Hotelling's product differentiation model extended

University of Amsterdam

Bachelor thesis

Hotelling’s product

differentiation model extended

Author:

Paul Broek

Supervisor:

T. A. Makarewicz

Contents

1 Introduction 2

2 Model specification 4

2.1 Original model . . . 4

2.2 General model . . . 6

2.2.1 Deriving general demand functions . . . 6

3 Cases 8 3.1 Case 1: offline firms located at sides . . . 8

3.1.1 Example equilibrium . . . 11

3.2 Case 2: offline firms located at quartiles . . . 12

3.2.1 Example equilibrium . . . 14

3.3 Average distance calculations . . . 16

1

Introduction

People can buy any product that meets their preferences in physical stores, however buying goods over the Internet is becoming far more popular. In-ternet retail sales in western Europe are expected to grow at an rate of 11%, from e 112 billion in 2012 to e 191 billion by 2017 [1]. For the Netherlands this would mean its online retail is 7% of its economy by 2017. In the US this proportion would be 10% by 2017 and its ecommerce sales are expected to grow from e 181 billion in 2012 to a staggering e 290 billion by 2017. This emerged and growing retail market has altered the traditional dynam-ics between customers and firms.

For consumers, the decision process of buying a product has become more complex. Factors like logistic costs and delivery time are now involved. These can be added to the traditional considerations of price and distance. This traditional enterprise setup was first studied by Hotelling in 1929 [2]. He stated that duopolistic competition in differentiated products leads to the minimal differentiation principle. Two firms were assumed to play a game in which first they choose a location and then a price for their goods. His model assumed uniformly distributed consumers, linear transportations costs and a location of firms within a straight line. Since then, a lot of modifications and related theory to this model appeared. The assumed linear transporta-tion cost functransporta-tion was replaced by quadratic functransporta-tion to show that a price equilibrium exists everywhere (d’Aspremont, 1979 [3]). However, this result was relaxed by Economides [4] in the sence that an equilibrium only exists when the curvature of the utility function is sufficiently high. He stated that not all equilibria are at maximal product differentiation. Also, when a third firm is added to the model, an equilibrium in pure strategies for location and price game no longer exists. Shaked [5] showed that firms will choose the same mixed strategy.

In the present we see one retail market consisting of two distinguishable sectors. For the average consumer it means that both sectors are equally accessible, however sometimes unequally preferred because of utility distor-tions. A good can be needed instantly or it must be inspected physically by the consumer. For this reason an offline two-firm model can still be considered. Cho and Lee [6] implemented online demand in the traditional Hotelling model, both firms can sell online and offline. In their symmetric case, they found that, depending on the tradeoff between delivery cost and transport cost, this will result in a welfare loss to consumers.

In-ternet player to examine the effects of its virtual presence on the price and location game. When we look at a market in a two-firm model with con-sumers located not too far away the online sector can be modelled as just one player because it does not have a physical location within the game but in the first instance only a price aspect to compete in. This principle was integrated in the former Hotelling model by Hu, Wei, Li and Xiao (2014, [7]). They introduce online demand in an extended Hotelling model. The effects of logistic costs, waitings costs and the ratios of online demands to the whole demands on the shops location competition are examined. Logistic costs ap-pear to be an incentive for firms to move away from the centre equilibrium. Including a third player to the original model is comparable to using the former model with an outside good. The decision not to buy a good with the two physical stores is equivalent to buying it for that reservation price from a third party. Such spatial competition with an outside good and distributed reservation prices is was first studied by B¨ockem (1994, [8]). She extended the quadratic transport costs model by adding reservation prices which are uniformly distributed as well. Under these circumstances, firms still locate within the city limits. This was furthur examined by Woeckener (2002, [9]). He showed that in equilibrium firms will always locate at the first and third quartile of the consumer distribution. These are the socially optimal loca-tions.

This thesis studies the effect of adding an external player to the two-firm Hotelling model on the price competition game. With the new player, a vast utility is obtained by consumers, as opposed to the variable utility related to the transport costs of purchasing goods at the physical firms. For given locations of the physical firms, we seek for price equilibria in a 3-player setup. Delivery cost is modelled as a fixed added cost to the internet commodity price.

