Pro…tability of Pay-what-you-like Pricing
Jose M. Fernandez and Babu Nahata
Department of Economics, College of Business
University of Louisville, Louisville, KY 40292.
emails jmfern02@louisville.edu; nahata@louisville.edu
1
Pro…tability of Pay-what-you-like Pricing
In this paper, we analyze a pricing strategy where consumers choose their own price
including zero. Our analysis shows that in this pay-what-you-like pricing strategy even
though consumers have an option to free-ride, not all will free-ride and some pay a positive
price. Even more surprising, we show that, under low marginal cost and for a low demand
elasticity range, pay-what-you-like pricing can achieve higher pro…ts for the seller compared
to uniform pricing. When price setting is costly and price discrimination is not feasible,
the pay-what-you-like option can serve as an alternative method to practice a form of price
discrimination that is similar to …rst-degree, but without incurring the cost associated with
identifying consumer types.
Keywords: pay-what-you-like pricing; price discrimination; microeconomics
1
Introduction
Pay-what-you-like pricing (PWYL) allows consumers to choose their own price, including
zero, and the …rm accepts whatever price is paid for a product or service. Although unorthodox, this form of pricing has existed for quite some time (e.g., village doctors and priests).
It has recently gained more attention in the popular press after the British band Radiohead
o¤ered their album In Rainbows to consumers on a pay-what-you-like basis in 2007. Between
October 1-29, 2007, 1.2 million people worldwide visited the website and among those who
downloaded the album, 38 percent worldwide and 40 percent in the U.S., willingly paid.
Free-riders were as prevalent in the U.S. as in the rest of the world, but in the U.S. a paying
customer paid $8.05 compared to $4.64 paid by his international counterpart.1 The success
of the album In Rainbows encouraged other …rms to o¤er their products using this form of
pricing. For example, the publisher of Paste magazine also adopted the same pricing policy
1
See http://www.inrainbows.com; and http://comscore.com/press/release.asp?press=1883; and the
Wall Street Journal, October 3, 2007, p.C14. The band also included a $2.00 transaction fee.
2
and subscribers can pay what they like for a year’s subscription to the magazine. Today,
we observe this pricing strategy among musicians, co¤ee houses, restaurants, theatres, and
magazines, but the economic rationale for its use compared to more traditional forms of
pricing has mostly remained unexplored.
The two most relevant questions for both theoretical and empirical analyses related to
PWYL pricing are: (1) what motivates consumers to pay when they have an option to
free-ride?2 and; (2) recognizing that the PWYL option may possibly result in losses, what
motivates the seller to o¤er such a pricing option?3 We provide some answers to both these
questions based on a theoretical economic model supported by real life anecdotes.
In the economics literature, such pricing strategies are analyzed as problems of multidimensional screening where information about willingness to pay is asymmetric (Rochet and
Stole 2003). From screening considerations, PWYL pricing and bu¤et pricing (or ‡at-fee
pricing) represent the two polar extremes. Under PWYL, the buyer decides how much to
pay for a given quantity. The opposite is the case with bu¤et pricing, where the buyer decides how much to consume for a given …xed fee (Nahata et al. 1999) . Under bu¤et pricing,
in spite of consumers having an option of unlimited consumption, they consume only a …nite
amount. Similarly, under PWYL pricing, even with a free-ride option, not all buyers select
to free-ride, as was the case with the Radiohead o¤er. Although not based on the traditional
pro…t-maximization assumption, both types of pricing strategies can generate higher pro…ts
than commonly used pricing strategies based on pro…t maximization. These strategies rely
on consumer self selection. Consumers freely self-select the quantity consumed in bu¤et
2
On CNN’s American Morning, John Roberts asked the question to the owner of the Java Street Cafe
in Kettering Ohio, who uses PWYL pricing: what prevents a customer to either free ride or pay a very
low price? The owner responded, “...When someone’s at the counter and you say, you get to pay what
you think is fair, very few people are going to take advantage of that situation.” (CNN March 17, 2009,
http://www.cnn.com/2009/US/03/017/lippert.quanda/#cnnSTCText
3
Wall Street Journal (August 28, 2007, p. B8) reports that the motivation for the owner of Terra
Bite Lounge in Kirkland, Washington for doing away with set prices was that PWYL pricing can be both
pro…table and charitable way of doing business. Further, “marketing buzz such a scheme generates can help
stand out from the pack.”
3
pricing and the payment under PWYL pricing.
Historically, pay-what-you-like pricing has existed for certain types of services. For example, in most Indian villages, the village priest accepts whatever the host pays for many
ceremonies he performs such as naming a newborn, performing a marriage, or other religious
services. This tradition still continues. Doctors in rural India are still paid based on how
much a patient can a¤ord. In the United States, it is common place to observe church parishioners practicing PWYL pricing when providing contributions (donations) to the church in
support of the services/programs o¤ered by the church.4
In 2005, the New Yorker magazine reported that in the Village, a quite popular place for
both the visitors and the residents of New York City, a restaurant named Babu used a menu
without any price. After …nishing their meals consumers paid what they liked.5 The owner
found that most consumers did pay, and some even paid considerably more than what the
owner expected.6 There were also a few cases of free-riders. Eventually, the owner did switch
to a menu with listed prices. On the other hand, since 1979, a small (maximum capacity
of about 10 people) and exclusive Japanese restaurant, Mon Cheri, in an expensive area of
Fukuoka City, Japan, has consistently maintained the PWYL pricing for dinners. The small
and intimate environment of this restaurant together with personal interactions could be the
reasons for this sustained use of PWYL pricing.
