Uniform and Targeted Advertising with Shoppers and

Asymmetric Loyal Market Shares

Michael Arnold,∗Chenguang Li†and Lan Zhang‡

October 9, 2012

Preliminary and Incomplete

Keywords: informative advertising, targeted advertising, uniform advertising, asymmetric

markets, shoppers, loyal customers

Abstract

This paper explores the strategic tradeoff between advertising and pricing strategies when

firms have asymmetric loyal market segments and also can compete for shoppers who purchase

at the lowest advertised price. Three advertising structures consistent with real world settings

are considered. In the first setting firms are limited to advertising campaigns that reach the

entire market and present all consumers with a uniform price. The analysis is then extended

to allow firms to target ads to specific market segments (either with both firms targeting ads

or with only one firm having the ability to target). In all cases our analysis demonstrates how

asymmetric loyal market segments impact the equilibrium tradeoff between advertising intensity

and price competition in interesting ways not revealed by the existing literature which largely

assumes firms compete in a symmetric environment.

∗Department of Economics, Alfred Lerner College of Business and Economics, University of Delaware, Newark,DE 19716, marnold@udel.edu†JP Morgan Chase, chenguang.x.li@chase.com‡Research Institute of Economics and Management, Southwestern University of Finance and Economics, Chengdu,

China, zhanglan@swufe.edu.cn

1

1 Introduction

A critical question facing firms when making advertising and pricing decisions in order to capture

shoppers is how the advertising and pricing strategy adopted by the firm impacts overall price

competition and equilibrium profits. In markets with loyal customers and shoppers, advertising a

low price to capture the shopping segment of the market may result in Bertrand price competition

and zero profits for all firms in the market. More strategic alternatives require firms to seek

a balance between advertising intensity and price competitiveness. An increase in advertising

intensity (holding one’s pricing strategy fixed) might induce increasingly competitive pricing from

other players in the market. Similarly, a change in one’s pricing strategy (to a less aggressive pricing

distribution, for example), might impact the competition’s equilibrium advertising intensity. This

paper explores the strategic trade-off between advertising and pricing strategies when firms have

asymmetric loyal market segments and also can compete for shoppers who purchase at the lowest

advertised price. Three advertising structures consistent with real world settings are considered. In

the first setting firms are limited to “uniform”advertising campaigns that reach the entire market

and present all consumers with the same price. We then consider options to target ads to specific

segments (either with both firms targeting ads or with only one firm having the ability to target). In

all cases our analysis demonstrates how the presence of asymmetric loyal segments has interesting

implications for the equilibrium trade-off between advertising intensity and price competition.

The main contribution of our analysis stems from incorporating asymmetric firms in an envi-

ronment requiring informative price advertising. Most of the literature on informative advertising

and pricing is restricted to the case of symmetric loyal market shares. The literature on promo-

tional pricing strategies includes analyses of asymmetric firms, but typically does not consider the

role of informative advertising to communicate prices to consumers. Rather, price competition

occurs with some consumers (“shoppers”) knowing all prices and others (“uninformed”or “loyal”

consumers) only observing prices charged by a subset of the firms in the market. Our analysis

enhances understanding of informative advertising and price competition by simultaneously con-

sidering asymmetries consistent with real-world market structures with the mechanism by which

price information is communicated to consumers. For example, our analysis demonstrates that if

consumers learn about pricing through uniform price advertisements distributed to all consumers,

2

then the firm with the larger loyal market will price less competitively but advertise more aggres-

sively than the firm with the smaller loyal share. However, if a firm can target advertising to its

own loyal segment and the shopping segment of the market, then the firm with the large loyal share

advertises less intensely and prices more aggressively than the firm with the smaller share.

There is a large literature exploring price competition without advertising in both symmetric

and asymmetric settings. For example, in Varian’s (1980) model of sales symmetric firms compete

in price in a market with both informed and uninformed consumers. The uninformed consumers are

allocated equally across firms, and the informed consumers purchase from the firm with the lowest

price. Varian demonstrates that firms adopt a symmetric mixed pricing strategy and earn positive

expected profits in equilibrium. Baye, Kovenock and de Vries (1992) extend Varian’s model to

the case of n > 2 firms and show that while there are multiple, asymmetric Nash equilibria, there

is a unique symmetric subgame perfect equilibrium. Narasimhan (1988) was among the first to

consider price competition between asymmetric firms. He modeled a duopoly market in which the

firms have asymmetric loyal market segments and also compete for shoppers who purchase at the

lowest price. His analysis finds that the firm with the larger loyal share prices less competitively

in equilibrium than the firm with the small loyal share. In particular, the firm with the large

loyal share is less likely to offer a discount off the monopoly price, but offers the same average

discount when it does discount from the monopoly price. Our analysis demonstrates that when

consumers only learn of a firm’s price through uniform advertising, the firm with the larger share

prices less aggressively and may offer a smaller average discount off the monopoly price, when it

does discount, than the firm with the smaller share. This occurs because the firm with the smaller

share may adopt a mixed advertising strategy which makes it less of a competitive threat than

in Narasimhan’s model in which firms do not make advertising decisions. Others have built on

Narasimhan’s work in a price competition framework with no advertising. Deneckre, Kovenock

and Lee (1992) consider a two period price leadership duopoly game and show that the firm with

the smaller loyal share prefers to be the follower. Kocas and Kiyak (2008) consider n > 2 firms

and find that only the firms with the two smallest loyal markets compete for shoppers by offering

discounts off the monopoly price. All other firms charge the monopoly price and sell only to their

loyal customers. Jing and Wen (2008) show how Narasimhan’s results change if loyalty to the firm

with the larger share is suffi ciently limited (so that consumers loyal to the larger firm switch to the

3

competitor if the price differential exceeds a threshold). None of these extensions consider the role

of informative advertising to communicate price information.

Another strand of literature focuses on informative price advertising in a symmetric setting.

This literature contains two branches. One considers the presence of a gatekeeper through which

prices can be advertised to the shopping segment of consumers who only purchase if the product is

advertised, while loyal customers always purchase the product from their preferred firm regardless

of whether an advertisement is posted through the gatekeeper. Baye and Morgan (2001) consider

a clearinghouse model with n identical firms which each serve a local market and can compete

for shoppers by advertising through a gatekeeper. Equilibrium is characterized by firms adopting

symmetric advertising and mixed pricing strategies and by persistent price dispersion. In a similar

setting with symmetric firms Renhoff and Serfes (2009) show that display advertising (which func-

tions similarly to the gatekeeper in the Baye and Morgan model) can correspond to increased price

competition consistent with observations from scanner panel data. Arnold, Li, Saliba and Zhang

(2011) analyze an asymmetric, duopoly version of the gatekeeper model and show that the firm with

the smaller loyal market share advertises more intensely but prices less competitively than the firm

with the large loyal market. A second branch of this literature considers informative advertising

in which customers are only aware of the product, and the firms selling the product, if a price is

advertised. Three cases of informative price advertising are of interest. Under uniform advertising,

firms are restricted to a choice between advertising a single price to all consumers in the market or

not advertising at all (and therefore, not participating in the market). Targeted advertising, on the

other hand, enables firms to target advertisements either to the firm’s own loyal market segment

or the loyal segment and the shoppers, without having to expend resources advertising to the com-

peting firm’s loyal customers. Targeted advertising may or may not allow for price discrimination

across the two segments. Iyer, Soberman and Villas-Boas (2005) analyze advertising strategies in a

duopoly model with symmetric loyal market shares when competing firms can target advertising to

different groups of consumers. In equilibrium firms adopt symmetric mixed advertising and pricing

strategies and advertise more intensely to loyal customers than to shoppers. In contrast, we show

that with asymmetric loyal market shares, at least one of the two firms always advertises to the

shoppers with probability one.

