Consumer Panic Buying and Quota Policy under Supply Disruptions

Transcription

1 Submitted to Manufacturing & Service Operations Management manuscript Consumer Panic Buying and Quota Policy under Supply Disruptions Biying Sou Dept. of Management Sciences, City University of Hong Kong, Huacun Xiong Dept. of Matematical Sciences, Tsingua University, Zuojun Max Sen Department of Industrial Engineering and Operations Researc, University of California, Berkeley, We study consumer panic buying under supply disruptions and investigate ow te retailer sould adjust inventory and quota policies to respond to panic buying. First, we derive a tresold for consumer product valuation above wic a consumer will stockpile products for future consumption. Ten, we conduct rational expectations equilibrium analysis to caracterize te retailer s inventory policy and sow ow it is affected by consumers risk attitude. Furtermore, we evaluate te cost of ignoring consumer beavior. Interestingly, we find tat te cost is te most significant wen supply reliability is intermediate (more specifically, as it approaces a critical ratio). Finally, we examine te effectiveness of limiting consumer purcasing quota for te retailer wit limited capacity, and sow wen quota policy is beneficial to te retailer. Key words : panic buying; supply disruption; strategic consumer beavior; quota policy History : 1. Introduction Supply disruptions, wic refers to te interruption of normal product supply witin supply cains, ave been frequently observed around te world (Snyder et al. 2010). Disruption could be caused by various reasons, suc as natural disasters, labor strikes, terrorist attacks, and canges in government regulation, etc. Supply disruption is considered as one of te top concerns for supply cain management. As a result, recent years ave observed an increasing amount of researc on ow firms sould deal wit supply disruptions, e.g., via multi-sourcing, supplier certification, etc. Little researc, owever, as considered te effect of supply disruption on consumer beavior. Because of te increasing access to information (e.g., via Internet and TV), many consumers also receive timely information about disruptive events. But te actual consequences of tese disruptive events are often uncertain. For example, consumers may receive extreme weater warning (e.g., urricane and snowstorm); owever, weter it would actually lead to severe supply disruption is yet to know. Some consumers take active actions to reduce teir potential loss due to supply 1

2 2 Article submitted to Manufacturing & Service Operations Management; manuscript no. disruption. Indeed, consumer panic buying associated wit disruptive events as been frequently observed, were many consumers buy unusually large amounts of product to avoid possible future sortage. For example, wen Hurricane Sandy stroke te New York City in 2012, many consumers rused to store to purcase unusually large amounts of goods (Bloomberg 2012). Wen Hurricane Katrina disabled most of te oil drilling facilities in te U.S. Gulf coast region in 2005, consumer oarding beavior and long lines were observed at gasoline stations (Wall Street Journal, 2005). Wen rice production in Australia reduced by 98% due to a long period of drougt in 2008, fears of rice sortages spread around te globe. Regions suc as Vietnam, India, and Hong Kong saw consumers rusing to stores and exausting te rice supply (New York Times, 2008). Amid te eartquake and nuclear crisis in Japan in 2011, worried soppers stripped stores of salt in Beijing, Sangai, San Francisco, and oter cities (Cina Daily, 2011). Tese consumers believed tat iodized salt would protect against radiation poisoning and tat tere may be potential salt sortage as a result of radiation pollution. Supply disruption, firm disruption management decisions, and consumer panic buying are closely intertwined: (1) Supply disruption can lead to consumer panic buying. Consumer panic buying can be viewed as a way to edge supply disruption. (2) Consumer panic buying can furter exaggerate te consequences of supply disruption. Abnormally ig demand leads to substantial stock-outs, tus inducing more panic buying. (3) Retailer decisions (suc as increasing price, limiting supply, and imposing purcasing quotas) can greatly affect consumer beaviors. Altoug tese practices normally repress demand, tey may sometimes actually increase consumer anxiety about supply sortage and make panic buying worse. As consumer beavior and retailer decisions under supply disruption are closely related to eac oter, it would seem natural to study tem togeter. However, te current studies on tese aspects are largely disjoint. Consumer panic buying as been mainly studied by economists, psycologists, and sociologists wo sare a common focus on qualitative studies based on empirical data. Supply disruption as mainly been studied in te supply cain management literature, were strategic consumer responses are ignored. In tis paper, we aim at addressing te gap in understanding strategic consumer beavior and retailer decisions under supply disruptions. We try to provide some rational explanations to consumers seemly irrational panic buying beavior. We also try to develop optimal strategies for retailing firms wic face great callenges on bot supply side (disruption) and demand side (consumer panic buying). We focus on two common disruption management strategies, safety inventory and quota policy. We also evaluate te potential loss to retailing firms if e ignores te consumer beavior canges. Our key contributions are summarized as follows:

