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Analysis of Exogenous Switching Cost on Firms’ Behavior in Hotelling Model

ERASMUS UNIVERSITY ROTTERDAMErasmus School of EconomicsDepartment of Economics

Supervisor: Dr. J. J. A. Kamphorst

Name: H.HanishaExam number: 366103E-mail address: [email protected]

Table of Contents

1. Introduction 32. Literature review

a. Spatial Competition and Hotelling Model 4b. Switching Costs 6

3. Modelsa. Hotelling Model period 1 7b. Hotelling Model period 2: switching costs 8

4. Model Solvinga. Hotelling model

i. Model 10ii. Model analysis 11

b. Case 1: Assumption firm 1 locates to the left of firm 2i. Market base of firm 1 11

ii. Market base of firm 2 12iii. Analysis 13

c. Case 2: Assumption firm 1 locates to the right of firm 2i. Market base of firm 1 18

ii. Market base of firm 2 19iii. Analysis 20

5. Resultsa. Analysis 25b. Results 29

6. Conclusion 317. Appendix 338. Bibliography 48

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SECTION 1: INTRODUCTION

This paper builds upon the existing literature of a duopoly price competition on the Hotelling

line with quadratic transportation costs. This paper analyses the presence of switching costs

on the location choice and profits of the firms. Additionally, it is analysed to what extent it is

profitable for the firm to induce consumer switching by reducing the prices charges or the

relocation of the firm. Most of the existing literature on switching costs on Hotelling model

covers the analysis of differing price elasticity and pre-commitment to prices, with less

attention being paid to the choice of relocation and price reduction, which will be the focus of

this paper.

Latest researches in the field of economics have given increasing importance on the concept

of switching costs. The concept of switching costs is best seen in the cases of consumer

electronics. A very common example would be in the sector of smartphones and laptops. The

case of Microsoft versus Apple is the perfect example to illustrate the phenomenon of

switching costs incurred by consumers. A survey conducted found that 21 percent of Apple

users would never switch from Apple platform to Microsoft or any other platform, regardless

of any level of discounts or promotions. For the rest of the consumers, more than half of them

would not switch platform unless they are granted a discount of more than 30 percent. In this

case, the discount of 30 percent represents the switching cost; if granted, they are willing to

switch to another platform if compensated for the cost of switching. It is estimated that the

average switching cost (in discount form) from Apple platform to any other platform (in

monetary terms) per user is around 49 percent – a very high level of switching costs, making

Apple a very successful company in maintaining and expanding its user base (Hughes, 2012).

Various researches have shown that the presence of switching costs has a tendency to affect

the behaviour loyalty. In their research, Heide and Weiss (1995) stated that high-technology

markets tend to involve unique problems and therefore pose issues for firms competing in the

market. The problems are related to uncertainty and the presence of switching costs. Their

research concluded that the presence of vendor-related switching costs restricted the buyer’s

choice behaviour (Heide & Weiss, 1995). The presence of switching costs can also be used as

an entry deterrent strategy. Various sources of switching costs, both psychological and in

monetary terms have proven successful in maintaining a competitive edge and preventing

new competition from entering the market (Worthington, 2005).

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The presence of large switching costs locks in the buyer after the initial purchase of the

product (Klemperer & Farrell, Coordination and Lock In: Competition with Swtiching Costs

and Network Effects, 2007). This makes it harder for other firms in the market to expand

their customer base after the initial purchases by the consumers. The existence of switching

costs segments the market (which otherwise would have been undifferentiated). Firms would

focus on maintaining their established customer base and not compete aggressively for the

customers of their rivals (Klemperer & Farrell, 2007).

The existence of segmented market due to switching costs gives rise to opportunities of price

discrimination. The most common practice is poaching, where firms offer low prices,

discounts or other inducements only to the customers of the rival firms in order to make them

switch (Fudenberg & Tirole, 2000). However, poaching comes at a cost to the firms and after

a certain point, once the switching costs of the most loyal consumers of the rival firm is too

high, it becomes too costly for the firm to induce switching.

The structure of the paper is as follows: the introduction is followed by literature review.

Next, in the model section, the main models of this paper are solved and analysed, The

following results section presents the analyses of the model, followed by the conclusion

section where the overall conclusion, limitations and further suggestions is stated.

SECTION 2: LITERATURE REVIEW

The structure of this section is as follows: first, literature review on both the spatial

competition and the Hotelling model is presented, followed by the literature review on

switching costs.

A. Spatial Competition and Hotelling Model

The analysis of spatial competition has its roots in the classic model of Hotelling. Since its

publishing in 1929, the Hotelling model has faced various criticisms regarding the

assumptions made in the model. Additionally, variations of the basic Hotelling model have

been used in analysis of various topics, from collusion to price discrimination. In this section,

literature review is done discussing the various assumptions and variations of the spatial

competition and Hotelling model.

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Spatial competition is of two basic types, which are mill pricing and spatial price

discrimination. Under mill pricing, the transportation costs are charged to the consumers and

under spatial price discrimination, the transportation costs are paid for by the firm (Hobbs,

1986). The Hotelling model assumes mill pricing, that is, the consumers bear the

transportation costs (Phlips & Thisse, 1982).

In his paper, Hotelling showed that through horizontal differentiation, stability could be

reached between small numbers of firms. The notion behind the model is that price reduction

for one of the variety will not necessarily lead to consumers switching due to product

differentiation, which implies that customers prefer certain varieties over others (Phlips &

Thisse, 1982). Although the Cournot and Bertrand models have been expanded to included

cases of differentiated goods, the Hotelling model is still widely used when investigating

cases of horizontal differentiation. In the case of the Hotelling model, the differentiation is

introduced in terms of transportation costs (from the location of the buyer to the location of

the seller). The main outcome of the analyses is that there is a tendency for the firms to

agglomerate in the middle of the market, i.e. minimal differentiation (Hotelling, 1929).

One of the main criticisms regarding the model and the results of Stability in Competition

was discussed by D’Aspermont, Gabszewicz and Thisse. Specifically, it was the assumption

of linear transportation cost that was criticized. It was shown that when the transportation

cost is linear, there exists no equilibrium price solution. The paper concluded that contrary to

the original result by Hotelling, no conclusion can be made about the tendency for both the

sellers to agglomerate at the centre of the market. Additionally, by changing the assumption

of linear costs and assuming quadratic costs instead, they concluded that there is a tendency

for both sellers to locate at the end point, thereby maximizing their differentiation

(d'Aspremont, Gabszewicz, & Thisse, 1979).

For the purpose of this paper, transportation costs are assumed to be quadratic, in line with

the assumption made by D’Aspermont, Gabszewicz and Thisse. Switching cost is introduced

in the second period model in addition to the transport costs.

B. Switching Costs

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The existence of switching costs allows firm to segment their market into their own

customers and that of the rivals’ on the bases of the preferences of their consumers (Shaffer

& Zhang, 2000). In the presence of switching costs, rational consumers exhibit brand loyalty

when they are given the choice between identical products (Klemperer, 1987). In this section

of the literature review, various papers that have analysed switching costs under various

different assumptions are compared and analysed.

Switching costs can be categorized as either exogenous or endogenous. Exogenous switching

costs are the ones that are not created by the producer, whereas endogenous switching costs

are the ones created by the producers themselves (Haucap, 2003). There is vast amount of

literature that paid attention to the analyses of exogenous switching costs, not so much on the

endogenous switching costs. For the purpose of this paper, the switching costs are treated as

exogenous – the switching costs arise automatically as soon as the consumer buys the product

in the first period.

Klemperer (1987), using a two period model, showed that once a consumer is locked in to a

particular firm, her or his sensitivity to a rival firm’s price cut decreases with the presence of

switching costs. After the first period purchase, the market gets segmented into two groups

with differing price elasticities (Klemperer, 1987). In this paper, similar to Klemperer, the

introduction of the switching cost at the beginning of the second period will cause the market

to be segmented into two. However, this model is based on Hotelling instead of Cournot

model, which was the case in Klemperer (1987). Additionally, Klemperer also took into

account reservation prices. In this paper, the reservation price is assumed to be low enough

that every consumer buys one product from either firm.

In their paper, Shaffer and Zhang (2000) analysed the strategies of pay-to-switch or pay-to-

stay to investigate which strategy will result in the firms maximizing their profits. Their paper

studies third degree price discrimination (preference based) with two firms competing in

prices for consumers with varying loyalties. Their findings suggest that the presence of price

discrimination may lead to less competition between firms instead of intensifying the

competition. Additionally, it was found that when the demand is symmetric, charging a lower

price to a rival’s customer is the optimal strategy for a firm. In case the demand is

asymmetric, it may be profitable to charge a lower price to one’s own customer base. The

overall result is that price discrimination leads to overall reduction of prices to all the

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consumers in the market (Shaffer & Zhang, 2000). In the paper, the authors introduced

asymmetry in switching cost, analogous to what will be done in this paper. The switching

cost introduced to both the firms will differ in value; in other words, it will be more costly to

one group of consumers to switch firms than the other group. However, one major difference

between this paper and Shaffer & Zhang (2000) is the assumption of the nature of the

switching cost: in this paper, the firms are not aware that switching cost will arise (myopia)

and do not have control over the level of the switching cost (exogenous switching cost)

whereas in Shaffer and Zhang (2000), the switching cost was endogenous and the firms are

able to create loyalty-inducing arrangements.

Most of the existing literature focuses on analysing the profits and behaviour of consumers in

the presence of switching costs. The focus lies much on pre-commitment pricing and

endogenous switching costs such as discounts and vouchers. This paper focuses more on the

strategies of the firms themselves and what happens to the intensity of competition in the

market when the firm chooses the strategy of price reduction and relocation along the market

in the case of exogenous switching costs.

