Optimal Export Policy under Bertrand Competition

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By recognizing the importance of changing product characteristic and cost ... the third stage, firms play Bertrand competition in a third-country commodity market. .... Totally differentiating (7) with respect to xi, respectively, we can derive the ...... Equilibrium in Monopolistic Competition,” (CORE Discussion Paper 8506). Maggi ...
Optimal Export Policy under Bertrand Competition with Horizontal Differentiation and Asymmetric Costs Wen-Jung Liang Department of Industrial Economics Tamkang University Tamsui, Taipei County 25137, Taiwan, ROC and Chao-Cheng Mai Department of Industrial Economics Tamkang University and Sun Yat-Sen Institute for Social Sciences and Philosophy Academia Sinica Current Version: February 15, 2007

JEL Classification: F12; L13 Key Words: Optimal Export Policy; Bertrand Competition; Horizontal Differentiation; Cost Asymmetry

1

Abstract This paper examines the optimal export policy under Bertrand competition when the products exhibit horizontal differentiation and production costs are asymmetric. The focus of this paper is on the differentiation-changing effect in the determination of the optimal export policy. It shows that when the cost difference is positive and is sufficiently large such that the foreign firm locates inside of the line segment, the optimal export policy of the home country under Bertrand competition is an export subsidy rather than an export tax. Therefore, the result of Eaton and Grossman (1986) turns out to be a special case of this paper.

Optimal Export Policy under Bertrand Competition with Horizontal Differentiation and Asymmetric Costs

Ⅰ. Introduction In a seminal paper, Brander and Spencer (1985) develop a “third-market” model in which one home firm and one foreign firm produce homogeneous products and compete in a third-country market. They find that under Cournot quantity competition if the home country’s government can credibly pre-commit itself to pursue a particular policy before firms make production decisions, then an export subsidy is optimal. Nevertheless, Eaton and Grossman (1986) show that the optimal export policy is an export tax rather than an export subsidy, when firms play Bertrand price competition in the commodity market and the products are substitutes. Since then, the theory of strategic trade policy has been extended to several directions, including: De Meza (1986) and Mai and Hwang (1988) demonstrate that government should offer higher subsidies to the more efficient firm under Cournot competition; Qiu (1994) studies the optimal export policy under asymmetric information on the production cost; Neary (1994) explores the optimality of export subsidies in oligopolistic markets, when home and foreign firms have different costs and the social cost of public funds exceeds unity; Maggi (1996) examines the optimal export policies by taking into account the capacity constraint; Bandyopadhyay et al. (2000) discusses the optimal export policies when there exists labor union; Zhou et al. (2002) investigates strategic trade and joint welfare-maximizing incentives towards investment in the quality of exports by an LDC and a developed country; and Yang and Hwang (2003) study the optimal trade policy under Bertrand competition with homogeneous products. It should be noticed that Eaton and Grossman’s result is based on the premise that the extent of the product differentiation between the two firms remains 1

unchanged. However, as time goes by, firms are generally capable of changing the characteristics of the products. In doing so, the price competition between firms becomes more severe if the products get to be closer, while lessened if more differentiated. This gives an incentive for the government to influence the choices of the characteristics of the firms via trade policies. A numerous of literature, such as MacLeod et al. (1985), Lederer and Hurter (1986), Greenhut et al. (1987), Anderson and De Palma (1988), De Fraja and Norman(1993), B&o&kem (1994), and Shimizu (2002), have suggested that the spatial model can be interpreted as a model of product location in characteristics space. Specifically, a consumer’s location can be interpreted as his optimal product specification, and the firm’s location as the amount of characteristic. The distance between the firm’s location and the consumer’s location represents the difference of the characteristics between the firm and consumer. When the distance is not nil, the consumer will suffer a disutility which can be represented by the transportation cost. By recognizing the importance of changing product characteristic and cost asymmetry, the purpose of this paper is to incorporate cost asymmetry into the product differentiation model and to re-explore the optimal export policy under Bertrand price competition when the product characteristic is endogenously determined. The model involves a three-stage game. In the first stage, the domestic government determines its optimal export policy to maximize domestic welfare; in the second stage, firms select their locations of characteristic to maximize their profits; in the third stage, firms play Bertrand competition in a third-country commodity market. It is found that the optimal policy for the government of the high cost firm is an export subsidy, when the cost difference is sufficiently large. This surprising result can be derived from the following channel: The location of the characteristic of the 2

low cost firm moves to the inside of the line segment, while that of the high cost firm is at the endpoint in this case. The government of the high cost firm can shrink the cost difference to force the rival firm moving farther apart via taking a subsidy export policy. Consequently, the market share of the high cost firm is enlarged and the price charged can be higher due to mitigated price competition. The profits and domestic welfare are thus improved. The remainder of the paper is organized as follows. Section Ⅱ explores the firms’ optimal locations of characteristics. Section Ⅲ analyzes the optimal export policies under Bertrand competition. The final section concludes the paper.

