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International trade policy towards domestic monopolies and

domestic oligopolies

Praveen Kujal and Juan Ruiz∗

Universidad Carlos III de Madrid

This version: October 29th, 2002

Abstract

International trade policy has focused mostly on policy choice in the presence of homogeneous

good domestic monopolies. It is shown that in a differentiated goods oligopoly, where firms invest

in process innovation and later compete in the market, optimal trade policy and welfare outcomes

are strikingly different. Welfare can increase over free trade and taxing (or subsidizing) output

may even become a dominant strategy for both countries. The prisoners dilemma nature of

policy games under domestic monopolies is never observed for domestic oligopolies. Policy choice

is determined by the number of firms in both countries and the degree of product differentiation.

When a country has one domestic firm, (and increasing the number of foreign firms) the choice

of policy instrument is always a subsidy or to remain inactive. However, if one increases the

number of domestic firms to two, then a country taxes, subsidizes, or may not promote R&D

or output depending on the number of firms in the other country and the degree of product

differentiation in a non-linear way. Further, the results are robust to Cournot or Bertrand

competition.

∗Mailing Address:

+(34) 91 624 9875.

+(34) 91 624 9652, e-mail:

http://www.eco.uc3m.es/~jruiz/numbers.pdf.

grant number 00/0064/2000.

Universidad Carlos III de Madrid, Calle Madrid 126, 28903 Getafe, SPAIN. Fax:

Kujal: Tel:+(34) 91 624 9651,e-mail:

jruiz@eco.uc3m.es Updated versions of this paper can be downloaded from

Both authors aknowledge support from Comunidad de Madrid

kujal@eco.uc3m.es.Ruiz:Tel:

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1. Introduction:

The model of two (domestic) monopolies competing in a third market (Brander and Spencer (1985),

henceforth BS) is based on the premise that the behavior of international oligopolies and the

profit shifting argument can be easily studied in such a framework. The main results arising from

this model are that unilateral policy choice is beneficial. While, bilateral policy choice has the

characteristics of a prisoners dilemma as both the countries are worse off. Extensions of this model

have shown that it is sensitive to the nature of market competition (Eaton and Grossman, 1986)

and to the distribution of firms across the exporting countries (Dixit,1984). Others have meanwhile

criticized the model for being a mercantilist model (see Helpman and Krugman, chapter 5, 1994).

Perhaps the most damaging criticism arises due to the sensitiveness of the policy to assumptions on

market structure. A policy maker using these models as a guide towards policy faces the baffling

question about the nature of market competition and what policy to choose1.

The sensitiveness of the BS model to market structure has been addressed by Cabral and Kujal

(1999) and Bagwell and Staiger (1994). Arguing that investing in a strategic variable prior to the

market competition stage captures the notion of entry barriers in oligopolistic industries Cabral and

Kujal show that in a such a scenario, and under product differentiation, both exporting governments

subsidize domestic monopolies under both price and quantity competition. They argue that in the

framework of their model the Eaton Grossman reversal is explained by non-committal on the part

of the government. Whether governments choose the policy instrument taking into account, or

ignoring, firm investment in the strategic variable (a cost reducing technology) determines the

policy reversal under price and quantity competition. If a government is non-committal then the

policy outcome is the Eaton-Grossman tax and if the government commits2to a policy then the

policy outcome is a subsidy. This is true under both price and quantity competition3. Bagwell

and Staiger, though not directly addressing the role of commitment or policy reversal under output

subsidies, show that the nature of R&D policy is independent of market structure. In their model

R&D outcomes are uncertain and they do not directly address the question of equilibrium unilateral

and bilateral strategies for exporting countries, nor explicitly model product differentiation. It

would seem that the degree of product differentiation could be an important determinant for policy

makers as it is often seen that policies reflect the degree of competition the domestic industry faces

1Helpman and Krugman, for example, conjecture that a reasonable policy should be taxing under both Cournot

and Bertrand competition (see chapter-5, 1994).

2A government is assumed to be credible.

3Maggi (1996) studies the choice of strategic trade policy instruments in a model of endogenous market competition.

He shows how Cournot, or Bertrand, outcomes may be a function of whether investment in capacity is credible (if

capacity expansion is costly), or non-credible (if capacity expansion is not costly). If investment in capacity is

non-credible then Bertrand outcomes are observed, and Cournot outcomes occur if the converse is true. The Eaton-

Grossman policy reversal is thus ruled out in such a scenario as price and quantity competition do not occur under

the same cost assumptions.

