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
1
Zhou (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 products.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. Typically adverts can only contain information on a very small number of prices. Although 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
2
Many 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.
3
Changes in the composition of demand, in particular the mix of low- and high-valuation consumers, also plays an important role in understanding the effects of search diversion by an intermediary (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 advertised 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 charge H 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, costrelated 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 consumers 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 advertising, 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 (unadvertised) 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 ‘cherrypick’ firms’ advertised specials.
The rest of the paper proceeds as follows. Section 2 begins by solving a benchmark model without advertising, whilst Section 3 characterizes a monopolist’s optimal 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.
4
In 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.
5
Unlike 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).
6
Most papers which distinguish between advertised and unadvertised prices, predict that unadvertised 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 let pm =
arg max (p − c) [1 − F (p)] denote the classic monopoly price, 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 , . . . , pen )
about the price of each product. A consumer then visits the firm if and only if
P
her expected surplus nj=1 max vj − pej , 0 exceeds the search cost s. We look for
7
Bagnoli 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.
P
We begin by noting that any price vector satisfying nj=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
P
manner. To this end let us now suppose that nj=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 ) =
Zb
pi
f (vi ) Pr
n
X
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 = argmax (pi − c) Di (pi ; pe ) we obtain Lemma 1.
pi
Lemma 1 The equilibrium price of unadvertised good i satisfies
Di (pi = pei ; pe ) − (pei − c) f (pei ) Pr
X
j6=i
8
max vj − pej , 0 ≥ s
!
=0.
(2)
v2
b
Search and buy
product 2 only
Search and buy
both products
pe2 +s
pe2
Marginal
consumers for
product 1 who
search
a
a
Search and
buy product
1 only
v1
pe1
pe1 +s
b
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. Secondly1however, the firm loses
margin pei − c on each consumer who stops buying the good, and they have mass
P
e
f (pei ) Pr
max
v
−
p
,
0
≥
s
. Intuitively, consumers who stop buying good
j
j
j6=i
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
P
the remaining n − 1 goods, j6=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 because 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. The
9
thick line in Figure 1 represents these consumers. We can now state that
Proposition 1 If the number of products stocked 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 the classic monopoly price pm .
Once we take seriously the multiproduct nature of retailing, the ‘no search problem’ 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, 2−4 2 . 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 equilibria, 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
8
We 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, consumers 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. Fortunately 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 . Secondly 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
X
tj ≥ s
j6=i
|
{z
!
e
e
e
[1 − F (p ) − (p − c) f (p )] +Pr
j=1
}
|
Price-sensitive consumers
9
n
X
tj ≥ s
!
− Pr
X
tj ≥ s
j6=i
{z
Price-insensitive consumers
!
}
(3)
This 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 ). Proposition 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.
10
Forward 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
=0,
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
P
caused by a small change in pi around pe . Note that only consumers with nj=1 tj ≥ s
search, and they naturally subdivide into two groups. Firstly some consumers have
P
j6=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 pricesensitive and give rise to the first term in equation (3). Secondly other consumers
P
P
have nj=1 tj ≥ s > j6=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).
P
n
t
≥
s
is log-supermodular in s and n,
We prove in the appendix that Pr
j=1 j
. P
P
n
n
Pr
or equivalently, Pr
j=2 tj ≥ s is increasing in s. This means
j=1 tj ≥ s
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 valuations 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 consumers
12
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 valuations 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 generally 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, because 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
11
As 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 , . . . , pen )
about the price of each product. Note that pen = pan because advertising is truthful
P
by assumption. Consumers search if and only if nj=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
12
In 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 distinguishes 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 unadvertised 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 unadvertised prices. Consumers therefore rationally anticipate that a higher advertised
price on one product, leads to higher (often strictly higher) prices on other products as well. Consumer heterogeneity is crucial for obtaining this result. If instead
consumers all had the same valuation v̄ for an unadvertised good (as is effectively
the case in Lal and Matutes 1994 and Ellison 2005), the firm would charge v̄ 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, Brown
15
(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 ) =
Zb
pa
n
f (vn ) Pr
n−1
X
!
max vj − pej , 0 + vn − pan ≥ s dvn .
j=1
(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. Secondly 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 therefore 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 the classic monopoly
price 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 .
