Toshiki Kodera

Discriminatory Pricing and Spatial Competition in Two-Sided Media Markets

De Gruyter | Published online: January 15, 2015

Abstract

This study describes a spatial model of price discrimination in two-sided media markets. Given that media platforms offer a uniform price for consumers and either a uniform or discriminatory price for advertisers, we compare a platform’s profit and welfare under these two different pricing schemes. In contrast to the well-known result that price discrimination based on a consumer’s location leads to lower profits, if consumers have a strong aversion to advertising, we show that a platform’s profit is better off under price discrimination. In addition, if consumers rather dislike advertising, we show that price discrimination is detrimental to both a platform’s profit and the consumer’s welfare.

1 Introduction

A media platform differs from traditional firms in that it has two revenue sources. One is advertising revenue, and the other is revenue from charging consumers. Moreover, in media markets, there are indirect network effects between advertisers and consumers. Advertisers receive positive network effects from consumers of the same platform because advertisers are interested in the number of consumers, which helps them to determine whether to place advertisements. However, consumers may dislike advertising. Therefore, advertisements in a media platform may yield negative network effects for consumers. Accordingly, media platforms adjust prices for advertisers and consumers to internalize these network effects.

Media platforms may be able to charge different agents different prices. The Economist, one of the most well-known weekly news magazines, price discriminates based on regions, sizes and colors in advertising rates. [1] For example, for the same sizes and colors, advertising rates in North America are more than twice that in the Asia Pacific region. Some Japanese newspaper publishers charge higher advertising fees for public firms than for private firms. Gil and Riera-Crichton (2011) investigated the Spanish TV market, in which TV stations price discriminate their advertisers and viewers. Moreover, publishing companies can sell advertising columns at different prices depending on the various characteristics of advertisers (such as industry and location). [2] The purpose of this paper is to consider whether this price discrimination in media markets is beneficial for media platforms, consumers and advertisers.

Oligopolistic price discrimination on the Hotelling model has previously been studied by Thisse and Vives (1988), Bester and Petrakis (1996) and Liu and Serfes (2004). [3] These analyses of price discrimination without indirect network effects suggest that a firm’s profit from price discrimination is below that from uniform pricing. [4] When a firm can price discriminate to account for a given rival’s prices, the firm extracts a surplus from local consumers. However, the firm sets low prices for distant consumers. Conversely, when a rival firm uses the same price schedule, price discrimination exacerbates competition; therefore, consumers benefit when firms engage in price discrimination. We examine how these well-known results change in media markets.

To investigate a spatial model of price discrimination in two-sided media markets, we extend the perfect spatial price discrimination model developed by Thisse and Vives (1988). [5] Two platforms are located at each end of a line, and two groups of agents, advertisers and consumers, are uniformly located along the line. There are asymmetric network effects between these groups. We consider a sequential game in which platforms first set prices for consumers, and then for advertisers. The reason that consumers choose first is the accuracy of marketing. [6] Recent developments in information technology improve the accuracy of marketing. Therefore, publishers can strictly predict the number of copies or consumer market share. Media platforms compete for advertisers using these consumer data.

In the first stage, platforms offer uniform prices for consumers deciding to join a platform. Platforms cannot price discriminate among consumers, as consumers can resell magazines. [7] In the second stage, both platforms simultaneously offer prices for advertisers. If price discrimination is feasible, both platforms price discriminate for each advertiser. Otherwise, platforms offer uniform prices for advertisers.

As advertisers select a platform in the final stage, similar to one-sided markets, price discrimination reduces the platform’s profit from advertisers. However, platforms adjust uniform prices for consumers in response to network effects. As price discrimination intensifies price competition in the final stage, advertisers can easily shift to a platform with a large consumer market share. Therefore, network effects are more influential in determining a platform’s profit from consumers under price discrimination than under uniform pricing. As negative network effects relax price competition, if consumers strongly dislike advertisements, a platform’s profit under price discrimination is higher than that under uniform pricing.

We also investigate consumer surplus. If negative network effects are moderate, platforms cannot compensate for price competition for advertisers by extracting a surplus from consumers. Consequently, despite the finding that price discrimination leads to reduced platform profit, consumers are worse off when platforms employ price discrimination. [8]

Our paper relates to the literature on two-sided markets using the Hotelling framework. [9] The seminal work of Armstrong (2006a) investigates the role of elasticities of demand in two-sided markets. Anderson and Coate (2005) analyze competition between media platforms in the presence of asymmetric network effects. However, these studies are limited to uniform pricing for each group of agents.

The recent paper by Liu and Serfes (2013) analyzes a Hotelling model of price discrimination in two-sided markets with positive network effects. They demonstrated that price discrimination and strong network effects increase a platform’s profit under the condition that the price schedule is limited to a non-negative price. While their model is related to ours, we demonstrate that platform profits increase in the presence of negative network effects under certain conditions. [10]

The remainder of this paper is organized as follows: Section 2 presents the model. Section 3 analyzes the equilibrium outcomes. We check the robustness of our model in Section 4. Section 5 concludes the paper.

2 Model

There are two homogenous platforms (1 and 2). Each platform is located at the end of a unit line. There are two groups of agents: advertisers (sellers, S) and consumers (buyers, B). Each agent is uniformly distributed along the unit line and selects only one platform. [11] To join the platform, each agent incurs transportation costs per unit of distance t k > 0 , k = S , B . The number of agents who join the platform i is n i k . When an agent joins the platform, the agent obtains network benefits, b k n i j , j k , affected by the number of agents who join the same platform from the other side. To focus on media markets, we assume that b S > 0 and b B < 0 . [12] The greater the number of consumers that select platform i, the greater the number of advertisers that place advertisements on the same platform. However, we assume that consumers dislike advertisements. The utility of group k’s agent, who is located at distance x k [ 0 , 1 ] from platform 1, is

u 1 k = α k + b k n 1 j p 1 k t k x k ,
where p 1 k is the price to join platform 1. The agent who joins platform i has sufficiently large benefits α k to cover the market. α B is an intrinsic benefit for a consumer from watching a TV program or reading a magazine. An intrinsic benefit for an advertiser, α S , arises from advertising that increases the advertiser’s reputation. [13]

Platform i’s profit is

[1] π i = ( p i B c ) n i B + p i S n i S .
The platform faces marginal costs c for each consumer. To simplify the analysis, we assume that the marginal cost for advertisers is zero. We assume the following inequality to satisfy the second-order condition under uniform pricing and price discrimination: 2 t S t B > ( b S ) 2 . [14]

The game is as follows: First, the two platforms simultaneously set a uniform price for consumers. Consumers decide whether to join platform 1 or 2. In the second stage, we consider two price regimes. Under the first, both platforms set uniform pricing for advertisers, and under the second, both platforms price discriminate among advertisers.

2.1 Uniform price

We consider that each platform charges uniform prices to both groups. In the second stage, each platform offers a uniform price for advertisers. The number of advertisers who join each platform is given by

[2] n i S = t S + b S ( n i B n j B ) ( p i S p j S ) 2 t S .

Using the platform’s profit function and the number of advertisers, we derive the price for advertisers. Proposition 1 summarizes prices and profits π i U P when each platform sets uniform prices for both groups.

