Should a monopolist sell before or after buyers know

their demands?∗

Marc Moller† Makoto Watanabe‡

Abstract

This paper explains why some goods (e.g. airline tickets) are sold cheap to

early buyers, while others (e.g. seasonal products) offer discounts to those who

buy late. We derive the profit maximizing selling strategy both for a monopolist

who is capacity constrained and for one who can choose his capacity ex ante. We

assume that (1) buyers face uncertainty about their individual demand and (2)

buyers are heterogeneous. So far the literature on dynamic monopolistic pricing

has considered these two aspects in separation resulting in clear cut predictions.

We show that none of these predictions survives in a model that combines both

aspects.

JEL classification: D42, D82, L12

Keywords: Dynamic monopolistic pricing, individual demand uncertainty, in-

tertemporal price discrimination

∗We thank Jeremy Bulow, Harrison Chen, Guido Friebel, Dan Kovenock, Diego Moreno, VolkerNocke, Katharine Rockett, Manuel Santos, Alan Sorensen, and participants at various seminars andconferences for valuable discussions and suggestions. The second author acknowledges support fromthe Spanish government research grant SEJ 2006-11665-C02.

†Department of Economics, University Carlos III Madrid, Calle Madrid 126, 28903 Getafe Madrid,Spain. Email: mmoller@eco.uc3m.es, Tel.: +34-91624-5744, Fax: +34-91624-9329.

‡Department of Economics, University Carlos III Madrid, Calle Madrid 126, 28903 Getafe Madrid,Spain. Email: mwatanab@eco.uc3m.es, Tel.: +34-91624-9331, Fax: +34-91624-9329.

1

1 Introduction

Many goods and services can be purchased in advance, i.e. long before their actual date

of consumption. Common examples are airline travel, theater performances, the right

to participate in sports events or trade fairs, and seasonal produts like the newest skiing

equipment. In these examples consumers often face a trade–off between buying early

and buying late. When capacity is restricted, buying early minimizes the consumers’

risk to become rationed. On the other hand, when individual demands are uncertain,

buying late maximizes the consumers’ information.

In this paper we study the implications of this trade–off for a monopolist’s optimal

selling strategy. We are motivated by the fact that examples which share the above

characteristics often show systematic differences in pricing patterns. For instance,

while airline ticket prices increase steadily until departure, theater tickets can often be

bought at half price on the day of the performance. For some products introductory

offers are employed whereas others are sold by use of a clearance sale. In this paper we

explain these differences in pricing patterns using a model with two main ingredients;

(1) individual demand uncertainty and (2) buyers’ heterogeneity.

So far the literature on dynamic monopolistic pricing has considered these two

aspects in separation. One branch of the literature assumes that buyers are hetero-

geneous but know their individual demands, see for example Conlisk, Gerstner and

Sobel (1984), Harris and Raviv (1981), Lazear (1986), Nocke and Peitz (2007), and

Van Cayseele (1991). In all of these papers the monopolist’s profit maximizing price

schedule is non–increasing, i.e. introductory offers are never optimal. When consumers

know their demands in advance the above trade–off is absent. Consumers have a clear

preference for buying early as it lowers their risk to become rationed. As a consequence

the above papers are unable to explain the existence of increasing price paths.

A second branch of the literature allows for individual demand uncertainty but

assumes that buyers are homogeneous ex ante, see for example Courty (2003b), and

DeGraba (1995). These papers find that the monopolist maximizes his profits by selling

exclusively either before or after individual demand uncertainty has been resolved. As

customers are homogeneous ex ante, intertemporal price discrimination has no bite.

2

In this paper we argue that none of these predictions survives in the presence of

both, individual demand uncertainty and buyers’ heterogeneity. For this purpose we

consider a simple two period model. In the first period buyers face individual demand

uncertainty whereas in the second period all demand uncertainty has been resolved.

Buyers differ in their expected valuations and valuations might turn out to be high

or low. A monopolist faces ex ante uncertainty about the buyers’ types and ex post

uncertainty about their realised valuations.

As the buyers’ trade-off between buying early and buying late hinges on their risk to

become rationed, the monopolist’s capacity is a crucial variable in our model. In some

of the above examples exogeneous restrictions prevent the seller from adjusting his

capacity. For instance, due to logistic restrictions, organizers of the London Marathon

have been unable to increase their capacity for several years even though demand has

soared. Similarly, the capacity of airline travel between two specific destinations is

often restricted by the number of landing slots available. In Section 3 we therefore

start our analysis by considering the case of a capacity constrained monopolist. In

Section 4 we then extend our analysis to the case of a monopolist who can choose his

capacity ex ante.

Our results are as follows. When the monopolist is capacity constrained his profit

maximizing selling strategy depends crucially on his capacity. When capacity is small

the monopolist’s profits are maximized by selling exclusively after buyers have learned

their demands. When capacity is large and the fraction of good types is small, it is

optimal for the monopolist to sell exclusively before buyers have learned their demands.

In all remaining cases selling in both periods at either increasing or decreasing prices

maximizes the monopolist’s profits. However, a necessary condition for decreasing

prices to be optimal is that capacity is smaller than potential demand so that consumers

face a positive risk to become rationed.

With respect to the monopolist’s optimal choice of capacity we find that the mo-

nopolist never has an incentive to restrict his capacity in order to be able to implement

a clearance sale. If the monopolist wishes to screen buyers, he does so by use of an

introductory offer as it comes without the implied cost of a capacity restriction.

The main insight of this paper is that in the presence of individual demand uncer-

3

tainty and buyers’ heterogeneity a monopolistic seller faces a trade–off. On the one

hand the existence of individual demand uncertainty gives the monopolist an incentive

to sell exclusively either before or after demand uncertainty has been resolved. On the

other hand, accounting for buyers’ heterogeneity requires some form of intertemporal

price discrimination and hence the sale in both periods. We are the first to study this

trade–off and its implications for the monopolist’s optimal selling strategy.