The present paper combines the price competition and informative advertising streams of lit-

4

erature by developing a duopoly model of informative advertising with asymmetric loyal customer

shares in which customers only consider purchasing the product if a price is advertised.1 Loyal

consumers only purchase if their preferred firm advertises and shoppers buy from the firm offering

the lowest price. Each firm makes two strategic decisions, namely whether or not to advertise

and, conditional on advertising, the price to post. In contrast to earlier literature, we find that

with asymmetric loyal segments, one of the firms always adopts a pure advertising strategy. Under

uniform advertising, the firm with the larger loyal customer base advertises with probability one,

both firms adopt mixed pricing strategies, and the firm with the smaller loyal share prices more

aggressively. With positive advertising costs, the firm with more loyal consumers advertises with

probability one to ensure sales to its large loyal base but prices less competitively (on average) to

avoid selling at a discount to its loyal customers. When both firms can target advertising and sets

a uniform price, each firm will, at a minimum, advertise to its own loyal segment. The decision

to also compete for the shopping segment requires more competitive pricing (on average) but also

increases the number of potential customers. The firm with the larger loyal share manages this

trade-off with a mixed advertising and pricing strategy while the firm with the smaller loyal share

advertises to shoppers with probability one. In balancing the equilibrium trade-off between adver-

tising and price competition, the firm with the smaller loyal share advertises more intensely, but

adopts a less competitive pricing strategy in order to avoid selling to its loyal customers at a low

price. The firm with the larger loyal share advertises to shoppers less intensely so as not to incent

deeper discounts by the competing firm and to avoid selling at a lower price to its larger share of

loyal customers. However, when it does advertise, it prices aggressively to increase the chance that

it wins the shoppers in order to justify selling to its loyal customers at a lower price.

The remainder of the paper is organized as follows. The model is presented in section 2.

Sections 3 and 4 analyze advertising and pricing behavior under uniform advertising and targeted

advertising, respectively. Finally, we conclude in section 5.

1Two interesting papers by Chiovenau (2009) and Eaton MacDonald and Meriluoto (2010) show how asymmetricloyal shares arise endogenously in a model of persuasive (or existence) advertising in which firms advertise to attractloyal market shares. Both papers consider a subgame perfect equilibrium of a game in which the second stage entailsprice competition with no advertising after asymmetric shares are established in the first stage.

5

2 The Model

Consider a market with two firms, i = 1, 2 which produce a product at a constant marginal cost

ci which is assumed to be zero without loss of generality. The market is comprised of a unit mass

of consumers who are only aware of product availability at a given firm if that firm advertises its

product. In the analysis below the firms are initially restricted to a uniform advertising strategy

under which a decision to advertise results in a single price being advertised to all consumers. We

then extend the analysis to allow for targeted advertising which enables a firm to limit its advertising

to a specific segment of consumers. The consumers, all of whom have the same reservation price r

for one unit of the product, are segmented into three groups. A fraction h1 > 0 of the consumers

are loyal to firm 1 in the sense that they will never purchase from firm 2. Similarly, a fraction

h2 < h1 are loyal to firm 2. The remaining fraction s ≡ 1− h1 − h2 > 0 are shoppers who purchase

from the firm advertising the lowest price (and do not purchase if neither firm advertises). We

refer to consumers in the segment h1 as type 1 consumers, to those in the segment h2 as type 2

consumers, and to those in the segment s as shoppers. The existence of three distinct segments

could arise for several reasons. For example, location differences and travel costs could lead to three

segments —those who work and live near firm 1 and will not travel to firm 2, those work and live

near firm 2 and will not travel to firm 1, and those who live near one firm and work near the other

who purchase from the firm offering the lowest price. The segmentation is also consistent with

a product differentiation scenario in which some consumers have a strong preference for unique

attributes of firm 1’s product, others have a strong preference for the unique attributes of firm 2’s

product, and still others are indifferent between the two products. We assume buyers are willing

to pay a maximum of r for one unit of the product.2

The cost to advertise to the entire market is A for both firms. When advertising can be targeted

to particular segments in the market, we assume that the cost to advertise to each segment is linearly

related to its size. Therefore, if firm 1 is able to target advertising, the cost to advertise to its loyal

consumers is h1A and to shoppers is sA.

2We also have analyzed the case in which some proportion α of each consumer type purchase the product evenwithout advertising. The results are not qualitatively different, so our model is consistent with promotional strategiesfor both new products (search goods) and experience goods.

6

2.1 Uniform Price Advertising without Targeting

Suppose that neither firm has the ability to advertise to specific segments of the market. Under

uniform advertising, a firm can choose either to reach the entire market at a cost A, or to not

advertise at all. Denote the minimum price a firm will ever consider advertising by pi, i = 1, 2.

This price equates the profit pi(hi + s)−A the firm can achieve by advertising a price suffi ciently

low to sell to its loyal customers and the shoppers with the profit hir − A the firm can achieve by

advertising a price of r and selling only to its loyal customers, so

pi= hir/ (hi + s) for i = 1, 2. (1)

These minimum prices play an important role in the analysis. Let p ≡ max{p1, p2

}, and note that

h1 > h2 implies p = p1= h1r/ (h1 + s) . In any equilibrium p is the minimum price that either firm

would advertise.3

Suppose that the advertising cost A is suffi ciently low that both firms advertise with probability

one in equilibrium (the condition for this is specified in Proposition 1 below). In this case, the

equilibrium pricing decisions for each firm are equivalent to those in the model considered by

Narasimhan (1988). Thus, if both firms advertise with probability 1, then Narasimhan’s results

imply that each firm adopts mixed pricing strategies in equilibrium.4 Furthermore, each firm must

attain at least the profit hir−A that it can achieve by advertising and selling to its loyal segment

at the monopoly price r. However, firm 2 can exceed this profit by advertising p1(with probability

1). By advertising p1 firm 2 captures the shoppers and earns a profit of (rh1/ (h1 + s)) (h2 + s)−A

which exceeds rh2 −A because h1 > h2.