PROPOSITION 1When each platform charges uniform prices for both the groups, equilibrium prices and profits are determined as follows:

[3] p i U P S = t S ,
[4] p i U P B = c + t B b S ( 2 t S + b B ) 3 t S ,
[5] π i U P = t S + t B 2 b S ( 2 t S + b B ) 6 t S .
Proof. See Appendix. ■

In symmetric equilibrium, each platform has half the market share of each group. The market share of consumers is determined in the first stage. In the second stage, all advertisers receive the same network benefits b S / 2 . Therefore, similar to the standard Hotelling model, each advertiser selects a platform while accounting for prices for advertisers and transportation costs. The transportation cost faced by advertisers represents the platform’s market power over advertisers. When t S is lower, both platforms fiercely compete for advertisers. A consumer’s price is composed of marginal costs, transportation costs and the term b S ( 2 t S + b B ) / ( 3 t S ) , resulting from network effects. The third term is divided into two components: one is external profits from positive network effects, and the other is external losses from negative network effects. Because of positive network effects each additional consumer draws additional advertisers. Moreover, the platform that obtains the additional consumer extracts an additional surplus from current advertisers. These external profits provide an incentive for the platform to increase price competition for consumers. In contrast, when an additional consumer attracts additional advertisers, the presence of additional advertisers drives out certain consumers due to negative network effects. External losses provide an incentive for the platform to relax price competition for consumers.

2.2 Price discrimination

We assume that in the second stage, two platforms can price discriminate among advertisers. Then, platforms can observe the location of each advertiser. Each platform controls the territory near its own location, i.e., platform 1’s territory is [ 0 , 1 / 2 ] , and platform 2’s territory is [ 1 / 2 , 1 ] . When an advertiser is located in x S , platform 1 sets a price of p 1 S to prefer platform 1 to platform 2 given platform 2’s price,

p 1 S + t S x S b S n 1 B p 2 S + t S ( 1 x S ) b S n 2 B .
Similarly, platform 2 sets a price of p 2 S . If the sum of transportation costs and network benefits is indifferent to an advertiser who is located in x ˆ S , both platforms charge their marginal costs to an advertiser, i.e., a price equal to zero, because each platform has no cost advantage over the rival’s platform. [15] Then, platform 1 (resp. 2) has a cost advantage for advertisers who are located in an area smaller (resp. larger) than x ˆ S . Prices for advertisers are given by
p 1 S = t S ( 1 2 x ˆ S ) + b S ( n 1 B n 2 B ) , p 2 S = 0 , i f x ˆ S t S + b S ( n 1 B n 2 B ) 2 t S ,
p 1 S = 0 , p 2 S = t S ( 2 x ˆ S 1 ) + b S ( n 2 B n 1 B ) , i f x ˆ S t S + b S ( n 1 B n 2 B ) 2 t S ,

When platforms price discriminate, each platform’s profit π i P P D is given by

[6] π 1 P P D = 0 ( t S + b S ( n 1 B n 2 B ) ) / 2 t S t S ( 1 2 x ˆ S ) + b S ( n 1 B n 2 B ) d x ˆ S + ( p 1 B c ) n 1 B ,
[7] π 2 P P D = ( t S + b S ( n 1 B n 2 B ) ) / 2 t S 1 t S ( 2 x ˆ S 1 ) + b S ( n 2 B n 1 B ) d x ˆ S + ( p 2 B c ) n 2 B .

Proposition 2 summarizes prices and profits when each platform price discriminates for advertisers.

PROPOSITION 2When each platform price discriminates in the second stage, equilibrium prices and profit are determined as follows:

[8] p 1 P P D S = t S ( 1 2 x S ) , p 2 P P D S = 0 , i f x S 1 2 ,
[9] p 1 P P D S = 0 , p 2 P P D S = t S ( 2 x S 1 ) , i f x S 1 2 ,
[10] p i P P D B = c + t B b S b S b B t S ,
[11] π i P P D = t S + 2 t B 2 b S 4 b S b B 2 t S .
Proof. See Appendix. ■

Similar to the case of uniform pricing, network effects do not affect prices for advertisers. The equilibrium prices for advertisers are identical to those in Thisse and Vives (1988). Prices for consumers include terms stemming from network effects. These terms under price discrimination are qualitatively similar to those under uniform pricing.

3 Comparison

We study the symmetric equilibrium prices with and without price discrimination to compare profits. First, we compare prices for advertisers in the second stage. As each platform sets symmetric prices in the first stage, each platform obtains one half of the consumers. There is no difference in an advertiser’s network benefits under the two pricing schemes. Therefore, prices for advertisers are identical to those in Thisse and Vives (1988) on one-sided markets. In the second stage, a platform is profitable when price discrimination is feasible for a given rival’s prices. However, the rival will perform the same action. Therefore, price competition for advertisers is more intense when platforms can price discriminate. In the second stage, uniform pricing is more profitable than price discrimination.

Next, we consider prices for consumers in the first stage. Suppose platforms offer a symmetric price to consumers. Then, platform 1 slightly reduces p 1 B . Certain consumers switch from platform 2 to platform 1. The additional consumer attracts additional advertisers due to positive network effects, i.e., a direct effect. Suppose that the marginal advertisers are x S ( n 1 B , n 2 B ) , including optimal prices for advertisers. This marginal change in market share under uniform pricing is x S / n 1 B = b S / 6 t S , while it is b S / 2 t S under price discrimination. Therefore, platform 1 attracts more advertisers under price discrimination than under uniform pricing. This demonstrates that under price discrimination, platforms have a greater incentive to reduce consumer prices. However, this marginal change in market generates a feedback effect, due to negative network effects. Suppose that x B ( n 1 S , n 2 S ) represents the marginal consumers who anticipate the market share in advertisers. The feedback effect of market share under uniform pricing is ( x B / n 1 S ) ( n 1 S / n 1 B ) = b S b B / 12 t S t B , while it is b S b B / 4 t S t B under price discrimination. Therefore, platform 1 loses more advertisers under price discrimination than under uniform pricing. Under price discrimination, platforms have less incentive to reduce consumer prices.

We investigate how these two effects impact consumer prices under price discrimination.

( π 1 ) p 1 B | P P D ( π 1 ) p 1 B | U P = b S ( t S + 2 b B ) 6 ( t S t B b S b B ) .
When b B < t S / 2 , under price discrimination, platforms have a greater incentive to soften the price competition for consumers. Therefore, when b B < t S / 2 , p P P D B p U P B = b S ( t S + 2 b B ) / 3 t S > 0 , the condition b B < t S / 2 indicates that the platform’s profit from consumers increases under price discrimination. If consumers strongly dislike advertising and/or the transportation cost faced by advertisers is low, platforms can extract a sufficiently large surplus from consumers under price discrimination to cover price competition for advertisers. As a result, a platform’s profit is greater under price discrimination. Summarizing this analysis,

PROPOSITION 3Price discrimination is more profitable than a uniform price, if and only if

b B < t S ( 3 t S + 2 b S ) 4 b S .

Proof. See Appendix. ■

When a consumer joins a platform, the consumer anticipates that the platform will attempt to attract more advertisers in the second stage. If consumers strongly dislike advertising, in the first stage, the consumer has less incentive to join the platform, because the feedback effect is larger than the direct effect. This implies that the consumer is price inelastic. Therefore, the platform sets higher prices for consumers. However, this effect is present under both price discrimination and uniform pricing. In addition to this effect, when the platforms can price discriminate, a platform (e.g., 1) can attract more advertisers when platform 1 obtains consumers from platform 2. However, platform 2 cannot reduce prices for advertisers located at the middle of the line due to marginal cost pricing. As a consequence, price discrimination reinforces the above effect. Therefore, consumer prices are higher under price discrimination than under uniform pricing.

In the Hotelling model of one-sided markets, price discrimination intensifies price competition (see Thisse and Vives 1988). However, in our model that considers two-sided markets, if the negative network effects are sufficiently large, a platform’s profit under price discrimination is higher than that under uniform pricing. [16] Our result extends the findings of Liu and Serfes (2013), who analyzed a spatial price discrimination model considering two-sided markets with positive network effects and non-negative prices. In their model, if positive network effects are strong and prices are required to be non-negative, the profit of a platform is greater under price discrimination. In contrast to their result, we can derive our result with asymmetric network effects and without price regulation.

