The Effect of Privacy Regulations and Endogenous Marketing

Research on Targeted AdvertisingI

1st Paper (A Working Paper)

Stephen Bruestle∗

Department of Economics, Black School of Business,Penn State University, Erie, PA 16563, USA

Abstract

I develop a monopolistic competitive logit model where advertisers purchase noisy signalsabout consumers tastes through marketing research to explore how changing the cost ofinformation affects consumer welfare, marketing research revenue, and advertiser platformrevenue. Advertisers choose the number of noisy signals so that the marginal cost of anadditional signal equals the marginal benefit of better targeting of their advertisements. Ishow that advertising and marketing research can be analyzed in a similar way to productioninputs, and I find that the substitution between advertising and marketing research behavessimilar to the classical findings on the substitution between production inputs. In addition,I find that marketing research costs have an ambiguous affect on consumer welfare. Highermarketing research costs decrease welfare by inducing a smaller, less-targeted selection ofproducts. Yet higher marketing research costs increase welfare by inducing fewer annoyingads, even without any pricing effects.

Keywords: targeted advertising, marketing costs, mass advertising, privacy regulations, adannoyance, information substitution

JEL Classification: M37, D83, M31, L51

II thank Simon Anderson (University of Virginia), Federico Ciliberto (University of Virginia), NathanLarson (American University), Regis Renault (Universite de Cergy-Pontoise), Justin Johnson (Cornell Uni-versity), Christopher Ruebeck (Lafayette College), Rick Harbaugh (Kelley School of Business, Indiana Uni-versity), Daniel Shiman (Federal Communications Commission), and an anonymous reviewer for their helpfulcomments. I also thank the participants of the 10th Annual International Industrial Organization Confer-ence Session 94 (2012 Arlington), the Fifth Annual Economics of Advertising and Marketing (2012 Beijing),the 34th ISMS Marketing Science Conference Session SB 11 (2012 Boston), and a seminar presentation atthe Federal Communications Commission (2013 D.C.) for their helpful comments. In addition I thank theBankard Fund for Political Economy for its generous financial support.∗Tel.: +1 609 540 1861E-mail address: sdb8g@virginia.edu (S. Bruestle).

Preprint submitted to Peers for Review and for Job Applications November 16, 2015

1. Introduction

Advertisers choose how much information they gather on consumers through marketingresearch. They do so by aggregating many independent, noisy signals about each con-sumer’s tastes. Each additional signal gives them a more accurate aggregate signal on thatconsumer’s taste. Advertisers use these signals to determine which consumers to target theirads. Targeted advertising is the sending of ads to a subset of consumers based on thesesignals. Yet each additional signal adds to the expense of doing more marketing research.Therefore advertisers choose the accuracy of this information, so that the marginal market-ing research cost of an additional signal equals the marginal benefit of better targeting oftheir advertisements.

Consumers, including myself, appreciate advertisers being better informed about ourtastes, because we get informed about a greater selection of more personalized products.I appreciate getting ads on products like DVDs and books, instead of ads on junk food orsenior living, because I am more likely to buy DVDs and books. Therefore consumers benefitfrom advertisers getting more accurate aggregate signals about their tastes.

Despite this benefit, the Federal Trade Commission and the European Parliament havestruggled with how to regulate targeted advertising to protect consumers’ personal informa-tion. Goldfarb and Tucker (2011) showed that these privacy regulations in the EuropeanUnion decreased the effectiveness of online ads. If better targeting of advertising is beneficialto consumers, then why would consumers want to protect their information from advertisers?

It could be that people value privacy because they want to protect against the criminaluse of their information. Yet even with identity theft protection, regulation, and insurance,people feel uncomfortable with businesses knowing too much of their personal information.It could be that people value privacy because it contains some psychological benefit. Thiswould make privacy a final good, instead of an intermediate. Yet Google has consumers optout of their targeted advertising program, when the personal information used to target isstored on the consumer’s personal computer. It could be that people value privacy becausethey want to avoid price discrimination. This possibility was explored when firms targetadvertise by Bergemann and Bonatti (2011).

Johnson (2013), the preliminary work of Shiman (1997), and this paper explore anotherexplanation for why consumers value privacy with advertisement annoyance. When poten-tial advertisers get too much information about consumers, too many advertisers enter themarket, pestering consumers with too many annoying ads. If advertisers know that I liketo read murder mystery novels, then I will be deluged with ads for murder mystery novels.Similarly if the viewers of a television program are very homogeneous, then they will bepestered with more advertisements. Personally, I dislike receiving junk mail and spam emailmore than I am worried about companies misusing my credit card numbers.1 Especially,because my credit card provider protects me with account monitoring, password protection,and a guarantee to cover stolen funds.

1 Don’t tell the criminals.

2

In the previous literature on targeted advertising, advertisers get an exogenous noisysignal on consumer tastes. Privacy regulations directly increases the noise of this signal,therefore giving advertisers less information on consumers. In this paper, I explore how ad-vertisers endogenously choose the accuracy of the information that they receive on consumersthrough marketing research. Privacy regulations indirectly affect the noise of the aggregatesignal on a consumer’s tastes by increasing the cost of gathering additional information (i.e.marketing research).

The cost of marketing research could be determined in a number of different ways, inaddition to a government regulator choosing privacy regulations to protect consumer welfare.The costs of marketing research could be determined by the current level of technology.Therefore as technology improves, marketing research should get less costly, which shouldinduce advertisers to gather more information on consumers. Another possibility is that thecosts of marketing research are determined by marketing research firms. These marketingresearch firms set the prices of their information to maximize their own profits. In this paper,I explore the implications of these three different mechanisms for determining marketingresearch costs.

Marketing research is an input into the advertising decision of the firm. An advertiserchooses how much information to gather on consumers (the quantity of marketing research)and how many consumers to send its ad (the quantity of advertisements). Advertisementsand marketing research are both complements in that marketing research increases the effec-tiveness of each ad. Yet they are both substitutes in that a firm could increase the number ofads it sends out to substitutes for doing less marketing research. This can be thought of asthe substitution between the quality of targeting versus quantity of advertising, or it can bethought of as the substitution rate between information. I will refer to this as informationsubstitution. In section 4.2, this will lead me to discover something similar to the MarginalRate of Technical Substitution (MRTS) for marketing research and advertising, which I callthe Marginal Rate of Information Substitution (MRIS).

Increasing the marginal cost of marketing research is similar to increasing the marginalcost of a production input. When the marginal cost of a production input increases, itnot only increases a firm’s own cost of production, it increases a firm’s competitor’s costof production. Similarly, when the marginal cost of marketing research increases, it notonly increases an advertiser’s own cost of gathering information, it increases an advertiser’scompetitor’s cost of gathering information. In this way, increasing the costs of marketingresearch could give advertisers profit and lead to market entry. This is because firm’s are ina version of prisoner’s dilemma where they would all be mutually better off advertising lessbut they are each individually better off advertising more. Increasing the marginal cost ofmarketing research reduces this negative externality.

2. Literature Review

This paper is most related to Johnson (2013). In Johnson (2013), improving the accuracyof the signal firms get on consumer tastes increases product selection and ad annoyance facedby consumers. In this paper, advertisers endogenously choose the accuracy of their signal

3

firms on consumer tastes through marketing research.. Johnson (2013) further explores theimpact of endogenous ad avoidance or ad blocking. In Johnson (2013), an improvement insignal accuracy encourages firms to advertise to more consumers, which encourages moreconsumers to block advertisements. Although in this paper, ad avoidance is exogenous. Iconsider the impact of endogenous ad avoidance at the end of the paper (to follow).

This paper is also related to Bergemann and Bonatti (2011). Both our papers analyze howimproving the accuracy of the signal consumers get on consumer tastes improves consumerwelfare through offering them a better selection of products. The biggest difference betweenour models is that Bergemann and Bonatti (2011) has no ad annoyance and the accuracyof the signal is exogenous. In his model, consumers are delivered a fixed number of ads.Firms buy these ads through perfect competition or from a monopolist publisher. Thisinduces firms to advertise in fewer markets and increase the price of their products. Whilethe product pricing effects are beyond the scope of this paper,2 when I explore changing theprice of advertising in the comparative statics sections of the paper (to follow), I show anadditional effect on welfare through product selection (similar to Bergemann and Bonatti,2011) and through ad annoyance. For a complete discussion on the product pricing effect,refer to Bergemann and Bonatti (2011).

This paper is also related to the preliminary work of Shiman (1997). Although Shimanabandoned the project, because there was very little interest at the time for targeted adver-tising in economics and the mathematics became too difficult, it is the first paper to explorehow improving the accuracy of the signal firms get on consumer tastes increases productselection and ad annoyance faced by consumers. The biggest differences between our papersis in the purchasing decision and that I allow for endogenous marketing research. In Shiman(1997), a consumer’s purchase decision is independent of what other products he bought. Inthis paper, I only allow a consumer to buy one product. In addition, I am able to exploreadvertisement prices, marketing costs, and the sharing of information between firms.

This paper is also related to Iyer et al. (2005). Both analyze firms decisions to eithermass advertise or target advertise.3 In Iyer et al. (2005), the cost of targeted advertisingis some exogenous constant, and I allow for a marketing cost of gathering information (abetter signal). Firms would pay for an equilibrium level of information, if they are using itto target, and not pay for any information, if they are mass advertising. In this way I ammaking the fixed targeting cost of Iyer et al. (2005) an endogenous constant.

Both Iyer et al. (2005) and this paper find that for a low per-consumer advertising costor a high marketing cost, firms will mass advertise, and for a high per-consumer advertisingcost or a low marketing cost, firms will target advertise.4 I extend this to show that for a low

2 there are plenty of papers that examine how consumer information can be used to price discriminate(see for example: Esteves, 2009; Goldfarb and Tucker, 2011; Villas-Boas, 1999, 2004), and several papersthat extend this to targeted advertising (see for example: Galeotti and Moraga-Gonzalez, 2004; Iyer et al.,2005).

3 Iyer et al. (2005) analyzes the case of a duopoly, and I analyze the case of monopolistic competition.4 Iyer et al. (2005) also show that for an intermediate per-consumer advertising cost or an intermediate

marketing cost, one firm will target advertise and the other will mass advertise. There results depend on the

4

signal accuracy, firms mass advertise, and for a high signal accuracy, firms mass advertise.This paper is also related to Esteban et al. (2001). In their paper, the specialization of a

magazine or other advertising medium is equivalent to signal accuracy. Consumers readingmore specialized magazines are more likely to buy the product. Esteban et al. (2001) showthat if a social planner picks product price and degree of advertisement specialization, thentargeted advertising is preferable to mass advertising. I get a similar result. I find that ifa social planner picks the marketing research cost, then targeted advertising is consumerwelfare improving over mass advertising.

Esteban et al. (2001) also show that a firm might choose to specialize its advertising toomuch. This might make targeted advertising less desirable than mass advertising, becausethe firm may choose to increase its prices too much.5 My paper extends this discussionby the addition of ad annoyance costs. I find that if firms are getting too accurate or tooinaccurate information on consumers, then targeted advertising might be less desirable thanmass advertising, because consumers face additional advertisement annoyance costs.

3. Model

There is a sufficiently large number N of profit maximizing firms. Each firm may poten-tially enter the market at an entry cost F > 0 and produce a product at a constant marginalcost normalized to zero. There is a unit mass of utility maximizing consumers. Half of theconsumers are type 0 and the other half of the consumers are type 1. Initially, each firmonly knows the aggregate distribution of types of consumers, but does not know the typesof individual consumers nor does it have any initial signal on individual consumer types.

Each entrant j will choose the quantity zj of signals to collect from each consumer, whichI interpret as its marketing research. Each signal is a private, i.i.d. signal that is true withprobability 1

1+φand false with a probability φ

1+φ, where φ ∈ [0, 1] is the error level of an

individual signal. A φ = 0 would be a perfect signal about consumers’ types and a φ = 1would be a meaningless signal about consumers’ types.

Entrant j pays a constant per signal marketing research cost of m ≥ 0. Consumers andfirms treat m as an exogenous constant. m could be many factors, including the level ofprivacy regulations, the tax or fee on information quality, and the efficiency of informationtechnology. And it could be determined by a government regulator choosing privacy regu-lations, a marketing research firm choosing the price of its information, an ad platform thatboth sells ad space and sells marketing research, or even the current level of technology.

After choosing marketing research, each entrant j simultaneously chooses its producttype tj ∈ {0, 1}, which consumers to send its ad to, and its single product price pj tomaximize its own profit Πj. For each consumer whom it chooses to advertise to, entrant j

fact that consumers who would be willing to buy either product (the comparison shoppers) have the samevalue r for both products. If both firms were to advertise to the comparison shoppers, then firms would bothset a product price of zero in Bertrand style price competition.

5 He gets this result when mass advertising wastes few ads and when product demand is sufficientlyinelastic.

5

pays a constant per consumer advertising cost of c. If firm j enters, buys zj signals fromeach consumer, advertises to vj consumers, and sells qj units of its good then it gets a profitof Πj ≡ pjqj − cvj −mzj − F .

