Optimal firm behavior with consumer social image
concerns and asymmetric information
Alexander Sebald
∗and Nick Vikander
†March 28, 2018
Abstract
This paper explores how consumers’ belief-dependent social image concerns can
affect firm strategic choices in a product market setting. We consider a theoretical
framework with incomplete information where a profit-maximizing monopolist sets a
price for its product, taking into account that consumers care about the belief that
others hold about the product’s popularity. Throughout our analysis, we highlight
the close connection between our dynamic psychological game and the literature on
network effects. We show in particular that belief-dependent social image concerns
generate equilibrium price distortions that do not arise in a network effect setting, and
we explore the implications for consumer demand and firm profits.
Keywords: social image, optimal firm behavior, consumer search
JEL-Classifications: D91, D21, L11, D83
1
Introduction
In recent years, more and more game theorists have started to incorporated insights from
psychology into models of strategic behavior [see e.g. Rabin (1998) for an overview]. One of ∗University of Copenhagen, Department of Economics. E-mail: [email protected]
those insights is that people are emotional. It is for example acknowledged that people often
behave reciprocally towards others [see e.g. Rabin (1993) and Dufwenberg and Kirchsteiger
(2004)], feel anger [see e.g. Battigalli and Smith (2015)] or are guilt averse which impacts
their behavior in strategic interactions [Battigalli and Dufwenberg (2007)].
One prominent way to model these emotions is psychological game theory [see
Geanakop-los et al. (1989) and Battigalli and Dufwenberg (2009)]. This approach assumes that people’s
utilities directly depend on their (higher-order) beliefs. Theories of belief-dependent
reci-procity, for example, assume that people use their beliefs about the beliefs of others to get a
sense of their intentions [see Rabin (1993), Dufwenberg and Kirchsteiger (2004) and Sebald
(2010)]. Theories of belief-dependent guilt aversion, on the other hand, assume that people
form beliefs about other people’s expectations (i.e. beliefs) and are averse to letting these
others down [see Battigalli and Dufwenberg (2007)].
In contrast to much earlier work on psychological games, which looks at how emotions
like reciprocity and guilt aversion affect cooperation, gift exchange, contributions to public
goods, etc, we consider how consumers’ belief-dependent preferences affect firm strategic
choices (i.e. pricing) in a product market setting. The strategic environment we analyze is
a dynamic psychological game with asymmetric information [see Battigalli and Dufwenberg
(2009, section 6.2)]. More specifically, we consider a setting with incomplete information in
which a profit-maximizing monopolist sets a price for its product. Subsequently consumers
motivated by belief-dependent image concerns choose whether or not to buy the product,
taking into account what other consumers will believe regarding the product’s popularity
ex-post.1
Consumers in our model are either shoppers or non-shoppers. Both are aware of the
product’s existence and hold beliefs about the beliefs of others regarding the product’s
pop-ularity. Shoppers and non-shoppers differ, however, with regard to what they know about
the price set by the firm. Shoppers automatically learn the price, whereas non-shoppers only
1One can think of these consumers as being motivated by concern for social esteem, which depends on
learn the price if they pay a strictly positive search cost.2
In this strategic setting, we investigate the firm’s optimal behavior when consumers care
about whether other consumers believe that the product is popular. We consider the case of
conformists and snobs whose willingness to pay is respectively increasing/decreasing in their
2nd-order beliefs about product popularity. The results show that social image concerns
leads to distortions that push the equilibrium price up if consumers are conformists and
down if consumers are snobs. The magnitude of this effect is non-monotonic in the fraction
of shoppers. It can also be non-monotonic in the strength of social image concerns, but
only if consumers are conformists and some non-shoppers search. We also show that if
consumers are snobs, then the downward distortion on prices due to social image concerns
can counteract the upward distortion that arises due to consumer search, which can benefit
both the firm and consumers.
Our paper adds to a literature looking at how consumer concerns about product
popu-larity can affect firm strategic choices. Becker (1991) shows that a capacity-constrained firm
facing conformist consumers may want to set a price that yields excess demand. Amaldoss
and Jain (2005a) also consider pricing, and demonstrate that demand in a market comprised
of both conformists and snobs can be upward sloping. Amaldoss and Jain (2005b) show that
these same ‘social effects’ only affect the market shares of duopolists if the firms offer
prod-ucts of different qualities. In Buehler and Halbheer (2012), increased consumer conformity
results in larger asymmetries in duopolists’ choice of price and persuasive advertising.
The key difference with our paper is that consumers’ behavior in this literature only
depends on their 1st-order beliefs about demand. In fact, work in this literature is often
non-commital about whether willingness to pay depends on 1st- or 2nd-order beliefs, because the
distinction does not matter in their settings; consumers have symmetric information about
their environment and the firm’s strategic choices, and so hold the same beliefs of every
order. Thus, ‘social effects’ in this literature are entirely equivalent to network effects that
2We think of non-shoppers as infrequent shoppers for whom searching for information regarding the
depend directly on quantity sold.3 In contrast, the distinction between 1st- and 2nd-order
beliefs, and that between network effects and social image concerns, is crucial in our setting,
because some consumers are more informed than others.
Our paper also contributes to the literature on consumer search, which typically assumes
that consumers only observe a product’s price and have the option to buy if they first
pay a strictly positive search cost.4 Typically, in this literature, some consumers end up
better informed than others, simply because they find it optimal to search.5 Our approach
of considering both non-shoppers, who face search costs, and shoppers, who do not, is also
common (see, e.g., Stahl (1989), Stahl (1996), Janssen and Moraga-Gonz´alez (2004), Janssen
and Parakhonyak (2017)), as is our focus on equilibrium pricing. Our paper is the first to
introduce social image concerns into a setting with consumer search, and to explore how
their interaction affects market outcomes.
Our paper also relates to work on social image (or social status) and conspicuous
con-sumption that assumes consumers differ in their type, type is unobservable, and consumers
care directly about what others believe about their type, conditional on their purchase (see,
e.g., Ireland (1994), Bagwell and Bernheim (1996), Corneo and Jeanne (1997); more recently
Yoganarasimhan (2012), Rayo (2013), Friedrichsen (2016)). A similarity with our paper is
that a consumer’s payoff from buying in this literature depends directly on other consumers’
beliefs. However, this literature has only considered settings where all consumers are fully
informed about their environment and the firm’s strategic choices, and therefore hold the
same beliefs (of every order) about who will buy particular products. As a result, the novel
price distortions we identify in our analysis do not occur, because a consumer’s willingness
to pay depends only on her 1st-order beliefs: not about demand for the product, as with
3With this interpretation, the network effect experienced by conformists is positive, and that experienced
by snobs is negative.
4We do not attempt to review this vast literature here. For seminal contributions, see, e.g. Diamond
(1971), Burdett and Judd (1983), Stahl (1989).