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from foreign competitors4 5.

In this paper we address the issue of equilibrium unilateral and bilateral strategic trade policy

for exporting oligopolies selling a differentiated good. We argue that though domestic monopolies

competing in a third market reflect oligopolistic competition, they do not capture export promo-

tion policy towards domestic oligopolies that compete in international markets. Given that many

markets are characterized by oligopolies it is of interest to see how policy and welfare may change

under such a scenario and whether results differ from the special case of domestic monopolies.

This is a reasonable question for many reasons. First, given that a domestic monopoly is a

special case (in terms of market structures) the question of the choice of strategic trade policy in

the presence of domestic oligopolies is completely ignored by the exporting domestic monopolies

model (though it captures oligopolistic competition in the foreign market). If the choice of trade

policy instruments is sensitive to market structures (a domestic oligopoly and/or monopoly) then

policy makers should be aware of this aspect. Using the domestic monopoly case as a guide for

policy choice towards other market structures could thus be misleading. Further, if changing the

market structure alters the structure of payoffs (and hence the resultant equilibrium of the game)

then the prisoners’ dilemma nature of policy choice in these models (implicitly assumed by most)

may have to be reconsidered6.

Secondly, product differentiation may play an important role in the choice of a policy instrument.

Given that the degree of product differentiation reflects the competitiveness of a market, it would

be reasonable to argue that the policy instrument may crucially depend upon this. For example,

the profit shifting argument may be less relevant if the goods sold by two firms are substantially

differentiated. In fact, and as we see in the paper, the incentive to subsidize is a decreasing function

levels of product differentiation, and reaches zero for the case of local monopolies. Further, the

incentive for the government to use output, or R&D, subsidies may depend on the degree of product

differentiation.

The issue of looking at domestic oligopolies is directly related to the numbers critique of the

BS models7. The numbers critique only argues that trade policy instruments are sensitive to the

4The exception being Cabral and Kujal. Bagwell and Staiger (1994) do not explicitly model product differentiation,

they have a Hotelling model for price competition and a homogenous goods model for quantity competition.

5For example, Cabral, Kujal and Petrakis (1998) show that if a country has to impose a Voluntary Export Restraint

it will do so depending on the degree of product differentiation. For low levels of product differentiation the home

country may completely want to shut out foreign imports, however, if the goods are significantly differentiated then

the optimal quota will correspond to the free trade level of imports.

6In fact, a standard introduction to such models presents the choice of strategic trade policy as a prisoners’

dilemma game.

7The relative asymmetry in the number of firms across countries determine the choice of trade policy instruments.

In particular if the difference between home and foreign firms is (less) greater than one then the optimal policy is a

tax (subsidy) (see the example in Bhagwati, Panagariya and Srinivasan, 1998). If however, the number of firms at

home and abroad are the same then there is no policy change. Also, Krishna and Thursby (1991) look at optimal

policies in the presence of n firms.

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relative distribution of firms in the foreign and home market8. In this context we put the numbers

critique in a different light. We frame it as policy choice towards a domestic monopoly, or an

oligopoly. We then show that trade policy instruments are sensitive to the number of firms inside a

country and to the degree of product differentiation in a non-linear way. Further, across the various

market structures that can emerge the monopoly case is shown to be a special case.

Moving away from domestic monopolies to domestic oligopolies, the optimal trade policy can

change in two ways. First, increasing the number of firms at home and abroad symmetrically alters

the optimal trade policy instrument for both domestic and foreign governments in the same way.

Note that this is in contrast to the result where the policy instrument does not change if the number

of domestic and foreign firms increases proportionately across both exporting countries. This is

important as it shows that the choice of the policy instrument is also sensitive to the absolute

number of firms in a country and not the relative difference across the countries (as shown by the

numbers critique). Second, in the standard numbers critique, policy instruments change depending

on the relative difference in the number of firms at home and in the foreign country. That is, if nf

is the number of foreign firms and nhis the number of domestic firms, the sign of the equilibrium

subsidy is equal to9the sign of nf+1−nh. We show that in equilibrium, a country may or may not

engage in active trade policy (an output or a R&D subsidy) and the choice of policy will depend

on the degree of product differentiation. If it decides to have an active trade policy, a monopolistic

market structure is always subsidized (R&D or output subsidies), while an oligopolistic market

structure can be either taxed or subsidized depending on the degree of product differentiation.

