cost asymmetries and industrial policy in vertically related markets

COST ASYMMETRIES AND INDUSTRIAL POLICY INVERTICALLY RELATED MARKETS*manc_2246 1..17

byYASUSHI KAWABATA†

Nagoya City University

In this paper we examine how the conventional finding from de Meza(Canadian Journal of Economics, Vol. 19 (1986), pp. 347–350) and Neary(Journal of International Economics, Vol. 37 (1994), pp. 197–218) that thecountry with the lowest-cost firm provides the highest subsidy modifies ina model of vertically related markets characterized by Cournot compe-tition. We show that the country where the sum of the costs of final-goodproduction and intermediate-good production are the lowest providesthe largest production subsidies to the final good and/or the intermediategood.

1 Introduction

In the Brander and Spencer (1985) model where a home firm and a foreignfirm compete in a Cournot duopoly in a third-country market, de Meza(1986) and Neary (1994) demonstrated that the country with the lowest-costfirm provides the largest export subsidy.1 In both these analyses, however,vertical industrial relationships are not considered. How then will consider-ation of imperfectly competitive intermediate-good and final-good industriesmodify the conventional result in de Meza (1986) and Neary (1994)? Forinstance, will the country where the costs of producing the final (respectivelyintermediate) good are the lowest provide the highest subsidies to the final(respectively intermediate) good?

World trade in intermediate goods has been expanding rapidly. Accord-ing to the Ministry of Economy, Trade and Industry, Japan (2008), the tradevalues of parts have increased 11.8 times from 1980 through 2005. Due to theactivation of the intraregional trade in intermediate goods, production

* Manuscript received 28.10.09; final version received 18.9.10.† I am grateful to Hiroshi Ohta, Masayuki Okawa, Kaz Miyagiwa, Takao Ohkawa, Toru

Kikuchi, Hiroshi Kurata, Toshihiro Ichida, Rune Jansen Hagen and an anonymous refereefor helpful comments. I am responsible for any remaining errors.

1Collie and de Meza (2003) generalized the result in de Meza (1986) and Neary (1994) byconsidering the possibility of export taxes and/or subsidies in the Nash equilibrium. Theyshowed that the absolute value of the export subsidy or tax given by the country with thelowest-cost firm is larger than that of the other country. Clarke and Collie (2006) extendedthe result in Collie and de Meza (2003) to the case of a Bertrand duopoly, and showed thatthe country with the lowest-cost firm imposes the largest export tax. Bandyopadhyay (1997)showed that the result in de Meza (1986) and Neary (1994) is reversed for inelastic demandbut that the equilibrium is unstable in policy space.

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1

networks have formed in East Asia and the European Union. Thus, indesigning industrial policies it is necessary for policy-makers to take thevertical industrial and trade structure into account.

The purpose of this paper is to examine the impact of the cost asymmetryof final-good production and the cost difference in intermediate-good pro-duction on the production subsidies provided to the final good and/or theintermediate good by the home and the foreign government in a model withvertically related markets.2 We construct a model where the home and foreignfinal-good firms are Cournot competitors in a third-country market andhome and foreign intermediate-good firms operate as Cournot competitors inthe home and foreign intermediate-good markets.

In our model, a production subsidy to the final (respectively intermedi-ate) good has the following effects: (i) a horizontal profit-shifting effectbetween the home and the foreign final-good (respectively intermediate-good) firms, (ii) a vertical profit-shifting effect between the final-good firmsand the intermediate-good firms, and (iii) a vertical efficiency gain effectarising from an increase in intermediate-good (respectively final-good) pro-duction. Depending on the relative magnitudes of these effects, the produc-tion subsidy given by the home government may be larger or smaller thanthat given by the foreign government.

We provide the following outcomes. The country where the sum of thecosts of final-good and intermediate-good production are the lowest providesthe highest production subsidies to the final good and/or the intermediategood. In other words, if the home intermediate-good firm’s costs are suffi-ciently higher than the costs of the foreign intermediate-good firm, the homecountry will provide the lowest production subsidy to the final good, even ifthe home final-good firms have the lowest costs. In addition, if the homefinal-good firm’s costs are much higher than the foreign final-good firm’scosts, the home country will offer the lowest production subsidy to theintermediate good, even if the home intermediate-good firms have the lowestcosts. The result in our model is that the country with the lowest-cost final-good (respectively intermediate-good) firm may give the lowest productionsubsidy to the final (respectively intermediate) good. This is in contrast to theconventional result in de Meza (1986) and Neary (1994). In our model, thereis not only the horizontal effect of the subsidy but also a vertical effect, andwhere the vertical effect dominates the horizontal effect, the conventionalresult reverses.

The remainder of the paper is organized as follows. Section 2 describesthe model and derives the market equilibrium. Section 3 analyses three indus-trial policy games: (i) a game where the home and the foreign government

2Studies concerning strategic trade policy in vertically related markets include Bernhofen (1995,1997), Ishikawa and Spencer (1999), Chang and Sugeta (2004), Hwang et al. (2007) andKawabata (2010).