Kim, Natter and Spann (2009) o¤er the …rst empirical study of PWYL pricing. Based
on three …eld experiments in Germany, the authors report several interesting …ndings about
consumers’responses to this form of pricing. In this study, three di¤erent sellers o¤ered three
di¤erent products for sale and consumers could choose any price they like to pay, including
4
Some donations are made out of guilt. Gruber (2003) shows that religious attendance and religious
givings are substitutes.
5
See Rebecca Mead, the New Yorker, March 21, 2005, for other details. Cabral (2000, p.185) also
mentions a restaurant in London that does not list prices in the menu; and each customer is asked to pay
what he or she thinks the meal was worth. Recently, many cafes and restaurant in the US have also adopted
this practice.
6
Lynn (1990) argues that some customers in a restaurant pay more to avoid the impression of looking
cheap.
4
zero. Although there was a wide distribution of payments made by consumers, surprisingly
no one did free-ride and all consumers paid a positive price. Based on the study, the authors
conclude that a consumer’s willingness to pay depends mainly on two factors: (i) an internal
reference price for each consumer; and (ii) a proportion of consumer surplus a consumer is
willing to share with the seller. Our analysis formally incorporates these two factors.
Except for the above mentioned …eld study, we are not aware of any other empirical
or theoretical analyses related to this particular form of pricing, although there is a vast
body of literature on public goods. The striking similarity between the PWYL o¤er and
the private provision of a public good is that in both cases no one can be excluded from
consuming the good. In spite of this similarity, the two situations are very di¤erent. First, a
public good will not be provided by the market unless there is some mechanism to pay for its
provision. Certainly, this is not the case with the goods that are being o¤ered using PWYL
option. Firms have an incentive to o¤er the good under both the PWYL option or use some
other form of pricing. Second, the choice of PWYL is endogenous as the …rm compares
the expected pro…ts with some other pricing strategy that is based on a pro…t-maximizing
solution, for example, uniform pricing. No such pro…t incentive exists in the provision of a
public goods.
Kim et al. (2009) present several behavioral factors that may dissuade consumers from
free-riding. Based on the literature from psychology, marketing, and experimental economics,
they posit four factors a¤ecting consumers’decision to pay. The most common reason given
in favor of paying, as opposed to free riding, is social-norms, which has also been considered as
one of the reasons for tipping.7 The other factors that may dissuade free riding are: avoiding
the appearance of looking cheap; fairness and reciprocity; and altruism.8 Further, based on
the estimation results, the authors conclude that the …nal prices paid were in‡uenced by (a)
7
Tipping for services (e.g, taxi, waiter etc.) is not considered a social norm in Japan and hence tipping
is almost non-existent in Japan. See also Conlin et al. (2003).
8
Kim, Natter and Spann (2009) survey the literature quite extensively. To conserve space, we avoid the
duplication and urge the interested readers to refer to their paper and the references included.
5
fairness, (b) satisfaction, (c) market price awareness, and (d) net income (p. 53).
Our main contribution is to derive the conditions under which PWYL pricing can achieve
higher pro…ts than a pro…t-maximizing uniform price. We derive these conditions under two
di¤erent sets of assumptions about consumer behavior: consumer guilt and repeat buyers.
To the best of our knowledge, no such criteria for pro…t comparison has been put forward
for PWYL pricing.9 In addition, we also explore how PWYL pricing becomes an attractive
alternative compared to other pricing strategies when price setting is quite costly. Wernerfelt (2008) explains the practice of “class pricing” where many products within a “class”
are priced the same (e.g., a dollar store) when costs of setting price of each good is quite
signi…cant.
Our conditions (Proposition 2) show that when consumers have some guilt from paying
a price lower than some reference price; and the marginal cost is either zero or very small
and the price elasticity of demand evaluated at the pro…t maximizing uniform (reference)
price is within a narrow range (between one and two) PWYL can be more pro…table than
uniform pricing.
The rationale behind this result is simple and appealing. Under a uniform price, the
maximum level of pro…ts occurs when the marginal cost of production is zero. At this price,
the elasticity of demand equals its lower limit of one. Further, the linear demand case yields
the special result that both the consumer surplus and the deadweight loss are equal to each
other and each is one-half of the maximum pro…t. This signi…cant amount of deadweight loss
cannot be captured as pro…ts using commonly observed and analyzed linear or nonlinear
pricing strategies, especially in the presence of large degree of consumer heterogeneity. However, PWYL pricing creates an opportunity for eliminating the deadweight loss by sharing
it between the consumers and the seller, thus making it pro…table for the seller. At the
9
As far as pro…tability of bu¤et pricing is concerned, several papers have argued that when the savings
from the transaction cost (e.g., cost of monitoring the consumption etc.) can o¤est the cost of extra production then bu¤et pricing can be more pro…table than a pro…t-maximizing two-part tari¤. More formally,
the criterion proposed states that if the total cost under two-part tari¤ exceeds the net total cost (total
production cost less the savings from transaction cost) then bu¤et pricing is more pro…table than a two-part
tari¤ (Proposition 1 in Nahata et al. 1999).
6
same time elimination of deadweight loss increases the consumer surplus for the existing
consumers and creates surplus for those who were previously excluded from the market. For
consumers who bought at the uniform price, the PWYL option (which allows them to pay
even lower than the uniform price) are unambiguously better o¤ because their surplus will
increase. But, because of loss in pro…ts from the existing consumers, the seller becomes
worse o¤.
On the other hand, the PYWL option opens up the market by now including those
consumers who were previously excluded from the market under the uniform price. This
opening up of the market creates an opportunity for both the seller and entrants to share
the deadweight loss. These new entrants gain surplus by paying a price lower than their
willingness to pay. The seller gains additional pro…ts from payments made in excess of the
marginal cost. Compared to uniform pricing, the seller’s pro…t will be higher under PWYL
if the pro…t earned from the elimination of the deadweight loss is large enough to o¤-set
the revenue lost from existing consumers. The logic is similar under positive but still very
small marginal cost (see Fig. 1). Thus, whenever PYWL pricing is pro…table, it is also
Pareto improving. It is socially optimal when the marginal cost is zero, but under positive
marginal cost because the total consumption exceeds the social optimal, it is not the …rst
best outcome.