Ⅱ. Optimal Characteristics Consider a three-country framework, in which domestic firm d competes with foreign firm f in the third country. Products are assumed to be consumed in the third country only, and their marginal costs are asymmetric that can be exhibited by the difference between cd and cf. Products differ with respect to a one dimensional characteristic. The distribution of characteristics is analogous to the Hotelling (1929)-type linear city model as shown in Figure 1. The amount of characteristic is measured by xi, where the characteristics of the firms locate at xd and xf with xd ≤ xf, along a line segment within the interval [0, 1]. Consumers’ ideal characteristics are uniformly distributed in the line segment with unit density. Each point of the line segment represents the amount of a consumer’s ideal characteristic.

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xf xd • 0

• d

• f

• 1

Figure 1. Characteristic Line

Based on the well-known framework described by D’Aspremont et al. (1979), consumers have unit demands, and consumption yields a positive constant surplus r. Thus, each consumer buys if and only if the net utility generated from consumption is non-negative:1

u = r − pi − td 2 ≥ 0, i = d , f ,

(1)

where r is the reservation price and is assumed to be large enough for total demand to be always equal to one; pi is the price of product i; td2 measures the disutility of the distance d between the amount of the characteristic product i contains and consumer’s ideal product characteristic; and t denotes the transport rate per unit of distance per unit of product i. When firms’ characteristics are set up at xd ≤ xf, the marginal consumer, who is indifferent between purchasing from either firm, is located at xˆ as given by: ⎛ xd + x f xˆ = ⎜⎜ ⎝ 2

⎞ ⎛ p f − pd ⎞ ⎟. ⎟⎟ + ⎜ ⎜ 2t ( x − x ) ⎟ f d ⎠ ⎠ ⎝

(2)

Using equation (2), we can derive total demand for final products for firm d and f, respectively, as: xˆ

q d = ∫ 1dx = xˆ ,

(3.1)

0

1

B&o&kem (1994) and Lambertini (1997) have employed the similar form of utility function. 4

1

q f = ∫ 1dx = 1 − xˆ ,

(3.2)



where qi, (i = d, f ) denotes firm i’s total demand. Each firm’s profit function can be expressed, respectively, as:

π d = [ p d − (τ + c d )]xˆ,

(4.1)

π f = [ p f − c f ](1 − xˆ ),

(4.2)

where πi (i = d, f ) denotes firm i’s profits, and τ is the specific tax rate imposed by domestic government. The model in this paper consists of three stages. The domestic government determines its optimal export policy to maximize domestic welfare in the first stage, the two firms then simultaneously select their optimal characteristics of products to maximize their profits in the second stage, and finally these firms play Bertrand price competition in the third country market in the third stage. The sub-game perfect equilibrium of the model is solved by backward induction, beginning with the final stage. Maximizing each firm’s profits with respect to its price and solving these first-order conditions, we obtain: p d = (1 / 3){2(τ + c d ) + c f + t ( x f − x d )(2 + x f + x d )},

(5.1)

p f = (1 / 3){(τ + c d ) + 2c f + t ( x f − x d )(4 − x f − x d )}.

(5.2)

Substituting (5) into (2) and (3) obtains:

[

]

q d = 1 / 6t ( x f − x d ) {− (τ + c d − c f ) + t ( x f − x d )(2 + x f + x d )},

[

]

q f = 1 − 1 / 6t ( x f − x d ) {− (τ + c d − c f ) + t ( x f − x d )(2 + x f + x d )}.

(6.1) (6.1)

We now turn to the second-stage problem with respect to the firms’ optimal characteristics. Making use of the equilibrium in the third stage, the profit functions of firms d and f can be specified as follows: 5



1

⎤ 2 ⎥ − (τ + c d − c f ) + t ( x f − x d )(2 + x f + x d ) , ⎣⎢18t ( x f − x d ) ⎦⎥ ⎡

1

]

(7.1)

⎤ 2 ⎥ (τ + c d − c f ) + t ( x f − x d )(4 − x f − x d ) . ⎢⎣18t ( x f − x d ) ⎥⎦

(7.2)

πd = ⎢

πf =⎢

[

[

]