Consumers don’t always see or remember every ad sent to them. Consumers see, remem-ber, or retain ads with at an i.i.d. probability of α ∈ [0, 1] (the ad retention probability). Iwill refer to those products whose ads are both sent to the consumer and retained by theconsumer as the consumer’s known products. These are the products that are available forthe consumer to buy. And I will refer to those products whose ads are either not retainedby the consumer or not sent to the consumer as the consumer’s unknown products. Theseare the products that are unavailable for the consumer to buy.

Each consumer i may buy one unit of his choice from his known products, or he maychoose to take an outside option.6 The utility consumer i would get from buying good jis uij ≡ uij + εij − A(n), where uij ≡ R + bλij − pj, R > 0 is the benefit from consumingany advertised good, b > 0 is the additional benefit from buying a good personalized tothe consumer’s type, λij is one if consumer i’s type τi ∈ {0, 1} matches product j’s typetj ∈ {0, 1} and zero otherwise, pj is the price of good j, εij is an i.i.d. stochastic shock

to consumer i’s value for good j with a c.d.f. of H(ε) = e−e−( εµ+γ)

, γ is Euler’s constant(≈ 0.5772),7 µ > 0 controls the variance of ε,8 and A(n) is the ad annoyance suffered by aconsumer from retaining ads from n different firms. Consumers are not annoyed by thoseads that are not retained. I assume that:

(A1) A is twice differentiable and continuous

(A2) A′(n) > 0 for all n ≥ 0

Assumption (A1) rules out irregular special cases like jump discontinuities and allowsus to use first and second order conditions. Assumption (A2) means consumers are alwaysannoyed by more ads.

The utility consumer i would get from buying the outside option is ui0 = ui0 + εi0−A(n),where ui0 = 0 and εi0 is an i.i.d. stochastic shock to consumer i’s value for the outside optionwith a c.d.f. of H(ε).

4. Equilibrium

Consider consumer i of type τi ∈ {0, 1}. Let ωij = 1 if product j is one of consumeri’s known products and ωij = 0 otherwise. And let ωi0 = 1, because every consumer knowsabout the outside option. Then the probability that consumer i will buy product j is the

6 This may simply be the option not to buy any good.7 The addition of the Euler’s constant γ makes E[εij ] = 0 and Hε(0) = .5.8V AR(ε) = π2

6 µ2.

6

well known multinomial logit result given by:9

Prbuy(i, j) = ωijeR+λijb−pj

µ

Ki

(1)

where Ki ≡ 1 +∑N

h=1 ωiheR+λihb−ph

µ .One way of thinking about Ki is that it is the price index for consumer i. A higher value

of Ki means that known product j faces more competition, either more price competitionor more competitors. In addition a higher value of Ki means that consumer i has moreconsumer surplus from the product that he buys, CSi.

10

Recall that consumer i only retains α ∈ [0, 1] of the ads sent to him. Therefore if firmj sends consumer i its ad, then the probability that consumer i will retain the ad is α.Therefore, if firm j sends its ad to consumer i, then the probability of a sale is:

Prsale(i, j) = αeR+λijb−pj

µ

Ki

(2)

Firm j does not know consumer i’s type, so it does not know λij in equation (2). Recallthat each entrant j does marketing research in the form of buying a quantity zj of signalsto collect from each consumer. Recall that each signal is a private, i.i.d. signal that is truewith probability 1

1+φand false with a probability φ

1+φ. To simplify the language, let’s call

the signals that match the firm’s type “matched signals” and the signals that do not matchthe firm’s type “unmatched signals”. Let y be the number of matched signals and let x bethe total number of unmatched signals. Therefore for any given consumer i of type τi andentrant j of type tj, the number of type 1 − tj signals xij is a binomial distribution (andtherefore yij = zj − xij), where:

Pr(xij = k|τi = tj) =(zjk

)( 1

1 + φ

)zj−k ( φ

1 + φ

)k(3)

Pr(xij = k|τi = 1− tj) =(zjk

)( φ

1 + φ

)zj−k ( 1

1 + φ

)kNote that we cannot look at the limit case for large number of signals zj. By the law

of large numbers, when zj is large, xij/zj = φ1+φ

if τi = tj and xij/zj = 11+φ

if 1 − τi = tj.Therefore for a large number of signals, the advertiser gets a perfect signal for each consumer’stype.11 For this reason, it would be inappropriate to approximate equation (3) with normal

9This result was first proved by Holman and Marley. See Luce and Suppes (1965) or Anderson et al.(1992).

10 Anderson et al. (1992, p. 60-61) shows that this relation is CSi = µ lnKi.11 This is also a problem is you allow φ to approach 1 as zj approached ∞. This limit case would avoid

the problem with the law of large numbers, because limφ→1φ

1+φ = limφ→11

1+φ = 12 . Yet the variance of the

7

distributions using the de Moivre-Laplace theorem.

For notational simplicity let’s define the function g(z, x) ≡(zx

) (1

1+φ

)z−x (φ

1+φ

)x. Then

the conditional probability mass functions Pr(xij = k|τi = tj) = g(zj, k) and Pr(xij = k|τi =1−tj) = g(z−k, zj) = θijg(z, k), where θij ≡ φzj−2k. Therefore the probability mass function

of xij is1+θij

2g(zj, xij).

Note that the distribution of xij is a symmetric bimodal distribution with a mean andmedian of xij =

zj2

, and modes of xij = zj1

1+φand xij = zj

φ1+φ

. Therefore its probability

mass function1+θij

2g(zj, xij) is symmetric with a mean and median of xij =

zj2

, and modes

of xij = zj1

1+φand xij = zj

φ1+φ

.

Applying Baye’s Theorem, I find the probability of type τi ∈ {0, 1} given θij:

Pr(τi = tj|θij) =1

1 + θij(4)

Pr(τi = 1− tj|θij) =θij

1 + θij

The proofs of mathematical equations like this is in Appendix Appendix A which startson page 29. I interpret θij as entrant j’s aggregate signal about consumer i’s type. A θij = 0would mean that there is a 100% probability that consumer i is type tj. A θij = 1 wouldmean that the signal is meaningless; there is a 50% probability that consumer i is type tjand a 50% probability that consumer i is type 1− tj. And a θij =∞ would mean there is a100% probability that consumer i is type 1− tj.

Therefore, if firm j sends its ad to consumer i, then the probability of a sale conditionalon consumer i’s signals is:

Prsale(i, j|θij) = α[

11+θij

eR+b−pj

µ

Ki,tj

+

θij1+θij

eR−pjµ

Ki,1−tj] (5)

Firm j does not know consumer i’s two possible price indices, Ki,0 or Ki,1, because theonly information firm j knows about consumer i comes from firm j’s private signal from itsmarketing research. Therefore firm j uses the expected value of the probabilities given byequation (5) to determine the probability of a sale. Therefore, if firm j sends consumer i itsad, the average probability that consumer i will buy product j conditional on consumer i’ssignals is:

Prsale(i, j|θij) = α[

11+θij

eR+b−pj

µ

Ktj

+

θij1+θij

eR−pjµ

K1−tj] (6)

where Kl is the harmonic mean12 of the price indices of consumers of type l ∈ {0, 1}. Because

probably distribution of xij would increase in zj , so both distributions would approach each other.12 Kl

−1 =∫ 1

01/2

(Ki)−1 ∗ 1{τi = l}di, where τi is consumer i’s type.

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signals to a firm are independent, each firm faces the same K1 and the same K0. Thereforeeach firm treats K1 and K0 as market constants. I will later show in Lemma 3 that K1 mustequal K0, but for now let’s allow K1 6= K0.

Note that equation (6) is independent of i and j. The only variables that matter are: thefirm’s price pj, the firm’s product type tj ∈ {0, 1}, and its aggregate signal θij. Thereforethe marginal expected profit π for an entrant of type t ∈ {0, 1} with a product price p for aconsumer with an aggregate signal of θ is π(p, t, θ) given by:

π(p, t, θ) = p ∗ α[1

1+θeR+b−pµ

Kt

+θ

1+θeR−pµ

K1−t]− c (7)

In this paper I analyze the case of monopolistic competition. Similar to the basic logitmonopolistic competition model presented in Anderson et al. (1992, p. 221-226), this marketstructure is the limit case where there are so many firms that an individual firm’s decisionsdo not impact the market variables K1 and K0, or an individual firm does not consider itsimpact on the the market variables K1 and K0 (as in Dixit and Stiglitz, 1977). Thereforefirms treat K1 and K0 as exogenous constants. This market structure not only makes mymodel more tractable, but more realistically reflects many advertising markets, includingonline advertising, billboard advertising, and telemarketing.

Optimizing any firm j’s profit over its price pj, I find that for every entrant j:

pj = µ (8)

Equation (8) finds that all entrants, independent of who they send their ads and howmuch marketing research, will always choose price p = µ to maximize their profits. Thisis not a limitation of the model, but a feature. In this paper, I examine the effects on theadvertising side of the market. There are plenty of papers that examine product pricingeffects of targeted advertising (see for example: Bergemann and Bonatti, 2011; Galeotti andMoraga-Gonzalez, 2004; Iyer et al., 2005).

Substituting p = µ into equations (7), I find that the marginal expected profit π for anentrant of type t for a consumer with an aggregate signal of θ is π(t, θ) given by:

π(t, θ) = αµ[1

1+θeR+b−µ

µ

Kt

+θ

1+θeR−µµ

K1−t]− c =

1

1 + θπtt +

θ

θ + θπ1−tt (9)

where πτitj ≡ αµ eR+bλij−µ

µ

Kτi− c is firm j’s additional expected profit from sending its ad to a

consumer i. Note that firm j would treat π00, π1

0, π01, and π1

1 as market constants.

Lemma 1. For every entrant j there exists one threshold xj ∈ (0, 1], such that:

(a) if xij ≤ xj, then entrant j will send its ad to consumer i, and

(b) if xij > xj, then entrant j will not send its ad to consumer i.

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Proof: Consider an arbitrary entrant with product type t. By equation (9), I have:∂∂θπ(t, θ) R 0 ↔ −(πtt−π

1−tt )

(1+θ)2R 0 ↔ π1−t

t R πtt. Note that this does not depend on θ. I

have three possibilities: (i) π1−tt < πtt, (ii) π1−t

t = πtt, or (iii) π1−tt > πtt.

Note that an entrant must send ads to at least one consumer or else it would notchoose to enter the market, due to the fixed entry cost F > 0. Therefore xj = 0 is notpossible.

Consider two arbitrary consumers, one who gives x type 1 − t signals and one whogives x′ type 1− t signals, where x′ < x.

Further note that ∂∂xθ(x) = ∂

∂x(φzj−2x) = −2ln(φ) ∗ φzj−2x > 0. Therefore θ(x′) <

θ(x).Case i: π1−t

t < πtt ↔ ∂∂θπ(t, θ) < 0:

Then by equation (9), if π(t, θ(x)) > 0 then π(t, θ(x′)) > 0. Because the choice of xand x′ < x is arbitrary, if π(t, θ(x)) > 0 then π(t, θ(x′)) > 0 for any x′ < x. Therefore iffirm j enters, there exist one threshold xj that satisfies the lemma.Case ii: π1−t

t = πtt ↔ ∂∂θπ(t, θ) = 0:

Therefore the expected profit from sending an ad to any consumer is the same,independant of the signal. Because the firm must make profit on at least one consumerto enter, it must make a profit on all consumers. Therefore we have the lemma withxj = zj.Case iii: π1−t

t > πtt ↔ ∂∂θπ(t, θ) > 0:

Then by equation (9), if π(t, θ(x′)) > 0 then π(t, θ(x)) > 0. Because the choice of xand x′ < x is arbitrary, if π(t, θ(x′)) > 0 then π(t, θ(x)) > 0 for any x′ < x. Therefore iffirm j enters, there exist one threshold xj such that if xij ≥ xj, then entrant j will sendits ad to consumer i, and if xij < xj, then entrant j will not send its ad to consumer i.

Note that because π1−tt > πtt ↔ αµ e

R−µµ

K1−t− c > αµ e

R+b−µµ

Kt− c ↔ e

R−µµ

K1−t> e

R+b−µµ

Kt. ↔

eR−µµ

K1−t> e

R−µµ

Kt.

For all x ∈ [zj/2, zj], I have that θ(x) ≥ 1 and consequentially 11+θ(x)

≤ θ(x)1+θ(x)

↔ 11+θ(x)

eR−µµ

Kt< θ(x)

1+θ(x)eR−µµ

K1−t↔

11+θ(x)

eR+b−µ

µ

Kt+

θ(x)1+θ(x)

eR−µµ

K1−t<

11+θ(x)

eR−µµ

Kt+

θ(x)1+θ(x)

eR+b−µ

µ

K1−t↔

π(t, θ) < π(1− t, 1− θ), which is the profit entrant j would get as type 1− t sending thead to the same consumer. Therefore for each and every consumer x ∈ [zj/2, zj], firm jwould make more profit as a type 1− t firm, instead of as a type t firm.

If xj ≥ zj/2, then entrant j would make more profit as type 1 − t entrant sendingits ad to the same set of consumers. This contradicts the assumption that entrant j isa type t firm, so let us assume that xj < zj/2.

