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Multiproduct Retailing
Andrew Rhodes∗
November 2013
We study the pricing behavior of a multiproduct retailer, 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. 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. Low-price advertising can therefore be used to build a ‘low price image’. In
equilibrium, retailers randomize over both their advertised and unadvertised prices.
The model therefore generates price dispersion in a simple way.
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 retailer’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 retailer 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 retailer from stocking a wider range of products? How do pricing incentives
change when a firm sells more products? Some retailers eschew advertising and
charge ‘everyday low prices’ (EDLP) across the whole store. Others use a so-called
HiLo 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
retailer’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 range of
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. Retailers 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
retailer, 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 have only 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 small 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.
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 retailers
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 be an
equilibrium in which some consumers search, because retailers would always hold
consumers up by charging more than they expected. This happens irrespective of
how many retailers there are in the market.
We show that this ‘no search problem’ disappears when retailers sell multiple
3
products.1 Intuitively in the single-product case, only consumers with a high valuation decide to search, so retailers exploit this and charge a high price. However
in the multiproduct case, somebody with a low valuation on one product may still
search because she has a high valuation on another. Retailers 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 retailer attracts more consumers, who are more representative of
the general population, and who therefore have a higher number of low product valuations. Consequently, the consumers who search a larger retailer tend to be more
price-elastic.2
We then characterize a multiproduct retailer’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 retailer’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
inform consumers about one of its prices. We show that when a firm cuts its ad1
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 retailers’ 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.
2
Two other papers which look at the relationship between product range and price, are VillasBoas (2009) and Zhou (2012). Villas-Boas shows that a monopolist charges a higher price when
it stocks more substitute products, because it is more likely to provide consumers with a better
match. In Zhou’s model firms sell two independent products, and matches are independent across
firms. He also finds that multiproduct firms compete more aggressively. However unlike in our
model, the reason is that firms find it more valuable to stop consumers searching further.
4
vertised price, some new consumers with relatively low valuations decide to search
it. The firm then finds it optimal to also reduce its unadvertised prices, in order to
sell more products to these new searchers. Therefore consumers (rationally) expect
a positive relationship between a firm’s advertised and unadvertised prices. Whilst
firms cannot commit in advance to their unadvertised prices, they can indirectly
convey information about them via their advertised price. This implies that a low
advertised price on one product can build a store-wide ‘low price-image’, even on
completely unrelated products.
This result is related to but different from Lal and Matutes (1994) and Ellison
(2005), in which two firms are located on a Hotelling line and sell two products but
advertise only one of them. In Lal and Matutes all consumers have an identical
willingness to pay H for one unit of each good. In Ellison all consumers value the
advertised (base) product the same, but have either a high or a low valuation for the
unadvertised (add-on) product. As in Diamond’s model, the unadvertised price is
driven up to H in Lal and Matutes, and (typically) up to the high-types’ willingness
to pay in Ellison’s model. Firms use their advertised price to compete for store
traffic, but unlike in our model, cannot use it to credibly convey information about
their unadvertised price. The difference arises because in our paper, valuations are
heterogeneous and continuously distributed. Therefore when a firm changes its
advertised price, the pool of searchers also changes and this alters the firm’s pricing
incentives on its other products. In another line of work, Simester (1995) also
finds that prices are positively correlated, but this is due to a different, cost-related
mechanism. In his model, a low-cost firm may charge a low advertised price in order
to signal its cost advantage to consumers.
Our model also provides a simple explanation for price dispersion, especially
dispersion of unadvertised prices. We show that sometimes a retailer charges a high
‘regular’ price on each product, and does not advertise. At other times, it decides
to hold a promotion. During the promotion, it advertises a low but random price on
one of its products.3 Moreover as explained above, unadvertised prices are positively
3
The retailer also chooses at random which product to promote. Lal and Rao (1997) reach the
same conclusion in a different model. However DeGraba (2006) shows that when some consumers
5
related to a firm’s advertised price.4 Hence our model generates dispersion, even in
the prices of products which neither firm is advertising. This is unlike existing papers
such as Lal and Matutes, Robert and Stahl (1993), and Baye and Morgan (2001),
which also differentiate between advertised and unadvertised products, but which all
predict that unadvertised prices should be non-dispersed. Our model also predicts
the coexistence of small (unadvertised) and large (advertised) price discounts. As
such it suggests that price distributions should have a mass point (at the regular
price) as well as at least two gaps. This differs from other models such as Varian
(1980) and Baye and Morgan, which predict no gap, and from Narasimhan (1988)
and Robert and Stahl, which predict a single gap.
Finally we extend our results in several directions. This includes relaxing the
assumption that consumers are interested in buying all of a retailer’s products. In
particular we consider a richer setting, where the retailer has categories of products,
and consumers wish to buy at most one good from each category. As in the main
model, we show that when a firm has more product categories, its prices are lower.
However we also show that when a firm stocks more products within each category,
its prices are not necessarily lower. This allows us to derive more precise predictions
about the relationship between product range and price, and we then offer some
empirical evidence in their favor.
The rest of the paper proceeds as follows. Section 2 outlines the main model,
which has only one retailer. Section 3 characterizes prices when the firm decides
not to advertise, whilst Section 4 characterizes the optimal advertising strategy.
Section 5 repeats these steps for multiproduct duopolists, and demonstrates a close
are more profitable than others, firms should disproportionately advertise products which are (primarily) bought by those consumers. Chen and Rey (2012) find that when retailers are asymmetric,
competition for one-stop shoppers causes them to offer their best deal on different products.
4
McAfee (1995) and Hosken and Reiffen (2007) find that multiproduct firms’ prices are negatively correlated, whilst Shelegia (2012) finds them to be uncorrelated. These papers differ from
us, because they look at markets in which all prices are unadvertised, but some consumers can
search costlessly. On the other hand, Bliss (1988) and Ambrus and Weinstein (2008) suppose that
consumers are informed about all of a retailer’s prices. Bliss shows that firms use Ramsey pricing,
and as such may charge low prices on some products but high prices on others. However Ambrus
and Weinstein prove that this is only possible when there are certain demand complementarities.
6
connection with the monopoly analysis. Section 6 then extends the model in several
directions, and Section 7 concludes.
2
Main model
There is a unit mass of consumers interested in buying 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.5 The
marginal cost of producing each product is c, where 0 ≤ c < b. We also define
pm = arg max (p − c) [1 − F (p)]; for expositional simplicity we focus on the case
pm > a, although our results also hold when pm = a.
There is one retailer, who has a local monopoly over the supply of n ≤ N of the
products. (This assumption is relaxed later on.) Consumers have full information
about which n products the retailer stocks. Each consumer also knows her individual
valuations (v1 , v2 , ..., vn ), but initially does not know any of the firm’s prices. In order
to buy anything consumers must travel to the store at cost s > 0; once inside the
store they learn all the retailer’s prices. We can therefore interpret s as both a
search and a shopping cost.
The model has three stages. In the first stage the retailer privately chooses the
prices of every product, which we denote by p = (p1 , p2 , .....pn ). It also has the
opportunity to inform consumers about the price of one its products, by buying an
advertisement at cost κ > 0. The advert must be truthful, and is received by all
consumers. In the second stage consumers observe whether or not an advert was sent
(and if so, learn the advertised price) and then form expectations pe = (pe1 , pe2 , .....pen )
about the prices of each product. Consumers then decide whether or not to incur
the search cost and visit the retailer. Finally in the third stage, consumers who have
5
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
travelled to the store learn actual prices and make their purchases.
3
No advertising
We first analyze the benchmark case, where the retailer has decided not to advertise.
3.1
Solving for equilibrium prices
Consumers learn that the firm has not advertised, and form expectations pe =
(pe1 , pe2 , . . . , pen ) about the price of each product. Consumers then visit the retailer if
P
their expected surplus nj=1 max vj − pej , 0 exceeds the search cost s.
As a preliminary step, we note that there is a continuum of equilibria whose prices
P
satisfy nj=1 max b − pej , 0 ≤ s. Consumers anticipate these very high prices, and
therefore do not search. The retailer consequently earns zero profit, and is therefore
willing to set p = pe . Moreover when n = 1 this is the unique equilibrium outcome;
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 pe < b − s, in which case all consumers with v1 ≥ pe1 + s
would find it worthwhile to visit the retailer. The problem is that since all the
consumers in the store would be willing to pay at least pe1 + s for the product, the
retailer 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 retailer 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 retailer. Once a consumer has searched, she buys
any product for which her valuation exceeds the price. Therefore we can write the
8
demand for product i as
Di (pi ; pe ) =
Zb
f (vi ) Pr
pi
n
X
j=1
!
e
max vj − pj , 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 be correct, so we require
that pei = argmax (pi − c) Di (pi ; pe ). Imposing this condition, we have the following
pi
lemma. (Note that proofs generally appear in the appendices.)
Lemma 1 The equilibrium price of unadvertised good i satisfies
!
Di (pi = pei ; pe ) − (pei − c) f (pei ) Pr
X
max vj − pej , 0 ≥ s
=0
(2)
j6=i
Equation (2) can be interpreted as follows. Suppose the firm charges slightly more for
product i than consumers were expecting. The retailer gains extra revenue from the
consumers who buy the product, and they have mass equal to demand. The retailer
loses margin pei −c on each consumer who stops buying the good, and they have mass
P
e
,
0
≥
s
. Intuitively, each consumer who stops buying
max
v
−
p
f (pei ) Pr
j
j
j6=i
the good a). has a marginal valuation vi = pei for it, and b). has searched. Any
consumer who is marginal for good i only searches if her expected surplus on the
P
remaining n − 1 goods, j6=i max vj − pej , 0 , exceeds the search cost s. Figure 1
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
definitely 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. Thickness of demand for product 1 is given by the thick line,
that is consumers with v1 = pe1 and v2 ≥ pe2 + s.
Proposition 2 If n is sufficiently large, there exists at least one equilibrium in
which consumers search and the retailer makes positive profit.
9
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
pe1
pe1 +s
v1
b
Figure 1: Search behavior when n = 2.
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 retailer increases p1 it will
lose the demand of these marginal consumers. Now, when the firm’s product range is
large, many consumers who are marginal for product 1, find it worthwhile to search.
Therefore the firm’s incentive to hold-up buyers of product 1 is weak, and this allows
us to support a rational expectations equilibrium with search. Notice that consumer
heterogeneity plays an important role in this argument. If consumers had the same
valuation v̄ for each product, the firm could (and would) extract all the surplus
from any consumer who decided to search. Therefore consumer heterogeneity and
multiple products are both important features of our set-up.6
6
We are also assuming here that the search cost does not increase in store size. However
this assumption is not crucial for the result. Let us define a general search cost s (n), and let
Rb
µpm ≡ pm (x − pm ) f (x) dx be average per-product surplus when price is pm . We can prove that
Proposition 2 still holds, provided there exists δ > 0 and an n0 , such that s (n) /n ≤ µpm − δ for all
10
Table 1: Number of products required for an equilibrium with search.
Support of (uniformly distributed) valuations
c=
c=
1
4
1
2
[0, 1]
[1/2, 3/2]
[1, 2]
[0, 2]
[1, 3]
[0, 3]
s = 10−6
3
2
2
3
2
3
s = 1/2
12
6
4
6
3
4
s=1
20
9
6
9
5
7
s=2
36
15
9
15
8
11
s = 10−6
4
3
2
3
2
3
s = 1/2
22
8
5
7
4
4
s=1
39
13
7
12
6
7
s=2
73
22
11
20
9
12
According to Proposition 2 only retailers with sufficiently many products, are
able to attract consumers and make a positive profit. We can show that the critical
number of products needed for this to happen, varies with underlying parameters
of the model in a natural way (provided f (v) increases or does not decrease too
rapidly). Firstly we can show that more products are required when either the
search or production cost is higher. Intuitively fewer marginal consumers search
when s is higher, so the firm’s incentive to hold-up consumers is greater. Similarly
marginal consumers are less valuable to the firm when c is higher, so raising prices
is again more attractive. The retailer must therefore stock more products, in order
to offset its increased desire to hold-up consumers. Secondly suppose valuations
are drawn from [a + q, b + q] using distribution function F (v − q). q can loosely
be interpreted as a measure of product quality. We can also show that when q is
higher, the hold-up problem is less severe, so the retailer need not stock so many
products. Table 1 gives the critical number of products referred to in Proposition 2,
for different combinations of parameters, when valuations are uniformly distributed.
For example a retailer with two or more products can make positive profit, when
valuations are uniform on [1, 2], and c = 1/2 and s = 10−6 . However the critical
n ≥ n0 . Therefore a search-equilibrium exists even when s0 (n) > 0, so long as the average search
cost is not too large.
11
number of products can also exceed two, and varies with s, c, and q as described
above.
Henceforth we assume that n is large enough to support a search-equilibrium.
We now want to gain more understanding of how a retailer chooses its prices. To do
this, it is convenient to introduce the notation tj ≡ max vj − pej , 0 for the expected
P
surplus on good j. Only consumers with nj=1 tj ≥ s actually visit the store, and
can therefore respond to changes in the actual price of product i. These consumers
naturally split into two groups. We borrow some terminology from the existing
literature, and use the term ‘shoppers for product i’ to denote those consumers
P
with
j6=i tj ≥ s. They search irrespective of how much they value product i.
We use the term ‘Diamond consumers for product i’ to denote those people with
Pn
P
j=1 tj ≥ s >
j6=i tj . They only search because they anticipate a strictly positive
surplus on product i. Equation (2), which is the first order condition for product i,
can be rewritten as:
!
!
!
n
X
X
X
Pr
tj ≥ s [1 − F (pei ) − (pei − c) f (pei )] +Pr
tj ≥ s − Pr
tj ≥ s = 0
j=1
j6=i
|
{z
}
Probability of a shopper
|
j6=i
{z
}
Probability of a Diamond consumer
(3)
The lefthand side of this equation decomposes into two parts, the change in profits
caused by a small change in pi around pei . Firstly shoppers for product i search
irrespective of their vi , which therefore continues to be distributed on [a, b] with the
usual density f (vi ). Consequently profits on shoppers are simply (pi − c) [1 − F (pi )]
(the same as in a standard zero-search-cost monopoly problem), and so small changes
in pi around pei affect profits by 1 − F (pei ) − (pei − c) f (pei ). Secondly Diamond
consumers for product i have vi − pei > 0, so they would all buy product i even if the
price were slightly higher than expected. Consequently a small increase in pi above
pei , causes profits on Diamond consumers to increase by 1. Just like consumers at a
single-product retailer, their demand is locally perfectly inelastic.
Recall from earlier that pm = arg max (p − c) [1 − F (p)] is the standard monopoly
price which arises when there is no search cost.
Remark 3 The equilibrium price of any unadvertised good i strictly exceeds pm .
12
Equation (3) can only be satisfied if 1 − Fi (pei ) − (pei − c) f (pei ) < 0. Using the
quasiconcavity of (pi − c) [1 − F (pi )], this is equivalent to saying that pei > pm .
Intuitively only consumers with relatively high valuations search, and the retailer
will naturally try to somewhat exploit this ex post. Hence there is still a partial
hold-up problem, and this drives equilibrium prices above pm .
3.2
Equilibrium multiplicity and selection
As discussed earlier our model has a whole continuum of high-price equilibria, in
which no consumer finds it worthwhile to search. However a priori we rule out
these equilibria, for the following two reasons. Firstly these equilibria are Paretodominated, because they generate zero surplus for both consumers and the retailer.
Secondly if the firm has to pay a small fixed cost, it would not enter the industry
if it expected to play one of these equilibria. A simple forward induction argument
(see below for a formal discussion) therefore suggests that when a retailer enters,
consumers should infer they are not playing one of these no-search equilibria.
This leaves us with search-equilibria, whose existence was proved in Proposition
2. Henceforth when we talk about ‘equilibrium’, we are referring to one in which
consumers search. However, there is no reason why the model should have a unique
equilibrium. Intuitively, suppose that we have already found one equilibrium price
vector p0 . If consumers suddenly expect a different price vector p00 , the pool of
searchers changes. As such the firm’s optimization problem also changes, and in
principle its new optimal price vector could be p00 , in which case p00 is also an
equilibrium. Fortunately if multiple equilibria exist, there is a natural way to select
amongst them. In particular consider two equilibrium price vectors p0 and p00 ,
and suppose for a moment that p00j < p0j for each j = 1, . . . , n (the relevance of
this supposition will become clear shortly). Consumers are clearly better off if the
‘lower’ equilibrium p00 is played, and surprisingly so too is the firm. Intuitively this
is because prices are already too high from the firm’s perspective. More formally if
13
the p0 equilibrium is played, the firm’s total profit is


