Selling to Strategic Consumers When Product Value is Uncertain:

The Value of Matching Supply and Demand

Robert Swinney
Graduate School of Business, Stanford University, Stanford, CA, swinney@stanford.edu
May, 2008. Last Revised February, 2011.
Abstract
We address the value of quick response production practices when selling to a forward-looking
consumer population with uncertain, heterogeneous valuations for a product. Consumers have
the option of purchasing the product early, before its value has been learned, or delaying the
purchase decision until a time at which valuation uncertainty has been resolved. While individual
consumer valuations are uncertain ex ante, the market size is uncertain to the rm. The rm
may either commit to a single production run at a low unit cost prior to learning demand, or
commit to a quick response strategy which allows additional production after learning additional
demand information. We nd that the value of quick response is generally lower with strategic
(forward-looking) customers than with non-strategic (myopic) customers in this setting. Indeed,
it is possible for a quick response strategy to decrease the prot of the rm, though whether this
occurs depends on various characteristics of the market; specically, we identify conditions under
which quick response increases prot (when prices are increasing, when dissatised consumers
can return the product at a cost to the rm) and conditions under which quick response may
decrease prot (when prices are constant or when consumer returns are not allowed).
1 Introduction
Long production and shipping leadtimes are cited as key causes of supply-demand mismatches,
particularly on products manufactured in an oshore fashion (e.g., produced in Asia and exported to
the US or Europe). Due to these long leadtimes, demand forecasts and production decisions must be
made far in advance of the selling season, when uncertainty concerning nal demand is high. Thus,
if leadtimes could be reducedvia, for example, localized production, improved information systems
and forecast updating, multi-channel production and emergency supply sources, and expedited
shipping methodsallowing for a rapid response to updated demand information closer to (or during)
the selling season, supply and demand could be more closely matched, reducing or eliminating costly
shortages and wasteful overproduction. Such techniques to mitigate demand uncertainty (which
we refer to hereafter as quick response systems) can be costly due to IT expenses and expedited
1
production or transportation costs, but are known to provide signicant value to rms by better
matching supply with uncertain demand (Fisher and Raman 1996, Eppen and Iyer 1997).
Most existing work on this subject analyzes quick response practices using a xed demand
model; that is, market demand (i.e., consumers) does not react when a rm adopts quick response
capabilities. However, the consequences of a quick response systemparticularly lower demand
variability and potentially greater costshave a tangible impact on consumers by inuencing mea-
sures that they directly experience, most notably the ll rate (fraction of demand that is satised).
Consumers, in turn, may take changes in these measures into account when making their own
purchasing decisions, and as a result market demand may change along a variety of dimensions
such as the timing of purchases by consumers. For example, Richtel (2007) describes how con-
sumers learned about the general scarcity of the Nintendo Wii (low ll rates) and modied their
own behavior (buying any available units) in response; ODonnell (2006) describes how consumers
learn about and react to inventory availability and markdown frequency in the fashion industry;
and Rosenbloom (2009) describes how luxury goods retailers have inuenced consumer behavior
by reducing inventory.
Thus, an important issue is how individual consumers may respond to the adoption of quick
response techniques, and indeed whether rms can continue to exploit the benets of quick response
production once the market reacts optimally to this practice. Previous work (Cachon and Swinney
2009, Cachon and Swinney 2011) has shown that consumers may react to quick response capabil-
ities in a way that benets the rm. Specically, if consumers are strategic in the sense that
they anticipate potential future price reductions by a rm and time their purchases accordingly,
adopting quick response can reduce the probability of clearance sales (by increasing the accuracy
of demand forecasts) and hence induce consumers to pay full price for a product. In this paper, we
examine a scenario in which the market may react to the adoption of quick response practices in
a negative waythat is, by employing quick response, a rm inuences inventory availability such
that consumers will optimally react in a manner that reduces total rm demand and prot. This
setting is characterized by a key feature: consumer value for the product is initially uncertain and
is learned over time.
Valuation uncertainty may arise in consumer goods in a number of dierent ways. Parents
increasingly participate in the unfortunate ritual of trying to identify the hot holiday toy for
2
their children (Slatalla 2002), trading o the risk of buying early and facing uncertain value for the
product (i.e., possibly buying a toy that turns out to be a dud or that their child does not want)
with the risk of buying late and facing uncertain availability for the product (i.e., experiencing a
stock-out). Consumer value may also be uncertain if the product is a new or innovative item (e.g.,
a complex product such as a Nintendo Wii, an Apple iPhone, or an automobile), a media item
(such as books, movies, music, or video games), or if the consumers requirements for the item are
uncertain (e.g., snow skis for a potential weekend trip in two months when weather is unknown).
A common feature of all of these examples is that over time, consumers learn more information
about the product and gain a better sense of its value; for example, via channels such as professional
product reviews from web sites and magazines, the reviews of fellow consumers (e.g., from online
retailers such as Amazon.com), the experiences of friends and family who may have purchased the
same product, or via the resolution of intrinsic uncertainty in product value (e.g., the weather
aecting the value of a pair of skis is known the day of the ski trip). Hence, consumers may
recognize that future learning will occur and may choose to delay a purchase until they have more
information about a products value.
In the spirit of these examples, our model consists of consumers that initially have uncertain
value for a product, but who know that information about product value will be learned at some
point in the future. Individual consumers in our model thus make a decision on when and whether
to purchase the product: the later the customer waits to buy, the more information she will have
about product value and the greater the risk of a stock-out. Specically, in our model consumers
choose to either purchase earlyprior to learning their value for a productor purchase late, after
learning their value. Hence, consumers in our model may strategically delay a purchase to learn
more about product value rather than obtain a product of known value at a lower price, as in
Cachon and Swinney (2009, 2011). While the mechanisms driving strategic consumer purchasing
behavior are similar in the two settingsdelaying a purchase to potentially increase net surplusthe
consequences for the rm, particularly in how the rm values quick response practices, are very
dierent.
We show that when consumers experience this type of time dependent learning, greater avail-
ability resulting from an improved matching of supply and demand encourages consumers to delay
purchasing the product: by reducing the likelihood of a stock-out, the rm decreases the risk of
3
waiting to learn more information about product value. As more consumers strategically delay
purchasing, total demand can decrease (via mitigation of an eect known as advance selling, Xie
and Shugan 2001) and prot can be reduced despite the fact that the rm can better match supply
and demand via quick response. Thus, in contrast to previous work showing that the interplay
of quick response and consumer behavior can lead to greater value for rms in some settings, our
model demonstrates that when consumers learn about product value over time quick response ca-
pabilities can actually inuence consumer behavior in a way that is detrimental to rm prots. We
further show that whether this occurs (and to what degree it occurs) depends heavily on several
characteristics of the selling environment; specically, the price path (increasing or decreasing over
time) and the consumer return policy (whether refunds are oered to dissatised consumers) play
a major role in determining the value of matching supply and demand via quick response.
2 Literature Review
Quick response production as a vehicle to help mitigate and respond to demand uncertainty has
received a signicant amount of attention in the literaturesee, e.g., Fisher and Raman (1996) and
Eppen and Iyer (1997). Typically, these papers model quick response as leading to a reduction
in leadtimes and hence a decrease in demand uncertainty. Mechanisms involved in this reduction
of demand uncertainty may include external revelation of information (exogenous fashion trends,
etc.), marketing or information collection eorts by the rm (e.g., focus group testing), or even
early sales data based on sample production in selected markets. Specically, we adopt a stylized,
two-stage approach in which the rm is allowed an initial order (long before the selling season
and subject to high demand uncertainty) and a second order (at a higher marginal cost close to
the selling season with demand uncertainty virtually eliminated), similar to Cachon and Swinney
(2009), Caro and Martnez-de-Albniz (2010), and Krishnan et al. (2010). In addition, much like
our model, a number of these papers analyze the impact of quick response on non-operational
aspects of the rm such as competition (Ha and Li 2008, Caro and Martnez-de-Albniz 2009) and
retailer sales eort (Krishnan et al 2010). The primary distinction between our model and previous
work in the quick response literature is that we explicitly model the consumer purchasing decision
subject to consumer valuation uncertainty.
4
A number of recent papers have incorporated models of inter-temporal consumer purchasing
decisions into traditional operational models. Examples include Su and Zhang (2008), Liu and van
Ryzin (2008), and Aviv and Pazgal (2008), all of which consider consumers that strategically time
purchases because prices change over time. In contrast, we consider consumers that time purchases
because information about product value will be revealed over time. Several papers consider a
similar scenario. DeGraba (1995) demonstrates that, in the absence of demand uncertainty, a
rm may intentionally understock to induce consumers to purchase when valuations are uncertain
and learned over time. Xie and Shugan (2001) demonstrate that selling to consumers prior to the
determination of value and consumption (e.g., with advance ticket sales) can substantially increase
rm prots. Alexandrov and Lariviere (2006) consider the problem of a restaurant choosing
whether to oer reservations (guaranteed seats) to customers who may or may not value dining on
a given night, demonstrating when reservations increase the prot of the rm. Akan et al. (2007)
discuss optimal pricing to screen heterogeneous consumers whose values are revealed over time. In
these papers, in contrast to our model, inventory (or capacity) is either innite, exogenously set,
or xed throughout the selling season, and hence issues of inventory replenishment after receiving
updated demand information are not considered. An exception is Prasad et al. (2010), who analyze
a newsvendor selling to consumers who may or may not know their valuations and show that the
optimality of advance selling (selling to consumers before they know their valuations) depends on
operational measures such as the degree of demand uncertainty.
Finally, a few recent papers bridge both streams of research by considering the impact of con-
sumer purchasing behavior on the value of quick response. In Cachon and Swinney (2009) and
Cachon and Swinney (2011), consumers may delay purchasing in order to obtain the product at
a lower price during an end-of-season clearance sale. Li and Zhang (2010) analyze the related
problem of accepting pre-orders to generate early demand information (and hence improve forecast
accuracy to better match supply and demand). In all three papers, consumers have known value
for the product in question and potentially delay a purchase to pay a lower price; by contrast,
our model focuses on valuation uncertainty as the driving mechanism of strategic consumer behav-
ior. To summarize, our model is the rst, to our knowledge, which considers the interaction of
strategic consumer purchasing behavior with the value of quick response production practices when
consumers learn about product value over time.
5
3 Model
3.1 The Firm and Consumers
A rm sells a single product at an exogenous price j to a consumer population of size over a single
selling season. There are two potential production opportunities for the rm: early production (far
in advance of the selling season) and late production (very close to the start of the season). Early
production is far enough in advance of the season that market size is unknown, though the rm
does possess some forecast of demand; thus, during the early production opportunity, is assumed
to be a random variable with positive support, distribution function 1 () and density 1 (). The
late production opportunity is close enough to the start of the selling season that market size is
known perfectly.
1
Production during the early opportunity incurs a unit cost c
1
, while production
during the late opportunity incurs a higher unit cost c
2
_ c
1
due to, e.g., expedited production and
shipping costs. Production at either point in time is assumed to be uncapacitated, and production
during the late opportunity is assumed to have a short enough leadtime that all units arrive before
the start of the selling season.
The rm thus operates in one of two potential regimes: the single procurement regime (SP) or
the quick response regime (QR). In the single procurement regime, all production occurs during
the early production opportunity, while in the quick response regime, production may occur at
both times. Excess inventory remaining at the end of the selling season has zero value. In both
operating regimes, we denote the early production quantity by c (the late production quantity in
the QR regime is assumed to be the prot-maximizing quantity), and the rm chooses production
levels to maximize total expected prot.
While the rm faces market size uncertainty, consumers initially face uncertainty about their
own private valuations for the product. Nature moves rst (prior to the start of the game) and
decides the type of each consumer: a fraction 0 of the population has positive value j for the
item, while a fraction 1 0 has zero value, where 0 and are common knowledge. If a consumer
possesses value for the product, we refer to her as a high type consumer, whereas if she possesses
1
In reality, forecast updating and renement may be the the result of an endogenous process that may continue
even during the selling season, e.g., monitoring early sales and imputing total demand, or performing market research.
To avoid issues outside the scope of this analysise.g., demand estimation based on stochastic arrivalswe assume
that the revelation of ` is exogenous and perfect and occurs just prior to the start of the season.
6
High Value Low Value
High Signal 0c (1 0) (1 c)
Low Signal 0 (1 c) (1 0) c
Table 1. The four possible combinations of signal and consumer value, and the probability of each for a
given signal strength.
zero value for the product, we refer to her as a low type consumer.
At the start of the selling season, consumers do not know their private valuation for the product
(their type). At a random time during the selling season (i.e., uniformly distributed throughout
the season), each consumer exogenously learns her value for the product (via, for instance, product
reviews from professionals and other consumers, experiences with demonstration units in the store,
etc.). While consumers do not know their individual valuations at the start of the selling season,
they are not completely ignorant: each consumer receives a noisy private signal that is an indication
of her type. We dene c to be the quality of the signal, i.e., the probability that the signal is
correct. For example, a high type consumer receives a signal of high product value with probability
c, and a low type consumer receives a signal of low product value with the same probability. Thus,
there are four possible consumer segments (corresponding to pairings of the two possible signals
and the two possible values), summarized in Table 1.
Consumers are heterogeneous in the quality of their private signals in the sense that c is
distributed among the population (independently of consumer type) according to the continuous
distribution G() and density o () with support on the interval (12. 1). Such heterogeneity in the
quality of the signal may represent, for example, domain expertise of the population in the product
category (e.g., some consumers are highly technical and capable of accurately judging the quality
of a new, high tech product, while some less sophisticated consumers receive more noisy signals
that leave them less sure of product value). Thus, the total number of consumers in each segment
depicted in Table 1 is found by integrating the probabilities in that table over the distribution of
signal strengths. The resulting distribution of consumer segments arising from this information
structure is summarized in Table 2.
After receiving their private signals, consumers arrive at the rm at the start of the selling
season. Each consumer updates her beliefs of product value via Bayes rule and calculates the
expected utility of purchasing early (before knowing product value) and the expected utility of
7
Signal Consumer Type Fraction of Population
High Value High Value (Correct Signal) 0