16
Secondly, a reduction in the advertised price encourages more consumers who like
good n to search. Once inside the store, these new visitors also buy some (highermargin) 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
17
lower prices. For example Devine and Marion (1979) publicized the prices of some
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
13
14
Bester (1994) shows that a single-product monopolist charges less when it advertises.
The 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
Advertising Gains
0.06
0.05
0.04
0.03
0.02
0.01
0.00
5
10
15
20 n
Figure 2: Non-monotonicity of the gains from advertising
behavior of large firms:
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
19
enough that the small firm decides to advertise. We can prove that when nb is
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 discuss what additional insights emerge when firms sell more than two products.) As
before valuations for these products are known to consumers, and are drawn independently 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, consumers 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
15
The relationship between product range and unadvertised prices is therefore qualitatively the
same as in Proposition 2, although it is no longer monotonic.
16
Suppose 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
no preference for where they purchase their goods.
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 knowing 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 information, 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 product(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 searchequilibrium 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 π u
denote the corresponding profit. In addition when a monopolist advertises price q
17
For 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.
18
A numerical example was provided on page 10, in which a search-equilibrium exists for n = 2.
21
on one product, let φ (q) be the (unique equilibrium) price that it charges for its
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 the search cost is sufficiently small, and the fraction of loyal consumers
is 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 implications 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 product. Non-loyals only consider shopping at the firm with the lowest advertised price
19
Lemma 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
because, recalling the properties of φ (q), that firm is expected to be cheaper on the
unadvertised product as well. Thirdly suppose only one firm has advertised. Nonloyals 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 onestop 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 the search cost is sufficiently small, and the fraction of
loyal consumers is 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 the classic monopoly price pm , and is 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 randomize 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 randomly. 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
24
neither firm is advertising. As discussed in the introduction, our model therefore
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.
20
21
Price dispersion is extremely widespread in practice. See Baye et al (2006) for a survey.
Related 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
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
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 randomization. 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
22
If, 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).
23
Our 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
denote the firms’ advertised prices, firm A’s profit is
h
γ/2 + (1 − γ) 1paA1 Rpa
B1
i
π (paA1 ) − κ ,
(6)
where 1paA1 Rpa is an indicator function, taking the value 0 if paA1 > paB1 , the value
B1
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 p̃ ∈ (c, pm ). Therefore using (6), a firm’s expected profit in
this putative equilibrium is γπ (p̃) /2 − κ. Now suppose firm A deviates from the
putative equilibrium, and advertises product 2 at a price p̃. There exists a unique
stage-one equilibrium, in which firm A charges φ (p̃) 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 ≥ p̃ + s. Consequently A’s expected profit
following its deviation is
γπ (p̃) /2 + (1 − γ) (p̃ − c) [1 − F (p̃ + s)] − κ .
(7)
The deviation is profitable because (7) exceeds γπ (p̃) /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 different 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 P̃ denote the set of prices which maximize (8). When s is small each member
of P̃ lies in (c, pm ). Suppose firm B plays along with the putative equilibrium, and
advertises a price paB2 ∈ P̃ on product 2. However suppose firm A deviates from the
27
putative equilibrium, and also advertises product 2, but at the lower price of paB2 −ǫ.
There exists a unique stage-one equilibrium, in which firm A charges φ (paB2 − ǫ) for
product 1. Recalling our earlier discussion, non-loyals now either stay at home or
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 − ǫ) − κ .
(9)
It is straightforward to show that the deviation is profitable, because there exists
an ǫ > 0 such that (9) exceeds (8) when evaluated at any paA1 ∈ P̃ . 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 advertising 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
5.1
Extensions and Discussion
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
28
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 faced
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 allow 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
m
p = argmax
x
Z
θ
θ
g (θ) (x − c) P −1 (x − θ) dθ .
29
(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 is
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 relationship 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 advertised 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 equilibrium 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 interesting question is the following: if products are not symmetric, which one of them
should a monopolist advertise? Empirical evidence shows that firms may advertise discounts on products with higher demand or popularity, such as tuna at Lent,
30
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 advertising costs, or because certain products are popular with high-spending consumers
(DeGraba 2006).