Let Fi(p) denote the equilibrium mixed pricing strategy cumulative distribution function for

firm i. Noting that the probability that firm i captures the shoppers when advertising a price p is

3Well known arguments imply it is never optimal for firm 2 to charge p2 ∈ [p2, p1).

4The intuition for this result is straightforward. A pure pricing strategy with an advertised price less than r isonly optimal if the firm captures the shopping segment of the market. However, if one firm captures the shopperswith a price less than r, then the other firm should either undercut that firm or forego the shoppers and charge thereservation price r. In either case, the price less than r is not an equilibrium for the first firm. Furthermore, neitherthe monopoly solution (r, r) nor the competitive outcome (0, 0) constitute an equilibrium because in the former caseone firm can undercut its the other to ensure it captures the shoppers, and in the later case, either firm can makestrictly positive profits by advertising pi = r and selling only to its loyal market segment.

7

(1− Fj (p)) , the following conditions characterize the equilibrium pricing strategies:

p (h1 + (1− F2 (p)) s)−A = h1r −A (2)

and

p (h2 + (1− F1 (p)) s)−A = h1rh2 + s

h1 + s−A (3)

where equation (2) equates firm 1’s expected profit from advertising a price p < r with the profit

from advertising r and selling only to its own loyal segment, and equation (3) equates firm 2’s

expected profit from advertising p < r with the profit from advertising the minimum price p1and

capturing the shoppers.5 These conditions generate

F1(p) =

0, p < h1r

h1+s,

1 + h2s − rh1

(h2+s)(h1+s)sp

, h1rh1+s

≤ p < r,

1, p ≥ r.

(4)

and,

F2(p) =

0, p < h1r

h1+s,

1− h1(r−p)sp , h1r

h1+s≤ p ≤ r,

1, p ≥ r.

(5)

The results above assume that both firms advertise with probability 1. However, if A >

(h1 + s) r, then neither firm will advertise because the maximum profit from advertising is strictly

negative. If (h1 + s) r > A > (h2 + s) r, then firm 2 will never advertise and firm 1 will advertise

a price of r with probability 1. Similarly, if A is suffi ciently low, then both firms will advertise

with probability 1. Using the equilibrium strategies above, firm profits are π1 = h1r −A and π2 =

(h1r/ (h1 + s)) (h2 + s)−A. Because h1 > h2, if A < (h1r/ (h1 + s)) (h2 + s) , then both firms accrue

strictly positive profit by advertising. If A is in the interval [(h1r/ (h1 + s)) (h2 + s) , (h2 + s) r],

then firm 2’s expected profit from advertising with probability 1 is negative given firm 1 also adver-

tises with probability 1. However, if firm 2 does not advertise, then firm 1 will advertise a price of

5Conditions (2) and (3) are identical to those in Narasimhan (1988) but also include the advertising cost A.Because the advertising cost cancels in these expressions, the equilibrium pricing strategies are identical to thosefound by Narasimhan.

8

r, whence firm 2 could profitably advertise a price of r− ε. Thus, if (h1r/ (h1 + s)) (h2 + s) < A <

(h2 + s) r, then firm 2 must adopt a mixed advertising strategy as well as a mixed pricing strategy.

Let βi denote the probability that firm i advertises. If firm 2 advertises with probability β2 < 1, then

firm 1’s equilibrium strategy is not clear. If h2 + s > h1, then for A suffi ciently close to (h2 + s) r,

firm 1 would earn negative profit if it advertised the monopoly price r and did not capture the

shoppers. However, as demonstrated in Proposition 1, for (h1r/ (h1 + s)) (h2 + s) < A < (h2 + s) r

the equilibrium mixed advertising strategy adopted by firm 2 ensures that the probability that firm

1 captures the shoppers even when it advertises a price of r is suffi ciently high so that firm 1 always

earns positive expected profit and advertises with probability β1 = 1.

Proposition 1 In a market with uniform advertising,

(1) if A ≤ h1rh2+sh1+s

, then firms advertise with probability β1 = β2 = 1 and employ the mixed

pricing strategies defined by equations (4) and (5). Expected profits are π1 = rh1 − A and π2 =

h1r (h2 + s) / (h1 + s)−A.

(2) if h1r h2+sh1+s< A < (h2 + s)r, then firm 1 advertises with probability β1 = 1, and employs the

mixed pricing strategy

F1(p) =

0 p < A/ (h2 + s) ,

1− A−h2psp p ∈ [ A

h2+s, r),

1 p ≥ r,

(6)

and firm 2 advertises with probability β2 = 1 −(h1+sh2+s

A− h1r)/rs < 1 and employs the mixed

pricing strategy

F2(p) =

0 p < A

h2+s,

1− r−pp

A(h2+s)r−A p ∈ [ A

h2+s, r],

1 p > r

(7)

when it does advertise. Expected profits are π1 = A(h1−h2h2+s

), and π2 = 0.

(3) if (h2 + s)r ≤ A ≤ (h1 + s)r, then firm 1 advertises the reservation price r with probability

β1 = 1, firm 2 does not advertise. Expected profits are π1 = r (h1 + s)−A and π2 = 0.

(4) if A > (h1 + s)r, then neither firm advertises.

Proof. See appendix.

9

It is interesting to analyze the case when firm 2 does not advertise with probability one. We

interpret the probability with which firms advertise as the intensity of advertising within a planning

period as in Iyer, Soberman and Villas-Boas. We also assume that each period is independent

and that there are no carry-over effects for consumers. Unlike previous models with symmetric

loyal market shares in which both firms employ symmetric advertising and pricing strategies, we

demonstrate how firms adopt asymmetric pricing and advertising strategy in equilibrium when

loyal consumer segments differ in size. Our results stand in contrast with Iyer, Soberman and

Villas-Boas in which both firms mix their advertising strategies symmetrically. Firm 1 adopts a

higher advertising intensity than firm 2 in our model because firm 1 can extract more surplus from

its high loyal customers upon advertising.

Two features of price distributions for case (1) are noticeable, first, F1(r) = 1 − A−h2rsr . This

implies F1(p) will have mass point at r equal to A−h2rsr , which is consistent with the analysis in

Narasimhan (1988). Second, F2(p) converges to the symmetric results analyzed by Iyer, Soberman

and Villas-Boas as h1 → h2. It is easy to check that ∂Fi(p)/∂A < 0 and ∂Fi(p)/∂r < 0 for both

firms, thus, the expected price is increasing in the advertising cost and in the buyer’s reservation

price, as in symmetric case.

Proposition 2 In the case of uniform advertising, if A < (h2 + s) r, so that both firms advertise

with positive probability, then the (random) price p1 charged by firm 1 is first-order stochastically

larger than the (random) price p2 charged by firm 2.

