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Comparative Advertising: The role of prices
Stuart Baumann∗
University of Edinburgh
June 19, 2017
Abstract
In markets where firms sell similar goods to their competitors, firms may be able
to free-ride off the costly price signalling of competitor firms by engaging in price
comparative advertising. As the goods are similar, consumers can reason that if one
good is high quality (revealed through price signalling) then so is the other. This
paper models this phenomenon and finds that in equilibrium there will be firms
price signalling as well as freeriding firms that signal through price comparative
advertising. Welfare is strictly higher in markets where advertising firms are active
relative to pure price signalling markets. In some cases advertising markets can be
even more efficient than full information markets as advertisers surrender market
power to avoid costly price signalling.
JEL Codes: D43, D82, D83, M37
Keywords: Comparative advertising, Price Signalling
∗For useful discussions I would like to thank Philipp Kircher, Ludo Visschers, Jan Eeckhout, Margaryta
Klymak, R´egis Renault, Sandro Shelegia, Alessandro Spiganti and Ina Taneva. The author was visiting
at Universitat Pompeu Fabra for part of the writing of this article. This work was supported by the
ESRC postgraduate funding scheme. Notwithstanding any advice I have received any errors here are my
own.
1
1 Introduction
In many markets firms have more information regarding the quality of their goods than
potential consumers. As a result firms with high quality goods can signal the true quality
of their goods to consumers with prices and warranties being common instruments for
this. In many of these markets some of this uncertainty is common to goods offered
by different firms. An example is cable television where two providers may offer sets of
channels with substantial overlap. An alternate example is package holidays where the
utility from holidays to the same location from different but similar tour companies are
likely to be similar. In instances like this it may be possible for firms to earn greater
profits from free-riding on the costly signalling of other firms.
This paper investigates the potential for firms to engage in this kind of signal free-riding
through comparative advertising. Comparative advertising occurs when a firm advertises
by contrasting the price and features of its good as compared to those of rival firms. In the
previous economic literature comparative advertising has been seen as directly informing
consumers of the difference in vertical quality between two goods (Barigozzi et al. 2009)
or of the difference in horizontal (seller specific) match utility a consumer would get if
he bought a good from a competing firm as compared to buying from the advertising
firm (Anderson & Renault 2009). The role of the disclosure of the prices of rivals in this
context has received less attention.1
There are many examples of comparative advertising however where the disclosure
of price information is a major part of the message. A basic example is offered in the
advertisements of Progressive Direct, an American auto insurance company that gives
prospective consumers the prices offered by competitors for comparable insurance plans
(Yu 2013). They also air advertisements promising “we compare our direct rates side
by side to find you a great deal, even if its not with us”. Three, a major UK phone
carrier, similarly advertisers with a webpage that asks shoppers on their website to “see
how our prices compare” with a price comparison of Three against the prices of all of the
1To the best of my knowledge this is the first paper to examine the informational content of prices
as a component of comparative advertising or the firm strategy of providing competitor prices to their
consumers more generally.
2
other major phone carriers in the market (Three Mobile 2016). The online travel agent
Skyscanner allows visitors to automatically replicate their flight searches on competing
services such as Expedia and eDreams. Another example is provided by Book Depository,
a leading UK online bookseller (Charlton 2009) which for every product sold presented a
link to the corresponding Amazon page for that item.2
This paper examines a vertically differentiated market in which consumers cannot
directly observe the quality of goods. In such a market when high quality firms have
higher costs, there is a literature that shows these firms can separate themselves from
low quality firms by proposing a high price, often above the monopoly price they would
charge if their quality were known. This is called in the literature “price signalling” as
the price transmits information about the underlying quality of the good. The problem
for high quality firms in such markets is that the high price that enables signalling may
result in lower profits than would be available by pricing at the (lower) monopoly price in
a full information setting. In this setting we show that price comparative advertising has
a clear role: By showing an identical product from a rival firm at a high price, a firm can
signal to its customers that its product must also be of high quality, even if the firm itself
does not charge such a high price. In doing so a firm may be able to increases profits by
pricing closer to its monopoly price and freeriding on the rival firm’s price signal.3
Clearly the logic of this argument does not immediately carry over to equilibrium
behaviour. If all firms revert to pricing at the lower level rather than price signalling,
free-riding possibilities may be lost. One of the questions in this paper is whether this
simple logic extends in some form to an equilibrium setting. Indeed, only some insights of
the previous setting carry over once equilibrium forces come into play. In equilibrium there
will be firms that engage in price comparative advertising and firms that remain pricing
2A further example is provided by Amazon itself which has a marketplace which allows competitors to
compete against Amazon on Amazon’s website. Amazon also receives a portion of the revenues from these
external sellers on their website. Whilst these revenues no doubt play some role in Amazon’s decision to
operate this marketplace it is also true that this allows Amazon customers to compare Amazon’s prices
with other vendors.
3The analogue in the traditional Spence (1973) signalling model of the labour market is that an
uneducated worker goes to a firm with his educated identical twin. As both individuals are identical and
this is observable the firm will reason that if one sibling is educated and has a high capability then so
does the other.
3
at the signalling level. The advertising firms face increased price competition from the
firms they advertise against and thus many will price at levels below the monopoly price.
Thus while in normal price signalling markets asymmetric information has the effect of
increasing prices, the equilibrium exhibiting price comparative advertising will have some
firms pricing above and some below the monopoly level. Total welfare is improved by
price comparative advertising relative to the asymmetric information equilibrium where
no advertising is allowed. Firms do not earn higher profits however due to this additional
price competition and the additional surplus is all accrued by consumers through lower
prices. In some cases the welfare of an asymmetric information market with price com-
parative advertising can be greater than under full information as firms surrender market
power in order to achieve more efficient signalling.
A number of extensions of the basic model are examined. When there is heterogeneity
in marginal costs among high quality firms I find the intuitive result that it will be the
lower cost firms that will be pricing lower while engaging in price comparative advertising.
This result suggest another avenue through which price comparative advertising increases
welfare as high price (and higher cost) advertisers lose sale quantity to low price (and
lower cost) advertisers. An alternate extension is the special case where all high quality
firms source their goods from a monopoly supplier.4Here it was found that the possibility
of advertising can induce the monopoly supplier to reduce the price they charge reselling
firms in the hope of increasing the quantity they sell. These extensions all point to
increased welfare from the use of price comparative advertising. These results on welfare
are considerably more positive for comparative advertising than the previous literature
that examined the comparative advertising of product attributes and found that total
welfare could be decreased when there was a large vertical quality difference between rival
firms (Anderson & Renault 2009).
Comparative advertising used to be relatively uncommon in developed countries but
was legalised in the United States and Europe in 1979 (Federal Trade Commission 1979)
and 1997 (Council of European Union 1997) respectively. Elsewhere in the world however
4For instance there is a monopoly supplier of Samsung smartphones and Iron Maiden albums.
4
comparative advertising is still subject to restrictions. In China and Hong Kong it is
banned while in Japan it is allowed but seldom used due to it being perceived as impolite
by Japanese consumers (Singh 2014). While comparative advertising has recently been
legalised in Turkey (G¨urkaynak et al. 2015) it remains banned in Saudi Arabia and Kuwait.
In other countries such as Qatar the situation surrounding comparative advertising is
simply ambiguous with no regulations or case law to determine its legality (Bradley 2014).
The policy implications of this paper are clear: comparative advertising delivers lower
prices to consumers, is welfare improving and should be supported and encouraged by
governments and regulators.
This paper first provides an outline of the surrounding literature in section 2. The
model is then presented in section 3 with welfare implications examined in section 4 and
model extensions in section 5 before section 6 concludes.
2 Literature Review
This paper has strong links to two key literatures: the price signalling literature and the
literature on comparative advertising.5In addition this paper is related to the marketing
literature on reference pricing as well as recent work on search deterrence.