The study is organized as follows. In section 2, a former model is speci-fied and extended. General demand functions are derived and for two cases they are given explicitly in section 3. Also, Nash equilibria are derived within these cases, for a fixed transport cost parameter. In the end, this study is concluded in section 4.

2

Model specification

2.1

Original model

In the original model for horizontal product differentiation, Hotelling (1929) [2] used his example of a linear city. L consumers are uniformly distributed on an interval [0, L]. Every consumers purchases one commodity of this homogenous good. The demand for the good is assumed to be inelastic. Two firms compete with price (p1,p2) and location (x1,x2), with 0 ≤ x1 ≤ x2 ≤ L.

The firms can produce cost free, but to consumers there is a linear transport cost, so total commodity cost to a consumer located at x is pi+ |x − xi|.

The linear model of Hotelling was later proven not to have an equilibrium for the location game. With the use of quadratic transport costs it has been shown by d’Aspremont and others (1979 [3]) that equilibrium exists for the location game at the ends of the line. The location equilibrium of this two-stage game will be used later in the discussed cases. This is the solution in a two-firm model with quadratic transport costs, called maximum product differentiation:

x∗₁ = −0.5, x∗₂ = 0.5 p∗₁ = p∗₂ = L2 = 1 q₁∗ = q₂∗ = L/2 = 1/2 π₁∗ = π₂∗ = L3/2 = 1/2

Figure 1 presents min(p∗₁+ (d(x∗₁, x))2, p∗₂+ (d(x∗₂, x))2) for an infinite amount of consumers, thus depicting the minimal commodity cost per located con-sumer. The graph can also be interpreted geographically in the horizontal axis, as we will stick with a horizontal line. From now on L is normalized to 1. Throughout this study, the minimal cost to consumers for the offline firms is depicted as one function, possibly intersected by the vast commodity costs of a third player, as will be anounced in the next section.

This equilibrium, shown in figure 1, is unfavourable for consumers in the middle of the line. With the entry of an online firm it is expected that offline firms move towards the centre in such a way that consumers are evenly spread over the two offline firms. This is the case with x1 = −0.25, x2 = 0.25, since

we are using the optimum derived by Woeckener (2002 [9]), who extended the former model with an outside option with uniformly distributed reservation prices. This gives the equilibrium prices p∗₁ = p∗₂ ≈ 0.3, shown in figure 2.

Figure 1: Minimal commodity cost per located consumer for firms located at ends

Figure 2: Minimal commodity cost per located consumer for firms located at -0.25 and 0.25

2.2

General model

In my model I assume that consumers are uniformly distributed on [−0.5, 0.5]. We choose to have a mass of consumers to allow for smooth demand functions. For simplicity, this intervallength will be normalized to 1. Every consumer buys one unity of the homogeneous product. The demand remains inelastic. The offline firms locations are x1 and x2, with −0.5 ≤ x1 ≤ x2 ≤ 0.5. The

third firm, called Online, is not located on our interval but does compete with a price p3, known to the offline firms.

Firms do not have production costs, but offline firms do have quadratic trans-port costs to consumers. The delivery price is therefore pi + (x − xi)2 for

consumers on location x, buying product i = 1, 2. The online firm has a fixed delivery cost (¯γ) added to its price, which brings the total commodity cost of buying at Online to: p3 + ¯γ, constant to all consumers.

Now we postulate a simultaneous game of three players. Given a location [−0.5, 0.5] of the offline players, the firms simultaneously choose a price for their product. The third player is first considered as external and thus given. It cannot set its price, but it ensures the consumers a direct obtainable fixed amount of utility, similar to that of a reservation price. We solve this game using best response functions.

The utility of any consumer n with money m and who buys product yi at

price pi is:

Un(m, yi, pi) = m − pi− g − (d(xi, x))2

d(xi, x) depicts the Euclidian distance in the product space

g is the utility of product yito consumer n, for i = 3 (Online) we have g = g0

For now we take g = 0 for i = 1, 2 and g = ¯γ for i = 3. 2.2.1 Deriving general demand functions

In order to derive demand functions for the general model used in this thesis, we seek for the point of indifference between buying at firm 1 and buying at firm 2. Due to symmetricity of these firms (x1 = −x2, p1 = p2) this point

will always represent the vertical axis in x = 0.

x1 = x2 = 0.5(x1+ x2) + 0.5(p2 − p1)/(x2− x1) symm.