5The idea that some consumers are better informed about prices than others is also a common feature of
network effects, but about the distribution of types who buy.
The previous overview over the existing related literature highlights the close connection
between our dynamic psychological game and the literature on network effects. Throughout
our analysis, we will therefore compare our results when consumers are motivated by social
image concerns with results that would arise in the absence of image concerns but if the
product exhibited network effects. Specifically, Section 2 presents the model, and Section 3
contains the analysis and results. Section 4 then concludes.
2
Model
In what follows, we characterize the model which forms the basis for our analysis presented
in Section 3. We first describe the firm’s characteristics and then turn to the consumers who
are assumed to be partially motivated by belief-dependent social image concerns.
Consider a monopolist that faces a measure one of consumers who have unit demand for
its product. The firm sets a price p in order to maximize its profits. Marginal production costs are constant and normalized to zero.
Consumers in our setting differ in two dimensions. First, they are either shoppers or
non-shoppers. Second, they differ in terms of their intrinsic valuation of the product. Specifically,
a fraction α P p0,1q of consumers are shoppers, who observe the firm’s choice of price and simply decide whether or not to buy the product. A fraction p1 αq of consumers are
non-shoppers, who only observe the price, and can only buy the product, if they decide to
search. A non-shopper who searches must pay cost c ¡ 0. Let θi represent the intrinsic
payoff of consumer i from buying the product. We assume θ Up0,1q for both shoppers and non-shoppers.
consumerj in the market, wherej’s beliefs about product popularity affects the social image of i. Payoffs are then realized and the game ends.
Let Qje denote the expectation of consumer j about demand for the product, given the information at her disposal at t 3. If consumer i buys the product, and is then matched with j, her payoff is equal to
θi
loomoon Intrinsic Payoff
λQje
loomoon Social Image Payoff
loomoonp
Price
, (1)
whereλ is a parameter whose magnitude measures the strength of social image concerns. If λ ¡ 0, then we will say that consumers in this market are conformists. In this case consumer i’s social image payoff from buying is larger if consumer j believes demand was high. If instead λ 0, we will say that consumers in this market are snobs. In this case consumeri’s social payoff from buying is larger if consumerj believes demand was low. Consumers who do not buy take an outside option of zero.6 Throughout the analysis, we will
assume thatλis ‘not too large’ in magnitude, whenλ¡0, to rule out corner solutions where all shoppers buy. Specifically, holding fixed αP p0,1q, we allow for any λ minp21α,1q.7 At some points we will consider comparative statics with respect to α, and letα be arbitrarily close to 1. There we can think of λ 12 for these comparative statics, to guarantee interior solutions for all αP p0,1q.
To summarize, the firm’s strategy is a choice of pP R . The strategy of a shopper is a choice whether or not to buy, for each possible value of p. The strategy of a non-shopper is
6Consumers do not know who they will be matched with, so they condition their purchase decision on
their belief about the average belief in the population regarding the popularity of the product. The belief-dependent social image payoff is realized only after random matching with another consumer, who serves as a social contact. This formulation means that each consumer never observes aggregate product sales, but only whether their social contact purchased the product. For a broadly similar approach, where consumers match with one another and then experience a social payoff that depends on their earlier purchase decisions, see Pesendorfer (1995) and Yoganarasimhan (2012).
7Formally, when stating our results, we also assumeλ0, so that social image concerns are not completely
a choice whether to search, and, conditional on searching, a choice whether to buy for each
value ofp.
We solve for a Perfect Bayesian Equilibrium, where (i) the firm sets the price p at t
0 which maximizes profits, given consumer equilibrium strategies; (ii) shoppers and
non-shoppers who search maximize their expected payoff by buying/not buying the product
at t 2, given the price p they observe, and their beliefs about the expectation of their partner about demand att3; (iv) the search decisions of non-shoppers at t1 maximize their expected payoff, given their beliefs about the price, about their purchase decision
conditional on searching, and about the expectation of their partner at t3; (v) all beliefs and expectations are consistent with consumers’ equilibrium strategies; (vi) for consumers
who observep, their beliefs and expectations are consistent with the price they observe; (vii) for consumers who do not observe p, their beliefs and expectations are consistent with the firm’s equilibrium strategy.8
3
Analysis
As mentioned in the Introduction, our analysis investigates how belief-dependent social image
concerns affect the equilibrium market outcome, in particular the equilibrium price. We first
describe the incentives of shoppers to buy the product, the incentives of non-shoppers to
search, and the incentives of non-shoppers to buy, conditional on searching. Following this,
we derive comparable results for a setting where social image concerns are absent but where
the product exhibits network effects. We then briefly compare the two before moving on to
equilibrium pricing.
Social image concerns: A consumer’s payoff from buying is increasing in her type, by
(1), so consumer behavior must follow a threshold structure. In particular, there exists a
critical value θn P r0,1s such that a non-shopper of type θ will search if and only if θ ¥θn.
8Notice that we are solving for a standard PBE, but in a context where consumers have belief-dependent
Given this critical value, a fraction 1θn of non-shoppers search, since type is uniformly distributed on r0,1s.
Given this, a shopper of type θ who observes pricep will buy if and only if
θp λ
αEspQseq p1αqp1θnqEspQen|seq p1αqθnEspQne|noq
¥0, (2)
where EspQseq denotes her belief about what others shoppers will believe about demand at
t 3, EspQn|se
e q denotes her belief about what non-shoppers who search will believe about
demand at t 3, and EspQne|noq denotes her belief about what non-shoppers who do not
search will believe about demand at t 3. A key feature of (2) is that willingness to pay depends on 2nd-order beliefs about demand.
To simplify (2), notice that all shoppers are fully informed, as they observe the price.
Non-shoppers who search will become fully informed by t 2, as they observe the price after searching. Shoppers and non-shoppers who search will therefore hold the same 1st-order
beliefs about demand att3, and these beliefs will be confirmed, as long as consumers follow their equilibrium strategies: Qs
e Q n|se
e Q. Moreover, all consumers realize this is the
case, so shoppers will correctly anticipate these 1st-order beliefs, EspQseq EspQne|seq Q.
All consumers will also correctly anticipate the 1st-order beliefs of non-shoppers who do not
search, EpQne|seq Qne|se. Given (2), a shopper of type θ will therefore buy if
θp λ
α p1αqp1θnq Q p1αqθnQne
¥0. (3)
A non-shopper of type θ who searches will also buy if (3) holds, since she holds the same information as shoppers after searching, and her search costs are sunk.
A non-shopper’s decision whether to search at t 1 depends on her expectation about the price, pe. Given type θ, she will search if and only if
θpe λ
αEnpQseq p1αqp1θnqEnpQen|seq p1αqθnEnpQne|noq
c¥0, (4)
where EnpQs
eq denotes her belief about what shoppers will believe about demand at t 3,
EnpQn|se
att 3, and EnpQn|no
e q denotes her belief about what non-shoppers who do not search will
believe about demand at t3.