An early paper to examine the economic consequences of consumers judging quality
by price is that of Scitovszky (1944). Since that time, the advent of signalling theory has
led to a number of papers applying price signalling to markets with imperfect information
regarding product quality. A common mechanism for signalling is to have higher quality
firms producing at a higher marginal cost than lower quality firms (Wolinsky 1983, Bagwell
& Riordan 1991, Daughety & Reinganum 2008). In this way high quality firms have a
higher optimal price than low quality firms which makes it comparatively less expensive
for them to charge a high price, thus allowing signalling. This paper will use this feature of
marginal cost increasing in product quality to model a market exhibiting price signalling.
Moving onto the comparative advertising literature, Barigozzi et al. (2009) argue that
5The comparative advertising literature can further be considered a branch of the informative adver-
tising literature. See Renault (2016) for a survey.
5
firms engage in comparative advertising as a means of signalling the vertical quality of
their good. By directly informing consumers of the vertical quality of their good compared
to a competitor’s good the firm opens itself up to litigation expenses if claims made about
the comparison of goods are unreasonable. Only a firm with a high quality good would
engage in this practice as low quality firms would face a high expected loss from litigation.
In this way comparative advertising serves as a signal for good quality as well as a vehicle
for direct disclosure of quality. While they do not consider the impact of comparative
advertising on total welfare they conclude that it should be supported by regulators as it
allows easier market access for new entrants. The authors note that this mechanism for
comparative advertising is reliant on an effective court system which suggests comparative
advertising would be less effective in countries with weaker institutions.
Another comparative advertising paper is that of Anderson & Renault (2009) who
examine a duopolistic market where each firm sells a horizontally and vertically differen-
tiated good. Vertical quality is readily apparent to consumers and so advertising focuses
on horizontal differentiation rather than signalling vertical quality. Firms can engage
in advertising by disclosing the match utility a consumer would get with their firm or
comparative advertising by disclosing the match utility a consumer would get from both
firms. They find that in cases where there is a small difference in vertical quality both
firms will engage in advertising of their own match utility to benefit from the higher price
generated by additional product differentiation. When there is a large quality differential
however weaker firms will generally be the firms using comparative advertising to disclose
both matches in order to increase their demand. They find in this setting that consumers
and lower quality firms are better off when comparative advertising is allowed. So much
damage is done to the higher quality firm profits however that total welfare falls as a result
of comparative advertising. Related ideas are examined by Koessler & Renault (2012)
and Celik (2014) who look at the conditions under which disclosure of quality attributes
will occur.
This paper is different from the previous literature on comparative advertising in
that it is the disclosure of prices of competitor firms rather than the direct disclosure
6
of horizontal or vertical product attributes that is important. Indeed to highlight this
channel, the model I present will exhibit firms advertising against competing firms offering
identical products with no horizontal differentiation existing.6Before introducing this
model however I will briefly mention two other literatures that are related to this paper.
The reference pricing strand of the marketing literature is related to the notion of price
signalling. An example of reference pricing is a $200 price crossed out and replaced with
“$100 for a limited time only!”. The idea is that this $200 is suggestive of the quality of the
good however the key problem with this strategy is credibility (Grewal & Compeau 2002,
Compeau et al. 2002, Kan et al. 2013). Consumers will often doubt that the reference
price is ever charged or is representative of the quality of the item. While this paper uses
an economic signalling approach, its message might otherwise be motivated by external
offers providing credible reference prices to consumers.
Finally the mechanism described by this paper represents an interesting contrast to
the mechanism described by Armstrong & Zhou (2015) in a paper on search deterrence.
That paper has a model where a shopper is presented with a good of known quality but
has an unknown outside option. It is shown that where possible a seller can increase
profits by committing to an exploding offer where the consumer has to buy before seeing
the outside option. This is in contrast to the current paper where the additional external
information can efficiently signal the quality of the current good and thereby increase
seller profits. When the good is high quality, this provision of information is good for the
selling firm as they can freeride on the price signalling of other firms.
3 The Model
There is a high quality good and a low quality good in the market each of which is sold by
a unit mass of firms. The quality of a good is observable to firms but not to consumers.
Firms selling low quality goods have a marginal cost of cLwhile firms selling high quality
6Of course in the real world firms may advertise against competing products that are differentiated
but share some common value component. In this case it is likely that comparative advertising plays the
price-comparative role outlined in this paper along other roles of comparative advertising explored in the
literature.
7
goods have a higher marginal cost of cHwith cH> cL. A fraction λof firms sell high
quality goods.
There is a unit mass of consumers, each of whom gets a utility equal to Q−Pffor
buying good Q∈ {H, L}from firm fat price Pf. Consumers have a heterogeneous outside
option denoted by Ωkfor customer k. This outside option is logconcave distributed with
a pdf denoted γ(Ω) and cdf denoted Γ(Ω).
The timing of this singleshot game is as follows. A firm is approached by one random
consumer. The firm offers that consumer a price and the consumer can either buy at
that price or leave the firm in favour of their outside option. Denoting the consumer’s
perceived quality level by ˆ
Q, the condition for a consumer to buy is:
ˆ
Q−Pf≥Ωk(1)
Thus the maximum Ωkconsumer that will buy the good will have an Ωkvalue of ˆ
Q−Pf
and hence from the firm’s perspective the probability of a sale is given by Γ( ˆ
Q−Pf).
Signalling equilibria are refined by the intuitive criterion (Cho & Kreps 1987). Beliefs
are formalised by a function µ(P) that gives the believed probability of a good being
high quality given a price of P. Finally as this paper examines the potential use of price
comparative advertising7as a signalling tool alongside price signalling, I will consider only
fully separating equilibria.
3.1 Separating Equilibrium without price comparative advertis-
ing
In this section the fully separating equilibrium for this basic model lacking price compar-
ative advertising will be examined. First defining the equilibrium concept:
Definition 1 PBNE (without advertising). A pure strategy Perfect Bayesian Nash
Equilibrium (PBNE) in this model without advertising will be described by low and high
7To be defined in terms of the model in section 3.2.
8
firm pricing strategies PL,PHas well as a belief function µ(P), such that:
πL(PL)≥πL(P)∀P∈ <+(2)
πH(PH)≥πH(P)∀P∈ <+(3)
and the belief function µ(P)is derived in accordance with bayes rule and player strategies
for all prices charged with positive probability in equilibrium.
As in a standard price signalling separating equilibrium we get the result that firms
selling low quality goods will price at their monopoly price (to be denoted PL) which
maximises their profit:
PL= arg max
P∈<+
(P−cL)Γ(L−P) (4)
With the corresponding profit being denoted πM
L= (PL−cL)Γ(L−PL). The prices that
high quality firms can charge to differentiate themselves from the low quality firms are
those prices P∗that satisfy:
πM
L≥(P∗−cL)Γ(H−P∗) (5)
The price which makes this expression bind with equality (The Riley (1979) price) is
denoted by PS. With the intuitive criterion applied beliefs will satisfy µ(PS) = 1. A
natural way to extend these beliefs over all prices is thus:
µ(P) =
1P≥PS
0P < P S
(6)
9
In any fully seperating equilibrium, high quality firms will charge the maximum of PSor
the high firm’s monopoly price (which will be denoted PM
H) which maximises their profit
in the absence of asymmetric information:
PM
H= arg max
P∈<+
(P−cH)Γ(H−P) (7)
With the corresponding profit being denoted πM
H= (PM
H−cH)Γ(H−PM
H). Henceforth
we assume that this market is one where price signalling is costly and hence PM
H< P S.
The profit of the high firm when signalling is denoted by πS
Hand can be expressed as:
πS
H= (PS−cH)Γ(H−PS)< πM
H(8)
To restrict attention to separating equilibria we assume that high firms prefer signalling
to being mistaken for low firms. This condition is
πS
H>(P−cH)Γ(L−P)∀P∈ <+(9)
Finally we can show that PM
H> PLby first obtaining an expression for the optimal
monopolist price from first order conditions of π(P)=(P−cQ)Γ(Q−P). This results in:
P=cQ+Γ(Q−P)
γ(Q−P)(10)
As Γ(x) is logconcave there is a well-defined solution to this problem and Γ(x)
γ(x)is a mono-
tonically increasing function (Bagnoli & Bergstrom 2005). From this we can see that the
solution price is increasing in cost and in perceived quality. Hence we will always have
PM
H> PL.
