Refundable Advance Purchase Tickets and Advance

Purchase Restrictions under Aggregate Demand

Uncertainty

Volodymyr Bilotkach∗

The University of California, Irvine

May 2009

Abstract

This paper rationalizes the use of discounted refundable advance purchase tickets

and advance purchase restrictions in context of models with costly capacity and ag-

gregate demand uncertainty. We show how a monopolist on such markets can improve

upon pricing mechanisms, previously proposed in the related literature for markets for

perishable products, characterized by uncertain demand and costly capacity (which

include but are not limited to such important industries as airlines, hotels, restaurants,

car rentals, and live performances). Our model provides a simple link between two

strains of literature (one modeling non-refundable discounted tickets with demand un-

certainty specified through states of demand, and the other analyzing fully or partially

refundable contracts while treating demand uncertainty at the customer level).

Keywords: demand uncertainty, capacity pricing models, refundable tickets, yield manage-

ment

JEL: D21, D42, L20

∗The author wishes to thank Jan Brueckner and Amihai Glazer for useful comments and guidance; theusual disclaimer applies. E-mail: vbilotka@uci.edu

1

1 Introduction

Under demand uncertainty, a seller whose capacity is fixed in the short-run can employ

various pricing strategies. He can either sell some or all of the products (from this point on,

let us consider the product in question to be a ticket, to be in line with the prior literature that

considered problems of this kind) in advance at a discount; or pre-commit to full or partial

refunds to the customers who choose not to use the tickets purchased in advance; and/or

place advance purchase restrictions on the discounted tickets to correspond to consumers’

learning about their demand. Markets where such strategies can be potentially applied

include (but are not limited to) such important and visible industries as airlines, hotels,

restaurants, car rentals, and live performances.

As an example, the cheapest tickets offered by a typical airline on the US market require

at least 21-days advance purchase, and are non-refundable1; whereas the most expensive

economy class tickets a customer can purchase are fully refundable, available to customers

at virtually any time before the flight2, and allow the consumer to change his itinerary

without paying a fee. In between the two extremes, airlines practice two different strategies:

either offering a series of non-refundable tickets with varying advance purchase restrictions

(in addition to 21-day one mentioned above, typically employed restrictions are 14-day, 10-

day, 7-day and 3-day advance purchase requirements), or writing out a menu of options, each

of which specifies the fare, advance purchase restrictions, and the refundability conditions

(which are typically the laxer the less stringent the refundability conditions are). Some

contracts may also specify other restrictions, such as Saturday night, minimum and maximum

stay; however, those are becoming a rarity on the US domestic airline market. Strangely

enough, theoretical models offered by the industrial organization economists have not as of

yet embraced this kind of pricing in its entirety.

Specifically, we can find two basic treatments of demand uncertainty in the literature.

Some papers treat uncertainty in terms of states of demand, also known as aggregate de-

mand uncertainty, usually considering high and low states (Prescott, 1975; Dana, 1998,

1999, 2001; Gale and Holmes, 1992, 1993); studies taking this approach always assume any

advance purchase tickets (effectively, tickets sold before the true state of demand is realized)

are non-refundable. Escobari and Jindapon (2008) is different in this respect, allowing the

monopolist to sell both refundable and non-refundable tickets to risk-averse consumers in ad-

1As well as non-transferable and changeable only for a fee.2Overbooking practiced by the airlines makes seat availability constraint less important than it seems.

2

vance of the travel date. Other papers treat demand uncertainty at the individual customer

level (assigning each consumer probability of travel, going to a concert, etc.). In these mod-

els (examples are Courty, 2003a; Courty and Li, 2000; Ringbom and Shy, 2004a,b,c; Akan

et al., 2009), consumers are typically characterized by different travel probabilities and/or

valuations; the issue then becomes that of determining the optimal contract(s) specifying

the purchase price and the size of refund a consumer will get should he decide to return

the ticket to the seller3. The firms thus maximize expected profit, subject to customers’

participation and incentive compatibility constraints. In these models (with the exception of

Courty (2003a)), all tickets (whether partially, fully, or not at all refundable) are offered for

sale before demand uncertainty is realized - therefore, an important segment of the market

is not modeled. Akan et al. propose the closest approximation to the naturally occurring

pricing mechanism, by suggesting a menu of expiring refund contracts as monopolist’s op-

timal pricing scheme in an environment where there is correlation between customer types

and the time at which demand uncertainty is realized.

The contribution of this paper is in putting refundable advance purchase contracts into

the context of models treating demand uncertainty via states of demand (i.e., models of

aggregate demand uncertainty). In our simple setup (which resembles that of Dana (1998)

in terms of the customer types and states of demand, and is put into the context of air

travel), we have a ‘leisure‘ customer whose demand is certain, and two ’business’ customers

exhibiting uncertain demand; as in Dana (1998), there is a negative correlation between

the demand uncertainty and valuation. The state of demand is defined by the number of

‘business‘ customers undertaking the trip (however, the identity of a customer traveling

under the low demand state is not revealed beforehand). We show that by expanding the

(usual for such models, as outlined above) menu of advance purchase non-refundable tickets

and the so-called ‘spot‘ tickets (sold after demand uncertainty has been realized) with the

non-refundable tickets requiring advance purchase, the monopolist can increase its profit,

provided such tickets are rationed. Assumption that business travelers receive private signals,

which refine (but do not resolve) their travel uncertainty at some (known to the seller)

time before uncertainty is completely resolved easily rationalizes different advance purchase

restrictions for the non-refundable and refundable tickets sold in advance.

We thus offer a simple model which offers a more realistic (even though a simplistic)

than what has been offered previously view of the pricing on markets characterized by

3Courty (2003a) specifies a model with some tickets sold in advance, and some - after demand uncertaintyhas been realized; refunds are not permitted in his model, but resale may be allowed.

3

costly fixed capacity; aggregate demand uncertainty; and perishable products. Some of the

previous models (while generally specifying a realistic structure of demand uncertainty) did

not allow for refundable tickets to be sold on the advance purchase markets; while other

studies, analyzing fully or partially refundable contracts, did not allow any products to be

offered for sale after demand uncertainty has been sorted out, operating with a demand

structure that can be considered not entirely realistic.

The rest of the paper is organized as follows. Next two sections discuss the model

in detail, starting with the general description of its setup, followed by the treatment of

different pricing mechanisms, two of which have been considered in the previous models

of this kind, and one which is our innovation; we then characterize monopolist’s optimal

choice among the three strategies. This is followed by the section rationalizing the advance

purchase restrictions within our model’s framework. Before providing concluding comments,

we discuss generalization of our simple setup; potential shortcomings of our model; and

directions for future research.

2 Model Setup

Setup of our model is very simple. A monopolist faces three consumers of two types; and

each individual consumer’s demand is dichotomous. Each consumer demands either one or

zero units of the good, which in context of the travel market is interpreted as traveling or

staying at home. There are two consumers of the type h, each valuing the service at V .