1
1/2
co (c) dc
Low Value (Incorrect Signal) (1 0)

1
1/2
(1 c) o (c) dc
Low Value High Value (Incorrect Signal) 0

1
1/2
(1 c) o (c) dc
Low Value (Correct Signal) (1 0)

1
1/2
co (c) dc
Table 2. A summary of the distribution of signals and consumer types amongst the population.
delaying her purchase until she learns the value of the product, based on her private signal and
individual signal strength. In order to evaluate the expected surplus of delaying a purchase, a
consumer must also consider the probability that she will be able to obtain a unit at some later
point in the selling season, i.e., the consumer must form a belief about the ll rate, denoted

c.
(Further details of this belief will be discussed in in the next section.)
After consumers learn their value, they purchase if and only if they have positive surplus and the
product is in-stock, and any consumer who does not obtain a unit receives zero surplus. Consumers
are risk-neutral expected utility maximizers who choose the purchasing strategy (before or after
learning product value) that maximizes their total expected surplus (expected product value minus
purchase price). We assume that customers who are indierent between the two strategies purchase
before learning product value. To summarize, each consumer knows:
1. Her private signal of product value (high or low) and her individual signal strength c;
2. The common valuation distribution and its parameters (i.e., that a fraction 0 of the population
has value );
3. The purchase price j;
4. Her belief about the future availability of the product,

c.
To simultaneously model both strategic (forward-looking) and non-strategic (myopic) cus-
tomers, we introduce a parameter c 0. 1 that is analogous to a discount factor: if c = 0,
customers do not anticipate the opportunity to purchase after learning product value, while if
c = 1, they do.
8
3.2 The Consumer Decision: Wait or Buy?
We now analyze the consumer decision: whether to wait or buy. In analyzing the consumer
decision, the relevant unit of analysis is a consumer who arrives at the start of the selling season,
nds a unit in-stock,
2
and considers purchasing the product immediately (which ensures that a unit
will be obtained, but not that value will be high) or delaying the purchase decision until she learns
her valuation (which ensures that the consumer will only purchase if she has high value for the
product, but does not ensure that she will successfully obtain a unit).
3
The expected surplus of
an immediate purchase is
c
(c) j, where
c
(c) is the posterior probability that the consumer
has high value for the product, conditional on a signal : |. / (i.e., low or high value) and signal
strength c. For a consumer receiving a high value signal, this posterior probability is