Oligopolies are always taxed for high levels of product differentiation, and this is true regardless

of the relative asymmetry in the distribution of firms. For oligopolistic market structures in both

countries, the equilibrium policy is to subsidize R&D or output only for low levels of product

differentiation. Further, departing from the case of domestic monopolies, engaging in active policy

increases welfare over free trade for the active country. In the case where only one country is active

in equilibrium, the inactive country may also increase its welfare over free trade, although in most

cases the inactive country obtains a lower welfare than under free trade depending on the degree

of product differentiation. This result is robust to price or quantity competition.

Our results are of interest because several results from the monopoly model have been generally

accepted as applying to strategic trade policy choice.

consensus that the choice of subsidies has the characteristics of a prisoners’ dilemma. We show that

in the case of policy towards R&D this only occurs for the special case of two domestic monopolies,

since in that case the equilibrium outcome (both countries subsidize R&D or output) generates a

lower welfare than free trade. If the countries use output subsidy, we still get a prisoners’ dilemma

but only for a low degree of product differentiation.10Further, countries may, or may not, choose

For example, there has been a general

8The other critiques of this model are: policy reversal under Bertrand competition (Eaton and Grossman, 1986),

the inexistence of domestic consumers, the lack of distinction distinction between short-run and long-run variables,

etc. (see Grossman (1988) for a critique of these models).

9See Bhagwati et al.(1998 p. 397).

10To further check the robustness of our model, we need to extend it to include domestic consumption. Note that

Helpman and Krugman (1994) (chapter 5) criticize the BS model as a pure mercantilist model as it excludes domestic

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policy bilaterally. Bilateral, or unilateral, policy choice will depend on the number of firms in the

other country and the degree of product differentiation. Depending upon the degree of product

differentiation a government may either subsidize or tax oligopolies and government intervention

can be welfare improving. Further, the results are robust to the nature of market competition1112.

We use a model where a domestic oligopoly (or monopoly) compete against a foreign oligopoly

(or monopoly) selling a differentiated good in a third market. In Section-II we solve the specific

model under free trade. In section-III we analyze government incentives to impose a tax/subsidy

on R&D unilaterally, or bilaterally, both under Cournot and Bertrand competition13.14Section-IV

concludes. The appendix contains the results for the case of output subsidies.

2. Free Trade

We use a third-country model to consider the case of many firms located in two different countries

and producing a differentiated good which they sell in a third country. Denote by n the total number

of firms in the world, composed of nhfirms in the home country and nf in the foreign country.

Let H and F be the set of home and foreign firms, respectively. There is a competitive numeraire

sector. Firms operate under constant returns to scale and initially have the same marginal costs of

production c. Firms can invest in a cost saving technology prior to engaging in market competition

and are able to reduce its marginal cost by ∆ by spending∆2

functions. If firm i is in the home country (i ∈ H), then15

2. All firms face symmetric demand

xi(¯ p) =

1

(1 − γ)(1 + (nh+ nf− 1)γ)

a(1 − γ) − pi(1 + γ(nh+ nf− 2)) + γ

?

j∈{H−i}

pjh+

?

j∈F

(2.1)

pjf

.

consumption. We are already working on extensions that include these possibilities plus asymmetric country sizes.

Further, the effect of a large number of firms on welfare (see, Krishna and Thursby (1991)) is also of interest but

beyond the scope of this paper due to the complexity of the problem.

11As in Celia and Kujal (1999) and Bagwell and Staiger (1994).

12This is true as the strategic complementarity (substitutability) of prices (output) is not clear in a two stage with

endogenous sunk costs. See Maggi (1996) for a discussion on this.

13Note that we have also analyzed results under output subsidies. Given that the general results are similar to the

R&D subsidy case we only present results for R&D subsidy.

14Note, the strategic complementarity between prices and the strategic substitutability between output is the

reason a policy reversal is observed from Bertrand to Cournot games. This is generally a part of the introduction

to a discussion on strategic trade policy (see for example, Brander, 1995, Helpman and Krugman, 1994, Bhagwati,

Panagariya and Srinivasan, 1998 among others).

15These are the demand functions of a consumer with utility u(x1,x2,...,xn) = a(?n

−1

2

i?=j

of goods. Resulting inverse demand is pi = a − xi− γ(?

i=1xi)

??n

i=1x2

i+ 2γ

??

xixj

??

+m with m representing money, generalizing Dixit (1979) for an arbitrary number

h?=ixh).

