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Multiproduct Retailing Andrew Rhodes * 4 July 2014 We study the pricing behavior of a multiproduct firm, when consumers must pay a search cost to learn its prices. Equilibrium prices are high, because consumers understand that visiting a store exposes them to a hold-up problem. However a firm with more products charges lower prices, because it attracts consumers who are more price-sensitive. Similarly when a firm advertises a low price on one product, consumers rationally expect it to charge somewhat lower prices on its other products as well. We therefore find that having a large product range, and advertising a low price on one product, are substitute ways of building a ‘low price image’. Finally, we show that in a competitive setting each product has a high regular price, with firms occasionally giving random discounts that are positively correlated across products. Keywords: retailing, multiproduct search, advertising JEL: D83, M37 * Contact: [email protected] Toulouse School of Economics, Manufacture des Tabacs, Toulouse 31000, France. Earlier versions of this paper were circulated under the title “Multiproduct Pricing and the Diamond Paradox”. I would like to thank Marco Ottaviani and three referees for their many helpful comments, which have significantly improved the paper. Thanks are also due to Mark Armstrong, Heski Bar-Isaac, Ian Jewitt, Paul Klemperer, David Myatt, Freddy Poeschel, John Quah, Charles Roddie, Mladen Savov, Sandro Shelegia, John Thanassoulis, Mike Waterson, Chris Wilson and Jidong Zhou, as well as several seminar audiences including those at Essex, Manchester, Mannheim, McGill, Nottingham, Oxford, Rochester Simon, Toulouse, UPF, Washington and Yale SOM. Funding from the Economic and Social Research Council (UK) and the British Academy is also gratefully acknowledged. All remaining errors are mine alone. 1

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Multiproduct Retailing

Andrew Rhodes∗

4 July 2014

We study the pricing behavior of a multiproduct firm, when consumers must

pay a search cost to learn its prices. Equilibrium prices are high, because consumers

understand that visiting a store exposes them to a hold-up problem. However a

firm with more products charges lower prices, because it attracts consumers who are

more price-sensitive. Similarly when a firm advertises a low price on one product,

consumers rationally expect it to charge somewhat lower prices on its other products

as well. We therefore find that having a large product range, and advertising a low

price on one product, are substitute ways of building a ‘low price image’. Finally, we

show that in a competitive setting each product has a high regular price, with firms

occasionally giving random discounts that are positively correlated across products.

Keywords: retailing, multiproduct search, advertising JEL: D83, M37

∗Contact: [email protected] Toulouse School of Economics, Manufacture des Tabacs,

Toulouse 31000, France. Earlier versions of this paper were circulated under the title “Multiproduct

Pricing and the Diamond Paradox”. I would like to thank Marco Ottaviani and three referees for

their many helpful comments, which have significantly improved the paper. Thanks are also

due to Mark Armstrong, Heski Bar-Isaac, Ian Jewitt, Paul Klemperer, David Myatt, Freddy

Poeschel, John Quah, Charles Roddie, Mladen Savov, Sandro Shelegia, John Thanassoulis, Mike

Waterson, Chris Wilson and Jidong Zhou, as well as several seminar audiences including those at

Essex, Manchester, Mannheim, McGill, Nottingham, Oxford, Rochester Simon, Toulouse, UPF,

Washington and Yale SOM. Funding from the Economic and Social Research Council (UK) and

the British Academy is also gratefully acknowledged. All remaining errors are mine alone.1

1 Introduction

Retailing is inherently a multiproduct phenomenon. Most firms sell a wide range

of products, and consumers frequently visit a store with the intention of purchasing

several items. Whilst consumers often know a lot about a firm’s products, they

are poorly-informed about the prices of individual items unless they buy them on a

regular basis. Therefore a consumer must spend time visiting a firm in order to learn

its prices. The theoretical literature has mainly focused on single-product retailing.

However once we take into account the multiproduct nature of retailing, a whole new

set of important questions arises. For example, what are the advantages to a firm

from stocking a wider range of products? How do pricing incentives change when a

firm sells more products? Some firms eschew advertising and charge ‘everyday low

prices’ across the whole store. Others use a so-called ‘high-low’ strategy, charging

high prices on many products but advertising steep discounts on others. When

is one strategy better than the other? In addition, advertisements usually can only

contain information about the prices of a small proportion of a firm’s overall product

range. This naturally raises the question of how much information consumers can

learn from an advert, about the firm’s overall pricing strategy. Related, when an

advertisement contains a good deal on one product, should consumers expect the

firm to compensate by raising the prices of other goods?

In order to answer these questions, we begin by looking at the pricing behavior

of a multiproduct monopolist. This delivers all of the first-order intuition. Later

in the paper, we generalize our results to the case of duopolists who sell the same

products. In our model, consumers have different valuations for different products,

and would like to buy one unit of each. Every consumer is privately informed about

how much she values the products, but does not know their prices. Firms can inform

consumers about one of their prices by paying a cost and sending out an advert.

Consumers use the contents of these adverts to form (rational) expectations about

the price of every product in each store. Consumers then decide whether or not to

search, and in the case of duopoly also decide where to search. After searching a

firm, a consumer learns actual prices and makes her purchase decisions. We think

2

this set-up closely approximates several important product markets. For instance,

local convenience shops and drugstores sell mainly standardized products which do

not change much over time. A consumer’s main reason for visiting them is not to

learn whether their products are a good ‘match’, but instead to buy products that

she already knows are suitable. In addition, as surveyed by Monroe and Lee (1999),

studies have repeatedly found that consumers only have a limited recall of the prices

of products which they have bought, and so in many cases, consumers are unlikely

to be well-informed about prices before they go shopping.

We first examine the relationship between the size of a firm’s product range, and

the prices it charges on individual items. There is lots of evidence, both empirical

and anecdotal, that larger stores generally charge lower prices. For example in

the United Kingdom, the two major supermarket chains charge higher prices in

their smaller stores (Competition Commission 2008). A similar negative relationship

between store size and prices is also documented by Asplund and Friberg (2002) for

Sweden, and by Kaufman et al (1997) for the United States. Our model provides

a novel explanation for this phenomenon, that is unrelated to production costs or

store differentiation. We show that larger stores should charge lower prices, because

they attract consumers who on average have more low product valuations.1

This result is best understood within the context of the “Diamond Paradox”

(Diamond 1971). Suppose for a moment that firms sell just one product. Also

suppose that consumers have unit demand for that product, and that travelling

to each firm costs some amount s > 0. Under these assumptions no consumer

will search, and firms gets zero profit. The reason is that if consumers expect

firms to charge a price pe, only people with a valuation above pe + s will search.

Therefore consumers will buy in the first store they visit without searching further,

so long as that store does not charge a price exceeding pe + s. Hence there cannot

1Zhou (2014) finds that larger firms charge lower prices, but this is due to a different, ‘joint

search’ effect. In his model multiproduct firms charge less than single-product firms, because they

have more incentive to stop consumers from searching competitors. Also related is the literature

on shopping malls (e.g. Dudey 1990 and Non 2010), which shows that when single-product firms

with substitute products co-locate, competition intensifies and prices fall. We show that prices fall

even when a monopolist ‘co-locates’ independent products together.

3

be an equilibrium in which some consumers search, because firms would always hold

consumers up by charging more than they expected. This happens irrespective of

how many firms there are in the market.

We show that this ‘no search problem’ disappears when firms sell multiple prod-

ucts.2 Intuitively in the single-product case, only consumers with a high valuation

decide to search, so firms exploit this and charge a high price. However in the

multiproduct case, a consumer with a low valuation on one product may still search

because she has a high valuation on another. Firms then have an incentive to charge

lower prices, in order to sell products to low-valuation consumers. Nevertheless -

and crucially for our results - the hold-up problem persists, albeit in a weaker form.

In particular only consumers with relatively high valuations search, and this leads

to high equilibrium prices. This explains the importance of product range. A larger

firm attracts more consumers, who are more representative of the general population,

and who therefore have a higher number of low product valuations.3 Consequently,

the consumers who search a larger firm tend to be more price-elastic.

We then characterize a multiproduct firm’s optimal advertising strategy. Typ-

ically adverts can only contain information on a very small number of prices. Al-

though there is a large literature on advertising (see Bagwell 2007), an important

but much-neglected question, is how a firm’s few (but typically very low) advertised

prices are related to the prices that it charges on its other products. In order to

answer this question, we introduce costly advertising whereby a firm can directly

2Many papers have offered other resolutions. For example, consumers might learn prices from

several firms during a single search (Burdett and Judd 1983). Alternatively some consumers may

enjoy shopping, or know all firms’ prices before searching (Varian 1980, Stahl 1989). In both cases

competition for the better-informed consumers leads to somewhat lower prices. Secondly firms

might send out adverts which commit them to charging a particular price, and thereby guarantee

consumers some surplus (Wernerfelt 1994, Anderson and Renault 2006). Thirdly consumers might

only learn their match for a product after searching for it. In this case, Anderson and Renault

(1999) show that the market does not collapse provided there is enough preference for variety.3Changes in the composition of demand, in particular the mix of low- and high-valuation con-

sumers, also plays an important role in understanding the effects of search diversion by an inter-

mediary (Hagiu and Jullien 2011), and in understanding why the entry of a new platform may

increase advertising on existing platforms (Ambrus et al 2014).

4

inform consumers about one of its prices. We show that when a firm cuts its ad-

vertised price, some new consumers with relatively low valuations decide to search

it. The firm then finds it optimal to also reduce its unadvertised prices, in order to

sell more products to these new searchers. Therefore consumers (rationally) expect

a positive relationship between a firm’s advertised and unadvertised prices. Whilst

firms cannot commit in advance to their unadvertised prices, they can indirectly

convey information about them via their advertised price. This implies that a low

advertised price on one product can build a store-wide ‘low price-image’, even on

completely unrelated products.

This result is in sharp contrast to the existing literature. In Lal and Matutes

(1994) firms sell two products, and consumers are willing to pay H for one unit of

each product. Firms advertise a low price on one product, but chargeH for the other.

As such, a firm’s advertised price conveys no information about its unadvertised

price. In Ellison (2005) firms are also unable to use a lower advertised price for

their base product, to credibly convince consumers that their add-on product will

be cheaper. We obtain a different result because in our paper, consumer valuations

are heterogeneous and continuously distributed. Therefore when a firm changes

its advertised price, the pool of searchers also changes and this alters the firm’s

pricing incentives on its other products. In another line of work, Simester (1995)

also finds that prices are positively correlated, but this is due to a different, cost-

related mechanism. In his model, a low-cost firm may charge a low advertised price

in order to signal its cost advantage to consumers.

In our model, having a large product range and using low-price advertising are

substitute ways of convincing consumers that overall prices are low. Consequently a

firm with a large product range follows an ‘everyday low pricing strategy’: it eschews

advertising but charges low prices across the whole of its store. On the other hand

a smaller firm is more likely to follow a ‘high-low’ strategy: it charges higher prices

on most products, but advertises a large discount on some others.

Finally, we introduce a second multiproduct firm into the model, and allow con-

sumers to multi-stop shop. As a simplification we assume that each firm sells the

same two products. Several new insights emerge. We show that in equilibrium,

5

each firm now randomizes over whether or not advertise. Conditional on adver-

tising, a firm also chooses at random which product to promote.4 The firm then

advertises a large but random discount on that product, and offers a smaller (un-

advertised) discount on its other product. As with the monopoly case, the sizes

of these two discounts are positively related.5 Hence, in contrast to the existing

literature, the paper provides a new explanation for the dispersion of unadvertised

prices that is commonly observed in practice.6 The equilibrium also provides an

endogenously-determined ‘regular price’, and allows for a rich pattern of consumer

shopping behavior, including a positive probability that some consumers ‘cherry-

pick’ firms’ advertised specials.

The rest of the paper proceeds as follows. Section 2 begins by solving a bench-

mark model without advertising, whilst Section 3 characterizes a monopolist’s op-

timal advertising strategy. Competition is then introduced in Section 4, whilst

Section 5 discusses our results and provides various extensions. Section 6 concludes.

All proofs appear in the appendices.

4In Lal and Rao (1997) the advertised product is also chosen at random. However in Lal and

Matutes (1994) both firms advertise the same product, whilst in Chen and Rey (2012) competition

for one-stop shoppers causes asymmetric firms to offer their best deal on different products.5Unlike us, McAfee (1995) and Hosken and Reiffen (2007) find that multiproduct firms’ prices

are negatively correlated, whilst Shelegia (2012) finds them to be uncorrelated. However they

consider a different setting where all prices are unadvertised, and some consumers can search

costlessly. Ambrus and Weinstein (2008) consider the opposite case where all prices are (effectively)

advertised. They show that each product is priced at marginal cost, unless demands are elastic

and satisfy certain complementarities. See also Bliss (1988).6Most papers which distinguish between advertised and unadvertised prices, predict that un-

advertised prices should be non-dispersed e.g. Robert and Stahl (1993), Lal and Matutes (1994)

and Baye and Morgan (2001). In Janssen and Non (2008, 2009) unadvertised prices are dispersed,

but the mechanism (as well as their single-product set-up) differs from ours.6

2 Benchmark model with no advertising

We begin by solving a benchmark model in which there is no advertising. There

is one firm who has a local monopoly over the supply of n products. These n

products are neither substitutes nor complements, and consumers demand at most

one unit of each. We let vj denote a typical consumer’s valuation for product

j. Each vj is drawn independently across both products and consumers, using

a distribution function F (vj) whose support is [a, b] ⊂ R+. The corresponding

density f (vj) is strictly positive, continuously differentiable, and logconcave.7 The

marginal cost of producing each product is c, where 0 ≤ c < b. We also define

pm = arg max (p− c) [1− F (p)] and for expositional simplicity focus on the case

pm > a.

