Advertising in a two-sided duopoly when media and product preferences are correlated Marco A. Haan∗

Ryanne M. van Dalen†

March 1, 2013

Abstract We study a two-sided duopoly with media outlets and producers. Media outlets set advertising levels. Producers need advertisements to inform consumers about their product. Consumer preferences for media and products are correlated in the sense that listeners to station A are more likely to prefer product 1, and listeners to station B are more likely to prefer product 2. We find that both the amount and price of advertising decrease when correlation between preferences increases. Media stations are worse off with higher correlation, while consumers are better off. The effect on producer profits and welfare are ambiguous.

Keywords: Advertising; two-sided markets; media markets; targeting. JEL Codes: L82; M37

1

Introduction

Media markets are fundamentally different from most other industries. Rather than selling products directly to consumers, media companies often generate a substantial share of their revenues by selling advertising space. Lower advertising rates and larger audiences will increase demand for advertising, but ∗

IEEF, Faculty of Economics and Business, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands. E-mail: [email protected] † CPB Netherlands Bureau for Economic Policy Analysis, P.O. Box 80510, 2508 GM The Hague, The Netherlands. E-mail: [email protected]

1

more advertising will often decrease the size of the audience. This two-sided nature of such markets has spawned a considerable literature over the last decade, starting with the seminal paper of Anderson and Coate [2005]. Needless to say, models in this literature often become complicated very quickly: one needs to model demand for producers as well as demand for media companies. Therefore, as a shortcut, it is routinely assumed that media are homogeneous from the point of view of advertisers. That is, advertisers are only interested in the size of the audience that a media platform can deliver, but not in its composition. That is obviously a stretch. In the Netherlands, for example, Radio 1 and Radio 3 FM have a market share of 7.7% and 6.6%, respectively.1 However, the fraction of listeners older than 55 is 77% at Radio 1, and only 13% at Radio 3 FM. An advertiser that sells products aimed at senior citizens will thus be much more interested in advertising on Radio 1 than in advertising on Radio 3 FM - despite the latter’s higher market share. In this paper, we allow for the possibility that preferences of consumers in the media market are correlated with their preferences in the product market. More precisely, in a two-sided duopoly, we study how the amount and the price of advertising are affected if consumers that prefer to watch or listen to station A rather than station B, are also more likely to prefer product 1 over product 2. We also determine how profits of media companies and advertisers are affected. One way to think of our model is that it can address not only the question to what extent media companies should differentiate their product, but also in which dimension they should differentiate: along the same dimension as their advertisers are differentiated (which would imply a high correlation between media and product preferences) or rather in a dimension perpendicular to that in which advertisers are differentiated (which would imply a low correlation). In our model, two media outlets are located at the endpoints of a Hotelling line. There are two producers that want to advertise to inform consumers about their product. A sale is realized if an advertisement is received by 1

In 2005, according to Nielsen Adware Radio Reporter.

2

a consumer that is interested in the advertised product. Consumers are differentiated in two dimensions: with respect to their media preferences, and with respect to the extent to which they are likely to be interested in the products of either advertiser. We allow these dimensions to be correlated. Thus, a consumer that has a relative preference for media A over media B, is also more likely to be more interested in product 1 rather than in product 2, and vice-versa. We do not model correlation explicitly but rather impose weak conditions that any measure of correlation should satisfy. The fraction of the audience that hears an advertisement is increasing in the amount of advertising on a station, but at a decreasing rate. Given the audience composition of a media station, we can derive the demand function of a producer for advertisements at a station. Given these demand functions, media stations set advertising levels to maximize profits. We find that advertising levels and advertising prices decrease with the extent of correlation. With more correlation, advertising is more effective in the sense that producers need fewer advertisements to reach the same number of potential consumers. This reduces the demand for advertising messages. As advertisements become less profitable, media stations are more reluctant to carry them. Advertising prices decrease, as producers are less eager to use them. With lower prices and lower quantities, media stations obviously are worse off with higher correlation. For producers, profits increase for low correlation, but decrease with higher correlation. Our results thus suggest that media stations lose out when media preferences are more closely correlated with product preferences. In turn, this suggests that media stations prefer to attract a broad audience, in the sense that they prefer their stations not to be differentiated along the same dimension as advertisers. Comparative statics with respect to the extent of media differentiation and the extent of advertising aversion is similar to that in related literature. As consumers become more annoyed by hearing advertisements, media stations become more reluctant to carry them, resulting in lower advertising in equilibrium, and higher advertising prices. As media stations become less differentiated, they become more reluctant to carry advertising, as consumers switch channels more easily. Again, this implies less advertising in 3

equilibrium, and higher advertising prices. The welfare effects of having more correlation are ambiguous. Our model builds on two papers in particular. First, Anderson and Coate [2005] also consider two media stations located at the endpoints of a Hotelling line but assume a continuum of heterogenous firms. Second, Dukes [2004] studies a model with both a finite number of media outlets as well as a finite number of producers, both located on two separate Salop circles. Product market competition ensues along the lines of Grossman and Shapiro [1984]. Yet, Dukes [2004] assumes that preferences in the media and product dimensions are independent. Other literature examines a wide range of other related topics such as endogenizing the location of media stations [Peitz and Valetti, 2008, Gabszewicz et al., 2004], allowing for subscription fees Crampes et al. [2009], studying free-to-air versus pay TV [Choi, 2006, Peitz and Valetti, 2008], assuming price-setting behavior [Kind et al., 2009], or introducing a public broadcaster [Kind et al., 2007, Pan, 2009]. The effect of correlated preferences, however, has only received little attention. Exceptions include Pousset and Sonnac [2008] who analyze a model for magazines with either a general or a specialized readership. In their model, a monopolist firm advertises in both the general and the specialized magazine, but price discriminates between readerships. Second, Bergemann and Bonatti [2011] look at targeted advertising in a model with multiple product and advertising markets that are perfectly competitive. Also, they do not take the two-sided nature of media markets into account. The remainder of this paper is organized as follows. The next section introduces the model and describes the behavior and objectives of the consumers, media stations, and advertisers. In section 3, the general model is analyzed. Comparative statics are presented in section 4. Section 5 contains two parameterizations of the model in which the correlation parameter is explicitly modeled. The final section concludes.

4

2

Model

We have three types of agents: consumers, media stations, and producers. Two media stations, A and B, are located at the endpoints on a Hotelling line with station A located at 0 and station B located at 1. In standard Hotelling fashion, a unit mass of consumers differ in their preference for media stations. Two producers, 1 and 2, want to inform consumers about their product using informative advertising. Consumers who receive no advertising message are uninformed about any product and do not make a purchase. Consumer differ in their preference for products 1 and 2. This preference is captured by a parameter β ∈ [0, 1]. Consumers are thus distributed on a unit square; the horizontal dimension reflecting their media preferences, and the vertical dimension reflecting their product preferences. The timing of the model is as follows. In the first stage, stations set advertising levels CA and CB . Based on these, the indifferent consumer λ can be determined. Given λ, we can determine the individual advertising demand functions of both producers on both stations. Equilibrium then requires that advertising prices pA and pB are such that total demand for advertising equals the advertising levels set by the stations. We will describe the behavior of all market participants (consumers, producers, and media stations) in more detail below.

2.1

Consumers

Overview A unit mass of consumers is distributed on a unit square. The horizontal dimension x ∈ [0, 1] reflects the media preferences of the consumer. Consumers with a low x have a preference for station A, while those with high x have a preference for station B. The vertical dimension β ∈ [0, 1] reflects the product preferences of a consumer. With probability β a consumer has willingness to pay 1 for product 2 and willingness to pay 0 for product 1; with probability 1 − β it is the other way round. Consumers only buy a product if they are informed about it through advertising. Consumers only tune in to one media station or in other words single-home.