!
b
Z
n
n
X
X

0
max vj − p0j , 0 ≥ s dvi 
(pi − c) × f (vi ) Pr
i=1
p0i
(4)
j=1
When consumers expect the p00 equilibrium they search if
Pn
00
,
0
≥ s,
max
v
−
p
j
j
j=1
and then buy product i provided that it gives positive surplus. Therefore if the firm
‘deviates’ and sets its prices equal to p0 rather than to p00 , its total profit equals


!
b
Z
n
n
X
X

0
max vj − p00j , 0 ≥ s dvi 
(5)
(pi − c) × f (vi ) Pr
i=1
p0i
j=1
which strictly exceeds (4). However when consumers expect the p00 equilibrium, by
definition the firm prefers to set prices equal to p00 rather than to p0 . So by revealed
preference profits in the lower p00 equilibrium must exceed (5), and therefore also
exceed profits in the p0 equilibrium.
In fact our model always has a lowest equilibrium. It is simple to prove that,
since valuations are drawn from the same distribution, each equilibrium has pe1 =
... = pen . Existence of a lowest equilibrium then follows immediately. However it is
possible to relax the assumption of identical distributions, and still have a lowest
equilibrium. Suppose for a moment that each vi is drawn independently according
to its own (strictly positive, continuous and logconcave) density fi (vi ). At least
one equilibrium exists whenever n is sufficiently large, although of course different
products will generally have different prices. Nevertheless provided the search cost
is sufficiently small, we can prove that one equilibrium is lower than all the others
(and therefore is also Pareto dominant).
We will select the lowest-price equilibrium. One justification is that this equilibrium is Pareto-dominant, and therefore it is natural that everyone would coordinate
on it. A second justification is that the firm might signal its intention to play this
equilibrium. In particular, suppose that the firm must pay a cost F to enter the
market (or otherwise stay out, and earn zero). Suppose further that F ∈ (Π0 , Π00 )
where Π00 is profit in the lowest equilibrium, and Π0 is the highest profit achievable in
any other equilibrium. There is one subgame perfect equilibrium in which the firm
14
enters and gets Π00 − F ; in all other subgame perfect equilibria the firm stays out
of the market and gets zero. Therefore in the spirit of forward induction, once the
firm enters, consumers should rationalize this as a signal that the firm will play the
lowest-price equilibrium. Since Π00 − F > 0 it is in the firm’s interest to enter and
send this signal. Numerical simulations also suggest that Π00 − Π0 can be relatively
large, so this argument may apply widely.
3.3
Comparative statics
We now study how the (lowest) equilibrium price p∗ is affected by changes in the
search cost and product range. To do this recall that tj ≡ max (vj − p∗ , 0) is the
(equilibrium) expected surplus on good j. Substituting pej = p∗ for j = 1, 2, . . . , n
into equation (3) and then rearranging, p∗ satisfies
P
P
n
n
Pr
j=2 tj ≥ s
j=1 tj ≥ s − Pr
P
1 − F (p∗ ) − (p∗ − c) f (p∗ ) +
=0
n
t
≥
s
Pr
j
j=2
(6)
The final term in equation (6) is the ratio of Diamond consumers to shoppers for
a single product. We use Karlin’s (1968) Variation Diminishing Property to prove
. P
P
n
n
t
≥
s
is increasing in s, or
t
≥
s
Pr
in the appendix that Pr
j=2 j
j=1 j
P
n
equivalently that Pr
j=1 tj ≥ s is log-supermodular in s and n. This means that
as search becomes costlier, the ratio of Diamond consumers to shoppers increases,
and demand for each product becomes less elastic. It is therefore intuitive that:
Proposition 4 The equilibrium price (i). increases in the search cost and (ii).
decreases in the number of products stocked.
According to part (i). of the proposition, a multiproduct retailer charges higher
prices when search is more costly.7 This is a very natural result, which 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
7
As discussed earlier, we are assuming that the shopping cost is independent of the number of
products. However the result would still hold for a general search cost s (n), provided that it did
not increase too rapidly in the size of the firm’s product range.
15
a better deal, which gives firms more market power. However the mechanism which
drives Proposition 4 is different. In our model, when s increases some consumers
stop searching. Importantly, those who stop searching are people with relatively
low valuations. Consequently the retailer’s potential customers now have, in some
sense, higher average valuations than before. This makes it optimal for the retailer
to raise its prices.
According to part (ii). of the proposition, a retailer charges lower prices when
it has a larger product range. In more detail, suppose that initially n < N i.e. the
retailer does not stock all the products which consumers wish to buy. Then suppose
that it adds an extra product to its range, which is not available elsewhere. We
expect that, ceteris paribus, some new consumers will 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. This means that if
the firm now reduces p1 (say), demand for product 1 will increase (proportionately)
more than it would have before. Equivalently, a larger retailer faces a more elastic
demand curve on each product that it stocks; consequently it charges lower prices.
As noted in the introduction, empirical evidence suggests that larger retailers do
generally have lower prices. One explanation for this might be that they have lower
costs - for example due to buyer power or economies of scale. However Proposition 4
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. Along similar lines, Shankar and Krishnamurthi (1996) show
that EDLP stores attract consumers who are more sensitive to unadvertised prices;
Ellickson and Misra (2008) then demonstrate that the EDLP format is favored by
larger retailers.
Now consider the relationship between product range and profitability. When
the retailer sells n0 products and the equilibrium price is p0 , the profit from any
16
individual product is
(p0 − c) ×
Zb
0
f (vi ) Pr
!
n
X
max (vj − p0 , 0) ≥ s dvi
(7)
j=1
p0
Now suppose instead that the firm sells n00 > n0 products, and that the corresponding
equilibrium price is p00 < p0 . Consumers expect prices to be p00 and therefore search
on that basis. However once inside the store, they will buy any product which they
value more than its price. Therefore if the firm ‘deviates’ and charges p0 , it earns
(p0 − c) ×
Zb
00
f (vi ) Pr
p0
n
X
!
max (vj − p00 , 0) ≥ s dvi
(8)
j=1
on each product. Clearly this strictly exceeds (7). However by definition, when
consumers expect p00 the retailer’s optimal response is to charge p00 . Therefore by
revealed preference, when n = n00 the profit earned on each product must exceed (8)
and therefore also exceed (7). Equivalently, the model predicts that a larger retailer
earns more profit on each product that it sells. Intuitively even though it charges
lower prices, it is also searched by more consumers and therefore makes higher sales.
Since equilibrium prices exceed pm , selling more units at a lower price is good for
profits. On the other hand smaller retailers are ‘trapped’ into charging high prices
and earning relatively low profits.
This may help explain why firms co-locate near to each other, for example in
shopping malls and highstreets. In particular we could interpret the multiproduct
monopolist as a cluster of n single-product firms, each of which sells a different
and unrelated product. Dudey (1990) and others have already shown that sellers
of similar or identical products may cluster together. This commits the firms to
fiercer competition, which then encourages more consumers to search, and ultimately
increases the profits of each firm in the cluster. However our model suggests a
stronger result: even single-product firms that sell completely unrelated goods, can
also commit to charging lower prices (and therefore earn higher profits) by clustering
together. This is despite the absence of any competition between firms in the cluster.
17
4
Advertising
We now consider the case where the retailer has decided to advertise one of its
products. Without loss of generality suppose that the firm advertises the price of
product n, and denote it by pan . The timing is as follows. Firstly the firm decides on
pan and sends the advert. The retailer then chooses its other prices p1 , ..., pn−1 , but
cannot communicate these to consumers.8 Secondly consumers receive the advert,
and form price expectations (pe1 , ..., pen ). Notice that pen = pan because advertising
is (by assumption) truthful. Consumers then decide whether or not to search, and
make their purchases.
Our first task is to understand how the retailer’s choice of pan , affects the prices
which it charges for its other products. Clearly consumers still search if and only
P
if nj=1 max vj − pej , 0 ≥ s, therefore demand for any unadvertised product i < n
remains
Di (pi ; pe ) =
Zb
f (vi ) Pr
pi
n
X
j=1
!
max vj − pej , 0 ≥ s vi dvi
(1)
Since the retailer again wants to maxpi (pi − c) Di (pi ; pe ), the equilibrium price of
any product i < n satisfies
!
Di (pi = pei ; pe ) − (pei − c) f (pei ) Pr
X
max vj − pej , 0 ≥ s
=0
(2)
j6=i
Consequently we again infer that when n is sufficiently large, there exists a rational
expectations equilibrium with search. Unadvertised prices still weakly exceed pm ,
because the firm attracts consumers with predominantly high valuations. Moreover
fixing a particular pan there is no guarantee that the equilibrium price vector is
8
We could instead assume that the firm chooses unadvertised prices at the same time as (or even
before) it chooses pan . We would then need to specify consumers’ beliefs in more detail. In particular
consumers would now expect a particular pan to be played, and would have associated beliefs about
(p1 , ..., pn−1 ); consumers would also have off-equilibrium beliefs about (p1 , ..., pn−1 ), in case the
firm advertised a price that was different from what they expected. Clearly the equilibrium price
vector pe1 , ..., pen−1 , pan which we identify in the main text, would still be an equilibrium. However
freedom in how to choose off-path beliefs, would generate additional equilibria as well. Nevertheless
In and Wright (2012) argue that a natural way to deal with this multiplicity, would be to reorder
the timing of the game as we have chosen to do here.
18
unique. However there again exists a lowest-price equilibrium. Consumers clearly
prefer playing this equilibrium. However the retailer’s preferences are 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 < 0. Nevertheless
we can show that total profits are highest in the lowest-price equilibrium, if either
(i). the search cost is small, or (ii). pan −c is not too negative. Under these conditions
we have a Pareto dominant equilibrium, and we therefore select it.
Lemma 5 The equilibrium price of products (1, ..., n − 1) is an increasing function
of the advertised price pan .
This result is explained as follows. Firstly note that when pan ≤ a−s all consumers
find it worthwhile to search. There is no hold-up problem, and the retailer charges
p1 = ... = pn−1 = pm . Secondly note that when pan > a − s not everybody searches,
and unadvertised prices strictly exceed pm . A small increase in pan will ceteris paribus
cause fewer consumers to search. Moreover those who stop searching, are people
with predominantly low product valuations. Therefore the composition of the pool
of searchers is now different. In particular the retailer now attracts consumers who
on average have higher valuations. Consequently it has more incentive to hold
them up by charging higher prices. Consumers anticipate this, and therefore expect
a positive relationship between advertised and unadvertised prices. Notice that
consumer heterogeneity is crucial for 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 retailer would charge v̄ for it, irrespective of
what it charged for its advertised product.
Lemma 5 shows how a low advertised price on one product, ‘signals’ a retailer’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 should use store flyers to update
their beliefs about a retailer’s overall price level. Consistent with this, Brown (1969)
surveyed consumers, and found that they associated low-price advertising with a low
19
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 retailer’s overall offering.
Our next task is to understand how the retailer optimally chooses its advertised
price. Demand for the advertised product itself is
Dn (pan ; pe ) =
Zb
f (vn ) Pr
n−1
X
!
max vj − pej , 0 + vn − pan ≥ s dvn
(9)
j=1
pa
n
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 retailer and buy this product precisely because they know
its price is lower. Comparing with equation (1), only the first of these effects is
present when the product is not advertised. Therefore demand becomes more elastic, when a product’s price is advertised. 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.
Proposition 6 The optimal advertised price is strictly below pm .
Proof. Let Πj (pan ; pe ) be the equilibrium profit earned on good j. Differentiating
P
total profit nj=1 Πj (pan ; pe ) with respect to pan and using the envelope theorem, we
get a first order condition:
dΠn (pan ; pe )
dpan
n−1
X
+
i=1
+
n−1
X
i=1
(pei − c)
Zb

