Revenue Management Without Commitment:

Dynamic Pricing and Periodic Fire Sales∗

Francesc Dilme† Fei Li‡

October 23, 2016

Abstract

A profit-maximizing monopolist seller has a fixed number of identical goods to sell before

a deadline. Over time, buyers privately enter the market, and they strategically time their

purchases. In the unique Markov perfect equilibrium, the seller sporadically holds fire sales

to lower the stock of goods. This increases future buyers’ willingness to pay, but it also

lowers the willingness to pay of buyers who arrive early in the game. On the path of play,

only one unit is offered in each fire sale, so buyers may be rationed. Interestingly, at the

limit where the probability of a buyer being rationed vanishes, the seller gets rid of all but

one unit of the good during a “big” initial fire sale.

Keywords: revenue management, commitment power, dynamic pricing, fire sales, inat-

tention frictions.

JEL Classification Codes: D82, D83

∗We are grateful to George Mailath and Mallesh Pai for their guidance and advice. We also thank Aislinn

Bohren, Heski Bar-Isaac, Gary Biglaiser, Simon Board, Luis Cabral, Eduardo Faingold, Hanming Fang, George

Georgiadis, Johannes Horner, Kyungmin Kim, Stephan Lauermann, Anqi Li, Qingmin Liu (Discussant), Steven

Matthews, Benny Moldovanu, Guido Menzio, Alessandro Pavan, Andrew Postlewaite, Maher Said (Discussant),

Yuliy Sannikov, Philipp Strack, Tymon Tatur, Can Tian, Rakesh Vohra, and Jidong Zhou for valuable comments.

Thanks also to participants at numerous conferences and seminars. Any remaining errors are ours.†University of Bonn, fdilme@uni-bonn.de‡University of North Carolina, Chapel Hill, lifei@email.unc.edu

1 Introduction

Some markets share the following characteristics: (1) goods for sale must be consumed before

a certain time, (2) the initial number of goods for sale is fixed in advance, and (3) consumers enter

the market over time and choose the timing of their purchases. Examples are the airline, cruise

line, hotel, sport ticket, and entertainment industries. In such industries, computer systems

based on revenue management algorithms are often employed, which is typically viewed in the

revenue management literature as a commitment device used by sellers. In reality, however,

revenue managers frequently adjust prices based on their information and personal experience,

instead of strictly following the pricing strategy suggested by the algorithm.1 Hence, assuming

perfect commitment by the seller may not be without loss, opening up the question of how the

seller’s lack of commitment affects the price dynamics. This paper studies revenue management

under the assumption that sellers are endowed with no commitment power. We highlight an

important effect of (the lack of) commitment power in such an environment: if the arrival of

buyers is slower than expected, the seller finds it optimal to hold a fire sale to lower his inventory

and thereby sell at a high price in the future, resulting in a highly fluctuating price path.

We consider the profit-maximizing problem faced by a monopolist (male) seller who has a

finite number of identical goods to sell before a deadline. Over time, (female) buyers privately

arrive. Each buyer has a single-unit demand, privately knows her willingness-to-pay, and can

time her purchasing time. At any time, the seller posts a price and a quantity (or capacity

control) but cannot commit to future offers. If the price crosses a given lower threshold, a third

party advertises the low price, referred to as a fire sale, to a broader market as well as the arrived

buyers.2 As a result, arrived buyers may be rationed with some probability in a fire sale.

The model has a unique Markov perfect equilibrium, in which the price exhibits rich dynamics.

At the deadline, if the seller has not sold all of the goods, he holds a fire sale to obtain some

revenue from the sale of the remaining units. For most of the time before the deadline, the

seller posts the highest price that makes buyers willing to purchase immediately upon arrival.

1In the article Confessions of an Airline Revenue Manager on FoxNews.com, Goerge Hobica said,“The com-

puter adjusts fares all the way up until the departure time, but as a revenue manager, I can go in and adjust

things based on information that the computer may not know. For example, are there specific events taking place

at a destination? Are there certain conditions at the departure airport that will allow more than the desired

amount of seats to go empty such as weather?”2In practice, the advertisement is a deal-alert email (or text message) from a third party (such as Kayak,

Orbitz, etc, in the airline industry) or the online real-time bidding (RTB) advertisement that facilitates the

seller’s ability to target and track “window shoppers.” The time windows for these fire sale offers are typically

very short compared to the length of the transaction season.

2

Occasionally, he makes fire sale offers to sell some goods.3 Posting a fire sale generates a low

revenue from the units sold but, by reducing the inventory, it induces future buyers to accept a

higher price. Intuitively, fire sales allow the seller to “commit” to future high prices.

The option to hold fire sales has an additional (and perverse) equilibrium effect for the seller:

his inability to commit to not holding them in the future lowers the current buyers’ willingness

to pay. If the seller could commit, he would commit not to hold fire sales after certain histories in

order to increase the buyers’ willingness to pay. Still, given that the arrival of buyers is stochastic,

the seller might have the incentive to deviate: if fewer than expected buyers arrived in the past,

fire sales would increase the seller’s revenue. Since buyers correctly anticipate this incentive and

they have the option to wait to (try to) obtain the good at a bargain price, the presence of fire

sales forces the seller to lower the price in equilibrium.

The price fluctuates over time due to the changes in the inventory size: the price decreases

slowly over time until a transaction happens, and then the price jumps up. This generates a price

path that fluctuates over time and is consistent with extensive empirical evidence.4 Such price

fluctuation results from the seller’s lack of commitment and buyers’ intertemporal concerns.

Intuitively, at the deadline, if the seller still holds some units, he sells unsold goods through

fire sales (also called “last-minute deals”).5 Before the deadline, because a last-minute deal is

expected to be posted soon, the reservation price of the buyers is low. However, waiting for the

final fire sale is risky because there is a positive probability that a buyer arrives; additional, at

the fire sale time, there is a positive probability that she does not obtain the good. Therefore,

to avoid future competition, a buyer is willing to purchase a good immediately at a price higher

than the discounted (fire sale) price. The reservation price of a buyer decreases over time because

the arrival of other buyers becomes less likely. Also, her reservation price is high when the

current inventory size is low because she expects to obtain a good through a fire sale with a low

3This equilibrium prediction on the transaction pattern is consistent with surveys in many retail industries

such as Consumer Reports, “America’s Bargain-Hunting Habits,” April 30th, 2014. According to those surveys,

60% of consumers “wait for a sale to buy what they want.”4A highly fluctuating path of realized sales prices is in line with actual observations in many relevant industries.

See, for example, McAfee and Te Velde (2006), who find that the fluctuation of airfares is too high to be

explained by the standard monopoly pricing models. Recently, Escobari (2012) finds that the airline price declines

conditional on the inventory size and jumps up after transactions. Aguirregabiria (1999) shows the important

role of inventories by using retail market data. In secondary markets for Major League baseball tickets, Sweeting

(2012) finds that the seller cuts prices dramatically as the deadline approaches.5Last-minute deals or clearance sales are optimal in many dynamic pricing settings. See Nocke and Peitz

(2007) as an example. In practice, the last-minute deal strategy is commonly used in many industries. See Wall

Street Journal, March 15, 2002, “Airlines now offer ‘last-minute’ fare bargains weeks before flights,” by Kortney

Stringer.

3

probability.

Fire sales further contribute to the price fluctuation. Remarkably, except maybe at the initial

time, the seller never holds a fire sale for more than one unit on the path of play. To see this,

assume that the seller holds two units, and he only sells them through a fire sale at the deadline.

When a buyer arrives close to the deadline and there are still two units unsold, her willingness to

pay is low: if she rejects the current price, it is unlikely that another buyer will arrive before the

deadline, and therefore she is likely to obtain one of the two goods in the final fire sale. When,

instead, a buyer arrives and the seller holds only one unit, she is less likely to obtain a good in

the final fire sale due to the competition the broader market, as a result, the seller can charge

a high price. Hence, even though not holding a fire sale prior to the deadline would be better

for the seller if he knew that two or more buyers would arrive, holding a fire sale is preferable

if only one buyer arrives. Since, as the deadline becomes close, the probability of the arrival of

two or more buyers becomes negligible compared to the probability of the arrival of one buyer,

the seller prefers to sell one of the units at a very low price (by holding a fire sale) but maximize

the probability of selling the other unit at a high price. This intuition can be easily generalized:

there is a sequence of isolated “cutoff times” before the deadline, such that if they are reached

and the seller has a given inventory size, he holds a fire sale.

To illustrate the role of fire sales and the broader market, we consider two limits of our model.

If buyers are increasingly unlikely to obtain a good in a fire sale, the seller’s commitment problem

disappears: he charges a high price, and at the deadline, he holds a fire sale for the remaining

units. If, instead, buyers are very likely to obtain a good in a fire sale, the commitment problem

of the seller becomes very severe. In this case, even though the expected number of arrived

buyers is high, he holds a fire sale at the beginning of time for all but one unit, and tries to sell

the remaining unit to the first buyer to arrive (if any). These results are reminiscent of the Coase

conjecture, albeit with the opposite effect: the more severe is the commitment problem, the less

efficient is the trade outcome. In our model, such an inefficiency arises from the misallocation of

goods instead of the delay in the trade.

There is a large body of revenue management literature that has examined markets with

sellers who need to sell a finite number of goods before a deadline to short-lived buyers who arrive

over time.6 One of the common assumptions in this literature is that buyers are short-lived, and

therefore cannot strategically time their purchases. However, as Besanko and Winston (1990)

argue, mistakenly treating forward-looking customers as myopic may have an important impact

on sellers’ revenue, highlighting the need to incorporate strategic buyers in revenue management

6See Gallego and Van Ryzin (1994) and Talluri and Van Ryzin (2006). Gershkov and Moldovanu (2009) extend

the benchmark model to address the case of heterogeneous objects.

4

models.

Similar to us, Board and Skrzypacz (2016) characterize a dynamic market where a seller sells

a finite number of goods before a deadline to agents who arrive in the market over time. They,

instead, study the revenue-maximizing mechanism under the assumption that the seller can

commit. In the continuous time limit, the optimal mechanism is implemented via a price-posting

mechanism with an auction for the last unit at the deadline.7 We analyze the non-commitment

case: the seller cannot commit to not lower the price if the realized demand is low. As a result,

we not only examine the critical role of the commitment power of the seller but also provide a

theory of fire sales: selling units at very low prices increases the price of the remaining units.