5

Media preferences The utility of a consumer located at (x, β) from listening to station A is U (x, β, A) = v − tx − γCA , with CA the number of commercials on station A, γ the degree of disutility from advertising, t the traditional Hotelling transportation costs, and v is the gross benefit from media consumption which is assumed to be large enough to assure that the market is covered. In our context, it is more natural to refer to the transportation costs t as the extent of media differentiation. We will do so in what follows. In similar fashion, the utility from listening to station B is equal to U (x, β, B) = v − t(1 − x) − γCB . Indifferent consumers are thus located at x = λ, with 1 γ(CB − CA ) (1) λ= + 2 2t Product preferences For simplicity we assume that consumers are either interested in product 1 or in product 2. A consumer that is interested in product 1 has a willingness to pay for product 1 that is normalized to 1, and a willingness to pay for product 2 that equals 0. Similarly, a consumer that is interested in product 2 has a willingness to pay for product 2 that is 1, and a willingness to pay for product 1 that equals 0. The parameter β reflects the probability that a particular consumer is interested in product 2. Thus if β = 0.6, there is a 60% probability that she is interested in product 2 and a 40% probability that she is interested in product 1. More generally, consumers with a high β are most likely to be interested in product 2, while those with low β are most likely to be interested in product 2. These assumptions imply that a producer will always find it profit-maximizing to set a price equal to 1. In turn, this implies that a consumer always has zero surplus from consumption. Hence she is indifferent between hearing advertisements from producer 1 or producer 2 as she does not receive any informational benefit from an advertisement. Hence, consumers will not let their choice of radio station be influenced by how likely it is to hear advertisements for a certain product on that station. Distribution of preferences We allow for correlation between media preferences and product preferences. The joint density function is given 6

by f (x, β; ρ), where ρ is some parameter that reflects correlation. We allow ρ to take any value between 0 and 1. We extend on this below. As R1 always, we denote the marginal distributions as fX (x; ρ) = 0 f (x, β; ρ)dβ R1 and fβ (β; ρ) = 0 f (x, β; ρ) dx. We make the following assumptions on this distribution function Assumption 1 Consumers are uniformly distributed in the media dimension: fX (x; ρ) = 1 ∀x, ρ ∈ [0, 1]. This assures that we have standard Hotelling competition in the media dimension, with consumers being uniformly distributed in that dimension. Naturally, we always need that the two media stations and producers are a priori identical. In other words, we need that producers and stations are anonymous, in the sense that we can freely exchange the labels ‘1’ and ‘A’ labels with the ‘2’ and ‘B’ labels. For the distribution of preferences, this implies that we need symmetry: the number of, say, consumers located at 0.2 that are interested in product 1, should equal the number of consumers located at 0.8 is interested in product 2. Hence Assumption 2 f (x, β; ρ) = f (1 − x, 1 − β; ρ) ∀x, β, ρ ∈ [0, 1]. Some notation Denote the (expected) number of consumers located at x that are interested in product i as Di . Hence 1

Z

1

D (x; ρ) ≡

(1 − β) f (x, β; ρ)dβ,

(2)

βf (x, β; ρ)dβ.

(3)

0 2

Z

D (x; ρ) ≡

1

0

Suppose that consumers located at x = λ are indifferent between listening station A and B. Any consumer with x < λ then strictly prefers station A, while any x > λ strictly prefers B. If we denote the expected number of consumers listening to station j that is interested in product i as Dji (λ; ρ),

7

we have Z

i DA (λ; ρ)

λ

= Z0 1

i DB (λ; ρ) =

Di (x; ρ) dx, Di (x; ρ) dx,

(4)

λ

From this, we immediately have, using Assumption 1: 1 DA

(λ; ρ) +

2 DA

1

Z

Z

λ

f (x, β; ρ) dxdβ = FX (λ; ρ) = λ,

(λ; ρ) = 0

0

This is obvious: the total fraction of consumers listening to station A is λ, and each of these is either interested in consuming product 1, or in consuming product 2. Similarly, 1 DB

(λ; ρ) +

2 DB

1

Z

1

Z

f (x, β; ρ) dxdβ = 1 − FX (λ; ρ) = 1 − λ.

(λ; ρ) = 0

λ

Moreover, note that 1 DA

(λ; ρ) +

1 DB

Z

1

Z

(λ; ρ) = 0

0

1

1 (1 − β)f (x, β; ρ) dxdβ = E(1 − β) = , 2

where the last equality is due to symmetry. Again, this is obvious: In total, a mass 1/2 of consumers is interested in product 1. In equilibrium, they either listen to station A or to station B, regardless of λ. A similar equality also holds for consumers interested in product 2. Correlation To deal with correlation, we could use an explicit probability distribution for f (x, β; ρ). We prefer not to do so. Rather, we take a more axiomatic approach and impose conditions that any reasonable measure of correlation should satisfy. As we will show, this is enough to derive comparative statics results with respect to the extent of correlation ρ. As an illustration, Section 5 sketches two parametrizations that satisfy the conditions we impose. Absent correlation, we have: Assumption 3 With ρ = 0, consumers are uniformly distributed on the unit 8

square. When ρ > 0, we assume without loss of generality that consumers listening to station A are more likely to be interested in product 1. A consumer that has a stronger preference for station A, should also be more likely to be interested in product 1. Hence: Assumption 4 A consumer with a stronger preference for station A is more likely to prefer product 1: ∂D1 (x; ρ) ≤ 0, ∂x with strict equality if and only if ρ = 0. Also, the stronger the extent of correlation (and hence the larger ρ), the stronger this effect will be. This implies Assumption 5 A given consumer with a preference for station A is more likely to prefer product 1 as the extent of correlation increases: ∂D1 (x; ρ) ≥ 0, ∀x ≤ 1/2. ∂ρ Symmetry implies that at λ = 1/2, the number of consumers interested in product 1 always equals the number interested in product 2, regardless of ρ. In other words: Lemma 1

∂D1 (1/2; ρ) = 0. ∂ρ

Proof. First note   Z 1       Z 1 1 1 1 1 2 1 D ;ρ = (1 − β) f , β; ρ dβ = βf , β; ρ dβ = D ;ρ . 2 2 2 2 0 0 The second equality follows from integration by substitution, the third equal

ity from symmetry (Assumption 2). As, by construction, D1 12 ; ρ +D2 21 ; ρ = 1, this implies D1 (1/2; ρ) = 1/2.

9

2.2

Producers

Producers are interested in the number of potential consumers that they reach by their advertisements. Producers can advertise on both channels and thus multi-home. Production costs are normalized to zero. Advertising is informative: only listeners that hear an advertisement can purchase a product. Denote by φij the fraction of listeners to station j that hears an advertisement from producer i. We assume that φij is increasing in the number of advertisements that producer i broadcasts on station j but at a decreasing rate: as the fraction of informed consumers increases, the probability that a new advertisement reaches an uninformed consumer becomes smaller. This is a standard approach in the literature on informative advertising that dates back to Butters (1977), see also. e.g. Grossman and Shapiro (1984) or Dukes (2004). We thus assume φij = Φ(Cji ), with Φ(0) = 0; Φ0 ≥ 0; Φ00 ≤ 0, and Φ(C) ∈ [0, 1] for all C ≥ 0. With Dji (λ; ρ) the number of consumers listening to station j that are interested in product i, total profits for producer i then equal πi =

X


Φ(Cji )Dji (λ; ρ) − pj Cji .

(5)

j∈{A,B}

2.3

Media stations

The two media stations simultaneously and noncooperatively set advertising levels to maximize profits. Broadcasting costs are normalized to zero. Media stations only receive advertising revenue: we assume that stations are not able to charge subscription fees to consumers. We assume that media stations cannot price discriminate between producers. Suppose that the price at station j is given by pj . Profits for station j then equal Πj = pj Cj = pj

X i∈{1,2}

10

Cji .

