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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Pay-What-You-Want Pricing Can It Be Profitable