SECTION 3: MODELS FOR PERIOD 1 AND 2

A. Hotelling Model Period 1

This section analyses the first period Hotelling model, where the firm simultaneously chooses

a location in the market and sets a price that maximizes their profits. The firms make

decisions regarding the location and prices while taking into account their rivals actions. Both

the firms are assumed to be myopic, that is, the firms do not foresee that switching costs will

arise in the beginning of period 2 and therefore make their decisions based on information

available to them in period 1 only (the firms are short-sighted).

The basic assumptions for this model follows the assumptions made by Hotelling in his

model, with only one of the assumptions changed: instead of linear transportation cost, it is

assumed instead that the transportation cost is quadratic.

The model consist of two firms (a duopoly situation) where each firm sell similar products

with the same marginal cost, denoted by c. Transportation costs are borne by the consumers

and is denoted by t (x i−x¿)2, where x¿ denotes the location of the consumer and x idenotes the

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location of firm, where i∈ {0 , 1 }. The products available to consumers are differentiated by

location only.

In the first stage, the firms choose their respective locations on a linear spectrum, denoted by

x i∈[0 , 1], i∈ {0 , 1 }. In the second stage, the firms set the price, denoted by Pi, where

i∈ {0 ,1 }.

The cost of obtaining the product consists of two parts: the transportation cost t (x i−x¿)2 and

price charged by the firm Pi. Therefore, the cost incurred by the consumer is Pi+t (xi−x¿)2,

where t is the cost of transportation per unit. While making the purchase decision, the

consumer purchases the product at the firm where the overall cost is minimized.

B. Hotelling Model Period 2: switching costs

As mentioned before, an important assumption made in this paper is that the firms in period

one are assumed to be myopic, as in, the firms do not foresee that switching costs will arise in

the beginning of period 2 and therefore make their decisions based on information available

to them in period 1 only (the firms are short-sighted).

The next step of the analysis is to introduce the switching costs at the beginning of period

two. At the end of period 1, it was found that consumers located between 0 and 0.5 will buy

from firm 1, and consumers located between 0.5 and 1 will buy from firm 2. Once the

consumers buy from a certain firm, the consumers are “locked-in”, that is, the consumer

would incur switching costs in order to buy from the other firm. Therefore, a consumer that

would switch and buy from different firm than previously will incur transportation cost, price

charged by the firm for the product itself and the switching cost.

The firms are aware of this in the beginning of period two and therefore will again make

location and price decisions that will maximize their profits in period two in the presence of

switching costs. While making the choice of their new locations at the beginning of period

two, it is assumed at this stage of the analysis that the firms are completely mobile and the

relocation costs of the firm from their optimal location in period one to their optimal location

in period two is zero.

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In this model, an additional variable, namely the switching cost is introduced into the

equations. The switching cost is denoted by si .Switching cost siis interpreted as the switching

cost incurred by consumers of firm i to switch to the other firm.

Due to the presence of switching costs, the market is segregated into two different markets:

the consumers located between 0 and 0.5 form the market base of firm 1 and consumers

located between 0.5 and 1 form the market base of firm 2. Therefore, the process of finding

marginal consumers and the consequent steps performed in Model 1 in the previous section

will be done for both the market bases separately. Consequently, the domain for the values of

market share in each market base is [0, ½].

Additionally, since the firms can only choose one location, the firms will maximize their

respective profit functions in both the markets combined (total profit of market base of firm 1

and market base of firm 2 will be added and it is the derivative of this joint profit function

that will be used to find the optimal location, and not the profit function of the respective

firms in the individual market).

Furthermore, the assumptions of the location of the firms are important. In the first set of

calculations, the model assumes that firm 1 is located to the left of firm 2. The second set of

calculations involves changing this assumption and assuming that firm 1 is located to the

right of firm 2.

In this part of the model, y j ,k is interpreted as follows: y represents the variable (price,

location, etc.), j represents the firm (either firm 1 or 2) and k represents the market base

(market base of firm 1 or firm 2). The marginal consumer is denoted by xm,iwhere i

represents the market base of the firm (either firm 1 or firm 2)

SECTION 4: MODEL SOLVING

A. Period 1: Hotelling Model

The first step in solving the model involves calculating the marginal consumer, which is

defined as the consumer at the point x who is indifferent between purchasing the product at

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either of the firms. The marginal consumer is calculated by equating the total cost of

purchasing at firm 1 and 2 as follows:

P1+t (x1−x¿)2=P2+t (x2−x¿)2

After calculating the marginal consumer, the profit functions of both the firms are calculated

by multiplying the market share of the firm with the profit obtained by the firm per unit. The

price strategy of the firms can be obtained by taking the first-order partial derivative of the

profit functions with respect to the price of both the firms respectively (to find out how the

profit changes as the price charged by the firm changes) 1. After obtaining the price strategy

of both the firms, the Nash equilibrium price strategy of both the firms can be found by

simultaneously solving the price strategy of both the firms. The Nash equilibrium prices

charged by the firms are as follows:

p1¿=2 t

3 ( x2−x1)+ t3 (x2

2−x12 )+c

p2¿=4 t

3 ( x2−x1 )− t3 (x2

2−x12 )+c

By substituting the Nash Equilibrium price strategies of both the firms in the equation for the

marginal consumer, the marginal consumer can be calculated2. The consumers to the right of

the marginal consumer will buy from one of the firms and the consumers to the left of the

marginal consumer will buy from the other firm. The profit functions of the firms can then be

calculated by multiplying the market share with the price strategies (in Nash) of the

respective firms3.

Firms maximize their profits with respect to their locations given the price strategies in the

Nash Equilibrium. To investigate the change in profits as the choice of location changes, the

first-order partial derivative of the profit functions of firms 1 and 2 with respect to their

locations (x1 , x2 respectively) are calculated and are as follows:

δ π1

δ x1=−1

18(3 x1−x2+2)(x1+x2+2)

δ π2

δ x2=−1

18(x1−3 x2+4)(x1+x2−4)

1 The profit functions and the partial derivatives can be found in appendix A.22 The location of marginal consumer and the market shares for both the firms can be found in appendix A.33 The profit functions can be found in appendix A.4

10

As can be seen from the derivatives of the profit functions, the profit for firm one is strictly

decreasing and profit for firm 2 is strictly increasing in the domain [0,1]. Therefore, firm 1

will locate at 0, whereas firm 2 will locate at 1 in order to maximize their respective profits.

By plugging in the values of 0 and 1 for x1and x2 in the equation to find the marginal

consumer, it is found that the marginal consumers is located at ½ and at equilibrium, the

firms will both charge a price of t+c.

B. Case 1: Assumption firm 1 locates to the left of firm 2

B.1. Market Base of Firm 1 (consumers located between 0 and 0.5)

As in the original Hotelling mode, the marginal consumer in market base of firm 1 is

calculated by equating the total cost of purchasing at firm 1 and 2. The switching cost is

incurred by the consumers that would buy from firm 2; therefore, for consumers that buy

from firm 2 in this market, the total cost incurred is the total of price paid for the product,

transportation cost and the switching cost paid to switch from firm 1 to firm 2.

To be more specific, consumers buying from firm 1 in this market base will pay

P1,1+t (x1−x¿)2; consumers buying from firm 2 would pay P2,1+t (x2−x¿)2+S1. Due to the

addition of the switching cost incurred by customers of firm 1 in this market, the equation to

calculate the marginal consumers is as follows:

P1,1+t (x1−x¿)2=P2,1+t(x2−x¿)2+S1

By performing the same steps taken when solving the Hotelling model period 1, the

following Nash equilibrium price strategy of the firms are obtained4:

p1,1¿ = t

3 ( x2−x1 )+ t3 ( x2

2−x12 )+c+

s1

3

p2,1¿ =2 t

3 ( x2−x1 )− t3 (x2

2−x12 )+c−

s1

3

By substituting the Nash Equilibrium price strategies of both the firms into the equation for

the marginal consumer, the marginal consumer can be calculated5. The consumers to the left

of the marginal consumer will buy from firm 1 and the consumers to the right of the marginal

4 The full calculations can be found in appendix B.25 The full calculations for this can be found in appendix B.3

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consumer will buy from firm 2. The profit functions of the firms can then be calculated by

multiplying the market share with the price strategies (in Nash) of the respective firms.

B.2. Market Base of Firm 2 (consumers located between 0.5 and 1)

Similar to solving the previous model, the marginal consumer in market base of firm 2 is






P1,2+t (x1−x¿)2+S2; consumers buying from firm 2 would pay P2,2+t (x2−x¿)2. Due to the



P1,2+t (x1−x¿)2+S2=P2,2+ t(x2−x¿)2



p1,2¿ = t

3 ( x22−x1

2)− s2

3

p2,2¿ =t ( x2−x1)− t

3 (x22−x1

2 )+ s2

3

By substituting the Nash Equilibrium price strategies of both the firms into equation for the

marginal consumer, the marginal consumer can be calculated7. The consumers to the left of

the marginal consumer will buy from firm 1 and the consumers to the right of the marginal

consumer will buy from firm 2. The profit functions of the firms can then be calculated by

multiplying the market share with the price strategies (in Nash) of the respective firms.

B.3. Analysis of Model with the introduction of switching costs with assumption firm 1

locates to the left of firm 2

I. Market Share

In market base of firm 1, the following marginal consumer is found:

6 The full calculations can be found in appendix C.27 The full calculations for this can be found in appendix C.3

12

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

The market share of firm 1 and 2 respectively are as follows:

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

(0.5−xm,1 )=13−1

6 ( x2+x1 )−s1

6 t ( x2−x1 )

In other words, the market share of firm 1 is the consumers located between 0 and xm ,1, and

the market share of firm 2 is the consumers located between xm ,1and 0.5.

The market share of firm 1 increases (and vice versa for firm 2) as the value of the switching

cost relative to transport cost increases. This would be logical as when the value of switching

cost increases when compared to the unit transportation cost, it is harder for firm 2 to induce

switching and therefore firm 2 will see its market share decrease even if it manages to locate

closer to the consumers to reduce the transportation cost.