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© 2011 The AuthorThe Manchester School © 2011 Blackwell Publishing Ltd and The University of Manchester

only use a production subsidy to the final good, (ii) a game where the twogovernments only use a production subsidy to the intermediate good, and (iii)a game where both governments use production subsidies to the intermediateand the final good. Section 4 provides a conclusion.

2 The Model

We consider two vertically related activities in two countries, a home and aforeign country. In each country, there exist n identical firms producing ahomogeneous final good and m identical firms producing a homogeneousintermediate good. The home and the foreign final-good firms export all oftheir output to a third-country final-good market where they engage inCournot competition. The home and the foreign intermediate-good firms actas Cournot competitors in supplying the intermediate good to integratedhome and foreign markets.3 The home and the foreign government subsidizeproduction of the final good and/or the intermediate good.

A three-stage game characterizes the model. In Stage 1, the home and theforeign government set their production subsidies to the final good and/or theintermediate good. In Stage 2, the home and the foreign intermediate-goodfirms choose their supplies of the intermediate good. In Stage 3, taking theprice of the intermediate good as given, the home and the foreign final-goodfirms choose their output of the final good.4 The intermediate-good price issimply the market-clearing price. The solution concept employed is asubgame perfect equilibrium, obtained by a process of backward induction.

2.1 The Final-good Market

The home and the foreign final-good firms produce outputs y1 and y2, respec-tively. In the third country, the price p of the final good is determined by theinverse demand function, p = a - Y (a > 0), where Y = Si=1,2nyi. The produc-tion of one unit of the final good requires one unit of the intermediate good.The cost of transforming one unit of the intermediate good into one unit ofthe final good is c1 for home final-good firms and c2 for foreign final-good

3For a point of comparison, we deal with segmented home and foreign markets for the interme-diate good in Appendix B.

4The assumption that final-good firms take the price of the intermediate good as given is madein Bernhofen (1995, 1997), Ishikawa and Spencer (1999), Hwang et al. (2007) and Kawa-bata (2010). However, this assumption is open to the criticism that the final-good firms haveno market power as buyers of the intermediate good, even though they have market poweras sellers of the final good. See Ishikawa and Spencer (1999) for a detailed discussion of thejustification for this assumption. For example, the market power of the final-good firms asbuyers of the intermediate good becomes negligible when the number of final-good firms, n,in each country is large.

Cost Asymmetries and Industrial Policy 3


firms.5 Letting w denote the price of the intermediate good in the integratedhome and foreign markets, the final-good firm’s marginal costs are w + ci (i =1, 2). The technological difference between the home and the foreign final-good firms accounts for the differences in marginal costs.

The profit functions of the home and the foreign final-good firm aregiven by

Π i i i ip w c s y i= − − +( ) = 1 2, (1)

where s1 and s2 are the specific production subsidies to the final good providedby the home and the foreign government, respectively.

We first set up the conditions determining the Cournot–Nash equilibriumin Stage 3. Given w, the first-order conditions for profit maximization by thehome and the foreign final-good firms under the Cournot assumption are

∂∂

= − − + + ′ = =Π i

ii i i

yp w c s y p i0 1 2, (2)

Solving these conditions simultaneously, we obtain the Cournot–Nash equi-librium outputs of the home and the foreign final-good firms:

y w s sn

w n c s n c s i j i ji i j i i j j, , , ,( ) =+

− − +( ) −( ) + −( )[ ] = ≠1

2 11 1 2α

(3)

2.2 The Intermediate-good Market

The home and the foreign intermediate-good firms produce outputs x1 and x2,respectively. The total supply to the intermediate-good market is given by X≡ Si=1,2mxi.

In Stage 2, the home and the foreign intermediate-good firms anticipatethe derived demand for the intermediate good arising from the Cournot–Nash equilibrium in Stage 3. From the market-clearing condition in theintermediate-good market, i.e. X = Si=1,2nyi(w, s1, s2), we derive the inversedemand function for the intermediate good:

w X s sn

n n X n c s n c s, ,1 2 1 1 2 21

22 2 1( ) = − +( ) − −( ) − −( )[ ]α (4)

where ∂w/∂X = -(2n + 1)/2n.The home and the foreign intermediate-good firms have constant mar-

ginal costs of producing the intermediate good, k1 and k2, respectively. Theprofit functions of the home and the foreign intermediate-good firm are givenby

5We can also assume that one unit of the intermediate good together with one unit of a secondinput is required to produce one unit of the final good, and that, in the home (foreign)country, the second input is supplied to the final-good firms at an exogenously given pricec1(c2).