Proposition 1. Without the possibility of advertising, a PBNE exists where Land H
firms price at PLand PSrespectively and consumer belief formation is as per equation 6.
Proof. Equations 4, 5 and 9 ensure that high and low firms cannot get higher profits from
deviating. The belief function described in equation 6 is consistent with this equilibrium.
10
Figure 1: Profits in the no advertising fully separating equilibrium
The prices charged and profits earned by each firm in equilibrium along with the high
firm full information profits (that are unattainable under asymmetric information) are
shown in figure 1.
3.2 Separating Equilibrium with price comparative advertising
In this section firms are allowed to engage in price comparative advertising at no cost.
As motivation for this, consider that a high quality firm deviates from the no advertising
signalling equilibrium described in the preceding section. They charge the monopolist
price PM
H. At this price however the consumer cannot tell if the good is of high or low
quality. The firm can show the offer of another firm offering this same (high quality)
good. This other price will be PSas all other firms selling the high quality good charge
this price. The consumer knows that both goods are the same and so will be convinced
11
that the good is high quality after seeing this other price. The firm will earn πM
H> πS
H
and hence this is a profitable deviation. This section considers how the possibility of this
profitable deviation changes the equilibrium outlined in proposition 1.
It is assumed that when the consumer enters the market and meets a firm, that firm
can costlessly show the consumer an offer from a competing firm that sells the same
good.8As both offers concern the same good if one is high quality then so is the other.
Henceforth the term advertiser will be used to describe a firm that provides an external
offer. Firms that do not provide an external offer will be referred to as non-advertisers.
The assumed timing is that each firm decides on their own price and whether or not
to show a competitors price simultaneously. A key assumption is made that advertising
is undirected - firms cannot condition their advertising on the price offered by the firm
they advertise against. This assumption can be simply motivated by considering firms to
commit separately and simultaneously to their strategy of prices and advertising.9
For simplicity an advertising firm only shows one price from an external firm. A
consumer observes a price from the external firm but not the advertising strategy of this
external firm. A consumer at this point can decide to buy from either the advertising firm
or the external firm (at no extra cost). The tie-breaking rule is adopted that when both
firms price at an equal amount is that it is assumed that the consumer will buy from each
with 50% probability.
The beliefs of a consumer who has seen one price will be represented by the function
µ(P) while the beliefs of consumer who has been shown two prices will be given by a
function taking two arguments, µ(P, PE). The convention will be maintained that the
first price is that of the advertising firm and the second price is the price from an external
firm. This paper will examine a fully separating mixed pricing PBNE. The equilibrium
price domain of type Q∈ {L, H}firms with strategy s∈ {A, N }for advertising and
8There is an assumption here prohibiting a high/low firm from showing an offer from a low/high firm.
This can be justified by the presence of many goods in the real world (while in the model there is one
high and one low quality good). Thus a consumer shown an unrelated good cannot use this information
to infer information about the good at hand and there is no incentive for firms to show these unrelated
goods to the consumer.
9This assumption of undirected price comparison seems to be a good representation of progressive
direct; three; skyscanner and book depository from the introduction.
12
non-advertising will be denoted by the set DQ,s.
Definition 2 PBNE (with advertising). A fully separating PBNE in this model with
advertising will be described by pricing strategies and equilibrium profits denoted ˆπL,ˆπH,
such that:
πH,N (P) = ˆπH≥πH,N (P0)∀P∈DH,N , P 0∈ <+\DH,N (11)
πH,A(P) = ˆπH≥πH,A(P0)∀P∈DH,A , P 0∈ <+\DH,A (12)
πL,N (P) = ˆπL≥πL,N (P0)∀P∈DL,N , P 0∈ <+\DL,N (13)
πL,A(P) = ˆπL≥πL,A(P0)∀P∈DL,A, P 0∈ <+\DL,A (14)
The belief functions µ(P), µ(P, PE)are in accordance with the intuitive criterion, Bayes
rule and player strategies for all information sets reached with positive probability in equi-
librium.
The logic behind simple price signalling for a firm that does not advertise still applies
and hence µ(P) will be as set in equation 6. Beliefs when advertising is undertaken shall
be in accordance with Bayes rule at all points within the domain of prices charged in
equilibrium. Thus:
µ(PA, PE) = λProb.(PA, PE|Good is H)
λProb.(PA, PE|Good is H) + (1 −λ)Prob.(PA, PE|Good is L) (15)
As we are restricting attention to fully separating equilibria we restriction attention
to the case where there is no price point PS> P > PLthat is charged by a positive mass
of high firms and low firms.
Lemma 1. In any fully separating equilibrium, no low firms offer a price P > PL.
Proof. We can first show there will be no advertisers pricing above PL. Consider a putative
equilibrium where there were advertisers pricing above PLwith price dispersion. The
highest pricing advertiser would make no sales and be better off monopolising at PL.
Consider a putative equilibria where there are low quality advertisers at a masspoint
above PL. As there are no high firms at this price (by the restriction to fully separating
13
equilibria), these firms will be seen as being of low quality. Hence beliefs cannot worsen
from undercutting and there is a profitable deviation for a firm to undercut this masspoint.
For low firms that do not advertise, as pricing at PLdominates pricing higher given the
beliefs in equation 6 there will be no low firms pricing above PLeither.
This lemma makes it possible for a high firm to signal high quality by advertising
whilst pricing at less than PSbut more than PL. This leads to the first proposition which
states that in any equilibrium there will exist advertising firms.
Proposition 2. In any equilibrium there will be a positive mass of high firm advertisers.
Proof. In the event of all mass of high firms being non-advertisers at PS, consider a high
firm’s option of deviating to price at PM
Hand advertising against another firm selling the
same good. If the customer accepted this firm was high quality the deviating firm would
be better off. By contrast a low firm attempting to emulate high quality by doing the same
would be worse off even if they were believed to be high quality as they would lose the
sale to another low firm (that would be pricing at PL). Thus this is a profitable deviation
as the high firm could use the intuitive criterion to price closer to their monopoly price
whilst convincing consumers of their high quality. As this profitable deviation remains
whilst there is a zero measure of advertisers we get the proposition.
One implication of this proposition is that the equilibrium described in Proposition
1 is no longer an equilibrium where advertising is allowed. Whilst beliefs of µ(P, PE) =
0∀P, PEcould support this equilibrium such beliefs would not be supported by the intu-
itive criterion.
Lemma 2. At all prices P > PL, the equilibrium price distributions of high quality
advertisers is atomless.
Proof. If there were an atom at a price exceeding marginal cost then one of those firms
could undercut the others. With a similar intuitive criterion argument as presented in
the proof of proposition 2, the undercutting firm could convince consumers of their high
quality and hence could get a discontinuous jump in expected profits. If there were an
14
atom at a price equal to marginal cost (Bertrand 1883) then profits are zero and the firms
at this price would be better off monopolising to earn πS
H.
This lemma is similar in spirit to Varian (1980, Proposition 3) or Stahl (1989, Lemma
1) and reflects the fact that if there were a mass of firms offering a certain price then one of
those firms could get a discontinuous jump in expected profits by undercutting the others.
Note that non-advertisering firms are monopolists and hence there is no similar restriction
on mass points in the pricing distribution of these firms. This lemma is truncated to prices
above PLto reflect the possibility of adverse beliefs for advertisers pricing at or below PL
which would prevent undercutting.
We have now established that no low firms will price above PL. The next two lemmas
show that all low firms will in fact not be advertising and will be charging a price of PL.
Lemma 3. If the equilibrium price distribution of high firm advertisers is atomless at
prices P≤PLthen in equilibrium there will be a mass of non-advertising low firms
offering a price of PL.
Proof. For low firms that do not advertise, Pricing at PLdominates any other price.