f (vi ) 
n−1
X
∂ Pr
P
∂pek
k=1
pei

!
X

tj = 0
(pei − c) −f (pei + s) Pr
j6=i
n
j=1 tj
∂pei
+
∂pan

≥ s ∂pe
k
dvi
a
∂pn
Zb
∂ Pr
f (vi )
pei
P
n
j=1 tj
∂pan
≥s


dvi  = 0
(10)
20
We claim that the first term in (10) is negative when pan ≥ pm . To prove this, first
note that for any i = 1, . . . n − 1, ∂Πn /∂pei < 0 and ∂pei /∂pan > 0. Then, to prove
Pn−1
that ∂Πn /∂pan < 0, define T = j=1
max vj − pej , 0 and note that
Z s
Z ∞
a
a
a
[1 − F (pn )] dµ (y)
(11)
[1 − F (pn + s − z)] dµ (z) +
Πn = (pn − c)
s
0
where µ is a measure over T with mass at zero. Since f (p) is logconcave, it is
straightforward to prove that (p − c) [1 − F (p + s − z)] is (weakly) decreasing in p
whenever p ≥ pm . Therefore dΠn /dpan < 0 as claimed. Since the remaining terms
in (10) are also negative, (10) cannot hold for any pan weakly above pm . Therefore
profit must be maximized at some pan < pm .
The firm chooses pan to maximize profits across the whole of its product range.
There are three different forces which lead the firm to choose pan < pm . Firstly the
search cost deters consumers from visiting the retailer. This reduces demand for the
advertised good, and (as explained above) also makes it more price-elastic. Therefore
even if the firm cared only about profits from good n, it would optimally choose some
pan < pm . Secondly the search cost also depresses demand for unadvertised products.
A low pan encourages more consumers who like good n to search; once inside the store,
these extra consumers also buy some of the (high-margin) unadvertised goods.9
Thirdly, recall that unadvertised prices are high because of the hold-up problem.
The retailer would be better off, if it could convince consumers that its unadvertised
prices were lower. According to Lemma 5, a low pan helps it to somewhat achieve
this goal, and build a ‘low price image’.
The retailer therefore uses a low advertised price (below pm ) to attract consumers
into the store, and then charges them 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 models such as those by Hess and
Gerstner (1987) and Lal and Matutes (1994). However there are several differences
with our analysis. Firstly in our model consumer valuations are heterogeneous,
9
Ailawadi et al (2006) offer some empirical evidence in favor of this second channel. Using data
for a U.S. drug store, they show that a promotion in one category leads to higher sales in other
categories.
21
whereas these other papers assume them to be homogeneous. Secondly therefore,
unadvertised prices are in some sense lower in our model. More precisely, in our
model consumer surplus from the unadvertised products is strictly positive; however
in the other papers, unadvertised prices are so high that consumers are left with
zero surplus. Thirdly, our paper offers an additional explanation for low advertised
prices, namely the desire to build a store-wide low price image (the third force, in
the previous paragraph). This is absent from the other papers.
4.1
When should the retailer advertise?
The retailer must decide whether or not to pay the cost κ and send out an advert.
When it does not advertise, it plays the equilibrium in Section 3 and charges a price
vector (pu , ..., pu ) where pu > pm . When it does advertise, it chooses pan < pu , and
therefore by revealed preference earns higher gross profits. Consequently the retailer
has an incentive to advertise whenever the cost κ is not too high.10
Of more interest, is the question of how a retailer’s product range affects its
advertising strategy. For example fixing κ, is a firm with a broader product range
more likely to advertise? How does the optimal advertised price vary with the number of products stocked? It turns out that neither question has a clear-cut answer.
Figures 2 and 3 show by example that the optimal advertised price and the gains
from advertising, are both non-monotonic in the size of the firm’s product range.
Intuitively advertising ‘works’ because (i). it commits the retailer to charging lower
prices, and (ii). it increases store traffic, which leads to sales of unrelated products.
Now on the one hand, larger n means that consumers who search, end up buying
more products. This suggests that the retailer should try harder to boost store
traffic i.e. advertising is more desirable, as is a lower advertised price. On the other
hand, larger n also means that even in the absence of advertising, more consumers
would search anyway (and unadvertised prices would be lower). This suggests that
10
Notice that the retailer’s decision of whether to advertise, and at what price, is deterministic.
This contrasts with Bester’s (1994) model, in which a single-product monopolist can choose how
many consumers receive its advert. He shows that the retailer might then randomize over both its
price and advertising reach.
22
Optimal pan
0.87
0.86
0.85
0.84
0.83
5
10
15
20 n
Figure 2: Non-monotonicity of the optimal advertised price.
Advertising Gains
0.06
0.05
0.04
0.03
0.02
0.01
0.00
5
10
15
Figure 3: Non-monotonicity of the gains from advertising
23
20 n
the retailer has less need to further boost store traffic i.e. advertising is less desirable, as is a lower advertised price. This tension causes the non-monotonicity in
Figures 2 and 3.
We can also say something more precise about the advertising strategy of a large
retailer. We can prove that for any > 0 there exists an n̄ such that for all n > n̄,
(i). the gains from advertising are less than , and (ii). the optimal advertised price
exceeds pm − . Intuitively when the firm’s product range is large, most consumers
search anyway, so advertising is of limited value. The model therefore predicts that
a large retailer should follow an ‘everyday low-price’ (EDLP) 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 retailer
is more likely to follow a ‘Hi-Lo’ strategy: it should charge generally high prices,
but advertise a very low price on one product (or more generally on a few products
- see below). Empirical evidence largely supports these predictions. 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 EDLP stores, are more responsive to regular (i.e. nonpromotional) price changes; this is again consistent with our model. On the other
hand Shankar and Bolton (2004) show that after controlling for whether a store is
EDLP or HiLo, larger stores are more likely to price-promote. However this result is
difficult to interpret, because according to our model store size and pricing strategy
are highly correlated.
Finally drawing together some earlier results, our model predicts that advertising
is associated with low prices:
Remark 7 When a retailer chooses to advertise, it charges strictly lower prices on
every product.
Proof. Without advertising the firm charges some pu > pm on every product. With
advertising it chooses some pan < pu (c.f. Proposition 6), and therefore by Lemma 5
unadvertised prices are less than pu as well.
24
In his survey article Bagwell (2007) says there is “substantial evidence” that advertising leads to lower prices. For example Devine and Marion (1979) publicized
the prices of some randomly selected grocery products in local newspapers; retailers
responded by reducing their prices on those products. However our model makes a
stronger prediction: even non-advertised prices should be lower. To the best of our
knowledge, almost no empirical work tests this prediction. Indeed Bagwell says that
“the effect of advertising on the prices of advertised and non-advertised products
warrants greater attention”. One exception is Milyo and Waldfogel (1999), which
looks 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.
A comment on multiple advertised prices Suppose the retailer can now
advertise up to k of its prices, each at a cost of κ > 0. For example, advertising more
products requires a longer (and therefore more expensive) flyer or TV commercial,
but consumers only have time to inspect k products. To make the model tractable,
we focus only the case of a very small search cost. Consider the equilibrium price of
unadvertised product i. We can show that as s → 0, pei → p̄ei ≥ pm where
1
(p̄ei − c) f (p̄ei )
Q
=
e
1 − F (p̄i )
1 − j6=i F pej
(12)
Moreover a small increase in pan (whilst fixing any other advertised prices) leads the
retailer to charge more on each of its unadvertised products. Optimal advertised
prices are also below pm , for the reasons given earlier. In addition, if κ were arbitrarily small the retailer would choose to advertise as many prices as it could, because
it would then have more control over them. However fixing κ, we can show that a
very large firm would not find it worthwhile to advertise even one price. Therefore
again firms with very large product selections find an EDLP strategy to be optimal;
small retailers are more likely to follow a HiLo strategy and advertise several low
prices, but also have higher prices on their remaining unadvertised goods.
25
5
Competition
We now suppose there are two retailers A and B, each of whom sells the same
n products. In the spirit of Varian (1980), we divide consumers into two groups.
A fraction γ of consumers are ‘loyals’. They have a strong affinity towards one
(randomly selected) retailer, and make any purchases there. The remaining fraction
1−γ of consumers are ‘non-loyals’. They have no preference for where they purchase
their goods, and are able to multi-stop shop. All consumers must pay a cost s > 0 in
order to search one retailer. Non-loyal consumers may then visit the other retailer,
by paying an additional s. As explained below, we have in mind a situation where s
is not too large, such that some non-loyal consumers do end up multistop shopping.
We also make the standard assumption that non-loyals have passive beliefs, and we
allow them to go back and purchase from a previously-searched retailer, by paying
a small cost.
The timing is now as follows. Firstly retailers simultaneously decide whether or
not to pay κ and send out an advertisement. Conditional on sending an advert, the
retailer picks one product to advertise and chooses its price. Secondly adverts and
their contents are observed by all agents. Retailers then set unadvertised prices,
whilst consumers form (rational) expectations about those prices. Consumers then
search and make purchases. We believe this timing is appropriate because in practice
retailers usually decide their promotions in advance; other prices are chosen later
and are made conditional on any promotional activity (see for example Chintagunta
et al 2003). As we demonstrate below, there is a close connection with the previous
monopoly analysis. To this end we introduce the following notation. We let pu be
the price that a non-advertising monopolist charges, and let π u be the corresponding
profit. For simplicity we assume π u > 0 for all n ≥ 2. If a monopolist advertises
one product at price q, we let φ (q) be the price it charges on each of the remaining
n − 1 products, and let π (q) be the total profit it earns. Recall from earlier that
φ (q) is increasing and that by definition pu = φ (pu ).
26
5.1
Two-product case
We first solve for equilibrium when the retailers sell two products.
Lemma 8 At the second stage of the game, there exists an equilibrium where (i). if
a firm does not advertise, it charges pu on both products and (ii). if a firm advertises
a price q < pm on one good, it charges φ (q) on the other.
As usual if a firm chooses not to advertise, there exist equilibria in which consumers expect it to charge very high prices and therefore do not search it. Excluding
this possibility, Lemma 8 gives the unique pure strategy equilibrium. Interestingly
unadvertised prices are the same under both monopoly and duopoly. This is reminiscent of Diamond (1971) - the difference being that our result holds in a multiproduct
context, and allows for one of the products to be advertised.11
The intuition for why the prices in the lemma constitute an equilibrium, is as
follows. Firstly suppose that neither firm has advertised. If consumers expect both
P
firms to charge pu , they search if and only if 2j=1 max (vj − pu , 0) ≥ s: loyals go to
their favorite store, whilst non-loyals choose one at random. Therefore in terms of
the joint distribution over (v1 , v2 ), consumers who search a duopolist look exactly
the same as those who search a monopolist. Hence (by definition) neither firm can do
better than charge pu . Secondly suppose that both firms have advertised the same
product. Suppose also that consumers expect a firm with advertised price q, to
charge φ (q) on its other product. Since φ (q) is increasing, one firm is then expected
to be weakly cheaper on both products. Therefore all non-loyals who search, will
go to that retailer. It is again easy to see that, in terms of the joint distribution
over (v1 , v2 ), each firm attracts the same mix of consumers as it would if it were
a monopolist. Hence it cannot do better than charge φ (q) on its unadvertised
good, as consumers were expecting. A similar logic can be used to understand
11
To simplify the exposition, the lemma only deals with the case in which advertised prices are
below pm . Our approach is to look for first-stage equilibria in which this is true, and then verify
that no firm wishes to deviate by advertising a price that exceeds pm . It turns out that sometimes,
such a deviation induces firms to charge unadvertised prices which differ from those given in the
lemma. Further details are provided in the appendix.
27
equilibrium pricing when only one of the two retailers has chosen to advertise. Third
and finally, suppose that the retailers have advertised different products. If a firm
with advertised price q < pm charges unadvertised price φ (q), there exists a search
cost s sufficiently small, such that non-loyal consumers ‘cherry-pick’ each retailer’s
advertised product. This implies that in the putative equilibrium, a firm only sells
its unadvertised product to loyal consumers. Conditional on this it is clear that
charging φ (q) on the unadvertised product is optimal. However since some nonloyal consumers visit both retailers, a firm could sell its unadvertised good to some
of them, if it reduces the price by a sufficiently large amount. Nevertheless such
a deviation is not profitable whenever the fraction of loyals γ is sufficiently large.
Thus charging φ (q) on the unadvertised good is indeed an equilibrium.
We are now in a position to analyze the first stage of the game. We first note that
under certain conditions, there can exist equilibria in which only one retailer chooses
to advertise. However empirically, Ellickson and Misra (2008) find that competing
retailers tend to coordinate on their broad advertising strategy, as measured by
their choice between EDLP and HiLo pricing. For this reason we focus attention on
equilibria in which retailers advertise with the same (positive) probability.
Our first result is a negative one. It is not an equilibrium for retailers to advertise
the same product. To gain some intuition, suppose to the contrary that both firms
advertise product 1 with probability one. It follows from Lemma 8 that whichever
firm has the lowest advertised price, captures all non-loyal consumers. Consequently
the first-stage of the game does not have a pure strategy equilibrium. Instead
each firm should draw its advertised price from an interval p, p using an atomless
distribution function, where c < p̄ < pm whenever the search cost is relatively small.
The expected profit from following this strategy is γπ (p) /2, because a firm with
advertised price p sells only to loyal consumers. However a firm can do even better
than this, by deviating and advertising product 2. For example, suppose firm A
advertises product 2 at a price p̄. According to Lemma 8 it will then charge φ (p̄)
for product 1. Therefore after deviating, firm A can (i). still earn γπ (p) /2 from its
loyal consumers, but (ii). now has the lowest price for product 2, and so can earn
28
additional profit by selling it to non-loyals. Consequently it is not an equilibrium
for the two retailers to advertise the same product.
Alternatively retailers might try to differentiate themselves by advertising different products. For example A could advertise good 1 and B could advertise good 2.