A single type l consumer has valuation αV where α < 1. Consumers arrive randomly (a

setting known as proportional rationing). An alternative arrangement, known as parallel

rationing (whereby type h consumers show up first) is trivial, as we will see in the next

section. Consumer types, as is usual in models of this kind, differ in terms of their travel

uncertainty. Specifically, the type l consumer’s demand is certain, whereas the demand of

type h consumers depends on realization of one of the two equally likely states of demand.

In the state H both type h consumers demand the good (choose to travel); whereas in the

state L only one of the two will want to travel, and identity of the type h consumer traveling

in the state L is not known ex ante. Note that ex ante travel probability for a type h

consumer is in this case equal to 3/4. This demand specification is very similar to that

proposed (and subsequently generalized) in Dana (1998); the type l consumer can be said

to roughly represent a leisure traveler; while the type h consumer is a more or less typical

business traveler. Generalization to arbitrary numbers of consumers (say, Nl type l and Nh

4

type h travelers) is feasible, even though algebra involved in this is not always pleasant, as

we will discuss further in this paper. The main point of our analysis is to introduce different

advance purchase contracts into the model, so we will do it in the most parsimonious way

possible.

On the supply side, we have a monopolist who knows consumer types’ valuations, but is

unable to differentiate between the types ex ante. Further, the monopolist has to commit

to prices and capacity before demand uncertainty is realized. The only cost our firm incurs

is the capacity cost of k per unit (such that αV > k), which the monopolist has to pay

whether the capacity is used or not. The entire capacity is delivered at a certain point in

time (e.g., according to the published schedule); if a unit of capacity is not used at the time

of delivery, the opportunity to earn money out of it is lost forever. In other words, we have

a market with uncertain demand, fixed and costly capacity, and a perishable good. These

are characteristics of a number of important and highly visible industries, such as airlines,

hotels, rental cars, restaurants, and live concert performances.

To be in line with most of the literature, we keep our example simple by assuming

consumers are risk neutral4 and that there is no intertemporal discounting. Further, we

prohibit any resale, so that consumers can only buy the tickets from the monopolist. Ideally,

our monopolist would like to sell the good at the price αV to the type l customer, and at the

price V to the type h customers. This is, however, easier said than done, since consumer’s

type remains unobservable to the monopolist.

As we briefly mentioned in the introductory section of this paper, in models similar to

ours, the goods (e.g., tickets) can be released for sale either after the uncertainty has been

realized (this setting is usually called the spot market), or before the type h consumers know

their demand (we will refer to this as the advance purchase market). The monopolist can

either sell all of the tickets on the spot market (think of a restaurant choosing not to accept

advance reservations), release all tickets for sale in advance (as is common on the market

for live concert performances), or offer a portion of tickets for sale on the advance purchase

market, holding the remainder for sale on the spot market5.

The next section gives detailed description of the various pricing strategies our monopolist

can employ. The option of releasing all the tickets on the advance purchase market is clearly

dominated, since the monopolist can count on selling to at least one type h consumer on the

4Escobari and Jindapon (2008) analyze the risk averse traveler’s choice in a different setting from ours.5In a typical airline’s pricing strategy, the highest fully refundable fares are made available at any time;

however, they tend to be purchased closer to the departure date

5

spot market at the full price; thus, we end up considering the following pricing strategies:

selling all the tickets on the spot markets; and selling some of the tickets on the spot market,

offering either only non-refundable tickets on the advance-purchase market (and rationing

the number of tickets offered), or both non-refundable and a limited number of refundable

tickets in advance. The former two strategies have been previously considered, and the latter

is this paper’s innovation.

3 Pricing Strategies

3.1 Spot Market

Let us start with the spot market case. The monopolist knows that on the spot market

there will be one type l consumer and either one or two type h customers. If the firm offers

one seat priced at αV , and two seats at V , then (assuming proportional rationing, and with

consumers purchasing the cheaper ticket first), the firm’s total expected profit will be:

πs1 =1

2

(αV +

1

2V

)+

1

2

(αV +

4

3V

)− 3k = V

(α +

11

12

)− 3k (1)

The first part of expression (1) represents expected profit in the low demand state (where

the type l consumer gets the cheaper ticket with probability 1/2, and the more expensive

ticket is consequently sold with probability 1/2), and the second part writes out expected

profit in the high demand state. In this case, the lower priced seat and one higher priced

seat sell with certainty; whereas the second high price seat only sells with probability 1/3,

since there is 2/3 chance a type h consumer bought the low priced seat.

This pricing leads to the monopolist offering three seats, of which either one, two or all

three will be occupied. One seat will be sold in case the type h consumer purchases the

low-priced ticket in state L (this happens with probability 1/4). Two tickets are sold if the

αV priced ticket is sold to either the type l customer in state L or to a type h traveler in state

H. The former happens with probability 1/4, the latter with probability 1/36. Then, two

seats are occupied with probability 7/12. Finally, three seats are occupied with probability

1/6 (three seats are only occupied in state H, and only if the αV priced ticket is sold to the

type l customer: probability of state H is 1/2, and probability that the type l customer is

6In state H - which happens with probability 1/2, two seats are sold with probability 2/3; and in stateL the monopolist is as likely to sell one seat as two

6

the first one to arrive in this state is 1/3). Then, expected load factor under this pricing

strategy will be 23/36, or slightly lower than 2/3.

Alternatively, the monopolist may choose to only offer two seats priced at V each (this

is what the monopolist will always do in the parallel rationing setup). In this case, the type

l consumer is priced out of the market; one seat is sold with certainty, and the second seat

is sold with probability 1/2, giving total profit equal to:

πs2 =3

2V − 2k (2)

Note that when our firm commits to selling fewer tickets, it provides lower total capacity.

In this case, one or two seats are sold with equal probability; and the expected load factor

is 3/4.

The following Lemma specifies the monopolist’s choice in case the firm decides to dis-

tribute all the tickets on the spot market.

Lemma 1 If the monopolist distributes all tickets on the spot market, and consumers arrive

in random order, the firm will choose to offer one ticket at price αV and two tickets at the

price V each, provided:

α ≥ k

V+

7

12(3)

Otherwise, the monopolist will offer only two tickets at price V each, pricing the type l

consumer out of the market.

Proof. The result in (3) follows trivially from comparison of (1) and (2).

Next, let us see how offering all tickets for sale on the spot market compares to other possible

pricing arrangements.

3.2 Advance Purchase Discounts - Non-Refundable Tickets

The trouble with using only the spot prices is that type l consumer is not guaranteed the

lower priced ticket (if such a ticket is offered in the first place). In Dana (1998), this problem

is solved by offering the lower priced tickets for sale before demand uncertainty is realized.

In our example, this strategy may require adjusting the spot prices; at the same time, we

can show that by selling some of the tickets in advance in a fashion similar to what was

proposed by Dana, our monopolist may be able to increase its profit.