Other papers analyze dynamic markets where a seller with no commitment power sells to

forward-looking customers. For example, Horner and Samuelson (2011) and Chen (2012) analyze

a seller who sells goods to a fixed number of buyers who strategically time their purchases.8 They

show that the seller either replicates a Dutch auction or posts an unacceptable price until the

deadline and then charges a static monopoly price. In line with most of the revenue management

literature, we maintain the assumption that buyers arrive stochastically, and we show that such

an arrival generates fluctuating price dynamics and the possibility of fire sales.

Alternatively, Deb and Said (2015) show that, in a two-period dynamic-screening problem

where a seller without capacity constrains sells goods to buyers who (non-stochastically) arrive

in each period, the seller may deliberately postpone the transaction with some buyers in order

to alter the demand in the second period, and therefore make charging a high price credible. In

our model, instead, the seller holds fire sales not to change the future (stochastic) demand, but

to lower the stock of goods in order to increase the competition (and the price) in the future.

Conlisk, Gerstner, and Sobel (1984) consider a durable good model with the arrival of buyers.

In their model, the price fluctuates due to the seller’s trade-off between rent extraction from the

buyers with a high valuation and the revenue from selling the goods to accumulated buyers with

a low valuation.9 In our model, instead, the seller holds fire sales to increase the scarcity of the

goods (and a high price) after low realizations of the demand.

The rest of this paper is organized as follows. In Section 2, we present the model setting

7Also see Mierendorff (2016) and Pai and Vohra (2013). Gershkov, Moldovanu, and Strack (2014) consider a

model in which the seller learns the arrival process of the buyer over time.8Aviv and Pazgal (2008) consider a two-period case where the seller lacks commitment, and so do Jerath,

Netessine, and Veeraraghavan (2010). Anton, Biglaiser, and Vettas (2014) illudstrate dynamic price competition

in a similar setting. Recently, Liu, Mierendorff, and Shi (2015) consider a dynamic auction problem in which the

seller posts the reserved price of an auction in each period.9In a recent paper, Garrett (2016) shows that similar price dynamic takes place in a model where the seller

has perfect commitment power, buyers arrive privately, and buyers’ valuations change stochastically over time.

5

and define the solution concept we are going to use. In Section 3, we analyze equilibria. Section

4 discusses some modeling choices and possible extensions of the baseline model. Section 5

concludes. Appendix A provides the formal model, while Appendix B includes the proofs omitted

in the main text.

2 Model

In this section, we describe our model. In order to prevent technicalities that may make the

presentation unnecessarily tedious, this section avoids some of the notation-intensive definitions,

and presents the economically relevant components of our model. Appendix A provides the

formal version of the model.

Environment. We consider a dynamic pricing game between a single (male) seller who has

K ∈ N identical and indivisible goods for sale, and many (female) buyers. Time is continuous:

t ∈ [0, 1]. Goods are valuable up to time 1, and deliver no value afterward.

Seller. At every time t, the seller offers a regular price Pt ∈ R and a supply (or capacity

control) Qt ∈ {0, 1, 2, ..., Kt}, where Kt is the amount of units not sold in [0, t). After offering

the regular price, he can make a “fire sale” offer. This will be discussed in more detail later. The

seller has a zero reservation value on each good, and he does not discount. He is a risk-neutral

expected-utility maximizer, so his payoff equals the expected summation of all transaction prices.

Buyers. At time 0, there is no buyer. Over time, buyers arrive privately and stochastically at a

rate λ > 0. Each buyer has a single-unit demand and values the good at vH > 0. A buyer who

buys a good at price p gets a payoff equal to vH − p, and her payoff is 0 if she does not obtain

any good. If a buyer purchases a good at time t she leaves the game. Otherwise, she enters

in a waiting state, and she is referred to as an accumulated buyer. An accumulated buyer faces

inattention frictions : at each moment t, she can decide to pay attention to the current offer by

paying a cost of c > 0.10

If, at a given time t, Nt > 0 buyers arrive or pay attention to the regular price offer of the

seller, the game proceeds as follows. They all observe Kt, the price Pt, and the quantity offered

10The inattention frictions assumption implies that an accumulate buyer will at most sample the price infor-

mation for a finite number of times. It captures the real-world observation that accumulated buyers only check

the price occasionally in a short time period because checking is costly. For example, when a consumer is waiting

for a discount flight, she may only check the price at agencies’ websites once or twice per day. The fact that all

our results are independent of the value of c (as long as it is positive) gives robustness to our model.

6

Qt and decide whether to purchase or not. Nature uniformly randomly orders the buyers who

decide to purchase a good, and allocates the good to the first Qt of them (to all of them if there

are fewer than Qt buyers willing to buy a good).

Fire sales. Within each instant, after the first stage, the seller can hold a fire sale by offering

QDt units of the good at price vL ∈ (0, vH). In particular, we assume that a fire sale offer

automatically triggers an advertisement being sent by a third party, which attracts the attention

of accumulated buyers as well as additional demand from a broader market. Such additional

demand can be interpreted as a pool of shoppers (or uninterested buyers) whose value are vL.

If, in a given period t, the seller offers QDt ≥ 1 units in a fire sale, the game proceeds similarly

as for the regular price offer, although now with the possibility that some goods are purchased by

shoppers. First, nature uniformly randomly orders the buyers (if any). If there is no buyer, all

goods are assigned to shoppers. If there is at least one buyer, with some probability β ∈ (0, 1),

the first good is assigned to the first buyer, and with probability 1− β to a shopper. After the

first unit is assigned, if QDt > 1, the second unit is assigned analogously: if no buyer remains, it

is assigned to a shopper, while if there is at least one buyer, the unit is assigned to the first of

the remaining buyers with probability β, and with probability 1− β to a shopper. This process

is iterated until there is no unit is left. Note that β is a measure of how “attentive” buyers are

with respect to shoppers.

Information. At time t, the seller observes the previous path of the stock, and of the prices

and quantities offered, (Kt, Qt, Pt, QDt )t

−s=0, but he does not observe the arrivals of buyers, nor

does he see their actions. A buyer arriving at time t only observes (t,Kt, Pt, Qt). Finally, an

accumulated buyer at time t observes Ks, Ps, and Qs in her previous attention times s (including

her arrival time), and whether she tried to purchase the good, as described above.

A heuristic description of the timing within an instant t is depicted as follows. First, nature

decides whether a buyer arrives at time t. Then, the seller decides what quantity and price to

offer. Next, accumulated buyers decide whether to pay the inspection cost at time t. After this,

goods are offered to the buyers who pay attention at time t and accept the offer (following the

procedure described above if many of them pay attention). Finally, the seller decides how many

units to offer through a fire sale, and nature assigns them according to the procedure described

above.

Solution Concept. We focus on Markov equilibria with state variables (t,Kt). That is,

1. the continuation strategy of the seller after time t depends only on (t,Kt), and

7

2. when a buyer checks the price and observes (t,Kt), her acceptance decision (that is, the set

of price and quantity offers she accepts) and next attention time depend only on (t,Kt).

We focus on pure-strategy Markov perfect equilibria (MPE), which we refer to as just “equilib-

ria”.11

2.1 Discussion of Assumptions

With the goal of studying the role of commitment and fire sales in a revenue-management

environment, our model makes some rather unconventional modeling choices. These modeling

choices ensure that a unique Markov-Perfect equilibrium with (t,Kt) as state variable exists while

keeping the main tradeoffs that we want to study, which allows us to provide a full characteri-

zation of it and some comparative static results. Before proceeding further, we want to discuss

the role of these assumptions.12

Homogeneous Buyers. We assume that arriving buyers have the same valuation for the good,

vH , so they only differ because of their different private history. This assumption ensures that, in

the unique MPE, the seller does not accumulate buyers: the regular price is set so that arriving

buyers purchase immediately upon arrival. This simplifies the equilibrium beliefs of the buyers

regarding the previous history and makes (t,Kt) the only payoff-relevant variable at time t.

Furthermore, as we show in Section 4.1, the model is not fragile in this assumption: as long as

the valuations of the buyers are not too spread out, the seller sets the regular price to attract

them. Greater buyer heterogeneity would make the model intractable, since accumulation would

happen on the path of play. As a result, the set of possible histories and buyers’ beliefs would

be very large.

Inattention Frictions. We introduce an arbitrary small cost of rechecking the price. This

is a simple and flexible modeling device to lower the buyers’ willingness to monitor the price.

Because, in equilibrium, a buyer does not keep tracking the price, the seller has a limited ability

to influence the beliefs of the buyers by changing the price offered, keeping the analysis of the

11We allow both the seller and buyers to deviate to non-Markov strategies.12Besides, we also make some other minor simplifying assumptions, which can be relaxed without undermining

our key results. For example, we assume that consumers are “forced” to accept a fire-sale offer (at price vL). This

assumption is made for simplicity, and since it is suboptimal for the seller to charge a regular price lower than

vL, it would naturally be part of equilibrium behavior if buyers were allowed to reject the offer. Also, we assume

that holding a fire sale is free. Assuming that offering such a deal would involve a small cost c′ > 0 (coming,

for example, from the cost of advertising) would not change our results. Therefore, we include advertising as a

way for the seller to lower his stock and increase competition among future buyers. Also see Ory (2014) for a

discussion of the role of such advertisements in a stationary model.

8

off-the-equilibrium-path incentives tractable. As a result, the seller’s commitment problem is

accentuated: there is no equilibrium where the seller charges an unacceptable high price until

the deadline. On the contrary, the seller immediately serves arriving buyers.