Unfortunately this is also unlikely to be an equilibrium. To see this, let paA1 , paB2 < pm
be the two advertised prices. When s is sufficiently small, non-loyal consumers can
cherry-pick from each store, and therefore retailer A earns
γ
π (paA1 ) + (1 − γ) (paA1 − c) [1 − F (paA1 + s)] − κ
2
(13)
Let p∗ be the price which maximizes (13), and which therefore also maximizes B’s
analogous profit function. Notice that c < p∗ < pm whenever the search cost is
relatively small. Then consider the following deviation: suppose that A advertises
product 2 at a price just below p∗ . We know from Lemma 8 that A will charge φ (p∗ )
on product 1 and will undercut its rival. Compared to advertising p∗ on product 1,
A will therefore (i). earn the same profit on its loyal customers, but (ii). will sell two
products to non-loyals instead of just one. This deviation is therefore profitable, so
long as the search cost is not too large. We cannot usually have an equilibrium in
which retailers deliberately advertise different goods, because it is too easy for one
of them to deviate and steal demand from the other.
We now turn to our main result:
Proposition 9 For any α ∈ [0, 1) there exists an advertising cost κ such that
(i). With probability α, a retailer does not advertise. It charges pu on all products.
(ii). With probability 1 − α, a retailer sends out an advert, containing the price of
one randomly chosen product. This advertised price q is strictly below pm and drawn
from a distribution. The firm charges φ (q) on its other product.
In this equilibrium each retailer randomly chooses which product to advertise. Once
one firm randomizes in this way, the other cannot do better than follow the same
strategy. Firms also randomize over their advertised price. This is necessary because with positive probability the two firms advertise the same product, in which
case all non-loyal consumers buy from whomever has the lowest advertised price.
29
Randomization prevents a rival firm from undercutting and stealing these non-loyal
consumers. This intuition is familiar from single-product models such as Varian
(1980).
However our multiproduct framework also provides some new insights. Firstly, it
offers a simple explanation for price dispersion, especially dispersion of unadvertised
prices. As discussed in the introduction, existing models that differentiate between
advertised and unadvertised prices, predict no dispersion of unadvertised prices.
However in our model, such dispersion arises very naturally. In particular, for the
same reasons as in the monopoly model, there is correlation between a firm’s advertised and unadvertised prices. Since a retailer must randomize over its advertised
price, this generates indirect dispersion even in prices which it does not advertise.
Second and related, our model predicts an aggregate price distribution which has
both a mass point and multiple gaps. According to Proposition 9 products have a
‘regular price’ pu , and stay at that level for a fraction α of the time. Retailers then
randomly hold a promotion, during which time they mark down the price of one
product by a random amount; the discount on this product exceeds pu − pm and
can therefore be relatively large. At the same time, retailers also reduce their other
prices by a smaller yet still discrete amount. Hence within the same framework, we
have both high regular prices, as well as the coexistence of small (unadvertised) and
large (advertised) price discounts.
Consistent with Proposition 9, Pesendorfer (2002) and Hosken and Reiffen (2004)
document that grocery products have a regular price, and stay at that price for long
periods of time; deviations from this regular price are generally downward, and potentially of sizeable magnitude. Klenow and Malin (2010) also stress the prevalence
of small price changes, across a wide range of studies. There is also some support for
our prediction that retailers randomly choose which product to promote. 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 is consistent our intuition that rivals
30
should find it hard to predict which products a retailer is going to promote. Finally,
an additional feature of our model is that some consumers multistop shop, and end
up cherry-picking advertised specials from each retailer. 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, and that
cherry-pickers buy significantly more advertised and discounted products.12
5.2
Many-product case
We again begin by solving for unadvertised prices in the second stage of the game.
Lemma 8 can be straightforwardly generalized, except when the firms have advertised the same product. In particular we can prove the following. (1). If neither
firm has advertised, it is an equilibrium for both to charge pu on all products. (2).
If both firms have advertised the same product, it is an equilibrium for a firm with
advertised price q to charge φ (q) on its remaining n − 1 products. (3). If only one
firm advertised: it is an equilibrium for the retailer with advertised price q to charge
φ (q) on its remaining products, and for the non-advertising firm to charge pu on
all products. Although we have not been able to prove that these equilibria are
unique, nor have we been able to construct alternative equilibria. The remaining
scenario to consider is (4). where the retailers have advertised different products.
Unlike in the n = 2 analysis, we expect that now some non-loyal consumers will
buy a product which neither firm has promoted. As a result the retailers face quite
different demand curves compared to when n = 2, and so equilibrium unadvertised
prices might be quite different. We therefore simplify the problem, by assuming that
if non-loyal consumers search, they first visit the retailer with the lowest advertised
price (and can then pay s and visit the other firm, if they wish).13 We are then able
12
Earlier we mentioned the empirical paper by Milyo and Waldfogel (1999). They find that
when one firm advertises, a non-advertising firm does not change its prices. This is consistent with
Proposition 9.
13
There are equilibria which do not require this additional assumption. For example when n is
sufficiently large, there exists a continuum of equilibria in which the two retailers price-match on
the n − 2 products which neither of them has advertised. However we think these equilibria are
less natural than the one given in Lemma 10. Further details are available upon request.
31
to prove:
Lemma 10 Suppose the firms have advertised prices pl and ph ∈ [pl , pm ) on different products. There exists an equilibrium in which (i). firm h charges φ (ph ) on all
its products, (ii). firm l charges between pm and φ (ph ) on its unadvertised products,
and (iii). if non-loyal consumers multistop shop and are indifferent about where to
buy a good, they buy it at store l.
Non-loyal consumers usually buy most products at the first store they visit. They
may then travel to the other retailer and cherry-pick its advertised special. Hence
this second retailer understands that it can only sell its unadvertised products to
loyal consumers, and therefore it acts like a monopolist when choosing the prices of
those products.
We are now in a position to characterize first-stage behavior. As a preliminary
step, we investigate conditions under which no advertising occurs. Notice that if
neither firm advertises during the first stage, we are in scenario (1). At the second
stage both firms charge pu on all products and earn a profit of π u /2 each. If one firm
deviates during the first stage and advertises, we are in scenario (3). Clearly the
best deviation would be to advertise the same price as would a monopolist, namely
the solution to equation (10) from earlier. Denote this optimal advertised price by
q ∗ < pm . At the second stage the deviant captures all non-loyal consumers, and
earns a total profit of (1 − γ/2) π (q ∗ ) − κ. Hence there is a first-stage equilibrium
in which neither retailer advertises, provided that
1
1−γ
(π (q ∗ ) − π u ) +
π (q ∗ ) ≤ κ
2
2
(14)
An interesting question is how product assortment affects this condition. Recall that
when a monopolist chooses to advertise, its profits increase by π (q ∗ ) − π u . Since
this is very small when n is large, we concluded that a monopolist with a large
product range is unlikely to advertise. The intuition is that when a retailer faces
little or no competition, it draws its customers from a fixed pool. When its product
range is large, most consumers search anyway and advertising is not very effective.
However it is clear from (14) that under duopoly there is a countervailing effect,
32
because π (q ∗ ) 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 nonloyal consumers from its rival. Since the profit earned on these extra consumers is
increasing in n, so are the incentives to advertise. Our model therefore suggests that
the degree of competition, as proxied by the fraction of non-loyal consumers, is also
crucial for determining whether a retailer uses an EDLP or a HiLo pricing strategy.
Now consider equilibria in which advertising occurs with positive probability.
We can again show that, in general, it is not possible to have an equilibrium where
firms either advertise the same product, or where they deliberately advertise different
products. The explanation is very close to the one we gave for the two-product case,
and so for brevity is omitted. Moreover
Remark 11 There exists an equilibrium which is qualitatively the same as that in
Proposition 9. Retailers randomly choose whether to advertise, which product to
advertise, and what price to advertise.
With some probability, a retailer again decides not to advertise, and charges a regular
price pu on each product. With complementary probability it offers a (potentially
large) discount on one advertised product, and also offers smaller discounts on its
other products. Note that the regular price pu , is exactly what a non-advertising
monopolist would charge. Therefore according to Proposition 4 from earlier on, a
larger retailer has a lower regular price. In addition this suggests that, at least for
very large competing retailers, their regular price will already be close to pm , such
that the discounts they offer on unadvertised goods will typically be quite small.
The main role of advertising is therefore to try and attract customers away from the
rival firm.
6
6.1
Extensions
Substitutes
Thus far we have assumed that a retailer sells n independent products. However
in practice, some of a firm’s products might be substitutes for one another. For
33
example a grocery store might stock several different types of milk and detergent.
Although a consumer’s demand for milk and detergent may be roughly independent,
her demand for two different detergents would not be. The aim of this section is to
show that many of our earlier results, generalize to this sort of environment.
In particular we extend our main model as follows. The retailer now stocks n
independent product categories, each of which contains z substitute products. Consumers wish to buy at most one product from each category. We let vi,j denote
a typical consumer’s valuation for product i in category j; each vi,j is drawn independently, using the same distribution function F (vi,j ). We also assume that all
products within a given category have the same price.14 Notice that we can interpret
n as measuring the breadth and z as measuring the depth, of the retailer’s product
selection.
Given our assumptions, a consumer who buys from category j gets a gross surplus
of max {v1,j , ..., vz,j }. This surplus has distribution function G (v; z) ≡ F z (v) and
corresponding logconcave density function g (v; z) ≡ zF z−1 (v) f (v). Without search
costs, the firm would charge a monopoly price pm (z) = arg max (p − c) [1 − G (p; z)],
and this can be shown to monotonically increase in z. Notice that in general, the
retailer will behave as if it sells n independent products, whose valuations are drawn
from G (v; z). Therefore all of our earlier analysis can be straightforwardly applied.
In particular firstly, a search-equilibrium fails to exist if the retailer stocks only one
category of products.15 Secondly however, such an equilibrium does exist, if the
retailer offers products in a sufficiently large number of categories. Thirdly, unadvertised prices exceed pm (z), increase in the search cost, but decrease in n i.e. in
14
Shankar and Bolton (2004) and Draganska and Jain (2006) note that in practice, differentiated
versions of a product or brand often have the same price.
15
This holds in a much more general setting, in which the retailer stocks one category, and
valuations (v1,1 , ..., vz,1 ) are drawn from an arbitrary distribution. This is because if consumers
expect prices pe1,1 , ..., pez,1 the firm can always charge pi,1 = pei,1 + s on each product; consumers
who search will buy precisely the product they intended to, but at a higher price. Consumers again
anticipate this hold-up problem, expect pei,1 ≥ max vi,1 − s and therefore do not search. Notice
that since we place no restrictions on the distribution of valuations, this nests for example the case
where v1,1 =, ..., = vz,1 i.e. perfect substitutes.
34
the breadth of the product range. Moreover fourthly, when the firm offers discounts
on one product category, it optimally reacts by charging less on other, unrelated,
product categories.16 Thus our main insights are robust to having substitute products.
We can also gain some new insights, into the relationship between price and
product depth. To do this we focus on the limiting case s → 0, and also assume
that the density f (v) is either increasing or not decreasing too rapidly. We can then
prove existence of a threshold n̂, such that a search-equilibrium exists if and only if
n > n̂. Moreover:
Proposition 12 There exists a critical product breadth ñ such that (i). if n ∈ (n̂, ñ)
the equilibrium price is U-shaped in product depth, and (ii). if n > ñ the equilibrium
price is strictly increasing in product depth.
Intuitively an increase in z has two opposing effects. Firstly at the level of an
individual category, it improves a consumer’s match value. This is a force towards
higher prices, since the retailer can take advantage of this. Secondly however, because category-level matches are better, ceteris paribus more consumers will search.
By the usual intuition these additional searchers have relatively low valuations, and
this is a force towards lower prices. Proposition 12 demonstrates that this second
effect dominates when n and z are small, whilst the first effect dominates otherwise.
Therefore our model offers two predictions concerning the relationship between product range and prices. Firstly, retailers with a broad product range should have low
prices. Consistent with this, Leibtag et al (2010) have UPC-level data on grocery
prices, for both traditional retailers such as supermarkets and drugstores, and also
16
Unfortunately it is unclear what would happen to prices, if the firm advertised only one product
in one category. This stems from the fact that even if n = 1 and there is no search cost, the prices
of individual products are not always positively related. Nevertheless, somewhat in the spirit of
our earlier results, Konishi and Sandfort (2002) show in a related model with substitute products,
that a firm might advertise a low price on one product, hoping that after searching, consumers
will buy a more expensive substitute instead. We also note that there are several other interesting
issues which we do not pursue here, including limited availability of advertised products (Rosato
2013), and obfuscation of some substitute products (Petrikaitė 2013).
35
for non-traditional outlets such as supercenters. The latter usually stock a broader
range of products, and also have significantly lower prices. Secondly, our model suggests that medium-sized retailers which offer little within-category variety, might
also have low prices. Consistent with this, Cleeren et al (2010) report that so-called
hard discounters stock only a limited selection of products, but also offer significantly
lower prices as compared with traditional retailers.
6.2
Downward-sloping demands
