Price competition in a model of arms trade∗

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Price competition in amodel of arms tradeMary Carmen García‐Alonso a

a Department of Economics , UniversidadAutónoma de Barcelona , Barcelona, 08193,SpainPublished online: 19 Oct 2007.

To cite this article: Mary Carmen García‐Alonso (1999) Price competition ina model of arms trade , Defence and Peace Economics, 10:3, 273-303, DOI:10.1080/10430719908404927

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Defence and Peace Economics, Vol. 10, pp. 273-303 © 1999 OPA (Overseas Publishers Association) N.V.Reprints available directly from the publisher Published by license underPhotocopying permitted by license only the Harwood Academic Publishers imprint,

part of The Gordon and Breach Publishing Group.Printed in Malaysia.

PRICE COMPETITION IN AMODEL OF ARMS TRADE*

MARY CARMEN GARCIA-ALONSO

Department of Economics, Universidad Autónoma de Barcelona,08193 Barcelona, Spain

(Received 4 March 1997; In final form 23 October 1997)

This paper presents a model of subsidized military production that examinesthe relationship between domestic procurement and arms exports. Weaponproducers satisfy the defence procurement in their own country and competein prices in the international market where weapons are imperfect substitutesfor each other. Importers are involved in an arms race situation and do nothave domestic military production. The model makes explicit the strategicinteraction between governments and firms in the export market. We thenanalyze the effect of a change in the most significant parameters on the equi-librium. The paper suggests an explanation for the evolution of the arms mar-ket in the past few years and highlights the important role of the demand andcost structures.

KEY WORDS: Arms trade; Defence economics; Product differentiation

1 INTRODUCTION

The end of the Cold War led to a precipitous drop in the demand formilitary procurement in the most important arms producing countriestogether with a drop in the world arms trade. Between 1986 and 1995,military expenditure decreased from 335048 to 238194 US $m. in theUS and from 527305 to 407738 US $m. for NATO (1990 prices, from

* This paper has been written in the context of the participation of the author in agroup working on arms trade, financially supported by the ESRC under the grantR000235685. I wish to thank Arcadi Oliveres, Pierre Regibeau and the members of thearms trade research group, especially Paul Levine, for their valuable comments. Also,I would like to thank Javier Coto for his support.

273

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274 M.C. GARCIA-ALONSO

SIPRI 1996). The real volume of the arms trade reached a peak in1984, fell for two years, rose in 1987 and then declined by over two-thirds up to 1993. World arms sales agreements also declined substan-tially from a peak of $73.3 billion in 1985 to $38.0 billion in 1994(US ACDA 1995).

The main reason for the decrease in the real volume of the armstrade is the reduction of superpower competition in various regions.This reduction has led to the elimination of arms aid by the USand Russia. Recipients must now pay the full price for their weaponssystems and countries exporting weapons compete commercially for ashare of the international market. Nonetheless, developed countriescontinue to be the major exporters of weapons, providing more than90% of world arms exports between 1983 and 1993. There is thus ahigh degree of concentration in this export market. Among the leadingrecipients of these exports are many countries involved in arm races,such as Turkey, Egypt, Saudi Arabia, Greece, India, Pakistan, Chinaand Taiwan.

In the last few years, the main exporters of weapons have expressedconcern about the situations of conflict outside their frontiers. In thecurrent structure of NATO, the Combined Joint Task Forces (CJTF)have been recently created in order to deal with operations of 'noArticle 5', that is, operations that do not have as an objective to defendcountries belonging to NATO but to impose or maintain peace, or togive human aid or evacuate residents, both inside or outside NATOcountries: indeed such outside operations are likely to become morefrequent in the near future. Moreover, most recent cases of conflict inthe world involve countries to which NATO had previously exportedweapons. Therefore, we see that exports can have a negative impact onthe perception of security of the main exporter countries.

Despite the end of the hostilities between the main weapon pro-ducers, the quality of the weapons produced has continued to increaseleading to a rise in unit costs of production (Kirkpatrick (1995)). Thisrise in unit costs of production together with the decrease in domesticdemand has led to a process of concentration in the military sectoreither nationally, as in the US1, or internationally, as in Europewhere rationalization has often taken the form of joint venturesbetween the companies of the different countries. This increase in

1 See Lichtenberg (1988) and (1989) for an analysis of the American case and Vestel(1995) for the European case.

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PRICE COMPETITION AND ARMS TRADE 275

concentration has exacerbated the oligopolistic character of the marketfor weapons.

A further complicating factor is that the relationship betweendefense companies, the so called 'national champions', and theirrespective governments is very close. Technologies are tightly con-trolled as strategic company assets and their transfer is closely watchedby governments. Moreover, most of these companies could not survivewithout the support of the state, its financing and access to its defensemarket.

This paper presents a model of the arms trade that incorporatesmost of the characteristics described above. Arms races among theimporters of weapons affect the perceived security of exporters. Thegovernments of exporter countries and their 'national champions' areindependent decision units linked by the government's defense pro-curement policy. Finally, we allow for economies of scale and marketpower on the part of the exporters.

We have ns exporting countries with one military firm each. Firmsproduce weapons of quality q which they sell in two different markets:the home market and the exports market. In the home market eachfirm is committed to satisfy the defense procurement required by thegovernment, in each of these national markets the home firm is theonly supplier. However, firms also participate in the exports marketwhere they compete with the other weapon producers. Countries in thedemand side of the international market do not have home weaponsproduction: hence, they depend on imports for their defense procure-ment. Importers perceive the weapons offered by the different sup-pliers as imperfect substitutes for each other. We then have horizontaldifferentiation in the international market.

