Optimal Industrial Policy in Vertically Related Markets

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Optimal industrial policy in vertically related markets

Siu-Kee Wong* and Yao Liu1

Nankai University, Department of International Economics and Trade,94 Wei Jin Road, Tianjin, 300071, China

(Received 27 May 2008; final version received 29 June 2009)

This paper examines the optimal industrial policy for an industry with a

vertical market structure. A home firm and a foreign firm both import anintermediate good from a third country to produce a final good. How thehome country government sets the optimal industrial policy has to takeaccount of both profit shifting between the two final good producers andbetween the intermediate good producer and the home firm. We explainhow the optimal industrial policy depends on the slope of the demandcurve, the levels of technology spillover and product differentiation. Inparticular, there exists a critical level of technology spillover at whichinvestments of the firms are neither strategic substitutes nor complementsand the optimal industrial policy is always investment tax.

Keywords: strategic trade policy; industrial policy; profit shifting;vertically related markets

JEL Classifications: F12; F13

1. Introduction

Studies of strategic trade policies often suggest that government intervention

can help a domestic firm to compete in the export market. However, whether

exports should be subsidized or taxed depends on the form of competition as

proven in Eaton and Grossman (1986). On the other hand, the result for

industrial policies seems to be more robust. Leahy and Neary (2001) showed

that for linear demand and cost functions, the optimal policy is an

investment subsidy in both the Bertrand and Cournot models. Moreover, an

investment subsidy is more practicable, as an export subsidy is prohibited

under the rules of WTO, while similar restrictions do not apply to subsidy of

R&D investment.

In this paper, we examine whether the government should subsidize the

cost-reducing investment of a firm that faces foreign competition in the

consumption good market and imports an intermediate good in the input

market. In the literature, many papers focus on the strategic effects of trade

The Journal of International Trade & Economic Development

Vol. 20, No. 5, October 2011, 631650


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policies in vertically-related markets. These include Spencer and Jones

(1992), Bernhofen (1997), Chen, Ishikawa and Yu (2004) and Hwang, Lin

and Yang (2007), among many others. Little has been done to consider the

industrial policy in vertically related markets and we try to fill this gap. To

see what the basic determinants of the optimal industrial policy are, many

simplifying assumptions will be made. In particular, we assume that the

intermediate good supplier is a monopoly that does not collude with any

final good producer. We also assume that all functions are linear and the

two firms involved in final good competition are symmetric, bar the absence

of government action in the foreign country.

The results of our analysis are sensitive to the timing of the model.

They depend on whether the intermediate good producer commits before

or after the final good firms make their investment decisions. If the

intermediate producer makes commitment prior to the investmentdecisions, it can be shown that the optimal industrial policy is always

investment subsidy. This basically echoes the result of Leahy and Neary

(2001). In contrast, if the pricing decision of the intermediate good firm

precedes the investment decisions, how the optimal investment policy is set

has to take account of both horizontal and vertical profit shifting. While

the investment subsidy can help the home firm to compete in the

consumption good market, it also raises the demand for intermediate

goods and worsens the terms of trade of the home country. Whether

the home government should subsidize or tax investment depends on therelative magnitude of horizontal and vertical profit shifting. Our main

results are not sensitive to whether the firms compete in a Cournot or a

Bertrand market. The determinants of the optimal industrial policy include

the slope of the demand curve, the level of technology spillover, and the

degree of product differentiation. In particular, there exists a critical level

of technology spillover at which the industrial policy is ineffective in

horizontal profit shifting. This happens when the positive technology

spillover of the foreign investment effect just cancels out the subsequent

negative effect arising at the stage of price or quantity competition. In thiscase, the optimal policy is always an investment tax.

The paper is organized as follows. We introduce the model and examine

the basic factors that determine the optimal industrial policy in Section 2.

Sections 3 and 4 consider Cournot and Bertrand competitions at the final

stage of the game. Section 5 considers the optimal policy when the

investment levels are decided before the intermediate good producer sets the

price. We draw some concluding remarks in Section 6.