7

Suppose the firm tries selling one non-refundable ticket in advance at price of αV (target-

ing type l consumer) - this strategy is consistent with Dana’s setup. The following Lemma

derives the spot prices the firm will be able to charge in this case.

Lemma 2 When selling one non-refundable ticket on the advance purchase market at price

αV , the monopolist will set the following prices for the two tickets sold on the spot market:

• If α > 34, the spot market tickets will be priced at V

• If α ≤ 34, the spot market tickets will be priced at 4

3αV

Proof. On the advance purchase market, the type h consumer’s expected value of the

non-refundable ticket priced at αV is:

3

4(V − αV )− 1

4αV = V

(3

4− α

)(4)

The first part of (4) represents ex ante value of the advance purchase ticket if it is used (which

happens with probability 3/4); while the second part shows the loss the type h consumer

expects in case it turns out he does not have to travel (recall that the αV priced ticket is

non-refundable).

Denote price of the ticket on the spot market via ps. Type h consumer’s ex ante value

of waiting for the spot market to purchase the ticket is 34

(V − ps). To ensure the type h

customer does not hunt for the non-refundable advance purchase ticket, the monopolist must

set ps so that:

V

(3

4− α

)≤ 3

4(V − ps) (5)

Analysis of (5) shows that if α > 34, the left-hand side is negative, hence ps = V will be the

highest price which will both satisfy the inequality and not deter the type h consumer from

traveling. When α ≤ 34, the highest value of ps that satisfies (5) is ps = 4

3αV .

Thus, when α > 34, the type h consumers will not be interested in purchasing in advance.

Then, the monopolist will be able to sell either two or three tickets, depending on the state

of demand; achieving expected load factor of 5/6, and expected profit of:

πa0 = V

(α +

3

2

)− 3k (6)

Note that the profit in (6) is as good as it can get for the monopolist, since all consumers

end up paying their willingness to pay.

8

In case α ≤ 34, the monopolist offering one ticket priced at αV in advance and two

tickets priced at 43αV on the spot will sell the lower priced ticket to the type l customer, and

higher priced ones to the type h customers, selling either one or two such tickets with equal

probability (again, this will correspond to the expected load factor of 5/6, in context of the

market for air travel). Then, the total profit is:

πa1 = αV +1

2

(4

3αV

)+

1

2

(2

4

3αV

)− 3k = 3αV − 3k (7)

The first part of (7) states that the monopolist can always count on selling the αV priced

ticket; the second part says that one spot market ticket at the price 43αV will be sold with

probability 1/2; finally, the third part of the expected revenue expression represents two spot

market tickets sold with probaibility 1/2.

The following Lemma compares profits the monopolist will obtain when selling one ticket

on the advance purchase market as a non-refundable contract with those the firm will get

when selling all tickets on the spot market.

Lemma 3 When α > 34, offering one non-refundable ticket on the advance purchase market

and two on the spot market strictly dominates offering tickets only on the spot market.

When α ≤ 34, offering one non-refundable ticket at the advance purchase market and two

on the spot market dominates offering three tickets on the spot market when;

α ∈[

11

24,3

4

](8)

and dominates offering two tickets on the spot market when

α ∈[

1

2+

k

3V,3

4

](9)

Proof. Comparing (6) with (1) and (2), we can easily see that πa0 > πs1 since 32> 11

12,

and πa0 > πs2 as k < αV by assumption. This completes analysis for α > 34.

For α ≤ 34, our results come out straightforwardly from direct comparison of (7) with (1)

and (2).

9

3.3 Advance Purchase Discounts - Non-Refundable and Refund-

able Tickets

The analysis performed up to now has simply put our setup into the context of previously

developed models. As we mentioned in the introductory section of this paper, in previously

considered models of costly capacity with aggregate demand uncertainty (i.e., uncertainty

specified in terms of demand states), all tickets released for sale on advance purchase mar-

kets have been assumed non-refundable (spot market tickets will be used if bought, since

uncertainty is realized and resale is prohibited).

In this subsection, we will investigate whether we can increase our monopolist’s profit

by expanding the advance purchase market in the following way. In addition to the non-

refundable ticket at the price of αV , the monopolist will sell a fully refundable ticket at the

price of 43αV . Note that only one such ticket will be offered. Finally, at the spot market, the

firm will sell one ticket at price V . Let us analyze what the outcome of this pricing scheme

will be.

Suppose α ≤ 34

(when α > 34, the refundable advance purchase ticket will have to be

priced above V , and will therefore not be sold). Any type h customer will clearly prefer

buying the fully refundable ticket in advance to waiting until the spot market opens up.

This is so since the ticket is refundable, hence the consumer has nothing to lose in case he

does not have to travel in the state L: he either uses the ticket he bought at below his

valuation, or returns it to the monopolist. On the spot market, at the same time, only the

ticket priced at V will be available. The type h consumer will also be indifferent between

the refundable and the non-refundable tickets offered in advance (inequality (5) applies here,

and turns into equality at these prices)7. It is also true that if the refundable ticket is not

available, the type h consumer will prefer the non-refundable ticket to purchasing on the

spot market (again, see inequality (5)). Thus, all three consumers will want to buy tickets

in advance.

Previous literature tends to make rationing assumptions more clearly for the spot market

than for the advance purchase one; therefore, we can consider ourselves to be more or less

free to choose the customer arrival order on the advance purchase market. The analysis

will be different depending on whether or not there is a positive probability that the type

l consumer can fail to purchase the non-refundable ticket. For instance, if we maintain the

7Strict preference of the refundable ticket can be assured via risk aversion or allowing the monopolist toset the refundable fare slightly below 4

3αV .

10

proportional rationing assumption, this probability will be equal to 1/3. For the type l

customer not to obtain the non-refundable ticket, he has to be the last one to show up (even

if the type l customer shows up second, the non-refundable ticket will still be available, since

the first type h customer that will have shown up before him will always grab the refundable

ticket).

While maintaining the proportional rationing assumption appears to make our modeling

exercise ‘internally consistent‘; it is also true that the type l consumer, being rational, may

try to make sure he is not the last one trying to purchase a ticket on the advance purchase

market (this also appears consistent with what we know about this aspect of the market for

air travel). We therefore will consider two possible rationing arrangements: one where the

type l consumer is never the last one to arrive on the advance purchase market, as well as

the previously maintained proportional rationing.

Suppose now the monopolist knows that the type l consumer is either the first or the sec-

ond to arrive8. This arrangement ensures that the type l consumer buys the non-refundable

advance purchase ticket.