The market share of firm 1 also increases (and vice versa for firm 2) when the distance

between the firms decrease. This is also logical as if the firms are very close to each other, the

difference in the transportation costs for a consumer from buying from firm 1 and 2 will be

less compared to the switching cost incurred and therefore the consumers would not switch

and firm 1 would have a bigger market share (and vice versa).

The domain for the values of market share for this market base is [0, ½ ]. However, as the

values of s1 increases, the market shares of the firm approaches infinitely positive or

infinitely negative. To be more specific, when the value of s1> t , firm 1 will capture the entire

market base of firm 1 and firm 2 will refrain from serving this market8. Therefore, for the

purpose of calculating the profit function, when s1> t , firm 1 will obtain ½ as the market

share and firm 2 will obtain 0.


xm,2=12+ 1

6 ( x2+ x1 )−s2

6 t ( x2−x1)

8 The calculation for this critical value can be found in appendix D.1

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(x¿¿m ,2−0.5)=16 ( x2+x1)−

s2

6 t ( x2−x1 )¿

(1−xm, 2 )=12−1

6 ( x2+x1)+s2

6 t ( x2−x1 )

In other words, the market share of firm 1 is the consumers located between 0.5 and xm,2, and

the market share of firm 2 is the consumers located between xm,2and 1.

Similar to the case of firm 1 in its market base, the market share of firm 2 increases (and vice

versa for firm 1) as the value of the switching cost relative to transport cost increases. This

would be logical as when the value of switching cost increases when compared to the unit

transportation cost, it is harder for firm 1 to induce switching and therefore firm 1 will see its

market share decrease even if it manages to locate closer to the consumers to reduce the

transportation cost.

In the case of the distance between the firms, it is similar to the previous case of firm 1. The

market share of firm 2 also increases (and vice versa for firm 1) when the distance between

the firms decrease for the reason explained above.

Similar to the market base of firm 1, when the value of s2> t , firm 2 will capture the entire

market share base of firm 2 and firm 1 will refrain from serving this market9. Therefore, for

the purpose of calculating the profit function, when s2> t , firm 2 will obtain ½ as the market


II. Price Strategy

The price strategies of firm 1 and 2 in Nash equilibrium respectively in the market base of

firm 1 are as follows:

p1,1¿ = t

3 ( x2−x1 )+ t3 ( x2

2−x12 )+c+

s1

3

p2,1¿ =2 t

3 ( x2−x1 )− t3 (x2

2−x12 )+c−

s1

3

The price function for firm 1 increases as the value of the switching cost increase and vice

versa for firm 2. This would be logical as when the switching cost increase, firm 1 can benefit

9 The calculation for this critical value can be found in appendix D.2

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by increasing the price since the consumers are not very sensitive to the price increase due to

the presence of switching cost. Alternatively, for firm 2, as the value of the switching cost

increase, the price charged would be reduced in order to compensate for the presence of

switching cost in order to induce switching.

As can be seen from the equation, the price function of firm 1 is always positive. However,

for firm 2, the price function is not always positive: for large enough values of s1relative to t ,

the price can fall below marginal cost (or even negative price). Since the model is only two

period model, the firms aim to maximize profit in the current period and therefore will not

charge price lower than the marginal cost. Therefore, the domain for the values of the price

charged by the firm would be [c, ∞].

More specifically, in the market base of firm 1, if the value of s1> t, the price function of firm

2 would be below marginal cost and therefore the firm would charge consumers the marginal

cost c10. Combined with the fact that the firm would not have a market share (see analysis :

market share) and the price charged by the firm is below marginal cost, firm 2 would refrain

from serving this market base if the value of the switching cost is high enough relative to the

unit transport cost.



p1,2¿ = t

3 ( x22−x1

2)− s2

3+c

p2,2¿ =t ( x2−x1)− t

3 (x22−x1

2 )+ s2

3+c

Similar to market base of firm 1, for large enough values of s2 , the price function for firm 1

can be below marginal cost and approaches -∞. Specifically, in the market base of firm 2, if

the value of s2> t, the price function of firm 1 would be below marginal cost and therefore the

firm would charge consumers the marginal cost c11. Combined with the fact that the firm

would not have a market share (see analysis : market share) and the price charged by the firm

is below marginal cost, firm 1 would refrain from serving this market base if the value of the

switching cost is high.

10 The calculation for this critical value can be found in the appendix D.3.11 The calculation for this critical value can be found in the appendix D.4.

15

III. Profit Functions

The market share and price strategy of the firms depends on the values of t , s1 and s2. The

profit functions are calculated by multiplying the price charged by the firm minus the

marginal cost with the market share12.

IV. First-order partial derivatives of profit functions 13

A. When s1<t and s2<t

The first-order partial derivatives for both the firms are neither strictly increasing nor strictly

decreasing for the values of s1<t and s2<t . Therefore, there exist infinite amount of

equilibria, depending upon the values of s1 , s2 and t . The equilibrium location for x1 and x2for

a specific value of s1 , s2 and t can be found by equating the derivatives of both the firms to 0

and solving them simultaneously.

When s1<t and s2<t , both the firms will operate in both market bases. However, if the firms

are located close enough to each other in equilibrium, it could very well be the case that the

price charged by firm 1 in the market base of firm 2 (and vice versa) be below the marginal

cost. It might also be very well the case that the firm does not have any market share in the

competitor’s market base. In this situation, the firm 1 would maximize a different profit

function that would arise when the entire market base of firm i is captured by the firm i14.

B. When s1<t and s2>t

Similar to the previous case, the first-order partial derivatives for both the firms are neither

strictly increasing nor strictly decreasing for the values of s1<t and s2>t . For this case, there

exist infinite amount of equilibria, depending upon the values of s1 and t . The equilibrium

location for x1 and x2 for a specific value of s1 and t can be found by equating the derivatives

of both the firms to 0 and solving them simultaneously.

When s1<t and s2>t , firm 1 will not serve the market base of firm 2 (consumers from 0.5 –

1). However, the market base of firm 1 will be served by both the firms. However, similar to

12 The profit functions for both the firms under different values of s1 , s2and t can be found in appendix D.513 The partial derivatives of all the profit functions can be found in appendix D.6.14 Profit function when s1>1∧s2>1.

16

the previous case, if in equilibrium the firms are located close enough with each other, it

could very well be the case that the price charged by firm 2 in the market base of firm 1 be

below the marginal cost, and the market share of firm 2 could be zero. In this situation, firm 1

would maximize a different profit function that would arise when the entire market base of

firm i is captured by the firm i (similar to the previous case).

C. When s1>t and s2<t

In this case, the first-order partial derivatives for both the firms are neither strictly increasing

nor strictly decreasing as well. For this case, there exist infinite amount of equilibria,

depending upon the values of s2 and t . The equilibrium location for x1 and x2 for a specific

value of s2 and t can be found by equating the derivatives of both the firms to 0 and solving

them simultaneously.

When s1>t and s2<t , firm 1 will obtain the entire market share in the market base of firm 1

(consumers between 0 and 0.5) and will also serve in the market base of firm 2 (consumers

between 0.5 and 1). In this scenario, similar to the previous case, it needs to be checked

whether the firms are located close enough with each other, and whether it is the case that the

price charged by firm 1 in the market base of firm 2 is below the marginal cost, and whether

the firm would in fact gain market share. In case the price charged is below marginal cost or

firm 1 does not have any market share in the market base of firm 2, firm 1 would maximize a

different profit function that would arise when the entire market base of firm 1 is captured by

the firm 1 and the entire market base of firm 2 is captured by firm 2.

D. When s1>t and s2>t

The partial derivative of the profit function of firm 1 is strictly decreasing with respect to x1

and does not depend on the values of s1 , s2 and t . Therefore, firm 1 would want to locate at 0.

Firm 1 would get the entire market share in its own market base (consumers between 0 and

0.5) and will not get any market share in the market base of firm 2 (consumers between 0.5

and 1)

The partial derivative of the profit function of firm 2 is strictly increasing with respect to x2.

Similar to firm 1, it does not depend on the values of s1 , s2 and t . Therefore, firm 2 would

want to locate at 1. Firm 2 would get the entire market share in its own market base

17

(consumers between 0.5 and 1) and will not get any market share in the market base of firm 1

(consumers between 0 and 0.5).

A necessary condition is that x1<x2 at all times. If this condition is not fulfilled, the profit

functions would differ (analysed in the next section).

In the case that either of the conditions of s2<t or s1<t is fulfilled, if the firms are located too

close together, both the firms will not get market share in their competitor’s market base. In

this case, they would optimize the profit function in the case of s2>t and s1>t ,which would

result in firm 1 being located at 0 and firm 2 being located at 115. Therefore, after finding the

equilibrium location for x1 and x2, it needs to be checked first whether there exist a market

share and price function value above marginal cost for that particular set of s1 , s2 and t values.

In case it does not, the firms maximize the profit functions as in the case of s2>t and s1>t .

C. Case 2: assumption firm 1 locates to the right of firm 2

C.1. Market Base of Firm 1 (consumers located between 0 and 0.5)

As done in the previous section, the marginal consumer in market base of firm 1 is calculated

by equating the total cost of purchasing at firm 1 and 2. The switching cost is incurred by the

consumers that would buy from firm 2; therefore, for consumers that buy from firm 2 in this

market, the total cost incurred is the total of price paid for the product, transportation cost and

the switching cost paid to switch from firm 1 to firm 2.


P1,1+t (x1−x¿)2; consumers buying from firm 2 would pay P2,1+t (x2−x¿)2+S1. Due to the



P1,1+t (x1−x¿)2=P2,1+t(x2−x¿)2+S1



15 Firms will optimize the profit functions when s2>1and s1>1.16 The full calculations can be found in appendix E.2

18

p1,1¿ =−2 t

3 ( x2−x1)+ t3 ( x2

2−x12)+c+

s1

3

p2,1¿ =−t

3 ( x2−x1 )− t3 ( x2

2−x12 )+c−

s1

3

By substituting the Nash Equilibrium price strategies of both the firms into the equation for

calculating the marginal cost, the marginal consumer can be calculated17. The consumers to

the right of the marginal consumer will buy from firm 1 and the consumers to the left of the

marginal consumer will buy from firm 2. The profit functions of the firms can then be

calculated by multiplying the market share with the price strategies (in Nash) of the

respective firms.

