pay-what-you-want pricing can it be profitable
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Pay-What-You-Want Pricing:
Can It Be Profitable?
Yong Chao, Jose Fernandez, and Babu Nahata
April, 2014
Abstract
Using a game theoretic framework, we show that not only can pay-what-
you-want pricing generate positive profits, but it can also be more profitable
than charging a fixed price to all consumers. Further, whenever it is more
profitable, it is also Pareto-improving. We derive conditions in terms of two
cost parameters, namely the marginal cost parameter for the seller, and the
social-preference parameter of a consumer to incorporate behavioral consid-
erations for paying too little compared to her reference price.
JEL codes: C70, D03, D21, D42
1 Introduction
Several field experiments using pay-what-you-want (PWYW) pricing, for exam-
ple, by musicians (Radiohead band), coffee houses (Mosaic Coffee House in Seat-
tle, Washington ), restaurants (Just around the Corner in London and Mon Cheri in
Fukuoka, Japan), a movie theatre (near Frankfurt Germany (Kim et al.[6]), and an
on-line magazine (Paste), have attracted attention in both economics and market-
ing literature. Under PWYW pricing, the seller does not set the price. Consumers
may choose any price to pay including zero.1 Those who practice this form of
pricing may not receive enough revenue to cover their costs. This problem isDepartment of Economics, College of Business, University of Louisville.1The significant difference between PWYW and name-your-own-price strategy is that, unlike
under PWYW pricing, in the name-your-own-price strategy, the seller can always refuse a buyer’s
final offer even when the buyer is willing to pay the required minimum price.
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exacerbated when either a large fraction of consumers free ride or the voluntary
payments they receive are below the marginal cost of production, thus potentiallymaking it an unprofitable pricing practice compared to charging a fixed price.
This paper considers if PWYW pricing can generate positive profits and also
earn profits in excess of those earned by using a fixed price. The paper makes the
following contributions to the existing literature. First, we endogenize the choice
of pricing strategies—PWYW price vs. fixed price. Thus rather than solely focus-
ing on the profitability of PWYW pricing, we evaluate its profitability vis-a-vis
uniform pricing. To the best of our knowledge this has not been done so far the-
oretically. Second, we specify consumer utility to account for both economic and
behavioral considerations. We show that when marginal cost is low and behav-
ioral considerations are strong, then PWYW pricing can exploit the deadweight
loss present under the uniform price to gain additional profit at the cost of servingsome free riders. Therefore, PWYW pricing can be more profitable than charging
a fixed price especially when the marginal cost is low and the deadweight loss is
high. Third, we demonstrate PWYW pricing is more attractive when the cost of
price setting is considerable or the market size is small.
The empirical evidence examining PWYW pricing comes mainly from field
experiments. Kim, Natter and Spann (2009)[6] conducted field experiments in a
medium-sized town near Frankfurt, Germany, in which three firms used PWYW
pricing. All three sellers (a lunch buffet in a middle-priced restaurant, a deli-
catessen serving twelve different types of hot beverages and a multiplex cinema
consisting of eight different movie theaters) reported receiving payments from
all customers (no free riders). To explain such payment patterns, the authors posit
that behavioral factors play an important role and how consumers react to a pricing
practice may not be solely rational. Based on the experimental data they conclude
that when buyers and sellers interact face-to-face, buyers will not free ride and
will pay a positive price. Unlike the online offering by Radiohead, in all their ex-
periments the interactions were face-to-face.2 Behavioral considerations such as
2The British band Radiohead offered their album In Rainbows to consumers online, where
the interaction was anonymous. The PWYW experiment resulted in both paying customers (38
percent worldwide and 40 percent in the U.S. willingly paid) and free riders, who were as prevalent
in the U.S. as in the rest of the world. From October 10~29, 2007, 1.2 million people worldwide
downloaded the album from Radiohead ’s Website. The average paying consumer in the US paidconsiderably more, $8.05 compared to $4.64, than her international counterpart. The band did
require a 45 pence minimum payment as a transaction fee. See http://www.inrainbows.com; and
http://comscore.com/press/release.asp?press=1883; and the Wall Street Journal, October 3, 2007,
p.C14.
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fairness, altruism, satisfaction and loyalty affect a consumer’s reference price and
that in turn influences the payment made by the consumer. These concerns be-come more significant when the interaction is face-to-face such that the consumer
will not free ride.3
Gneezy et al. (2012)[3] introduce two additional factors that also play an im-
portant role in assessing the viability of PWYW pricing. Based on the results from
three field experiments, they show that consumers would avoid free riding under
PWYW pricing, in part, because consumers want to maintain their self-image of
being fair. Since both free riding and not paying a “fair” price create a negative
self-image, to protect self-image, buyers rather prefer to forego purchasing from
the firm using PWYW pricing in favor of the firm who uses a declared fixed-price.
This no-purchase outside option, although helping to maintain the self-image,
also results in fewer purchases under PWYW pricing. The authors conclude that“...choosing whether to purchase a product or service, and how much to pay for
it, has a self signaling value. People feel bad when violating the norm and thus
would avoid the situation by choosing not to buy the product or service. If they do
choose to purchase the product or service, they often choose to pay a “fair” price
that does not have a negative effect on their self-image (p.7240).” Using online
laboratory experiments, Schmidt, Spann and Zeithammer (2014)[11] also reach
the same conclusion when firms compete and consumers have an outside option.