The only papers that combine individual demand uncertainty with buyers’ hetero-

geneity are Courty and Li (2000) and Gale and Holmes (1993). Courty and Li (2000)

allow the monopolist to screen types by offering a menu of refund contracts before

demand uncertainty has been resolved. Offering one contract with zero refund and

price P1 and another contract with full refund and price P2 is equivalent to selling the

product at price P1 before demand uncertainty has been resolved and at price P2 after.

Note that in order to screen types it has to hold that P1 < P2 as otherwise all types

would prefer the contract with full refund. Hence using refund contracts the monopolist

is unable to implement decreasing price paths. However, as in Courty and Li (2000)

the monopolist’s capacity is unconstrained, this inability has no consequence. In the

absence of rationing risk, buyers have a clear preference for buying late which implies

that decreasing price paths can never be optimal. In this paper we abstract from the

possibility of refund contracts. Moreover, in order to cover the above examples we

allow the monopolist to be capacity constrained and show that in this case decreasing

price paths may become optimal.

Gale and Holmes (1993) consider a monopolistic airline which aims to divert demand

from a peak time flight to an off–peak time flight. Customers learn over time which

flight they prefer and differ in their disutility from taking the wrong flight. Gale and

Holmes offer the insight that the airline should offer a discount for the off–peak flight

together with an advance purchase requirement in order to benefit from customers’

demand uncertainty. As Gale and Holmes restrict the degree of individual demand

uncertainty and buyers’ heterogeneity, intertemporal price discrimination is always

optimal. In our model we allow for an arbitrary level of demand uncertainty and

heterogeneity and derive the conditions under which intertemporal price discrimination

is dominated by selling exclusively before or after buyers know their demands.

4

Finally our work is also related to Lewis and Sappington (1994) who study a mo-

nopolist’s incentive to supply potential customers with information (e.g. a test drive).

They identify general settings in which it is optimal for the monopolist to supply either

perfect information or no information at all. Translated into our setup this amounts

to selling exclusively either before or after buyers have learned their demands. As we

allow buyers to be heterogeneous ex ante the monopolist may find it optimal to supply

intermediate levels of (aggregate) information by selling to some buyers before they

know their demands and to others after.

The plan of the paper is as follows. Section 2 describes our setup. In Section 3

we take capacity as exogeneously given and characterize the monopolist’s profit maxi-

mizing selling strategy in dependence of the size of individual demand uncertainty and

the monopolist’s capacity. We distinguish between the case where the monopolist’s

capacity constraint is binding (i.e. capacity is smaller than potential demand) and the

case where it is slack. Section 4 considers the monopolist’s optimal choice of capacity.

In Section 5 we show that our results remain valid when the monopolist is unable to

commit to price schedules ex ante. Section 6 concludes. All proofs are contained in

the Appendix.

2 The setup

We consider a monopolistic seller who faces a continuum of buyers with mass normal-

ized to one.1 Buyers have unit demands and trade might take place in two periods;

in an advance purchase period 1 and in the consumption period 2. We assume that

demand exhibits the following two characteristics.

Individual demand uncertainty : Ex ante, buyers are uncertain about the consump-

tion value they will derive from the good. A buyer’s valuation might take two values;

a high value normalized to 1 and a low value u ∈ (0, 1).2 Valuations are distributed

independently and buyers privately learn their own valuation between period 1 and

period 2.

1A model with N buyers leads to similar results but is less tractable.2An alternative interpretation is that ex ante buyers value the good at 1 but face a risk that their

consumption value is reduced to u.

5

Heterogeneity : There are two types of buyers. A fraction g ∈ (0, 1) has good

type and a fraction 1 − g has bad type. Bad types face a greater risk to have a

low consumption value than good types. In particular, we assume that a bad type’s

likelihood of having a low valuation is r ∈ (0, 1) while for good types it is ar with

a ∈ (0, 1). We let UG ≡ 1 − ar + aru and UB ≡ 1 − r + ru denote the expected

valuations of good and bad types respectively. Buyers’ types are private knowledge.

Note that our assumptions on demand imply that a bad type’s consumption value

might turn out to be higher than a good type’s expected valuation. We will see that

this assumption is crucial for the optimality of increasing price schedules.

Selling strategies We assume that the monopolist is able to commit to a selling

strategy ex ante. We consider two possible cases. In Section 3 we focus on the case of a

capacity constrained monopolist by assuming that capacity k > 0 is fixed exogeneously.

We assume that the monopolist optimizes over selling strategies of the form (P1, P2, k1).

A selling strategy specifies the prices P1 and P2 at which the product can be purchased

in period 1 and 2 respectively, as well as a first period capacity limit k1 ∈ [0, k].3

In Section 4 we remove the monopolist’s capacity constraint and assume instead that

capacity can be chosen endogeneously by allowing the monopolist to commit to selling

strategies of the form (P1, P2, k1, k).

Every selling strategy together with the implied buyers’ behaviour belongs to one

of five categories which we define as follows:

Definition 1 We say that a selling strategy implements

• Clearance Sales if P1 > P2 and sales are positive in both periods.

• Introductory Offers if P1 < P2 and sales are positive in both periods.

• Constant Prices if P1 = P2 and sales are positive in both periods.

• Advance Selling if sales are positive in period 1 and zero in period 2.

• Spot Selling if sales are positive in period 2 and zero in period 1.

3Capacity limits are frequently employed in the sale of airline tickets. Airlines partition theiravailable capacity into fare classes and different fares are released into the electronic booking systemsat different pre–determined points in time.