I
(c) =
Pr (High Type and High Signal)
Pr (High Signal)
=
c0
c0 + (1 c) (1 0)
. (1)
Note that
I
(c) is increasing in c. Similarly, if the consumer receives a signal indicating that
the product is low value, the posterior probability is
|
(c) =
(1c)0
(1c)0+c(10)
. Note that
|
(c) is
decreasing in c. If
|
(c) j 0 for some c, consumers receiving a low signal may receive positive
surplus from an early purchase, whereas if
|
(c) j < 0, all low signal consumers receive negative
surplus. In the following analysis, we assume that the latter case holds for all c.
4
Due to this
assumption, all consumers receiving a low signal have negative expected surplus from purchasing
before learning their valuation. It follows that all such consumers will delay purchasing until
after learning their valuations, and only those consumers who receive a high signal will consider
2
If any consumer nds the rm out-of-stock, the game is essentially over; due to our assumption that the rms
QR order arrives prior to the start of the selling season, if a consumer nds the rm out-of-stock, all subsequent
consumers will as well, regardless of the operating regime.
3
Technically, the consumer chooses between purchasing before learning her value and after learning her value, both
of which could potentially be at any time during the selling season. However, conditional that a consumer decides
to purchase before learning her value, the optimal time to purchase is immediately at the start of the season (as this
minimizes the risk of a stock-out). Similarly, conditional on purchasing after learning product value, the optimal
purchase time is at the moment she realizes her value for the product, as this too minimizes the risk of a stock-out.
Hence, the consumer eectively chooses between an immediate purchase at the start of the season and a purchase
at the moment she learns her valuation. Note that subgame perfection of the consumer strategy is not an issue, as
consumers do not observe inventory directly and hence cannot update beliefs about demand, supply, or ll rates as
the season progresses.
4
Equivalently, 0 j < 0, i.e., a customer with a completely non-informative signal will not purchase the product
before learning its value. This assumption allows us to ignore customers who a receive a low value signal in all
further equilibrium discussion, as their dominant action is to delay purchasing. If we relax this assumption, we must
account for low signal customers in each equilibrium, but the qualitative eects of the model remain unchanged.
9
a purchase prior to learning their valuations. The expected surplus for a high signal consumer
from an early purchase is
I
(c) j, while the expected surplus from a delayed purchase is
c

c
I
(c) ( j). Note that if
I
(c) = 1 early surplus is greater than late surplus (since c.

c _ 1),
whereas if
I
(c) = 0 late surplus is strictly greater than early surplus. It intuitively follows
that consumers purchase early if
I
(c) is high, a fact that leads to our rst preliminary result
characterizing consumer actions in any possible equilibrium in which consumers have identical
beliefs about the ll rate:
Lemma 1 In any equilibrium with identical consumer beliefs

c there exists a unique critical c

such that all consumers who receive a high value signal and have c _ c

purchase before learning

product value while all consumers with c < c

wait until after learning product value.

Proof. All proofs appear in the appendix.
Lemma 1 shows that, in any equilibrium (the precise form of which we have not yet specied
beyond requiring identical beliefs

c), consumers who receive a signal of high product value and
who have high signal quality (accurately judge product value) will purchase before learning their
valuations, while consumers who have low signal quality (poorly judge product value) will delay
until after learning product value. Thus, we may characterize the equilibrium behavior of the
consumer population by a single parameter, the critical signal strength c

. In equilibrium, the
critical signal strength is determined by calculating the surplus from an immediate purchase by
a consumer who arrives at the store and nds a unit in-stock and equating that surplus with the
expected surplus of delaying the purchase until learning product value, solving for c and yielding
c

=
(1 0) j
(1 0) j + 0 ( j)

1 c

c
. (2)
4 The Inventory-Purchase Timing Game
Consumers and the rm thus take part in a game: consumers choose when to purchase (either
before or after learning product value) and the rm chooses how much inventory to produce (either
in the single early production opportunity in the SP regime, or in both production opportunities in
the QR regime). We rst analyze the SP and QR regimes separately, then consider rm prot in
10
Figure 1. Sequence of events with quick response.
each scenario to determine rm preference between the two operational capabilities. The sequence
of events in the quick response regime is summarized in Figure 1. The sequence in the SP regime
is identical, except there is no second production opportunity.
We assume that the rm cannot credibly commit to an inventory level; that is, consumers do
not directly observe the inventory level of the rm prior to making their purchasing decisions.
This is a typical assumption (Su and Zhang 2008, Cachon and Swinney 2009) reective of the fact
that precise inventory information is often obscured from common consumers and, moreover, it
is dicult for the rm to credibly convey information about inventory (e.g., the rm always has
incentive to tell consumers there is less inventory than actually is available in order to engender a
sense of scarcity). Similarly, we assume that the consumer population cannot credibly commit to
a critical signal strength that determines equilibrium purchase timing, hence the rm must form
beliefs about the critical signal strength (which we label c) and make optimal inventory decisions
given these beliefs. Such beliefs may derive from past experience with similar products, from
marketing research, or from a detailed understanding of the consumer valuation structure. From
a modeling perspective, this means the game is one of simultaneous moves between the consumer
population and the rm. In other words, consumers optimally time their purchases given a xed
belief about inventory availability (

c) while the rm optimizes inventory given a xed belief about

consumer purchase timing ( c), and the simultaneous solution to these optimization problems forms
the Nash equilibrium to the game.
As a nal step, we assume that both rm and consumer expectations are rational, i.e., consistent
with the equilibrium outcomes (Su and Zhang 2008; Cachon and Swinney 2009; Huang and Van
11
Mieghem 2009). Rational expectations are a result of both consumers and the rm possessing
beliefs that do not systematically deviate from reality; more discussion of this assumption follows
below. The equilibrium will thus be characterized by values of c (the rms inventory level)
and c (the critical signal strength of the consumer population). Let the superscript + denote a
generic equilibrium parameter (replacing + with :j or cr when referring specically to the single
procurement or quick response case). We then formally dene the equilibrium as follows:
Denition 1 A Nash equilibrium (c

. c

) with rational expectations to the game between the rm

and the consumer population satises:
1. The rm chooses an initial inventory level c

(and, in the QR regime, a second inventory

procurement) to maximize total expected prot, conditional on beliefs about consumer behavior,
c;
2. The consumer population determines the critical signal strength c