The search literature often assumes that a retailer sells only one product, but that
each consumer wishes to buy multiple units of it, i.e. consumers have a downwardsloping demand. This differs from but is clearly related to our assumption, that
consumers have a unit-demand for multiple products. We believe that our modelling
choice is more appropriate in many retail contexts. However to show how our results
fit into the wider search literature, we now allow for downward-sloping demands.
As a first step consider the following alternative model, which is a monopoly
version of Stiglitz (1979). There is a single retailer who sells one product. Consumers are identical, and faced with a price p, wish to buy 1 − F (p) units of
the product. If consumers were informed about price, the retailer 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 2 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. Indeed if this were not the case, the
retailer would be able to extract all surplus, and a search-equilibrium would fail to
exist. As explained earlier, consumer heterogeneity has at least two important implications. Firstly, prices in our model (almost always) strictly exceed pm . Secondly,
equilibrium prices respond to changes in s (and also product range and advertising
36
strategy) in a way that is both intuitive and economically interesting. This does
not happen in Stiglitz’s model - consumers are homogeneous, and equilibrium price
always equals pm .
To illustrate this point further, we now depart from Stiglitz’s approach, and allow
for heterogeneous downward-sloping 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
differentiable, 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
Z θ
m
p = argmax
g (θ) (x − c) P −1 (x − θ) dθ
x
(15)
θ
Henceforth assume that there is a search cost, that the retailer sells n ≥ 1 products,
and that (for simplicity) their prices are unadvertised. 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. Intuitively, a relatively small amount of
heterogeneity does not prevent the firm from standard monopoly pricing. Secondly
however when s > 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 charge strictly more than pm .
Proposition 13 When s ∈ (sn , sn ) there exists an equilibrium in which some consumers search and the price strictly exceeds pm . The firm charges (i). more when
search is costlier, but (ii) less when its product range is larger.
Once we introduce heterogeneity, we get results which differ from Stiglitz, but which
are essentially the same as those from our earlier unit-demand model. The intuition
37
is by now familiar. Consumers are heterogeneous, and only high-types find it worthwhile to search. The retailer therefore exploits this by charging prices which exceed
pm . However a reduction in the search cost or an expansion of the product range,
induces new lower-type consumers to search. The retailer’s demand becomes more
elastic, and so it charges lower prices. This again highlights the important role which
consumer heterogeneity plays in our results. (For completeness, when s ≥ sn the
search cost is too high to support a search-equilibrium.)
6.3
Alternative retailing strategies
Thus far we have assumed that firms use simple pricing and advertising strategies.
We now discuss whether a monopolist should use alternative strategies.
A retailer could choose to bundle its products. To maintain tractability, we ask
whether or not it should use pure bundling. Note that when a firm bundles its
P
products in this way, it effectively sells one good whose valuation is nj=1 vj ; this
means that if consumers anticipate bundling, they will not search. Now suppose
that the retailer is expected not to bundle. Consumers believe that prices will
be the same as they were in Sections 3 and 4, and therefore search provided that
Pn
e
,
0
≥ s. After consumers have sunk the search cost, the firm has
max
v
−
p
j
j
j=1
two options. Firstly it could set pi = pei on each product as expected. But secondly
it could deviate and use bundling, in which case it would choose the bundle price pb
to
max (pb − nc) Pr
pb
n
X
j=1
!
n
X
v j ≥ pb max vj − pej , 0 ≥ s
(16)
j=1
We can show that this deviation is not profitable whenever average valuations are
relatively low, and the number of products stocked is relatively high. Intuitively
under these conditions the retailer would not sell many units of the bundle, unless
it set pb very low. However in that case it would do better to just sell the products
separately, as consumers were expecting.
A firm might also charge consumers a fee τ ≥ 0 to enter its store and browse its
products. In this case it is convenient to split the search cost into two parts: a cost
s̃ ∈ (0, s) of physically travelling to the store, and a further cost s − s̃ of going inside
38
and learning prices. The timing of the game is then as follows. Firstly consumers
form an expectation of the entry fee τ e and of the price pe of each product. They
then decide whether or not to travel to the store. Secondly if a consumer goes to
the store, she learns the actual entry fee τ . At this point she can update her belief
about the price, which we denote by p̃e . She then decides whether or not to pay τ
and go inside the store. Thirdly after entering the store, a consumer observes actual
prices and makes her purchases.
Remark 14 There exists an equilibrium in which the retailer sets an entry fee
τ (= τ e ) ≥ 0, and charges pe on each product where pe solves
Zb
f (vi ) Pr
pe
n
X
j=1
!
max (vj − pe , 0) ≥ s + τ e vi dvi
!
− (pe − c) f (pe ) Pr
X
max (vj − pe , 0) ≥ s + τ e
= 0 (17)
j6=i
Proof. Consider the following strategy on the part of consumers. (a). Travel to
P
the retailer if and only if nj=1 max (vj − pe , 0) ≥ s + τ e . (b). After travelling to
the retailer, if τ ≤ τ e then set p̃e = pe , otherwise expect price to be some p̃e which
P
satisfies nj=1 max (b − p̃e , 0) ≤ s − s̃ + τ . (c). Then enter the store if expected
surplus exceeds s − s̃ + τ . Given this strategy, the retailer (weakly) prefers to set
τ = τ e . Demand for any product is exactly the same as in equation (1) except that
s is replaced by s + τ e . Equation (17) then follows.
The pricing condition (17) is the same as equation (2) except that s is replaced by
s + τ e . Intuitively this is because the entry fee acts like an extra search cost. There
is always an equilibrium where τ e = 0; however typically there exist other equilibria
with τ e > 0 as well. Nevertheless in practice, a retailer who imposes a positive
entry fee would find it hard to prevent consumers from reselling its products. This
arbitrage problem suggests that in practice, τ e = 0 is the natural outcome.
Finally, we have focused on price advertising. However an alternative promotional strategy is to send consumers a coupon, which gives them a small discount
τ 0 < s off their shopping bill. This coupon acts like a negative entry fee, so the
equilibrium price would solve equation (17) but with τ e replaced by −τ 0 . Applying
39
Proposition 4 from earlier, prices are decreasing in the size of the coupon. Therefore
in theory, coupons provide an alternative way to commit to lower prices. However
again in practice, coupons might be subject to an arbitrage problem. Our intuition is
that if the arbitrage problem could be solved, smaller retailers would be more likely
to use coupons compared to large retailers. However it is unclear (and not possible
to show analytically) whether or not coupons would dominate price advertising.
7
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 retailer’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 retailer’s overall price image. This price image then
influences consumer search, and ultimately also purchase behavior. We noted that
retailers with broader 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.
However we also proved that product depth has an ambiguous impact on prices. As
such, our model suggests that prices are lowest at stores which sell a wide range of
products, but which offer relatively little choice within individual product categories.
We also demonstrated that the breadth of a retailer’s product range, determines
how it chooses to position itself. In particular, it influences whether a firm chooses
an ‘every day low pricing’ strategy or whether it uses more of a HiLo pricing strategy.
We showed that when a monopolist has a broader product range, it is more likely
to eschew advertising. To the extent that it does advertise, it offers little or no
discount on any of its products. We also demonstrated that this result could change
in a competitive market, but only if the number of non-captive consumers was large
enough.
Usually a retailer can only inform consumers about a small number of its prices.
40
Nevertheless, we showed that price-advertising can still be informative. In contrast
to much previous work, we demonstrated that if a firm advertises a low price on
one product, this induces consumers to rationally anticipate somewhat lower prices
on other products as well. Competing firms try to prevent their rival from secondguessing their advertising strategy. This leads them to randomize over whether
to hold a promotion, which product to promote, and what price to charge for it.
Therefore an interesting feature of our model, is that it generates equilibrium price
dispersion, even for products which no firm is advertising. This again contrasts with
lots of previous work, which finds that unadvertised prices should be non-dispersed.
We believe that our paper demonstrates some of the advantages that can be
gained, by explicitly modelling the multiproduct nature of a retailer’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 retailers’ 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. We plan to think more about these and
other related issues, in future work.
41
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47
A
A.1
Main Proofs
Useful lemmas
Lemma 15 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 0 (p) /f (p), which is strictly increasing in p for all p > c. This
follows because f (p) is logconcave and therefore f 0 (p) /f (p) decreases in p.
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 0 (p̄) /f (p̄) >
0. Step 1 then implies that −2/ (p − c) − f 0 (p) /f (p) > 0 for all p ∈ [p̄, pei ).
Step 3. When pi < pei the derivative of (21) with respect to pi is
[−2f (pi ) − (pi − c) f 0 (pi )] Pr (T ≥ s)
(18)
Step 2 implies that (18) is strictly positive for every pi ∈ [p̄, pei ). However by
definition (21) is zero when pi = pei , so (21) is strictly negative for every pi ∈ [p̄, pei ).
Given the definition of (21), 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 16 (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 =
P
j6=i
max vj − pej , 0 as defined earlier.
Proof. We must show that (pi − c) Di (pi ; pe ) strictly increases in pi when pi ∈
[a, pei ), and strictly decreases in pi when pi ∈ (pei , b]. Equivalently we must check
that (21) is strictly positive when pi ∈ [a, pei ) and strictly negative when pi ∈ (pei , b].
48
Condition 1. Rewrite (21) 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)] (19)
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 0 (pi )] Pr (T ≥ s)
≤ Pr (T ≥ s) [−2f (pi ) − (pi − c) f 0 (pi )]
which is strictly negative because 1 − F (p) − (p − c) f (p) is strictly decreasing in
p by assumption.17 Since by definition the first term in (19) is zero when pi = pei ,
it must be strictly positive when pi ∈ [a, pei ) and strictly negative when pi ∈ (pei , b].
The second term of (19) is 0 when pi ≤ pei , and negative otherwise. Therefore (21) is
strictly positive when pi ∈ [a, pei ) and strictly negative when pi ∈ (pei , b], as required.
Condition 2. Note that (19) 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 )
(20)
The second term is again 0 when pi ≤ pei , and negative otherwise. The derivative of
the first term with respect to pi is
− Pr (T + max (pi − pei , 0) ≥ s)−Pr (T ≥ s)−
Di (pi ; pe ) f 0 (pi )
Di (pi ; pe )
<
−1+
≤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 0 (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 (20) is strictly
decreasing in pi , means that (21) is strictly positive when pi ∈ [a, pei ) and strictly
negative when pi ∈ [a, pei ).
Lemma 17 In a search-equilibrium all unadvertised prices are the same.
17
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 (19) is strictly decreasing in pi .
49
Proof. Suppose to the contrary that there exists an equilibrium where two different
goods g and h have prices satisfying peg < peh . Applying Lemma 15 1 − F (peh ) −
peh f (peh ) is (weakly) less than 1 − F peg − peg f peg , which is itself negative since (as
P
P
proved in the text) peg > pm . Clearly also Pr
t
≥
s
>
Pr
t
≥
s
.
j
j
j6=h
j6=g
Therefore equation (3) cannot hold for both i = g and i = h - a contradiction.
A.2
Proofs for Section 3 (no advertising)
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 =
(21)
e
,
0
. Substitute pi = pei and set (21) to zero. Lemma 16
max
v
−
p
j
j
j6=i
P
(above) provides sufficient conditions under which (pi − c) Di (pi ; pe ) is quasiconcave
in pi .
Proof of Proposition 2. Let p̃ be the unique solution to 1−F (p)−pf (p) /2 = 0,
P
n
and ñ be the smallest n such that Pr
j=2 max (vj − p̃, 0) ≥ s > 1/2. According
to Lemma 17 (above) we must 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
=0
(22)
j6=i
The lefthand side of (22) is strictly positive when evaluated at pe = pm , be
P
e
cause Di (pi = pe ; pe ) > [1 − F (pe )] Pr
max
(v
−
p
,
0)
≥
s
and 1−F (pm )−
j
j6=i
(pm − c) f (pm ) = 0. Assuming n ≥ ñ, the lefthand side of (22) is strictly negative
when evaluated at pe = p̃, because Di (pi = pe ; pe ) ≤ 1 − F (pe ) and 1 − F (p̃) −
p̃f (p̃) /2 = 0. Therefore since the lefthand side is continuous in pe , there exists (at
least one) pe ∈ (pm , p̃) which solves (22). According to Lemma 16 the profit on each
P
n
e
good is quasiconcave because Pr
max
(v
−
p
,
0)
≥
s
> 1/2.
j
j=2
Proof of Proposition 4. Equation (6) is copied here for convenience
P
n
Pr
t
≥
s
j
j=1
P
−1=0
φ (p∗ ; s, n) = 1 − F (p∗ ) − (p∗ − c) f (p∗ ) +
n
Pr
t
≥
s
j=2 j
where tj ≡ max (vj − p∗ , 0). Firstly fix n and decrease s from s0 to s1 < s0 . Let
p∗s0 denote the lowest solution to φ (p∗ ; s0 , n) = 0. Lemma 18 (below) shows that
50
φ (p∗ ; s, n) increases in s, so we conclude that φ p∗s0 ; s1 , n ≤ 0. Since φ (pm ; s1 , n) >
0 and φ (p∗ ; s1 , n) is continuous in p∗ , this implies that when s = s1 the lowest
equilibrium price is (weakly) below p∗s0 . So the lowest equilibrium price increases in
s. Secondly fix s and increase n from n0 to n1 . Let p∗n0 be the lowest solution to
φ (p∗ ; s, n0 ). Lemma 19 (below) shows that φ (p∗ ; s, n) decreases in n, so we conclude
that φ p∗n0 ; s, n1 ≤ 0. This again implies that when n = n1 , the lowest equilibrium
price is (weakly) below p∗n0 , so the lowest equilibrium price falls in n.
Lemma 18 φ (p∗ ; s, n) increases in s.
Define ṽj = vj − p∗ , and note that ṽj has a logconcave
h i
density function f˜ (ṽj ) defined on the interval ã, b̃ where ã = a − p∗ and b̃ = b − p∗ .
Proof of Lemma 18.
Recall that the ṽj are iid and that tj ≡ max (ṽj , 0).
. P
P
n
n
∗
Pr
Define Ω (s, n) = Pr
j=2 tj ≥ s . φ (p ; s, n) increases in s if
j=1 tj ≥ s
and only if Ω (s, n) increases in s. We now prove that Ω (s, n) increases in s.
Begin with n = 2 Consider Pr (t1 + t2 ≥ x) for some x > 0. If ṽ1 ≥ x then
t1 + t2 ≥ x since t2 ≥ 0; if ṽ1 ∈ (0, x) then t1 + t2 ≥ x if and only if t2 = ṽ2 ≥ x − ṽ1 ;
if ṽ1 ≤ 0 then t1 + t2 ≥ x if and only if t2 = ṽ2 ≥ x. Therefore
Rx
Pr (ṽ1 ≤ 0) Pr (ṽ2 ≥ x) + 0 f˜ (z) Pr (ṽ2 ≥ x − z) dz + Pr (ṽ1 ≥ x)
Pr (t1 + t2 ≥ x)
=
(23)
Pr (t2 ≥ x)
Pr (ṽ2 ≥ x)
Z x
Pr (ṽ2 ≥ x − z)
= Pr (ṽ1 ≤ 0) +
f˜ (z)
dz + 1
Pr (ṽ2 ≥ x)
0
which increases in x. This is because ṽ2 is logconcave and so has an increasing
hazard rate, which means that Pr (ṽ2 ≥ x − z) / Pr (ṽ2 ≥ x) increases in x.
Now proceed by induction We show that if Ω (w, n − 1) increases in w, then
Ω (s, n) increases in s. To prove this, let k > 1 and write:
P
P
P
n
n
n
Pr
t
≥
s
Pr
t
≥
s
−
k
Pr
t
≥
s
j=1 j
j=2 j
j=1 j
P
−k =
P
Ω (s, n) − k =
n
n
Pr
Pr
j=2 tj ≥ s
j=2 tj ≥ s
(24)
Then using the same principles used to derive equation (23), expand only the top of
51
equation (24) to get
P

