Advertising and Fixed Sample Search ∗ Alexei Parakhonyak and Vladimir Pavlov

Advertising and Fixed Sample Search∗
preliminary and incomplete
Alexei Parakhonyak†and Vladimir Pavlov‡
February 15, 2014
Abstract
We consider a duopoly model of informative advertising, in which consumers are
engaged in fixed sample search. We derive three types of equilibria in our model. In the
first one consumers do not search, firms either charge a high price and do not advertise
it, or play a mixed strategy over prices and advertise. In the second equilibrium
there is an active search by consumers, but no advertising. Finally, if advertising and
search costs are sufficiently low, there is an equilibrium with advertising and search.
There is no consistent relation between advertised and non-advertised prices in this
equilibrium. We find out that higher advertising costs increase firms profits by reducing
the competition.
Keywords: consumer search, advertising.
JEL-Codes: D40, D83, L13, M37
∗
Support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.
†
National Research University Higher School of Economics, Moscow, Russia. E-mail: [email protected].
‡
University of Oxford.
1
1
Introduction
A well known feature of consumer markets is a presence of information frictions. Often,
consumers do not have full information about the offers available: prices, quality of the
goods, delivery conditions, etc. A transaction occurs when one of the parties makes an effort
to reveal or obtain such information. If this party is consumers, we speak about search, if this
party is firms, we speak about advertising. In this paper we study a model of oligopolistic
competition with advertising and fixed sample search.
There is a vast literature on advertising considering different edges of this phenomenon:
price, quality and availability advertising; signalling effects; possible entry deterrence. An
informative advertising, which credibly reveals information about the price charged by advertising firm to consumers, was studied, among others, in seminal paper by Butters (1977)
and is the key focus of our paper. Despite the fact that both consumer search and advertising
are two ways to obtain information in a market with frictions, the literature, which studies
these two phenomena together is quite limited. Robert and Stahl (1993) develop the model
by Butters (1977) by allowing consumers to search sequentially. They show, that either
firms charge a high price, which is not advertised, or play a mixed strategy over prices and
advertise. Janssen and Non (2008) consider a sequential search model with heterogeneously
informed consumers based on Stahl (1989). They find, that there is no clear distinction
between the prices charged by advertising and non-advertising firms, and that the prices
are not necessarily monotone in search costs. Janssen and Non (2009) consider a model of
consumer search and availability advertising. Finally, Haan and Moraga-Gonzalez (2011)
study a model of advertising for attention, in which the order of search is affected by the
advertisement.
In our paper we consider an informative advertisement in a duopoly setting with fixed
sample search, which distinguishes our work from all the aforementioned papers. This search
protocol implies that consumers first decide how many firms they want to visit, obtain prices
2
from those firms and buy at the best price. There are two reasons for our choice of the
search protocol. Firstly, although theoretically sequential search might be preferable to the
fixed sample search, recent empirical findings reject sequential search hypothesis in favour
of fixed sample search. De los Santos et al. (2012) using an excessive dataset on online
shopping behaviour in books market find that it is consistent with fixed sample search, while
the assumption of sequential search is rejected. In particular, they find that the decision
to continue searching (sample another firm) does not depend significantly on the observed
prices. This behaviour is consistent with fixed sample search, when people decide how
many firms to sample beforehand, but this is not consistent with sequential search. Recent
empirical work on the U.S. auto insurance market by Honka and Chintagunta (2013) also
support fixed sample (or “simultaneous” as they call it) search protocol. Secondly, Stahl
(1989) type of model, which is used in Janssen and Non (2008), has exogenous fraction of
shoppers, i.e. consumers who know all the prices in the market. Moreover, the rest of the
consumers obtain the first price quotation for free and only then engage in the costly search
process, which we find to be inconsistent. With fixed sample search model similar to Burdett
and Judd (1983) we endogenise both the decision about the first search and the decision to
become shopper.
There are three types of equilibria in our model. There is an equilibrium without search,
in which firms either set a high price and do not advertise it, or set lower prices and advertise.