Machado and Sinha (2013)[8] specify a utility function to explicitly control
for three behavioral factors, namely fairness, reciprocity and consumers’ bias to-
ward a fixed-price strategy. Employing both laboratory and field experiments they
explore when these three behavioral factors could make PWYW a viable pric-
ing option. The utility function under PWYW pricing includes disutility from not
paying a “fair” price, and a positive utility because of reciprocity consideration. In
the absence of any posted or anchor price, a consumer’s internal reference price
plays the main role in determining the “fair” price. They conclude that PWYW
pricing has the potential to expand the market size because all buyers participate
and thus it could serve as an effective mechanism to price discriminate. In their
model specification, not only could PWYW pricing increase the market size, but
because of reciprocity concerns it may lead to an unusual result of consumers
paying more than their reference prices.
Regner and Barria (2009)[10] analyze consumers’ payment patterns at the on-line music label Magnatune, where consumers can pay what they wish within a
3Kim et al. (2009)[6] provide an extensive literature review that provides reasons why con-
sumers might pay when they have an option not to pay.
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specified price range of $5-$18. They find that, on average, customers paid $8.20,
far more than the suggested minimum price of $5, and even higher than the rec-ommended price of $8. The authors conclude that PWYW pricing could serve as
a viable alternate pricing option because such open contracts encourage customers
to make voluntary payments. They argue that voluntary payments can be due to
reciprocity, “warm glow”—acts of kindness, or experiencing a large enough guilt
from not paying a “fair” price. Repeated interactions or loyalty is another plausi-
ble explanation. Since the focus of their analysis is mainly on payment patterns
of consumers, no definitive conclusions can be drawn about the profitability of
PWYW pricing.
Regner (2010)[9] uses the Magnatune data to find which behavioral determi-
nants have the strongest affect on consumer payments. Consumers are categorized
into three groups: Low payers who paid near the minimum price of $5 (15%), av-erage payers who paid near the recommended price of $8 (60%), and generous
payers who paid substantially more than the recommended price (25%). The au-
thor identifies reciprocity and fairness/guilt considerations as the primary drivers
for generous payments and social norms as the driver for payments made around
the recommended price.
The extensive literature in behavioral economics, marketing and psychology
studying PWYW pricing strongly suggests that many behavioral considerations
play a significant role. For the profitability of PWYW pricing, it may not be
possible to identify the single most significant determinant of both how much
consumers would like to pay and how the profits are affected. Experimental stud-
ies also show that, in spite of the option to free ride, not all consumers free ride.
However, in the case of Radiohead’s online experiment about 68 percent did not
pay at all. We argue that regardless of which behavioral factor is a dominant fac-
tor in deciding whether to pay or not and how much to pay, a theoretical analysis
should not, a priori, rule out free riding by focussing on specific behavioral fac-
tors. Further, one cannot conclude that positive revenue under PWYW pricing
implies higher profits compared to charging a fixed price without considering the
magnitude of marginal cost and what fraction of buyers paid above or below the
marginal cost. Even when free riding is ruled out and one focuses primarily on
the payments, marginal cost still remains a relevant factor.
The main motivation for the paper is to provide a plausible theoretical expla-nation incorporating both the economic and the behavioral considerations to two
important questions. Not ruling out free-riding a priori based on some specific
behavioral factors, when would some consumers pay and some free ride? Un-
der what conditions could PWYW pricing generate higher profits than charging a
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fixed price to all consumers? We provide answers to both these questions based
on two parameters, namely the marginal cost of production for the seller and a“catch-all” social-preference parameter that serves as a proxy for all relevant be-
havioral factors.
We use a stylized game theoretical model based on profit maximization to en-
dogenize the choice of pricing strategies between PWYW pricing and uniform
pricing. Our basic framework relies upon the growing body of literature related to
social preferences in experimental and behavioral economics. Fair-minded con-
sumers are modeled to maximize net utility, where the utility function is com-
prised of two parts: (1) consumers wish to maximize consumer surplus (defined
as the difference between consumers’ private values for the good and the amount
paid); and (2) consumers also wish to minimize transaction utility. Transaction
utility incorporates the effects of social preferences that are typically ignored instandard models of utility maximization but quite relevant under PWYW pric-
ing. For tractability reasons, it is impossible to explicitly incorporate every single
social-preference factor, (e.g., fairness, warm glow, self-image, reciprocity etc.),
into a consumer’s utility function. Such a specification will make a closed form
solution for the demand functions highly complex, perhaps even impossible, as
shown in Machado and Sinha (2013)[8]. However, by including a single “catch-
all” social-preference parameter for the consumer that serves as a proxy for the
behavioral factors mentioned above we get additional insight about the profitabil-
ity of PWYW pricing that cannot be captured fully by experimental studies.
Based on social preferences, a consumer experiences disutility whenever the
voluntary payment made for the good is below some asked (fixed) reference price.
The social-preference parameter measures the relative importance of the transac-
tion utility within the net utility. Our tractable model is based on a profit maxi-
mization assumption and states conditions based on two parameters—–a social-
preference parameter encompassing the behavioral considerations for the buyers
and a cost parameter for the seller. We extend the results for the case when the
market size is small and price-setting is costly. Finally we allow random reference
prices for the consumers.