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 < λ̄, where λ̄ < 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 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
.
(11)
=
λi [1 − F (qi )]
1 − F pe3−i
The lefthand side of (11) increases in pe3−i , such that a low λ̄ translates into a high
unadvertised price. Intuitively when λ̄ is small, only a small number of consumers
who are marginal for the unadvertised product find it worthwhile to search. Consequently 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
24
This places an upper bound on the profits earned from an any unadvertised product.
31
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
(and thus be more profitable, per interested consumer), when it is the less popular 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 π u (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 π u (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
25
If firm A advertises a price q < pu (n) which satisfies φ (q; n) = pu (n), we must drop the
32
insight is that because φ (y; n) increases in y, firm A is expected to charge a lower
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, provided that
κ≥
maxq π (q; n) − π u (n) (1 − γ) maxq π (q; n)
+
.
2
2
(12)
To interpret condition (12) note that when a monopolist chooses to advertise, its
profits increase in proportion to maxq π (q; n) − π u (n). Since maxq π (q; n) − π u (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 product 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 consumers 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
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.
26
Some 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
33
independent product categories, each category contains z substitute products, and
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
expensive substitute instead. Rosato (2013) explores the incentive of a firm to have only limited
availability of advertised products, whilst Petrikaitė (2013) shows that a firm may wish to obfuscate
some of its product range.
34
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
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 particular 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 theoretical. 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 empirical research on this topic. We hope that our paper will generate renewed empirical
interest, in understanding how pricing and advertising on one product category, affects 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
A.1
Main Proofs
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) ,
where T =
P
j6=i
(A.1)
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 p̃ be the unique solution to 1−F (p)−(p − c) f (p) /2 =
P
n
0, and ñ be the smallest integer n such that Pr
max
(v
−
p̃,
0)
≥
s
> 1/2.
j
j=2
Assume that n ≥ ñ. 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
X
max (vj − pe , 0) ≥ s
j6=i
!
=0.
(A.2)
The lefthand side of (A.2) is strictly positive when pe ≤ pm , because 1 − F (pe ) −
P
e
(pe − c) f (pe ) ≥ 0 and Di (pi = pe ; pe ) ≥ [1 − F (pe )] Pr
max
(v
−
p
,
0)
≥
s
j
j6=i
with at least one inequality strict. Given that n ≥ ñ, the lefthand side of (A.2) is
strictly negative when pe = p̃, because Di (pi = pe ; pe ) ≤ 1 − F (pe ) and 1 − F (p̃) −
(p̃ − c) f (p̃) /2 = 0. Therefore since the lefthand side of (A.2) is continuous in pe ,
there exists a pe ∈ (pm , p̃) which solves (A.2). According to Lemma A.2 (below) the
P
n
e
profit on each good is quasiconcave because Pr
max
(v
−
p
,
0)
≥
s
> 1/2.
j
j=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 proportional 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 p̄ ∈ [pm , pei ) such that −2/ (p̄ − c)−f ′ (p̄) /f (p̄) >
0. Step 1 then implies that −2/ (p − c) − f ′ (p) /f (p) > 0 for all p ∈ [p̄, 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 ∈ [p̄, pei ). However by
definition (A.1) is zero when pi = pei , so (A.1) is strictly negative for every pi ∈ [p̄, pei ).
Given the definition of (A.1), this means that (pi − c) Di (pi ; pe ) is strictly decreasing
in pi for every pi ∈ [p̄, pei ) - so the firm would do better to charge some pi ∈ [p̄, 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 =
e
max
v
−
p
,
0
as defined earlier.
j
j
j6=i
P
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 ,
27
Note 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 ; pe )
− (pi − c) Pr (T ≥ s) +(pi − c) [Pr (T ≥ s) − Pr (T + max (pi − pei , 0) ≥ s)] .
f (pi )
(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)
Di (pi ; pe )
Di (pi ; pe ) f ′ (pi )
< −1+
≥ s)−Pr (T ≥ s)−
≤0,
1 − F (pi )
f (pi )2
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:
!