This result has similar flavour as Robert and Stahl (1993), with continuous advertising
intensity substituted by a probabilistic all-or-nothing advertising decision. However, there
is a fundamental difference between their and our results. Since consumers do not search in
our equilibrium, non-advertising firms do not participate in the market, we have a Diamond
(1971) paradox for non-advertising firms. If search costs are sufficiently low, there are two
equilibria with consumer search. One of the equilibria exists, when the advertising costs
are above a certain threshold. High advertising costs remove incentives to advertise, and
3
the equilibrium is similar to Burdett and Judd (1983). If advertising costs are below this
threshold, there is an equilibrium with search, advertising and price dispersion of both types
of the firms. Similar to Janssen and Non (2008) we do not find consistent pattern in prices:
depending on the parameters of the model, the expected advertised price can be both higher
and lower than the expected price charged by non-advertising firm. We found out that the
level of advertising costs has no effect on firms profits in equilibrium with advertising and
without search, and increases equilibrium profits of the firms when search and advertising
coexist. The reason for the first result is that higher level of advertising costs leads to both
lower probability of advertising and higher expected prices, and these two effects exactly
compensate the increase in advertising costs. In the equilibrium with search and advertising,
the effect of the raise of advertising costs is relatively weaker. This difference occurs due
to the fact, that in the first equilibrium when firms do not advertise, consumers leave the
market, while in the second one consumers engage in costly search, which leaves a chance that
the firm will obtain some monopoly power over these consumers even without advertising.
The rest of the paper is organised as follows. In Section 2 we present the assumptions
of the model, the analysis is presented in Section 3, Section 4 concludes. All the proofs are
given in Appendix A, and comparative statics results are derived in Appendix B.
2
Model
There is a market for homogeneous good. There is a unit mass of consumers in the market.
Consumers have unit demand for the good, all consumers have the same valuation of the
good, which we denote as v.
We assume that there are two firms in the market. Firms compete in prices, we normalize production costs to zero. Firms can choose to advertise their prices to consumers.
Advertising is all-or-nothing decision: firms can either advertise to all consumers at a cost
A or not to advertise at all. We assume that A < v, i.e. the cost of advertising is less than
4
the total revenue can be generated in the market.
Consumers are searching for the best price. They can either learn price(s) from advertisement, or engage in costly search. We assume the fixed sample search protocol: consumers
has to decide in advance how many firms they want to sample (zero, one or two in our case).
If consumers have received an advertisement from both firms, they naturally prefer to buy
at the lowest price. If only one advertisement have been received, consumers can condition
their search decision on the price in this message. Search is costly and we denote a cost of
sampling one firm as c > 0. We treat c as a cost of obtaining an information about the
price rather than transportation costs. This implies, that if the price was learned from the
advertisement, there is no further cost c of buying the good.
The timing of the model is as follows:
1. Firms simultaneously decide on their advertising and pricing strategies.
2. Consumers decide on their search strategies, conditional on the information they obtain
from the advertisement.
3. Consumers buy at the best price from the sampled firms and the payoffs are realized.
We denote the strategy of the firms as (α, F0 (p), F1 (p)), where α is the probability that
a firm advertises its price, F0 (p) is the (possibly degenerate) price distribution played by a
non-advertising firm, F1 (p) is the (possibly degenerate) price distribution played by a firm,
conditional on the fact that this firm advertises its price. Let pi (pi ) be the lower (upper)
bound of the support of Fi (p). For consumer who has not received an advertisement, we
denote the probability that she samples i firms by qi . If a consumer receives just one price
through the advertisement, her optimal search rule is characterised by the reservation price
property: if the current price is below some r it is optimal to stop, otherwise it is optimal
to sample the remaining firm.
5
3
Analysis
We start with the analysis of consumers’ behaviour. When the consumer receives an ad from
one of the firms, she has to make a decision, whether to search for another price or to stop.
Conditional on the fact that only one ad was received, it is known, that the other firm plays
pricing strategy is F0 (p). Let r be a price, at which consumer is indifferent between buying
now and continuing to search. Then, it is defined by
r = F (r)E0 (p|p < r) + (1 − F (r))r + c
Assuming that F0 (p) is continuously differentiable (we verify it in equilibrium) this reduces to
Z
r
F0 (p)dp = c
(1)
p0
Now consider the case when none of the firms decided to advertise. In this case the
consumer has to decide how many price to search. Consumer’s behaviour is summarised by
the following Lemma.
Lemma 1. In equilibrium it is either q0 = 1 or q1 + q2 = 1 with 1 > q1 > 0.
Note, that for q0 = 1 a Diamond (1971) type of equilibrium exists, but the market does
not brake down due to the presence of advertising option. This equilibrium is summarised
in the following Proposition.
Proposition 2. For all values of parameters there is an equilibrium, in which consumers
do not search, firms advertise with probability α =
v−A
v
advertise, and mix over an interval [A, v] according to
F1 (p) =
p−A v
p v−A
6
and play p0 = v when they do not
when they advertise.