Fire Sales. We assume there are two channels through which the seller can sell goods: “regular”

sales are accessible to arriving buyers only, and “fire” sales are accessible by both arriving buyers

and consumers in a broader market. The existence of these channels makes the equilibrium

effect of the lack of commitment power not trivial. First, fire sales give the seller the option of

increasing the competition between buyers (and the regular price) by lowering the stock. Second,

the inability to commit to not hold fire sales lowers the buyers’ willingness to pay, as they know

that they may get a good at a low price in the future with a positive probability. For the sake

of clarity, we assume that the broader market is stationary. Incorporating a history-dependent

population of low-value shoppers would not only complicate the analysis, but also prevent the

existence of a Markov equilibrium: buyers would differ on their beliefs about the past history

(which would also depend on their arrival time), and therefore also in their decision on which

prices to reject.

Observability of Inventory. We assume that the remaining inventory size Kt is observable.

This is necessary to avoid the usual signaling problem: the game when the seller has private

information and sets the price is a signaling game, with the usual multiplicity of equilibria and

off-the-equilibrium-path beliefs. It is worthwhile to point out, however, that using a simple

unraveling argument shows that if costless and verifiable, the seller always reveals the remaining

stock.13

3 Analysis

We begin with a lemma that will simplify our analysis. This lemma establishes that if a buyer

does not accept the offer that the seller makes at her arrival time, she does not check the price

again.

13The assumption of observable stock is irrelevant if either the seller has perfect commitment power or consumers

are short-lived or non-strategic. In practice, buyers may observe some imperfect but informative signals of the

inventory in some markets, providing the seller some incentive to hold a fire sale. For example, in the airline

industry, airlines sometimes report online the amount of remaining available seats. Even though the advertised

number of available seats may not coincide with the actual inventory (airlines sometimes block some seats for elite

passengers), it is an informative proxy. Escobari (2012) uses the number of available seats as a proxy for the real

inventory and empirically studies the price patterns in the airline industry. He finds that the price significantly

increases as the number of available seats decreases. This suggests that the number of available seats as an

imperfect signal of inventory is interpreted as credible.

9

Lemma 1. In any equilibrium, if a buyer rejects the price at her arrival time, she never rechecks

the price again.

The reasoning behind Lemma 1 is similar to the standard Diamond paradox reasoning. As-

sume that a buyer rejects the price offer at her arrival time. Such a buyer rechecks the price

at some future time only if the expected payoff of doing so compensates the rechecking cost c.

However, once she rechecks the price, the rechecking cost is sunk, and her willingness to pay is as

high as a newly arrived buyer who does not pay the rechecking cost. Hence, in any equilibrium,

if the seller offers a price intended to be accepted by the arriving buyers, he charges a price equal

to the buyers’ (common) continuation value. As a result, the gain from rechecking the price does

not compensate its cost, and a buyer never rechecks the price.

Lemma 1 implies that a buyer either makes the purchase upon her arrival or at a fire sale.

As a result, the probability that two or more buyers observe a given price offer at a given time

t ∈ [0, 1] is 0. Thus, in equilibrium, the seller is indifferent on the quantity offered Qt ≥ 1 at

any time t. Since the seller can also ensure that no good is bought by offering Pt > vH , and

buyers purchase the good only if the price is below their reservation price (independently of the

value of Qt), one can assume, without loss of generality, that Qt = Kt for all t ∈ [0, 1]. Still,

even though each buyer enjoys some monopsony power, she competes with other buyers inter-

temporally owing to the fact that the inventory is finite: she knows that if she rejects an offer,

other buyers may obtain the goods, and she may end up with no good at all.

3.1 Single-Unit Case

We first analyze the case where K0 = 1. We begin with a result stating that, when there is

a single unit to sell, a fire sale happens at the deadline and only then.

Lemma 2. If K = 1, in all equilibria, the seller holds a fire sale at time t if and only if t = 1.

Proof. We begin by noting that, in any equilibrium, if the good has not been sold before at the

deadline, the seller holds a fire sale. This is because, conditional on the good not having been

sold in [0, 1] at the regular price, a fire sale offers him a payoff of vL > 0.

Notice now that, at any time, a buyer accepts for sure an offer equal to min{vL + c, vH −β(vH − vL)} > vL. The reason is that rejecting such an offer implies either waiting until a fire

sale takes place (and obtaining an expected payoff of at most β(vH − vL)) or paying c at some

time in the future and receiving a price offer no lower than vL (with a probability lower than

1). As a result, the seller never holds a fire sale before the deadline. Indeed, if he was supposed

to hold a fire sale at some time t < 1, he could deviate to not holding a fire sale at time t and

10

posting a price equal to min{vL + c, vH − β(vH − vL)} until the deadline (and then holding a fire

sale), which provides with him an expected payoff higher than vL.

The “if” part of the lemma is an important consequence of the seller’s inability to commit: if

the unit has not yet been sold, the seller prefers to obtain some revenue by offering a last-minute

deal. This lowers the ability of the seller to extract rents from the buyers, since they know that

they can wait until the deadline and obtain the good at a low price with a positive probability.

Still, owing to the intertemporal competition between buyers and, at fire sale times, between

buyers and shoppers, the seller can keep the price above vL before the deadline. As a result, we

obtain the “only if” part: the lowest acceptable price before the deadline is higher than vL, so it

is optimal to only hold a fire sale at the deadline.

As a consequence of Lemma 2, the seller only serves buyers before t = 1. Hence, it is optimal

to charge the highest price that they are willing to accept at any moment t < 1. Formally,

Proposition 1. Suppose K0 = 1. There is an essentially unique Markov perfect equilibrium,14

in which

1. for any seller’s history, the seller posts a regular price equaling the buyers’ reservation price

p1(t), which is given by

vH − p1(t) = e−λ(1−t)(vH − vL)β , (1)

2. the seller holds a fire sale only at time t = 1, and

3. a buyer never rechecks the price, and accepts the price at an attention time t ∈ [0, 1] if and

only if it is less than or equal to p1(t).

Proposition 1 establishes that the equilibrium price smoothly declines over time. The intuition

is simple. The left-hand side of equation (1) represents the buyer’s payoff if she arrives at time

t and purchases the good at the equilibrium price p1(t), while the right-hand-side represents

her continuation value of rejecting the offer. The buyers’ reservation price p1(t) makes them

indifferent between making the purchase upon the arrival and rejecting the equilibrium offer.

According to Lemma 1, a buyer never rechecks the price. Consequently, by waiting, she will get

the good at a fire sale price vL only if (1) no other buyer arrives in (t, 1] and (2) the good is

assigned to her (instead of to a shopper) at the final fire sale. The former event takes place with

probability e−λ(1−t), and the probability of the latter event is β, conditional on the former event.

14By “essentially” unique, we mean that all equilibria generate the same distributions of transaction timing

and prices, as the only multiplicity comes from the choice of Qt.

11

Intuitively, as the deadline approaches, the probability that new buyers will arrive shrinks, so

a buyer faces less competition; and thus her reservation price decreases. In addition, at the

deadline, p1(1) = (1 − β)vH + βvL > vL. This gap between p1(1) and vL reflects the buyer’s

willingness to avoid competition with shoppers.

Notice that, because the fire sale only occurs at the deadline, the good is not allocated to a

shopper unless no buyer arrives. As a result, the allocation rule is efficient. This will not be true

in a multi-unit case.

3.2 The Two-Unit Case

Before considering the general multi-unit case, it is worthwhile to study the case where the

seller has two units to sell. We first prove the necessity of a fire sale before the deadline. We

then use this result to characterize the equilibrium time of such a fire sale, which determines the

unique equilibrium.

No Accumulation

We begin with a useful result which applies to any Kt > 0:

Lemma 3. In any equilibrium, at any history where Kt > 0, the price equals the buyers’ (com-

mon) reservation price, and buyers purchase a good at their arrival time.

Lemma 3 establishes that, in equilibrium, the seller sets a price such that buyers purchase

immediately when they arrive. The reason for this is that, by Lemma 1, if a buyer does not

buy a good when she arrives, she never rechecks. Therefore, from the seller’s point of view,

the presence of an accumulated buyer is irrelevant, as goods sold in fire sales are sold at vL

independently of the presence of a buyer. On the contrary, the belief about the number of buyers

accumulated in the past is relevant for arriving buyers. The reason is that accumulated buyers

increase the competition in future fire sales. Still, even though the seller could be willing to

commit to accumulating buyers in order to increase the reservation price of future buyers, the

non-observability of the previous history makes accumulating buyers not incentive compatible,

as early buyers have a higher willingness to pay.

In order to formalize the previous intuition, fix some (t,Kt), with t < 1 and Kt > 0, and

assume that, in equilibrium, the seller posts a price in [t, t′] which buyers reject, for some t′ > t.

If t′ = 1 then the incentive to deviate is clear: the willingness to pay of a buyer arriving in [t, 1]

is higher than vL (recall that, by Lemma 1, the payoff of the seller would be KtvL in this case).

Also, it is clearly suboptimal to hold a fire sale in [t, t′] since, again, the willingness of buyers

12

arriving in [t, t′] is higher than vL. Finally, assume that t′′ > t′ is the first time in which the seller

holds a fire sale if there is no sale in [t, t′′]. Since the reservation price of a buyer is constant in

[t, t′] (because, given that the seller accumulates buyers in this interval, the chance of obtaining

the good in a fire sale if the offer is rejected is independent of the arrival time in [t, t′]) and

decreasing in [t′, t′′] (since, as the deadline approaches, the buyer is more likely to obtain a good

in a fire sale if the offer is rejected), it is optimal for the seller to deviate and offer the reservation

price of the buyers in [t, t′].

Pre-Deadline Fire Sale

By Lemma 1, the continuation payoff of the seller at any history is independent of the number

of buyers accumulated in the past. This is because an accumulation of buyers in the past is

unobservable to future buyers, and accumulated buyers only obtain the good at fire-sale times.

As a result, independently of the previous history, once the first unit is sold, the equilibrium is

characterized by Proposition 1. Therefore, the equilibrium described in Section 3.1 is the unique

continuation play that is sequentially rational.

Now, Lemma 2 can be restated to claim that the seller never holds a fire sale for both goods

before the deadline. Clearly, if Kt = 2 for t < 1, the seller’s payoff is at least vL + Π1(t) > 2vL,

where

Π1(t) ≡∫ 1

t

λe−λ(s−t)p1(s)ds+ e−(1−t)λvL

= vH − e−λ(1−t)(vH − vL)(1 + β(1− t)λ) (2)

is the expected profit at time t associated with the unique equilibrium in Section 3.1. This is the

case because the seller can hold a fire sale at time t and obtain a continuation expected payoff

of Π1(t). Therefore, our goal is to characterize the optimal strategy of the seller when Kt = 2.

