period 2, while if  , war is welfare reducing, and never occurs; and, in the latter case, F and L receive payo↵s V F 2 (w 2 = 1) and V L 2 (w 2 = 1) + k⇧ 2 in period 2.
Proof. Let that is the share that leaves F indi↵erent between starting a war in period 1, or not (note that, by the results summarised above, F always receives payo↵ V F 2 (w 2 = 1) in period 2). By backward induction and tie-breaking rule 1, in a SPNE, it must be that If B = 0, let ⌘ . If  <, the share that leaves L indi↵erent between o↵ering that share (and it being accepted) and starting a war must be equal to since, by the results summarised above, there is a war in period 2. Using (11), (87) can be re-written as The term k is the joint welfare gain (expressed as a share of period 1 pie) from delaying war by one period in period 1. As explained above, it is equal to k. By the results summarised above, since  < , k < 0. Then, war is welfare increasing in period 1. Since the share that leaves L indi↵erent is less than s F 1 a L 1 , a F 1 , by backward induction and tie-breaking rule 2, in a SPNE, it must be that where s 2 [0, 1] and s 0 2 ⇥ 0, s F 1 a L 1 , a F 1 . Given (86), (88) and (89), we have w 1 = 1. If B = 1, there are two cases. If  < , the share that leaves L indi↵erent between o↵ering that share (and being accepted) and starting a war, still defined as in (87), can now be re-written (using 11) as The term is the joint welfare gain (expressed as a share of period 1 pie) from delaying war by one period in period 1. As explained above, it is less than k, which by the results summarised above, is negative (since  < ). Then, war is welfare increasing in period 1. Since the share that leaves L indi↵erent is less than s F 1 a L 1 , a F 1 , by backward induction and tie-breaking rule 2, in a SPNE, actions must be as in (88) and (89). Given (86), (88) and (89), w 1 = 1. If  , the share that leaves L indi↵erent between o↵ering that share (and it being accepted) and starting a war, must now be equal to since, by the results summarised above, if war does not occur in period 1, it does not occur anymore. Using (11), (90) can be re-written as The term is the joint welfare gain (expressed as a share of period 1 pie) from permanently delaying war in period 1. To determine its sign, we use another result established earlier in this Online Appendix. In the proof to Lemma 5, it was shown that k is continuously increasing in , and increases from 0 to 1 + (1 ) (1 ⌘) ↵ /⇡ 1 as  increases from  to 1. It follows that the RHS of (91), which is continuously increasing in k, is continuously increasing from as  increases from  to 1. Then, there exists 2 (, 1) such that, if  <, the RHS of (91) is negative, and war is welfare increasing in period 1. In addition, since the share that leaves L indi↵erent is less than s F 1 a L 1 , a F 1 , by backward induction and tie-breaking rule 2, in a SPNE, actions must be as in (88)  . The threshold was defined in Lemma 2. As shown in the proof to that lemma, it is , where  2 (0, 1) is a threshold defined earlier in this Online Appendix. Then,   implies  . It follows that, if war does not occur in period 1, the unique balanced growth path SPNE of the subgame starting in period 2 is as described in Lemma 7. In such an equilibrium, as shown in the proof to that lemma (see in particular the Preliminaries), for t 2, it is the case that w t = 0, a L t = a L t w and a F t = a L t w , and F receives payo↵ V F t (w t = 1). ⌅ Proposition 1. If w 1 = 0, by Lemma 3, F receives payo↵ V F 2 (w 2 = 1) in period 2; as for L's payo↵ in period 2, given  !, it converges to V L 1 (w 1 = 1). Then, (where s 2 [0, 1]) follow from backward induction, and from tie-breaking rule 1). It also follows from backward induction that ⇥ w L 1 (a L 1 , a F 1 ) ⇤ ⇤ converges to Finally, given ⇥ w F t a L 1 , a F 1 , s F 1 ⇤ ⇤ , ⇥ s F 1 (a L 1 , a F 1 ) ⇤ ⇤ and ⇥ w L 1 (a L 1 , a F 1 ) ⇤ ⇤ , it follows from backward induction that (a L 1 ) ⇤ and (a F 1 ) ⇤ converge to If it were the case that (q F 1 ) w = q F 2 w , this could be written as Then, there is an L-led war if and only if the condition in the proposition holds. ⌅ Proposition 