Similar to Diamond (1971) non-advertising firms do not participate in the market. However, although consumers are not engaged in active search in this equilibrium, the possibility
of advertising generates a price dispersion in this model. Moreover, as A approaches zero,
the equilibrium outcome approaches perfect competition: firms advertise and set price equal
to zero.
This equilibrium has features similar to Robert and Stahl (1993): firms either charge
high price and do not advertise it, or choose randomly a lower price and advertise it with
certain probability (or intensity in Robert and Stahl (1993) setting). However, due to the
absence of search, non-advertising firms do not sell in our equilibrium.
In the subsequent analysis we concentrate on the case with q1 + q2 = 1 with with 1 >
q1 > 0. This is similar to high-intensity equilibrium in Janssen and Moraga-Gonzalez (2004).
In order to simplify notation we denote q ≡ q1 with q2 = 1 − q. Now we proceed with the
analysis of equilibrium candidates.
Consider an equilibrium candidate, when both firms always advertise their prices, i.e.
α = 1. In this case consumers always compare both prices. This implies, that firms compete
a la Bertrand, and the equilibrium price equals to zero. Therefore, firms are better off by
not spending A on advertisement. We can put it formally as the following lemma.
Lemma 3. In any symmetric equilibrium α < 1.
Now, consider an equilibrium candidate with α = 0. This equilibrium derived in the
similar way to Burdett and Judd (1983), with the only exception that we have to control for
firms not deviating to advertising. The equilibrium profit of the firms is defined by:
1
π = pq + p(1 − q)(1 − F0 (p))
2
where the first term comes from consumers who decided to search once (they are distrib7
uted equally between the firms), the second term comes from consumers who compare two
prices, and buy from the current firm provided that the price is lower than in competing
firm. Note, that the upper bound of the price distribution must be equal to v. Firms lose
all customers by setting p0 > v. Firms sell only to those who searched once by setting price
at the upper bound. These consumers buy at any price up to v, so the upper bound must
be equal to v.
q
F0 (p) = 1 −
2(1 − q)
qv
π0 =
2
v−p
p
(2)
(3)
Now we turn our attention to consumer’s problem. Due to Lemma 1 consumers must be
indifferent between searching one or two firms. This implies that
Z
v
C1 ≡
Z
v
pd(1 − (1 − F0 (p))2 ) ≡ C2
pdF0 (p) =
p0
p0
This indifference condition reduces to
g(q) ≡
) − 2(1 − q))
q(ln( 2−q
q
2(1 −
q)2
=
c
v
(4)
Denote q ∗ = arg max g(q). Since, for example, g(1/2) = v(ln 3 − 1) > 0, it is clear that
g(q ∗ ) > 0. Let c ≡ vg(q ∗ ). Then for any c < c there exists a solution of (4). The benefits of
additional search (for v normalized to 1) are plotted in Figure 1. Note, that in general there
are 3 equilibria: low q, high q and a corner solution of q = 1. The latter case cannot be
a part of equilibrium due to Lemma 1. As pointed out by Fershtman and Fishman (1992),
high value of q is not a stable solution. Indeed, a small downward perturbation of q means
that search benefits are higher than the search costs, so consumers want to search more,
and q tend to decrease. If there is a small upward perturbation of q then search benefits are
8
Figure 1: Benefits of additional search (humped shaped curve) vs search costs(horizontal
line)
less than the search costs and consumers tend to reduce their search activity, which leads to
further raise in q. Thus, we are going to work only with the lowest root.
Now let’s consider a possible deviation to advertising. Here and throughout the paper
we assume passive out-of-equilibrium beliefs, i.e. consumers, upon observing a deviation by
one of the firms, believe that the other firm, whose actions are not (yet) observed, sticks to
equilibrium strategy.
It is not optimal to set a price below r while deviating to advertising: when a firm sets
r and advertises (and the other firm does not advertise), it sells to the whole market. Then
deviating to p < r will not increase demand, but will decrease the margin.
To proceed, we will need to prove the following lemma.
Lemma 4. In equilibrium with α = 0, for every q there exists a unique r, which solves (1),
9
such that v ≥ r ≥ qv. r is implicitly defined by
r
2−q
(2 − q)r − qv ln = 2(1 − q)c + qv 1 + ln
v
q
(5)
Consider a firm, which deviates by advertising. If it sets a price p = r it covers the whole
market. If it sets a price p > r, then the deviation profit is given by π d = (1 − F0 (p))p − A =
q
(v
2−q
− p) − A, which is decreasing in p. Thus, the most profitable deviation is to set p = r.