While the quality of weapons is in general a function of the R&Dinvestment, we assume that this investment is exogenous. This makesthe quality exogenous as well. Nonetheless, governments can imposean upper limit on the quality of weapons exported. This limit ismodelled as a proportion of the maximum quality q available to thefirm. The relationship between governments and firms is very close.Apart from the restriction on the quality of exports, each governmentpays for the domestic defense procurement the price that secures theexistence of a national military firm by not allowing for negativeprofits. We call the domestic firm a 'national champion' in order toremind us of the characteristics of the domestic market in our model.

The interaction between governments of producer countries andnational champions is modeled as a two stage game. In the first stage,

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governments choose policy instruments (domestic defense procurement)taking as given other governments' policies. In the second stagenational champions compete in prices, taking as given the policiesestablished in the first stage. A reasonable consistency requirementthat we impose on the outcome of this two-stage game is that govern-ments acting in the first stage take into account the effects of theirpolicies on the second-stage outcome.2

Both the countries that import weapons and the governments ofcountries with national champion care about security and private con-sumption. Nevertheless, they have different perceptions of security.There is a number, nr, of arms races between importers, each involv-ing two countries, hence we have 2nr importers whose perception ofsecurity is determined by their own arms race situation. Though thegovernments of exporter countries are not involved in an arms racewith other countries, exports to countries involved in arms races havea negative effect on the exporter's security, and this includes not onlythe domestic firm's exports but also the foreign firm's exports.

A number of recent papers have developed models that attempt todescribe the arms market in the post Cold War framework. Levine,Sen and Smith (1994) and Levine and Smith (1995) provide a dynamicmodel of the arms trade market which allows for competing forwardlooking suppliers whose welfare depends on both the economic bene-fits from the sales and the security repercussions of the recipient'sbehavior.

Levine and Smith (1995), introduces a demand side in which coun-tries are involved in arm races with their neighbors. On the supply sideof the international arms market a number of countries are involved inoligopolistic competition for the arms market. Constant returns toscale are assumed in the production of weapons. There is only onedecision maker in each of these countries (firms and governments arefused in the same entity which has both a political and economicalconcern). The seller's objective function depends on firm profits and afunction of exports which represents the security concern of the sellerfor the weapons exported.

The contribution of these papers differs significantly from the pres-ent one in that they ignore the relationship between governments anddefense companies and the strategic interaction between them, they

2 This assumption that governments move first is common to the literature on govern-ment policy in international markets. See, e.g., Brander and Spencer (1985) andDixit (1984).

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also assume that the government does not participate as a buyer inthe arms market. Finally, since weapons are simplified to be a homo-geneous good, there is no room for the analysis of the effect of qualityrestriction or the degree of weapons differentiation on firms profits aswell as on security.

This paper modifies the framework set in the two previous papers inorder to develop the supply side of the arms market. In contrast withLevine and Smith (1995), governments and firms are independent deci-sion entities and the weapons sold by the different countries are imper-fect substitutes and have an exogenous quality. National championscompete in prices for the exports market under the increasing returnsto scale assumption and governments decide the amount of defenceprocurement the domestic firms must provide anticipating the effecttheir decisions have on firms behaviour. This structure introducessome of the elements commonly found in the strategic trade literature3

in which governments anticipate the influence their decisions have onthe domestic firms position in the exports market. In this literaturegovernments are usually only concerned about firms profits and con-sumer surplus. We introduce the security concern on the side of thegovernment and the arms race situation between importers: these fea-tures which are new to the strategic trade literature create a situationin which firms have downward sloping reaction functions in prices.This in turn, can reverse some of the standard conclusions of the stra-tegic trade literature.

Our model yields a rich set of comparative static results. As expected,when importers of weapons increase their interest in security exportsincrease together with prices; when the weapons sold by the differentcountries become closer substitutes exports increase induced by thedecrease in prices; an increase in the fixed R&D costs of weapons pro-duction discourages both domestic procurement and exports; finally,a shift in the exporters' priorities towards consumption decreases thequantity of weapons exported and domestic procurement. More sur-prisingly, an increase in the awareness of the negative impact exportshave on the exporter country's security or an increase in the restrictionon the quality allowed to be exported encourage both the quantity ofweapons exported and domestic procurement. The paper provides adetailed explanation for these results based on the structure of thearms trade market that we present.

3 For a review of this literature see Helpman and Krugman (1989).

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The paper is organized as follows. Section 2 presents the model forthe international market for weapons. Subsection 2.1 describes thedemand side of the international market, the situation of the arm racebetween the importers and it derives the aggregated demand functionfor each type of weapon exported. Subsection 2.2 describes the twostage problem and stresses the nature of the strategic interactionbetween national champions and governments on the supplier side ofthe international market. Section 3 summarizes the equilibrium andcomments on the results obtained. Finally, section 4 presents the mainconclusions and suggests future lines of research.

2 THE MODEL

Let us summarize the main assumptions of the model. We consider aworld with ns national champions exporting xu x2, • • •,xns weapons atprices PuP2,...,Pni and procuring gug2 gns weapons to theirrespective governments. National champions are the only suppliers ofthe home government. Importers do not have home weapon produc-tion, consequently, they depend completely on foreign firms for theirdefense procurement. Besides, they perceive the weapons produced inthe different countries as imperfect substitutes for each other (see thefigure below for the 1 arms race 2 exporter case).

EXPORTER COUNTRY 1 EXPORTER COUNTRY 2

Government 1tgi

Firm 1 Xi-f IMPORTERt IMPORTER

1]

Government 2t

gi

• <— x2 Firm 2

The arms market with one arms race and 2 exporter countries.

The quality of the weapons produced, q, is a function of the R&Dinvestment that we assume to be exogenous and equal for all weaponproducers. Therefore, all weapons produced have the same given qual-ity. However, firms are not necessarily allowed to export state of theart weaponry; the quality of exported weapons, q, is a proportion,(1 — r), of the maximum available quality:

(1)

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2.1 The Importer Country Problem

The representative importer does not have weapon production and itis involved in an arms race with a rival. There are nr (we use the sub-script V for recipients of exports) two-importer arms races and thus 2«r

importers. The utility function of the representative importer dependson consumption, Cr, and security, Sr - i.e. U=U(Cr, Sr).