Totally differentiating (7) with respect to xi, respectively, we can derive the profit-maximizing conditions for the two firms as follows: ⎤ ∂π d ⎡ xˆ =⎢ ⎥{− (τ + c d − c f ) − t ( x f − x d )(2 + 3 x d − x f )}, ∂x d ⎢⎣ 3( x f − x d ) ⎥⎦ ∂π f ∂x f

⎡ 1 − xˆ ⎤ =⎢ ⎥{− (τ + c d − c f ) + t ( x f − x d )(4 − 3x f + x d )}. ⎣⎢ 3( x f − x d ) ⎦⎥

(8.1)

(8.2)

By the assumptions xd ≤ xf and xi ≤ 1, the second term of the right-hand side in the brace of (8.1) is negative while the second term of (8.2) is positive. This second may be named as the competition effect. It indicates that as the locations of the two firms become farther away, their transport costs (i.e., disutility) at any site become dissimilar and therefore the competition is lessened under Bertrand price competition. The first term of (8) denotes the cost difference effect. When the cost difference τ +cd – cf > ( 0. This arises because the cost difference effect is so weak that the competition effect dominates. Thus, the two firms select to locate at the endpoints of the line segment. As a result, the Principle of Maximum Differentiation

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occurs.2 This result can be expressed as: x d* = 0, if τ + c d − c f is sufficiently small,

(9.1)

x *f = 1, if τ + c d − c f is sufficiently small.

(9.2)

When the cost difference τ +cd – cf < 0 and its absolute value is sufficiently large, it can be shown from (8) that ∂πd /∂xd = 0 and ∂πf /∂xf > 0. Hence, as the foreign firm has severe cost disadvantage, both cost difference and competition effects attract it to locate as far away from the domestic firm as possible. Consequently, it selects to locate at the right endpoint. On the other hand, the domestic firm tends to get closer to the foreign firm due to severe cost advantage it owns, while take apart to mitigate competition. The equilibrium characteristic of the domestic firm is determined by the balance of these two oppose effects. It can be inside of the line segment, which is derived by setting (8.1) equal to zero and xf* = 1. Accordingly, we obtain: x d* = (1 / 3){1 − [1 + 3(τ + c d − c f + t ) / t ]1 / 2 } ≥ 0,

x *f = 1,

if τ + c d − c f < 0 and its absolute value sufficiently large,

(10.1)

if τ + c d − c f < 0 and its absolute value sufficiently large.

(10.2)

Similarly, when the cost difference τ +cd – cf > 0 and is sufficiently large, it follows from (8) that ∂πd /∂xd < 0 and ∂πf /∂xf = 0. The domestic firm has severe cost disadvantage, and selects to locate at the left endpoint of the line segment. On the other hand, the equilibrium characteristic of the foreign firm can be inside of the line segment by setting (8.2) equal to zero and xd* = 0. Thus, we have: x d* = 0, if τ + c d − c f > 0 and sufficiently large,

x *f = (1 / 3){2 + [4 − 3(τ + c d − c f ) / t ]1 / 2 } ≤ 1, if τ + c d − c f > 0 and sufficiently large. 2

(11.1)

(11.2)

Liang and Mai (2006) show that this result can be overturned when the cost difference between firms is sufficiently large. 7

Based on the above analysis, we can establish:

Proposition 1. (i) When the cost difference τ +cd – cf is sufficiently small, the Principle of Maximum Differentiation holds. (ii) When the cost difference τ +cd – cf < 0 and its absolute value is sufficiently large, the foreign firm has severe cost disadvantage and selects to locate at the right endpoint, while the domestic firm could locate inside of the line segment. (iii) When the cost difference τ +cd – cf > 0 and is sufficiently large, the domestic firm has severe cost disadvantage and selects to locate at the left endpoint, while the foreign firm could locate inside of the line segment.

We are now in a position to examine the effects of domestic government’s export tax on the equilibrium locations of the firms. Differentiating (9) - (11) with respect to domestic government’s export tax, we yield: ∂x d* / ∂τ = 0,

[

if τ + c d − c f is sufficiently small,

= (-1 / 2t ) 1 + 3( τ + c d − c f + t ) / t

]

−1 / 2

< 0,

(12.1)

if τ + c d − c f < 0 and its absolute value sufficiently large, if τ + c d − c f > 0 and sufficiently large,

= 0, ∂x *f / ∂τ = 0,

if τ + c d − c f > 0 is sufficiently small,

= 0,

if τ + c d − c f < 0 and its absolute value sufficiently large,

[

= (-1 / 2t ) 4 − 12t(τ + c d − c f ) / t

]

