Endogenous Monopoly Market Segmentation

David McAdams, Duke University∗

November 23, 2009

Abstract

The standard monopoly pricing problem is re-considered when the buyer’s type

(e.g. age, income, experience) can be credibly disclosed at some cost. In the optimal

sales mechanism with costly disclosure, the seller posts a “sticker price” available to

any buyer, as well as a schedule of discounts available to those who disclose certain

types. Unambiguous welfare implications are available in the limiting case when

the buyer’s type is fully informative. (i) The buyer is better off and the monopolist

worse off when disclosure is more costly. (ii) When discounts are sufficiently rare,

total welfare is strictly less than if the seller could not offer discounts.

1 Introduction

Market segmentation (or “third-degree price discrimination”) is the pervasive practice

of offering different prices for the same good to buyers having different characteristics

∗Email: david.mcadams@duke.edu, Post: A416, Duke Fuqua School of Business, One Towerview Rd,

Durham, NC 27708. I thank numerous colleagues at MIT and Duke for comments and suggestions.

1

(or “types”). In some settings, it is natural to view buyer types as being known to the

seller a priori, e.g. employee discounts, costlessly observable by the seller, e.g. “ladies’

night” discounts at a nightclub, or costlessly disclosable by the buyer, e.g. the Kama’aina

rate offered only to Hawaiian residents. If so, the monopoly pricing problem reduces to

finding an optimal uniform price to offer buyers in each separate segment. However, in

practice sellers sometimes offer customized prices on the basis of information that is not

freely available.

This paper develops a theory of optimal monopoly pricing when information about

the buyer can be credibly disclosed, at a cost to the seller and/or the buyer. The optimal

sales mechanism in this case takes the form of what I call a “sticker-price mechanism”.

Any buyer who does not disclose faces a take-it-or-leave-it “sticker price”, while those

who disclose certain pre-specified types qualify for a customized discount. The seller’s

information about the buyer is endogenous in this mechanism. In particular, the seller

does not learn about those buyers who do not purchase its product, nor about those who

pay sticker price. This allows the seller to economize on disclosure costs.1

The findings here qualify some well-known welfare comparative statics. Consider the

extreme case in which the buyer’s type is his true willingness to pay (or “value”) for

the good. When the buyer’s value is known to the seller a priori, the seller’s optimal

scheme is efficient, perfect price discrimination (or “first-degree price discrimination”).

Thus, social welfare is higher when the buyer’s type is perfectly informative and known,

compared to when his type is totally uninformative. Suppose now that the buyer’s value

is not known a priori, but can be disclosed at some cost. As long as the cost of disclosure

is in an intermediate range, so that the buyer’s value is disclosed with a small enough but

1All disclosure costs paid by the buyer are ultimately passed through to the seller in the form of lower

revenues. Thus, the seller internalizes all disclosure costs.

2

positive probability in the optimal mechanism, expected social welfare is lower than if

his type were totally uninformative. In other words, as long as discounts are sufficiently

rare in the optimal mechanism, a regulator could increase total welfare by forcing the

seller to offer a uniform price.

The fact that disclosure is costly is essential to these results. There are many reasons

why it may be costly for the buyer’s type to be disclosed.

First, direct and credible communication between the buyer and seller may be costly.

Consider financial aid. Middlesex School, an elite private high school in Massachusetts,

charged tuition of $35,450 for its day students (and $44,320 for its boarders) in 2009-

2010. However, approximately 30% of the student body received some financial aid, with

an average tuition reduction of about $32,000 for those receiving aid.2 The process to

determine each student’s financial aid can be costly, both to parents who must reveal

(possibly painfully) private information and to the school which must evaluate aid ap-

plications and provide other customer services. With the rising cost of pharmaceuticals,

drug manufacturers have also begun to offer financial aid. For instance, Genentech offers

lower prices to patients who cannot afford the $50,000 price-tag for its cancer-fighting

drug Avastin, through its Avastin Access Solutions program. To assess financial need,

Genentech maintains a staff of counselors who meet with patients to review their financial

situation and insurance coverage.

When direct communication is not feasible or credible, the seller may pay an inter-

mediary for information about a buyer. For example, when a consumer is interested in

purchasing a new car but is not sufficiently credit-worthy to qualify for loan pre-approval,

the seller will access consumer information maintained in third-party databases as part of

its review of the consumer’s credit history, to decide whether to offer credit and on what

2 See http://www.mxschool.edu/podium/default.aspx?t=100033, accessed November 16, 2009.

3

terms.3 Internet websites such as Keycode.com provide another sort of example along

these lines. At this website, consumers can generate and print a customized coupon based

on information that they provide. Customer usage of such coupons then provides addi-

tional information that can help to personalize future offers. As stated on their website,

“We generate sales in-store and online for retail clients on a pay-per-sale basis. [We help

our clients] through a unique (and patent-pending) form of dynamic offer generation”.

Or, the buyer may pay an intermediary to convince the seller to offer him a dis-

count. For example, Restaurants.com charges $10 for coupons worth $25 at its partner

restaurants,4 while many hotels in the United States offer discounts to anyone who has

purchased AAA or AARP membership.

Finally, in addition, offering or administering a customized price may be costly even

when communication is not. For one thing, price discrimination may generate consumer

complaints about fairness.5 Best Buy faced bad press and an investigation of its pricing

practices in 2007, when it was discovered that prices offered in its brick-and-mortar stores

differed from those offered on the internet (Marco (2007)). Indeed, price discrimination

may even expose a seller to litigation risk. In the United States, a wholesaler may invoke

the Robinson-Patman Act to sue a supplier who has offered a discount to one of its

competitors. For that supplier, each sale at a customized price creates an incremental

3Unlike in the other examples, the buyer’s type in the retail credit context – his creditworthiness –

is relevant for predicting the seller’s cost rather than the buyer’s value. This paper’s analysis applies to

such situations, when we re-interpret value as the buyer’s net value = [buyer value - seller cost].4Restaurants.com does not charge its partner restaurants, but rather provides value-added services

such as customers’ e-mail addresses, online feedback, and reservations support.5Kimes and Wirtz (2003) argue: “Demand-based pricing is underused in many service industries

because customers are believed to perceive such pricing as unfair”. In the context of restaurant pricing

in Singapore, Sweden, and the United States, they find that customers view coupons, time-of-day pricing,

and lunch/dinner pricing as fair, but view weekday/weekend pricing and table location pricing as unfair.

4

litigation risk.

The rest of the paper is organized as follows. The introduction continues with a

discussion of some related literature. Section 2 presents the model. Section 3 considers

the illustrative special case in which the buyer’s type is fully informative of his value,

while Section 4 presents the main analysis. Section 5 concludes with some comments and

directions for future research. Proofs are in the Appendix.

Related literature. The most closely related literature is that on classic market seg-

mentation (Schmalensee (1981) and Varian (1985)). The difference here is that the seller’s

information about the buyer is costly and endogenously determined. Also, since the mo-

nopolist’s profits are always higher when the buyer’s type can be disclosed than under

uniform pricing, making it possible for information about the buyer to be credibly re-

vealed can be viewed as a rent-seeking activity (Tullock (1967), Posner (1975), and Fisher

(1975)). Thus, any welfare gains that might arise from such information revelation could

be diminished or reversed by the cost of rent-seeking.

A complementary literature is that on monopoly menu pricing (“second-degree price

discrimination”). The monopolist in that literature extracts more of the total surplus

from trade by allowing the buyer to choose among different goods, sorting different types

of buyers on volume (e.g. Wilson (1993)), delay (e.g. Chiang and Spatt (1982)) and/or

quality (e.g. Deneckere and McAfee (1996)). The main difference here is that the buyer

can reveal information about himself directly rather than indirectly through product

choice. Some examples such as movie ticket pricing combine elements of menu pricing

(e.g. matinee discounts) with elements of sticker pricing (e.g. senior citizen and student

discounts). In other settings, the menu that is offered to a buyer depends on what he

has disclosed about himself.6 Characterizing the profit-maximizing pricing mechanism is

6 For example, in June 2005, The Baptiste Power Vinyasa Yoga Institute offered an unlimited monthly

5

more general setting is an important area for future research. However, to isolate what

is new, this paper focuses on the special case of a single indivisible perishable good in

which it is well known that menu pricing is unprofitable.7

While similar in spirit, this paper is very different from the literatures on disclosure

(e.g. Grossman (1981) and Milgrom (1981)) and mechanism design with partially verifi-

able information (e.g. Green and Laffont (1986) and Bull and Watson (2005)). In these

literatures, all messages are costless. Here, “credible messages” are costly to send.