If we ignore guilt, but instead consider that consumers are loyal to the …rm, then we
can show that under su¢ ciently low marginal cost and a su¢ ciently high proportion of loyal
customers PWYL pricing becomes a viable and pro…table alternative to uniform pricing.
Generally, in a small community local …rm is frequented by many loyal customers and when
in some cases the proportion of loyal customers is large enough, PWYL may become more
pro…table.
The PWYL option also allows …rms to practice price discrimination when it is typically
considered infeasible. For example, when consumer heterogeneity is large and …rms cannot
separately identify consumer types (or doing so is very costly), then uniform pricing becomes
7
the most appealing pricing option because screening consumers to practice either second- or
third-degree price discrimination is infeasible. But, under uniform pricing the deadweight
loss remains signi…cantly high compared to either second or third degree price discrimination. Since PWYL pricing cannot exclude consumers based on price, this option makes the
potential level of market participation the highest. Further, since consumers choose how
much to pay for the good or service themselves, the seller does not have to incur any cost
associated with …nding potentially pro…table and at the same time easily implementable
screening mechanism. Lastly, the payment schedules of these consumers mimics an indirect
version of …rst-degree price discrimination.
The rest of the paper is organized as follows. § 2 presents a model based on the assumption
that consumers derive disutility from paying a price lower than some reference price and
compares pro…ts under PWYL with uniform pricing. We extend the model by incorporating
the cost of price setting. In § 3, we present another model based on repeated interactions
between consumers and the …rm and obtain similar results. § 4 contains discussions and
summarizes the results.
2
Model
In this section, we present a game theoretical model considering “guilt” or “fairness”
in a consumer’s utility function and demonstrate how guilt, resulting from paying below a
known reference price, can make consumers pay even in the presence of a free-ride option
(Proposition 1). Our Proposition 2 states the necessary conditions for PWYL pricing to
generate higher expected pro…ts than uniform pricing. We extend the model by incorporating
the costs of price setting. When price setting is costly, there is an incentive to use pay-whatyou like pricing when the market is small and/or pricing cost is large (Proposition 3).
8
2.1
Model with Guilt
Traditional economic models view consumers as self-interested agents.
Under PWYL
pricing, these consumers would select a price of zero ignoring any cost these actions could
impose on others. Yet, there is substantial evidence from the behavioral economics literature
suggesting that consumers do care for the welfare of others (Thaler 1992; Camerer 2003) and
of local …rms (Duncan and Olshavsky 1982). For this reason, we consider a model incorporating consumer guilt. Let there be N consumers in a market for a highly di¤erentiated
good. Consumer i’s utility function is
Ui =
vi
vi
pi
pi
(pu
(vi
pi )2 vi pu
pi )2 vi < pu
(1)
where vi is consumer i’s private value for the product, pi is the price paid by consumer i,
pu is some reference price which is assumed to be equal to the uniform price, and the “loss”
parameter
price.
captures the degree of guilt associated with paying a price below the uniform
Consumers’ values are assumed to be distributed uniformly and have a support
vi 2 [0; v]. Further, let G(v) =
v
v
be the cumulative density function over consumer values.
In our formulation, the …rst term in (1) captures a consumer’s surplus. The second term
internalizes the loss from guilt and also highlights the importance of the reference price.
In our speci…cation of consumer’s utility, the loss parameter is identical for all consumers
(we relax this assumption later) and has a support
self-interested when
2 [0; 1): A consumer is said to be
= 0: A consumer experiences guilt when
> 0; and she pays an
amount less than the uniform price (or the private value), pi < min[vi ; pu ]. The cost of guilt
is proportional to the additional amount of surplus received from paying below the uniform
price.
In the case of consumers who are previously excluded from the market, this cost
is proportional to their gross surplus as they would have received a surplus of zero under
the prevailing uniform price. Note, a consumer will never pay an amount greater than the
uniform price (or their private value). If vi
pu , then a consumer can always purchase the
9
product without guilt at the uniform price. If vi < pu ; then the consumer’s contribution
cannot be greater than their willingness to pay, vi :
2.2
Structure of the Game
The game is played in two stages. First, the …rm chooses a pricing strategy: uniform
pricing or PWYL pricing.
Second, consumers observe the pricing strategy and choose
to participate. Under uniform pricing, all consumers with private values greater than the
uniform price enter into the market and pay the uniform price, vi
pu . Under PWYL pricing
all consumers enter into the market and choose a price that maximizes their utility.
We
assume consumers make payment decisions independent of other consumers. They neither
observe the private values of other consumers nor the …rm’s cost of production. The reason
for this structure is that, in reality, consumers do not formally or informally cooperate with
other consumers when making a purchase. Consumers simply observe the pricing options
and make choices that maximize their utility. We seek a subgame-perfect equilibrium in pure
strategies under asymmetric information: the …rm’s payo¤ is taken over the expectation of
consumers’ strategies.
The equilibrium of the game is found using backward induction.
The …rm solves the consumer’s problem under both pricing strategies and then chooses the
pricing strategy that yields the highest expected pro…ts.