π i i i iw k v x i= − +( ) = 1 2, (5)

where v1 and v2 are the specific production subsidies to the intermediate goodprovided by the home government and the foreign government, respectively.The first-order conditions for profit maximization by the home and theforeign intermediate-good firms under Cournot behaviour are

∂∂

= − + +∂∂

= =π i

ii i i

xw k v x

wX

i0 1 2, (6)

Solving these first-order conditions simultaneously, we obtain the equilib-rium outputs of the intermediate good:

xn

m nc s c s m k v m k vi i i j j i i j j=

+( ) +( )− −( ) − −( ) − +( ) −( ) + −( )

2 1 2 12 2 1 2α[[ ]

= ≠i j i j, ,1 2(7)

From (4) and (7), the equilibrium price of the intermediate good is

wm

c s c s m k v m k v=+( )

− −( ) − −( ) + −( ) + −( )[ ]12 2 1

2 2 21 1 2 2 1 1 2 2α (8)

Substituting (8) into (3), we have the equilibrium outputs and price of thefinal good:

ym n

m mn m n c s

mn n c s

i i i

j j

=+( ) +( )

− + + +( ) −( )[

+ + +( ) −

12 2 1 2 1

4 4 4 2 1

4 2 1

α

(( ) − −( ) − −( )]= ≠

2 21 2

m k v m k vi j i j

i i j j

, ,

(9)

pm n

m n mn c s mn c s

mn k v

=+( ) +( )

+ +( ) + −( ) + −( )[

+ −(

12 1 2 1

2 2 1 2 2

2

1 1 2 2

1 1

α

)) + −( )]2 2 2mn k v(10)

3 Industrial Policy Game

This section examines the industrial policies of the home and the foreigncountry. In Stage 1, the home and the foreign government independently andsimultaneously set their production subsidies to the final good and/or theintermediate good to maximize their own welfare, realizing the effects of theirintervention on the firms’ decisions in Stages 2 and 3. We analyse threeindustrial policy games: (i) a game where the home and the foreign govern-ment only use a production subsidy to the final good, (ii) a game where thetwo governments only use a production subsidy to the intermediate good,and (iii) a game where both governments use production subsidies to theintermediate and the final good.



The welfare of each country is the total profits of the final-good and theintermediate-good firms, less the cost of the production subsidies:

W n m s ny v mx n p w c y m w k x ii i i i i i i i i i i= + − − = − −( ) + −( ) =Π π 1 2, (11)

3.1 Industrial Policy Game with Production Subsidy to the Final Good

We begin with the case where the home and the foreign government subsidizefinal-good production.6 In the Nash equilibrium, each government maximizesits welfare given the production subsidy to the final good set by the othergovernment. Thus, the first-order conditions for the Nash equilibrium are

∂∂

= − −( ) ∂∂

+∂∂

− −( ) ∂∂

+ −( ) ∂∂

Ws

n p w cys

nyps

ny mxws

m w kxi

ii

i

ii

ii i

ii

i

ssi

i

=

=

0

1 2,(12)

The first term is the horizontal profit-shifting effect: the production subsidyto the final good provided by the home government increases the output ofthe home final-good firms and thereby shifts profits from the foreign to thehome final-good firms. The second term is the terms of trade effect on exportsof the final good: the production subsidy to the final good reduces the exportprice of home final-good firms and worsens the terms of trade. The third termis the vertical profit-shifting effect: the production subsidy to the final goodraises the price paid to intermediate-good firms, and this shifts profits fromhome final-good firms to the foreign intermediate-good firms if the homecountry is a net importer of the intermediate good (i.e. ny1 - mx1 > 0).7 Thefinal term is the vertical efficiency gain effect: the production subsidy to thefinal good increases the output of the home intermediate-good firms andthereby reduces the inefficiency arising from the double marginalization invertical oligopolies.

Using the comparative static results from (A1) in Appendix A, togetherwith (2) and (6), solving (12) yields the Nash equilibrium production subsidiesof both countries:

sm

n mn m nny n x

mmn n

mn n

i i i=+ + +( )

+ +( )[ ]

=+ +( )

+ +( )

24 4 2 1

2 2 1

44 2 1

4 2 1Ω

22 2 1

2 4 4 2 1 0 1 2

m n c k

mn mn m n c k i j i j

i i

j j

+ +( ) − +( )[

+ + + +( ) +( )] > = ≠

α λ

, ,

(13)

6In this subsection, we assume that the production subsidy to the intermediate good is zero (i.e.v1 = v2 = 0).

7The production subsidy to the final good provided by the home government raises the pricereceived by the intermediate-good firm and this shifts profits from the foreign final-goodfirms to the home intermediate-good firms if the home country is a net exporter of theintermediate good (i.e. ny1 - mx1 < 0).



where W ≡ 8n(2n + 3)m2 + 2(4n + 1)(2n + 1)m + (2n + 1)2 and l ≡ 8n(n + 2)m2

+ 2(3n + 1)(2n + 1)m + (2n + 1)2. As shown, the Nash equilibrium productionsubsidies to the final good are positive in both countries. Using (13) tocompare the Nash equilibrium production subsidies yields

s sm

mn nc k c k

mmn n

c c k k1 2 2 2 1 1 2 1 1 24

4 2 14

4 2 1− =

+ ++( ) − +( )[ ] =

+ +−( ) − −( ))[ ]

(14)

If the sum of the marginal costs, w + ci, of producing the final good andthe marginal costs, ki, of producing the intermediate good are lower in thehome country than in the foreign country (i.e. c1 + k1 < c2 + k2), then the homegovernment will provide the largest production subsidy to the final good (i.e.s1 > s2). If the home intermediate-good firms produce at sufficiently highermarginal costs than the foreign intermediate-good firms, the home countrymay give the lowest production subsidy to the final good, even if the homefinal-good firms have the lowest marginal costs. This leads to the followingproposition.