3 Article submitted to Manufacturing & Service Operations Management; manuscript no. 3 First, we examine te impact of different factors on consumer decisions and identify te main drivers for consumers panic buying beavior. We sow tat a consumer would stockpile product for future consumption if is valuation of te product is above a tresold. Moreover, te tresold value as monotone property wit regards to te price of te product, te olding cost of te consumers, and te risk attitude of te consumers. Second, we use rational expectations equilibrium analysis to derive te retailer s optimal inventory policy in response to consumer panic buying. We sow ow te optimal level of safety inventory is dependent on te risk attitude of te consumers, te product valuations as well as te risk of supply disruption. More specifically, wen consumers are risk neutral, te retailer sould carry enoug safety inventory to cover all demands in future period if bot te supplier reliability and consumers desirability of te product is lower tan certain tresold values; oterwise, e will carry no safety inventory at all. On te oter and, wen consumers are risk averse, keeping safety inventory to satisfy only partial of te demand is optimal under some conditions. Tird, we investigate te cost of ignoring consumer panic buying. Interestingly, we find tat te cost of ignoring consumer panic buying is te most prominent (as muc as 25% of profit loss) wen supply reliability is intermediate (more specifically, as it approaces a specific critical ratio). Tis result is in opposition to our intuition tat te cost of ignoring consumer beaviors sould be more significant wen supply reliability is lower. Finally, we study te impact of limited capacity and te effectiveness of quota policies for te retailer. We sow tat limiting quota on consumer purcases benefit te retailer only wen te retailer s capacity is below certain level. We also sow tat a quota policy is more attractive to te retailer wen consumer product valuation is ig, consumer olding cost is low, retailer olding cost is low, or consumer is risk averse. Furtermore, our numerical analysis demonstrates tat attractiveness of te fixed quota policy increases in supply reliability and selling price wen supply reliability is low, and it decreases in supply reliability and selling price wen supply reliability is sufficiently ig. Te rest of te paper is organized as follows. Section 2 reviews te related literature. Section 3 presents our model. Section 4 analyzes consumer panic buying beavior. Section 5 derives te retailer s inventory policy. Section 6 evaluates te cost of ignoring consumer beavior cange. Section 7 examines te effectiveness of a quota policy for te retailer wit capacity constraint. Finally, Section 8 concludes te paper. 2. Literature review Our work is related to te existing literature on strategic consumer beavior (e.g., Aviv and Pazgal 2008, Liu and van Ryzin 2010, van Ryzin and Liu 2008, Sen and Su 2007, Su and Zang 2008,

4 4 Article submitted to Manufacturing & Service Operations Management; manuscript no. Su 2010, and Allon and Bassamboo 2011). Sen and Su (2007) reviews customer beavior models in te revenue management and auction literatures. Aviv and Pazgal (2008) analyzes a model wit a Poisson stream of strategic customers and a single price reduction point. Te autors sow numerically tat neglecting strategic customer beavior may lead to substantial losses. Liu and van Ryzin (2010) studies te effects of capacity rationing, and finds tat competition may inder te profitability of capacity rationing. Su (2010) studies dynamic pricing problem were consumers may stock up te product at current prices for future consumption. Te autor develops a solution metodology based on rational expectations, and sows tat in equilibrium te seller may eiter carge a constant fixed price or offer periodic price promotions at predictable time intervals. Our paper differs from te above literature as we focus on consumer panic buying beavior under supply disruptions. Te beavior of a consumer wo wants to avoid future supply sortages significantly differs from te beavior of a consumer seeking lower prices. Also note tat our study is very different from te stream of literature on buying frenzies (e.g., Courty and Nasiry 2012) wic examines te impact of te uncertainty of product valuations on consumer purcasing beaviors. Our paper is also related to te researc on supply uncertainty. Tere are tree approaces to modeling supply uncertainty: random lead-time, random yield, and supply disruption. Te random lead-time model assumes tat te order lead-time is a random variable (e.g., Song and Zipkin 1996). In a random yield model, te realized supply level is a random function of te order/capacity level (see Yano and Lee 1995, Gercak and Parlar 1990, Parlar and Wang 1993, Swaminatan and Santikumar 1999, Federgruen and Yang 2008, Kazaz 2008, Wang et al. 2010). Supply disruption models te uncertainty of supply as one of two states: up or down (e.g.,tomlin (2006), Yang et al. (2009)). Te orders are fulfilled on time and in full wen te supplier is up, and no order can be fulfilled wen te supplier is down. Snyder et al. (2010) provides a compreensive review of te existing supply disruption researc. Tere are very few papers in te literature tat explicitly consider te impact of supply uncertainty on customer demand. Rong et al. (2008) considers a one-sot game interaction between an unreliable supplier and multiple retailers. Te retailers may inflate teir order quantities in order to obtain teir desired allocation from te supplier, a beavior known as te rationing game. It is sown tat no Nas Equilibrium of te retailers order quantities exists wen te supplier applies te well known proportional allocation rule. Furtermore, if te retailers make reasonable assumptions about teir competitors beavior, ten bot te bullwip effect and reverse bullwip effect occur. Rong et al. (2009) investigates tree different pricing strategies under supply disruption, namely naive, one-period correction (1PC), and regression approaces. Te autors sow tat in fact te naive pricing strategy is superior to eiter of te more sopisticated ones, bot in terms