When the seller holds two units, he may try to sell them at regular prices and hold a fire sale

at the deadline for the units that have not been sold. This strategy maximizes the probability

of selling both units at the regular price, which, as we will see, is above vL. Nevertheless, the

price offer for the first unit must be low, since a buyer who arrives and observes that the seller

still holds two units believes that she is likely to obtain one of them in a fire sale. Alternatively,

the seller may hold a fire sale for one of the units (i.e., he may sell it at vL) and then offer the

one-unit price (1) during the remaining time. In this case, the seller’s revenue from the sale of

the first unit is low, but the revenue from the sale of the second unit is potentially large. The

next lemma establishes that, when the time is close to the deadline, since the arrival of two

buyers is very unlikely, the second strategy dominates, so the seller never holds two units until

the deadline.

13

Lemma 4. There exists a unique t∗2 ∈ [0, 1), such that the seller holds a single-unit fire sale

(QDt = 1) at time t < 1 if and only if Kt = 2 and t ≥ t∗2.

In order to gather some intuition for the previous result, assume, for the sake of contradiction,

that the seller never holds a fire sale before the deadline. Consider a buyer who arrives at time

t and observes Kt = 2, for some t ∈ [0, 1]. In this case, she is willing to accept some price p only

if it is lower than p2(t), satisfying

vH − p2(t) = e−λ(1−t)(vH − vL)β2︸ ︷︷ ︸no buyer arrives

+ e−λ(1−t)λ(1− t)(vH − vL)β︸ ︷︷ ︸one buyer arrives

,

where β2 = β + (1− β)β is the probability for a buyer of obtaining a good if there is a fire sale

for two goods (if there is no other accumulated buyer). A simple comparison of the previous

equation and equation (1) shows that p2(1) < p1(1). Indeed, if a buyer arrives and observes that

the seller still holds two units, she assigns a larger probability to obtaining one of them in a fire

sale than if the seller held only one unit.

For retaining the two units until the deadline to be an optimal strategy for the seller, it has

to be better than holding a fire sale at any time t and then offering p1(s) described in Section

3.1 at all s > t. If t = 1− ε, for ε > 0 small, such a condition is given by:

fire sale only at t = 1︷ ︸︸ ︷e−λε2vL︸ ︷︷ ︸no arrival

+ e−λελε(p2(1)+vL)︸ ︷︷ ︸one arrival

+O(ε2) ≥fire sale at t = 1− ε and t = 1︷ ︸︸ ︷

e−λε2vL︸ ︷︷ ︸no arrival

+ e−λελε(p1(1) + vL)︸ ︷︷ ︸one arrival

+O(ε2) .

Note that the probability of two or more buyers arriving in [t− ε, 1] is O(ε2). Hence, comparing

the first-order terms it is easy to see that this equation cannot hold.

Remark 1. In our model, the fire sale serves as a device to “advance” (and therefore increase

the probability of a sale at) high prices. Indeed, when the time is close to the deadline, the seller

can only charge a (relatively) high price when the stock of goods is low. As the deadline becomes

close, the probability of being able to sell all remaining goods at the regular price decreases. As

a result, the fire sale ensures not only higher prices for the remaining units, but also a higher

probability that they are sold at the regular price.

Equilibrium Characterization

Let us now characterize the equilibrium time at which the first fire sale happens. To do this,

we use t∗2 ∈ [0, 1) to denote the time in which buyers expect the seller to hold the fire sale if he

14

arrivals probability fire sale at 1 fire sale at 1− ε0 e−λε 2vL 2vL

1 λε+O(ε2) vL + p2(1) +O(ε) vL + p1(1) +O(ε)

> 1 O(ε2) p2(1) + p1(1) +O(ε) vL + p1(1) +O(ε)

Table 1: Probability and payoffs for the seller for different events that may happen in (1− ε, 1),

comparing holding a fire sale at 1 or at 1− ε for some small ε > 0. Because the payoff when no

buyer arrives is independent of when the first fire sale takes place, and the probability of two or

more buyers arriving is very small, the relevant event is the arrival of one buyer. In this case,

since p2(1) < p1(1), the seller prefers to hold a fire sale at t = 1− ε.

still has two units. Such time determines the reservation price of the buyers, which we denote

with p2(t; t∗2). When t < t∗2 we have the following:

vH − p2(t; t∗2) =

no arrival in [t, t∗2]︷ ︸︸ ︷e−λ(t∗2−t)

(β + (1− β)e−λ(1−t∗2)β

)(vH − vL)

+ e−λ(t∗2−t)λ(t∗2 − t)e−λ(1−t∗2)β(vH − vL)︸ ︷︷ ︸one arrival in [t, t∗2]

. (3)

The first term of the right-hand side of the previous equation corresponds to the event that no

other buyer arrives in (t, t∗2]. In this case, the buyer may obtain the object at time t∗2 (with

probability β) or, if no other buyer arrives in (t∗2, 1], she may obtain it at time 1 (again with

probability β). The second term on the right-hand side corresponds to the event that exactly

one other buyer arrives in (t, t∗2], and no buyer arrives in (t∗2, 1].

For a fixed time of the first fire sale t∗2, the expected profit for the seller from deviating and

holding a fire sale at time t < t∗2 is given by∫ t

0

e−λsλ[p2(s; t∗2) + Π1(s)]ds+ e−λt[vL + Π1(t)]

In equilibrium, anticipating the fire sale (that is, choosing t < t∗2) is suboptimal, which provides a

lower bound on the value of t∗2. The proof of Proposition 2 shows that by imposing the condition

that delaying the fire sale is also suboptimal uniquely determines t∗2. The reason is that, if t∗2 is

set too high, the seller has the incentive to deviate and hold an early fire sale owing to the fact

that buyers arriving after t∗2 have a low reservation value (this intuition is similar to the one for

Lemma 4). Then, when t > t∗, arriving buyers expect an immediate fire sale, so the price that

makes a buyer indifferent between accepting or rejecting the offer is p2(t; t).

15

t

Π1,Π2

1

Π1

Π2

Π2

vL2vL

t∗2t2

Figure 1: Illustration of the continuation expected profits of the seller. If there was no fire sale

until the deadline, the profit from holding two units at t ∈ (t2, 1) (given by Π2(t)) is lower than

the profit from immediately holding a fire sale (given by Π1(t) + vL), so the seller benefits from

deviating in this range (see Lemma 4). In equilibrium, the seller holds a fire sale at t∗2 (see

Proposition 2), which generates a profit from holding to units equal to Π2.

For a fixed t∗2, the problem of finding the optimal time for the fire sale can be formulated as

an optimal stopping time problem. In this context, the optimality condition can be written as a

smooth-pasting condition (obtained in the proof of Proposition 2) given by

Π2(t∗2) = Π1(t∗2) . (4)

The two other equations needed to characterize the equilibrium are the value-matching condition

and the Hamilton-Jacob-Bellman (HJB) equation for Π2(t) when t < t∗2 (also obtained in the

proof of Proposition 2) given by:

Π2(t∗2) = vL + Π1(t∗2) , (5)

Π2(t) = −λ[p2(t) + Π1(t)− Π2(t)], ∀t ∈ [0, t∗2) . (6)

Equation (6) is the standard HJB equation when there is neither discounting nor flow payoff.

Also, its intuition is clear: if p2(t) + Π1(t) > Π2(t) (as it is the case in equilibrium for all t < t∗2),

the arrival of a buyer is (strictly) preferable than no buyer arriving, so the continuation value

decreases over time if there are no arrivals.

Proposition 2. Suppose K0 = 2. There exists a unique Markov perfect equilibrium. In this

equilibrium, there exists some t∗2 ∈ [0, 1) such that

16

1. the seller posts a regular price pKt(t) equal to the buyers’ reservation price, where p1 is

given in equation (1), and p2(t) is given by p2(t; t∗2) when t ≤ t∗2, and by p2(t) = p2(t; t)

when t > t∗2, where p2(·; ·) solves (3),

2. the seller posts a single-unit fire sale offer (QDt = 1) at time t if Kt = 2 and t ∈ [t∗2, 1), and

for the remaining units at the deadline, and

3. a buyer accepts a price at time t ∈ [0, 1] if and only if the price is lower than or equal to

pKt(t).

Simple algebra shows that p2(t) < p1(t) for any t. This is intuitive: the larger the current

inventory size, the less future competition a buyer faces; thus she is willing to pay less to avoid

additional competition. Hence, when the seller has a large inventory, he has the incentive to

hold a single-unit fire sale. By doing so, he can immediately sell one unit, so if a buyer arrives

at t′ > t she is willing to pay p1(t′) rather than p2(t′). The fire sale is a commitment device that

endows the seller with some commitment power to charge a higher price in the future. This effect

compensates the opportunity cost of holding a fire sale: the seller may ration future (high-value)

buyers by selling now to (low-value) shoppers. It is easy to show that, for some realizations of

the arrival of the buyers, the seller would prefer to not hold a fire sale, in order to sell both goods

at a relatively high price. However, as the deadline approaches, the probability that multiple

buyers arrive shrinks, so such an opportunity cost becomes small. Consequently, a fire sale is

optimal when the deadline is close.

3.3 The Multi-Unit Case

We now consider the general case, where K0 = K is an arbitrary natural number. Even

though the intuitions are similar to the case where K0 = 2, the extension of the results is not

trivial. The variety of deviations by the seller is larger now as, for example, the seller may hold

fire sales for multiple units.

The construction of the unique equilibrium is done recursively. As the previous results prove,

there is a unique equilibrium when K = 1 and K = 2. A similar result can be proven when

K = 3. In this case, there is some t∗3 ∈ [0, t∗2) where the first fire sale takes place (if Kt∗3= 3)

and, if Kt < 3, then the continuation play is as described in Propositions 1 and 2. In general,

for each K ≥ 3, the induction argument assumes that for all k = 2, ..., K − 1, we have that

1. there is some t∗k ∈ [0, 1] such that if Kt = k the seller holds a fire sale if and only if t ≥ t∗k,

2. t∗1 = 1 and t∗k ≤ t∗k−1, with strict inequality if t∗k > 0, and

17

3. buyers buy upon arrival and never recheck.