Each consumer has full information about which n products the firm stocks, and

knows her individual valuations for each of these products. However a consumer

can only learn the prices of these products, by incurring a cost s > 0 and travelling

to the firm’s store. Once inside the store, a consumer can buy whichever products

she wants. We therefore interpret s as both a search and a shopping cost. The

benchmark model has three stages. At stage one, the firm chooses a price for each

product which we denote by p = (p1, p2, . . . , pn). At stage two, each consumer

decides whether or not to incur s and visit the firm. Note that since the firm does

not advertise, consumers do not know prices when they make this decision. At

stage three, consumers who have travelled to the store learn prices and make their

purchases.

2.1 Solving for equilibrium prices

Consumers must decide whether or not it is worthwhile for them to search the

firm. In order to do this, at stage two they form expectations pe = (pe1, pe2, . . . , p

en)

about the price of each product. A consumer then visits the firm if and only if

her expected surplus∑n

j=1 max(vj − pej , 0

)exceeds the search cost s. We look for

7Bagnoli and Bergstrom (2005) show that many common densities (and their truncations) are

logconcave. Logconcavity ensures that the hazard rate f (vj)/ (1− F (vj)) is increasing.

7

rational expectations equilibria, in which consumers’ belief about prices is correct.

We begin by noting that any price vector satisfying∑n

j=1 max(b− pej , 0

)≤ s is

an equilibrium. If consumers expect such a price vector, none of them choose to

search. The firm then expects to earn zero profit, and is therefore willing to set

its prices equal to whatever consumers were expecting. Hence we have p = pe,

as is required for a rational expectations equilibrium. Moreover when n = 1 there

are no other equilibria; in particular, we cannot construct an equilibrium in which

consumers search. The intuition for this result can be traced back to Diamond

(1971) and Stiglitz (1979). Suppose to the contrary that pe1 < b − s, in which case

all consumers with v1 ≥ pe1 + s would find it worthwhile to visit the firm. The

problem is that since all the consumers in the store would be willing to pay at least

pe1 + s for the product, the firm would hold them up by charging p1 > pe1. This

hold-up problem can only be avoided when pe1 ∈ [b− s, b], i.e. when the expected

price is so high that no consumer wishes to search.

Clearly in order to construct a theory of retailing, firms must ordinarily make

positive sales. What we now show is that when a firm stocks multiple products,

the above ‘no search problem’ can be resolved in a very natural and straightforward

manner. To this end let us now suppose that∑n

j=1 max(b− pej , 0

)> s, in which case

some consumers visit the firm. Once a consumer has searched, she buys any product

for which her valuation exceeds the price. Therefore we can write the demand for

product i as

Di (pi;pe) =

b∫pi

f (vi) Pr

(n∑j=1

max(vj − pej , 0

)≥ s

∣∣∣∣∣ vi)dvi (1)

The firm chooses the actual price pi to maximize its profit (pi − c)Di (pi;pe) on good

i. In equilibrium consumer price expectations must also be correct. After imposing

pei = argmaxpi

(pi − c)Di (pi;pe) we obtain Lemma 1.

Lemma 1 The equilibrium price of unadvertised good i satisfies

Di (pi = pei ;pe)− (pei − c) f (pei ) Pr

(∑j 6=i

max(vj − pej , 0

)≥ s

)= 0 (2)

8

pe1 pe1+sa b

pe2

pe2+s

a

bv2

v1

Search andbuy product

1 only

Marginalconsumers forproduct 1 who

search

Search and buyboth products

Search and buyproduct 2 only

Figure 1: Search and purchase behavior when n = 2.

To interpret Lemma 1, suppose the firm charges slightly more for product i than

consumers were expecting. In equilibrium the net impact on profits should be

zero. Firstly, the firm gains extra revenue from consumers who continue to buy

the product, and they have mass equal to demand. Secondly however, the firm loses

margin pei − c on each consumer who stops buying the good, and they have mass

f (pei ) Pr(∑

j 6=i max(vj − pej , 0

)≥ s)

. Intuitively, consumers who stop buying good

i must be marginal for it (i.e. have vi = pei ) and must also have searched. However

any consumer who is marginal for good i only searches if her expected surplus on

the remaining n− 1 goods,∑

j 6=i max(vj − pej , 0

), exceeds the search cost s. Figure

1 further illustrates this when n = 2 and consumers rationally anticipate prices pe1

and pe2. Consumers in the top-right corner have both a high v1 and a high v2, and

therefore search and buy both products. Consumers in the bottom-right corner have

v2 ≤ pe2 and therefore do not expect to buy product 2; however they still search be-

cause v1 ≥ pe1 + s and therefore product 1 alone is expected to give enough surplus.

Demand for product 1 therefore equals the mass of consumers located in those two

regions. Consumers who respond to small changes in the price of product 1, have a

marginal valuation and yet also search i.e. they have v1 = pe1 and v2 ≥ pe2 + s. The9

thick line in Figure 1 represents these consumers. We can now state that

Proposition 1 If n is sufficiently large, there exists at least one equilibrium in

which consumers search and the firm makes positive profit. The price of each good

strictly exceeds pm.

Once we take seriously the multiproduct nature of retailing, the ‘no search prob-

lem’ which we discussed earlier, can be resolved. Intuitively a multiproduct firm

attracts a broad mix of consumers, not all of whom have high valuations for every

product in the store. In particular some consumers who are marginal for product 1

(say), search because they want to buy other goods. If the firm increases p1 it will

lose the demand of these marginal consumers. Furthermore when the firm’s product

range is large, many consumers who are marginal for product 1, find it worthwhile to

search. This weakens the firm’s incentive to hold-up buyers of product 1, and allows

the existence of a rational expectations equilibrium with search.8 Nevertheless only

consumers with relatively high valuations decide to search. The firm is unable to

commit not to exploit this, and hence ends up charging prices which are higher than

pm.

The critical number of products required for such an equilibrium with search is

not necessarily large. For instance two products are sufficient when c = 0, valuations

are uniformly distributed on [1/2, 3/2] and s ∈(

0, 2−√

24

). More generally, the

critical number of products depends upon parameters of the model in a natural way.

In particular more products are required when either the search cost or marginal

production cost is higher, assuming that f (v) does not decrease too rapidly.

Henceforth we assume that n is large enough to support an equilibrium with

search. As discussed earlier, there also exist other high-price equilibria in which

consumers do not search. However we immediately rule out these no-search equi-

libria, for the following two reasons. Firstly these equilibria are Pareto-dominated,

because they generate zero surplus for both consumers and the firm. Secondly if

the firm has to pay a small fixed cost, it would not enter the industry if it expected

8We are assuming that the search cost does not depend upon store size. However Proposition 1

holds for a general search cost s(n), provided that the average search cost s (n) /n is not too large.10

to play one of these equilibria. Therefore when a firm does enter the industry, con-

sumers can use forward induction to infer that they are not playing a no-search

equilibrium. However this does not necessarily leave us with a unique equilibrium

prediction, because there could be more than one search-equilibrium. Intuitively

the optimal price of one product, depends (via the consumer search decision) on the

expected prices of all other products; this can create multiplicity of equilibrium. For-

tunately there is a natural way to select amongst different search-equilibria. Firstly

in any equilibrium, each product has the same price i.e. pe1 = pe2 = . . . = pen. Sec-

ondly the firm’s profit is higher in equilibria which have a lower price, because more

consumers search and buy its products. Consumer surplus is also higher when the

equilibrium price is lower. Therefore equilibria can be Pareto-ranked.9 Henceforth

we will select the lowest-price (i.e. Pareto-dominant) equilibrium, since it seems a

natural point on which to coordinate.10

2.2 Characterizing equilibrium prices

We now provide further intuition about how the equilibrium price is determined.

Substituting pek = pe for k = 1, 2, . . . , n into equation (2) and then rearranging, the

equilibrium price pe satisfies

Pr

(∑j 6=i

tj ≥ s

)︸ ︷︷ ︸

Price-sensitive consumers

[1− F (pe)− (pe − c) f (pe)] +Pr

(n∑j=1

tj ≥ s

)− Pr

(∑j 6=i

tj ≥ s

)︸ ︷︷ ︸

Price-insensitive consumers

= 0

(3)

9This argument extends to the case where valuations are not drawn from the same distribution.

Suppose that each vj is drawn independently according to its own density function fj (vj). Propo-

sition 1 still holds. Moreover provided the search cost is sufficiently small, one equilibrium has a

lower price vector, higher profit and greater consumer surplus compared to any other equilibrium.10Forward induction may also select this equilibrium. Denote stage-one equilibrium prices by

p(I) < p(II) < p(III) < . . . and the associated profits by π(I) > π(II) > π(III) > . . .. Suppose

the firm must pay F to enter the market, and play the game described in the text. When F ∈(π(II), π(I)

)and the firm enters, consumers should rule out playing anything other than the Pareto-

dominant equilibrium ‘I’, since any other continuation play yields negative profit after entry.

11

where tj ≡ max (vj − pe, 0) denotes a consumer’s expected surplus from good j.

The lefthand side of equation (3) decomposes into two parts, the change in profits

caused by a small change in pi around pe. Note that only consumers with∑n

j=1 tj ≥ s

search, and they naturally subdivide into two groups. Firstly some consumers have∑j 6=i tj ≥ s. They search irrespective of how much they value product i. Hence

the profits earned on these consumers is (pi − c) [1− F (pi)], and small changes in

pi around pe affect profits by 1−F (pe)− (pe − c) f (pe). These consumers are price-

sensitive and give rise to the first term in equation (3). Secondly other consumers

have∑n

j=1 tj ≥ s >∑

j 6=i tj. They only search because they anticipate a strictly

positive surplus on product i. Since vi − pe > 0, they would all buy product i even

if its price were slightly higher than expected. Consequently a small increase in pi

above pe, causes profits on these consumers to increase by 1. These consumers are

price-insensitive and give rise to the second term in equation (3).

We prove in the appendix that Pr(∑n

j=1 tj ≥ s)

is log-supermodular in s and n,

or equivalently, Pr(∑n

j=1 tj ≥ s)/

Pr(∑n

j=2 tj ≥ s)

is increasing in s. This means

that when search is costlier, the ratio of price-insensitive to price-sensitive consumers

is larger. Therefore when the search cost increases, the firm’s demand becomes less

price-elastic. Hence it is intuitive that:

Proposition 2 The equilibrium price increases in the search cost, and decreases in

the number of products stocked.

The proposition shows that a multiproduct firm charges higher prices when

search is more costly. This natural result is also found in single-product search

models such as Anderson and Renault (1999). The usual intuition is that higher

search costs deter consumers from looking around for a better match, which gives

firms more market power. However the mechanism which drives Proposition 2 is

different. In our model, when s increases some consumers with relatively low valu-

ations stop searching. This causes the firm’s customers to have, in a certain sense,

higher average valuations. As such the firm optimally raises its prices.

Proposition 2 also shows that a firm charges lower prices when it has a larger

product range. To be precise, what we have in mind is a situation where consumers12

are interested in buying N products, and the firm has a monopoly supply over n < N

of them. Suppose the firm then adds an extra product to its range, which is not

available elsewhere. Ceteris paribus some new consumers search because they like

the n+ 1th product. Intuitively these new searchers must have relatively low valua-

tions for the other n goods, or they would have already been searching. Therefore

if the firm now reduces p1 (say), demand for product 1 increases (proportionately)

more than it would have before. Equivalently, a larger firm faces a more elastic

demand curve on each product that it stocks; consequently it charges lower prices.11

A larger firm also earns higher profit on every product that it sells, for two reasons.

Firstly fixing prices, demand for each product is higher because the search cost

is spread more widely. Secondly a larger firm can commit to charge lower prices.

As explained earlier, this also increases per-good profits, because lower (expected)

prices induce more consumers to search and make purchases.

As noted in the introduction, empirical evidence suggests that larger firms gen-

erally charge lower prices. One possible explanation is that they have lower costs,

for example due to buyer power or economies of scale. However Proposition 2 says

that even after controlling for costs, larger stores should still charge lower prices, be-

cause they have more elastic demand. Hoch et al (1995) offer some evidence which

is consistent with this. They find that after controlling for demographic factors,

stores with higher sales volumes (which they use to proxy store size) have more

elastic demand. Meanwhile Shankar and Krishnamurthi (1996) find that everyday

low-price stores attract consumers who are more sensitive to unadvertised prices,

whilst Ellickson and Misra (2008) show that this format is favored by larger firms.

3 Advertising

We now extend the model from the previous section by including an additional stage,

during which the firm can advertise one of its prices. In particular at stage zero,

the firm now has the option to choose a price pan for good n, and send consumers an

11As in footnote 8 this result holds for a general search cost s (n), provided that the search cost

does not increase too rapidly in the size of the firm’s product range.

13

advert informing them about pan. This advert costs κ > 0 and cannot subsequently

be reneged upon. At stage one, the firm chooses a price for each product which it

is not advertising. At stage two, each consumer either receives the advert or learns

that the firm did not send one. Each consumer then decides whether or not to incur

s > 0 and visit the firm. Finally at stage three, consumers who have travelled to

the store learn prices and make their purchases. The timing captures the fact that a

firm must prepare its advertising campaign in advance, but can adjust other prices

at any time, up until the point where a consumer enters its store (Chintagunta et

al 2003).12 Notice that if the firm chooses not to advertise during stage zero, all n

prices are chosen together at stage one, such that the game becomes identical to the

one already solved for in the previous section.

3.1 Pricing with advertising

Suppose the firm chooses to advertise during stage zero. We start by fixing its

advertised price pan and solving for the associated equilibrium unadvertised prices.

This then allows us to also solve for the firm’s optimal advertised price.