Proof. This follows directly from the fact that 1− F1 (p) ≥ 1− F2 (p) for the distributions F1 (p)

and F2 (p) for all price p and advertising costs A in the relevant ranges defined in parts (1) and (2)

of proposition 1. �

The results of proposition 2 imply that with asymmetric loyal segments and uniform price

advertising firm 2, with the smaller loyal segment, prices more aggressively than firm 1 when it

does advertise. Combined with proposition 1, these results present an interesting picture of the

impact of asymmetric loyal market segments on competitive advertising and pricing decisions. A

well known result for the symmetric case is that equilibrium profit for a firm is the maximum of

the profit hir−A that the firm can realize by advertising the monopoly price to its loyal segment,

and the profit 0 from not advertising. Our results indicate that with asymmetric loyal market

10

segments firms can earn profit in excess of these amounts. In particular, if the advertising cost A

is low, then both firms advertise with probability 1 and firm 2 earns strictly more than the profit

it would achieve by advertising the monopoly price and selling only to its loyal customers. This

occurs because the larger loyal market segment of firm 1 reduces firm 1’s incentive to compete for

shoppers. As a result, firm 2 adopts the more aggressive pricing strategy of the two firms, but is still

able to generate profit in excess of what is earned by selling only to its loyal segment at the price r.

Furthermore, firm 2 will advertise with probability 1 even if the advertising cost exceeds the revenue

h2r that can be generated by selling to its loyal segment at the price r (for h2r < A < h1rh2+sh1+s

).

Once the advertising cost A increases beyond h1r (h2 + s) / (h1 + s) , then firm 2 earns zero profit in

equilibrium and adopts a mixed advertising strategy, where the advertising probability is decreasing

in A and increasing in r. This strategy applies until A reaches (h2 + s) r at which point firm 2 will

cease advertising because the return does not justify the expense even if all shoppers purchase from

firm 2 at the reservation price r.

The results of propositions 1 and 2 also contrast starkly with the advertising and pricing be-

havior of firms with asymmetric market shares that compete for the shopping segment through an

information gatekeeper but can sell to their loyal segment at no cost. As Arnold et. al. (2011)

demonstrate, in that environment the firm with the small loyal segment is more likely to advertise,

but it prices less aggressively. Because loyal customers always purchase if the price does not exceed

the reservation value r in the gatekeeper setting, the large firm poses more of a competitive threat.

As a result, the firm with the smaller loyal segment manages the trade-off between competing

through advertising frequency versus aggressive pricing by advertising aggressively but pricing less

aggressively than the firm with the large loyal segment. The firm with the large loyal segment,

on the other hand, limits competition in the market by advertising less intensively (has a lower

probability of advertising), but it prices aggressively when it does advertise to ensure that the price

reduction offered to loyal customers is justified with a high probability of capturing the shoppers.

If, as in our current setting, firms must advertise to attract loyal and shopping customers alike,

then the large firm has a strong incentive to advertise because advertising is required in order to

capture even its loyal segment. Therefore, the large firm advertises more aggressively. However, in

order to mitigate competition, it prices less aggressively than firm 2.

In the symmetric model, h = h1 = h2, so any change in s causes the same change in h1 and h2.

11

In our asymmetric model, h1 and h2 are different and since Fi(p) is function of two of the three

market segment size variables, h1, h2 and s, it is possible for us to explore the effects of market

segment sizes on price distributions. Our results show that expected prices for both firms increase

as h1 goes up whether h2 goes down (s is constant) or s goes down (h2 is constant). However, if

firm 1 hold its market share h1 constant, the expected price for firm 1 increases when s goes up (h2

goes down) and decreases when s goes down (h2 goes up). When s goes up holding h1 constant,

the lowest price firm 2 charges does not change, while the lowest price firm 1 charges goes down.

This gives firm 1 more incentive to charge a higher price. As a result, the expected price for firm

1 increases when s goes up (h2 goes down).

Table 1: Market segment sizes effect on price for uniform advertising

h1 h2 s F1 F2+ - 0 + ++ 0 - + +- + 0 - -- 0 + - -0 - + +0 + - -

As suggested by intuition, ∂β∗

∂A < 0 and ∂β∗

∂r > 0, so firm 2’s advertising probability is decreasing

with advertising cost and increasing with reservation price. The relation between β∗ and s is more

interesting. Firm 2 will advertise less if the increased shoppers are from its loyal segment, h2.

However, if the increased shopper are coming from firm 1’s loyal segment, h1, it advertises more

only when h1 > h2 + s. Basically, there are two effects when shoppers increase. First,it increases

the fraction of firm 2’s demand that it has to compete with firm 1, which lowers its advertising

frequency. Second, it increases total demand available to firm 2 when it is price is lower than firm

1,which increases its frequency to advertise. In case when increased shoppers are from its loyal

segment, the second effect disappears, so firm 2 will advertise less. However, if increased shoppers

are from h1, Firm 2 will advertise more only if the first effect dominated by the second effect, which

is when h2 + s is less than h1.

12

2.2 Targeted Advertising with Uniform Prices

We now analyze firms’ advertising and pricing behavior when firms have the ability to target

particular segments of the market. In contrast to the case of uniform advertising in which the firm

advertises to the entire market at a cost of A, we now assume that an individual segment hi, or

s can be reached at a cost of hiA or sA. In this setting a given firm i would never advertise to

the segment hj of consumers loyal to the rival firm. Furthermore, firm i can achieved a profit of

πi = hi (r −A) by advertising only to its own loyal segment. With targeted advertising, a firm will

always advertise to its own loyal segment provided A < r, while if A > r, then neither firm would

ever choose to advertise. For the remainder of the analysis, we assume A < r. We also assume

that while firms can selectively choose to advertise to only its own loyal customers, or to both its

loyal customers and the share s of shoppers in the market, the firm is not able to price discriminate

between these two segments. We maintain our assumption that h1 > h2.

Proposition 3 Suppose that both firms can target their advertising to specific segments of the

market. Then in equilibrium, with probability

β1 =

(1− A

r

)h2 + s

h1 + s

firm 1 advertises to both its own loyal segment h1 and the shopping segment s and adopts the mixed

pricing strategy

F1(p) =

0 p < h1r+sA

h1+s,

1− h1r+As(r−A)s

r−pp

h1r+sAh1+s

≤ p ≤ r,

1 p > r,

and with probability 1− β1 firm 1 advertises the price r to its loyal segment h1 only;

with probability β2 = 1 firm 2 advertises to both its loyal segment h2 and the shopping segment s

and adopts the mixed pricing strategy

F2(p) =

0 p < h1r+sA

h1+s,

1 + h1s −

h1r+Assp

h1r+sAh1+s

≤ p < r,

1 p ≥ r.

13

The equilibrium profits are π1 = h1(r −A) and π2 = (h1r+sAh1+s−A)(h2 + s) = h1(r −A)h2+sh1+s

.

Proof. See appendix.

Under uniform advertising, the lowest price p1that firm 1 is willing to advertise is less than that

of firm 2 because firm 1 has a larger consumer base to assume the advertising cost A. However,

under targeted advertising, firm 1 is less willing to lower its price in order to compete for comparison

shoppers because it can extract more surplus by targeting its loyal consumers. As a result, when

both firms are able to target their advertising, the lowest price p2that firm 2, with the smaller

loyal share, is willing to charge is less than that of firm 1. By standard arguments, advertising to

both its loyal segment and the shoppers is firm 2’s strictly dominant strategy.