Att1, non-shoppers believe that the firm has set the price as expected, ppe, by the
very definition of pe. Thus, non-shoppers believe at t 1 that all consumers will hold the
same beliefs about demand att 3, regardless of whether they will have observed the price. Each non-shopper’s 2nd-order beliefs at t 1 therefore coincide with her 1st-order beliefs:
EnpQs
eq EnpQ n|se
e q EnpQen|noq Qne. Given (4), a non-shopper of type θ will therefore
search if
θpe λQne c¥0. (5)
Network effects: Now suppose that social image concerns are absent, but that the product exhibits network effects. That is, the payoff from buying, for a consumer of type θi, is
θi
loomoon Intrinsic Payoff
λQ
loomoon Network Payoff
loomoonp
Price
. (6)
To facilitate comparison with social image concerns, we again use the parameter λ, but the magnitude of this parameter now captures the strength of the network effect. We again allow
λto take on both positive and negative values, withλ minp21α,1q. Comparing (6) with (1) shows that the network payoff depends on actual demand, whereas social image depended
on other consumers’ beliefs about demand.
From (6), a shopper of type θ who observes price pwill buy if and only if
θp λQse ¥0, (7) where Qs
e denotes the expectation of shoppers about demand. Since shoppers are fully
informed, they can correctly predict demand as long as consumers follow their equilibrium
strategies, so (7) amounts to
θp λQ¥0. (8)
A non-shopper of type θ who expects price pe at t1 will search if and only if
θpe λQne c¥0, (9)
where Qne denotes the 1st-order beliefs of non-shoppers about demand.
To compare consumer behavior with social image to that with network effects, first notice
that (5) is identical to (9). That is, the incentive of non-shoppers to search is the same in
these two settings. What effectively matters for searching, in the end, is a non-shopper’s
1st-order beliefs about demand, at the moment when she must choose whether to search.
Second, notice that (3) differs from (8). The incentive of consumers to buy, conditional
on observing the price, is not necessarily the same with social image concerns as it is with
network effects. The two only coincide if all non-shoppers choose to search. This suggests
that the impact of social image concerns (relative to network effects) on firm pricing will
very much depend on the search behavior of non-shoppers. We explore this in the following
two subsections: the first where we assume that search costs are large, and the second where
we assume that search costs are small.
3.1
Large Search Costs
In this subsection, we assume that search costs c are sufficiently large that both (5) and (9) are violated in equilibrium, for all types θ P r0,1s. This implies that all non-shoppers choose not to search, so that all buyers will be shoppers. Given this, what is demand under
belief-dependent social image concerns and network effects?
Social image concerns: In a setting with social image concerns, where non-shoppers do not search, a shopper of type θ will buy if and only if (3) holds when evaluated at θn 1. This condition is equivalent toθ ¥pλrαQ p1αqQnes. Since type is uniformly distributed onr0,1s, and there are a totalα of shoppers, demand Qsatisfies
This is equivalent to
Qα
1p λp1αqQn e
1λα2
, (10)
which describes how demand depends on the (1st-order) beliefs of non-shoppers.
Non-shoppers know that demand satisfies (10), and believe the firm has set the price as expected,
by the definition of pe. Taking expectations of both sides of (10) yields
Qne α
1pe λp1αqQne
1λα2
.
Non-shoppers therefore believe that demand is equal to
Qne α
1pe
1λα
.
Substituting into (10) yields
Qs.i. α
1p λp1αqα 1pe
1λα
1λα2
. (11)
where the subscript s.i. stands for social image.
Network effects: In a setting with network effects, a shopper of type θ will buy if and only if (8) holds, or equivalently θ ¥ pλQ. Since type is uniformly distributed on r0,1s, and there are a total α of shoppers, this means that demand Q satisfies
Qαp1p λQq,
which implies
Qn.e. α
1p
1αλ
, (12)
where the subscript n.e. stands fornetwork effects.
Comparing (11) with (12) shows the two expressions for demand are equal if and only
if the firm sets the price that consumers expect, p pe. Thus, in an equilibrium with
a particular price p, demand under social image concerns will be equal to demand under network effects (on the equilibrium path), and is given by (12). That is, we can write this
demand as
Qppe pq α
1p
1αλ
However, demand under social image concerns will differ from demand under network effects
off the equilibrium path.
This allows us to formulate the following result regarding equilibrium pricing in the case
of high search costs.
Proposition 1. Suppose that c ¥ c max 12 11αλ,2αλ1α2λ
(
. Then the equilibrium
price under network effects is pn.e. 1{2, and the equilibrium price under social image
concerns is
ps.i. 1α
2λ
2αλα2λ,
where ps.i. ¡ pn.e. if consumers are conformists pλ ¡ 0q and ps.i. pn.e. if consumers are
snobs pλ 0q. The difference between ps.i. and pn.e. is increasing in the magnitude of λ,
B|ps.i.pn.e.|
B|λ| ¡ 0, and is non-monotonic in the proportion of shoppers, with limαÑ0 |ps.i.
pn.e.| limαÑ1|ps.i.pn.e.| 0.
Proof: Appendix 1
When search costs are high, belief-dependent social image concerns generally lead to
a different equilibrium price than the firm would set in a situation with network effects.
Specifically, if consumers are conformists, the equilibrium price will be higher under social
image concerns than under network effects. The reverse is true if consumers are snobs.
Intuitively, under social image concerns, the firm can fool non-shoppers about demand
for its product by deviating in its choice of price. If it deviates to a higher price, then
shoppers will observe the deviation but realize that non-shopper do not. Shoppers therefore
realize that non-shoppers will not adjust their 1st-order beliefs about demand following the
price increase, so that shoppers’ 2nd-order beliefs do not change by very much. If consumers
are conformists, this means that the social payoff from buying can remain high following
the deviation, which makes demand less sensitive to an increase in price. Thus, it is the
firm’s temptation to set a higher price than expected that pushes up the equilibrium price
if consumers are conformists. Similarly, it is the firm’s temptation to set a lower price than
present under network effects, because willingness to pay then only depends on shoppers’
1st-order beliefs.
The size of this effect is increasing in the magnitude of λ because the social image payoff of buyers is proportional to this parameter. When λ is large in magnitude, the firm is more tempted to fool non-shoppers about demand, since fooling them then has a large impact on
the social image of buyers. The size of the effect is non-monotonic in the fraction of shoppers
because there are two opposing effects. On the one hand, when there only few shoppers,
this mean that there are many consumers in the market who can be fooled. However, when
there are few shoppers, a deviation in price will only have a small impact on demand, so
non-shoppers are not fooled by very much.