The national (balanced trade) budget of the importer forces it tofind the optimal share of consumption and military expenditure:

fr = Gr+Gr = Cr+ £(MV+P,)JH,. (2)

Here, Yr is the national income in terms of the numeraire (which willbe the consumption good), non-military consumption is denoted Cr

and Gr is military expenditure. In general, one would expect somesubstitutability between labour and weapons in the military sector sothat Gr=wrLr+YH'=1Pimi, where L, is the sum of military personnelneeded to manage each kind of weapon and wr is the wage rate in themilitary sector. However, in the interest of tractability we assumethe number of people needed to manage weapons4 coincides with thenumber of weapons imported from each of the firms; L;=m;. Hence,we can express the total defence expenditure of the representativeimporter as GI.=^"s

=1(wr+Pi)Wi-Security is given by:

S, = aLT?t+firK-K*. (3)

This definition of security reflects the arms race situation. It is simi-lar to that found in the existing literature on arms races (Sandier andHartley (1995)) where security is positively related with the home mili-tary capability, denoted K, and negatively related with the adversary'smilitary capability, denoted K*. The variable benefits of defence, ftr inthe above equation, have been used in the arms trade literature(e.g. Levine and Smith, 1995) and they are assumed to be bigger thanone. Finally, we have the fixed benefits of defense, <x,Yr, which arethe product of a positive constant, ar, and national income, Yr. Theinclusion of national income in the definition is a way of normalizingsecurity for the different countries. It is intuitive to assume that as

4 The one to one proportion is of no importance: the units of labor can always bedefined so that it holds.

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national income increases the perceived security is higher independentlyof the weapons this country has.5

The important difference with respect to the previous literature liesin the precise definition of military capability, K. Usually, the militarycapability of a country is understood as the number of weapons it has.However, in our model K is an aggregate of the arms the countryimports from the different arm producers. This aggregate takes intoaccount both the quality of those weapons and the fact that they areimperfect substitutes for each other in the imports market:

). (4)

Here, m; are the imports of the arms from producer i and (1/(1 — <r))is the elasticity of substitution between any two quality-adjustedweapons. We assume that the elasticity is greater that one (weaponsare imperfect substitutes).6 The military capability of the rival, K*,which also enters the security function is defined in a symmetric way.The problem of the rival is defined in the same way so that we have areciprocal threat which constitutes a negative externality that the twocountries involved in an arms race impose on each other.7

Finally, let us assume that the utility function takes a Cobb Douglasform: U(Cr, 5

lr)=C™r5r

1~tOr. Here, <or, 0<cor<l, represents the weightof consumption in the utility function. The representative importerchooses the amount of weapons to be imported from each weapon pro-ducer that maximizes its utility function (using log U):

Max a>rlog(Fr-G,) + (l-ov)logSrr, (5)

{"•! m,,}

subject to the national budget constraint, Eq. (2), and given the im-ports of the rival, mf, i = l,...,n3 (this implies that the military capa-bility of the rival, K*, is given).

5 One could also argue that in some cases having a higher income makes the countrya better target, the case of Kuwait is one example. However, we consider that in generala country feels safer when it has a higher income.

6 This is a 'quality adjusted' Dixit-Stiglitz utility function. See Dixit and Stiglitz (1977).7 The arms race leads to an individually optimal amount of weapons imported which

is suboptimal from the two countries point of view. Countries are crowding-outconsumption and buying too many weapons. This problem has been extensively treatedin the arms race literature.

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The corresponding first order conditions of the maximization prob-lem presented in Eq. (5) can be derived using Eqs. (2) to (4):

(6)

Under the assumption of symmetry over the utility function and in-come of the countries involved in the arms race we have that in the sym-metric Nash equilibrium for the arms race, all countries import the sameamount of each weapon, w(=mf, i=l,...,ns. Using this we obtain thedemand of the representative country for each weapon as a function ofprices.8 For firm 1 the demand of the representative importer is:

where:

i = l

Its

= 2

1

For the case of wr — 1, ns = 2, q = \ the individual demand functionfor the weapons produced by firm 1 is much simpler:

where:

Using the above expression, it can be shown that (1—ca^crC1 "f}r—corP1ar>0 is a sufficient condition for the demand of firm 1 from therepresentative importer to be decreasing in its own price, Pu anda necessary and sufficient condition for it to be increasing in the

8 See Appendix for the derivation of the representative importer's demand function.

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competitor's price,9 P2- We assume that the equivalent condition forthe general demand function if fulfilled. For this it is enough to imposea sufficiently low ar.

In order to obtain the derivatives of the general demand function(Eq. (7)) we have performed simulations assuming reasonable values forthe different parameters by using the stylized facts about the arms mar-ket10 for a sufficiently low <xr. From those simulations, we conclude thatthe demand function depends negatively on own price and positively onthe competitor's price, as it would have been expected since weaponsproduced by different countries are imperfect substitutes for each other.We also find that the demand function is convex. This means that, whenthe quantity of weapons imported is high and prices are low demand ismore sensitive to variations in prices. Somewhat more surprisingly, wealso find that an increase in the competitors weapon's price makes thefirm's demand function steeper (when one of the firm imposes a highprice, a small decrease in its competitors' price will cause a big increasein the demand of the competitor's product). As we will see later, thisproperty of the demand function will have important implications forthe interaction between firms.