(6)

3

Solving the model

Overview In our model, first, media stations set advertising levels CA and CB . Then, consumers decide which station to listen to: a fraction λ will tune in to A, the remaining 1 − λ will tune in to B. Given λ, producers decide on their demand function for advertising at both stations. Demand and supply for advertising at a station then determines its equilibrium price. We solve using backward induction. Producer behavior First consider the behavior of producers. For a given λ, the amount of advertising Cji that producer i wants to put out on station j is then such that its profits as given in (5) are maximized, i.e. ∂Φ ∂π i = · Dji (λ; ρ) − pj = 0, i ∂Cj ∂Cji for j ∈ {A, B}, i ∈ {1, 2}. These can be rewritten as Cji

0−1

=Φ



pj i Dj (λ; ρ)

 .

Total demand for advertisements at media station j is then given by: Cjd (pj )

=Φ

0−1



pj 1 Dj (λ; ρ)

 +Φ

0−1



pj 2 Dj (λ; ρ)



For the sake of tractability, we assume that the advertising reach function has the following form: 1 Φ(C) = 2C(1 − C) 2 for C ∈ [0, 1] . Note that indeed Φ0 > 0 and Φ00 < 0. We then have ∂π i = (2 − 2Cji )Dji (λ; ρ) − pj = 0 ∂Cji

11

for j ∈ {A, B}, i ∈ {1, 2}. Total demand for advertisements at station j are then given by Cjd

 (pj ) = 1 −

pj 1 2Dj (λ; ρ)



 + 1−

pj 2 2Dj (λ; ρ)

 (7)

Consumer behavior An indifferent consumer obtains the same utility from listening to station A as she does from listening to station B. From (1), this is only determined by a consumer’s media preference x, and not by her product preference β. Any indifferent consumer thus has x = λ, with λ=

1 γ(CB − CA ) + . 2 2t

(8)

Station behavior Media station j sets advertising level Cj to maximize profits, given the demand function for advertising, the behavior of consumers, and the amount of advertising set by its competitor. First note that the equilibrium advertising price pj follows from equating Cj with demand for advertising at station j, given by (7). This yields pj = Hj (λ; ρ) (2 − Cj ) where Hj (λ; ρ) ≡

1 Dj1 (λ;ρ)

2 +

1

(9)

(10)

Dj2 (λ;ρ)

is the harmonic mean of Dj1 (λ; ρ) and Dj2 (λ; ρ). For the profits of media station j, we then have Πj = pj Cj = Hj (λ; ρ) (2 − Cj )Cj .

(11)

For station A, the FOC is ∂ΠA ∂HA ∂λ = HA (2 − 2CA ) + (2 − CA )CA = 0. ∂CA ∂λ ∂CA

12

(12)

Note from (8) that ∂λ/∂CA = γ/2t. If we define ηj (λ; ρ) ≡

∂Hj (λ; ρ) /∂λ , Hj (λ; ρ)

we can thus write ∂ΠA γ = (2 − 2CA ) − ηA (λ; ρ) (2 − CA )CA = 0. ∂CA 2t

(13)

γ ∂ΠB = (2 − 2CB ) + ηB (λ; ρ) (2 − CB )CB = 0. ∂CB 2t

(14)

Similarly

In equilibrium, we have CA = CB ≡ C ∗ , and the first order conditions collapse into2 γ (2 − 2C ∗ ) − η (ρ) (2 − C ∗ )C ∗ = 0. (15) 2t
where η(ρ) ≡ ηA 21 ; ρ . We can now solve for the equilibrium amount of advertising: s  2 2t 2t ∗ − 1+ C (ρ) = 1 + . (16) γη (ρ) γη (ρ) From (9), the equilibrium advertising price is then given by p∗ (ρ) = H (ρ) (2 − C ∗ (ρ)),

(17)


with H(ρ) ≡ HA 21 ; ρ , while from (11) equilibrium profits of each media station can be written as Π∗ (ρ) = p∗ (ρ) C ∗ (ρ) = H (ρ) (2 − C ∗ (ρ))C ∗ (ρ) The profit for producer A is 1 1 1 1 πA = 2CA1 (1 − CA1 )DA + 2CB1 (1 − CB1 )DB − pA C A − pB C B 2 2 2

Note that ηB

1 2; ρ




= −ηA

1 2; ρ




.

13

(18)

1 1 Using DA + DB = 21 , equilibrium profits thus equal

 1 ∗ π (ρ) = C (ρ) 1 − 2  1 ∗ = C (ρ) 1 − 2 ∗

4

 1 ∗ C (ρ) − Π∗ (ρ) 4  1 ∗ C (ρ) − H (ρ) (2 − C ∗ (ρ))C ∗ (ρ) 4

(19)

Comparative statics

4.1

Preliminaries

Before deriving the comparative statics in our model, it is useful to derive some preliminary results. Without loss of generality, we do so for station A. Of course, similar results apply for B. For simplicity and ease of exposition,
1 1 1 1 1 ; ρ . we denote the equilibrium value of DA (λ; ρ) as DA . Thus DA ≡ DA 2 We can then show the following; Lemma 2 In equilibrium we have 1 ≤ 12 ; ≤ DA 1 ∂DA 2. ∂λ 1 = 12 ;

1.

1 4

λ= 2

1 1 ); (1 − 2DA 3. H (ρ) = 2DA 1 1 4. ∂HA∂λ(λ;ρ) 1 = 1 − 4DA (1 − 2DA ) = 1 − 2H(ρ); λ= 2

5. η (ρ) = 6.

∂η(ρ) ∂ρ

=

1 1−2D 1 1−4DA ( A) 1 1−2D 1 2DA ( A)

∂ ∂ρ



;

1 1 1−2D 1 2DA ( A)

 =

1 −1 4DA

2

1 1−2D 1 2(DA ( A ))

1 ∂DA . ∂ρ

Proof. Proofs of all lemmas and theorems are in the Appendix. 1 Since by construction ∂DA /∂ρ > 0, we directly have from 1 and 6 in Lemma 2 that Lemma 3

∂η(ρ) ∂ρ

> 0 for ρ > 0. 14

4.2

Advertising

We can now establish Theorem 1 The equilibrium amount of advertising is decreasing in the intensity of advertising aversion γ, increasing in the extent of media differentiation t, and decreasing in the extent of correlation ρ. That is, dC ∗ dC ∗ dC ∗ < 0; > 0; < 0. dγ dt dρ When consumers have a stronger dislike for advertising, or when they perceive stations as less differentiated, they will switch stations more easily if the amount of advertising on one station increases. Hence, ceteris paribus, media stations are more reluctant to carry advertising. This is a standard result in this literature. These effects are similar to the comparative static effects on price in a standard Hotelling model, if we interpret the amount of advertising in our model as the implicit price listeners have to pay to tune in. The effect of an increase in correlation is less straightforward. Whereas a change in either t or γ will affect the willingness of media station to carry advertisements and hence the supply of advertising, a change in ρ will affect demand for advertising. If ρ increases, producers now need fewer advertisements to reach the same number of potential consumers. This can be seen as follows. First, suppose there is no correlation, so ρ = 0. The equilibrium then has stations A and B each attracting exactly half of the potential consumers of product 1. To reach, say, half of these potential consumers, producer 1 has to reach half of the listeners on both station A and station B. Now consider the other extreme, where potential consumers of product 1 only listen to station A, so ρ = 1. In that case, producer 1 only has to reach half of the listeners of A. At equal prices, it is obviously cheaper to do so. Higher correlation thus effectively boils down to an increase in the productivity of advertising. At equal advertising levels, the price producers are willing to pay is lower. In turn, this implies that media stations are less willing to carry

