Advance Selling by a Newsvendor Retailer

Advance Selling by a Newsvendor Retailer

Ashutosh Prasad, Kathryn E. SteckeSchool of Management, The University of Texas at Dallas, Texas 75083, USA

[email protected], [email protected]

Xuying ZhaoDepartment of Management, University of Notre Dame, Notre Dame, Indiana 46556, USA [email protected]

Retailers often face a newsvendor problem. Advance selling helps retailers to reduce demand uncertainty. Consumers,however, may prefer not to purchase in advance unless given a discount because they are uncertain about their valuation

for the product in advance. It is then unclear whether or when advance selling to pass some uncertainty risk to consumers isoptimal for the retailer. This paper examines the advance selling price and inventory decisions in a two-period setting, wherethe first period is the advance selling period and the second is the selling (and consumption) period. We find that an advanceselling strategy is not always optimal, but is contingent on parameters of the market (e.g., market potential and uncertainty)and the consumers (e.g., valuation, risk aversion, and heterogeneity). For example, we find that retailers should sell inadvance if the consumers’ expected valuation exceeds consumers’ expected surplus when not buying early by a certainthreshold. This threshold increases with the degree of risk aversion but decreases with stock out risk. If the degree of riskaversion varies across consumers, then a retailer should sell in advance if the probability for a consumer to spot buy is lessthan a critical fractile.

Key words: advance selling; newsvendor; consumer valuation; uncertainty; pricingHistory: Received: September 2008; Accepted: October 2009 by George Shanthikumar; after 2 revisions.

1. IntroductionAdvance selling is the sale of a product to customersin advance of delivery. It is often implemented in themanufacturing and retailing sectors by firms facingnewsvendor problems. A newsvendor problem ariseswhen a retailer needs to purchase its inventory beforea short selling period with uncertain demand. In suchcases, advance selling is a recommended method toreduce the inventory risk, because the advance ordersare precommitted. Consider the following examples:� Tang et al. (2004) note that bakeries in Hong Kong

encourage consumers to book moon cakes in ad-vance for the Chinese mid-autumn festival.Orders paid for a month prior to the festival en-joy a 25% price discount. The advance orders arefulfilled a week before the festival.

� In the classical newsvendor context of magazineand newspaper retailers, a major goal of market-ing is to increase subscriptions, which is a form ofadvance selling. We also see that retailers of newnovels, DVDs, and video games, which haveshort selling seasons, encourage consumers toplace precommitted orders before the release.Booksellers such as Amazon.com often offer ad-vance purchase discounts for soon to be releaseditems. For example, Amazon offered a 32% pre-

release discount on the paperback book NewMoon: The Official Illustrated Movie Companion(Twilight Saga) before its release date of October6, 2009.

A second benefit of advance selling is that the sell-ing season demand can be more accurately forecasted,because orders received in the advance selling periodmay be correlated with the demand in the sellingseason. A firm can update its demand forecast andadjust the order quantity accordingly to reduce excessinventory and stock out risks. With books, for exam-ple, a large number of precommitted orders onAmazon might indicate high popularity for a bookand prompt its publisher to print more copies for theselling season.

Finally, in addition to reducing inventory risks, ad-vance selling takes advantage of changes inconsumers’ product valuation uncertainty over time.Xie and Shugan (2001) and Shugan and Xie (2004) inthe case of services marketing, and Gundepudi et al.(2001) in the case of information goods show that inthe presence of consumer valuation uncertainty, ad-vance selling can increase demand and profits. Thereason is that consumers realize their valuation onlyin the selling season. If it is lower than the price of theproduct, there is no sale from those consumers in theselling season. However, the same consumers might

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PRODUCTION AND OPERATIONS MANAGEMENTVol. 20, No. 1, January–February 2011, pp. 129–142ISSN 1059-1478|EISSN 1937-5956|11|2001|0129

POMSDOI 10.1111/J.1937-5956.2010.01133.x

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have made a purchase during the advance selling pe-riod when they were uncertain about their futureproduct valuations (e.g., Swinney 2009, Xie andShugan 2001). It may also be the case that whereasconsumers are uncertain about whether they will havehigh or low eventual valuations, their heterogeneity invaluations during the advance period is lower, per-mitting a greater extraction of surplus by the seller atthat time. The following examples illustrate this.� According to usfireworks.biz, about 95% of US

fireworks sales occurs between May 15 and July 4.It takes four to six weeks to ship the fireworksfrom China, where most of the fireworks aremade, to the United States (Quint and Shorten2005). Given the short selling season and the longtransportation lead times, fireworks companiesface a newsvendor problem. Consumer valuationof the fireworks on July 4 is negatively affected byrain but lacking weather information in advancethey would still have bought them.

� In October, the parent of a young child may facethe dilemma of whether to buy a Christmas toythat the child currently seems to desire and thatmay face a stock out during the Christmas seasonor to wait in case the child changes his mind(Swinney 2009).

The arguments in favor of advance selling seemcompelling. But we do not observe advance selling inall situations. What are possible arguments againstthis strategy? Two issues that warrant considerationserve to motivate our investigations in this paper.

The first is that a significant, and ultimately un-profitable, early purchase discount may have to beoffered to consumers to compensate them against therisks of realizing negative surplus in the future. Forexample, consumers’ valuation of a vacation dependson a host of variables, such as health and mood, fi-nances, work schedule, and the weather at the time ofthe vacation, all of which can raise or lower valuation(Shugan and Xie 2004). In other examples, the con-sumer, by buying early, foregoes the opportunity toevaluate the product during the purchase period, suchas browsing the new novel or magazine or newspaper,or sampling the new CD, or trying out the new videogame in the store and then deciding whether or not topurchase it. However, it is true that by buying in ad-vance, the consumer gets the security of avoiding astock out. But in general, the seller offers a compensa-tion to consumers to overcome their uncertainty incommitting to an advance purchase. Thus, advanceselling may reduce the firm’s profit margin. This resultsin a tradeoff between the possible reduced profit marginand the aforementioned benefits from advance selling.

The second issue is the matter of implementation.From the discussion above, a large number of factorsshould be taken into account when making the de-

termination of the depth of the advance sellingdiscount and the retailer’s order quantity. We incor-porate the advantages from advance selling, includingthe nature and forecasting of demand, and consumercharacteristics, including valuation uncertainty andheterogeneity. We also allow that the consumer maynot even be aware that the product is available for salein an advance period. By incorporating aspects ofconsumer behavior into the retailer’s strategy, weprovide answers to the following four questions.When should a retailer adopt (or not) an advanceselling strategy? What should be the advance sellingprice? How much inventory should the firm preparefor the selling season? How do the retailer’s advanceselling decisions change with consumer characteris-tics such as heterogeneity, foresightedness, and riskaversion? It is known that structural models, whichexplicitly look at the decisions of actors, tend to pro-vide robust policy formulations as compared withreduced form models where the demand function isposited without, for example, accounting for con-sumer decisions. As related research questions havebeen variously examined in previous research, in sec-tion 2 the literature is reviewed and the contributionsof this paper to it are clarified.