Then, our goal is to prove that the same is true for K.

For the strategy profile (informally) described above, we can compute the corresponding

prices and profits. If for a given time t and stock Kt ∈ {1, ..., K} we have t ≤ t∗Kt , then the seller

offers a price pKt(t) which makes buyers indifferent between accepting it and rejecting it. He

does so until either there is a sale at the regular price or t∗Kt is reached, where the seller holds a

fire sale for one unit. As a result, the following equations hold:

vH − pKt(t) =

∫ t∗Kt

t

λe−λ(s−t)(vH − pKt−1(s))ds

+e−λ(t∗Kt−t)(β(vH − vL) + (1− β)(vH − pKt−1(t∗Kt))

), (7)

ΠKt(t) =

∫ t∗Kt

t

λe−λ(s−t)(pKt(s) + ΠKt−1(t))ds+ e−(t∗Kt−t)λ(vL + ΠKt−1(t∗Kt)) . (8)

If, instead, t∗k′+1 ≤ t < t∗k′ for some k′ < Kt, then the seller immediately holds a fire sale for

Kt − k′ units. Then, the following equations apply:

vH − pKt(t) = βKt−k′(vH − vL) + (1− βKt−k′)(vH − pk′(t)) , (9)

ΠKt(t) = (Kt − k′)vL + ΠKt−k′(t) , (10)

where βk denotes the probability that a given buyer in the waiting state obtains the good at

time t if k goods are offered and she is the only buyer in the waiting state at time t. Notice that

βk is increasing in both β and k, for all h ≥ 1. Finally, as in the two-unit case, t∗k is found by

ensuring that the seller does not have the incentive to change the timing of the first fire sale. As

in Section 3.2, this implies the smooth-pasting condition – that is, the equilibrium profit function

for k units is smooth at t∗k:

limt↘t∗k

Π′k(t) = limt↗t∗k

Π′k(t) = Π′k−1(t∗k) . (11)

The following proposition characterizes the unique equilibrium:

Proposition 3. There exists a unique Markov perfect equilibrium. In this equilibrium, there

exist price functions (pk(·))Kk=1 and a decreasing sequence of threshold times (t∗k)Kk=1, with t∗1 = 1

and t∗k < t∗k−1 if t∗k−1 > 0 for all k ∈ {2, ..., K}, such that

1. the strategy of the seller consists of:

(a) if t < t∗Kt the price satisfies equation (7) and the seller does not hold a fire sale,

18

t

Πk

1

Π1 − vL

Π2 − 2vL Π3 − 3vL

Π4 − 4vL

t∗2t∗3t∗4

Figure 2: Illustration of the profits function. As we see, the smooth-pasting condition holds at

fire sale times.

(b) if t∗k ≤ t < t∗k−1, for k < Kt, the price satisfies equation (9) and the seller holds a fire

sale offering the lowest number of units necessary to guarantee that Kt+ < k,15

(c) if t = 1 then the seller holds a fire sale for all remaining units,

2. a buyer never rechecks the price, and accepts a price at time t ∈ [0, 1] if and only if it is

less than or equal to pKt (t) (so, on the path of play, she purchases the good immediately),

and

3. t∗k satisfies equation (11).

Figure 3 provides some simulated price paths. Conditional on the inventory size, the price

declines as the deadline approaches. However, transactions occur over time, and each transaction

triggers an uptick in price.16 As a result, the price goes up or down depending on the arrival of

buyers.

3.4 The Role of Competition in Fire Sales

In our model, the access to a broader market in a fire sale has a dual role. First, it serves as

an “outside option” to the seller, as he can immediately sell some units (at a low price). Second,

it increases the buyers’ willingness to pay, as they know that obtaining the good at a fire sale is

not guaranteed. The former leads to the last-minute fire sale, while the latter allows the seller to

15If Qt units are sold through regular price at time t, the seller offers QDt = max{Kt −Qt − k − 1, 0} units as

a deal.16Escobari (2012) finds a similar price pattern in the airline industry.

19

t

pk

11t∗2t∗3t∗4t∗5t∗6

p1

p2p3p4

p5

t

pk

11t∗2t∗3t∗4t∗5t∗6

vL

(a) (b)

Figure 3: (a) depicts pK in [0, t∗K ] for different values of K. (b) depicts a possible realization of

the price path, where the dots denote transaction prices.

use fire sales as an endogenous commitment device to increase the buyers’ willingness to pay. In

order to further understand the importance of the broader market, consider some comparative

statics results for β.

Consider the case where β is large.This is the case where, conditional on a fire sale, a buyer

(if any) obtains the good with a very high probability. One can interpret a bigger β as either

(1) a smaller amount of demand in the boarder market, or (2) that buyers have bigger attention

advantage. The following result claims that the seller sells most of the goods through an initial

fire sale due to his inability to commit to future high prices:

Proposition 4. There exists some β < 1 such that, if β > β, then t∗k = 0 for all k > 1.

Proposition 4 establishes that when buyers are very likely to obtain a good in a fire sale,

the seller can only generate intertemporal competition for one unit. In equilibrium, the seller

holds a fire sale at time 0 for K0 − 1 units and keeps only one unit after then. This is indeed

incentive-compatible: if, for example, a buyer arrives at some time t ∈ (0, 1) and Kt = 2, she

believes that the seller is going to hold a fire sale immediately and therefore she is going to obtain

the good with a high probability. Thus she has a very low willingness to pay. Even if λ is large

(that is, if there is a high chance that more buyers are going to arrive in the future, and therefore

there is a high intertemporal competition), this belief depresses the expected profit of the seller,

so he has the incentive to hold fire sales at the beginning.

To gather some intuition about Proposition 4, assume, for the sake of contradiction, that

20

arrivals probability fire sale at t∗2 fire sale at t∗2 − ε0 e−λε vL + Et[p1(t)|t>t∗2] vL + Et[p1(t)|t>t∗2]

1 λε+O(ε2) vL +O(ε) + Et[p1(t)|t>t∗2] vL + p1(t∗2) +O(ε)

> 1 O(ε2) vL + p1(t∗2) +O(ε) vL + p1(t∗2) +O(ε)

Table 2: Probability and payoffs for the seller for different events that may happen in (t∗2− ε, t∗2),

comparing holding a fire sale at t∗2 or at t∗2− ε, assuming t∗2 > 0 and β = 1, for some small ε > 0.

Because the payoff when no buyer arrives is independent of when the first fire sale takes place,

and the probability of two or more buyers arriving is very small, the relevant event is the arrival

of one buyer. Since Et[p1(t)|t>t∗2] < p1(t∗2), the seller prefers to hold a fire sale at t = 1− ε.

t∗2 > 0 when β = 1. Consider a history such that K(t∗2 − ε) = 2 for some ε > 0 small. Recall

that p2(t∗2 − ε) = vL +O(ε) because, if a buyer arrives at t∗2 − ε and rejects the offer, she almost

surely obtains the good at t∗2 (since the probability that another buyer arrives in (t∗2 − ε, t∗2] is

O(ε)). This implies that, if a buyer arrives in (t∗2 − ε, t∗2], the payoff of the seller is

vL + e−λ(1−t∗2)E[p1(t)

∣∣ t > t∗2]

+(1− e−λ(1−t∗2)

)vL +O(ε) . (12)

where the expectation is over the time t where the next buyer arrives. The seller can instead

hold a fire sale at t∗2 − ε. In this case, if a buyer arrives in (t∗2 − ε, t∗2], the payoff of the seller is

vL+p1(t∗2)+O(ε), which is above the payoff in (12), given that p1(·) is decreasing. So, conditional

on one buyer arriving in (t∗2 − ε, t∗2], the seller prefers holding a fire sale at t∗2 − ε than at t∗2. If,

instead, no buyer arrives in (t∗2− ε, t∗2], the deviation does not change the seller’s payoff. Finally,

it is very unlikely that two or more buyers arrive in (t∗2 − ε, t∗2] (its probability is O(ε2)). Then,

the expected gain from deviating is positive, which shows that the initial assumption t∗2 > 0 is

not valid.

Proposition 4 is another consequence of the lack of commitment of the seller and the stochastic

arrival of buyers. When β is large, a buyer arriving shortly before a fire sale has a low willingness

to pay: she knows that she is very likely to obtain the good at a very low price. Furthermore,

since future arriving buyers believe that there was no accumulation of buyers in the past, they

have a low reservation price, so the seller has an incentive to reduce the price in the future. As

a result, the seller has further incentives to reduce the current price. If β is large enough, such

reinforcing effects incentivize an initial “big” fire sale to, at least, sell one unit at a high price.

Proposition 4 has some resemblance to the outcome in the Coase conjecture literature (Gul,

Sonnenschein, and Wilson 1986 and Ausubel and Deneckere 1989) when we interpret the prob-

ability that a buyer does not obtain the good at a fire sale as a friction: the seller is unable to

commit to not cuting the price in the future. As β → 1, the good today and the good tomorrow

21

becomes a perfect substitute for the buyer. Consequently, the multi-self intertemporal competi-

tion of the seller instantaneously drives down the price. However, the implication of Proposition

4 differs from that of the Coase conjecture literature in the following aspects.

First, in our model, the seller still enjoys positive rent as β → 1, since

limβ→1

ΠK(0) = (K − 1)vL + Πβ=11 (0),

where Πβ=11 (0) is the payoff of the firm when K = 1 and β = 1 in equation (2). Although the

seller cannot obtain any rent from the first K0 − 1 units, Π1(0) > vL since λ > 0: buyers still

face inter-temporal competition from future arriving buyers, which endows the seller some rent.

As a result, the price remains high after t = 0. There is a delay in the transaction of the last

unit until a buyer arrives. On the other hand, the Coase conjecture implies that (1) no delay

of any transaction, and (2) the seller’s profit converges to the lower bound of the buyer’s value

when he makes offers arbitrarily frequently.