Suppose the firm has chosen some arbitrary advertised price pan. Consumers

receive the advert and learn pan, whereupon they form expectations (pe1, pe2, . . . , p

en)

about the price of each product. Note that pen = pan because advertising is truthful

by assumption. Consumers search if and only if∑n

j=1 max(vj − pej , 0

)≥ s, and

therefore much of the analysis from the previous section can be straightforwardly

applied. Equation (1) gives the demand for an unadvertised product, whilst equation

(2) implicitly defines its equilibrium price. Adapting Proposition 1, an equilibrium

with search exists whenever n is sufficiently large. Unadvertised prices weakly exceed

pm, because the firm attracts consumers with predominantly high valuations. For

any fixed advertised price pan, there is no guarantee that the vector of associated

equilibrium unadvertised prices is unique. However as before, each equilibrium has

12In some markets a firm may choose all its prices at once. This suggests an alternative timing,

whereby all actions currently taken during stages zero and one, are instead taken together. The

equilibrium which we solve for in the paper, remains an equilibrium under this alternative timing.

In and Wright (2012) also argue that this equilibrium is the reasonable one to focus on.

14

a unique unadvertised price i.e. pe1 = pe2 = ... = pen−1. The difference with the

previous section is that now, the firm’s preference over equilibria with different

unadvertised prices is more subtle. On the one hand, profits from unadvertised

goods are maximized by playing the lowest-price equilibrium. On the other hand,

playing this equilibrium also maximizes sales of the advertised product, and this

is bad for profits if pan < c. Overall the firm prefers playing the equilibrium with

the lowest unadvertised price, provided either the search cost is small or pan − c is

not too negative. Under either of these conditions it follows that we have a Pareto

dominant equilibrium, and we therefore select it. This allows us to then state:

Lemma 2 When the advertised price increases, so does the equilibrium price of each

unadvertised product.

This result plays an important role in the remaining analysis, and also distin-

guishes our model from others. The result is explained as follows. Firstly when

pan ≤ a − s all consumers search, and therefore the firm charges pm on each un-

advertised product. Secondly when pan > a − s not everybody searches, and the

unadvertised price strictly exceeds pm. When the advertised price increases, some

consumers with relatively low valuations no longer find it worthwhile to search. As

such the consumers who continue to search, have on average higher valuations. The

firm then has more incentive to hold-up consumers, and charge them higher unad-

vertised prices. Consumers therefore rationally anticipate that a higher advertised

price on one product, leads to higher (often strictly higher) prices on other prod-

ucts as well. Consumer heterogeneity is crucial for obtaining this result. If instead

consumers all had the same valuation v for an unadvertised good (as is effectively

the case in Lal and Matutes 1994 and Ellison 2005), the firm would charge v for it,

irrespective of what it charged for its advertised product.

The model therefore explains how a low advertised price on one product can

‘signal’ a firm’s intention to charge relatively low prices on other products. This

happens even though products are completely independent, and there is no private

information about production costs. Accordingly, consumers can use adverts to

update their beliefs about a firm’s overall price level. Consistent with this, Brown15

(1969) surveyed consumers, and found that they associated low-price advertising

with a low general price level. Cox and Cox (1990) constructed some supermarket

circulars; they too found that consumers associate reduced price specials with low

overall store prices. More broadly, Alba et al (1994) show that a flyer’s composition

can affect consumer perceptions about a firm’s overall offering.

We now consider the firm’s optimal choice of advertised price at stage zero.

Demand for the advertised product is

Dn (pan;pe) =

b∫pan

f (vn) Pr

(n−1∑j=1

max(vj − pej , 0

)+ vn − pan ≥ s

)dvn (4)

A small reduction in pan affects demand for the product in two ways. Firstly some

marginal consumers who would have searched anyway, now buy the product. Sec-

ondly some above-marginal consumers who would not have searched, now find it

worthwhile to visit the firm and buy this product. Comparing with equation (1),

only the first effect is present when the product is not advertised. The model there-

fore suggests that when a product is advertised, its demand becomes more elastic.

Kaul and Wittink (1995) argue on the basis of previous empirical studies that this

is true, and call it an ‘empirical generalization’. Chintagunta et al (2003) similarly

find that price sensitivity rises for promoted items.

The advertised price is chosen to maximize joint profits across all products. Let

pe (pan) denote the equilibrium unadvertised price which is set at stage one; Lemma

2 shows it to be an increasing function of pan. Also let Πi (pe (pan) , pan) denote the

profit earned from good i. At stage zero the firm chooses pan to maximize

(n− 1)× Π1 (pe (pan) , pan) + Πn (pe (pan) , pan) (5)

Proposition 3 The optimal advertised price is strictly below pm.

There are three effects which drive this result. Firstly, the search cost reduces

demand for the advertised product, and makes its demand more elastic. Therefore

even if the firm cared only about profits from good n, it would optimally set pan < pm.

Secondly, a reduction in the advertised price encourages more consumers who like

16

good n to search. Once inside the store, these new visitors also buy some (higher-

margin) unadvertised products. For example Ailawadi et al (2006) find empirically

that a promotion in one category leads to higher sales in other categories. This

makes it optimal to further reduce the advertised price. Thirdly, advertising acts as a

commitment device. A lower advertised price on one product, persuades consumers

that other prices will also be lower. This also has a positive impact on profits,

because unadvertised prices are too high to begin with. As such it provides the firm

with an incentive to reduce its advertised price yet further.

The firm therefore optimally uses a low advertised price (below pm) to attract

consumers into the store, but then charges a high price (above pm) on its remaining

products. Depending upon parameters it may even set its advertised price below

marginal cost, i.e. engage in loss-leader pricing. A similar pattern of low advertised

but high unadvertised prices, also arises in other papers such as Hess and Gerstner

(1987) and Lal and Matutes (1994). However there are some important differences

with our analysis. Firstly in our model consumer valuations are heterogeneous,

whereas these other papers assume them to be homogeneous. Secondly therefore,

in our model unadvertised products give positive consumer surplus. This contrasts

with the other papers, where unadvertised products have such high prices that they

leave consumers with no surplus. Thirdly, our paper offers an additional explanation

for low advertised prices, namely the desire to build a store-wide ‘low price image’

i.e. the third effect in the previous paragraph. This is absent from the other papers.

Corollary 1 When the firm advertises, it decreases the price of every product.

When the firm does not advertise, it charges a price pu (n) > pm on each product.

When the firm does advertise, it could choose pan = pu (n) such that by equation (2)

it would also charge pu (n) for each of its unadvertised products. However from

Proposition 3 the firm’s optimal advertised price is strictly below pu (n), so by

Lemma 2 equilibrium unadvertised prices are also below pu (n). Hence advertising

leads to lower prices across the firm’s whole product range. In his survey article

Bagwell (2007) writes that there is “substantial evidence” that advertising leads to

lower prices. For example Devine and Marion (1979) publicized the prices of some17

randomly selected grocery products in local newspapers; firms responded by reducing

their prices on those products. However our model makes a stronger prediction:

even non-advertised prices should be lower.13 To the best of our knowledge, almost

no empirical work tests this prediction. Indeed Bagwell writes that “the effect of

advertising on the prices of advertised and non-advertised products warrants greater

attention”. One exception is Milyo and Waldfogel (1999), who look at the lifting

of a ban on alcohol advertising in Rhode Island. They find that some stores began

advertising, and that their other unadvertised prices fell but not by a statistically

significant amount (Table 5, page 1091). However the authors note that during their

sample period, relatively few stores advertised in newspapers, and that the long-run

impact of advertising could be different.

3.2 Under what conditions will the firm advertise?

At stage zero the firm must decide whether or not to advertise. Advertising strictly

increases gross profit, because it gives the firm more control over the prices that it

charges. However sending an advert is costly, and so the firm advertises if and only

if the cost κ is small enough. How is this decision affected by the size of the firm’s

product range? To gain some insights, consider an example in which [a, b] = [0, 1],

f (v) = 1, c = 3/4 and s→ 0. Figure 2 shows that the extra gross profit generated

by advertising, varies non-monotically with product range. The reason is that in

this example, a search-equilibrium exists if and only if n > 5. Therefore when n ≤ 5

the profit earned from not advertising is zero, whilst the profit from advertising is

positive and clearly increasing in n. Once n > 5 the firm can earn positive profit even

without advertising, and this profit grows sufficiently quickly, that the gains from

advertising decrease with n.14 Other numerical examples suggest a similar pattern,

although since unadvertised prices are only implicitly defined, we are unable to

show this analytically. Analytic results can however be obtained by focusing on the

behavior of large firms:

13Bester (1994) shows that a single-product monopolist charges less when it advertises.14The firm’s optimal advertised price is also non-monotonic, reaching a minimum when n = 5

and approaching pm = 7/8 as n becomes large.18

n

Advertising Gains

0.00

0.01

0.02

0.03

0.04

0.05

0.06

5 10 15 20

Figure 2: Non-monotonicity of the gains from advertising

Proposition 4 (i) for any κ > 0 there exists an n′ such that a firm with n ≥

n′ products does not advertise. (ii) for any δ > 0 there exists an n′′ such that

conditional on advertising, a firm with n ≥ n′′ products charges more than pm − δ

for its advertised product, and less than pm + δ for its unadvertised products.

Proposition 4 is explained as follows. Even if the firm does not advertise, most

consumers find it worthwhile to search when n is large. Indeed the number of

consumers who do not search decreases exponentially in n. Therefore a large firm

gets little benefit from advertising, because advertising induces only a small number

of new consumers to search and buy other products. Of course a large firm does

advertise if the cost κ is sufficiently small, but it won’t choose an advertised price

that is significantly below pm. Intuitively a low advertised price would sacrifice a

lot of profit on the advertised good, relative to the (small) increase in profits earned

from other products.

Finally Proposition 4 also enables us to characterize the relationship between

product range and overall retail strategy. Compare two local monopolists, a ‘small’

one with ns products and a ‘big’ one with nb > ns products. Suppose κ is low

enough that the small firm decides to advertise. We can prove that when nb is19

sufficiently large, the big firm charges a lower price on each product which the two

firms have in common and which neither is advertising.15 Therefore the model

predicts that a large firm should follow an ‘everyday low-pricing’ strategy: it should

charge low prices, but either eschew advertising completely, or else barely offer any

discount on its advertised product. The model also predicts that a smaller firm is

more likely to follow a ‘high-low’ strategy: it should charge higher prices on most

products, but also advertise a very low price on one product (or more generally, on

a few products16). Empirical evidence largely supports this prediction. For example

Ellickson and Misra (2008) find that larger US grocery stores, are more likely to

report being an everyday low-pricer. Shankar and Krishnamurthi (1996) also find

that people who shop at everyday low-price stores, are more responsive to regular

(i.e. non-promotional) price changes; this is consistent with our model.

4 Competing retailers

This section introduces retail competition into our set-up. There are now two firms

A and B, each of whom sells the same two products. (In the next section we dis-

cuss what additional insights emerge when firms sell more than two products.) As

before valuations for these products are known to consumers, and are drawn inde-

pendently using a common density function f (v). For simplicity we now assume

that (p− c) [1− F (p)] is concave. Independent of their product valuations, con-

sumers divide into two different groups. A fraction γ ∈ (0, 1) are ‘loyal’ to one firm,

and make any purchases there. This loyal consumer base is split equally between

the two firms. The remaining fraction 1− γ of consumers are ‘non-loyals’, and have

no preference for where they purchase their goods.

15The relationship between product range and unadvertised prices is therefore qualitatively the

same as in Proposition 2, although it is no longer monotonic.16Suppose the firm can pay κ′ and advertise m prices. Assume the search cost is relatively small.

A large firm eschews advertising but charges a low price on all products. A small firm charges

high prices on most products but, for suitable values of κ′, advertises low prices on m products.

Moreover for a fixed κ′, if the firm could choose how many prices to advertise, it would select

m = n. This would yield the highest possible profit, gross of the advertising cost.

20

The timing closely follows that of the previous section. At stage zero, each firm

has the opportunity to choose a price for one of its products, and send consumers

an advert informing them about this, for a cost κ > 0. At stage one, each firm

simultaneously chooses a price for each of its unadvertised products, without know-

ing what its rival did at the previous stage. Then at stage two, consumers either

receive a firm’s advert, or learn that it did not send one. Using only this informa-

tion, each consumer then decides whether to incur s > 0 and search a firm; without

loss of generality, we assume that loyals may only search their favorite firm. At

stage three, consumers who searched a firm learn its prices, and may buy any prod-

uct(s) from it. Non-loyals may also then incur an additional s > 0 and search the

other firm, whereupon they also learn its prices, and can buy any product(s) from

it. Non-loyals may also go back and buy more products from the first firm, for a

return cost r ∈ (0, s). As explained below, we assume that s is not too large, such

that non-loyal consumers sometimes multistop shop i.e. buy each product from a

different firm. Finally, as in the previous section, our results are robust to other

natural assumptions about timing.17

We also impose some technical assumptions. Firstly if a consumer is indifferent

about which firm to search first, we assume that she picks one at random. Secondly

at stage two, consumers must form an expectation about each firm’s prices. We

assume that if a consumer searches one firm and finds that its prices are different

from what she expected, she does not change her belief about the other firm’s prices.

This ‘no-signaling-what-you-don’t-know’ assumption is standard within the search

literature. Thirdly to make the problem more interesting, we assume that a search-

equilibrium exists under monopoly, even when the firm does not advertise.18 Finally,

it is convenient to introduce the following notation. Let pu denote the (Pareto

dominant equilibrium) price that a non-advertising monopolist charges, and let πu

denote the corresponding profit. In addition when a monopolist advertises price q

on one product, let φ (q) be the (unique equilibrium) price that it charges for its

17For example, suppose that stages zero and one are collapsed, such that a firm must choose both

prices together, and at the same time decide whether to advertise one of them. The equilibrium

which we solve for in Proposition 5 (below) remains an equilibrium.18A numerical example was provided on page 10, in which a search-equilibrium exists for n = 2.

21

other product, and let π (q) denote its total profit. It is straightforward to prove

that (i) φ (q) = pm when q ≤ a− s, (ii) φ (q) is continuous and strictly increasing in

q for all q ∈ [a− s, pu], and (iii) φ (pu) = pu.