Holdingλfixed, Proposition 1 applies to situations where search costs exceed a threshold value. An equivalent formulation is as follows: holding search costs fixed, Proposition 1
applies to situation where λ is below a threshold value. Specifically, Proposition 1 applies when λ ¤ minpλn.e., λs.i.q, with λn.e. 22cαc1 and λs.i. αp21cα1qc. For these small values of
λ, no non-shoppers have an incentive to search, either under network effects or social image concerns. In particular, this means that for fixed c ¡ 12, Proposition 1 applies for all λ ¤0 and also for valuesλ¡0 that are sufficiently small. In contrast, for fixedc 12, Proposition 1 only applies for values λ¤0 that are sufficiently large in magnitude.9
Proposition 1 compared equilibrium prices under network effects and social image
con-cerns. An additional result highlights the comparison of the equilibrium profit levels and
consumer payoffs.
Proposition 2. Suppose that c ¥ c max 12 11αλ,2αλ1α2λ
(
. Then profits under
network effects are strictly higher than under social image concerns, πn.e. ¡ πs.i.. Each
consumer’s payoff is higher under network effects than under social image concerns if λ¡0,
and the opposite is true if λ 0.
9In this sense, holding c fixed, Proposition 1 describes features of the equilibrium prices for values ofλ
Proof: Appendix 2
If the firm could commit to a particular price, to maximize its profits given demand
(13), then it would choose to set p 12. This is precisely equal to the equilibrium price under network effects. The equilibrium price under social image concerns always differs from
p 12 implying lower equilibrium profits. In a nutshell, social image concerns generate price distortions which negatively impact the firm’s profits, all because the firm cannot commit not
to fool non-shoppers. The result on consumers’ payoffs follows from the fact that consumers
always benefit from having a lower price, which is consistent with our earlier result that
demand is always downward sloping.
3.2
Small Search Costs
In the previous subsection, we assumed search costs were sufficiently high such that only
shoppers were willing to buy the product. In contrast, we now assume that search costs
c are sufficiently low so that both (5) and (9) hold in equilibrium, for at least some type
θ P r0,1s. This implies that at least some non-shoppers choose to search. Just as in the earlier analysis, we proceed to derive the demand function under social image concerns and
under network effects.
Social image concerns: Under social image concerns, a shopper of type θ will buy if and only if (3) holds, which is equivalent to θ¥pλ
α p1αqp1θnq Q p1αqθnQn e
.
Thus, demand from shoppers is
Qsα
1p λ
α p1αqp1θnq Q p1αqθnQne . (14) A non-shopper of type θ will search if and only if (5) holds, which is equivalent to θ¥pe
λQne c. This means that the number of non-shoppers who search isp1αqp1pe λQnecq.
Since condition (8) is weaker than (9) for all p sufficiently close to the expected pricepe, by
will buy at any pricepin that neighborhood. For any such price, demand from non-shoppers is
Qn p1αqp1pe λQne cq. (15)
We now derive expression for Qn
e and Qse, the (1st-order) beliefs of non-shoppers and
shoppers, which we will then substitute into (14) and (15) to obtain explicit expressions
for demand. At t 1, non-shoppers believe the firm has set the price as expected, by the definition of pe. They also know that demand, Qs Qn, is given by
Qα
1p λ
α p1αqp1θnq Q p1αqθnQne p1αqr1pe λQnecs, (16)
by (14) and (15). Taking the expectation of both sides of (16) gives
Qne α
1pe λ
α p1αqp1θnq Qne p1αqθnQne p1αqr1pe λQne cs.
The beliefs of non-shoppers are therefore
Qne 1pe p1αqc
1λ . (17)
Substituting (17) into (15) and simplifying gives demand from non-shoppers
Qn p1αq
1pe p1λαqc
1λ
. (18)
For pricespin a neighborhood ofpe, all non-shoppers who search will buy: p1αqp1θnq
Qn. The fraction of non-shoppers who search is therefore
1θn
1pe p1λαqc
1λ
. (19)
To obtain demand from shoppers, substitute QQs Qn into (14)
Qsα
1p λ
α p1αqp1θnq pQs Qnq p1αqθnQne . (20)
which implies
Qsα
1p λ
pα p1αqp1θnqqQn p1αqθnQne
1αλrα p1αqp1θnqs
with Qne given by (17), Qn by (18), andp1θnq by (19).
To summarize, we have derived demand from non-shoppers and shoppers under
belief-dependent social image concerns, given by (18) and (21) respectively, for price pin a neigh-borhood of the expected pricepe. Notice that demand from non-shoppers is perfectly inelastic
for prices in this neighborhood.
Network effects: Under network effects, a shopper of type θ will buy if and only if (8) holds, which is equivalent to θ¥pλQs
e. Thus, demand from shoppers is
Qs αp1p λQseq. (22)
A non-shopper of type θ will search if and only if (9) holds, which is equivalent to θ ¥ peλQne c. The number of non-shoppers who search is thereforep1αqp1pe λQnecq.
Since condition (8) is weaker than (9) for all p sufficiently close to the expected pricepe, by
c ¡0, this means there exists a neighborhood of pe such that all non-shoppers who search
will buy at any pricepin that neighborhood. For any such price, demand from non-shoppers therefore satisfies
Qn p1αqp1pe λQne cq, (23)
just as under social image concerns (see (15)).
We now derive expression for Qn
e and Qse, which we will substitute into (22) and (23) to
obtain explicit expressions for demand. At t 1, non-shoppers believe the firm has set the price as expected, by the definition of pe. They also know that demand, Qs Qn, is given
by
Qαp1p λQseq p1αqp1pe λQne cq, (24)
by (22) and (23). Taking the expectation of both sides of (24) gives
Qne αp1pe λQneq p1αqp1pe λQne cq,
or equivalently
Qne 1pe p1αqc
Substituting these beliefs into (23) and simplifying gives demand from non-shoppers
Qn p1αq
1pe p1λαqc
1λ
. (25)
This means that total demand, QQs Qn, is equal to
QQs p1αq
1pe p1λαqc
1λ
. (26)
Shoppers realize that demand satisfies (26), so their beliefs are
Qse Qs p1αq
1pe p1λαqc
1λ
.
Substituting these beliefs into (22) and solving for Qs gives the demand from shoppers
Qs
α
1αλ
p1pq p1αqλ
1λ r1pe p1λαqcs
. (27)
To summarize, we have derived demand from shoppers and from non-shoppers, under
net-work effects, for price p in a neighborhood of the expected price pe. Demand from
non-shoppers is given by (25), which is identical to that under social image concerns. Demand
from shoppers is given by (27), which is not. However, demand from shoppers will coincide
in the two settings if the firm sets pe p. Thus, on the equilibrium path (so evaluating at
pep), with network effects and with social image concerns, demand from non-shoppers is
Qnppe pq p1αq
1p p1λαqc
1λ
, (28)
demand from shoppers is
Qsppe pq α
1p p1αqλc
1λ
, (29)
so that total demand, Qn Qs, on the equilibrium path, is
Qppe pq
1p p1αqc
1λ . (30)
We are now in a position to state our result about equilibrium pricing for situations in
Proposition 3. Suppose that c c min 12 11αλ,2αλ1α2λ
(
. Then the equilibrium
price under network effects is
pn.e. r1cp1αqsp1αλq
p1λqα p1αλq .