Because the probability mass function is symmetric about zj/2, for each consumerx1 ∈ [xj, zj/2) there exists a consumer x2 = zj − x1 ∈ (zj/2, zj − xj], and for eachconsumer x2 ∈ (zj/2, zj − xj] there exists a consumer x1 = zj − x2 ∈ [xj, zj/2). Also byequation (9), I have the symmetry π(t, θ(x1)) + π(t, θ(x2)) = π(1− t, 1− θ(x1)) + π(1−

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t, 1 − θ(x2)), which means that entrant j would make the same expected profit on allthe consumers x ∈ [xj, zj − xj] as a type 1− t firm. Because I have already shown thatentrant j would make a higher expected profit on all the consumers x ∈ (zj − xj, xj], Ihave that entrant j’s total expected profit would be higher as a type 1 − t firm. Thiscontradicts the assumption that entrant j is a type t firm, which proves the lemma.

Lemma 1 shows that any entrant will always send ads to the consumers with the leastunmatched signals. In order to enter the market as type t: (1) the marginal profit fromsending ads to the consumers with the least unmatched signals has to be positive, and (2)it has to be weakly more profitable to be type t, which rules out strategies were the firmtargets consumers with the most unmatched signals.

Therefore the total expected profit Π for an entrant of type t ∈ {0, 1} that pays for zsignals and sends ads to each consumer with no more than x unmatched signals is:

Π(t, z, x) =x∑x=0

π(t, θ)[1 + θ

2g(z, x)]− F −mz (10)

where θ = φz−x and 1+θ2g(z, x) is the probability mass function of x.

In order to simplify equation (10), I introduce notation for the cumulative density functionfor equation (3), which is given by:

Gρ(z, k) ≡ 1

2

k∑x=0

ρz−2x ∗(zx

)( 1

1 + φ

)z−x(φ

1 + φ

)x(11)

where Pr(xij = x|τi = tj) and Pr(xij = x|τi = 1−tj) are given by equation (3). Therefore theconditional probability density functions are G1(z, k) = Pr(xij ≤ k|τi = tj) and Gφ(z, k) =1−G1(z, z − k − 1) = Pr(xij ≤ k|τi = 1− tj).13

Note that the total profit function for entrant j can be rewritten as:

Π(t, z, x) =1

2G1(z, x)πtt +

1

2Gφ(z, x)π1−t

t − F −mz (12)

Because it is easier to interpret the number of consumers sent ads v that a firm sendsthan the maximum number of unmatched signals x, I introduce a transformation of equation(12). Because half of the consumers are type t and the other half of the consumers are type1− t, I have that:

v =G1(z, x) +Gφ(z, x)

2(13)

Consider how v relates to x. The ∂v∂x

would be total additional consumers sent ads fromincreasing the maximum number of unmatched signals from x − 1 to x. Therefore it is

13 Gρ can also be represented in terms of the regularized incomplete beta function, which is included inmany statistical packages, including Stata and Matlab.

11

the number of consumers with exactly x unmatched signals. There are 12g(z, x) matched

consumers with exactly x unmatched signals. And there are 12θg(z, x) unmatched consumers

with exactly x unmatched signals. Therefore I have:

∂v

∂x=

1 + θ

2g(z, x) > 0 (14)

Consider increasing z to z + 1, while holding x constant. If a consumer already hasmore than x unmatched signals, then with an additional signal, it will still have more thanx unmatched signals. If a consumer already has less than x unmatched signals, then anadditional signal will not give it more than x unmatched signals. So an additional signal willonly screen out those consumers who currently have x unmatched signals. Because thereare 1

2g(z, x) matched consumers, and because matched consumers will give an additional

unmatched signal with a probability of φ1+φ

, the advertiser would lose 12

φ1+φ

g(z, x) matched

consumers. Because there are 12θg(z, x) unmatched consumers, and because unmatched

consumers will give an additional unmatched signal with a probability of 11+φ

, the advertiser

would lose 12

11+φ

θg(z, x) matched consumers. Therefore the change in the total number ofconsumers sent ads is:

v(z + 1, x)− v(z, x) = −1

2

θ + φ

1 + φg(z, x) < 0 (15)

Therefore v is strictly increasing in x and strictly decreasing in z. Therefore we can invertequation (13) to solve for x(z, v). I have that x(z, v) is strictly increasing in v. In order toincrease z, holding v constant, I must have 0 = dv

dxdx+ dv

dzdz. This becomes:

dx

dz=

φ+ θ

(1 + φ)(1 + θ)> 0 (16)

Therefore x(z, v) is also strictly increasing in z. Therefore I transform equation (12) to:

Π(t, z, v) = v[1

1 + θ∗(z, v)πtt +

θ∗(z, v)

1 + θ∗(z, v)π1−tt ]− F −mz (17)

where θ∗(z, v) ≡ Gφ(x(z,v),z)

G1(x(z,v),z).

In order to rule out empty market or a monopoly equilibria, I assume that:

µ

2(F + c)− 1

α[eR+b−µ

µ + eR−µµ ]

> 1 (18)

which is sufficient to show that at least two firms enter the market.Price competition will make at least one firm enter as type 0 and one firm enter as type

1, which I show in Lemma 2.

12

Lemma 2. At least one firm enters as each product type.

Proof: Suppose not. Then by equation (18), I have that there are two firms of the sametype. Let us call this type, type t ∈ {0, 1}, and the two firms, firms 0 and 1. Note thatboth (z0, v0) and (z1, v1) must be optimal choices of (z, v) for Π(t, z, v), but (z0, v0) isnot necessarily equal to (z1, v1).

Kt = 1 + 2α[ 11+θ∗(z0,v0)

v0 + 11+θ∗(z1,v1)

v1]eR+b−µ

µ

> 1 + 2α[ θ∗(v0,z0)1+θ∗(z0,v0)

v0 + θ∗(z1,v1)1+θ∗(z1,v1)

v1]eR−µµ = K1−t.

Because 11+θ∗(z0,v0)

> θ∗(z0,v0)1+θ∗(z0,v0)

, I have:

Π(t, z0, x0) = v0[ 11+θ∗(z0,v0)

πtt + θ∗(z0,v0)1+θ∗(z0,v0)

πt1−t]− F −mz0

= v0[ 11+θ∗(z0,v0)

(αµ eR+b−µ

µ

Kt− c) + θ∗(z0,v0)

1+θ∗(z0,v0)(αµ e

R−µµ

K1−t− c)]− F −mz0

< v0[ 11+θ∗(z0,v0)

(αµ eR+b−µ

µ

K1−t− c) + θ∗(z0,v0)

1+θ∗(z0,v0)(αµ e

R−µµ

Kt− c)] − F − mz0 = Π(1 − t, z0, v0)

Therefore it is more profitable for firm 0 to be type 1 − t, which means all firms can’tbe type t.

Lemma 2 rules out empty product types. Therefore by free entry, the profit from beinga type 0 and a type 1 firm should be equal. This ensures that the price indices K0 = K1,given by:

Lemma 3. K0 = K1

Proof: For any arbitrary type t ∈ {0, 1}, let (zt, vt) denote any optimal choices of (z, v)for Π(t, z, v). From Lemma 2, I have that there is at least one type t firm and one type1− t firm. Because of free entry, I have Π(t, zt, vt) = Π(t, z1−t, v1−t). Because (z1−t, v1−t)is an optimal choices of (z, v) for Π(1−t, z, v), I have Π(1−t, z1−t, v1−t) ≥ Π(1−t, zt, vt).Therefore Π(t, zt, vt) ≥ Π(1− t, zt, vt)↔ vt[

11+θ∗(zt,vt)

πtt+θ∗(zt,vt)

1+θ∗(zt,vt)π1−tt ]−F −mzt ≥ vt[

11+θ∗(zt,vt)

πt1−t+θ∗(zt,vt)

1+θ∗(zt,vt)π1−t

1−t]−F −mzt

↔ vt[1

1+θ∗(zt,vt)(αµ e

R+b−µµ

Kt− c) + θ∗(zt,vt)

1+θ∗(zt,vt)(αµ e

R−µµ

Kt− c)]− F −mzt

≥ vt[1

1+θ∗(zt,vt)(αµ e

R+b−µµ

K1−t− c) + θ∗(zt,vt)

1+θ∗(zt,vt)(αµ e

R−µµ

K1−t− c)]− F −mzt

↔ 11+θ∗(zt,vt)

eR+b−µ

µ

Kt+ θ∗(zt,vt)

1+θ∗(zt,vt)eR−µµ

K1−t≥ 1

1+θ∗(zt,vt)eR+b−µ

µ

K1−t+ θ∗(zt,vt)

1+θ∗(zt,vt)eR−µµ

Kt

↔ Kt ≤ K1−t (because 11+θ∗(zt,vt)

≥ θ∗(zt,vt)1+θ∗(zt,vt)

and eR+b−µ

µ > eR−µµ ).

Therefore for any arbitrary type t ∈ {0, 1}, Kt ≤ K1−t. Choosing type t = 0, I haveK0 ≤ K1. Choosing type t = 1, I have K1 ≤ K0. Therefore K0 = K1.

Therefore there is one common price index for both sub-markets, which I define to beK ≡ K0 = K1. Because competing firms only influence each other through the price index K,see equations (7) and (10), entrants face symmetric pricing and marketing research choices on

13

either side of the market. Later I will show that there is one unique symmetric equilibrium,but for now let’s entertain the possibility of multiple equilibria and asymmetric equilibria.

Therefore I define the marginal profit for a consumer with a signal of θ as π(θ) ≡ π(t, θ) =π(1 − t, θ). I define the marginal profit for a consumer that matches the firm’s type asπM ≡ πtt = π1−t

1−t. I define the marginal profit for a consumer that does not match the firm’stype as πU ≡ π1−t

t = πt1−t. And I define profit as Π(z, v) ≡ Π(t, z, v) = Π(1− t, z, v).If there is positive profit, then more firms will enter, until profit is zero. If there is negative

profit, then firms will leave, until profit is zero. This gives us a zero-profit condition. Usingsimilar reasoning to the proof of Lemma 3 and this zero-profit condition, I rule out thepossibility of multiple potential price indices:

Lemma 4. There is one unique potential K.

Proof: Let K0 and K1 be any two arbitrary price indices. For any arbitrary ` ∈ {0, 1}, let(z`, v`) denote any optimal choices of (z, v) under price index ` ∈ {0, 1}, and let Π`(z, v)denote a firm’s profit under price index ` ∈ {0, 1}. Because of the zero profit condition,I have that Π`(z`, v`) = Π1−`(z1−`, v1−`) = 0. Because (z1−`, v1−`) is an optimal choicesof (z, v) for Π1−`(z, v), I have Π1−`(z1−`, v1−`) ≥ Π1−`(z`, v`).Therefore Π`(z`, v`) ≥ Π1−`(z`, v`)

↔ v`[ 11+θ∗(z`,v`)

(αµ eR+b−µ

µ

K` − c) + θ∗(z`,v`)1+θ∗(z`,v`)

(αµ eR−µµ

K` − c)]− F −mz`

≥ v`[ 11+θ∗(z`,v`)

(αµ eR+b−µ

µ

K1−` − c) + θ∗(z`,v`)1+θ∗(z`,v`)

(αµ eR−µµ

K1−` − c)]− F −mz`

↔ 11+θ∗(z`,v`)

eR+b−µ

µ

K` + θ∗(z`,v`)1+θ∗(z`,v`)

eR−µµ

K` ≥ 11+θ∗(z`,v`)

eR+b−µ

µ

K1−` + θ∗(z`,v`)1+θ∗(z`,v`)

eR−µµ

K1−` ↔ K` ≤ K1−`

Therefore for any arbitrary price index ` ∈ {0, 1}, K` ≤ K1−`. Choosing price index` = 0, I have K0 ≤ K1. Choosing price index ` = 1, I have K1 ≤ K0. ThereforeK0 = K1.

Next I will consider the three possible cases: all entrants do no marketing research (sec-tion 4.1), all entrants do marketing research (section 4.2), and some entrants do marketingresearch while other entrants do no marketing research (section 4.3).

4.1. Mass Advertising

In this section, I consider the case where all entrants do no marketing research. In orderfor a firm to enter, the expected marginal profit π needs to be positive for at least oneconsumer. If an entrant does no marketing research, then its expected marginal profit fromeach consumer is identical, and therefore positive. Therefore I have that all firms massadvertise, which I define as:

Definition 1. mass advertise: to send an ad to all consumers.

Therefore each entrant will get the mass advertising profit given by:

ΠMA ≡ Π(z = 0, v = 0) =1

2πM +

1

2πU − F (19)

14

Note that mass advertising is equivalent to v = 1 and z = 0.Let n0 be the number of entrants of type 0 and n1 be the number of entrants of type 1.

Then each entrant j would make the expected marginal profit of π = 12[πM + πU] on each

consumer. Therefore by free entry, I have the zero profit condition F = π, which solving forn0 and n1 becomes:

nMA ≡ n0 = n1 =µ

2(F + c)− 1

α[eR+b−µ

µ + eR−µµ ]

(20)

Note that (20) simplifies equation (18) to NMA > 1. Therefore I am assuming that themarket can at least sustain a mass advertising equilibrium.