= 0

Price of internet commodity denotes the indifference between buying at firm i = 1, 2 and not buying the good of either two:

ˆ

a = p3+ γ = (x − xi)2+ pi

This is very similar to using a reservation price, like Woeckener [9] did. Sum-marized, a consumer located at xk will choose the cheapest commodity:

Shop 1 : (xk− x1)2+ p1

Shop 2 : (xk− x2)2+ p2

Internet : γ + pint

This leads to the following general demand functions

D1 =        0.5 if p1 < min(p2+ x22− x21, p3+ γ) 0.5 + x1+ √ p3+ γ − p1 if 2 points of indifference 0.5 + x1+ 2 √ p3+ γ − p1 if 4 points of indifference 0 if p1 > min(p2+ x22− x21, p3+ γ) D2 =        0 if p2 > min(p1+ x22− x21, p3+ γ) 0.5 − x2+ 2 √ p3+ γ − p2 if 4 points of indifference 0.5 − x2+ √ p3+ γ − p2 if 2 points of indifference 0.5 if p2 < min(p1+ x22− x21, p3+ γ) D3 = Dint=    1 if p3+ γ < min(p1, p2) 1 − D1− D2 if p3+ γ different 0 if p3+ γ > min(p1+ x21, p2+ x22)

The demand function heavily depend on the number of intersections between the minimical cost function of the offline firms and the constant commodity cost line of Online. Both situations will occurr in the next section. For 2 points of indifference see figure 5, for 4 points of indifference see figure 8.

3

Cases

For two cases I will have a closer look what happens to the prices of offline firms when Online enters the market. It can be concluded from the previous section that the demand functions of offline firms are equal, as in both up-coming cases x1 = −x2 holds. This results in furthur symmetricity in price,

profit function and best responses.

3.1

Case 1: offline firms located at sides

Here I will derive new price equilibria for the 3-firm game, with fixed locations from the 2-firm quadratic transport costs equilibrium x1 = −0.5, x2 = 0.5.

I assume that firm 1 and firm 2 have equal demand functions due to sym-metricity of price and location. Demand functions are now:

D1 = D2 =    0.5 if ul ≤ p3+ γ √ p3+ γ − p1 else 0 if p1 = p2 ≥ p3+ γ (1) Dint= 1 − D1− D2 = 1 − 2 √ p3+ γ − p1 (2)

With ul the maximum of the minimal commodity cost of firms 1 and 2, in

this case ul= p1+ x21

So we allow p3 to differ from p1 = p2. In order to derive best response

functions, profit functions are derived first.

π1(p1, p3, γ) = π2(p1, p3, γ) = p1∗ D1 = p1 √ p3+ γ − p1 (3) π3(p1, p3, γ) = p3∗ (1 − 2 ∗ D1) = p3− 2p3 √ p3+ γ − p1 (4)

First order conditions: dπ1 dp1 =√p3+ γ − p1− p1 2√p3+ γ − p1 = 0 (5) dπ3 dp3 = 1 − 2√p3+ γ − p1− p3 √ p3+ γ − p1 = 0 (6) Deriving the best response function for firms 1 and 2:

√ p3+ γ − p1− p1 2√p3+ γ − p1 = 0 ⇐⇒ p1 = 2(p3 + γ − p1) ⇐⇒ BR1(p3, ¯γ) = BR2(p3, ¯γ) = 2(p3+ ¯γ) 3 (7)

In order to derive the best response function for Online, we rewrite (6) and square both sides

1 − 2√p3+ γ − p1− p3 √ p3+ γ − p1 = 0 ⇐⇒ 12 ₌  2√p3+ γ − p1+ p3 √ p3+ γ − p1 2 =4(p3+ γ − p1) + p2 3 p3+ γ − p1 + 4p3 ⇐⇒ p3+ γ − p1 =4(p3+ γ − p1)2 + p23+ 4p3(p3+ γ − p1)