There is a unique equilibrium price ps.i. under social image concerns, where ps.i. ¡ pn.e. if
λ ¡ 0 and ps.i. pn.e. if λ 0. The difference between ps.i. and pn.e. is non-monotonic
in the proportion of shoppers, with limαÑ0 |ps.i.pn.e.| limαÑ1|ps.i. pn.e.| 0. When
α¤ 12, this difference is non-monotonic in λ over the domain λP p0,1q, with limλÑ0 |ps.i.
pn.e.| limλÑ1|ps.i. pn.e.| 0. When α ¡
1
2, this difference is non-monotonic in λ over
the domain λ P p0,21αq if search costs are sufficiently small, with limλÑ0 |ps.i. pn.e.|
limpλ,cqÑp1 2α
,0q|ps.i.pn.e.| 0.
Proof: Appendix 3
Equilibrium prices with network effects and social image concerns differ from their
cor-responding counterparts in Proposition 1, but the same logic as before applies: depending
on whether we look at conformists or snobs, either equilibrium prices with social image
con-cerns are higher than with network effects (conformists, λ ¡ 0) or the other way around (snobs, λ 0). The reason is again that social image concerns make the firm tempted to fool non-shoppers who do not search about product popularity, either by deviating to a
marginally higher price if consumers are conformists, or by deviating to a marginally lower
price if consumers are snobs.
More importantly, the difference between the two prices can now be non-monotonic in
the strength of social image concerns, if consumers are conformists. A large λ ¡ 0 means that the firm has a larger incentive to fool non-shoppers who do not search, by deviating
from the expected price. But large λ also has a countervailing effect for two reasons. First, largeλmeans that more non-shoppers will search and observe the price, so there are fewer of them to fool. When there are relatively many shoppers, and when search costs are small, this
many non-shoppers may not search; however, in this limit, strategic complementarities are
so strong that they push the equilibrium price to the same value, namely p1 p1αqc, under both network effects and social image concerns (see expression (30)).
Holding search costs fixed, Proposition 3 applies to situations whereλexceeds a threshold value, specifically when λ ¡maxpλn.e., λs.i.q, with λn.e. 22cαc1 and λs.i. αp21cα1qc. For these
large values ofλ, some non-shoppers have an incentive to search, both under network effects and social image concerns. In particular, this means that for fixed c 12, Proposition 3 applies for all λ ¥ 0, and also to values λ 0 that are sufficiently small in magnitude. In contrast, for fixed c ¡ 12, Proposition 3 only applies for values λ ¡ 0 that are sufficiently large.10
Again, an additional result can be obtained for the equilibrium profits and consumer
payoffs.
Proposition 4. Suppose that c c min 12 11αλ,2αλ1α2λ
(
. Then for any given
λ ¡ 0, profits under network effects are strictly higher than under social image concerns.
However, if c 12, then there exists λ 0 such that profits under social image concerns are
strictly higher than under network effects whenever λ P pλ,0q. Each consumer’s payoff is
higher under network effects than under social image concerns if λ¡0, and the opposite is
true if λ 0.
Proof: Appendix 4
Unlike with large search costs, Proposition 4 shows that profits under social image
con-cerns can exceed profits under network effects. The reason is that commitment issues due to
consumer search push the price up, compared to the level that would be optimal (conditional
on serving a strictly positive measure of non-shoppers) if the firm could commit, which is
p 1p12αqc. This effect is typical in a setting with consumer search; non-shoppers pay
10In this sense, holding c fixed, Proposition 3 describes features of the equilibrium prices for values ofλ
the search cost before observing the price, which leaves the firm tempted to charge a higher
price than expected. The presence of social image concerns pushes up the price further still,
relative to the price under network effects, if consumers are conformists, but pushes down
the price if consumers are snobs.
The two opposing effects when consumers are snobs opens up the possibility that profits
under social image concerns can exceed profits under network effects. That is precisely what
occur when λ 0 is small in magnitude, as both prices then exceed the optimum, but the price under network effect does so to a larger degree.11 In terms of consumer payoffs, we
again have the result that a price increase makes consumers suffer. Thus, we have a range
of parameter values,λ 0 but small in magnitude, for which all players in the game have a (weakly) higher payoff under social image concerns than under network effects.
Figure 1 presents equilibrium prices ps.i. and pn.e. as a function of λ, for different values of the search cost cand of the fraction of informed consumers α. The firm serves a strictly positive measure of non-shoppers under network effects when λ ¡ λn.e. 22cαc1, and does
so under social image concerns when λ ¡ λs.i. αp21cα1qc. Proposition 1 applies when λ
minpλn.e., λs.i.q, and Proposition 3 applies when λ¡maxpλn.e., λs.i.q.12
Figure 1 displays a number of features of the equilibrium prices, over and above those
stated in Proposition 1 and 3. First, it shows that prices are increasing monotonically in λ, including under social image concerns when λ ¡λs.i.. The fact that the difference in prices
ps.i. pn.e. is eventually decreasing in λ, which reflects the non-monotonicity described in Proposition 3, is simply because the price under network effects then increases more quickly
than the price under social image concerns. Second, this non-monotonicity of ps.i. pn.e.
occurs when α ¡ 12 for relatively large values of c, not just for search costs very close to zero. Third, ps.i. ¡ pn.e. holds whenever λ ¡ 0, and ps.i. pn.e. holds whenever λ 0, even for values of λ for which neither Proposition 1 nor Proposition 3 applies, i.e. for
minpλn.e, λs.i.q λ maxpλn.e, λs.i.q. Fourth, frame (a) shows that profits under social image
11Notice that as λ 0 increases in magnitude, Proposition 1 will eventually apply, so that profits under
network effects will exceed those under social image concerns.
12Recall that we assume throughout thatλ minp 1
Figure 1: Equilibrium prices as a function of λ
ps.i
pn.e
-1.0 -0.5 0.0 0.5 1.0λ 0.4 0.5 0.6 0.7 0.8 p
c=0.4,α=0.4
λs.i. λn.e.
(a)
ps.i
pn.e
-1.0 -0.5 0.0 0.5 λ 0.4 0.5 0.6 0.7 0.8 p
c=0.4,α=0.6
λs.i.λn.e.
(b)
ps.i
pn.e
-1.0 -0.5 0.0 0.5 1.0λ 0.4 0.5 0.6 0.7 0.8 p
c=0.5,α=0.4
λs.i.=λn.e.