Lemma 1 states the necessary conditions for making voluntary payments by
consumers even in the presence of a free ride option. Proposition 1 states the
sufficient and necessary conditions for PWYW pricing to generate positive prof-its when social preference considerations are included. Proposition 2 states the
sufficient and necessary conditions for PWYW pricing to generate higher profits
compared to charging a fixed price. Proposition 3 states that when PWYW pricing
is more profitable, it is also Pareto improving compared to charging a fixed price.
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Qualitatively speaking, we find that in our framework, PWYW pricing could
generate higher profits than charging a fixed price when marginal cost is suffi-ciently low and social preference considerations are strong enough. The intuition
is that when the social preference considerations become significant, the volun-
tary payments from consumers who were excluded from the market under the
fixed price could generate sufficient revenue to compensate for both the free rid-
ers and the additional production cost. This increased revenue and low marginal
cost could result in higher profits.
The rest of the paper is organized as follows. Section 2 describes the setting
of the model. Section 3 describes consumer and firm behavior when a fixed price
is used. Section 4 presents the main results of the paper. Section 5 offers two
extensions to the model. In the first extension, we incorporate potential cost saving
associated with price setting a price and allow for the market size to affect theoutcome. In the second extension, we relax the assumption of a constant anchor
price for all consumers and show how profits under PWYW pricing are affected
by random anchor/reference prices. Section 6 discusses the results and limitations
with some recommendations for the direction of future research.
2 Model Setting
We consider a monopolist serving a continuum of heterogenous consumers
with a constant marginal cost of production c, 0
c < 1, who chooses between
charging the uniform price or letting consumers choose what they want to pay.Each consumer demands a single unit of the good. Consumers’ valuations or
willingness-to-pay (WTP) v are assumed to be distributed uniformly with a sup-port [0; 1]. Consumers make purchasing decisions independently. A consumer’sutility function under the uniform price (UP) is given as
U u = v pu
when she pays pu, and 0 otherwise.Under PWYW pricing, following Thaler (2004)[13], consumers are assumed
to be motivated by two considerations: (1) to maximize the consumption utility;
and (2) to minimize the transaction utility which is a function of social prefer-ences. As observed in experiments using this form of pricing, consumers get
disutility based on social preferences when they do not pay at all or pay too little
when choosing a payment. If consumers derive disutility when their payments are
lower than some reference price, then this disutility can be captured by including a
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social-preference parameter into the traditional utility maximization problem. We
follow Fehr and Schmidt (1999)[2] and Thaler (1985)[12] to incorporate thesebehavioral considerations into the consumer’s utility function as follows.4
U pwyw = v p (R p)2. (1)
The first term in (1), v p, is the consumption utility, which is the same as underthe uniform price. The second term, (R p)2, represents the transaction utilitythat internalizes the disutility from not paying a “fair” price and also highlights the
importance of some reference price R.5 The net utility is the difference betweenthe positive consumption utility and the disutility from paying a price below the
reference price.
Let p be the voluntary payment by a consumer, and R the reference price.The social-preference parameter captures the degree of disutility experiencedby the consumer when her voluntary payment p is below her reference price R.For simplicity, we assume that is identical for all consumers and has a support 2 [0; 1). The social-preference parameter encompasses all the potential behav-ioral considerations. The social-preference parameter is increasing with respect to
fairness, self-image, reciprocity, and warm glow as each of these behavioral con-
siderations increases the importance of the transactional utility. Note, a consumer
can always avoid disutility from these social preferences by paying their reference
price.
Consumers may construct reference prices from a variety of sources (i.e., ad-
vertisements, the price of close substitutes, from private perception of seller’scost, from social preferences, from social norms, etc.). When consumers decide
how much to pay, some external anchor price denoted as pa matters. The refer-ence price cannot exceed a consumer’s reservation price—–the most a consumer
is willing to pay for the good. Thus, the reference price is:
R = minfv; pag.
To illustrate why R is the minimum of v and pa consider an example. A con-sumer may value Radiohead ’s newest music CD at $20, but the same CD may be
purchased at a market price of $10. In this case, although her valuation v is $20,
4In a recent paper based on a field experiment, Just and Wansnik (2011)[5] use a similar ap-
proach to analyze how variations in flat-fee charges affect consumers’ behavior at an all-you-can-
eat pizza lunch buffet.5In general, the transaction utility function can be any montonic concave function. We choose
the quadratic loss function for simplicity and to ensure a unique solution.
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her reference price is more likely to be $10, and not $20. On the other hand, a
different consumer’s valuation v for the CD is $8, and the anchor price pa
equalsthe market price of $10. Thus, she is likely to have a reference price of $8 rather
than $10 and will not buy the good at $10.
The game is played in two stages. In the first stage, the firm chooses a pricing
strategy: the uniform price or PWYW pricing. In the second stage, consumers
observe the pricing strategy and choose to participate and make payments cor-
respondingly. We seek a subgame-perfect Nash equilibrium, which is found by
using backward induction. The firm solves the consumer’s problem under both
pricing strategies and then chooses the pricing strategy that yields the highest
profits.
3 Benchmark: Uniform Pricing
The monopolist chooses a price, p, that maximizes profit. Only consumerswith v p buy the good. So the monopolist’s profit function, u, is
u = max p
( p c) (1 p). (2)
The expressions for the profit-maximizing uniform equilibrium price pu, quantityq , and profits are standard and are given below.
pu = 1 + c2
; q = 1 c2
; and u = 1 c2
2 . (3)It is important to note the presence of consumers who have a positive value
for the good v > 0, but are locked out of the market because v < pu. As we willsee, the voluntary payment pattern of these locked out consumers under PWYW
pricing is critical for the viability of PWYW pricing.