5 Article submitted to Manufacturing & Service Operations Management; manuscript no. 5 of te firm s profit and te magnitude of te customer s order variability. However, te autors do not model customer panic buying directly. Consumer panic buying is implicitly reflected by te difference between te order quantity and te underlying demand, wic is assumed to be a linear function of te anticipation of price cange. To summarize, tere are tree key differences between our paper and te existing literature: 1) we examine te impact of different factors on consumer decisions under supply disruption, and identify te main drivers for consumers panic buying beavior; 2) we caracterize te optimal safety inventory strategy and analyze te effectiveness of quota policy, wic are bot common mitigation strategies in practice; 3) we demonstrate te substantial profit loss for te retailer if it ignores consumer beavior canges under supply disruption. 3. Model We consider a monopolist retailer tat sells a staple product to a mass of consumers over two periods: Periods 1 and 2. At te beginning of Period 1, te retailer orders Q 1 units of product (a decision variable) to sell, and leftover product at te end of Period 1 incurs a olding cost and is carried over for sale in Period 2 (let H be te olding cost rate per unit product per period). At te beginning of Period 2, te retailer as an opportunity to replenis its inventory. However, te replenisment at te beginning of Period 2 is not always successful due to supply disruption. We assume tat supply disruption occurs wit probability 1 β, i.e., te retailer can successfully replenis its inventory at te beginning of Period 2 wit probability β 1. To focus on te impact of expectation of future supply uncertainty, rater tan actual supply uncertainty, we only allow supply disruptions in Period 2 and not in Period 1. Witout loss of generality, te procurement cost for te retailer is normalized to zero and product leftover at te end of Period 2 is assumed to ave zero salvage value. We assume tat te product retail price p is exogenous and constant over te bot periods. Te assumption is valid in many situations. For example, during Hurricane Sandy in 2012 te State of New Jersey warned tat price gouging during a state of emergency is illegal and tat complaints would be investigated by te attorney general. Specifically, mercants are barred from raising prices more tan 10 percent over teir normal level during emergency conditions (Yglesias 2012). Also, during te crisis of milk powder sortage in Hong Kong, all te major milk powder providers announced publicly tat te retail prices would be fixed. Furtermore, tis simplifying 1 In tis paper we assume tat β is common information to all players (i.e., te retailer and all consumers). We acknowledge tat in reality it is possible tat te retailer may be better informed of disruption risks tan te consumers; and some consumers may be better informed tan oter consumers. Te information asymmetry likely furter complicates te interactions of players and exaggerates te consequences of supply disruption. To facilitate te analysis in tis paper, we use tis simplifying assumption and leave te information asymmetry issue to future researc.

6 6 Article submitted to Manufacturing & Service Operations Management; manuscript no. assumption of constant price also elps to isolate te impact of price canges on consumer decisions and allow us to better demonstrate te effect of supply disruptions. Tere is a large population of (N) consumers, wo possess eterogeneous product valuations. Eac consumer valuation v is drawn from a common distribution F (v). Eac consumer demands at most one unit of product for consumption in eac period (te utility increase for consuming extra products is assumed to be 0). At te beginning of eac period, te consumers ave an opportunity to purcase from te retailer. However, due to supply disruption, consumer orders may not be satisfied in Period 2, and tus a loss of consumer utility is incurred. To avoid tis utility loss, a consumer may purcase 2 units in Period 1 (i.e., stockpile 1 unit for Period 2). Neverteless, at te end of Period 1, any unused product incurs a olding cost per unit for te consumer. Te cronology of events is as follows: 1. At te beginning of Period 1, te retailer decides ow muc to order initially Q 1 from its supplier. Tis order will be received in full by te retailer immediately. 2. Eac consumer decides is purcase quantity z 1 {0, 1, 2} in Period During Period 1, eac consumer uses min{1, z 1 } units of product. Te leftover product, if any, incurs a olding cost. 4. At te beginning of Period 2, te retailer decides ow muc to order Q 2 from its supplier. Due to supply disruption, tis second order may not be fulfilled. 5. Eac consumer decides is purcase quantity z 2 {0, 1} in Period 2. However, fulfillment of tese orders depends on te availability of retailer inventory. In particular, we assume te fulfillment rate at te beginning of Period 2 is α, wic depends on te probability of supply disruption and te retailer s inventory decision. We furter assume te all consumers ave rational expectations on tis fulfillment rate, wic is a common assumption in te existing literature (e.g., Liu and Ryzin 2008, Su and Zang 2009). 6. During Period 2, consumers consume any on and product. We assume te retailer is risk neutral, since in general retailers ave large and well-diversified product portfolios. We assume tat consumers are risk averse (te risk neutral case is considered as a special case). In particular, we assume a consumer wit valuation v as a utility function U v (x) = U(x r v ), were U( ) is an increasing concave function wit U(0) = 0 (te risk neutral case is a special case wit U(x) = x). We also assume tat U(x y) lim = 1, (1) x + U(x) for any real number y. Many increasing concave utility functions satisfy Eqn. (1), suc as U(x) = 1 exp( rx) (0 < r < + ), U(x) = x γ (0 < γ 1), etc. In U v ( ), r v is te reference point (please see Prospect Teory of Kaneman and Tversky 1979 for details). We assume tat { v p, if v > p, r v = 0, if v p. (2)