!
!
n
X
X
X
Pr
tj ≥ s [1 − F (pei ) − (pei − c) f (pei )] + Pr
tj ≥ s −Pr
tj ≥ s = 0 .
j6=i
j=1
j6=i
(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 negative by equation
P
P
(A.6). Clearly also Pr
t
≥
s
>
Pr
t
≥
s
. Therefore equation (A.6)
j
j
j6=h
j6=g
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
search we can rewrite equation (3) as
P
n
j=2 tj ≥ s
φ (pe ; s, n) = 1 − F (pe ) − (pe − c) f (pe ) +
Pr
Pr
> 0, in an equilibrium with
P
n
j=1 tj ≥ s
P
n
j=2 tj ≥ s
−1=0.
Firstly fix n. Start with a search cost s0 , and let pes0 denote the lowest solution
P
m
to φ (pe ; s0 , n) = 0. Lemma A.4 (below) shows that Pr
t
≥
x
is TP2 on
j
j=1
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 pen0 denote
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
P
m
j=1 tj
≥ x is TP2 on R × N.
Proof of Lemma A.4. Define ṽj = vj − pe , let F̃ (ṽj ) be its distribution function,
h i
f˜ (ṽj ) be its (logconcave) density function, ã, b̃ be the interval on which it is
defined, with ã = a − pe and b̃ = b − pe . Note that the ṽj are independent, and
P
m
recall that tj ≡ max (ṽj , 0). Also define Pm (x) = Pr
j=1 max (ṽj , 0) ≥ x , and
note that
Pm (x) =
Z
10≤y≤b̃ Pm−1 (x − y) f˜ (y) dy + Pm−1 (x) F̃ (0) ,
(A.7)
where 10≤y≤b̃ 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+1 (x)
F̃ (0) Ln (x) + F̃ (0) ,
(A.8)
= 1−
Pn (x)
Pn (x)
28
See Karlin (1968) for a definition.
45
where
R
Ln (x) = R
10≤y≤b̃ Pn (x − y) f˜ (y) dy
.
1
Pn−1 (x − y) f˜ (y) dy
0≤y≤b̃
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
Z
Z
˜
10≤y≤b̃ Pm (x − y) f (y) dy = Pm (z) f˜ (x − z) 10≤x−z≤b̃ dz ,
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)
=1+
P1 (x)
Z
x
0
1 − F̃ (x − z)
dz + F̃ (0) ,
f˜ (z)
1 − F̃ (x)
which is increasing in x because f˜ (y) is logconcave and hence has an increasing
hazard rate.
Lemma A.5 Pr
P
m
j=1 tj
P
m
≥ x / Pr
t
≥
x
decreases in m.
j
j=2
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
Z
10≤y≤b̃ [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
!
!
n
n
X
X
Φ (pe ; pan ) = Pr
tj ≥ s − Pr
tj ≥ s × [F (pe ) + (pe − c) f (pe )] = 0 .
j=1
j=2
Let k > 1 and rewrite the expression Pr
Z
"
pa
n
f (vn ) Pr
a
+
Z
pa
n +s
pa
n
n−1
X
tj ≥ s
j=1
"
f (vn ) Pr
n−1
X
!
− k Pr
P
n
j=1 tj
n−1
X
tj ≥ s
j=2
tj ≥ s + pan − vn
j=1
!
≥ s − k Pr
!#
(A.10)
n
j=2 tj ≥ s as
P
dvn
− k Pr
n−1
X
tj ≥ s + pan − vn
j=2
+
Z
b
!#
dvn
f (vn ) [1 − k] dvn . (A.11)
pa
n +s
Using a change of variables, (A.11) can in turn be written as
Z ∞
1a≤pan −y≤b f (pan − y) γ (y) dy ,
(A.12)
−∞


1−k
if y ≤ −s



P
P
n−1
n−1
if y ∈ (−s, 0)
where γ (y) =
Pr
j=2 tj ≥ s + y
j=1 tj ≥ s + y − k Pr


P
P

n−1
n−1
 Pr
if y ≥ 0
j=1 tj ≥ s − k Pr
j=2 tj ≥ s
and 1a≤pan −y≤b is an indicator function. Zero terms being discarded, γ (y) is piecewise
continuous and changes sign at most once, from negative to positive: this follows
from Lemma A.4 and the fact that k > 1. Since 1a≤pan −y≤b f (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.
e0
Start with pan = pa0
be the lowest solution to Φ (pe ; pan ) = 0, and define
n . Let p
k 0 = F pe0 + pe0 − c f pe0 .