4 PWYW Pricing
In this section, we first analyze consumers’ behavior under PWYW pricing,and then derive the PWYW firm’s profit function to derive the conditions under
which PWYW pricing yields a positive profit.
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4.1 Consumer Behavior under PWYW
Under PWYW pricing, consumers independently choose their voluntary pay-ment p including zero to maximize their utility,
U pwyw = v p (R p)2:
For simplicity, we assume the optimal uniform monopoly price represents the
anchor price. That is, pa = pu. As a result,
R = minfv; 1 + c2
g;
as shown in Figure 1 below.
A type-v consumer’s marginal utility when she chooses to pay p is
@U
@p = 1 + 2 (R p)
= 2 (R 12
p):
Note that the upper bound of R is 1+c2
. When 1+c2 1
2, or 1
1+c, then for any
p 0 we have R 12
p p 0. That is, for any positive payment, themarginal utility is non-positive and hence no one would pay anything in this case.
By contrast, when 12
< 1+c2
, or 11+c
< , consumers are segmented into three
groups. For consumers with v 12
, R = v because 12
< 1+c2
, thereby for any
p 0, R 12
p = v 12
p p 0. Thus, these consumers will pay0. The consumers with 1
2 < v 1+c
2 , R = v and will make a positive payment
p = v 12
so that @U @p
= 0. For those with 1+c2
< v 1, R = 1+c2
, and they will
pay p = 1+c2 1
2.
The market segmentations for the above two cases are shown in Figure 1 be-
low.
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Figure 1: Reference Point and Market Segmentations
Lemma 1 below states the conditions for consumers’ voluntary payments un-
der PWYW pricing.
Lemma 1 (Consumers’ Payments under PWYW) Under PWYW pricing no one
pays (i) when 11+c
, and (ii) when 11+c
< , a type-v consumer’s payment ruleis
p(v) =8 0, a consumer is never willing to make a voluntary payment inexcess of the market price or his own WTP. In particular, whenever the consumer
pays, she will downwardly adjust her payment from her reference price R by anamount equal to 1
2. As such, 1
2 turns out to be the cutoff value of consumer’s val-
uation below which the consumer will be a free rider. This condition highlights
that even when social preferences are taken into account, there still exists an in-
centive to free ride. But as the parameter increases, the number of free riders
will decrease. A firm, however, can lower the number of free riders by requiringa minimum payment, for example the $5 minimum set by Magnatune.6
6As noted by Regner and Barria (2009)[10], a price floor or a minimum payment requirement,
cannot rule out the free-rider problem entirely in the electronic music market due to widespread
availability in P2P network.
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The following corollary that follows from the Lemma 1 relates to how v and
affect p(v).Corollary 1 A consumer’s voluntary payment p(v) weakly increases with v aswell as the social-preference parameter .
An increase in the social-parameter , a consumer’s voluntary payment ap-proaches her reference price and thus an increase in encourages consumers topay more. Voluntary payments increase with private values for consumers who
were previously excluded from the market under the uniform price. However,
high-value consumers, 1+c2
< v 1, experience no change because their refer-ence price does not increase with their private values.
4.2 PWYW Firm’s Profit
Lemma 1 states that when 11+c
, the PWYW firm receives nothing from
any consumer but must incur the total cost c to serve the whole market causingprofit to be equal to c. When 1
1+c < , the PWYW firm still serves the whole
market, but receives voluntary payments from some consumers. Based on Lemma
1, the PWYW firm’s profit will be
pwyw = c +Z 1+c
2
1
2
v 1
2
dv +
1 + c
2 1
2
(1 1 + c
2 ) (4)
= 18 (3 6c c2 + 1 42 ).The PWYW firm’s profit function consists of three parts. The first term in (4)
represents the total cost of serving the entire market as no consumer can be turned
away. The second term accounts for the revenue obtained from some of those
consumers who would not have been served under the uniform price, i.e., v 1+c2
= pu. Only consumers with intrinsic valuations in excess of 12
will contribute
as the rest have an incentive to free ride. The last term is the revenue obtained
from consumers who would have participated in the market under the uniform
price, v > 1+c2
= pu. These consumers have different intrinsic valuations for thegood, but each makes the same payment, because their reference price is identical,
i.e., R = 1+c2
. The following lemma summarizes the PWYW firm’s profit.
Lemma 2 (PWYW Firm’s Profit) Under PWYW, the firm’s profits
(i) when 11+c
, pwyw = c.(ii) when 1
1+c < , pwyw is given by (4).
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An immediate result following from the lemma is that the PWYW firm’s profit
strictly decreases with the marginal cost c, but increases with the parameter .
Corollary 2 When 11+c
then @pwyw
@c 0.
Although an increase in marginal cost also increases the average reference
price, profits still decrease because the deadweight loss from the cost overrun
increases even more. PWYW pricing cannot exclude anyone from the market.
Therefore, the firm must pay the entire amount of the total cost c. Secondly, whenthe social-preference parameter is sufficiently high, an increase in this parameter
increases profits because the voluntary payments are weakly increasing in as
indicated in Corollary 1.Proposition 1 states the sufficient and necessary conditions when the PWYW
firm would earn a strictly positive profit.