Proposition 4 In the case of targeted advertising with uniform pricing, if A < r, so that both

firms advertise with positive probability in equilibrium, then the (random) price p2 charged by firm

2 is first-order stochastically larger than the (random) price p1 charged by firm 1.

The expected price increases with both reservation price r and advertising cost A as before. We

also have similar asymmetric market segment effects on pricing. The expected prices for both firms

decrease as h1 increases regardless of whether h2 decreases (and s remains constant) or s decreases

(and h2 remains constant). Similarly, the expected prices for both firms increase as h1 decreases

regardless of whether h2 increases (and s is constant) or s increases (and h2 is constant). However,

if h1 is held constant, both firms’expected price increase when s increases (and h2 decreases) and

decrease when s decreases (and h2 increases). Explanations for these results are similar to the case

of uniform advertising except that firm 1 and firm 2 have switched their roles.

Table 2: Market segment sizes effect on price for targeted advertising

h1 h2 s F1 F2+ - 0 - -+ 0 + - -- + 0 + +- 0 + + +0 - + + +0 + - - -

Similar to β∗2 in the case of uniform advertising, with targeted advertising, firm 1’s equilibrium

advertising probability β∗1, is decreasing with advertising cost and increasing with reservation price.

14

Firm 1 will advertise less if the increased shoppers are from its loyal segment, h1. However, in

contrast to the uniform advertising case, where firm 2 advertise less than firm 1 only if A is

suffi ciently large, with targeted advertising firm 1 will always advertise less even when the increased

shoppers are from its competitor’s loyal segment h2. Because firm 1 cannot price discriminate

between shoppers and its loyal consumers, the gain from lowering its price to capture shoppers is

always less than the loss from cutting the price for its loyal consumers. By the same reasoning,

firm 1 will advertise less if h1 goes up and h2 goes down (holding s constant).

Again, if we consider a monopolist advertising firm, the optimal advertising fee is A = r. The

profit r(h2+s+h2) from charging an advertising fee of r is greater than the profitA (h2 + s+ h1 + β1s)

that can be achieved from any A < r. Firm 1 advertises the price r to its loyal segment only, and

firm 2 advertises r to both its loyal segment and the shoppers with probability 1. Both firms earn

0 profit —when both firms have the ability to target advertising. A monopolist advertising firm

would extract all available surplus in the market.

2.3 Large firm adopts targeted advertising and small firm adopts uniform ad-

vertising

Given the ability to target advertising, firm 1 would eliminate its “wasted” advertising cost on

those consumers who only consider buying products from firm 2. Firm 2, on the other hand, pays

an advertising cost A and reaches the entire market if it advertises.

Proposition 5 Suppose that firm 1 has the ability to target its advertising while firm 2 is limited

to uniform advertising. Then

(1) if A < h2+s1+s r, then with probability β1 =

h2+sh1+s

r−Ar firm 1 advertises to both its loyal segment

and shoppers and employs the mixed pricing strategies

F1(p) =1

β1(1−

h1r+sAh1+s

(h2 + s)− h2psp

) =r

r −A

(1 +

h1s− h1r + sA

sp

)for p ∈ [h1r + sA

h1 + s, r], (8)

and with probability 1 − β1, firm 1 advertises a price r to to its loyal segment only; and firm 2

advertises to the whole market with probability β2 = 1 and employs the mixed pricing strategy

F2(p) = 1 +h1s− h1r + sA

spfor p ∈ [h1r + sA

h1 + s, r]. (9)

15

Equilibrium profits are π1 = h1(r −A) and π2 = (h2 + s)(h1r+sAh1+s

)−A.

(2) if A ∈ [h2+s1+s r, (h2 + s)r], then firm 1 advertises to both its loyal segment and shoppers with

probability β1 = 1 and employs the mixed pricing strategy

F1(p) = 1−A− ph2sp

for p ∈ [ A

h2 + s, r], (10)

and with probability β2 =(h2+s)r−A

rsh1+sh2+s

firm 2 advertises (uniformly) and employs the mixed

pricing strategy

F2(p) =1

β2(1−

A(h1 + s)(1 +h1h2+s

)− h1psp

) = 1− A (r − p)((h2 + s) r −A) p

for p ∈ [ A

h2 + s, r], (11)

and with probability 1 − β2 firm 2 does not advertise. Equilibrium profits are π1 = h1h1+sh2+s

A and

π2 = 0.

(3) if (h2 + s)r < A < r, firm 1 advertises the price r to both its loyal segment and shoppers

with probability β1 = 1, and firm 2 does not advertise. Equilibrium profits are π1 = (h1+ s)(r−A)

and π2 = 0.

Proof. See appendix.

Notice that firm 1 always advertises to its loyal consumers as long as A < r because it has

the ability to target specific segments, however firm 2 adopts a mixed advertising strategy when

the advertising cost is relatively high because it is unable to target its ads. The mixed advertising

strategy by firm 2 reduces competition in price for the shoppers by firm 1. This enables firm 2

to price less competitively which raises firm 2’s expected revenues by an amount just suffi cient to

cover the increased advertising cost.

2.4 Large firm adopts uniform advertising and small firm adopts target adver-

tising

Suppose that only the small firm 2 has the ability to target advertising. As usual, our equilibrium

results depend on the level of advertising cost.

Proposition 6 Suppose that firm 2 has the ability to target its advertising while firm 1 is limited

to uniform advertising.

16

(1) If A < (h1−h2)rh1+s

, then firm 1 advertises (uniformly) with probability β1 = 1 and employs the

mixed pricing strategy

F1(p) = 1 +h2s− (h2 + s)h1r(h1 + s) sp

for p ∈ [ h1rh1 + s

, r], (12)

and firm 2 advertises to both its loyal segment and shoppers with probability β2 = 1 and employs

the mixed pricing strategy

F2(p) = 1−h1(r − p)

spfor p ∈ [ h1r

h1 + s, r]. (13)

Equilibrium profits are π1 = h1r −A and π2 = ( h1rh1+s−A)(h2 + s).

(2) If A ∈ [ (h1−h2)rh1+s, h1+s1+s r], then firm 1 advertises with probability β1 = 1 using the mixed

pricing strategy

F1(p) = 1−h2r +As− ph2

psfor p ∈ [h2r + sA

h2 + s, r]; (14)

with probability β2 =h1+sh2+s

r−Ar firm 2 advertises to both its loyal segment and shoppers using the

mixed pricing strategy

F2(p) =(r −A) rs− (h2 + s) (r − p) r

(r −A) sp for p ∈ [h2r + sAh2 + s

, r],

and with probability 1 − β2 firm 2 advertises the price r to its loyal segment only. Equilibrium

profits are π1 = (h1 + s)h2r+sAh2+s−A and π2 = h2(r −A), respectively;

(3) A ∈ [h1+s1+s r, (h1 + s)r], firm 1 advertises with probability β1 =(h2+s)(h1r−A+rs)

(s+h1)rsusing the

mixed pricing strategy

F1(p) =(h1 + s) rp−Ar(h1 + s) rp−Ap

for p ∈ [ A

h1 + s, r]; (15)

firm 2 advertises to its loyal segment and shoppers with probability β2 = 1 using the mixed pricing

strategy

F2(p) = 1−A− h1psp

for p ∈ [ A

h1 + s, r].