C.2. Market Base of Firm 2 (consumers located between 0.5 and 1)

Similar to solving the previous model, the marginal consumer in market base of firm 2 is






P1,2+t (x1−x¿)2+S2; consumers buying from firm 2 would pay P2,2+t (x2−x¿)2. Due to the



P1,2+t (x1−x¿)2+S2=P2,2+ t(x2−x¿)2



p1,2¿ =−t ( x2−x1 )+ t

3 (x22−x1

2 )− s2

3

p2,2¿ =−t

3 (x22−x1

2 )+s2

3

By substituting the Nash Equilibrium price strategies of both the firms into the equation used

to calculate the marginal consumer, the marginal consumer can be calculated19. The

17 The full calculations can be found in appendix E.318 Full calculations for this can be found in appendix F.219 Full calculations for this can be found in appendix F.3

19

consumers to the right of the marginal consumer will buy from firm 1 and the consumers to

the left of the marginal consumer will buy from firm 2. The profit functions of the firms can

then be calculated by multiplying the market share with the price strategies (in Nash) of the

respective firms.

C.3. Analysis of Model with the introduction of switching costs assumption firm 1

locates to the right of firm 2

I. Market Share

The market share of firm 1 and 2 respectively in the market base of firm 1 (consumers

between 0 and 0.5) is as follows:

(x¿¿m ,1−0.5)=13−1

6 ( x2+x1 )−s1

6 t ( x2−x1 )¿

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

In other words, the market share of firm 1 is the consumers located between xm,1and 0.5, and

the market share of firm 2 is the consumers located between 0 and xm,1 .

Similar to the case with the previous location assumption, the market share of firm 1

increases (and vice versa for firm 2) as the value of the switching cost relative to transport

cost increases.. The market share of firm 1 also increases (and vice versa for firm 2) when the

distance between the firms decrease for the reason explained before.

The domain for the values of market share for this market base is [0, ½ ]. However, as the

values of s1 increases, the market shares of the firm approaches infinitely positive or

infinitely negative. To be more specific, when the value of s1> 2t , firm 1 will capture the

entire market share base of firm 1 and firm 2 will refrain from serving this market 20.

Therefore, for the purpose of calculating the profit function, when s1> 2 t, firm 1 will obtain

½ as the market share and firm 2 will obtain 0.

20 The calculation for this critical value can be found in appendix G.1

20

The market share of firm 1 and 2 respectively in the market base of firm 2 (consumers

between 0.5 and 1) is as follows:

(1−xm, 2 )=12−1

6 ( x2+x1)+s2

6 t ( x2−x1 )

(x¿¿m ,2−0.5)=16 ( x2+x1)−

s2

6 t ( x2−x1 )¿

In other words, the market share of firm 1 is the consumers located between xm,2and 1, and

the market share of firm 2 is the consumers located between 0.5 and xm ,2 .

Similar to the case of firm 1 in its market base, the market share of firm 2 increases (and vice

versa for firm 1) as the value of the switching cost relative to transport cost increases. The

market share of firm 2 also increases (and vice versa for firm 1) when the distance between

the firms decrease for the reason explained above.

Similar to the market base of firm 1, when the value of s2> 2 t, firm 2 will capture the entire

market share base of firm 2 and firm 1 will refrain from serving this market21. Therefore, for

the purpose of calculating the profit function, when s2> 2 t, firm 2 will obtain ½ as the market


II. Price Strategy



p1,1¿ =−2 t

3 ( x2−x1)+ t3 ( x2

2−x12)+c+

s1

3

p2,1¿ =−t

3 ( x2−x1 )− t3 ( x2

2−x12 )+c−

s1

3

Similar to the case with the previous location assumption, the price function for firm 1

increases as the value of the switching cost increase and vice versa for firm 2. Also in this

case, the price function of firm 1 is always positive. However, for firm 2, the price function is

not always positive: for large enough values of s1relative to t , the price can fall below

marginal cost (or even negative price). Since the model is only two period model, the firms

21 The calculation for this critical value can be found in appendix G.2

21

aim to maximize profit in the current period and therefore will not charge price lower than the

marginal cost. Therefore, the domain for the values of the price charged by the firm would be

[c, ∞].

More specifically, in the market base of firm 1, if the value of s1> 2t, the price function of

firm 2 would be below marginal cost and therefore the firm would charge consumers the

marginal cost c22. Combined with the fact that the firm would not have a market share (see

analysis : market share) and the price charged by the firm is below marginal cost, firm 2

would refrain from serving this market base if the value of the switching cost is high.



p1,2¿ =−t ( x2−x1 )+ t

3 (x22−x1

2 )− s2

3+c p2,2

¿ =−t3 (x2

2−x12 )+ s2

3+c

Similar to market base of firm 1, for large enough values of s2 , the price function for firm 1

can be below marginal cost and approaches -∞. Specifically, in the market base of firm 2, if

the value of s2> 2t, the price function of firm 1 would be below marginal cost and therefore

the firm would charge consumers the marginal cost c23. Combined with the fact that the firm

would not have a market share (see analysis : market share) and the price charged by the firm

is below marginal cost, firm 1 would refrain from serving this market base if the value of the

switching cost is high.

III. Profit Function

The market share and price strategy of the firms depends on the values of t , s1 and s2. The

profit functions are calculated by multiplying the price charged by the firm minus the

marginal cost with the market share24.

IV. First-order partial derivatives of profit functions 25

A. When s1<2 t and s2<2 t

22 The calculation for this critical value can be found in appendix G.3.23 The calculation for this critical value can be found in the appendix G.4.24 The profit functions for both the firms under different values of s1 , s2and t can be found in appendix G.525 The partial derivatives of all the profit functions can be found in appendix G.6.

22

The first-order partial derivatives for both the firms are neither strictly increasing nor strictly

decreasing for the values of s1<2 t and s2<2 t. Therefore, there exist infinite amount of

equilibria, depending upon the values of s1 , s2 and t . The equilibrium location for x1 and x2for

a specific value of s1 , s2 and t can be found by equating the derivatives of both the firms to 0

and solving them simultaneously.

When s1<2 t and s2<2 t, both the firms will operate in both market bases. However, if the

firms are located close enough to each other in equilibrium, it could very well be the case that

the price charged by firm 1 in the market base of firm 2 (and vice versa) be below the

marginal cost. It might also be very well the case that the firm does not have any market

share in the competitor’s market base. In this situation, the firm 1 would maximize a different

profit function that would arise when the entire market base of firm i is captured by the firm

i26.

B. When s1<2 t and s2>2 t

Similar to the previous case, the first-order partial derivatives for both the firms are neither

strictly increasing nor strictly decreasing for the values of s1<2 t and s2>2 t. For this case,

there exist infinite amount of equilibria, depending upon the values of s1 , and t . The

equilibrium location for x1 and x2 for a specific value of s1 and t can be found by equating the

derivatives of both the firms to 0 and solving them simultaneously.

When s1<2 t and s2>2 t, firm 1 will not serve the market base of firm 2 (consumers from 0.5

– 1). However, the market base of firm 1 will be served by both the firms. However, similar

to the previous case, if in equilibrium the firms are located close enough with each other, it

could very well be the case that the price charged by firm 2 in the market base of firm 1 be

below the marginal cost, and the market share of firm 2 could be zero. In this situation, firm 1

would maximize a different profit function that would arise when the entire market base of

firm i is captured by the firm i (similar to the previous case).

C. When s1>2 t and s2<2 t

In this case, the first-order partial derivatives for both the firms are neither strictly increasing

nor strictly decreasing as well. For this case, there exist infinite amount of equilibria,

26 Profit function when s1>1∧s2>1.

23

depending upon the values of s2 , and t . The equilibrium location for x1 and x2 for a specific

value of s2 and t can be found by equating the derivatives of both the firms to 0 and solving

them simultaneously.

When s1>2 t and s2<2 t , firm 1 will obtain the entire market share in the market base of firm

1 (consumers between 0 and 0.5) and will also serve in the market base of firm 2 (consumers

between 0.5 and 1). In this scenario, similar to the previous case, it needs to be checked

whether the firms are located close enough with each other, and whether it is the case that the

price charged by firm 1 in the market base of firm 2 is below the marginal cost, and whether

the firm would in fact gain market share. In case the price charged is below marginal cost or

no firm 1 does not have any market share, firm 1 would maximize a different profit function

that would arise when the entire market base of firm i is captured by the firm i.

D. When s1>2 t and s2>2 t

The partial derivative of the profit function of firm 1 is strictly increasing with respect to x1

and does not depend on the values of s1 , s2 and t . Therefore, firm 1 would want to locate at 1.

Firm 1 would get the entire market share in its own market base (consumers between 0 and

0.5) and will not get any market share in the market base of firm 2 (consumers between 0.5

and 1)

The partial derivative of the profit function of firm 2 is strictly decreasing with respect to x2.

Similar to firm 1, and does not depend on the values of s1 , s2 and t . Therefore, firm 2 would

want to locate at 0. Firm 2 would get the entire market share in its own market base

(consumers between 0.5 and 1) and will not get any market share in the market base of firm 1

(consumers between 0 and 0.5).

A necessary condition is that x1>x2 at all times. If this condition is not fulfilled, the profit

functions would differ (analysed in the previous section).

In the case that either s2<t or s1<t condition is fulfilled, if the firms are located too close

together, both the firms will not get market share in their competitor’s market base. In this

24

case, they would optimize the profit function in the case of s2>t and s1>t ,which would result

in firm 1 being located at 1 and firm 2 being located at 027.