2. The model

A home firm and a foreign firm compete in a consumption good market in a

632 S.-K. Wong and Y. Liu


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A home firm and a foreign firm compete in a consumption good market in a

The production of the final good requires the input of an intermediate good,

which is produced by a monopoly firm M. Firm Mdoes not locate in the

home country and operates independently from the final good producers.

Assume that discriminative treatment between firmsHandFis not feasible

due to the possibility of resale of the intermediate good. Firm M sells the

intermediate good to firmsHandFat a uniform pricew.2 It commits to its

strategy by setting the price of the intermediate good.3 The problem will be

modeled as a four-stage game. Figure 1 illustrates the timing of the game.

In the first stage, the home government determines the investment subsidy

rate s. For investment tax, s is negative. At the second stage, the upstream

producer sets the price of the intermediate goodw. At the third stage, the two

final good producers determine the levels of cost-reducing investment. At the

last stage, these two firms compete by setting the output or price levels of the

final good. We look at the optimal investment policy at the subgame perfectequilibrium. Letkandk* be the investment levels of firmsHandF, andAand

Bbe the actions of firmsHandFat the last stage, respectively. ActionsAand

Bwill be the outputs of the firms under Cournot competition or the product

prices under Bertrand competition. For simplicity, we assume that the two

firms are symmetric and the demand function, which will be spelled out in

Sections 3 and 4, is linear. Production of one unit of final good requires one

unit of intermediate good if the two firms do not invest. One unit of investment

reduces one unit of the production cost and also reduces the cost of the other

firm by f units due to technology spillover. Thereby, the unit cost ofproduction for firmsHandFare w7(k fk*) andw7(k* fk). The costof investment is assumed to bek2/2. Denote the prices of firmsHandFby p

andp*, respectively. Letxandybe the outputs of firmsHandF, respectively.

Without any subsidy or tax, the profits of the two firms are

p pwkjk xk2 2

p p wk jk yk2

21

We can write the profits of the firms as functions ofk, k*,A, B, wand s.Assume that the foreign government does not take any action to help firmF,

the profits of the firms are:

Pk; k;A;B;w; s p k; k; A; B; w sk 2

The Journal of International Trade & Economic Development 633


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and

p k; k; A;B;w 3

Solving by backward induction, we consider first the decisions of firmsH

andFat the last stage. The first order conditions for profit maximization are:

PAk; k; A;B;w; s pA k; k

;A;B;w 0 4

pB k; k;A; B; w 0 5

where all the subscripts signify partial differentiation. Equations (4) and (5)

implicitly define functions A(k, k*, w) and B(k, k*, w). Substitute these

functions into equations (2) and (3) to get

p k; k; w p k; k;Ak; k; w;Bk; k;w; w ;

Pk; k; w; s p k; k; w sk6

p k; k; w p k; k; A k; k;w ; B k; k; w ;w 7

Specifically, the first-order condition for firmHisp7w k fk* 7pxxfor Cournot competition and p7w k fk* 7x/xp for Bertrandcompetition. The reduced form profit function of firmH is

p k; k;w pxx k; k; w 2k2

2 for Cournot competition,

p k; k;w x p k; k;w ;p k; k; w 2=xpk2

2:for Bertrand competition

Similar conditions hold for firmF. Note that these functions still hold even if

the investment decisions are made before the intermediate good is priced.

The first-order conditions for optimal investments are:

Pk

k;

k

;w;

s

pk

k;

k

;w

s

0

8

pk k; k;w 0 9

Equations (8) and (9) can be used to obtain the reaction functions of firmsH

and Fat the investment stage. The two reaction functions can be used to

solve for the investment levels as functions of w and s. Substituting these

solutionsk(w, s) andk* (w, s) into equations (6) and (7), we get the reduced

form profit functions:

p w; s p kw; s; k w; s ; w ; P w; s p w; s skw; s 10



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2.1. Upstream firm decision

Firm Mproduces an intermediate good and sells it to firmsHand Fat a

uniform pricew. Assume that the cost of producing one unit of intermediate

good is constant atc. Firm Mchoosesw to maximize its profitpM.

pM w; s wc xy 12

where

x x k w; s ; k w; s ;w y y k w; s ; k w; s ;w

under Cournot competition and

x x p k w; s ; k w; s ;w ;p k w; s ; k w; s ;w y y p k w; s ; k w; s ;w ;p k w; s ; k w; s ;w

under Bertrand competition. The first-order condition for profit maximiza-

tion is

pMw x y w c xwyw xkykkw xkyk kw

0Cournot

pMw x y w cxpyp pwpw pkpkkw pkp

k k

w 0Bertrand 13

We can use equation (13) to solve for the price of the intermediate goodw(s).