Let us see what the monopolist’s expected profit will be under these circumstances. The

non-refundable ticket is sold and used with certainty. In the low demand state, the type

h consumer who realizes he has to travel will either use his refundable ticket bought in

advance, or will have to purchase one on the spot market at the price V (if the consumer

who bought the refundable ticket finds out he is not traveling, he simply returns the ticket for

full refund). In the high demand state, both type h consumers travel, but one of them has to

purchase the ticket at the spot price. In either case, the probability that the refundable ticket

purchased in advance will be used is the same as the probability that the ticket offered on

the spot market will be purchased (they are both equal to 3/4). Given these considerations,

monopolist’s expected profit will be:

πa2 = αV +3

4

(4

3αV

)+

3

4V − 3k = 2αV +

3

4V − 3k (10)

The following Lemma compares πa2 with profit the monopolist will obtain under other

arrangements considered up to now.

Lemma 4 Suppose α ≤ 34. Further, suppose that on the advance purchase market, the

8Or, we can suppose that all consumers show up simultaneously, in which case the type l consumer willbuy the non-refundable ticket, and both consumers of type h will attempt buying the refundable ticket (butonly one will succeed).

11

monopolist sells one non-refundable ticket at αV ; and one fully refundable ticket at 43αV ;

with one ticket priced at V sold on the spot market. Then, if consumers arrive in such an

order that the type l consumer obtains the non-refundable ticket with certainty:

• The setup with both the non-refundable and the refundable tickets sold on the advance

purchase market dominates that where only the non-refundable ticket is offered in ad-

vance (or, πa2 > πa1).

• Selling three tickets on the spot market is dominated by the setup proposed here when

α > 16.

• Selling two tickets on the spot market is dominated by the setup proposed here when

α > 38

+ k2V

Proof. All the results follow from direct comparison of (10) with (7), (1), and (2),

respectively.

Thus, selling one non-refundable and one fully refundable ticket with advance purchase

discount gives our monopolist higher profit than what he obtains by only selling a single

non-refundable ticket in advance, followed by the sale of two tickets on the spot market.

Obviously, when type l’s valuation is close to that of type h (α > 3/4); the monopolist

will be able to obtain the entire consumer surplus, since the type h consumers will not be

interested in buying the non-refundable tickets in advance.

The second result in the above Lemma allows us to establish that offering three tickets

on the spot market becomes a dominated strategy for our monopolist, with introduction of

the new pricing scheme. Clearly, while for α ∈(0, 1

6

)offering three tickets on the spot brings

higher profit than implementing the just discussed advance-purchase setup; it is also true

that for the same interval offering two tickets on the spot market brings even higher profit

(see corresponding Lemma).

Things will be different with the proportional rationing on the advance purchase market.

As we noted above, in this case the type l consumer will purchase the non-refundable ticket

with probability 2/3. When the non-refundable ticket is bought by a type h customer, the

type h customer will either use this ticket (which will happpen with probability of 3/4), or

lose it (with probability 1/4). The problems that the monopolist will face in this case are

that the spot market ticket priced at V goes unsold, and there is a chance (50 percent chance

in the low demand state, to be precise) that the refundable ticket is returned. In general,

if the type h customers obtain both the non-refundable and the refundable tickets on the

12

advance purchase market; the monopolist gets to keep αV for sure, and the refundable ticket

priced at 43αV is used with probability 3/4. This yields expected revenue of 2αV . Then,

given that probability of this state is 1/3, and that the revenue in case the non-refundable

ticket goes to the type l customer is given by the corresponding part of (10) (and is equal

to 2αV + 34V ), we can write the monopolist’s expected profit with our setup under the

proportional rationing as:

πa3 =2

3

(2αV +

3

4V

)+

1

3(2αV )− 3k = 2αV +

1

2V − 3k (11)

It is obvious (and in fact very intuitive) that πa2 > πa3; the monopolist offering advance

purchase discounts will clearly prefer that the lower-valuation lower-uncertainty traveler

obtain a ticket with certainty. At the same time, dominance of our setup with the advance

purchase refundable ticket over the one on which only the non-refundable ticket is sold in

advance is no longer assured under the proportional rationing assumption. This and other

relevant results are discussed in the following Lemma.

Lemma 5 Suppose α ≤ 34. Further, suppose that on the advance purchase market, the

monopolist sells one non-refundable ticket at αV ; and one fully refundable ticket at 43αV ;

with one ticket priced at V sold on the spot market. Then, if consumers arrive randomly at

the advance purchase market:

• The setup with both the non-refundable and the refundable tickets sold on the advance

purchase market dominates that where only the non-refundable ticket is offered in ad-

vance provided α < 12.

• Selling three tickets on the spot market is dominated by the setup proposed here when

α > 512

.

• Selling two tickets on the spot market is dominated by the setup proposed here when

α > 12

+ k2V

Proof. All the results follow from direct comparison of (11) with (7), (1), and (2),

respectively.

The first result in the above Lemma means that under the proportional rationing rule on

the advance purchase market, for α ∈(

12, 3

4

), the monopolist will prefer selling two tickets

priced at 43αV on the spot market to our setup with selling both the refundable and the

non-refundable tickets in advance.

13

The result that offering three tickets on the spot market becomes a dominated strategy

for our monopolist holds for the case of proportional rationing on the advance purchase

market. The monopolist will make more money using the advance purchase setup with the

non-refundable and the refundable tickets than selling three tickets on the spot market, if

α > 512

. At the same time, where α < 512

, offering three tickets on the spot market is

dominated by offering two tickets on the spot market.

The reader can wonder whether the monopolist could be better off by selling the refund-

able ticket priced at 43αV on the spot market rather than in advance (offering two tickets

on the spot market, the second one priced at V ). While this looks like an appealing setup;

serious incentive compatibility problems will prevent the firm from operationalizing it within

our framework. For one thing, as we have discussed above, such spot prices will not be incen-

tive compatible for the type h consumer, as expected spot price the type h consumer would

face in this case will be higher than the one making him indifferent between purchasing the

non-refundable ticket and waiting for the spot market (see inequality (5)). If the monopolist

could ensure (as we assumed above) that the lowest priced non-refundable ticket is purchased

by the type l customer; the monopolist’s expected profit in case the non-refundable ticket is

sold to the type l customer, and spot market features tickets priced at 43αV and at V (the

former will be sold with certainty and the latter - with probability 1/2) will be:

αV +4

3(αV ) +

1

2V − 3k =

7

3αV +

1

2V − 3k (12)

It is straightforward to show that expected profit in (12) is lower than πa29, but higher than

πa3. At the same time, to arrive at πa3 we had to assume positive probability that the type l

customer is not able to obtain the non-refundable advance purchase ticket, which is different

from the assumption we made in this paragraph. Whereas in case the type l customer gets

the non-refundable ticket with certainty, the monopolist is clearly better off selling the ticket

priced at 43αV as an advance purchase refundable contract rather than on the spot market.