Second, Proposition 4 has very different welfare implication from the Coase conjecture when

we interpret the consumers in the boarder market as shoppers whose valuation on the good is

vL: at the limit, the first K − 1 goods are purchased by shoppers in the boarder market, rather

than arriving buyers. Hence, there exists significant inefficiency due to the misallocation of most

of the goods. Remarkably, endowing the buyers with a bigger advantage in the fire sale (larger

β) hurts, in equilibrium, both the seller and buyers.

The next proposition considers the limit where β becomes small – that is, when the probability

that a buyer obtains the good in a fire sale shrinks. It claims that, in this limit, the commitment

problem of the seller disappears: he extracts all the rent from the buyers and holds fire sales

only at the deadline.

Proposition 5. As β → 0, we have that t∗K → 1 and pK(t)→ vH for all K ∈ N and t ∈ [0, 1].

This is very intuitive. At the limit where β → 0, a buyer expects zero continuation payoff by

waiting, so her willingness to pay goes to vH , and she behaves as a myopic player as in Gallego

and Van Ryzin (1994).

4 Further Discussion

This section discusses some possible extensions of our model.

22

4.1 Multiple Types

In our model, we assume that buyers are identical, so their valuation of the good is always

vH . This section relaxes this assumption by allowing the valuation of each arriving buyer to

be drawn from a distribution, which for simplicity is taken to be uniform. The following result

states that, as long as the buyers’ valuation is bounded away from vL and is not too sparse, there

are equilibria as described above.

Proposition 6. Fix vH > vL. Then, there exists some vH∗ > vH such that if the types of buyers

are uniformly distributed in [vH , vH ] for vH ∈ (vH , vH∗) then there is an equilibrium without

accumulation as described in Proposition 3.

Proof. First, consider the case where K0 = 1, and for notational convenience, let F (vH) =vH−vHvH−vH

denote the CDF of a uniform distribution in [vH , vH ]. Assume that there is an equilibrium as

described in Proposition 1, where now p1(t) is the price corresponding to the case where the

valuation of the buyers is vH . Assume that vH is sufficiently low that no type of buyer rechecks

the price. Therefore, if a buyer with valuation vH arrives at time t, the price she is willing to

pay equals

p1(t, vH) ≡ vH − e−λ(1−t)(vH − vL)β > vH − e−λ(1−t)(vH − vL)β = p1(t) .

Consider the seller’s gain from deviating. By increasing the price from p1(t) to p1(t) + ε the

seller increases the price conditional on acceptance, but lowers the probability of acceptance

conditional on arrival to 1− F (vH(t, p1(t) + ε)), where

p = vH(t, p)− e−λ(1−t)(vH(t, p)− vL)β ⇒ vH(t, p) = vL +p− vL

1− e−(1−t)λβ.

Consider the benefit from deviating at all s ∈ [t, t + ∆] to offering p1(s) + ε, for ∆ > 0 small.

This deviation is not profitable only if p1(t) is higher than (p1(t) + ε)(1−F (vH(t, p1(t) + ε))) for

all ε > 0. The derivative of this last expression with respect to ε gives

1− F (vH(t, p1(t) + ε))− (p1(t)+ε)F ′(vH(t, p1(t)+ε))

1−e(1−t)λβ< 1− vL

vH−vH.

If vH − vH is small enough, the last term to the right of the previous function is negative, so it

is optimal to choose ε = 0.

For a general K0, a similar argument applies. Indeed, at any given time, the reservation price

of a buyer with valuation is vH − bKt(t)(vH − vL), where bK(t) is the probability of obtaining the

23

object in the future (in a fire sale) if at time t the stock is K. So, in general, the derivative of

the payoff from offering a price equal to p1(t) + ε with respect to ε is

1− F (vH(t, p1(t) + ε))− (p1(t)+ε)F ′(vH(t, p1(t)+ε))

1−bKt(t)< 1− vL

vH−vH.

Hence, again, if vH−vH is sufficiently small, the right-hand side of the inequality of the previous

function is negative, so it is optimal to choose ε = 0.

The intuition for Proposition 6 is as follows. In a static model where a monopolist makes a

take-it-or-leave-it offer to a buyer, if the buyer is known to have a valuation uniformly distributed

in [vH , vH ] and vH ≤ 2vH , it is optimal for the monopolist to charge a price equal to vH . Indeed,

the decrease in the probability of trade that an increase of the price above vH generates is high

enough that the monopolist prefers to ensure the trade by setting the price equal to vH . The

intuition is similar in our dynamic model: increasing the price above the reservation price of the

buyers with the lowest valuation effectively implies losing them as potential buyers, since they

do not recheck the price in the future. Even though increasing the price increases the revenue if

a buyer with a high valuation arrives, as in the static model, the negative effect dominates the

positive one. This result can be generalized to distributions of types with a support bounded

away from vL and with a probability density function bounded away from 0.

4.2 Time-Dependent Arrival Rate

So far we have assumed that the arrival rate of buyers is constant over time. In some markets

this may be a restrictive assumption. In the airline industry, for example, it is typically observed

that buyers tend to “arrive” (or become arctive in the market) close to the flight departure time.

In this section we argue that none of our results relies on this assumption.

Assume that the arrival rate of buyers in the market is λ : [0, 1] → R++. Then, define the

“normalized time” τ as

τ(t) ≡ 1

λ

∫ t

0

λ(s)ds with λ ≡∫ 1

0

λ(s)ds .

Note that τ(0) = 0, τ(1) = 1 and τ(·) is differentiable and strictly increasing. Now let’s compute

the probability that exactly one buyer arrives in [t, t+ ε), for some t > 0 and small ε > 0. This

is given by

λ(t)ε+ o(ε) = λτ ′(t)ε+ o(ε) = λ(τ(t+ ε)− τ(t)) + o(ε) .

Hence, under normalized time, the arrival rate of buyers is constant and equal to λ, so all our

results hold identically when “physical time” is replaced by “normalized time.”

24

The result above implies that the buyers’ arrival process sets the tempo of our model. For

example, if the arrival rate of buyers is low at early times, normalized time runs slowly, which

implies that prices will change slowly, and few fire sales will take place. Analogously, if buyers

tend to arrive toward the deadline, then the fluctuations of the price (including fire sales) are

going to be more frequent shortly before the deadline.

4.3 Creating Scarcity by Discarding Inventory

In our model, the seller sometimes finds it optimal to create scarcity by holding fire sales.

Holding a fire sale is good for the seller because it reduces the available stock and allows him to

increase the price in the future. Nevertheless, since the seller cannot commit to the timing of the

fire sales and buyers may obtain goods through them, the prospect of future fire sales lowers the

reservation value of the buyers and also their willingness to pay. Thus, if the seller had (some)

commitment power, he might find it optimal to discard some of his stock, instead of holding fire

sales, in order to get rid of some units in the future without lowering the willingness to pay of

the current buyers.

Consider, for example, adding to our base model the possibility that the seller reduces the

stock (obtaining 0 revenue from the discarded units), but thereby ensuring that buyers would not

obtain these units. Given that our seller lacks commitment power and has access to the market

of shoppers, at the times when the seller would have discarded the goods, he would rather hold

fire sales since, as long as vL > 0, in doing so he would obtain some revenue.

4.4 Disappearing Buyers and Discounting

In our baseline model, we assume that a buyer leaves the market only when her demand is

satisfied. None of our results qualitatively change if buyers leave at a non-trivial rate over time.

Suppose, in the one-unit case, that a buyer in the waiting state leaves the market at a rate

ρ > 0, and her payoff by leaving the market without making a purchase is zero. Now, when

rejecting the current offer, a buyer needs to take into account three risks. As before, (1) another

buyer may arrive and purchase the remaining unit, and (2) if the deadline is reached, she may

not obtain the good due to the competition with shoppers. Now, in addition, (3) she may be

exogenously forced to depart. Her payoff is zero if any of these three events happens. This

increases her reservation price: the equilibrium price at each instant t is increasing in ρ. Thus,

the reservation price of a buyer (1) is now replaced by

vH − p1(t) = e−(λ+ρ)(1−t)(vH − vL)β .

25

Similar expressions can be found for the rest of the equilibrium prices.

Similarly, the effect of time discounting on buyers is the same as the effect of them leaving

the market.

4.5 Returned Goods and Overbooking

One of the reasons that fire sales can serve as an endogenous commitment device is that

transactions are irreversible in our baseline model: the inventory Kt weakly declines over time.

In reality, however, there exist mechanisms which allow Kt to occasionally increase.

One such possibility is that buyers may experience regret after having bought the good and

request to return the good. Imagine that returns may occur with a positive probability. In

such an event, Kt increases. Our model and equilibrium can easily accommodate to allowing

returning goods as long as the probability of a good being returned is independent of the value

that its buyer attached to it (at the purchase time). The reason is that, in this case, a Markov

equilibrium with (t,Kt) as the state variable remains: the seller holds a fire sale to get rid

of the returned goods instantaneously. Note that the possibility of these stock increases (and

corresponding random fire sales) lowers the buyers’ willingness to pay; as a result, the fire sale

as an endogenous commitment device is less effective.

When the seller is allowed to overbook and buy back sold goods, Kt may also go up. In

this case, the seller may endogenously buy back sold goods from low-type buyers and sell it to

high-value buyers. See Ely, Garrett, and Hinnosaar (2016) for an interesting discussion of this

topic. Obviously, the fire sale is less effective to push up the buyers’ willingness to pay, but the

mechanism remains as long as such reallocation is sufficiently costly.17

5 Conclusion

This paper studies the role of commitment power in revenue management with strategic

arriving buyers. When sales are low, the seller finds it optimal to hold fire sales to lower the

inventory and to increase future prices. Still, as buyers anticipate the possibility of obtaining a

good at a bargain price, the price decreases before fire sales, and jumps up immediately afterward.

The lack of commitment power, then, provides a new channel to generate price fluctuations and

fire sales. This insight can contribute to our understanding of the price fluctuations in industries

such as airlines, cruise-lines and hotel services.