SECTION 5: RESULTS AND EQUILIBRIUM ANALYSIS

Analysis

A. When s1>2 t and s2>2 t

If firm 1 decides to locate to the left of firm 2, firm 1 would decide to locate at 0 and firm 2

would locate at 1. The corresponding profit of firm 1 would be 3 ( s1+2t )

18 and profit of firm 2

would be 3 ( s2+2t )

18. If firm 1 decides to locate to the right of firm 2, firm 1 would decide to

locate at 1 and firm 2 would locate at 0. The corresponding profit of firm 1 would be 3 ( s1+ t )

18

and profit of firm 2 would be 3 ( s1+ t )

18.

Profit when (x1, x2¿ is (0,1 )is higher than if it is (1,0) for both the firms, therefore neither of

the firms has an incentive to deviate from (0,1 ) when s1>2 t and s2>2 t. Therefore, this is one

possible equilibrium.

B. When t <s1<2t and t <s2<2t

Firm 1 would strictly prefer to locate to the right of firm 2 if the following inequality is

fulfilled28:

s12>5 t2+t s1

This inequality is only fulfilled if s1>12((1+√21 ) t). Due to the constraint t <s1<2t , firm 1 will

never strictly prefer to locate to the right of firm 2. Firm 1 will prefer to locate to the right of

firm 2 if in equilibrium, both the firms are located close enough that the profit function of

firm 1 (when it locates to the right of firm 2) is higher than 3 ( s1+2t )

18. Therefore, in

equilibrium, firm 1 will either locate to the right of firm 2 or will locate at the endpoint at 0.

27 Firms will optimize the profit functions when s2>1and s1>1.28 The calculation for this critical value can be found in appendix H.1

25

Firm 2 would strictly prefer to locate to the left of firm 1 if the following inequality is

fulfilled29:

s22>5 t2+t s2


never strictly prefer to locate to the left of firm 2. Similar to firm 1, firm 2 will only prefer to

locate to the left of firm 1 if in equilibrium, both the firms are located close enough that the

profit function of firm 2 (when it locates to the left of firm 1) is higher than 3 ( s2+2t )

18.

Therefore, in equilibrium, firm 2 will either locate to the left of firm 1 or will locate at the

endpoint at 1.

C. When s1>2 t and t <s2<2t


fulfilled30:

(s2−2t)2>3 t 2

This inequality is only fulfilled if s22>(√3+2) t. Due to the constraint t <s1<2t , firm 1 will

never strictly prefer to locate to the right of firm 2. Similar to the previous case, firm 1 will

prefer to locate to the right of firm 2 only if in equilibrium, both the firms are located close

enough that the profit function of firm 1 (when it locates to the right of firm 2) is higher than

3 ( s1+2t )18

. Therefore, in equilibrium, firm 1 will either locate to the right of firm 2 or will

locate at the endpoint at 0.


fulfilled31:

s22>5 t2+t s2




29 The calculation for this critical value can be found in appendix H.130 The calculation for this critical value can be found in appendix H.231 The calculation for this critical value can be found in appendix H.2

26


18.


endpoint at 1.

D. When t <s1<2t and s2>2 t


fulfilled32:

s12>5 t2+t s1


never strictly prefer to locate to the right of firm 2. Firm 1 will prefer to locate to the right of

firm 2 if in equilibrium, both the firms are located close enough that the profit function of

firm 1 (when it locates to the right of firm 2) is higher than 3 ( s1+2t )

18. Therefore, in

equilibrium, firm 1 will either locate to the right of firm 2 or will locate at the endpoint at 0.


fulfilled33:

(s1−2 t)2>3 t 2

This inequality is only fulfilled if s12>(√3+2) t. Due to the constraint t <s2<2t , firm 2 will




18.


endpoint at 1.

E. Analysis

Therefore, one main equilibrium that exist is (0,1) when s1>2 t and s2>2 t. For instances

where one (or both) of the values of switching cost falls between t and 2 t, the firms compare

their profit functions (given the location assumption that firm 1 locates to the right of firm 2)

32 The calculation for this critical value can be found in appendix H.333 The calculation for this critical value can be found in appendix H.3

27

in equilibrium with 3 ( s1+2t )

18 for firm 1 and

3 ( s2+2t )18

for firm 2. If the profit function in

equilibrium for both the firms result in higher profits than the values mentioned above, then

the firms will locate along the line, depending on the values of s1 , s2 and t . If the profit

functions of one of the firms result in value lower than the abovementioned value, then that

firm will choose to locate at endpoint and will not have any incentive to relocate, regardless

of what the other firm does. Therefore, if this happens for firm 1, it will locate at 0 and for

firm 2 at 1. Given this situation, the other firm will have no choice but to also locate at

endpoint since firm 2 cannot locate itself to the left of firm 1 if firm 1 decides to locate at 0,

and firm 1 cannot locate itself to the right of firm 1 if firm 2 decides to locate at 1.

For any other instances, at equilibrium, each firm would compare their profit function values

under the two different assumptions using the values they obtained for x1 and x2 given a

specific value of s1 , s2 and t . The equilibrium values of x1 and x2 and the distance between

them are likely to be different, for the same values of s1 , s2 and t , under different location

assumption. Since the value of the profit function mainly relies on the distance between the

locations of the two firms, therefore the values of the profit functions are also likely to differ

under different location assumption. The firm chooses the profit function that yields the

highest value, given a specific value of s1 , s2 and t .

Therefore, when the switching costs are high enough, the firms will not switch their positions

(firm 1 will locate to the left of firm 2). However, in the case that the switching costs are low,

the firms might switch (firm 1 might locate to the right of firm 2).

Results

The equilibrium analysis revealed that when the ratio of switching cost and the unit transport

cost is high (more than 2), it is more profitable for the firm to locate at endpoints.

Specifically, the firms should remain in the same location that they were at in period 1.

However, since it is assumed to be the case that the switching cost differ, it could very well

be the case that the switching cost of one the firms is high and the other is low, making it

profitable for one firm to locate at end –point and the other firm not. In this case, the Nash

equilibrium still states that the firms are better off locating at end points (since the firm with

the higher switching cost will have no incentive to deviate; the other firm taking this into

28

account maximizes the profit function and will locate at end-point as well). Therefore, when

x1=0, x2 will never locate anywhere else but at 1 and vice versa.

When the ratio of the switching costs to unit transportation cost are not too high (less than 2),

at equilibrium, there are multiple equilibria that can exist; the firms could locate along the

line, maintaining the assumption of x1< x2, or it could also be the case that it is more

profitable for the firms to switch location assumption, i.e. x2< x1. However, it has to be

checked whether the values of x1 , x2 in equilibrium are not too close, as it might be the case

that there is negative market share and negative price, in which case the firms would again

locate at end points. Competition intensifies and firms locate closer together when the values

of switching costs relative to unit transportation costs are not too high. Intuitively, this is

logical. If the values of the switching costs are not too high, switching can still be profitably

induced and the firm and therefore competition intensifies. If the value is too high, firms

would not be able to induce switching profitably and therefore will just try to maintain its

existing market base.

Minimal differentiation is the situation where both the firms are located at 12 . The intuition is

as follows: when firms locate both in the middle, no switching will be induced. Since there is

no difference in transportation costs, the consumers will not switch because if they do, they

would incur the switching cost but will not have the benefit of reduced total transportation

cost. The prices charged by the firm will also be quite low in order to prevent switching.

Since the market share does not increase, at ½ , both the firms have an incentive to move

further apart from each other and will therefore end up either locating along the line but with

some distance between the firms or will just locate at end points. Therefore, minimal

differentiation will not occur in equilibrium.

For firm 1, the first order partial derivative of the profit functions with respect to s1 is strictly

increasing for all possible cases and values of s1 , s2 and t . The same is the case for firm 2: the

first order partial derivative of the profit functions with respect to s2 is strictly increasing for

all possible cases and values of s1 , s2 and t .

29

In the case of the market share, the market share is affected by two variables: the distance

between the firms and the switching cost. As the distance between the locations of the firms

decrease, the market share of the existing firm in the market base increases. This is logical as

the closer the firms are located with each other, consumers are less likely to switch to rival

firm as the value of the switching cost is higher than the reduction in the transportation costs

of buying from the rival firm and therefore will buy from the same firm they did in period 1.

As the switching cost increases, the market share of the existing firm in its market base

increases, as more consumers will find that the switching cost that must be paid is greater

than the possible reduction in the transportation costs while buying from the rival firm.

Similarly, the prices charged by the firms in equilibrium depend on the switching cost and the

distance between the firms. As the distance between the firms decrease, the price charged by

the rival to the consumers of the market base of the rival firm also decreases. This is also

logical as the firm that would like to poach the consumers of the rival firm would have to

reduce their prices compared to the rival as the difference in the transportation costs due to

the distance between the location of the firms is less than the switching costs and in itself is

not sufficient to induce switching (without price reduction).

The overall effect of the switching cost on the firms on the profit of the firms is ambiguous.

For lower values of switching costs, the profit of the firm is likely to be less than period 1

(when there are no switching costs). This is due to the fact that when switching costs are

lower, it is still feasible to induce switching and therefore firms will behave more

competitively. Therefore, the prices for consumers are likely to decrease and the firms are

likely to see their profits reduced due to the switching costs. However, for high values of

switching costs, it becomes harder to induce switching and becomes unprofitable for the

firms to do so. Therefore, in equilibrium, neither of the firms will try to expand their market

shares and will serve their own initial market base (locate at endpoints). When this is the

case, the profits of the firms will increase due to switching costs.

SECTION 6: CONCLUSIONS

This paper aimed at investigating the changes in the equilibrium price and location of two

firms in Hotelling model with quadratic transportation costs when switching costs are

introduced in period 2. In this paper, it is assumed that the firms are myopic, that is, the firms

30

do not foresee that in the beginning of period two, switching costs will arise. This paper

aimed at analysing to what extent it is profitable for the firms to relocate and/or undergo price

reduction in order to capture the rival’s customers. It was assumed that the relocations costs

are zero and the firms are perfectly mobile.