15

advertisements and the equilibrium number of advertisements decreases.3 More generally, it can be shown that for any reach function Φ that satisfies our assumptions the additional restriction that Φ000 ≥ 0, is sufficient to obtain this result.4 One family of functions that satisfies these constraints is given by Φ(C) = C a , for a < 1.5

4.3

Price

Comparative statics effects on the equilibrium advertising price are as follows: Theorem 2 The equilibrium advertising price is increasing in the intensity of advertising aversion γ, decreasing in the extent of media differentiation t, and decreasing in the extent of correlation ρ; dp∗ dp∗ dp∗ > 0; < 0; < 0. dγ dt dρ Note that the comparative statics on equilibrium advertising prices are exactly opposite to those for equilibrium advertising quantities. An increase in γ or a decrease in t again implies that media stations are more reluctant to carry advertisements, yielding higher equilibrium prices. An increase in ρ 3

More formally, for given prices p, demand for advertising from producer 1 on stations A and B respectively, is given by 1 2 − 2CA

=

1 2 − 2CB

=

Hence total demand is C1 = 1 −

p 1 DA p 1 DB

p p +1− 2DA 2DB

1 Without correlation DA = DB = 14 . With positive correlation, we can write DA = 41 + δ, 1 and DB = 4 − δ, with δ ∈ (0, 1/4) , where higher δ reflects more correlation. Total demand for advertising from producer 1 now equals   p p 2p

C1 = 1 − + 1 − = 2 1 − , 1 − 16δ 2 2 14 + δ 2 14 − δ

which indeed decreases in δ. 4 See appendix for proof. 5 Note that this is the reach function that Dukes (2004) uses.

16

again implies an increase in the effectiveness on advertising and hence a lower demand. Stations react to this by decreasing the amount of advertising that they carry, but not to the extent that the demand decrease is neutralized. Hence, equilibrium prices decrease. This is natural: as long as a market is not perfectly competitive a decrease in demand necessarily leads to a decrease in the profit maximizing price.

4.4

Media profits

Theorem 3 The equilibrium profits of media stations are decreasing in the intensity of advertising aversion γ, increasing in the extent of media differentiation t, and decreasing in the extent of correlation ρ. That is ∂Π∗ ∂Π∗ ∂Π∗ < 0; > 0; <0 ∂γ ∂t ∂ρ Both an increase in γ and a decrease in t imply that media stations have less market power, as consumers are less loyal and are more inclined to switch to the other station. This implies that equilibrium profits are lower. For an increase in ρ, we know that it leads to less advertising, and lower advertising prices. Hence media profits decrease. As demand for advertising shifts downwards with an increase in ρ, media stations make lower profits: as argued above, an increase in ρ implies that producers can reach potential consumers more easily and gain market power, which is bad news for media stations.

4.5

Producer profits

Theorem 4 The equilibrium profits of producers are decreasing in the intensity of advertising aversion γ, increasing in the extent of media differentiation t, and are increasing in the extent of correlation ρ for low enough ρ, but decreasing for high ρ. That is ∂π ∗ ∂π ∗ ∂π ∗ ∂π ∗ < 0; > 0; > 0 iff ρ < ρ∗ ; < 0 iff ρ > ρ∗ ∂γ ∂t ∂ρ ∂ρ

17

for some ρ∗ ∈ (0, 1) . Again, an increase in γ and a decrease in t imply that media stations have less market power vis-a-vis consumers. This implies that if they are still willing to broadcast advertisements, they are only willing to do so at a higher price. This hurts producers as it is now more costly to reach potential consumers. The effect of an increase in correlation ρ is more involved. As argued, an increase in ρ implies that advertising is more effective to producers, leading to lower costs to reach potential consumers. Yet, an increase in ρ also implies that media stations become more reluctant to provide advertising slots: C ∗ decreases. We thus have an efficiency effect (with the same number of advertisements, more consumers are reached), a price effect (advertisements are cheaper), and a quantity effect (a lower number of advertisements is supplied, hence at equal productivity, fewer consumers are reached). The first two effects increase producer profits, the last effect decreases it. For low correlation, the first two effects dominate; for high correlation, the last one does.

4.6

Industry profits

Looking at total industry profits, we have from (18) and (19), Π∗tot

 1 ∗ 1 ∗ ∗ C (1 − C ) − π − 2π ∗ =2 2 4 1 = C ∗ (1 − C ∗ ) 4 

and industry profits thus only depend on advertising. The comparative statics are thus similar as those given in Theorem 1. Theorem 5 The equilibrium total profits are decreasing in the intensity of advertising aversion γ, increasing in the extent of media differentiation t, and decreasing in the extent of correlation ρ. That is dΠ∗tot dΠ∗tot dΠ∗tot < 0; > 0; < 0. dγ dt dρ 18

The effects of a change in γ and t are obvious, as these go in the same direction for media stations and producers. As we saw above, as ρ increases, profits of media stations will decrease. The effect on producers is ambiguous. Yet the negative effect on station profits always outweighs the possibly positive effect on producers.

4.7

Consumer surplus

Consumers do not receive net utility from product purchases. As prices of products equal their willingness to pay, consumers do not obtain any surplus from purchasing the product. Therefore, we only have to consider the utility from media consumption. Consumers are indifferent between hearing an advertisement from producer 1 or producer 2 regardless of their product preferences. The net utility for a consumer located at x from listening to media station A is ν − tx − γC A . In equilibrium the first two terms are not affected by the level of advertising. For ease of exposition, we can thus write CS = −γC ∗ . Theorem 6 Equilibrium consumer surplus is increasing in the intensity of advertising aversion γ, decreasing in the extent of media differentiation t, and increasing in the extent of correlation ρ. That is dCS ∗ dCS ∗ dCS ∗ > 0; < 0; > 0. dγ dt dρ Proof. Again C ∗ ∈ [0, 1) and for any variable of interest v, we have ∗ dCS ∗ = −γ dC . The results then follow directly from Theorem 1. dv dt Note that these effects are directly opposite to those on advertising. That is obvious, as an increase in advertising unambiguously leads to a decrease in consumer welfare.

19

4.8

Total welfare

Total welfare consists of the profits of media stations, the profits of producers, and consumer welfare. That is   1 ∗ ∗ ∗ (20) W =C 1−γ− C . 4 Theorem 7 Equilibrium total welfare is decreasing in the intensity of advertising aversion γ for low γ, and increasing for high γ; increasing in the extent of media differentiation t for low γ, and decreasing for high γ; decreasing in the extent of correlation ρ for low γ and increasing for high γ. That is, dW ∗ dW ∗ < 0 iff γ < γ ∗ ; > 0 iff γ > γ ∗ dγ dγ ∗ dW dW ∗ > 0 iff γ < γ ∗∗ ; < 0 iff γ > γ ∗∗ dt dt ∗ dW ∗ ∗∗ dW < 0 iff γ < γ ; > 0 iff γ > γ ∗∗ dρ dγ Total welfare thus crucially depends on the degree of advertising aversion. This can be explained as follows. Industry profits and consumer surplus move in opposite directions for each of the parameters considered. So, only for high levels of advertising aversion given by γ ∗ the effects of consumer surplus will be stronger than the effects of industry profits. It is also interesting to look at the socially optimal level of advertising. A social planner would maximize total welfare and taking the FOC of (20) with respect to C, we obtain C opt = 2 (1 − γ) .

(21)

Note that this only makes sense for γ < 1; in the case that γ > 1, the welfare optimum would be a corner solution at C ∗ = 0. In that case any advertising is so much disliked by consumers that that effect is outweighed by any positive effect on producers and media stations. Also note that the socially optimal advertising level does not depend on either ρ or t.