After the literature review, the rest of the paper isorganized as follows. Section 3 introduces the prob-lem settings. Section 4 provides models and optimalsolutions for the retailer’s problem when there is noadvance selling. Section 5 studies a retailer’s advanceselling decisions when consumers are homogenous onrisk aversion. The results are extended in section 6 tothe case of heterogeneous consumers. Section 7 pro-vides a numerical study to show the sensitivity ofadvance selling strategies to consumer characteristics.Section 8 provides a summary and conclusions.

2. Literature ReviewMuch of the newsvendor problem literature has ex-amined the issues of order quantity and the effect ofuncertainty (e.g., Arrow et al. 1951, Edgeworth 1888).The newsvendor formula is a fundamental buildingblock of many models of supply chain coordination.Many extensions of the classical newsvendor modelhave been proposed (Khouja 1999). An important ex-tension is to incorporate advance selling strategies.Examples of advance selling can be given for bothmanufacturing firms (e.g., books and fireworks) andservice firms (e.g., airlines and hotels). We restrict ourattention to manufactured goods in order to focus onorder quantity and inventory risk issues. Manufac-tured products are sold to channel intermediaries andend consumers, and they have different characteris-tics. Therefore, similar to the distinction madebetween B2B and B2C marketing, we subdivide this

Prasad, Stecke, and Zhao: Advance Selling by a Newsvendor Retailer130 Production and Operations Management 20(1), pp. 129–142, r 2010 Production and Operations Management Society

usfireworks.biz

literature into papers dealing with (a) advance sellingfrom manufacturers to retailers and (b) advance sell-ing from retailers to consumers.

Most papers addressing advance selling in manu-facturing have studied advance selling frommanufacturers to retailers. For example, Cachon(2004) compares inventory risks under push, pull,and advance purchase discount contracts betweensuppliers and retailers. Cvsa and Gilbert (2002) andGilbert and Cvsa (2003) examine the manufacturer’swholesale price commitment decision and issues re-lated to sale timing. Taylor (2006) studies the saletiming for a manufacturer and decides when to sell toretailers. Boyaci and Ozer (2010) study when to startand stop advance selling in order to acquire enoughdemand information from retailers for capacity plan-ning in a manufacturing company.

Our paper focuses on advance selling from retailersto consumers, where there is relatively less research.Weng and Parlar (1999) determine the optimal pricediscount rate and inventory order quantity, assumingthat the probability for a consumer to buy early is anincreasing function of the price discount rate. Tang etal. (2004) examine the benefits from the demand fore-cast update after advance selling. Consumers buyearly with a probability that is increasing with theprice discount. McCardle et al. (2004) extend Tanget al. (2004) to consider brand competition amongretailers.

While building on a similar framework, our paperstudies advance selling from a different point of viewand with different results compared with the previousliterature. We study how a newsvendor retailer’sadvance selling strategy depends on consumer char-acteristics such as consumer valuation for products,consumer risk preferences, and stocking out proba-bility in the selling season. That is, we assumestrategic consumer choice decisions based on anexplicit characterization of their utility functions. Asdifferent consumers behave differently, we also allowfor heterogeneity on multiple dimensions of the utilityfunction.

From this consumer choice perspective, we find thata retailer should sell in advance if consumer expectedvaluation exceeds consumer expected surplus whennot buying early by a certain threshold at least. Alsowe show that, independent of the stocking out prob-ability, high risk aversion hurts a retailer’s profits andprevents it from adopting advance selling because ofconsumers’ valuation uncertainty risks when buyingin advance. These insights are new because previousresearch generally modeled nonstrategic consumerdemand in the aggregate, as a prespecified stochasticarrival process.

Several studies consider intertemporal pricing withstrategic consumers. Su (2007) studies a dynamic

pricing problem for durable products over an infinitetime horizon, considering that strategic consumersmay stock for future consumption. Su and Zhang(2008) study a newsvendor problem considering stra-tegic consumers who anticipate future sales at the endof a selling season. Su and Zhang (2009) study anewsvendor problem considering strategic consumerswho anticipate the likelihood of stock outs and maynot visit the retailer during the selling season. Yin etal. (2009) study a display mode choice problem wherea retailer can show all available units or show onlyone unit at a time, considering that strategic consum-ers may wait for a postseason clearance sale. Mak(2008) examines how inventory constraints andbuyer/seller time preferences in a two-period sellingseason affect a seller’s profit when consumers arestrategic. Liu and Van Ryzin (2008) study quantitydecisions in a capacity rationing model with strategiccustomers. All of the above papers focus on strategiccustomer behavior after the selling season starts. Inaddition, we consider the tradeoff between consumervaluation uncertainty and price discount for a strate-gic customer before the selling season starts. As far aswe know, we are one of two papers (the other is Zhaoand Stecke 2010) that considers strategic consumerbehavior in the first period and where a newsvendorretailer places the order after realizing the first periodsales.

Notable papers that have considered consumer val-uation uncertainty are Xie and Shugan (2001), Shuganand Xie (2004), Gundepudi et al. (2001), and Yu,Kapuscinski, and Ahn (2009). These papers considerconsumer valuation uncertainty and strategic con-sumer behavior in advance selling, with or without asupplier’s capacity limitation. When capacity is lim-ited, capacity risk and capacity allocation is analyzed.The major difference between this study and theabove four papers is demand stochasticity. While theabove papers consider that the total market size isfixed, our model includes the newsvendor setting,namely, that there is stochastic market size and in-ventory risk. In the introduction, we provide severalexamples where stochasticity is present in the mar-ketplace. We find that with demand stochasticity, theresults can be quite different, even reversed in someinstances.

This paper is a companion to the study in Zhao andStecke (2010), which focuses on the impact of con-sumer loss aversion on a retailer’s advance sellingstrategies. Although both papers study advance sell-ing strategies, they have different problem scenariosand results. First, Zhao and Stecke (2010) classifiesconsumers based on loss aversion, while this studyconsiders different information availability to con-sumers and risk preferences of consumers. Lossaversion means that people are more averse to losses

Prasad, Stecke, and Zhao: Advance Selling by a Newsvendor RetailerProduction and Operations Management 20(1), pp. 129–142, r 2010 Production and Operations Management Society 131

than attracted to the equivalent gains. That is, con-sumers lose more utility to a negative surplus thanthey would gain from the same amount of positivesurplus. Loss averse consumers may even be riskseeking under some circumstances. Second, retailersin Zhao and Stecke (2010) may not sell in advance orsell in advance with either a moderate or deep dis-count. Each of the three strategies is presented withtheir optimal conditions under which to use. In thispaper, retailers may sell in advance without any pricediscount or even with a premier price because ofconsumer risk aversion of stocking out in the sellingseason.