More closely related is the literature following Townsend (1979) on costly state ver-

ification. For example, Border and Sobel (1987) consider optimal taxation when the

taxation authority can verify (“audit”) a citizen’s wealth at some cost.8 A key feature of

the optimal taxation scheme is that the taxation authority sometimes takes the citizen’s

cheap talk claims “on faith”, thereby economizing on audit costs. The analogous question

in monopoly pricing, of how to design an optimal sales mechanism when the seller can

conduct a costly audit to learn the buyer’s type, is interesting and important but (to the

best of my knowledge) remains open.

This paper can be viewed as restricting attention to sales mechanisms in which the

seller is constrained, as a function of buyer cheap talk, to audit the buyer with probability

zero or one. Such a restriction is natural in some settings, such as (i) when direct

communication between the buyer and seller is not possible but the seller can purchase

pass for $99 to anyone who identified themselves as a “raw beginner”, whereas all others had to pay $12

per class. www.baronbaptiste.com, accessed June 24, 2005. (Baptiste is the dominant provider of “hot

yoga” in the Boston area. Its pricing policy has since changed.)7Non-linear pricing schemes which screen the buyer with different probabilities of getting the good

are not profitable. See e.g. Wilson (1993).8I am unaware of any papers that consider optimal pricing when the seller can verify the buyer’s type

at some cost. Severinov and Deneckere (2006) consider a monopoly pricing context in which the buyer

can misrepresent his private information at some cost.

6

information about the buyer from a third party, or (ii) when the most relevant costs arise

from offering / administering a customized price, as when some customers view price

discrimination as being unfair. Indeed, fairness concerns provide one possible explanation

for why random audits appear to be relatively rare in practice, even though in theory

they should be able to generate greater expected profit than the non-random schemes

explored here. For example, one imagines that Genentech most likely screens all patients

in its Avastin Access Solutions program equally, in part to avoid the unpleasant public

scrutiny that could arise were it to charge different prices to critically ill patients having

verifiably identical financial need.

2 Model

A risk-neutral monopoly seller has a single, indivisible, perishable good and faces a

single, risk-neutral buyer. This buyer has willingness to pay (or “value”) v ∈ V ⊂ R+

and “type” t ∈ T . As shorthand, I will refer to a buyer with value v and type t simply

as “buyer (v, t)”. The buyer’s value is distributed according to c.d.f. F (⋅∣t) conditional

on type t. For simplicity, I assume that v∣t has a well-defined, continuously differentiable

p.d.f. f(⋅∣t). The buyer’s type also has density g(.) over measurable type-space T . (See

discussion point (a) below.) Initially, (v, t) is the buyer’s private information while the

seller only knows its joint distribution.

Assumption 1. v − 1−F (v∣t)f(v∣t) is strictly increasing in v for all t.9

The buyer participates in a sales mechanism with costly disclosure. In such a mech-

anism, buyer (v, t) sends a message m ∈ Mall ∪Mt, receives the object with probability

9Assumption 1 implies that d[p(1−F (p∣t))]dp = −f(p∣t)

(p− 1−F (p∣t)

f(p∣t)

)is decreasing in p for all t. Stan-

dard “ironing” techniques can be adapted to extend the analysis when Assumption 1 fails.

7

q(m), and pays z(m). Mall is a set of messages that can be sent by any buyer, while Mt is

a set of messages that can only be sent by type-t buyers. If the buyer sends a message in

Mt, I will say that the buyer has “disclosed type t”. Such disclosure is costly, potentially

to both the buyer and seller. In particular, the buyer pays CB ≥ 0 and the seller pays

CS ≥ 0 whenever the buyer discloses his type. C = CB + CS > 0 is the total cost of

disclosure. (See points (b,c) as well as further discussion in Section 5.)

Most of the analysis in the paper will focus on a sub-class of sales mechanisms with

costly disclosure, called “sticker-price mechanisms”.

Definition 1 (Sticker-price mechanism). A sticker-price mechanism is a sales mechanism

with costly disclosure in which, for some D ⊂ T and p : (D∪{all})→ R+: (i) the cheap-

talk message-space Mall = V × {all} and type-t disclosing message-space Mt = V × {t};

(ii) allocation probability q(v, all)) = 1 if v > p(all), q(v, t) = 1 if t ∈ D and v > p(t)+CB,

and q(m) = 0 otherwise; (iii) payment z(v, all)) = p(all) if v > p(all), z(v, t) = p(t) if

t ∈ D and v > p(t)+CB, and z(m) = 0 otherwise; and (iv) prices satisfy p(t) ≤ p(all)−CB

for all t ∈ D.

Interpretation: Any sticker-price mechanism can be implemented as a buyer best

response to the following sort of scheme.10 The seller commits to a schedule of take-it-or-

leave-it prices, including a “sticker price” p(all) to any buyer who chooses not to disclose

his type and, for all t ∈ D ⊂ T , a “customized price” (or “discount”) p(t) ≤ p(all)−CB

to any buyer who discloses type t. Buyers having type t ∈ D always at least weakly prefer

to disclose and pay p(t) for the good than to pay p(all) without disclosing. However,

10If the buyer’s cost of disclosing CB > 0, then this best response is unique. Otherwise, if CB = 0, the

buyer will sometimes be indifferent between disclosing or not but, when he is, the seller strictly prefers

for him not to disclose. This may help explain why a seller may prefer to make applying for a discount

at least a little inconvenient, since this inconvenience can serve to sort for “serious” customers.

8

should such a buyer’s value be less than p(t) + CB, he prefers to not disclose and not

buy. On the other hand, buyers having type t ∈ D = T ∖ D do not have the option of

receiving a discount should they disclose. Such buyers never disclose and buy the good

iff v > p(all). To simplify the exposition, I will typically refer to this more informal

characterization of a sticker-price mechanism in terms of a disclosing set of types and a

schedule of prices with and without disclosure, rather than to the underlying message

spaces, allocation rule and payment rule, etc.

Discussion: (a) All results extend to settings in which T is endowed with any prob-

ability measure, including the case of finitely many types. The main additional compli-

cation that arises in this more general setting is that Theorem 2’s “iff” conditions on

the optimal disclosing set need no longer hold. In particular, the seller may prefer to

induce some types to disclose at a loss in order to extract more revenue from remaining

non-disclosers. However, this paper’s analysis and conclusions can be re-interpreted to

apply to settings with finitely many types, as follows. First, imagine for the moment that

the buyer can disclose an uninformative “label” drawn uniformly from [0, 1] as well as

one of finitely many payoff-relevant types. Since a density now exists over the enlarged

type-space T × [0, 1], this paper characterizes the optimal sales mechanism, which takes

the form of a sticker-price mechanism. Finally, as is shown in the proof of Theorem 2,

the seller is never indifferent between offering a sticker price or a customized price to any

type of buyer in the optimal sticker-price mechanism. Consequently, buyers having the

same payoff-relevant type but different labels must either all disclose or all not disclose

their types and labels in the optimal mechanism. In particular, this mechanism remains

optimal in the original model of interest with finitely many types but no labels.

(b) The model abstracts from any fixed costs associated with the infrastructure of

disclosure. For example, to allow the buyer to reveal more fine-grained information about

9

himself, the seller may need to hire an expert to interview the buyer. Or, the seller may

have to pay a “listing cost” that depends on the complexity of its sales mechanism (e.g.

on the number of distinct price / allocation probability pairs (p(m), q(m)) specified by

the mechanism). In the face of such fixed costs, the seller may prefer to “ignore” some of

what the buyer discloses. These issues appear to be first-order concerns when it comes

to understanding real-world pricing policies, and would be an interesting direction for

future research.