4.1 Solving for an equilibrium

To begin with we fix play at stage zero, and solve for the associated equilibrium

unadvertised prices:

Lemma 3 Suppose that any advertised price chosen at stage zero is strictly below

pm. Provided s is sufficiently small and γ sufficiently large, at stage one there is an

equilibrium in which:

(i) A firm that did not advertise, charges pu on both products.

(ii) A firm that advertised q on one product, charges φ (q) for the other.

Lemma 3 is a natural extension of our earlier results. It says that conditional on

how a firm plays at stage zero, there exists an equilibrium in which its unadvertised

price(s) are the same regardless of whether it is a monopolist or a duopolist. In

particular the unadvertised price(s) set by one firm, do not depend on how often its

competitor advertises, or on the distribution from which its competitor’s advertised

prices are drawn. Lemma 3 is therefore reminiscent of Diamond (1971), the difference

being that we prove it in a multiproduct context where firms can advertise. Although

we are not able to prove that this equilibrium is unique, we are also unable to

construct any alternative (Pareto-undominated) equilibria.19

Before providing an intuition for Lemma 3, it is useful to first comment on its im-

plications for non-loyal consumers. Firstly suppose that neither firm has advertised.

Non-loyals expect both firms to charge the same prices, and hence are indifferent

about where to shop. Secondly suppose both firms have advertised the same prod-

uct. Non-loyals only consider shopping at the firm with the lowest advertised price

because, recalling the properties of φ (q), that firm is expected to be cheaper on the

19Lemma 3 is only stated for advertised prices which are strictly below pm. We will look for an

equilibrium of the whole game in which this is true, and then verify that no firm wishes to deviate

and advertise a price above pm. Further details are provided in the appendix.22

unadvertised product as well. Thirdly suppose only one firm has advertised. Non-

loyals only consider shopping at the advertising firm because, given the properties

of φ (q), it is expected to be strictly cheaper on both products. Notice that in all

three cases, non-loyal consumers one-stop shop in equilibrium. Fourthly suppose

the firms advertised different products. According to Lemma 3 advertised prices are

strictly below pm whilst unadvertised prices exceed pm. Therefore provided that s

is sufficiently small, a non-loyal consumer who intends to buy both products, does

better to multistop shop and ‘cherry-pick’ each firm’s advertised product. Hence in

this fourth case, a firm never sells its unadvertised product to a non-loyal consumer.

With this in mind we now provide an intuition for Lemma 3. The main idea is

that a firm never faces any ‘real’ competition on its unadvertised product(s). To

see this more clearly, start with part (i) of the lemma. Suppose that firm A has not

advertised. Some of the time, its competitor has advertised: non-loyal consumers do

not search firm A because they (correctly) anticipate that it will be more expensive.

As such there is no competition between the two firms. The rest of the time, the

competitor has not advertised either: some non-loyals visit firm A, and having sunk

the search cost, are willing to pay slightly higher prices than they expected, rather

than search a second time. Again, there is no real competition between the two

firms. As such, firm A exploits its de facto monopoly position over any consumer

who enters its store, by charging the monopoly price pu. Now consider part (ii)

of the lemma. Suppose that firm A has advertised product 1. Some of the time,

its competitor has advertised product 2: non-loyal consumers multi-stop shop, and

don’t buy unadvertised products. The rest of the time, non-loyal consumers one-

stop shop: they visit firm A if and only if they expect it to have the weakly lowest

price for both products. Once these non-loyal consumers are inside A’s store, it is

costly for them to go to the competitor. This again gives firm A (local) monopoly

power when choosing its unadvertised price.

Having solved for an equilibrium at stage one of the game, where firms choose

their unadvertised prices, we now solve for an equilibrium at stage zero, where firms

have the opportunity to send out an advert.

23

Proposition 5 Provided s is sufficiently small and γ sufficiently large, there exists

an equilibrium of the whole game in which:

(i) With probability α, at stage zero a firm does not advertise. At stage one, the

firm charges pu on both products.

(ii) With probability 1− α, at stage zero a firm sends out an advert containing the

price of one product. Each product is equally likely to be included in the advert. The

advertised price is strictly below pm and drawn from an atomless distribution. At

stage one, if the firm’s advertised price was q, it charges φ (q) for its other product.

(iii) There exist two thresholds satisfying 0 < κ < κ. α = 0 when κ ≤ κ. α is a

continuous and strictly increasing function of κ when κ ∈ [κ, κ]. α = 1 when κ ≥ κ.

Proposition 5 characterizes a natural equilibrium where each firm randomizes

at stage zero. Firstly, depending upon the advertising cost κ, a firm may random-

ize over whether or not to send an advert. As expected, when the cost of sending

an advert increases, each firm is less likely to send one. Secondly, when a firm

does advertise, it chooses at random which product to promote. We discuss this

in more detail in Section 4.2. For now we simply note that if one firm picks its

advertised product at random, its competitor cannot do better than also pick ran-

domly. Thirdly, each firm randomizes over what price to charge for its advertised

product. Intuitively, with positive probability firms advertise the same product,

such that non-loyal consumers one-stop shop at the firm with the lowest advertised

price. This creates a tension: each firm wants to advertise a lower price than its

competitor, and thereby capture non-loyal consumers, but also wants to advertise a

high price and exploit its loyal customer base. This tension is resolved when firms

randomize over their advertised price.

We now discuss a number of interesting implications from Proposition 5. First,

one of the key insights from our analysis is the positive relationship between a firm’s

advertised and unadvertised prices. Since in a competitive setting each firm must

randomize over its advertised price, it indirectly randomizes over its unadvertised

price as well. Thus ex post, there can be price dispersion even for a product which

neither firm is advertising. As discussed in the introduction, our model therefore

24

provides a new explanation for dispersion of unadvertised prices.20

Second and related to this, Proposition 5 makes a novel prediction about the

aggregate price distribution. Suppose that κ ∈ (κ, κ) i.e. the advertising cost is

intermediate. With probability α ∈ (0, 1) a firm does not advertise, and charges

a ‘regular price’ pu on both its products. With probability 1 − α a firm holds a

promotion, and advertises a random discount on one product which exceeds pu−pm.

At the same time, the firm offers a smaller discount on its other product, which is

less than pu − pm but bounded away from zero. Hence within the same framework,

we have both high regular prices, as well as the coexistence of small (unadvertised)

and large (advertised) random price reductions. As such, Proposition 5 suggests

that the aggregate price distribution should have a mass point at the regular price,

as well as at least two gaps. Compared to other theoretical papers, this prediction

is richer and more consistent with empirical evidence.21 For instance Pesendorfer

(2002) and Hosken and Reiffen (2004a) document that grocery products have a

regular price, and stay at that price for long periods of time. They show that

deviations from this regular price are generally downward, and potentially of sizeable

magnitude. Meanwhile Klenow and Malin (2010) also stress the prevalence of small

price changes, across a wide range of studies.

Third, Proposition 5 has each firm randomly choosing which product it is going

to promote. Consistent with this, Rao et al (1995) conclude that promotions on

ketchup and detergent are independent across competitors, as are the promotional

prices themselves. Lloyd et al (2009) have data on UK supermarket prices, and find

that over a two and a half year period, roughly two-thirds of the products in their

sample experience a sale. This suggests that sales are applied across a wide range of

products, and accords with the idea that rivals should find it hard to predict which

products a firm is going to promote.

Finally, an important feature of our set-up is that non-loyal consumers may

multi-stop shop. In equilibrium, with positive probability these consumers end up

20Price dispersion is extremely widespread in practice. See Baye et al (2006) for a survey.21Related single-product models include Varian (1980) and Baye and Morgan (2001) which

predict no gap, and Narasimhan (1988) and Robert and Stahl (1993) which predict a single gap.

25

cherry-picking advertised specials from each firm. Consistent with this, Smith and

Thomassen (2013) find that around 60% of UK grocery shoppers visit more than one

store a week. Meanwhile Fox and Hoch (2005) document significant heterogeneity

in household cherry-picking behavior. Using a rather strict definition they find that

8% of shopping trips involve cherry-picking. As in our model, they also document

that cherry-pickers buy significantly more advertised and discounted products.22

4.2 Must the advertised product be chosen at random?

To complete this section, we now discuss whether a firm needs to pick its advertised

product at random. Proposition 5 already solved for an equilibrium with this ran-

domization. Here we ask whether there is another equilibrium, where either (i) both

firms advertise the same product, or (ii) each firm advertises a different product.

We assume throughout that at stage one of the game, firms play the equilibrium

in Lemma 3. To simplify the exposition, we assume that κ is relatively small, such

that each firm advertises with probability one.23

Firstly, does there exist an equilibrium where both firms advertise the same

product? Consider a putative equilibrium in which both firms advertise product 1.

Recalling the discussion after Lemma 3, non-loyals either stay at home or one-stop

shop at the firm with the lowest advertised price. Therefore letting paA1 and paB1

denote the firms’ advertised prices, firm A’s profit is[γ/2 + (1− γ) 1pa

A1Rpa

B1

]π (paA1)− κ (6)

22If, in contrast to this evidence, non-loyals cannot multi-stop shop, the model becomes more

complicated to solve. To simplify the problem we therefore focus only on advertised prices below

pm, and assume that each firm can observe its rival’s advertising decision, before it chooses its own

unadvertised prices. Under the same hypotheses as Proposition 5, and assuming c is not too large,

we can construct a candidate equilibrium in which (i) each firm randomly decides to advertise,

(ii) it randomizes over what product to advertise, and at what price, and (iii) there is a positive

relationship between a firm’s advertised and unadvertised prices. However we are only able to

verify that a firm’s profit function is locally concave in its unadvertised price(s).23Our results also apply when κ is larger, provided the firms advertise with the same probability.

However for certain values of κ there also exist asymmetric equilibria, where one firm doesn’t

advertise, whilst the other firm picks a product at random and advertises it.26

where 1paA1Rp

a

B1is an indicator function, taking the value 0 if paA1 > paB1, the value

1/2 if paA1 = paB1, and the value 1 if paA1 < paB1. As usual there is a tension: the

firm would like to advertise a relatively high price and extract surplus from its γ/2

loyal customers, but advertise a relatively low price and capture the 1− γ non-loyal

customers. This tension is resolved when firms mix, and draw their advertised price

from the same atomless distribution. When s is small the upper support of this

distribution is a price p ∈ (c, pm). Therefore using (6), a firm’s expected profit in

this putative equilibrium is γπ (p) /2 − κ. Now suppose firm A deviates from the

putative equilibrium, and advertises product 2 at a price p. There exists a unique

stage-one equilibrium, in which firm A charges φ (p) for product 1. Recalling our

earlier discussion, non-loyal consumers now cherry-pick and purchase only advertised

products. Hence they search firm A if and only if their expected surplus on A’s

advertised product exceeds s i.e. v2 ≥ p + s. Consequently A’s expected profit

following its deviation is

γπ (p) /2 + (1− γ) (p− c) [1− F (p+ s)]− κ (7)

The deviation is profitable because (7) exceeds γπ (p) /2 − κ. As such, there does

not exist an equilibrium where both firms advertise the same product.

Secondly then, does there exist an equilibrium where each firm advertises a dif-

ferent product? Consider a putative equilibrium in which firm A advertises product

1 at a price paA1 < pm, and firm B advertises product 2 at a price paB2 < pm. Recalling

our discussion after Lemma 3, non-loyals cherry-pick and purchase only advertised

products. Therefore firm A’s profit is

γπ (paA1) /2 + (1− γ) (paA1 − c) [1− F (paA1 + s)]− κ (8)

Let P denote the set of prices which maximize (8). When s is small each member

of P lies in (c, pm). Suppose firm B plays along with the putative equilibrium, and

advertises a price paB2 ∈ P on product 2. However suppose firm A deviates from the

putative equilibrium, and also advertises product 2, but at the lower price of paB2−ε.

There exists a unique stage-one equilibrium, in which firm A charges φ (paB2 − ε) for

product 1. Recalling our earlier discussion, non-loyals now either stay at home or27

one-stop shop at firm A, because they (correctly) anticipate that A is cheaper on

both products. Consequently A’s profit following its deviation is

[1− γ/2]π (paB2 − ε)− κ (9)

It is straightforward to show that the deviation is profitable, because there exists

an ε > 0 such that (9) exceeds (8) when evaluated at any paA1 ∈ P . As such, there

does not exist an equilibrium where each firm advertises a different product.

The intuition behind these two negative results is as follows. Competition is

fierce when firms advertise the same product, because non-loyals only purchase

from the firm with the lowest advertised product. However, competition is soft

when firms advertise different products because non-loyals cherry-pick and therefore

purchase from both firms. Firstly then, when firms are supposed to advertise the

same product, there is an incentive to soften competition by deviating, and adver-

tising a different product. Secondly, when firms are supposed to advertise different

products, it is (too) easy to steal demand away from the competitor, by deviating

and advertising the same product as the competitor, but at a lower price. Overall

then, an important feature of equilibrium is that a firm should randomize over which

product it advertises, such that its promotional activity cannot be second-guessed

by its competitor.

5 Extensions and Discussion

5.1 Elastic demands

Some of the search literature assumes that a firm sells only one product, but that

each consumer wishes to buy multiple units of it, i.e. individual consumers have

an elastic demand. This differs from but is clearly related to our assumption, that

consumers have a unit-demand for multiple products. We believe that our modelling

choice is more appropriate in many retail contexts. However to show how our results

fit into the wider literature, we briefly consider elastic demands.

As a first step, consider the following monopoly version of Stiglitz (1979). There

is a single firm who sells one product. Consumers are identical, and when faced28

with a price p, wish to buy 1 − F (p) units of the product. If consumers were

informed about the price, the firm would charge pm = arg max (p− c) [1− F (p)].