6

Timing. The timing of events is as follows. First nature determines the buyers’

types and the monopolist commits to a selling strategy. In period 1, all buyers decide

simultaneously whether to buy the good at price P1. If demand exceeds the capacity

limit k1 buyers are rationed randomly. After period 1 and before period 2 buyers learn

their valuations. Finally, all capacity that has remained unsold is offered in period 2.

Buyers who have not bought in period 1 decide simultaneously whether to buy the

good at price P2. Again, if demand exceeds the remaining capacity buyers are rationed

randomly. Note that we do not allow for the possibility of resale.

Payoffs. We assume that production and capacity costs are zero so that the mo-

nopolist’s payoff, Π, is identical to his revenue. Buyers are risk–neutral and have

quasi–linear preferences. In particular, when a buyer purchases the good his payoff

is equal to the difference between his (expected) consumption value and the price.

Otherwise his payoff is zero.4

3 Capacity constraints

We begin our analysis by considering the case of a capacity constrained monopolist.

In particular, we take capacity k > 0 as given exogenously and derive the monopolist’s

profit maximizing selling strategy (P ∗

1 , P ∗

2 , k∗

1). The monopolist’s optimal capacity

choice will be the topic of the next section.

As capacity is usually harder to adjust than prices, our results in this section might

be interpreted as a description of the monopolist’s behaviour in the short run. However,

in the introduction we gave some examples in which exogeneous restrictions prevent a

monopolist from adjusting his capacity even in the long run. In the presence of such

exogenous capacity restrictions our results in this section are relevant also in the long

run.

It is immediate that for k > 1 the monopolist’s profit maximizing selling strategy

is identical to the one for the case k = 1. We can therefore restrict our attention to

the case where k ∈ (0, 1) and attain the monopolist’s optimal selling strategy for the

4We assume throughout that a buyer demands the good when indifferent between buying and notbuying.

7

remaining cases by considering the limit as k → 1. We start by discussing optimality

for the five categories of selling strategies defined above.

Spot Selling

Consider the possibility that the monopolist sells exclusively after demand uncertainty

has been resolved by choosing k1 = 0. Note first that Spot Selling with P2 = u cannot

maximize the monopolist’s profits as it is dominated by (P1, P2, k1) = (UB, 1, k). For

Spot Selling to be optimal it therefore has to hold that P2 = 1. If the monopolist’s

capacity k does not exceed the expected second period demand, g(1−ar)+(1−g)(1−r),

then Spot Selling with P2 = 1 clearly maximizes the monopolist’s profits. Otherwise,

the monopolist can increase his profits by setting (P1, P2, k1) = (UB, 1, k − g(1− ar)−

(1 − g)(1 − r)). Defining

rS ≡1 − k

1 − g(1 − a)(1)

we therefore have the following result:

Lemma 1 Suppose that k ∈ (0, 1). Spot Selling maximizes the monopolist’s profits if

and only if k ≤ g(1− ar)+ (1− g)(1− r) ⇔ r ≤ rS. When Spot Selling is optimal, the

monopolist sells his entire capacity k at price P2 = 1 and his profits are ΠS ≡ k.

Clearance Sales

Clearance Sales are employed for a broad variety of products ranging from bargain

holidays to theater tickets. By definition a Clearance Sale requires that P2 < P1 and

that sales are positive in both periods. A Clearance Sale therefore requires that P2 < 1.

Also note that a Clearance Sale (P1, P2, k1) with P2 ∈ (u, 1) is dominated by the selling

strategy (P1, 1, k1) which increases P2 without lowering neither second nor first period

demand. Hence for a Clearance Sale to be optimal it has to hold that P2 = u.

Now consider the first period price. Given that P2 = u implies a positive option

value to buyers who postpone their purchase until period 2, a P1 < UB is necessary

to induce first period demand from both types of buyers. Hence a Clearance Sale

8

that induces first period demand from both types is dominated by the selling strategy

(P1, P2, k1) = (UB, 1, k) which sells the same quantity but at strictly higher prices.

This implies that for Clearance Sales to be optimal, the monopolist has to use the

two selling–periods to screen his customers. In particular, P1 has to be chosen such that

good types purchase before and bad types purchase after demand uncertainty has been

resolved.5 In order to induce first period demand from only good types given P2 = u and

k1 ≤ min(g, k) the monopolist optimally sets P1 = P C1 (k1) ≡ UG − k−k1

1−k1

(1− ar)(1−u).

The term k−k1

1−k1

(1 − ar)(1 − u) represents the good types’ option value of delaying the

purchase until period 2. Note that P C1 − u = (1− ar)(1− u) 1−k

1−k1

which implies that in

order to implement a Clearance Sale it has to hold that k < 1. When deciding whether

to buy in advance or not, buyers trade off a lower risk to be rationed when buying

early, with a price discount and better information when buying late. For k ≥ 1 the

risk to be rationed is absent so that first period demand is zero for any decreasing

price schedule. This implies that the monopolist might have an incentive to restrict

his overall capacity in order to be able to screen buyers through a Clearance Sale. We

will consider this incentive in Section 4 where we allow the monopolist to choose his

capacity, k.

Finally, consider the monopolist’s optimal choice of k1. For any k1 ≤ min(g, k)

the monopolist’s profits under the selling strategy (P C1 (k1), u, k1) are given by Π =

k1PC1 (k1)+ (k−k1)u. As P C

1 > u and∂P C

1

∂k1

= 1−k(1−k1)2

(1−ar)(1−u) > 0 the monopolist

optimally chooses k1 = min(g, k). Note that this implies that for k ≤ g Clearance Sales

are dominated by the Advance Selling strategy (P C1 (k), u, k) which leads to the profits

Π = kUG. The next lemma summarizes these findings.