3.4 Optimal Pricing Strategies

Up to now, we have considered several ways the monopolist can price tickets on the market

with two consumer types which differ in terms of their valuation and travel uncertainty (with

9This is so because when the refundable discounted ticket is offered on the advance purchase market,there is a 1/4 probability (when the realized demand state is L, and the customer holding the discountedrefundable ticket finds out he is not traveling) that the monopolist will be able to sell the full fare ticket instate L; which is not possible if the discounted refundable ticket is offered on the spot market.

14

negative correlation between the two); two demand states, and costly capacity, with the firm

precommitting to prices and capacity before demand is realized. We have provided a number

of pairwise profit comparisons; it is now time to determine the monopolist’s optimal pricing

strategies.

Determining the optimal pricing strategy when α > 34

is an easy task. We will formally

do this via the following Proposition.

Proposition 1 When α > 34, the monopolist’s optimal strategy will involve selling one non-

refundable ticket at αV in advance, and two tickets priced at V on the spot market.

Proof. Lemma 3 demonstrates that when α > 34, the firm will be able to make higher

profit by selling the non-refundable ticket on the advance purchase market and the remaining

tickets on the spot market as compared to selling all the tickets on the spot market.

The strategy involving both the non-refundable (priced at αV ) and the refundable (priced

at 43αV ) tickets will not be operationalizable when α > 3

4; and selling the refundable advance

purchase ticket at V is equivalent to offering the same ticket on the spot market.

Restricting our attention to the case α ≤ 34, we can identify the following four pricing

options in our model:

1. Sell all three tickets on the spot market (i.e., when customers’ demand uncertainty has

been realized), offering one seat at αV , and two seats at V .

2. Offer two seats at price V on the spot market. This will price the low valuation

consumer out of the market, at the same time producing cost savings for the monopolist

due to lower offered capacity.

3. Offer the ticket priced at αV for sale on the advance purchase market (before type h

consumers’ uncertainty has been realized) as a non-refundable ticket; and two tickets

priced at 43αV on the spot market.

4. Offer two tickets on the advance purchase market: one priced at αV and non-refundable,

and the other one priced at 43αV and fully refundable. At the spot market, offer one

ticket priced at V .

For the latter of the four strategies outlined above, we have determined the monopolist’s

profit under two different ‘rationing‘ options on the advance purchase market; the difference

between the two is in whether the non-refundable ticket is purchased by the type l customer.

15

Our analysis of the optimal pricing strategies will also be conducted separately for each of

these rationing assumptions.

If it can be assured that the type l customer obtains the non-refundable ticket10, we have

shown that two of the four pricing strategies outlined above are dominated (for α ∈ [0, 3/4]):

selling three tickets on the spot market is dominated by either selling two tickets on the spot

market, or by offering both the non-refundable and the refundable tickets on the advance

purchase market (Lemmas 1 and 4), while selling only the non-refundable ticket in advance is

dominated by selling both the non-refundable and the refundable tickets in advance (Lemma

4). The monopolist’s optimal strategy will in the end depend on the value of α in the following

way.

Proposition 2 Suppose α ≤ 34

and consumers on the advance purchase market arrive in

such a way that the type l customer will obtain the non-refundable ticket (if offered) with

certainty. Then, the monopolist’s optimal pricing strategy is defined as follows:

• For α ∈[0,min

{34, 3

8+ k

2V

}], price the type l consumer out of the market, and only

sell two tickets on the spot market to the type h consumers, at price V per ticket.

• For α ∈(min

{34, 3

8+ k

2V

}, 3

4

], offer both the non-refundable (priced at αV ) and the

refundable (priced at 43αV ) contracts on the advance purchase market; offer one ticket

on the spot market, at the price V .

Proof. Of the four pricing strategies, two can be dropped as dominated (see discussion

above), leaving either offering both non-refundable and refundable advance purchase tickets,

or pricing the type l consumer out and selling two tickets on the spot market as possible

optimal pricing strategies. Lemma 4 defines the cut-off value of α (specifically, 38

+ k2V

) above

which the pricing strategy involving both the non-refundable and the refundable tickets yields

higher profit as compared to only selling two tickets on the spot market.

Under proportional rationing on the advance purchase market (which implies there is

1/3 probability that the type l customer is unable to purchase the non-refundable advance

purchase ticket), the arrangement with both the non-refundable and the refundable advance

purchase tickets is actually never the monopolist’s optimal strategy in our parsimonious

setup. More specifically, we can characterize the monopolist’s optimal pricing strategies

under the proportional rationing rule on the advance purchase market in the following way.

10We have suggested this option is in line with known stylized facts about purchasing patterns in theairline and similar industries, where leisure travelers act more or less strategically by trying to purchase inadvance.

16

Proposition 3 Suppose α ≤ 34; however, assume proportional rationing on the advance

purchase market. Then, the monopolist’s optimal pricing strategy is defined as follows:

• For α ∈[0,max

{12

+ k3V, 3

4

}], price the lower demand consumer out of the market,

and only sell two tickets on the spot market to the type h consumers, at price V per

ticket.

• For α ∈(min

{12

+ k3V, 3

4

}, 3

4

], offer one non-refundable (priced at αV ) contract on the

advance purchase market; offer two tickets on the spot market, at the price 43αV .

Proof. Selling three tickets on the spot market is a dominated strategy (see discussion

following Lemma 5). Next, Lemma 5 actually tells us that under proportional rationing on

the advance purchase market, the pricing strategy involving selling both the non-refundable

and the refundable tickets on the advance purchase market is dominated (by selling two

tickets on the spot market for α < 12

+ k2V

and by selling only the non-refundable ticket

on the advance purchase market for α > 12). Finally, Lemma 3 gives the cut-off value of

α below which selling two tickets on the spot market yields higher profit than selling one

non-refundable ticket on the advance purchase market and two tickets priced at 43αV on the

spot (note also that 12

+ k3V∈(

12, 1

2+ k

2V

)).

We have to caution the reader at this point that our result that the pricing strategy

involving both the non-refundable and the refundable advance purchase tickets is ‘never‘ the

monopolist’s optimal choice is not generalizable to the case of the arbitrary number of type

h and type l consumers. We will discuss this issue later on in the paper.

Note that results we obtained are very intuitive. For low values of α the type l customer is

too unimportant for the monopolist (α technically defines incentive compatibility constraints

for the type h consumers, so the lower the α, the harder it is to change high prices to the type

h customers). Then, as α increases, the monopolist realizes that taking the type l customer

into account can actually yield higher profit, if prices on the advance purchase market are

set right (the optimal pricing strategies to achieve this higher profit do vary depending on

the assumed customer arrival order on the advance purchase market). Finally, when α gets

larger than 3/4, the incentive compatibility constraint that type l customer used to impose

on the type h consumers stops binding the monopolist.

So, we have shown that the pricing strategy we proposed11 can yield higher profit for the

monopolist. We do need to make certain assumptions on customer arrival sequence on the

11Technically, the strategy of offering refundable tickets in advance has been discussed in the literature,but in the context of a different class of models.