In the symmetric equilibrium of the whole game Pi=P, mi—m, gt=g,<f>i=\, n=ns{wr+P) and i=qnlla. In this case, Eq. (5) gives the symmet-ric Nash equilibrium demand for each good from each country, m:

f,m = —z — qn\>°

9 Let us note that this condition is weaker than the condition for the individualdemand to be positive, hence, there is the possibility that the demand of a firm from therepresentative country decreases when the competitor's price increases. In a differen-tiated oligopoly, the increase of a competitor's price has two effects on the demand of thefirm: a substitution effect which makes the demand increase (consumers decrease thedemand for the good which has become more expensive) and an income effect (realincome diminishes due to the increase in the competitor's price and consumers buy lessof both goods). It is standard to assume that the substitution effect dominates so thatdemand is increasing with the competitor's price (in this case the two goods are said tobe 'gross subtitutes'). However, we have a third effect: the arms race effect. This effect isdue to the fact that the decrease in real income makes the rival country in the arms racedecrease its purchase of weapons as well, this makes it possible for both countries toachieve the same level of security buying less weapons. Because of this effect, an increasein the price of the competitor can decrease the firm's demand even if the substitutioneffect is stronger than the income effect. In order to be consistent with the traditionaloligopoly theory we assume that the substitution effect is sufficiently strong.

10 See Appendix for a description of the values of the parameters we have used inorder to perform these simulations.

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We can now analyze the effect of some parameter changes onthe intensity of the arms race by simply differentiating the demandfunction and see that, for given prices, the individual demand foreach weapon, m, increases as the exported quality, q, increases or adecreases. That is to say, increased quality and/or greater productdifferentiation exacerbates the arms race in the sense that the equilib-rium number of weapons bought from each firm increases in the coun-tries involved in the arms race.

An increase in quality makes the different weapons more attractivefrom the point of view of the importers so they are ready to buy moreof each kind of weapon for a given price. However, in the extreme casewhere the elasticity of substitution is unity, a=0, demand is indepen-dent of quality.11

A decrease in a is equivalent to a decrease in the elasticity ofsubstitution between weapons exported by the different sellers. Thishas a positive effect on the demand of each weapon for given prices.

2.2 The Exporter Country Problem

We have two active agents in the exporter countries: the militaryfirm and the government. Firms are national champions in their owncountry (i.e. the national champion is the only supplier of the domesticgovernment). Also, they participate in the international arms marketcompeting in prices with the other suppliers.

The final outcome in the arms market is the result of a two stagegame. In the first stage the governments of exporter countries committo the amount of weapons to be procured. The price they pay for themis the minimum necessary to keep the national champion in the market(i.e. the price that ensures non-negative profits). This assumptioncorresponds with the fact that governments subsidize their nationalchampions because they are interested in having national military pro-duction. Also, in the first stage the restriction to the quality exported isgiven as an exogenous parameter. In the second stage, firms competein prices in the exports market.

We solve for the subgame perfect equilibrium of the game. That is tosay, in the first stage of the game, governments take into account the

1 1 In this case, ns

1/σ tends to infinity as long as we have more that one seller(ns > 1), and

Y,1 —

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influence their decisions have over both firms' behaviour in the secondstage of the game. However, since governments make their decisionssimultaneously they do take each others actions as given. In the secondstage, firms maximize profits taking the decisions of the governmentsas given. Also, as firms decide simultaneously they also take eachothers' actions as given.

2.2.1 The national champion problemIn the representative exporter country, the government first commitsto a level of procurement, gt, and the price to be paid for the procure-ment, pi. The national champion satisfies the domestic defense pro-curement at the price given by the government and competes in priceswith the other firms in the exports market. Therefore, firms maximizeprofits, nt:

x ( )2 -A (8)

with respect to the exports price, Pt, and taking domestic procure-ment, gi, the price that the government pays for procurement, pu andthe other firms' export prices, Pj,j¥=i, as given.

In the above expression, D are the fixed R&D costs, c is a positiveconstant and d will be assumed to be a negative constant in order togenerate increasing returns to scale in production costs. Finally, xt isthe aggregated demand for exporter / and it is given by:

Xi=2nrmi, (9)

where mt represents the demand that exporter i has from each buyer(Eq. (7)). Let us note that (3xI/3/>i)=2nr(3wi/5i'i), consequently the ag-gregate demand function has the same basic properties as the individualdemand function which was studied in the previous section.12

The first order conditions for the maximization of firms' profits arethen given by:

Xi))~ = 0, i = l,...,/»,. (10)

Equation (10) is also the reaction function of firm / to the prices ofthe other firms. The solution to the system of reaction functions givesthe equilibrium prices as a function of the decisions of the governmentsand the rest of the parameters. Notice that the first order condition isindependent of the price that the domestic government pays to the

12See Appendix for the derivation of mi/ Pi.

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national champion per unit of defense procurement, p f : it onlydepends on the amount of defense procurement, gt, through its effecton marginal cost of production. Therefore, the price that the govern-ment pays for domestic procurement does not have a direct effect onthe decisions of the firm.

Let us now state the problem for the case when we have a duopolyin the exports market (ns=2), with 2 firms we already capture thestrategic effects we want to highlight. In order to derive the influenceof the domestic demand for weapons over the equilibrium exportprices, we use the Implicit Function Rule. The two subscripts in F]',i = l,2 refer to the variables with respect to which the profit func-tion is differentiated firstly and secondly. Therefore, noting fromEq.(10)thatrip2gl=0:

(12)

From Eq. (10) we get:

Assuming that the profit function is concave, flpiPi*^' and that owneffects dominate cross effects13 implies that the denominator Eqs. (11) and(12) must be positive. Therefore, dPJdgi is of the same sign as \\plgl

while the sign of dPildgi same as the sign of rip^i Fli^PrThe sign of FTp, Pl depends on the elasticity of the demand func-

tion. For the case of increasing returns to scale (henceforth IRTS),if demand becomes more elastic when the competitor's price

13 This amounts to assuming that the oligopoly equilibrium is globally stable.

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increases,14 then the competitor will respond to an increase in thefirm's price by cutting its price. Hence, fjpi p2

1S negative which impliesthat the reaction functions of the firms competing in prices are nega-tively sloped. That is, prices are strategic substitutes in the arms exportmarket.15

In the presence of IRTS, Y\plgl is negative. Intuitively, because ofIRTS, an increase in government procurement decreases, the marginalcost of the home firm thereby increasing its aggressiveness in theexport market. Consequently, an increase in gt leads to a lower equi-librium value for Pt.