(c) The model also abstracts from marginal costs that the buyer and/or seller pay

before buyer values are realized. For example, direct-mail marketers send catalogs to lists

of customers meeting certain criteria, many of whom will turn out not to be interested in

their products. Similarly, grocery stores pay to place coupons in a newspaper regardless

of whether its reader takes the time to clip them. From the perspective of the analysis

here, such costs are sunk. This paper’s model rather captures additional costs incurred

per customized transaction.

3 Example: perfectly informative disclosure

The cleanest intuitions and crispest results are available in the limiting case in which

types are perfectly informative. In particular, all results of this section apply to a setting

in which the buyer’s type is simply his value, t = v, so that T = V . Strictly speaking,

this limiting case is not consistent with the model of Section 2: Assumption 1 fails since

v∣t does not have a well-defined density. To proceed, in this section I will maintain a

similar, alternative assumption on the unconditional distribution of bidder values.

Assumption 1’. v − 1−F (v)f(v)

is strictly increasing in v.

In this section, I will characterize the optimal sticker-price mechanism in this envi-

10

ronment,11 develop intuition, and provide some welfare implications of optimal sticker

pricing. Recall that a sticker-price mechanism is fully described by (i) a set of “disclosing

types” D ⊂ V , (ii) a sticker price p(all) available to any buyer who chooses not to dis-

close, and (iii) a customized price p(t) ≤ p(all)−CB available to any buyer who discloses

type t ∈ D.

Proposition 1. The optimal sticker-price mechanism has disclosure set D∗ = (C, p∗(all)),

customized prices p∗(v) = v − CB for all v ∈ D∗, and sticker price

p∗(all) ∈ arg maxp

(∫ ∞p

f(v)

(v − 1− F (v)

f(v)

)dv +

∫ p

min{p,C}f(v)(v − C)dv

). (1)

Corollary 1. Suppose that 1−F (v)f(v)

is strictly decreasing in v. Then the optimal sticker

price p∗(all) is uniquely determined by 1−F (p∗(all))f(p∗(all))

= min{p∗(all), C}.

Intuition. In the standard monopoly pricing problem, the seller is forced to exclude all

buyers having values less than the uniform price. Given the possibility of disclosure, the

seller may now instead offer such buyers the opportunity to qualify for a discount. Let p∗

be the optimal uniform price. Whenever C > p∗, the optimal sticker price p∗(all) = p∗,

reflecting the fact that the seller finds it unprofitable to offer any discount that would

induce excluded buyers to disclose their values.

By contrast, whenever C < p∗, the seller can strictly improve upon uniform pricing

by offering customized discounts to some buyers. In particular, let MR(v) = v − 1−F (v)f(v)

denote the “marginal revenue”12 that the seller receives when selling to type-v buyers

at the sticker price, and let NR(v) = v − C denote the “net revenue” that she receives

when selling to such buyers at the highest customized price consistent with disclosure.

11The proof of Theorem 1 can be adapted to show that the optimal sticker-price mechanism is optimal

among all sales mechanisms with costly disclosure.

12Let q(v) = 1− F (v) be the probability of sale at price v. d[vq(v)]dq = − 1

f(v)d[v(1−F (v)]

dv = v − 1−F (v)f(v) .

11

1− F (v)

price

C

1/2

p∗(all) = 1− C

1

C 1/2 1− C 1

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Figure 1: Optimal sticker price given linear demand.

To maximize expected revenue, the seller will induce disclosure from the buyer whenever

NR(v) > max{MR(v), 0}. In Figure 1, the seller’s expected revenue is illustrated graph-

ically as the area under the (bolded) upper envelope of marginal revenue, net revenue,

and the x-axis.

Example 1 (Linear demand). Suppose that v ∼ U [0, 1]. The optimal uniform price

p∗ = 12, given which the seller’s expected revenue is 1

4while the buyer’s expected surplus

is 18. For any disclosure cost C ≥ 1

2, uniform pricing remains optimal. When C < 1

2, by

contrast, the optimal sticker price p∗(all) = 1− C > p∗ and the buyer receives the good

at a customized price that leaves him with zero surplus when v ∈ (C, 1 − 2C). In this

case, the seller’s expected revenue is 12− C + C2 > 1

4while the buyer’s expected surplus

is C2/2 < 18.

12

C

welfare

1/2

3/81/3

1/6 1/3 1/2

Figure 2: Total welfare depending on the cost of disclosure C.

Welfare implications. If disclosure is sufficiently costly, a profit-maximizing seller

simply offers a uniform price with no opportunities to qualify for a discount. On the other

hand, when disclosure is costless, the optimal sticker-pricing mechanism here reduces to

costless, perfect price discrimination. Thus, if C = 0, total welfare under optimal sticker

pricing is greater than that under optimal uniform pricing. More generally, how does

welfare under optimal sticker pricing vary with the cost of disclosure, and compare with

that under optimal uniform pricing? Proposition 2 shows that the buyer and seller have

conflicting interests to raise or lower the cost of disclosure, and that total welfare under

optimal sticker pricing is less than that under optimal uniform pricing whenever the cost

of disclosure is in an intermediate range so that the seller sometimes but sufficiently rarely

offers discounts.

Proposition 2. The seller’s expected profit is non-increasing in C while the buyer’s ex

post surplus is non-decreasing in C. Further, as long as the buyer is offered a discount

sufficiently rarely, total welfare is strictly lower under optimal sticker pricing than under

optimal uniform pricing.

13

Intuition: For an intuition, consider the case of linear demand illustrated in Figure 3. As

the cost of disclosure decreases, the optimal sticker price p∗(all) = 1−C increases along

with the set of buyer-types who receive customized discounts. However, since buyers’

types fully reveal their values, any buyer who receives a discount earns zero surplus in

the optimal sticker pricing scheme. Thus, every buyer-type is at least weakly worse off

as the cost of disclosure decreases. Overall, as the cost of disclosure decreases from 12

to

any level C < 12, expected buyer welfare decreases by an amount equal to the highlighted

trapezoid in Figure 3.

By contrast, for all buyers having values v ∈ (C, 1 − C), the seller is able to more

profitably conduct a sale that generates net revenue v − C than in the case of uniform

pricing, when such sales would have generated either marginal revenue 1−2v or else zero

revenue. (v−C > max{1−2v, 0} for all v ∈ (C, 1−C).) Overall, as the cost of disclosure

decreases from 12

to any level C < 12, expected seller profit increases by an amount equal

to the highlighted triangle in Figure 3.

As C → 12, the seller’s expected profit increase is only second-order, while the buyer’s

expected welfare loss is first-order. Consequently, there is a net expected welfare loss

whenever the cost of disclosure is close enough to 12

to make customized discounts suffi-

ciently rare under optimal sticker pricing.

Example 1 continued. As shown earlier, total welfare under optimal sticker pricing is

1/2− C + 3C2/2, which is (i) minimized at C = 1/3, (ii) maximized at C = 0, and (iii)

equals the total welfare under optimal uniform pricing at C = 1/2 and C = 1/6. Overall,

total welfare under optimal sticker pricing is less than under optimal uniform pricing iff

C ∈ (1/6, 1/2). Since the buyer’s value is uniformly distributed on [0, 1] and he receives

a discount iff v ∈ (C, 1−C) (and never if C > 1/2), this welfare finding can be re-stated

in terms of the probability that the buyer receives a discount. Namely, optimal sticker

14

1− F (v)

price

1/2

1− C

1

1/2 1− C 1

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Figure 3: Welfare effects of optimal sticker pricing.

pricing generates less total welfare than optimal uniform pricing iff the buyer receives a

discount less than 2/3 of the time.

4 Sales mechanisms with costly disclosure

In this section, I will characterize the expected profit-maximizing incentive-compatible

sales mechanism with costly disclosure (or, simply, “optimal mechanism”), which turns

out to be a sticker-price mechanism.