Proposition 1: The country where the total marginal costs of producing thefinal good and the intermediate good are the lowest provides the largestproduction subsidy to the final good. In other words, country i, whosefinal-good firms have the lowest costs, provides the lowest production subsidyto the final good if and only if the cost difference between country i’s andcountry j’s intermediate-good firm is larger than the cost difference betweencountry j’s and country i’s final-good firm (i.e. ki - kj > cj - ci).

We can explain the intuition behind this proposition as follows. Supposethat the home final-good firms have a cost advantage (i.e. c1 < c2) and thehome intermediate-good firms have a cost disadvantage (i.e. k1 > k2). When c1

< c2, the price–cost margin of the home final-good firm is larger than that ofthe foreign final-good firm, so the horizontal profit-shifting effect of the homeproduction subsidy to the final good is greater than that of the foreignfinal-good subsidy.8 This ‘horizontal’ effect induces a stronger incentive bythe home government to subsidize final-good production than the foreigngovernment. On the other hand, when k1 > k2, the price–cost margin of thehome intermediate-good firm is smaller than that of the foreign intermediate-good firm, so the home country’s vertical efficiency gain effect is smaller thanin the foreign country. In addition, the home country that is a net importer ofthe intermediate good has a negative vertical profit-shifting effect because ofthe increase in the price of the intermediate good, but the foreign country that

8When c1 < c2, home final-good firms, to borrow Neary’s (1994) phrase, possess a ‘comparativeadvantage in profit-shifting’, so the pay-off to subsidizing the home final-good firms isgreater than that of subsidizing the foreign final-good firms.



is a net exporter has a positive vertical profit-shifting effect.9 These ‘vertical’effects entail a weaker incentive for the home government to provide aproduction subsidy to the final good than the foreign government. If thedifference in intermediate-good production costs is larger than the differencein final-good production costs (i.e. k1 - k2 > c2 - c1), the vertical effect of theproduction subsidy to the final good outweighs the horizontal effect. Conse-quently, the home government offers the lowest subsidy to final-good pro-duction, even if the home final-good firms have a cost advantage.

From (14), the final-good subsidy differential is decreasing in the numberof final-good firms, but it is increasing in the number of intermediate-goodfirms.

The intuition behind this result can be explained as follows. Suppose thatc2 - c1 > k1 - k2 > 0. Then s1 > s2. As the number of final-good firms increases,the reduction in the price of the final good due to the production subsidy tothe final good becomes larger, and the negative terms of trade effect onfinal-good exports becomes greater. This greatly reduces the incentive for thehome country that exports the final good more as compared with the foreigncountry to subsidize final-good production. Thus, an increase in n decreasesthe difference between s1 and s2. As the number of intermediate-good firmsincreases, the increase in the price of the intermediate good due to the final-good subsidy becomes smaller. Thus, the negative vertical profit-shiftingeffect of the home final-good subsidy becomes weaker, and the positivevertical profit-shifting effect of the foreign subsidy becomes smaller. Thisstrengthens the incentive for the home country to subsidize final-good pro-duction and weakens the foreign country’s incentive. Therefore, an increasein m increases the difference between s1 and s2.

3.2 Industrial Policy Game with Production Subsidy to theIntermediate Good

We next turn to the situation where both governments subsidizeintermediate-good production.10 The first-order conditions for the Nash equi-librium in production subsidies to the intermediate good are

∂∂

= − −( ) ∂∂

+∂∂

− −( ) ∂∂

+ −( ) ∂∂

Wv

n p w cyv

nypv

ny mxwv

m w kxi

ii

i

ii

ii i

ii

i

vvi

i

=

=

0

1 2,(15)

The first term is the vertical efficiency gain effect: the production subsidy tothe intermediate good provided by the home government increases the output

9When both governments pursue a laissez-faire policy, country i’s net imports of the intermediategood is given by nyi - mxi = [n/2(2n + 1)][(2n + 1)(cj - ci) + 2m(ki - kj)] (i, j = 1, 2, i � j). Ifcj - ci > 0 and ki - kj > 0, then nyi - mxi > 0 and nyj - mxj < 0.

10In this subsection, we assume that the production subsidy to the final good is zero (i.e. s1 = s2

= 0).



of home final-good firms, thereby reducing the inefficiency associated withthe double marginalization. The second term is the terms of trade effect onexports of the final good: the production subsidy to the intermediate goodlowers the price of the final good, thereby worsening the terms of trade. Thethird term is the vertical profit-shifting effect: the production subsidy to theintermediate good reduces the price of the intermediate good bought by thefinal-good firms, and this shifts profits from the foreign intermediate-goodfirms to the home final-good firms if the home country is a net importer of theintermediate good (i.e. ny1 - mx1 > 0).11 The final term is the horizontalprofit-shifting effect: the production subsidy to the intermediate goodincreases the output of the home intermediate-good firms, thereby shiftingprofits from the foreign to the home intermediate-good firms.