3. Problem SettingsConsistent with the literature, a retailer who faces anewsvendor problem is modeled over two periods.The first period is the advance selling period and thesecond period is the selling season. The selling seasonis also the consumption period. There are two types ofcustomers, referred to as the informed and uninformedtypes depending on whether they are aware aboutadvance purchase offerings or not. For both types,each customer buys at most one unit of the productand the number of customers is stochastic.

This section details the assumptions about the re-tailers and consumers. Sections 4, 5, and 6 providemodels based on these assumptions. Table 1 lists thenotation used in this paper.

3.1. Retailer SettingThe retailer pays c dollars to procure one unit of theproduct and charges a selling price of p dollars perunit in the selling season. For any quantity left unsoldat the end of the selling period, the retailer gets s dol-lars per unit as the salvage value. It is reasonable thatp4c4s. The first inequality ensures that the retailerorders a positive quantity. The second ensures a finitequantity.

In the advance selling period, which ends before theselling season starts, the retailer may allow consumersto purchase, or equivalently, commit to purchase, at aprice of X dollars per unit. The retailer commits tofulfilling advance purchase orders in the selling sea-son, guaranteeing these consumers against a stockout. The decision on the advance selling price X, madeat the beginning of the advance selling period, stays ineffect until the end of this period.

At the beginning of the selling season, the retailermakes a decision on the quantity of products to pro-cure. This decision, because it is made after theconclusion of the advance selling period, is informedby sales in the advance selling period. The number ofconsumers who buy in the advance period is denotedby the random variable N1 and the number who buyin the selling season by the random variable N2. n1 is

the realization of N1 at the end of the advance sellingperiod. Then the order quantity decision can be rep-resented as Q1n1, where n1 products are used tofulfill precommitted orders immediately. The remain-ing Q products are used for the new demand in theselling season. As shown in section 5, the advanceselling information can be used to update the demandforecast for the selling season and generate a betterdecision on Q.

3.2. Consumer SettingWhen making early purchase decisions, the informedconsumers compare the expected utilities from ad-vance purchase and not. That is, we assume thatconsumers are strategic. In this section, the consumervaluation of a product is introduced, consumer sur-plus is analyzed, and the consumer utility function ispresented. We use the standard definition that con-sumer valuation is the maximum dollar value that aconsumer is willing to pay for the product. Consumersurplus from choosing the product is the differencebetween the consumer valuation and the price of theproduct. In practice, information about consumer val-uation can be obtained in several ways (Voelckner2006). For example, consumer surveys can give theretailer an idea of consumer valuation of the product.

Table 1 Notation

Parameters/variables concerning a retailer

c 5 Procurement cost per unit of product for the retailer

p 5 Market price per unit of product sold in the selling season

s 5 Salvage price per unit of product unsold at the end of the selling season

P5 Total expected profit achieved by the retailer, a variable to be maximized

Parameters/variables concerning consumers and market

V 5 Consumer valuation for a product, a random variable with a probability

density function f ðvÞ, cumulative distribution function F ðvÞ, mean mv ,

and standard deviation sv ; F ðvÞ � 1� F ðvÞNj 5 Number of potential type j consumers, a random variable with a normal

distribution, mean mj, and standard deviation sj, j 2 fi ; ugr5 Correlation coefficient between the number of informed consumers (Ni)

and the number of uninformed consumers (Nu)

N1 5 Number of consumers who buy in the advance selling period, a random

variable with mean m1 and standard deviation s1

N2 5 Number of consumers who buy in the selling season, a random variable

with mean m2 and standard deviation s2

n1 5 Realized value of the random variable N1 at the end of the advance

selling period

Uw 5 Expected utility of a consumer when not buying in advance

Ua 5 Expected utility of a consumer when buying in advance

b5 Risk index to indicate how averse a consumer is to risks; b � 0

Z5 Stock out risk (or probability) in the selling season

Decision variables

Q 5 Quantity of products that the retailer should order for the selling season

X 5 Price per unit in the advance selling period


Also, conjoint analysis can obtain the valuations of theproduct’s multiple attributes.

Following the literature, it is assumed that consum-ers are uncertain about their own valuation for theproduct in the advance selling period (Shugan and Xie2004, Xie and Shugan 2001). The uncertainty can becaused by many factors as noted in the introduction.

Consumer valuation is denoted by the randomvariable V. It has expected value mv, standard devia-tion sv, and a probability density function f(v) withfinite support on [l, h], where l40.

Consumer surplus is contingent on whether a con-sumer buys in the advance selling period or not. Thesurplus is derived under two scenarios. Consumerswho buy in advance pay a price X, so their surplus isV�X. The expected surplus must be high enough forthem to make the decision to purchase. But there ex-ists the possibility that the realized surplus in theselling season will turn out to be negative becausevaluation V is a random variable. If consumers do notbuy in advance, they can decide whether not to buyduring the selling season. If their realized valuation isgreater than or equal to the market price p, the con-sumers buy and get a surplus of V� p. Otherwise, theconsumers do not buy at all, generating a surplus of 0.Combining these together, the consumer surplus inthe selling season is maxfV � p; 0g.

Consumer heterogeneity is introduced by classify-ing consumers into two groups, informed and unin-formed, as in Varian (1980). Here the differentiatinginformation is about the availability of purchasing theproduct in advance. Informed consumers know aboutthe advance sale through TV, Internet, or some othermedium. Uninformed consumers have no informationin that they do not know that they can buy the prod-uct before the selling season. Therefore they may buyonly in the selling season. Note that the two segmentsof informed and uninformed consumers have thesame valuation probability density function f(v). Acase can be made that high-valuation consumers maybe more likely to be informed. For example, Applefans keep themselves updated about Mac news. Wesuggest that heterogeneous consumer valuation dis-tributions be explored in future research.

Let random variables Ni and Nu represent, respec-tively, the number of informed and uninformedconsumers in the market. We allow these random vari-ables to be correlated. The joint distribution of Ni and Nu

is bivariate normal with means mi and mu, standard de-viations si and su, and correlation coefficient r 2 ð�1; 1Þ.The bivariate normal is degenerate for r ¼ �1.

The introduction of heterogeneity requires us todifferentiate the decisions for the different consumers.An informed consumer may need to make a decision ineach period. Should they buy during the advance sell-ing period? If not, should they buy in the selling season?

Uninformed consumers face only the second decision,which is simpler. To make the second decision, con-sumers need only compare their realized valuationagainst the market price p. If v � p, then the consumerbuys in the selling season. Otherwise, they do not buy.