Equilibrium profits are π1 = 0 and π2 = (h2 + s)( Ah1+s

−A) = h2(h2+s)h1+s

A.

(4) A ∈ [(h1 + s) r, r], firm 1 does not advertise, and firm 2 advertises the price r to its loyal seg-

ment and shoppers with probability β2 = 1, and firm profits are π1 = 0, and π2 = (h2 + s) (r −A) .

17

Proof. See appendix.

3 Concluding remarks

In this paper,we analyze duopoly advertising and pricing strategies with asymmetrical loyal con-

sumers. We extend results of Narasimhan showing that in a market with informative advertising and

asymmetric loyal market shares firms not only price asymmetrically but also advertise asymmetri-

cally. When firms have asymmetric loyal customer shares, the nature of advertising competition is

critical to determining optimal advertising and pricing strategies. Firms can balance the incentive

to compete in price by competing less intensively in advertising. Similarly, a firm can successfully

advertise to expand its market if it prices less aggressively (i.e., by adopting a puppy dog strategy).

However, whether a firm should advertise intensely and price cautiously or price aggressively and

advertise cautiously depends on the advertising mechanism available. If firms are limited to uniform

advertising strategies, then the firm with the larger loyal market will advertise more aggressively

and price less competitively than the firm with the smaller loyal market share. These results are

reversed if both firms are able to implement targeted advertising strategies.

18

4 Appendix

Proof of Proposition 1. The equilibrium strategies for A ≤ h1rh1+s

(h2 + s) and A > (h2 + s) r

were verified in the paragraph preceding Proposition 1. Also note that the minimum price that firm

2 would ever advertise must satisfy p2(h2 + s) − A ≥ 0, or p2 ≥ A/ (h2 + s) . Because firm 1 can

capture all shoppers by advertising p2, firm 1’s profit satisfies π1 ≥ A

h2+s(h1 + s)−A = A(h1−h2)

h2+s> 0,

so firm 1 will advertise with probability β1 = 1 in any equilibrium in which it is optimal for firm

2 to advertise. Now consider the case of h1rh1+s

(h2 + s) < A < (h2 + s) r. First note that there

is no equilibrium in which firm 2 does not advertise. If firm 2 did not advertise, then firm 1

would advertise r and get a share h1 + s of the consumers. However, if firm 1 advertises r,

then firm 2 could advertise r − ε and sell to the share h2 + s of consumers and earn a profit of

π2 = (r − ε) (h2 − s) − A > 0 for ε > 0 suffi ciently small. Similarly, if β2 = 1, then the pricing

strategies (4) and (5) imply π2 < 0 if A > h1rh1+s

(h2 + s) . Therefore, 0 < β2 < 1 must hold in any

equilibrium with h1rh1+s

(h2 + s) < A < (h2 + s) r, and, if β2 ∈ (0, 1) , then π2 = 0. (If π2 > 0, then

β2 = 1 is optimal and if π2 < 0, then β2 = 0 is optimal.) Because firm 1 advertises with probability

β1 = 1, and π2 = 0, firm 1’s equilibrium mixed pricing strategy must satisfy

h2p+ (1− F1(p))sp−A = 0 (16)

for any p ∈ [p, r), which implies

F1(p) = 1−A− h2psp

.

Setting F1(p)= 0 yields p = p

2= A

h2+s, and F1 (r) = (r (h2 + s)−A) /rs > 0 because A <

r (h2 + s) by assumption. This implies firm 1’s equilibrium mixed pricing strategy has a mass

point at r. Because firm 1 can capture the share h1 + s of customers by advertising p, firm 2’s

equilibrium advertising and pricing strategies must satisfy

p (h1 + (1− β2)s+ β2(1− F2(p))s)−A =A

h2 + s(h1 + s)−A (17)

for any p ∈ [p, r). Noting that at most one firm’s equilibrium pricing strategy can have a mass

point at r, so F2 (r) = 1, and evaluating equation (17) at p = r yields β2 = 1−(h1+sh2+s

A− h1r)/rs.

Substituting this expression for β2 into equation (17) yields F2(p) = 1− r−pp

(A

(1−h1)r−A

). Finally,

19

setting F2(p)= 0 yields p = A

h2+s. �

Proof of Proposition 3. Profit for firm i from advertising only to its loyal segment is

hi (r −A) .Firm i will advertise to both its loyal segment and the shoppers if (hi + s)(p−A

)≥

hi (r −A) or

p ≥ rhi +As

hi + s.

This implies p1≥ h1r+sA

h1+s, so the lowest profit firm 2 can earn by advertising to its loyal segment

and the shoppers is

(h2 + s)

(h1r + sA

h1 + s−A

)= h1

h2 + s

h1 + s(r −A) > h2 (r −A)

because r > A and h1 > h2 by assumption. Because firm 2 earns strictly greater profit by advertis-

ing to both its loyal segment and the shoppers than by advertising to its loyal segment alone, β2 = 1.

Given β2 = 1, the expected profit firm 1 achieves by advertising any price p in the support of its

mixed pricing strategy to both its loyal segment and the shoppers must equal the profit h1 (r −A)

it can achieve by advertising r to its loyal segment alone. This implies firm 2’s equilibrium mixed

pricing strategy must satisfy

p (h1 + (1− F2(p))s)− (h1 + s)A = h1(r −A) (18)

which implies

F2(p) = 1 +h1s− h1r + sA

sp.

Setting F2(p)= 0 yields p = h1r+sA

h1+s. Notice that F2(p) has a mass point of magnitude A/r at the

monopoly price r. Given p = h1r+sAh1+s

, firm 2’s expected profit from advertising p to its loyal segment

and the shoppers is(h1r+sAh1+s

−A)(h2 + s) . Equating this expected return with firm 2’s expected

return from advertising any price p implies that firm 1’s equilibrium advertising and mixed pricing

strategies must satisfy

p (h2 + (1− β1)s+ β1(1− F1(p))s)− (h2 + s)A =(h1r + sA

h1 + s−A

)(h2 + s) . (19)

20

Noting that F1 (r) = 1 and evaluating equation (19) at p = r yields

β1 =

(1− A

r

)h2 + s

h1 + s.

Substituting this expression for β1 into (19) yields

F1(p) = 1−h1r +As

s(r −A)r − pp

.