20

Theorem 8 From a welfare point of view, we have the following: 1. For γ < 1/2, there is always too little advertising. 2. For γ ∈ (1/2, 1) , there exists a level of correlation ρˆ ∈ (0, 1) for which the market provides a level of advertising that is socially optimal. For any ρ < ρˆ there is too much advertising, whereas for any ρ > ρˆ, there is too little advertising. 3. For γ ≥ 1, the socially optimal level of advertising is zero, hence the market always provides too much advertising. Thus, for small γ there is always too little advertising, whereas for large γ there is always too much. For intermediate values, there would be too much advertising when preferences are independent, but we get closer to the social optimum as the extent of correlation ρ increases. When correlation is very high, however, we would switch to a case of underadvertising. The intuition is as follows. Advertising levels are set by media stations. When doing so, they fail to take into account the positive externality that advertising has on producers (more advertising means that they can reach more consumers) and the negative externality it has on consumers (more advertising lowers their utility of listening). When the advertising aversion parameter γ is very low, the positive externality outweighs the negative one, hence there is too little advertising. When γ is high, the opposite is true, and there is too much advertising. In the case of intermediate γ, note that producers can target consumers more effectively if ρ is high, which also implies that the market demand for advertising is more quickly saturated and more advertising yields fewer positive externalities. Hence in that case there is also too much advertising, while there is too little for low ρ.

4.9

Overview of comparative statics

The comparative statics are summarized in Table 1. This table shows that are some ambiguous results which depend on the level of correlation and/or advertising aversion. The comparative statics for the degree of advertising 21

and the degree of media differentiation are comparable to the standard results in the media literature. The effects of correlation are somewhat more involved and sometimes counterintuitive. Correlation is not preferred by the two types of firms, that is media stations and producers, at least for mildly correlated consumer preferences. Consumer surplus is increasing in correlation as this decreases their exposure to advertising messages. For highly concentrated audiences, producer profits increase. However this increase is never so substantial that it makes up for the loss in media profits as total industry profits are a decreasing function of correlation. The welfare effects depend crucially on the level of advertising aversion with underadvertising for low advertising aversion and overadvertising for high advertising aversion. Moreover, there exists a socially optimal level of correlation for intermediate values of advertising aversion. Table 1: Comparative γ t C∗ − + p∗ + − Π∗ − + ∗ π − + ∗ Πtot − + CS ∗ + − W ∗ +/− +/−

5

Statics ρ − − − +/− − + +/−

Two parametrizations

In the analysis above, we derived the comparative statics in our model, provided that some weak conditions on correlation are satisfied. In this section we provide two possible specifications, and show that for those cases, the restrictions we impose on correlation are indeed satisfied and hence our analysis applies. We thus provide two specific models for which our analysis holds. In both cases it is impossible to provide simple analytic closed-form solutions for the equilibrium variables of interest. Hence, we restrict attention to

22

merely showing that the required assumptions are indeed satisfied in these specifications.

5.1

Parametrization 1

Note that from our definition in section 2.1 that higher correlation is interpreted as consumers that favor product 1 to be more likely to listen to station A, and consumers that favor product 2 to be more likely to listen to station B. If we think of consumers being located in (x, β)-space, one natural way to implement this is to think of higher correlation as consumers being more concentrated around the diagonal that runs from (0, 0) to (1, 1). For simplicity, we therefore assume that consumers are uniformly distributed on a subset of the unit square that runs parallel to the main diagonal. More precisely, suppose that they are not located in the triangle at the top left of the square (where consumers with a strong preference for station A and a high probability of being interested in product 2 would be located) and not located in a triangle at the bottom right of the square (where consumers with a strong preference for station B and a high probability of being interested in product 1 would be located), see Figure 1. The extent of correlation ρ can now be interpreted as the length of the legs of the right-angled triangles. We thus assume that consumers are uniformly distributed on the area enclosed by the lines β = 1 − ρ + x and β = 1 − ρ (see Figure 1). We will distinguish between two cases which are low correlation, 0 < ρ < 1 , and high correlation, 21 < ρ < 1. 2 5.1.1

Low correlation

Suppose that ρ ∈ (0, 12 ). The indifferent consumer is given by (1). Note from Figure 1 that for x ≤ ρ, a mass 1 of consumers is uniformly distributed on a line of length 1−ρ+x. This implies that for x on the interval [0, ρ] the density 1 of consumers is given by f (x, β; ρ) = 1−ρ+x . Similarly, for x on [1 − ρ, 1] we 1 have f (x, β) = 2−ρ−x , while for x in [ρ, 1 − ρ] the density is trivially given by f (x, β, ρ) = 1. Using (2) – (4), for λ close to 1/2, total potential demand for

23

Figure 1: Distribution of preferences on the unit square

product j from listeners to station i is then given by 1 DA

Z

ρ

Z

ρ

Z

1−ρ+x

= 0

Z

0

1−β dβdx + 1−ρ+x

1−ρ+x

Z

λ

Z

λ

Z

ρ

Z

λ 2 DB =

Z

0 1−ρ

Z

1

Z

1

Z

1−ρ 1

βdβdx + λ

0

1−ρ

x−1+ρ

1 1 (1 − β)dβdx = λ + ρ2 2 4

1

1 1 βdβdx = λ − ρ2 2 4

0

β dβdx + 1−ρ+x 0 0 ρ Z 1−ρ Z 1 Z 1 Z 1 1 (1 − β)dβdx + DB = 2 DA =

1

0

x−1+ρ

1−β 1 1 dβdx = (1 − λ) − ρ2 2−ρ−x 2 4

β 1 1 dβdx = (1 − λ) + ρ2 2−ρ−x 2 4

To show that this parametrization indeed satisfies the assumptions of our model, we have to show that Assumptions 1–5 are satisfied. First, we already established Assumption 1. Second, Assumption 2 is trivially satisfied

24

for x ∈ [ρ, 1 − ρ] . For x ∈ [0, ρ], we have 1 − x ∈ [ρ, 1], hence f (1 − x, 1 − β; ρ) =

1 1 = = f (x, β; ρ). 2 − ρ − (1 − x) 1−ρ+x

For x ∈ [ρ, 1], we have 1 − x ∈ [0, ρ] , hence f (1 − x, 1 − β; ρ) =

1 1 = = f (x, β; ρ), 1 − ρ + (1 − x) 2−ρ−x

establishing Assumption 2. Third, it is immediate that we get the uniform for ρ = 0, establishing Assumption 3. Fourth, note that  R 1−ρ+x 1−β dβ = 12 (1 + ρ − x) if x ≤ ρ   R0 1−ρ+x 1 D1 (x; ρ) = (1 − β) dβ = 12 if ρ < x ≤ 1 − ρ 0   R1 1−β 1 dβ = 1 − 2 (ρ + x) if x > 1 − ρ x−1+ρ 2−ρ−x so we immediately have ∂D1 (x; ρ) ≤ 0; ∂x 1 ∂D1 (x; ρ) ≥ 0 ∀x ≤ , ∂ρ 2 confirming Assumptions 4 and 5. 5.1.2

High correlation

Now suppose that ρ ∈ ( 12 , 1). For x ∈ [0, 1 − ρ] a mass 1 of consumers is now uniformly distributed on a line of length 1 − ρ + x. Thus for x ∈ [0, 1 − ρ] we 1 have f (x, β; ρ) = 1−ρ+x . For x ∈ [1 − ρ, ρ] the density is given by f (x, β; ρ) = 1 1 while for x ∈ [ρ, 1] we have f (x, β; ρ) = 2−ρ−x . Hence, for λ close to 2(1−ρ)