Theorem 1. The optimal mechanism is a sticker-price mechanism.

For the remainder of this section, I will focus on sticker-price mechanisms. The seller’s

objective is to select the set of types D ⊂ T who will be eligible for a discount and the

schedule of prices {p(all), p(t) : t ∈ D} so as to maximize expected profit subject to

15

the constraint that buyers with types t ∈ D must at least weakly prefer to disclose and

buy at price p(t) than not disclose and buy at price p(all). That is, the seller selects

{D, p(all), p(t) : t ∈ D} so as to maximize expected profit

Π(D; p(.)) =

∫t∈D

(p(t)− CS) (1− F (p(t) + CB∣t)) g(t)dt+

∫t∈D

p(all) (1− F (p(all)∣t)) g(t)dt

subject to the disclosure constraint p(t) ≤ p(all) − CB for all t ∈ D. (Recall that

D = T ∖ D is the set of non-disclosing types, who would not be offered a customized

price even if they were to disclose.)

Theorem 2. The set of disclosing types D∗ and the schedule of prices p∗(.) in an optimal

sticker-price mechanism satisfy:

p∗(all) = arg maxp

∫t∈D∗

p (1− F (p∣t)) g(t)dt (2)

p∗(t) = arg maxp

(p− CS) (1− F (p+ CB∣t)) for all t ∈ D∗ (3)

t ∈ D∗ ⇔ p∗(all) (1− F (p∗(all)∣t)) > maxp≤p∗(all)−CB

(p− CS) (1− F (p+ CB∣t)) (4)

Discussion: The sticker price (2) is the optimal monopoly price against the endoge-

nous set of buyer types who do not disclose. Similarly, the customized discount (3)

offered to each type t ∈ D∗ is the optimal monopoly price against a type-t buyer, in an

environment in which the buyer has value distributed as (v − CB)∣t and the seller has

constant marginal cost equal to CS. (Disclosure costs are not sunk when the seller sets

the customized price for type-t buyers nor when the buyer decides whether to buy the

good.)

While natural, these properties of the sticker price p∗(all) and customized prices

{p∗(t) : t ∈ D} are not obvious. For example, since the incentive to disclose depends

on the sticker price, one might expect the optimal sticker price to be constrained by

the customized prices being offered. To see why this does not arise, suppose for the

16

sake of contradiction that p∗(t) = p∗(all) − CB for some t ∈ D∗. Type-t buyers are

indifferent between paying p∗(all) without having to disclose, or disclosing and paying

p∗(t). However, the seller is not indifferent between these two modes of purchasing the

product, since she sells to the same set of buyers (those of type t with values greater

than p∗(all)) but receives C more per buyer when they purchase at sticker price. Thus,

the seller would have been better off excluding type-t buyers from receiving a discount.

Least obvious is how to construct the optimal set of types to make eligible for a

discount. (4) establishes an important property of the optimal disclosing set D∗ that

characterizes it in terms of the optimal sticker price. In particular, type-t buyers will be

induced to disclose iff, against type-t buyers only, the customized price (3) with disclosure

is strictly more profitable than the sticker price (2) without disclosure. In this sense, the

problem of whether to offer customized discounts is separable across types.

An interesting implication of (4) is that there does not exist any (positive measure)

set of types such that the seller is marginally indifferent between offering each of them a

customized discount p∗(t) or only the sticker price p∗(all). The reason is that, if such a

set of types existed, the seller would strictly prefer to induce all of them not to disclose.

Intuitively, switching such buyers to the sticker price causes only a second-order profit

loss because the seller is marginally indifferent between selling at the sticker price or not,

but leads to a first-order profit increase as the sticker price can then be “re-optimized”

to maximize expected revenue from the now-enlarged set of non-disclosing types.

Computing the optimal sticker-price mechanism. Theorem 2 suggests a numeri-

cal approach to compute the optimal sticker-price mechanism. For any candidate sticker

price p(all), (4) uniquely determines the set of types D(p(all) that do not disclose if

p(all) is the optimal sticker price. Conversely, given a disclosing set D, first-order con-

17

dition (2) uniquely determines what the sticker price p(D) must be if D is the optimal

disclosing set. All together, the optimal sticker price and optimal disclosing set must

satisfy the fixed-point condition that p(D(p)) = p and D(p(D)) = D. While this is a

necessary condition of the optimal sticker-price mechanism, it is not sufficient. In prin-

ciple, one must identify all such fixed points. The optimal mechanism corresponds to

whichever fixed point generates the greatest expected profit for the seller.

Example 2. Suppose that the buyer’s type t ∼ U [0, 1], the buyer’s value v∣t ∼ U [0, t]

conditional on type t, and the buyer pays disclosure cost CB = C while the seller pays

nothing.

For any given sticker price p(all), what is the optimal set of types D(p) to be offered

a customized price p(t) ≤ p(all) − C? (If p(t) > p(all) − C, then every type-t buyer

will prefer to not disclose and buy the good at sticker price.) If pooled at the sticker

price, type-t buyers generate expected profit p(all)(1 − F (p(all)∣t)) = p(all)(t−p(all))t

. On

the other hand, if offered customized price p(t), they generate expected profit of at

most maxp(t)≤p−C p(t)(1 − F (p(t) + C)∣t)) = maxp(t)≤p−Cp(t)(t−p(t)−C)

t. As can be easily

checked, the optimal customized price p∗(t) = maxp(t)≤p−Cp(t)(t−p(t)−C)

t= t−C

2when

t−C2< p−C; otherwise, customized price revenue is at most (p(all)−C)(1−F (p(all)∣t)) <

p(all)(1− F (p(all)∣t)), in which case the seller strictly prefers to sell at the sticker price

only. So, consider the case in which t−C2

< p(all) − C or, equivalently, t < 2p(all) − C.

By (4), the seller offers type-t buyers a discounted price p∗(t) = t−C2

iff

(t− C)2

4t>p(all)(t− p(all))

t⇒ t < 2p(all) + C − 2

√p(all)C. (5)

In particular, the set of buyer-types not offered a discount in the optimal sticker-price

mechanism is an increasing interval of the form [t∗, 1], where t∗ = 2p∗(all) + C −

2√p∗(all)C.

18

Finally, by (2), p∗(all) is determined by the first-order condition:

∫ 1

t∗

d[p∗(all)(t−p∗(all))

t∣t))]

dpg(t)dt = 0

⇔∫ 1

t∗

t− 2p∗(all)

tdt = (1− t∗) + 2p∗(all) ln t∗ = 0

⇔ p∗(all) =1− t∗

−2 ln t∗.

Proposition 3 summarizes these findings.

Proposition 3. In the optimal sticker-price mechanism in Example 2, the seller offers

sticker price p∗(all) as well as customized prices p∗(t) = t−C2

to buyers who disclose types

t < t∗, where (p∗(all), t∗) solve the following system of equations:

p∗(all) =1− t∗

−2 ln t∗(6)

t∗ = 2p∗(all) + C − 2√p∗(all)C (7)

5 Concluding Remarks

Standard monopoly pricing models of a single, indivisible, perishable good take as given

what the monopolist knows about the distribution of buyer values: either values are per-

fectly known (perfect price discrimination), some payoff-relevant characteristic is known

(market segmentation), or nothing is known (uniform pricing). This paper endogenizes

what the monopolist knows about buyers when setting prices, in a setting with costly

disclosure of a buyer characteristic. The optimal sales mechanism takes a familiar form:

the seller offers a “sticker price” to any buyer, as well as a pre-specified list of discounts

to qualifying buyers (Theorem 1).

This optimal sales mechanism bears a close resemblance to standard, optimal monopoly

market segmentation. In particular, the optimal sticker price is equal to the optimal

19

monopoly price against the endogenous segment of buyers who choose not to disclose

their type (Theorem 2). However, there are important differences. For one thing, since

disclosure is costly, the practice of perfect price discrimination need not increase total

welfare. Indeed, as long as the fraction of buyers receiving fully-extractive customized

prices is small enough, one may infer that total welfare is lower than if price discrimina-

tion were not possible (Proposition 2).