Now consider the first decision. Informed consum-ers who have the option of buying in advance facetwo lotteries. If they buy in advance, they take a riskthat their realized surplus can be lower than or higherthan expected. But if they decide to wait, the productmay be unavailable in the selling season. In general,consumers are risk averse. Thus their surplus shouldbe corrected for the cost of risk. We therefore adopta mean–variance utility formulation, where the ex-pected utilities in the advance selling period, for buy-ing and waiting, are denoted by Ua and Uw, respec-tively. Figure 1 shows the corresponding surpluses fordifferent purchase decisions.

When consumers buy in advance, their surplus isV�X. Therefore, the mean and variance of consumersurplus are mv�X and s2

v, respectively. So we have thefollowing expression for Ua.

Ua ¼ mv � X � bs2v; ð1Þ

where the parameter b is a nonnegative constant thatcaptures the degree of risk aversion. A larger b meansthat a consumer is more risk averse.

When consumers do not buy early, we need to con-sider a few different possibilities. In the selling season,there is a risk of stock out denoted Z. Thus, withprobability Z the consumers who waited until theselling season receive zero surplus. However, if thereis no stock out but if their realized valuation v issmaller than the selling price p, then the consumers donot make a purchase either and again receive zerosurplus. Only when v4p and the product is availabledoes the consumer buy, obtaining a surplus of v� p.Let mw and sw denote the mean and standard devia-tion, respectively, of consumer surplus when notbuying in advance. Then the mean is

mw ¼ ð1� ZÞZ h

pðv� pÞfðvÞdv;

and the variance is Zð0� mwÞ2 þ ð1� ZÞR p

l fðvÞdvð0�mwÞ2þð1� ZÞ

R hp ðv� p�mwÞ2fðvÞdv, which simplifies to

s2w ¼ ð1� ZÞ

Z h

pðv� pÞ2fðvÞdv� m2

w:

Therefore, considering the mean–variance formula-tion again, we can define Uw as follows:

Uw ¼ mw � bs2w: ð2Þ

Assuming consumers are strategic then if and only ifUa � 0 and Ua � Uw is a consumer willing to buy inadvance.


4. No Advance SellingNext the baseline scenario is analyzed, where the firmdoes not implement an advance selling strategy. Inthis case, demand in the advance selling period iszero, i.e., N1 ¼ 0, and we need not consider this periodfurther in the analysis. Focusing on the selling season,a consumer makes a purchase if v � p. Aggregatingover consumers, the total demand N2 follows a bino-mial distribution. We assume that the market size islarge enough so that the binomial distribution can beapproximated by a normal distribution. Given that themarket size Ni þNu ¼ n and let FðpÞ � 1� FðpÞ, wehave

EðN2jNi þNu ¼ nÞ ¼ nFðpÞ;VarðN2jNi þNu ¼ nÞ ¼ nFðpÞFðpÞ:

Note that the market size Ni þNu is a linear combi-nation of Ni and Nu. As Ni and Nu are distributedbivariate normal, their linear combinations also follow anormal distribution. Therefore, the market size Ni þNu

has a normal distribution with mean and variance givenby mu þ mi and ðsuÞ2 þ ðsiÞ2 þ 2susir, respectively.

Therefore, we have EðN2Þ ¼ EðEðN2jNi þNu ¼ nÞÞ¼ ðmu þ miÞFðpÞ. As VarðN2Þ ¼ EðN2

2Þ � E2ðN2Þ, as longas EðN2

2Þ is known, we can obtain VarðN2Þ.

EðN22Þ ¼EðEðN2

2 jNi þNu ¼ nÞÞ

¼E½nFðpÞFðpÞ þ n2F2ðpÞ�

¼EðnÞFðpÞFðpÞ þ ½VarðnÞ þ E2ðnÞ�F2ðpÞ¼ ðmu þ miÞFðpÞFðpÞ þ ½ðsuÞ2 þ ðsiÞ2

þ 2susir�F2ðpÞ þ ðmu þ miÞ2F

2ðpÞ:

In summary, the demand N2 follows a normal dis-tribution with the following mean and standarddeviation.

m2 ¼ ðmu þ miÞFðpÞs2 ¼ ½ðmu þ miÞFðpÞFðpÞ þ ½ðsuÞ2 þ ðsiÞ2

þ 2susir�F2ðpÞ�1=2:

ð3Þ

Let Pb denote the optimal expected profit from thebaseline model with no advance selling. Then

Pb ¼ maxQ�0EN2f�cQþ p minfQ;N2g

þ s maxfðQ�N2Þ; 0gg:

The first term on the right-hand side, cQ, is the pro-curement cost. The second term represents the sellingrevenue and the last term is the salvage revenue forproducts unsold at the end of the selling season.

The retailer needs to determine the optimal orderquantity that maximizes the total expected profit. Thesolution is a traditional newsvendor problem withnormally distributed demand (e.g., Silver et al. 1998).Following the standard solution method, the optimalorder quantity Q and the optimal expected profit are

Qb ¼ m2 þ ks2; ð4Þ

Pb ¼ ðp� cÞm2 � ðp� sÞfðkÞs2; ð5Þ

where k ¼ F�1ðp�cp�sÞ, and Fð:Þ and fð:Þ are the distri-

bution and the density functions of the standardnormal distribution, respectively.

5. Advance Selling to ConsumersIn this section, we study a retailer’s optimal advanceselling strategy. A precise threshold is presented to

Figure 1 Corresponding Surpluses for Different Purchase Decisions


guide retailers on whether or not to sell in advance. If aretailer should sell in advance, the optimal advanceselling price is also suggested. The impact of consumerrisk preference on a retailer’s advance selling strategyis analyzed. Although stocking out risk motivates con-sumers to buy in advance, negative surplus riskdiscourages consumers from purchasing early. Weshow that consumers being risk averse has an overallnegative impact on a retailer’s advance selling profits.

Backward induction is adopted to solve the prob-lem. First, start with the selling season, where aretailer faces a standard newsvendor problem anddetermines the optimal order quantity Q, given thatdemand from informed consumers during the ad-vance selling period is realized as n1. Based on theoptimal order quantity Q, the stocking out probabilityin the selling season Z is determined. Second, a retailerdecides the advance selling price X to attract all in-formed consumers to buy early.