2.2. Government policies

Following Brander and Spencer (1983) and Leahy and Neary (2001), we

assume that the home government imposes an investment subsidy that isproportional tok. This specification has an advantage of preserving all the

linearity and symmetry of the result. The downside of this specification is

that, for smallk, the investment subsidy exceeds the investment cost and the

policy becomes very unreasonable. However, we can prove that this

undesirable property would not arise at an equilibrium. See Section A7 in

the Appendix.4 As the home country exports all the output, the welfare

of the home countryWis just the difference between the profit of firmHand

the cost of subsidy.

W s P w s ; s sk w s ; s 14



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14

Differentiate the welfare function with respect to s to get the first-order

condition for welfare maximization:

dW

ds p

k k

ww

sk

s p

k k

ww

sk

s pwws 0From equation (8), we obtain

s pk k

s pk k

w pw

ws

kskwws15

The term pk is the effect of foreign investment on the profit of firmH.

Foreign investment can be friendly pk >0 or unfriendly pk dkdk F where dkdk H and dkdk F are the slopes ofthe reaction curves of the home and foreign countries respectively.5 The

stability condition allows us to determine the sign ofks kwws.

Lemma 1. Under the stability condition,ks kwws 4 0.

Proof: For the proof, see the Appendix.

If the intermediate good market is competitive,w would not change. The

optimal policy in this case, which is identical to Leahy and Neary (2001), can

be obtained by settingws 0.

s pk k

s

ks pk

pkkpkk

pkpkk

pkk

Under the assumptions of symmetry and linear demand function, pkk and

pk always have the same sign and pkk is negative by the second-order

condition. Thereby, an investment subsidy should be used at the optimum if

w remains constant. However, when the subsidy triggers a change inw, the

indirect effect captured by the term pk kw pw

ws

kskwws in equation(15) can change the sign of s. In the following sections, we will derive the

exact forms of expression (15) under Cournot and Bertrand competition in

the final good market.

3. Cournot competition

Assume that firms H and F produce a homogeneous good and they decide



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Assume that firmsHandFproduce a homogeneous good and they decide

good is determined by the inverse demand function p a7b (x y).Substitute the demand function into equation (1). FirmsHand Fmaximize

profit by choosing outputs. Use the first-order conditions (4) and (5) to

obtain the optimal outputs

xaw 2j k 2j1 k

3b 16

ya w 2 j k 2j1 k

3b 17

Substitute equations (16) and (17) into the profit functions to get^P k; k;w; s and p k; k; w and differentiate with respect tok and k*. Ingeneral, investment by firmFhas two effects on the profit of firmH. First,

foreign investment lowers the production cost of firmHthrough technology

spillover. Second, a rise ink* raises the industry output and thus lowers the

price of the final good. For a linear demand function, these two effects also

determine whether k and k* are strategic complements or strategic

substitutes. There is a critical level of technology spillover, denoted byj,

at which these two effects cancel out each other. Setting pk 0 orequivalently pkk 0, it can be shown that j 1=2 under Cournot

competition. The second-order condition requires

pkk 1 2j1

3

2j

3b 1< 0

and the stability condition is

b> 23

2 j 1 j for j< j

b> 29

2 j 1 j for j> j(The term pkk is negatively related tob. The first-order conditions (8) and

(9) can be used to derive the following expressions:

k29

2j aw bs2

b 29

2 j 1 j

bs2

b 23

2 j 1 j 18

k 29

2j aw bs2

2

bs2

219



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kb 2 2 1

b 2 2 1

19

As shown in equation (13), differentiatingpM with respect to w yields the

following equation.

pM

w xy wc xwyw xkyk kw xkyk kw 0

Substitute equations (16)(19) into the equation above and totally

differentiate to get

ws xk xk ksk

s

2 2xw xkxk kwkw

1j4

20

Substitute equations (16) and (17) into equations (6) and (7) anddifferentiate with respect tok* and w to get pk 2pxxk x