If all firms were advertising with price dispersion the firm with the highest price would
get no profit and hence would be better off not advertising whilst setting a price of PL.
If all low firms were advertising at a certain price less than PLthen from equation 15
(and the lack of high firm masspoints below PL) they would be considered low quality
and would be better off not advertising at a price of PL.
Lemma 4. If the equilibrium price distribution of high firm advertisers is atomless at
prices P≤PLthen in equilibrium all low firm will be non-advertisers at a price of PL.
Proof. Lemma 1 shows that no low firm will price at P > PLand not-advertising at PL
dominates not-advertising at any other price. Thus it suffices to show no low firms will
advertise at any price P≤PL. Considering the possibility of advertising at a price P≤PL
the possible price distributions are a continuous distribution of prices, a masspoint of low
firm advertisers at certain discrete prices or a distribution with both of these features.
We shall show that these distributions are not sustainable in equilibrium.
15
We can first show that no low firm advertiser masspoints can survive in equilibrium
at prices less than PL. If there were an advertiser masspoint at a price P < PLthen from
equation 15 (and the lack of high firm masspoints below PL) they would be considered low
quality and would earn strictly higher profits undercutting other firms in this masspoint.
Thus no masspoints of low firm advertisers can survive in equilibrium.
Now consider a putative equilibria where there is no atom of high firm advertisers
at PLbut a mass of low firm non-advertisers at PL(as per lemma 3) and a distribution
of low firm advertisers on the domain [ ¯
P , P ] with PL≥¯
P > P . The total mass of low
firm advertisers is some number 0 <∇<1. Consider the advertiser pricing at ¯
P. It
matches with a firm pricing at PLwith probability (1 − ∇) and matches with another
(lower priced) advertiser with probability ∇. When PA=¯
P,PE=PLare subbed into
the beliefs equation 15 the numerator of this expression P(¯
P , PL|Good is H) will be zero
as there is no mass of high firms charging PL. The denominator is nonzero as there is a
mass of low firms offering this price. Thus the advertiser at ¯
Pis recognised as being low
quality and earns:
πL,A(¯
P) = (1 − ∇)¯
PΓ(L−¯
P)< πL,M (PL) (16)
Thus the advertiser would be strictly better off being a non-advertiser at PL.
In addition note that due to the assumed tiebreaking rule an advertiser at PLis worse
off than a non-advertiser at the same price. Thus if the equilibrium price distribution of
high firm advertisers is atomless at a price P≤PLthere can be no low firm advertisers
in equilibrium.
We have now established that if the equilibrium price distribution of high firm adver-
tisers is atomless at a price P≤PLthen all low firms will not advertise while setting a
price of PL. We can now shift attention to examining the behaviour of high firms. As-
suming an atomless high firm pricing distribution, As no low firm will ever price at a price
other than PLhigh firms will be able to use the intuitive criterion to price at any level
that represents a profitable deviation for them and still be recognised as high quality. We
16
can thus write the profit function for high firm advertisers when the price distribution of
high firm advertisers is atomless:
πH, Advertiser(P)=(P−cH)Γ(H−P) [(1 −η) + ηG(P) + ηG(P)] (17)
Where G(P) is the survival function of the prices charged by high quality advertisers and
ηis the proportion of high quality advertisers. The terms in the square brackets account
for the probabilities of matching with a non-advertiser, matching with an advertiser and
an external advertiser matching with the firm respectively.
Now denoting g(P) to be the pdf of the advertiser price distribution (which is de-
fined when the advertiser pricing distribution is atomless) we can show that the pricing
distribution of high firms will be gapless:
Lemma 5. If the equilibrium price distribution of high firm advertisers is atomless then
there are no equilibria exhibiting gaps of positive measure in the price distribution of
advertising high firms. That is for any P,P+with > 0and with g(P)>0,g(P+)>0
then RP+
g(p)dp > 0.
Proof. Note from Lemma 4 that atomlessness in equilibrium implies low firms never ad-
vertise and an advertiser will always be recognised as high quality. Consider the case
if such a gap did form between prices Pand P+with > 0. Consider in particular
the firm pricing at P. This firm could increase its price to P+which would increase
(P−cH)Γ(H−P) whilst not changing [(1 −η) + ηG(P) + ηG(P)]. Thus there is a prof-
itable deviation.
We can now find that there will be high firm non-advertisers and in equilibrium all
high firms will earn the same profits they would make price signalling at PS. Intuitively
this occurs because the possibility of not advertising while charging PSputs a lower bound
on how much Bertrand competition among advertisers can reduce their profits.
Lemma 6. If the equilibrium price distribution of high firm advertisers is atomless then
there is a positive mass of high firms selling at the signalling price PSand not providing
additional offers to consumers.
17
Proof. If all mass of high firms were advertising in an atomless distribution then the top
pricing firm would make no profits and be strictly better off offering a price of PSwithout
advertising.
Corollary 7. If the equilibrium price distribution of high firm advertisers is atomless then
in equilibrium all high firms earn profits of πS
H.
Proof. Immediate from lemma 6, proposition 2 and definition 2.
We can now use the profit function from equation 17 and the equilibrium profit from
Corollary 7 to find the domain of high firm advertiser pricing and the proportion of
advertisers.
Lemma 8. If the equilibrium price distribution of high firm advertisers is atomless, The
bottom pricing advertiser will charge PBwhere PBis the solution to:
(PB−cH)Γ(H−PB) = πS
H
1 + η(18)
Proof. Substituting equilibrium profit and that for the bottom pricing firm G(PB)=1
into equation 17 yields this expression.
Lemma 9. If the equilibrium price distribution of high firm advertisers is atomless then
the top pricing advertiser will charge PM
H.
Proof. This comes from equation 17. In the event the top pricing firm had a price less
than PM
Hthey could raise their price and increase (P−cH)Γ(H−P) whilst the fraction
lost to other firms [(1 −η)+2ηG(P)] stayed the same.
Lemma 10. If the equilibrium price distribution of high firms is atomless, the proportion
of advertising firms in equilibrium is:
η= 1 −πS
H
πM
H
(19)
Proof. To see this consider equation 17 for the advertiser offering a price of PM
H. As
G(PM
H) = 0, their profit is πM
H(1 −η) which from lemma 7 must be equal to πS
Hin
18
expectation for a firm to price at this level. The lemma follows immediately from this
equality.
Now now split the analysis into two different cases. The first is where PB> PLand
hence the atomlessness of the high firm advertiser pricing distribution is assured from
lemma 2. The second is the complementary case where atomlessness is not assured.
3.2.1 Case A: PB> PL
From Lemma 8 all high firm advertisers will price at least at PB> PL. Thus from lemma
2 the advertiser price distribution will be atomless and all of the preceding results are
applicable.
Proposition 3. There will exist a PBNE for this game. All low firms will price at PLand
earn πL= (PL−cL)Γ(L−PL)whilst all high firms will earn πS
Has defined by equation 8.
A proportion ηas defined by equation 19 of high firms will advertise with a pdf of prices
as given by:
g(P) = πS
H
2ηΓ(H−P)−(P−cH)γ(H−P)
(P−cH)2Γ(H−P)2PB≤P≤PM
H(20)
A proportion 1−ηof high firms will not advertise and will set a price of PS.
Beliefs in this PBNE will satisfy:
µ(P) =
1P≥PS
0P < P S
µ(P, PE) =
1for P > PLand ∀PE
0for P≤PLand ∀PE
Proof. This pricing distribution can be obtained by noting that in equilibrium the price
19
distribution G(P) must satisfy:
(P−cH)Γ(H−P) [1 −η+ 2ηG(P)] = πS
H
[1 −η+ 2ηG(P)] = πS
H
(P−cH)Γ(H−P)
G(P) = 1
2ηπS
H
(P−cH)Γ(H−P)−1 + ηPB≤P≤PM
H
(21)
The associated pdf can be found by differentiating the survival function and multiplying
by negative one. Equation 21 is a valid survival function being decreasing in price with
endpoints of G(PM
H) = 0 and G(PB) = 1, thus this price distribution is feasible.