In period one, the firms choose location and therefore, set the prices, based on information

that is available to them in period one (firms do not foresee the switching costs and make

their location choices independent of the existence of the switching costs). Consistent with

the findings of d'Aspremont, Gabszewicz, & Thisse, the firms locate at the endpoints (e.i.

maximum differentiation) and set their prices as t+c in equilibrium.

When the switching cost arise in the beginning of period two, the equilibrium location of the

firms change. When the ratio of the switching costs to the unit transportation cost of both the

firms are not too high, the firms will move closer together (move away from their initial

endpoint location) and the competition will intensify. However, when the ratio of switching

costs to unit transportation costs are too high, the firms will not deviate from their initial

location and will remain at endpoints. When this happens, neither of the firms will be able to

increase their market shares, however, they increase their profits by the value s1

3.

When the firms are not located too far apart at equilibrium and the ratio of the switching costs

to the unit transportation costs are not too high, the compeition intensifies and firms might

see their profits reduced. Due to increased competition, attempt at increasing their own

market share by poaching the rival’s consumes and to prevent poaching by their rivals, the

firms would lower the price charged to the consumers, therefore the consumers might overall

benefit from price redeuction. The firms might see their market share increasing (success at

poaching their rival’s consumers), however, the overall effect on the profit is ambiguous,

depending on how much the firms reduce their prices to maintain and/or increase their market

share.

There are several shortcomings of this paper. First, the firms are assumed to be myopic, that

is, the firms are not aware that switching costs will arise in the beginning of period 2. This

assumption was crucial in this paper, as the strategy of the firms and the results of the paper

heavily relied on this. However, this assumption is unrealistic. A suggestion to overcome this

31

limitation is to use backward-induction method in solving the model. In this method, the

firms will consider the last optimal strategy that they will have in period 2 and taking this

information into account, the firms will have an optimal strategy in period 1. The firms are

aware that the switching cost will arise in the beginning of the second period and therefore

will have their strategies in such a way in period 1 that would maximize their profits, given

their optimal strategy in period 2.

Another limitation is that this model is only a two period model. The firms operate for only

two periods and therefore, the firms maximize their profits only in two periods, which is

another unrealistic assumption. This assumption is the reason that in case of high switching

costs, poaching does not happen as it is not profitable for the firms to do so in period 2 (as

they will not operate in future periods). Changing this assumption will most likely change the

optimal strategy of the firm in period 2 as at one point. It is logical to expect that it might be

profitable for the firm to lower the price below marginal cost in order to increase their overall

market share, after which they will have a larger market base and recover their losses and

increase future profits. Therefore, a possible suggestion to overcome this limitation is to take

make the assumption that the firms will operate indefinitely and take into account profits

from future periods.

The third and final limitation is that it is assumed that there is no relocation cost (per distance

moved) incurred by the firm when they move from their optimal location in period 1 to the

optimal location in period 2. This is another unrealistic assumption and this limitation can be

overcome by incorporating relocation costs when finding the optimal location in period 2, as

the optimal decision of the firms are likely to change when this additional cost is taken into

account. It is expected that the intensity of competition might decrease (compared to the

situation when there are no relocation costs with the same s1 , s2and t) as now the firms will

have to take into account the costs of relocation which would reduce the profitability of

inducing switching. Therefore, firms might not locate as close to each other as they would

have been in the situation without the relocation costs.

SECTION 7: APPENDIX

A. Hotelling Model

32

A.1 Marginal Consumer

xm=

p2−p1

t+(x2

2−x12 )

2 ( x2−x1 )

A.2 Market share, profit function price charged by the firms in Nash

Firm 1:

Market share of firm 1: consumers to the left of the marginal consumer (between 0 and xm¿

xm=

p2−p1

t+(x2

2−x12 )

2 ( x2−x1 )

Profit function of firm 1:

π1=(p1−c )(

p2−p1

t+( x2

2−x12)

2 ( x2−x1 ))

By taking the first order partial derivative of the profit function with respect to price, the

following price charged by firm 1 is found:

p1=P2

2+ c

2+ t

2(x2

2−x12)

Firm 2:

Market share of firm 2: consumers to the right of the marginal consumer (betweenxmand 1).

1−xm=2 ( x2−x1 )−

p2−p1

t−(x2

2−x12 )

2 ( x2−x1 )

Performing the same steps, the following profit functions and price function for firm 2 is

found:

π2=(p2−c )¿

p2=p1

2+ c

2− t

2 (x22−x1

2 )+t (x2−x1)

Solving both the equations above simultaneously yielded the following price functions in

Nash equilibrium:

p1¿=2 t

3 ( x2−x1)+ t3 (x2

2−x12 )

p2¿=4 t

3 ( x2−x1 )− t3 (x2

2−x12 )+c

A.3 Marginal consumer and market shares of firm 1 and 2

33

By substituting the price functions for both the firms in the equation for the marginal

consumer, the following marginal consumer is calculated:

xm=13+ 1

6(x¿¿2+x1)¿

Therefore market share of firm 1 is:

xm=13+ 1

6(x¿¿2+x1)¿

and firm 2 is :

(1−xm )=23−1

6(x¿¿2+x1)¿

A.4 Profit functions and derivatives

By multiplying the market shares and price functions in Nash of the respective firms, the

following profit functions are calculated:

π1=2t9 ( x2

2− x12 )+ 2 t

9 ( x2−x1 )+ t18

(x22−x1

2 )2

x2−x1

π2=−4 t

9 ( x22−x1

2)+ 8 t9 ( x2−x1 )+ t

18( x2

2−x12)2

x2−x1

To maximise the profit functions with respect to their locations, the following first-order

derivatives of the profit function with respect to the firms’ locations are found:

δ π1

δ x1=−1

18(3 x1−x2+2)(x1+x2+2)

δ π2

δ x2=−1

18(4+x1−3 x2)¿

B. Hotelling Model with Switching Cost Market Base Firm 1 (Assumption x1<x2¿

B.1 Marginal consumer

The marginal consumer is calculated by solving the following equation:

P1,1+t (x1−xm,1)2=P2,1+ t(x2−xm, 1)

2+S1

xm,1=

p2,1−p1,1+s1

t+ (x2

2−x12 )

2 ( x2−x1 )

B.2 Market share, profit function price charged by the firms in Nash

Consumers to the left of the marginal consumer forms the market share of firm 1:

xm,1=

p2,1−p1,1+s1

t+ (x2

2−x12 )

2 ( x2−x1 )

34

Consumers to the right of the marginal consumer forms the market share of firm 2:

0.5−xm,1=( x2−x1 )−

p2,1−p1,1+s1

t−(x2

2−x12 )

2 ( x2−x1 )

Profit and price function of firm 1:

π1,1=( p1,1−c)(

p2,1−p1,1+s1

t+(x2

2−x12 )

2 ( x2−x1 ))

p1,1=P2,1

2+ c

2+ t

2 (x22−x1

2 )+ s1

2


π2,1=( p2,1−c)¿ p2,1=p1,1

2+ c

2− t

2 (x22−x1

2 )+ t2 ( x2−x1 )−

s1

2

Price functions in Nash equilibrium:

p1,1¿ = t

3 ( x2−x1 )+ t3 ( x2

2−x12 )+c+

s1

3

p2,1¿ =2 t

3 ( x2−x1 )− t3 (x2

2−x12 )+c−

s1

3

B.3 Marginal consumer and market shares of firm 1 and 2

Marginal consumer:

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

Market share of firm 1 is:

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

and firm 2 is :

(0.5−xm )=13−1

6 ( x2+x1 )−s1

6 t ( x2−x1 )

C. Hotelling Model with Switching Cost Market Base Firm 2 (Assumption x1<x2¿

C.1 Marginal consumer


P1,2+t (x1−xm,2)2+S2=P2,2+ t(x2−xm, 2)

2

35

xm ,2=

p2,2−p1,2−s2

t+(x2

2−x12 )

2 ( x2−x1)

C.2 Market share, profit function price charged by the firms in Nash


xm,2−0.5=

p2,2−p1,2−s2

t+(x2

2−x12 )−(x2−x1)

2 ( x2−x1 )


1−xm, 2=2 ( x2−x1 )−

p2,2−p1,2−s2

t−(x2

2−x12 )

2 ( x2−x1 )


π1,2=( p1,2−c)(

p2,2−p1,2−s2

t+(x2

2−x12 )−(x2−x1)

2 ( x2−x1 ))

p1,2=P2,2

2+ t

2 ( x22−x1

2)− t2 ( x2−x1 )−

s2

2+ c

2


π2,2=( p2,2−c)¿p2,2=p1,2

2− t

2 (x22−x1

2 )+t ( x2−x1 )+s2

2+ c

2

Price functions in Nash equilibrium:

p1,2¿ = t

3 ( x22−x1

2)− s2

3+c

p2,2¿ =t ( x2−x1)− t

3 (x22−x1

2 )+s2

3

C.3 Marginal consumer and market shares of firm 1 and 2

The following marginal consumer is calculated:

xm,2=12+ 1

6 ( x2+ x1 )−s2

6 t ( x2−x1)


(x¿¿m ,2−0.5)=16 ( x2+x1)−

s2

6 t ( x2−x1 )¿

and firm 2 is :

(1−xm, 2 )=12−1

6 ( x2+x1)+s2

6 t ( x2−x1 )

36

D. Hotelling Model with Switching Cost Analysis (Assumption x1<x2¿

D.1 Market base of firm 1 analysis


xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )


xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

(0.5−xm,1 )=13−1

6 ( x2+x1 )−s1

6 t ( x2−x1 )

Both firms will serve in this market base if 0<xm, 1<12 and 0<0.5−xm,1<

12 . Minimum value

of 16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 ) is when x1is 0 and x2 is 1. To find out the critical value after

which xm ,1 is greater than ½ , the following inequality is solved:

s1

6 t+ 1

3> 1

2

Rearranging the terms gives the critical value s1>t

D.2 Market base of firm 2 analysis


xm,2=12+ 1

6 ( x2+ x1 )−s2

6 t ( x2−x1)


(x¿¿m ,2−0.5)=16 ( x2+x1)−

s2

6 t ( x2−x1 )¿

(1−xm, 2 )=12−1

6 ( x2+x1)+s2

6 t ( x2−x1 )

Both firms will serve in this market base if 0<(x¿¿m, 2−0.5)< 12¿ and 0<1−xm,2<

12

.