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where xiis the output produced by firm i ∈ H, p•hand p•fare the prices charged for the home

and foreign varieties of the good, respectively and ¯ p = {p1h,p2h,...,pnhh,p1f,p2f,...,pnff} is the

vector of prices16. The parameter γ measures the degree of product differentiation, and we assume

it between zero and one17. As γ approaches zero each firm becomes a local monopolist and as γ

approaches one, goods become almost perfect substitutes.

Firms play a two-stage game. In stage one, firms simultaneously decide how much to invest in

cost saving R&D (∆i). In stage two, given the reduced unit cost, firms simultaneously compete in

prices, or quantities. In this context, investment in R&D has a commitment value, as firms can use

R&D strategically to improve their position in the subsequent market competition stage. We look

for the subgame perfect equilibria of the game.

Note that our interest is to capture a fundamental aspect of entry barriers in oligopolistic

industries. We do this by modelling firm investment in innovation in an earlier stage that has

strategic value for both firms18. Firm investment in a strategic variable prior to market competition

captures firm investment in a long run variable (see Grossman, 1988 and Herguera, Kujal and

Petrakis, 1997). As argued by Grossman (1988), firm investment in quality, innovation (or any

such variable) has commitment value and should have an important effect not only on market

competition in the later stages but also on the choice of trade policy instruments. This aspect of

modeling oligopolies has not been extensively studied by international trade theorists.

We analyze both quantity competition and the price competition cases.

2.1. Cournot competition

2.1.1. The output choice stage

Firm i chooses xito maximize profits, given inverse demand pi= a − xi− γ(?

reduced unit costs (c − ∆i). Firm i’s problem is:

j∈{H∪F−i}xj) and

max

xi

(a − xi− γ

?

j∈{H∪F−i}

xj

− (c − ∆i)

xi−∆2

i

2

(2.2)

with xj∈{H∪F−i}and ∆itaken as given. Each firm’s reaction function is thus given by:

16Note that there are nh varieties of the home good and nf varieties of the foreign good, each with a (potentially)

different price.

17This is a sufficient condition to assure concavity of the utility function.

18Note that unlike justifying an oligopoly by exogenously imposing a fixed cost (that has no strategic value in the

market competition stage), we endogenize sunk costs (in the sense that the choice of the strategic variable now plays

an important role in the market competition stage).

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xi(x−i) =1

2

a − c + ∆i− γ

?

j∈{H∪F−i}

xj

,for all i ∈ {H ∪ F}

(2.3)

It is easy to see from the reaction function that the slope of each reaction function is negative

decreasing in the degree of product differentiation. Given that the profit transfer effect depends

on the output shifting effect it is easy to see that the derivative

γ = 0. This simple intuition tells us that we can expect the output shifting effect of a policy to be

smaller and hence the incentive to subsidize decreases as γ gets smaller.

dxi

dx−idecreases in γ and zero for

The intersection of the n = nh+ nfreaction functions gives us the vector of equilibrium quan-

tities x = {x1h,x2h,...,xnhh,x1f,x2f,...,xnff}, each chosen given the output of the other firm.

Equilibrium output and profits (as a function of first-stage R&D expenditures) are, respectively:

ˆ xi(∆;γ) =

(a − c)(2 − γ) + (2 + γ(nf+ nh− 2))∆i− γ

??

j∈{H∪F−i}∆j

?

(2 − γ)(2 + (nf+ nh− 1)γ)

(2.4)

ˆ πi(∆;γ) =

(a − c)(2 − γ) + (2 + γ(nf+ nh− 2))∆i− γ

??

j∈{H∪F−i}∆j

?

(2 − γ)(2 + (nf+ nh− 1)γ)

2

−∆2

i

2.

(2.5)

2.1.2. R&D stage

Firm i, given ∆−i,chooses ∆ito maximize its profits (defined above). Reaction Functions in R&D

expenditures are given by:

∆i(∆−i) =

2[2 + γ(n − 2)]

?

(a − c)(2 − γ) − γ

??

j∈{H∪F−i}∆j

??