Here µ2(F+c)

is the number of firms that would enter each sub-market if there were nooutside option. It is also the effect of the entry cost F and the advertising cost c on the

entry. And [α(eR+b−µ

µ + eR−µµ )]−1 is the number of firms discouraged from entering each

sub-market due to the outside option. Using (20) to solve for K, I have:

KMA =αµ

2(F + c)[e

R+b−µµ + e

R−µµ ] (21)

Therefore I have that marketing research costs would have no marginal effect on the priceindex when all firms are not doing any marketing research.

Lemma 5. K ≥ KMA

Proof: Suppose by way of contradiction that KMA > K ↔ αµ2(F+c)

[eR+b−µ

µ + eR−µµ ] > K

↔ αµ2eR+b−µ

µ +eR−µµ

K− F − c > 0

Therefore entrants could make a positive profit from mass advertising. Therefore allentrants would choose to mass advertise, which would mean that KMA = K.

If πU > 0, then an entrant would send its ads to every consumer, because it expects amarginal profit on each ad it sends out. Therefore I assume: πU < 0, which is equivalent to:

F

c<

1

2(eb/µ − 1) (22)

4.2. Targeted Advertising

In this section, I consider the case where all entrants do marketing research. When afirm does any marketing research (i.e. z > 0), then it must be sending its ads to some butnot all consumers (i.e. 0 < v < 1). Otherwise it would be more profitable not to do anymarketing research. Therefore by Lemma 1, I have that all entrants target advertise, whichI define as:

Definition 2. target advertise: to send an ad to all and to only those consumers with nomore than x unmatched signals, where 0 ≤ x < z.

15

Note that by definition, if a firm is target advertising, then z > 0.If the firm sends one more ad v then it is sending an additional ad to a threshold consumer,

who has a θ = θ. Therefore with a probability of 11+θ

it will send an ad to a matched

consumer, each of whom yields an expected marginal revenue of αµ eR+b−µ

µ

K. And with a

probability of θ1+θ

it will send an ad to a unmatched consumer, each of whom yields an

expected marginal revenue of αµ eR−µµ

K. Therefore the expected marginal revenue from sending

an additional ad is:

MRv = αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K> 0 (23)

If the firm sends one more signal without changing x, the maximum number of unmatchedsignals, then it will be advertising to fewer consumers. If a consumer already has more thanx unmatched signals, then with an additional signal, it will still have more than x unmatchedsignals. If a consumer already has less than x unmatched signals, then an additional signalwill not give it more than x unmatched signals. So an additional signal will only screen outthose consumers who currently have x unmatched signals.

There are 12g(z, x) matched consumers that currently have a signal of x, each of whom

yields an expected marginal revenue of αµ eR+b−µ

µ

K. An additional signal would mean that

the firm will not send an at to φ1+φ

of these matched consumers. And there are 12θg(z, x)

unmatched consumers that currently have a signal of x, each of whom yields an expected

marginal revenue of αµ eR−µµ

K. An additional signal will mean that the firm will not send an

at to 11+φ

of these unmatched consumers. Therefore if the firm sends out an additional signal

without changing x, it will send out 12[ φ1+φ

+ 11+φ

θ]g(z, x(v, z)) fewer ads and lose the revenue

of αµ2g(z, x(z, v))

φ1+φ

eR+b−µ

µ + 11+φ

θeR−µµ

Kfrom sales.

Note that having the firm send out one more signal without changing the number of adscan be split into two steps: (Step 1) the firm sends out an additional signal, while holdingx constant. This causes the firm to advertise to 1

2[ φ1+φ

+ 11+φ

θ]g(z, x(v, z)) fewer consumers.

(Step 2) the firm sends advertises to an additional 12[ φ1+φ

+ 11+φ

θ]g(z, x(v, z)) consumers.These consumers would be threshold consumers, so it would earn a marginal revenue ofMRv for each of these additional consumers. Therefore the marginal revenue from sendingan additional signal is:

MRz = −αµ2g(z, x(z, v))

φ1+φ

eR+b−µ

µ + 11+φ

θeR−µµ

K+

1

2[φ

1 + φ+

1

1 + φθ]g(z, x(v, z))MRv

=αµ

2

1− φ1 + φ

θ

1 + θg(z, x(z, v))

eR+b−µ

µ − eR−µµ

K> 0 (24)

When an advertiser gathers an additional signal about consumers tastes, then each ad-vertisement is more likely to reach a matched consumer and less likely to reach an unmatched

16

Figure 1: Cost Minimization Problem

v

zTC3TC2

TC1 < TC2 < TC3

TC1

slope = −mc

chea

per

Equation:TC = F +mz + cv

(a) Iso-Cost Lines

v

z

vPT

TCmin

•

z∗

v∗

•

z

v

Uneconomic Region

•vNT

TR

slope = −MRISzv

•1 Mass Advertising

(b) Iso-Revenue Curve

consumer. Therefore the advertiser does not have to send out as many ads to get the samesales revenue. This substitution between marketing research and advertising is given by themarginal rate of information substitution (MRISzv), which I define as the number ofadditional ads v needed in order to retain the same Total Revenue if the firm gathers onefewer signal z about consumers tastes.

MRISzv =MRz

MRv

=1

2

1− φ1 + φ

θg(z, x(z, v))eb/µ − 1

eb/µ + θ> 0 (25)

MRISzv is always positive, because the firm would always need to do more advertisingto get the same Total Revenue with less marketing research. The MRISzv can be thoughtof as the substitution rate between the quality of targeting versus quantity of advertising, orit can be thought of as the substitution rate between information. It is the rate of how manymore consumers does an advertiser need to inform about its product if it is less informedabout consumers’ tastes.

Figure 1 illustrates the cost minimization problem of a firm that is targeted advertising.Figure 1 (1a) illustrates the cost of marketing research z (on the horizontal axis) and

advertising v (on the vertical axis). It shows three iso-cost lines, TC1, TC2, and TC3, whichare the combinations of marketing research z and advertising v that cost the same. I havethat each iso-cost line has a slope of −m

c. And I have that a closer iso-cost line to the origin

represents a cheaper Total Cost. Therefore TR1 < TR2 < TR3.Figure 1 (1b) illustrates an iso-revenue curve, which is the combination of marketing

research z and advertising v that yield the same Total Revenue. The slope of the iso-revenuecurve would be −MRISzv.

The cost minimizing level of marketing research z and advertising v would be the pointon the iso-revenue curve that is on the lowest possible iso-cost line. This happens when the

17

iso-cost line is tangent to the iso-revenue curve. Therefore the cost-minimizing conditionwould be MRISzv = m

c. Or equivalently that MRz

m, which is the marginal revenue per dollar

spent on marketing research, is equal to MRvc

, which is the marginal revenue per dollar spenton advertising.

I will show in Lemma 6 that MRISzv is not always decreasing. When MRISzv is in-creasing, I have that the iso-revenue curve is concave, and therefore MRISzv = m

cwould

mean that the firm would be cost maximizing. I will show in Lemma 6 that the potentialprofit minimizing (and therefore cost maximizing) point occurs when x < φ

1+φ, and I will

show that the profit maximizing (and therefore cost minimizing) point occurs when x > φ1+φ

.

This creates the potential uneconomic region shown in 1 (1b).I will next solve for the profit maximizing level of advertising v and marketing research z.

I will start by solving for the zero-profit condition. Each entrant j would make the expectedmarginal profit of πM on each of the 1

1+θ∗(z,v)v consumers that it advertises to who match its

type and πU on each of the θ∗(z,v)1+θ∗(z,v)

v consumers that it advertises to who do not match itstype. Therefore by free entry, I have the zero-profit condition:

F +mz = v[1

1 + θ∗(z, v)πM +

θ∗(z, v)

1 + θ∗(z, v)πU] (26)

If a firm is targeted advertising, then it is optimizing over its choice of v, the total numberads. The profit maximizing level of advertising v would occur when the marginal revenueMRv from advertising is equal to the marginal cost c of advertising. This is equivalent tothe marginal profit π(θ) = 0. If the marginal profit π(θ) > 0, then it would be profitable forthe firm to increase v. If the marginal profit π(θ) < 0, then it would be profitable for thefirm to decrease v. Therefore the first-order condition for v is given by:

π(θ) = 0 (27)

↔ θ =πM

−πU

> 0

where θ ≡ θt = θ1−t.This defines a unique v∗, because θ(z, v) is strictly increasing in v. Note that equation

(22) guarantees πU < 0, and that I must have πM > 0 for any firm to enter the market.Therefore θ > 0.

Likewise, if a firm is targeted advertising, then it is optimizing over its choice of z, thetotal number of signals. The profit maximizing level of marketing research z would occurwhen the marginal revenue MRz from marketing research is equal to the marginal cost mof marketing research. Therefore the first-order condition for z is given by:

m =αµ

2

1− φ1 + φ

θ

1 + θg(z, x(z, v))

eR+b−µ

µ − eR−µµ

K(28a)

↔ m =1

2(1− φ1 + φ

)g(z, x(z, v))πM (28b)

18

where z∗ is the optimum z.Note that there are potentially two solutions to equation (28b), because the distribution of

g(x, z) is inverse-u-shaped in z (think the probability mass function of a negative binomialdistribution). The lower of the two potential values would be the the z that minimizesprofit. And the higher of the two potential values would be the the z that maximizes profit.Therefore the advertiser would choose the higher of the two values. It can be shown thatequation (28a) only has one potential solution, which is equal to the higher of the two valuesof z that solves equation (28b).

This alone does not guarantee that the combination of z∗ and v∗ is unique, becauseequations (27) and (28b) both depend on z and v. Therefore to rule out the possibility ofsaddle points, I use the fact that θ is fixed to prove that the combination of z∗ and v∗ isunique:

Lemma 6. For a given K, there is at most one optimal (z∗, v∗).

Proof: Because K is a unique constant (Lemma 4), I have that πM−πU

is a unique constant.

Therefore, by equation (27), I have that θ = φz−2x(z,v) is a unique constant. Therefore Iconsider increasing z, while holding θ (or equivalently z − 2x) constant.↔ dΠ

dz= [MRz −m] + x

dz∂v∂x

[MRv − c]↔ dΠ

dz= [MRz −m] + 2 ∂v

∂x[MRv − c] (because z − 2x is constant)

↔ By equations (14), (24), and (24):

dΠdz

= [αµ2

1−φ1+φ

θ1+θ

g(z, x(z, v)) eR+b−µ

µ −eR−µµ

K−m] + 21+θ

2g(z, x)[αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K− c]

→ Because θ is held constant:

d2Πdz2

=([αµ

21−φ1+φ

θ1+θ

eR+b−µ

µ −eR−µµ

K] + 21+θ

2[αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K− c]

)ddzg(z, x(z, v))

→ d2Πdz2

(z∗, v∗) =m ddzg(z,x(z,v))

g(z,x(z,v))(because dΠ

dz= 0 and dΠ

dv= 0 in equilibrium)

↔ sgn(d2Πdz2

(z∗, v∗)) = sgn( ddzg(z, x(z, v))) (because m > 0 and g(z, x(z, v)) > 0)

↔ Because z − 2x is constant:sgn(d2Π

dz2(z∗, v∗)) = sgn([g(z + 1, x)− g(z, x)] + 2[g(z, x)− g(z, x− 1)])

↔ By Lemma A-1 on pg. 34:sgn(d2Π

dz2(z∗, v∗)) = sgn([( φ

1+φg(z, x− 1) + 1

1+φg(z, x))− g(z, x)] + 2[g(z, x)− g(z, x− 1)])

↔ sgn(d2Πdz2

(z∗, v∗)) = sgn((2− φ1+φ

)[g(z, x)− g(z, x− 1)])

↔ sgn(d2Πdz2

(z∗, v∗)) = sgn(g(z, x)− g(z, x− 1)) (because 2 > φ1+φ

)

Note that g(z, x) is the probability mass function of the binomial distribution of x (witha success rate of 1

1+φ), which is increasing in x when x < φ

1+φz and decreasing in x

when x > φ1+φ

z. Because θ = φz−2x → x = z−ln(θ)/ ln(φ)2

, I have that: x R φ1+φ

z ↔z−ln(θ)/ ln(φ)

2R φ

1+φz ↔ z R ln(θ)/ ln(φ)

2[ 12− φ

1+φ]

19

Therefore I have that d2Πdz2

> 0 when z < ln(θ)/ ln(φ)

2[ 12− φ

1+φ]

and d2Πdz2

< 0 when z > ln(θ)/ ln(φ)

2[ 12− φ

1+φ].

Therefore there is one unique (z∗, θ), which means that there is one unique (z∗, v∗).

This Lemma 6 and Lemma 3 ensure that all firms that do marketing research have thesame number of signals, z∗, and the level of advertising, v∗.

Note that these conditions only hold for interior values of z∗ and v∗, or equivalentlywhen a firm is targeted advertising. Therefore I need conditions for when I have a boundarycondition.