Proposition 1 (When pwyw > 0) The profits under PWYW pricing is positive pwyw > 0 if and only if
0 c 2 +
p c2 + 6c + 1
3 6c c2 : (R2)
The first condition places an upper bound on the marginal cost. Since underPWYW pricing, the firm cannot exclude any consumers, the monopolist serves the
entire market and incurs the total cost c. This implies that even if the monopolistis capable of capturing the entire area under the demand curve (consumers’ total
willingness to pay), any value of the marginal cost greater than c > 1=2 wouldstill lead to negative profits.7 Further, the presence of free riders implies that the
marginal cost is strictly bounded below 1=2. To be more precise, the exact cut-off in (R1) accounts for the presence of free riders as well as the difference between
the voluntary payments made by consumers and the reference price described in
Lemma 1. The condition (R2) places a lower bound on the parameter . Recall
from Lemma 1 that, all consumers will free ride when 1=(1 + c). So PWYWpricing may not generate positive profit even when marginal cost is equal to zeroif social preference considerations are not very important. In the absence of a
7Note that the inverse demand curve in our model is p = 1 q following from uniformdistribution on [0; 1]. The total area under the demand curve is 1=2.
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strong, consumers have an incentive to free ride, resulting in zero profits. The two
conditions (R1) and (R2) are shown in Figure 3 below.
Figure 2: When pwyw > 0
4.3 Profits Comparison
In this section, by comparing the monopolist’s profits under the uniform priceand PWYW pricing, we derive the sufficient and necessary conditions for PWYW
pricing to be more profitable than the uniform price in the equilibrium.
For any 0 c 0 , it followsthat u > pwyw for the case when 1
1+c. But, when 1
1+c < , pwyw > u is
equivalent to the following inequality
1
8 (3 6c c2 + 1 4
2 ) >
1 c
2
2;
which can be reduced to
2 p 3c2 + 2c + 3 > 1
: (5)
Since > 0, for (5) to hold, we need:
2 p
3c2 + 2c + 3 > 0; (6)
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and
>
1
2 p 3c2 + 2c + 3 : (C1)Because 0 c 1 = 1
1 + c jc=0 :
Thus, we have1
2 p 3c2 + 2c + 3 > 1
1 + c;
for all 0 c < 13
. This implies > 11+c
given (C1) and (C2) hold.
The following proposition summarizes the sufficient and the necessary condi-
tions for PWYW to be more profitable than the uniform price.
Proposition 2 (When pwyw > u) Compared to the uniform price, profits under PWYW pricing will be higher pwyw > u if and only if (C1) and (C2) hold.
(C2) states that for PWYW pricing to be more profitable than the uniformprice, the marginal cost c cannot be too high for two reasons. First, Corollary 1shows that when c is sufficiently large, pwyw u.The exact upper bound of c = 1
3 is specific to the linear demand because of
the normalized choke-price equal to one in our model. In general, the intuition
behind an upper bound on the marginal cost is simple. Under the uniform price,
the lower the marginal cost, the higher is the deadweight loss. The deadweight
loss represents the potential lost profit that cannot be realized by charging theuniform price. The maximum size of the deadweight loss is reached when the
8 @pwyw
@c = 3+c
4 14
.
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marginal cost is zero. The deadweight loss can be viewed as the potential addi-
tional profit that can be captured using PWYW pricing. However, the magnitudeof the deadweight loss under uniform pricing decreases as marginal cost increases.
Hence, the potential gains from using PWYW pricing decrease as marginal cost
increases. Further, the cost associated with free riding increases with marginal
cost. This trade-off places an upper bound on the marginal cost. PWYW pric-
ing should only be adopted when the marginal cost is relatively low compared to
the choke price (demand price intercept). In other words, PWYW pricing is only
suitable for high mark-up items.
(C1) demonstrates that there is a lower bound of for pwyw > u. (C1) indi-cates that the value of must increase with the marginal cost for PWYW pricingto be more profitable than the uniform price, as the term 1
2
p 3c2+2c+3
is increasing
in c. The simple intuition is that with a higher marginal cost, the marginal profitloss from free riders must increase and consequently, not only is a higher value
of required for more consumers to contribute, but also to contribute more whenthey pay. In particular, a higher has two effects. First, it decreases the num-ber of free riders. Recall that consumers with private values v 1
2 always free
ride. The costs that these free riders impose is equal to c2
. Therefore, an increase
in the parameter can reduce the number of free riders to offset the increase inthe marginal cost. Second, recall from Lemma 1 that, for those who are paying
under PWYW pricing, their voluntary payment p(v) is adjusted downward by theamount 1
2 from their reference point R. Thus, a higher will make consumers’
payments closer to their reference point, thus compensating for the increase in
marginal cost.
Proposition 2 implies that for a fixed , PWYW pricing is less likely to beobserved for products with a high marginal cost of production. The two conditions
(C1) and (C2) are shown in Figure 3 below.
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Figure 3: When pwyw > u
From the bounds on placed by conditions (C1) and (C2), it is easy to ver-ify that as the marginal cost approaches 1
3, the minimum level of necessary for
PWYW pricing to be more profitable approaches infinity (as c ! 13
, 12
p 3c2+2c+3
!
1). The bound on
will be satisfied only when
c <
1
3
.