n−1
≥ s − k Pr
t
≥
s
j
j=2

P
Ω (s, n) − k = Pr (ṽn ≤ 0) 
n
Pr
j=2 tj ≥ s
h
i
R b̃
Rs
Pn−1
Pn−1
˜
f˜ (z) [1 − k] dz
f
(z)
Pr
t
≥
s
−
z
−
k
Pr
t
≥
s
−
z
dz
+
j
j
j=1
j=2
s
0
+
Pn
Pr
j=2 tj ≥ s

Pr
P
n−1
j=1 tj
(25)
Since t1 and tn are iid, the first term in (25) simplifies to Pr (ṽn ≤ 0) [1 − k/Ω (s, n − 1)],
which is weakly increasing in s because of the inductive assumption that Ω (w, n − 1)
increases in w. The second term in (25) is proportional to
"
!
!#
Z b̃
Z s
n−1
n−1
X
X
˜
f˜ (z) [1 − k] dz
dz +
f (z) Pr
tj ≥ s − z − k Pr
tj ≥ s − z
0
j=1
s
j=2
Using the change of variables y = s − z, this can be rewritten as
"
!
!#
Z 0
Z s
n−1
n−1
X
X
f˜ (s − y) [1 − k] dy +
f˜ (s − y) Pr
tj ≥ y − k Pr
tj ≥ y
dy
s−b̃
0
j=1
j=2
and then written more compactly as
Z ∞
10≤s−y≤b̃ f˜ (s − y) Γ (y) dy
(26)
−∞
where 10≤s−y≤b̃ is an indicator function taking value 1 when 0 ≤ s − y ≤ b̃, and 0
otherwise;
 and where