−1 / 2

(12.2)

< 0,

if τ + c d − c f > 0 and sufficiently large. When the cost difference τ +cd – cf is sufficiently small, the cost difference effect is so weak that an increase in the domestic export tax is unable to balance the 8

competition effect. Hence, the Principle of Maximum Differentiation still holds. When the cost difference τ +cd – cf < 0 and its absolute value is sufficiently large, the cost difference effect is strong enough to attract the domestic firm to move closer to the foreign firm due to cost advantage. An increase in the domestic export tax reduces this cost difference effect, and leads to the result that the domestic firm moves farther away from the foreign firm. On the other hand, the foreign firm still selects to locate at the right endpoint although its cost disadvantage lessened. Similarly, when the cost difference τ +cd – cf > 0 and is sufficiently large, an increase in the domestic export tax enhances the foreign firm’s cost advantage. This will make the foreign firm moving closer to the domestic firm due to a stronger cost difference effect. However, the domestic firm still locates at left endpoint due to a worsened cost disadvantage. Accordingly, we have:

Proposition 2. An increase in the domestic export tax has the following effects on the firms’ choices of characteristics: (i) When the cost difference τ +cd – cf is sufficiently small, the Principle of Maximum Differentiation still holds. (ii) When the cost difference τ +cd – cf < 0 and its absolute value is sufficiently large, the foreign firm selects to locate at the right endpoint, while the domestic firm moves farther away from its rival. (iii) When the cost difference τ +cd – cf > 0 and is sufficiently large, the domestic firm selects to locate at the left endpoint, while the foreign firm moves closer to its rival.

Ⅲ. Optimal Export Policy

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In this section, we are going to explore the first-stage problem in which the domestic government determines the optimal export policy. Since the products are consumed in the third country market only, the domestic welfare is composed of domestic firm’s profits and government’s tax revenue. This can be derived by considering (2), (3), and (5) – (8) as following: wd = π d ( x d , x f ,τ ) + τq d ( x d , x f ,τ ),

(13)

where wd denotes the domestic welfare. The first-order condition for welfare maximization can be derived by differentiating (13) with respect to τ: ∂wd ⎡⎛ ∂π d = ⎢⎜⎜ ∂τ ⎢⎣⎝ ∂x d

⎞⎛ ∂x d ⎞ ⎛ ∂π d ⎟⎟⎜ ⎟ + ⎜⎜ τ ∂ ⎠ ⎝ ∂x f ⎠⎝

⎡⎛ ∂q + q d + τ ⎢⎜⎜ d ⎢⎣⎝ ∂x d

⎞⎛ ∂x f ⎟⎜ ⎟⎜ ∂τ ⎠⎝

⎞ ⎛ ∂π d ⎟⎟ + ⎜ ⎠ ⎝ ∂τ

⎞⎤ ⎟⎥ ⎠⎥⎦

⎞⎛ ∂x d ⎞ ⎛ ∂q d ⎞⎛ ∂x f ⎞ ⎛ ∂q d ⎞⎤ ⎟⎜ ⎟⎟ + ⎜ ⎟⎟⎜ ⎟ + ⎜⎜ ⎟ ⎥ = 0. ⎜ ∂ ∂ τ x ⎠ ⎝ f ⎟⎠⎝ ∂τ ⎠ ⎝ ∂τ ⎠⎥⎦ ⎠⎝

(14)

In order to calculate the optimal export tax, we need the following comparative-static effects by differentiating (6) and (7): ⎤ ∂q d ⎡ 1 2 =⎢ ⎥ − (τ + c d − c f ) + t ( x f − x d ) , ∂x d ⎢⎣ 6t ( x f − x d ) 2 ⎥⎦

(15.1)

⎤ ∂q d ⎡ 1 (τ + c d − c f ) + t ( x f − x d ) 2 , =⎢ 2 ⎥ ∂x f ⎣⎢ 6t ( x f − x d ) ⎦⎥

(15.2)

∂q d −1 = < 0. ∂τ 6t ( x f − x d )

(15.3)

{

}

{

∂π d ⎛ q d =⎜ ∂x f ⎝ 3

⎞⎧⎪⎛⎜ τ + c d − c f ⎟⎨⎜ xf ⎠⎪⎩⎝

}

⎫ ⎞ ⎟ + t (2 + 3 x f )⎪⎬, ⎟ ⎪⎭ ⎠

∂π d ⎛ − 2 ⎞ =⎜ ⎟q d < 0. ∂τ ⎝ 3 ⎠

(15.4)

(15.5)

In what follows, we discuss the determination of the optimal export tax under the above-mentioned three cases.