7 Article submitted to Manufacturing & Service Operations Management; manuscript no. 7 Table 1 Notation H () Retailer s (consumers ) olding cost rate per unit per period p Regular selling price of te product N Number of consumers v A consumer s valuation to te product F (v) Distribution of consumer valuation v Te igest valuation wen F (v) is assumed to be uniform distribution S Supply status; S = 0 means supply disruption wile S = 1 means full order fulfillment β Supply reliability (te probability of S = 1) z 1 (z 2 ) A consumer s purcasing quantity in Period 1 (Period 2) Q 1 (Q 2 ) Te retailer s order quantity in Period 1 (Period 2) Q 1 (Q 2) Te retailer s equilibrium order quantity in Period 1 (Period 2) U v ( ) Te utility function of a consumer wit valuation v r v Te reference point of a consumer wit valuation v γ A parameter measures consumers degree of risk aversion α (α c, α ) Te fulfill rate (consumers belief on te fulfill rate, te equilibrium fulfill rate) in Period 2 u( ) A consumer s expected utility T (p,, α c ) Te tresold of consumer valuation v for panic buying D 1 Total consumer demand in Period 1 D 2 Total consumer demand in Period 2 Π( ) Π Te retailer s (optimal) expected profit under te basic model Q i1 (Q i2 ) Te retailer s order quantity at te beginning of Period 1 (Period 2) under Case i Π i ( ) Π i Te retailer s (optimal) expected profit under Case i i = I: te case of ignoring consumer beaviors; i = f: te case of fixed quota; P L Te retailer s profit loss of ignoring consumer beaviors K Te retailer s capacity δ Te retailer s capacity over demand of Period 1 θ Te percentage of Type-2 consumers served in Period 1 wit uniform arrival assumption x Te yield proportion in te random proportional yield model µ Te mean of x, E(x) = µ B(x) Te distribution of x All of tese parameters and distribution functions are common knowledge to te retailer and consumers. We define te notation F (x) = 1 F (x) and x + = max{x, 0}. Table 1 provides a summary of te notation in tis paper. 4. Consumers purcasing decisions In tis section we analyze consumers panic buying beavior. For a given belief about te fulfill rate in Period 2, we derive consumers optimal purcasing decisions in eac period. Note tat based on te rational expectations assumption, all of te consumers beliefs are te same and consistent wit te realized fulfillment rate in equilibrium. Hence, we only need to consider te case were consumers ave omogeneous beliefs over te fulfillment rate. Let α c be te consumers belief about te fulfillment rate and S be a random variable indicating weter a consumer receives is order in Period 2. S = 1 means tat te consumer receives is order in Period 2 and S = 0 means e does not. Consider a consumer wit valuation v. Recall tat