5.1. The Selling SeasonIn the selling season, uninformed consumers arriveand a consumer is willing to buy if and only if hervaluation is larger than the price p. Therefore, demandin the selling season N2 follows a binomial distribu-tion. When the market size Nu is large enough, such abinomial distribution can be approximated by a nor-mal distribution with the following mean andstandard deviation.

m2 ¼ muFðpÞ;

s2 ¼ ½muFðpÞFðpÞ þ s2uF

2ðpÞ�1=2:

At the end of the advance selling period, the demandN1 from informed consumers is realized as n1. Giventhat Ni and Nu follow binomial distributions with cor-relation coefficient r, the retailer can update its forecastfor market size Nu and then demand N2. After the up-date, the market size Nu will have a new mean

m0u ¼ mu þ rðn1 � miÞsusi

and a new standard deviation s0u¼ su

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

p(Bickel and Doksum 1977). Correspond-

ingly, the demand N2 during the selling season has anew mean and standard deviation as follows:

m02 ¼ ðmu þ rðn1 � miÞsusiÞFðpÞ;

s02 ¼ ½ðmu þ rðn1 � miÞsusiÞFðpÞFðpÞ þ s2

uð1� r2ÞF2ðpÞ�1=2:

Let Pa denote the total expected profit with advanceselling. Thus we have

Pa ¼ EN1fðX � cÞN1 þmaxQ�0EN2jN1¼n1

½�cQ

þ p minfQ;N2g þ s maxfQ�N2; 0g�g;

where ðX � cÞN1 is the profit from the advance sellingperiod and the remaining term is the expected profitfrom the selling season.

The retailer needs to decide the quantity Qa thatmaximizes the expected profit from the selling season.Because the retailer faces a standard newsvendorproblem during the selling season, the optimal orderquantity Qa and the corresponding profit Pa can beexpressed as follows:

Qa ¼ m02 þ ks02; ð6Þ

Pa ¼ ðX � cÞmi þ ðp� cÞEðm02Þ � ðp� sÞfðkÞEðs02Þ; ð7Þ

where Eðm02Þ � EN1ðm02Þ ¼ muFðpÞ and Eðs02Þ � EN1

ðs02Þ �½muFðpÞFðpÞ þ s2

uð1� r2ÞF2ðpÞ�1=2.Such an optimal newsvendor stocking level Qa en-

sures a service level (in-stock probability) p�cp�s, which is

also the newsvendor critical ratio. Therefore, thestocking out probability Z in the selling season should

be 1� p�cp�s. Note that the stocking out probability in the

selling season remains the same whether informedconsumers buy early or not because it only dependson p, c, and s.

5.2. The Advance Selling PeriodAt the beginning of the advance selling period, aninformed consumer compares the expected utilitygiven by Equation (1) from buying in advance withthe expected utility given by Equation (2) from notbuying in advance. The informed consumers buy inadvance if and only if Ua � Uw. This implies that

X � mv � b s2 � ð1� ZÞZ h

pðv� pÞfðvÞdv;

where s2 ¼ s2v � s2

w.The retailer charges the highest price

X ¼ mv � b s2 � ð1� ZÞZ h

pðv� pÞfðvÞdv

that induces all informed consumers to buy in ad-vance. That is, the demand during the advance sellingperiod is N1 ¼ Ni.

5.3. Comparing Advance Selling to the BenchmarkCaseFor comparison purposes, denote the standard devi-ation of demand N2 in (3) when there is no advanceselling as s2, i.e.,

s2 � ½ðmu þ miÞFðpÞFðpÞ þ ½ðsuÞ2 þ ðsiÞ2

þ 2susir�F2ðpÞ�1=2:

Obviously, s2 is larger than Eðs02Þ in Equation (7).According to (6), the optimal order quantity Qa in-

cludes two parts: m02 and ks02. The second part ks02 canbe identified as safety stock to hedge against demanduncertainty. Comparing expressions (4) and (6), onecan conclude that advance selling helps a retailer to


better predict demand and thus lowers the expectedsafety stock.

To further appreciate the tradeoffs of selling in ad-vance, the profit Pa from advance selling is comparedwith the profit Pb with no advance selling. Compar-ing the two profit expressions from (7) and (5),we find that the net benefit from advance selling can beexpressed as the sum of three terms:

benefit from increased demands ¼ ðX � cÞFðpÞmi; ð8Þ

loss from reduced profit margin ¼ ðp� XÞFðpÞmi; ð9Þ

benefit from reduced inventory risk

¼ ðp� sÞfðkÞðs2 � Eðs02ÞÞ:

ð10Þ

Define a threshold fDmv, which defines the boundarybetween consumer expected valuation and consumerexpected surplus when not buying early. If and only ifmv � mw � fDmv, the net benefit from advance selling isnonnegative. Therefore, Theorem 1 follows.

THEOREM 1. There exists a threshold fDmv ¼ �KþeX þ b s2, where K ¼ ðp� sÞfðkÞðs2 � Eðs02ÞÞ=mi andeX ¼ pFðpÞ þ cFðpÞ. If and only if mv � mw � fDmv, then aretailer should sell in advance. The optimal advance selling

price X ¼ mv � b s2 � mw.

(All proofs are in Appendix A.)

The threshold fDmv increases with marginal cost c.When c becomes larger, advance selling is less attrac-tive to the retailer. From another point of view,Theorem 1 implies that there exists another, equiva-lent threshold ec for the marginal cost, where

ec ¼ mvþK�bs2�R h

pvfðvÞdvþZ

R h

pðv�pÞfðvÞdv

FðpÞ . If and only if c � ec,

a retailer should sell in advance. Both thresholds, fDmv

and ec, are equivalent because they both provide a re-tailer with the same decision on her advance sellingstrategy.

The result that retailers should not sell in advance iffaced with above-threshold marginal costs can becompared with the results from the simpler advanceselling model in Xie and Shugan (2001). Xie andShugan (2001) found that no retailer should sell inadvance if marginal cost exceeds the lower bound ofcustomer valuation. In that model, the benefit of ad-vance selling is to extract surplus from all of the exante homogenous customers, vis-a-vis ex post eithercharging a high price and serving only the high-val-uation segment, or charging a low price and servingboth segments but losing margin. If serving the low-valuation segment is unprofitable due to high mar-

ginal cost, it is better not to sell in advance and toserve only the high-valuation segment in the sellingperiod. Thus, in the Xie and Shugan (2001) model,zero marginal cost implies advance selling and in-creasing marginal cost starts to favor spot selling.

The threshold on marginal cost in this paper alsofinds that higher marginal cost favors spot selling.However, our threshold is different from the one inXie and Shugan (2001) because our model deals withadditional consumer and market characteristics aswell as the newsvendor and forecasting issues. As aconsequence, the threshold in Xie and Shugan (2001)does not apply to several examples provided in theintroduction section that serve to motivate this paper.Our contribution lies in deriving the threshold foradvance selling problems with stochastic demand.

After considering demand stochasticity, the ad-vance selling strategy recommended to a retailerbased on Theorem 1 can be the opposite from thestrategy suggested in Xie and Shugan (2001). For asimple example, Xie and Shugan (2001) show that noretailer should sell in advance if marginal cost c reachesthe lower bound l of consumer valuation. Becauseconsumers are risk neutral in Xie and Shugan (2001),the b value is 0. Then it is easy to show that ec4l ormv � mw4fDmv if c 5 l. That is, Theorem 1 suggests thata retailer should always sell in advance if c 5 l.