23

2j1 xand pw 2pxxwx

23

x. Differentiate equation (19) with respect towand

s to obtain ks and kw. Substitute these results into equation (15) and the

optimal investment policy is given as follows:

sQ

Rx 21

where

Q49

2j2j12b

b 23

2j1j

1j

2 b

2

92j2

R 3kskwws b

2

92j1j

By Lemma 1 and the stability condition,R 4 0. So far we have not imposed

any non-negativity constraint in our maximization problems and it is not

obvious whether the first-order conditions gives a positivex. The followinglemma makes sure that x is positive and hencesign (s) sign (Q).

Lemma 2. Under the second order condition for welfare maximizationd2Wds2

0 and sign (s) sign (Q).


By Lemma 2, the sign of s only depends onj and b. Under the stability

condition, a high b reduces the importance of profit shifting between the

downstream firms relative to the effect of the change inw in equation (15).

Specifically, a rise in b decreases pk and pw in the same proportion, has no



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Specifically, a rise inb decreases pk and pw in the same proportion, has no

investment tax for large b. For j j 1=2; pk 0 ands pwws= ks kwws < 0. The following result can be easily obtained bydifferentiatingQ with respect to j and b.

Lemma 3. Under the stability condition, @Q@j


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As shown in Figure 2, the left and right halves of the set fors 4 0 are

not symmetric. Roughly speaking, the optimal policy is more likely to be a

subsidy forj 5 1/2 because a higherj makes the supply of consumption

good more sensitive to investment subsidy or tax. This induces a larger rise

in the price of the intermediate good and aversely affects the profit of firm

H. Moreover, when j is close to 1/2, the optimal policy is more likely to

be an investment tax. Consider j 1=2; pkk 0 and investments areneither strategic complements nor substitutes. Following the argument in

Eaton and Grossman (1986), the government should take action only if

the domestic firm does not take the foreign firms reaction into

consideration. When j 1/2, any change in k will not affect k* andeven if k* changes, it will not influence the profit of firm H. Any

government effort to shift profit between firms H and F is bound to fail.

Hence for j 1/2, the only consideration for the optimal policy is profitshifting between firm H and firm M. A fall in investment reduces the

industry output and induces firmMto cut w. So an investment tax can be

used to improve the terms of trade of the home country. Letrc (j) be the

value ofb associated withj such that the equationQ 0 and the stabilitycondition hold. The following proposition summarizes the results in this

section.

Proposition 1. For Cournot competition, ifj 1/2, the optimal industrial

policy must be investment tax. For any j 6 1/2, the optimal policy isinvestment tax if and only ifb 4 rc (j).

4. Bertrand competition

Next we consider Bertrand competition where the goods produced by firms

H and F are not homogeneous. The change in the form of competition

would not change the results much. Let the demand functions bex a b(p ep*) and y a b (p* ep) where e 2[0,1) is the coefficient of

substitutability between the goods. Substitute the demand functions into theprofit functions.

p pwk jk a bpep k2

2

p p wk jk abp ep k2

2

Using the first-order conditions for pricing at stage four, the first-

order condition for investments at stage three, and the profit

maximizing condition for firm M at stage two, we can derive the exact

formula for s. Differentiate to get pk 2 xppk xp p

k

x and



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formula for s. Differentiate to get pk x xppk xp pk

x and

we set pkk 2e2 je

4e2 2x 0 and findj e

2e2. The second-order condition

for investment at stage three is:

bpkk 22e2 ej2b4 e22

1< 0:

which is always satisfied under the following stability condition:

1b>

2 2e2 ej

1e 1j

4e2

2 e ; for j< j

1b>

2 2e2 ej

1e 1j

4e2 2e ; for j> j

8>>>:

Under the linearity assumption, we can again show thats is proportional tox:

sC

Dx

where

C

4

4e2 2 2e

2 ej

2e2

je 21

b

1b

24e2

2e2ej 1e 1j 2e

1j

2

1

b

2

4e2 2

2e2 ej 2" #

2e 1e

D 4e2

kskwws

1

b

2

4e2 2

2e2 ej

2e 1 e 1j

!> 0

Lemma 4. Under the second order condition for welfare maximizationd2W

ds2 0 and sign(s) sign (C).