Lemma 2 Suppose that k ∈ (0, 1). For Clearance Sales to maximize the monopolist’s

profits it has to hold that k ∈ (g, 1). When Clearance Sales are optimal, the monopolist

sets k1 = g and P1 = UG − k−g

1−g(1 − ar)(1 − u) > u = P2 and his profits are ΠC ≡

gP1 + (k − g)P2.

Using a Clearance Sale, the monopolist sells his entire capacity but pays information

5Note that good types have a stronger propensity to buy in advance than bad types. In particularthere is no selling strategy which induces bad types to buy in advance and good types to wait.

9

rents. He leaves the option value k−g

1−g(1− ar)(1− u) to good types and a rent of 1− u

to bad types who turn out to have a high consumption value.

Introductory offers

Empirical evidence for the use of introductory offers stems from ticket pricing of low

cost airlines and early registration discounts for academic conferences and sports events.

By definition an Introductory Offer requires that P1 < P2 and that sales are positive

in both periods. It is straight forward to see that for an Introductory Offer to be optimal

it has to hold that P2 = 1. However, for the optimal first period price there are two

possible candidates, P1 = UB and P1 = UG.

Given Lemma 1 we can restrict attention to the case where r > rS. In this case the

monopolist cannot expect to sell his entire capacity at a price of P2 = 1. However, by

implementing an Introductory Offer with P1 = UB and k1 ≥ ki1 ≡ 1− 1−k

r(1−g(1−a))∈ (0, k)

the monopolist can ensure to sell all remaining capacity k − k1 at P2 = 1 in period 2.

Under the selling strategy (UB, 1, k1) profits are given by

Π =

{

k1UB + (1 − k1)(g(1 − ar) + (1 − g)(1 − r)) for k1 < ki1

k1UB + k − k1 for k1 ≥ ki1.

(2)

Note that

∂Π

∂k1=

{

r(u − g(1 − a)) for k1 < ki1

−r(1 − u) for k1 ≥ ki1.

(3)

When the fraction of good types is sufficiently large, i.e. when g > u1−a

, the price dis-

count P1 = UB < 1 = P2 that is necessary to ensure sell out is too large in comparison

to the gain in sales and profits are maximized by setting k1 = 0. Hence for g > u1−a

an Introductory Offer with P1 = UB cannot maximize the monopolist’s profits. For

g ≤ u1−a

the optimal Introductory Offer with P1 = UB sells as much in period 1 as is

necessary to ensure sell out, by setting k1 = ki1 and the resulting profits are

Πi ≡ k −(1 − u)(k − 1 + r(1 − g(1 − a)))

1 − g(1 − a). (4)

An Introductory Offer with P1 = UG offers a smaller price discount than an Introduc-

toty Offer with P1 = UB, but as it only appeals to good types, expected demand in

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the second period is smaller. As a consequence, in order to ensure sell out at P2 = 1,

the monopolist has to sell a larger quantity kI1 ≡ k−1+r(1−g(1−a))

ra∈ (ki

1, k) in period

1. Moreover, Introductory Offers with P1 = UG offer less flexibility than Introductory

Offers with P1 = UB as only a maximum of k1 ≤ g can be sold in the first period.

This threshold becomes binding when k > g and r > 1−k1−g

in which case kI1 > g. For

k1 ∈ [0, min(g, k)], profits under the selling strategy (UG, 1, k1) are given by

Π =

{

k1UG + (g − k1)(1 − ar) + (1 − g)(1 − r) if k1 < kI1

k1UG + k − k1 if k1 ≥ kI1 .

(5)

As profits are increasing in k1 for all k1 < kI1 and decreasing in k1 for all k1 > kI

1 the

monopolist optimally ensures sell out whenever he can by setting

k1 =

{

g if k ≥ g and r ≥ 1−k1−g

kI1 ∈ (0, min(g, k)) otherwise.

(6)

The resulting profits are

ΠI ≡

{

gUG + (1 − g)(1 − r) if k ≥ g and r ≥ 1−k1−g

k − (1 − u)(k − 1 + r(1 − g(1 − a))) otherwise.(7)

Comparing the profits ΠI and Πi and defining

riI ≡(1 − k)(u − g(1 − a))

(1 − g(1 − a))(u − g(1 − a(1 − u)))(8)

we get the following result:

Lemma 3 Suppose that k ∈ (0, 1). When Introductory Offers are optimal the mo-

nopolist sets P1 = UB, P2 = 1, and k1 = ki ∈ (0, k) if g < u1−a(1−u)

and r > riI .

Otherwise he sets P1 = UG, P2 = 1, and k1 = min(g, kI1). The resulting profits are

Π = max(Πi, ΠI).

Note that when feasible, the monopolist adjusts first period capacity to ensure sell out.

By implementing an Introductory Offer the monopolist avoids to pay information rents

to those buyers whose valuation turns out to be high. Moreover, buyers’ option value

of postponing their purchase is zero. However, the elimination of information rents

11

and option values comes at a cost when the monopolist’s capacity is sufficiently large

and buyers are sufficiently likely to have low valuations. In this case the monopolist

faces a high risk to become rationed and in order to ensure sell out he has to offer a

large capacity with a large introductory discount.

Advance Selling

Consider the possibility that the monopolist sells exclusively before demand uncertainty

has been resolved.6 He can achieve this by setting P2 > 1 and k1 = k. For the optimal

first period price there are two candidates, P1 = UB and P1 = UG. Choosing P1 = UB

the monopolist receives the profits Π = kUB. Note however, that this Advance Selling

strategy is dominated by an Introductory Offer with P1 = UB which sells the same

quantity but at higher prices. Similarly, Advance Selling with P1 = UG leads to the

profits Π = gUG but is dominated by an Introductory Offer with P1 = UG. We therefore

have the following result:

Lemma 4 Suppose that k ∈ (0, 1). Advance Selling cannot maximize the monopolist’s

profits.