These beliefs will be satisfied in this equilibria as all high firms price at more than PL
and all low firms price at PL. There is no profitable deviations for high firms who earn πS
H
at any point in the pricing domain and cannot earn higher profits outside this domain.
Likewise low firms cannot profitably deviate.
From its construction we can note that the equilibrium pricing distribution in propo-
sition 3 is unique in the class of fully separating equilibria.10 This equilibrium can be
seen in figure 2 where the advertising support line shows the prices and corresponding
sale quantities of advertising firms. At prices close to PH
Mthe advertising support line sits
beneath the high firm demand curve, Γ(H−P), as they lose a share of their customers to
the firms they advertise against. At lower prices it sits above the non-advertiser demand
curve as they retain most of their customers and also gain customers from competing ad-
vertising firms. While the advertisers at PM
Hretain a proportion of 1 −ηof the firms they
encounter, those pricing closer to cHretain or win a greater total proportion of 1+ηfirms.
This demonstrates the demand shifting taking place with the lowest pricing advertisers
selling (1+η)Γ(H−PB)
(1−η)Γ(H−PM
H)times more goods than the highest pricing advertiser. All advertisers
earn the same equilibrium profits however with expected profit the same as for the price
signalling non-advertiser firms.
10Although this equilibrium pricing distribution could be sustained by different out of equilibrium
beliefs.
20
Figure 2: Equilibrium with advertising
3.2.2 Case B: PB≤PL
Considering the case when PB< PLis more problematic as lemma 2 cannot be used to
obtain atomlessness of the pricing distribution. The equilibrium explored in this section
will be one where the price distribution of high firms is guessed to be atomless at all
prices (this guess will be verified later on). As a result of this however no claim can be
made as to the uniqueness of the resulting equilibrium among the class of fully separating
equilibria. Note that this assumption of atomlessness at any price is sufficient such that
all of the previous supporting lemmas hold. In particular this conjecture results in an
equilibrium closely related to the equilibrium expressed in proposition 3. This equilibrium
will have the same functions to describe firm pricing decisions however PLis cut out of
this distribution
Proposition 4. There will exist a PBNE for this game. All low firms will price at PL
21
and earn πL= (PL−cL)Γ(L−PL)whilst all high firms will earn πS
Has defined by equation
8. A proportion ηas defined by equation 8 of high firms will advertise with a pdf of prices
given by:
g(P) =
πS
H
2ηhΓ(H−P)−(P−cH)γ(H−P)
(P−cH)2Γ(H−P)2ifor PB≤P≤PM
H, P 6=PL
0for P=PL
(22)
A proportion 1−ηof high firms will not advertise and will set a price of PS.
Beliefs in this PBNE will satisfy:
µ(P) =
1P≥PS
0P < P S
(23)
µ(P, PE) =
1for P > PLand ∀PE
1for P≤PLand PE6=PL
0for P≤PLand PE=PL
(24)
Proof. The proof of this proposition largely follows that of proposition 3. The key differ-
ence is the omission of PL(a zero measure set) from the advertiser pricing function. The
pricing distribution is atomless (which verifies the atomless guess) and all of the previous
lemmas hold. Thus low firms will all not advertise while setting a price of PLas per
lemma 4 and beliefs for advertisers are the same as in proposition 3 for any set price.
While this is an equilibrium with beliefs that are robust to the intuitive criterion it
relies on discontinuous beliefs at a certain point. This may be less credible as a model
for some markets than the previous case where PB> PL. For instance this equilibrium
may not be robust if the marginal cost of low firms is continuously heterogeneous in some
interval as there would then be a continuum of PLprices.
22
4 Producer and Consumer Surplus
The total surplus generated in a separating equilibria when a consumer visits a firm is
Q−cQif that consumers buys the good and is that consumer’s outside option otherwise.
The problem for the consumer on the other hand is to choose from the maximum of Q−Pf
and Ωk. Clearly this implies that the closer is price to marginal cost the greater surplus
generated in the market.
In order to evaluate the impact of advertising we can note that the low firms do not
change their price from the full information case or the no advertising case (presented
in proposition 1) and so welfare is unchanged for consumers visiting low firms. When
consumers visiting high firms are considered, equilibrium advertiser prices are less than
both the full information monopoly price or the signalling price. Specifically we can write
the following expressions for the surplus generated from consumers visiting high firms in
the full information (FI) equilibrium, pure price signalling (PS) equilibrium as well as the
surplus from a single advertiser (SA) and surplus from the advertising equilibrium (AE)
respectively:
SFI = (H−cH)Γ(H−PM
H) + Z¯
Ω
H−PM
H
Ωγ(Ω)dΩ (25)
SPS = (H−cH)Γ(H−PS) + Z¯
Ω
H−PS
Ωγ(Ω)dΩ (26)
SSA = (1 −η)ZPM
H
PB"(H−cH)Γ(H−P) + Z¯
Ω
H−P
Ωγ(Ω)dΩ#g(P)dP (27)
+ηZPM
H
PB"(H−cH)Γ(H−P) + Z¯
Ω
H−P
Ωγ(Ω)dΩ#¯g(P)dP
SAE = (1 −η)SPS +ηSSA (28)
Where the distribution ¯g(P) is the pdf of the first order statistic from two draws from
the advertiser pricing distribution. This arises when a consumer has prices from two
advertisers and will pick the lower price.
Here it can be seen that the surplus generated by a single advertiser is higher than the
surplus generated by a price signalling firm or a firm in the full information equilibrium
23
as the price an advertiser offers is always lower. This implies that when the fraction of ad-
vertisers is sufficiently high, it is possible for the advertising equilibrium to deliver greater
surplus than the corresponding full information equilibrium. As the above expressions
are analytically intractable this is shown numerically but first stating these implications
in a proposition.
Proposition 5. In any asymmetric information market, surplus is always weakly greater
in the fully separating equilibrium exhibiting advertising relative to the fully separating
equilibrium where no advertising is allowed. In some cases surplus can be higher in asym-
metric information markets with advertising than in the corresponding full information
markets.
Proof. Proof for the first statement is provided by the fact consumers always choose the
maximum of Ωkand H−Pf, whilst surplus is maximised by them taking the maximum
of Ωkand H−cH. The price distribution in the price signalling case weakly stochastically
dominates the price distribution where advertising occurs and hence delivers weakly lesser
surplus (strictly if πS
H< πM
H). The proof of the second statement is by example 1.
Example 1. We define a uniform distribution of outside options in the space [0,1]. Thus
we have Γ(x) = xin this domain. We assume that H= 1,cL= 0 and L=8
17 .11 We can
use these to get the following expressions for monopoly prices and profits in terms of cH:
PL=4
17 PM
H=1 + cH
2PS=16
17
πL=16
289 πM
H=1−cH
22
πS
H=1
17(16
17 −cH)
From these prices and profits expressions for the proportion of advertisers and the bottom
11The values given H,cLare chosen so that the demand curve is downward sloping across all feasible
prices. The value given Lis chosen as it is approximately halfway up the interval and exploits the
pythagorean triple (8,15,17) to get a rational signalling price. In general a higher Lvalue leads to full
information delivering higher surplus for all cHwhilst lower Lvalues lead to the advertising equilibrium
being more efficient for all cH. Figures 1 and 2 were constructed with these parameters and cH= 0.4.
24
price can be obtained:
η= 1 −πS
H
πM
H
PB=1 + cH
2−q(1 −cH)2−4πS
H
1+η
2
Finally we can write the survival function and pdf of the advertiser pricing distribution
as well as an expression for ¯g(P):12
G(P) = πS
H
2η(P−cH)(1 −P)+1
2−1
2ηg(P) = πS
H(1 + cH−2P)
2η(P−cH)2(1 −P)2
¯g(P)=2G(P)g(P)
Now examining the bounds of feasible cHvalues we focus on the case of a separating
equilibria with costly signalling. Hence the maximum cHvalue we consider is 15
17 as at
this cost level the signalling price is equal to the monopoly price for the high firm. The
lower limit cHvalue is zero as by construction it is at a marginal cost of cLwhen a firm
is indifferent to selling with an expected value of Lat the low monopoly price or Hat the
signalling price.