The effect of a change in gx on P2 is only indirect and dependson how firm 2 responds to firm l's more aggressive behavior. Whenthe reaction functions are downward sloping, ]1P2J>I<0>

ant^ t n e

inward shift in firm l's reaction function leads to a higher equilibriumlevel of P2.

In other words, firm l's more aggressive behavior induces firm 2 to'back off and behave less aggressively in the export market. This, inturn, creates an incentive for a government to expand defence procure-ment in order to give its national champion a strategic advantage inthe export market.16

2.2.2 The government problemAs the importer country, the utility function of the government of theexporter country (we will use the lsV subscript for seller country i),U(Csi,Ssi), depends on consumption, Csi, and security, Ssi.

The national (balanced trade) budget of the representativeexporter is:

Ys=Csi+Gsi=Csi+(.ws+Pi)gi. (13)

Here, Gsi={pr\-ws)gi is military expenditure where, pt is the pricepaid by the government that ensures that firms have non-negativeprofits, it is defined as:

i+xd + djgi+x^ + D-PiXi

14 As was said earlier, the simulations performed show that this is indeed the behaviorpresented by our demand function.

15 For an explanation of this terminology see Bulow, Geanakoplos and Klemperer(1985).

16 This type of analysis is frequent in the 'strategic trade literature'. See for instance:Eaton and Grossman (1986) and Krugman (1984).

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Let us recall that because of the way the procurement price is set,firms make zero profits. This means that firm's profits are not part ofthe government's objective function. Therefore, we will not have theusual 'profit shifting' incentives so typical of the 'strategic trade litera-ture'. Instead there will be a 'scale incentive' apart from the securityconcern in governments that we now present.

The security perceived by exporter /, Ssi, is defined as:

Ssi=ocs?s+qgi-psXs. (15)

Here, /?5 captures the weight of exports in the security function ofthe exporters, a5 Fs are the fixed benefits of defense and q is the qualityof the defence procurement defined in Eq. (1) which does not coincidewith the quality of the weapons exported whenever there is a positiverestriction to the quality exported (r>0).

This definition of security reflects the fact that seller countriesare not directly involved in an arms race with any country, indeedthe weapons the other sellers keep for their own procurement do notaffect their perception of security. However, they do perceive the 'de-stabilizing' effects that exports, both own and others' exports, have c .the arms races in which importers are involved. There are manyevidences of this 'de-stabilizing' effects: the Gulf War is a well knownexample, a most recent one is the situation of conflict in Central Africabeing fuelled with the exports of countries like Spain or France.

However, it is not exports themselves but exported military capa-bility which affects the exporters' security in a negative way: Xs is theaggregate of the exports:

(16)

As we did on the international demand side of the market, we assumethe Cobb Douglas form for the utility function of the seller countries,this is, Ul(Csi,Ssi) = C^Sli~

a: Where, cos captures the weight of con-sumption in the utility function of the seller countries, 0<cus<l. Therepresentative government maximizes its utility with respect to thedomestic defense procurement country, gt. The problem is redefined as:

Max loglf, = coslog(Csj) + (1 -oiJlogS',,

subject to the national budget constraint, Eq. (13).As in the importer countries, the governments of the exporter coun-

tries move simultaneously. However, countries move first with respect

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to all of the national champions so that they perceive the effect of theiractions not only on their domestic national champion but on the otherones as well.

Hence, the first order condition for government i can be stated as:

dlogl/,- cos dCsi (1 — cos) dSsi—5 • ~F<—J ' o A = ^ > ( '

dgi Csi agi Ssi agi

where, using Eqs. (13) to (16):

,dgi

dS,,-dgi

Here, dxi/dgj reflects the effect that a variation in the amount ofweapons the domestic firm must produce for its government has on theamount of weapons exported by each of the countries involved in com-petition for the international market. At this stage the governmenttakes into account the effect of governmental decisions on the deci-sions taken by firms in stage 2. The strategic interactions between thedecisions of the two governments are complex.

First, a variation in domestic procurement in exporter country iaffects consumption: the variation in domestic procurement changesthe expenditure in procurement for given procurement prices; theincrease in procurement also affects the price the government pays forthis procurement. The procurement price changes both becausedomestic procurement increases and because exports and exportsprice change as a result of the increase in domestic procurement. Thissecond effect comes from the fact that the governments move first andforesee the impact of their actions on the firms' decisions. The strategicinteraction between governments and firms comes through the effect ofan increase in domestic procurement on the cost function of the firm:indeed, the properties of the cost function will have a decisive rolewhen we analyze the comparative statics.

Second, governments perceive the global effect of their actions ondomestic security, they take into account not only the positive directeffect of an increase in domestic procurement but also, the indirect im-pact that an variation in domestic procurement has on security throughits effect on the national champion's and the other firms' exports.

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Since governments move first with respect to all of the firms, they do seethe strategic interaction not only with the domestic firm but with theforeign firms as well. Indeed, though there does not exist a direct armsrace between exporter countries we see that there exists an indirect ri-valry which is generated by exports. Other exporter countries' decisionshave an impact on the domestic security function, not through theweapons they buy from their national champions but through theweapons they export to the rest of the world.