25

1/2, we have 1 DA

Z

1−ρ+x

1−ρ

Z

1−ρ

Z

ρ

Z

0 1−ρ+x

ρ

Z

= 0

0 2 DA =

Z 0

1 DB =

Z λ

2 = DB

Z λ

1−β dβdx + 1−ρ+x

1−ρ+x

−1+ρ+x 1−ρ+x −1+ρ+x

λ

Z

1−ρ λ

Z

Z Z

1−ρ+x

−1+ρ+x 1−ρ+x

1 1 1−β dβdx = λ2 + (1 − ρ)2 2(1 − ρ) 2 4

β β 1 1 dβdx + dβdx = λ − λ2 − (1 − ρ)2 1−ρ+x 2 4 1−ρ −1+ρ+x 2(1 − ρ) Z 1Z 1 1−β 1−β 1 1 dβdx + dβdx = (1 − λ2 ) − (1 − ρ)2 2(1 − ρ) 2 4 ρ −1+ρ+x 2 − ρ − x Z 1Z 1 β β 1 1 dβdx + dβdx = (1 − λ)2 + (1 − ρ)2 2(1 − ρ) 2 4 ρ −1+ρ+x 2 − ρ − x

Again, Assumptions 1 and 3 are satisfied by construction. The proof that Assumption 2 is satisfied goes along the same lines as in the case of low correlation. Third, note that  R 1−ρ+x 1−β dβ = 12 (1 + ρ − x) if x ≤ 1 − ρ  1−ρ+x  R0 1−ρ+x 1−β D1 (x; ρ) = dβ = 1 − x if ρ < x < 1 − ρ −1+ρ+x 2(1−ρ)   R1 1−β dβ = 1 − 21 (ρ + x) if x > 1 − ρ −1+ρ+x 2−ρ−x hence ∂D1 (x; ρ) ≤ 0; ∂x 1 ∂D1 (x; ρ) ≥ 0 ∀x ≤ , ∂ρ 2 confirming Assumptions 4 and 5.

5.2

Parametrization 2

This section provides another example of how our measure of correlation ρ can be parameterized. Suppose that the distribution of consumers is as follows f (x, β) = 1 + ρ − 2ρx − (2ρ − 4ρx)β which implies that there as many consumers that are interested in product 1 as in product 2 when there is no correlation. The effect of introducing cor26

relation in this parametrization can be shown by analyzing this distribution at x = 0 and at x = 1. At x = 0 the distribution of consumers is as follows: f (0, β) = 1 + ρ − 2ρβ which is a straight line when ρ = 0 and runs from 2 to 0 when ρ = 1. Hence, the higher the extent of correlation, the more this distribution is skewed towards β = 0. This implies that consumers at x = 0 are equally divided in terms of product preferences when there is no correlation. On the other hand, there are more consumers interested in product 1 when we introduce correlation as more consumers are then located towards β = 0. The total number of consumers at x = 0 is equal to 1

Z

((1 + ρ) − 2ρβ) dβ = 1 0

A similar analysis for the distribution of consumers at x = 1 gives: f (1, β) = (1 − ρ) + 2ρβ which is also a straight line when ρ = 0 but now runs from 0 to 2 when ρ = 1. Hence, the higher the extent of correlation, the more this distribution is skewed towards β = 1. The total number of consumers at x = 1 then equals Z 1

((1 − ρ) + 2ρβ) dβ = 1 0

This distribution has the most extreme outcomes at the endpoints of the Hotelling line and it rotates in between. The total potential demand function

27

can be derived in the following way: Z

λ

Z

1

1 (1 + ρ − 2ρx − (2ρ − 4ρx)β)(1 − β)dβdx = λ(ρ(1 − λ) + 3) 6 0 0 Z λZ 1 1 2 (1 + ρ − 2ρx − (2ρ − 4ρx)β)βdβdx = λ(ρ(λ − 1) + 3) = DA 6 Z0 1 Z 0 1 1 1 (1 + ρ − 2ρx − (2ρ − 4ρx)β)(1 − β)dβdx = (λ − 1)(λρ − 3) DB = 6 Zλ 1 Z0 1 1 2 (1 + ρ − 2ρx − (2ρ − 4ρx)β)βdβdx = (1 − λ)(λρ − 3) DB = 6 0 λ 1 DA

=

Again, we have to establish that Assumptions 1–5 are satisfied. First note Z

1

(1 + ρ − 2ρx − (2ρ − 4ρx)β) dβ = 1,

fX (x) = 0

confirming Assumption 1. Second, f (1 − x, 1 − β) = 1 + ρ − 2ρ (1 − x) − (2ρ − 4ρ (1 − x)) (1 − β) = 1 + ρ − 2ρx − (2ρ − 4ρx)β = f (x, β) , which confirms Assumption 2. Third, it is immediate that f (x, β)=1 for ρ = 0, establishing Assumption 3. Fourth, note that 1

Z

1

(1 − β) (1 + ρ − 2ρx − (2ρ − 4ρx)β) dβ =

D (x; ρ) = 0

so we immediately have ∂D1 (x; ρ) 1 = − ρ < 0; ∂x 3 1 ∂D (x; ρ) 1 1 = (1 − 2x) > 0 ∀x ≤ . ∂ρ 6 2 confirming Assumptions 4 and 5.

28

1 1 + ρ (1 − 2x) 2 6

6

Conclusion

This paper has examined the implications of consumer preferences for media and products that are not independent. The model provides a very general way to introduce correlation in preferences by assuming that a consumer’s location in media space is related to her location in product space. The effects of advertising aversion and the degree of media differentiation are straightforward if we interpret advertising as an implicit price that consumers pay for tuning in to a media station. However, the effect of the degree of correlation is less obvious. We find that advertising levels and advertising prices decrease with the extent of correlation. With more correlation, advertising is more effective in the sense that producers need fewer advertisements to reach the same number of potential consumers. This reduces the demand for advertising messages. As advertisements become less profitable, media stations are more reluctant to carry them. Advertising prices decrease, as producers are less eager to use them. With lower prices and lower quantities, media stations obviously are worse off with higher correlation. The welfare results are ambiguous. The level of advertising that is optimal from a welfare point of view can be either higher or lower than the level provided by the market. Yet, the socially optimal level is independent of the extent to which preferences are correlated. As more correlation leads to fewer advertising on a decentralized market, more correlation is good from a welfare perspective if the original situation had too much advertising, but bad if it had too little. Consumers are unambiguously better off from more correlation, as it decreases their disutility from advertising. For producers, an increase in correlation implies that advertising is more effective, leading to lower costs to reach potential consumers. Yet, it also implies that media stations become more reluctant to provide advertising slots. We thus have an efficiency effect (with the same number of advertisements, more consumers are reached), a price effect (advertisements are cheaper), and a quantity effect (a lower number of advertisements is supplied, hence at equal productivity, fewer consumers are reached). The first two effects 29

increase producer profits, the last effect decreases it. For low correlation, the first two effects dominate; for high correlation, the last one does. From a technical viewpoint, our model introduces a way to deal with the correlation of consumer preference between two types of product. This approach may be useful, not only in the context of media advertising, but also when studying other models of two-sided markets where platforms compete and consumers also have preferences for firms on the other side of the market. A possible drawback of our current model is that consumers receive no informational benefit from advertising. This is driven by the assumption that producers are always able to capture the entire consumer surplus. Relaxing this will not qualitatively change our results as long as the informational benefit is smaller than the disutility consumers experience from advertising. However, if the current model would be enriched by incorporating competition between (multiple) producers the outcomes may differ. Then a purchase would not only depend on the probability of a consumer’s interest in the advertised product but also in the price of the product. Producers then need to take into account the price of competing products because purchasing decision of consumers that receive multiple advertisements will also depend on prices of the products of which they have become informed. Another venue is to allow media stations to price discriminate between producers. In our model, prices are equalized across media stations and producers such that the market for advertising messages clears. If media stations can price discriminate they can set different prices depending on the concentration of different types of consumers that are present in their audience. We leave these extensions for further research.