17In fact, the overbooking exercise is very costly in the airline industry. When a passenger is involuntarily

bumped, the airline is required to pay four times the cost of the one-way fare.

26

A Formal Model

We present here the formal version of our model.

Players: The players in the model are a seller and a unit mass of buyers, indexed by τ ∈ [0, 1].

Buyer states: At each time t ∈ [0, 1], a buyer τ ∈ [0, 1] can be in one of the following three

states: inactive, active or served, denoted θτt ∈ {I,A, S}. The activation process is denoted N(t),

which is a (right-continuous) Poisson jump process with arrival rate λ. Buyer τ is not inactive

at time t only if τ ≥ t and N(τ)−N(τ−) = 1.18 In particular, if τ > t, buyer τ has not arrived,

so θτt = I.

Actions: At each instant t, the seller decides the price Pt ∈ R+ and the quantity offered

Qt ∈ {0, ..., Kt}, where Kt is the remaining stock of units at time t. At any time t, an active

buyer τ < t decides whether to pay attention to the price or not. Let Jt be the set of (active)

buyers who pay attention at time t, which includes τ = t if N(τ)−N(τ−) = 1 and the remaining

active buyers who decide to pay attention. Then, within the instant t, the game proceeds as

follows:

1. Each of the buyers j ∈ Jt, after observing (Kt, Pt, Qt), decides to accept the offer (ajt = 1)

or not (ajt = 0). Let J ′t be the set of buyers who accepted it.

2. Nature orders buyers in J ′t uniformly randomly.

3. The first min{Qt, |J ′t|} buyers in J ′t obtain the good.

Histories: A total history of the game up to time t ∈ [0, 1] is as follows:

1. A history of the inventory {Ks}s∈[0,t), sales at normal prices {SNs }s∈[0,t), and sales at deal

times {SDs }s∈[0,t), such that Ks −Ks+ = SNs+ − SNs + SDs+ − SDs for all s ∈ [0, t).

2. A history of prices and quantities, {Ps, Qs}s∈[0,t), and a finite set of deal times, Dt ⊂ [0, t)

with quantities {QDs }s∈Dt , where SDs+ − SDs = QD

s for all s ∈ [0, t).

3. A jump (arrival) process {Ns}s∈[0,t), and a set Tt ⊂ [0, t) of times where N jumps.

4. If Nτ − Nτ− = 1 for τ ∈ [0, 1], a time Tτ ∈ [τ, t) ∪ {∅} determining the date at which

the τ -buyer has been served (∅ indicates that she has not been served), a finite set of

observation times Oτ,t ⊂ [τ, t), with τ ∈ Oτ,t, and a set of observations (P τs , K

τs )s∈Oτ,t and

the corresponding decisions to accept the offer or not, (aτs)s∈Oτ,t . We use |Oτ,t| to denote

the (jump) process which indicates the number of observation times of buyer τ prior to

time t.18Nt records the number of buyer arrivals until time t. Without loss of generality, we restrict ourselves to

histories where there is at most one arrival at each instant.

27

5. For each time s ∈ [0, t), let J ′s ≡ {τ ≤ s|s ∈ Oτ,t ∧ aτt = 1}. If |J ′s| > 0 then |{τ ∈ J ′s|Tτ =

s}| = min{|J ′s|, Qs} = SNs+ − SNs .

Information: Given a history until time t, the seller observes {Ks, Ps, Qs, SNs , S

Ds }s∈[0,t). A

buyer τ who is not inactive observes {Ks, Ps, Qs}s∈Oτ,t .Strategies: A strategy for the seller is a left-continuous ‘regular offer’ process (Pt, Qt) and

a (fire sale) jump process SDt , both measurable with respect to his information filtration and

satisfying the conditions above. The strategy of buyer τ is right-continuous (“observation”)

jump process |Oτ,t| and a stopping-time process Aτ,t (indicating the time of acceptance of an

offer), both measurable with respect to his information filtration and satisfying the conditions

above.

Payoffs: Given a total history of the game, the payoff of the seller is∫ 1

0PtdS

Nt +

∫ 1

0vLdSDt ,

while the payoff of a buyer τ is 0 if she does not purchase the good, vH − Pt − (|Oτ,t| − 1)c if

she obtains the good at the regular price at time t and vH − vL − (|Oτ,t| − 1)c if she obtains the

good through a fire sale at time t.

28

B Omitted Proofs

The Proof of Lemma 1. For a given pair (t,Kt), let a(t,Kt)(·) be the acceptance strategy of a

buyer with attention time at t, where a(t,Kt)(Pt, Qt) equals 1 if the buyer accepts the price and

quantity pair (Pt, Qt), and 0 otherwise. Let p(t,Kt) be the supremum of the offers which are

accepted at state (t,Kt). Clearly, in equilibrium, either the seller makes an unacceptable offer,

or p(t,Kt) is offered (and accepted for sure).

Given that buyers follow the same Markov strategy, they have the same willingness to pay

for the good, so p(t,Kt) is the amount which makes them indifferent on accepting or not the offer.

As a result, an accumulated buyer does not expect to obtain any rent after having paid the

rechecking cost c > 0 at time t, independently of the observed (t,Kt). As a result, a buyer is

better off without rechecking the price.

The Proof of Proposition 1. Obvious given the argument in the main text.

The Proof of Lemma 3. By Lemma 1, if a buyer does not buy at her arrival time, she never

rechecks again. As a result, the continuation payoff of the seller at time t is only a function of

(t,Kt), so denote it Π(t,Kt).

Assume that at time t < 1 the stock is Kt > 1 and that, in equilibrium, the seller posts a

price in [t, t′] which buyers reject, for some t′ > t. First, notice that it is suboptimal to hold a fire

sale in [t, t′]. To prove this, and to keep the proof simple, assume that a single fire sale for one

unit takes place in [t, t′] (the argument is easily generalizable to multiple fire sales or fire sales

for multiple units). By Lemma 1, the willingness to pay of a buyer arriving at any time in [t, t′]

is higher than vL, since rejecting the offer allows her to obtain the good in a future fire sale with

a probability strictly lower than 1. Then, the seller can deviate to offering the reservation price

of the arriving buyers in [t, t′], holding a fire sale at t′ if there is no sale in [t, t′], and posting an

unacceptable price in (t′′, t] if a sale takes place at t′′. This deviation is unobservable to future

buyers, so the continuation payoff of a buyer arriving after t′ is not affected. Also, it is clear

that the expected revenue for the unit sold in [t, t′] is higher than vL, which makes the deviation

profitable.

Then, there two possible cases:

1. If t′ = 1 then the incentive to deviate is clear: if the seller follows the equilibrium strategy, he

obtains a payoff of KtvL, while holding a fire sale of Kt−1 units at time t (and then following

the unique continuation play specified in Proposition 1) gives him (Kt−1)vL+Π1(t) > KtvL.

2. Assume then that t′ < 1, and assume that the price is acceptable (and therefore equal to

the reservation of the buyers) and no fire sale takes place in [t′, t′′], for some t′′ > t′. Then,

29

the payoff of the seller is∫ t′′

t′e−λ(s−t′)(p(s) + Π(s,Kt−1)

)λds+ (1− e−λ(t′′−t′))Π(t′′, Kt) .

where p(·) is the reservation value of the buyers and Π(s,Kt− 1) is the continuation payoff

if a unit is sold at time s. Note first that, since the price is acceptable for all s ∈ (t′, t′′),

optimality requires that p(s) + Π(s,Kt − 1) ≥ Π(s,Kt). Note also that p(s) is strictly

decreasing for s ∈ (t′, t′′), since buyers who arrive early face a higher future competition

that buyers who arrive late. Note finally that Π(s,Kt−1) is (weakly) decreasing in s given

that the seller has the option to offer unacceptable prices. Then, p(s) + Π(s,Kt − 1) is

strictly decreasing, so we have p(t′+) + Π(t′+, Kt − 1) > Π(t′′, Kt) and Π(t′, Kt) < p(t′+) +

Π(t+, Kt−1). So, if the seller deviates to offering p(t′+) in [t, t′] (and an unacceptable price

until t′ if there is a transaction before t′), he obtains:

e−λ(t′−t)(p(t+) + Π(t′, Kt − 1))

+ (1− e−λ(t′−t))Π(t′, Kt) > Π(t′, Kt)

Therefore, such a deviation is profitable.

The Proof of Lemma 4. Let’s call t∗2 ∈ [0, 1] the latest time such that, in any equilibrium and

history, if t > t∗2 and Kt = 2 then the seller immediately holds a fire sale.

Assume first that t∗2 = 1. This implies that, for any ε > 0, there exists an equilibrium where

the seller does not hold a fire sale before time 1− ε on the path of play. Lemma 1 implies that

the continuation value of a buyer who draws attention time at 1 − ε′, for some ε′ > 0 small

enough, and rejects the offer is (vH − vL)β2 +O(ε′) independently of the continuation play of the

seller (assuming that, if there are units left at the deadline, he holds a fire sale for them).19 So,

if ε� ε′, the seller’s payoff at time 1− ε′ is given by

e−λε′2vL + (1− e−λε′)(vH − (vH − vL)β2 + vL) + o(ε′) .

Instead, the seller could hold a fire sale at time 1− ε′. In this case, he obtains a payoff equal to

vL + e−λε′vL + (1− e−λε′)(vH − β(vH − vL)) + o(ε′) .

The difference between the two payoffs is −(vH − vL)(β2 − β)(1− e−λε′) + o(ε′) < 0. Then, the

seller has a profitable deviation, which leads to a contradiction, so t∗2 < 1.

19Notice that if the seller holds a single fire sale for the two units at the deadline, the buyer obtains a good

with probability β2 +O(ε′), while if the seller holds two fire sales of one unit each, the buyer obtains a good with

probability β + (1− β)β +O(ε′) = β2 +O(ε′).