It can be seen in figure 3 that when cHis low it implies that πM
His high compared to
πSand thus there are many advertisers in the market. Intense competition between these
firms depresses prices closer to marginal cost. This results in surplus in the advertising
equilibrium being higher than the full information equilibrium.
On the other hand when cHis high πM
His not much higher than πSand thus there are
fewer advertisers in the market with less intense competition between them. In this case
surplus is higher in the full information equilibrium than the advertising equilibrium.
12For discussion on finding the pdf of an order statistic see for instance Blitzstein & Hwang (2015,
Theorem 8.6.4)
25
Figure 3: Surplus generated in each equilibrium case
5 Extensions
5.1 Cost heterogeneity in high firms
As noted in Shelegia (2012) even small differences in marginal cost can lead to firms
randomising over different ranges of prices in a mixed strategy pricing equilibria such as
those presented in this paper. This observation may have welfare implications in this
paper as advertising results in a shift of quantity from higher pricing advertisers to lower
price advertisers. If low cost firms are also low price firms this means that advertising can
boost aggregate surplus by awarding larger quantity to lower cost firms.
This section examines this possibility by augmenting the model of section 3 with high
firms with heterogeneous costs. Rather than having a homogeneous group of high firms
we assume half are αfirms with a marginal cost of cαand half are βfirms with a higher
marginal cost of cβ> cα. Both types of firms sell high quality goods and a firm cannot
selectively advertise against one of the two classes of firms. In all other aspects the model
is unchanged. As the analysis for the most part follows that performed in section 3 it is
deferred for appendix A but two key implications are here stated.
26
Proposition 6. In any fully separating equilibrium exhibiting advertising:
(a) All αfirms will price equal or less than all βfirms.
(b) βfirms will never earn more than their signalling profits however αfirms may earn
more.
Proof. See appendix A.
The intuition for the point (a) is that αfirms have a lower monopoly price which means
price signalling is relatively more expensive for them as compared to βfirms. This leads
them to be more likely to advertise and more likely to offer lower prices while advertising.
For a simple example of point (b) consider a case where cαis low enough such that the
monopoly price of αfirms, PM
α, is less than the marginal cost of βfirms, cβ. In this case
an αfirm could price at PM
αand advertise losing at most half of their consumers to other
αfirms and still gain some customers from advertising against βfirms. In some cases this
fraction of the monopoly profit will exceed the signalling profit.
Thus in the presence of cost heterogeneity, advertising can add to market efficiency as
it shifts demand away from higher cost firms towards lower cost firms. Firms with lower
costs will position themselves as lower pricing firms within the advertising equilibrium.
Thus there is an efficiency gain from demand being shifted towards firms with a lower
marginal cost.
5.2 Monopolist supplier for high quality good
Now we consider the special case where the common good sold in the market is provided
by a single supplier firm that behaves as a monopolist and produces the product costlessly.
This will often be the case where the product is copyrighted or patented. We will refer to
these supplying firms as suppliers and the firms that buy from the suppliers as merchants.
We assume full information exists between suppliers and merchants but consumers do not
know the quality level of a good unless it is signalled to them.
In analysing this case we will first write an expression for the cHlevel which equalises
the monopoly price of high firms (which does depend on cH) with the signalling price
27
(which does not depend on cH). This cost is denoted cSig and is defined as the cost level
which solves the following equation.
PS= arg max
P∈<+
(P−cSig)Γ(H−P) (29)
where PSis as defined in section 3.1. We can now write the profit function for the
supplier as a function of the price they charge merchants. In this expression we will write
the advertiser price pdf as gc(P), the bottom price as PB(c) and the monopoly price as
PM
H(c) reflecting the fact that these are affected by c:
πSupplier(c) =
cΓ(H−PM
H(c)) c>cSig
cΓ(H−PS)c≤cSig,without advertising
ch(1 −η)Γ(H−PS) + ηRPM
H(c)
PB(c)Γ(H−P)gc(P)dP ic≤cSig,with advertising
(30)
The benchmark price signalling model without advertising is first considered. We get
the result that suppliers will always price at least cSig such that PM
H≥PS.
Proposition 7. In markets where advertising is not allowed the price charged by suppliers
to merchants will never be less than cSig
Proof. Given that merchants will need to charge at least the signalling price to ensure
low quality firms will not emulate high quality, all firms will price at PSfor all levels
of cbelow a critical level to be denoted cSig. As the price charged by merchants is the
minimum of the signalling and their monopoly price, this critical level cSig is such that
these prices are equal as defined in equation 29.
Therefore in the absence of advertising a supplier will never charge less than cSig as
they would sell the same quantity of Γ(H−PS) whilst earning a lower price.
The intuition here is that the supplier profit strictly increases as c increases until c
reaches cSig. This is because merchants charge PSat all of these cost levels and hence as
the supplier increases cthe price the supplier receives increases whilst the quantity stays
28
constant.
If advertising is allowed when c≥cSig then PS≤PM
Hand hence no advertising will be
undertaken.13 On the other hand if advertising is allowed when c < cSig then PS> P M
H
and hence advertising will be undertaken. This observation raises the following possibility:
Proposition 8. In some markets where advertising is allowed it may be optimal for
suppliers to reduce their price below cSig and thus induce advertising to increase their sale
quantity.
Proof. Proof is by example 2.
Example 2. This example follows on from example 1 except for cHnow being endoge-
nously determined by a monopolist supplier who produces the good costlessly. All other
parameters and the outside option distribution are identical. We can note that in this case
setting PM
H=PSobtains cSig =15
17 :
The demand curve faced by the monopolist supplier in the advertising and no adver-
tising case along with areas representing optimal profits are shown in figure 4.14 It can
be seen that where no advertising is allowed the monopolist supplier prices at cSig. When
advertising is allowed however a lower cof approximately 0.41 is set as the greater sales
volume obtainable is sufficient to make up for the loss in margin.
6 Conclusion
This paper has examined comparative advertising from the perspective of disclosing dif-
ferences in prices. While at first glance this strategy seems counter-intuitive as it would
result in greater pricing competition it is found that it can act as an alternative method
of signalling quality than price signalling.
In the fully separating equilibria that this paper presents some firms will remain price
signalling whilst other firms will lower their price closer to the monopoly level and instead
13From equation 29 we have πS
H=πM
Hwhen c=cSig. As cSig increases above this level the signalling
profit drops below the monopoly profit. Hence there is no incentive to drop price and advertise.
14Note that equations for these demand curves are as described in equation 30 (once the cterm giving
the margin per item is removed)
29
Figure 4: Monopolist supplier demand curves and profits
signal by price comparative advertising against rival firms. While an advertiser may be
able to achieve greater sales quantity by reducing their price, they also face competition
from the firm they advertise against. Advertising has the effect of decreasing the equilib-
rium distribution of prices offered in the asymmetric information setting which increases
consumer surplus. Firm profits on the other hand do not change from the case where no
advertising is allowed. While firms do manage to price closer to their monopoly price any
additional profits are lost through increased competition with other advertising firms.
A number of extensions were examined including the possibility of high quality firms
having heterogeneous marginal costs. In this case advertising can play a role in shifting
demand from higher marginal cost firms to lower cost firms. The case of a monopoly
supplier who provides goods to many reselling firms was also examined. It is found that
advertising can result in suppliers reducing the price they charge reselling firms. As a
large fraction of reselling firms no longer need to price signal (at a high price with low
sale quantity) the supplier can reduce their price to incentivise advertising and boost sales
volume. If the increase in quantity makes up for the decrease in margin then this can be
more profitable.