17

advance purchase market for our result to work in our example (we have noted above this

result is not generalizable to the more general setup with arbitrary number of customers).

However, assuming the type l customer is not the last one to purchase in advance is not

something implausible; rather, it appears consistent with what is observed empirically on

the markets of the type our model applies to.

Depending on how advance purchase tickets are distributed among the customer types

(looking only at the case of α ≤ 34); the monopolist will offer on the advance purchase market

either only the non-refundable tickets, or both non-refundable and discounted refundable

fares. Note that in the former case the spot market fares will be lower than in the latter.

Interestingly, if we examine the pricing strategies currently used by the airlines on the US

domestic market; what we will find roughly corresponds to these two options. Moreover,

carriers which are typically described as being ’low cost’ or ’low fare’ (such as Southwest

Airlines, Airtran, and JetBlue Airways) are less prone to offering discounted refundable

advance purchase fares as compared to the traditional ‘full service‘ carriers. We will leave

more in-depth discussion of this and other issues to the corresponding section of the paper.

4 Advance Purchase Restrictions

In a ‘naturally occurring‘ world we observe firms on at least some of the markets of the

kind our model applies to offering advance purchase contracts entailing different prices and

refundability restrictions. Typically, lower priced tickets will have the most stringent restric-

tions, involving forfeiture of moneys if the ticket is not used, as well as the cancellation and

change fees. Additionally, the lowest priced and most restrictive advance purchase contracts

are less likely to be offered closer to the date of departure. These ‘fences‘ are typically

claimed to be manifestation of price discrimination by the airlines.

Let us actually slightly modify our model to see that such ‘fences‘ are not that inconsistent

with the application of scarcity pricing in ‘real life‘. Consider the case of an airline. As we

noted above in this paper, our type l consumer will roughly correspond to a ‘leisure traveler‘,

whose demand is certain, but valuation is lower. Type h consumer is then a typical ‘business

traveler‘. Our initial setup supposes business traveler’s uncertainty as to whether or not he

will travel is realized at some time before departure (denote this time through T1 - one can

read this notation as ‘T1 days before departure‘), and the airline knows at what time this

will happen. In this case, the airline will place an appropriate advance purchase restriction

on both the refundable and the non-refundable tickets (both will require purchase at least

18

T1 days prior to departure)12.

Let us now suppose that (as is most probably the case in real life) some time before

business traveler’s uncertainty is realized (say, at day T2 > T1) business travelers receive

private (not observable to the airline) signals which ‘refine‘ their ex ante probabilities of

travel (but do not resolve uncertainty completely). The airline only knows the signal is

received by the business travelers at time T2, but does not know what the signal is. We can,

however, see that, since both demand states involve at least one business traveler undertaking

the trip; at least one of the travelers will receive the signal which will change his or her ex

ante travel probability to something greater than 3/4.

Now recall that the refundable advance purchase contract was designed to make type h

consumer with ex ante travel probability of 3/4 indifferent between purchasing in advance

and waiting until the spot market. Having received the signal, at least one of the type

h ‘business‘ travelers will now prefer the non-refundable advance purchase contract to the

refundable one (given what we assumed about customer arrival, at time T2 > T1 there

is a positive probability that such a ticket is still available). Unless something is done,

the monopolist will face a non-zero chance of selling the non-refundable ticket to a type h

consumer, which is not what our firm’s intentions are. This can be prevented by putting the

appropriate advance purchase restriction on the non-refundable ticket (specifying it must be

purchased at least T2 days before departure). The firm will then face a possibility of not

selling the lowest priced non-refundable contract at all; however, a strategic type l customer

will approach the airline before that date13.

We thus have developed what is a typical scarcity pricing model, in which ‘fences‘ or ad-

vance purchase restrictions appear endogenously as part of the seller’s profit-maximizing

strategy. In our particular example, if otherwise customers arrive in such a way that

the type l traveler would be guaranteed the non-refundable ticket, and supposing α ∈(min

{34, 3

8+ k

2V

}, 3

4

], the seller will (subject to classifications made in the preceding para-

graph) maximize its profit by offering the following contracts:

• One advance purchase non-refundable contract, at the price αV , with restriction that

it must be purchased at least T2 days prior to departure;

12Note we do not allow the airline to practice ‘dynamic‘ pricing here, so that the carrier has to precommitto prices and is not allowed to make any changes

13At this time, we do not take our analysis any further, but one would be right to note that, as themonopolist knows that after time T2 there is a customer out there whose ex ante probability of travel is nowhigher than 3/4; it could have an incentive to change its pricing policy. Yet, with the signal being private(and accounting for the possibility of a signal that does not update the type h customer’s travel probability),we can leave our analysis at this point.

19

• One advance purchase fully refundable contract, at the price 43αV , which must be

purchased at least T1 < T2 days prior to departure; and

• One ‘spot‘ contract (refundability is not an issue since any travel uncertainty is assumed

to have been realized) at price V .

Also note that even though refundable and non-refundable tickets will have different

advance purchase restrictions attached to them; they will be offered simultaneously, as oth-

erwise a type h customer, should he arrive before a type l customer, will grab the non-

refundable ticket, viewing it as a better deal than waiting for the spot market and having

to pay V should he need to travel. Further, no matter when refundable ticket has been

purchased (either before time T2, or between T2 and T1), the type h customer who bought

it has all incentives to hold on to the ticket up until the time uncertainty has been realized

(the ticket is fully refundable, and if the customer returns it and discovers he has to travel,

the available alternative will be more expensive, so as long as the traveler has the refundable

ticket on hand and faces positive ex ante probability of travel, returning the refundable ticket

to the airline is not a rational thing to do).

5 Discussion

5.1 Generalization of the Model

This paper, by analyzing a very simple model, unites two strains of literature which analyze

price setting on the markets with uncertain demand, fixed capacity, and price precommit-

ment. Refundable (fully or partially) advance purchase contracts have been placed into the

framework of models treating demand uncertainty via the states of demand (aka aggregate

demand uncertainty) rather than at the individual consumer level. We have shown how the

monopolist can increase its profit by offering both refundable and non-refundable discounted

fares in advance of the date when the customers’ travel uncertainty has been realized. Fur-

thermore, we have demonstrated that the advance purchase restrictions (which are stricter

for the lower non-refundable fares than for the refundable discounted prices) can emerge

within our model’s framework with little modification to its structure.

Our model can be considered too simplistic; however, we tend to view our exercise as a

parsimonious description of an important problem which can be applicable to a number of

important and highly visible industries. In this sub-section, we will consider the simplest

20

way to complicate our model, which is to assume Nl type l and Nh type h customers. Not

to overburden the reader, we will perform a rather descriptive analysis instead of conducting

the full-scale investigation. At the same time, we will be able to make conclusions about the

generalizability of our results from the previous sections.

To give the reader an idea of how cumbersome our analysis will become with this seem-

ingly small modification, consider the simple extension with two demand states as before,

with 12Nh type h customers showing up in the state L, and all Nh traveling in the state H.