30

Assume then that t∗2 < 1, so for all ε > 0 there exists an equilibrium where the seller holds

a first fire sale at some time t ∈ (t∗2 − ε, t∗2]. Fix ε > 0 and assume that there is an equilibrium

where the first time where seller holds a fire sale is t∗2 − ε. Fix some ε′ > 0 and assume that

ε� ε′. Then, the payoff of the seller at time t∗2 − ε− ε′ if he follows the equilibrium strategy is

given by

(1− e−λε′)(p2(t∗2) + Π1(t∗2)

)+ e−λε

′(vL + Π1(t∗2))) + o(ε′)

= vL + Π1(t∗2) + λε′(p2(t∗2)− vL) + o(ε′) ,

where p2(t∗2) ≡ βvL + (1 − β)p1(t∗2) is the reservation price of a buyer at time t∗2. Instead, the

seller can deviate and hold a fire sale at t = t∗2 − ε − ε′, for ε′ > 0. In this case, he obtains

vL + Π1(t∗2)− ε′Π′1(t∗2) + o(ε′). Then, we have that p2(t∗2)− vL ≥ −Π′1(t∗2).

Alternatively, the seller can deviate by not holding a fire sale at time t∗2 − ε and, instead,

holding a fire sale at t∗2 + ε′. The seller’s payoff at time t∗2 from following this deviation is given

by

(1− e−λε)(p2(t∗2) + Π1(t∗2)

)+ e−λε(vL + Π1(t∗2 + ε))) + o(ε)

= vL + Π1(t∗2) + ε(λ(p2(t∗2)− vL) + Π′1(t∗2)) + o(ε) .

In this case, we have λ(p2(t∗2)− vL) + Π′1(t∗2) ≤ 0. As a result, we have Π′1(t∗2) = −λ(p2(t∗2)− vL).

Finally, note that from equation (3) we have

vH − p2(t∗2) =(β + (1− β)e−λ(1−t∗2)β

)(vH − vL) .

Using equation (2), we obtain that t∗2 solves

0 = (vH − vL)λ(1− β − e−(1−t2)λ(1− β2 − β(1− t2)λ

). (13)

When t2 = 1 the right-hand side of the previous equation is negative. Also, simple algebra shows

that the derivative of the right-hand side with respect to t2 of the has at most one zero, and

the second derivative at this point is positive, so it is a minimum. Therefore, if a solution for t2

exists, it is unique, and if it is positive, it is equal to t∗2 for the unique equilibrium. If, instead,

the solution is not positive or no solution for t2 exists, then t∗2 = 0 in the unique equilibrium.

The Proof of Proposition 2. This result is a particular case of Proposition 3 when K = 2.

The Proof of Proposition 3. The proof of Lemma 1 shows that buyers do not recheck the price

for any K. So it is only left to prove a result analogous to that of Lemma 4.

31

We conduct our proof by induction. We assume that for all k ≤ K − 1 there exists a unique

equilibrium as described in the statement of the proposition. This implies that after any history

the outcome is uniquely determined. We want then to prove that the result also applies to k = K.

In particular, the following properties are going to be important:

• If t ∈ (t∗k+1, t∗k) then Πk′(t) < (k′ − k′′)vL + Πk′′(t) for all k′ > k and k′′ < k′, and

Πk′(t) > (k′ − k′′)vL + Πk′′(t) for all k′ ≤ k and k′′ < k′, for all k ∈ {1, ..., K − 1}. In

particular, this implies Πk(t) > (k − k′)vL + Πk′(t) for all k′.

Proving t∗K < 1: Let us first prove that if t is close enough to 1 and Kt = K, then the

seller holds a fire sale. The argument is the same as in the proof of Lemma 4. Indeed, let’s

assume that a buyer draws attention time at t = 1 − ε, for some ε > 0 sufficiently small,

and she observes Kt = K. Her continuation value if she rejects the offer is no higher than

(vH − vL)βK +O(ε) independently of the continuation play of the seller (see footnote 19) where

βK = β∑K

k=1(1− β)k−1 So, the seller’s payoff if he does not hold a fire sale is

e−λεKvL + (1− e−λε)(vH − (vH − vL)βK + (K − 1)vL) + o(ε) .

Instead, the seller could hold a fire sale of K − 1 units at time t. In this case, he obtains

(K − 1)vL + e−λεvL + (1− e−λε)(vH − β(vH − vL)) + o(ε) .

The difference between both payoffs is −(vH − vL)(βK − β)(1− e−λε) + o(ε) < 0.

Proving that t∗K ≤ t∗K−1: We now prove that t∗K ≤ t∗K−1 and if t∗K−1 > 0 then t∗K < t∗K−1.

For the sake of contradiction, assume t∗K > t∗K−1, that is, the seller does not hold a fire sale before

t∗K−1 if Kt = K. Assume that q ≥ 1 units are offered in a fire sale at time t∗K if Kt∗K= K. Let

k < K−1 denote the unique natural number lower than K such that t∗K ∈ [t∗k+1, t∗k), so necessarily

then q = K − k (given that k has to maximize (K − k′)vL + Πk′(t) for all k′ ∈ {1, ..., K − 1} and

t ∈ (t∗k+1, t∗k)). Then, ΠK(t∗K) = (K − k)vL + Πk(t

∗K). The seller can deviate and hold a fire sale

of one unit at t∗K − ε, and then, if no extra unit is sold, make fire sale of K − k − 1 units at t∗K .

The additional payoff that such deviation provides is20

λε(pK−1(t∗K)− pK(t∗K)

)+ o(ε) .

Note that vH − pK(t∗K) = βK−k(vH − vL) + (1 − βK−k)(vH − pk(t∗K)) and vH − pK−1(t∗K) =

βK−k−1(vH − vL) + (1 − βK−k−1)(vH − pk(t∗K−1)) which implies pK(t∗K) < pK−1(t∗K). Then, this

is a profitable deviation, so t∗K ≤ t∗K−1. Assume now that t∗K−1 > 0 and t∗K = t∗K−1. Now, the

20Note that the only term that is linear in ε is the payoff if exactly one buyer arrives in [t∗K − ε, t∗K ].

32

seller can make a fire sale of one unit at time t∗K − ε if he still holds K units. In this case, the

difference between these strategies is, again,

λε(pK−1(t∗K)− pK(t∗K)

)+ o(ε) .

Now vH − pK(t∗K) = β2(vH − vL) + (1− β2)(vH − pK−2(t∗K)) and vH − pK−1(t∗K) = β1(vH − vL) +

(1− β1)(vH − pK−2(t∗K−1)). So, again, this is a profitable deviation.

Existence and uniqueness of an incentive-compatible t∗K . Let pK(t; t∗K) be the reser-

vation price of a buyer if the seller holds a fire sale at time t such that Kt = K if and only if

t ≥ t∗K . Note that follows equations (7) and (9). It is easy to show that when t < t∗K we have

∂pK(t, t∗K)

∂t∗K= −e−λ(t∗K−t)λ

((1− β)pK−2(t∗K)− pK−1(t∗K)− βvL

)< 0 .

Given that buyers follow a threshold strategy with a threshold equal to pK(t; t∗K), let ΠK(t; tK , t∗K)

be the payoff of the seller if Kt = K, he offers pK(s; t∗K) while Ks = K for s ≥ t and tK is the

first time he holds a fire sale if he has not yet sold any units. This is given by the following

expression:

ΠK(t; tK , t∗K) ≡

∫ tK

t

λe−λ(s−t) [pK(s; t∗K) + ΠK−1(s)] ds+ e−λ(tK−t) [vL + ΠK−1(tK)] . (14)

If we differentiate expression (14) with respect to tK we obtain

∂ΠK(t; tK , t∗K)

∂tK

∣∣∣∣tK=t∗K

=∂ΠK(t; t∗K , t

∗K)

∂t∗K

∣∣∣∣tK=t∗K

= e−λ(t∗K−t)(λ((1− β)pK−1(t∗K)− vL) + Π′K−1(t∗K)

)≡ e−λ(t∗K−t)f(t∗K) . (15)

Also, we have that the derivative with respect to t is given by

∂ΠK(t; t∗K , t∗K)

∂t

∣∣∣∣t=t∗K

= −(1− β)λ(pK−1(t∗K)− vL) < 0 .

This equation is valid independently of whether the derivative is taken from the left or from the

right of t∗K , so we have a smooth-pasting condition.

The part of this proof where we prove that t∗K ≤ t∗K−1 shows that f(tK) defined in (15) is

negative when tK is close to t∗K−1. If f(t∗K) is negative for all t∗K < t∗K−1 then clearly t∗K = 0 in

the unique equilibrium. Assume otherwise, and let t∗K be the lowest value such that f(t∗K) = 0.

33

Assume that there exists an equilibrium where the first fire sale time equals t∗K > t∗K . As we

showed before, pK(t∗K , t∗K) < pK(t∗K , t

∗K). Therefore, we have the following inequality:

vL =1

λΠ′K−1(t∗K) + pK(t∗K , t

∗K) >

1

λΠ′K−1(t∗K) + pK(t∗K , t

∗K)

= ΠK−1(t∗K)− ΠK−2(t∗K) .

This is a contradiction since, otherwise, holding a fire sale before t∗K would be optimal. Then,

if f(0) ≤ 0 we have t∗K = 0, while if f(0) > 0 there is a unique t∗K ∈ (0, t∗K−1) where it is

incentive-compatible for the seller to hold a fire sale. As a result, the equilibrium exists and is

unique.

The Proof of Proposition 5. Using equation (1) it is clear that the result is true for K = 1. Then,

one can proceed analogously to show limβ→0 t∗2 = 1 and limβ→0 p2(t) = vH for all t ∈ [0, 1]. The

reason is that when β is low, even when the time is close to the fire sale, the willingness to pay

of the buyers is again, close to vH , since obtaining the good in a fire sale is unlikely. As a result

the seller has the incentive to post-pone fire sales toward the deadline. This argument can be

applied iteratively for any K.

The Proof of Proposition 4. We only need to prove that there exists some β < 1 such that if

β > β then t∗2 = 0 for all k = 2. Recall that t∗2 is the solution of equation (13) in the proof of

Lemma 4 (if it exists and it is positive, it is 0 otherwise). The right hand side of the equation

(13) is −(vH − vL)e−λ when t2 = 0 and β = 1. So, by continuity, if β is close to 1 the right hand

side of equation (13) is negative for all t2 ∈ [0, 1], so t∗2 = 0 in the unique equilibrium.

34

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