Lemma 7. x(z∗, v∗) ∈ ( φ1+φ

z∗, 11+φ

z∗)

Proof: Lemma 6 shows θ is unique, thus z−2x is constant. From the proof of Lemma 6,I have sgn(d2Π

dz2(z∗, v∗)) = sgn(g(z, x) − g(z, x − 1)). By similar reasoning, I have that

sgn(d2Πdz2

(z∗, v∗)) = sgn(θ(z, x)g(z, x)−θ(z, x−1)g(z, x−1)) (see Lemma A-3 on pg. 35).

Therefore I have that sgn(g(z∗, x(z∗, v∗)) − g(z∗, x(z∗, v∗) − 1)) = sgn(θ(z, x)g(z, x) −θ(z, x − 1)g(z, x − 1)). Note that θ(z, x)g(z, x) is the probability mass function of thebinomial distribution of x (with a success rate of φ

1+φ), which is increasing in x when

x < 11+φ

z and decreasing in x when x > 11+φ

z. And that g(z, x) is the probability mass

function of the binomial distribution of x (with a success rate of 11+φ

), which is increasing

in x when x < φ1+φ

z and decreasing in x when x > φ1+φ

z.

Lemma 7 ensures that any boundary condition only occurs where z = 0. This will becomea condition on m to ensure a targeted advertising equilibrium once I solve for the targetedadvertising price index.

Next I will solve for the price index, using the zero-profit condition. Each entrant j wouldmake the expected marginal profit of πM on each of the 1

1+θ∗(z,v)v consumers that it advertises

to who match its type and πU on each of the θ∗(z,v)1+θ∗(z,v)

v consumers that it advertises to whodo not match its type. Therefore by free entry, I have the zero-profit condition:

F +mz = v[1

1 + θ∗(z, v)πM +

θ∗(z, v)

1 + θ∗(z, v)πU] (29)

Let n0 be the number of entrants of type 0 and n1 be the number of entrants of type 1.Solving for n0 and n1 equation (29) becomes:

nTA ≡ n0 = n1 =µ

2(F +mz∗ + cv∗)− 1

α2v∗[ 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ ]

(30)

where θ∗ ≡ θ∗(z∗, x(z∗, v∗)), v∗ is defined by equation (27), and z∗ is defined by equation(28b).

Here µ2(F+mz∗+cv∗)

is the number of firms that would enter each sub-market if there wereno outside option. It is also where entry cost F , marginal marketing research cost m, and

20

marginal advertising cost c effects entry. And 1

2αv∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]

is the number of

firms discouraged from entering each sub-market due to the outside option.Note that, unlike (20), this term depends on the amount of marketing research z∗ and the

amount of advertising v∗, because here firms are using marketing research to decide whichconsumers to show their ads.

Using (30) to solve for K, I have:

KTA =αµ

F +mz∗ + cv∗v∗[

1

1 + θ∗eR+b−µ

µ +θ∗

1 + θ∗eR−µµ ] (31)

From this I have:

∂KTA

∂z∗= 0 (32a)

∂KTA

∂v∗= 0 (32b)

I will show in section ??, that this is because the advertiser is choosing advertising v∗

and marketing research z∗ to maximize the price index K.All this assumes interior values of z∗ and v∗. Lemma 7 shows that the only possible

border solution occurs where the number of signals z = 0. Therefore for interior values ofz∗ and v∗, all I need is for m to be small enough so that the number of signals z ≥ 1. Thiscondition is given by:

m ≤12

1−φ(1+φ)2

[4(F + c/2) eb/µ1

1+φebµ+ φ

1+φ

− c]

1− 2 1−φ(1+φ)2

eb/µ1

1+φebµ+ φ

1+φ

(33)

Note that equation (33) is only the condition for KTA to exist. It does not determinewhether an entrant would choose to mass advertise or target advertise. That will be deter-mined by whether targeted advertising covers the fixed cost F or not. I will analyze this insection 4.4.

4.3. Mixed Advertising

In this section, I consider the case some entrants do no marketing research while someentrants do marketing research, or equivalently the case where some entrants mass advertiseand some entrants target advertise. By Lemma 3, I have that entrants are indifferent betweentargeted and mass advertising.

Let n0TA be the number of type 0 entrants that target advertise, n0

MA be the numberof type 0 entrants that mass advertise, n1

TA be the number of type 1 entrants that targetadvertise, and n1

MA be the number of type 1 entrants that mass advertise. Then by free entry,I have that both the mass advertising zero profit condition, which states F = 1

2[πM +πU], and

the targeted advertising zero profit condition, which states F +mz∗ = v∗[ 11+θ∗

πM + θ∗

1+θ∗πU],

21

would be satisfied. This reduces to:

nMA = nMA + 2v∗1

1+θ∗eb/µ + θ∗

1+θ∗

eb/µ + 1nTA = 2v∗

11+θ∗

eb/µ + θ∗

1+θ∗

eb/µ + 1nTA (34)

n1MA − n0

MA

n1TA − n0

TA

= −2v∗1

1+θ∗eb/µ − θ∗

1+θ∗

eb/µ − 1(35)

where θ∗ ≡ θ∗(z∗, x(z∗, v∗)), v∗ is defined by equation (27), and z∗ is defined by equation

(28b), nMA ≡n0MA+n1

MA

2, nTA ≡

n0TA+n1

TA

2, nMA is the number of entrants in a mass advertising

equilibrium, which is given by equation (20), and nTA is the number of entrants in a targetedadvertising equilibrium, which is given by equation (30).

Equation (34) bounds (nMA, nTA) to a line between (nMA, 0) and (0, nTA). This ensuresthat advertisers are indifferent between mass advertising and targeted advertising, whichresults in Lemma 8 (below). Equation (35) assures that the line between (n0

MA, n0TA) and

(n1MA, n

1TA) has a particular negative slope. This ensures that the price indices from each sub-

market are equal, and therefore no entrant is induced to switch product type. Interestingly,this allows one sub-market to have more mass advertisers while the other sub-market tohave more targeted advertisers. There is even one equilibrium where all entrants in onesub-market mass advertise and all entrants in the other market target advertise. For anexamination of equations (34) and (35), see Appendix C.

Lemma 8. If entrants are indifferent between targeted and mass advertising, then:

Kmixed = KMA = KTA

Proof: Part 1: Proof of Kmixed = KMA

From (34) I have: nMA = nMA+2v∗1

1+θ∗ eb/µ+ +θ∗

1+θ∗

eb/µ+1nTA↔ KMA = 1+αnMA[e

R+b−µµ +e

R−µµ ]

= 1 + αnMA[eR+b−µ

µ + eR−µµ ] + 2αnTAv

∗[ 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ ] = Kmixed

Part 2: Proof of Kmixed = KTA

From (34) I have: 2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1nTA = nMA + 2v∗

11+θ∗ e

b/µ+ θ∗1+θ∗

eb/µ+1nTA

↔ nTA = eb/µ+1

2v∗[ 11+θ∗ e

b/µ+ θ∗1+θ∗ ]

nMA + nTA ↔ KTA = 1 + 2αnTAv∗( 1

1+θ∗eR+b−µ

µ + θ∗

1+θ∗eR−µµ )

↔ KTA = 1 + αnMA[eR+b−µ

µ + eR−µµ ] + 2αnTAv

∗[ 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ ] = Kmixed

Lemmas 6 and 8 gives us that (z∗, v∗) must be unique.

4.4. Equilibrium Uniqueness

Putting the results from sections 4.1, 4.2, and 4.3 together, I have established a uniqueequilibrium:

Proposition 1. Equilibrium is determined by one unique KTA defined by equation (31) andone unique KMA defined by equation (21).

22

(a) When KMA > KTA, there exists one uniqe equilibrium, where:

� There are nMA entrants of each type, where nMA is uniquely defined by equation(20).

� Each entrant chooses its price p = µ, does no marketing research (z = 0), andmass advertises.

(b) When KTA > KMA, there exists one uniqe equilibrium, where:

� There are nTA entrants of each type, where nTA is uniquely defined by (30).

� Each entrant chooses its price p = µ, does z = z∗ marketing research and v = v∗

advertising, where z∗ is uniquely defined by equation (28b) and v∗ is uniquelydefined by equation (27).

(c) When KMA = KTA, there exist multiple potential equilibria. The necessary and suffi-cient conditions for an equilibrium are:

� The total number of entrants of each type solves equations (34) and (35).

� Each entrant chooses its price p = µ, chooses to either target advertise or massadvertise, and is indifferent between targeted and mass advertising.

� Those entrants that target advertise choose z = z∗ and v = v∗, where z∗ isuniquely defined by equation (28b) and v∗ is uniquely defined by equation (27).

Proof: I have established that there are three potential equilibria: (1) All entrants massadvertise, (2) All entrants target advertise, and (3) entrants are indifferent betweentargeted and mass advertising. And I have established that KTA is uniquely defined byequation (31) and KMA is uniquely defined by equation (21).

If all entrants mass advertise, then K = KMA and it must be the case that:v∗[ 1

1+θ∗πM + θ∗

1+θ∗πU ]− F −mz∗ ≤ 0 (or else entrants would target advertise)

↔ F +mz∗ + cv∗ ≥ αµv∗1

1+θ∗ eR+b−µ

µ + θ∗1+θ∗ e

R−µµ

KMA

↔ KMA ≥ αµF+mz∗+cv∗

v∗[ 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ ] ↔ KMA ≥ KTA

If all entrants target advertise then K = KTA and it must be the case that:

12πM + 1

2πU − F ≤ 0 ↔ 1

2[αµ e

R+b−µµ

KTA− c] + 1

2[αµ e

R−µµ

KTA− c]− F ≤ 0

↔ F + c ≥ αµ12

[eR+b−µ

µ +eR−µµ ]

KTA↔ KTA ≥ αµ

2(F+c)[e

R+b−µµ + e

R−µµ ] ↔ KTA ≥ KMA

Lemma 8 shows that if entrants are indifferent between targeted and mass advertis-ing, Kmixed = KMA = KTA.

23

Figure 2: The Dual Problem

v

z

TC

•

z∗

v∗

TR(K1)TR(K2)TR(K3)TR(K4)

large

r K

5. Equilibrium Analysis

In this section, I explore the equilibrium found by Proposition 1 with a particular em-phasis on the Targeted Advertising Equilibrium.

Until now, we have looked at the problem where advertisers choose marketing researchz and advertising v to maximize their profit given the price index K. Now we will belooking at an alternative way of looking at the advertiser’s problem, which will yield severaladditional comparative statics. Note that in equilibrium I must have the advertiser’s total

revenue TR(z, v,K) = αµv1

1+θ∗(z,v) eR+b−µ

µ +θ∗(z,v)

1+θ∗(z,v) eR−µµ

Kequal to its total cost TC(z, v) =

F +mz + cv, or equivalently the zero-profit condition given by equation (29). Further notethat ∂

∂KTR(z, v,K) < 0, which is equivalent to the iso-revenue curve shifting upward and

rightward with an increase in K, as depicted in Figure 2.Therefore in equilibrium, marketing research z, advertising v, and price index K solve

the dual problem:14

maxz,v,K

K (36)

subject to F +mz = v[1

1 + θ∗(z, v)πM +

θ∗(z, v)

1 + θ∗(z, v)πU] (37)

This is consistent with Currier (2002) who made a similar argument to show that thechoices of production inputs maximize K subject to the zero-profit condition in similarmonopolistically competitive models. This yields a number of interesting comparative staticsthat show how similar production substitution and information substitution are.

By the envelop theorem on the dual problem given by equations (36), I have the effect ofchanging the marginal marketing research cost m, the marginal advertising cost c, and the

14 This also works for the Mass or Mixed advertising equilibria.

24

fixed cost F on the price index K:

EK,m = − mz∗

F +mz∗ + cv∗< 0 (38a)

EK,c = − cv∗

F +mz∗ + cv∗< 0 (38b)

EK,F = − F

F +mz∗ + cv∗< 0 (38c)

where EK,ρ is the elasticity of the price index K with respect to changing ρ ∈ {m, c, F}.Therefore the elasticity of price index K with respect to changing any input price (market

research, advertising, or fixed cost) is equal to negative of the proportion of the total costfrom that input. Das (1981) and Currier (2002) found a similar result for the affect forproduction inputs in monopolistic competition.