Minimum value of 12−1

6 ( x2+ x1 )+s2

6 t ( x2−x1 ) is when x1is 0 and x2 is 1. To find out the

critical value after which xm is greater than ½ , the following inequality is solved:

12−1

6 ( x2+ x1 )+s2

6 t ( x2−x1 )> 1

2

37

Rearranging the terms gives the critical value s2>t

D.3 Price functions in market base of firm 1 analysis

In the market base of firm 1 (consumers between 0 and 0.5), the following are the price

functions in Nash equilibrium of firm 1 and 2 respectively:

p1,1¿ = t

3 ( x2−x1 )+ t3 ( x2

2−x12 )+c+

s1

3

p2,1¿ =2 t

3 ( x2−x1 )− t3 (x2

2−x12 )+c−

s1

3

The maximum value that 2t3 ( x2−x1 )− t

3 ( x22−x1

2)can have is t3 . If the value of the switching

cost is greater than t3 , firm 2 would have to charge a price below marginal cost. To find the

critical value, the following inequality needs to be solved:

t3−

s1

3>0

Therefore, if s1> t, firm 2 would have a price function that is below marginal cost.

D.4 Price functions in market base of firm 2 analysis

In the market base of firm 2 (consumers between 0.5 and 1), the following are the price


p1,2¿ = t

3 ( x22−x1

2)− s2

3

p2,2¿ =t ( x2−x1)− t

3 (x22−x1

2 )+ s2

3

The maximum value that t3 (x2

2−x12 )can have is

t3 . If the value of the switching cost is greater

than t3 , firm 1 would have to charge a price below marginal cost. To find the critical value,

the following inequality needs to be solved:

t3−

s2

3>0

Therefore, if s2>t , firm 1 would have a price function that is below marginal cost.

D.5 Profit functions

D.5.I Profit functions of firm 1

When s1<t

38

π1{( s1+t ( x22−x1

2 )+ t ( x2−x1 ))2+(s2−t ( x22−x1

2) )2

18 t ( x2−x1 ),∧s2<t

(s1+ t (x22−x1

2 )+t ( x2−x1 ))2

18 t ( x2−x1 ),∧s2> t

When s1>t

π1{3( s1+t ( x22−x1

2 )+ t ( x2−x1) )18

+( s2−t (x2

2−x12 ))2

18 t ( x2−x1 ),∧s2<t

3 (s1+ t (x22−x1

2 )+t ( x2−x1 ))18

S ,∧s2> t

D.5.II Profit functions of firm 2

When s2<t

π2{( s1+t ( x22−x1

2)−2 t ( x2−x1 ))2+( s2−t ( x22−x1

2)+3 t ( x2−x1 ))2

18 t ( x2−x1 ),∧s1<t

( s2−t (x22−x1

2 )+3 t ( x2−x1 ))2

18 t ( x2−x1 ),∧s1>t

When s2>t

π2{( s1+t ( x22−x1

2)−2 t ( x2− x1))2

18 t ( x2−x1 )+

3 (s2−t ( x22−x1

2 )+3 tS(x2−x1))18

,∧s1<t

3 (s2−t ( x22−x1

2 )+3t (x2−x1))18

,∧s1> t

D.6 Derivatives of profit functions

D.6.I First order partial derivative of firm1

When s1<t

δ π1

δ x1 { 118

(2 (s1−s2 )−t (1+6x12−2x2

2+4 x1+4 x1 x2)+s1

2+s22

t ( x2−x1)2 ),∧s2<t

s12+2 t s1(x1−x2)

2−t 2 ( x1−x2 )2(1+3 x12−x2

2+4 x1+2 x1 x2)

18t ( x2−x1)2 ,∧s2> t

When s1>t

39

δ π1

δ x1 {s22−2t s2(x1−x2)

2−t 2 ( x1−x2 )2(3+3 x12−x2

2+6 x1+2 x1 x2)

18t ( x2−x1)2 ,∧s2< t

t(−1−2x1)6 ,∧s2>t

D.6.II First order partial derivative of firm 2

When s2<t

δ π2

δ x2 { 118

(2 (s1−s2 )+t (13−2 x12−14 x2+4 x1 x2+6 x2

2 )− s12+s2

2

t ( x2−x1 )2) ,∧s1< t

−s22+2 t s2(x1−x2)

2−t2 ( x1−x2)2(9−x12+3 x2

2+2 x1 x2−12 x2)

18t ( x2−x1)2 ,∧s1> t

When s2>t

δ π2

δ x2 {−s12−2t s1 ( x1−x2 )2−t 2 ( x1−x2 )2(13−x1

2+3 x22+2x1 x2−14 x2)

18t ( x2−x1)2 ,∧s1< t

t (3−2x2)6 ,∧s1>t

E. Hotelling Model with Switching Cost Market Base Firm 1 (Assumption x1>x2¿

E.1 Marginal consumer


P1,1+t (x1−xm,1)2=P2,1+ t(x2−xm, 1)

2+S1

xm,1=

p2,1−p1,1+s1

t+ (x2

2−x12 )

2 ( x2−x1 )

E.2 Market share, profit function price charged by the firms in Nash


0.5−xm,1=( x2−x1 )−

p2,1−p1,1+s1

t−(x2

2−x12 )

2 ( x2−x1 )

xm,1=

p2,1−p1,1+s1

t+ (x2

2−x12 )

2 ( x2−x1 )


40

xm,1=

p2,1−p1,1+s1

t+ (x2

2−x12 )

2 ( x2−x1 )


π1,1=( p1,1−c)(( x2−x1 )−

p2,1−p1,1+s1

t−(x2

2−x12 )

2 ( x2−x1 ))

p1,1=P2,1

2+ c

2+ t

2 (x22−x1

2 )− t2(x2−x1)+

s1

2


π2,1=( p2,1−c)¿p2,1=p1,1

2+ c

2− t

2 (x22−x1

2 )− s1

2

By solving the equations simultaneously, the following price functions in Nash equilibrium is

obtained:

p1,1¿ =−2 t

3 ( x2−x1)+ t3 ( x2

2−x12)+c+

s1

3

p2,1¿ =−t

3 ( x2−x1 )− t3 ( x2

2−x12 )+c−

s1

3

E.3 Marginal consumer and market shares of firm 1 and 2


xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )


(0.5−xm )=13−1

6 ( x2+x1 )−s1

6 t ( x2−x1 )

and firm 2 is :

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

F. Hotelling Model with Switching Cost Market Base Firm 2 (Assumption x1>x2¿

F.1 Marginal consumer


P1,2+t (x1−xm,2)2+S2=P2,2+ t(x2−xm, 2)

2

xm ,2=

p2,2−p1,2−s2

t+(x2

2−x12 )

2 ( x2−x1)

41

F.2 Market share, profit function price charged by the firms in Nash


1−xm, 2=2 ( x2−x1 )−

p2,2−p1,2−s2

t−(x2

2−x12 )

2 ( x2−x1 )


xm,2−0.5=

p2,2−p1,2−s2

t+(x2

2−x12 )−(x2−x1)

2 ( x2−x1 )


π1,2=( p1,2−c)(2 ( x2−x1 )−

p2,2−p1,2−s2

t−(x2

2−x12 )

2 ( x2−x1 ))

p1,2=P2,2

2+ t

2 ( x22−x1

2)−t ( x2−x1 )−s2

2+ c

2


π2,2=( p2,2−c)¿p2,2=p1,2

2− t

2 (x22−x1

2 )+ t2 ( x2−x1 )+

s2

2+ c

2

By solving the equations simultaneously, the following price functions in Nash equilibrium is

obtained:

p1,2¿ = t

3 ( x22−x1

2)−t ( x2−x1 )−s2

3+c

p2,2¿ =−t

3 (x22−x1

2 )+ s2

3+c

F.3 Marginal consumer and market shares of firm 1 and 2


xm,2=12+ 1

6 ( x2+ x1 )−s2

6 t ( x2−x1)


(1−xm, 2 )=12−1

6 ( x2+x1)+s2

6 t ( x2−x1 )

and firm 2 is :

(x¿¿m ,2−0.5)=16 ( x2+x1)−

s2

6 t ( x2−x1 )¿

G. Hotelling Model with Switching Cost (Assumption x2<x1¿

42

G.1 Market base of firm 1 analysis


xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )


(0.5−xm,1 )=13−1

6 ( x2+x1 )−s1

6 t ( x2−x1 )

xm,1=16+ 1

6 ( x2+x1 )+s1

6 t ( x2−x1 )

Both firms will serve in this market base if 0<xm, 1<12 and 0<( 0.5−xm, 1)< 1

2

Since the location of firm 1 is to the right of firm 2, x1> x2 . The denominator is therefore less

than or equal to zero. Minimum value of 13−1

6 ( x2+x1 )−s1

6 t ( x2−x1) is when x1is 1 and x2 is 0.