References Simon Anderson and Stephen Coate. Market provision of broadcasting: A welfare analysis. Review of Economic Studies, 74(4):947–972, 2005. Dirk Bergemann and Alessandro Bonatti. Targeting in advertising markets: Implications for offline versus online media. RAND Journal of Economics, 42(3):417–443, 2011.

30

Gerard Butters. Equilibrium distribution of prices and advertising. Review of Economic Studies, 44:465–492, 1977. Jay Pil Choi. Broadcast competition and advertising with free entry: Subscription versus free-to-air. Information Economics and Policy, 18(2):181– 196, 2006. Claude Crampes, Carole Haritchabalet, and Bruno Jullien. Advertising, competition and entry in media industries. The Journal of Industrial Economics, 57(1):7–31, 2009. Anthony Dukes. The advertising market in product oligopoly. Journal of Industrial Economics, 52(3):327–348, 2004. Jean J. Gabszewicz, Didier Laussel, and Nathalie Sonnac. Programming and advertising competition in the broadcasting industry. Journal of Economics & Management Strategy, 13(4):657–669, 2004. Gene M. Grossman and Carl Shapiro. Informative advertising with differentiated products. Review of Economic Studies, 51(1):63–81, 1984. Hans Jarle Kind, Tore Nilssen, and Lars Sørgard. Competition for viewers and advertisers in a tv oligopoly. Journal of Media Economics, 20(3): 211–233, 2007. Hans Jarle Kind, Tore Nilssen, and Lars Sørgard. Business models for media firms: Does competition matter for how they raise revenue? Marketing Science, 28(6):1112–1128, 2009. Hui Pan. Content and advertising: Tv media competition in a mixed duopoly market. Journal of Economic Asymmetries, 6(2):137–154, 2009. Unpublished manuscript. Martin Peitz and Tommaso M. Valetti. Content and advertising in the media: Pay-tv versus free-to-air. International Journal of Industrial Organization, 26(4):949–965, 2008. Joanna Pousset and Nathalie Sonnac. New determinants of advertising rates and pricing strategy in the product market: An application to the press industry. 2008.

31

Appendix: Proofs Proof of Lemma 2 1 1 1. From Assumption 3, with ρ = 0 we have DA = 1/4. Hence DA > 1/4 if ρ > 0. The total mass of consumers that are interested in product in the market equals 1/2. In the most extreme case, all these consumers 1 listen to station A and DA = 1/2. Rλ 1 1 1 ∂DA ∂DA ∂ 1 2. Note that ∂λ = ∂λ 0 D (x; ρ) dx = D (λ; ρ) , hence ∂λ 1 = λ= 2
D1 12 ; ρ = 21 .

3. Note that HA (λ; ρ) =

1 1 (λ;ρ) DA


2 1 2 1 = DA (λ; ρ) λ − DA (λ; ρ) . 1 λ + λ−D1 (λ;ρ)

(22)

A

Evaluating this λ = 1/2 establishes the result. 4. From (22), we have   1 1

2 1 2 1 ∂DA 2 ∂HA (λ; ρ) 1 ∂DA 1 = DA 1 − − 2 DA + λ − DA λ − DA . ∂λ λ ∂λ λ ∂λ λ Evaluating this in λ = 1/2 and using result 2 establishes the result. ∂HA (λ;ρ)/∂λ|λ= 1

2 . The result then follows 5. Note that, by definition η (ρ) ≡ H(ρ) directly from 3 and 4.   ∂η(ρ) ∂ 1 , from which the result follows di6. Note that ∂ρ = ∂ρ 2D1 1−2D1 A( A) rectly.

Proof of Theorem 1 √ 2t It is convenient to write C ∗ = 1 + Ω − 1 + Ω2 with Ω ≡ γη(ρ) . It is easy ∗ ∗ to see that C = 0 if Ω = 0, that C is strictly increasing in Ω, and that limΩ→∞ C ∗ = 1 which implies the first two results. Next, we look a the comparative static effect of a change in the amount of correlation ρ. Using (15) and the implicit function theorem, we have dC ∗ =− dρ 2+

γ(2−C ∗ (ρ))C ∗ (ρ) 2t γ η 2t

∂η (ρ) (ρ) (2 − 2C ∗ (ρ)) ∂ρ 32

(23)

The first ratio is strictly positive, whereas from Lemma 3, ∂η(ρ)/∂ρ > 0. This establishes the third result. Proof that C is decreasing in ρ for any Φ000 ≥ 0 This can be seen as follows. Total demand for advertising equals     p p 1 C =φ +φ , DA DB where we have written φ ≡ Φ0−1 . Again denoting DA = 14 +δ and DB = 14 −δ, this implies     p p 1 C =φ 1 +φ 1 . +δ −δ 4 4 Compare this to a case of no correlation. Total demand for advertising is lower if       p p p +φ 1 < 2φ 1 , φ 1 +δ −δ 4 4 4
p but this exactly requires that φ D is strictly concave in D. We now show that that is indeed the case. First note that Φ0 is strictly decreasing (Φ00 < 0), which implies that Φ0−1 is strictly decreasing as well, so φ0 < 0. With Φ0 convex (Φ000 ≥ 0) and strictly decreasing, we necessarily have that φ is also convex, so φ00 ≥ 0. Next, note ∂ 2φ = −φ00 p2 D−4 + 2φ0 pD−3 . 2 ∂D With φ00 ≥ 0 and φ0 < 0, this expression is strictly negative, which indeed implies that φ is strictly concave in D. More generally, with δ 00 < δ 0 < 1/4, we need         p p p p φ 1 +φ 1 <φ 1 +φ 1 , + δ0 − δ0 + δ 00 − δ 00 4 4 4 4 which is immediate given that φ is decreasing and strictly concave.

33

Proof of Theorem 2 First note that dp∗ ∂C ∗ = −H (ρ) ; dγ ∂γ dp∗ ∂C ∗ = −H (ρ) . dt ∂t ∗

∗

From Theorem 1 we have dC < 0 and dC > 0. From results 1 and 3 in dγ dt Lemma 2 H(ρ) > 0, which establishes the first two results. For the third result, note that dp∗ dH dC ∗ ∗ = (2 − C (ρ)) − H (ρ) . dρ dρ dρ Dropping arguments for ease of exposition we have, substituting for dC ∗ : dρ

dH dρ

dp∗ ∂D1 1 = −2(4DA − 1)(2 − C ∗ ) A dρ ∂ρ γ(2−C ∗ )C ∗ 1 1
4DA −1 ∂DA 1 1 2t + 2DA 1 − 2DA . γ 1 1 2 ∂ρ η (2 − 2C ∗ ) 2 (DA 2 + 2t (1 − 2DA )) 1 /∂ρ > 0, this has the same sign as With D1A ≥ 1/4, C ∗ < 1 and ∂DA

−2 + 2

γC ∗ 2t

2+

γ η 2t

(2 −

1 2C ∗ ) 2DA

1 . 1 (1 − 2DA )

For this to be negative we require 1>

γC ∗ 2t

2+

γ η 2t

(2 −

1 2C ∗ ) 2DA

1 1 (1 − 2DA )

or

1 1 1 1 1 − 2DA > γC ∗ 8tDA 1 − 2DA + 2γη (2 − 2C ∗ ) DA Using results 3 and 5 in Lemma 2 and rearranging gives 8tH + 2γ(1 − 2H)(1 − C ∗ ) − γC ∗ > 0

34

and

Plugging in C ∗ from (16) gives 

s



2



2tH 2tH  + 1+ γ (1 − 2H) γ (1 − 2H)   s  2 2tH 2tH  > 0. − γ 1 + − 1+ γ (1 − 2H) γ (1 − 2H)