10

Case Ⅰ: The cost difference τ +cd – cf is sufficiently small. Recall from (12) that the terms ∂xd /∂τ = ∂xf /∂τ = 0 and from (9) that xd* = 0 and xf* = 1. Substituting these terms into (14), we have: ⎛ − 1 ⎞⎧⎛ ∂π d ⎞ ⎫ ⎟⎟⎨⎜ τ = ⎜⎜ ⎟ + q d ⎬, ⎭ ⎝ ∂q d / ∂τ ⎠⎩⎝ ∂τ ⎠

(16)

where ∂π d / ∂τ = (−2 / 3)q d < 0, and ∂q d / ∂τ = −1 / 6t < 0. We see from (16) that the optimal export tax is determined by the direct tax effect and the tax revenue effect, which are represented by the terms in the brace of the right-hand side of (16) in that order. The intuition behind these effects can be stated as follows. First of all, the direct tax effect is negative via increasing the cost of the home firm. This tends to subsidize home firm to increase its profits. Secondly, the tax revenue effect is positive due to a rise in tax revenue. Consequently, the optimal export tax is determined by these two effects. We find that the tax revenue effect outweighs the direct tax effect.3 Consequently, the optimal trade policy is an export tax, i.e., τ * > 0. This result restates the Eaton and Grossman (1986) argument for an optimal export tax policy. Figure 2 illustrates this result. Representative iso-profit loci of the home firm are indicated as πdB, and πdτ. Higher curves correspond to higher profits. The Bertrand reaction curves are depicted by Rd for the home firm and Rf for the foreign firm. We show from (A.1) and (A.2) in the Appendix that both curves are positively sloping and Rd is steeper than Rf. The Bertrand equilibrium without policy intervention is at point B, which is the intersection of the two curves. The corresponding domestic iso-profit curve is πdB. We find from (A.5) and (A.6) that an increase in export tax shifts the home reaction curve rightward to Rdτ due to a decrease in the intercept, while the foreign reaction curve remains unchanged. The 3

This can be proved by summing the two terms in the numerator, which equals (1/3)qd, being positive. 11

new equilibrium point is at S, which yields a higher profit and welfare. This new outcome indicates that similar to Eaton and Grossman (1986), the domestic government has an incentive to help its firm act as a Satckelberg leader. (Insert Figure 2 here) Substituting

the

terms

xd *

=

0

and

xf*

=

1

into

(6.1)

yields

q d = (1 / 6t ) {− (τ + c d − c f ) + 3t }. We can thus solve (16) as:

τ * = [3t − (c d − c f )]/ 4 > 0.

(16.1)

It follows from 916.1) that the optimal export tax gets to be lower, if the extent of the cost disadvantage is larger and the transport rate is lower. Consequently, we can establish the following proposition:

Proposition 3. When the cost difference τ +cd – cf is sufficiently small such that the two firms locate at the opposite endpoints of the line segment, the optimal export tax must be positive under Bertrand competition. Moreover, the larger the extent of the cost disadvantage and the lower the transport rate, the lower the optimal tax will be.

Eaton and Grossman (1986) find that the optimal trade policy is an export tax under Bertrand competition assuming the extent of the differentiation between the two products being unchanged. In this paper, we obtain the same result when τ +cd – cf is sufficiently small. The Principle of Maximum Differentiation holds, i.e., the location of each firm’s characteristic lies at the endpoints of the line segment irrespective of the change in the export tax. Hence, the extent of the differentiation between the two products keeps unchanged in this case. Case Ⅱ: The cost difference τ +cd – cf < 0 and its absolute value sufficiently large. We recognize from (8) and (12) that the terms ∂πd /∂xd = 0 and ∂xf /∂τ = 0 in this 12

case. Accordingly, (14) can be reduced to: ⎛ − 1 ⎞⎧⎛ ∂π ⎞ ⎫ τ * = ⎜⎜ ⎟⎟⎨⎜ d ⎟ + q d ⎬, ⎭ ⎝ ∆ n ⎠⎩⎝ ∂τ ⎠ ⎛ ∂q where ∆ n = ⎜⎜ d ⎝ ∂x d

(17)