Behavioral considerations could potentially create an opportunity to eliminate
the deadweight loss that can be shared between the consumers and the seller. The
elimination of the deadweight loss increases the consumer surplus for the existing
consumers and creates a surplus for those consumers who were previously ex-
cluded from the market. These new entrants gain surplus by paying a price lower
than their willingness to pay. The seller gains additional profits from payments
made in excess of the marginal cost. Compared to the uniform price, the seller’s
profit will be higher under PWYW pricing if the profit earned from the elimina-
tion of the deadweight loss is large enough to off-set the revenue lost from the
existing consumers and the additional cost in production (see Figure 4). That is
why we never see luxury cars or private jets sold using PWYW pricing. Mostproducts under PWYW pricing—–hot beverages, menu items in a delicatessen
and music CDs, used in the field experiments have a low marginal cost compared
to the consumers’ willingness to pay, so the condition c < 13
is most likely to be
satisfied.
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Figure 4 illustrates the results of this section. Since consumers’ WTPs are
drawn from a uniform distribution on [0; 1], the monopolist faces a downwardsloping demand curve q = 1 p, shown by the blue solid line. The payment sched-ule of consumers under PWYW pricing is indicated by the red solid line. Profit
under the uniform price is the sum of Area I and Area II. Profit under PWYW
pricing is Area II + Area III - Area IV. Therefore, the difference in profits is Area
III - Area I - Area IV. If the additional profit 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 market extension (Area IV), then PWYW pricing is more
profitable than the uniform price. It is worth noting that (i) as guilt increases (a
higher ), Areas I and IV decrease but Area III increases allowing the firm to cap-ture more of the deadweight loss; (ii) as marginal cost approaches zero, c
! 0,
there is no deadweight loss in equilibrium. This happens without incurring anycost for expanding the market (Area IV disappears).
Figure 4: If Area III - Area I - Area IV >0, then pwyw > u.
4.4 Welfare Comparison
The proposition below follows immediately from the discussion above.
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Proposition 3 (Pareto Improvement) Whenever PWYW pricing is more profitable
compared to the uniform price, it is also Pareto-improving.
The simple intuition for the above proposition is as follows. The consumers
who previously bought at the uniform price can still pay the same price, but they
choose to pay a lower price so they must be better off. The consumers who were
previously excluded under the uniform price are better off because they get some
surplus previously not enjoyed. The seller’s profits are also higher. Since no one
is worse off, it is Pareto-improving.
5 Extensions
5.1 Costly Price Setting and Market Size
PWYW pricing option offers a source of cost savings by minimizing the
transaction cost of price setting (e.g., market research). Since under PWYW pric-
ing the seller does not incur the cost associated with price setting, by allowing a
fixed cost F under the uniform price, we incorporate the cost savings between thetwo pricing options. Additionally, we relax the assumption that the market size
is equal to unity and let N be the market size. The equilibrium monopoly priceunder the uniform price and the fixed cost is not affected by the market size. How-
ever, the monopolist’s profit under the uniform price with market size N and fixed
cost F isu = N
1 c2
2 F: (7)
For PWYW pricing, we focus only on the interesting case when profit under
PWYW pricing can be positive. That is, when > 11+c
. With market size N , theprofit now becomes
pwyw = N
3 6c c2
8 +
1 482
: (8)
The following proposition states the sufficient and the necessary conditions
for PWYW pricing to yield higher profits than the uniform price.
Proposition 4 (Costly Pricing) Given > 11+c
, price setting cost F and market size N ,
Case (i): if F N
> 3c2+2c+38
, then pwyw > u.
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Case (ii): if F N 3c2+2c+3
8 , then pwyw > u if and only if
8><>:
0 c < 2p 1+6F N 1
3
> 12
p 3c2+2c+38 F
N
:
Proposition 4 provides insight into why some of the firms mentioned earlier
use PWYW pricing. An example of Case (i) is the exclusive restaurant in Japan
with a small (low N ) niche market and has a modest marginal cost of production.Under the uniform price, it may experience a large cost of price setting (high
F ). In this case, the average cost savings F N
are large enough to offset the effects
of marginal cost such that F N > 3c2
+2c+38 . Additionally, the intimate nature of the restaurant creates a face-to-face interaction (high ); thereby, ensuring therestaurant meets the second condition of > 1
1+c.
On the other hand, Case (ii) resembles the case of Radiohead . Given Radio-
head’s lack of pricing experience or lack of having a distribution network, the
price setting costs could arguably be high, but the global fan base would tend
to make cost savings small F N
. Therefore, the band must rely on a strong and
loyal fan base (high ) as well as a low marginal cost of production, because themusic is traded digitally to meet the requirements of higher profitability stated
in Proposition 4. The example of Radiohead can be extended to other forms of
digital media where the producers face a significant cost of price setting, but can
deliver the good with a low marginal cost (e.g. Linux allows for contributions, but
downloads are free).
The cost savings described in this section can also affect entry. On the margin,
a firm only enters into a market if the expected profits are positive. One can argue
that firms who would not have entered the market because a uniform price would
have resulted in negative profits (u < 0) but may enter under PWYW pricingas the cost saving without market research may be significant enough to result in
positive profits ( pwyw > 0).
