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

Wen-Jung Liangand

Chao-Cheng Mai

October 08, 2008

Overview

• Introduction• Basic Model and Optimal Characteristic• Optimal Export Policy• Concluding Remarks

Introduction

• Brander and Spencer (1985): three-country model, Cournot competition, the optimal export policy is an export subsidy to help home firm act as a Stackelberg leader.

• Eaton and Grossman (1986): The optimal export policy is an export tax rather than an export subsidy, when firms play Bertrand price competition and the products are differentiated.

The Premise of E-G model

The premise of E-G model: The degree of product differentiation remains unchanged – This is a short-run scenario.

Firms are able to change their characteristics of the products (i.e., changing the degree of horizontal differentiation) in the long run

The Definition of Horizontal Differentiation

• According to Ferreira and Thisse (1996, p. 486), two products are said to be horizontally differentiated when both products have a positive demand whenever they are offered at the same price. Neither product dominates the other in terms of characteristics, and heterogeneity in preferences over characteristics explains why both products are present in the market.

Motivation• The effect of horizontal differentiation among

products on optimal strategic trade policy has not been touched upon. This topic is interesting because:

• 1. Products exhibited horizontal differentiation is commonly existed in the real world, such as different brands of sedans, for example BENZ and BMW, designed clothes, and perfumes…

• 2. Firms can change their characteristics of (i.e., horizontal differentiation among) products in the long run, for example, BENZ, Smart and TOYOTA, Lexus.

• As is common in horizontally differentiated models, consumers are characterized by their most preferred products and by the disutility incurred when they buy a non-preferred product.

• As in Hotelling (1929), firms model this disutility as transport cost in a Hotelling linear city model. This disutility is determined by the distance in the characteristic line between that product and the most-preferred product of the consumer.

• Specifically, a consumer’s location can be interpreted as his most-preferred product specification, and the firm’s location as the characteristic of the product it produces.

• The distance between the firm’s location and the consumer’s location represents the difference of the characteristics between the firm and the consumer.

• The consumer will suffer a disutility which can be represented by the transport cost if the distance is not nil.

The Relationship of Price Competition and Horizontal Differentiation

• The price competition between firms becomes more severe if the products get to be more homogeneous, while lessened if more differentiated

• This gives government an incentive to influence the choices of the characteristics of firms for mitigating price competition via trade policies.

The Effect of Cost Asymmetry

• Liang and Mai (2006) show that there exists a cost-advantage (cost-disadvantage) effect for low (high) cost firm when the two firms compete in the Hotelling model.

• This effect attracts the low cost firm to move closer to its rival for capturing larger market share, while forces the high cost firm to get farther away from its rival.

The Purpose of the Paper

• The purpose of the paper is to integrate Brander and Spencer’s three-country model and Hotelling’s linear city model, in which cost asymmetry and horizontal differentiation are taken into account to explore the optimal export policy under Bertrand price competition.

The Main Finding of the Paper

• The optimal export policy imposed by the government of the high cost firm is an export subsidy rather than export tax under Bertrand competition, when the characteristics of firms are endogenously determined.

The Intuition Behind the Finding• 1. Imposing an export subsidy for the high cost

firm can improve its cost disadvantage, and then force the rival to move farther away by the cost difference effect. This enlarges the high cost firm’s market share and its output.

• 2. By forcing the rival to move farther away makes the two firms locate more distantly, which means they are more differentiated. Thus, price competition is lessened, and the prices charged are higher.

A Three-stage Game• In the first stage, the home government

determines its optimal export policy to maximize domestic welfare;

• In the second stage, each firm simultaneously selects a characteristic to maximize its profit;

• In the third stage, firms play Bertrand competition in a third-country market.

Literature Review• De Meza (1986): offer subsidies to efficient firm,• Qiu (1994): considers asymmetric information on cost,• Maggi (1996): introduces capacity constraint,• Bandyopadhyay et al. (2000): incorporate labor union,• Zhou et al. (2002): take into account quality investmen

t of LDC and developed country,• Miller and Pazgal (2005): take into consideration the d

elegated game.