, conditional on beliefs
about product availability

c;
3. Firm beliefs are rational, i.e., c = c

.
4. Consumer beliefs are rational, i.e.,

c = c(c

. c

), where c(c. c) is the ll rate given initial

inventory c and critical signal strength c.
We emphasize here that while we have explicitly modeled beliefs and imposed rationality on
those beliefs, the end result is identical to the Nash equilibrium of a simultaneous move game with
full information. This also implies that we implicitly assume consumers are aware of the operating
regime of the rm (either single procurement or quick response) as, in general, these two regimes will
have dierent equilibrium ll rates. This assumption is motivated by several considerations. First,
our primary focus in this paper is on how the value of quick response is impacted by consumers
responding optimally to this practice. This is not to say that its impossible for rms to mask
operational capabilities from consumers, but rather that the regime of interest for this analysis is
precisely the scenario in which consumers do correctly infer and optimally respond to the rms
operational capabilities.
Second, while the precise manner in which consumers learn about rm operational capabilities
is outside the scope of this paper, quick response has tangible outcomes for consumers (e.g., the
12
impact on product availability) and one might easily imagine that consumers repeatedly patronizing
rms may learn over time that Firm A has a greater inventory availability than Firm B (i.e., Firm
A uses quick response while Firm B does not). Thus it is reasonable to assume that, even if a
consumer does not explicitly know that a rm uses quick response per se, the consumer becomes
aware of the consequences of this strategy by observing measures tangible to her (like ll rates).
5
In addition to anecdotal evidence, recent empirical studies suggest that consumersor at least,
some fraction of any particular consumer populationare both forward-looking and capable of
developing rational expectations thereby correctly inferring future rm actions, even if such actions
are probabilistic. Israel (2005), using data from the automotive insurance industry, estimates that
about 20% of individuals are forward-looking and form rational expectations of future insurance
prices. Osadchiy and Bendoly (2010), in an experimental setting, determine that about 38% of
subjects are forward-looking, and the extent of strategic behavior increases as more information
about future probabilistic events is given to consumers. Soysal (2008) assumes that consumers
do form rational expectations of prices and in inventory availability in a fashion apparel setting,
and then uses structural estimation to derive a demand model and determine that expectations of
future inventory availability plays a signicant role in current period sales. Chevalier and Goolsbee
(2009), in an empirical investigation of the college textbook market, nd strong support that
consumers are capable of forming rational expectations of the probability that a new edition of a
textbook is released (which impacts the resale value of the current edition of a text). While none
of these papers empirically demonstrates that consumers form rational expectations in precisely
our setting,
6
the results do provide evidence that consumers are both forward looking and capable
of forming such expectations in general, be it regarding price (Israel 2005, Osadichy and Bendoly
2010), inventory availability during clearance sales (Soysal 2008), or other probabilistic rm actions
(such as introducing new products, Chevalier and Goolsbee 2009).
Third, a number of papers in the literature employ similar assumptions in which consumers
5
For instance, consumers have come to expect that video game manufacturer Nintendo is incapable of rapid
inventory replenishment to meet demand (Richtel 2007) and hence future availability is low. On the other hand,
consumers have come to expect that General Motors will satisfy demand on hit products and hence future availability
is high, a belief that GM is now actively trying to change (Stoll 2007). More formally, see Su and Zhang (2009) and
Kalai and Lehrer (1993) for a discussion and analysis of how consumer learning over time in a similar setting can
converge to the equilibrium of a single-shot game with rational expectations.
6
Indeed, an interesting direction for future research would be to empirically verify whether consumers form rational
expectations of rm operational capabilities such as quick response.
13
are aware of high-level rm strategy (quick response or no quick response; display all inventory or
display one unit at a time; obtain advance demand information or not) while consumers cannot
directly observe low-level tactical decisions like precise inventory amounts. In these models, and
in ours, the intent is to consider the equilibrium consumer response to an operational practice once
consumers have become aware of, and reacted to, that practice.
5 Equilibrium and the Value of Quick Response
Having dened the equilibrium that we will analyze, we now proceed to calculate the equilibrium
and explore the value of quick response in the next section. We must rst prove that an equilibrium
to the game exists (and that such an equilibrium is unique) in order to discuss its properties; the
following lemma accomplishes this for the SP regime.
Lemma 2 When the rm operates in the single procurement regime, an equilibrium (c
cj
. c
cj
) exists
and is unique. The equilibrium total demand to the rm is
1 =

0 + (1 0)

1
c
sp
(1 r) o (r) dr

. (3)
From (3), the equilibrium demand of the rm is decreasing in c
cj
. It is apparent, then, that
the rm prefers more consumers to purchase early as this increases total demand. This result is
sometimes referred to as the advance selling phenomenonsee Xie and Shugan (2001)in which a
rm exploits consumer valuation uncertainty by inducing some consumers to purchase the product
before learning their value that will ultimately be dissatised (have low valuation).
We next move to the game in which the rm operates in the QR regime. Recall that when
determining the number of units to produce using quick response, the rm chooses an inventory
level that maximizes total prot. As a result, if the rm has quick response capabilities, the
optimal action is to procure enough inventory in the quick response order to satisfy all demand,
conditional on a xed belief about consumer behavior (xed c). Because rm expectations about
c are rational, this means the rm procures enough inventory to satisfy all demand it receives, and
hence consumers believe that the ll rate at that rm is equal to 1; after learning the true value
of demand, the rm cannot credibly commit to satisfying anything less than the total demand it
14
receives. Quick response thus increases the expected surplus of consumers who delay their purchase
by increasing the expected ll rate, and so strengthens the incentive for consumers to wait. All
else being equal, this will shift demand to later times, which will in turn decrease the amount of
advance selling that occurs.
The story does not end with the eect of quick response on consumer behavior, however; QR
also oers value by better matching supply and demand under uncertainty. Thus, it remains to be
seen how QR aects the prot of the rm in equilibrium. Before we answer this question, we must
rst demonstrate that an equilibrium exists and is unique when the rm operates in the QR regime.
The following lemma does this, in addition to comparing the equilibrium outcomes (critical signal
strength and inventory level) to the single procurement regime.
Lemma 3 When the rm operates in the quick response regime, an equilibrium (c
ov
. c
ov
) exists
and is unique. In equilibrium, more consumers delay their purchases (c
cj
_ c
ov
) and the rm sets
a lower inventory level (c
ov
_ c
cj
) than in the single procurement regime.
Having demonstrated that equilibria exist and are unique in both regimes, we may now address
the value of quick response: the incremental increase in prot due to the adoption of a quick
response system. Our rst result demonstrates how the value of quick response is aected by
strategic customer behavior:
Theorem 1 (i) The incremental equilibrium value of quick response (
ov

cj
) is smaller if con-
sumers are strategic (c = 1) than if they are non-strategic (c = 0).
(ii)The incremental equilibrium value of quick response (
ov

cj
) is strictly decreasing in the
cost of quick response (c
2
), and if c
2
= j,
ov
_
cj
.
In other words, Part (i) of Theorem 1 shows that quick response yields less value to the rm
when consumers are strategic than when they are non-strategic. This is because strategic behavior
by consumers reduces the total demand to the rm: when customers are strategic (c = 1) all
individuals intentionally delay their purchase, and inevitably some of these customers will not buy
the product once they learn their valuation. As a result, the value of matching supply and demand
is lower (there is less potential demand to match).
15
Figure 2. The incremental value of quick response (
qr

sp
) as a function of the cost of an expedited
procurement (c
2
) when c = 1, separated into component factors. Matching supply and demand provides
positive value while shifting demand provides negative value.
A natural question to ask is: how much is the value of quick response reduced by strategic
behavior? Can it ever be negative? Part (ii) of Theorem 1 addresses this question, yielding a
surprising result: quick response may reduce the prot of the rm even if the marginal procurement
cost is strictly less than the selling price. This stands in contrast to the existing literature on
quick response: with non-strategic consumers (e.g., Fisher and Raman 1996) or with strategic
consumers in the absence of learning (Cachon and Swinney 2009), quick response always provides
non-negative value if the margin on a unit procured using quick response is weakly positive (i.e.,
if c
2
_ j). Theorem 1 shows that this need not be the case when consumers learn about their
valuations over time: it is possible for quick response to yield a positive margin on each unit sold
while simultaneously yielding lower expected prot to the rm than the single procurement regime.
The key to both theorems lies in the dual eects of quick response: shifting demand and
matching supply with demand. These two eects pull the equilibrium prot of the rm in opposite
directions. Shifting demand (from early purchases to later purchases) reduces prots by decreasing
the amount of advance selling. Matching supply with demand increases prots by eliminating lost
salesall demand is captured, albeit at a higher unit procurement costand reducing the chance of
overstock. Hence, the rm only values quick response so long as the cost of shifting demand is
exceeded by the gain from better matching supply with demand; see Figure 2.
7
7
In Figure 2 and all other graphical examples, = 18, j = 10, c1 = 5, 0 = 0.75, ` is gamma distributed with
mean 10 and standard deviation 5, and c follows a beta distribution with both parameters equal to 5 condensed to
lie in the interval (12. 1).
16
Figure 3. The incremental value of quick response (
qr