Let us now analyze the strategic interaction between governments andfirms. From Eq. (10) we see that the solution to the problem ofthe firms gives the equilibrium prices as functions of the government'sdemand for weapons. Therefore, since exports are function of firms' pri-ces, we also have them depending on the defense procurement of the ex-porter countries:

Governments perceive the influence they have on the decisions offirms through their decisions about how many weapons to acquire fromthe domestic firm. Therefore, the effect of an increase in domestic pro-curement on the procurement price is:

dp,_dpt dpidx, dptdP,dgi dgi dx( dgt dPt dgi

where

dx; = _&LaP1 dxt dPns

dgi 8P, dgi'"+dPn, dgi-

For the two seller case and government 1, using Eqs. (10), (11), (12) and(14), Eq. (20) can be rewritten as:17

dpi ^PiXj + dgl-cXi-dxl-D | xj, dxJdP2 dP2

dgi g\ gi dxJdPi dg!'

The first term in the RHS of Eq. (21) represents the direct effect ofan increase in domestic procurement on the price to be paid for pro-curement which ensures non-negative profits to the domestic firm: this

17 See Appendix for the technical details.

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effect is clearly negative when the national champion needs the domes-tic market in order to get positive profits.18 The reason is that an in-crease in domestic procurement decreases the fixed costs to be paid perunit of procurement and also reduces the marginal costs of productionbecause of the IRTS. The second term in the RHS represents the in-direct effect that an increase in domestic procurement has on domesticprice through its impact on exports and price of exports. Under theproperties of our demand function for exports this effect is negative aswell. An increase in domestic procurement has a positive effect on thedomestic firm's profits; also, it makes the home firm more aggressivein the exports market; as a result of the downward sloping reactionfunctions in prices, the other firms increase their exports prices whichagain increases the domestic firm's profits, and as a consequence of theincrease in home firm's profits, the procurement price decreases.

Finally, recalling Eq. (18) we can conclude that an increase in do-mestic procurement has a direct negative impact on consumption butthere is also a positive impact because the average price to be paid foreach weapon is lower now. Therefore, the overall effect is ambiguous.

The effect of an increase in domestic procurement on security ispresented in Eq. (19). For the two producers case, the indirect impacton security is:

Here, since (dxi/dgi)>0 and (dx1/dg2)<0 we have that an increasein domestic procurement has a negative impact on security through theincrease in domestic exports, xu and a positive impact through thedecrease in foreign exports, x2. Therefore, the perceived positive effectof an increase in domestic procurement on global exports becomessmaller due to the negative impact on the competitors exports.

Finally, there are different effects positive and negative from theincrease in procurement on the utility function. The results we obtainon the comparative statics will be determined both by the firms and

18 The first term in the RHS of the equation can be written in the following way:

D + (c-Pi)xi+dx\2 '

Clearly, if (Pi— c)x,— dx\—D<0 the above expression is negative. In other words,if the firm makes negative profits when there is no domestic market, an increase indefence procurement has a negative direct effect on the procurement price.

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importers' problem and the importance of these effects in the exportercountries' utility functions.

3 COMPARATIVE STATICS IN THE EQUILIBRIUM

The equilibrium of the two stage game is determined by the aggregatedemand function for weapons imports (Eqs. (7) and (9)), the first orderconditions of the firms (Eq. (10)) and governments (Eq. (17)), all evaluatedat the symmetric equilibrium where Pt=P, x,=x, gt=g. In Figs. (1) to(7), we present the results of comparative statics obtained through thedifferent simulations of the model. That is, the effect that a change inthe most relevant parameters in the model has on the final equilibriumof exports, domestic procurements and their prices. We start by doingcomparative statics on the parameters of the utility functions of impor-ters and exporters of weapons.

In Fig. 1 we see the effect of a variation in the weight that exportersassign to consumption. If the exporter countries get more interested in

0.147 0.148 0.149 0.150 5.25

2 3 4Domestic procurement

0.99Price of domestic procurement

Figure 1 Effect of a change in the weight on consumption of exporters.

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consumption, (i.e. if the weight on security in the utility function de-creases) exports and domestic procurement decrease and their price in-creases. We can establish a comparison with reality: in the last fewyears the 'interest' of the exporters in security has decreased due to theend of the Cold War in which most of them were involved. The predic-tion of the model is that this would decrease both domestic procure-ment and exports, which is what it has been observed in the last fewyears. The reason is that a decrease in procurement increases the mar-ginal costs of firms due to the IRTS assumption and this forces firmsto increase prices of exports, (firms become less aggressive on theexports market), as a consequence, the quantity demanded diminishes.

Figure 2 analyzes the effect of variation in the weight of consump-tion for the importers. As the importer countries get more interestedin security (i.e. if the weight of consumption in the utility functiondiminishes), exports increase, their price increases, domestic procure-ment also increases and its price decreases. Here, importers are willingto pay more for weapons since they have become more interested insecurity, exporters take advantage of this and increase prices. Both this

0.15 0.20Exports

0.700.25 5.0

0.85

0.80

0.75

rwn

5.5

/

/

6.0Price of exports

6.5 7.0

•

0.4 0.6 0.6Domestic procurement

0.5 1.0Price of domestic procurement

Figure 2 Effect of a change in the weight on consumption of importers.

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and the decrease in marginal costs, due to the increase in exports,make the price of procurement decrease and so the domestic govern-ment buys more weapons both because it does not have to subsidizethe firm so much and also in order to compensate for the negativeeffect that the increase in exports has on their security.

In Fig. 3 we see that if the variable benefit of defence for importers,Pr, increases, exports decrease, exports price increases, domestic pro-curement decreases and domestic price increases. The intuition is thatnow, buying the same amount of weapons the perception of securityof the importer increases and the effect is to diminish imports and in-crease consumption. However, the price for these weapons increasesdue to the increased marginal costs, and this constitutes a furtherincentive to decrease imports. Therefore, the domestic price of pro-curement has to compensate for these two effects in order to securenon-negative profits for the firms.