I conclude by discussing some potential directions for future work.

Listing costs. The optimal pricing mechanism derived here can be viewed as a list of

prices: a “sticker price” available to any buyer, as well as a schedule of discounts available

to the buyer depending on his disclosed type. An implicit assumption is that the seller

incurs no extra cost when adding another price to this list. Consequently, the optimal

mechanism exhibits a potentially unrealistic proliferation of discounts. A worthwhile

topic for future work would be to examine the impact of listing costs on what discounts

are offered – in particular, is there a clean characterization of the optimal partition of

types into discount-eligibility classes? – as well as on seller profits and buyer welfare.

Type-dependent disclosure costs and future benefits. The model here assumes

non-negative and type-independent disclosure costs, but this is not realistic in some

important settings. For example, sellers of experience goods and services often offer

first-time buyer discounts, e.g. the nationwide tanning salon L.A. Tan offers a “Free $50

tanning value” coupon to new customers only. To restrict such a discount to first-time

buyers, the seller needs to check and update a database listing all users of its product who

have ever claimed the first-timer discount. Updating such a database may provide future

benefits to the seller and hence correspond to a negative disclosure cost, if it enables the

seller to extract more revenue from its relationship with the buyer. Of course, if buyers

20

are rational, they will demand a sufficiently attractive discount today to undo any such

future revenue-extraction benefit enjoyed by the seller. In that case, total disclosure cost

would be positive. On the other hand, if the database allows the seller to provide more

valuable products and services and thereby increase total surplus in the relationship,

total disclosure costs would be negative.

As this example suggests, negative disclosure costs arise naturally when the seller

and/or buyer get some future benefit from disclosure today. Indeed, search engines,

social networks and other information intermediaries often provide their services for free,

in exchange for their users’ willingness to share information about themselves that can

then be used to customize advertisements or other product offerings. Yet such future

benefits naturally depend on the information that is being disclosed, i.e. on the buyer’s

type. While this paper’s approach can be generalized to accommodate negative and

type-dependent disclosure costs in a straightforward way, a deeper analysis is needed to

understand the role of future benefits in relationships with disclosure. For one thing,

whereas the buyer here must either reveal his type, or else reveal nothing at all, future

work could attempt to endogenize what information is shared, and when.

A Appendix

A.1 Proof of Proposition 1 and its corollary.

Proof. Suppose that the seller offers sticker price p(all). To induce type-v buyers to

disclose, the seller must offer customized price p(v) ≤ min{v − CB, p(all) − CB}, for

profit p(v) − CS ≤ min{v − C, p(all) − C}. In particular, the seller will not find it

profitable to induce disclosure from any buyer having value v ≥ p(all) or v ≤ C. On

the other hand, all buyers having values v ∈ (C, p(all)) refuse to pay the sticker price

21

but can be profitably induced to disclose. Further, the optimal customized price for any

such type is clearly that which extracts all of the surplus, i.e. p∗(v) = v − CB for seller

profit v − C on such types. All together, the seller’s expected profit given sticker price

p(all) = p and optimally-induced disclosure of buyer-types v ∈ (C, p) equals

Π(p) = p(1− F (p)) +

∫ p

min{p,C}f(v)(v − C)dv (8)

=

∫ ∞p

f(v)

(v − 1− F (v)

f(v)

)dv +

∫ p

min{p,C}f(v)(v − C)dv. (9)

This completes the proof of Proposition 1, since p∗(all) maximizes (9).

To establish the corollary, let p∗ = arg maxp p(1 − F (p)) be the optimal uniform

price. If C ≥ p∗, then the optimal disclosure set D∗ = ∅ and p∗(all) = p∗. Thus,

1−F (p∗(all))f(p∗(all))

= p∗(all). Otherwise, if C < p∗, the optimal disclosure set is non-empty and

dΠ(p)dp

= f(p∗(all))(

1−F (p∗(all))f(p∗(all))

− C)

. Thus, 1−F (p∗(all))f(p∗(all))

= C.

A.2 Proof of Proposition 2

Proof. Part I: Buyer surplus and seller profit. The set of buyer-types v ∈ [0, 1−C] that

receive zero ex post surplus is non-increasing in C, while (9) implies that the sticker-

price paid by all other buyer-types is also non-increasing in C. Thus, the buyer’s ex post

surplus is non-decreasing in C. Let Π(C) be the seller’s expected profit, viewed now as

a function of disclosure cost C. By the Envelope Theorem applied to (9),

dΠ(C)

dC= −

∫ p∗(all;C)

min{p∗(all;C),C}f(v)dv ≤ 0 (10)

so that the seller’s expected profit is non-increasing in C.

Part II: Total welfare. Let p∗ = arg maxp p(1 − F (p)) be the optimal uniform price.

For all C > p∗, p∗(all;C) = p∗ and ex post welfare is the same under sticker pricing

or uniform pricing. Suppose that the cost of disclosure decreases from Cℎ to Cl, for

22

any Cl < Cℎ ≤ p∗. There are three effects on total welfare. First, buyers having value

v ∈ (Cl, Cℎ) now receive the good (after disclosure at cost Cl), for an expected welfare gain

of at most (Cℎ − Cl)(F (Cℎ)− F (Cl)). Second, buyers having value v ∈ (Cℎ, p∗(all;Cℎ))

disclose at lower cost, for expected welfare gain (Cℎ−Cl)(F (p∗(all;Cℎ))−F (Cℎ)). Finally,

buyers having value v ∈ (p∗(all;Cℎ), p∗(all;Cl)) now disclose, for an expected welfare loss

Cl(F (p∗(all;Cl))− F (p∗(all;Cℎ))).

(p∗ −Δ)(F (p∗)− F (p∗(all; p∗ −Δ))).

Consider now Cℎ = p∗ and Cl = p∗ − Δ, where Δ > 0. Since p∗(all; p∗) = p∗,

the second effect disappears and the expected welfare gain associated with lowering the

disclosure cost from p∗ to p∗ −Δ is at most

Δ(F (p∗)− F (p∗ −Δ))− (p∗ −Δ)(F (p∗)− F (p∗(all; p∗ −Δ))). (11)

To prove that total welfare falls as disclosure costs fall from p∗ to p∗−Δ for small enough

Δ, it suffices to show that limΔ→0F (p∗(all;p∗−Δ))−F (p∗)

Δ> 0. Since F (.) has well-defined

density, this condition holds iff limΔ→0p∗(all;p∗−Δ)−p∗

Δ> 0.

By (9), p∗(all;C) satisfies necessary condition f(p∗(all;C))C = 1− F (p∗(all;C)) for

all C ≤ p∗. In particular, the total derivative d[f(p∗(all;C))C+F (p∗(all;C))]dC

= 0. Since F (.), f(.)

are assumed to have well-defined derivatives, dp∗(all;C)dC

= −f(p∗(all;C))f(p∗(all;C))+Cf ′(p∗(all;C))

< 0 exists.

We conclude that total welfare is strictly increasing in disclosure cost C, over the

range C ∈ (p∗ − Δ, p∗) for some Δ > 0. Let (C) = F (p∗(all;C)) − F (C) be the

probability that the buyer receives a discount. Equivalently, we have shown that total

welfare is strictly increasing in C whenever (C) < (p∗ −Δ).

23

A.3 Proof of Theorem 1

First, some definitions and preliminary results. Given the possibility of disclosure, the

standard Revelation Principle does not directly apply. Let M(v, t) be the set of messages

sent with positive probability by buyer (v, t).

Definition 2 (Non-random disclosure). A sales mechanism with disclosure has non-

random disclosure if Pr(M(v, t) ⊂Mall or M(v, t) ⊂Mt) = 1.

Lemma 1 (Non-random disclosure). The optimal mechanism has non-random disclosure.

Proof. By the Envelope Theorem, ∂(maxm S(m;v,t))∂v

= q(m(v, t)) so that, in particular,

buyer (v, t)’s probability of receiving the good is the same, q(m1) = q(m2) = q(v, t),

for all m1,m2 ∈ M(v, t). The buyer’s expected surplus S(v, t) =∫ v

0q(v′, t)dv′ + S(0, t).