sp
) as a function of the cost of an expedited
procurement (c
2
) when c = 0. Compared to Figure 2, in which c = 1, all the curves are shifted upwards.
Theorem 1 demonstrates that the value of both eects is higher when consumers are non-
strategic (c = 0) than when they are strategic (c = 1). When consumers are non-strategic, the
demand shifting eect is eliminated. Furthermore, total demand to the rm is higher, so the value
of matching supply and demandfor any given c
2
is higher than when consumers are strategic.
Thus, when c = 0, all three curves depicted in Figure 2 are higher, as Figure 3 demonstrates.
While we have shown that the value of quick response is lower if consumers are strategic and
learn about product value over time, this is not to say that quick response is always harmful to
the rm in this setting. As Theorem 1 and Figure 2 demonstrate, quick response can increase
the protability of the rm if, all else being equal, c
2
is small enough. Nevertheless, a result of
Theorem 1 is that it may be in the best interests of the rm to forgo quick response tactics and
the option to procure additional inventory, and further to ensure that consumers are aware of this
operating regime. Particularly in light of additional xed costs that inevitably accompany the
adoption of any quick response system (e.g., shipping and fulllment infrastructure, IT systems,
and production capacity or reservation costs), it is clear that the rm is less likely to benet from
a quick response system when customers are strategic and learn about product value over time.
This relates, in part, to the rationing risk results in the literature on strategic consumer purchas-
ing. In contrast to the mere reduction of inventory described in this literature, Theorem 1 implies
that the rm may be better o with an entirely dierent operating policy (Single Procurement
vs. Quick Response) when consumers are strategicby operating without quick response, the rms
17
inability to react to updated demand information in a timely and responsive way can benet the
rm by generating a credible mismatch between supply and demand and inducing more consumers
to purchase prior to learning their value.
6 Consumer Returns
The preceding analysis assumed that a consumer who purchased an item early had no recourse if
her value for that item turned out to be lowthat is, the possibility that a consumer could return
a product if she is dissatised was excluded. In some industries, this assumption is appropriate.
For example, with most types of media (e.g., movies, music, video games, or computer software)
returns are forbidden once an item has been opened (often due to fears of piracy), and Amazon.com
does not allow returns on large televisions due to the logistical challenges of return shipping. In
some cases, however, product returns are a common and important component of rm strategy.
Satisfaction guarantees abound in many settings (clothing, electronics, etc.), with rms encouraging
customers to try new products risk free while promoting generous return policies.
8
Such policies
increase the consumer incentive to purchase early by reducing the consequences of buying a product
which is not valued. Returns policies have received attention in the literature: see, for example,
Davis et al. (1995), Su (2009), Liu and Xiao (2008), and Schulman et al. (2009). These papers
do not consider the impact of consumer returns policies on a rms incentives to adopt a quick
response strategy, however.
We assume that returns are allowed throughout the selling season, and each return is for a full
refund minus a consumer restocking fee, r
c
_ 0 (i.e., the net refund is j r
c
). We present our
results for general r
c
to include the case in which the restocking fee is established by the norm of the
industry (e.g., no fee may be customary for competitive reasons), and we discuss the rms choice
of optimal restocking fee below. Returns occur immediately after a consumer who purchased early
learns her valuation (e.g., uniformly throughout the selling season). We assume that returned
products are resalablethat is, the rm may immediately repackage and resell any returns that it
receives. Furthermore, we assume that any consumer who wishes to purchase and nds the rm
out-of-stock costlessly waits to see if any returned products become available to purchase during
8
At both Amazon.com and the electronics retailer Best Buy, for example, returns are allowed for full refunds on
most items within a 30 day period; during the holidays this return window is extended up to a maximum of 90 days.
18
the selling season.
Consumers who make a return incur a hassle cost / _ 0 deriving from, for instance, the travel
cost of returning to a store. Returns are also costly to the rm, incurring an internal rm restocking
fee of r
;
_ 0 on each returned item (due to, for example, repackaging costs or the cost of employee
time). We assume that j / r
c
_ 0, i.e., a dissatised consumer benets from a return. This
implies that if
I
(c) j _ 0, then
I
(c) j + (1
I
(c)) (j / r
c
) _
I
(c) j _ 0,
i.e., with returns, high signal consumers have greater incentive to purchase early than without
returns. We assume also that returns are enough of a hassle (/+r
c
is large enough) that low signal
consumers still do not purchase before learning their valuations.
9
We are interested in how the addition of the described return policy changes the results of
5, specically the results provided in Theorem 1. By increasing expected surplus from an early
purchase, returns encourage more consumers to purchase before learning their values. While this
would seem to benet the rm, the increase in advance purchasing comes at a price: consumers
who purchase early and are dissatised can be costly to the rm, due to the fact that each returned
unit costs the rm the price of the refund minus the charged consumer restocking fee, j r
c
,
and the internal rm restocking fee, r
;
. Thus, the value of quick response practiceswhich as we
have already mentioned shift demand by lessening the availability risk associated with delaying a
purchasewill depend upon the magnitude of these restocking fees, as the following theorem shows.
Theorem 2 If consumer returns are allowed:
(i) If r
;
_ r
c
, equilibrium rm prot (in either regime) and the incremental value of quick
response are greater with strategic customers than with myopic customers, and the incremental
value of quick response is always positive.
(ii) Otherwise (if r
;
< r
c
), equilibrium rm prot (in either regime) and the incremental value of
quick response are greater with myopic customers than with strategic customers, and the incremental
value of quick response may be positive or negative
The preceding theorem yields several intriguing results. First, the theorem shows that under
consumer returns, if r
;
_ r
c
, rm prot in either regime is greater if customers exhibit strategic
behavior than if they are non-strategic. The key to this result lies in the fact that, if r
;
_ r
c
,
9
Specically, this implies 0 j + (1 0) (j / vc) < 0.
19
returns (a) are costly to the rm on a marginal basis and (b) ensure that no consumer who doesnt
value the product receives the product, thereby eliminating the value of advance selling eect and
guaranteeing that rm demand (net of returns) is always 0 regardless of the value of c. Thus,
there is no benet to selling a unit to a consumer who ultimately possesses low value for the
product; on the contrary, this is costly to the rm because of the restocking costs. The rm seeks
to minimize the number returns, and the number of returns is lower when consumers are strategic
(and hence wait to learn about product value before purchasing) than when they are non-strategic
(blindly purchasing before knowing their real valuation, only to return the item later). If, on the
other hand, r
;
< r
c
, then the rm charges customers more for a return than its own internal costs
associated with a return; in this case, the rm prots from each individual return and so, just as in
the model without consumer returns, prefers if customers purchase before learning their valuations.
Consequently, the rm prefers a non-strategic customer population that is more apt to purchase
early.
Theorem 2 also shows that if r
;
_ r
c
, quick response always increases rm prot. Just as in
part (i) of the theorem, the rm benets from minimizing the number of costly returnshence, the
tendency of quick response to shift demand also increases rm prot. When r
;
< r
c
, however,
this may or may not be the case; just as in the model without returns, the rm is hurt by demand
shifting as it reduces advance selling and protable returns. Finally, Theorem 2 shows that if
r
;
_ r
c
the result of Theorem 1 is reversed: the value of quick response is greater if customers
exhibit strategic behavior than if they are non-strategic. Intuitively, the ability of a quick response
system to induce demand shifting (which is protable if r
;
_ r
c
) is most eective when consumers
are strategic (indeed, when consumers are completely non-strategic, quick response induces no
demand shifting at all). Hence, the value of quick response is greatest under forward-looking
customer behavior. Alternatively, when r
;
< r
c
, we again have a result similar to Theorem 1:
quick response is less valuable when customers are strategic because it generates demand shifting
and causes the rm to lose protable returns.
The results of Theorem 2 are due to the inclination of consumers to hoard inventory: given that
returns are possible, a consumer would rather purchase an item early and run the risk of having
to return the product, as opposed to delaying the purchase and risking a stock-out. Two ways
to reduce hoarding are to increase availability (e.g., adopt quick response) and make consumers
20
strategic (increase c from 0 to 1). If r
;
_ r
c
, then hoarding is costly to the rm and so both
strategic behavior and quick response help to minimize this negative behavior. This implies that
if Figure 2 were plotted for the case of costly returns (r
;
_ r
c
), the demand shifting portion of the
graph would have positive value.
Lastly, consider the scenario if the rm is capable of choosing whether to oer returns and
may set the consumer restocking fee r
c
to maximize prot. Given our assumptions, the optimal
consumer restocking fee is r