Now suppose that the price can only be learnt by paying a search cost s > 0.

Stiglitz shows that provided s is not too large, there exists an equilibrium in which

all consumers search and the firm charges pm. Intuitively consumers all search,

because they anticipate earning positive surplus on inframarginal units. The firm

then faces the same demand as it would if s = 0, and so charges pm as expected.

Proposition 1 of our model is clearly related. We show that a search-equilibrium

exists in a multiproduct world, because surplus on one product attracts consumers

who are marginal for another. The difference is that in our model, consumers are

heterogeneous, and this has at least two important implications. Firstly, prices in

our model (almost always) strictly exceed pm. Secondly, equilibrium prices respond

to changes in advertising strategy (as well as changes in s and n) in a way that

is both intuitive and economically interesting. This does not happen in Stiglitz’s

model, because consumers are homogeneous, such that the equilibrium price always

equals pm.

To illustrate this point further, we now depart from Stiglitz’s approach, and al-

low for heterogeneous elastic demands. To do this as simply as possible, we model

consumer heterogeneity using a single parameter θ which is drawn from[θ, θ]

using

a strictly positive density function g (θ). A consumer’s demand for any one product

is p = θ+P (q) where q is the quantity consumed and P (q) is continuously differen-

tiable, strictly decreasing and concave. Consequently consumers with higher θ both

earn higher surplus and have less elastic demands. When there is no search cost

and types are not too different, the firm sells to everybody and charges the standard

monopoly price pm, which is now defined as

pm = argmaxx

∫ θ

θ

g (θ){

(x− c)P−1 (x− θ)}dθ (10)

Henceforth assume that there is a search cost and that the firm sells n ≥ 2 products.

To begin with, suppose the firm does not advertise. The nature of equilibrium

depends on two thresholds sn and sn which satisfy 0 < sn < sn. Firstly when s ≤ sn

there is an equilibrium where each good is priced at pm. Since the search cost is29

relatively small, every consumer finds it worthwhile to search. The firm then faces

the same problem as it does when s = 0, and consequently charges pm. This is

qualitatively the same as in Stiglitz. Secondly however when s ∈ (sn, sn), such an

equilibrium is no longer possible. This is because even if consumers expect to pay

pm on each product, those with low θ cannot earn enough surplus to make search

worthwhile. Since the firm is only searched by consumers with relatively high θ,

the firm has an incentive to hold them up. Instead there is an equilibrium in which

the price strictly exceeds pm, increases in the search cost, and decreases in product

range. This differs from Stiglitz, but is qualitatively the same as in our earlier

unit-demand model. Thirdly when s ≥ sn the search cost is too high to support a

search-equilibrium.

One of the main insights from our unit-demand model, was the positive relation-

ship between a firm’s advertised and unadvertised prices. With elastic demands, we

also find that:

Proposition 6 Suppose the firm advertises one product. An increase in the adver-

tised price, (weakly) increases the equilibrium price of each unadvertised product.

The intuition for this result is by now familiar. An increase in the advertised price

causes fewer consumers to search. Those who stop searching have a relatively low

θ. Consequently the firm’s customers become less price-sensitive, such that in equi-

librium it optimally charges a higher price for its unadvertised products.

5.2 Heterogeneous products

Throughout the paper we have assumed that products are symmetric. One inter-

esting question is the following: if products are not symmetric, which one of them

should a monopolist advertise? Empirical evidence shows that firms may adver-

tise discounts on products with higher demand or popularity, such as tuna at Lent,

or toys before Christmas (Chevalier et al 2003, Hosken and Reiffen 2004b). The

existing literature suggests that firms do this either to economize on their adver-

tising costs, or because certain products are popular with high-spending consumers

(DeGraba 2006).30

Here we briefly use our framework to provide some new perspectives on the

link between popularity and advertising. Suppose a consumer is ‘interested’ in

product i with probability λi. Conditional on being interested in product i, the

consumer’s valuation vi is drawn from [a, b] using a logconcave density function

f (vi). Conditional on being ‘not interested’ in product i, the consumer’s valuation

is vi = d. A consumer’s interest (and valuation) is drawn independently for each

product. We assume that a > c ≥ 0 > d, such that interested consumers are capable

of generating a profit, whilst uninterested consumers are not. Henceforth we will

interpret λi as a measure of product i’s popularity. It is convenient to order the

products such that 0 < λ1 < . . . < λn < λ, where λ < 1 is a constant which is

defined in the appendix.24 We also focus for expositional simplicity on the case

s→ 0, that is where the search cost is positive but very small.

Proposition 7 The monopolist maximizes its profits by advertising product n i.e.

the most popular product.

To gain some intuition suppose the monopolist sells only n = 2 products. When

the firm advertises product i at price qi, the equilibrium price pe3−i of the other

(unadvertised) product satisfies equation (2). More succinctly, as s→ 0 it satisfies(pe3−i − c

)f(pe3−i

)1− F

(pe3−i

) =1

λi [1− F (qi)](11)

The lefthand side of (11) increases in pe3−i, such that a low λ translates into a high

unadvertised price. Intuitively when λ is small, only a small number of consumers

who are marginal for the unadvertised product find it worthwhile to search. Con-

sequently demand for that product is relatively inelastic, and its price is high. On

the other hand, by definition, the firm is able to commit to charge a lower and

more profitable price on its advertised product. Consequently the profit earned per

interested consumer, is higher on the advertised good. This implies it is optimal to

ensure that the advertised good is also the one which interests the largest number

of consumers i.e. the most popular product. Moreover it is also clear from (11) that

since λ1 < λ2, ceteris paribus the firm’s unadvertised product will have a lower price

24This places an upper bound on the profits earned from an any unadvertised product.31

(and thus be more profitable, per interested consumer), when it is the less popu-

lar product. Intuitively by advertising the more popular product, the firm induces

more consumers who are marginal for the other product to search, thus making its

demand more elastic. Consequently another reason to advertise the more popular

product, is that it helps the firm to create a lower price image. As such we provide

two new reasons for why a firm should advertise popular products.

5.3 Relationship between product range and advertising

We have already shown that a monopoly firm has little incentive to advertise when

its product range is large. We now discuss whether this is also true when a firm

faces competition. In order to do this we reconsider the duopoly model of Section

4, but now assume that the firms sell n ≥ 2 (identical) products. At this point it

is convenient to modify slightly our earlier notation. In particular let pu (n) denote

the price charged, and πu (n) the profit earned, by a non-advertising monopolist.

Also when a monopolist advertises price q on one product, let φ (q;n) denote the

equilibrium price that it charges for its other products, and let π (q;n) be its total

profit.

Under what conditions does the duopoly model have an equilibrium where neither

firm advertises? Firstly, suppose that at stage zero neither firm advertises nor

expects its competitor to do so. It is straightforward to show that there is an

equilibrium at stage one in which each firm charges pu (n). Non-loyal consumers

are then indifferent about where to shop and so choose randomly. Each firm earns

a profit πu (n) /2. Secondly, suppose that at stage zero neither firm is expected to

advertise, but in fact firm A ‘deviates’ and advertises a price q < pu on one product.

It is straightforward to show that there is an equilibrium at stage one in which firm

A charges φ (q;n) on its remaining n − 1 unadvertised products.25 Again the key

insight is that because φ (y;n) increases in y, firm A is expected to charge a lower

25If firm A advertises a price q < pu (n) which satisfies φ (q;n) = pu (n), we must drop the

assumption made in Section 4 that non-loyals randomly choose where to shop when they are

indifferent. Instead we assume that they all shop at firm A, even if they intend to buy only

unadvertised products.

32

price on each product, and therefore attracts all non-loyal consumers. Hence firm

A’s best stage-zero deviation is to advertise a price in arg maxq π (q;n) (which by

Proposition 3 is below pm) and earn maxq π (q;n) (1− γ/2)− κ.

It follows that there exists an equilibrium in which neither firm advertises, pro-

vided that

κ ≥ maxq π (q;n)− πu (n)

2+

(1− γ) maxq π (q;n)

2(12)

To interpret condition (12) note that when a monopolist chooses to advertise, its

profits increase in proportion to maxq π (q;n)− πu (n). Since maxq π (q;n)− πu (n)

is small when n is large, we concluded in Section 3 that a monopolist with a large

product range is unlikely to advertise. The intuition was that when a firm faces

little or no competition, it draws its customers from a fixed pool. When its prod-

uct range is large, most consumers search anyway and so advertising is not very

effective. However it is clear from (12) that under duopoly there is a countervailing

effect, because maxq π (q;n) increases in n. Intuitively with duopoly, a firm’s pool

of consumers is not fixed. In particular by advertising a low price, a firm is able to

steal non-loyal consumers from its rival. Since the profit earned on these extra con-

sumers is increasing in n, so are the incentives to steal them. This suggests that the

degree of market competition, as proxied in our model by the fraction of non-loyal

consumers, is crucial for determining whether or not a large firm advertises.

5.4 Other extensions

The Online Appendix provides two other extensions. The first extension allows for

substitute products.26 In particular it considers a setting where a firm stocks n

independent product categories, each category contains z substitute products, and

26Some related papers which focus more on substitutes, include the following. Villas-Boas (2009)

shows that a firm should avoid stocking too many substitute products, otherwise consumers will

expect it to charge high prices, and consequently choose not to search it. Konishi and Sandfort

(2002) demonstrate that when consumers are uncertain about their match values, a firm might

advertise a low price on one product, hoping that after searching, consumers will buy a more

expensive substitute instead. Rosato (2013) explores the incentive of a firm to have only limited

availability of advertised products, whilst Petrikaite (2013) shows that a firm may wish to obfuscate

some of its product range.33

consumers wish to buy at most one product from each category. It is shown that

the main results from the paper continue to hold, in this more general set-up. The

second extension considers some alternative firm strategies. It points out that when

a firm bundles its products together, it effectively becomes a single-product firm,

and hence the market breaks down. However it also provides conditions, such that

a monopolist does not wish to bundle its products in this way. This extension

also shows, for example, that a firm can commit to charge lower prices by sending

consumers money-off vouchers. As such, these vouchers are a substitute to the (more

commonly-used) price advertising which we focus on in the paper.

6 Conclusion

Consumers are often poorly-informed about prices, and must spend time and effort

in order to learn them. Our paper has sought to understand how this affects a

multiproduct firm’s pricing and advertising strategy. We showed that consumers can

use cues, such as product range and information contained within adverts, to form

an impression about a firm’s overall price image. This price image then influences

consumer search, and ultimately also purchase behavior. We noted that firms with

larger product ranges generally charge lower prices. Our model provides a novel

explanation for this phenomenon, that is unrelated to production costs or store

differentiation. In particular, we showed that these firms charge less because they

are searched by consumers who are endogenously more price-sensitive.

We also demonstrated that the size of a firm’s product range determines whether

it positions itself as an ‘everyday low’ pricer or as a ‘high-low’ pricer. In particular a

monopolist with a large product range should either eschew advertising entirely, or

else advertise only very small discounts. In contrast a smaller firm should generally

charge higher prices, but advertise a large discount on a few of them. Moreover in

contrast to much previous work, we found a positive relationship between a firm’s

advertised and unadvertised prices. Thus when a firm advertises a low price on one

product, it is able to convince consumers that its unadvertised prices will also be

somewhat lower. As such, if a small firm did not advertise, its unadvertised prices

34

would be even higher.

Finally we showed that when two multiproduct firms compete, in equilibrium

each firm is unable to second-guess its competitor’s advertising strategy. In par-

ticular each firm randomizes over whether to hold a promotion, which product to

promote, and what price to charge for it. We also showed that compared to the

existing theoretical literature on multiproduct competition, our model provides a

new explanation for the dispersion of unadvertised prices, and has rich implications

for micro-level price distributions.

We believe that our paper demonstrates some of the advantages that can be

gained by explicitly modelling the multiproduct nature of a firm’s offering. We also

hope that it will encourage more research on this topic, both empirical and theoret-

ical. On the empirical side, our model generates a number of testable predictions.

One implication, is that advertising should lead to a reduction even in the prices of

unadvertised products. However as discussed earlier, there is currently little empir-

ical research on this topic. We hope that our paper will generate renewed empirical

interest, in understanding how pricing and advertising on one product category, af-

fects other unrelated categories. On the theoretical side, we think that our paper

opens up several avenues for future research. It would be interesting, for example,

to endogenize firms’ product selections, and investigate under what conditions they

choose to compete head-to-head. It would also be interesting to allow repeated

interaction between firms and consumers.

35

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A Main Proofs

A.1 Proofs for Subsection 2.1

Proof of Lemma 1. Differentiate (pi − c)Di (pi;pe) with respect to pi to get

Di (pi;pe)− (pi − c) f (pi) Pr (T + max (pi − pei , 0) ≥ s) (A.1)

where T =∑

j 6=i max(vj − pej , 0

). Substitute pi = pei and set (A.1) to zero. Lemma

A.2 (below) gives sufficient conditions for (pi − c)Di (pi;pe) to be quasiconcave in

pi.

Proof of Proposition 1. Let p be the unique solution to 1−F (p)−pf (p) /2 = 0,

and n be the smallest integer n such that Pr(∑n

j=2 max (vj − p, 0) ≥ s)> 1/2. It

is sufficient to look for a symmetric equilibrium in which pej = pe∀j. Using equation

(2) pe satisfies

Di (pi = pe;pe)− (pe − c) f (pe) Pr

(∑j 6=i

max (vj − pe, 0) ≥ s

)= 0 (A.2)

The lefthand side of (A.2) is strictly positive when evaluated at any pe ≤ pm, be-

cause Di (pi = pe;pe) > [1− F (pe)] Pr(∑

j 6=i max (vj − pe, 0) ≥ s)

and 1−F (pe)−

(pe − c) f (pe) ≥ 0 for all pe ≤ pm. Assuming n ≥ n, the lefthand side of (A.2) is

strictly negative when evaluated at pe = p, because Di (pi = pe;pe) ≤ 1−F (pe) and

1−F (p)−pf (p) /2 = 0. Therefore since the lefthand side of (A.2) is continuous in pe,

there exists a pe ∈ (pm, p) which solves (A.2). According to Lemma A.2 (below) the

profit on each good is quasiconcave because Pr(∑n

j=2 max (vj − pe, 0) ≥ s)> 1/2.