4tH + 2γ(1 − 2H) −

Rearranging yields   s s   2 2 2tH 2tH 2tH >0 −1 + − 1+ 2(1−2H) 1 + γ (1 − 2H) γ (1 − 2H) γ (1 − 2H) or

2tH 1 + γ(1−2H) 2(1 − 2H) > r  2 − 1. 2tH 1 + γ(1−2H)

(24)

√ Note that the function f (Ω) = (1 + Ω) / 1 + Ω2 has a maximum at Ω = 1, √ where f (Ω) = 2 < 2. This implies that the RHS of (24) is strictly smaller than 1. Using results 1 and 3 in Lemma 2, it is easy to see that H ∈ [0, 1/4], which implies that the LHS of (24) is strictly bigger than 1. Taken together, these observations imply that (24) is always satisfied, which establishes the final result. Proof of Theorem 3 First note that ∂C ∗ ∂Π∗ = H(2 − 2C ∗ ) ∂γ ∂γ ∗ ∂C ∗ ∂Π = H(2 − 2C ∗ ) ∂t ∂t ∗ ∂Π∗ ∂C ∂H = H(2 − 2C ∗ ) + (2 − C ∗ )C ∗ ∂ρ ∂ρ ∂ρ ∗

∗

We have H > 0 and (2 − 2C ∗ ) > 0 as C ∗ ∈ [0, 1). The signs of ∂C , ∂C , ∂γ ∂t ∂C ∗ ∂H and ∂ρ are established in Theorem 1. The sign of ∂ρ is negative. It can be

35

shown that in equilibrium 1 ∂H ∂H ∂DA = 1 ∂ρ ∂DA ∂ρ

=2 1− as

1 4

1 ≤ DA ≤

1 2

1 4DA

and by construction




1 ∂DA ∂ρ



1 ∂DA ∂ρ

 <0

> 0. This establishes the result.

Proof of Theorem 4 First note   ∂π ∗ 1 ∂C ∗ ∗ ∗ = (2 − C (ρ)) − H (ρ) (2 − 2C (ρ)) ; ∂γ 4 ∂γ   ∂π ∗ 1 ∂C ∗ ∗ ∗ = (2 − C (ρ)) − H (ρ) (2 − 2C (ρ)) ; ∂t 4 ∂t   1 ∂C ∗ ∂H ∂π ∗ ∗ ∗ = (2 − C (ρ)) − H (ρ) (2 − 2C (ρ)) − (2 − C ∗ (ρ))C ∗ (ρ) . ∂ρ 4 ∂ρ ∂ρ
1 ∈ 14 , 12 , hence H (ρ) ≤ 14 . Moreover, 0 ≤ From Lemma 2 we have DA C ∗ (ρ) ≤ 1 so 2 − 2C ∗ (ρ) < 2 − C ∗ (ρ). Therefore the first term of the derivatives is positive. This established the first two results. For the third result, dropping arguments for ease of exposition, and substituting from (23) and Lemma 2 we have !   γ(2−C ∗ )C ∗ 1 1 ∂DA 1 − 1 4D ∂π ∗ A 2t =− (2 − C ∗ ) − H(2 − 2C ∗ ) γ 1 2 1 ∂ρ 4 2 + 2t η (2 − 2C ∗ ) 2 (DA ∂ρ )) (1 − 2DA 1 + 2(2 − C ∗ )C ∗ (4DA − 1)

1 ∂DA ∂ρ

1 With D1A ≥ 1/4, C ∗ < 1 and ∂DA /∂ρ > 0, this has the same sign as

 −

 1 ∗ ∗ (2 − C ) − H(2 − 2C ) 4

γ 2t

2+

γ η 2t

1

1 1 2 (2 − 2C ∗ ) 2 (DA (1 − 2DA ))

! +2

or  −γ



2 1 ∗ ∗ 1 1 (2 − C ) − H(2 − 2C ) + 4 (4t + γη (2 − 2C ∗ )) DA 1 − 2DA 4

36

Using the expressions for η and H from Lemma 2, equals       2 1 1 − 2H 1 ∗ ∗ ∗ −γ (2 − C ) − H(2 − 2C ) + 4 4t + γ (2 − 2C ) H 4 H 2 which simplifies to 1 4tH 2 + 4γ (1 − H) (1 − C ∗ ) H − (2 − C ∗ )γ. 4 Plugging in C ∗ from (16) and rearranging yields   s 2  2tH 2tH  4tH 2 + 4 (1 − H) H − + γ2 + (1 − 2H) (1 − 2H)   s  2 1 2tH 2tH , − γ − + γ2 + 4 (1 − 2H) (1 − 2H) which simplifies to   s 2 1 Ht (1 − 8H) 1 2tH 2 − γ+ + 4 (1 − H) H − γ + 4 2 (1 − 2H) 4 (1 − 2H)

(25)

For H = 0, this expression equals − 21 γ < 0. For the highest possible value of q
H, which is H = 1/4, it equals 14 γ 2 + 41 t2 − 14 t + 41 γ . This expression is q positive whenever 14 γ 2 + 14 t2 > 14 t + 14 γ. Taking squares on both sides this
1 implies 16 (t − γ)2 + 2γ 2 + 2t2 > 0, which is always true. The derivative of (25) is given by s   2 1 (1 − 16H (1 − H)) t 2tH  + 4 (1 − 2H)  γ 2 + 2 2 (1 − 2H) (1 − 2H)  2  4t H   1 (1−2H)3 r + 4 (1 − H) H −  2 4 2tH γ 2 + (1−2H)

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which can be simplified to 



(1 − 16H (1 − H)) t  1 1  − tH q 2 2 2 2 (1 − 2H) 2 γ (1 − 2H) + (2tH) q  2 2 2 + 4 (1 − 2H) γ (1 − 2H) + (2tH) Numerical simulations show that this is strictly positive for any H ∈ (0, 1/4) . We thus have that the derivative of producer profits with respect to H are negative for H = 0, positive for H = 1/4, and strictly increasing for any H ∈ (0, 1/4) . In turn, this implies that the derivative of producer profits with respect to ρ are positive for ρ = 0, negative for ρ = 1, and strictly decreasing for ρ ∈ (0, 1) , which establishes the result. Proof of Theorem 5
∗ dΠ∗ . The Note that for any variable of interest x, we have dxtot = 12 1 − 12 C ∗ dC dx dC ∗ dC ∗ dC ∗ comparative statics for dγ , dt , and dρ follow from Theorem 1. Note that C ∗ ∈ [0, 1), so the first term on the RHS is always positive. This establishes the results. Proof of Theorem 7 Note that  dW ∗ = 1 − 2γ − dγ  dW ∗ = 1 − 2γ − dt  dW ∗ = 1 − 2γ − dρ

 1 ∗ dC ∗ C − 2C ∗ 2 dγ  1 ∗ dC ∗ C 2 dt  1 ∗ dC ∗ C 2 dρ

It is immediate that the bracketed term is positive for sufficiently low γ, and negative for sufficiently high γ; it is definitely negative for γ > 1/2. This implies the stated result.

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Proof of Theorem 8 There is too much advertising if the market equilibrium level, given by (16), is higher than the socially optimal level given by (21), hence if s  2 2t 2t 1+ − 1+ > 2 (1 − γ) γη (ρ) γη (ρ) which implies η (ρ) <

(2γ − 1) t. γ 2 (1 − γ)

Note that the RHS of this inequality is negative for γ < 1/2, so in that case the inequality is always satisfied, which establishes the first part of Theorem 8. At γ = 1/2, the RHS is zero, whereas for γ → 1, it goes to infinity. Hence, for values of γ ∈ (1/2, 1) , there is a η ∗ such that the condition holds with equality. As η(ρ) is increasing in ρ and η (ρ) → ∞ as ρ → 1, this establishes part 2. The last part of the Theorem follows directly from the discussion below (21).

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