5.2 Random Reference Prices
Information asymmetries across consumers may result in different anchor
prices for different consumers as such anchor prices are formed from advertise-
ments, past experiences, and word of mouth. Empirical research has found an-
chor/reference prices to vary among consumers (see Winer, 1986[14]). Therefore,
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we relax our assumption of a constant and equal anchor price for all consumers in
this section.Consumers are assumed to know their anchor price, but the seller is unsure
about consumers’ external anchor e pa. Thus the consumer’s reference price nowbecomes a random variable eR = minfv; e pag. Applying Lemma 1, the expectedvoluntary payment from a consumer becomes
E [ p(v)] = E [max
eR 12
; 0
] (9)
= E [ eR 12
eR > 1
2] Pr( eR > 1
2):
We assume that the random external anchor price e pa is uniformly distributed on[ pu ; pu + ], where pu + pu 1
2 and the equilibrium uniform price
pu = 1+c2
.10
Since the lower bound of the random reference price, pu , is greater than1
2, we have e pa > 1
2 for all e pa 2 [ pu ; pu + ]. So consumers with any e pa will
pay if v > 12
. We can write down the firm’s expected profit as
E [ pwyw] = E
eR 12
eR > 12
Pr(eR > 12
) c
= 1
2[
Z pu+ pu
Z e pa
1
2
(v 12
)dvd
e pa +
Z pu+ pu;
Z 1
e pa
(
e pa 1
2)dvd
e pa] c
= 1
8 (3 + 2c c2 + 1 4
2 ) 2
6 :
Note that when = 0, E [ pwyw] is reduced to the same profit function pwyw asin the full information case under PWYW pricing. Moreover, the expected profit
decreases as the variance of reference prices increases.Two major patterns in consumer payments are observed when the external
anchor or the reference price is random. First, a random reference price can result
in payments exceeding the uniform price. In our baseline model, the maximum
payment per consumer is strictly below the uniform price, i.e., p = pu 12
. Here
for > 1
2 , when the reference price is allowed to vary, some payment amountmay exceed the uniform price, i.e., e pa 12
> pu now becomes possible. This
10We have already assumed that a consumer’s willingness to pay, v , is uniformly distributed.Therefore, we need to assume a bivariate uniform distribution over the anchor price, e pa, and v forthe ordered statistic to be well defined under the expection of the minimum.
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result is consistent with the empirical findings by Regner and Barria (2009)[10]
who record payments exceeding the recommended price of $8.Second, an increase in the variance of reference prices is found to decrease
expected profit as shown by the last term in the equation above. A suggested
minimum price might reduce the variance in consumers’ external anchors and
thus their reference prices. This implies that a firm offering PWYW pricing may
benefit from advertising a suggested price to reduce the variance of the reference
prices among consumers.
6 Discussion and Limitations
By incorporating behavioral factors and using a game theoretical model, we en-dogenize the choice of pricing strategy between PWYW pricing and the uniform
price. We provide the sufficient and necessary conditions for PWYW pricing to
be more profitable than charging a uniform price. This significant result shows
that PWYW pricing could emerge as a more profitable alternative to charging a
fixed-price in equilibrium when the marginal cost is relatively low and some con-
sumers’ behavioral considerations are significant enough to encourage voluntary
payments. Further, whenever it is profitable, it is also Pareto-improving. Our
results are robust to costly pricing and random reference prices.
We discuss the following limitations of our model. First, for tractability, the
social-preference utility is specified as a simple quadratic disutility function which
is the difference between a consumer’s reference price and the voluntary payment.However, it can be generalized to any increasing and concave social-preference
utility function, G(R p), and keep the same qualitative results.11The second limitation of the model is that it does not allow for the volun-
tary payment to exceed the reference price. That is, the model does not account
for “warm glow”. PWYW pricing is sometimes used by charities who are sell-
ing goods to collect funds. These goods may be of little value to donors, but the
donors still provide generous voluntary payments for these goods in the form a do-
nation. The donation enhances the donor’s utility due to “warm glow” and causes
voluntary payments to approach or even exceed the consumer’s reference price,
11For example, given consumer’s utility as U = v p G(R p), it is easy to derive theoptimal voluntary payment p = max[R G01(1); 0]. Therefore, the voluntary payment is stillcharacterized by subtracting a fixed amount from the reference price. The relative magnitude
of the discount factor is dependent upon the consumer’s non-pecuniary utility based on social
preferences.
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generating higher revenue for the firm. Yet, our model finds that PWYW pricing
can still be viable and profitable even without the presence of “warm glow”. Inthis regard, the model is more conservative than the model that consider “warm
glow”.12 In addition, the extension making reference prices random does allow for
some buyers to have their voluntary payments exceeding the expected reference
price.
Another limitation relates to the role of fairness and self-image in using PWYW
pricing. The model captures fairness and self-image via the social-preference util-
ity function. Maintaining the self-image encourages payments closer to the con-
sumer’s reference price. However, as noted by Gneezy et al. (2012)[3], protecting
the self-image can also have a very strong opposite effect by discouraging market
participation when PWYW is used. That is, preserving the self-image, can on the
one hand encourage higher payments but, on the other hand, can discourage par-ticipation. The model presented does not capture this non-participation behavior
by the consumer. In the monopoly case modeled here, consumers are always bet-
ter off by participating in the market under PWYW pricing even when free riding
because their outside option is assumed to be equal to zero.
Schmidt, Spann and Zeithammer (forthcoming)[11] explore the effects of out-
side option by introducing competition. They consider a duopoly. A firm using
PWYW pricing competes with another firm that uses a fixed or a posted price.