In particular, the buyer’s payment net of buyer disclosure cost z(m1) − CB ∗ 1m1∈Mt =

z(m2)− CB ∗ 1m2∈Mt = vq(v, t)−∫ v

0q(v′, t)dv′ − S(0, t) for all m1,m2 ∈M(v, t).

Suppose f.s.o.c. that there exists m1,m2 ∈M(v, t) such that m1 ∈Mall and m2 ∈Mt.

The seller can increase expected revenue from buyer (v, t) inducing him to send only the

non-disclosing message m1: payment from the buyer will increase by CB, the seller will

avoid disclosure cost CS herself, and no other buyer has any new incentive to deviate

since the non-disclosing message m1 was already available to all buyers.

Definition 3 (Disclosure and non-disclosure sets). In a given mechanism, let Y ES =

{(v, t) : M(v, t) ⊂Mt} and NO = {(v, t) : M(v, t) ⊂Mall} be the sets of buyers that are

sure to disclose or sure not to disclose, respectively.

24

Lemma 2. In any incentive-compatible (IC) mechanism with non-random disclosure:

S(v, t) =

∫ v

0

q(v′, t)dv′ + S(0, t) (12)

v′ > v ⇒ q(v′, t) ≥ q(v, t) (13)

(v, t) ∈ NO ⇒∫ v

0

q(v′, t)dv′ + S(0, t) ≤∫ v

0

q(v′, t′)dv′ + S(0, t′) for all t′ ∈ T (14)

Proof. (12) was derived in the proof of Lemma 1. The monotonicity condition is also

standard: Buyer (v′, t)’s surplus when mimicking buyer (v, t) is S(m(v, t); v′, t) = S(v, t)+

(v′−v)q(v, t), so that incentive-compatibility requires (S(m(v, t); v′, t)−S(v, t))+(S(m(v′, t); v, t)−

S(v′, t)) = (v′ − v)(q(v′, t)− q(v, t)) ≥ 0. Namely, v′ > v implies S(v′, t) ≥ S(v, t). Next,

a buyer (v, t) ∈ NO does not disclose, so that any buyer (v, t′) having the same value

can mimic him and earn the same surplus. Thus, incentive-compatibility requires that

(v, t) ∈ NO ⇒ S(v, t) ≤ S(v, t′) for all t′. The disclosure condition (14) then follows

from (12).

Outline of proof: By Lemma 1, it is without loss to focus on mechanisms with non-random

disclosure, while Lemma 2 provides necessary (though not sufficient) conditions for such

a mechanism to be incentive-compatible. To proceed, I shall consider the relaxed problem

of finding the optimal mechanism with non-random disclosure satisfying the necessary

conditions (12-14). The solution to this relaxed problem is a sticker-price mechanism.

To complete the proof, I will show that this sticker-price mechanism is in fact incentive-

compatible, and thereby optimal among all IC mechanisms.

Preliminaries: virtual values and net virtual values. Assigning the object to buyer (v, t)

without disclosure has the direct effect of increasing expected total surplus by vg(t)f(v∣t)

while also, through (12), increasing expected buyer surplus by g(t)(1 − F (v∣t)). The

seller’s expected profit thus increases by the buyer’s “virtual value” V V (v, t) = v −

25

1−F (v∣t)f(v∣t) times the density g(t)f(v∣t) of such a buyer. Similarly, assigning the object to

buyer (v, t) after disclosure has the direct effect of increasing expected total surplus by

(v−C)g(t)f(v∣t) while again increasing expected buyer surplus by g(t)(1−F (v∣t)). The

seller’s profit now increases by the buyer’s “net virtual value” V V (v, t) − C times the

density g(t)f(v∣t).

The seller’s overall expected profit in any IC mechanism with non-random disclosure

can therefore be expressed as:∫(v,t)∈Y ES

q(v, t) (V V (v, t)− C) g(t)f(v∣t)dvdt+

∫(v,t)∈NO

q(v, t)V V (v, t)g(t)f(v∣t)dvdt− E[S(0, t)]

(15)

The next several steps of the proof will identify the disclosure set Y ES∗ ∈ V ×T , the

allocation probabilities {q∗(v, t) : (v, t) ∈ V × T} and the zero-value surpluses {S∗(0, t) :

t ∈ T} that maximize (15), subject to the monotonicity constraint (13) and the disclosure

constraint (14). Let S∗(v, t) = S∗(0, t) +∫ v

0q(v′, t)dv′ be the associated buyer surpluses,

implied by the buyer surplus (IC) constraint (12). Steps 1-2 identify properties of the

optimal disclosure set Y ES∗ taking as given the allocation probabilities and zero-value

surpluses; Step 3 shows that all buyers who disclose must receive the object; Step 4

establishes that zero-value buyers get zero surplus; Steps 5-6 then show that each type

of buyer can be viewed as facing a take-it-or-leave-it price; Step 7 concludes by showing

that the resulting mechanism is indeed an incentive-compatible sticker-price mechanism.

Step 1: (v, t) ∈ Y ES∗ iff S∗(v, t) > mint′ S∗(v, t′). Since disclosure is costly, the seller

clearly maximizes (15) subject to (14) by only inducing buyer (v, t) to disclose iff (14)

requires disclosure. That is, by (12), iff S∗(v, t) > mint′ S∗(v, t′).

Step 2: (v, t) ∈ Y ES∗ implies (v′, t) ∈ Y ES∗ for all v′ > v. Suppose for the sake

of contradiction that there exists v < v such that buyer (v, t) discloses his type in the

26

optimal mechanism, but not buyer (v, t). In particular, S∗(v, t) = mint′ S∗(v, t′) but

S∗(v, t) > mint′ S∗(v, t′). Since S∗(v, t) −mint′ S

∗(v, t′) is continuous in v and ∂S∗(v,t)∂v

≥∂ mint′ S

∗(v,t′)∂v

for all v such that q(v, t) = 1, there must exist some w ∈ (v, v) such that

q∗(w, t) < 1 but S∗(v, t) > mint′ S∗(v, t′) for all v ∈ [v, w].

Consider now a “marginal” change to the posited allocation probabilities q∗(., t),

whereby the allocation probability to type (v, t) is decreased by Δf(v∣t) and that to type

(w, t) is increased by Δf(w∣t) . (This modification keeps fixed the overall probability of

allocation conditional on type t.) Such a change increases seller profits, leading to a

contradiction, for a combination of reasons.

First, assuming that the disclosure constraint (14) remains satisfied, such a change

increases seller expected profit (15). Decreasing the allocation probability to buyer (v, t)

by Δf(v∣t) reduces profit by g(t) (V V (v, t)− C) since, by presumption, buyer (v, t) discloses.

On the other hand, increasing the allocation probability to buyer (w, t) by Δf(w∣t) increases

profit by g(t)(V V (w, t)− C ∗ 1(w,t)∈Y ES

)≥ g(t) (V V (v, t)− C) since, by assumption,

virtual value V V (v, t) is increasing in v.

Second, the disclosure constraint (14) remains satisfied with respect to the same

disclosure set. By the choice of w, S∗(v, t) > mint′ S∗(v, t′) for all v ∈ [v, w]. Thus, all

buyers (v, t) with v ∈ [v, w] already had to disclose in order to deter buyers of other types

from mimicking them. Next, since allocation probability is shifted from (v, t) to (w, t),

the surplus that must be given to buyer (v, t) does not change for all v ∕∈ [v, w]. Thus,

type-t buyers’ incentive to mimic buyers of other types remains unchanged, as do type-t

buyers’ incentives to mimic those of type t.