c
= j /, i.e., the greatest possible restocking fee which will induce
consumers to return the product. The rm will clearly not oer returns if r

c
< r
;
because returns
are individually costly and also result in a decrease in total sales. Thus, part (i) of Theorem 2
cannot hold if the rm can choose whether to oer returns, because clearly the rm will not oer
returns if they are costly.
10
The rm may oer returns if r

c
r
;
, in which case individual returns
are protable and part (ii) of the theorem holds. In either case, if the rm can chose whether
and how much to charge for returns, the model with consumer returns mirrors our base model,
supporting all of our original results.
The fact that in some cases strategic customer behavior can be good for the rm (and for the
value of quick response) runs contrary to the vast majority of the strategic consumer literature.
This is because, in our model, forward-looking behavior results in actions that benet customers
(due to the avoidance of hassle costs and consumer return fees) and the rm (due to the avoidance
of internal rm restocking costs). Thus, our model demonstrates how the interaction of two eects
consumer learning and costly product returnscan lead the rm to benet from both quick response
practices and a very strategic customer population.
7 Pricing
In this section, we endogenize pricing in our original model and address how the value of quick
response is aected. We consider two types of pricing: xed pricing (in which the retailer sets
a single price for the entire selling season) and introductory pricing (in which the retailer may
set a dierent price during the initialor introductoryrelease of the product, e.g., when consumer
10
Nevertheless, its important to keep in mind that in practice rms may oer returns policies even if returns are
individually costly; in many industries (e.g., retailing) the vast majority of returns are for full (or nearly full) refunds
due to competitive pressure, and are subsequently costly to rmssee Stock et al. (2006) for a discussion of how rms
actively attempt to minimize returns. If this is the case, part (i) of the theorem holds.
21
valuations are still unknown). Unlike the inventory level, price is directly observed by consumers,
and hence the rm acts as a leader in the price game. Thus, the model with xed pricing entails
a rst stage in which the rm sets the (constant) selling price, and a second stage which behaves
identically to the games analyzed in 35. As a result, given a particular price, the previous
results continue to hold (notably the equilibrium existence results) in the second stage of the game,
and we need only analyze the rms choice of the selling price by comparing expected prots in the
inventory/purchasing subgames using various price levels. The following theorem conrms that the
result of Theorem 1quick response may decrease rm protcontinues to hold even when the rm
may set a (constant) price level. In what follows, we use the subscript 1j to denote equilibrium
values (prots, quantities, signal strengths) in a model with xed endogenous pricing.
Theorem 3 The incremental equilibrium value of quick response with xed pricing is strictly de-
creasing in the cost of quick response (c
2
), and if c
2
= ,
ov
;j
_
cj
;j
.
The key to this result is the following: when prices are xed across time, regardless of the
optimal price level, adopting quick response increases the consumer incentive to wait and hence
decreases advance selling and rm prot. The freedom to set the price is of little value in the quick
response regime when c
2
is large, as the rms optimal price lies in the interval [c
2
. ]if the the price
is lower than c
2
, then quick response is never used, hence the rm essentially moves to the single
procurement regime. In the single procurement regime, the rm remains free to price anywhere
in the interval [c
1
. ]. When the cost of quick response is large, the quick response regime has two
detrimental eects to the rm: pricing is constrained and more consumers delay purchasing due to
higher availability. As a result, the single procurement regime becomes even more attractive than
in the exogenous price case. Thus, Theorem 3 mirrors the result of Theorem 1: it is possible for
quick response to decrease prot even when the margin is positive (c
2
_ j _ ).
In the introductory pricing case, we assume that the rm charges two dierent prices: an
introductory price and a regular price. The introductory price is valid only at the start of the selling
season (e.g., the rst week, or for pre-orders) when consumers make their initial decision on when
to purchase, while the regular price is valid thereafter. Consumers develop rational expectations of
future pricesthat is, they correctly anticipate the regular price (or, equivalently, the rm credibly
announces the regular price along with the introductory price). We rst note that if the rm
22
is free to set dierent prices but is constrained only to mark prices down over time, Theorem 3
continues to hold.
11
If the rm can raise prices over time, however, a dierent picture emerges.
Let j
1
and j
2
be the introductory price and the regular price, respectively. Note that the optimal
regular price is j
2
= ; all consumers know their values when purchasing at the regular price, and
possess values equal to or 0 for the product. Hence, the rm extracts all surplus from consumers
purchasing after learning the products value by charging the valuation of the high type consumers.
Consequently, all consumers have zero surplus from delaying a purchase (both high and low types,
regardless of whether they successfully procure a unit), and all consumers with positive expected
surplus from an early purchase will choose to buy before learning their valuations. In general,
the optimal introductory price satises j
1
_ , i.e., the rm charges a lower introductory price to
induce some advance selling among consumers.
Because all consumers have identically zero surplus from a delayed purchase, if the rm adopts
quick response and raises the consumer expectation of product availability (

c), the rm does not

raise the expected surplus to any consumers from a delayed purchase. Thus, quick response
no longer shifts demandthe only eect remaining is matching supply and demand, hence quick
response always has positive value. The following theorem summarizes this result.
Theorem 4 The incremental equilibrium value of quick response with introductory pricing is al-
ways positive if c
2
_ .
The key to Theorem 4 is that increasing prices over time provides consumers with greater
incentive to purchase early, shifting demand from later purchases to the earlier purchases. This
eect counteracts the tendency of quick response to shift demand in the opposite direction. Thus,
introductory pricing and quick response are complimentary in the sense that they enhance one
anothers value: increasing prices reduces costly demand shifting due to quick response, and quick
response eliminates costly supply/demand mismatches (mismatches which are particularly costly
under introductory pricing due to the higher regular price).
Due to the assumption that consumer values follow a two point distribution, introductory
pricing in the present model completely eliminates strategic waiting in the sense that all consumers
11
It is never optimal in the current model to set an introductory price that is higher than the regular pricethe
lower regular price would only encourage more consumers to delay purchasing and hence decrease the amount of
advance selling. Thus, a rm constrained to mark down over time chooses to set a constant price, and the model
reduces to the xed pricing case analyzed above.
23
receive zero surplus from a delayed purchase and hence consumers purchase early if and only if
they have positive expected surplus (e.g., as if they were non-strategic). Should consumers have
more than one positive valuation, in general introductory pricing will not eliminate all strategic
waiting. In that case, the adoption of quick response once again shifts demand to later times and
decreases advance selling; nevertheless, increasing prices over time continues to reduce the amount
of strategic waiting that occurs and hence minimizes the negative aspects of demand shifting due
to quick response. Thus, while strategic waiting will not in general be eliminated by adopting
introductory pricing if consumers have a more complicated valuation distribution, it will be reduced
by introductory pricing, a fact which increases the value of quick response relative to the xed
pricing case.
8 Discussion
Quick response systemsor, more generally, leadtime reduction and rapid inventory replenishment
are often suggested as potential panaceas to the ill eects of supply and demand mismatches. In
this paper, we show that such strategies are less valuable to the rm when consumers are forward-
looking and have uncertain value for a product about which they learn over time. Furthermore,
even if the xed cost of implementing a quick response system is zero, it is possible that the option
to receive additional inventory after a forecast update decreases the rms prot once the consumer
response to increased availability is taken into account. In that regard, our base model represents
a worst case scenario for the rm. In this scenario, strategic customers that are aware of and
respond optimally to the rms operating strategyquick responsecan decrease, and even make
negative, the value of this operational strategy. This is due to the fact that the tangible outcome
of quick response to consumers inuences purchasing incentives in precisely the wrong way (from
the rms point of view).
12
Managerially, these results are important for three reasons. First, a rm may not wish to
12
Indeed, even if the rm can credibly commit to any arbitrary ll rate in either operating regime, it is possible
to show that the optimal action is to commit to identical ll rates in the quick response and single procurement
regimes (details available from the author upon request). In this case, consumers are ignorant to the use of quick
response, and hence the demand shifting eect is eliminated; as a result, Part (ii) of Theorem 1 no longer holds (i.e.,
the value of quick response cannot be negative) but Part (i) continues to hold (the value of quick response is reduced
by strategic consumer behavior). So even in the best case scenario in which consumers are oblivious to the use of
quick response, its value is lowered due to forward looking consumer behavior when valuations are uncertain.
24
adopt quick response in this setting, due to the reduction in value caused by the optimal consumer
response to the rms operational capabilities. Second, our model illustrates that in certain cases
the rm may wish to mask its operational capabilities from consumers, i.e., to prevent them from
reacting optimally. Third, these results show that context mattersspecically, the characteristics
of consumers and of the productwhen a rm determines whether to invest in operational exibility
and whether to publicize that investment. It is also worth noting that quick response is an oper-
ational proxy for (more generically) information. Taken in that context, our results on the value
of quick response are essentially results on the value of accurate information concerning product
demand. The fact that quick response sometimes yields negative value supports the maxim that
ignorance can be bliss; the lack of accurate information about demand can serve as a commitment
mechanism to keep inventory scarce and increase advance selling.
These results provide insight into when a rm should adopt a fast supply chain that allows
action on improved demand information. Quick response is most valuable in settings with known
or easily determined product value, while it is less valuable if the product has dicult to ascertain
or uncertain value to consumers. This supports the application of quick response in industries
such as fashion apparel (Fisher and Raman 1996, Ferdows et al 2004) over industries with more
complicated or innovative products such as electronics (Fisher 1997, Krishnan et al 2010). Quick
response can be particularly valuable if paired with introductory pricing (7) but may be less
valuable if consumer returns are allowed and are costly to the rm (6). As we have shown, the
value of matching supply and demand depends not only on the reduction of lost sales and excess
inventory, but also on the strategic response of the rms customers to increased product availability.
This response can be harmful (if advance selling decreases as a result), benecial (if costly returns
are allowed and hoarding is an issue), and even diminished or eliminated by the appropriate pricing
strategy (increasing prices over time in the optimal manner).
Acknowledgements. Many thanks to the Department Editor, Associate Editor, and two
referees, as well as Grard Cachon, Marshall Fisher, Serguei Netessine, Senthil Veeraraghavan,
Arvind Tripathi, Tava Olsen, and seminar participants at the University of Pennsylvania, the
University of Texas at Dallas, the University of Rochester, the University of Washington, New York
University, London Business School, Washington University in St. Louis, Northwestern University,
the University of Chicago, Columbia University, Duke University, Stanford University, and the
25
INFORMS Annual Meeting in Seattle for numerous comments and suggestions.
A Proofs
Proof of Lemma 1. Consumers who receive a high value signal purchase early if
I
(c) j _ 0
and if the expected surplus from purchasing early is greater than the expected surplus from delaying,
i.e., if
I
(c) j _ c