(c)
ps.i
pn.e
-1.0 -0.5 0.0 0.5 λ 0.4 0.5 0.6 0.7 0.8 p
c=0.5,α=0.6
λs.i.=λn.e.
(d)
ps.i
pn.e
-1.0 -0.5 0.0 0.5 1.0λ 0.4 0.5 0.6 0.7 0.8 p
c=0.6,α=0.4
λs.i.
λn.e.
(e)
ps.i
pn.e
-1.0 -0.5 0.0 0.5 λ 0.4 0.5 0.6 0.7 0.8 p
c=0.6,α=0.6
λs.i.
λn.e.
concerns exceed profits under network effects for all value of λ 0 such that Proposition 3 applies.13
4
Conclusion
Our analysis has focused on how the belief-dependent social image concerns of consumers
affect firm behavior. We have shown that such social image concerns can lead to price
distor-tions with associated consequences for consumer demand and firm profits. We developed our
analysis as a direct comparison between the effects of network effects and the effects of social
image concerns. In comparing the two, we demonstrated that belief-dependent social image
concerns generate novel effects on equilibrium prices and profits, highlighting the subtle role
and importance of consumers’ 2nd-order beliefs.
We focus on a very simple setting with shoppers and non-shoppers. A fruitful direction
of future research might for example explore the impact of advertising on the interaction
between firms and consumers with social image concerns. By targeting advertising to specific
segments of consumers, a firm might be able to create informational asymmetries between
consumers, and in this way influence their belief about the believes of others and hence their
willingness to pay. At the same time, a commitment to advertise widely in equilibrium might
help the firm commit to a more profitable price. For a first step in this direction, see Sebald
and Vikander (2018).
13This follows from the fact that the optimal price under commitment in frame (a), conditional on serving
a strictly positive measure of non-shoppers, isp 1p12αqc 0.38, and both curves lie above this value for all λ ¡maxpλn.e., λs.i.q. Calculations show in fact that profits under social image concerns exceed those
1
Appendix: Proof of Proposition 1
Consider network effects, and a candidate equilibrium where all non-shoppers choose not to
search. Demand is then given by (12). Profits, πpQ, are therefore
π pα
1p
1αλ
.
These profits are strictly concave inp, so the unique equilibrium price is the value ofp that satisfies the first-order condition, dπdp 0. Thus,
α
12p
1αλ
0,
which implies
pn.e. 1
2 (31)
Substituting pn.e. into (12) gives equilibrium demand of
Qn.e. α
4p1αλq
We now verify that all non-shoppers will take their outside option, rather than search and
buy, given pn.e. and Qn.e.. A non-shopper of type θ who searches and buys will earn
θ λα
1pn.e.
1αλ
pn.e.c,
which is increasing inθ. Thus, this payoff is strictly negative for all θ P r0,1s if
p1pn.e.q λα
1pn.e.
1αλ
c 0,
or equivalently
1pn.e.
1αλ c 0,
which holds by pn.e. 12 and c¡ 12 11αλ.
Consider social image concerns, and a candidate equilibrium where all non-shoppers
choose not to search. Demand is then given by (11). Profits, π pQ, are therefore
π pα
1p λp1αqα 1pe
1λα
1λα2
These profits are strictly concave in p, so the equilibrium price is the value of pthat satisfies the first-order condition evaluated at pe p, i.e. dπdp|pep 0. This condition is
α
12p λp1αqα 1pe
1λα
1λα2
0.
Evaluating at pep and solving for pyields
ps.i. 1α
2λ
2α2λαλ, (32)
which is the unique equilibrium price.
We now verify that all non-shoppers will take their outside option, rather than search
and buy, given ps.i. , and given Qs.i., which is equal to (13) evaluated at p ps.i.. Recall that on the equilibrium path, demand under social image concerns is equal to demand under
network effects, for any givenpep. Thus following the same steps as above under network
effects, all non-shoppers will have an incentive to take their outside option if
1ps.i.
1αλ c 0
which holds by (32) and c¡ 2αλ1α2λ. From (31) and (32), write
ps.i.pn.e. 1α
2λ
2α2λαλ
1 2, or equivalently
ps.i.pn.e. λαp1αq
2p2α2λαλq, (33)
which shows that limαÑ0 |ps.i.pn.e.| limαÑ1|ps.i.pn.e.| 0. Moreover, sinceαP p0,1q,
it follows thatps.i. ¡pn.e.ifλ ¡0 andps.i. pn.e. ifλ 0. This also implies that|ps.i.pn.e.|
is non-monotonic in α, since |ps.i.pn.e. | 0, for any αP p0,1q. Differentiating (33) gives
B
Bλpp
s.i.pn.e.q
αp1αq2p2α2λαλq λαp1αq2pα2 αq
or equivalently
B
Bλpp
s.i.pn.e.q
αp1αq
p22α2λαλq2,
which is strictly positive by αP p0,1q. Combined with the fact thatps.i. ¡pn.e. ifλ ¡0 and
ps.i. pn.e. if λ 0, it immediately follows that B|ps.i.pn.e.|
B|λ| ¡0.
2
Appendix: Proof of Proposition 2
Whenpe p, demand under both network effects and social image concerns is given by (13).
This implies profits of
π α
1αλpp1pq.
These profits are strictly concave in p, and attain their maximum at p 12. Thus, the fact that pn.e. 12, and that ps.i. 21 for all λ0, implies πn.e. ¡πs.i. .
Let θ denote the type of shopper who is indifferent between buying and taking her outside option, given price pe p. Demand at this price is therefore αp1θq. We also
know that this demand is given by (13). Thus
αp1θq α
1p
1αλ
,
or equivalently
θppq pαλ
1αλ, (34)
where the notation makes explicit that θppq is a function of p.
By the definition of θppq, we have the following: all consumers of type θ θppq take their outside option and earn a payoff of zero. A consumer of type θ θppq is indifferent about buying and taking her outside option, and so also earns a payoff of zero. A consumer
of type θ ¡ θppq buys and earns a payoff of θθppq ¡ 0, which is the amount by which her intrinsic payoff from buying exceeds that of type θ.
Thus, when λ¡0, each consumer θ ¡θppn.eq earns a strictly higher payoff under network effects than under social image concerns, and each consumer θ ¤ θppn.eq earns zero under both network effects and social image concerns. When λ 0, each consumer θ ¡ θpps.iq
earns a strictly higher payoff under social image concerns than under network effects, and
each consumer θ¤θpps.iqearns zero under both social image concerns and network effects.