Using laboratory experiments via a computer network (no personal interactions)
they demonstrate that in the presence of competition not only does a significant
fraction of buyers turn away from the PWYW firm, but the voluntary payments
also are significantly lower than under monopoly. As a result, the market penetra-
tion may not be perfect but the PWYW firm ends up with a larger market share.
However, the firm using a posted price earns higher profits.
Chao, Fernandez and Nahata (2014)[1] analyze the role of competition in a
duopoly under Bertrand price competition. In their model one firm uses PWYW
and the other firm sets a uniform price. The outside option is endogenously deter-
mined. For example, consumers can obtain the good either from the PWYW firm
or from a different firm at some reference price without social-preference costs.
They show that for some value of the social-preference parameter consumers pre-
fer to purchase the good at the reference price from the firm using the uniform
price and forgo purchasing the good from the PWYW firm. This result supportsthe conclusion reached by Schmidt et al. (2014)[11] from their online labora-
12Isaac et al. (2010)[4] focus on a “warm glow” effect in a consumer’s utility function to explain
PWYW pricing for donations.
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tory experiments. The most novel result Chao et al. (2014)[1] obtain is that the
outside option segments the market in such a way that, under certain conditions,both firms end up earning positive profits instead of zero profits in equilibrium;
thus breaking the Bertrand Trap.13 However, the firm using the uniform price al-
ways earns higher profits—–a result similar to the one obtained by Schmidt et al.
(2014)[11] in their laboratory experiments.
In addition, we acknowledge that PWYW pricing may be adopted for reasons
other than those we consider in this model, such as cross-selling, loyalty effects,
and so on. For instance, in the music industry, bands may use PWYW pricing for
music albums to drive up the ticket sales for their live concerts in the future, rather
than a major revenue stream for its own sake. To model these factors, we need to
consider a multi-product dynamic setting, which is beyond the scope of this paper.
We mention two potential directions for future research. First, we adopt astatic reduced form model to examine PWYW pricing without any loyalty consid-
erations consumers may have for the seller. However, incorporating loyalty con-
siderations involve repeated interactions that can be best captured in a dynamic
setting. Exploring repeated interactions under some dynamic settings will be an
interesting extension.
Second, for comparison and simplicity we assume that consumers use the
firm’s actual profit-maximizing uniform price as an external anchor. An endoge-
nously determined external anchor price and therefore a reference point could be
another extension of our model (K ˝ oszegi and Rabin, 2006[7]).
13In Bertand price competition, when both firms set a fixed price the standard result is that bothfirms set price equal to marginal cost and earn zero economic profits.
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Appendix
Proof of Corollary 2. Based on Lemma 1,
@ pwyw
@c =
1 when 11+c
3+c4
when 11+c
< ;
so @pwyw
@c 12
, thereby @pwyw
@ > 0.
Proof of Proposition 1. From Lemma 2, when 11+c
, pwyw = c. So it isimpossible to have strictly positive profit in this case.
When 1
1+c < , pwyw
= 1
8 (3 6c c2
+ 14
2 ). Hence, pwyw
> 0 isequivalent to
1 42
> c2 + 6c 3: (10)Note that > 1
1+c > 1
2 follows from 0 c < 1. Thus, the left hand side (LHS)
of (10) is always negative. In order for (10) to hold, we must have the right hand
side (RHS) to be negative, too. That is,
c2 + 6c 3 u, we must haveCS pwyw > C S u, thereby T S pwyw > T S u.
Under the uniform price, a type-v consumer gets utility v pu. Thus, theconsumer surplus under the uniform price is
CS u =
Z 1
pu(v pu)dv = (1 c)
2
8 : (11)
Under PWYW pricing, when 11+c
, all consumers are free riders, thus the
firm would never adopt PWYW pricing in this case. So for the welfare compari-
son, only the region where the firm chooses PWYW pricing is relevant. That is,when 1
1+c < .
When 11+c
< , for a consumer with 0 v 12
, she pays nothing, and her
utility is v v2; for a consumer with 12
< v 1+c2
, she pays v 12
, and her
utility is v(v 12
)[v(v 12
)]2 = 14
; and for a consumer with 1+c2
< v 1,
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she pays 1+c2 1
2, and her utility is v (1+c
2 1
2) [1+c
2 (1+c
2 1
2)]2 =
v 1+c
2 + 1
4 . Hence, the consumer surplus under PWYW is
CS pwyw =
Z 12
0
(v v2)dv +Z 1+c
2
1
2
1
4 dv +
Z 1
1+c2
(v 1 + c2
+ 1
4)dv
= 1
8 [(1 c)2 + 2
1
32]: (12)
Thus,
CS pwyw CS u = 18
(2 13
)
> 0 ( because in the relevant region, > 11 + c > 12).
From Proposition 2 pwyw > u, and we have shown above that C S pwyw >CS u, therefore we must have T S pwyw > T S u within the relevant region of andc.Proof of Proposition 4. From (7) and (8), pwyw > u can be reduced to
1 42
> 3c2 + 2c 1 8 F N
(13)
Denote A
3c2 + 2c
1
8
F N
. Note that > 11+c
> 12
, which implies that the
LHS of (13) is negative. So in order for (13) to hold, we require A 3c2 + 2c + 3, then (14) is true for any > 11+c
. In
this case, A 12
p 3c2+2c+38 F
N
. Recall that we also require A < 0, which is equivalent to
c < 2
p 1+6 F
N 1
3 . Moreover, it is easy to verify that 1
2p
3c2+2c+38 F N
> 11+c
in this
case.
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