In Fig. 4, Ps should be understood as the impact of exports onthe exporters' security. If ft increases, exports increase and theexports price decreases, domestic procurement increases and its price

0.25 3.5 4.0 4.5 5.0 5.5

0.6 0.8Domestic procurement

10 0.85 0.90 0.95 1.00Price of domestic procurement

Figure 3 Effect of a change in the importers variable benefit of defence.

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0.1465 5.390 5.395 5.400 5.405

0.4 0.6Domestk: procurement


Figure 4 Effect of a change in the weight on exports of exporters.

decreases. That is to say, if exporters are more aware of the negativeimpact exports have on security, they will buy more weapons in orderto increase security: paradoxically this will encourage exports, because,with the increase in domestic procurement, the marginal costs ofproduction decrease and this allows firms to decrease the price ofexports.

Figures 5 to 7 incorporate the comparative statics on the para-meters of the cost function and the restriction to the exported quality.Figure 5 shows that, if weapons exported become closer substitutes,(i.e. if c increases) exports increase while their price decreases anddomestic procurement decreases while its price increases. The intuitionis simple, as weapons become more substitutes, the arms marketbecomes more competitive and so prices decrease which encouragesexports. But, also firm profits diminish and so the price of procure-ment increases in order to subsidize firms. As a consequence it is lessattractive for the domestic government to buy weapons.

In Fig. 6 we see that an increase in the fixed costs of weapons produc-tion discourages exports and domestic procurement. That is, if fixedcosts are higher the price the government has to pay in order to make

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0.13 0.14 0.15 0.16 0.17 3

0.4Domestic procurement


Figure 5 Effect of a change in the elasticity of substitution.

0.45Domestic procurement

0.50 0.0 0.5 t.oPrice of domestic procurement

Figure 6 Effect of a change in fixed R&D costs.

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0.3 0.4Domestic procurement

1.2 1.4Price of Domestic procurement

Figure 7 Effect of a change in the restriction on the quality of weaponsexported.

sure that firms stay in the market has to increase: this discouragesdomestic procurement, and so increases marginal costs of production,consequently, export prices increase and so imports decrease which re-inforces the first effect.

Figure 7 shows that a tighter restriction on the quality allowed to beexported (represented by the parameter r in the model) encouragesthe amount of weapons exported and decreases their price while itdecreases domestic procurement and increases its price. Now, since thequality importers are receiving is lower they are ready to pay less; alsohome governments decrease domestic procurement because, with theincrease in the restriction, they get the same security by buying lessweapons. This also results in a decrease in the price of exports whichhas to be compensated with an increase in the price of domesticprocurement for firms to have non-negative profits. The reason whythe amount of weapons exported increases is that, in our model,quality and quantity are perfect substitutes for each other. In otherwords, weapons are measured in quality units and as a consequence

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a decrease in exported quality is compensated with a higher amountof imports.

4 SUMMARY AND CONCLUSIONS

We have presented a model which reflects the manner in which themajor arms trade has been working after the Cold War. There is asmall number of firms which compete in prices for the export market.The domestic market of producers is characterized by highly subsi-dized firms: in our model we have assumed there is only one firm serv-ing the domestic market. The governments in the exporter countriescare about the effect exports have on security and this is reflected inthe way their utility functions are presented. Also, we use the importsdemand function characteristic of countries which are involved in armraces. We have seen this demand function has some properties whichmake the problem interesting for the trade literature in general, Thesespecial properties together with the assumption of IRTS in the produc-tion of weapons make firms reaction functions in prices have a nega-tive slope instead of positive one as is usually the case in that literature.

In the comparative statics we see this model explains some of thestylized facts in the evolution of the arms trade market like thedecrease in exports and domestic procurement in the exporter coun-tries in the past few years. One of the strengths of this model is thefact that it incorporates the strategic interaction between firm andgovernment in the exporter countries. This, in turn, reveals complexstrategic interactions between weapon exporters. This set-up also helpsto integrate the arms trade literature with the literature on strategictrade. From this point of view, our analysis shows that the additionof security concerns can significantly affect the comparative statics ofthe model.

In our model, the quality of exported weapons was taken asexogenous. This could be modified in two ways. Firstly, the maxi-mum quality of a firm's weapons could be modelled as the outcome ofan endogenous R&D process. Secondly, one could allow home govern-ments to determine the effective quality of weapons for sale on the ex-port market through restrictions on the transfer of strategictechnology.19

19 These extensions are the object of current research.

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298 M.C. GARC1A-ALONSO

REFERENCES

Brander, J.A. and Spencer, B. (1985) Export subsidies and internationalmarket share rivalry. Journal of International Economics 18, 83-100.

Bulow, J.I., Geanakoplos, J.D. and Klemperer, P.D. (1985) Multimarketoligopoly: strategic substitutes and complements. Journal of PoliticalEconomy 93(3), 488-511.

Dixit, A.K. (1984) International trade policy for oligopolistic industries.Economic Journal (supplement) 16, 1-69.

Dixit, A.K. and Stiglitz, J.E. (1977) Monopolistic competition and optimalproduct diversity. American Economic Review June, 297-308.

Eaton, J. and Grossman, G.M. (1986) Optimal trade and industrial policyunder oligopoly. Quarterly Journal, of Economics May, 383-406.

Garcia-Alonso, M.C. and Levine, P. (1996) Domestic procurement, subsidiesand the arms trade. Mimeo. University of Surrey.

Hartley, K. and Sandier, T. (1995) The Handbook of Defence Economics.M. North-Holland, Amsterdam.

Helpman, E. and Krugman, P.R. (1989) Market Structure and Foreign Trade.Cambridge, MA: MIT Press.

Helpman, E. and Krugman, P.R. (1989) Trade Policy and Market Structure.Cambridge, MA: MIT Press.