Step 3: (v, t) ∈ Y ES∗ implies q∗(v, t) = 1. By Step 2, buyers (v′, t) must disclose for

all v′ > v when buyer (v, t) discloses. Thus, although allocating the good to buyer

(v, t) increases the buyer surplus awarded to buyers (v′, t) for all v′ > v, no buyers

27

(v′, t′) now prefer to mimic such buyers (v′, t) (since, by assumption, disclosure of type

t simply cannot be mimicked). Thus, the only effect of seller profit of increasing the

allocation probability to buyer (v, t) is that captured by virtual value. So, inducing

buyer (v, t) to disclose and then allocating the object to him with probability q∗(v, t)

generates expected profit if q∗(v, t)V V (v, t)− C. Since the seller could have guaranteed

zero profit by not inducing him to disclose and not allocating him the object, it must be

that q∗(v, t)V V (v, t) ≥ C and hence V V (v, t) > 0. But, in that case, profit is maximized

by setting q∗(v, t) = 1.

Step 4: Buyers with zero value get zero surplus. Clearly, mint S∗(0, t) = 0 since otherwise

the seller can maintain incentives while increasing profit by reducing all buyers’ surplus.

Suppose S(0, t̂) = 0. If S(0, t) > 0 for some t ∈ T , buyer (0, t) must disclose his type

to dissuade buyer (0, t̂) from mimicking him. By Steps 2-3, buyer (v, t) must discloses

and receives the good with probability one for all v ≥ 0. However, since buyer (v, t)

has a negative virtual value for all v ≈ 0, the seller can increase profits by excluding

some type-t buyers from receiving the good, while continuing to induce all such buyers

to disclose.13

Step 5: All buyers who disclose face a “take-it-or-leave-it price”. For each type t ∈ T ,

let v(t) = inf{v : S∗(v, t) > 0}. By (12-13) and the fact that S∗(0, t) = 0, v(t) is the

threshold value below which a buyer of type t receives the object with probability zero.

If v(t) < maxt′ v(t′), then S∗(v(t) + ", t) > mint′ S∗(v(t) + ", t′) and buyer (v(t) + ", t)

must disclose for all " > 0. By Step 3, all buyers (v, t) such that v > v(t) must disclose

13Inducing buyer (v, t) not to disclose has the direct benefit of saving the disclosure cost C, but could

have indirect costs if other buyers need to be offered more surplus to dissuade them from attempting

to mimic (v, t). I side-step this unclear trade-off by showing that the posited mechanism is sub-optimal

even when the disclosure set is left unchanged.

28

and receive the object with probability one. On the other hand, since buyer (v, t) never

receives the good and gets zero surplus for all v < v(t), the seller maximizes profit from

such types by not inducing them to disclose. Finally, since S∗(v(t), t) = 0 and q∗(v, t) = 1

for all v > v(t), buyer net payment znet(v, t) = v(t) for all v > v(t) and znet(v, t) = 0 for

all v < v(t). In other words, type-t buyers face a take-it-or-leave price p∗(t) = v(t)−CB.

Step 6: All buyers who do not disclose face a “take-it-or-leave-it price”. In Step 5, we

considered those buyer types t for which the threshold value v(t) < maxt′ v(t′), all of

whom I showed must disclose whenever they win. By Step 3, remaining buyer types can

potentially be of two kinds: (i) T 1 = {t : v(t) = maxt′ v(t′)} and buyer (v, t) discloses

for all v > v(t); (ii) T 2 = {t : v(t) = maxt′ v(t′)} and there exists " > 0 such that

buyer (v, t) does not disclose for all v ∈ (v(t), v(t) + "). By (14), non-disclosure requires

that S∗(v, t) = S∗(v, t′) and hence (by (12)) that q∗(v, t) = q∗(v, t′) = q∗,no(v) for all

v ∈ (v(t), v(t) + ") and all t, t′ ∈ T 2.

Increasing all allocation probabilities {q∗(v, t) : t ∈ T 2} by Δ increases seller profit

by ΔE[V V (v, t)∣t ∈ T 2]. By presumption, q∗(v, t) > 0 for all v > v(t) and all t ∈ T 2;

thus, E[V V (v(t), t)∣t ∈ T 2] ≥ 0. Since virtual value V V (v, t) is strictly increasing in v

for all t, E[V V (v, t)∣t ∈ T 2] > 0. We conclude that q∗(v, t) = 1 for all v ∈ (v(t), v(t) + ")

and all t ∈ T 2. Since such buyers’ surplus increases at the fastest possible rate (by (12),

dS∗(v, t)/dv = q∗(v, t) ≤ 1 for all (v, t)), t2 ∈ T 2 implies that S∗(v, t2) ≥ S∗(v, t1) for all

v ∈ (v(t2), v(t2) + ") and all t1 ∈ T 1. On the other hand, buyer (v, t2) cannot get strictly

greater surplus than buyer (v, t1), since then he would have to disclose. We conclude that

q∗(v, t1) = 1 for all v ∈ (v(t2), v(t2) + ") and all t1 ∈ T 1, so that S∗(v, t1) = S∗(v, t2) for

all such values. In particular, by this logic buyer (v, t1) does not disclose for any value

in a neighborhood of maxt′ v(t′), whereby we conclude that T 1 = ∅.

Put differently, we conclude that q∗(v, t) = 1 and buyer (v, t) does not disclose for

29

all t ∈ T 2 and all v in a neighborhood of v(t). Indeed, extending this logic, one can

easily show that q∗(v, t) = 1 and buyer (v, t) does not disclose for all t ∈ T 2 and all

v > maxt′ v(t′). (Straightforward details omitted to save space.) Again, buyer net

payment znet(v, t) = maxt′ v(t′) for all v > maxt′ v(t′) and znet(v, t) = 0 for all v <

maxt′ v(t′). In other words, buyers of any type t ∈ T 2 face a take-it-or-leave price

p∗(all) = maxt′ v(t′)− CB.

Step 7: Optimal IC mechanism is a sticker-price mechanism. The previous steps have

shown that the mechanism that maximizes seller profit, subject to the IC necessary

conditions (12-14), is a sticker-price mechanism. (The sticker price p∗(all) = maxt′ v(t′)

and is paid by all buyers of types t ∕∈ T 2 whenever they buy the object; for each t ∈ T 2 =

D that discloses whenever they buy, the customized price p∗(t) = v(t).) To complete

the proof, we need to check that this sticker-price mechanism is incentive-compatible.

Consider first a buyer of type t ∈ T 2. Since p∗(t) + CB ≤ p∗(all), he never prefers

to buy the good at sticker price, and only prefers to disclose and then purchase it at

his customized price when v > p∗(t) + CB. Finally, consider a buyer of type t ∕∈ T 2.

Purchasing the good at a customized price is not an option for him, since (by assumption)

he cannot mimic a buyer of another type, and he only prefers to purchase the good at

sticker price when v > p∗(all).

A.4 Proof of Theorem 2

Proof. Discount condition (3). Consider any sticker-price mechanism with sticker price

p(all) and disclosing set D ∋ t. Since disclosure costs CB, CS are incurred by the buyer

and seller when a sale occurs to a buyer of type t, but not otherwise, the seller’s problem

when setting the customized price p(t) is equivalent to that of a monopolist facing a buyer

30

whose value is distributed as (v − CB)∣t given marginal cost CS and a price ceiling of

p(all)−CB. (If the price ceiling is violated, no type-t buyer will ever choose to disclose, a

contradicting the presumption that t ∈ D, a contradiction.) Thus, if p(all)−CB > p∗(t),

then p∗(t) is the optimal customized price to charge type-t buyers.

Finally, in any optimal sticker-price mechanism, p∗(all)−CB > p∗(t) for any t ∈ D∗.

Suppose to the contrary that p∗(all)−CB ≤ p∗(t). The seller can increase expected profit

by revoking type-t buyers’ eligibility for a discount and forcing them to pay sticker price:

maxp≤p∗(all)−CB

(p− CS)(1− F (p+ CB∣t)) = (p∗(all)− CB − CS)(1− F (p∗(all)∣t)) (16)

< p∗(all)(1− F (p∗(all)∣t)). (17)

To verify this, note that d[(p−CS)(1−F (p+CB ∣t))]dp

= −f(p+CB∣t)(p+ CB + 1−F (p+CB ∣t)

f(p+CB)− C

).