In this case, profit of the monopolist offering Nl tickets priced at αV and Nh tickets at V on

the spot market (analogous to (1)) ‘simplifies‘ to:

πs1 = αV Nl +V N2

h

8

[3Nh + 5Nl(

12Nh +Nl

)(Nh +Nl)

]− k (Nl +Nh) (13)

The reader can easily verify that if we substitute values for the case worked out before

(Nh = 2, Nl = 1), (13) will turn into (1). Relevant expressions for some of the other pricing

strategies are easier to arrive at, and will be presented below. Before we do this, note that

α = 34

remains the cutoff above which the type h customers will prefer purchasing tickets

at the spot market at the price V to getting the non-refundable discounted ticket at αV in

advance. Also, for α ≤ 34, the monopolist offering only the non-refundable tickets on the

advance purchase market will have to price the spot market tickets at 43αV (inequality (5)

defined the incentive compatibility constraint for each individual type h customer); and if

the refundable tickets are offered at a discount in advance, the quantity of such tickets will

be restricted to 12Nh.

Now, let us write out expressions for the monopolist’s profit in our setup, and see if we

can learn anything from them. If the type l consumers are priced out of the market (so that

the monopolist only offers Nh tickets on the spot market priced at V ), the profit is:

πs2 =3

4V Nh − kNh (14)

Offering both Nl non-refundable tickets on the advance purchase market, and Nh tickets on

the spot market at 43αV (assuming again α ≤ 3

4), the monopolist obtains:

πa1 = αV (Nl +Nh)− k (Nl +Nh) (15)

If the monopolist chooses the pricing strategy involving Nl non-refundable and 12Nh refund-

21

able advance purchase tickets (priced at 43αV ), followed by 1

2Nh tickets sold on the spot

market at the price V , and if consumers arrive in such an order that the type l customers

will grab all the non-refundable tickets14, the firm’s expected profit will be:

πa2 = αV

(Nl +

1

2Nh

)+

3

8V Nh − k (Nl +Nh) (16)

It is easy to verify that profit implied by (16) is higher than that under (15) when α < 34; that

is, generalizing our model to the arbitrary composition of customers preserves the dominance

of setup with both refundable and non-refundable tickets sold in advance over that with only

the non-refundable advance purchase tickets, so that the new pricing strategy introduced in

this paper will remain the monopolist’s optimal choice over some interval of α ≤ 34

with

arbitrary number of consumers of either type.

The remaining case with the proportional rationing on the advance purchase market

where the monopolist offers both the refundable and the non-refundable tickets is the most

cumbersome one to generalize. On the advance purchase market, we have Nl +Nh consumers

competing for Nl + 12Nh tickets, Nl of which are non-refundable. Every time fewer than Nl

type l customers are included into the group of Nl + 12Nh lucky travelers who will be able

to purchase the cheapest tickets available, some of the non-refundable tickets end up being

purchased by the type h travelers. More specifically, the probability that the type l customers

will purchase i ≤ Nl non-refundable discounted tickets will be given by:

Pr(i) =Ci

Nl+12Nh∗ CNl−i

12Nh

CNl+

12Nh

Nl+Nh

(17)

Expression (17) only makes sense for Nl − i ≤ 12Nh - otherwise, Pr(i) = 0. Denominator

in (17) is simply the number of ways to construct sample of Nl + 12Nh customers who will

end up buying tickets on the advance purchase market out of Nl +Nh total customers. The

numerator measures the number of combinations which include i type l customers into the

group lucky enough to snatch the cheapest tickets, and Nl − i type l customers into the

group unable to purchase tickets in advance.

Next, let us see what the monopolist’s profit will be. This will actually depend on

the number of non-refundable tickets purchased by the type h consumers; furthermore, the

14Recall that when choosing between refundable and non-refundable ticket the type h customer will selectthe former; therefore, to ensure that the type l customers obtain all non-refundable tickets, the last suchconsumer must be at most Nl + 1

2Nhth customer to show up.

22

example we considered in the previous sections of the paper deals with one specific case,

which means, as we will shortly see, that generalizing our model to include the arbitrary

composition of consumers may lead to our setup with the non-refundable and refundable

advance purchase tickets potentially becoming the monopolist’s optimal choice even when

we have proportional rationing on the advance purchase market.

The problem we had in our example was that under proportional rationing on the advance

purchase market the monopolist was unable to sell the full fare ticket on the spot market

should the type h consumer get hold of the only non-refundable ticket available. In fact,

if we suppose the type h consumers purchase Nl − i non-refundable tickets on the advance

purchase market; no spot market tickets will be sold in the high demand state provided

Nl−i+ 12Nh ≥ Nh, which clearly was true in our case (Nl−i = 1, Nh = 2). Otherwise, the type

h customers will have to purchase 12Nh−Nl +i full fare tickets on the spot market. Generally,

given that Nl non-refundable tickets are always sold, refundable advance purchase tickets are

used with probability 3/4, and there is a possibility that the type h customers may have to

purchase the full fare tickets on the spot market; the monopolist’s profit with proportional

rationing on the advance purchase market will be given by the following expression:

πa3 = V

max{0,Nl− 12Nh}∑

i=0

Pr(i) ∗[α

(Nl +

1

2Nh

)+

1

2max

{0,

1

2Nh −Nl + i

}]− k (Nl +Nh)

(18)

Recall that Pr(i) is the probability that the type l customers purchase i non-refundable

tickets, and is calculated according to (17). Clearly, expression (18) is very involved; we

can nevertheless suggest an implication that stems from it, without doing formal analysis.

Specifically, for this pricing strategy to have any chance of beating (in terms of profit for

the firm) the one with only non-refundable tickets sold on the advance purchase market,

inequality Nl − i+ 12Nh < Nh must hold for at least one value of i (in this case some of the

full fare spot market tickets will be sold). Alternatively, we can conclude that (unlike what

we found for the case where the type l customers purchase all non-refundable tickets) our

previous conclusion that under the proportional rationing on the advance purchase market

the setup with non-refundable and refundable tickets is dominated by that with only the

non-refundable advance purchase contracts is not generalizable; it is possible that (if we have

sufficiently more type h than type l customers) it will be optimal for the monopolist to sell

both non-refundable and refundable tickets in advance for both rationing rules assumed for

the advance purchase market.

23

5.2 Practical Considerations

In this subsection, we will discuss how our model and its results correspond to both other

modeling approaches proposed in the literature, and to empirically observed firms’ pricing

strategies. This will help us identify where our model improves upon our understanding of

markets with costly capacity, perishable product, and uncertain demand; as well as issues

we have not been able to successfully address yet.