c
I
(c) ( j). Because c

c _ 1, it is true that expected surplus from a delay

is increasing in c at a slower rate than expected surplus from an immediate purchase. Furthermore,
if c = 1, then early surplus is j while late surplus is c

c( j), i.e., early surplus is weakly

greater than late surplus. If c = 12, the opposite relationship holds (from our assumption that

|
(12) j < 0). Thus, there exists some (unique) critical c

such that, for all c c

, the
inequality above is strict, while for c < c

, the inequality is violated.

Proof of Lemma 2. Given the rms belief c concerning the critical signal strength,early
demand is composed of two consumer segments: those with high value and correct signals, and
those with low value and incorrect signals. Let :
1
( c) = 0

1
b c
ro (r) dr+(1 0)

1
b c
(1 r) o (r) dr.
The total demand from these consumers is thus :
1
( c). All consumers with signal strengths
less than c delay purchasing, and only those with high value will purchase the product. Late
demand is thus consumers who have high value and received a low value signal, and consumers
who have high value , received correct signals, and chose to delay their purchase. Let :
2
( c) =
0

1
1/2
(1 r) o (r) dr + 0

b c
1/2
ro (r) dr, such that the total demand from these consumer segments
is :
2
( c). The total rm demand is thus 1 = : ( c), where : ( c) = :
1
( c) + :
2
( c). The
rms expected prot is (c) = E[j min(c. 1) c
1
c], which is a concave function of c yield-
ing an optimal inventory level satisfying Pr (1 < c) = (j c
1
) j. Substituting for 1, we see
that the best reply function is c ( c) = : ( c) 1
1

jc
1
j

. We may now derive the equilibrium

by imposing the rational expectations hypothesis, which implies c = c
cj
and

c = c(c
cj
. c
cj
).
With a random allocation rule, the actual second ll rate for any (c. c) is given by c(c. c) =
E

min

(c : (c) )
+
. : (c)

: (c)

. Substituting the rms optimal inventory level, we see

that in any equilibrium, c(c. c) =
1
1

pc
1
p

0
1 (r) dr+

1
1
1

pc
1
p

1
1

pc
1
p

a
a

1 (r) dr, which

is independent of c. Next, note that the left hand side of (2) is increasing in c with a slope of
1, and the right hand side is increasing independent of c in any equilibrium because c(c. c) is
26
independent of c in equilibrium. Hence, there clearly exists a unique c which satises (2), and
thus the equilibrium to the game is unique.
Proof of Lemma 3. Because the rm operates in the QR regime, the only rational belief of
the consumer population is that

c = 1; because the quick response procurement is subgame per-
fect, the rm will satisfy all demand. Hence, the consumer best reply is independent of any rm
actions, and is dictated by the solution to (2) with

c = 1, which implies c
ov
=
(10)j
(10)j+0(j)(1c)
. It
is follows that c
cj
_ c
ov
for any equilibrium ll rate in the single procurement regime. The rms
prot function is (c) = E[j1 c
1
c c
2
(1 c)
+
], where 1 = (:
1
+ :
2
) and :
1
and :
2
as are in
the proof of Lemma 1. It follows that the rm best reply exists and is unique, given by c (c
ov
) =

0 + (1 0)

1
c
qr
(1 r) o (r) dr

1
1

c
2
c
1
c
2

, hence the equilibrium existence and uniqueness re-

sults follow. This furthermore implies (1 0)

1
c
sp
(1 r) o (r) dr _ (1 0)

1
c
qr
(1 r) o (r) dr,
and it follows that total equilibrium demand to the rm is greater in the SP regime than in the
QR regime, yielding c
ov
_ c
cj
.
Proof of Theorem 1. (i) Let =
ov

cj
be the incremental equilibrium value of
quick response. Recall that : (c
ov
) is the equilibrium total demand in the QR regime, while
: (c
cj
) is the demand in the SP regime, where : (c

) = 0 + (1 0)

1
c

(1 r) o (r) dr and
c

=
(10)j
(10)j+0(j)(1cc

)
. From the expression for c

, c
cj
c=0
= c
ov
c=0
=
(10)j
(10)j+0(j)
_ c
cj
c=1
_ c
ov
c=1
,
where, e.g., c
cj
c=0
denotes the equilibrium critical signal strength in the SP regime when c = 0. Be-
cause :
0
(c

) < 0, :

c
ov
c=1

_ :

c
cj
c=1

_ :

c
ov
c=0

= :

c
cj
c=0

. The equilibrium rm prot is, in

the SP regime,
cj
= : (c
cj
) E

j min

. 1
1

jc
1
j

c
1
1
1

jc
1
j

, and in the QR regime,

ov
= : (c
ov
) E

j min

. 1
1

c
2
c
1
c
2

c
1
1
1

c
2
c
1
c
2

+ (j c
2
)