⎞⎛ ∂x d ⎞ ⎛ ∂q d ⎞ ⎟⎟⎜ ⎟+⎜ ⎟. ⎠⎝ ∂τ ⎠ ⎝ ∂τ ⎠

Note that ∂qd /∂xd > 0 from (15.1), ∂xd /∂τ < 0 from (12.1), and ∂qd /∂τ < 0 from (15.3). We then yield that ∆n < 0. Moreover, equation (15.5) indicates that ∂πd /∂τ = (-2/3)qd < 0. Equation (17) shows that the optimal export tax is determined by both the direct tax effect and the tax revenue effect. Since ∂πd /∂τ +qd = (1/3) qd > 0, the tax revenue effect outweighs the direct tax effect. The optimal export policy is also an export tax. (Insert Figure 3 here) This result can be illustrated by Figure 3. The Bertrand equilibrium without policy intervention is at point B. We find from (A.7) and (A.8) that an increase in export tax shifts the home reaction curve rightward to Rdτ due to a decrease in the intercept, while the foreign reaction curve leftward due to a rise in the intercept. Intuitively explanation for the shift of the reaction curves can be stated as follows. We have proved in Proposition 2 that the degree of product differentiation between the two firms is higher in this case, as the export tax rises. The price competition can be accordingly lessened. Hence, each firm can charge a higher price given its rival’s price. This results in a rightward shift to the home reaction curve, and a leftward shift to the foreign reaction curve. The new equilibrium point is at T, which yields a higher profit and welfare. We can thus establish:

Proposition 4. When the cost difference τ +cd – cf < 0 and its absolute value sufficiently large such that the home firm locates inside of the line segment, the 13

optimal export policy is an export tax under Bertrand competition.

Although this result is similar to that in Proposition 3, the impact channels are different. In this case, a rise in export tax enlarges the differentiation of the products via reducing the cost advantage of home firm. This will mitigate the price competition and raises prices. Consequently, profits and welfare of the domestic country are improved. In contrast, the product differentiation remains unchanged in Proposition 3 and Eaton and Grossman (1986). Case Ⅲ: The cost difference τ +cd – cf > 0 and is sufficiently large. We recognize from (12) that the terms ∂xd /∂τ = 0 and ∂xf /∂τ < 0 in this case. Accordingly, (14) can be reduced to: ⎛ − 1 ⎞⎧⎪⎡⎛ ∂π d ⎟⎨⎢⎜ τ * = ⎜⎜ ⎟ ⎢⎜ ∂x ∆ p ⎠⎪⎩⎣⎝ f ⎝

⎛ ∂q where ∆ p = ⎜ d ⎜ ∂x ⎝ f

⎞⎛ ∂x f ⎟⎜ ⎟⎜ ∂τ ⎠⎝

⎞⎛ ∂x f ⎟⎜ ⎟⎜ ∂τ ⎠⎝

⎞ ⎛ ∂π d ⎟⎟ + ⎜ ⎠ ⎝ ∂τ

⎫⎪ ⎞⎤ ⎟⎥ + q d ⎬, ⎠⎥⎦ ⎪⎭

(18)

⎞ ⎛ ∂q d ⎞ ⎟⎟ + ⎜ ⎟. ⎠ ⎝ ∂τ ⎠

Note that ∂qd /∂xf > 0 from (15.2), ∂xf /∂τ < 0 from (12.2), and ∂qd /∂τ < 0 from (15.3) in this case. We then yield that ∆p < 0. Meanwhile, we find that ∂πd /∂xf > 0 from (15.4), and ∂πd /∂τ < 0 from (15.5). Accordingly, equation (18) shows that in addition to the direct tax effect and the tax revenue effect in (16), there is an extra effect, the differentiation-changing effect, which is denoted by the first term in the brace of the right-hand side of (18). As an increase in the export tax attracts foreign firm to get closer to the domestic firm due to enlarging the home firm’s cost disadvantage, this will reduce home firm’s profits via shrinking its output. Thus, the differentiation-changing effect is negative. As a result, the domestic government tends to subsidize home firm. Substituting ∂qd /∂xf, ∂xf /∂τ, and ∂qd /∂τ along with (12) into (18), we can 14

rewrite (18) as: ⎛



q τ * = ⎜⎜ d ⎟⎟[( J − 1)], ⎝ 3∆ p ⎠

(18.1)