2.3
Consumer Behavior
Consumer behavior under uniform pricing is de…ned over a discrete strategy space. Consumers choose either to participate or remain out of the market.
when vi
Consumers participate
pu and pay pi = pu : Regardless of the value of ; consumers do not experience
guilt under uniform pricing as the reference price is equal to the uniform price. It is important to highlight the presence of consumers who have positive value for the good vi > 0,
but are locked out of the market because vi < pu : If the …rm could price discriminate, then
pro…ts would increase by including consumers whose values are greater than the marginal
10
cost, vi > c:
Consumer behavior under PWYL pricing is de…ned in a continuous strategy space. All
consumers participate in the market, but each consumer pays a di¤erent price, pi .
The
optimal contribution amount maximizes utility conditional on guilt. A consumer’s marginal
utility with respect to price, pi ; is given by the following equation.
@U
=
@pi
1
2 (pi
(2)
min [pu ; vi ]) = 0.
The optimal price, pi ; which maximizes utility, is
pi =
pu
vi
1
2
1
2
vi pu
vi < p u
(3)
where pi is strictly increasing with respect to the guilt parameter, :
From Equation (3) we state the following proposition.
Proposition 1.
In the presence of guilt, pay-what-you-like pricing results in free riding
behavior or pi = 0 8 i when
1
2pu
= . Consumers with private values vi 2 0; 21
always
free-ride:
Remark. The cuto¤ value
represents the minimum level of guilt required in order to
make positive contributions under PWYL pricing. In the presence of guilt, there still exists
an incentive to free-ride. The participation constraint requires Ui
0. Given the functional
form of consumer’s utility function, free-riders have positive utility when their private values
are represented by vi 2 0; 1 : Among these consumers, those with vi <
1
2
and consumers with vi
1
2
will free-ride
are better o¤ making a small contribution in accordance with
Equation (3), but all consumers within this bound do participate as the minimum utility
gained from free riding is zero. A …rm can e¤ectively stop free riding by requiring a minimum
contribution equal to
1
2
.
11
2.4
Firm Behavior
The …rm chooses between two pricing strategies: uniform pricing or PWYL pricing.
Under uniform pricing, the …rm chooses a price, pu ; that maximizes pro…t. The expected
pro…t function, E [
u] ;
is
E[
u]
= maxN [1
G (pu )] (pu
pu
c)
(4)
F
where c is the marginal cost of production and F is the …xed cost.
The expressions for
equilibrium price, quantity, and pro…ts under uniform pricing are well known and are given
below.
pu =
v+c
8 i; q = N
2
v
c
2v
; and
u
=
N
(v
4v
c)2
F.
(5)
Note, the …rm’s strategy space is continuous as it chooses a single price on the support
pu 2 R+ . In the case of PWYL pricing, the …rm’s decision is reduced to a discrete choice
of entry or exit. The roles of the …rm and the consumer are reversed under PWYL pricing.
Furthermore, the …rm chooses to relinquish its property rights for the good instead of selling
those rights away.
The expected pro…t function under PWYL pricing has the following form
E[
where revenue, R =
N
P
pwyl ]
= E [R]
cN
(6)
F
pi ; remains uncertain and total costs are pre-determined as no con-
i=1
sumer can be turned away.
Having previously de…ned consumer behavior under PWYL
pricing. The expected total revenue is:
2
6
6
E [R] = N 6[1
4|
G (pu )] pu
{z
vi pu
1
2
}
+ G (pu )
|
G
1
2
Z
pu
1
2
vi
{z
vi <pu
12
1
2
G0 (vi )
G(pu ) G( 21
3
7
7
dvi .7 (7)
)
5
}
The expected revenue function consists of two parts. The …rst term measures the revenue
obtained from consumers who would have otherwise participated in the market under uniform
pricing. Note that these consumers have di¤erent private values for the good, but each makes
the same contribution.
Furthermore, the amount paid cannot exceed the uniform price
implying that PWYL pricing can only increase welfare for these consumers. The uniform
price creates a lower bound on the utility function as no consumer can be made worse o¤
under PWYL pricing compared to uniform pricing.
The second term in (7) accounts for the revenue obtained from those consumers who
would not have been served with uniform pricing. Only consumers with private values in
excess of
1
2
will contribute as the rest have an incentive to free-ride. Payments made by
these individuals can be viewed as an indirect form of …rst degree price discrimination. Each
consumer pays an amount which is a monotonic transformation of their true private values.
Therefore, PWYL pricing allows a …rm to practice …rst degree price discrimination without
the knowledge of consumers’ true willingness to pay. However, such pricing could become
more costly as no consumer can be turned away. The expected revenue function simpli…es
to
E [R] = N
h
(4v
2pu
1)(2pu
8v
2
1)
i
where revenues are increasing both with respect to the uniform price,
and guilt,
2.5
@E[R]
@
=
N
2
1
1
2v
(8)
@E[R]
@pu
=N 1
pu
v
0;
0.
Pro…ts Comparison
In this section, we derive conditions to compare pro…ts under uniform and PWYL pricing
in terms of three parameters namely the guilt parameter ; valuation v; and the marginal
cost of production c such that PWYL pro…ts exceed pro…ts under uniform pricing. PWYL
pro…ts dominate uniform pro…ts,
pwyl
>
u;
when the following inequality is satis…ed
13
N
(4v
2pu
1) (2pu
1)
2
8v
c
F >
N
(v
4v
c)2
F.
(9)
In order for this inequality to hold, the lower and the upper bounds on the guilt parameter
can be founds as:
2v
p
1
2c2 +4cv+6v 2 8vpu +4p2u
<
< 1:
(10)
Derivation is included in appendix. Substituting for the uniform price, pu =
bound on
v+c
,
2
the above
simpli…es to the following inequality
2v
In essence, the bound on
p
1
3c2 +2cv+3v 2
<
< 1:
(11)
indicates that the level of guilt must increase with marginal
cost for PWYL pricing to be more pro…table than uniform pricing. For this reason, PWYL
pricing is less likely to be observed for products with a high marginal cost of production.