The deviation profit in this case π d = r − A, which has to be smaller than
qv
,
2
making the
deviation unprofitable. Let us denote r∗ (c) and q ∗ (c) the stable solution of the system of two
equations (4) and (5). Lemma 4 guarantees that this solution always exists and is unique.
Denote
A∗ (c) ≡ r∗ (c) −
q ∗ (c)v
2
(6)
The deviation to advertising is not profitable if and only if A > A∗ (c). This allows us to
formulate the following Proposition.
Proposition 5. For every c < c the equilibrium with α = 0 exists if and only if A > A∗ (c). In
this equilibrium firms play mixed pricing strategies according to distribution (2). Equilibrium
strategy of consumers is defined by (4) and (5).
Now we proceed with the analysis of equilibria with α > 0. We start our analysis with
fixing up the upper bounds of the support of the equilibrium distribution.
Lemma 6. In any symmetric equilibrium with α > 0, the upper bounds of the supports of
equilibrium price distributions are p¯0 = v, p¯1 ≥ r.
We are going to construct an equilibrium in which p¯1 = r. Other equilibria may exist.
The equilibrium profits in our case are defined by
10
π0 (p) = (1 − α)
q
+ (1 − q)(1 − F0 (p)) p
2
π1 (p) = (1 − α + α(1 − F1 (p)))p − A
(7)
(8)
Equation (7) differs from (3) only by multiplier (1 − α). Note, that due to the fact that
p1 = r non-advertising firm sells only when the other firm does not advertise, which happens
with probability 1 − α. Advertising firm (equation (8)) sells to the full market when another
firm does not advertise (this happens with probability 1 − α) and competes with this firm
when the latter decides to advertise. Moreover, there is a cost of advertising A. Now, the
boundary conditions dictate that
qv
2
π1 (p) = π1 (r) = (1 − α)r − A
π0 (p) = π0 (v) = (1 − α)
(9)
(10)
This allows us to derive the equilibrium distribution functions:
q
v−p
2(1 − q) p
(1 − α)(r − p)
F1 (p) = 1 −
αp
F0 (p) = 1 −
(11)
(12)
qv
with the supports [ 2−q
, v] and [(1 − α)r, r] respectively. Note, that F0 (p) is exactly the
same as in the case with α = 1. The reason is that in our model non-advertising firm
competes only with the same type of firm and does not make any profit when the other firm
advertises. Finally, in equilibrium firms must be indifferent between advertising and not.
This gives
11
(1 − α)
qv
= (1 − α)r − A
2
Thus,
α=1−
A
r − qv
2
Therefore, the equilibrium with α > 0 exists if and only if A < r −
(13)
qv
.
2
This condition
implies, that the equilibrium with advertising exists only when:
• the advertising costs are sufficiently low;
• the reservation price is high enough, i.e. consumers are willing to accept relatively high
prices from advertising firm;
• there is sufficient search in the case when nobody advertises (q is low), making a
strategy not to advertise less attractive.
Note, that as the equilibrium distribution of non-advertising firm F0 (p) is the same as
in the case with α = 0, the equilibrium strategy of consumers (q, r) is defined by the same
equations (4), (5). This implies that condition α > 0 can be reduced to A > A∗ (c), where
A∗ (c) is defined by (6).
We summarise our analysis in the following proposition.
Proposition 7. For every c < c the equilibrium with α > 0 exists if and only if A < A∗ (c).
In this equilibrium firms play mixed pricing strategies according to distributions (11) and
(12), and advertise with probability α given by (13). Equilibrium strategy of consumers is
defined by (4) and (5).
Note, that Propositions 5 and 7 together imply that the parameter space is split by A∗ (c)
into two parts: for large value of advertising costs only equilibrium without advertising exists.
12
Figure 2: A∗ (c) (the thick blue line), c = c¯ (red vertical line)
If the advertising costs are below this threshold, then the only equilibrium is when the firms
advertise with positive probability. These two parts of parameter space (for v normalized to
1) are depicted in Figure 2.