I use this to solve for how z and x change with marketing research cost:

Ez∗,m =

1+θe−b/µ

2EπM ,m + 1+θ

3−θ (1− EπM ,m)

z∗ lnφ(12− φ

1+φ)

< 0 (39a)

Ex,m =

φ1+φ

1+θe−b/µ

2EπM ,m + 1

21+θ3−θ (1− EπM ,m)

x lnφ(12− φ

1+φ)

< 0 (39b)

where the elasticity of πM with respect to marketing research cost EπM ,m = −EK,m πM+cπM∈

(0, 1), and θ ∈ (0, 3) (by Lemma A-9 on page 35). Notice that when m changes, x changesfor two reasons: (1) z∗ is changing, and (2) v∗ is changing. A percentage increase in m wouldcause an Ez∗,m change in z∗, which would cause a Ex,zEz∗,m change in x, where Ex,z is theelasticity of x with respect to z (holding v constant). Also a percentage increase in m wouldcause an Ev∗,m change in v∗, which would cause a Ex,vEv∗,m change in x, where Ex,v is theelasticity of x with respect to v (holding z constant). Therefore Ex,m = Ex,zEz∗,m + Ex,vEv∗,m,which becomes:

Ev∗,m = Ev,x(Ex,m − Ex,zEz∗,m) (40)

=1 + θ

2g(z, x)

[ φ1+φ− φ+θ

(1+φ)(1+θ)]1+θe−b/µ

2EπM ,m + [1

2− φ+θ

(1+φ)(1+θ)]1+θ3−θ (1− EπM ,m)

v∗ lnφ(12− φ

1+φ)

where dxdm

< 0 is given by equation (39b), ∂x∂z> 0 is given by equation (16), and dz∗

dm< 0 is

given by equation (39a).Equation (39a) shows that z has the classic downward sloping demand curve, despite the

fact that increasing the cost of marketing research indirectly increases the value of advertisingby encouraging less competition. Equation (40) shows that advertising can be either asubstitute or a complement to marketing research: If an advertiser is sending ads to a small

25

targeted group of consumers (which needs to be sufficiently less than half of the consumers),then increasing the cost of marketing research could induce the advertiser to substitutemore ads for less marketing research. However, if the advertiser is sending ads to a lesstargeted group of consumers, then increasing the cost of marketing research would inducethe advertiser to not only do less marketing research but do less advertising as advertisingwould be less effective.

Equations (39a) and (40) also show that: (1) for small m: increasing m increases theamount an advertiser spends on advertising v and marketing research z, and (2) for largem: increasing m decreases the amount an advertiser spends on advertising v and marketingresearch z. This is important because the number of advertisers can be rewritten as:

Targeted andMass AdvertisingEquilibria

: n =µ

2TC(z∗, v∗)[1− 1

K] (41)

Mixed AdvertisingEquilibria

: TC(1, 1)nMA + TC(z∗, v∗)nTA =µ

2[1− 1

K] (42)

Therefore I have that (1) for small m: increasing m decreases n (because TC is increasing)and (2) for large n: increasing m increases n (because TC is decreasing. This is importantbecause the number n of firms affects the welfare analysis of consumers, advertiser platforms,and marketing research firms.

6. Conclusion

In conclusion, I have found several similarities between the advertising-marketing trade-off and the trade-off between production inputs. The Marginal Rate of Information substitu-tion is downward sloping and convex (where firms optimize), similar to the Marginal Rate ofTechnical Substitution. As a result, the cost minimization and profit maximization problemsare similar. I found similar properties of the Price Index for a monopolistically competitivefirm deciding how much advertising and marketing research to use and a monopolisticallycompetitive firm deciding how much inputs to use. This includes the discovery that both thequantities of advertising and marketing research maximize the price index, which is similarto a discovery by Currier (2002) for production inputs.

In future papers, could this be extended to an advertising-production input trade-offor a marketing-production input trade-off? Possibly. Although, changing the quantitiesof advertising and marketing research shift the demand for a product, and changing thequantities of a production input would move us along the demand for a product (as it onlychanges quantity supplied). More likely, this could be extended to an examination of thesubstitution between all demand shifters, which may prove useful as some demand shifters(like advertising and marketing research) are controllable by the firm and other demandshifters (like prices of other goods) are controllable by other entities.

26

Also I done some preliminary work on the welfare implications of changing m. In par-ticular, I found that in industries where the marketing research is controlled by a single adplatform (like Google), privacy regulations would always hurt consumers. The ad platform’sincentive for higher profits would impose stricter privacy controls than a government regu-lator trying to maximize consumer welfare. This is particularly important because many adplatforms act as a monopoly bottleneck to consumers.

Also I have shown that there is one unique Equilibrium (either a mass advertising Equi-librium or a Targeted advertising Equilibrium), with the exception of the razor-edge casewhere marketing research costs are at a threshold value (which yields a continuum of possiblemixed advertising Equilibra). This is important because multiple eqilibria would complicatecomparative static analysis and policy analysis.

Also I have created a very flexible framework that could be used to look at differentaspects of targeted advertising. I have created a monopolistically competitive logit modelwhere it is easy to look at imperfect targeted advertising. In a previous version of this paper,I used a simpler version of the model, where advertisers get one noisy signal on consumers(if they choose to target advertise), to look at the effects of ad taxes, endogenous advertisingprices, ad retention, and shared information on a noisy targeted advertising market. Thispaper is a narrow look at one of the extensions in that previous version of this paper.

Also I have developed the binomial distribution as a way of aggregating multiple signals.Most of the literature on discrete economics focuses on a zero-one decision (for example:Anderson et al., 1992). In this paper, I explored how we can use the binomial distribution toaggregate many zero-one decisions to make an integer decision. While much of the analysisusing the binomial distribution was similar, there were some differences. For example, insteadof first-order equations, I used first-difference equations.

Also this paper emphasizes the need for empirical and theoretical developments of the adannoyance function. This is critical in understanding how privacy regulations affect consumerwelfare. Are consumers more annoyed by the first ad or the second? This empirical questionneeds to be answered for us to understand whether we want to have an intermediate-levelsignal accuracy, or an extreme. And it needs to be answered for us to understand whetherprivacy regulations should limit the information firms can gather or make it more expensivefor firms to gather additional information.

Ad annoyance is a possible explanation for why we value privacy, why we don’t want firmsto know too much information about us. While we may never be able to test empiricallywhy people value privacy, we shouldn’t claim that it has to be purely an intermediate goodor purely from the fear of the criminal use of our information.

Also there is a need to study different marketing research technologies than the binomialmarketing research technology that I have discussed here. For example, if firms were to stopgathering signals from consumers after a fixed number of unmatched signals and advertise toconsumers after a fixed number of matched signals, then the negative binomial distributionwould be more appropriate than the binomial distribution. This would better fit situationswhere firms go through a search process for consumers than gather a fixed number of signalsfrom every consumer. Also continuous signals distributed normally would aggregate to a nor-

27

mal distribution, which might prove easier to handle. Also additional spending on marketingresearch does not necessarily yield additional information on a consumer through additionalsignals. It could be that additional marketing research yields better quality signals. Underthis interpretation any distribution could be used as long as additional spending reduces theerror of a signal.

Also there is potential in adapting this model to other market structures, like monopoly,perfect competition, and oligopoly. Although there has been other research in targetedadvertising in other market structures, the flexibility of this model may prove helpful infurther studies of other markets.

References

Anderson, S. P., de Palma, A., Thisse, J.-F., 1992. Discrete Choice Theory of ProductDifferentiation. MIT Press.

Bergemann, D., Bonatti, A., 2011. Targeting in advertising markets: implications for offlineversus online media. RAND J of Economics 42 (3), 417–443.

Currier, K. M., 2002. Long-run equilibrium in a monopolistically competitive industry. Jour-nal of Economic Research 7 (1), 77–89.

Das, S. P., 1981. Long run behavior of a monopolistically competitive firm. InternationalEconomic Review 22 (1), 159–165.

Dixit, A., Stiglitz, J., 1977. Monopolistic competition and optimum product diversity. Amer-ican Economic Review 67 (3), 297–308.

Esteban, L., Gil, A., Hernandez, J. M., 2001. Informative Advertising and Optimal Targetingin a Monopoly. The Journal of Industrial Economics 49 (2), 161–180.

Esteves, R. B., 2009. Customer Poaching and Advertising. The Journal of Industrial Eco-nomics 57 (1), 112–146.

Galeotti, A., Moraga-Gonzalez, J. L., May 2004. A model of strategic targeted advertising,cESifo Working Paper No. 1196.

Goldfarb, A., Tucker, C., 2011. Search engine advertising: Channel substitution when pricingads to context. Management Science.

Iyer, G., Soberman, D., Villas-Boas, J. M., 2005. The Targeting of Advertising. MarketingScience 24 (3), 461 – 476.

Johnson, J. P., 2013. Targeted Advertising and Advertising Avoidance, forthcoming RANDJ of Economics.

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Luce, R. D., Suppes, P., 1965. Preference, utility, and subjective probability. In: Luce, D.,Bush, R. R., Galanter, E. H. (Eds.), Handbook of Mathematical Psychology. Vol. 3. Wiley,New York, Ch. 19, pp. 249–410.

Shiman, D. R., 1997. The impact of firms‘ increased information about consumers on thevolume and targeting of direct marketing, sSRN Working Paper.

Silberberg, E., 1974. The theory of the firm in “long-run” equilibrium. The American Eco-nomic Review 64 (4), 734–741.

Villas-Boas, J. M., 1999. Dynamic competition with customer recognition. RAND J. Econom30, 604631.

Villas-Boas, J. M., 2004. Price cycles in markets with customer recognition. RAND J.Econom 35, 486501.

Appendix A. Proofs of Equations

The following are the algebraic proofs for the equations in the text as needed.

Proof of equation (4): By Baye’s Theorem: Pr(τi = tj|xij = k) =12g(zj ,k)

1+θij2

g(zj ,k)= 1

1+θij

↔ Pr(τi = 1− tj|θij) = 1− Pr(τi = tj|θij) =θij

1+θij

Proof of equation (8): From equation (7), I have that the first-order condition ∂∂pjπ(pj, tj, θij) =

0 is equivalent to pj = µ:

∂∂pjπ(pj, tj, θij) = 0 ↔ α[

11+θij

eR+b−pj

µ

Ktj+

θij1+θij

eR−pjµ

K1−tj]− p

µ∗ α[

11+θij

eR+b−pj

µ

Ktj+

θij1+θij

eR−pjµ

K1−tj] = 0

↔ 1− pjµ

= 0 ↔ pj = µ

Proof of equation (16): 0 = dvdxdx+ dv

dzdz ↔ dx

dz= − dv/dz

dv/dx

↔ dxdz

= −−12θ+φ1+φ

g(z,x)

1+θ2g(z,x)

↔ Equation (16) (by equations (14) and (15))

Proof of equation (17): 12G1(z, x) = 1

2[G1(z, x) +Gφ(z, x)] G1(z,x)

G1(z,x)+Gφ(z,x)

= v G1(z,x)G1(z,x)+Gφ(z,x)

= v 11+θ∗(z,v)

(by equation (13))

12Gφ(z, x) = 1

2[G1(z, x) +Gφ(z, x)]

Gφ(z,x)

G1(z,x)+Gφ(z,x)

= vGφ(z,x)

G1(z,x)+Gφ(z,x)= v θ∗(z,v)

1+θ∗(z,v)(by equation (13))

29

Proof of equation (18): µ2(F+c)

− 1

α[eR+b−µ

µ +eR−µµ ]

> 1 ↔ µ2(F+c)

> 1+α[eR+b−µ

µ +eR−µµ ]

α[eR+b−µ

µ +eR−µµ ]

↔ 2(F+c)µ

< α[eR+b−µ

µ +eR−µµ ]

1+α[eR+b−µ

µ +eR−µµ ]↔ F + c < µ

2α[e

R+b−µµ +e

R−µµ ]

1+α[eR+b−µ

µ +eR−µµ ]

↔ αµ[12eR+b−µ

µ + 12eR−µµ

1+α[eR+b−µ

µ +eR−µµ ]

]− F − c > 0.

Let’s assume that one firm enters as type 1 − t, then it is sufficient to show that itis profitable for another firm to enter as type t to prove the lemma. Lets call the firmthat potentially enters as type t firm t and the firm that enters as type 1− t firm 1− t.Let z1−t and v1−t be firm 1− t’s choice of z and v.

If firm t were to send ads to all consumers, then

Kt = 1 + α[eR+b−µ

µ + 2θ∗(z1−t,v1−t)1+θ∗(z1−t,v1−t)

v1−teR−µµ ] < 1 + α[e

R+b−µµ + e

R−µµ ], and

K1−t = 1 + α[eR−µµ + 2

1+θ∗(z1−t,v1−t)v1−te

R+b−µµ ] < 1 + α[e

R+b−µµ + e

R−µµ ].

Then Π(t, 0, 0) = αµ[12eR+b−µ

µ

Kt+

12eR−µµ

K1−t]− F − c > αµ[

12eR+b−µ

µ + 12eR−µµ

1+α[eR+b−µ

µ +eR−µµ ]

]− F − c, which I

have shown is greater than 0. Therefore firm t would enter the martet. And by symmetryfirm 1− t would enter the martet.

Proof of equation (20): By Lemma 3, K0 = K1

↔ 1 + α(n0eR+b−µ

µ + n1eR−µµ ) = 1 + α(n0e

R−µµ + n1e

R+b−µµ ) ↔ n0 = n1 ≡ nMA.