8 + γ[(4 − γ2)γ + n2γ[2 − (4 − γ)γ] − 16] − 2n[(4 − γ)(2 − γ)γ − 4]

where n = nh+ nf. Solving the system of n reaction functions for R&D and using symmetry we

can derive the equilibrium level of R&D spending, output, price and profits for each firm:

∆∗(γ) =

2(a − c)[2 + (n − 2)γ]

4 − γ[8 − 6n − 2γ(n − 3)(n − 1) + γ2(n − 1)2],

(2.6)

x∗(γ) =

(a − c)(2 − γ)[2 + (n − 1)γ]

4 − γ[8 − 6n − 2γ(n − 3)(n − 1) + γ2(n − 1)2],

(2.7)

p∗(γ) =a(n − 1)γ2− c(2 − γ)(1 − (n − 1)γ)(2 − (n − 1)γ)

4 − γ[8 − 6n − 2γ(n − 3)(n − 1) + γ2(n − 1)2]

.

(2.8)

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Firms’ profits are then given by

π∗(γ) = (a − c)2((a − c)(2 − γ)[2 + (n − 1)γ])2− 2((a − c)[2 + (n − 2)γ])2

(4 − γ[8 − 6n − 2γ(n − 3)(n − 1) + γ2(n − 1)2])2

.

(2.9)

One should note that a firm has more incentive to invest in cost-reducing R&D under Cournot

competition than under a pure cost-minimizing strategy, since there is a positive strategic effect of

R&D on profits.

2.2. Bertrand competition

2.2.1. The price choice stage

Firm i chooses piso as to maximize profits:

max[pi− (c − ∆i)]xi(¯ p).

(2.10)

with ¯ p−i= {pj}j∈{H∪F−i}and ∆itaken as given. This defines each firm’s reaction function:

pi(¯ p−i) ≡

a(1 − γ) + (c − ∆i)(1 + (nf+ nh− 2)γ) + γ

??

j∈{H∪F−i}pj

?

2(1 + (nf+ nh− 2)γ)

(2.11)

Once more, the intersection of the nf+nhreaction functions determines the vector of equilibrium

prices ¯ p, each chosen given the price of the other firms. As before we can see the effect on the slope

of the reaction function as γ becomes smaller. Differentiating pi with respect any p−i we see

that,

dp−i=

(d2pi

dp−iγ=

us some intuition on how policy outcomes may be affected by γ and n19.

dpi

γ

2(1+(nf+nh−2)γ). Note that the cross price derivative is positive and increasing in γ

1

2(1−2γ+γn)2) and negative and decreasing in n (d2pi

dp−in= −

γ2

2(1−2γ+γn)2). This again gives

Equilibrium prices and profits, as a function of first-stage R&D expenditures are:

ˆ pi(∆;γ) =

(2 + γ(2n − 3))(a(1 − γ) + c(1 + (n − 2)γ)) − (1 + (n − 2)γ)

?

∆i(2 + (n − 2)γ) + γ

??

j∈{H∪F−i}∆j

??

(2 + γ(n − 3))(2 + γ(2n − 3))

(2.12)

and

ˆ πi(∆; γ) =

?

(a − c)(1 − γ)(2 + [2n − 3]γ) + [2 + 3(n − 2)γ + (n2− 5n + 5)γ2]∆i− γ(1 + (n − 2)γ)

??

j∈{H∪F−i}∆j

??2

(1 + (n − 2)γ)

(1 − γ)[2 + γ(n − 3)]2(1 + (n − 1)γ)[2 + γ(2n − 3)]2

−

∆2

i

2

(2.13)

19Note that in a stage game these derivatives alone do not determine the policy outcome. In a two stage game

where firms first invest in a strategic variable it is the net effect from the two stages that determines policy outcomes.

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where n = nf+ nh.

2.2.2. The R&D stage

Firm i, given ∆−i,chooses ∆ito maximize its profits (defined above). From the first-order condi-

tions and symmetry we obtain optimal R&D spending, output, price and profits for each of the n

firms:

∆∗(γ) =2(a − c)[1 + (n − 2)γ](2 + γ[5γ − 6 + n(3 + [n − 5]γ)])

D1(γ)

,

(2.14)

x∗(γ) =

[a[4 + 14(n − 2)γ + (77 − (77 − 18n)n)γ2+ 5(n − 2)(11 + n(2n − 11))γ3+

+(89 − n[178 + n(−119 − 2(n − 15)n)])γ4+ 2(n − 3)(n − 2)(n − 1)(2n − 3)γ5]+

+c(2 + (n − 3)γ)(1 + (n − 2)γ)(1 − (n − 1)2γ2)(2 + (2n − 3)γ)]

(1 − γ)(1 + (n − 1)γ)D1(γ)

p∗(γ) =

c(2 + (n − 3)γ)(1 + (n − 2)γ)(1 + (n − 1)γ)(2 + (2n − 3)γ)+

+a(n − 1)γ2[3 + 5(n − 2)γ + (n − 3)(2n − 3)γ2]

D1(γ)

(2.15)

Firms’ profits are then given by

π∗(γ) =

(a − c)2(1 + (n − 2)γ)×

[8 + 32(n − 2)γ + (216 − 216n + 50n2)γ2+ 2(n − 2)(100 + n(19n − 100))γ3−

(439 − n[878 + n{−603 + 2(82 − 7n)n}])γ4+ (n − 2)(139 + n[−278 + n(179 + 2(n − 20)n)])γ5− (3 − 2n)2(n − 3)2(n − 1)γ6]

D1(γ)2

.