3
Appendix: Proof of Proposition 3
Consider network effects, and a candidate equilibrium where some non-shoppers choose to
search. Then for p in a neighborhood of pe, profits are π ppQn Qsq, given (25) and
(27). The equilibrium price p must satisfy the first-order condition for profit maximization, evaluated at pep, i.e. dπdp|pep 0. That is
Qnppe pq Qsppepq p
d
dppQn Qsq 0
Demand atppe, Qppe pq Qnppe pq Qsppepq, is given by (30), which implies
1p p1αqc
1λ p
d
dppQn Qsq 0.
Differentiating (25) and (27) with respect to pand substituting gives 1p p1αqc
1λ
αp
1αλ 0, (35)
and solving for p yields
pn.e. r1cp1αqsp1αλq
1 αp12λq . (36)
To show that (36) is indeed the equilibrium price, we now rule out any non-marginal
devi-ations in price. First consider a deviation to a price ppe such that all non-shoppers who
searched choose to buy. By the same argument used in Section 3.2,Qn is then given by (25)
and Qs by (27), so thatQn Qs is linear inp. Profitsπ ppQn Qsqare therefore strictly
cannot be profitable, since ppn.e. satisfies the first-order condition. More generally, when
pepn.e., a deviation to any price such that Qn is given by (25) andQs by (27) will not be
profitable.
Let θppeqdenote the type of non-shopper who is indifferent about searching and buying,
and taking her outside option of zero, when p pe. That is, θppeq peλrQn Qspp
peqs c, where Qspp peq is (27) evaluated at p pe. After searching and observing
p pe, this type will buy if and only if θppeq p λpQn Qsq c ¡ c, or equivalently
pppeqλrQsQspppeqs ¤c. By (27), this is in turn equivalent topppeqp1 1αλαλq ¤ c.
Thus, all non-shoppers who search will buy following the deviation if p¤pe cp1αλq, in
which case the deviation cannot be profitable.
Now consider a deviation to p ¡ pe cp1αλq, so at least some non-shoppers who
search will not buy. A consumer will then buy, regardless of whether she is a shopper or a
non-shopper, if and only if her type exceeds θppq pλQppq, where Qppq denotes demand at this price. Thus, demand is equal to that in a situation where all consumes are shoppers,
and is therefore given by (27) evaluated at α1, so Qppq 11pλ. Direct comparison shows this demand is equal to Qn Qs, given (25) and (27), when evaluated atppe cp1αλq,
and is strictly lower thanQn Qsfor any higher price. Thus, profits following a deviation to
p¡pe cp1αλqare strictly lower than ifQn were given by (25), and Qs by (27), following
the deviation. It follows that such a deviation cannot be profitable.
We now verify that a strictly positive fraction of non-shoppers will search and buy in
equilibrium, with pricepn.e. given by (36), and demandQn.e.ppepqgiven by (30) evaluated
atppn.e.. A non-shopper of type θ who searches and buys will earn
θ λ
1pn.e. p1αqc
1λ
pn.e.c,
which is increasing inθ. Thus, this payoff is strictly positive for at least some θ P r0,1sif 1pn.e. λ
1pn.e. p1αqc
1λ
c¡0,
That is,
or
1
r1cp1αqsp1αλq
1 αp12λq
cp1αλq ¡0.
This is equivalent to c 12 11αλ, which holds by assumption.
We now verify that a strictly positive fraction of shoppers will not buy in equilibrium,
with pricepn.e.given by (36), and demand Qn.e.ppepqgiven by (30) evaluated atppn.e..
From (29), we require
1pn.e p1αqλc
1λ
1,
or equivalently
pn.e p1αqλc¡λ.
Using (36) to substitute for pn.e, and rearranging, gives the equivalent condition
1αλ
1 αp12λq
r1 p1αqcs ¡0,
which holds by α 1 and λ 1.
Now consider social image concerns, and a candidate equilibrium where some
non-shoppers choose to search. Then profits are π ppQn Qsq, given (18) and (21). The
equilibrium price must satisfy the first-order condition for profit maximization, evaluated at
pep, i.e. dπdp|pep 0. Following the exact same steps as above, under network effects, the
condition is
1p p1αqc
1λ p
d
dppQn Qsq 0.
Differentiating (18) and (21) with respect to pand substituting gives 1p p1αqc
1λ
αp
1αλrα p1αqp1θnqs 0, (37)
with p1θnq P p0,1s given by (19) evaluated at pe p:
1θn
1p p1λαqc
1λ
. (38)
image concerns, we need to rule out any non-marginal deviations in price. First consider a
deviation to p pe such that all non-shoppers who searched choose to buy. Then, by the
same argument used in Section 3.2, Qnis given by (18) and Qsby (21), evaluated at pricep.
Profitsπ ppQn Qsqare therefore strictly concave for any suchp. This means that, when
peps.i., a deviation to any such price cannot be profitable, sincepps.i. then satisfies the
first-order condition. More generally, when peps.i., a deviation to any price such that Qn
is given by (18) and Qs by (21) will not be profitable.
Following the same steps as under network effects, all non-shoppers who search will buy
after a deviation to price p pe if and only if pppeq λrQsQspp peqs ¤ c, with Qs
and Qspp peq given by (21) evaluated at price p and price pe respectively. By (21), this
condition is equivalent to pppeqp1 1αλrα pαλ1αqp1θ
nqsq ¤c. Thus, all non-shoppers who
search will buy following the deviation if p ¤ pe c
1αλrα p1αqp1θnqs
1αλrα p1αqp1θnq1s p
1, in which
case the deviation cannot be profitable.
Now consider a deviation to p ¡ p1, so at least some non-shoppers who search will not buy. A consumer will then buy, regardless of whether she is a shopper or a non-shopper, if
and only if here type exceeds θppq pλQppq, where Qppq denotes demand at this price. Thus, demand is equal to what it would be if a fractionα1 α p1αqp1θnqof consumers were shoppers, who observed pricep, and a fraction 1α1 of consumers were non-shoppers, who do not search, and expect pricepe. That is, demand is given by (11) but withαreplaced
byα1. Direct comparison shows this demand is equal to Qn Qs, given (18) and (21), when
evaluated at p p1, and that it is strictly lower than Qn Qs for any higher price. Thus,
profits following a deviation to p¡p1 are strictly lower than ifQn were given by (18) , and
Qs by (21), following the deviation. It follows that such a deviation cannot be profitable.
that p1θnq P p0,1q holds when evaluated at ppn.e. It follows that the equilibrium price under social image concerns must satisfy ps.i ¡pn.e. if λ ¡0, and must satisfy ps.i pn.e. if
λ¡0.
The left-hand sides of (35) and (37) are also continuous inα,λ, andc, sopn.e.andps.i.must also be continuous in these parameters. Conditions (35) and (37) coincide when evaluated
at α 0 and α 1. Thus, we must have limαÑ0pn.e limαÑ0ps.i. and limαÑ1pn.e
limαÑ1ps.i..