Furthermore, although Theorem 1 may be equiva-lently stated as a threshold on the customer valuationor as a threshold on the marginal cost, both interpre-tations have some merit depending on the context. Forexample, a book retailer sells different books that haveall been procured at relatively similar costs but some ofthem are sold in advance whereas others are not. Thisphenomenon may be justified by considering thethreshold on the difference between consumer expectedvaluation and consumer expected surplus when notbuying early. Note that only the more anticipated, high-valuation books should be offered for advance sale. Onthe other hand, suppose that two booksellers sell thesame book. One seller is large enough to avail of quan-tity discounts and thus lower its cost structure whereasthe other seller is smaller and cannot. An argument canbe made based on the threshold on cost for why theformer sells the book in advance whereas the latter doesnot. In this case, the costs are variable or controllable, sothe appropriate threshold to consider is cost.

From Theorem 1, the threshold fDmv increases in

consumer risk aversion b because @ eDmv@b ¼ s240. Note

that s2 is the difference between variances of Ua andUw. As Uw has a truncated distribution compared toUa, it has a smaller variance and thus s240.

Theorem 1 also shows that if and only ifmv � fDmv þ mw, a retailer should sell in advance. Onecan observe that fDmv þ mw increases in stocking out


possibility (Z). This is because a retailer can increasethe advance selling price X without driving consum-ers away from buying early when the probability ofstocking out in the selling season is high. From Equa-tions (8) and (9), the benefits from advance sellingincrease while losses decrease. Therefore, a retailer ismore encouraged to sell early when the stocking outprobability increases.

Note that, in some circumstances, consumers buy inadvance even without a price discount. In other words,the optimal advance selling price X is not always lessthan p. This can happen when consumers want to secureresources early in order to avoid stock out risks. When ithappens that the optimal advance selling price X4p,consumer valuation expectation mv must be larger thanfDmv þ mw. This is because X4p means that mv4pþ b s2 þ ð1�ZÞ

R hp ðv� pÞfðvÞdv4�K þ eX þ b s2 þ ð1� ZÞR h

p ðv� pÞfðvÞdv ¼ fDmv þ mw. Hence, when consumers

are willing to buy in advance without discounts, a retailershould always sell in advance.

We can show that a demand forecast update re-duces inventory risks and thus improves profits for aretailer when selling in advance. Without a demandforecast, a retailer expects optimal profit ðX � cÞmiþðp� cÞFðpÞmu � ðp� sÞfðkÞs2, which is strictly lessthan the expected profits as expressed in (7) with de-mand forecast when r 6¼ 0. Thus we provide moreprecise suggestions on whether or not a retailershould sell in advance than previous studies withoutdemand forecast update. That is, the threshold fDmv islower after considering the demand forecast update.

Now consider two special cases. In the first, con-sumers act as if they are myopic. They buy in advanceas long as the expected surplus is nonnegative. Thiscase provides a lower bound and upper bound, re-spectively, for threshold fDmv þ mw and optimaladvance selling price X. The second special case isthat consumers are strategic but risk neutral. Whenmaking advance purchase decisions, they comparethe expected surpluses from purchasing in advanceand not. Risks, both stocking out risk and negativesurplus risk, do not affect their decisions. Comparingadvance selling strategies for risk neutral consumers(b5 0) and for risk averse consumers (b40) helps tofurther illustrate the impact of consumer risk prefer-ence on a retailer’s advance selling strategy.

5.3.1. When Consumers are Myopic. When my-opic consumers make decisions on whether to buyearly or not, they do not compare the expected utili-ties from the two periods. Instead, myopic consumersbuy in advance if their expected utility from earlypurchase is nonnegative, i.e., EðV � XÞ � 0.

Therefore, the advance selling price X must be lessthan or equal to mv in order to motivate myopic in-

formed consumers to buy in advance. As theobjective is to maximize profits, a retailer shouldcharge X ¼ mv when consumers are myopic. Thiscircumstance is equivalent to setting b5 0 and Z5 1.That is, if consumers are neutral to valuation uncer-tainty risks and if the stocking out probability in theselling season is 100%, then they act the same asmyopic consumers. From Theorem 1, the followingcorollary for myopic consumers is given.

COROLLARY 1. When consumers are myopic, a retailer

should sell in advance if and only if mv � fDmvmyopic,

where fDmvmyopic ¼ �K þ eX. The optimal advance selling

price Xmyopic ¼ mv.

The threshold fDmvmyopic in Corollary 1 also serves asthe maximum lower bound for fDmv þ mw in Theorem1 because Z 2 ½0; 1�, b � 0, and mw decreases in Z. Forsimilar reasons, Xmyopic provides the minimum upperbound for the optimal advance selling price X inTheorem 1.

5.3.2. When Consumers are Strategic but RiskNeutral. In this special case, we study a retailer’sadvance selling strategy when consumers are riskneutral. Through the comparison of advance sellingstrategies for risk neutral (b5 0) and risk averse(b40) consumers, we gain insights on the impact ofconsumer risk attitude on a retailer’s advance sellingstrategy. When consumers are strategic but riskneutral, based on (1) and (2), the following corollaryfor a retailer’s optimal advance selling strategy isgiven, which can also be generated from Theorem 1.

COROLLARY 2. When consumers are strategic but riskneutral, a retailer should sell in advance if and only if

mv � mw � fDmvneutral, where fDmvneutral ¼ �K þ eX. The op-timal advance selling price Xneutral ¼ mv � mw, which is nolarger than the advance selling price Xmyopic.

To show the impact of consumer risk preferences ona retailer’s advance selling strategy, the results of ad-vance selling strategies and the retailer’s profits arecompared back to back. Let fDmvaverse;Xaverse, and Paverse

represent the threshold on mv � mw, optimal advanceselling price, and optimal profits, respectively, whenconsumers are risk averse, i.e., b40. From Theorem 1and Corollary 2, we obtain the following:

fDmvaverse � fDmvneutral ¼ b s2; ð11Þ

Xaverse � Xneutral ¼ �b s2; ð12Þ

Paverse �Pneutral ¼ �b s2mi: ð13Þ


When consumers are more risk averse, a retailer hasto offer a lower advance selling price to attract themto buy early. That is, XaverseoXneutral according to (12).Correspondingly, PaverseoPneutral from (13). There-fore, a retailer is more cautious about advance

selling, i.e., fDmvaverse4fDmvneutral according to (11).Therefore, although the stock out risk encourages arisk averse consumer to buy in advance, the negativesurplus risk prevents consumers from buying early.The overall impact of consumers’ risk averse pref-erences is negative on the profit of a retailer whosells in advance.

To summarize, a retailer’s advance selling strategywhen consumers are risk neutral is analyzed. Com-pared with the strategy for risk averse consumers,we find that advance selling is less attractive whenconsumers are risk averse.