The simple intuition is that as the marginal cost increases, the marginal pro…t loss from
free-riders must also increase. Therefore the number of free-riders must decrease in order to
compensate for this pro…t loss. Consumers with private values vi 2 0; 21
always free-ride.
1
2
c: Therefore, an
The costs that free-riders impose on the …rm are equal to N Pr vi <
increase in the guilt parameter ( ) is required to reduce the number of free-riders and o¤set
the increase in marginal cost. But, when the uniform price increases, the required level of
guilt needed for PWYL pricing to be more pro…table decreases. The uniform price (reference
price) represents the opportunity cost of consumer in participating in the pay-what-you-like
option.
A higher uniform price has two e¤ects on consumers: (1) the opportunity cost
of purchasing the product “guilt free” increases and; (2) at a higher uniform price more
consumers are excluded from the market. These two e¤ects encourage the use of the PWYL
in order to attract these consumers.
In Proposition 2, we state the necessary conditions for PWYL pricing to be more profitable than uniform pricing.
14
Proposition 2. If pro…ts under pay-what-you-like pricing were to exceed the pro…ts under
uniform pricing then: (i) the marginal cost c <
v
3
or; (ii) the price elasticity of demand at
the pro…t-maximizing uniform price in equilibrium must be less than 2 .
PROOF.
From the bounds on
(Equation. 11), it is easy to see that as the marginal cost ap-
proaches v3 , the minimum level of guilt for PWYL pricing to be more pro…table approaches
in…nity c =
v
3
=)
will be satis…ed only when c < v3 : The
= 1 . The upperbound on
second part of the proposition follows from the Lerner’s Index which at the equilibrium
pro…t-maximizing uniform price can be written as
1
=
pu c
,
pu
number) is the price elasticity of demand. Noting that pu =
the price elasticity of demand
where
v+c
2
(de…ned as a positive
and c < v3 , it follows that
< 2:
The results from this section can be explained by Figure 1. Pro…t under uniform pricing
is the sum of Area I and Area II. Pro…t under pay-what-you-like pricing is Area II + Area
III - Area IV. Therefore, the di¤erence in pro…ts is Area III - Area I - Area IV. If the
marginal pro…t gained from the new entrants (Area III) is greater than the revenue lost from
the existing consumers (Area I) and the associated cost due to expanding the market (Area
IV), then PWYL pricing is more pro…table than uniform pricing. It is worth noting that (i)
as guilt increases (an increase in ); Areas I and IV decrease but Area II increases allowing
the …rm to capture more of the deadweight loss with an increase in consumer guilt; (ii)
as marginal cost approaches zero, c ! 0; PWYL pricing o¤ers the Pareto e¢ cient outcome
and there is no deadweight loss in equilibrium. This happens without incurring any cost for
expanding the market (Area IV disappears).
15
Figure 1: Comparing pro…t if Area III - Area I - Area IV >0, then
2.6
pwyl
>
u.
Costly Price Setting
Thus far we have assumed that setting a uniform price or the use of any other pricing
strategy is costless. However, an optimal price setting requires demand estimation which in
turn requires market research. According to IBISWORLD, the US market research industry
generated over $14 billion of revenue in 2009.10
The cost of pricing is very relevant in
particular for experience goods or for launching a new product. For example, an aspiring
musicians may have little or no information about their fans’willingness to pay, or the quality
of their own product. Suppose consumers’ values are uniformly distributed v ! U (0; a),
but a may take on only two values a 2 f1; 2g where Pr(a = 1) = :5 and Pr(a = 2) = :5: In
this case, the …rm is uncertain about both consumers’valuations for the good and whether
to serve high-demand consumers (a = 2) or low-demand (a = 1) consumers. Since it may
be quite costly to determine the optimal price given the uncertainty about consumer values
and the value of a, …rms may choose the less costly option of using PWYL pricing to
10
http://www.ibisworld.com/industry/default.aspx?indid=1442
16
reduce uncertainty. Thus a potential source of savings associated with PWYL pricing is the
reduction in the cost related to setting a price because now consumers, and not the …rm,
bare the cost of pricing.
We demonstrate this cost savings by allowing the …xed cost to vary between the two
pricing options. In Equation (9) let the …xed cost associated with PWYL pricing be F =
, where
F
is the cost of pricing . As in the previous section, we solve for the minimum
level of guilt such that
pwyl
>
u
as a function of the marginal cost, consumer valuation
v; the market size N , and the cost of pricing : Incorporating the cost of pricing reduces
the minimum level of guilt for pro…tability and increases the likelihood of the use of PWYL
pricing. The lower bound on
is given by
>
2v
p
1
3c2 +2cv+3v 2
8v
N
.
(12)
Proposition 3. When price setting is costly then there exists an incentive to use paywhat-you like pricing when the market is small (low N) and/or pricing cost are large (high
q
2
< 1.11
). The upper bound on marginal cost is c < 3 v(vNN+6 ) v3 = c for
Proof. See appendix.
This result provides insight into why some of the …rms mentioned earlier use PWYL
pricing. The exclusive restaurant in Japan is quite small (low N) thereby amplifying the
cost savings associated with price setting. On the other hand, in the case of Radiohead,
both consumer heterogeneity and the market size are quite large. Yet, PWYL pricing is
adopted because both the costs of market research and the screening cost for price setting
are signi…cant considering that music is an experience good (high ).
These same cost savings could also a¤ect entry.
market only if the expected pro…ts are positive.
On the margin, a …rm enters into a
One can argue that a …rm that would
not have entered the market because a uniform price would have resulted in negative pro…ts
11
Note that
@c
@
=2
q
v
N (N v+6 )
> 0 and
@c
@N
=
N
q
17
v
N (N v+6 )
< 0.