Equilibrium described in Proposition 5 is well studied in the literature (see, e.g. Burdett
and Judd (1983), Fershtman and Fishman (1992)). Thus, we concentrate on the analysis
of the properties of equilibrium with advertising, described in Proposition 7. The expected
prices of advertising and non-advertising firms in this equilibrium are given by
q
2−q
log
2(1 − q)
q
1−α
1
E1 (p) =
r log
α
1−α
E0 (p) =
(14)
(15)
Comparative statics of the model are summarised in the following table. Results indicated
13
Table 1: Comparative Statics
Eq. w/o search
α π E0 p E1 p
Eq. w/o advertising Eq. with advertising and search
q π
E0 p
r q α
π E0 p
E1 p
d
dc
0
0
0
0
+ +
+
+ + +∗
+∗
+
+∗
d
dA
−
0
0
+
0
0
0
−
+
0
+
0
0
by astronim are derived using numerical techniques. Proofs of comparative statics results
can be found in Appendix B.
In equilibrium with q0 = 1, described in Proposition 2 advertising activity negatively
depends on the advertising costs. Profit does not change with advertising costs, since it must
be equal to zero in equilibrium. Non-advertising firms charge price equal to v, advertising
firms tend to charge higher prices for higher A, since in this case there is less probability
that the competitor advertises, i.e. less competition. Since there is no search, the level of c
does not play any role in this equilibrium (provided that it is positive).
In equilibrium without advertising, which exists for sufficiently small levels of search and
advertising costs, search activity is decreasing in c, which implies higher prices and profits
of the firms.
Finally, we consider an equilibrium with advertising and search. The comparative statics
with respect to c are quite intuitive. Higher search costs tend to raise reservation prices
for situations, when one of the firms advertises and reduce search activity in the absence of
advertisement. The effect on advertising activity is not that obvious. On one hand, firms
have incentives to advertise more, since in the situation when only one firm advertises they
have more opportunities to exploit consumers due to higher r. On the other hand, firms
have more monopoly power over consumers in the case when they do not advertise due to
lower q. It turns out that the former effect dominates the latter and firms advertise more for
higher levels of search costs. In this case advertisement plays a role of substitute for search:
14
the lower the search activity of consumers, the higher the advertising activity of the firms.
This finding is similar to Janssen and Non (2008).
The effect of search costs on equilibrium profits comes from two sources. On the one hand
higher search costs reduce the search activity of the customers, and therefore increase the
prices. One the other hand, as we have just learned, the lower search activity is compensated
by higher advertising efforts. This compensation is not sufficient and equilibrium profits are
increasing with the search costs. The same holds for expected prices of both advertising and
non-advertising firms. This result is in contrast with findings by Janssen and Non (2008),
who found that there is a non-monotone effect of search costs on the expected price of
non-advertising firm.
As consumers’ strategy (r and q) depends only on the pricing strategy of non-advertising
firm (F0 (p)), which does not depend on A, these variables do not depend on A either.
Advertising activity naturally decreases with respect to advertising costs. Surprisingly this
is not true for the equilibrium profits. There are two effects: higher costs lower the profits
per se, but they also reduce advertising activity, leading to lower competition and higher
prices. The latter effect dominates the former and profits are increasing in A. There is no
effect on prices of non-advertising firms (though firms choose not to advertise more often),
and prices of advertising firms are increasing in A due to lower competition.
Now we are going to compare prices between advertising and non-advertising firms
in equilibrium with search and advertisement. Note, that since the distribution of nonadvertising firm in equilibrium with α > 0 is the same as the distribution in a model, when
advertisement is prohibited, we can directly access the impact of advertising on expected
price in a market.
˜ < A∗ (c), such that:
Proposition 8. For every c < c¯, there exists A(c)
˜ < A < A∗ (c), E1 (p) > E0 (p),
• if A(c)
˜
• if A ≤ A(c),
E1 (p) ≤ E0 (p).
15
˜ (the purple line), v = 1
Figure 3: A∗ (c) (the blue line) and A(c)
Thus, when the advertising costs are low enough, expected prices of advertising firms
are lower. This implies, that if the advertisement is prohibited, it can decrease the expected
˜
prices only for A ∈ [A(c),
A∗ (c)], i.e. when advertising costs are relatively small. Figure 3
shows, that the domain of parameters for which the presence of advertisement implies lower
prices constitutes around 78% of the area, where the equilibrium with advertising exists
(assuming uniformly distributed parameters).
Finally, we look at global comparative statics by considering cases when either search or
advertising costs are going to zero.
Proposition 9.
1. Suppose, that A > 0. Then, if c → 0 equilibrium with advertising
stops existing and equilibrium without advertising approaches perfect competition.