Setting F1(p)= 0 yields p = h1r+sA

h1+s. Thus, the equilibrium mixed pricing strategies are defined

by F1 and F2, firm 2 advertises with probability β2 = 1, and firm 1 advertises with probability

β1 < 1. �

Proof of Proposition 5. Firm 1 can earn h1 (r −A) by targeting its loyal customers, so the

lowest price p1that firm 1 will ever advertise to shoppers satisfies h1(r−A) = (h1+s)p1−(h1+s)A

or p1 = h1r+sAh1+s

. Given this price, firm 2 must earn at least (h2+s)(h1r+sAh1+s

)−A in any equilibrium.

This return is strictly positive if A < h2+s1+s r. Thus, if A < h2+s

1+s r, then firm 2 earns a strictly greater

profit by advertising than by not advertising, so β2 = 1. Given β2 = 1, the expected profit firm 1

achieves by advertising any price p in the support of its mixed pricing strategy to both its loyal

segment and the shoppers must equal the profit h1 (r −A) it can achieve by advertising r to its

loyal segment alone. This implies firm 2’s equilibrium mixed pricing strategy must satisfy

p (h1 + (1− F2(p))s)− (h1 + s)A = h1(r −A) (20)

which implies

F2(p) = 1 +h1s− h1r + sA

sp.

Setting F2(p)= 0 yields p = h1r+sA

h1+s. Because firm 2 earns a positive expected profit from

advertising (and a profit of 0 by not advertising), the equilibrium pricing and advertising strategies

of firm 1 must ensure that firm 2 earns the same expected profit from any price it advertises which

impliesh1r + sA

h1 + s(h2 + s)−A = p (h2 + s (1− β1F1 (p)))−A. (21)

or F1(p) = 1β1(1−

h1r+sAh1+s

(h2+s)−ph2ps ). Noting that firm 2’s equilibrium pricing strategy has a mass

21

point at r, so F1 (r) = 1, and substituting p = r into equation (21) yields β1 = 1−r(h1−h2)+A(h2+s)

(h1+s)r.

The resulting equilibrium profits are π1 = h1(r −A) and π2 = h1r+sAh1+s

(h2 + s)−A.

If h2+s1+s r < A < (h2 + s)r, then firm 2 would achieve a negative expected return from the

strategies above, so the minimum price firm 2 would advertise equates the profit (h2 + s)p2− A

from advertising the minimum price with the profit of 0 from not advertising. This implies p2= A

h2+s

which is greater than the minimum price p1 = h1r+sAh1+s

that firm 1 would ever charge. By advertising

the price p2= A

h2+sto its loyal segment and the shopping customers firm 1 can achieve an expected

profit of (h1 + s) (A/ (h2 + s)−A) = (h1 + s)h1A/ (h2 + s) > (h1+s)h1r1+s > h1 (r −A) where the two

inequalities follow from the fact that A > h2+s1+s r. Because h1 (r −A) is the profit firm 1 achieves

by advertising only to its loyal segment, firm 1 will advertise to its loyal segment and the shoppers

with probability β1 = 1. Given β1 = 1, the expected profit firm 2 achieves by advertising any

price p in the support of its mixed pricing strategy must equal the profit of 0 it can achieve by not

advertising. This implies firm 1’s equilibrium mixed pricing strategy must satisfy

p (h2 + (1− F1(p))s)−A = 0 (22)

or F1(p) =p(h2+s)−A

sp . Setting F1(p)= 0 implies p = A

h2+s= p

2. Similarly, because β1 = 1, any

price firm 1 advertises must satisfy

(h1 + s)

(A

h2 + s−A

)= p (h1 + s (1− β2F2 (p)))− (h1 + s)A

which implies F2(p) = 1β2sp

(h1+sh2+s

(h2p−A+ sp)). Noting that firm 1’s equilibrium pricing strategy

has a mass point at r, so F2 (r) = 1, implies β2 =1rsh1+sh2+s

((h2 + s) r −A) . The resulting equilibrium

profits are π1 = (h1 + s)(

Ah2+s

−A)and π2 = 0.

Finally, if (h2 + s)r < A < r, then firm 2 will not advertise, and firm 1 advertises to its loyal

customers and shoppers with probability β1 = 1 and charges r in equilibrium. �

Proof of Proposition 6 The proof is similar to the proof of Proposition 5 with the roles of

firms 1 and 2 reversed. Firm 2 can earn h2 (r −A) by targeted advertising to its loyal customers

only, so the lowest price p2that firm 2 will ever advertise to shoppers satisfies

h2(r −A) = (h2 + s)p2 − (h2 + s)A (23)

22

or p2= h2r+sA

h2+s. Given this price, firm 1 must earn at least (h1+s)

(h2r+sAh2+s

)−A in any equilibrium.

This return is strictly positive if A < h1+s1+s r. Thus, if A < h1+s

1+s r, then firm 1 earns a strictly greater

profit by advertising than by not advertising, so β1 = 1. Similarly, the lowest price firm 1 will ever

advertise satisfies6

h1r −A ≤ (h1 + s) p1 −A

or p1≥ h1r/ (h1 + s) . It follows that if A < (h1−h2)rs

h1+s, then p

1> p

2and firm 2 is strictly better off

targeting both loyal and shopping customers than targeting loyal customers only, so β2 = 1

Assume A < (h1−h2)rsh1+s

, so β1 = β2 = 1. Then firm 1’s equilibrium pricing strategy must satisfy

p (h2 + (1− F1(p))s)− (h2 + s)A = p (h2 + s)− (h2 + s)A. (24)

Suppose that F1 (r) = 1. Then equation (24)implies p = h2r/ (h2 + s) < p2which can not occur. It

follows that F1 (r) < 1, so firm 1’s strategy has a mass point at r and F2 (r) = 1. Given F2 (r) = 1,

firm 2’s equilibrium pricing strategy must satisfy

p (h1 + s (1− F2 (p)))−A = h1r −A

which implies

F2 (p) = 1−h1 (r − p)

ps.

Setting F2 (p) = 0 implies p = h1r/ (h1 + s) . Substituting this value for p into equation (24) yields

F1(p) = 1 +h2s− h1r (h2 + s)

(h1 + s) ps.

Setting F1(p)= 0 yields p = h1r/ (h1 + s). Firm profits are π1 = h1r−A and π2 = h2+s

h1+s(h1 (r −A)−As) .

Now assume (h1−h2)rsh1+s

≤ A < h1+s1+s r. Note that β1 = 1 still applies, so equation (24) , and

consequently F1 (r) < 1 and F2 (r) = 1, must still hold in equilibrium. However, it is no longer

6Note that this equation does not necessarily hold with strict equality because the probability β2 that firm 2advertises impacts firm 1’s expected return from (uniformly) advertising the price r, and it is possible that β2 < 1.

23

necessary that β2 = 1, so firm 2’s equilibrium pricing and advertising strategies must satisfy

p (h1 + s (1− β2F2 (p)))−A = r (h1 + (1− β2) s)−A (25)

which implies

F2 (p) =β2rs− (h1 + s) (r − p)

β2sp.