Lemma A.1 If pei is an equilibrium price, then 1 − F (p) − (p− c) f (p) is weakly

decreasing in p for all p ∈ [pm, pei ).

Proof. We prove the contrapositive of this statement. Suppose that 1 − F (p) −

(p− c) f (p) is not weakly decreasing in p for all p ∈ [pm, pei ).

Step 1. Note that for all p > c, the derivative of 1− F (p)− (p− c) f (p) is propor-

tional to −2/ (p− c) − f ′ (p) /f (p), which is strictly increasing in p for all p > c.

This follows because f (p) is logconcave and therefore f ′ (p) /f (p) decreases in p.42

Step 2. Since by assumption 1−F (p)−(p− c) f (p) is not weakly decreasing in p for

every p ∈ [pm, pei ), there will exist a p ∈ [pm, pei ) such that−2/ (p− c)−f ′ (p) /f (p) >

0. Step 1 then implies that −2/ (p− c)− f ′ (p) /f (p) > 0 for all p ∈ [p, pei ).

Step 3. When pi < pei the derivative of (A.1) with respect to pi is

[−2f (pi)− (pi − c) f ′ (pi)] Pr (T ≥ s) (A.3)

Step 2 implies that (A.3) is strictly positive for every pi ∈ [p, pei ). However by

definition (A.1) is zero when pi = pei , so (A.1) is strictly negative for every pi ∈ [p, pei ).

Given the definition of (A.1), this means that (pi − c)Di (pi;pe) is strictly decreasing

in pi for every pi ∈ [p, pei ) - so the firm would do better to charge some pi ∈ [p, pei )

rather than charge pei . This implies that pei cannot be an equilibrium price.

Lemma A.2 (pi − c)Di (pi;pe) is quasiconcave in pi if either:

1. 1 − F (p) − (p− c) f (p) is strictly decreasing in p for all p ∈ [a, b]. Densities

which do not decrease too rapidly - such as the uniform - satisfy this.

2. Pr (T ≥ s) > 1/2, where T =∑

j 6=i max(vj − pej , 0

)as defined earlier.

Proof. We must show that (pi − c)Di (pi;pe) strictly increases in pi when pi ∈

[a, pei ), and strictly decreases in pi when pi ∈ (pei , b]. Equivalently we must check

that (A.1) is strictly positive when pi ∈ [a, pei ) and strictly negative when pi ∈ (pei , b].

Condition 1. Rewrite (A.1) as

Di (pi;pe)− (pi − c) f (pi) Pr (T ≥ s)

+ (pi − c) f (pi) [Pr (T ≥ s)− Pr (T + max (pi − pei , 0) ≥ s)] (A.4)

The derivative of the first term with respect to pi is

−f (pi) Pr (T + max (pi − pei , 0) ≥ s)− [f (pi) + (pi − c) f ′ (pi)] Pr (T ≥ s)

≤ Pr (T ≥ s) [−2f (pi)− (pi − c) f ′ (pi)]

which is strictly negative because 1 − F (p) − (p− c) f (p) is strictly decreasing in

p by assumption.27 Since by definition the first term in (A.4) is zero when pi = pei ,

27Note that the first term is differentiable everywhere except at the point pi = pei + s (assuming

pei + s < b), because Di (pi;pe) kinks at that point. However this does not affect the argument

that the first term of (A.4) is strictly decreasing in pi.43

it must be strictly positive when pi ∈ [a, pei ) and strictly negative when pi ∈ (pei , b].

The second term of (A.4) is 0 when pi ≤ pei , and negative otherwise. Therefore

(A.1) is strictly positive when pi ∈ [a, pei ) and strictly negative when pi ∈ (pei , b], as

required.

Condition 2. Note that (A.4) is proportional to:[Di (pi;p

e)

f (pi)− (pi − c) Pr (T ≥ s)

]+(pi − c) [Pr (T ≥ s)− Pr (T + max (pi − pei , 0) ≥ s)]

(A.5)

The second term is again 0 when pi ≤ pei , and negative otherwise. The derivative of

the first term with respect to pi is

−Pr (T + max (pi − pei , 0) ≥ s)−Pr (T ≥ s)−Di (pi;pe) f ′ (pi)

f (pi)2 < −1+

Di (pi;pe)

1− F (pi)≤ 0

where the first inequality follows because Pr (T ≥ s) > 1/2 (by assumption), and

because vi has an increasing hazard rate and this implies that −f ′ (pi) /f (pi)2 ≤

1/ [1− F (pi)]. The second inequality follows because Di (pi;pe) ≤ 1−F (pi). Using

the same arguments as for Condition 1, the fact that the first term of (A.5) is strictly

decreasing in pi, means that (A.1) is strictly positive when pi ∈ [a, pei ) and strictly

negative when pi ∈ [a, pei ).

Lemma A.3 In a search-equilibrium all unadvertised prices are the same.

Proof. Rewrite equation (2) as follows:

Pr

(∑j 6=i

tj ≥ s

)[1− F (pei )− (pei − c) f (pei )] + Pr

(n∑j=1

tj ≥ s

)−Pr

(∑j 6=i

tj ≥ s

)= 0

(A.6)

Suppose counter to the claim, that there exists an equilibrium where products

g and h 6= g have prices satisfying peg < peh. Applying Lemma A.1 1 − F (peh) −

(peh − c) f (peh) is (weakly) less than 1−F(peg)−(peg − c

)f(peg), which is itself nega-

tive by equation (A.6). Clearly also Pr(∑

j 6=h tj ≥ s)> Pr

(∑j 6=g tj ≥ s

). There-

fore equation (A.6) cannot hold for both i = g and i = h - a contradiction.

44

A.2 Proofs for Subsection 2.2

Proof of Proposition 2. Since Pr(∑n

j=2 tj ≥ s)> 0, in an equilibrium with

search we can rewrite equation (3) as

φ (pe; s, n) = 1− F (pe)− (pe − c) f (pe) +Pr(∑n

j=1 tj ≥ s)

Pr(∑n

j=2 tj ≥ s) − 1 = 0

Firstly fix n. Start with a search cost s0, and let pes0 denote the lowest solution

to φ (pe; s0, n) = 0. Lemma A.4 (below) shows that Pr(∑m

j=1 tj ≥ x)

is TP2 on

R × N.28 Therefore we conclude that φ(pes0 ; s1, n

)≤ 0 for any s1 ∈ (0, s0). Since

φ (pm; s1, n) > 0 and φ (pe; s1, n) is continuous in pe, this implies that the lowest

solution to φ (pe; s1, n) = 0 is (weakly) below pes0 . Therefore the equilibrium price

increases in s. Secondly fix s. Start with a product range n0, and let pen0denote

the lowest solution to φ (pe; s, n0) = 0. Lemma A.5 (below) shows that φ (pe; s, n)

decreases in n. Therefore we conclude that φ(pen0

; s, n1

)≤ 0 for any n1 > n0, such

that the lowest solution to φ (pe; s, n1) = 0 is (weakly) below pen0. Therefore the

equilibrium price falls in n.

Lemma A.4 Pr(∑m

j=1 tj ≥ x)

is TP2 on R× N.

Proof of Lemma A.4. Define vj = vj− pe, let F (vj) be its distribution function,

f (vj) be its (logconcave) density function,[a, b]

be the interval on which it is

defined, with a = a − pe and b = b − pe. Note that the vj are independent, and

recall that tj ≡ max (vj, 0). Also define Pm (x) = Pr(∑m

j=1 max (vj, 0) ≥ x)

, and

note that

Pm (x) =

∫10≤y≤bPm−1 (x− y) f (y) dy + Pm−1 (x) F (0) (A.7)

where 10≤y≤b is an indicator function. We proceed by induction. Suppose Pm (x) is

TP2 on R× {n− 1, n}. When Pn (x) > 0 we may use equation (A.7) to write

Pn+1 (x)

Pn (x)=

[1− Pn−1 (x)

Pn (x)F (0)

]Ln (x) + F (0) (A.8)

28See Karlin (1968) for a definition.

45

where

Ln (x) =

∫10≤y≤bPn (x− y) f (y) dy∫

10≤y≤bPn−1 (x− y) f (y) dy

Since Pm (x) is TP2 on R× {n− 1, n}, the first term in (A.8) is weakly increasing.

Ln (x) is also weakly increasing. To see this, use a change of variables to rewrite∫10≤y≤bPm (x− y) f (y) dy =

∫Pm (z) f (x− z) 10≤x−z≤bdz

and notice that this is TP2 because all functions in the second integral are TP2,

and TP2 is preserved under composition (Karlin 1968, p.17). Thus Ln (x) is weakly

increasing in x, such that Pm (x) is TP2 on R× {n, n+ 1}.

Therefore to complete the proof, we need only prove that Pm (x) is TP2 on

R× {1, 2}. To do this, note that when P1 (x) > 0 we may write

P2 (x)

P1 (x)= 1 +

∫ x

0

f (z)1− F (x− z)

1− F (x)dz + F (0)

which is increasing in x because f (y) is logconcave and hence has an increasing

hazard rate.

Lemma A.5 Pr(∑m

j=1 tj ≥ x)/Pr

(∑mj=2 tj ≥ x

)decreases in m.

Proof of Lemma A.5. Recalling the definition of Pm (x), it is sufficient to show

that for any m ≥ 2,Pm+1 (x)

Pm (x)≤ Pm (x)

Pm−1 (x)(A.9)

Using the definition of Pm (x) in equation (A.7) to substitute out the lefthand side,

(A.9) is equivalent to∫10≤y≤b [Pm (x)Pm−1 (x− y)− Pm (x− y)Pm−1 (x)] f (y) dy ≥ 0

which is true, because from Lemma A.4 Pm (x) is TP2.

A.3 Proofs for Section 3

Proof of Lemma 2. Suppose that pan ∈ (a− s, b], and denote expected surpluses

by tj = max (vj − pe, 0) for j < n, and by tn = max (vn − pan, 0). Using equation (3)46

the equilibrium unadvertised price pe satisfies

Φ (pe; pan) = Pr

(n∑j=1

tj ≥ s

)− Pr

(n∑j=2

tj ≥ s

)× [F (pe) + (pe − c) f (pe)] = 0

(A.10)

Let k > 1 and rewrite the expression Pr(∑n

j=1 tj ≥ s)− k Pr

(∑nj=2 tj ≥ s

)as

∫ pan

a

f (vn)

[Pr

(n−1∑j=1

tj ≥ s

)− k Pr

(n−1∑j=2

tj ≥ s

)]dvn

+

∫ pan+s

pan

f (vn)

[Pr

(n−1∑j=1

tj ≥ s+ pan − vn

)− k Pr

(n−1∑j=2

tj ≥ s+ pan − vn

)]dvn

+

∫ b

pan+s

f (vn) [1− k] dvn (A.11)

Using a change of variables, (A.11) can in turn be written as∫ ∞−∞

1a≤pan−y≤bf (pan − y) γ (y) dy (A.12)

where γ (y) =

1− k if y ≤ −s

Pr(∑n−1

j=1 tj ≥ s+ y)− k Pr

(∑n−1j=2 tj ≥ s+ y

)if y ∈ (−s, 0)

Pr(∑n−1

j=1 tj ≥ s)− k Pr

(∑n−1j=2 tj ≥ s

)if y ≥ 0

and 1a≤pan−y≤b is an indicator function. Zero terms being discarded, γ (y) is piece-

wise continuous and changes sign once from negative to positive: this follows from

Lemma A.4 and the fact that k > 1. Since 1a≤pan−y≤bf (pan − y) is TP2, the Variation

Diminishing Property (Karlin 1968, p.21) says that (A.12) changes sign at most once

in pan, with the sign change from negative to positive.

Start with pan = pa0n . Let pe0 be the lowest solution to Φ (pe0; pan) = 0, and define

k0 = F(pe0)

+(pe0 − c

)f(pe0)

Using equation (A.10), we know that (A.12) is zero when evaluated at pan = pa0n , k =

k0 and pe = pe0. Therefore using the above, we also deduce that (A.12) is (weakly)

negative when evaluated at pa1n < pa0

n , k = k0 and pe = pe0 i.e. Φ (pe0; pa1n ) < 0. Since

Φ (pe; pan) is continuous in pe, we conclude that the lowest solution to Φ (pe; pa1n ) = 0

is below pe0. Thus the equilibrium unadvertised price increases in pan.

47

Proof of Proposition 3. Suppose pan ∈ [pm, b]. Firstly we can write profit from

the advertised product as

Πn (pe (pan) , pan) = (pan − c)(∫ s

0

[1− F (pan + s− z)] dµ (z) +

∫ ∞s

[1− F (pan)] dµ (y)

)(A.13)

where µ is a measure over T ≡∑n−1

j=1 max (vj − pe (pan) , 0). Note that Pr (µ < s) > 0.

Since f (p) is logconcave, (p− c) [1− F (p+ w)] is decreasing in p whenever w ≥ 0.

It follows from equation (A.13) that fixing pe, an increase in pan (directly) reduces

profit on good n. Moreover Πn is clearly decreasing in pe whilst pe (pan) is increasing

by Lemma 2. We can therefore conclude that Πn (pe (pan) , pan) is decreasing in pan for

pan ∈ [pm, b].