Next, we consider welfare. As all agents participate in platform 1 or 2, both pricing schemes have identical total welfare, which is composed of a platform’s profit and agent groups’ surplus. Therefore, under the condition of Proposition 3, the sum of both agent groups’ surplus under price discrimination is lower than under uniform pricing. To determine which pricing schemes affect each group, we compare each group’s surplus under price discrimination with that under uniform pricing. When each platform charges uniform prices for both groups, advertiser and consumer surplus are as follows:

[12] C S U P S = 2 0 1 / 2 α S + b S 2 t S t S x S d x S = α S + b S 2 5 t S 4 ,
C S U P B = 2 0 1 / 2 α B + b B 2 c t B + b S ( 2 t S + b B ) 3 t S t B x B d x B
[13] = α B c + b B 2 + b S ( 2 t S + b B ) 3 t S 5 t B 4 .
Similarly, when both platforms price discriminate among advertisers, advertisers and consumers surplus are as follows:
[14] C S P P D S = α S + b S 2 3 t S 4 ,
[15] C S P P D B = α B c + b B 2 + b S ( t S + b B ) t S 5 t B 4 .
When platform profit is lower under price discrimination than under uniform pricing, we compare agent groups’ surplus under uniform pricing with that under price discrimination. Then, we can derive Proposition 4.

PROPOSITION 4When platform profit under price discrimination is lower than that under uniform pricing,

  • (i)

    the advertisers’ surplus under price discrimination is always larger than that under uniform pricing;

  • (ii)

    the consumers’ surplus under price discrimination is smaller than that under uniform pricing, if and only if

t S ( 3 t S + 2 b S ) 4 b S < b B < t S 2 .

Proof. See Appendix. ■

The preceding studies on one-sided markets, such as Thisse and Vives (1988), reveal that price discrimination reduces a firm’s profits and improves consumer surplus. In contrast, our model finds that price discrimination is detrimental not only to a platform’s profit but also to consumer surplus. [17] If negative network effects are moderate, platforms cannot compensate for engaging in price competition for advertisers by extracting surplus from consumers. Consequently, despite that a platform’s profit under price discrimination is lower than that under uniform pricing, consumers are worse off when platforms employ price discrimination.

This result suggests that policy action is necessary for agents on two-sided media markets. In our model, competition authorities may approve not only consumer surplus but also a platform’s profit as a policy objective. Platforms set a uniform price for consumers, resembling resale price maintenance (RPM). [18] RPM is used to protect publishing culture. Therefore, a platform’s profit becomes one of the policy objectives. Under the condition of Proposition 4, price discrimination is detrimental to a platform’s profit and consumer surplus. Competition authorities should ban price discrimination for advertisers. However, it may be difficult to implement the policy, as advertisers negotiate on their prices in a closed-door room. In contrast, competition authorities can observe a platform’s profit and prices for consumers. When competition authorities notice low profits and high prices for consumers, they could infer large advertiser surplus. Then, competition authorities should examine the prices for advertisers to protect a platform’s profit and consumer surplus.

Our result also suggests that competition authorities do not always have to ban price discrimination for advertisers. Under the condition of Proposition 3, a platform’s profit under price discrimination is higher than that under uniform pricing. However, price discrimination for advertisers is harmful to consumers under the condition. If competition authorities prefer to protect platforms, they permit price discrimination for advertisers. If b B > t S / 2 , then price discrimination for advertisers is harmful to platforms, but it is beneficial to consumers. If competition authorities prefer consumer surplus to a platform’s profit, they permit price discrimination for advertisers under the condition.

In contrast to our result, Liu and Serfes (2013) demonstrate that when price discrimination reduces a platform’s profit, both groups’ welfare improves. In their model, as both agent groups simultaneously join platforms, the two network effects equally affect prices and profits. In contrast, in our model, consumers select platforms before the platforms offer prices to advertisers. Therefore, the two network effects differently affect prices and profits.

4 Discussion

In this section, we discuss four important assumptions about our basic model to check its robustness. [19] First, network effects are asymmetric. In Section 4.1, we consider a model with positive network effects for both sides. Second, the agent groups sequentially select a platform. We extend our model to the simultaneous case in Section 4.2. In the simultaneous game, platforms can potentially set negative prices for agents by internalizing network effects. Third, platforms can price discriminate among advertisers. We adapt our model to include price discrimination on both sides in Section 4.3. Finally, we discuss multihoming in Section 4.4. After reviewing these assumptions, we find that the simultaneous game, discriminating on both sides and multihoming do not intrinsically affect our main result. Therefore, the assumption of asymmetric network effects plays a key role in our main results. Moreover, we can compare our model with that Liu and Serfes (2013) by relaxing our assumptions except for multihoming. [20] Therefore, negative network effects differentiate our model from theirs. Moreover, they limit the price schedule to a non-negative price. However, our model intrinsically permits negative prices.

4.1 Positive network effects on both sides

In this subsection, we investigate how the direction of network effects affects our main result. We extend our basic model to include positive network effects for both groups, i.e., suppose that b B > 0 . As the condition of Proposition 3 does not hold, a platform’s profit is lower under price discrimination than under uniform pricing. Similarly, as t S + 2 b B is positive, consumer surplus under price discrimination is larger than that under uniform pricing. Therefore, asymmetric network effects play an important role in our model.

4.2 Simultaneous game

We extend our model to a simultaneous game in which platforms set prices for consumers and advertisers simultaneously. We demonstrate that the results of this extension are qualitatively identical to those of the basic model.

First, we assume that platforms set uniform prices for both sides in the simultaneous game. This game is identical to that in Armstrong (2006a). Therefore, the equilibrium prices are p U P i = c i + t i b j . These equilibrium prices indicate that the equilibrium price of the standard Hotelling model is adjusted by the platform’s external benefit from obtaining an additional agent.

In the simultaneous game, platforms price discriminate among advertisers and set uniform prices for consumers. First, we consider prices for advertisers following Liu and Serfes (2013). Given platform 2’s prices for advertisers and consumer market share, platform 1 charges prices for advertisers who prefer platform 1 to platform 2:

p 1 S = t S ( 1 2 x S ) + b S ( n 1 B n 2 B ) + p 2 S , i f x S t S + b S ( n 1 B n 2 B ) 2 t S .

We investigate platform 2’s prices for advertisers. For the advertisers who prefer platform 1, platform 2’s prices are equivalent to platform 2’s external benefit of attracting an additional advertiser. [21] When platform 2 attracts an additional advertiser, platform 2 loses a further b B / t B consumers. Platform 2 loses the external benefits p B b B / t B from consumers. This change in market share, b B / t B , has a feedback effect on advertisers. Therefore, platform 2 loses 2 b S b B ( 1 x S ) / t B in additional revenue from infra-marginal advertisers. As these two external benefits must satisfy the connectedness property, platform 2’s prices for advertisers are [22]

p 2 S = b B ( p 2 B + 2 b S ( 1 x S ) ) t B , i f x S t S + b S ( n 1 B n 2 B ) 2 t S .
However, if platforms set these prices for advertisers in equilibrium, the platforms may have an incentive to deviate. If platforms deviate from the equilibrium, price competition arises because the platforms can price discriminate among advertisers. Therefore, the equilibrium prices and platform profits under price discrimination are
p 1 S = t S ( 1 2 x S ) b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) ,
p 2 S = b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) , i f x S 1 2 ,
p 1 S = b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) ,
p 2 S = t S ( 2 x S 1 ) b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) , i f x S 1 2 ,
p 1 B = p 2 B = t B b S 2 ( b S ) 2 b B t S t B b S b B ,
π i P D = t B b S b B 2 + t S 4 + ( b S ) 2 b B ( b B t B ) t B ( t S t B b S b B ) .
We compare profits under two different pricing schemes.
π U P π P D = t S t B ( t S t B b S b B ) 4 ( b S ) 2 b B ( b B t B ) 4 t B ( t S t B b S b B ) .
If the condition b B < t B { t S b S + ( t S 2 b S ) 2 + 16 ( t S ) 2 } / 8 b S holds, profits under uniform pricing are lower than under price discrimination. Therefore, when our model is extended to the simultaneous game, the main result does not intrinsically change. We can obtain this result not to assume that prices for agents are non-negative. The assumption of non-negative prices differentiates our model from that of Liu and Serfes (2013).