To determine the equilibrium values for p and β2, first note that p ≥ p2= h2r+sA

h2+s. Suppose

p > h2r+sAh2+s

. This can only occur if p1> p

2which implies the equilibrium values of p and β2

satisfying equation (25) result in p > h2r+sAh2+s

. Evaluating (25) at p implies p = r − rsβ2(h1+s)

which is

greater than h2r+sAh2+s

if and only if β2 <(r−A)(h1+s)r(h2+s)

≤ 1 where the second inequality follows from

A ≥ r(h1−h2)h1+s

. However, if p > h2r+sAh2+s

, then β2 = 1 is optimal for firm 2. But β2 = 1 implies

p = h1r/ (h1 + s) < p2, a contradiction. Thus, p ≤ h2r+sA

h2+sin equilibrium. Because firm 2 will

never establish a minimum price less than h2r+sAh2+s

, it follows that p = h2r+sAh2+s

and β2 =(r−A)(h1+s)(h2+s)r

in equilibrium. Using these equilibrium values for p and β2 implies

F1 (p) =s (p−A)− h2 (r − p)

ps,

and

F2 (p) =(r −A) rs− r (h2 + s) (r − p)

(r −A) sp .

It is easy to verify that these are well defined cumulative distribution functions for p ∈ [h2r+sAh2+s, r]

and that firm 1’s strategy has a mass point of 1−A/rs at p = r. The resulting equilibrium profits

are π1 = h1(r −A) and π2 = h2r+sAh2+s

(h2 + s)−A = h2r −A (1− s).

Assume h1+s1+s r < A < (h1 + s)r. Note that p

1must satisfy p

1(h1 + s) − A ≥ 0, so p ≥

A/ (h1 + s) , and A > h1+s1+s r implies A/ (h1 + s) >

h2r+sAh2+s

= p2.This implies firm 2 earns a strictly

greater profit by targeting shoppers and its loyal customers at the price p than by targeting its

loyal customers alone, so β2 = 1. Given β2 = 1, the equilibrium pricing strategy for firm 2 must

satisfy p (h1 + s (1− F2 (p))) − A = (h1 + s)p − A, so F2 (p) =(h1+s)(p−p)

ps . Suppose F2(r) =

1. Then p = rh1/ (h1 + s) < A/ (h1 + s) , where the inequality follows from A > h1+s1+s r, which

contradicts p ≥ A/ (h1 + s) . Thus, F2 (r) < 1 and F1 (r) = 1 in equilibrium. Next, note that firm

24

1’s equilibrium pricing and advertising strategies must satisfy

p (h2 + s (1− β1F1 (p)))− (h2 + s)A = r (h2 + (1− β1) s)− (h2 + s)A (26)

which implies

F1 (p) =rsβ1 − (r − p) (h2 + s)

psβ1

F1(p)= 0 implies p = r− β1rs

h2+s. which is greater than A

h1+sif and only if β1 >

(h2+s)(r(h1+s)−A)(h1+s)rs

To

determine the equilibrium values for p and β1, suppose p >A

h1+s. This implies β1 = 1 is optimal for

firm 1. But if β1 = 1, then equation (26) implies p =rh2

(h2+s)< A/ (h1 + s) , a contradiction. Thus,

p = Ah1+s

and β1 =(h2+s)(r(h1+s)−A)

rs(h1+s)in equilibrium. Using these values for p and β1 implies

F1 (p) =(h1 + s) rp−Ar(h1 + s) rp−Ap

,

and

F2 (p) =(h1 + s) p−A

sp.

It is straightforward to verify that these are valid cumulative distribution functions for p ∈[

Ah1+s

, r]

and that firm 2’s pricing strategy has a mass point of r(h1+s)−Ars < 1 at p = r. Equilibrium profits

are π1 = 0 and π2 = As+h1

(s+ h2) (1− s− h1) = h2(h2+s)h1+s

A.

Finally, if (h1 + s)r < A < r, then β1 = 0 because firm 1 is unable to make positive profits by

advertising. Therefore, firm 2 advertises the price r to its loyal customers and shoppers. Equilibrium

profits are π1 = 0 and π2 = (h2 + s) (r −A). �

25

References

[1] Arnold, M., Li, C., Saliba, C., and Zhang, L. 2011 “Asymmetric Market Shares, Advertising,

and Pricing: Equilibrium with an Information Gatekeeper,”Journal of Industrial Economics,

59, 63-84.

[2] Baye, M., and Morgan, J. 2009. “Brand and Price Advertising in Online Markets,”Manage-

ment Science, 55, 1139-1151.

[3] Baye, M., and Morgan, J. 2001. “Information gatekeepers on the internet and the competi-

tiveness of homogeneous product markets,”American Economic Review, 91, 454-474.

[4] Baye, M. Kovenock, D., and de Vries, C. 1992. “It Takes Two to Tango: Equilibria in a Model

of Sales,”Games and Economic Behavior, 4, 493-510.

[5] Bagwell, K. 2007. “The economic analysis of advertising,” in Armstrong, M., and Porter, R.

(eds.) Handbook of Industrial Organization, vol 3. North-Holland, Amsterdam, 1701-1844.

[6] Chioveanu, I. 2008. “Advertising, brand loyalty, and pricing,”Games and Economic Behavior,

64, 68-80.

[7] Deneckere R., Kovenock D., and Lee R. 1992. “A model of price leadership based on consumer

loyalty,”Journal of Industrial Economics, 40, 147-156.

[8] Eaton, B. C., MacDonald, I. and Meriluoto, L. 2010. “Existence Advertising, Price Com-

petition and Asymmetric Market Structure,” The B.E. Journal of Theoretical Economics,

Contributions, 10, article 39.

[9] Galeotti, A., and Moraga-González, J. 2008. “Segmentation, Advertising and Prices,” Inter-

national Journal of Industrial Organization, 26, 1106-1119.

[10] Iyer, G., Soberman, D., and Villas-Boas, J. M. 2005. “The Targeting of Advertising,”Marketing

Science, 24, 461-476.

[11] Jing, B. and Wen, Z. 2008. “Finitely Loyal Customers, Switchers, and Equilibrium Price

Dispersion,”Journal of Economics and Management Strategy, 17, 683-707.

26

[12] Kocas, C. and Kiyak, T. 2006. “Theory and Evidence on Pricing by Asymmetric Oligopolies,”

International Journal of Industrial Organization, 24, 83-105.

[13] LeBlanc, G. 1998. “Informative Advertising Competition,” Journal of Industrial Economics,

46, 63-77.

[14] Narasimhan, C. 1988. “Competitive promotional strategies,”Journal of Business, 61, 427-449.

[15] Rennhoff, A., and Serfes, K. 2009. “Retailer Price Distributions and Promotional Activities,”

Economics Letters, 103, 91-95.

[16] Roy, S. 2000. “Strategic Segmentation of a Market,”International Journal of Industrial Orga-

nization, 18, 1279-1290

27