Our modeling exercise showed how different advance-purchase contracts can be combined

by the monopolist with pricing on the spot market under aggregate demand uncertainty in

the very simple setting. Models treating uncertainty at consumer level have been unable

to incorporate both ‘rich‘ advance purchase and traditional spot markets. Courty (2003a)

modeled both advance purchase and spot markets under consumer level demand uncertainty:

this was done at the expense of excluding refundable advance purchase contracts (Courty’s

model provides for the possibility of resale, however). Escobari and Jindapon (2008) consid-

ered refundable and non-refundable advance purchase contracts in the context of aggregate

demand uncertainty; however, their model requires consumers to be risk averse, neglects the

issue of costly capacity, and does not consider the spot market formally. Though our model’s

setup is simple, we have shown that some of the results are generalizable.

While modeling demand uncertainty as aggregate (in terms of states of demand) seems

to be a more appealing approach than considering individual consumer level ambiguity;

it is also true that our framework does not account for some of the strategies firms use

on the kinds of markets our model applies to. Most notably, we have not incorporated

such important demand and revenue management strategy as overbooking (neither has any

other similar model treating demand uncertainty at the aggregate level). We believe that

modeling overbooking successfully within our framework will require introducing an element

of individual uncertainty, and this could be an interesting topic for future research.

Another area for improvement is treatment of customer arrival on the advance purchase

market. We have considered two setups; one generally allowing some consumers to act more

or less strategically (the rationing mechanism whereby the type l consumers purchase all

of the available non-refundable tickets can be rationalized by assuming they will make sure

to purchase well in advance). Useful insights in this area can be drawn from the queueing

theory (Hassin and Haviv (2003) is a recent book that provides a thorough overview of this

approach, which is more commonly applied in the operations research than in economics

literature). The queueing literature typically considers two kinds of setups: the difference

between them is in whether the customer’s priority in queue is out of his/her control (see,

24

for instance, Glazer and Hassin (1986)), which is in line with the two rationing mechanisms

discussed in our model.

Next, while our framework allows for advance purchase restrictions to arise endogenously

(subject to the safe assumption of gradual realization of demand uncertainty); we have thus

far been able to only remotely approach the multi-tier pricing schemes actually employed by

some of the firms in the relevant industries. We believe the key to arriving at an even more

realistic model is in further generalizing the model to include multiple consumer types, as

well as in allowing the consumers to act strategically on the advance purchase market. We

mentioned that Akan et al. (2009) have made an important step in this direction, in the

framework of individual-level uncertainty model.

Our discussion up to now has been motivated by (and all our ’real-world’ examples have

referred to) pricing in the airline industry, even though we clearly indicated that our analysis

is potentially applicable to a number of very important and visible industries. The ’practical’

issue is that of all the markets with similar characteristics, the airline industry has been the

most successful in implementing the pricing scheme similar to the one we outline in this paper.

In this industry, firms can easily prevent resale by making tickets non-transferable; the airline

industry has always been a technologically intensive one, which presumably made it easier for

the firms to embrace sophisticated revenue management techniques, and facilitated learning

about demand.

Hotels and rental car companies are slowly approaching the airlines’ level of sophistica-

tion in pricing; however, advance purchase refundable contracts15 are still more common in

those industries than advance purchase discounted rates that are non-refundable. In these

industries preventing resale is relatively easy, so slower adoption of airline-type pricing tech-

niques by hotels and rental cars seems puzzling, and is probably related to relatively lower

degree of technological intensity and lower sale volumes (hence, less demand learning) by

individual players in these industries16.

Full service restaurants (admittedly, constrained capacity is not a very important concept

for fast food restaurants) differ in both whether they accept reservations (in context of our

model reservations are equivalent to refundable advance purchase contracts17), and in how

15Booking in advance and paying only at the time service is rendered (a common practice in hotel andrental car industries) is observationally equivalent to a fully refundable advance purchase contract.

16Most hotels and many rental car locations are franchises: maybe if every aircraft were independentlyowned and franchised to a particular airline, we could see less sophistication in airline prices as well.

17The difference between restaurants and hotels is that hotels do have the right to charge you if you donot honor your reservation.

25

(sometimes, whether) they accommodate walk-ups (equivalent to the spot market). We can

say that while full service restaurants do share some of the features of our model; in this

industry customers’ demand is not exactly dichotomous, which probably limits applicability

of our analysis to this market (apart, probably, from restaurants offering prix fixe menu).

In the live performance (concert) industry, the typical pricing strategy employed is selling

most tickets as non-refundable advance purchase contracts, and subsequently allowing resale

(even where technically resale is not allowed, little is done to effectively prevent it18). Courty

(2003b) provides a model of ticket resale, in which he makes a convincing argument that

promoters will be unable to deter brokers or scalpers from entering the market where resale

cannot be stopped; it is not clear, however, why promoters are either unwilling or unable to

take actions to address the root of the problem by preventing resale of concert tickets in the

first place.

6 Concluding Comments

Broadly, this paper’s contribution is in uniting two strains of literature on revenue man-

agement by a firm selling perishable product under constrained capacity and demand un-

certainty (industries that have such characteristics include airlines, hotels, car rentals, live

performances, and full service restaurants). We add a pricing strategy (selling refundable

tickets on the advance purchase market) which was previously considered in the context of

models treating uncertainty at the individual consumer level to the set of strategies available

to the monopolist facing aggregate demand uncertainty. We show that our monopolist can

indeed extract more of the consumers’ surplus with this additional strategy as compared

to what previous models of aggregate demand uncertainty supposed. In the end, we pro-

vide what looks like a more realistic framework for modeling revenue management in certain

industries, as compared to what has been previously proposed in the economics literature.

Our model is admittedly simplistic, and we show that some of the results of the simplest

setup are not necessarily generalizable; we leave more general treatment of our framework

for future research endeavors.

An important contribution of our study is in showing how ‘fences‘ or advance purchase

restrictions can arise endogenously within the framework of capacity pricing models. Pre-

vious similar models treating demand uncertainty at the aggregate level have an advance

18Anti-scalping laws are in place in some jurisdictions; ticket brokers tend to bypass those by charging‘fees‘ on top of published ticket prices, which is not expressly prohibited.

26

purchase restriction ‘wired‘ into their structure: advance purchase discounts end at the time

demand uncertainty is realized. We also have this restriction in our framework. However,

we add another ‘fence‘ that comes in effect before complete realization of demand uncer-

tainty; and this advance purchase restriction is only possible if the monopolist offers both

non-refundable and refundable contracts on the advance purchase market.

Actual structure of demand uncertainty in industries our model applies to is definitely

more complicated than any of the models in economic literature has been able to describe.

The firms have to deal with both aggregate and individual level uncertainty (leading, for

instance, to high full fares on one hand, and overbooked flights on the other); and customers

do not necessarily arrive in random order (leisure travelers can strategically choose to make

plans well in advance, as an example). We believe this study, while not addressing all the

relevant issues, advances our understanding of pricing decisions firms have to make in a very

complex environment of costly capacity and uncertain demand.

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