(
u
< 0) may enter under PWYL pricing as the cost saving without market research may
be signi…cant enough to result in positive pro…ts (
pwyl
> 0).
Further, PWYL pricing is an experiment for “learning.”A …rm could learn the demand
for the product by observing the sequence of consumer payments. Recall that each payment
is a monotonic function of the consumer’s true value. The …rm can deduce a consumer’s true
value from her payment or use a Bayesian method to update beliefs on consumer’s willingness
to pay. For this reason some …rms may abandon the PWYL option as they become more
established and have learned the demand for their product. This may have been the case
with the restaurant Babu in the Village that subsequently switched to a uniform price after
initially practicing PWYL pricing.
Our results and arguments are similar to those made for class pricing, which also attempts
to reduce costs associated with pricing. Wernerfelt (2008) motivates the use of “class pricing”
as a method to reduce the cost associated with selecting individual prices for similar products
(i.e., one tuition fee for a variety of university classes). Class pricing is preferred in markets
with a small number of buyers, a large number of product versions, and a small variance in
costs. Class pricing and PWYL pricing share some of the same characteristics in reducing
the cost associated with pricing within small markets (low N), but di¤er with respect to
the ex ante di¤erence in demand.
PWYL pricing is preferred when there is a great deal
of consumer heterogeneity where as class pricing is preferred when there is little ex ante
di¤erence in demand between versions.
So far, we have assumed that every consumer has the same level of guilt or sense of
fairness, : Dubner and Levitt (2004) show consumers have varying beliefs about guilt
and/or fairness. The article describes payment di¤erences between employees who purchase
bagels based on the honor system. In this case, high paid executives are less likely to pay
than lower paid employees within the same …rm. For this reason, we relax the assumption
of identical guilt among consumers. Instead assume Pr ( = 0) = 1
and Pr
=b = :
Adding uncertainty to the guilt parameter unambiguously decreases pro…t under pay-what-
18
you-like pricing, but does not a¤ect pro…t under uniform pricing.
Expected pro…t under
pay-what-you-like pricing is
E
h
2
i
( )pwyl = N 4
4vb
2pub
1
2
8vb
2pub
1
3
c5
F
(13)
where the …rm still serves the whole market, but revenues fall as the percentage of “guilt free
consumers” increases. The minimum level guilt required for pay-what-you like pricing to
be pro…table is b >
2v
r
1
2N (v+c)2
8v +N (v c)2
N
where
@b
@
< 0 for
2 (0; 1]: The minimum level
of guilt must rise to o¤-set both the cost associated with expanding the market and the
presence of two free-rider types: (1) strategic free-riders who have vi <
free-riders caused by
3
1
2
; and (2) random
who are located over the entire distribution of private values, vi :
Model of Loyal Consumers
In addition to the earlier model based on guilt, we now present a model that considers
repeated interactions between the consumer and the …rm. In this model payments are made
as a result of a dynamic game played between the …rm and its loyal consumers. Loyal
(repeat) customers have an incentive to support PWYL pricing as this pricing mechanism
o¤ers a higher level of life time consumer surplus than paying uniform prices in every period.
Although the underlying assumptions for these two models are very di¤erent, both models
predict a similar payment behavior on the part of consumers.
We derive the necessary
conditions under which PWYL pricing gives lower expected pro…ts than uniform pricing.
Let there be N in…nitely lived consumers with a single period utility function given
as: Ui = vi
pi : Suppose the proportion of repeat (loyal) buyers is
repeat buyer has a discount factor
2 [0; 1] and each
2 [0; 1): A consumer maximizes their lifetime utility by
selecting a contribution sequence. If the …rm chooses a uniform price, then each consumers
19
has a lifetime utility
Vu (vi ; ) =
vi
1
pu
(14)
where a consumer chooses to participate if vi > pu : The consumer’s value function under
pay-what-you-like pricing for repeat buyers is
V (vi ; ) =
vi pi
1
s.t: E [
pwyl ]
The above expression simpli…es to V (vi ; 0) = vi
E[
u] .
(15)
pi for non-repeat consumers ( = 0).
Therefore, a non-repeat consumer always has an incentive to free-ride under the PWYL
option.
Consider the following game.
In the …rst stage, the …rm chooses a particular pricing
strategy: uniform or PWYL. In the second stage, consumers choose to participate and, in
the case of PWYL pricing, choose how much to pay. The game is repeated subject to the
…rm earning positive pro…ts.
If the …rm’s pro…t under PWYL pricing are less than the
pro…t under uniform pricing, then the …rm will punish consumers by choosing the uniform
price in all future periods. Given the setup of the game, consumers will choose a payment
schedule such that the …rm is indi¤erent between uniform and PWYL pricing. Consumers
capture the entire surplus in this game, but there exists an in…nite number of possible
payment schedules that satisfy the constraint E [
pwyl ]
E[
u]
in equilibrium.12 Further,
the payment schedule must satisfy the following consumer participation constraint.
vi pi
1
vi + min
(vi
1
pu )
;0 .
(16)
Consumers will not choose to free ride under PWYL pricing if their lifetime utility when
committing to a positive price exceeds lifetime utility where consumers free ride in the …rst
12
For example, two potential equilibria result in either dividing the net surplus (after payment) among
all consumers evenly or giving all the net surplus to one consumer. At that point, no consumer can gain
additional surplus without triggering the …rm to use uniform price, thereby making everyone else worse o¤.
20
period and pay the uniform price in all subsequent periods. The participation constraint
can be simpli…ed to
pi =
min [vi ; pu ]
(17)
pi
where the maximum contribution amount, pi ; leaves the consumer indi¤erent between freeriding and committing to a positive price.
The maximum amount a consumer is willing to pay is strictly less than her private value,
but is proportional to min [vi ; pu ].