To find out the critical value after which xm,1 is greater than ½ , the following inequality is

solved:

s1

6 t+ 1

6> 1

2

Rearranging the terms gives the critical value s1>2 t

G.2 Market base of firm 2 analysis


xm,2=12+ 1

6 ( x2+ x1 )−s2

6 t ( x2−x1)


(1−xm, 2 )=12−1

6 ( x2+x1)+s2

6 t ( x2−x1 )

(x¿¿m ,2−0.5)=16 ( x2+x1)−

s2

6 t ( x2−x1 )¿

Both firms will serve in this market base if 0<1−xm,2<12 and 0 <(x¿¿m ,2−0.5)< 1

2¿

Since the location of firm 1 is to the right of firm 2, x1> x2 . The denominator is therefore less

than or equal to zero. The minimum value of 16 ( x2+x1 )−

s2

6 t ( x2−x1 ) is when x1is 1 and x2 is

43

0. To find out the critical value after which xm,2 is greater than ½ , the following inequality is

solved:

s2

6 t+ 1

6> 1

2− 1

6

Rearranging the terms gives the critical value s2>2 t

G.3 Price functions in market base of firm 1 analysis

In the market base of firm 1 (consumers between 0 and 0.5), the following are the price


p1,1¿ =−2 t

3 ( x2−x1)+ t3 ( x2

2−x12)+c+

s1

3

p2,1¿ =−t

3 ( x2−x1 )− t3 ( x2

2−x12 )+c−

s1

3

The maximum value that −t3 ( x2−x1 )− t

3 (x22−x1

2 )can have is 2t3 . If the value of the switching

cost is greater than 2t3 , firm 2 would have to charge a price below marginal cost. To find the


2t3

−s1

3>0

Therefore, if s1>2 t, firm 2 would have a price function that is below marginal cost.

G.4 Price functions in market base of firm 2 analysis

In the market base of firm 2 (consumers between 0.5 and 1), the following are the price


p1,2¿ = t

3 ( x22−x1

2)−t ( x2−x1 )−s2

3+c

p2,2¿ =−t

3 (x22−x1

2 )+ s2

3+c

The maximum value that t3 ( x2−x1 )−t (x2

2−x12 )can have is

2t3 . If the value of the switching

cost is greater than 2t3 , firm 1 would have to charge a price below marginal cost. To find the


2t3

−s2

3>0

Therefore, if s2>2 t, firm 1 would have a price function that is below marginal cost.

44

G.5 Profit functions

G.5.I Profit functions of firm 1

When s1<2 t

π1{( s1+t ( x22−x1

2 )−2 t ( x2−x1 ))2+(s2−t ( x22−x1

2)+3 t ( x2−x1 ))2

18 t ( x1−x2 ),∧s2<2 t

( s1+t ( x22−x1

2 )−2 t ( x2−x1 ))2

18 t ( x1−x2 ),∧s2>2 t

When s1>2 t

π1{3( s1+t ( x22−x1

2 )−2 t ( x2−x1 ))18

+( s2−t (x2

2−x12 )+3 t ( x2−x1 ))2

18 t ( x1−x2 ),∧s2<2 t

3 (s1+ t (x22−x1

2 )−2t ( x2−x1 ))18

,∧s2>2t

G.5.II Profit functions of firm 2

When s2<2 t

π2{( s1+t ( x22−x1

2)+tt ( x2−x1 ))2+(s2−t (x22−x1

2 ))2

18 t ( x1−x2),∧s1<2t

( s2−t (x22−x1

2 ))2

18 t ( x1−x2),∧s1>2 t

When s2>2 t

π2{( s1+t ( x2−x12)+t(x2−x1))

2

18 t ( x1−x2 )+

3 (s2−t ( x22−x1

2 ) )18

,∧s1<2 t

3 (s2−t ( x22−x1

2) )18

,∧s1>2 t

G.6 Derivatives of Profit functions 1

G.6.I First order partial derivatives for profit functions of firm 1

When s1<2 t

45

δ π1

δ x1 { 118

(2 (s2−s1 )+t (13+6 x12−2 x2

2−20 x1+4 x1 x2)−s1

2+s22

t ( x2−x1 )2) ,∧s2<2 t

−s12+2t s1(x1−x2)

2−t 2 ( x1−x2 )2(4+3 x12−x2

2−8 x1+2 x1 x2)

18 t ( x2−x1 )2,∧s2>2 t

When s1>2 t

δ π1

δ x1 {−s22−2t s2 ( x1−x2 )2−t 2 ( x1−x2 )2(15+3 x1

2−x22−18 x1+2 x1 x2)

18 t ( x2− x1 )2,∧s2<2 t

t (1−x1)3 ,∧s2>2 t

G.6.II First order partial derivatives for profit functions of firm 2

When s2<2 t

δ π2

δ x2 { 118

(2 (s2−s1 )−t (1−2x12+4 x2+4 x1 x2+6 x2

2 )+ s12+s2

2

t ( x2−x1 )2) ,∧s1<2 t

s22+2t s2 ( x1−x2 )2−t2 ( x1−x2 )2(−x1

2+3 x22+2 x1 x2)

18t ( x2−x1 )2 ,∧s1>2 t

When s2>2 t

δ π2

δ x2 {s12−2t s1 ( x1−x2 )2−t 2 ( x1−x2 )2(1+x1

2+3x22−2 x1 x2+10 x2)

18 t ( x2−x1 )2,∧s1<2 t

−t x2

3,∧s1>2t

H. EQUILIBRIUM ANALYSIS

H.1 Equilibrium when t <s1<2t and t <s2<2t

Firm 1 strictly prefers x2< x1 over x1< x2if:

( s1+t ( x22−x1

2)−2 t ( x2−x1 ))2+( s2−t (x22−x1

2)+3 t ( x2−x1 ))2

18 t ( x1−x2 )>

3 ( s1+2 t )18

The minimum value of LHS is (s1+ t )2

18 t. To find the critical value, the following inequality is

solved:

(s1+ t )2

18 t >3 ( s1+2 t )

18

Rearranging the terms, the following inequality is obtained:s12>5 t2+t s1

46


( s1+t ( x22−x1

2)+t ( x2−x1 ))2+(s2−t ( x22−x1

2) )2

18 t ( x1−x2 )>

3 ( s2+2 t )18

The minimum value of LHS is (s2+t )18 t

2

. To find the critical value, the following inequality is

solved:

(s2+t )18 t

2

>3 ( s2+2t )

18

Rearranging the terms, the following inequality is obtained: s22>5 t2+t s2

H.2 Equilibrium when s1>2 t and t <s2<2t


3( s1+t ( x22−x1

2 )−2 t ( x2−x1 ))18

+( s2− t (x2

2−x12 )+3 t ( x2−x1 ))2

18 t ( x1−x2 )>

3 (s1+2t )18

The minimum value of the LHS is 3 ( s1+ t )

18 +(s2−2 t )2

18 t. Rearranging the terms, to obtain the

critical value, the following inequality is obtained and solved:

(s2−2 t )2

18 t >3 t18

Rearranging the terms, the following inequality is obtained: (s2−2t )2>3 t 2.


( s2−t (x22−x1

2 ))2

18 t ( x1−x2)>

3 (s2+2t )18

The minimum value of LHS is (s2+t )18 t

2

. To find the critical value, the following inequality is

solved:

(s2+t )18 t

2

>3 ( s2+2t )

18


H.3 Equilibrium when t <s1<2tand s2>2 t


( s1+t ( x22−x1

2)−2 t ( x2−x1 ))2

18 t ( x1−x2 )>

3 ( s1+2 t )18

47

The minimum value of the LHS is (s1+ t )2

18 t. Rearranging the terms, to obtain the critical value,

the following inequality is obtained and solved:

(s1+ t )2

18 t >3 t18



( s1+t ( x2−x12)+t(x2−x1))

2

18 t ( x1−x2 )+

3 (s2−t ( x22−x1

2 ) )18

>3 ( s2+2 t )

18

The minimum value of the LHS is 3 ( s2+ t )

18 +(s1−2 t )2

18 t. Rearranging the terms, to obtain the

critical value, the following inequality is obtained and solved:

(s1−2 t )2

18 t >3 t18


48

Section 8: Bibliography

Caminal, R., & Matutes, C. (1990). Endogeneous Switching Costs in a Duopoly Model.

International Journal of Industrial Organization , 353-373.

d'Aspremont, C., Gabszewicz, J. J., & Thisse, J. F. (1979). On Hotelling's "Stability in

Competition". Econometrica , 1145-1150.

Fudenberg, D., & Tirole, J. (2000). Customer Poaching and Brand Switching. The RAND

Journal of Economics , 31, 634-657.

Haucap, J. (2003). Endogenous switching costs and exclusive systems applications . Review

of Network Economics , 29-35.

Heide, J. B., & Weiss, A. M. (1995). Vendor Consideration and Stiching Behaviour for

Buyers in High-Technology Markets. Journal of Marketing , 59, 30-43.

Hobbs, B. F. (1986). Mill Pricing Versus Spatial Price Discrimination Under Bertrand and

Cournot Spatial Competition. The Journal of Industrial Economics , 173-191.

Hotelling, H. (1929). Stability in Competition. The Economic Journal , 41-57.

Hughes, N. (2012, June 29). 21% of iOS users say they wouldn't leave Apple at any price .

Retrieved June 2, 2015, from AppleInsider:

http://appleinsider.com/articles/12/06/29/21_of_ios_users_say_they_wouldnt_leave_apple_at

_any_price

Klemperer, P. (1987). Markets With Consumer Switching Costs. Quarterly Journal of

Economics , 375-394.

Klemperer, P., & Farrell, J. (2007). Coordination and Lock In: Competition with Swtiching

Costs and Network Effects. In Handbook of Industrial Organization (Vol. 3, pp. 1970-1994).

Phlips, L., & Thisse, J. F. (1982). Spatial Competition and the Theory of Differentiatied

Markets: An Introduction. Journal of Industrial Economics , 1-9.

Shaffer, G., & Zhang, Z. J. (2000). Pay To Switch of Pay To Stay : Preference Based Price

Discrimination in Markets with Swtiching Costs. Journal of Economics and Management

Strategy , 397-424.

Worthington, S. (2005, May). The Effect of Endogeneous Switching Costs on Consumer

Choice: A Case Study - Apple’s iPod and iTunes Music Store. Retrieved June 2, 2015, from

Pugetsound:

http://www.pugetsound.edu/files/resources/1359_TheEffectsofEndogneousSwitching.pdf

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