2. Suppose, that c > 0. Then, if A → 0 equilibrium without advertising does not exist,
and equilibrium with advertising approaches perfect competition.
16
These results are quite intuitive and similar in nature to Janssen and Non (2008). As it
is easy to see in Figure 2 for any positive A only equilibrium without advertisement exists
for c small enough. As c approaches zero in this equilibrium, consumers tend to compare two
prices, which results in perfect competition. Similar situation is with A approaching zero. As
Figure 2 suggests, only equilibrium with advertisement exists in this case. The probability
of advertising approaches one in this case, so consumers again compare two prices almost
surely.
4
Conclusions
In this paper we consider a model of advertising in a consumer search environment. By
considering fixed sample search protocol, instead of sequential search, which is common in
the literature, we achieve several goals. Firstly, instead of making assumption about free
first search, we endogenise participation decision of the customers. Secondly, we do not need
exogenous fraction of shoppers in the model, the decision to become shopper is endogenous.
Finally, recent empirical findings reject the hypothesis of sequential search in favour of fixedsample search protocol.
We derive three equilibria in our model. First equilibrium exists for all parameter values
and is characterised by abstention of consumers from search. For non-advertising part of
the market this equilibrium is similar to Diamond (1971). However, the presence of advertising option changes the situation: firms advertise with certain probability and play mixed
strategies over prices.
If search costs are sufficiently low, there are two other types of equilibria, which are
characterised by active search of consumers. If the advertising costs are sufficiently high,
there is an equilibrium similar to Burdett and Judd (1983). If the advertising costs are low,
there is an equilibrium with search and advertisement. Consumers search in this equilibrium
only when they do not receive an advertising message. Surprisingly, an increase advertising
17
costs leads to higher profits of the firms.
Appendix A: Proofs
Lemma 1. In equilibrium it is either q0 = 1 or q1 + q2 = 1 with q1,2 > 0.
Proof. Firstly, suppose, that q1 = 1. In this case consumers observe just one price quotation, so it is the best response to charge p = v. Then, for customers it is not optimal to
search. Secondly, suppose that q2 = 1. Then all the consumers compare both prices and the
equilibrium strategy of the firms is to charge p = 0. Then, consumers should not waste the
search costs and sample only one firm. Case with q0 + q1 = 1, q0 , q1 > 0 is similar to the first
one. Case with q0 + q2 = 1, q0 , q2 > 0 is similar to the second one. Finally, consider the case
with qi > 0, i = 0, 1, 2. Then, it must be the case that
0 = v − E0 (p) − c = v − E0 (min(p1 , p2 )) − 2c
now note that
Z
v−E0 (min(p1 , p2 ))−2c = v−2c−
p0
Z
2
p0
pd(1−(1−F0 (p)) = v−2c−2E0 (p)+2
p0
pf0 (p)F0 (p)dp
p0
Now, since E0 (p) + c = v we get
Z
p0
v − E0 (min(p1 , p2 )) − 2c = 2
pf0 (p)F0 (p)dp − v
p0
Using integration by parts we obtain
Z
p0
Z
p0
pf0 (p)F0 (p)dp = p0 −
p0
2
Z
p0
pF0 (p) dp −
p0
pf0 (p)F0 (p)dp
p0
18
This gives
Z
p0
v − E0 (min(p1 , p2 )) − 2c = p − v −
pF0 (p)2 dp < 0
p0
which cannot be true, as p ≤ v (firms earn zero profits by charging prices higher than
v).
Proposition 2. For all values of parameters there is an equilibrium, in which consumers
do not search, firms advertise with probability α =
v−A
v
and play p0 = v when they do not
advertise, and mix over an interval [A, v] according to
F1 (p) =
p−A v
p v−A
when they advertise.
It is clear, that if non-advertising firms charge p = v, the best response of the customers
is q0 = 1. Moreover, any price up to v is accepted if only one firm advertises. Now, nonadvertising firm does not sell to anybody, obtaining a profit of zero. Advertising firm, which
plays a mixed strategy over price gets a profit
π1 (p) = ((1 − α) + α(1 − F1 (p)))p − A
The upper bound of the price distribution must be equal to v (since at the upper bound
a firm sells only when the competitor does not advertise). Equal profit condition implies
that
α=
v−A
v
and
F1 (p) =
p−A v
p v−A
19
with the support p ∈ [A, v].