We consider four different values ofe at 0, 0.25, 0.5, and 0.75 and plot

the results in Figures 3(a)(d). The results for Bertrand competition is

qualitatively the same as those for Cournot competition. As in Figure 2, the

upper boundary of the shaded area is the s 0 locus and the lowerboundary is derived from a sufficient condition for the second-order

condition for welfare maximization at stage one (see Note 6). The lowest

curve is given by the stability condition at the investment stage. The optimal

industrial policy is investment subsidy for j and 1/b in the shaded area.



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industrial policy is investment subsidy forj and 1/b in the shaded area.

Figures 3(a)(d), a rise ine moves the critical levelj to the right. There are

two effects of foreign investment on the profit of firmH. On the one hand,

technology spillover lowers the production cost of firm H. On the other

hand, a rise ink* lowersp* relative top and makes firmHless competitive.

The second effect will be stronger if the two goods are close substitutes. To

keep foreign investment from affecting the profit of firmH,jhas to increase

with e. Ife is zero, two goods are completely independent. There would not

be any substitution between the two goods when the prices change. In this

case,k* would not affect the profit of firmHonly ifj is zero. Letrb (j) be

the value of 1/b associated with j such that the equation C 0 and thestability condition hold. The following proposition summarizes the results in

this section.



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Proposition 2. For Bertrand competition, ifj e2e2

, the optimal industrial

policy must be investment tax. For any j6 e2e2

, the optimal policy is

investment tax if and only if 1/b 4 rb (j).

It would be interesting to use the same demand functions to consider

Cournot competition with product heterogeneity. It can be shown that the

results remain mostly intact. See Section A5 in the Appendix for details.

5. Alternative timing

One may argue that cost-reducing investment is a long term commitment

and is often known before the intermediate good producer sets its price. It

turns out that the assumption on the timing of the model greatly influences

our results. Now suppose firmsHandFinvest before firmMsets the price

of the intermediate good. The sequence of decision making is as shown inFigure 4.

The results derived in the previous sections will be substantially changed.

Under this alternative assumption, the main result in Leahy and Neary

(2001) still holds in our model. The first-order conditions of the firms at

different stages would allow us to solve for s. The reduced form profit

functionsbp k; k; s andbp k; k; s are obtained by substituting thesolutions at the last two stages into equations (2) and (3). The formula for

optimal subsidy is the same as the one in Leahy and Neary (2001), except

that the firms have already considered the change inw when they make theirinvestment decisions:

s pkdk

dk pk

pkkpkk

where pk pk pBBk pwwk . By the second-order condition, pkk


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5. Conclusion

The results in this paper again confirm that the effectiveness of an economic

policy in an open economy is often influenced by trade in inputs. If the move

of the intermediate good producer precedes the investment decisions, the

optimal industrial policy is not necessarily subsidy. The failure of firmHto

anticipate the changes ink* andwleaves room for remedial action to be taken

by the home country. In general, the government has to weigh the benefit of

horizontal profit shifting from subsidizing investment against the adverse

change in the terms of trade in the input market. The relative importance of

these two effects depends on the slope of the demand curve, the technological

spillover coefficient, and the degree of product differentiation. The optimal

policy is more likely to be investment tax for high value ofbor 1/b. We also

find that there is a critical level of technology spilloverj at which the profit offirmH is independent ofk* and the optimal policy must be investment tax.

Similar to the results in Leahy and Neary (2001), the results are robust to the

change in the form of competition in the final good market.

As is well known, the applicability of the results in a strategic trade policy

model should be considered with caution. Active pursuit of strategic industrial

policy may cause retaliatory actions from other countries. Finally, our analysis

is greatly simplified by a few assumptions. In this paper, the intermediate

good is supplied by an independent firm. If the upstream and downstream

firms are vertically integrated, the pricing and outputs decisions will be quitedifferent. Moreover, our results are obtained under the assumption of linear

functions. It is well known that many results in the strategic trade policy

literature change once the assumption of linear functions is dropped. In

particular, production and R&D are often subject to increasing returns in hi-

tech sectors. Extension of the model in these directions will be desirable.