Constant Prices

Consider a selling strategy that implements positive sales in both periods at constant

prices P1 = P2 = P . Since in period 1 no buyer is willing to pay a price P1 > UG such

a strategy requires that P ≤ UG. While Constant Prices with P = u are dominated by

(P1, P2, k1) = (UB, 1, k), for P ∈ (u, UG] Constant Prices are dominated by the selling

strategy (P1, P2, k1) = (P, 1, k) which increases prices without decreasing sales. We

therefore have the following result:

Lemma 5 Suppose that k ∈ (0, 1). Constant Prices cannot maximize the monopolist’s

profits.

6A recent example of Advance Selling is the 2006 FIFA Football Worldcup. Ticket sale started onFebruary 1st 2005 but the composition of the 32 finalists was not known until November 30th. Manytickets were sold before buyers knew whether they would be able to watch their team.

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3.1 Characterization

As the monopolist’s optimal selling strategy has to fall in one of the five categories

above, Lemmas 1 to 5 together with a direct comparison of the profits under Clearance

Sales, ΠC , with the profits under Introductory Offers, max(Πi, ΠI) lead to a character-

ization of the monopolist’s profit maximizing strategy. Defining the thresholds

rCi ≡(1 − g)2 + (k − g)ga

(1 − g)2 + ga(k(1 − g) − ga(1 − k))(9)

rCI ≡(1 − g)2 + (k − g)(g − u)

(1 − g)2 + (k − g)ga(1 − u)(10)

we have:

Proposition 1 Suppose the monopolist’s capacity constraint is binding, i.e. k <

1. Spot Selling maximizes the monopolist’s profits for all (k, r) ∈ S ≡ {(k, r) ∈

(0, 1)2|r ≤ rS}. Clearance Sales are optimal for all (k, r) ∈ C ≡ {(k, r) ∈ (0, 1)2|r ≥

max(rCi, rCI)}. For all remaining (k, r) ∈ I ≡ (0, 1)2 − C − S, Introductory Offers

constitute the monopolist’s optimal selling strategy. Note that S 6= ∅ and I 6= ∅ while

C 6= ∅ if and only if g < u1−a(1−u)

.

Figure 1 provides a graphical representation of the monopolist’s optimal selling

policy. The intuition for this result has two parts. First, the monopolist faces the

usual negative relation between price and demand. Spot Selling implements the highest

price P2 = 1 but results in the lowest expected demand g(1 − ar) + (1 − g)(1 − r).

Introductory Offers with P1 = UG decrease the price for good types thereby increasing

expected demand to g +(1− g)(1− r). Introductory Offers with P1 = UB decrease the

price even further in order to match expected demand with the monopolist’s capacity.

Finally, Clearance Sales implement the lowest prices resulting in demand from the

entire unit mass of buyers.

It is intuitive that when buyers’ expectations are high already, boosting demand

through price discounts cannot be profitable. Only if the fraction of good buyers is

sufficiently small, i.e. g < u1−a(1−u)

, the monopolist’s capacity is sufficiently large,

i.e. k > g, and buyers’ face a relatively high risk r of having a low consumption

13

g g(1-a)

riI

rS

rCi

rCI

I

r

1

0 1 k

C

S

Figure 1: Characterization of the monopolist’s profit maximizing selling strategy(P ∗

1 , P ∗

2 , k∗

1) in dependence of the monopolist’s capacity k ∈ (0, 1) and the buyers’likelihood r ∈ (0, 1) of having a low consumption value. The bounds rS, riI , rCi, andrCI are defined by (1), (8), (9), and (10) respectively. The areas for which Spot selling,Introductory Offers, or Clearance Sales are optimal are separated by bold lines. Thefigure shows the case where g < u

1−a(1−u). For g ≥ u

1−a(1−u)the set C is empty.

value, Clearance Sales or Introductory Offers with P1 = UB maximize the monopolist’s

profits. In contrast, when capacity is sufficiently small, i.e. k ≤ g(1 − a), or r is

relatively low, Spot Selling is optimal. In an intermediate range, Introductory Offers

with P1 = UG maximize the monopolist’s profits.

Second, the monopolist’s choice between Clearance Sales and Introductory Offers

with P1 = UB hinges on the comparison of ex ante versus ex post information rents.

Under Introductory Offers with P1 = UB, information rents stem from the monopolist’s

inability to distinguish between buyers of distinct types ex ante. The good types of

those ki1 buyers who buy in advance receive an information rent of UG − UB. In

contrast, the information rents under Clearance Sales originate from the unobservability

of buyers’ consumption values ex post. Bad types have an expected information rent

of (1 − 1−k1−g

)(1 − r)(1 − u) while good types receive (1 − 1−k1−g

)(1 − ar)(1 − u).

14

Under both selling strategies, information rents increase with k. Note however,

that ex post information rents depend crucially on the buyers’ risk to become rationed

in the second period, 1−k1−g

. For k → g buyers expect to be rationed with certainty.

Hence ex post information rents become neglegible and Clearance Sales are the most

profitable way to implement a sell–out. On the other hand, for k → 1 the buyers’

risk to become rationed in a Clearance Sale converges to zero. Hence buyers prefer to

postpone their purchase until period 2 and the monopolist has to let P C1 → u in order

to implement a Clearance Sale. When capacity is sufficiently high, Clearance Sales are

therefore dominated by Introductory Offers with P1 = UB.

By considering the limit as k → 1 we can derive the monopolist’s optimal selling

strategy for the remaining case k ≥ 1. For k → 1 Clearance Sales are no longer feasible.