Basic Model and Optimal Characteristic

The basic model is an integration of Brander and Spencer’s three-country model and Hotelling’s linear city model.

• Firms’ marginal costs, cd and cf, are asymmetric.• Products differ with respect to a one dimensiona

l characteristic. The characteristics are measured by xd and xf, where xd xf.

• The distribution of characteristics is analogous to the Hotelling (1929)-type linear city model,

• Consumers’ ideal characteristics are uniformly distributed in the line segment with unit density.

• Each consumer buys one unit of the product.• Each point of the line segment within the

interval [0, 1] represents the consumer’s ideal characteristic.

• The disutility of the distance between firm’s characteristic and consumer’s ideal characteristic can be represented by a quadratic transport cost function.

(1)

• The utility function:

• The marginal consumer’s location:

(2)• The aggregate demand and profit for firms d and f:

(3.1)(3.2)

(4.1)(4.2)

,,,0)()( 2 fdixxtprxu ii

.)(22

ˆ

df

dffd

xxtppxx

x

,ˆ1ˆ

0xdxq

x

d ,ˆ11

1

ˆxdxq

xf ,ˆ)]([ xcp ddd

),ˆ1]([ xcp fff

Solve the three-stage game by backward induction starting with the third stage:

※ In stage 3, solving each firm’s profit-maximizing conditions, we obtain:

(5.1)

(5.2)

)},2)(()(2){3/1( dfdffdd xxxxtccp

)}.4)((2)){(3/1( dfdffdf xxxxtccp

※ In stage 2, maximizing the reduced profit function with respect to xd and xf, respectively, we derive:

(10.1)

(10.2)＊ The cost difference effect.＊ The competition effect.

,)32)(()()(3

ˆfddffd

dfd

d xxxxtccxx

xx

.)34)(()()(3

ˆ1dfdffd

dff

f xxxxtccxx

xx

Optimal Characteristic

• The cost difference effect: as the cost difference +cd – cf > (<) 0, the characteristic of the home firm tends to stay away from (get closer to) that of the foreign firm due to its cost disadvantage (advantage), while the behavior of the foreign firm is reversed.

• The competition effect: as the two firms’ characteristics become farther apart, they become more differentiated and therefore mitigates the competition under Bertrand competition.

Solve (10), we have:

(11)

(12)

(13)

.)34( if

,1}]/)(34[2){3/1( and ,0 2/1**

t/cct

tccxx

fd

fdfd

,)( )34(if

,1 and ,0}]/)(34[1){3/1( *2/1*

-tcct/-

xtccx

fd

ffdd

. if ,1and ,0* tcc-t xx fd*fd

Proposition 1. Assuming the export tax rate be equal to zero initially (i) When the cost difference lies in between t < cd – cf

(4t/3), the home firm has severe cost disadvantage and selects to locate at the left endpoint, while the foreign firm could locate inside of the line segment.

(ii) When the cost difference lies in between (-4t/3) cd – cf < (-t), the foreign firm has severe cost disadvantage and selects to locate at the right endpoint, while the home firm could locate inside of the line segment.

(iii) When the cost difference lies in between (-t) cd – cf t, the Principle of Maximum Differentiation holds.

Differentiating (11) – (13) with respect to yields:

(14.1)

(14.2)

, )( if 0,

),(- )34( if

0 /)(34)2/1(

),34( if 0,

21

0

*

tcc-t

tcct/-

tccτt-

t/cc tx

fd

fd

/fd

fdd

o

t. 0 (-t) if 0,

),(- )34( if ,0

),34( if

0,/)34)2/1( 21

0

*

fd

fd

fd

/fd

f

cc

tcct/-

t/cc t

tcc(τt-x

o

Proposition 2. The effects of an increase in the home export tax on the firms’ choices of characteristics evaluated at the initial value of the export tax rate equaling zero are as follows:

(i) When the cost difference lies in between t < cd – cf (4t/3), the home firm selects to locate at the left endpoint, while the foreign firm moves closer to its rival.