Lemma 4. In equilibrium with α = 0, for every q there exists a unique r, defined by (1),
such that v ≥ r ≥ qv. r is implicitly defined by
r
2−q
(2 − q)r − qv ln = 2(1 − q)c + qv 1 + ln
v
q
Proof. First we need to show that r as a solution to (1) exists for any given q. Plug F0 (p)
in equation (1) and get
2−q
r
(2 − q)r − qv ln = 2(1 − q)c + qv 1 + ln
v
q
(16)
qv
The LHS is a function of r with a local minimum at rmin = 2−q
. If we plug it back, we’ll see
and is lower than the RHS. The
that the minimal value of the LHS equals qv −1 + ln 2−q
q
LHS of 5 goes to infinity as r goes to infinity. Therefore, there always exists a solution to (5).
Note, that rmin = p0 so there always exists a solution which is above the lower bound of the
support. Moreover, since the LHS is increasing in r for all r > rmin this solution is unique.
Now we need to show that the solution belongs to the interior of the support, i.e. r ≤ v. By
plugging r = v into LHS of (5) we get that it simplifies to (2 − q)v. By substituting c with
the expression from (4) we obtain the the RHS equals to
q(ln( 2−q
) − 2(1 − q))
q
1−q
2−q
v + qv 1 + ln
q
As v cancels out we have both LHS and RHS to be functions only of q, and it is can be
shown, that the LHS is larger than the RHS. This implies, that the solution of (5) is lower
than v.
Now let’s prove that r > qv. If we plug r = qv in the LHS of 5, we will get that the LHS
is lower than the RHS ∀q ∈ (0, 1). As the LHS is increasing in r, then r > qv.
20
Lemma 6. In any symmetric equilibrium with α > 0, the upper bounds of the supports of
equilibrium price distributions are p¯0 = v, p¯1 ≥ r.
Proof. Let us begin with non-advertising firms. First of all, it is never optimal to charge any
price above v, as it leads to zero demand.
Second, let us prove that p¯0 can not be smaller than v. Suppose in equilibrium p¯0 < v.
If the other firm does not advertise, at p = p¯0 we sell only to half of consumers, who search
only once (i.e. the demand is 2q ). Then the profit is
q p¯0
.
2
But these consumers are willing
to accept any price below v. Then a firm can deviate to playing v instead of p¯0 and strictly
increase its profit as
qv
2
>
q p¯0
.
2
Therefore, in equilibrium it must be the case that p¯0 = v.
Now consider advertising firms. Let us prove that p¯1 < r is not possible in equilibrium.
If p¯1 < r, then at p = p¯1 an advertising firm will sell to all the consumers if the other firm
does not advertise and will sell to nobody if the other firm advertises. If the firm deviates
to playing p = r, the demand will not change, but the margin will increase, so the profit will
strictly increase. Then it must be the case that p¯1 ≥ r in equilibrium.
˜ < A∗ (c), such that:
Proposition 8. For every c < c¯, there exists A(c)
˜ < A < A∗ (c), E1 (p) > E0 (p),
• if A(c)
˜
• if A ≤ A(c),
E1 (p) ≤ E0 (p).
˜
Proof. Define A(c)
as a solution to the equation, which equalizes the expected price of
advertising firm to the expected price of non-advertising firm, i.e.
qv
2−q
1−α
1
log
=
r log
2(1 − q)
q
α
1−α
(17)
We know that if limA→A∗ α = 0. Then the RHS of (17) approaches r. The RHS of (17) is
increasing in A, while the LHS is constant (see the comparative statics). Therefore, for any
˜ E1 (p) > E0 (p). The only thing which has to be proved is that A(c)
˜ is well-defined,
A > A(c)
21
˜
i.e. A(c)
≤ A∗ (c). This is always the case if r ≥
r=
qv
2(1−q)
qv
2(1−q)
log 2−q
. To prove that, we plug
q
log 2−q
into the LHS of (5). The LHS is increasing in r, so now we need to prove
q
the following:
2−q
2−q
qv ln
− qv ln
2(1 − q)
q
q
2−q
ln
2(1 − q)
q
2−q
≤ 2(1 − q)c + qv 1 + ln
q
Using (4) this inequality can be transformed into:
ln( 2−q
) − 2(1 − q)
q
(1 − q)
q
2−q
≥
ln
− ln
2(1 − q)
q
q
2−q
ln
2(1 − q)
q
−1
The difference between LHS and RHS of this equation is decreasing in q and approaches
˜ ≤ A∗ (c).