1−k
if y ≤ 0
P
P
Γ (y) =
n−1
n−1
 Pr
if y > 0
j=2 tj ≥ y
j=1 tj ≥ y − k Pr
P
n−1
Now let S be an interval in R++ such that Pr
t
≥
s
> 0∀s ∈ S, and
j=1 j
choose s0 , s1 ∈ S where s1 > s0 . Also choose a constant k0 such that (26) is zero
when evaluated at s = s0 and k = k0 . (Zero terms being discarded) Γ (y) has one
sign-change from negative to positive: this follows directly from the definition of
k0 (> 1) and the inductive assumption that Ω (w, n − 1) increases in w.
Using the definition in Karlin (1968) §1.1, 10≤s−y≤b̃ is totally positive of order 2
(T P2 ) in (s, y). Since f˜ (z) is logconcave, f˜ (s − y) is also T P2 in (s, y). Therefore
10≤s−y≤b̃ f˜ (s − y) is also T P2 in (s, y).
52
Applying Karlin’s Variation Diminishing Property (Karlin §1.3, Theorem 3.1),
(26) changes sign once in s (and from negative to positive) when it is evaluated at
k = k0 . Since by assumption (26) is zero when evaluated at s = s0 and k = k0 , it is
positive when evaluated at s = s1 and k = k0 . Therefore returning to equation (25)
it follows that Ω (s1 , n) − k0 ≥ Ω (s0 , n) − k0 , or equivalently that Ω (s, n) increases
in s. In summary we have shown that if Ω (w, n − 1) increases in w, then Ω (s, n)
increases in s. Since Ω (w, n) increases in w for n = 2, Ω (s, n) increases in s for
any integer-valued n.
Lemma 19 φ (p∗ ; s, n) decreases in n.
Proof of Lemma 19.
φ (p∗ ; s, n) decreases in n if and only if Ω (s, n) decreases
in n. To show that Ω (s, n) decreases in n, write out Ω (s, n) as
P
P
Rs
n−1
n−1
t
≥
s
t
≥
s
−
ṽ
dṽ
+
Pr
(ṽ
≤
0)
Pr
Pr (ṽn ≥ s) + 0 f˜ (ṽn ) Pr
j
j
n
n
n
j=1
j=1
P
P
Rs
n−1
n−1
t
≥
s
t
≥
s
−
ṽ
dṽ
+
Pr
(ṽ
≤
0)
Pr
Pr (ṽn ≥ s) + 0 f˜ (ṽn ) Pr
j
j
n
n
n
j=2
j=2
This is less than Ω (s, n − 1) because Pr
P
n−1
j=1 tj
≥x
.
Pr
P
n−1
j=2 tj
≥ x exceeds
1 and (from Lemma 18) is increasing in x. Therefore we know that Ω (s, n) ≤
Ω (s, n − 1) for any n, or alternatively Ω (s, n) decreases in n.
A.3
Proof of Lemma 5 (effect of advertising on unadvertised
prices)
Proof. Let p0 be the equilibrium unadvertised price. Let tj = max (vj − p0 , 0) for
j < n, and tn = max (vn − pan , 0). Then using equation (6) again, p0 satisfies
P
n−1
Pr
t
+
t
≥
s
n
j=1 j
P
−1=0
Φ (p0 ; pan ) = 1 − F (p0 ) − (p0 − c) f (p0 ) +
n−1
Pr
j=2 tj + tn ≥ s
We focus on the smallest p0 which solves Φ (p0 ; pan ) = 0. Note that Φ (p0 ; pan ) is
continuous in p0 , and that Φ (pm ; pan ) ≥ 0. Therefore as with comparative statP
. P
n
n
ics in s and n, if Pr
Pr
increases in pan , the lowj=1 tj ≥ s
j=2 tj ≥ s
est p0 (that solves Φ (p0 ; pan ) = 0) also increases in pan . To prove that ω (pan ) =
53
Pr
P
n
j=1 tj
≥s
.
Pr
P
n
j=2 tj
ω (pan ) − k =
Pr
≥ s increases in pan , write:
P
P
n
≥ s − k Pr
t
≥
s
j=2 j
P
n
Pr
j=2 tj ≥ s
n
j=1 tj
(27)
Just as in the proof of Lemma 18, the numerator of (27) can be rewritten as
"
!
!#
Z pan
n−1
n−1
X
X
f (vn ) Pr
tj ≥ s − k Pr
tj ≥ s
dvn
a
j=1
Z
pa
n +s
"
f (vn ) Pr
+
pa
n
j=2
n−1
X
!
tj ≥ s + pan − vn
n−1
X
− k Pr
j=1
!#
tj ≥ s + pan − vn
dvn
j=2
Z
b
f (vn ) [1 − k] dvn (28)
+
pa
n +s
Using the change of variables y = pan − z (28) can be written as
Z ∞
1a≤pan −y≤b f (pan − y) γ (y) dy
(29)
−∞


1−k
if



P
P
n−1
n−1
where γ (y) =
if
Pr
j=2 tj ≥ s + y
j=1 tj ≥ s + y − k Pr



 Pr Pn−1 t ≥ s − k Pr Pn−1 t ≥ s
if
j=2 j
j=1 j
a
and 1a≤pan −y≤b is again an indicator function that is T P2 in (pn , y).
y ≤ −s
y ∈ (−s, 0)
y≥0
Now choose
a,1
a,0
a,1
pa,0
n , pn ∈ [a − s, b] such that pn < pn . Also define k0 such that (29) is zero when
evaluated at pan = pa,0
n and k = k0 . (Zero terms being discarded) γ (y) is piecewise
continuous and changes sign once from negative to positive: this follows from the
P
. P
n−1
n−1
definition of k0 (> 1), and because Pr
t
≥
x
Pr
t
≥
x
increases
j
j
j=2
j=1
in x (from Lemma 18). Karlin’s variation diminishing property says that (29) is
a,0
single-crossing from negative to positive in pan . Therefore ω (pa,1
n ) − k0 ≥ ω (pn ) − k0 ,
or equivalently ω (pan ) increases in pan as required.
A.4
Proof of Proposition 9 (competition)
Proof. First modify the timing of the game as follows. Stage 1: firms randomly
choose a product to advertise and pick an advertised price. Stage 2: Nature decides
whether a firm can advertise or not (advertising happens with probability 1 − α, and
is chosen independently for the two firms). If a firm advertises, the advert contains
54
the price chosen at stage 1. If a firm does not advertise, the price from stage 1
is abandoned and never observed by anybody else. Stage 3: adverts are observed
by all parties, unadvertised prices are chosen, then search and purchase decisions
taken. We first prove that advertised prices in Stage 1 are chosen using a mixed
strategy. We then prove that during Stage 2, a firm is indifferent about advertising
and therefore happy to (let Nature) randomize.
Suppose the two retailers have chosen advertised prices qA and qB where qA , qB <
pm . Suppose also that during Stage 2 Nature decides that firm i can advertise.
Provided s is sufficiently small to permit multistop shopping, firm i’s profit is

γ


−κ
+
+
α
(1
−
γ)
π (qi ) + Z + 1−α
(1 − γ) π (qi ) if qi < q−i

2
2

y
1−γ
Πi (qi , q−i ) −κ + γ2 + α (1 − γ) π (qi ) + Z + 1−α
π (qi )
if qi = q−i
2
2