(ii) When the cost difference lies in between (-4t/3) cd – cf < (-t), the foreign firm selects to locate at the right endpoint, while the home firm moves farther away from its rival.

(iii) When the cost difference lies in between (-t) cd – cf t, the Principle of Maximum Differentiation still holds.

Optimal Export Policy The domestic welfare function is defined as:

(15)

Totally differentiating (15) with respect to and evaluating it at o = 0, we obtain:

(16)

),,,(),,( fddfddd xxqxxw

.0

ddf

f

dd

d

dd qx

xx

xddw

o

Case : The Cost Difference Lies in BetweenⅠ t < cd – cf (4t/3).

(18)

＊ The product-differentiation effect＊ The direct tax effect＊ The tax revenue effect

.0

ddf

f

dd qx

xddw

o

• The product-differentiation effect: a rise in the export tax attracts foreign firm to get closer due to worsening the home firm’s cost disadvantage. This creates two impacts: (1) The products become less differentiated and prices fall due to stronger price competition; (2) the market share of the home firm is reduced due to a worsened cost disadvantage.

• Thus, the product-differentiation effect is negative.

• The direct tax effect is also negative via increasing the cost of the home firm.

• The tax revenue effect is positive due to the rise in tax revenue.

• The optimal tax is jointly determined by the three effects.

Substituting (14) into (18) we can rewrite (18) as:

•

• (18.1)

• It follows from (11) that the term (J - 1) is greater than zero. This implies that the product-differentiation effect and the direct effect outweigh the tax revenue effect. The optimal policy is to subsidize.

,130

Jqd

dw dd

o

0

xd xf

1

pd

pf

Fig.2. A Case of Optimal Export Subsidy Case

E

ES

Sdp

Sfp

x̂

x̂Sx̂

Sfx

Proposition 3. When the cost difference lies in between t < cd

– cf (4t/3) and the initial value of the export tax rate is assumed to be zero, a rise in the export subsidy (or a fall in the export tax) increases home firm’s profits and hence welfare via enlarging the degree of the horizontal differentiation between the two products. Thus, the optimal export policy under Bertrand competition is an export subsidy.

Case : The Cost Difference Lies in Between Ⅱ(-4t/3) cd – cf < (-t).

(19)＊ The direct tax effect＊ The tax revenue effect＊ The tax revenue effect outweighs the direct t

ax effect＊ The optimal export policy is an export tax

.0

ddd q

ddw

o

0 xd

xf

1

pd

pf

Fig.3. A Case of Optimal Export Tax Case

xd

E

E

dp

fp

x̂

x̂

Proposition 4. When the cost difference lies in between (-4t/3)

cd – cf < (-t) and the initial value of the export tax rate is assumed to be zero, a rise in the export tax increases home firm’s profits and welfare via enlarging the degree of the horizontal differentiation between the two products. Thus, the optimal export policy is an export tax under Bertrand competition.

Case : The Cost Difference Lies in Between Ⅲ(-t) cd – cf t.

(20)＊ The tax revenue effect outweighs the direct t

ax effect.＊ The optimal export policy is to tax.

.0

ddd q

ddw

o

Proposition 5. When the cost difference lies in between (-t)

cd – cf t such that the two firms are located at the opposite endpoints of the line segment and the initial value of the export tax rate is assumed to be zero, the horizontal differentiation between the two products remains unchanged irrespective of the export policy imposed by the government. As a result, the optimal trade policy is an export tax under Bertrand competition.

Concluding Remarks1.The focus of this paper is on the product-differ

entiation effect in the determination of the optimal export policy.

2. When the cost difference lies in between t < cd – cf (4t/3) such that the foreign firm is located closer to the home firm due to its cost advantage, the optimal export policy of the home country under Bertrand competition is an export subsidy.

Concluding Remarks3. Given the horizontal differentiation of the

products been endogenously determined, our paper shows that as the cost advantage gained by the home firm is switched into cost disadvantage sufficiently, the outcome of the game moves from Eaton-Grossman tax to Brander-Spencer subsidy under Bertrand price competition.

4. The result of Eaton and Grossman (1986) turns out to be a special case of this paper.