Revenue Management Without Commitment:...
Transcript of Revenue Management Without Commitment:...
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, [email protected]‡University of North Carolina, Chapel Hill, [email protected]
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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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.
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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
References
Aguirregabiria, V. (1999): “The dynamics of markups and inventories in retailing firms,”
The review of economic studies, 66(2), 275–308.
Anton, J. J., G. Biglaiser, and N. Vettas (2014): “Dynamic price competition with
capacity constraints and a strategic buyer,” International Economic Review, 55(3), 943–958.
Ausubel, L. M., and R. J. Deneckere (1989): “Reputation in bargaining and durable goods
monopoly,” Econometrica: Journal of the Econometric Society, pp. 511–531.
Aviv, Y., and A. Pazgal (2008): “Optimal pricing of seasonal products in the presence of
forward-looking consumers,” Manufacturing & Service Operations Management, 10(3), 339–
359.
Besanko, D., and W. L. Winston (1990): “Optimal price skimming by a monopolist facing
rational consumers,” Management Science, 36(5), 555–567.
Board, S., and A. Skrzypacz (2016): “Revenue management with forward-looking buyers,”
Journal of Political Economy, forthcoming.
Chen, C.-H. (2012): “Name your own price at priceline. com: Strategic bidding and lockout
periods,” The Review of Economic Studies, 79, 1341–69.
Conlisk, J., E. Gerstner, and J. Sobel (1984): “Cyclic pricing by a durable goods mo-
nopolist,” The Quarterly Journal of Economics, pp. 489–505.
Deb, R., and M. Said (2015): “Dynamic screening with limited commitment,” Journal of
Economic Theory.
Ely, J., D. Garrett, and T. Hinnosaar (2016): “Overbooking,” The Journal of the Euro-
pean Economic Association, forthcoming.
Escobari, D. (2012): “Dynamic Pricing, Advance Sales and Aggregate Demand Learning in
Airlines,” The Journal of Industrial Economics, 60(4), 697–724.
Gallego, G., and G. Van Ryzin (1994): “Optimal dynamic pricing of inventories with
stochastic demand over finite horizons,” Management science, 40(8), 999–1020.
Garrett, D. (2016): “Intertemporal Price Discrimination: Dynamic Arrivals and Changing
Values,” American Economic Review, forthcoming.
35
Gershkov, A., and B. Moldovanu (2009): “Dynamic revenue maximization with heteroge-
neous objects: A mechanism design approach,” American Economic Journal: Microeconomics,
1(2), 168–198.
Gershkov, A., B. Moldovanu, and P. Strack (2014): “Dynamic allocation and learning
with strategic arrivals,” Unpublished manuscript, Hebrew University of Jerusalem.
Gul, F., H. Sonnenschein, and R. Wilson (1986): “Foundations of dynamic monopoly and
the Coase conjecture,” Journal of Economic Theory, 39(1), 155–190.
Horner, J., and L. Samuelson (2011): “Managing Strategic Buyers,” Journal of Political
Economy, 119(3), 379–425.
Jerath, K., S. Netessine, and S. K. Veeraraghavan (2010): “Revenue management with
strategic customers: Last-minute selling and opaque selling,” Management Science, 56(3), 430–
448.
Liu, Q., K. Mierendorff, and X. Shi (2015): “Auctions with limited ommitment,” Working
Paper, Columbia University.
McAfee, R. P., and V. Te Velde (2006): “Dynamic pricing in the airline industry,” forth-
coming in Handbook on Economics and Information Systems, Ed: TJ Hendershott, Elsevier.
Mierendorff, K. (2016): “Optimal dynamic mechanism design with deadlines,” Journal of
Economic Theory, 161, 190–222.
Nocke, V., and M. Peitz (2007): “A theory of clearance sales,” The Economic Journal,
117(522), 964–990.
Ory, A. (2014): “Consumers on a Leash: Advertised Sales and Intertemporal Price Discrimi-
nation,” Unpublished manuscript, Yale University.
Pai, M. M., and R. Vohra (2013): “Optimal dynamic auctions and simple index rules,”
Mathematics of Operations Research, 38(4), 682–697.
Sweeting, A. (2012): “Dynamic pricing behavior in perishable goods markets: Evidence from
secondary markets for major league baseball tickets,” Journal of Political Economy, 120(6),
1133–1172.
Talluri, K. T., and G. J. Van Ryzin (2006): The theory and practice of revenue management,
vol. 68. Springer Science & Business Media.
36