6. Advance Selling when b VariesAcross Consumers

So far we have looked at cases where consumers arehomogenous with respect to b. In practice, differentconsumers may have different risk aversion parame-ters. In this section, the effect of different consumershaving different b values on the retailer’s advanceselling decisions is examined.

Suppose that b follows a distribution g(b). That is,each consumer has a b value drawn from g. Whenconsumers are risk averse and strategic, they buy inadvance if X � mv � b s2 � ð1� ZÞ

R hp ðv� pÞfðvÞdv

from section 5. In other words, given an advance sell-ing price X, a consumer buys in advance if and only ifb � eb, where

eb ¼ mv�X�ð1�ZÞR h

pðv�pÞfðvÞdv

s2 :

Let W1 be the proportion of informed consumerswho buy early, given advance selling price X. Then

W1 ¼Rmaxfeb; 0g

0 gðbÞdb. Therefore, demand in the

advance selling period is N1 ¼W1Ni, which followsa normal distribution with m1 ¼W1mi and standarddeviation s1 ¼W1si.

Let M denote the market size in the selling season.As M ¼ Nu þ ð1�W1ÞNi, which is a linear combina-tion of ðNu;NiÞ, it follows a normal distribution withmean

mM ¼ mu þ ð1�W1Þmi

and standard deviation

sM ¼ ½s2u þ ð1�W1Þ2s2

i þ 2ð1�W1Þsisur�1=2:

According to similar reasons from section 4, de-mand N2 during the selling season can be approxi-

mated by a normal distribution with mean andstandard deviation

m2 ¼ mMFðpÞ;

s2 ¼ ½mMFðpÞ þ s2MF

2ðpÞ�1=2:

At the end of the advance selling period, demandN1 is realized as n1. In order to use this demand in-formation to update the demand forecast in the sellingseason, the retailer first needs the covariance betweenN1 and the forecasted market size M in the sellingseason.

CovðN1;MÞ ¼E½W1NiðNu þ ð1�W1ÞNiÞ�� E½W1Ni�E½Nu þ ð1�W1ÞNi�

¼W1ðEðNiNuÞ � EðNiÞEðNuÞÞþW1ð1�W1ÞðEðN2

i Þ � E2ðNiÞÞ¼W1rsisu þW1ð1�W1Þs2

i :

Therefore, using demand information n1, the fore-casted market size M in the selling season can beupdated, with the new mean and new standard de-viation given by

m0M ¼ mM þ ðn1 � mMÞCovðN1;MÞ

s1sM

sM

s1;

s0M ¼ sM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� CovðN1;MÞ

s1sM

s:

After updating the market size forecast in the sell-ing season, the retailer can further update the demandforecast in the selling season as follows:

m02 ¼ m0MFðpÞ;

s02 ¼ ½m0MFðpÞ þ s02MF2ðpÞ�1=2:

Let Qh and Ph denote the optimal inventory quan-tity to hold for the selling season and the optimalprofits achieved. By using the standard newsvendorresults, we have

Qh ¼ m02 þ ks02; ð14Þ

Ph ¼ ðX � cÞW1mi þ ðp� cÞEðm02Þ� ðp� sÞfðkÞEðs02Þ;

ð15Þ

where Eðm02Þ ¼ mM and Eðs02Þ � ½mMFðpÞ þ s02MF2ðpÞ�1=2.

Comparing profits (15) and (5), the last term in (15)is smaller than that in (5) because s24Eðs02Þ .That is, inventory risk is reduced after selling in ad-vance. Furthermore, we obtain the following theorem.

THEOREM 2. A retailer should always sell in advance if theprobability for a consumer to buy in the selling season is

less than a critical fractile mv�mw�cp�c , i.e., FðpÞomv�mw�c

p�c .


1. When FðpÞomv�mw�cp�c , selling in advance with X ¼eX � pFðpÞ þ cFðpÞ generates a lower bound PhðX ¼eXÞ of the optimal profit Ph.

2. This lower bound PhðX ¼ eXÞ is always greater thanPb, which is the optimal profit without advanceselling.

3. Advance selling always improves profits over notadvance selling at least by ðp� sÞfðkÞðs2 � Eðs02ÞÞ.

To summarize, in this section, a retailer’s advanceselling strategy when consumers are heterogeneouson risk aversion is studied. A critical fractile that doesnot require any information other than selling price,unit production cost, unit salvage value, and con-sumer valuation distribution is provided. When thepossibility of a consumer to buy in the selling seasonis no larger than the critical fractile, a retailer shouldsell in advance.

7. Sensitivity Analysis of a Retailer’sAdvance Selling Strategy

In this section, how a retailer’s advance selling stra-tegy changes with consumer valuation uncertainty,consumer risk preferences, correlation between num-bers of informed and uninformed consumers, andprofit margin during the selling season is examined.To do so, numerical examples are constructed toillustrate the optimal profits from each strategy. Theretailer should choose the strategy that generates thehighest profit.

We use the following initial values: p ¼ $100; c ¼$70; s¼$25; mu¼mi¼50; su¼si¼5; r¼0:4; mv¼98, andsv ¼ 10. The consumer valuation V is uniformlydistributed with ðmv; svÞ. Consumer risk averse indexis set at 0.1 initially. The stocking out risk in the sellingseason is calculated as 1� ðp� cÞ=ðp� sÞ ¼ 0:6.

To examine the impact of the standard deviation ofconsumer valuation sv on a retailer’s advance sellingdecision, sv is increased from 5 to 15 in 0.5 increments.Figure 2 illustrates the optimal profit improvement ineach case. The curve with squares represents the

optimal profit improvement from advance sellingwhen consumers are myopic. The curve with trianglesillustrates the optimal profit improvement fromadvance selling when consumers are strategic butrisk neutral. The third curve represents percentages ofprofit change caused by advance selling when con-sumers are risk averse. Figure 2 shows that asconsumer valuation uncertainty sv increases, advanceselling is less attractive.

Next, to examine the impact of risk averse index bon a retailer’s advance selling decisions, sv is fixed at10 while b varies from 0.02 to 1.42 in 0.02 increments.As profits are independent of the risk averse indexwhen consumers are myopic or risk neutral, we focuson the case where consumers are risk averse. Figure 3shows that profit improvement from advance sellingdecreases as consumers are more averse to valuationuncertainty risks. Therefore, a retailer should be morelikely to sell in advance if consumers are less riskaverse, which is consistent with our findings insection 5.

Third, the impact of the correlation between thenumbers of informed and uninformed consumers ona retailer’s advance selling strategy is examined. Inthe numerical examples, sv ¼ 10; b ¼ 0:1, and Z ¼ 0:6.r is varied from � 0.98 to 0.98. Figure 4 shows thatprofit improvement from advance selling increases asthe correlation coefficient r increases.