4.3 Discrimination on both sides

Third, we explore the case in which platforms can price discriminate on both sides. In this extension, platforms sequentially attract both sides similarly to our basic model. As platforms set prices for advertisers in the final stage, the discriminatory prices for advertisers are identical to those in our basic model, i.e., eqs [8] and [9]. We consider discriminatory prices for consumers. Given the market share of advertisers, platforms select discriminatory prices for consumers near platform 1 as follows: [23]

p 1 B = t B ( 1 2 x B ) b S b B t S ( 1 2 x B ) b S ,
p 2 B = b S , i f x B 1 2 .
Similarly, we can derive discriminatory prices for consumers near platform 2. Therefore, the platform profit under price discrimination is
π P D = t S + t B 2 b S 4 b S b B 4 t S .
When platforms set uniform prices on both sides, their profit is eq. [ 5]. We compare platform profits under uniform pricing on both sides and under price discrimination:
π U P π P D = 3 t S ( t S + t B ) + 2 t S b S + b S b B 12 t S .
In this extension of our model, if the condition b B < ( 3 t S ( t S + t B ) + 2 t S b S ) / b S holds, profits are greater under price discrimination than under uniform pricing. Therefore, when a platform can price discriminate on consumers and advertisers, the main result does not intrinsically change.

In this extension, consumer surplus is as follows:

C S P P D b o t h B = α B + b B 2 + b S ( t S + b B ) t S 3 t B 4 .
We compare this consumer surplus to eq. [ 15]:
C S P P D b o t h B C S P P D B = t B 2 b S b B 2 t S > 0.
Consumer surplus under price discrimination on both sides is larger than consumer surplus under price discrimination on only the advertiser side. Moreover, platform profits are lower under price discrimination on both sides than under price discrimination on only the advertiser side, as advertiser prices are identical under sequential participation.

4.4 Multihoming

In this subsection, we consider the extension of our model in which advertisers have the possibility to multihome. Similar to the result of our basic model, we find that the platform profit under price discrimination rises when consumers strongly dislike advertisements. First, we consider the prices for advertisers when advertisers can multihome. Suppose that θ with θ ( 0 , α S ) is the additional reservation utility of shifting singlehoming to multihoming.

In the basic model, we assume that α S is sufficiently large to cover the advertiser side. In this case, all advertisers can multihome. Then, platforms relax price competition to advertisers. As multihoming relaxes price competition for advertisers, price discrimination explicitly raises platform profits relative to the singlehoming case.

Next, we consider the case in which certain advertisers are singlehoming and other advertisers are multihoming. Then, the equilibrium prices for advertisers are

p 1 S = t S ( 1 2 x S ) , p 2 S = 0 , f o r x S 2 t S 2 θ b S 2 t S ,
p 1 S = θ + b S 2 t S x S , p 2 S = θ + b S 2 t S ( 1 x S ) , f o r x S 2 t S 2 θ b S 2 t S , 2 θ + b S 2 t S ,
p 1 S = 0 p 2 S = t S ( 2 x S 1 ) f o r x S 2 θ + b S 2 t S .
Here, suppose that x 10 S = ( 2 t S 2 θ b S ) / 2 t S , and we differentiate x 10 S with respect to θ :
x 10 S θ = 1 t S < 0.
The multihoming threshold for platform 1 decreases with θ . Therefore, when the additional reservation utility increases, platforms can attract a larger number of multihoming advertisers.

We compare platform profits from advertisers under price discrimination with those under uniform pricing. When advertisers can multihome under price discrimination, a platform can extract all surpluses from advertisers joining the platform. However, under uniform pricing, as each platform sets a monopoly price for advertisers, advertisers enjoy some surplus. Therefore, the platform profit from advertisers increases under price discrimination, which differs from the result of our basic model.

Next, we consider the prices for consumers in the first stage by adapting the intuition from Section 3. When platform 1 has an additional consumer, platform 1 has b S / 2 t S additional advertisers under uniform pricing, while it has b S / t S additional advertisers under price discrimination. Therefore, platforms employing price discrimination have a greater incentive to reduce prices for consumers due to positive network effects, while they have less incentive to reduce prices for consumers due to negative network effects. We compare the prices for consumers under two pricing schemes to investigate these two effects, p U P B p P D B = b S ( 4 t S + 2 b B 2 θ b S ) / 4 t S . When b B < ( 2 θ + b S 4 t S ) / 2 , the price for consumers under price discrimination is higher than that under uniform pricing. Therefore, the platform profit under price discrimination increases whenever this condition holds.

We consider agents’ surplus when advertisers can multihome. In the singlehoming case, advertiser surplus is greater under price discrimination, because it produces fierce competition. However, in the multihoming case, advertiser surplus is lower under price discrimination. However, consumer surplus does not change qualitatively irrespective of whether advertisers multihome. When negative network effects are sufficiently low, consumer surplus is lower under price discrimination.

5 Conclusion

We have investigated a spatial model of price discrimination in two-sided media markets. The result is that if negative network effects are sufficiently large, a platform’s profit under price discrimination is higher than that under uniform pricing. We also demonstrate that platforms and consumers are worse off when platforms engage in price discrimination if negative network effects are moderate. This result concerning agents’ welfare indicates that policymakers may need to reconsider policies in two-sided markets.

We take a preliminary step in studying a model in which platforms can price discriminate on the locations of agents. In media markets, platforms can price discriminate on the size or quantity of advertising. The extension of our model to include second-degree price discrimination should be addressed in future research.

Appendix A

Proof of Proposition 1

We consider prices for advertisers in the second stage. Using eqs [1] and [2], the first-order condition is

[16] π i U P p i S = t S + p j S 2 p i S + b S ( n i B n j B ) 2 t S = 0.
Solving these equations, we obtain prices for advertisers and the number of advertisers:
[17] p i S = t S + b S ( n i B n j B ) 3 , n i S = 1 2 + b S ( n i B n j B ) 6 t S .
Next, we consider prices for consumers in the first stage. Using eq. [ 17], the number of consumers who join each platform is given by
[18] n i B = 3 t S ( p j B p i B ) b S b B n j B + 3 t S t B 6 t S t B b S b B .
Therefore, we obtain the number of consumers:
[19] n i B = 3 t S ( p j B p i B ) + 3 t S t B b S b B 2 ( 3 t S t B b S b B ) .
Substituting eq. [ 19] for prices for advertisers and the number of advertisers, platform’s profit is
π i U P = t S + t S b S ( p j B p i B ) 3 t S t B b S b B 1 2 + b S ( p j B p i B ) 2 ( 3 t S t B b S b B )
+ ( p i B c ) 3 t S ( p j B p i B ) + 3 t S t B b S b B 2 ( 3 t S t B b S b B ) .
The first-order condition is
3 t S ( p j B 2 p i B ) + 3 t S t B b S b B 2 t S b S + 3 t S c 2 ( 3 t S t B b S b B ) t S ( b S ) 2 ( p j B p i B ) ( 3 t S t B b S b B ) 2 = 0.
We also derive the second-order condition:
2 π i U P p i B 2 = t S { ( b S ) 2 9 t S t B + 3 b S b B } ( 3 t S t B b S b B ) 2 < 0.
Solving the first-order conditions, the equilibrium price for consumers is
p i B = c + t B b S ( 2 t S + b B ) 3 t S .
Using price for consumers, we obtain the equilibrium price for advertisers and platform’s profit:
p i S = t S ,
π i U P = t S + t B 2 b S ( 2 t S + b B ) 6 t S .