0 as q approaches to 1. Thus, A(c)
Proposition 9.
1. Suppose, that A > 0. Then, if c → 0 equilibrium with advertising
stops existing and equilibrium without advertising approaches perfect competition.
2. Suppose, that c > 0. Then, if A → 0 equilibrium without advertising does not exist,
and equilibrium with advertising approaches perfect competition.
Proof. Note, that A∗ (c) is decreasing in c and approaches zero as c → 0. This implies, that
for any positive A there is c small enough, such that the equilibrium with advertising does
not exist. Equation (4) implies that limc→0 q ∗ (c) = 0, and, thus, F0 (p) → I{p≥0} .
Exactly by the same reasoning for any c > 0 there is A small enough, that the equilibrium
without advertising does not exist. As limA→0 α = 1, F1 (p) → I{p≥0} .
22
Appendix B: Comparative Statics Results
First consider the equilibrium without search. From Proposition 2 it is clear that
dα
dA
< 0.
Non-advertising firms always set p0 = v. The expected price of advertising firms equal to
E1 p =
Av
v−A
ln Av , which is increasing in A. Finally, the equilibrium level of profits equals to 0
and does not depend on A.
Since q ∗ (c) is defined as the lower root of (4) it is clear that
dq
dc
> 0. This holds in both
equilibria with search. From (7) it follows that profits are increasing in q, and thereby, c.
The expected price is equivalent to the average industry profit in this case.
Now consider
dr
.
dq
We apply the implicit function theorem to (5):
r + v ln vr − 2c + v + v ln 2−q
−
∂r∗
q
=
v
∂q ∗
2 − q(1 + r )
First, note the useful fact that by definition r ≥ p0 + c =
2v
2−q
qv
2−q
(18)
+ c which is smaller than
v due to Lemma 4. From this follows that the denominator of (18)is positive. Now consider
the numerator.
r
2−q
2v
r(2 − q)
qv
c(2 − q)
r +v ln −2c+v +v ln
−
= r +v ln
−2c−
≥ v ln (1 +
)−c
v
q
2−q
vq
2−q
vq
The last inequality is obtained using r ≥
qv
2−q
+ c. The RHS of this expression is positive
if and only if
c
c (2 − q)
≤ ln (1 +
)⇔q≤
v
v q
2
ec/v −1
c/v
+1
The RHS is decreasing in c, so the tightest restriction would be if c = c¯. If we plug it in,
we’ll get q ≤ 0.97, which is always satisfied, given that we consider the smaller root of (4).
Now we can proceed with the analysis of comparative statics.
23
dr∗
∂r∗ ∂r∗ dq ∗
=
+ ∗
dc
∂c
∂q dc
The second summand it positive. By applying the implicit function theorem to (5) we
get that the first summand is positive as well. Thus,
dr∗
dc
> 0.
(4) implies that q does not depend on A. Since none of the elements of (5) depends on
A we get
dr∗
dA
= 0.
dα
Now we turn our attention to α. From (13) it is clear that dA
< 0 and the sign of dα
dc
qv
d
is determined by dc
r − 2 . Numerical evaluation for all values of (A, c) such that c < c,
A < A∗ (c) shows that
dr
dc
>
v dq
,
2 dc
therefore
dα
dc
> 0. It is clear from (13) that
dα
dA
< 0.
Now consider the effect of parameters on equilibrium profits. From (9) we get
dα qv
dπ
=−
>0
dA
dA 2
and
dπ
dα qv
v dq
=−
+ (1 − α)
dc
dc 2
2 dc
We evaluate the latter expression numerically and show that
dπ
dc
> 0, i.e. search effect is
stronger than the advertising effect.
Now we consider the expected prices.
∂E0 (p)
=
∂c
The multiplier before
dq
dc
1
2−q
1
log
+
(1 − q)2
q
(2 − q)(1 − q)
is always positive, so
∂E0 (p)
∂c
> 0 as
dq
dc
dq
dc
> 0.
dr
(1 − α) − ∂α
r(2 − α)
∂E1 (p)
r
∂c
= − dc
log
(1
−
α)
+
∂c
α2
α
24
This expression we evaluate numerically and it turns out that
∂E1 (p)
>0
∂c
.
The effect of advertising costs on the expected prices is rather straightforward:
∂E0 (p)
= 0
∂A
∂α∗ 1
1
(1 − α)2
∂E1 (p)
= −r
( 2 log
+
)>0
∂A
∂A α
1−α
α
25
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