Kirkpatrick, D.L. (1995) The rising unit cost of defence equipment - thereasons and the results. Defence and Peace Economics 6, 263-288.

Krugman, P. (1984) Import protection as export promotion: internationalcompetition in the presence of oligopolies and economies of scale.Monopolistic Competition in International Trade. Edited by Kierzkousky.Oxford University Press, Oxford.

Levine, P., Sen, S. and Smith, R. (1994) A model of the international armsmarket. Defence and Peace Economics 5, 1-18.

Levine, P. and Smith, R. (1995) The arms trade and arms control. EconomicJournal 105, 471-484.

Lichtenberg, F.R. (1988) The private R&D response to federal design andtechnical competitions. American Economic Review 78(3), 550-559.

Lichtenberg, F.R. (1989) How elastic is the government's demand forweapons? Journal of Public Economics 40, 57-78.

Sandier, T. and Hartley, K. (1995) The Economics of Defense. CambridgeUniversity Press, Cambridge.

SIPRI Yearbook 1996. Oxford University Press, Oxford.Tirole, J. (1987) The Theory of Industrial Organization. Cambridge, MA: MIT

Press.Vestel, P.D. (1995) Defence Markets and Industries in Europe: Time for

Political Decisions? Chaillot Papers 21. Institute for Security StudiesWestern European Union, Paris.

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APPENDIX

Section 2: Derivation of the Import Demand Function of Weapons

Recall Eq. (6) which gives the first order condition for maximization ofthe utility function of the countries involved in the representative armrace (we drop Y subscripts):

This condition can be stated as:

-1 coSq (l-co)j?r ( f r -G) "

Under the assumption of symmetry over the utility function andincome of the countries involved in the arm race the RHS of thisexpression is independent of i.

Hence,

is independent of /,i.e.

W + Pi

or

q=

where <j>i — <t>i[Pi,Pi], no te </>i = l .The above expression gives the relative demand of producer i

weapons with respect to producer 1 weapons.Substituting back into Eq. (6) we obtain:

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In a symmetric Nash equilibrium for the arm race, m^nif andK=K*. Let us note that:

where fi, £ depend on q and P{.Then putting i=\ in Eq. (6):

This equation can be solved for m( in terms of Pi, q, fi and ( obtain-ing the individual demand for weapons produce by firm 1:

Section 2: The Simulations

Since the model is very complex we have decided not to do a directcalibration. Instead we have used the perfect competition and homo-geneous weapons version of the model in order to obtain reasonableassumptions for the main parameters (see: Garcia-Alonso, Levine(1996)). Based on them are the weights of consumption: cwr = a>s=0.85,the parameters of fixed benefits of defense: ar = 0.12, as=0.14, theparameter c of the cost function: c — 1, the wage price ratios in importerand exporter countries: wr/P=0.15, wslp=0A9 and the relationbetween importers and exporters' national income: 2nrYr/nsYs=0.6.Also, as in the simpler model, the variable benefits of defense for theimporter are supported by conventional wisdom: J?r = 3. The impor-tance of exports on the exporters security, /?s, is assumed to be l/«s = l/2,the idea is that each seller is concerned about a proportion of

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exports equal to its market share which is \lns because firms solvesymmetric problems. However, our simulations must introduceassumptions on the characteristics that the simpler model does nothave. In the baseline equilibrium, we have assumed unitary qualityand no exports restriction: q = \ and r = 0. The IRTS assumption isreflected in a negative parameter d of the cost function: d= —0.005,from the assumption of weapons as imperfect substitutes in the exportsmarket we set an arbitrary elasticity of substitution, 1/(1 — a), of1/(1—0.3). Finally, we assumed the following fixed R&D costs:D=0.65.

The different assumptions on parameter values were adjusted inorder to obtain reasonable values for military expenditure/nationalincome ratios for importers and exporters: GrIYr=0.075, Gs/Ys=0.066and fixed R&D costs/national income ratio: 0.065. This values arequite close reasonably closed to reality. Military expenditure is in real-ity higher for the main importer countries which are involved in armsraces with rivals but below 10% of the national income. The proportionof R&D expenditure is below our 6.5%, total R&D expenditure(including civil R&D) is around 3% of national income for the mainexporters, hence, our percentage overestimates military R&D but, it iswe think still reasonably close given the complexity of the model.

Section 2: Derivation of dmJdPi

Write m^ulv. Consider i^2.Then:

dm, _ v(duldP,)-u(dvldP,)~dP~~ V2 '

First note that, for z#l:

d < f > , _ 1 b8P~ l-a\w+Pt) (w + P,)2'D

ownl

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rsity

of

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Hence, from Eq. (14):

~dPi=Yr

Section 2: The Strategic Interaction in the Problemof the Government

Recall Eqs. (18) and (19) that express the effect of an increase in do-mestic procurement both on consumption and security for the repre-sentative seller /:

Here governments know the problem of the firms and so theyhave the firms solution as a function of all governments domesticprocurements:

So we have:

dxi _ dxt dPi dxi dPns

dgt dgt dxf dgi dPt dgt

_ PiXj+dgf - cXi - dxf - D c+2</(gj+*,)-/>; d^ xt dP{

gf Si dgt gtdgi'

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Now, taking Eq. (10) into account, this is, firms are profit maxi-mizers, we substitute in dpildgf.

x{dxt dPt

dgi ~ gf gi \dXildPi dgi

where substitution of dxj/dgj yields:

dpi_Pixl+dgf-cxi-dxf-Dgf

( ( ^L^\__+ g,[dxtldPXdPx dgi

+ '"+ dPns dgi) dgj•

Let us take the 2 producers case and state the expression forproducer 1:

d p t

gj

Xiffdi dxJdP2 dP

Finally, this expression can be rewritten in the following way:

j—cxi—dx\—Dg\

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Price competition in a model of arms trade∗