Proof of Proposition 2

We consider prices for consumers in the first stage. Since platforms charge uniform prices for consumers, the number of consumers is

n i B = t B + b B ( n i S n j S ) ( p i B p j B ) 2 t B .
Using the number of advertisers, we simplify the number of agents who join platform i,
[20] n i B = t S ( p j B p i B ) + t S t B b S b B 2 ( t S t B b S b B ) ,
[21] n i S = b S ( p j B p i B ) + t S t B b S b B 2 ( t S t B b S b B ) .
Substituting eqs [ 20] and [ 21] for eqs [ 6] and [ 7], we can derive the first-order and second-order conditions:
1 2 t S b S ( t S t B b S b B b S ( p i B p j B ) ) 2 ( t S t B b S b B ) 2 + t S ( p j B 2 p i B ) + t S c 2 ( t S t B b S b B ) = 0 ,
t S ( ( b S ) 2 + 2 b S b B 2 t S t B ) 2 ( t S t B b S b B ) 2 < 0.
Solving the first-order conditions, we obtain the equilibrium price for consumers and platform’s profit:
p i B = c + t B b S b S b B t S ,
π i P P D = t S + 2 t B 2 b S 4 b S b B 2 t S .
The number of consumers who join platform i is 1 / 2 . Therefore, the equilibrium prices for advertisers are
p 1 S = t S ( 1 2 x S ) , p 2 S = 0 , x S 1 2 ,
p 1 S = 0 , p 2 S = t S ( 2 x S 1 ) , x S 1 2 .

Proof of Proposition 3

We compare platform’s profits:

[22] π i U P π i P P D = 3 ( t S ) 2 + 2 t S b S + 4 b S b B 12 t S .
Therefore,
π i U P < π i P P D i f b B < t S ( 3 t S + 2 b S ) 4 b S .

Proof of Proposition 4

We compare consumer surplus using eqs [13] and [15]:

[23] C S U P B C S P P D B = b S ( t S + 2 b B ) 3 t S .
Therefore,
C S U P B > C S P P D B i f b B < t S 2 .

Appendix B

In this supplemental section, we provide a detailed analysis of Section 4. As it is easy to verify whether there exist positive network effects on both sides, we discuss three cases: a simultaneous game, discrimination on both sides and multihoming.

Simultaneous game

We extend our model to a simultaneous game and compare platform profits under uniform and discriminatory pricing. [24] In this extension, platforms simultaneously set prices for both groups. The main difference between the simultaneous and sequential games is the prices offered to advertisers. In the simultaneous game, platforms must account for the feedback loop created by the network effect in advertiser price. First, we explore the uniform pricing case in the simultaneous game, which is the same as that of Armstrong (2006a). Therefore, equilibrium prices and profits are

[24] p U P i = t i b j ,
[25] π U P = t S + t B b S b B 2 .
Next, we consider the case in which platforms simultaneously set discriminatory prices for advertisers and uniform prices for consumers. [25] The equilibrium prices are
p 1 S = t S ( 1 2 x S ) b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) ,
p 2 S = b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) , i f x S 1 2 ,
p 1 S = b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) ,
p 2 S = t S ( 2 x S 1 ) b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) , i f x S > 1 2 ,
p 1 B = p 2 B = t B b S 2 ( b S ) 2 b B t S t B b S b B .
We describe how to obtain the equilibrium prices. First, we consider the prices for advertisers. Given the consumer market share, when an advertiser at location x ˆ S is indifferent between platforms 1 and 2, advertisers for x S x ˆ S = ( t S + b S ( n 1 B n 2 B ) ) / 2 t S prefer platform 1 to platform 2. Then, platform 1 sets p 1 S given platform 2’s prices as follows:
[26] p 1 S = t S ( 1 2 x S ) + b S ( n 1 B n 2 B ) + p 2 S , f o r x S t S + b S ( n 1 B n 2 B ) 2 t S .
For x S ( t S + b S ( n 1 B n 2 B ) ) / 2 t S , platform 2’s prices for advertisers are equivalent to platform 2’s benefit from obtaining an additional advertiser.

We explore platform 2’s prices for advertisers for x S ( t S + b S ( n 1 B n 2 B ) ) / 2 t S . Suppose that platforms set equilibrium prices. If platform 2 deviates for x S < ( t S + b S ( n 1 B n 2 B ) ) / 2 t S and obtains an additional advertiser from platform 1, platform 2 loses an additional b B / t B consumers. As platform 2 sets an equilibrium uniform price p 2 B for consumers, platform 2 incurs the additional loss p 2 B b B / t B from consumers. Moreover, the change in consumer market produces a feedback effect concerning advertisers. Therefore, platform 2 also suffers the additional loss from advertisers. As platform 2 has 1 x S market share of advertisers, platform 2 loses 2 b S b B ( 1 x S ) / t B in additional revenue from infra-marginal advertisers. [26] Therefore, platform 2’s total loss from obtaining an additional advertiser is b B ( p 2 B + 2 b S ( 1 x S ) ) / t B . We can adapt this discussion to the advertiser’s price for x S ( t S + b S ( n 1 B n 2 B ) ) / 2 t S . Prices for advertisers are given by

[27] p 1 S = t S ( 1 2 x S ) + b S ( n 1 B n 2 B ) b B ( p 2 B + 2 b S ( 1 x S ) ) t B ,
p 2 S = b B ( p 2 B + 2 b S ( 1 x S ) ) t B , i f x S x ˜ S .
[28] p 1 S = b B ( p 1 B + 2 b S x S ) t B ,
p 2 S = t S ( 2 x S 1 ) + b S ( n 2 B n 1 B ) b B ( p 1 B + 2 b S x S ) t B , i f x S x ˜ S ,
x ˜ S = t S + b S ( n 1 B n 2 B ) 2 t S .
Here, we investigate whether these prices offered to advertisers achieve equilibrium following Liu and Serfes (2013). In the symmetric equilibrium, each platform has own territory, e.g., platform 1 obtains all advertisers on [ 0 , 1 / 2 ] . [27] Platform 1 must provide for advertisers who are connected because we exclude the deviation that the platform obtains disjoint advertisers. Given prices and market shares on the consumer side, we differentiate platform 1’s prices for advertisers (eqs [ 27] and [ 28]) with respect to x S :
p 1 S x S = 2 t S + 2 b S b B t B < 0 i f x S x ˜ S ,
p 1 S x S = 2 b S b B t B > 0 i f x S x ˜ S .