The last equation shows a polynomial z(p3) of degree 2, so we look at its

discriminant

0 = 9p2₃ + (12γ − 12p1 − 1)p3 + 4γ2 + 4p21 + p1 − γ − 8p1γ (8)

with a = 9, b = 12γ − 12p1− 1 and c = 4γ2+ 4p21+ p1− γ − 8p1γ, this means

that:

4 > 0 for 0 ≤ p1 ≤ γ +

1

12 (9)

So the polynomial has two real number roots, which means we can write p3

in terms of p1 and γ. p3 = 1 18(1 − 12γ + 12p1± p 1 + 12γ − 12p1) (10)

Only the higher root is used, since for p1 < γ − ₁₂1, (10) will be negative for

the lower root. Together with (7), this gives the following lemma

Lemma 3.1 For all three firms, best response function BRi(pj|¯γ) gives the

best response price of player i to the price of player j, with i 6= j and γ given. BR1(p3|¯γ) = BR2(p3|¯γ) = 2(p3+ ¯γ) 3 BR3(p1|¯γ) = 1 18(1 − 12¯γ + 12p1+ p 1 + 12¯γ − 12p1)

For an equilibrium we require BR3(p1) = BR1−1(p1). Easily, we rewrite

BR1(p3) in terms of p1:

p3 = BR−11 (p1) =

3p1− 2γ

2 (11)

Now we equate (10) and (11): BR3 = 3p1−2γ₂ and express p1 in terms of γ

p1 =

1 + 10γ ±√1 + 20γ

25 (12)

My main result is summarized in a proposition.

Proposition 3.2 For given ¯γ the simultaneous Nash equilibrium of the price competition game is given by

p∗₁ = p∗₂ = 1 + 10¯γ ± √ 1 + 20¯γ 25 p∗₃ = BR−1₁ (p∗₁) = 3 − 20¯γ ± 3 √ 1 + 20¯γ 50

This solution is feasible if 0 ≤ p∗₁ ≤ ¯γ +₁₂1 , as shown in (9).

3.1.1 Example equilibrium

For a given value of γ I will show where the equilibrium of the higher p∗₁(¯γ) lies. We will be using ¯γ = 0.10 for simplicity and the upcoming contrast be-tween our cases. From the previous proposition it follows that in equilibrium: p∗₁ = p∗₂ = 2+ √ 3 25 ≈ 0.149 and p ∗ 3 ≈ 0.124

Figure 4: Graph of best response functions, p3 against p1 for ¯γ = 0.1

Figure 5: Graph of minimal commodity cost to consumers in equilibrium with p∗₁ = p∗₂ = 0.149 and p∗₃ = 0.124 for ¯γ = 0.1

So the offline firms make a profit of π1 = π2 = 0.041 and Online makes

π3 = 0.056 profit. Online also attains a market share of

1 − 2√0.124 + 0.1 − 0.149 = 1 − 2 ∗ 0.274 = 45, 2% against shares of 27,4% for each physical store. Compared to the initial 2-firm Nash equilibrium on page 4, this is a drop of 0.5−0.041_0.5 = 91.8% in profits for the offline firms.

3.2

Case 2: offline firms located at quartiles

Now I will look at the 2-firm with outside goods equilibrium. Does it have a price equilibrium in my 3-firm model? x1 = −0.25, x2 = 0.25, p1 = p2

D1 = D2 =    0.5 if ul ≤ p3+ γ 2√p3+ γ − p1 else 0 if p1 = p2 ≥ p3+ γ (13) Dint= 1 − D1− D2 = 1 − 4 √ p3+ γ − p1 (14) Profit functions π1 = π2 = p1∗ D1 = 2p1 √ p3+ γ − p1 (15) π3 = p3∗ (1 − 4 ∗ D1) = p3− 4p3 √ p3+ γ − p1 (16)

First order conditions dπ1 dp1 = 2√p3+ γ − p1 − p1 √ p3+ γ − p1 = 0 (17) dπ3 dp3 = 1 − 4√p3+ γ − p1− 2p3 √ p3+ γ − p1 = 0 (18) Deriving the best response functions for the offline firms