Using the comparative static results from (A2), together with (2) and (6),solving (15) yields the Nash equilibrium production subsidies:

vn m

ny n x

m mn m nm n mn

i i i=+( )

+ +( )[ ]

=+ + +( )

+ +( ) − +

12 1

2 2 1

14 2 6 1

2 2 1 2 2α mm n c k

n m c k i j i j

i i

j j

+ +( ) +( )[

+ +( ) +( )] > = ≠

4 1

2 1 0 1 2, ,

(16)

The Nash equilibrium production subsidies to the intermediate good arepositive for both countries. Using (16) to compare the Nash equilibriumproduction subsidies yields

v vm

c k c km

k k c c1 2 2 2 1 1 2 1 1 21 1

− = +( ) − +( )[ ] = −( ) − −( )[ ] (17)

If the sum of the marginal costs of the intermediate-good firm and thefinal-good firm are lower in the home country than in the foreign country(i.e. c1 + k1 < c2 + k2), then the home government will provide the largestproduction subsidy to the intermediate good (i.e. v1 > v2). If the home final-good firms are significantly more inefficient than the foreign final-goodfirms, the home country may provide the lowest production subsidy to theintermediate good even if the home intermediate-good firms are more cost-competitive than the foreign intermediate-good firms. This leads to the fol-lowing proposition.

Proposition 2: The country where the total marginal costs of theintermediate-good firm and the final-good firm are the lowest provides thelargest production subsidy to the intermediate good. In other words, country

11The production subsidy to the intermediate good provided by the home government lowers theintermediate-good price received by the intermediate-good firms and thereby shifts profitsfrom the home intermediate-good firms to the foreign final-good firms if the home countryis a net exporter of the intermediate good (i.e. ny1 - mx1 < 0).



i with the lowest-cost intermediate-good firm provides the lowest productionsubsidy to the intermediate good if and only if the cost differential betweencountry i’s and country j’s final-good firm is greater than the cost differentialbetween country j’s and country i’s intermediate-good firm (i.e. ci - cj > kj -ki).

We can explain the intuition behind the above proposition as follows.Suppose that the home intermediate-good firms have the lowest costs (i.e. k1

< k2) and the home final-good firms have the highest costs (i.e. c1 > c2). Whenk1 < k2, the horizontal profit-shifting effect of the home production subsidy tothe intermediate good is larger than that of the foreign intermediate-goodsubsidy. This ‘horizontal’ effect leads the home country to have a greaterincentive than the foreign country to subsidize intermediate-good produc-tion. On the other hand, when c1 > c2, the home country’s vertical efficiencygain effect is smaller than the foreign country. Moreover, the home countrythat is a net exporter of the intermediate good has a negative vertical profit-shifting effect, but the foreign country that is a net importer has a positivevertical profit-shifting effect. These ‘vertical’ effects cause the foreign coun-try’s incentive for a production subsidy to the intermediate good to outweighthat of the home country. If the final-good production cost differential islarger than the intermediate-good production cost differential (i.e. c1 - c2 > k2

- k1), the vertical effect of the production subsidy to the intermediate gooddominates the horizontal effect. Therefore, the home government offers thelowest subsidy to intermediate-good production even if home intermediate-good firms have the lowest costs.

From (17), the intermediate-good subsidy differential is decreasing in thenumber of intermediate-good firms.

The intuition for this result can be explained as follows. Suppose that k2 -k1 > c1 - c2 > 0. Then v1 > v2. As the number of intermediate-good firms increases,the decrease in the price of the intermediate good due to the production subsidyto the intermediate good becomes greater. Thus, the negative vertical profit-shifting effect of the home intermediate-good subsidy becomes larger, and thepositive vertical profit-shifting effect of the foreign subsidy becomes greater.This decreases the incentive for the home country to subsidize intermediate-good production and increases the foreign country’s incentive. As a result, anincrease in m decreases the differential between v1 and v2.

3.3 Industrial Policy Game with Production Subsidies to the Final and theIntermediate Good

We now consider the case where the home and the foreign governmentssubsidize both final-good and intermediate-good production. The first-orderconditions for the Nash equilibrium in production subsidies to the final andthe intermediate good are



∂∂

= − −( ) ∂∂

+∂∂

− −( ) ∂∂

+ −( ) ∂∂

Ws

n p w cys

nyps

ny mxws

m w kxi

ii

i

ii

ii i

ii

i

ssi

= 0

∂∂

= − −( ) ∂∂

+∂∂

− −( ) ∂∂

+ −( ) ∂∂

Wv

n p w cyv

nypv

ny mxwv

m w kxi

ii

i

ii

ii i

ii

i

vvi

i

=

=

0

1 2,(18)