where J = (1 / 2cx f )[(τ + c d + c f ) + tx f (2 + 3 x f )][4 − 3(τ + c d + c f ) / t ] −1 / 2 . We can obtain from (11.2) that [4 − 3(τ + c d − c f ) / t ] −1 / 2 = (3 x f − 2) −1 ≥ 0. 4 Calculating (18.1) along with the above relationship, we yield that J-1 = [2txf (3xf – 2)]-1[(τ +cd – cf) +txf (6-3xf)] > 0 for 2/3 ≤ xf ≤ 1. As ∆p < 0, we can derive from (18.1) that the optimal export tax is negative. Contrasting to the result derived in the first case, the differentiation-changing effect and the direct tax effect outweigh the tax revenue effect in this case. Intuitively, an export subsidy can force foreign firm to select a characteristic of its product farther away from that of the home firm, and then increase home firm’s output via reducing the cost disadvantage faced by home firm. More differentiated products mitigate the price competition between the two firms. The prices charged are higher. Thus, the home firm’s output and profits increase, and the welfare of the domestic country improves. Moreover, we have shown in Proposition 3 that the larger the cost disadvantage, the lower the optimal tax will be. These two impacts indicate that the optimal export policy is an export subsidy when the cost disadvantage is large enough. Likewise, Figure 3 can illustrate this result. The Bertrand equilibrium without policy intervention is at point B. We find from (A.9) and (A.10) that a fall in export tax shifts the home reaction curve rightward to Rdτ due to a decrease in the intercept, while the foreign reaction curve leftward due to a rise in the intercept. This arises because the differentiation between firms gets to be more distant and then mitigates the price competition in this case. The new equilibrium point is at T, which yields a 4

It can be verified from (11.2) that xf ≥ 2/3. 15

higher profit and welfare. We thus have:

Proposition 5. When the cost difference τ +cd – cf > 0 and is sufficiently large such that the foreign firm locates inside of the line segment, the optimal export policy under Bertrand competition is an export subsidy.

This result is in sharp contrast to that derived in Eaton and Grossman (1986). The key factor is that an export subsidy is capable of forcing foreign firm to produce a more differentiated product, which will increase the output of home firm and mitigate the price competition between the two firms. In contrast, the differentiation between products remains unchanged in Eaton and Grossman (1986).

Ⅳ. Concluding Remarks

This paper has examined the optimal export policy under Bertrand competition when the products exhibit horizontal differentiation and production costs are asymmetric. The focus of this paper is on the differentiation-changing effect in the determination of the optimal export policy. This effect demonstrates that a fall in the export tax attracts foreign firm to move farther away from the domestic firm arising from the reduction of the home firm’s cost disadvantage. This raises home firm’s profits by increasing its output and charging a higher price due to producing more differentiated products, which mitigates the price competition between firms. More specifically, several striking results are derived as follows. First of all, we show that when the cost difference τ +cd – cf > 0 and is sufficiently large such that the foreign firm locates inside of the line segment, the optimal export policy of the home country under Bertrand competition is a subsidy policy. Secondly, when the cost difference τ +cd – cf < 0 and its absolute value 16

sufficiently large such that the home firm locates inside of the line segment, the optimal export policy of the home country under Bertrand competition is to tax. its exports Thirdly, when the cost difference τ +cd – cf is sufficiently small such that the two firms locate at the opposite endpoints of the line segment, the optimal export tax must be positive under Bertrand competition. Fourthly, contrary to the argument that the government should favor the firms, who have lower cost structure under Cournot competition, we show that the government should help the higher cost firm via a subsidy policy under Bertrand competition. Lastly, we conclude that when the cost difference equals zero as that in Eaton and Grossman (1986), the product differentiation remains unchanged and the optimal export policy is a tax policy. Accordingly, the result of Eaton and Grossman (1986) turns out to be a special case of this paper.

17

References

Anderson, S.P., and De Palma, A., 1988, “Spatial Price Discrimination with Heterogeneous Products,” Review of Economic Studies 55, 573-92. Bandyopadhyay, S., S.C. Bandyopadhyay, and E.S. Park, 2000, “Unionized Bertrand Duopoly and Strategic Export Policy,” Review of International Economics 8, 164-74. B&o&kem , S., 1994, “A Generalized Model of Horizontal Product Differentiation,”

Journal of Industrial Economics 42, 287-98. Brander, J.A., and B.J. Spencer, 1985, “Export Subsidies and International Market Share Rivalry,” Journal of International Economics 18, 83-100. D’Aspremont, C., J. Gabzewicz, and J.-F., Thisse, 1979, “On Hotelling’s Stability in Competition,” Econometrica 47, 1145-50. De Fraja, G., and Norman G., 1993, “Product Differentiation,Pricing Policy and Equilibrium,” Journal of Regional Science 33,343-63. De Meza, D., 1986, “Export Subsidies and High Productivity: Cause or Effect?” Canadian Journal of Economics, 347-50. Eaton, J., and G.M. Grossman, 1986, “Optimal Trade and Industrial Policy under Oligopoly,” Quarterly Journal of Economics 101, 383-406. Greenhut, M.L., G. Norman, and H. Hung, 1987, The Economics of Imperfect Competition: A Spatial Approach (Cambridge: Cambridge University Press). Hotelling, H., 1929, “Stability in Competition,” Economic Journal 39, 41-57. Lambertini, L., 1997, “Optimal Fiscal regime in a Spatial Duopoly,” Journal of Urban Economics 41, 407-20. Lederer, P.J., andA.P. Hurter, 1986, “Competition of Firms: Discriminatory Pricing 18