As b B is negative, p 1 S decreases with x S over the interval [ 0 , x ˜ S ] and increases with x S over the interval [ x ˜ S , 1 ] . Then, if platform 1 raises prices for advertisers near to x ˜ S and lowers advertiser prices near the right end of the line without changing the market share on the advertiser side, it is possible for platform 1 to increase its profits. When platform 1 pursues this deviation, we consider price competition for advertisers. Here, suppose that both platforms set the symmetric equilibrium price for consumers, p 1 B = p 2 B = p ˜ B , and have the same market shares on the consumer side, i.e., n 1 B = n 2 B = 1 / 2 . p 1 S ( x ˜ S ) indicates that platform 1 sets the price for an advertiser located at x S = x ˜ S . We have p 1 S ( x ˜ S ) from eq. [27]:

[29] p 1 S ( x ˜ S ) = b B ( p ˜ B + b S ) t B .
Moreover, p 1 S ( 1 ) indicates that platform 1 sets a prices for an advertiser located at x S = 1 ,
[30] p 1 S ( 1 ) = b B ( p ˜ B + 2 b S ) t B .
Because p 1 S ( x ˜ S ) < p 1 S ( 1 ) , platform 1 has an incentive to follow the above-mentioned deviation. Then, platform 2, which predicts the deviation, reduces the price for the advertiser located at x S = 1 to recover the market share on x S > x ˜ S . This behavior continues until the price offered to the advertiser is equal to p 1 S ( x ˜ S ) . The same thing holds for advertisers located on [ x ˜ S , 1 ] . Therefore, platform 1 sets p 1 S ( x ˜ S ) for advertisers located on [ x ˜ S , 1 ] . Similarly, platform 2 sets p 2 S ( x ˜ S ) = b B ( p ˜ B + b S ) / t B for advertisers located on [ 0 , x ˜ S ] . Therefore, prices for advertisers are
p 1 S = t S ( 1 2 x S ) + b S ( n 1 B n 2 B ) b B ( p 2 B + b S ) t B ,
[31] p 2 S = b B ( p 2 B + b S ) t B , i f x S x ˜ S .
p 1 S = b B ( p 1 B + b S ) t B ,
[32] p 2 S = t S ( 2 x S 1 ) + b S ( n 2 B n 1 B ) b B ( p 1 B + b S ) t B , i f x S x ˜ S .
Next, we prove that platform 1 has no incentive to pursue other deviations: The first deviation is that platform 1 decreases its market share on the advertiser side by changing the price for advertisers located on x S x ˜ S ; the second deviation is that platform 1 contracts with more advertisers on x S x ˜ S . [28] Suppose that platform 1’s profit is π 1 s i m when the platforms set p ˜ B for consumers. We demonstrate that platform 1 has no incentive to engage in the first deviation. Suppose that the deviation price, p 1 d S , is the minimum deviation price for losing the advertisers. For x S x ˜ S , the deviation price p 1 d S decreases with x S ; p 1 d S / x S = 2 t S < 0 . If platform 1 pursues the deviation, platform 1 initially loses an advertiser near to one half. Therefore, platform 1 serves advertisers who are connected on [ 0 , x ˜ S ] . Suppose that platform 1 loses an extremely small number δ > 0 of advertisers due to the deviation and has market share x _ S = 1 / 2 δ on the advertiser side. Then, the market share for consumers changes by b B δ / t B . Platform 1’s deviation profit π 1 s i m d is
π 1 s i m d = 0 x _ S t S ( 1 2 x S ) 2 b S b B δ t B b B ( p ˜ B + b S ) t B d x S + p ˜ B 1 2 b B δ t B .
Therefore, platform 1’s additional profit is
π 1 s i m d π 1 s i m = ( t S t B 2 b S b B ) δ 2 t B < 0.
It is not profitable for platform 1 to decrease its advertiser market share.

We also demonstrate that platform 1 has no incentive to pursue the second deviation. Suppose that the deviation price, p 1 d S , is the minimum deviation price that allows it to obtain advertisers. Then, platform 1 contracts with an additional δ advertisers on x S 1 / 2 , and platform 1’s advertiser market share is x ˉ s = 1 / 2 + δ . Then, the consumer market share changes by b B δ / t B . Platform 1’s deviation profit π 1 s i m d is

π 1 s i m d = 0 1 / 2 t S ( 1 2 x S ) + 2 b S b B δ t B b B ( p ˜ B + b S ) t B d x S 1 / 2 x s b B ( p B + b S ) t B d x S + p ˜ B ( 1 2 + b B δ t B ) .
Therefore, platform 1’s additional profit is
π 1 s i m d π 1 s i m = 0.
It is not profitable for platform 1 to increase its market share of advertisers. On the advertiser side, platforms have no incentive for deviations in which they attract additional or fewer advertisers.

We investigate consumer prices. From eqs [31] and [32], platform 1’s profit is

([33] π 1 P D = 0 ( t S + b S ( n 1 B n 2 B ) ) / 2 t S t S ( 1 2 x S ) + b S ( n 1 B n 2 B ) b B ( p 2 B + b S ) t B d x S + p 1 B n 1 B .
Because platforms set uniform consumer prices, the market shares of platform i are equal to eqs [ 20] and [ 21]. We can derive the first-order conditions for consumer prices:
[34] π i P D p i B = t S + t S b S ( p j B p i B ) t S t B b S b B b B ( p j B b S ) t B b S 2 ( t S t B b S b B ) + t S t B b S b B + t S ( p j B 2 p i B ) 2 ( t S t B b S b B ) = 0.
Solving the equations, the equilibrium prices and profit are
p 1 S = t S ( 1 2 x S ) b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) ,
p 2 S = b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) , i f x S 1 2 ,
p 1 S = b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) ,
p 2 S = t S ( 2 x S 1 ) b B + 2 ( b S b B ) 2 t B ( t S t B b S b B ) , i f x S 1 2 ,
p 1 B = p 2 B = t B b S 2 ( b S ) 2 b B t S t B b S b B ,
[35] π i P D = t B b S b B 2 + t S 4 + ( b S ) 2 b B ( b B t B ) t B ( t S t B b S b B ) .
Note that 2 ( t S t B b S b B ) ( b S ) 2 > 0 is necessary to satisfy the second-order condition.

Finally, we compare profits under two different pricing schemes using eqs [25] and [35].

π U P π i P D = t S t B ( t S t B b S b B ) 4 ( b S ) 2 b B ( b B t B ) 4 t B ( t S t B b S b B ) .

As the second-order condition holds, profits under uniform pricing are lower than under price discrimination if and only if b B < ( t B { t S b S + ( t S 2 b S ) 2 + 16 ( t S ) 2 } ) / ( 8 b S ) . Therefore, if negative network effects are sufficiently large, this condition holds. When our model is extended to the simultaneous game, the main result does not change intrinsically.

Discrimination on both sides

In this subsection, we extend our basic model to price discrimination on both sides. Platforms can price discriminate not only among advertisers but also among consumers. Analogously to the basic model, platforms sequentially set prices for both groups, i.e., consumers are charged in the first stage and advertisers are in the next. Suppose that n ˆ i B is platform i’s market share on the consumer side determined in the first stage. Price discrimination among advertisers in this extension is the same in our basic model,

p 1 S = t S ( 1 2 x ˆ S ) + b S ( n ˆ 1 B n ˆ 2 B ) , p 2 S = 0 , i f x ˆ S t S + b S ( n ˆ 1 B n ˆ 2 B ) 2 t S ,
p 1 S = 0 , p 2 S = t S ( 2 x ˆ S 1 ) + b S ( n ˆ 2 B n ˆ 1 B ) , i f x ˆ S t S + b S ( n ˆ 1 B n ˆ 2 B ) 2 t S .
Then, platform 1’s market share of advertisers is n ˆ 1 S x ˆ S (resp. n ˆ 2 S 1 x ˆ S ).