2√p3+ γ − p1 − p1 √ p3+ γ − p1 = 0 ⇐⇒ p1 = 2(p3+ γ − p1) ⇐⇒ BR1(p3, ¯γ) = BR2(p3, ¯γ) = p3+ ¯γ 3

Like the first case, in order to derive the best response function for Online, we rewrite (18) and square both sides

1 − 4√p3+ γ − p1− 2p3 √ p3+ γ − p1 = 0 ⇐⇒ 12 =  4√p3+ γ − p1+ 2p3 √ p3+ γ − p1 2 =16(p3+ γ − p1) + 4p2₃ p3+ γ − p1 + 16p3 ⇐⇒ p3+ γ − p1 = 16(p3+ γ − p1)2+ 4p23+ 16p3(p3+ γ − p1) (19)

The last equation shows a polynomial z(p3) of degree 2, so we look at its

discriminant:

0 = 36p2₃ + (48γ − 48p1 − 1)p3 + 16γ2 − γ + p1 + 16p21 − 32γ ∗ p1 (20)

with a = 36, b = 48γ − 48p1− 1 and c = 16γ2− γ + p1+ 16p21− 32γ ∗ p1 this

means that:

4 > 0 for 0 ≤ p1 ≤ γ +

1

48 (21) So the polynomial has two real number roots, which means we can write p3

in terms of p1 and γ. p3 = BR3(p1) = 1 72(1 − 48γ + 48p1± p 1 + 48γ − 48p1) (22)

Only the higher root is used, since for p1 < γ − ₄₈1, (22) will be negative for

the lower root. Together with (19), this gives the following lemma

Lemma 3.3 For all three firms, best response function BRi(pj|¯γ) gives the

best response price of player i to the price of player j, with i 6= j and γ given. BR1(p3|¯γ) = BR2(p3|¯γ) = 2(p3+ ¯γ) 3 BR3(p1|¯γ) = 1 18(1 − 12¯γ + 12p1+ p 1 + 12¯γ − 12p1)

For an equilibrium we require BR3(p1) = BR1−1(p1). Easily, we rewrite

BR1(p3) in terms of p1:

p3 = 3p1− γ (23)

Now we equate BR3 and BR−11 : BR3 = 3p1− γ and express p1 in terms of γ

p1 =

1 + 28γ ±√1 + 56γ 196

Proposition 3.4 For given ¯γ the simultaneous Nash equilibrium of the price competition game is given by

p∗₁ = p∗₂ = 1 + 28¯γ ± √ 1 + 56¯γ 196 p∗₃ = BR−1₁ (p∗₁) = 3 − 112¯γ ± 3 √ 1 + 56¯γ 196

This solution is feasible if 0 ≤ p∗₁ ≤ ¯γ +₄₈1 , as shown in (21).

Figure 6: Graph of p1 against γ under condition BR3 = BR1−1

3.2.1 Example equilibrium

We find an equilibrium for ¯γ = 0.1: p∗₁ = p∗₂ = 3.8+

√ 6.6

196 ≈ 0.0325 and

p∗₃ ≈ −0.0025

Figure 7: Graph of best response functions, p3 against p1 for ¯γ = 0.1

So with this scenario the online firm would make a small negative amount of profit, which makes it more likely for him not to enter the market. Algebraic

Figure 8: Graph of minimal commodity cost to consumers in equilibrium with p∗₁ = p∗₂ = 0.0325 and p∗₃ = −0.0025 for ¯γ = 0.1 Clearly, Online does not have a market share

calculations show that for γ = ₃₂3 ≈ 0.094, Online exactly has a p3 = 0 and

thus π3 = 0. So for γ < ₃₂3 it would be rewarding for Online to enter the

market.

Figure 9: Response functions, p3 against p1 with γ = γindiff= 0.094, resulting

3.3

Average distance calculations

For each case we can calculate the average distance to consumers by inte-grating the minimal commodity cost to consumers function. This is shown for γ = 0.05, since then both cases show nonzero equilibria. First graphs are shown of the integrated area, then integrals are derived and compared.