30
These implications for welfare are substantially more clearcut and supportive of com-
parative advertising than previous papers that model it as the disclosure of differences
in product features. This may indicate that the form of comparative advertising matters
from a policy perspective. Anderson & Renault (2009) find that comparative advertising
of horizontal good attributes can deteriorate total welfare when there is a sufficiently large
quality gap between rival firms. By contrast this paper implies that price comparative ad-
vertising will increase total welfare and in addition no agent’s surplus is decreased by the
legalisation of comparative advertising. In the basic model this increase in welfare comes
entirely from lower prices to consumers and more efficient signalling for firms. When the
extensions are considered however there are additional vehicles for surplus to increase
including the shifting of quantity to lower cost producers of the high quality good and the
inducing of a monopolist supplier into decreasing the price they change reselling firms. All
of these insights present the clear and unambiguous implication that price comparative
advertising is beneficial for welfare and should be supported by legislators and regulators.
References
Anderson, S. & Renault, R. (2009), ‘Comparative advertising: disclosing horizontal match
information’, RAND Journal of Economics 40(3), 558–581.
Armstrong, M. & Zhou, J. (2015), ‘Search deterrence’, The Review of Economic Studies
83(3), 1–32.
Bagnoli, M. & Bergstrom, T. (2005), ‘Log-concave probability and its applications’, Eco-
nomic Theory 26(2), 445–469.
Bagwell, K. & Riordan, M. (1991), ‘High and declining prices signal product quality’,
American Economic Review 81(1), 224–239.
Barigozzi, F., Garella, P. & Peitz, M. (2009), ‘With a little help from my enemy: Compar-
ative advertising as a signal of quality’, Journal of Economics and management strategy
18(4), 1071–1094.
31
Bertrand, J. (1883), ‘Book review of theorie mathematique de la richesse sociale and
of recherches sur les principles mathematiques de la theorie des richesses’, Journal de
Savants 67, 499–508.
Blitzstein, J. & Hwang, J. (2015), Introduction to Probability, 1st edn, CRC Press.
Bradley, C. W. (2014), ‘Navigating Qatari law’, World Trademark Review April/May.
Celik, L. (2014), ‘Information unravelling revisited: Disclosure of horizontal attributes’,
Journal of Industrial Economics 62(1), 113–136.
Charlton, G. (2009), ‘Q&A: Kieron smith of the book depository’, https://
econsultancy.com/blog/3497-q-a-kieron-smith-of-the-book-depository.
Cho, I.-K. & Kreps, D. (1987), ‘Signaling games and stable equilibria’, Quarterly Journal
of Economics 102, 179–221.
Compeau, L., Grewal, D. & Chandrashekaran, R. (2002), ‘Comparative advertising: Be-
lieve it or not’, The Journal of Consumer Affairs 36(2), 284–294.
Council of European Union (1997), ‘Directive 97/55/EC of European Parliament and of
the Council of 6 October 1997 amending Directive 84/450/EEC concerning misleading
advertising so as to include comparative advertising’.
Daughety, A. & Reinganum, J. (2008), ‘Imperfect competition and quality signalling’,
RAND Journal of Economics 39(1), 163–183.
Federal Trade Commission (1979), ‘Statement of policy regarding compar-
ative advertising’, https://www.ftc.gov/public-statements/1979/08/
statement-policy-regarding-comparative-advertising.
Grewal, D. & Compeau, L. (2002), ‘Comparative price advertising: Informative or decep-
tive?’, Journal of Public Policy & Marketing 11(1), 52–62.
G¨urkaynak, G., Yilmaz, I. & Yeilaltay, B. (2015), ‘Turkey green-lights comparative
advertising’, http://www.mondaq.com/turkey/x/444016/advertising+marketing+
branding/Turkey+GreenLights+Comparative+Advertising.
32
Kan, C., Lichtenstein, D. R., Grant, S. J. & Janiszewski, C. (2013), ‘High and declining
prices signal product quality’, Journal of Consumer Research 40, 1078–1096.
Koessler, F. & Renault, R. (2012), ‘When does a firm disclose product information’,
RAND Journal of Economics 43(4), 630–649.
Renault, R. (2016), Advertising in markets, in S. Anderson, D. Stromberg & J. Waldfogel,
eds, ‘Handbook of Media Economics’, Elsevier.
Riley, J. G. (1979), ‘Informational equilibrium’, Econometrica 47(2), 331–359.
Scitovszky, T. (1944), ‘Some consequences of the habit of judging quality by price’, The
Review of Economic Studies 12(2), 100–105.
Shelegia, S. (2012), ‘Asymmetric marginal costs in search models’, Economic Letters
116, 551–553.
Singh, M. (2014), ‘Comparative advertising effectiveness with legal and cross culture
framework’, International journal for research in management and pharmacy 3(3).
Spence, M. (1973), ‘Job market signalling’, The Quarterly Journal of Economics
87(3), 355–374.
Stahl, D. (1989), ‘Oligopolistic pricing with sequential consumer search’, American Eco-
nomic Review 79(4), 700–712.
Three Mobile (2016), ‘Three store - pay as you go price plans’, http://www.three.co.
uk/Store/Pay_As_You_Go_Price_Plans.
Varian, H. R. (1980), ‘A model of sales’, The American Economic Review 70(4), 651–659.
Wolinsky, A. (1983), ‘Prices of product quality’, Review of Economic Studies 50(4), 647–
658.
Yu, E. (2013), ‘Price anchoring to optimize your pricing strat-
egy’, http://www.priceintelligently.com/blog/bid/181199/
Price-Anchoring-to-Optimize-Your-Pricing-Strategy.
33
Appendices
A Heterogeneity Main Results
Following on from the extension to the model introduced in section 5.1, equilibrium in
this context can be defined as:
Definition 3. A fully separating PBNE in this model will be described by pricing strategies
and equilibrium profits ˆπL,ˆπα,ˆπβ, such that:
πL,N (P) = ˆπL≥πL,N (P0)∀P∈DL,N , P 0∈ <+\DL,N (A.1)
πL,A(P) = ˆπL≥πL,A(P0)∀P∈DL,A, P 0∈ <+\DL,A (A.2)
πα,N (P) = ˆπα≥πα,N (P0)∀P∈Dα,N , P 0∈ <+\Dα,N (A.3)
πα,A(P) = ˆπα≥πα,A(P0)∀P∈Dα,A, P 0∈ <+\Dα,A (A.4)
πβ,N (P) = ˆπβ≥πβ,N (P0)∀P∈Dβ ,N , P 0∈ <+\Dβ,N (A.5)
πβ,A(P) = ˆπβ≥πβ,A (P0)∀P∈Dβ,A , P 0∈ <+\Dβ ,A (A.6)
The belief functions µ(P)and µ(P, PE)are derived in accordance with Bayes rule and
player strategies for all information sets reached with positive probability in equilibrium.
It can be seen that all of the results from lemma 1 to lemma 6 carry over without
modification to this case. Thus all low firms will not advertise while pricing at PLand all
advertisers will be believed as being of high quality. It can be noted that the signalling
price, PSof both firms will be identical and as described in equation 5. Profits at the
signalling price will be denoted by πS
αand πS
β. The profit of a type d∈ {α, β}advertiser
can be written as:
πd,A(P)=(P−cd)Γ(H−P) [1 −η+ 2ηG(P)] (A.7)
34
Where ηis the proportion of all firms that are advertising and G(P) is the advertiser
price distribution. For brevity in some proofs this will be written as:
πd,A(P)=(P−cd)Q(P) (A.8)
with Q(P) = Γ(H−P) [1 −η+ 2ηG(P)].
The monopoly prices of αand βfirms are denoted PM
α,PM
βrespectively with the
corresponding profits denoted πM
αand πM
β. To ensure that from lemma 2 the advertiser
price distribution is atomless we assume that the bottom pricing advertiser will price
above PL. This condition will be formalised later on.
Lemma 11. No equilibrium exhibits an αfirm not advertising while a βfirm does adver-
tises.