2. The first part of the proof is identical to that of the proof to Proposition 1. It is the case that Using (47), (48), (46) and (9), and imposing s F 1 a L 1 , a F 1 > 1 as a necessary and su cient condition for an F -led war, and s F 1 a L 1 , a F 1 < 0 as a necessary and su cient condition for a L-led war, we obtain the two conditions in the proposition. ⌅ Corollary 1. Consider first the case ↵/↵ = / . This can be divided into two subcases, ↵/↵ = / = 1 and ↵/↵ = / > 1. If ↵/↵ = / = 1, the LHS of (30) can be written as which, using (48) and (46), and (9), can be re-written as The last expression was shown in the Preliminaries of the Proof to Lemma 7 to lie between 0 and 1, implying that there is peace. Next consider the case ↵/↵ = / > 1. The LHS of (30) can now be written as (98) As shown in numerical example 4 in Online Appendix C, this expression can be greater than 1, implying that there can be a F -led war. Next, (98) is greater than zero, implying that there cannot be a L-led war. To see this, note that one can always simultaneously decrease ↵ and (while keeping ↵) and ( constant) so as to obtain ↵/↵ = / = 1. Such a change unambiguously decreases (98), making it equal to (97), or to an expression which we have shown to be greater than zero. The result then follows.
Second, consider the case ↵/↵ < / . That there can be peace (i.e., that the RHS of 30 can lie between zero and one) follows from continuity, since: if ↵/↵ = / = 1, the LHS of (30) lies between zero and one; the case ↵/↵ < / can be obtained starting from ↵/↵ = / = 1, and increasing by a small amount; and the LHS of (30) is continuous in . Similarly, that there can be a F -led war (i.e. that condition 30 can hold) follows from continuity, since the case ↵/↵ < / can be obtained starting from ↵/↵ = / > 1, and increasing by a small amount. There cannot be a L-led war (i.e. the LHS of 30 cannot be negative). To see this, note that one can always decrease so as to obtain ↵/↵ = / 1. Such a change unambiguously decreases the LHS of (30), making it equal to either (97) or (98), or two expressions which were shown to be greater than zero. The result then follows.
Finally, consider the case ↵/↵ > / . Again, that there can be peace or an F -led war (i.e. that the LHS of 30 can take value between 0 and 1, or be greater than 1) follows from continuity, since this case can be obtained starting from ↵/↵ = / = 1 or from ↵/↵ = / > 1, and increasing ↵ by a small amount. There can also be a L-led war (i.e. the LHS of 30 can be less than 0), as shown in numerical example 5 in Online Appendix C. ⌅ If B = 0, the second group of assumptions are as follows. First, it should be the case that where the endogenous threshold is defined as in (50) (recall from the proof to Lemma 2 that  =  in this case). Next, it should be the case that m J t w,u  y J t i J t for J 2 {L, F } and t 1: in both countries, the national endowment of the industrial input should be enough to produce the unconstrained optimal arming level (see Section 3.3). The next four conditions, derived using (35), (36), and (2), ensure that this is true, respectively, in L in t = 1, in L in t 2, in F in t = 1, and in F in t 2: If B = 1, the second group of assumptions are as follows. First, it should be the case that where the endogenous threshold is now defined as the value that makes the RHS of (91) equal to zero. Next, we need to make sure that m L t w,c  y L t i L t for t 1: L's endowment of the industrial input should always be enough to produce the constrained optimal arming level. 39 The next two conditions, derived using (43), (44), and (2), ensure that this is true, respectively, in 39 Note that, for F , it is m F (1 ) ↵ 2(t 1) Also, it should be the case that x F t  m F t w,u for t 1: in F , the national endowment of raw materials should never be enough to produce the unconstrained optimal arming level (see Section 3.3). The next two conditions, derived using (35), (36) and (2), ensure that this is true for t = 1 and t > 1 respectively: Finally, one additionally needs to check that Assumptions 1 and Assumption 3 hold. This requires , which clearly satisfy (99)-(106).