Moreover, the optimal Introductory Offers with P1 = UB converges to Advance Selling

since the capacity ki1 that is sold in advance strictly increases in k with limk→1 ki

1 = 1.

We therefore get the following result:

Proposition 2 Suppose that the monopolist’s capacity constraint is slack, i.e. k ≥ 1.

If g < u1−a(1−u)

then Advance Selling with P1 = UB, P2 > 1, and k1 = 1 maximizes

the monopolist’s profits. Otherwise an Introductory Offer with P1 = UG, P2 = 1, and

k1 = g is optimal.

One can summarize our results so far by saying that whether a monopolist will sell

before or after buyers have learned their demands, depends crucially on his capacity.

When capacity is low, the monopolist will sell exclusively after buyers have learned their

demands. When capacity is high, the monopolist will sell exclusively before buyers have

learned their demands if the fraction of good buyers is sufficiently low. For intermediate

values of capacity the monopolist’s will employ some form of intertemporal screening

by selling in both periods at either increasing or decreasing prices depending on the

buyers’ distribution of types and valuations.

15

4 Capacity choice

In the last section we have seen that for Clearance Sales to be feasible, buyers must

face a positive probability to become rationed. For k ≥ 1 the threat to be rationed is

absent and a Clearance Sale can no longer be implemented. It is therefore important

to understand whether the monopolist has an incentive to restrict his overall capacity

in order to be able to screen buyers by use of a Clearance Sale. In order to answer this

question using our framework we now follow Nocke and Peitz (2007) by assuming that

the monopolist can choose his capacity k ex ante at zero cost. We find the following

result:

Proposition 3 If the monopolist can choose his capacity ex ante at zero cost then his

profits are maximized by setting k = 1. His optimal selling strategy (P ∗

1 , P ∗

2 , k∗

1) is as

stated in Proposition 2.

It is immediate that except for Clearance Sales the monopolist has no incentive to re-

strict his capacity to some k < 1. Moreover, restricting capacity in order to implement

a Clearance Sale can only be optimal when the increase in the price paid by good types,

outweighs the loss in sales to bad types. This is the case when the fraction of good

types is sufficiently high, i.e. when g ≥ u1−ar(1−u)

. It turns out however, that in this

case the optimal capacity choice under Clearance Sales is smaller than the expected

sales under an Introductory Offer with P1 = UG and k = 1. As prices are strictly

higher under the Introductory Offer, Clearance Sales are never optimal.

Proposition 3 differs from the main result of Nocke and Peitz (2007). In their setup

Introductory Offers can never be optimal and the monopolist’s profits are maximized

by selling at a uniform price (as under Advance Selling) or by implementing a Clearance

Sale. This difference stems from the fact that in Nocke and Peitz (2007) buyers know

their individual demands ex ante but aggregate demand is uncertain. When buyers

know their demands ex ante, they prefer to buy in period 1 in order to minimize

their risk to become rationed. As a consequence it cannot be optimal to offer a price

discount to those who buy early. In our setup, buyers do not know their demands ex

ante and, given the risk to be rationed is absent, i.e. for k = 1, prefer to buy in period

2. Hence in our model it cannot be optimal to give a price discount to those who buy

16

late. The comparison of our results with those of Nocke and Peitz (2007) suggests

that Introductory Offers are likely to be used for products whose value is uncertain

to buyers at the time of purchase (e.g. airline tickets) while Clearance Sales can be

expected to be employed when buyers know their valuations but aggregate demand is

uncertain ex ante (e.g. for seasonal products).

In order to relate our results to the literature which assumes that buyers are homo-

geneous ex ante (Courty (2003b) and DeGraba (1995)) we now consider the limiting

case where a → 1. We find the following:

Corollary 1 Suppose that buyers are homogeneous. If the monopolist is capacity con-

strained and k < 1 then profits are maximized by Spot Selling if r ≤ 1 − k and by

Introductory Offers with k1 = 1 − 1−kr

∈ (0, k) otherwise. If the monopolist can choose

his capacity he sets k = 1 and implements Advance Selling.

The second part of Corrolary 1 is similar to the “buying frenzies” of DeGraba (1995)

and Courty’s (2003b) result that selling both before and after buyers have learned

their demands can never be optimal. The intuition for this result is that in the absence

of a capacity constraint the monopolist prefers to sell ex ante in order to avoid the

information rents he would have to pay ex post. Corollary 1 however shows that

the monopolist’s profit maximizing selling strategy differs from the one obtained by

DeGraba (1995) and Courty (2003b) when the monopolist is capacity constrained. In

this case the monopolist avoids information rents by setting P2 = 1 and optimally

sells either in both periods or exclusively after buyers have learned their valuations.

Selling in both periods does not serve the purpose of sreening buyers (as buyers are

homogeneous) but enables the monopolist to avoid rationing risk in period 2. He uses

his first period capacity restriction to make second period capacity match second period

demand.

5 Price commitment

In this section we discuss the robustness of our results with respect to an important

assumption. As in Courty (2003b), Harris and Raviv (1981), and Van Cayseele (1991)

17

we have assumed that the monopolist can commit to a price schedule in advance.7

In the absence of such commitment the monopolist faces a time consistency problem

similar to the one studied in the durable goods literature (see Bulow (1982), Coase

(1972), Gul, Sonnenschein and Wilson (1986), and Stokey (1981)). After selling to

good types in the first period, the monopolist might have an incentive to adjust his

second period price to second period demand.