Proof. To see this consider the case where a βfirm advertisers at some price P∈Dβ,A
while an αfirm is not advertising. For the βfirm we must have for some advertising price
P:
(P−cβ)Q(P)≥(PS−cβ)Γ(H−PS)
(P−cα)Q(P)+(cα−cβ)Q(P)≥(PS−cα)Γ(H−PS)+(cα−cβ)Γ(H−PS)
(P−cα)Q(P)−(PS−cα)Γ(H−PS)≥(cα−cβ)Γ(H−PS)−Q(P)
Note that as cα< cβand Γ(H−PS)< Q(P) the right hand side is positive. This putative
case also implies for αfirms:
(PS−cα)Γ(H−PS)≥(P−cα)Q(P)
(P−cα)Q(P)−(PS−cα)Γ(H−PS)≤0
A contradiction. Thus no equilibrium exhibits an αfirm not advertising while a βfirm
does advertise.
Corollary 12. In any equilibrium there will be a positive mass of αfirms advertising.
35
Proof. From proposition 2 there must be a positive mass of advertisers. From lemma
11 these advertisers cannot all be βfirms unless all firms are advertisers which would
contradict lemma 6.
Lemma 13. In any equilibrium:
1. A positive mass of βfirms will not advertise while setting a price at PS;
2. βfirms earn πS
βin equilibrium. Thus ˆπβ=πS
β.
Proof. From lemma 6 in any equilibrium there exists a positive mass of αnon-advertisers
and/or βnon-advertisers and one or both of the following equalities will hold:
πα=πS
απβ=πS
β(A.9)
Application of lemma 11 shows that no equilibrium with only αadvertisers can exist.
Thus in equilibrium there must be a positive mass of βfirms not advertising while setting
a price of PS. As a result of this all βfirms must earn πS
β.
Proposition 9. In any equilibrium all αfirms will price lower than all βadvertisers.
Proof. First considering the case for advertisers. Consider a putative equilibrium where
βfirms weakly prefer pricing at Pthan pricing at P0and αfirms weakly prefer pricing
at P0than Pwith P < P 0. Then for βfirms:
(P−cβ)Q(P)≥(P0−cβ)Q(P0) (A.10)
(P−cα)Q(P)+(cα−cβ)Q(P)≥(P0−cα)Q(P0)+(cα−cβ)Q(P0) (A.11)
(P−cα)Q(P)−(P0−cα)Q(P0)≥(cα−cβ) [Q(P0)−Q(P)] (A.12)
Note that as cβ> cαand Q(P)> Q(P0) the right hand side is positive. Now consider the
case of the αfirm:
(P0−cα)Q(P0)≥(P−cα)Q(P) (A.13)
(P−cα)Q(P)−(P0−cα)Q(P0)≤0 (A.14)
36
A contradiction. Hence there is no equilibrium where αfirms advertise at a price higher
than βfirms. Now considering non-advertisers from lemma 11 there can never be α
non-advertisers whilst there are βfirms advertising.
At this point we introduce the bottom advertising price (analogously to PBin equation
18).
Lemma 14. The bottom price will be PB,α which will be defined by:
(PB,α −cα)Γ(H−PB,α) = πα
1 + η(A.15)
Proof. From proposition 9 the bottom pricing advertiser will be an α. The bottom price
is the lowest price that delivers this firm the equilibrium profit level for αfirms.
The condition for all advertisers to price more than PLis thus PB,α > PL. From this
point onwards we focus on proving the existence of equilibrium in the special case where
there are some βfirms advertising.15
Lemma 15. If πM
β≥2πS
βthen there will be a positive mass of βadvertisers in equilibrium.
Proof. If this did not hold then a βfirm could price at PM
βand would win against all
other βfirms to earn profits of at least πM
β
2. With the assumption πM
β≥2πS
βthis is strictly
more than the profits attainable by not advertising with a price of PS.
Lemma 16. If πM
β≥2πS
βthe top advertiser price will be PM
β
Proof. The top pricing advertiser is a βfirm. With similar arguments to lemma 9 they
will charge their monopoly price.
Lemma 17. If πM
β≥2πS
βthen ηis given by:
η= 1 −πS
β
πM
β
(A.16)
15Whilst there is in principle no impediment to analysing the alternate case where all βfirms (and
potentially some αfirms) monopolise, this restriction is in order to show the notable result where αfirms
earn above signalling profits as discussed in section 5.1.
37
Proof. The top pricing βfirm charges PM
βwhere G(PM
β) = 0 and must earn πS
β. The
expression follows immediately from substituting these factors into equation A.7.
Lemma 18. If πM
β≥2πS
βthen there is a unique price charged by both types of advertisers
¯
P≡Dβ,A ∩Dα,A which is given by:
¯
P=cβˆπα−cαˆπβ
ˆπα−ˆπβ
(A.17)
Proof. At the unique price ¯
Pwhere there are advertisers from both types of firm the
profits are:
πβ,A(¯
P)=(¯
P−cβ)Γ(H−¯
P)[1 −η+ 2ηG(¯
P)] (A.18)
πα,A(¯
P)=(¯
P−cα)Γ(H−¯
P)[1 −η+ 2ηG(¯
P)] (A.19)
And thus:
πβ(¯
P)
πα(¯
P)=¯
P−cβ
¯
P−cα
(A.20)
Rearranging this equation and noting at this point they make their equilibrium profits
yields the lemma.
Proposition 10. If πM
β≥2πS
βthen αfirms will earn more than their signalling profits
in equilibrium.
Proof. First recounting equation A.20 and noting that at ¯
Pboth high firm types earn
their equilibrium profits.
ˆπβ
ˆπα
=¯
P−cβ
¯
P−cα
(A.21)
38
Now examining signalling profits:
πS
β= (PS−cβ)Γ(H−PS) (A.22)
πS
α= (PS−cα)Γ(H−PS) (A.23)
And thus:
πS
β
πS
α
=PS−cβ
PS−cα
(A.24)
As P−cβ
P−cαis a monotonic function for of Pin the region [cβ,1].
ˆπβ
ˆπα
<πS
β
πS
α
(A.25)
And substituting in ˆπβ=πS
β
πS
α<ˆπα(A.26)
So profits above signalling profits are made.
At this point all of the results presented in section 5.1 have been shown to hold. The
last remaining task is to show that an equilibrium will exist.
The profit functions for αand βfirms are:
πβ,A(¯
P)=(¯
P−cβ)Γ(H−¯
P)[1 −η+ 2ηG(¯
P)] (A.27)
πα,A(¯
P)=(¯
P−cα)Γ(H−¯
P)[1 −η+ 2ηG(¯
P)] (A.28)
And after rearranging to get the required G(P):
G(P) =
1
2ηhπβ
(P−cβ)Γ(H−P)−1 + ηifor ¯
P < P ≤PM
β
1
2ηhπα
(P−cα)Γ(H−P)−1 + ηifor Pα,B ≤P≤¯
P
(A.29)
39
Proposition 11. If πM
β≥2πS
βthen the equilibrium described by a proportion ηof β
firms (as described in equation A.16) monopolising at PSand all other firms advertising
at prices described by the survival functions in equation A.29 and the beliefs described by
µ(P) =
1P≥PS
0P < P S
(A.30)
µ(P, PE) =
1for P > PLand ∀PE
0for P≤PLand ∀PE
(A.31)
is a PBNE.
Proof. Similar arguments as were made in lemma 4 show that these beliefs will be robust
in equilibrium and no low firms will attempt to emulate high quality.
The G(P) function described in equation A.29 is feasible, being decreasing in price
and ranges between 0 and 1 when price changes from PM
βto Pα,B.
The case of low firms is unchanged to that described in proposition 3 with them unable
to convincingly advertise. Hence there is no profitable deviation for these firms.
All αfirms earn the same profit at any price P∈Dα,A ≡Pα,B,¯
P. From proposition
10 they earn more than their signalling profit. From proposition 9 they also earn more
than is possible at any point in Dβ,A and hence there are no profitable deviations for these
firms.
All βfirms earn the same profit at any price P∈Dβ,A ≡¯
P , P M
β. From proposition
13 they earn their signalling profit. From lemma 9 they also earn more than is possible
at any point in Dα,A and hence there are no profitable deviations for these firms.
40