Notes

1. At the time of print publication of this article, author Yao Lius affiliation had

changed to Dongbei University of Finance and Economics, 217 Jian Shan Street,Shahekou District, Dalian, 116025, China.

2. If the home country imposes an investment subsidy, it is possible that firmMcanhave a higher profit by foreclosing firm F. With the possibility of foreclosure, aninvestment subsidy helps firm Hto become a monopoly. It does not follow thatsuch a policy is optimal because rent shifting between firms H and M calls foran investment tax. A complete analysis needs to compare the profits of firm Hwith and without foreclosure. The results may be very different from those inthis paper. The optimal policy under Cournot competition may be dominatedby an investment subsidy aimed at inducing firm M to foreclose firm F.

3. It is immaterial whether firmMcommits to its price or output but we assume

that firm M sets the price for convenience of presentation. However, choosingoutput by the intermediate good producer may have a more appealinginterpretation since it usually takes a long time to change production capacity



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interpretation since it usually takes a long time to change production capacity

4. A subsidy that partially defrays the cost of investment (17s)k2/2 where s 5 1could be a better specification. Obviously, the subsidy never exceeds the cost ofinvestment. We can show that our results remain intact under this specificationbut the derivations of the results are more tedious. The analysis will be providedupon request.

5. See the appendix for the exact form of the stability condition.6. For eachj 2 [0,1],j 6 1/2, there are two values ofb satisfyings 0. However,

only one of these solutions satisfies the stability condition and the unstablesolution is not shown in the diagram.

7. The derivation of the sufficient condition of the second-order condition ofwelfare maximization under Cournot competition as well as the correspondingcondition for Bertrand competition will be provided upon request.

References

Bernhofen, Daniel M. 1997. Strategic trade policy in a vertically related industry.

Review of International Economics 5, no. 3: 42933.Brander, James A., and Barbara J. Spencer. 1983. International R&D rivalry and

industrial strategy. Review of Economic Studies 50: 70722.Chen, Yongmin, Ishikawa Jota, and Zhihao Yu. 2004. Trade liberalization and

strategic outsourcing. Journal of International Economics 2004, no. 63: 41936.Eaton, Jonathan, and Gene M. Grossman. 1986. Optimal trade and industry policy

under oligopoly. Quarterly Journal of Economics May: 383405.Hwang, Hong, Yan-Shu Lin, and Ya-Po Yang. 2007. Optimal trade policies and

production technology in vertically related markets. Review of InternationalEconomics15, no. 4: 82335.

Leahy, Dermot, and Peter J. Neary. 2001. Robust rules for industrial policy in open

economies. Journal of International Trade & Economic Development 10, no. 4:393409.

Spencer, Barbara J., and Ronald W. Jones. 1991. Vertical foreclosure andinternational trade policy. Review of Economic Studies 58, no. 1: 15371.

Spencer, Barbara J., and Ronald W. Jones. 1992. Trade and protection in verticallyrelated markets. Journal of International Economics 1992, no. 32: 3155.

Appendix

A1. Stability condition at the investment stage

Differentiate equations (8) and (9) totally to get

bPkk bPkkpkk p

kk

dk

dk

bPkwpkw

dw

bPkspks

ds 0 22

Keep dw 0 and ds 0 to derive the reaction curves of firmsHandF. The slopes of

these curves are dk

dk

H

bPkkbPkk

pkkpkk

for firm H and dk

dk

F

pk k

pk k

for firm F,

where pkk


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dk

dk

k k k k k k dk

A2. Proof of Lemma 1

SubstitutebPks1 and pk s0 into equation (22) to getbPkk bPkkpkk pkk !

ks

ks 10 ; bPkk bPkkpkk pkk ! kw

kw bPkwpkw !By the stability condition,bPkkpkk bPkk pkk >0 and by the second-order

condition for optimum investment, pkk 0; kw

bPkk pkwbPkwpkkbPkkpkk bPkk pk k 0kwwsj j

xk xk ks ks

4 xw

kwxk xk

< xk xk2 xwkw

xk xk

ks


17/21

j 23

Using equations (16), (18), (19), and (23), we get

x l Zs 24

where

l1

6b ac 2j

b 29

2j 1j > 0

Z1

2

1j6

b 29

2j 1j

1j

b 23

2j 1j

" #> 0

By equations (21) and (24), x lRRZQ. Since

dWds

QR l Zs s

ks kwws , the

second-order condition is @2W@s2

QRZ 1 ks kwws < 0 orR ZQ 4 0. AsR and lare both positive, x 4 0 if and only if the second-order condition holds.