By free entry, I have the zero profit condition

↔ F = 12αµ e

R+b−µµ +e

R−µµ

K− c ↔ F + c = 1

2αµ e

R+b−µµ +e

R−µµ

1+αnMA[eR+b−µ

µ +eR−µµ ]

↔ 2(F+c)µ

= α[eR+b−µ

µ +eR−µµ ]

1+αnMA[eR+b−µ

µ +eR−µµ ]↔ µ

2(F+c)= 1

α[eR+b−µ

µ +eR−µµ ]

+ nMA ↔ (20)

Proof of equation (21): By equation (20), I have KMA = 1 + αnMA[eR+b−µ

µ + eR−µµ ]

= 1 + α

(µ

2(F+c)− 1

eR+b−µ

µ +eR−µµ

)[e

R+b−µµ + e

R−µµ ] ↔ (21)

Proof of equation (24):

MRz = −αµ2g(z, x(z, v))

φ1+φ

eR+b−µ

µ + 11+φ

θeR−µµ

K+ 1

2[ φ1+φ

+ 11+φ

θ]g(z, x(v, z))MRv

= −αµ2g(z, x(z, v))

φ1+φ

eR+b−µ

µ + 11+φ

θeR−µµ

K+ 1

2g(z, x(z, v))[ φ

1+φ+ 1

1+φθ]∗αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K

= −αµ2g(z, x(z, v))

φ1+φ

eR+b−µ

µ + 11+φ

θeR−µµ

K+ αµ

2θ+φ1+φ

g(z, x(z, v))1

1+θeR+b−µ

µ + θ1+θ

eR−µµ

K

30

= αµ2g(z, x(z, v))[( θ+φ

1+φ1

1+θ− φ

1+φ) ∗ e

R+b−µµ

K+ ( θ+φ

1+φθ

1+θ− θ

1+φ) ∗ e

R−µµ

K]

= αµ2g(z, x(z, v))[ (θ+φ)−φ(1+θ)

(1+φ)(1+θ)eR+b−µ

µ

K+ θ(θ+φ)−θ(1+θ)

(1+φ)(1+θ)eR−µµ

K] ↔ Equation (24)

Proof of equation (25): MRISzv = MRzMRv

=αµ2

1−φ1+φ

θ1+θ

g(z,x(z,v)) eR+b−µ

µ −eR−µµ

K

αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K

↔ (25)

Proof of equation (27): MRv = c ↔ αµ1

1+θeR+b−µ

µ + θ1+θ

eR−µµ

K= c ↔ π(θ) = 0 and

θ = πM−πU

Proof of equation (28a): m = αµ2

1−φ1+φ

θ1+θ

g(z, x(z, v)) eR+b−µ

µ −eR−µµ

K

↔ m = αµ2

1−φ1+φ

πMπM−πU

g(z, x(z, v)) eR+b−µ

µ −eR−µµ

K(by equation (27))

↔ m = 12

1−φ1+φ

πMπM−πU

g(z, x(z, v))(πM − πU) ↔ (28b)

Proof of equation (30): By Lemma 3, K0 = K1 = 1+2αv∗( 11+θ∗

n0eR+b−µ

µ + θ∗

1+θ∗n1e

R−µµ )

= 1 + 2αv∗( 11+θ∗

n1eR+b−µ

µ + θ∗

1+θ∗n0e

R−µµ ) ↔ n0 = n1 ≡ nTA.

By free entry, I have the zero profit condition:

F +mz∗ = v∗[ 11+θ∗

πM + θ∗

1+θ∗πU] = v∗[ 1

1+θ∗(αµ e

R+b−µµ

K− c) + θ∗

1+θ∗(αµ e

R−µµ

K− c)]

↔ F +mz∗ + cv∗ = αµv∗1

1+θ∗ eR+b−µ

µ + θ∗1+θ∗ e

R−µµ

K= αµv∗

11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ

1+2αnTAv∗( 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ )

↔ µ2(F+mz∗+cv∗)

= nTA + 1

2αv∗[ 11+θ∗ e

R+b−µµ + 1

1+θ∗ eR−µµ ]↔ (30)

Proof of equation (31):

By equation (30), I have: KTA = 1 + 2αnTAv∗( 1

1+θ∗eR+b−µ

µ + θ∗

1+θ∗eR−µµ )

= 1+2α( µ2(F+mz∗+cv∗)

− 1

2αv∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]

)v∗( 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ )↔ (31)

Proof of equations (32a) and (32b):

Let K(x, z) ≡ αµF+mz+cv

v[ 11+θ∗(z,v)

eR+b−µ

µ + θ∗(z,v)1+θ∗(z,v)

eR−µµ ].

By equation (24):

∂∂zK(x, z) = αµ

12

1−φ1+φ

θ(z,x)

1+θ(z,x)g(z,x(z,v))(e

R+b−µµ −e

R−µµ )∗(F+mz+cv)−v[ 1

1+θ∗(z,v) eR+b−µ

µ +θ∗(z,v)

1+θ∗(z,v) eR−µµ ]∗m

(F+mz+cv)2

31

= K(z,v)F+mz+cv

[MRz −m] → By the first-order condition of z: (32a)and by similar reasoning (32b).

Proof of equation (33): By Parts:Part 1: Characteristic of an Interior SolutionKTA exists if and only if (v∗, z∗) is an interior solution. From Lemma 7, I have that thesolution for the first-order condition for v, which is given by equation (27), is always aninterior solution. Therefore KTA exists if and only if z∗ is an interior solution for z, orequivalently when equation (28b) is satisfied for at least one z∗ ≥ 1.Part 2: At least one threshold value of m exists

Let K(z, v) ≡ αµF+mz+cv

v[ 11+θ∗(z,v)

eR+b−µ

µ + θ∗(z,v)1+θ∗(z,v)

eR−µµ ]. By equations (32a) and (32b),

I have that ∂∂mK(z > 0, v) < 0, so ∂

∂mπM(K = K(z > 0, v)) > 0.a By l’Hopital’s

Rule: limm→∞πMm

= limm→∞dπMdm

= 0. Therefore m > 12(1−φ

1+φ)g(z, x(z, v))πM for all

(z, v), when m → ∞. Also when marketing research is free (i.e. m = 0), I have thateach entrant will buy as many free signals as possible (i.e. z = ∞), or equivalentlym < 1

2(1−φ

1+φ)g(z, x(z, v))πM for all (z, v).

Therefore by the intermediate value theorem, there exists at least one threshold valueof m, which is the border between values of m where KTA exists and values of m whereKTA does not exist.Part 3: Characteristic of a threshold value of mFrom Lemma 7, I have that any such threshold value of m would be where z = 1 andx = 0 (i.e. v = 1

2). For this value of m, KTA = K(z = 1, v = 1

2)

= αµF+m+c/2

12[ 11+φ

eR+b−µ

µ + φ1+φ

eR−µµ ], because all firms would set z = 1 and v = 1

2. Also

by the first order condition of z, I have: m = 12(1−φ

1+φ)g(z = 1, x = 0)πM = 1

2(1−φ

1+φ) 1

1+φπM

= 12(1−φ

1+φ) 1

1+φ[αµ e

R+b−µµ

K(z=1,v= 12

)− c] = 1

2(1−φ

1+φ) 1

1+φ[αµ e

R+b−µµ

αµF+m+c/2

12

[ 11+φ

eR+b−µ

µ + φ1+φ

eR−µµ ]− c]

↔ m =

12

1−φ(1+φ)2

[4(F+c/2) eb/µ

11+φ

ebµ+φ

1+φ

−c]

1−2 1−φ(1+φ)2

eb/µ

11+φ

ebµ+φ

1+φ

.

Because this is unique, there is one threshold value. When m = 0, I have shown thatKTA exists, and when m→∞, I have shown that KTA does not exist. Therefore I havethat KTA exists if and only if equation (33) is satisfied.

a I will examine this result more in section ??.

Proof of equations (34) and (35): Part 1: Proof of (34)

I have that K0 = 1 + α[n0MAe

R+b−µµ + n1

MAeR−µµ ] + 2αv∗[ 1

1+θ∗n0TAe

R+b−µµ + θ∗

1+θ∗n1TAe

R−µµ ]

and that K1 = 1 + α[n1MAe

R+b−µµ + n0

MAeR−µµ ] + 2αv∗[ 1

1+θ∗n1TAe

R+b−µµ + θ∗

1+θ∗n0TAe

R−µµ ].

From Lemma 3, I have that:

32

K = K0 = K1 = K0+K1

2= 1 + αnMA[e

R+b−µµ + e

R−µµ ] + 2αnTAv

∗[ 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ ].

The free entry condition of the mass-advertising entrants is:

F = 12[πM + πU] ↔ F + c = αµ

12

(eR+b−µ

µ +eR−µµ

K)

↔ F + c = αµ12

(eR+b−µ

µ +eR−µµ )

1+αnMA[eR+b−µ

µ +eR−µµ ]+2αnTAv∗[ 1

1+θ∗ eR+b−µ

µ + θ∗1+θ∗ e

R−µµ ]

↔ µ2(F+c)

=1+αnMA[e

R+b−µµ +e

R−µµ ]+2αnTAv

∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]

α[eR+b−µ

µ +eR−µµ ]

↔ µ2(F+c)

= 1

α[eR+b−µ

µ +eR−µµ ]

+ nMA + 2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1nTA

↔ nMA + 2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1nTA = nMA, where nMA is defined in equation (20).

The free entry condition of the targeted-advertising entrants is:

F +mz∗ = v∗[ 11+θ∗

πM + θ∗

1+θ∗πU] ↔ F +mz∗ + cv∗ = αµv∗

11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ

K

↔ F +mz∗ + cv∗ = αµv∗1

1+θ∗ eR+b−µ

µ + θ∗1+θ∗ e

R−µµ

1+αnMA[eR+b−µ

µ +eR−µµ ]+2αnTAv∗[ 1

1+θ∗ eR+b−µ

µ + θ∗1+θ∗ e

R−µµ ]

↔ µ2(F+mz∗+cv∗)

=1+αnMA[e

R+b−µµ +e

R−µµ ]+2αnTAv

∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]

2αv∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]

↔ µ2(F+mz∗+cv∗)

= 1

2αv∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]

+ eb/µ+1

2v∗[ 11+θ∗ e

b/µ+ θ∗1+θ∗ ]

nMA + nTA

↔ nMA + 2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1nTA = 2v∗

11+θ∗ e

b/µ+ θ∗1+θ∗

eb/µ+1nTA, where nTA is defined in equation

(30).Part 2: Proof of (35)

K0 = K1

↔ 1 + α[n0MAe

R+b−µµ + n1

MAeR−µµ ] + 2αv∗[ 1

1+θ∗n0TAe

R+b−µµ + θ∗

1+θ∗n1TAe

R−µµ ]

= 1 + α[n1MAe

R+b−µµ + n0

MAeR−µµ ] + 2αv∗[ 1

1+θ∗n1TAe

R+b−µµ + θ∗

1+θ∗n0TAe

R−µµ ]

↔ 0 = (n1MA−n0

MA)[eR+b−µ

µ −eR−µµ ]+(n1

TA−n0TA)2v∗[ 1

1+θ∗eR+b−µ

µ − θ∗

1+θ∗eR−µµ ]↔ (35)

Proof of equations (38a) to (38c): By the envelop theorem on the dual problem givenby equations (36) and (29), I have that dKTA

dm= ∂KTA

∂m.

Equation (31) → dKTAdm

= −z∗ αµ(F+mz∗+cv∗)2

v∗[ 11+θ∗

eR+b−µ

µ + θ∗

1+θ∗eR−µµ ] ↔ Equation (38a)

and by symmetry Equations (38b) and (38c).

Proof of equations (39a) and (39b): I have that EπM ,m = EπM ,K∗EK,m = −αµ eR+b−µ

µ

K2KπM∗

EK,m = −EK,m πM+cπM

> 0

and E−πU ,m = αµ eR−µµ

K2KπU∗ EK,m = EK,m πU+c

πU= −θe−b/µEπM ,m < 0

33

and dπUdm

= −αµ eR−µµ

K2 ∗ −ϕmKTAm

= e−b/µ dπMdm

> 0.

Therefore: Eθ,m = EπM ,m − E−πU ,m = (1 + θe−b/µ)EπM ,m > 0 (by equation (27))

θ = φz∗−2x → Eθ,m = lnφ ∗ ( dz∗

d lnm− 2 dx

d lnm)

↔ 1+θe−b/µ

2 lnφEπM ,m = 1

2z∗Ez∗,m − xEx,m < 0 (S1)

Equation (28b) → EπM ,m = 1− Eg(z∗,x),m = 1− 1g(z∗,x)

dg(z∗,x)d lnm

= 1− 1g(z∗,x)

([g(z∗ + 1, x)− g(z∗, x)] dz∗

d lnm+ [g(z∗, x)− g(z∗, x− 1)] dx

d lnm)

By Lemma A-1 on page 34:EπM ,m = 1− 1

g(z∗,x)([( 1

1+φg(z∗, x)+ φ1+φg(z∗, x−1))−g(z∗, x)] dz∗

d lnm+[g(z∗, x)−g(z∗, x−1)] dxd lnm)

= 1 + lnφ3−θ1+θ

( φ1+φ

dz∗

d lnm− dx

d lnm) (by Lemma A-9 on page 35)

↔ 1+θlnφ(3−θ)(EπM ,m − 1) = φ

1+φz∗Ez∗,m − xEx,m (S2)

(S1)− (S2)↔ 1+θe−b/µ

2 lnφEπM ,m + 1+θ

lnφ(3−θ)(1−EπM ,m) = (12− φ

1+φ)z∗Ez∗,m ↔ Equation (39a)

φ1+φ

(S1)− 12(S2) ↔ φ

1+φ1+θe−b/µ

2 lnφEπM ,m − 1

21+θ

lnφ(3−θ)(1− EπM ,m) = (12− φ

1+φ)xEx,m

↔ Equation (39b)

Note that by equations (38a) and (38b): −EK,m < 1 + EK,m < 1 − cπM+c

= πMπM+c

↔1 > −EK,m πM+c

πM= EπM ,m

Proof of equation (40): Ev∗,m = Ev,x(Ex,m − Ex,zEz∗,m) = ∂v∂x

xv∗

(Ex,m − ∂x∂z

z∗

xEz∗,m)

= 1+θ2g(z, x) x

v∗(

φ1+φ

1+θe−b/µ2

EπM,m+ 12

1+θ

3−θ(1−EπM,m)

x lnφ( 12− φ

1+φ)

− φ+θ

(1+φ)(1+θ)z∗

x

1+θe−b/µ2

EπM,m+ 1+θ

3−θ(1−EπM,m)

z∗ lnφ( 12− φ

1+φ)

)

(By equations (14), (39b), (16) , and (39a))↔ Equation (40).