(2.16)

where, D1(γ) = 4+γ(14n−24+2(3n−7)(3n−4)γ+[−61+n(108+n(10n−59))]γ2+(n−3)2(n−

1)(2n − 3)γ3).

It should be noted that a firm has less incentive to invest in cost-reducing R&D under price

competition than under a pure cost-minimizing strategy, since there is a negative strategic effect of

R&D on profits: as a response to firm i’s reduction of unit costs, its rival decreases its price, thus

shifting i’s demand inwards. Firm i then has to reduce its price in order to sell the same output.

By lowering its R&D expenditures beyond the cost minimizing level, a firm can commit to softer

competition in the subsequent market game.

It can now be clearly seen that firms invest more in R&D under Cournot than under Bertrand

competition. Further, due to the competitive nature of the Bertrand game, Bertrand competition

results in higher output and lower profits than under Cournot competition.

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2.3. Relating n and γ

Given that firms bear sunk costs prior to market competition it is clear that we can obtain a

relationship between product differentiation and the number of firms from the profit equations for

both price and quantity competition. This relationship is obtained by solving for the locus of n

and γ that give zero profits. We show that the maximum number of firms that can be supported in

the industry (n), decreases as the good produced becomes increasingly homogeneous. This result is

important as it shows that if sunk costs are endogenous then homogeneous goods industries cannot

support a large number of firms in the market. For an increasing degree of product differentiation

we see that a greater number of firms (local monopolies) can be accommodated in the market. This

relationship is shown in figures 2.1 and 2.2.

00.2 0.40.60.81

g

20

40

60

80

100

n

Figure 2.1: Cournot Competition: Maximum number of firms (n) that can be sustained for a

given degree of product differentiation (γ)

3. R&D subsidies

We look at government policy for R&D subsidies towards domestic monopolies and duopolies under

output competition. It is shown that a country, if it decides to engage in active R&D policy, will

tax a duopoly if the other country has a monopoly. If the other country also has a duopoly then, in

equilibrium, the home country taxes R&D for high levels of product differentiation only20. We show

20Helpman and Krugman (1992) ask the question on what is the right policy? They argue that the best policy

is the one that maximizes welfare. They further argue (p. 102) “that the case for export subsidies is very fragile

indeed.” Our results agree with what they argue when there are domestic oligopolies in both the countries.

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00.20.40.60.81

g

20

40

60

80

100

n

Figure 2.2: Bertrand Competition: Maximum number of firms (n) that can be sustained for a

given degree of product differentiation (γ)

that increasing the number of firms at home and abroad, symmetrically or asymmetrically, changes

the optimal trade policy instrument.21However, we show that taxing and welfare improvement

(over free trade) is not just an artifact of Bertrand competition. We show this to be true for

both quantity and price competition. Though, as shown by Cabral and Kujal (1999) a domestic

monopoly is always subsidized if the other country also has a monopoly. Further, even though

policy reversal is not to be observed, the prisoners’ dilemma nature of policy choice is still observed

in the case of domestic monopolies. However, as we will see in the paper this aspect of strategic

trade policy does not occur if we consider duopolies.

In this section we first present the results for R&D policy when we have a monopoly in each

country under Cournot competition in the third market. Following this we present the case of two

duopolies. We finish with the effects of introducing asymmetry in the model by looking at the

case of a monopoly in one country and a duopoly in the other. Note that we only present the

case of foreign and domestic duopoly. However, similar results are obtained under a symmetric

distribution of firms.

3.1. Cournot competition22

21Thus domestic monopolies do not capture trade policy towards domestic (and international) oligopolies.

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First we briefly present the results from Cabral and Kujal (1999) for domestic monopolies. Their

main results are that if firms invest in a strategic variable prior to the market competition stage

and the government takes into account this investment (credible commitment on the part of the

government) then the policy reversal (as shown by Eaton and Grossman, 1986) is not observed.