Conditions (35) and (37) also coincide when evaluated at λ 0, so that limλÑ0pn.e
limλÑ0ps.i.. First suppose that α¤ 12, so that minp 1
2α,1q 1. Thus, we have interior
solu-tions for allλP p0,1q. Conditions (35) and (37) directly imply limλÑ1pn.e.limλÑ1ps.i.
1 p1αqc, since otherwise the left-hand side of these conditions would increase or decrease without bound in this limit. Combined with the fact that limλÑ0pn.e limλÑ0ps.i., and that
|pn.eps.i.|is bounded way from zero, for any fixedλP p0,1q, for allc, we have the following:
|pn.eps.i.| non-monotonic inλ, over the domain λ P p0,1q.
Now suppose instead that α ¡ 12, so that minp21α,1q 21α. Thus, we have interior solutions for all λ P p0,21αq. Direct substitution shows that both (35) and (37) hold when evaluated at c 0 and p λ 21α, where 1θn 1 for those parameter values. Thus, for c 0, we have limλÑ 1
2α
pn.e limλÑ1
2α
ps.i. Since both pn.e. and ps.i. are continuous
in c, we also have limpλ,cqÑp1 2α
,0qp
n.e limpλ,cqÑp1 2α
,0qp
s.i. Combined with the fact that
limλÑ0pn.e limλÑ0ps.i., and that |pn.e ps.i.| is bounded way from zero, for any fixed
λ P p0,21αq, for all c, we have the following: for c ¡ 0 sufficiently small, |pn.eps.i.| non-monotonic in λ, over the domain λP p0,21αq.
Notice that the left-hand side of (35) is strictly positive when evaluated at p 0, by
c 12 11αλ 11α. It is strictly decreasing inp, and equals zero at a unique price, namely
ppn.e ¡0, given by (36). Moreover, the left-hand side of (37) must also be strictly positive atp0, since conditions (35) and (37) coincide at that price.
(37). Multiplying both sides of (37) by 1θn P p0,1s gives the equivalent condition
1αλrα p1αqp1θnqs r1p p1αqcs p1λqαp0. (39) The left-hand side of (39) is quadratic in p, by (38). If λ 0, then the coefficient of p2
is positive. Moreover, if λ 0, we know that the left-hand side of (37) does not exceed the left-hand side of (35). Thus, there must be a unique price, over the intervalpP p0, pn.eq, that satisfies (39). We can therefore conclude there is a unique price p that satisfies (37), with
p pn.e..
If λ ¡ 0, then the coefficient of p2 is negative. Moreover, if λ ¡ 0, we know that the
left-hand side of (37) exceeds the left-hand side of (35). Thus, there must be a unique price
p that satisfies (37) and (39), with p¡pn.e..
We still have to show that, at the equilibrium price under social image concerns, which
satisfies (37), at least some non-shoppers will search. Following the same steps as above
under network effects, the condition is
1ps.i.cp1αλq ¡0,
or
ps.i. 1cp1αλq.
To establish this, it is sufficient to show that the left-hand side of (37) is strictly negative
when evaluated at p1cp1αλq.
Notice, from (38), that 1θn 0 when p 1cp1αλq. Thus, the left-hand side of (37) reduces to
1p p1αqc
1λ α
p
1α2λ.
Substituting p1cp1αλqand simplifying gives the condition
αcα
1cp1αλq
1α2λ
0.
4
Appendix: Proof of Proposition 4
We first prove the result regarding consumer payoffs. Letθsdenote the type of shopper who is indifferent between buying and taking her outside option, given price pe p. Demand
from shoppers at this price is therefore αp1θq. We also know that this demand is given by (29). Thus
αp1θq α
1p p1αqλc
1λ
,
or equivalently
θsppq p p1αqλc
1λ , (40)
where the notation makes explicit that θsppq is a function of p. Let θn denote the type
of non-shopper who is indifferent between buying and taking her outside option, given price
pep. Demand from non-shoppers atpe p, p1αqp1θnq, is given by (28). Thus
p1αqp1θnq p1αq
1p p1λαqc
1λ
,
which simplifies to θnppq θsppq c.
By the definition of θsppq, we have the following: all shoppers of type θ θsppq take their outside option and earn a payoff of zero. A shopper of type θ θsppq is indifferent
about buying and taking her outside option, and so also earns a payoff of zero. A shopper of
typeθ ¡θsppqbuys and earns a payoff ofθθsppq ¡0, which is the amount by which her
intrinsic payoff from buying exceeds that of type θsppq. The same holds for non-shoppers, but with θsppq replaced by θnppq θsppq c.
We know from Proposition 3 that pn.e ps.i if λ ¡ 0, and that pn.e ¡ ps.i if λ 0. It therefore follows from (40) that θspp
n.eq θspps.iq if λ ¡ 0, and θsppn.eq ¡ θspps.iq
if λ 0. Thus, when λ ¡ 0, each shopper of type θ ¡ θsppn.eq, and each non-shopper of type θ ¡ θspp
n.eq c, earns a strictly higher payoff under network effects than under
social image concerns; and each shopper of typeθ ¤θsppn.eq, and each non-shopper of type
θ ¤ θspp
n.eq c earns zero under both network effects and social image concerns. When
earns a strictly higher payoff under social image concerns than under network effects; and
each shopper of typeθ ¤θspp
s.iq, and each non-shopper of type θ ¤θspps.iq cearns zero
under both social image concerns and network effects.
We now turn to the results on profits. When pe p, demand is given by (30), with
profits
π
1p p1αqc
1λ
p1pq.
These profits are strictly concave in p, and attain their maximum at p 1p12αqc. Thus, these profits are strictly decreasing in p whenever p¡ 1p12αqc, and strictly increasing in p
whenever p 1p12αqc. From Proposition 3, pn.e.¡ 1p12αqc is equivalent to
r1cp1αqsp1αλq
p1λqα p1αλq ¡
1 p1αqc
2 .
Simplifying shows that this inequality indeed holds, for any fixed λ, by α 1.
Proposition 3 shows that λ ¡ 0 implies ps.i. ¡ pn.e. , and hence πn.e. ¡πs.i. . Proposition 3 also shows that λ 0 implies ps.i. pn.e. . This will in turn imply πs.i. ¡πn.e. , as long as
ps.i. ¡ 1p12αqc.
Suppose that c c min 12 11αλ,2αλ1α2λ
(
when evaluated at λ 0, so that Proposition 3 applies for at least some values λ 0. This condition is equivalent to c 1
2.
We know thatps.i. and pn.e. are continuous in λ, and in the limit as λapproaches zero, they both tend to the same value, namely 1p11ααqc. Since 1p11ααqc ¡ 1p12αqc, it must be that
ps.i. ¡ 1p12αqc holds, forλ 0 sufficiently close to zero. That is, there existsλ 0 such that profits under social image concerns are strictly higher than under network effects whenever
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