This can also be shown analytically. Comparing theoptimal profits with advance selling and withoutadvance selling, two advantages and one disadvan-tage of advance selling are quantified in expressions(8), (9), and (10). Only the benefit from reducedinventory risk (10) is relevant to the correlation r.Furthermore, this benefit increases with r because itsfirst order derivative with respect to r is alwayspositive.

That profit improvement from advance sellingincreases as r increases can be explained by thefollowing reasons. If the numbers of informed anduninformed consumers are positively correlated, byusing the realized advance selling information, a

Figure 2 Profit Changes as Standard Deviation of Consumer Valuation Changes


retailer can forecast demand in the selling seasonmore accurately when r is larger. Therefore, thebenefit of reduced inventory risk increases as r ispositive and increases. If demands are negativelycorrelated, because of demand pooling and variancecancellation when there is no advance selling,inventory risk gets smaller as the correlation becomesstronger, i.e., r is smaller. Therefore, the difference ininventory risk between the cases with and withoutadvance selling decreases as r is negative anddecreasing.

8. Summary and ConclusionsAdvance selling from a retailer facing a newsvendorproblem to heterogeneous consumers is studied.Advance selling not only reduces inventory risk butalso tends to bring more demand for a retailer.Advance orders can also be used to update theforecast of demand in the selling season. Consumersare strategic. When making purchase decisions, theycompare expected utilities from advance purchaseand not. So the retailer considers a consumer’sdecision making process when making her optimaladvance selling decisions.

We find that advance selling is not always anappropriate strategy for a retailer. There exists a

threshold on the difference between consumerexpected valuation and consumer expected surplusof not buying in advance, which provides a normativeguide to retailers on whether or not they should sell inadvance. Such a threshold is delineated, whichconsiders the tradeoffs of advance selling for anewsvendor retailer facing stochastic demand as wellas the tradeoffs of advance purchase for a consumer.

The impact of consumer risk preference on aretailer’s advance selling strategy is analyzed. Whenconsumers are risk averse, stock out risk motivatesconsumers to purchase early but negative surplus riskprevents them from doing so. It is shown analyticallythat the overall impact of a consumer’s risk aversepreference is negative on the profit of a retailer whosells in advance. Advance selling is less attractive to aretailer when consumers are more risk averse.

Multiple consumer settings are studied and resultsare compared. The threshold and advance selling pricefor myopic consumers provide the greatest lowerbound of threshold and least upper bound of advanceselling price, respectively, for strategic consumers.

The scenario where risk aversion varies acrossconsumers is studied. There exists a critical fractile,expressed as consumer expected valuation minusmarginal cost and consumer expected surplus whennot buying early and then divided by the profit

Figure 3 Profit Changes as Consumer Risk Attitude Changes

Figure 4 Impact of Demand Correlation on Profits


margin in the selling season. If the possibility for aconsumer to purchase in the selling season is less thanthis critical fractile, a retailer should sell in advance.

For future research, it is worthwhile to considerboth advance selling price and spot price as decisionvariables. In this research, the advance selling price Xis derived. An exogenous selling price p is assumed.The exogenous price assumption is not problematicbecause p can be different for different products. Ingeneral, p is determined through a careful considera-tion prior to period 1. For example, the considerationthat consumers should be left with a certain surplus(reservation price minus selling price) could deter-mine the selling price. This is done in value-in-usepricing. Defense for exogenous parameters vs. endo-genous parameters is given by Shugan (2004). There aretwo caveats. First, if the surplus depends on strategiccompetitors, there is an issue of why only p should besubject to competitive forces and not X. Thus, explicitlyincorporating competition should be looked at in futureresearch. Second, the seller may want to change theprice p after observing the advance purchases.However, a plausible reason for the seller committingto price is reputation and advertising costs of changingthe price after having made it public.

Another aspect that deserves to be analyzed in thefuture is the effect of product returns with full orpartial refund. Although retailers may not allowconsumers to return open-boxed video games, soft-ware, and movie DVDs, other products such as booksmay possibly be returned. Thus, if the return policy isso liberal that customers can order in advance andwithout cost return the product after receiving it, thenthe advance selling discount may need to be reducedto protect the firm’s profits. Future advance sellinganalysis should consider the design of a productreturn policy.

AcknowledgmentThe authors thank the department editor and anonymousreviewer for their helpful suggestions.

Appendix APROOF OF THEOREM 1: A retailer should sell in advance ifand only if the benefits from advance selling are not lessthan the losses, i.e., (8)–(9)1(10) � 0. From expressions(8) and (9), we have that (8)–(9) ¼ ½X � c� ðp� cÞR h

p fðVÞdV�mi ¼ ½X � pR h

p fðVÞdV � cR p

l fðVÞdV�mi. As

the optimal advance selling price is X ¼ mv�b s2 � ð1� ZÞ

R hp ðv� pÞfðvÞdv, then the sufficient and

necessary condition for a retailer to sell in advance is as

follows: mv � mw � fDmv ¼ �ðp� sÞfðkÞðs2 � Eðs02ÞÞ=miþpR h

p fðVÞdV þ cR p

l fðVÞdV þ b s2 � �K þ eX þ b s2.

PROOF OF COROLLARY 1: Corollary 1 can be obtaineddirectly from Theorem 1 by setting b ¼ 0 and Z ¼ 1.

PROOF OF COROLLARY 2: Corollary 2 can be obtaineddirectly from Theorem 1 by setting b ¼ 0 and 0oZo1.

PROOF OF THEOREM 2: First we show that if mv4eX, thereexist informed consumers who buy in advance whenX ¼ eX. That is, selling in advance with price X ¼ eX isa feasible advance selling strategy and thus provides alower bound for the optimal profit under an optimaladvance selling strategy. Next we prove that thislower bound is always larger than the optimal profitwithout advance selling.

Suppose that a retailer sells in advance with price

X ¼ eX. Then W1 ¼Rmaxfeb; 0g

0 gðbÞdb; where eb ¼ 1s2½mv �eX � mw� . If mv4eX þ mw, which can be transformed to

FðpÞomv�mw�cp�c after some algebra, then eb40 and thus

W140.As W140, selling in advance with X ¼ eX is a fea-

sible advance selling strategy. It generates a lowerbound on the optimal profit under an optimal ad-vance selling strategy. That is, Ph4PhðX ¼ eXÞ.

According to (15) and (5), PhðX ¼ eXÞ �Pb ¼ðp� sÞfðkÞðs2� Eðs02ÞÞ, which is greater than zeroconsidering that 0oW1 � 1. That is, the lower boundof the optimal profit from advance selling is greaterthan the optimal profit without advance selling.

Therefore, the optimal profit with advance selling islarger than the optimal profit without advance selling

if FðpÞomv�mw�cp�c . A retailer should sell in advance.

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Advance Selling by a Newsvendor Retailer