Note: φ1+φ− φ+θ

(1+φ)(1+θ)= φ(1+θ)−(φ+θ)

(1+φ)(1+θ)= −1−φ

1+φθ

1+θ< 0

and θ Q 1 ↔ (1− φ)(1− θ) R 0 ↔ (1 + φ)(1 + θ) R 2(φ+ θ) ↔ 12R φ+θ

(1+φ)(1+θ)

Proof of equations (41) and (42): If targeted advertising equilibrium:

Equation (30) ↔ nTA = µ2TC(z∗,v∗)

− µ/K

2αµv∗[ 11+θ∗ e

R+b−µµ + θ∗

1+θ∗ eR−µµ ]/K

↔ Equation (41) (By the zero-profit condition)

If mass advertising equilibrium:Equation (20) ↔ nMA = µ

2TC(z∗=0,v∗=1)− µ/K

2αµ[eR+b−µ

µ +eR−µµ ]/K

↔ Equation (41) (By the zero-profit condition)

34

If mixed advertising equilibrium:Equation (34)↔ (F +c)nMA = (F +c)nMA+(F +mz∗+cv∗)nTA = (F +mz∗+cv∗)nTA↔ Equation (42) (By what I just found for targeted and mass advertising equilibria)

Appendix B. Additional Proofs

Lemma A-1. g(x, z) = φ1+φ

g(x− 1, z − 1) + 11+φ

g(x, z − 1)

Proof: g(x, z) =(zx

) (1

1+φ

)z−x (φ

1+φ

)x= [(z − 1x− 1

)+(z − 1x

)](

11+φ

)z−x (φ

1+φ

)x= φ

1+φ∗(z − 1x− 1

) (1

1+φ

)(z−1)−(x−1) (φ

1+φ

)x−1

+ 11+φ∗(z − 1x

) (1

1+φ

)(z−1)−x (φ

1+φ

)x= φ

1+φg(x− 1, z − 1) + 1

1+φg(x, z − 1)

Lemma A-2. g(x, z)− g(x− 1, z) =(z+1)−x/ φ

1+φ

(z+1)−x g(x, z)

Proof: g(x, z) =(zx

) (1

1+φ

)z−x (φ

1+φ

)x↔ g(x− 1, z) =

(z

x− 1

) (1

1+φ

)z−(x−1) (φ

1+φ

)x−1

= 1φ

xz−(x−1)

g(x, z)

↔ g(x, z)− g(x− 1, z) = [1− 1φ

xz−(x−1)

]g(x, z) =(z+1)−x/ φ

1+φ

(z+1)−x g(x, z)

Lemma A-3. sgn(d2Πdz2

(z∗, v∗)) = sgn(θ(z, x)g(z, x)− θ(z, x− 1)g(z, x− 1))

Proof: Define g(z, x) ≡ θ(z, x)g(z, x) = Pr(xij = x|τi = 1− tj).In the proof of Lemma 6, I showed that:

d2Πdz2

=([αµ

21−φ1+φ

θ1+θ

eR+b−µ

µ −eR−µµ

K] + 21+θ

2[αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K− c]

)ddzg(z, x(z, v))

=([αµ

21−φ1+φ

11+θ

eR+b−µ

µ −eR−µµ

K] + 21+θ

2θ[αµ

11+θ

eR+b−µ

µ + θ1+θ

eR−µµ

K− c]

)ddzg(z, x(z, v))

→ sgn(d2Πdz2

(z∗, v∗)) = sgn(m ddzg(z,x(z,v))

g(z,x(z,v))) (because dΠ

dz= 0 and dΠ

dv= 0 in equilibrium)

↔ sgn(d2Πdz2

(z∗, v∗)) = sgn( ddzg(z, x(z, v))) (because m > 0 and g(z, x(z, v)) > 0)

↔ Because z − 2x is constant:sgn(d2Π

dz2(z∗, v∗)) = sgn([g(z + 1, x)− g(z, x)] + 2[g(z, x)− g(z, x− 1)])

↔ By Lemma A-1 on pg. 34:sgn(d2Π

dz2(z∗, v∗)) = sgn([( 1

1+φg(z, x− 1) + φ

1+φg(z, x))− g(z, x)] + 2[g(z, x)− g(z, x− 1)])

↔ sgn(d2Πdz2

(z∗, v∗)) = sgn((2− 11+φ

)[g(z, x)− g(z, x− 1)])

↔ sgn(d2Πdz2

(z∗, v∗)) = sgn(g(z, x)− g(z, x− 1)) (because 2 > 11+φ

)

Lemma 9. I have that:

35

(a) h(z, v) ≡ g(z,x(z,v))−g(z,x(z,v)−1)g(z,x(z,v))

= lnφ(1−θ1+θ

mMRz

+ 21+θ

)

(b) h∗ ≡ h(z∗, v∗) = lnφ3−θ1+θ

< 0

(c) θ(z∗, v∗) ∈ {0, 3}

where m ≡ 12(1−φ

1+φ)g(z, x(z, v))πM.

Proof: Because θ = φz−2x:∂θ∂v

= θ ∗ −2 lnφ ∗ ∂x∂v

= −2θ lnφ1+θ2g(z,x)

> 0 (by equation (14))

∂θ∂z

= θ ∗ lnφ ∗ [1− 2∂x∂z

] = θ lnφ[1− 2 φ+θ

(1+φ)(1+θ)] = θ lnφ1−φ

1+φ1−θ1+θ

(by equation (16))

Equation (23) → ∂∂zMRv = −αµ

1(1+θ)2

(eR+b−µ

µ −eR−µµ )

K∗ θ lnφ1−φ

1+φ1−θ1+θ

= αµ1

(1+θ)(eR+b−µ

µ −eR−µµ )

K∗ −θ lnφ1−φ

1+φ1−θ

(1+θ)2= −θ[αµ e

R−µµ

K−MRv] ∗ lnφ1−φ

1+φ1−θ

(1+θ)2

= lnφ1−φ1+φ

1−θ(1+θ)2

∗ πMNote that πM = πM in equilibrium by the first-order condition of v and equation (27).Equation (24) →∂∂vMRz = αµ

21−φ1+φ

1(1+θ)2

g(z, x) eR+b−µ

µ −eR−µµ

K∗ −2θ lnφ

1+θ2g(z,x)

+αµ2

1−φ1+φ

θ1+θ

∂∂vg(z, x(z, v)) e

R+b−µµ −e

R−µµ

K

= MRz[−2 lnφ

(1+θ)2

2g(z,x)

+ h ∗ ∂x∂v

] = MRz1+θ2g(z,x)

[−2 lnφ

1+θ+ h] (by equation (14))

By Young’s theorem ∂∂zMRv = ∂

∂vMRz ↔ lnφ1−φ

1+φ1−θ

(1+θ)2∗ πM = MRz

1+θ2g(z,x)

[−2 lnφ

1+θ+ h]

= lnφ1−θ1+θ

mMRz

= −2 lnφ

1+θ+ h

Note that m = m and therefore MRz in equilibrium by equation (28b).

↔ h = lnφ(1−θ1+θ

mMRz

+ 21+θ

). In equilibrium m = MRz → h∗ = lnφ3−θ1+θ

, which has to beless than zero by Lemma 7.

Lemma A-4. d2KTAdm2 > 0

Proof: Currier (2002, p. 79-82)Silberberg’s (1974) primal-dual formulation of the dual problem given by equations (36)and (29) would be:

maxz,v,K,F,m,c

K −K(F,m, c)

subject to TR(z, v,K)− [F +mz∗ + cv∗] = 0

where K(F,m, c) would be the optimal value of K as a function of F , m, and c. Notethat for a fixed K, z, and v, that this problem is the same as minimizing K(F,m, c)subject to a linear constraint. Because dK

dm< 0 and in a similar fashion dK

dc< 0 and

36

dKdF

< 0, K(F,m, c) must be locally convex in (F,m, c) at the equilibrium values of z∗

and v∗.

Appendix C. Examination of equations (34) and (35)

During the special case where KMA = KTA, then and only then can I have a mixtureof mass advertisers and targeted advertisers. Equations (34) and (35) define the sufficientand necessary conditions for the number of each type of advertiser in each sub-market.It describes the trade-offs between mass and targeted advertiser in each sub-market andbetween each sub-market. It is straightforward to show that the slopes of these trade-offsare not equal.

Lemma B-1. 2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1< 2v∗

11+θ∗ e

b/µ− θ∗1+θ∗

eb/µ−1

Proof: θ∗ < 1 ↔ 2eb/µ θ∗

1+θ∗< 2eb/µ 1

1+θ∗

↔ (eb/µ − 1)( 11+θ∗

eb/µ + θ∗

1+θ∗) < (eb/µ + 1)( 1

1+θ∗eb/µ − θ∗

1+θ∗) ↔ The Lemma

From this, I have that the number of each type of advertiser in each sub-market canbe represented by Figure C.1(a). There is a line between (nMA, 0) and (0, nTA), defined

by equation (34). This line has a slope of −2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1. The point (nMA, nTA), which

represents the average number of each type of advertiser in each sub-market, must be on thisline. How much (n0

MA, n0TA) and (n1

MA, n1TA) deviate from (nMA, nTA) is defined by equation

(35). This defines a line (given by the dashed line) that runs through (nMA, nTA) with a

slope of −2v∗1

1+θ∗ eb/µ− θ∗

1+θ∗

eb/µ−1. By Lemma B-1, the slope of this line must be steeper than the

slope of the line defined by equation (34).This creates a trapezoidal range of possible (n0

MA, n0TA) and (n1

MA, n1TA) shown in Figure

C.1(b). Given (n0MA, n

0TA), there is one unique (n1

MA, n1TA) (and visa versa). Both points

must be on the same line with a slope of −2v∗1

1+θ∗ eb/µ− θ∗

1+θ∗

eb/µ−1, given by the dashed lines. And

both points must be equidistant from the solid black line through (nMA, 0) and (0, nTA).One extreme combination is (2nMA, 0) and (0, 2nTA). where all entrants in one sub-

market mass advertise and all entrants in the other market target advertise.

nTA =e2b/µ − 1

4v∗[ 11+θ∗

e2b/µ − θ∗

1+θ∗]nMA (C.1)

nMA = 2v∗1

1+θ∗eb/µ − θ∗

1+θ∗

eb/µ − 1nTA >

nMA

2(C.2)

37

Proof: From equation (34), I have: nMA = nMA + 2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1nTA.

From equation (35), I have: − nMA

nTA= −2v∗

11+θ∗ e

b/µ− θ∗1+θ∗

eb/µ−1↔ nMA = 2v∗

11+θ∗ e

b/µ− θ∗1+θ∗

eb/µ−1nTA.

Therefore: nMA = 2v∗1

1+θ∗ eb/µ− θ∗

1+θ∗

eb/µ−1nTA + 2v∗

11+θ∗ e

b/µ+ θ∗1+θ∗

eb/µ+1nTA = 4v∗

11+θ∗ e

2b/µ− θ∗1+θ∗

e2b/µ−1nTA

↔ Equation (C.1)

↔ nMA = 2v∗1

1+θ∗ eb/µ− θ∗

1+θ∗

eb/µ−1∗ nTA

=

11+θ∗ e

b/µ− θ∗1+θ∗

eb/µ−11

1+θ∗ eb/µ− θ∗

1+θ∗

eb/µ−1+

11+θ∗ e

b/µ+ θ∗1+θ∗

eb/µ+1

nMA >nMA

2(by Lemma B-1)

38

Figure C.1: Possible Mixed Advertising Equilibrium Number of Entrants of Each Type

nMA, nMA, n0MA, n

1MA

nTA, nTA, n0TA, n

1TA

•nMA

•nTA

•

•

•

slope = −2v∗1

1+θ∗ eb/µ− θ∗

1+θ∗

eb/µ−1

slope = −2v∗1

1+θ∗ eb/µ+ θ∗

1+θ∗

eb/µ+1

n0MA

n1MA

nMA

∆nMA

∆nMA

n0TA n1

TAnTA

∆nTA∆nTA

(a) One Possible Equilibrium

nMA, nMA, n0MA, n

1MA

nTA, nTA, n0TA, n

1TA

•nMA

•nTA

•2nMA

•2nTA

•

nTA

nMA

(b) Range of Possible Equilibria

39