Under any screeing policy, the monopolist’s lack of commitment therefore requires

second period prices to maximize second period revenue. In Section 3 we have shown

that for an Introductory Offer with P1 = UB to be optimal the monopolist chooses

the first period capacity ki1 ∈ (0, k) so as to ensure the sale of his remaining capacity

k − ki1 at price P2 = 1. Hence in the parameter range for which an Introductory

Offer with P1 = UB is optimal, it does not require any commitment power. However,

Introductory Offers with P1 = UG fail to sell out when k ≥ g and r ≥ 1−k1−g

. In this case

an Introductory Offer with P1 = UG is feasible in the absence of commitment if and

only if P2 = 1 maximizes second period revenue, i.e. when (1− g)(1− r) ≥ (k− g)u ⇔

r ≤ ru ≡ 1− k−g

1−gu. In turn, a Clearance Sale does not require commitment if and only

if r ≥ ru. Note that 1−k1−g

< ru < rCI for all k ∈ (g, 1).

This implies that in the parameter range where Clearance Sales are optimal they

do not require commitment power. The monopolist’s optimal pricing strategy only

changes for parameter values in a strict subset of I which is depicted as the shaded

area in Figure 2. For all remaining parameter values the monopolist’s optimal pricing

strategy without commitment is identical to the optimal strategy with commitment as

stated in Proposition 1. As for k = 1 commitment power is not required, Propositions

2 and 3 hold even if the monopolist cannot commit to prices in advance.

7This practise is common in the organisation of events such as sports or music contests and tradeexhibitions. For example, the Madrid Marathon states on its website that participation in the racetaking place in April will cost 55 Euros if purchased until the end of December, 60 Euros if purchasedfrom January till March, and 85 Euros if purchased thereafter. Although such explicit commitmentis absent in the sale of airline or theater tickets it is implicitely achieved through the repeated natureof the transactions.

18

g g(1-a)

riI

rS rCi

rCI

I r

1

0 1 k

C

S ru

Figure 2: The monopolist’s profit maximizing selling strategy without price commit-ment. The shaded area depicts the parameter set for which the optimal selling policywithout commitment differs from the one with commitment. For parameter values inthis area, Introductory Offers with P1 = UG are not feasible in the absence of commit-ment and will be substituted by a selling policy that implements sell out.

6 Conclusion

In this paper we have considered a monopolist’s profit maximizing selling strategy

when buyers are heterogeneous and uncertain about their individual demands. Our

analysis fills an important gap in the literature on dynamic monopolistic pricing. So

far this literature has concentrated exclusively on the effects of either individual demand

uncertainty or buyers’ heterogeneity. As a consequence authors have either found that

it is optimal to sell exclusively before or after demand uncertainty has been resolved

or that prices should be decreasing. We have shown that in a model which combines

the two effects, none of these predictions survives.

From an applied viewpoint our model is particularly suitable to study the trade–off

between increasing and decreasing price schedules. The model’s simplicity suggests the

introduction of further ingredients that may help to explain why for instance airline

19

tickets are sold cheap to early buyers, while theater tickets offer discounts to those who

buy late. For example, an important issue concerning advance sales problems is the

presence of third parties, so called brokers or middlemen. Brokers stock an inventory

in advance and stand ready for buyers to sell close to the consumption date. The

existence of brokers will therefore influence the consumers’ trade–off between buying

early versus buying late and hence the monopolist’s optimal selling strategy. A closer

look at this issue is left for future research.

Appendix

Proof of Lemma 1

In text. Note that rS decreases linearly in k with limk→1 rS = 0 and rS = 1 for k = g(1−a) ∈(0, g).

Proof of Lemma 2

In text.

Proof of Lemma 3

Compare the profits in (4) with those in (7). As 1− g(1− a) ∈ (0, 1) it is immediate that forΠi > ΠI to hold it is necessary that k ≥ g and r ≥ 1−k

1−gso that ΠI = gUG + (1 − g)(1 − r).

Πi > ΠI is therefore equivalent to r > riI . Moreover riI < 1 for some k ∈ (0, 1) if and onlyif g < u

1−a(1−u) . Note that riI decreases linearly in k with limk→1 riI = 0 and riI = 1 for

k = gu−g(1−a)(1−a(1−u))

u−g(1−a) ∈ (g, 1).

Proof of Lemma 4 and 5

In text.

Proof of Proposition 1

Follows directly from Lemmas 1–5 and a direct comparison of ΠC with max(Πi,ΠI). Westart by comparing ΠC with ΠI . As P I

1 = 1 − ar + aru ≥ 1 − ar + aru − V = PC1 and

P I2 = 1 > u = PC

2 a necessary condition for ΠC > ΠI is that the monopolist is rationedusing an Introductory Offer which happens only if k ≥ g and r ≥ 1−k

1−g. In this case ΠI =

20

g(1− ar + aru) + (1− g)(1− r) and ΠC = g(1− ar + aru− k−g1−g

(1− ar)(1−u)) + (k − g)u so

that ΠC > ΠI ⇔ r > rCI . Note that rCI < 1 for some k ∈ (0, 1) if and only if g < u1−a(1−u) .

In this case

∂rCI

∂k=

(1 − g)2[g(1 − a(1 − u)) − u]

[(1 − g)2 + (k − g)ga(1 − u)]2< 0 (11)

and limk→1 rCI = 1−u1−g(1−a(1−u)) ∈ (0, 1) and rCI = 1 for k = g. Also note that ∂2rCI

∂2k> 0

implies that there exists a unique k∗ ∈ (g u−g(1−a)(1−a(1−u))u−g(1−a) , 1) such that riI(k∗) = rCI(k∗)

and riI(k) > rCI(k) if and only if k < k∗.Now compare ΠC with Πi. It holds that Πi > ΠC if and only if r < rCi. Note that

∂rCi

∂k=

(1 − a)ag2(1 − g)(1 − g(1 − a))

[(1 − g)2 + ga(k(1 − g) − ga(1 − k))]2> 0 (12)

and ∂2rCi

∂2k< 0 with limk→1 rCi = 1. Also note that rCi(k∗) = riI(k∗) as for (k, r) =

(k∗, riI(k∗)) it holds that Πi = ΠI and ΠC = ΠI which implies that ΠC = Πi. We havetherefore shown that the monopolist’s profit maximizing selling strategy looks as depicted inFigure 1.