We investigate discriminatory prices for consumers. Suppose that a consumer at location x ˆ B is indifferent between joining platform 1 or 2. Given platform 2’s consumer prices, platform 1 charges prices for consumers at x B x ˆ B as follows:

p 1 B = t B ( 1 2 x B ) + b B ( n ˆ 1 S n ˆ 2 S ) + p 2 B
= t B ( 1 2 x B ) + b S b B ( n ˆ 1 B n ˆ 2 B ) t S + p 2 B f o r x B x ˆ B = t S t B + b S b B ( n ˆ 1 B n ˆ 2 B ) 2 t S t B .
Here, we consider platform 2’s prices for consumers at x B x ˆ B . Platform 2’s prices for consumers at x B x ˆ B is equivalent to platform 2’s benefit from obtaining an additional consumer. When platform 2 steals an additional consumer from platform 1, the utility of each advertiser who joins platform 2 increases by 2 b S . As platform 2 has n ˆ 2 S market share of advertisers, platform 2 earns 2 b S n ˆ 2 S in additional revenue from advertisers. Moreover, when platform 2 steals an additional consumer from platform 1, platform 2 obtains b S / t S additional advertisers and platform 1 loses b S / t S advertisers. This change in market share increases the utility of each consumer joining platform 2 by 2 b B b S / t S . Therefore, as platform 2 has 1 x B market share on the consumer side, platform 2 earns 2 b S b B ( 1 x B ) / t S in additional revenues from consumers. Similarly, we can derive prices for consumers located on [ x ˆ B , 1 ] . Therefore, the prices for consumers are
p 1 B = t B ( 1 2 x B ) + b S b B ( n ˆ 1 B n ˆ 2 B ) t S 2 b S n ˆ 2 S + 2 b S b B ( 1 x B ) t S ,
p 2 B = 2 b S n ˆ 2 S + 2 b S b B ( 1 x B ) t S , i f x B t S t B + b S b B ( n ˆ 1 B n ˆ 2 B ) 2 t S t B ,
p 1 B = 2 b S n ˆ 1 S + 2 b S b B x B t S ,
p 2 B = t B ( 2 x B 1 ) + b S b B ( n ˆ 2 B n ˆ 1 B ) t S 2 b S n ˆ 1 S + 2 b S b B x B t S , i f x B t S t B + b S b B ( n ˆ 1 B n ˆ 2 B ) 2 t S t B .
Suppose that x B n ˆ 1 B and 1 x B n ˆ 2 B . The prices for consumers are rewritten as follows:
[36] p 1 B = t B ( 1 2 x B ) b S + b S ( b S ( 2 x B 1 ) + b B ( 4 x B 3 ) ) t S ,
p 2 B = b S + b S ( b S ( 2 x B 1 ) 2 b B ( 1 x B ) ) t S , i f x B 1 2 ,
[37] p 1 B = b S b S ( b S ( 2 x B 1 ) + 2 b B x B ) t S ,
p 2 B = t B ( 2 x B 1 ) b S b S ( b S ( 2 x B 1 ) + b B ( 4 x B 1 ) ) t S , i f x B 1 2 .
Similar to the simultaneous game, we explore whether platforms provide for consumers who are connected. We differentiate platform 1’s prices for consumers with respect to x B :
p 1 B x B = 2 t B + 2 b S ( b S + 2 b B ) t S i f x B 1 2 ,
p 1 B x B = 2 b S ( b S + b B ) t S i f x B 1 2 .
We focus on the case in which consumers strongly dislike advertisements, i.e., b B < b S . Therefore, p 1 B decreases with x B over the interval [ 0 , 1 / 2 ] and increases with x B over the interval [ 1 / 2 , 1 ] . If platform 1 raises prices for consumers close to 1 / 2 and lowers consumer prices near the right end of the line without changing its market share on the advertiser side, it is possible for platform 1 to increase its profits. When platforms pursue the above-mentioned deviation, we consider the effects on price competition for consumers. Using eq. [ 37], p 1 B ( 1 / 2 ) indicates that platform 1 sets a price for a consumer located at x B = 1 / 2 as follows:
p 1 B ( 1 / 2 ) = b S b S b B t S .
Similarly, p 1 B ( 1 ) indicates that platform 1 sets a price for a consumer located at x B = 1 as follows:
p 1 B ( 1 ) = b S b S ( b S + 2 b B ) t S .
As p 1 B ( 1 / 2 ) < p 1 B ( 1 ) , platform 1 pursues the above-mentioned deviation in which it decreases its market share on x B < 1 / 2 by increasing the price for a consumer close to 1 / 2 and increases its market share on x B > 1 / 2 by lowering the price for a consumer located at x B = 1 . Platform 2 predicts this deviation and reduces the price for a consumer located at x B = 1 to recover its market share on x B > 1 / 2 . This behavior continues until the price for a consumer located at x B = 1 is equal to p 1 B ( 1 / 2 ) . The same holds for consumers in the interval [ 1 / 2 , 1 ] . Moreover, we can adapt this discussion for consumers in [ 0 , 1 / 2 ] . Therefore, the prices for consumers are
[38] p 1 B = t B ( 1 2 x B ) b S + 2 b S b B ( x B 1 ) t S ,
p 2 B = b S b S b B t S , i f x B 1 2 ,
[39] p 1 B = b S b S b B t S ,
p 2 B = t B ( 2 x B 1 ) b S + 2 b S b B x B t S , i f x B 1 2 .
Platform 1’s profit under price discrimination is
π 1 P D = 0 1 / 2 t B ( 1 2 x B ) b S + 2 b S b B ( x B 1 ) t S d x B + 0 1 / 2 t S ( 1 2 x S ) d x S
[40] = t S + t B 4 b S 2 3 b S b B 4 t S .
When platforms set these prices, they may have an incentive to pursue other deviations: The first deviation is that platform 1 contracts with more consumers on x B > 1 / 2 ; the second deviation is that platform 1 decreases its market share on the consumer side by changing the price for consumers located on x B < 1 / 2 . We consider the first case. Suppose that platform 1 sets a minimum deviation price p 1 d B for consumers on x B > 1 / 2 . Then, platform 1 acquires an extremely small number δ of consumers on x B > 1 / 2 , and its market share of consumers is x ˉ B = 1 / 2 + δ . Moreover, platform 1 obtains an additional δ b S / t S through the deviation. Platform 1’s deviation profit is
π 1 P D d = 0 1 / 2 t B ( 1 2 x B ) b S + 2 b S b B ( x B 1 ) t S d x B b S + b S b B t S δ + 0 1 / 2 + b S δ / t S t S ( 1 2 x S ) 2 b S δ d x S .
We compare the deviation profit with eq. [ 40]:
[41] π 1 P D d π 1 P D = b S δ ( b S δ b B ) t S > 0.
As the delta is sufficiently small, π 1 P D d > π 1 P D . Therefore, when platform 1 pursues the deviation and acquires an additional consumer on x B > 1 / 2 , platform 1 obtains an additional profit, b S b B / t S . Moreover, platform 2 predicts this deviation and reduces its price for the consumer to recover its market share. This behavior continues until platform 1’s price for consumers on x B > 1 / 2 is equal to p 1 B = b S . We can adapt the discussion for consumers on [ 0 , 1 / 2 ] . The prices for consumers and the platform profits are
[42] p 1 B = t B ( 1 2 x B ) b S + b S b B ( 2 x B 1 ) t S ,
p 2 B = b S , i f x B 1 2 ,
[43] p 1 B = b S ,
p 2 B = t B ( 2 x B 1 ) b S + b S b B ( 1 2 x B ) t S , i f x B 1 2 ,
[44] π 1 P D = t S + t B 4 b S 2 b S b B 4 t S .
Finally, we prove that platforms have no incentive to pursue the deviation in which platform 1 decreases its market share on the consumer side by changing the price for consumers located on x B < 1 / 2 . Suppose that platform 1 sets a minimum deviation price p 1 d B for consumers on x B < 1 / 2 . Then, platform 1 loses an extremely small number δ of consumers on x B < 1 / 2 , and its consumer market share is x B = 1 / 2 δ . Moreover, platform 1 loses an additional δ b S / t S through the deviation. Platform 1’s deviation profit is
π 1 P D d = 0 1 / 2 δ t B ( 1 2 x B ) b S + b S b B ( 2 x B 1 ) t S d x B + 0 1 / 2 b S δ / t S t S ( 1 2 x S ) 2 b S δ d x S .