Figure 10: Integrated area of minimal commodity cost to consumers with ¯

γ = 0.05 and offline firms located at x1 = −0.5, x2 = 0.5.

Figure 11: Integrated area of minimal commodity cost to consumers with ¯

In the first case, the graphs of offline and online minimal commodity cost intersect twice, at x1,2 = ±0.5 −

√

p3 + γ − p1, so the total area is

I1 = 2 Z 0.5 0.5−√p3+γ−p1 (0.5 − x)2+ p1dx + (p3+ γ)(0.5 − √ p3+ γ − p1)  = 2 (p3+ γ)  1 2 − √ p3+ γ − p1  + 1 4+ p1  x − 1 2x 2 ₊1 3x 3₎      1 2 1 2− √ p3+γ−p1 ! ≈ 2(0.0453 + 0.0327) = 0.156 for p1 = 0.117, p3 = 0.125 and γ = 0.05

In the second case, the graphs of offline and online minimal commodity cost intersect 4 times, at x1,2,3,4 = ±0.25 ±

√

p3+ γ − p1, so the total area is

I2 = 2 Z 0.25+ √ p3+γ−p1 0.25−√p3+γ−p1 (0.25 − x)2+ p1dx + 2(p3+ γ)(0.25 − √ p3+ γ − p1) ! = 2 2(p3+ γ)  1 4− √ p3+ γ − p1  + 1 16+ p1  x −1 4x 2₊ 1 3x 3₎      1 4+ √ p3+γ−p1 1 4− √ p3+γ−p1 ! ≈ 2(0.0052 + 0.0156) = 0.0416 for p1 = 0.0222, p3 = 0.0166 and γ = 0.05

This result shows that in case 1, with price and profit cuts for offline firms, also the average distance to consumers is reduced by 1.08333−0.156_1.08333 ≈ 86%. For the second case we cannot compare this result with an equilibrium for two firms, since it doesn’t exist. But one can see analytically from the graphs that the entry of Online sleeks the minimal cost function, and therefore the average distance to consumers.

4

Conclusions

In this thesis I discussed a model of horizontal product differentiation, com-bining a fixed location with a price game of three players. A standard model for such differentiation was discussed and extended with a third player, who is participating in the price game but not having a physical location on a line. For this model I derived demand functions for two specific cases. With the use of profit maximization under first order conditions I showed that two equilibria exist for a given delivery cost in each of the two cases.

The strength of this study lies in the discussed cases of section 3. Where it is more common to model this kind of problem with an outside option, I tried to look at aspects that make a difference for an external internet player whether or not to enter the market.

The main result in this paper is about stability in price competition equilib-ria of two discussed cases. With the entry of an online firm to an isolated offline market the prices and profits of the offline firms drop significantly. The social optimum derived by Woeckener [9] deserves special evaluation. For a certain γ I showed that entering the market for the online firm is no longer profitable. This can be explained as follows. In the first case our firms were located at the ends of the line, meaning that for people in the middle it is rather expensive to buy the commodity. The offline firms are more willing to give up market share here because lowering prices would drastically cut their profits. In the second case, however, the offline firms have minimized their average distance to consumers. The area of consumers that are less worth supplying to is now smaller, resulting in a lower minimal required γ for Online. For a given γ = ¯γ we can see that in the first case all three players make a fair profit, whereas in the second case it is no longer profitable for the internet firm to stay in the market. If we impose the restriction of positive prices, the third firm would in this case not enter.

Also, analysis of average distance to consumers shows that in both cases the average distance to consumers drops significantly. Since Online is not physically located on the line, it has a big impact on the average distance to consumers. This is another welfare aspect that contributes positively to the utility obtained by consumers.

The major analysis given in this study is based on the comparison of equilib-ria. In the first case one standard solution was taken as starting point, and if the third player would decide to enter, this new result was compared to the former solution. This gives some interesting results, since for many γ’s,

the equilibrium prices will drop significantly. However, the 1-to-1 trade-off between transport cost and online delivery cost is far from realistic. Consid-erations like delivery time and search costs should also be taken into account to achieve a more profound model.

References

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