8 8 Article submitted to Manufacturing & Service Operations Management; manuscript no. eac consumer demands at most one unit of product for consumption during eac period. Clearly, if v p, ten te consumer will not buy any product in Periods 1 or 2. Wen v > p, a consumer s purcasing quantity at te beginning of Period 1, z 1, is eiter 1 or 2. If e purcases one unit in Period 1, ten e will also purcase one unit in Period 2; if e purcases two units in Period 1, ten e will not purcase anyting in Period 2. In oter words, is purcasing quantity at te beginning of Period 2, z 2, equals 2 z 1. Te consumer s payoff is { 2(v p) (z1 1), if S = 1, π = (v p)z 1 (z 1 1), if S = 0. (3) In te case of S = 1, since te consumer gets wat e purcases in bot periods, te utility from consumption is always 2(v p) regardless of z 1. In te case of S = 0, te utility from consumption is (v p)z 1. Te term (z 1 1) is te consumer s olding cost. For a given belief about te fulfillment rate α c, te consumer s expected utility wit z 1 and z 2 = 2 z 1 is ( ) ( ) u(z 1 ) = E S U v (π r v ) = α c U (v p) (z 1 1) + (1 α c )U (v p )(z 1 1). (4) We identify te optimal purcasing policy for consumers in te following teorem. Teorem 1. For a given belief about te fulfill rate in Period 2 α c, tere exists a tresold T (p,, α c ) p + wit U(T (p,, α c ) p ) U(T (p,, α c ) p) = α c, (5) suc tat te optimal purcasing policy for te consumer wit valuation v is as follows: 1. If v p, ten z1(α c ) = z2(α c ) = 0; 2. If p < v T (p,, α c ), ten z1(α c ) = z2(α c ) = 1; 3. If v > T (p,, α c ), ten z1(α c ) = 2, z2(α c ) = 0. Proof. Case 1 is straigtforward, so we focus on te proofs of Cases 2 and 3. By Eqn. (4), te consumer s expected utility wen purcasing one unit in bot periods and two units in Period 1 only are respectively u(1) = α c U(v p), u(2) = U(v p ). Tus, u(1) > (<)u(2) is equivalent to U(v p ) U(v p) < (>)α c.

9 Article submitted to Manufacturing & Service Operations Management; manuscript no. 9 Let g(v) = U(v p )/U(v p). g(v) is continuous and monotonically increasing in v since U(x) is an increasing concave function. In addition, we ave g(p + ) = 0 and lim v + g(v) = 1 by Assumption (1). Terefore, tere exists a number T (p,, α c ) p + wit U(T (p,, α c ) p ) U(T (p,, α c ) p) = α c, suc tat te consumer will purcase two units in Period 1 if v > T (p,, α c ), and e will purcase one unit during eac period if p < v T (p,, α c ). Teorem 1 identifies a tresold policy for consumers, wic is quite intuitive. Wen te consumer valuation is low, e will not buy any product. Wen te consumer valuation is intermediate, e will only buy one unit product in eac period if available. Wen te consumer valuation is ig, e will stockpile one unit of product in Period 1. Next we examine some properties of te tresold T (p,, α c ). Te proofs of te following two propositions are found in te appendix. Proposition 1. T (p,, α c ) is increasing in p, and α c. Proposition 1 indicates tat consumers are more likely to stockpile product wen te selling price is low, te consumer olding cost is low, or consumer confidence about availability in Period 2 is low. Next we investigate ow te tresold T (p,, α c ) canges wit te degree of consumer risk aversion. According to Pratt (1964), an increase in te degree of risk aversion can be represented by an increasing concave transformation. Let W ( ) = ϕ ( U( ) ), were ϕ is an increasing concave function wit ϕ(0) = 0. Tis utility function W ( ) represents a iger degree of risk aversion. Let T U, T W proposition. be te tresolds corresponding to U( ), W ( ) respectively. We obtain te following Proposition 2. T U T W, tat is, a consumer wit a iger degree of risk aversion is more likely to purcase 2 units at te beginning of Period 1. Proposition 2 implies tat more risk averse consumers are more likely to stockpile product. 5. Retailer s Optimal Inventory Policy We conduct equilibrium analysis in tis section to study te retailer s optimal inventory policy and te cost of ignoring consumer panic buying beavior. To study strategic interactions between te retailer and consumers, and strategic interactions among te consumers, we will adopt te framework of rational expectations (RE) equilibrium. Tis concept of RE was originally proposed by Jon F. Mut (1961) and ave been widely used in economics models to study ow a large number of individuals, firms and organizations make coices under uncertainty. Te basic idea of RE is tat te outcomes of te model do not differ systematically from wat players in te model