By Assumption 1, p + CB + 1−F (p+CB ∣t)f(p+CB)

is increasing in p. In particular, the seller’s

expected profit is strictly increasing in p for all price-levels up to p∗(t), including all

p ≤ p∗(all)− CB ≤ p∗(t).

Sticker-price condition (2). Consider any sticker-price mechanism with disclosing set D,

sticker price p(all), and discounts p(t) ≤ p(all)−CB for all t ∈ D. Increasing the sticker

price from p(all) to p(all) + " is feasible and profitable for small enough " > 0 unless∫t∈D

d[p(all)(1− F (p(all)∣t))]dp

g(t)dt ≤ 0. (18)

Decreasing the sticker price from p(all) to p(all) − " is feasible as well, as long as each

discount p(t) is also decreased to min{p(t), p(all)− "−CB} so as to maintain disclosure

incentive-compatibility for all types t ∈ D. Such a pricing change is profitable for all

small enough " > 0 unless∫t∈D and t∈D:p(t)=p(all)−CB

d[p(all)(1− F (p(all)∣t))]dp

g(t)dt ≥ 0. (19)

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However, as shown in the proof of (3), p∗(t) < p∗(all) − CB for all t ∈ D∗. Thus, the

sticker price p∗(all) in any optimal sticker-price mechanism must satisfy∫t∈D∗

d[p∗(all)(1− F (p∗(all)∣t))]dp

g(t)dt = 0. (20)

Note that (20) is the standard first-order condition of a monopoly seller who faces a

buyer having value randomly distributed as v∣t ∈ D∗, and uniquely identifies p∗(all) as

the solution to (2). (Uniqueness follows from Assumption 1.)

Disclosing-set condition (4). “⇐”. Suppose that type-t buyers are induced to disclose

in the optimal sticker-price mechanism, i.e. t ∈ D∗. As shown in the proof of (3),

p∗(t) < p∗(all) − CB, so that maxp≤p∗(all)−CB(p− CS) (1− F (p+ CB∣t)) is realized at

customized price p∗(t). Suppose for the sake of contradiction (f.s.o.c.) that p∗(all)(1 −

F (p∗(all)∣t)) > (p∗(t)− CS) (1− F (p∗(t) + CB∣t)). If so, the seller can increase expected

profit by revoking type-t buyers’ eligibility for a discount, contradicting the presumption

of optimality.

“⇒”. Suppose that t ∈ D∗ and let p̂(t) = arg maxp≤p∗(all)−CB(p− CS) (1− F (p+ CB∣t)).

Suppose f.s.o.c. that p∗(all) (1− F (p∗(all)∣t)) < (p̂(t)− CS) (1− F (p̂(t) + CB∣t)). If so,

the seller can increase expected profit by inducing type-t buyers to disclose their type

with a customized price p(t) = p̂(t), contradicting the presumption of optimality. Finally,

define

Z = {t ∈ T : p∗(all) (1− F (p∗(all)∣t)) = (p̂(t)− CS) (1− F (p̂(t) + CB∣t))} . (21)

To complete the proof, it suffices to show that Z ∩D∗ = ∅ since then the seller strictly

prefers not to induce any of the types t ∈ D∗

to disclose in the optimal sticker-price

mechanism.

As shorthand, let p∗(all;D) denote the optimal sticker price given non-disclosing

set D ⊂ T ; in particular, the optimal sticker price p∗(all) = p∗(all;D∗) and p∗(all; t) =

32

arg maxp p(1−F (p∣t)) denotes the optimal monopoly price without disclosure when faced

with a buyer known to be type t.

Suppose f.s.o.c. that Z ∩ D∗ ∕= ∅. If so, the seller can increase expected profit by

(i) inducing every type t ∈ Z ∩ D∗ to disclose with a customized price of p̂(t) and

(ii) changing the sticker price from p∗(all) to p∗(all;D∗ ∖ Z). By (21), inducing types

t ∈ Z ∩ D∗ to disclose has no effect on the seller’s expected profit from such buyers.

Further, “re-optimizing” the sticker price for all other types D∗ ∖ Z if anything allows

the seller to increase expected profits from those non-disclosing types that remain. Thus,

such a two-fold modification to the original sticker-price mechanism must weakly increase

seller expected profit if it does not violate any disclosure IC constraints, i.e. as long as

p∗(all;D∗ ∖Z) ≥ p̂(t) +CB for all t ∈ D∗∪Z. To complete the proof, it therefore suffices

to show that p∗(all;D∗ ∖ Z) ≥ p∗(all) since then all disclosure IC constraints become

more slack.

First, t ∈ Z implies p∗(all; t) < p∗(all). Suppose f.s.o.c. that p∗(all; t) ≥ p∗(all). If so,

(p̂(t)− CS)(1− F (p̂(t) + CB∣t)) < p̂(t)(1− F (p̂(t)∣t))

≤ p∗(all)(1− F (p∗(all)∣t)),

violating the definition of Z. The first inequality follows directly from CB, CS ≥ 0 and

CB + CS > 0, while the second follows indirectly from the presumption that p∗(all; t) ≥

p∗(all). Namely, (i) d[p(1−F (p∣t))]dp

> 0 for all p < p∗(all; t) by Assumption 1, (ii) p̂(t) ≤

p∗(all) by the definition of p̂(t), and (iii) p∗(all) ≤ p∗(all; t) by presumption. By this con-

tradiction, we conclude p∗(all; t) < p∗(all) and in particular that d[p∗(all)(1−F (p∗(all)∣t))]dp

< 0

for all t ∈ Z. Consequently,∫t∈D∗∖Z

d [p∗(all) (1− F (p∗(all)∣t))]dp

g(t)dt > 0.

33

so that p∗(all;D∗ ∖ Z), the optimal sticker price against buyer-types D

∗ ∖ Z, is strictly

greater than p∗(all). This completes the proof.

References

Border, K., and J. Sobel (1987): “Samurai accountant: a theory of auditing and

plunder,” Review of Economic Studies, 54(4), 525–540.

Bull, J., and J. Watson (2005): “Hard evidence and mechanism design,” Mimeo.

Chiang, R., and C. Spatt (1982): “Imperfect price-discrimination and welfare,” Re-

view of Economic Studies, 49(2), 153–181.

Deneckere, R., and R. McAfee (1996): “Damaged goods,” Journal of Economics

and Management Strategy, 5(2), 149–174.

Deneckere, R., and S. Severinov (2006): “Screening when some agents are non-

strategic: does a monopoly need to exclude?,” RAND Journal of Economics, pp.

816–841.

Fisher, F. (1975): “The social costs of monopoly and regulation: Posner reconsidered,”

Journal of Political Economy, 83, 322–340.

Green, J., and J.-J. Laffont (1986): “Partially verifiable information in mechanism

design,” Review of Economic Studies, 53, 447–456.

Grossman, S. (1981): “The informational role of warranties and private disclosure

about product quality,” Journal of Law and Economics, 24(3), 461–483.

Kimes, S., and J. Wirtz (2003): “Has Revenue Management become acceptable?,”

Journal of Service Research, 6, 125–135.

34

Marco, M. (2007): “Best Buy’s secret ‘Employee Only’ in-store website shows different

prices than public website,” consumerist.com, February 9.

Milgrom, P. (1981): “Good news and bad news: representation theorems and applica-

tions,” Bell Journal of Economics, 12, 380–391.

Posner, R. (1975): “The social costs of monopoly and regulation,” Journal of Political

Economy, 83, 322–340.

Schmalensee, R. (1981): “Output and welfare implications of third-degree price dis-

crimination,” American Economic Review, 71, 242–247.

Townsend, R. (1979): “Optimal contracts and competitive markets with costly state

verification,” Journal of Economic Theory, 21, 265–293.

Tullock, G. (1967): “The welfare costs of tariffs, monopolies, and theft,” Western

Economic Journal, 5, 224–232.

Varian, H. (1985): “Price-discrimination and social-welfare,” American Economic Re-

view, 75(4), 870–875.

Wilson, R. (1993): Nonlinear pricing. Oxford University Press, New York.

35