Secondly consider profits on the unadvertised goods, and assume for simplicity

that search occurs even when pan = b. When the advertised price is pa′n the equilibrium

unadvertised price is pe′, and the profit earned on each unadvertised good is

(pe′ − c)×

[∫ b

pe′f (v1) Pr

(n−1∑j=1

max (vj − pe′, 0) + max (vn − pa′n , 0) ≥ s

)dv1

](A.14)

When the advertised price is pa′′n < pa′n , let pe′′ ≤ pe′ denote the equilibrium unadver-

tised price. The firm could deviate by charging pe′ on each unadvertised product, in

which case its profits on each unadvertised product would be

(pe′ − c)×

[∫ b

pe′f (v1) Pr

(n−1∑j=1

max (vj − pe′′, 0) + max (vn − pa′′n , 0) ≥ s

)dv1

](A.15)

However when the advertised price is pa′′n , by revealed preference the firm prefers

to charge pe′′ on each unadvertised good, and hence its profit must weakly exceed

(A.15), which by inspection strictly exceeds (A.14). Hence profit on the advertised

products strictly decreases in pan when pan ∈ [pm, b]. Combining both parts of the

proof, the optimal advertised price must be strictly below pm.

Proof of Proposition 4. Without advertising the firm earns

πna = n× [pe (n)− c]×∫ b

pe(n)

f (v1) Pr

(n∑j=1

max (vj − pe (n) , 0) ≥ s

)dv1 (A.16)

48

where pe (n) > pm is the equilibrium price. By revealed preference (A.16) exceeds

the profit earned from deviating, and setting actual price equal to pm. Therefore

πna ≥ (pm − c) [1− F (pm)]× n× Pr

(n∑j=2

max (vj − pe (n) , 0) ≥ s

)(A.17)

With advertising the firm’s profit πa cannot exceed (pm − c) [1− F (pm)]×n. Hence

πa − πna ≤ (pm − c) [1− F (pm)]× n× Pr

(n∑j=2

max (vj − pe (n) , 0) < s

)(A.18)

Proposition 2 proves that pe (n) decreases in n, therefore ∀n ≥ n0:

πa − πna ≤ (pm − c) [1− F (pm)]× n× Pr

(n∑j=2

max (vj − pe (n0) , 0) < s

)(A.19)

Notice that the random variable max (vj − pe (n0) , 0) is distributed on [0, b− pe (n0)],

and let µ > 0 denote its mean. When n0 is sufficiently large that s/ (n0 − 1) < µ,

Theorem 2 of Hoeffding (1963) allows us to rewrite the righthand side of (A.19),

such that ∀n ≥ n0:

πa − πna ≤ (pm − c) [1− F (pm)]× n× exp

−2 (n− 1)

(µ− s

n0−1

)2

(b− pe (n0))2

(A.20)

The righthand side of (A.20) decreases monotonically in n when n is large, and also

tends to zero as n→∞. This establishes part (i) of the proposition. For part (ii),

note that we can also write profits with advertising as

πa ≤ Πan (pan) + (n− 1)× (pm − c) [1− F (pm)] (A.21)

where Πan (pan) denotes the profit earned on product n. The firm’s advertised price

pan must at least ensure that πa ≥ πna; using (A.17) and (A.21) this implies

Πan (pan) ≥ (pm − c) [1− F (pm)]×

[1− nPr

(n∑j=2

max (vj − pe (n) , 0) < s

)](A.22)

For n sufficiently large the righthand side is positive and therefore (A.22) implies

that pan > c. Since pan > c we can also state that Πan (pan) ≤ (pan − c) [1− F (pan)], in

which case (A.22) also implies that

(pan − c) [1− F (pan)] ≥ (pm − c) [1− F (pm)]×

[1− nPr

(n∑j=2

max (vj − pe (n) , 0) < s

)](A.23)

49

We know from above that we can place a lower bound on the righthand side of (A.23),

which for large n is monotonically increasing in n and tends to (pm − c) [1− F (pm)]

as n → ∞. Since the lefthand side of (A.23) is strictly increasing in pan for all

pan < pm, the first bit of part (ii) of the proposition follows. For the remainder of

part (ii), it is straightforward to adapt the proof of Proposition 1 and show this is

true for a non-advertising n-product firm. Then notice that an n-product firm with

an advertised price below pm, charges less than a non-advertising n-product firm.

A.4 Proofs for Section 4

Proof of Lemma 3. Suppose that at stage zero, firm B advertises with probability

1− α ∈ [0, 1], let Xi be the (conditional) probability with which it advertises good

i = 1, 2, and let µ (paBi) be a measure over B’s advertised price in case it decides

to advertise product i. Suppose that at stage one, firm B follows the strategy in

Lemma 3. We now prove that firm A best responds at stage one, by also playing

the strategy in Lemma 3. We use pij to denote the actual price charged by firm

i = A,B on product j = 1, 2, peij to denote consumers’ expectation of that price,

and paij to denote that the price is advertised.

Consider part (i). A is searched by loyals, and by half the non-loyals in case B

does not advertise. Firstly notice that if∑2

j=1 max (pAj − pu, 0) < s, non-loyals who

come to A prefer to stay rather than search the competitor, such that A’s profit is

γ + (1− γ)α

2∑i=1

(pAi − c)∫ b

pAi

f (vi) Pr

(2∑j=1

max (vj − pu, 0) ≥ s

)dvi (A.24)

Equation (A.24) is proportional to equation (1), and so is maximized by setting

pA1 = pA2 = pu. Secondly notice that if∑2

j=1 max (pAj − pu, 0) ≥ s, some non-

loyals who come to A might search again, and thus will not buy from A any product

which is (expected to be) cheaper at B. Therefore given any pA1, pA2 ≥ c which

satisfy∑2

j=1 max (pAj − pu, 0) ≥ s, A’s profit is less than equation (A.24), which

itself is globally maximized at pA1 = pA2 = pu. Therefore it cannot be profitable to

charge any {pA1, pA2} 6= {pu, pu}.

Consider part (ii). Suppose A advertises product i. A is searched by loyals.

In addition, (a) non-loyals search A if B has not advertised. They then buy A’s50

unadvertised product if and only if pA−i ≤ pu + s. (b) non-loyals search A if B also

advertises good i, but at a weakly higher price. They buy A’s unadvertised product

if and only if pA−i ≤ φ (paBi) + s. (c) suppose B advertises good −i. Non-loyals with

vi ≥ paAi + s and v−i ≥ paB−i + s intend to multi-stop shop; those who come to A

first, buy product −i if and only if pA−i ≤ paB−i + s. Non-loyals with vi ≥ paAi + s

and v−i < paB−i+s intend to one-stop shop at firm A; they will only consider buying

product −i from A, if pA−i < paB−i + s.

Firstly demand for A’s advertised product is independent of pA−i. Secondly when

pA−i ∈ [φ (paAi)− ε, φ (paAi) + s] firm A’s profit from its unadvertised product is

Y × (pA−i − c)b∫

pA−i

f (v−i) Pr

(2∑j=1

max(vj − peAj, 0

)≥ s

)dv−i (A.25)

where Y =γ

2+ (1− γ)α + (1− γ) (1− α)Xi

[Pr (paAi < paBi) +

Pr (paAi = paBi)

2

]where ε, s > 0 are sufficiently small that when B is advertising product −i, A doesn’t

sell product −i to any non-loyals. Equation (A.25) is proportional to equation (1),

and so is maximized by setting pA−i = φ (paAi). Thirdly when pA−i ≥ φ (paAi) + s,

A’s profit from its unadvertised product is weakly less than (A.25), which itself

is globally maximized at pA−i = φ (paAi). Hence it is not profitable to charge any

pA−i ≥ φ (paAi)+s. Fourthly when pA−i ≤ φ (paAi)−ε, A’s profit from its unadvertised

product is

Y × (pA−i − c)b∫

pA−i

f (v−i) Pr

(2∑j=1

max(vj − peAj, 0

)≥ s

)dv−i

+ Y (pA−i − c)∫µ(paB−i

) [1

2

∫ b

paB−i+s

f (v−i) dv−i +

∫ paB−i+s

pA−i

f (v−i) dv−i

]dpaB−i

(A.26)

where Y = (1− γ) (1− α) (1−Xi) Pr (vi ≥ paAi + s), and 1 is an indicator function

taking value 1 if pA−i ≤ paB−i + s and zero otherwise. Since paB−i + s < φ (paAi) for

s sufficiently small, (A.26) is maximized at pA−i = φ (paAi) provided γ is sufficiently

large.29

29Alternatively we can also rule out downward price deviations, by assuming that when a con-51

Proof of Proposition 5. We state (without proof) that for any p ≤ pu both φ (p)

and π (p) are continuous. It is convenient to define the following function

Z (p, α) = π (p)[γ

2+ (1− γ)α

]+

(1− γ) (1− α)

2(p− c) [1− F (p+ s)] (A.27)

It is also convenient to define a function ψ (p, α). (i) Let ψ (p, α) = w > 1 for all

prices p satisfying π (p) ≤ 0, and (ii) let ψ (p, α) be defined as follows, whenever

π (p) > 0:

ψ (p, α) =maxy Z (y, α)− (1−γ)(1−α)

2(p− c) [1− F (p+ s)]

π (p)(A.28)

Let p′ be the smallest p such that π (p) = 0, and define

Ψ (p, α) = minz∈[p′,p]

ψ (z, α) (A.29)

The proof of Proposition 5 is constructive. In particular we hypothesize that con-

ditional on α < 1, when a firm advertises, it draws its advertised price from a

cumulative distribution function G (p):

G (p) =

0 when p ≤ p

1− 2(1−γ)(1−α)

[Ψ (p, α)− γ

2− (1− γ)α

]when p ∈

[p, p]

1 when p ≥ p

(A.30)

where p is the lowest p which solves Ψ (p, α) = γ2

+ (1−γ)(1+α)2

, and p is the lowest p

which maximizes Z (p, α). We also assume (then verify later) that if a firm deviates

and advertises a price q > p it (is expected to) charges φ (q) on its other good.

Finally we also hypothesize that

κ = maxyZ (y, 0)− γ πu

2and κ = max

yZ (y, 1)− πu

2(A.31)

When κ ∈ [κ, κ], α is the unique solution to

maxyZ (y, α)− πuγ + (1− γ)α

2= κ (A.32)

We now show that the strategies in Proposition 5, along with equation (A.30) and

the rules (A.31) and (A.32) which govern α, constitute an equilibrium.

sumer enters a store, she only checks the prices of products that she either intends to buy, or would

buy if her belief about the price turned out to be slightly too high.52

Step 1. The unadvertised prices in parts (i) and (ii) follow from Lemma 3. It is also

straightforward to show that if a firm deviates and advertises a price q > p, there is

an equilibrium at stage one where it charges φ (q) for its other product.

Step 2. G (p) is a valid cumulative distribution function. First p ∈ (p′, p) exists

because Ψ (p, α) is a continuous decreasing function of p, satisfying Ψ (p′, α) > γ2

+

(1−γ)(1+α)2

> Ψ (p, α). Second G (p) is continuous and increasing, because Ψ (p, α)

is continuous and decreasing in p for all p ∈[p, p], whilst limp→p

+G (p) = 0 and

limp→p− G (p) = 1. Third G (p) < 1 for all p < p because p is the smallest p which

maximizes Z (p, α). Fourth p > 0 whenever γ is sufficiently close to 1.

Step 3. Note that p < pm because π (p) and (p− c) [1− F (p+ s)] are both maxi-

mized at p < pm.

Step 4. The expected profit of a non-advertising firm is

πuγ + (1− γ)α

2(A.33)

Step 5. The expected profit from advertising good 1 at a price, which is drawn with

strictly positive probability from G (p), is

Z (p, α) + π (p)(1− γ) (1− α)

2[1−G (p)]− κ (A.34)

Any price p played with strictly positive probability must satisfy ψ (p, α) = Ψ (p, α).30

Using equation (A.27) to substitute out for Z (p, α), and equation (A.30) to substi-

tute out for G (p), equation (A.34) equals maxy Z (y, α)−κ. The profit earned from

advertising good 2 at a price drawn from G (p) with strictly positive probability,

also equals maxy Z (y, α)−κ. Hence conditional on advertising and drawing a price

from G (p), a firm is indifferent about which product to advertise.

Step 6. There is no strict incentive to advertise any price which is not drawn with

strictly positive probability from G (p). First advertising any p < p generates a

profit

Z (p, α) + π (p)(1− γ) (1− α)

2− κ (A.35)

30Suppose to the contrary that ψ (p, α) > Ψ (p, α). Then there exists some p < p such that

ψ (p, α) = Ψ (p, α) = Ψ (p, α). This implies that G (p) = G (p) contradicting the assumption that

p is drawn from G (.) with strictly positive probability.

53

Since ψ (p) ≥ Ψ (p) > Ψ(p)

for all p < p, (A.35) is easily shown to be strictly less

than maxy Z (y, α) − κ. Second any p ∈(p, p)

which is not drawn from G (p) with

strictly positive probability, satisfies ψ (p) ≥ Ψ (p). It is easily shown that the profit

from advertising any such p is weakly less than maxy Z (y, α)− κ. Third according

to Step 1 for any p > p, there is an equilibrium where the firm charges φ (p) on

its other product. It is straightforward to show that its profit is (weakly) less than

Z (p, α)− κ which by definition is (weakly) less than Z (p, α)− κ.

Step 7. Step 4 shows that the expected profit from not advertising is (A.33), and

Step 5 shows that the expected profit from advertising is maxy Z (y, α) − κ (this

holds even when α = 1). It follows that when κ < κ there is an equilibrium where

each firm strictly prefers to advertise, when κ > κ there is an equilibrium where each

firm strictly prefers not to advertise, and when κ ∈ [κ, κ] and one firm is expected

to advertise with probability 1− α which solves (A.32), then the other firm is also

indifferent about advertising or not, and so happy to advertise with probability 1−α.