10 10 Article submitted to Manufacturing & Service Operations Management; manuscript no. expected tem to be. Te RE concept is also commonly used in te recent emerging strategic consumer beavior literature in operations management (e.g., Liu and Ryzin 2008, Su and Zang 2009). Te implication of RE in our model will be discussed in Definition 1. From Teorem 1, we know tat given fulfillment rate belief α c, te total consumer demand in Period 1 is D 1 = NF (p) NF ( T (p,, α c ) ) + 2NF ( T (p,, α c ) = NF (p) + NF ( T (p,, α c ) ), (6) and in Period 2 is D 2 = NF (p) NF ( T (p,, α c ) ). (7) Tus, te retailer s expected profit from ordering Q 1 (Q 2 ) during period 1(2) is Π(Q 1, Q 2, α c ) = p min{d 1, Q 1 } H [ Q 1 D 1 ] + + βp min{q 2 + [ Q 1 D 1 ] +, D2 } + (1 β)p [ Q 1 D 1 ] +, (8) were β is te probability tat te supplier is up in Period 2 and T (p,, α c ) is defined in (5). In Eqn. (8), te first term represents te revenue in Period 1, te second term represents te olding cost, te tird term represents te expected revenue wen te supplier is up in Period 2, and te last term represents te expected revenue wen te supplier is down in Period 2. For any given α c, let Q 1(α c ), Q 2(α c ) be te optimal order quantities tat maximize (8). Because te total consumer demand over bot periods is 2NF (p), if D 1 > Q 1(α c ) ten te retailer sould order Q 2(α c ) = 2NF (p) D 1, and if D 1 Q 1(α c ) ten te retailer sould order Q 2(α c ) = 2NF (p) Q 1(α c ). In oter words, Q 2(α c ) = 2NF (p) max{d 1, Q 1(α c )}. Based on tis reasoning, we omit te decision Q 2 and only consider te decision Q 1 for te rest of tis paper. Next we introduce te definition of a RE equilibrium. Definition 1. A RE equilibrium satisfies te following conditions: (i) Given belief α c, te optimal purcasing policy for a consumer wit valuation v is z 1 = z 1(α c ), z 2 = z 2(α c ), were z 1(α c ) and z 2(α c ) are defined in Teorem 1. (ii) Given belief α c, te retailer s optimal order quantities satisfy Q 1 = Q 1(α c ) and Q 2 = Q 2(α c ) = 2NF (p) max{d 1, Q 1(α c )}. (iii) Te beliefs of all consumers α c are consistent wit te realized fulfillment rate α, i.e., { [Q 1(α ] + c ) D 1 } α c = α = β + (1 β) min, 1. (9) D 2 In Eqn. (9), te term [ Q 1(α c ) D 1 ] + represents te retailer s inventory at te beginning of Period 2, so te ratio between tis term and te total demand in Period 2 is te fulfillment rate

11 Article submitted to Manufacturing & Service Operations Management; manuscript no. 11 wen a supply disruption occurs. Since α c = α at te equilibrium, we use α to replace α c ereafter. Furtermore, we write Q 1(α c ) as Q 1 for simplicity. Next we describe te retailer s inventory policy in a RE equilibrium. First, we sow tat te retailer s initial order quantity Q 1 is not greater tan 2NF (p), te total demand of bot periods. Second, we sow tat te retailer s initial order quantity Q 1 sould be greater tan or equal to [ ] D 1, te total demand in Period 1. Tus, te retailer must coose Q 1 D 1, 2NF (p) to maximize profit. We see tat te RE equilibrium can be computed by solving te following optimization problem: max Q 1,T,α Π = 2pNF (p)β + pq 1(1 β) H[Q 1 D 1 ] (10) ] + { [Q 1 D 1 } s.t. α = β + (1 β) min, 1, (11) D 2 αu(t p) = U(T p ), (12) Q 1 [D 1, 2NF (p)]. (13) Te objective function (10) is te simplified form of te retailer s expected profit in Eqn. (8) based on te coices Q 1 [D 1, 2NF (p)] and Q 2 = 2NF (p) Q 1. Constraint (11) follows from Eqn. (9) of te RE equilibrium definition, constraint (12) comes from Eqn.(5), te definition of te tresold T, and constraint (13) is due to te discussion before Eqn. (10). We omit te constraint T [T (p,, β), + ) because it is implied by constraints (11)-(13). Substituting (11) and (12) into te objective function (10), we ave te following problem: [ max Π(T ) = 2pNF (p) N p Hβ ( T 1 β p H 1 β ) U(T p ) U(T p) ] [F (T ) F (p)] (14) s.t. T [ T (p,, β), + ). (15) Once te optimal solution to Problem (14)-(15), T, is obtained, we can easily calculate te realized fulfillment rate and te optimal order quantity Q 1: x γ α = U(T p )/U(T p), (16) Q 1 = α β 1 β N[F (T ) F (p)] + N[F (T ) + F (p)]. (17) We will assume for te rest of tis paper tat consumers possess a power utility function U(x) = (0 < γ 1), wic is a common utility function representing risk aversion in economics and operations management literature (e.g., Liu and Ryzin 2008). A lower value of γ represents a iger degree of risk aversion. In addition, we assume consumer valuations are uniformly distributed on [0, v], were te average consumer valuation is v/2. Te quantitity v can be interpreted as a