 −κ + γ + α (1 − γ) π (q ) + Z
if q > q
i
2
i
−i
(30)
where
Z=
1−α
(1 − γ) (qi − c) [1 − F (qi + s)]
2
(31)
Equation (30) follows directly from Lemma 8. Firstly i earns π (qi ) on its loyal
consumers, who have mass γ/2. Secondly i also earns π (qi ) on non-loyal consumers,
in the event that rival firm −i does not advertise. Thirdly, the Z term arises when
the retailers advertise different products, which happens with probability (1 − α) /2.
In this case non-loyal consumers cherry-pick, so firm i sells them its advertised good.
Fourthly with probability (1 − α) /2 the two firms advertise the same product. Nonloyal consumers are all captured by the firm with the lowest advertised price, or
shared if advertised prices are tied.
Suppose instead that during Stage 2 Nature decides that firm i cannot advertise.
i’s profits are then
Πni =
γ + α (1 − γ) u
π
2
(32)
because it charges pu and earns π u on its loyal consumers, and also on half of the
non-loyal consumers if the rival −i has also decided not to advertise.
During the first stage, retailer i chooses qi to maximize its expected profits
αΠni + (1 − α) Πyi . Equivalently since Πni is independent of qi , the retailer chooses
qi to maximize Πyi . The latter is discontinuous and so a pure strategy equilibrium
55
does not exist. We now use Dasgupta and Maskin (1986) to prove that qi and q−i
are chosen using a mixed strategy.
Let q be the solution to π (q) = 0 and for simplicity assume it is unique (this
can be easily relaxed). Define each firm’s action space as q, pm . Notice that
Πyi (qi , q−i ) is continuous except on a measure-zero subset of the action space, namely
where qi = q−i . Moreover whilst individual payoffs are discontinuous, Πyi (qi , q−i ) +
Πy−i (qi , q−i ) is continuous and therefore also upper semi-continuous. In addition
Πyi (qi , q−i ) is bounded and satisfies lim inf q̃i →qi− Πyi (q̃i , qj ) ≥ Πyi (qi , qj ). This means
that Πyi (qi , q−i ) is weakly lower semi-continuous in the sense described by Dasgupta
and Maskin (1986). Consequently applying Theorem 5 of their paper, there exists
a mixed-strategy Nash equilibrium. Since payoff functions are symmetric, applying
Lemma 7 of their paper, there exists a symmetric mixed strategy Nash equilibrium.
The lower support of this mixed strategy is above q. This is because if firm i
advertises a price qi < q, it must be true that qi < c in which case Πyi (qi , q−i ) < 0.
However the firm can earn strictly more than this by not advertising at all.
The upper support of this mixed strategy is strictly below pm . Firstly suppose
that (30) gives i’s profits for all qi , q−i , not just for qi , q−i < pm . It is straightforward
to argue that if qi ≥ pm then (i). each line of (30) is decreasing in qi , and (ii). the
first line strictly exceeds the second which strictly exceeds the third. Therefore it is
never optimal for i to advertise a price qi ≥ pm since by reducing it his profits must
strictly increase. Hence the mixed strategy that we have derived must have firms
randomizing over advertised prices which are strictly below pm .
Secondly, in fact if firm −i randomizes over prices strictly below pm and firm i
deviates to a price above pm , the third line of (30) provides an upper bound on i’s
profits. Hence firm i has no incentive to make such a deviation. We now prove that
profits are bounded in this way. Suppose firm i advertises a price qi ≥ pm . It is simple
to show that if either (1). firm −i does not advertise or (2). both firms advertise
the same product, then there is an equilibrium in which firm i charges φ (qi ) on its
unadvertised product. (3). Suppose instead that the two firms advertise different
products. (i). If qi > φ (q−i ) there is an equilibrium in which each firm charges
φ (.) on its unadvertised product, and firm i makes no sales to non-loyals. (ii). If
56
qi < φ (q−i ) − s there is an equilibrium in which non-loyal consumers who multi-stop
shop all start by searching firm i, and each firm charges φ (.) on its unadvertised
product. (iii). If qi ∈ [φ (q−i ) − s, φ (q−i )] there is an equilibrium in which i charges
φ (qi ) on its unadvertised product and sells it only to loyals, and firm −i charges
a price on [qi , φ (q−i )) on its unadvertised product. Firm i therefore sells less units
of its advertised product to loyals, than it would as given by Z in equation (31).
Bringing these three steps together, we conclude that i’s deviation profit is less than
the third line of (30), as originally stated.
It only remains to check that firms are willing to randomize during Stage 2.
This is easily done by noting that for any symmetric mixed strategy equilibrium,
conditional on advertising both firms earn the same expected payoff, and that this
payoff shifts one-for-one in κ (in other words, κ does not affect the distribution of
prices). Therefore for any α and associated mixed strategy equilibrium, we can find
a κ such that Πyi (qi , q−i ) = Πni .
57
B
B.1
Additional proofs
Proof of Lemma 10 (many-product competition)
Proof. Let pij be the price charged by firm i on product j, and let peij be consumers’
expectation of it. Also let Tj denote the expected surplus from shopping only at
store j, and TAB denote the expected surplus from shopping at both firms. Suppose
without loss of generality that A advertises good 1 at a price paA1 , and B advertises
good 2 at a price paB2 ∈ [paA1 , pm ). If firm B is expected to charge φ (paB2 ) on its
unadvertised products, only loyal consumers will buy them. These loyal consumers
search if and only if TB ≥ s. Therefore using our monopoly analysis, it is clear
that firm B cannot do better than charge φ (paB2 ) as expected. Now consider firm
A’s pricing problem. According to Lemma 10, A is more expensive on product 2
and therefore for sufficiently small s, will sell it only to loyal consumers. Loyals
search if and only if TA ≥ s and hence peA2 is determined exactly as in the monopoly
model. Retailer A sells its other unadvertised products to both loyal and non-loyal
consumers. Demand for a typical product j > 2 is
Z b
hγ
i
DAj =
f (vj )
Pr (TA ≥ s) + (1 − γ) Pr (max {TA , TAB } ≥ s) dvj
2
pAj
(33)
whenever pAj ≤ peBj . The derivative of profit with respect to the actual price pAj ,
evaluated at pAj = peAj , is
hγ
i
DAj |pAj =pe − peAj − c f peAj
Pr TA ≥ s|vj = peAj
Aj
2
− peAj − c f peAj (1 − γ) Pr max {TA , TAB } ≥ s|vj = peAj (34)
There are two possibilities to consider. Firstly equation (34) may be zero for some
peAj < peBj . In this case, peAj is the equilibrium price. Secondly (34) may be strictly
positive for all peAj < peBj . In this case the equilibrium price is peAj = peBj . This is
explained as follows. Some non-loyal consumers multistop shop because they like
both advertised products. If firm A charges more than peBj , all these multistop
shoppers will immediately buy product j from firm B. Hence a small price increase
triggers a discrete drop in demand. We can show that this makes it unprofitable
for A to charge more than peBj , whenever the number of products sold is sufficiently
58
large.18 Moreover when s is sufficiently small, it is straightforward to prove that
peA2 < φ (paB2 ).
B.2
Proof of Proposition 12 (substitute products)
We start with some preliminary results. Adapting equation (12) on page 25, the
first order condition is
1 − G (pe ; z) − (pe − c) g (pe ; z) 1 − G (pe ; z)n−1 = 0
Using the definition of G (v; z) this can be rewritten as
1 − F (pe )z
1
e
e
z−1 − (p − c) f (p ) = 0
z(n−1)
e
e
zF (p )
1 − F (p )
(35)
Lemma 20 A search-equilibrium exists if and only if n > n̂, where n̂ solves 1 −
z (n̂ − 1) (b − c) f (b) = 0. This equilibrium is unique, and its price decreases in n.
Proof. The first part of (35) decreases in pe whenever n ≥ 2, whilst the second
part of (35) decreases so long as f (v) is not decreasing too rapidly. Therefore if
an equilibrium exists it must be unique. Moreover since the lefthand side of (35) is
strictly positive when pe = pm (z), an equilibrium exists if and only the lefthand side
of (35) is negative when pe = b. Applying L’hôpital’s rule, the latter holds if and
only if 1 − z (n − 1) (b − c) f (b) < 0. Since the lefthand side of (35) is decreasing in
n, so must be the equilibrium price.
Lemma 21 There exists a rotation point F̄ such that after a small increase in z,
the lefthand side of (35) increases if F (pe ) < F̄ and decreases for F (pe ) > F̄ .
Moreover this rotation point F̄ is increasing in both n and z, and tending to 1 as
either n → ∞ or z → ∞.
18
It is straightforward to show that firm A’s profits on each product j > 2 are quasiconcave,
under very similar conditions to the monopoly model. To check that firm A’s profits on product 2
are also quasiconcave, we must verify that the retailer does not wish to set a price pA2 ≤ paB2 + s
and prevent non-loyals from travelling on to the rival. This is shown to be true if either (i). the
fraction of non-loyals is sufficiently small, or (ii). non-loyals do not check pA2 because they believe
they can get the product much more cheaply at the other store. Similarly retailer B’s profits
are shown to be quasiconcave if either (i). the fraction of non-loyals is sufficiently small, or (ii).
non-loyals who shop there, only check the price of the advertised special which they intend to buy.
59
Proof. Differentiating the lefthand side of (35) gives an expression which for all
F (pe ) ∈ (0, 1) is proportional to
−
−1 + F (pe )z + F (pe )z(n−1) − F (pe )zn
−1 + nF
(pe )z(n−1)
− (n − 1)F
(pe )zn
!
+ z ln F (pe )
(36)
Step 1. Note that (36) is strictly positive as F (pe ) → 0, and by L’hôpital’s rule
is strictly negative as F (pe ) → 1. We can also prove that (36) strictly decreases
in F (pe ). (The algebra is omitted and is available on request.) This immediately
implies that (36) is strictly positive for F (pe ) < F̄ and strictly negative for F (pe ) >
F̄ . This in turn implies existence of a unique rotation point.
Step 2. Since (36) is strictly decreasing in F (pe ), the rotation point F̄ increases in
n and z if and only if (36) increases in n and z. This is again easily shown to hold.
To prove that limn→∞ F̄ = limz→∞ F̄ = 1 simply take limits of (36) then set each
expression to zero and solve for F̄ .
Lemma 22 There exists a critical ñ (> n̂) such that the equilibrium price is above
F̄ if n < ñ, and below F̄ if n > ñ.
Proof. When n = n̂ the equilibrium price equals b whilst the rotation point is
strictly below b. Fixing z, as n increases the equilibrium price decreases (Lemma
20) whilst the rotation point increases (Lemma 21). Moreover as n → ∞ the rotation
point converges to F̄ = 1 whilst the equilibrium price converges to pm (z) < b. Thus
there is a unique ñ such that the equilibrium price p∗ satisfies F (p∗ ) = F̄ . For n 6= ñ
the relationship between p∗ and F̄ is as described in the lemma.
Proof of Proposition 12.
Note that if we substitute F (pe ) = F̄ into equation
(35), the resulting expression is decreasing in z. (1). Suppose to begin with n ∈
(n̂, ñ). According to Lemma 22 the equilibrium price exceeds the rotation point, so
by Lemma 21 the equilibrium price decreases in z. Since ñ depends positively on z,
eventually the equilibrium price p∗ satisfies F (p∗ ) = F̄ . Notice that for higher z, the
lefthand side of (35) is negative when evaluated at the rotation point; equivalently,
F (p∗ ) < F̄ . Thus by Lemma 21 the equilibrium price now increases monotonically
in z. (2). Suppose that to begin with n > ñ. Then a similar argument shows that
the equilibrium price increases monotonically in z.
60
B.3
Proofs of Proposition 13 (downward-sloping demand)
Define Q (z) ≡ P −1 (z) and note that Q0 , Q00 < 0. Also define Π (p; θ) = (p − c) Q (p − θ),
p∗ (θ) = arg maxx (x − c) Q (x − θ), and let CS (pj ; θ) be the consumer surplus of a
θ-type on one product when its price is pj .
Lemma 23 p∗ (θ) and CS (p∗ (θ) ; θ) are both strictly increasing in θ.
Proof. Differentiate (p − c) Q (p − θ) with respect to p to get a first order condition
Q (p − θ) + (p − c) Q0 (p − θ) = 0. Then totally differentiate with respect to θ to get
∂p∗ (θ)
Q0 (p∗ (θ) − θ) + (p∗ (θ) − c) Q00 (p∗ (θ) − θ)
=
∈ (0, 1)
∂θ
2Q0 (p∗ (θ) − θ) + (p∗ (θ) − c) Q00 (p∗ (θ) − θ)
Since ∂p∗ (θ)/ ∂θ ∈ (0, 1) and P 0 < 0 and (by definition) p∗ (θ) = θ + P (q ∗ (θ)), it
follows that ∂q ∗ (θ)/ ∂θ > 0. We can write consumer surplus as
Z
∗
q ∗ (θ)
Z
∗
q ∗ (θ)
[pj − p (θ)] dqj =
CS (p (θ) ; θ) =
[P (qj ) − P (q ∗ (θ))] dqj
0
0
which is strictly increasing in θ because P 0 < 0 and ∂q ∗ (θ)/ ∂θ > 0.
In equilibrium each good will have the same price - let p be the actual price
and pe the expected price. Consumers search if and only if θ ≥ θ̃ where θ̃ as
the minimum θ ∈ θ, θ such that n × CS (pe ; θ) ≥ s. Profit on good j is therefore
Rθ
g (θ) {(p − c) Q (p − θ)} dθ. Differentiating with respect to p and imposing p = pe ,
θ̃
we find that pe satisfies19
Z
θ
0
e
Z
g (θ) Π (p ; θ) dθ =
θ̃
θ
g (θ) {Q (pe − θ) + (pe − c) Q0 (pe − θ)} dθ = 0
(37)
θ̃
Since ∂Π0 (pe ; θ) /∂θ > 0 equation (37) implies Π0 pe ; θ̃ < 0, which further implies
Rθ
Rθ
that Π0 (pe ; θ) < 0∀θ ≤ θ̃. Consequently θ̃ g (θ) Π0 (pe ; θ) dθ ≥ θ g (θ) Π0 (pe ; θ) dθ.
Lemma 24 In any equilibrium with trade, pe ≥ pm .
Rθ
Proof. Proceed by contradiction and suppose that pe < pm . θ g (θ) Π0 (pe ; θ) dθ >
Rθ
0 because θ g (θ) Π (p; θ) dθ is concave and maximized at p = pm . This implies that
Rθ
Rθ
0
e
g
(θ)
Π
(p
;
θ)
dθ
>
0
because
(from
the
previous
paragraph)
g (θ) Π0 (pe ; θ) dθ ≥
θ̃
θ̃
19
Since Q0 , Q00 < 0 profit is quasiconcave in p provided that types are not too different.
61
Rθ
θ
g (θ) Π0 (pe ; θ) dθ. However this contradicts (37) and thus the supposition that
pe < pm .
Define sn = n × CS (pm ; θ) and sn = n × CS p∗ θ ; θ and note that sn < sn .20
It follows immediately that when s ≤ sn there is an equilibrium where the retailer
charges pm on every product.
Proof of Proposition 13.
Existence is proved as follows. Firstly since s > s0
then θ̃ > θ. Adapting the proof of Lemma 24 the lefthand side of equation (37) is
strictly positive when evaluated at pe = pm . Secondly since s < s0 consumers with
θ = θ search, and by continuity so do nearby types. Adapting the proof of Lemma
24 we can also conclude that Π0 p∗ θ ; θ < 0 for all θ < θ. Therefore the lefthand
side of equation (37) is strictly negative when evaluated at pe = p∗ θ . Thirdly
putting these two facts together, and using the fact that the lefthand side of (37)
is continuous in pe , there exists at least one pe ∈ pm , p∗ θ such that (37) holds.
Lemma 24 already shows that pe ≥ pm .
Now suppose that the number of products stocked increases from n0 to n1 (the
proof for an increase in s is virtually identical and is omitted). Let pe0 be the lowest
equilibrium price when n = n0 , and θ̃0,0 be the lowest type that searches when
n = n0 and price pe0 is expected. Let θ̃0,1 < θ̃0,0 be the lowest type who searches
when n = n1 but price pe0 is still expected. Adapting earlier arguments we know
that a). θ̃0,0 > θ and b). Π0 (pe0 ; θ) < 0 for all θ < θ̃0,0 . Therefore the lefthand side
of (37) when evaluated at pe = pe0 but n = n1 is
Z
θ
0
g (θ) Π
θ̃0,1
(pe0 ; θ) dθ
Z
θ
<
θ̃0,0
g (θ) Π0 (pe0 ; θ) dθ = 0
i.e. strictly negative. At the same time (37) is still weakly positive when evaluated
at pe = pm and n = n1 . Therefore (by continuity) the lowest solution to (37) when
n = n1 , must be strictly lower than the lowest solution when n = n0 .
It is straightforward to show that pm > p∗ (θ) which then implies CS (p∗ (θ) ; θ) > CS (pm ; θ).
Since CS p∗ θ ; θ exceeds CS (p∗ (θ) ; θ) by Lemma 23, sn exceeds sn .
20
62