and Location,” Econometrica 54, 623-40. Liang, W.J., and C.C. Mai, 2006, “Validity of the Principle of Minimum Differentiation under Vertical Subcontracting,” Regional Science and Urban Economics, forthcoming. MacLeod, W.B., G. Norman, and J.-F., Thisse, 1985, “Price Discrimination and Equilibrium in Monopolistic Competition,” (CORE Discussion Paper 8506). Maggi, G., 1996, “Strategic Trade Policies with Endogenous Mode of Competition,” American Economic Review 86, 237-58. Mai, C.C., and H. Hwang, 1988, “Optimal Export Subsidies and Marginal Cost Differentials,” Economics Letters 27, 279-82. Neary, J.P., 1994, “Cost Asymmetries in International Subsidy Games: Should Governments Help Winners or Losers?” Journal of International Economics 37, 197-218. Qiu, L.D., 1994, “Optimal Strategic Trade Policy under Asymmetric Information,” Journal of International Economics 36, 333-54. Shimizu D., 2002, “Product differentiation in spatial Cournot markets,” Economics Letters 76, 317-22. Yang, Y.P., and H. Hwang, 2003, “Optimal Trade Policy under Homogenous Bertrand Competition,” Avademia Economic Paper 31, 73-89. (in Chinese) Zhou, D., B.J. Spencer, and I. Vertinsky, 2002, “Strategic Trade Policy with Endogenous Choice of Quality and asymmetric Costs,” Journal of International Economics 56, 205-32.

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pf

Rd Rdτ πds πd

Rf

S

B

B pd Fig.2

pf

Rd Rdτ πd

τ

T

Rfτ R2

πdB

B pd Fig.3

20

Appendix

Maximizing each firm’s profits with respect to its price, we obtain each firm’s reaction as: p f = I d + 2 pd ,

(A.1)

p f = I f + (1 / 2) p d ,

(A.2)

[

]

where I d = − (τ + c d ) − t ( x 2f − x d2 ) < 0 denotes the intercept of the reaction curve of

[

]

home firm, and I f = (1 / 2) c f + 2t ( x f − x d ) − t ( x 2f − x d2 ) > 0 is the intercept of the reaction curve of foreign firm. Differentiating the intercepts with respect to the export tax, we have:

[

]

I dτ = −1 − 2t x f (∂x f / ∂τ ) − x d (∂x d / ∂τ ) ,

[

(A.3)

]

I fτ = t (1 − x f )(∂x f / ∂τ ) − (1 − x d )(∂x d / ∂τ ) .

(A.4)

In what follows, we discuss the impact of the export tax on the intercepts in the three cases. Case Ⅰ: The cost difference τ +cd – cf is sufficiently small. It has been shown that ∂xd /∂τ = ∂xf /∂τ = 0 in this case. Equations (A.3) and (A.4) can be rewritten as: I dτ = −1,

(A.5)

I fτ = 0.

(A.6)

Case Ⅱ: The cost difference τ +cd – cf < 0 and its absolute value sufficiently large. Recall that the terms ∂xd /∂τ < 0, ∂xf /∂τ = 0, and xf = 1 in this case. Equations (A.3) and (A.4) can be rewritten as: I dτ = −1 + 2cxd (∂x d / ∂τ ) < 0,

(A.7)

I fτ = −t (1 − x d )(∂x d / ∂τ ) > 0.

(A.8)

21

Case Ⅲ: The cost difference τ +cd – cf > 0 and is sufficiently large. Note that the terms ∂xd /∂τ = 0, and ∂xf /∂τ < 0 in this case. Equations (A.3) and (A.4) can be rewritten as: I dτ = −1 − 2cx f (∂x f / ∂τ )

{

[

= (2 / 3) − 1 + 4 − 3(τ + c d − c f ) / t

] }, −1 / 2

I fτ = t (1 − x f )(∂x f / ∂τ ) < 0.

(A.9)

(A.10)

With the help of the term x *f = (1 / 3){2 + [4 − 3(τ + c d − c f ) / t ]1 / 2 } ≤ 1 in this case, it can be easily calculated that Idτ > 0.

22