Solving (18) using (2), (6), (A1) and (A2) yields the Nash equilibrium pro-duction subsidies:

sm

n mn m nny n x

n mn m nm n

i i i=+ + +( )

+ +( )[ ]

=+ + +( )

+ +( )

2 2 2 12 2 1

14 6 6 3

2 2 1 αα − + + +( ) +( )[

+ + + +( ) +( )] > = ≠

2 2 2 1

2 2 2 1 0 1 2

mn m n c k

mn m n c k i j i j

i i

j j , ,

(19)

vn

n mn m nny n x

nmn mn m n

i i i=+

+ + +( )+ +( )[ ]

=+

+ + +( )

2 12 2 2 2 1

2 2 1

2 12 4 6 6 3

22 2 1 2 2 2 1

2 2 2 1 0

m n mn m n c k

mn m n c k i

i i

j j

+ +( ) − + + +( ) +( )[

+ + + +( ) +( )] >

α

,, ,j i j= ≠1 2

(20)

The Nash equilibrium production subsidies to the final and the intermediategood are positive for both countries.12 The differential between the two Nashequilibrium production subsidies to the final good (19) is

s sn

c k c kn

c c k k1 2 2 2 1 1 2 1 1 21 1

− = +( ) − +( )[ ] = −( ) − −( )[ ] (21)

The differential between the Nash equilibrium production subsidies to theintermediate good (20) is

v vnmn

c k c knmn

k k c c1 2 2 2 1 1 2 1 1 22 12

2 12

− =+

+( ) − +( )[ ] =+

−( ) − −( )[ ] (22)

If the sum of the final-good firm’s marginal costs and the intermediate-good firm’s marginal costs is lower in the home country than in the foreigncountry (i.e. c1 + k1 < c2 + k2), then the home government will provide thelargest production subsidies to both the final and the intermediate good (i.e.s1 > s2 and v1 > v2). This leads to the following proposition.

12From (19) and (20), it follows that if 2n + 1 > (<)2m, vi > (<)si. That is, if the number ofintermediate-good firms is small (large) relative to the number of final-good firms, theproduction subsidy to the intermediate good is higher (lower) than the production subsidyto the final good.



Proposition 3: The country where the total marginal costs of producing thefinal and the intermediate good are the lowest provides the largest subsidiesto final-good and intermediate-good production.

From (21), the final-good subsidy differential is decreasing in the numberof final-good firms. From (22), the intermediate-good subsidy differential isdecreasing in the number of final-good firms and in the number ofintermediate-good firms.13

4 Conclusion

We have investigated the conventional finding of de Meza (1986) andNeary (1994) that the country with the lowest-cost firm provides the highestsubsidy changes in a model of vertically related markets characterized byCournot competition. We show that the country where the sum of the final-good firm’s costs and the intermediate-good firm’s costs are the lowest pro-vides the largest production subsidies to the final good and/or theintermediate good. In other words, if the costs of home intermediate-good(respectively final-good) firms are sufficiently greater than foreignintermediate-good (respectively final-good) firms, the home country willprovide the lowest production subsidy to the final (respectively intermedi-ate) good, even if the home final-good (respectively intermediate-good)firms have the lowest costs.

We deal with segmented home and foreign markets for the intermedi-ate good in Appendix B. We find the results to be similar to the analysis ofintegrated markets in that whether the production subsidy given by thehome country is higher than that of the foreign country depends on boththe cost asymmetry of final-good production and the cost difference inintermediate-good production. However, the effect of the cost asymmetryof intermediate-good production on the production subsidies to the finaland the intermediate good is greater in segmented-markets than inintegrated-markets.

13The relationship between the intermediate-good subsidy differential and the number of final-good firms can be explained as follows. Suppose that c2 - c1 > k1 - k2 > 0. Then v1 > v2. Asthe number of final-good firms increases, the reduction in the price of the final good due tothe production subsidy to the intermediate good becomes larger. Thus, the negative termsof trade effect on exports of the final good becomes greater. This greatly reduces theincentive for the home country that exports the final good more compared with the foreigncountry to provide a production subsidy to the intermediate good. Consequently, anincrease in n decreases the difference between v1 and v2.



Appendix A

Comparative Static Results for the Effects of Production Subsidies to theFinal and the Intermediate Good

Using (7)–(10), the effects of a production subsidy to the final good are

∂∂

=∂∂

=∂∂

=∂∂

=+( ) +( )

∂∂

=∂∂

=+

xs

xs

xs

xs

nm n

ws

ws m

1

1

1

2

2

1

2

2 1 22 1 2 11

2 2 11( )

∂∂

=∂∂

=+ + +

+( ) +( )∂∂

=∂∂

= −+y

sys

mn m nm n

ys

ys

mn1

1

2

2

1

2

2

1

4 4 2 12 2 1 2 1

4 2nnm n

++( ) +( )

12 2 1 2 1

(A1)

∂∂

=∂∂

= −+( ) +( )

ps

ps

mnm n1 2

22 1 2 1

From (7)–(10), the effects of a production subsidy to the intermediate good are

∂∂

=∂∂

=+( )