As the discount factor decreases consumers become
less patient and the incentive to free-ride increases. Since the marginal bene…t of future
consumption decreases with the discount factor, consumers are less willing to cooperate with
the …rm over the long-run. If we consider the discount factor to represent the probability
of a repeat visit in the future, then contributions are increasing as consumers become more
loyal. This theoretical result is supported by empirical evidence in the tipping literature.
Conlin, Lynn, and O’Donoghue (2003) …nd that repeat consumers, on average, provide larger
tips than one-time consumers.
Now we can obtain a necessary condition which when satis…ed makes pay-what-you-like
pricing always less pro…table than uniform pricing. First, we …nd the expected maximum
level of revenue that can be generated in the absence of free riding. From Equation (17) ; if
consumers have private values vi > pu ; then the maximum contribution is pi = pu , otherwise
pi = vi : The expected revenue function is given in Equation (18) where the …rst term in
the brackets is the average payment made by consumers with private values vi > pu and the
second term is the average payment by all other consumers.
max E [R] =
N [1
G (pu )] pu + G (pu )
Z
0
=
N pu 1
pu
2v
.
pu
vi
g (vi )
dvi
G (pu )
(18)
Next, we compute expected pro…ts under both pricing strategies, then solve for the value
of the discount factor such that uniform pricing is strictly more pro…table than PWYL,
21
pwyl
<
u:
Proposition 4 states the conditions.
Proposition 4. Given the uniform price pu =
v+c
.
2
Pro…ts under uniform pricing are
higher than pro…ts under pay-what-you-like pricing when: (i)
<
2(v+c)
(3v c)
; or (ii)
v
3
< c; 8 ;
< 23 ; 8 and 8c:
or (iii)
Proof. See appendix.
The magnitude of marginal cost again becomes critically important for the adoption of
PWYL pricing. A su¢ ciently low marginal cost and a high proportion of loyal consumers
can make PWYL pricing a viable pro…table alternative to uniform pricing. The intuition is
supported by what is observed in real life. For example, PWYL pricing is not practiced by
…rms serving mainly tourists because the proportion of repeat buyers is very low. On the
other hand, PWYL pricing may turn out to be a pro…table strategy in small communities
where consumers have a preference for locally owned businesses (high
and/or high ).
Finally, incorporating pricing cost relaxes the restriction on the discount factor. As
stated in Equation 17, contributions are function of the discount factor. The …rm can accept
lower payments from consumers (due to lower
) when the loss in revenue is o¤-set by a
corresponding increase in the cost associated with pricing. The upper bound on the discount
factor
is given by13
<
2(v+c)2 =N
(3v c)(v+c)
(19)
where where =N is the average amount of savings per consumer Again, the likelihood of
PWYL pricing increases as the market size decreases (low N ) and/or pricing cost increase
(high ).
The savings resulting from forgoing the cost of pricing allows the …rm to use
PWYL pricing while accepting lower payments.
13
For derivation see appendix
22
4
Conclusions and Discussion
Our analysis provides a theoretical economic framework that incorporates both the seller
and consumers behavior under the pay-what-you-like option. The model shows why a …rm
may allow consumers to choose their own price and why consumers would choose to pay
even when they have the option to free ride. The more pro…table choice between PWYL
or uniform pricing for the …rm depends on consumers behavior. When consumers derive
disutility because of guilt from paying lower than the uniform price then a …rm can exploit
the presence of guilt to expand the market by o¤ering PWYL pricing. By doing so, the …rm
can generate higher pro…ts compared to uniform pricing when marginal cost is small relative
to consumers’valuations and the demand elasticity is low. Our result provides support for
the empirical …ndings of Kim et al. (2009) by demonstrating that in equilibrium not all
consumers free-ride as was observed in their …eld study and also in many other examples
mentioned.
We extend the model by explicitly including the cost of price setting and show that
PWYL pricing can provide a signi…cant source of savings when the cost of setting a price
is high due to market research or uncertainty. In small markets, the savings in setting price
can motivate the use of PWYL pricing and class pricing. These two strategies di¤er in the
ex-ante di¤erence in demand. Class pricing is preferred when ex-ante di¤erences in demand
are small, but PWYL pricing is better suited when there is more consumer heterogeneity.
We further show that even in the absence of any guilt, PWYL can also be a pro…table
strategy when consumers are loyal and there are repeated interactions between consumers
and the …rm. In this case, when the discount factor is high, the proportion of loyal consumers is greater than 2/3, and the marginal cost is low, PWYL pricing may be sustained
as an equilibrium. Under these conditions, consumers can maximize their lifetime utility by
providing a positive payment instead of facing a uniform price in every period and they will
choose a payment schedule such that the …rm is left indi¤erent between PWYL pricing and
uniform pricing. Further, the net-surplus gained by using PWYL pricing can be distributed
23
among all the consumers resulting in a Pareto improvement.
Our theoretical model o¤ers some reasons as to why this unorthodox form of pricing is
rarely observed.
First, this pricing strategy requires a very low marginal cost relative to
consumers’valuations. Second, the elasticity of demand at the pro…t maximizing uniform
price must be within a narrow range (between one and two). Third, unlike a market serving
mainly tourists, the market for this pricing option must contain a captive set of consumers
who are driven either by loyalty or guilt. For this reason, PWYL pricing is most likely to be
observed for experience goods with small market size. If either the good or the seller of the
good is not su¢ ciently di¤erentiated (e.g., art, music, etc.), then pay-what-you-like pricing
is not suited and the …rm should compete in prices with other …rms. But, if the product is
su¢ ciently di¤erentiated or consumer heterogeneity is high, then PWYL pricing facilitates
a voluntary segmentation based on consumers’self selection facilitating an indirect form of
…rst-degree price discrimination without incurring the cost such practice generally requires.