12 12 Article submitted to Manufacturing & Service Operations Management; manuscript no. measure of consumers desirability for te product. To avoid trivial cases, we assume tat v > p. Based on Eqn.(5), we ave T (p,, α) = p + /(1 α γ 1 ), and Problem (14)-(15) becomes { 2Np(v p) ( T ) max Π(T ) = N min T v v, 1 p } { ( H + p H ) [ ( 1 1 ) γ ] } (18) v 1 β T p [ s.t. T p + ), +. (19) 1 β γ 1 Tere are two interesting questions: (1) Under wat conditions sould te retailer carry inventory at te end of Period 1 to increase te fulfillment rate in Period 2? If so, ow muc inventory sould te retailer carry over? (2) How do consumer risk preferences affect te retailer s inventory decision? Te following two teorems answer tese questions. Teorem 2. Suppose consumers are risk neutral, i.e., γ = 1. Te retailer s optimal order quantity Q 1 and te realized fulfillment rate α are (i) Q 1 = N[F (p) + F (p + 1 β )], α = β, if (β, v) Ω n 1 ; (ii) Q 1 = 2NF (p), α = 1, if (β, v) Ω n 2, were Ω = {(β, v) 0 β 1, p v + }, Ω n 2 = {(β, v) β < 1 H p, v < p(h+) H } and Ωn 1 = Ω Ω n 2. Proof. We consider two cases: β > 1 H/p and β 1 H/p. Case (a): β > 1 H/p. In tis case, we ave p < H/(1 β). It can be verified tat Π(T ) in (18) is decreasing wit respect to T, so it is optimal to coose T = T (p,, β). Tus α = β and Q 1 = N[F (T (p,, β)) + F (p)] = N[2 2p v max T 2Np(v p) v ]. (1 β)v Case (b): β 1 H/p. Problem (18)-(19) can be split into two subproblems: { ( Π 1 (T ) = H + p N(v p) v H 1 β )[ 1 (1 ]} )γ T p s.t. T [v, + ), (21) { 2Np(v p) N(T p) ( max Π 2 (T ) = H + p H )[ ( 1 1 ) γ ] } (22) T v v 1 β T p [ s.t. T p + ], v. (23) 1 β γ 1 First consider te optimal solution of Subproblem (20)-(21). It is clear tat Π 1 (T ) as defined in (21) is increasing wit respect to T. Tus, te optimal solution to Problem (20)-(21) is T 1 = + and te optimal value is Π 1 = Π 1 (T 1 ) = 2Np(v p) v (20) NH(v p). (24) v Now consider Subproblem (22)-(23). Taking first and second derivatives of Π 2(T ) in (22), we obtain Π 2(T ) = N v { H ( p H ) [ ( 1 ) γ 1 ( 1 1 β T p (1 γ) ) 1] }. (25) T p

13 Article submitted to Manufacturing & Service Operations Management; manuscript no. 13 and Π 2 (T ) < 0. Since γ = 1, we ave Π 2(T ) < 0 for all T > p +, wic means tat te optimal solution of Subproblem (22)-(23) is T2 = p +, and te optimal value is 1 β If p v < p + Π 2 = Π 2 (T 2 ) = 2Np(v p) v Np v. (26), ten Subproblem (22)-(23) vanises and te optimal solution is T = +. If 1 β p + v < p(h+), ten 1 β H Π 1 > Π 2, wic implies T = T1 = +. If v p(h+), ten H Π 1 Π 2 and T = T 2 = p + 1 β. Teorem 2 describes te retailer s optimal policy wen facing risk neutral consumers: wen bot te supply reliability and consumer product valuations are lower tan certain tresolds, te retailer sould order enoug inventory in period 1 to cover all demands in two periods. Oterwise, it will just order inventory to satisfy period 1 demand and carry no extra inventory over to period 2 at all. Fig. 1 provides an illustration wit retail price p = 5, te retailer s olding cost H = 1, and te consumers olding cost = 0.5. Te X-axis is supply reliability β and te Y-axis is maximum consumer product valuation v, wic reflects consumer desirability for te product. Te two-dimensional area of (β, v) is divided into two regions. In te lower left region, te retailer will carry inventory from Period 1 to fulfill all consumer demand in Period 2. In te oter region, te retailer will carry no inventory from Period 1 to Period 2. Te intuition of Teorem 2 is as follows. First, wen te supplier reliability β is ig, te retailer is most likely to receive is second inventory order to fulfill consumer demand in Period 2, tus it does not ave incentive to carry extra inventory. Second, wen te supplier reliability β is low but te consumers valuation of te product is ig, more consumers will coose to stockpile inventory in Period 1, wic again provides no incentive for te retailer to carry safety inventory into Period 2. It is wen bot te supplier reliability β and te consumers valuation of te product is low, will te retailer coose to carry safety inventory. Moreover, since in suc case te expected stockout cost always outweigs te retailer s olding cost, te retailer will carry enoug safety inventory to meet all demands in Period 2. Te following proposition sows tat te retailer sould not carry inventory wen supply reliability is ig, even under only mild conditions on te distribution of consumer valuation and te consumer utility functions. Proposition 3. For any consumer valuation distribution F (v) and any increasing concave utility function U(x), if (1 β)p < H ten te optimal order quantity and fulfillment rate are Q 1 = N[F (T (p,, β)) + F (p)] and α = β.