Using equation (A.10), we know that (A.12) is zero when evaluated at pan = pa0
n , k =
k 0 and pe = pe0 . Therefore using the above, we also deduce that (A.12) is (weakly)
a0
0
e
e0
e0 a1
negative when evaluated at pa1
n < pn , k = k and p = p i.e. Φ (p ; pn ) ≤ 0. Since
Φ (pe ; pan ) is continuous in pe , we conclude that the lowest solution to Φ (pe ; pa1
n ) = 0
is below pe0 . Thus the equilibrium unadvertised price increases in pan .
47
Suppose pan ∈ [pm , b]. Firstly we can write profit from
Proof of Proposition 3.
the advertised product as
Πn (p
e
(pan ) , pan )
=
(pan
− c)
Z
s
[1 − F
(pan
+ s − z)] dµ (z) +
0
where µ is a measure over T ≡
Pn−1
j=1
Z
∞
[1 − F
(pan )] dµ (y)
s
(A.13)
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
"Z
!
#
n−1
b
X
(pe′ − c) ×
f (v1 ) Pr
max (vj − pe′ , 0) + max (vn − pa′
.
n , 0) ≥ s dv1
pe′
j=1
(A.14)
a′
e′′
When the advertised price is pa′′
≤ pe′ denote the equilibrium unadvern < pn , let p
tised price. The firm could deviate by charging pe′ on each unadvertised product, in
which case its profits on each unadvertised product would be
"Z
!
#
n−1
b
X
(pe′ − c) ×
f (v1 ) Pr
max (vj − pe′′ , 0) + max (vn − pa′′
.
n , 0) ≥ s dv1
pe′
j=1
(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
π̃ na = n × [pe (n) − c] ×
Z
b
pe (n)
f (v1 ) Pr
n
X
j=1
48
!
max (vj − pe (n) , 0) ≥ s dv1 , (A.16)
,
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
!
n
X
π̃ na ≥ (pm − c) [1 − F (pm )] × n × Pr
max (vj − pe (n) , 0) ≥ s . (A.17)
j=2
With advertising the firm’s profit π̃ a cannot exceed (pm − c) [1 − F (pm )] × n. Hence
!
n
X
π̃ a − π̃ na ≤ (pm − c) [1 − F (pm )] × n × Pr
max (vj − pe (n) , 0) < s . (A.18)
j=2
e
Proposition 2 proves that p (n) decreases in n, therefore ∀n ≥ n0 :
π̃ a − π̃ na ≤ (pm − c) [1 − F (pm )] × n × Pr
n
X
max (vj − pe (n0 ) , 0) < s
j=2
!
. (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 :
π̃ a − π̃ na ≤ (pm − c) [1 − F (pm )] × n × exp


 −2 (n − 1) µ −
s
n0 −1
(b − pe (n0 ))2


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
π̃ a ≤ 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 π̃ a ≥ π̃ na ; using (A.17) and (A.21) this implies
"
!#
n
X
Πan (pan ) ≥ (pm − c) [1 − F (pm )] × 1 − n Pr
max (vj − pe (n) , 0) < s
.
j=2
(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 − n Pr
49
n
X
max (vj − pe (n) , 0) < s
j=2
(A.23)
!#
.
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
P
does not advertise. Firstly notice that if 2j=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
!