They further show that under low degrees of product differentiation both governments subsidize

R&D23, while output is always subsidized. Thus either instrument dominates depending on the

degree of product differentiation.

Then we present results for a R&D subsidy/tax, unilateral or bilateral, for a country with a

domestic monopoly or duopoly and solve for the equilibrium in the R&D policy game played in the

first stage24. We consider this to be the relevant case in our paper as under GATT ruling subsidies

to R&D are allowed25. We assume that there are firms in two countries selling a differentiated

good in a third market. We assume that in the first stage of the game the two governments decide

simultaneously whether to engage in active R&D policy, and if so, commit to a subsidy (or tax)

on R&D. Given the policy announcement of both governments, firms choose the profit maximizing

level of R&D in the second stage. In the third stage they compete in quantities, or prices26.

3.2. Domestic and Foreign monopoly

Cabral and Kujal (1999) show that if firms first invest in a strategic variable and governments

commit to a policy prior to this decision the policy reversal result case due to Eaton and Grossman

will not be observed. They present results for both output and R&D subsidy and show that

qualitatively similar policy results are obtained under both price and quantity competition.

We will focus on degrees of product differentiation for which we obtain an interior solution.

When both countries have just one firm, then under a R&D subsidy (and as shown earlier by

Cabral and Kujal) we need to restrict27to γ < 0.663916 for the case of unilateral policy, and to

γ < 0.585998 for the case of bilateral policy28. We thus see that the permissible value of γ decreases

under bilateral policy for a the case of two monopolies.

We show that (and as shown by Cabral and Kujal) active bilateral policy is observed only for

the case when γ < .427853. That is, for an increasing level of product differentiation both countries

22Results for Bertrand competition are available under request. As mentioned earlier the qualitative policy results

do not change.

23For high degrees of product differentiation, only one country engages in active R&D policy.

24At this point we would like to point out that in a two stage we cannot make the convienient symmetry assumption

on output that is conventional in one stage games. We need to explicitly solve for the reaction functions for each

firm and then solve the problem in the R&D stage. We were unable to achieve this due to analytical complexity. We

thus solve the problem only for domestic duopolies.

25Later on in the paper we discuss the qualitative differences between subsidizing R&D or output.

26Note, as the policy results do not change between quantity and price competition we only present the quantity

competition case in the paper. The price competition results are attached in the appendix.

27This range ensures that output and R&D choices of the firm in the other country are positive

28This restriction prevents imaginary roots for the equilibrium bilateral subsidy.

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find it profitable to engage in export promoting policies (R&D subsidies). Further we show that for

a range γ ∈ (0.427853,0.585998) there are two equilibria and in each, only one country subsidizes

R&D.

Proposition 1 (Equilibrium for R&D subsidies with (1,1) firms). If we restrict attention

to values of γ for which an interior solution exists (i.e. γ < 0.585998) then the equilibrium of the

policy game is as follows:

• For γ < 0.427853 both countries subsidize R&D.

• For 0.427853 < γ < 0.585998 we have two equilibria.

subsidizes R&D while the other does not engage in active trade policy.

In each equilibrium one country

Proof.

table 2 shows that if both countries engage in active policy, they both subsidize R&D. Note that if

country f is not subsidizing, country h prefers to unilaterally engage in active policy rather than

remaining on free trade (see figure a - Table 3). On the other hand, if country f subsidizes, country

h prefers to subsidize as well (bilateral subsidies) for γ < 0.427853, otherwise, country h prefers to

remain inactive (see figure a - Table 5).

Figure a in table 1 shows that countries want to unilaterally subsidize R&D. Figure a in

Proposition 2 (Welfare under R&D subsidies for (1,1) case). Restrict attention to values

of γ that generate an interior solution. If we compare the equilibrium of the policy game with the

outcome under free trade,

1. Welfare is lower for both countries when they bilaterally subsidize (γ < 0.427853)

2. Welfare is higher for the subsidizing country and lower for the inactive country in the case of

unilateral subsidies (0.427853 < γ < 0.585998).

Proof.

4). However, for 0.427853 < γ < 0.585998 we have an asymmetric equilibrium where one country

subsidizes and the other does not. In figure a of table 3 we can see that the country which subsidizes

is better that under free trade, but in figure 3.1 we can see that the country that does not subsidize

is worse off than under free trade.