+( ) +( )∂∂

=∂∂

= −+( )

xv

xv

n mm n

xv

xv

mnm

1

1

2

2

1

2

2

1

2 12 1 2 1

22 1 2nn +( )1

∂∂

=∂∂

= −+

∂∂

=∂∂

=∂∂

=∂∂

=+( ) +(

wv

wv

mm

yv

yv

yv

yv

mm n1 2

1

1

1

2

2

1

2

22 1 2 1 2 1)) (A2)

∂∂

=∂∂

= −+( ) +( )

pv

pv

mnm n1 2

22 1 2 1

Appendix B

The Case of Segmented Intermediate-good Markets

Consider segmented home and foreign markets for the intermediate good. Denote theprice of the intermediate good in the home country and the foreign country by w1 andw2, respectively. Sales by the home and the foreign intermediate-good firms in thehome market are denoted x1

1 and x21 , respectively. Sales in the foreign market are

denoted x12 and x2

2, respectively. The profit functions of the home and foreign final-good firm are given by

Π i i i i ip w c s y i= − − +( ) = 1 2,

The profit functions of the home and the foreign intermediate-good firm are

π i i i i ii

j i i ijw k v x w k v x i j i j= − +( ) + − +( ) = ≠, ,1 2

The welfare of each country is

W n m s ny v m x x n p w c y m w k x m wi i i i i i ii

ij

i i i i i ii

j= + − − +( ) = − −( ) + −( ) +Π π −−( )= ≠

k xi j i j

i ij

, ,1 2

Using procedures analogous to that in Section 2, the equilibrium values insegmented intermediate-good markets are given by



xn

m nn c s n c s m k v m ki

ii i j j i i j=

+( ) +( )− +( ) −( ) + −( ) − +( ) −( ) + −

2 1 2 11 1α vvj( )[ ]

xn

m nn c s n c s m k v m ki

ji i j j i i j=

+( ) +( )+ −( ) − +( ) −( ) − +( ) −( ) + −

2 1 2 11 1α vvj( )[ ]

wm

c s m k v m k vi i i i i j j=+

− −( ) + −( ) + −( )[ ]12 1

α

ym

m nn c s n c s k v k vi i i j j i i j j=

+( ) +( )− +( ) −( ) + −( ) − −( ) − −( )

2 1 2 12 2 1 2α[[ ]

pm n

m n mn c s mn c s

mn k v

i i j j

i i

=+( ) +( )

+ +( ) + −( ) + −( )[

+ −(

12 1 2 1

2 2 1 2 2

2

α

)) + −( )] = ≠2 1 2mn k v i j i jj j , ,

Industrial Policy Game with Production Subsidy to the Final Good. When the homeand the foreign government simultaneously and independently choose their produc-tion subsidies to the final good to maximize national welfare, the Nash equilibriumproduction subsidies are

smn n

n mx n m ny

mn nmn n

i ii

i=+( )

+( ) − − +( )[ ]

=+ +( )

+ +

12 1

2 2 1 2 2 1

12 2 1

4 2Φ

11 4 4 2 1 1

2 4 1 4 2 1

1

2

( ) − + +( ) +( )[

− + + +( ) + +( ) +

α μ

μ

m c mn m n c

mn m n mn n k

i j

i kk i j i jj ] = ≠, ,1 2

where F ≡ 2n(2n + 3)m + (2n + 1)(n + 1), m1 ≡ 2n(n + 2)m + (n + 1)2 and m2 ≡ 8n(n + 1)m2

+ 2n(4n + 5)m + (2n + 1)(n + 1). The Nash equilibrium production subsidy to the finalgood can be negative (a production tax) for one of the two countries.14 Comparing theNash equilibrium production subsidies of the two countries yields

s smn n

m c c m k k1 2 2 1 1 21

2 12 2 1− =

+ +−( ) − +( ) −( )[ ]

Suppose the home intermediate-good firms have the highest costs (i.e. k1 > k2). Ifthe home final-good firms are greatly more efficient than the foreign final-good firmsare, the home country will provide the largest production subsidy to the final good.Otherwise, it will give the lowest production subsidy or impose a production tax. Thisleads to the following proposition.

Proposition A1: Country i, with the most efficient final-good firms (i.e. ci < cj), pro-vides the smallest production subsidy to the final good (i.e. si < sj) if and only if cj - ci

< [(2m + 1)/2m](ki - kj).

14Using xii > 0 (i = 1, 2), we can obtain s1 + s2 > 0. Consequently, there are no cases where the Nash

equilibrium production subsidies to the final good are negative for both countries.