We compare the deviation profit with eq. [44]:

[45] π 1 P D d π 1 P D = δ 2 ( t S t B b S b B ( b S ) 2 ) t S < 0.
As b B < b S , π 1 P D d < π 1 P D . Therefore, the platforms have no incentive to pursue the above-mentioned deviation.

From our basic model, platform profits under uniform pricing are ( t S + t B ) / 2 b S ( 2 t S + b B ) / 6 t S . We compare profits under uniform pricing and under price discrimination:

π i U P π i P D = t S + t B 4 + b S 6 + b S b B 12 t S .
In this extension of our model, if the condition b B < ( t S ( 3 ( t S + t B ) + 2 b S ) ) / b S holds, the profit under price discrimination is larger than that under uniform pricing. Therefore, when platforms can price discriminate on both sides, the main result does not change intrinsically.

Multihoming

We investigate the extension of our basic model in which advertisers have the ability to multihome. We explore the case in which certain advertisers singlehome and other advertisers multihome. Similar to the result obtained from our benchmark model, we find that platform profits under price discrimination are larger than under uniform pricing when negative network effects are sufficiently large.

The utility of an advertiser located in x S [ 0 , 1 ] is given by

u i S = { α S + b S n 1 B p 1 S t S x S , i f t h e a d v e r t i s e r j o i n s p l a t f o r m 1 , α S + b S n 2 B p 2 S t S ( 1 x S ) , i f t h e a d v e r t i s e r j o i n s p l a t f o r m 2 , α S + θ + b S p 1 S p 2 S t S , i f t h e a d v e r t i s e r j o i n s b o t h p l a t f o r m s .
Suppose that θ ( 0 , α S ) is the additional reservation utility for an advertiser who is multihoming. In this case, the unit line of the advertiser side is divided into three intervals. Advertisers only join platform 1 near the left end. Near the middle, advertisers join both platforms. Advertisers only join platform 2 near the right end. To determine these intervals, we consider two marginal advertisers located at x 10 S and x 02 S ( 0 < x 10 S < x 02 S < 1 ). The location x 10 S (resp. x 02 S ) is indifferent between visiting platform 1 (resp. 2) and visiting both platforms. Similar to the case of singlehoming, we assume that 2 t S t B > ( b S ) 2 to satisfy the second-order condition under uniform pricing and price discrimination.

Uniform price

We study the case in which platforms offer uniform prices to both groups. The locations of the marginal advertisers are given by

x 10 S = t S + p 2 S θ b S ( 1 n 1 B ) t S ,
x 02 S = θ + b S ( 1 n 2 B ) p 1 S t S .
The number of advertisers joining platform 1 (resp. 2) is n 1 S = x 02 S (resp. n 2 S = 1 x 10 S ). Substituting these equations for the platform profit function and maximizing profits with respect to p i S , as consumers are singlehoming, we derive prices for advertisers and the market share as follows:
p i S = θ + b S n i B 2 , n i S = θ + b S n i B 2 t S .
Using these equations, the first-order conditions in the first stage are as follows:
π i U P p i B = 1 2 t S ( 2 p i B p j B c ) 2 t S t B b S b B b S ( 2 t S t B b S b B ) ( 2 θ + b S ) 2 t S b S ( p i B p j B ) 4 ( 2 t S t B b S b B ) 2 = 0.
We obtain equilibrium prices, the number of agents, and platform profits as follows:
p i S = 2 θ + b S 4 ,
p i B = c + t B b S ( 2 θ + b S + 2 b B ) 4 t S ,
n i S = 2 θ + b S 4 t S , n i B = 1 2 ,
[46] π i U P = t B 2 + ( 2 θ + b S ) ( 2 θ b S ) 4 b S b B 16 t S .

Price discrimination

We consider the case in which platforms can price discriminate among advertisers. We identify the location x 10 S (resp. x 02 S ) that is indifferent between visiting platform 1 (resp. 2) and visiting both platforms, when platform 2 (resp. 1) sets zero price. The locations of the marginal advertisers are given by

x 10 S = t S θ b S ( 1 n 1 B ) t S
x 02 S = θ + b S ( 1 n 2 B ) t S
The number of advertisers joining platform 1 (resp. 2) is also n 1 S = x 02 S (resp. n 2 S = 1 x 10 S ). The advertisers located at x S [ x 10 S , x 02 S ] are indifferent between joining only one platform and both platforms. Platform 1’s profit is given by
π 1 P D = 0 ( t S θ b S ( 1 n 1 B ) ) / t S t S ( 1 2 x S ) + b S ( n 1 B n 2 B ) d x S + ( t S θ b S ( 1 n 1 B ) ) / t S ( θ + b S ( 1 n 2 B ) ) / t S θ + b S ( 1 n 2 B ) t S x S d x S + ( p 1 B c ) n 1 B .
Similarly, we have platform 2’s profit. Solving these equations, the first-order conditions in the first stage are
π i P D p i B = 1 2 t S ( 2 p i B p j B c ) 2 ( t S t B b S b B ) t S b S ( t S t B b S b B ) b S ( p i B p j B ) 2 ( t S t B b S b B ) 2 = 0.

We obtain equilibrium prices, the number of agents and platform profit as follows:

p 1 S = t S ( 1 2 x S ) , p 2 S = 0 , f o r x S 2 t S 2 θ b S 2 t S
p 1 S = θ + b S 2 t S x S , p 2 S = θ + b S 2 t S ( 1 x S ) , f o r x S 2 t S 2 θ b S 2 t S , 2 θ + b S 2 t S ,
p 1 S = 0 p 2 S = t S ( 2 x S 1 ) f o r x S 2 θ + b S 2 t S
p i B = c + t B b S b S b B t S , n i B = 1 2 ,
[47] π i P D = t S + t B 2 b S 2 θ 2 + ( 2 θ + b S ) 2 2 b S b B 4 t S .
Here, we investigate the extent to which θ affects market share and profits:
x 10 S θ = 1 t S < 0 ,
π i P D θ = 2 θ + b S t S t S > 0.
As we assume that x 10 S < x 02 S , t S < 2 θ + b S . Profits under price discrimination increase with θ .

Comparison

Comparing profits under uniform pricing and price discrimination, we find that platform profits are larger under price discrimination than under uniform pricing, if and only if

b B < 8 t S ( t S 2 θ 2 b S ) + ( 2 θ + b S ) ( 6 θ + 5 b S ) 4 b S .
We have difficulty directly interpreting this condition. We divide platform profits under each pricing scheme into profits from advertisers and profits from consumers. First, we compare profits from advertisers under uniform pricing with those under price discrimination,
π U P S π P D S = 3 ( 2 θ + b S ) 2 8 t S ( 2 θ + b S ) + 16 ( t S ) 2 16 t S < 0.
When advertisers multihome, each platform behaves like a monopolist. If platforms can price discriminate, platforms extract all surplus (excluding transportation costs) from each advertiser. Therefore, platform profits from advertisers increase under price discrimination. Next, we compare profits from consumers under the two pricing schemes:
π U P B π P D B = b S ( 4 t S 2 θ b S + 2 b B ) 8 t S .
If b B < ( 2 θ + b S 4 t S ) / 2 , profits from consumers are larger under price discrimination than under uniform pricing.

Acknowledgments

I would like to thank an editor and two anonymous referees, Hiroshi Aiura, Makoto Hanazono, Toshihiro Matsumura, and various seminar audiences for helpful comments. Financial support from AICA Kogyo is gratefully acknowledged.

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Published Online: 2015-1-15
Published in Print: 2015-4-1

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