Free trade is always welfare improving over bilateral subsidies (see figure a - Table

Note that only for γ < 0.427853 we have the standard prisoners’ dilemma. Thus, even for the

case of a monopolistic market structure the classic prisoners’ dilemma may not be observed for all

γ.

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0.1 0.20.3 0.4 0.50.6

g

0.5

1

1.5

2

wfF.T.

wf unil

Figure 3.1: Welfare comparison for foreign country and R&D Subsidies: Free trade vs, inactive

unilateral subsidies, case nh= 1,nf= 1.

3.3. Unilateral and Bilateral R&D subsidies—the bilateral duopoly case

The interesting result here is that welfare under bilateral export promotion is greater than under

free trade. This result is totally different from the generally accepted prisoners’ dilemma nature of

these games under bilateral export promotion. Moreover, we observe welfare improvement under

quantity and price competition. Further, the optimal policy is a tax for both countries when both

of them engage in active policy, otherwise, if only one country is active, it subsidizes R&D.

Now, why does the policy instrument change from the case of two monopolies? To understand

this one has to look at the effect on R&D investment and the resulting effect on output (and hence

prices). We see that when both countries have two firms, a tax on R&D decreases R&D expenditure

for both domestic and foreign firms. As a result the overinvestment in R&D is softened and firms

lower output and increase the price, which consequently raises profits (and welfare). Note that a

similar intuition is put forth by Helpman and Krugman (1994). They argue that a tax achieves tacit

collusion between the firms as it increases prices and decrease output (and R&D expenditure in

our model). Further, as the profit transfer effect under bilateral choice is absent, domestic welfare

increases. This is precisely what we see under quantity competition in our model. For the case of

price competition our results are in the same direction29. Our result is of interest because it shows,

first, that qualitative results are robust to the nature of market competition. Further, that welfare

results are also independent of market structure and the prisoners dilemma nature of policy choice

may only be restricted to the case of bilateral monopolies.

We only present the results for the case of bilateral R&D subsidies/taxes.

ference from our predecessors is that welfare in the equilibrium where both countries subsidize

(γ < 0.514708) is still greater than under free trade. Further, policy choice is shown to be a func-

tion of not only whether the domestic market is monopolistic/duopolistic, it is also a function of

the degree of product differentiation.

The main dif-

29Results for price competition are available upon request.

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First we present the results where only the home country subsidizes/taxes. Then we present our

results in detail for the bilateral R&D subsidy case and we derive the equilibrium in the policy game.

As before, we focus only on the levels of product differentiation that provide interior solutions. For a

R&D subsidy and bilateral duopolies we need to restrict30to γ < 0.665703 for the case of unilateral

policy, and to γ < 0.586505 for the case of bilateral policy31.

3.3.1. Unilateral R&D Subsidies

The output choice stage

quantity competition stage we can write the profit maximization problem for the domestic firm as

The domestic country subsidizes, or taxes, its firm(s). Solving for the

max

xh

(a − xh− γ

?

j∈{H∪F−i}

xj

− (c − ∆h)

xh− (1 − zh)∆2

h

2

and for the foreign firm as,

max

xf

(a − xf− γ

?

j∈{H∪F−i}

xj

− (c − ∆f)

xf−∆2

f

2.

¿From the first order conditions we obtain the reaction functions for the domestic and the foreign

firm, respectively,

xi(x−i) =1

2

a − c + ∆i− γ

?

j∈{H∪F−i}

xj

,for all i ∈ {H ∪ F}

(3.1)

Note that as R&D subsidy only enters the first order conditions in the final stage, the reaction

functions under an R&D subsidy and under free trade are the same. As before, under free trade

the intersection of the nh+ nfreaction functions gives us the vector of equilibrium quantities x =

{x1h,x2h,...,xnhh,x1f,x2f,...,xnff}, each chosen given the output of the other firm. Equilibrium

output and profits (as a function of first-stage R&D expenditures) are, respectively:

ˆ xi(∆;γ) =

(a − c)(2 − γ) + (2 + γ(nf+ nh− 2))∆i− γ

??

j∈{H∪F−i}∆j

?

(2 − γ)(2 + (nf+ nh− 1)γ)

.

(3.2)

R&D choice

equilibrium R&D under unilateral subsidies. We obtain the reaction functions for the domestic and

foreign firms, respectively.

Substituting the equilibrium quantities into the profit equation we solve for the

30This range ensures that output, R&D and welfare are positive

31This restriction prevents imaginary roots for the equilibrium bilateral subsidy.

15