Industrial Policy Game with Production Subsidy to the Intermediate Good. Whenboth governments simultaneously and independently choose their production subsi-dies to the intermediate good to maximize national welfare, the Nash equilibriumproduction subsidies are

vn m

ny n x x

m m mn m n

i i ii

ij=

+( )+ +( ) +( )[ ]

=+( ) + + +( )

12 1

2 2 1

12 2 1 4 2 6 1

2 2mm n m c c

mn m n m k n m

i j

i

+ +( ) +( ) − +[

− + + +( ) +( ) + +(

2 1 2 1

2 2 2 4 1 2 1 4 2 1

1 2α θ θ

)) +( ) ] > = ≠m k i j i jj1 0 1 2, ,

where q1 ≡ 8(n + 1)m2 + 2(8n + 3)m + 2n + 1 and q2 ≡ 8nm2 + 2(4n - 1)m - 2n - 1. The Nashequilibrium production subsidies to the intermediate good are positive for both coun-tries. Comparing the Nash equilibrium production subsidies of the two countries yields

v vm m

m c c m k k1 2 2 1 1 21

2 12 2 1− =

+( )−( ) − +( ) −( )[ ]

Suppose that the home intermediate-good firms have the lowest costs (i.e. k1 <k2). If the home final-good firms are significantly more inefficient than the foreignfinal-good firms are, the home country will give the lowest production subsidy to theintermediate good; otherwise, the largest production subsidy. This leads to the fol-lowing proposition.

Proposition A2: Country i, with the lowest-cost intermediate-good firms (i.e. ki < kj),provides the smallest production subsidy to the intermediate good (i.e. vi < vj) if andonly if kj - ki < [2m/(2m + 1)](ci - cj).

Industrial Policy Game with Production Subsidy to the Final and the IntermediateGood. When both governments non-cooperatively choose their production subsidiesto the final and the intermediate good to maximize their country’s welfare, the Nashequilibrium production subsidies are

smn mn m n

m n mx n mx

mn m

i ii

ij=

+ + +( )+( ) +( ) − +( )[

− − +

14 4 4 3

4 3 2 1 2 1

2 2 2 22 nn ny

mnm m n mn c c m k

i

i j

+( ) ]=

+( )+ − −( ) +( ) − + −

11

4 12 4 2 2 1 4 1 22

1 2 3Ψα γ γ γ ii jk+[ ]2 4γ

vn

mn mn m nnmx n mx m ny

nm

i ii

ij

i=+

+ + +( )+ +( ) + +( )[ ]

=+

2 14 4 4 3

2 2 1 2 1

2 14 nn m

m n mn m c c

m k

i j+( ) +( )+ +( ) +( ) +( ) − +[

− +( )1 2 1

4 1 4 1 2 1 2 2

2 1

1 2

3

Ψα δ δ

δ ii jm k i j i j+ +( ) ] > = ≠2 1 0 1 24δ , ,

where

Ψ ≡ +( ) + + +( ) + +8 2 3 2 12 12 1 2 12 2n n m n n m n

γ 13 2 232 2 8 6 1 1 4 2 1 2 1≡ +( ) + +( ) +( ) − −( ) − −n n m n n m n m n



γ 23 232 1 8 6 5 8 1 2 1≡ +( ) + +( ) + +( ) + +n n m n n m n n m n

γ 32 2 216 2 4 8 10 1 8 12 3≡ +( ) + + +( ) + + +n n m n n m n n

γ 43 2 216 1 32 1 16 16 1 2 1≡ +( ) + +( ) + + +( ) + +n n m n n m n n m n

δ13 2 216 2 4 8 1 1 4 1 1≡ +( ) + +( ) +( ) + +( ) + +n n m n n m n m n

δ23 2 216 1 4 4 3 2 2 2 1 1≡ +( ) + +( ) − + +( ) − −n n m n n m n n m n

δ32 216 2 4 8 8 1 4 3≡ +( ) + + +( ) + +n n m n n m n

δ4 16 1 1 1≡ +( ) +( ) −mn m n

The Nash equilibrium production subsidies to the intermediate good are positive forboth countries. In contrast, the Nash equilibrium production subsidies to the finalgood can be negative (a production tax) for both countries. The differential betweenthe two Nash equilibrium production subsidies to the final good is

s smn

m c c m k k1 2 2 1 1 22

4 12 2 1− =

+−( ) − +( ) −( )[ ]

The differential between the Nash equilibrium production subsidies to the intermedi-ate good is

v vn

mn mm c c m k k1 2 2 1 1 2

2 2 14 1 2 1

2 2 1− =+( )

+( ) +( )−( ) − +( ) −( )[ ]

Suppose that the home intermediate-good firms have a cost disadvantage (i.e. k1

> k2). If the home final-good firm’s cost c1 is much lower than the foreign final-goodfirm’s cost c2, the home country will set the largest production subsidies to theintermediate and the final good (the lowest production tax on the final good whenboth countries tax final-good production). Otherwise, the home country will set thelowest production subsidy to the intermediate good and will set the lowest productionsubsidy or a production tax on the final good (the largest production tax with bilateralproduction taxation on the final good). This leads to the following proposition.

Proposition A3: If 2m(cj - ci) - (2m + 1)(ki - kj) < 0, country i sets the smallestproduction subsidies to the intermediate and the final good (the largest production taxon the final good when both countries tax final-good production).

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cost asymmetries and industrial policy in vertically related markets

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