1
1

c
2
c
1
c
2

. In
each expression, the term inside the bracket is the maximum expected prot without strategic cus-
tomers (i.e., a traditional newsvendor and a newsvendor with quick response, respectively). Denote
the bracket term in regime 1 by 1
1
, 1 = :j. cr. Note that 1
ov
_ 1
cj
. The incremental value of
QR is thus = : (c
ov
) 1
ov
: (c
cj
) 1
cj
. When c = 0, this implies
c=0
= :

c
cj
c=0

(1
ov
1
cj
),
and when c = 1,
c=1
= :

c
ov
c=1

1
ov
:

c
cj
c=1

1
cj
_ :

c
cj
c=1

(1
ov
1
cj
) _
c=0
, which proves
the result.
(ii) Dene
ov
(c) = E

j1 c
1
c c
2
(1 c)
+

, where 1 is the total demand at the rm (a

function of c
ov
). Let
ov
be the equilibrium prot of the rm with quick response, and let
cj
be the
27
equilibrium prot without QR. Dierentiating
ov
with respect to c
2
, we have
o
qr
oc
2
=
0
qr
(o)
0c
2

o=o
qr
= Pr (1 c
ov
) = 1 +
c
2
c
1
c
2
< 0. Thus, the equilibrium prot of the rm is decreasing in c
2
. In
the limit as c
2
j, the margin on each unit sold that is procured via QR goes to zero. The rms
prot eectively becomes the same as if it did not have QR capabilities, except in equilibrium,
more consumers will delay purchasing than if the rm did not have QR. Thus, lim
c
2
!j

ov
=

cj
[
c=c
qr _
cj
[
c=c
sp.
Proof of Theorem 2. We use the subscript r to denote equilibrium values with returns.
The proofs of equilibrium existence and uniqueness are similar to Lemmas 2 and 3, and are omit-
ted. With consumer returns, any consumers who purchase early and are dissatised with the
product will return the item. Because we assume that these products are resalable, the to-
tal demand to the rm is simply 0. Thus, the expected prot (without quick response) is

cj
v
(c) = E

j0 j (0 c)
+
c
1
c (r
;
r
c
) (1 0)

1
c
sp
r
(1 r) o (r) dr

, where c
cj
v
refers
to the equilibrium critical consumer signal strength with returns, determined by equating early
purchase and late purchase surplus, yielding c
cj
v
=
(I+vc)(10)
(I+vc)(10)+0(j)(1c
b
c)
. Dierentiating
cj
v
(c),
we see
o
sp
r
(o)
oo
= j (1 1 (c0)) c
1
and
o
sp
r
(o)
oo
= j1 (c0). Hence,
cj
v
(c) is concave in c and
yields an optimal inventory level equal to c
cj
v
= 01
1

jc
1
j

. Note that the optimal inven-

tory level is independent of the critical signal strength, c
cj
v
, and as a result so is the ll rate,
which we denote c
cj
v
. Thus,
o
sp
r
oc
= (r
;
r
c
) (1 0) j(1 c
cj
v
) o (c
cj
v
)
oc
sp
r
oc
. Because
oc
sp
r
oc
=
c
cj
v
0(j)c
sp
r
(I+vc)(10)+0(j)(1cc
sp
r )
0, it follows that
o
sp
r
oc
_ 0 if r
;
_ r
c
(and
o
sp
r
oc
_ 0 if r
;
_
r
c
). Similarly, in the quick response regime, as in the case without returns, quick response in-
duces

c = 1, hence c
ov
v
=
(I+vc)(10)
(I+vc)(10)+0(j)(1c)
and c
cj
v
_ c
ov
v
for any equilibrium belief con-
cerning the ll rate in the SP regime. The expected prot with quick response is
ov
v
(c) =
E

j0 c
2
(0 c)
+
c
1
c (r
;
r
c
) (1 0)

1
c
qr
r
(1 r) o (r) dr

. Dierentiating
ov
v
(c), we
see
o
qr
r
(o)
oo
= c
2
(1 1 (c0)) c
1
and
o
sp
r
(o)
oo
= c
2
1 (c0).
ov
v
(c) is thus concave in c and yields
an optimal inventory level equal to c
ov
v
= 01
1

c
2
c
1
c
2

. Again, the optimal inventory level is

independent of the critical signal strength, c
ov
v
, and as a result so is the ll rate, c
cj
v
. As before,
o
qr
r
oc
= (r
;
r
c
) (1 0) j(1 c
ov
v
) o (c
ov
v
)
oc
qr
r
oc
, where
oc
qr
r
oc
= c
ov
v
0(j)
(I+vc)(10)+0(j)(1c)
0, hence
o
qr
r
oc
_ 0 if r
;
_ r
c
and
o
qr
r
oc
_ 0 if r
;
_ r
c
. Next, let
cj
v
= E

j0 j (0 c
cj
v
)
+
c
1
c
cj
v

such
that
cj
v
=
cj
v
(r
;
r
c
) (1 0) j

1
c
sp
r
(1 r) o (r) dr, and let
ov
v
be dened analogously such
28
that
ov
v
=
ov
v
(r
;
r
c
) (1 0) j

1
c
qr
r
(1 r) o (r) dr. Note that
cj
v
and
ov
v
are the optimal prof-
its (without and with quick response, respectively) of a newsvendor facing demand 0, hence
ov
v
_

cj
v
and both are independent of c. Thus,
ov
v

cj
v
=
ov
v

cj
v
+(r
;
r
c
) (1 0) j

c
qr
r
c
sp
r
(1 r) o (r) dr.
If r
;
_ r
c
, then clearly
ov
v
_
cj
v
. Lastly, if c = 0, then c
cj
v
= c
ov
v
and thus
ov
v

cj
v
=
ov
v

cj
v
.
If c 0 and r
;
_ r
c
, then because c
cj
v
_ c
ov
v
,
ov
v

cj
v
_
ov
v

cj
v
. If c 0 and r
;
_ r
c
, then
because c
cj
v
_ c
ov
v
,
ov
v

cj
v
_
ov
v

cj
v
.
Proof of Theorem 3. The subscript 1j denotes equilibrium values with xed endogenous
pricing. The existence of an equilibrium is immediate, due to the fact that we have already shown
an equilibrium exists to the inventory/purchasing subgames and the rms expected payos are
bounded (by 0 and E ( c
1
)) and its strategy space is a compact interval [c
1
. ] in the pricing
game ([c
2
. ] when using quick responseif price is less than c
2
but greater than c
1
, the rm will
never use QR and reverts to the SP regime). Let
ov
;j
, j
ov
;j
, and c
ov
;j
be the equilibrium prot, price,
and inventory of the rm with quick response and xed pricing, and let
cj
;j
be the equilibrium prot
without QR. Dierentiating
ov
;j
with respect to c
2
, we have
o
qr
fp
oc
2
=
0
qr
fp
0c
2
+
0
qr
fp
0j
oj
qr
fp
oc
2
+
0
qr
fp
0c
oc
qr
fp
oc
2
_
0
qr
fp
0c
2
. Observe that either
0
qr
fp
0j
= 0 (the rm prices at an interior optimum) or
oj
qr
fp
oc
2
= 0 (the rm
prices on the boundary, i.e., c
2
or ). Unlike the case without pricing,
oc
qr
fp
oc
2
in general does not
equal zero. This is due to the fact that
oj
qr
fp
oc
2
_ 0 and
oc
qr
fp
oj
_ 0in other words, higher costs of quick
response lead to higher prices (a natural result) and higher prices lead to more consumers waiting,
see equation (2). Because
0
qr
fp
0c
_ 0 (the more consumers that wait, the lower the rms prots),
it follows that the
0
qr
fp
0c
oc
qr
fp
oc
2
_ 0. Finally, since
o
qr
fp
oc
2
_
0
qr
fp
0c
2
= Pr

1 c
ov
;j

= 1 +
c
2
c
1
c
2
< 0,
we nd that prot is decreasing in c
2
, precisely as in the case without pricing, and
ov
;j

cj
;j
is
similarly decreasing in c
2
. In the limit as c
2
, the rms optimal price with QR goes to , and
margin on each unit sold that is procured via QR goes to zero. Hence, the rms prot eectively
becomes the same as if it did not have QR capabilities, with two caveats: it is constrained to price
at (in the SP regime, the rm can price anywhere in the interval [c
1
. ]), and in equilibrium,
more consumers will wait than if the rm did not have QR due to the fact that QR naturally shifts
demand. In other words, if c
2
= ,
ov
;j

cj
;j
=
ov
[
j=
max
j2[c
1
.]

cj
_
ov
[
j=

cj
[
j=
_ 0,
where the last inequality follows from Theorem 1.
Proof of Theorem 4. Omitted; because consumers have zero surplus from a delayed purchase
they are essentially myopic.
29
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