Step 8. Finally let p∗ (α) denote the set of maximizers of Z (y, α). Applying Milgrom

and Segal’s (2002) envelope theorem maxy Z (y, α) is continuous and differentiable

almost everywhere in α, with left- and right-derivatives which exceed

minp∈p∗(α)

(1− γ) π (p)− 1− γ2

(p− c) [1− F (p+ s)] (A.36)

Hence the left- and right-derivatives of the lefthand side of (A.32) exceed

minp∈p∗(α)

(1− γ) π (p)− 1− γ2

(p− c) [1− F (p+ s)]− πu1− γ2

(A.37)

When s is sufficiently small, any element of p∗ (α) strictly exceeds c, and therefore

π (p) > (p− c) [1− F (p+ s)] + πu/2. Hence (A.37) is strictly positive, such that

for κ ∈ [κ, κ] the α which solves (A.32) is continuous and strictly increasing in κ.

54

B Online Appendix

B.1 Extension with elastic demands

Here we provide various proofs associated with Section 5.1 on elastic demands.

DefineQ (z) ≡ P−1 (z) and note thatQ′, Q′′ < 0. Also define Π (p; θ) = (p− c)Q (p− θ),

p∗ (θ) = arg maxx (x− c)Q (x− θ), and let CS (pj; θ) be the consumer surplus of a

θ-type on one product when its price is pj.

Lemma B.1 CS (p∗ (θ) ; θ) is strictly increasing in θ, whilst p∗ (θ) < pm < p∗(θ).

Proof. is omitted for brevity.

Given their expectations about prices, consumers search if and only if θ ≥ θ.

Hence the firm’s profit on good j is∫ θθg (θ) [(p− c)Q (p− θ)] dθ. Differentiating

with respect to p and imposing p = pe, we find that pe satisfies31∫ θ

θ

g (θ) Π′ (pe; θ) dθ =

∫ θ

θ

g (θ) [Q (pe − θ) + (pe − c)Q′ (pe − θ)] dθ = 0 (B.1)

It follows immediately that each unadvertised product has the same equilibrium

price. Moreover since ∂Π′ (pe; θ) /∂θ > 0 equation (B.1) implies Π′(pe; θ

)< 0,

which further implies that Π′ (pe; θ) < 0∀θ ≤ θ. Consequently∫ θθg (θ) Π′ (pe; θ) dθ ≥∫ θ

θg (θ) Π′ (pe; θ) dθ.

Lemma B.2 Suppose the firm does not advertise. (i) When s ≤ sn there is an

equilibrium in which each product costs pm. (ii) When s ∈ (sn, sn) there is an

equilibrium in which consumers search, price strictly exceeds pm and is increasing in

s but decreasing in n. (iii) The thresholds satisfy

sn = n× CS (pm; θ) and sn = n× CS(p∗(θ)

; θ)

Proof. (i) is easily verified. (ii) Firstly note that the equilibrium price must exceed

pm.32 Secondly, then, θ > θ because pe ≥ pm and s > sn. This implies that the left-

hand side of equation (B.1) is strictly positive when evaluated at pe = pm. Thirdly,

31Since Q′, Q′′ < 0 profit is quasiconcave in p provided that types are not too different.32This is proved by contradiction. Suppose that the (common) equilibrium price satisfies pe <

pm. Then∫ θθg (θ) Π′ (pe; θ) dθ > 0 because

∫ θθg (θ) Π (p; θ) dθ is concave and maximized at p = pm.

Using some earlier results, this implies that∫ θθg (θ) Π′ (pe; θ) dθ > 0, which is a contradiction.55

suppose pe = p∗(θ). Since s < s′ consumers with θ = θ search, and by continuity

so do nearby types. Thus θ < θ. We can also conclude that Π′(p∗(θ)

; θ)< 0 for

all θ < θ. Therefore the lefthand side of equation (B.1) is strictly negative when

evaluated at pe = p∗(θ). Fourthly putting these last two points together, and using

the fact that the lefthand side of (B.1) is continuous in pe, there exists at least one

pe ∈(pm, p∗

(θ))

such that (B.1) holds. A proof that pe increases in s and decreases

in n is straightforward and hence omitted.

Proof of Proposition 6. Given an advertised price pan and an expected unadver-

tised price pe, consumer surplus is (n− 1)×CS (pe, θ)+CS (pan, θ). Let p denote the

unadvertised price which makes a θ-type indifferent about searching. (i) If p < pm

we know that no equilibrium with search exists, so without loss set pe = p∗(θ). (ii)

If p ≥ pm then define the interval I =[pm,min

{p, p∗

(θ)}]

. For each p ∈ I let θp

denote the lowest type in[θ, θ]

who finds it worthwhile to search. Then for any

p ∈ I the lefthand side of (B.1) is∫ θ

θp

g (θ) Π′ (p; θ) dθ (B.2)

(a) If (B.2) is strictly positive for all p ∈ I then no equilibrium with search exists,

so without loss set pe = p∗(θ). (b) If (B.2) equals zero for some p ∈ I, pick the

lowest such price and select it as the equilibrium price. Now consider what happens

as pan increases. Firstly for each p ∈ I, observe that θp increases and thus, since

∂Π′ (pe; θ) /∂θ > 0, (B.2) increases. Hence the (lowest) equilibrium unadvertised

price increases in pan. Secondly notice that p decreases and thus I weakly shrinks.

Consequently as pan continues to increase, eventually there is no equilibrium with

search, and hence we can let the equilibrium unadvertised price jump to p∗(θ).

B.2 Extension with heterogeneous products

Proof of Proposition 7. We only provide a proof for n = 2. The proof for n > 2

is similar. Firstly suppose the firm advertises product 1 at price q. Using equations

(1) and (4) from the paper, as s→ 0 total profit tends to

λ1 (q − c) [1− F (q)] + λ2 (pe2 − c) [1− F (pe2)] (B.3)56

where pe2 solves equation (11) from the paper, for the case i = 1. Secondly suppose

the firm advertises product 2 at price q. As s→ 0 total profit tends to

λ1 (pe1 − c) [1− F (pe1)] + λ2 (q − c) [1− F (q)] (B.4)

where pe1 ∈ (pm, pe2) solves equation (11) from the paper, for the case i = 2. Thirdly

note that (pe1 − c) [1− F (pe1)] > (pe2 − c) [1− F (pe2)] and hence (B.4) certainly ex-

ceeds (B.3) if (q − c) [1− F (q)] ≥ (pe2 − c) [1− F (pe2)]. Moreover it is clear that in

order to maximize (B.3) the firm would choose q ∈ (a, pm). Thus a lower bound

on (q − c) [1− F (q)] is given by a − c. Moreover pe2 always exceeds the p which

uniquely solves(p− c) f (p)

1− F (p)=

1

λ

and hence an upper bound on (pe2 − c) [1− F (pe2)] is given by (p− c) [1− F (p)].

Hence by choosing λ in such a way that a − c ≥ (p− c) [1− F (p)], (B.4) exceeds

(B.3) and the firm’s profit is highest when it advertises the most popular product.

B.3 Extension with substitutes

In the paper we assume that a firm sells n independent products. However in

practice, some of a firm’s products might be substitutes for one another. For example

a grocery store might stock several different types of milk and detergent. Although a

consumer’s demand for milk and detergent may be roughly independent, her demand

for two different detergents would not be. To capture this, in this extension we

consider a monopolist that stocks n independent product categories, each of which

contains z substitute products. Consumers wish to buy at most one product from

each category. We let vi,j denote a typical consumer’s valuation for product i in

category j; each vi,j is drawn independently, using the same distribution function

F (vi,j). We also assume that all products within a given category have the same

price.33

33Shankar and Bolton (2004) and Draganska and Jain (2006) note that in practice, differentiated

versions of a product or brand often have the same price.

57

Given our assumptions, a consumer who buys from category j gets a gross surplus

of max {v1,j, ..., vz,j}. This surplus has distribution function G (v; z) ≡ F z (v) and

corresponding logconcave density function g (v; z) ≡ zF z−1 (v) f (v). Without search

costs, the firm would charge a monopoly price pm (z) = arg max (p− c) [1−G (p; z)],

and this can be shown to monotonically increase in z. Notice that in general, the

firm will behave as if it sells n independent products, whose valuations are drawn

from G (v; z). Therefore all of our analysis from the paper can be straightforwardly

applied. In particular firstly, a search-equilibrium fails to exist if the firm stocks

only one category of products.34 Secondly however, such an equilibrium does exist,

if the firm offers products in a sufficiently large number of categories. Thirdly,

unadvertised prices exceed pm (z), increase in the search cost, but decrease in n i.e.

in the number of categories. Moreover fourthly, when the firm offers discounts on one

product category, it optimally reacts by charging less on other, unrelated, product

categories.35 Thus our main insights are robust to having substitute products.

B.4 Extension with alternative retailing strategies

In the paper we assume that firms use simple pricing and advertising strategies. We

now discuss what happens when a monopolist employs some alternative strategies.

A firm could choose to bundle its products. To maintain tractability, we ask

whether or not it should use pure bundling.36 Note that when a firm bundles its

products in this way, it effectively sells one good whose valuation is∑n

j=1 vj; this

means that if consumers anticipate bundling, they will not search. Now suppose

34More generally, when the firm stocks only one category, a search-equilibrium fails to exist

regardless of how the valuations (v1,1, ..., vz,1) are distributed. This includes therefore the case of

perfect substitutes i.e. v1,1 =, ...,= vz,1.35Unfortunately it is unclear what would happen to prices, if the firm advertised only one product

in one category. This stems from the fact that even if n = 1 and there is no search cost, the prices

of individual products are not always positively related.36The market breaks down without advertising, if the firm can use mixed bundling. Let N be

the set of all possible combinations of the n products, and let pei be what consumers expect to

pay for combination i ∈ N. For ε sufficiently small, the firm can profitably hold-up consumers by

charging pi = pe1 + ε for each i ∈ N. Nevertheless in practice, ‘full’ mixed bundling is infeasible in

the markets we have in mind in the paper, because the cardinality of N grows very quickly in n.58

that the firm is expected not to bundle. Consumers believe that prices will be the

same as they were in Section 2 of the paper, and therefore search provided that∑nj=1 max

(vj − pej , 0

)≥ s. After consumers have sunk the search cost, the firm has

two options. Firstly it could set pi = pei on each product as expected. But secondly

it could deviate and use bundling, in which case it would choose the bundle price pb

to

maxpb

(pb − nc) Pr

(n∑j=1

vj ≥ pb

∣∣∣∣∣n∑j=1

max(vj − pej , 0

)≥ s

)(B.5)

We can show that this deviation is not profitable whenever average valuations are

relatively low, and the number of products stocked is relatively high. Intuitively

under these conditions the firm can not sell many units of the bundle, unless it sets

pb/n very low. However in that case it would do better to just sell the products

separately, as consumers were expecting.

A firm might also charge consumers a fee τ ≥ 0 to enter its store and browse its

products. In this case it is convenient to split the search cost into two parts: a cost

s ∈ (0, s) of physically travelling to the store, and a further cost s− s of going inside

and learning prices. Without advertising, the timing of the game is then as follows.

Firstly consumers form an expectation of the entry fee τ e and of the price pe of

each product. They then decide whether or not to travel to the store. Secondly if a

consumer goes to the store, she learns the actual entry fee τ . At this point she can

update her belief about the price, which we denote by pe. She then decides whether

or not to pay τ and go inside the store. Thirdly after entering the store, a consumer

observes actual prices and makes her purchases.

Remark B.1 There exists an equilibrium in which the firm sets an entry fee τ (= τ e) ≥

0, and charges pe on each product where pe solves

b∫pe

f (vi) Pr

(n∑j=1

max (vj − pe, 0) ≥ s+ τ e

∣∣∣∣∣ vi)dvi

− (pe − c) f (pe) Pr

(∑j 6=i

max (vj − pe, 0) ≥ s+ τ e

)= 0 (B.6)

Proof. Consider the following strategy on the part of consumers. (a) Travel to

the firm if and only if∑n

j=1 max (vj − pe, 0) ≥ s + τ e. (b) After travelling to the59

firm, if τ ≤ τ e then set pe = pe, otherwise expect price to be some pe which satisfies∑nj=1 max (b− pe, 0) ≤ s− s+τ . (c) Then enter the store if expected surplus exceeds

s− s+ τ . Given this strategy, the firm (weakly) prefers to set τ = τ e. Demand for

any product is exactly the same as in equation (1) from the paper, except that s is

replaced by s+ τ e. Equation (B.6) then follows.

The pricing condition (B.6) is the same as equation (2) from the paper, except that

s is replaced by s + τ e. Intuitively this is because the entry fee acts like an extra

search cost. There is always an equilibrium where τ e = 0; however typically there

exist other equilibria with τ e > 0 as well. Nevertheless in practice, a firm who

imposes a positive entry fee would find it hard to prevent consumers from reselling

its products. This arbitrage problem suggests that in practice, τ e = 0 is the natural

outcome.

Finally, in the paper we focused on price advertising. An alternative (but less

common) promotional strategy is to send consumers a coupon, which gives them

for example a small discount τ ′ < s off their shopping bill. This coupon acts like a

negative entry fee, so the equilibrium price would solve equation (B.6) but with τ e

replaced by −τ ′. Applying Proposition 2 from the paper, prices are decreasing in

the size of the coupon. Therefore in theory, coupons provide an alternative way to

commit to lower prices. However again in practice, coupons might be subject to an

arbitrage problem. Our intuition is that if the arbitrage problem could be solved,

smaller firms would be more likely to use coupons compared to large firms. However

it is unclear (and not possible to show analytically) whether or not coupons would

dominate price advertising.

References

[1] Draganska, M. and Jain, D. (2006): ‘Consumer Preferences and Product-Line

Pricing Strategies: An Empirical Analysis’, Marketing Science 25(2), 164-174.

[2] Shankar, V. and Bolton, R. (2004): ‘An Empirical Analysis of Determinants of

Retailer Pricing Strategy’, Marketing Science 23(1), 2004.

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