Z b
2
2
X
γ + (1 − γ) α X
×
(pAi − c)
f (vi ) Pr
max (vj − pu , 0) ≥ s dvi . (A.24)
2
p
Ai
i=1
j=1
Equation (A.24) is proportional to equation (1), and so is maximized by setting
P
pA1 = pA2 = pu . Secondly notice that if 2j=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
P
satisfy 2j=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’s
50
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 ) − ǫ, φ (paAi ) + s] firm A’s profit from its unadvertised product is
Y × (pA−i − c)
Zb
f (v−i ) Pr
pA−i
2
X
j=1
!
max vj − peAj , 0 ≥ s dv−i ,
(A.25)
γ
Pr (paAi = paBi )
a
a
where Y = + (1 − γ) α + (1 − γ) (1 − α) Xi Pr (pAi < pBi ) +
,
2
2
where ǫ, 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 )−ǫ, A’s profit from its unadvertised
product is
Y × (pA−i − c)
Zb
pA−i
f (v−i ) Pr
2
X
j=1
!
max vj − peAj , 0 ≥ s dv−i
#
" Z
Z paB−i +s
Z
b
1
f (v−i ) dv−i +
f (v−i ) dv−i dpaB−i ,
+ Ỹ (pA−i − c) µ paB−i
2 paB−i +s
pA−i
(A.26)
where Ỹ = (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
29
Alternatively 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)
hγ
i (1 − γ) (1 − α)
+ (1 − γ) α +
(p − c) [1 − F (p + s)] . (A.27)
2
2
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) = min
ψ (z, α) .
′
(A.29)
z∈[p ,p]
The proof of Proposition 5 is constructive. In particular we hypothesize that conditional on α < 1, when a firm advertises, it draws its advertised price from a
cumulative distribution function G (p):



0


2
G (p) =
1 − (1−γ)(1−α)
Ψ (p, α) − γ2 − (1 − γ) α



 1
where p is the lowest p which solves Ψ (p, α) =
γ
2
+
when p ≤ p
when p ∈ p, p̄
(A.30)
when p ≥ p̄
(1−γ)(1+α)
,
2
and p̄ 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 > p̄ it (is expected to) charges φ (q) on its other good.
Finally we also hypothesize that
κ = max Z (y, 0) −
y
γ πu
πu
and κ = max Z (y, 1) −
.
y
2
2
(A.31)
When κ ∈ [κ, κ], α is the unique solution to
max Z (y, α) − π u
y
γ + (1 − γ) α
=κ.
2
(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 > p̄, 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′ , p̄) exists
because Ψ (p, α) is a continuous decreasing function of p, satisfying Ψ (p′ , α) >
γ
2
+
(1−γ)(1+α)
2
> Ψ (p̄, α). Second G (p) is continuous and increasing, because Ψ (p, α)
is continuous and decreasing in p for all p ∈ p, p̄ , whilst limp→p+ G (p) = 0 and
limp→p̄− G (p) = 1. Third G (p) < 1 for all p < p̄ because p̄ is the smallest p which
maximizes Z (p, α). Fourth p > 0 whenever γ is sufficiently close to 1.
Step 3. Note that p̄ < pm because π (p) and (p − c) [1 − F (p + s)] are both maximized at p < pm .
Step 4. The expected profit of a non-advertising firm is
πu
γ + (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)
[1 − G (p)] − κ .
2
(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 substitute 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)
30
(1 − γ) (1 − α)
−κ.
2
(A.35)
Suppose to the contrary that ψ (p, α) > Ψ (p, α). Then there exists some p̃ < p such that
ψ (p̃, α) = Ψ (p̃, α) = Ψ (p, α). This implies that G (p̃) = 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, p̄ 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 > p̄, 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 (p̄, 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 > κ̄ there is an equilibrium where each
firm strictly prefers not to advertise, and when κ ∈ [κ, κ̄] 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
min
∗
p∈p (α)
(1 − γ) π (p) −
1−γ
(p − c) [1 − F (p + s)] .
2
(A.36)
Hence the left- and right-derivatives of the lefthand side of (A.32) exceed
min
∗
p∈p (α)
(1 − γ) π (p) −
1−γ
1−γ
(p − c) [1 − F (p + s)] − π u
.
2
2
(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)] + π u /2. Hence (A.37) is strictly positive, such that
for κ ∈ [κ, κ̄] the α which solves (A.32) is continuous and strictly increasing in κ.
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