Proof of Proposition 2

Note first that the monopolist’s profit maximizing selling strategy for k > 1 is identical tothe one for the case k = 1 as total demand is normalized and never larger than 1. Due tocontinuity we can attain the optimal strategy for k = 1 from Proposition 1 by letting k → 1.limk→1 rS = 0 and limk→1 rCi = 1 imply that for k = 1 neither Spot Selling nor ClearanceSales can be optimal. Moreover limk→1 riI = 0 together with Lemma 3 implies that theoptimal selling strategy depends on g. If g < u

1−a(1−u) , Lemma 3 implies that selling with

P1 = UB , P2 = 1 and k1 = ki1 = 1 − 1−k

r(1−g(1−a)) has to be optimal. As limk→1 ki1 = 1 this

selling strategy falls into the class of Advance Selling. For g ≥ u1−a(1−u) , Lemma 3 implies that

selling with P1 = UG, P2 = 1 and k1 = min(g, kI1) is optimal. As limk→1 kI

1 = 1−g(1−a)a

> 1 itholds that k1 = g so that this selling strategy constitutes an Introductory Offer.

Proof of Proposition 3

The only reason why choosing a k < 1 might be optimal for the monopolist is that byrestricting his capacity the monopolist increases the consumers risk to be rationed whichreduces the consumers option value of waiting until period 2. Except for Clearance Sales thiseffect is absent and the monopolist optimally chooses k = 1 as profits are increasing in k.

21

Under Clearance Sales, profits are ΠC = g(1− ar + aru− k−g1−g

(1− ar)(1− u)) + (k − g)u andfor k ∈ (g, 1)

∂ΠC

∂k= u −

g

1 − g(1 − ar)(1 − u). (13)

If u < g1−g

(1 − ar)(1 − u) the monopolist therefore has an incentive to restrict his capac-

ity. However as ΠC ≤ gUG Clearance Sales are dominated by Introductory Offers with(P1, P2, k1, k) = (UG, 1, g, 1). If instead u ≥ g

1−g(1 − ar)(1 − u) the monopolist has no incen-

tive to restrict his capacity in a Clearance Sale. It holds that ΠC ≤ u so that Clearance Salesare dominated by Advance Selling with (P1, P2, k1, k) = (UB , 1, 1, 1). This shows that themonopolist’s optimal capacity choice is k = 1 and his optimal selling strategy (P ∗

1 , P ∗

2 , k∗

1) istherefore as described in Proposition 2.

Proof of Corollary 1

We consider the limiting case a → 1 for which buyers become homogeneous. lima→1 rS = 1−k

implies that for r ≤ 1 − k Spot Selling is optimal. Moreover, for a → 1 the two types ofIntroductory Offers converge to one which sets P1 = 1 − r + ru, P2 = 1, and k1 = 1 − 1−k

r.

Finally, lima→1 rCi = 1 implies that Clearance Sales are dominated by Introductory Offers.This shows that when the monopolist is capacity constrained with k < 1 his profits aremaximized by Spot Selling for r ≤ 1 − k and by Introductory Offers with k1 = 1 − 1−k

r

otherwise. When the monopolist can choose k, Propositions 2 and 3 and the fact that fora → 1 it always holds that g < u

1−a(1−u) imply that the monopolist chooses k = 1 andimplements Advance Selling.

References

[1] Bulow, J. I. (1982) Durable Goods Monopolists. Journal of Political Economy 90(2),314-332.

[2] Coase, R. H. (1972) Durability and Monopoly. Journal of Law and Economics 15, 143-149.

[3] Conlisk, J., Gerstner, E., Sobel, J. (1984) Cyclic Pricing by a Durable Goods Monopolist.Quarterly Journal of Economics 99(3), 489-505.

[4] Courty, P., Li, H. (2000) Sequential Screening. Review of Economic Studies 67(4), 697-717.

[5] Courty, P. (2003a) Some Economics of Ticket Resale. Journal of Economic Perspectives

17(2), 85-97.

22

[6] Courty, P. (2003b) Ticket Pricing under Demand Uncertainty. Journal of Law and Eco-

nomics 46, 627-652.

[7] DeGraba, P. (1995) Buying Frencies and Seller–Induced Excess Demand. RAND Journal

of Economics 26(2), 331-342.

[8] Gale, I. L., Holmes, T. J. (1993) Advance Purchase Discounts and Monopoly Allocationof Capacity. American Economic Review 83(1), 135-146.

[9] Gul, F., Sonnenschein, H., Wilson, R. (1986) Foundations of Dynamic Monopoly andthe Coase Conjecture. Journal of Economic Theory 39(1), 155-190.

[10] Harris, M., Raviv, A. (1981) A Theory of Monopoly Pricing Schemes with DemandUncertainty. American Economic Review 71(3), 347-365.

[11] Lazear, E. (1986) Retail Pricing and Clearance Sales. American Economic Review 76(1),14-32.

[12] Lewis, T. R., Sappington, D. E. M. (1994) Supplying Information to Facilitate PriceDiscrimination. International Economic Review 35(2), 309-327.

[13] Nocke, V., Peitz, M. (2007) A Theory of Clearance Sales. Economic Journal, 117, 964-990.

[14] Stokey, N. L. (1981) Rational Expectations and Durable Goods Pricing. Bell Journal of

Economics 12(1), 112-128.

[15] Van Cayseele, P. (1991) Consumer Rationing and the Possibility of Intertemporal PriceDiscrimination. European Economic Review 35, 1473-1484.

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