Willig (1978) shows that a uniform price that exceeds marginal cost is Pareto dominated
by a nonlinear outlay schedule. He constructs a particular schedule that is preferred to the
uniform price by all consumers and the seller. We extend this consumer by showing that
PWYL schedule which is not a nonlinear outlay schedule can also can be Pareto dominating
compared to uniform price. Although the set of nonlinear Pareto-superior schedules is much
bigger, a nonlinear outlay is not necessary for Pareto domination. Further, under the PWYL
option, Pareto domination occurs in spite of free riding by some consumers and the seller
not covering the marginal cost of all additional quantities consumed. This interesting result
is the novelty of our paper.
Finally, we o¤er some suggestions for the future direction of empirical research. Although
conclusions from both models …nd similar restrictions on marginal cost and derive consumers’
payments as monotonic transformations of their private value for the good, the underlying
assumptions on consumer behavior are quite di¤erent. Future empirical work should focus
on designing a method that can separately identify the reasons for consumer payments, guilt
24
or loyalty, and in particular, the relative weights consumers place on these two factors when
making payments.
Appendix. Proofs
Derivation of equation (11)
In the guilt model,
pwyl
u;
2pu
(4v
N
>
when the following inequality is satis…ed.
1) (2pu
1)
c
2
8v
F >
N
(v
4v
c)2
F
This equation can be simpli…ed to
(4v
2pu
1) (2pu
8v
1)
2
c>
(v
c)2
4v
as …xed cost and market size are the same in both settings. We next solve the inequality
with respect to the guilt parameter, : The solution is found using Mathematica.
roots are found for
@ 1
@c
:
1
=
2
=
2v
p
1
2c2 +4cv+6v 2 8vpu +4p2u
2v+
p
1
2c2 +4cv+6v 2 8vpu +4p2u
@ 1
@c
< 0: Note, the di¤erence between pro…ts pwyl
u is decreasing with
@ ( pwyl
u)
@
respect to marginal cost,
< 0 and increasing with respect to guilt, @pwyl > 0:
@c
where
> 0 and
Two
Therefore, the second root of the guilt parameter, 2 ; is internally inconsistent as it implies
@ ( pwyl
u)
> 0: We conclude 1 is the unique solution.
that
@c
Proof of Proposition 3
The proof to proposition 3 is similar to the derivation of Equation (11) : We modify
Equation (9) to re‡ect the cost of pricing
N
(4v
2pu
1) (2pu
8v
1)
2
25
c +
>
N
(v
4v
c)2
where …xed cost under PWYL pricing is F = F
: We next set the uniform price pu =
v+c
2
and solve the inequality with respect to the guilt parameter, : The solution is found using
Mathematica. Two roots are found for
where
@ 1
@c
> 0 and
the solution.
@ 1
@c
1
=
2
=
:
2v
p
1
3c2 +2cv+3v 2
8v
N
2v+
p
1
3c2 +2cv+3v 2
8v
N
< 0: As in the derivation of Equation (11) ; the root
1
is select as
The bound on marginal cost is found by solving the denominator of
1
for
zero. In this case, the upper bound on marginal cost for PWYL to be more pro…table is
2
c<
3
which simpli…es to c <
v
3
r
v (vN + 6 )
N
v
3
when the cost of pricing is zero,
= 0:
Proof of Proposition 4
If consumer’s private values vi > pu , then the maximum she is willing to pay is
vi +
(vi
1
pi
(1
) vi + (vi
pu
pi
vi pi
1
vi
where vi
pu )
pu )
pu
vi
vi pi
1
vi
pi
(1
vi
pi
) vi
Next, we …nd the maximum possible revenue in the absence of free-riding. The expected
26
revenue function is
max E [R] =
N [1
G (pu )] pu + G (pu )
Z
pu
vi
0
=
where pu =
v+c
2
pu
2v
N pu 1
is the uniform price. Therefore, PWYL expected pro…ts take the form
E[
pwyl ]
=
N pu 1
=
N
pu
2v
v+c
2
cN
F
3v c
4v
and we compare these pro…ts to the case of uniform pricing,
expression E [
g (vi )
dvi
G (pu )
pwyl ]
<
u
simpli…es to
<
2(v+c)
:
(3v c)
cN
u
F
=
N
4v
<
2
3
c)2
F: The
v
3
the bound
Note, as marginal cost c !
on the discount factor approaches in…nity regardless of the value on
proportion of loyal buyers
(v
.
Further, if the
then for all values of marginal cost and the discount factor
the inequality will be satis…ed. Let marginal cost c = 0; then the bound simpli…es to
If
<
2
3
<
2
3
:
then the bound is always greater than 1, the discount factor cannot be greater than
1. Therefore, for all relevant values of marginal cost and the discount factor the bound is
satis…ed.
Derivation of equation (19)
We incorporate pricing cost into the repeat consumer game via the …rm’s pro…t function.
We adopt the notation of F = F
where F is …xed cost under PWYL pricing and
represents the cost of pricing. The …rm’s expected pro…t function under PWYL pricing is
then
E[
pwyl ]
=
N
v+c
2
3v c
4v
and we compare these pro…ts to the case of uniform pricing,
expression E [
pwyl ]
<
u
simpli…es to
<
2(v+c) N
(3v c)
cN
u
F
=
N
4v
(v
c)2
F: The
: Note, as marginal cost c !
v
3
the
bound on the discount factor approaches in…nity regardless of the value on . Further, if
27
the proportion of loyal buyers
<
2
N
3
then for all values of marginal cost and the discount
factor the inequality will be satis…ed. Let marginal cost c = 0; then the bound simpli…es to
<
2
N
3
:
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