A4. Proof of Lemma 4

Write x l0 Z0 s where l0 is the equilibrium output of firm Hwhen s 0 and

Z0 xppk xppk

ks kwws xppk xpp

k

ks k

wws

xp xp

pwws

The first-order condition for welfare maximization is

dW

ds

C

Dxs

ks kwws

C

D l0 Z0s s

ks kwws 0

So s l0C

DZ0 Cand x l0DDZ0C. The second-order condition is

@2W

@s2

C

DZ0 1

ks kwws < 0

or D7Z0C4 0. As D and l0 are both positive, the second-order condition holds ifand only ifx 4 0.

A5. Cournot competition with product differentiation

We can invert the demand functionsx a7b(p7ep*) and y a7b(p*7ep) to get

p 1e

1e2 b

1

1e2 b xey

p 1e

1e2 b

1

1e2 b yex

It can be shown that j e=2. So the critical level of technology spillover underCournot competition is lower than that under Bertrand competition The resultsare shown in Figures 5(a) to (d) For small e Cournot and Bertrand competitions



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are shown in Figures 5(a) to (d) For small e Cournot and Bertrand competitions

As shown in the diagrams, the s 0 locus has a higher kink but a lower verticalintercept under Cournot competition. The details for the derivation will beprovided upon request.

A6. Proof of Proposition 3

Under Cournot competition, if the functions are linear,

pk px xk xwwk py yk ywwk wk j

x

pwkjk xkxwwk

The first order condition gives p7w k jk* 7pxx, and thus

Figure 5. Optimal policy with Cournot competition and product differentation.



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Under Bertrand competition, the first order condition for firm Hat the pricingstage is (p7w k jk*) 7x/xp and thus

pk pk pwwk wk jx pwkjkxppkpwwk

xppk pwwk wk j p

k p

wwk xp=xpx 26

Under the assumption of linear functions, all the partial derivatives in equations (25)and (26) only involve exogenous variables. Equations (25) and (26) can be writtenrespectively as

pk Fj; bx

pk Gj;bx

Under Cournot competition and the symmetry assumption, pk Fj; by and

pkk pkk Fj; b

@y

@k

Moreover, @y@k Bk >0 under Cournot competition and thus

s pkpkkpkk

1

pkkFj; b 2

@y

@kx

0 27

As for Bertrand competition, @y@k bBk eAk . Since Bk* 5 0 and

jBk*j 4 jAk*j, Bk* eAk >0. Hence s 1p

k kGj;b 2@y@kx

0.

A7. The proof ofk2

2 sk> 0

The investment subsidy would not exceed the investment cost at equilibrium. ForCournot competition, bybPk 0, the net payment on investment can be written as

k2

2 sk k

2 x 1pyyk s

k

2

x 1pyyk

jxpyyk ks kwws

ks kwws

pwws

ks kwws

As pwwskskwws

ks kwws

12

b 19

2j 1j

b 2 2 j 1 j

b2

b 2 2 j 1 j



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2 b 2 2 j 1 j b 2 2 j 1 j

the net payment will be positive ifx 1pyyk

pk x 1pyyk

jxpyyk >0.Using equations (16)(19), we obtain

x 1pyyk jxpyyk

2

3 2j x

2

3 2j1 x

2

3 1j x> 0

Similarly, for Bertrand competition, bPk 0 implies ks xpwkjk xpp

k. By Pp 0, p 7 w k jk* 7x/xp. The net payment

on investment is

x pckjk xppk pk x 1

xp

xppk

x j

xp

xppk

x 1je

1j 1e=2

2e2=2

x 1j

2e2=2 2ee2

> 0



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Optimal Industrial Policy in Vertically Related Markets