14 14 Article submitted to Manufacturing & Service Operations Management; manuscript no. Figure 1 10 Te optimal inventory policy for te retailer facing risk neutral consumers Consumer Desirability Q1=Period 1 demand Q1=total demand Supply reliability Next we discuss te retailer s optimal order quantity wen consumers are risk averse. Before stating Teorem 3, we introduce some more notation. Let β t be te solution of H (p Let T o be te solution of Π 2(T ) = 0 in (25), i.e., ( H p H 1 β )[γβ1 1 γ + (1 γ)β 1] = 0. (27) H ) [ ( 1 ) γ 1 ( 1 1 β T p (1 γ) ) ] 1 = 0. (28) T p Denote and v t (β) = { p + T o p H {H + (p H 1 β p + p(1 β), if β β H(1 β 1γ t, ) )[1 (1 { } Ω = (β, v) 0 β 1, p v +, { Ω v 2 = (β, v) 0 β < 1 H } p, v < v t(β), { } Ω v 3 = (β, v) 0 β β t, v > v t (β), Ω v 1 = Ω ( Ω v 2 Ω v 3). T o p )γ ]}, if β < β t, (29) Teorem 3. Suppose te consumers are risk averse wit 0 < γ < 1. Te retailer s optimal order quantity and fulfillment rate follow: [ (i) If (β, v) Ω v 1, ten Q 1 = N F (p) + F ( p + (ii) If (β, v) Ω v 2, ten Q 1 = 2NF (p), α = 1; ) ], α = β;

15 Article submitted to Manufacturing & Service Operations Management; manuscript no. 15 (iii) If (β, v) Ω v 3, ten Q 1 = α β 1 β N[ F (T o ) F (p) ] + N [ F (T o ) + F (p) ] ( [ N F (p) + F (p + ] ) ), 2NF (p), 1 β γ 1 α = (T o p) γ (β, 1). (T o p ) γ Proof. Tere are tree cases: (a) β > 1 H p, (b) β t < β 1 H p, (c) 0 < β β t. For Case (a), we ave Q 1 = N[F (T (p,, β)) + F (p)] and α = β by Proposition 3. For Case (b), we ave β > β t and X(β, γ, p, H) = Π (p + ) = N 1 β γ 1 v we ave T o < p + {H (p H 1 β )[γβ1 1 γ + (1 γ)β 1]} < 0, (30). (It can be verified tat X(β, γ, p, H) is decreasing wit respect to β, lim β 0 X(β, γ, p, H) = + and X(1 H/p, γ, p, H) = NH/v < 0. Tus, β t is in (0, 1 H ).) Recall p tat Problem (18)-(19) can be split into Subproblems (20)-(21) and (22)-(23). By a similar argument to te proof of Teorem 2, te optimal solution to Subproblem (20)-(21) is T 1 = + and te optimal value is Π 1 in (24). Te optimal solution to Subproblem (22)-(23) is T 2 = p + and te optimal value is Π 2 = Π 2 (T2 ) = 2Np(v p) Np(1 β). v v(1 β 1γ ) If v < p +, ten Subproblem (22)-(23) vanises and te optimal solution is T = T1 = +. If p + v < v t (β), ten Π 1 > Π 2, wic indicates T = T 1 = +. If v v t (β), ten Π 1 Π 2, wic implies tat T = T 2 = p + In Case (c), it can be verified tat T o p +.. Te optimal solution of Subproblem (22)-(23) becomes T 2 = min{t o, v}. If v < T o, ten T 2 = v. Tus, Π 1 = Π 1 (+ ) > Π 1 (v) = Π 2 (v) = Π 2, wic indicates tat T = T 1 = +. If T o v < v t (β), ten T 2 = T o Π 1 = Π 1 (+ ) > Π 2 (T o ) = Π 2 (implied by T o v) and (implied by v < v t (β)). Tus, T = T 1 = +. If v v t (β), ten T 2 = T o and Π 1 = Π 1 (+ ) Π 2 (T o ) = Π 2. Tus, T = T 2 = T o. Teorem 3 describes te optimal inventory policy for risk averse consumers: wen te supply reliability is ig, te retailer sould not carry inventory at te end of Period 1; wen te supply reliability is low and consumer product valuations are low, te retailer sould carry inventory at te end of Period 1 to cover all demand in Period 2; oterwise, it sould carry inventory to cover partial demand in Period 2. Fig. 2 displays a numerical example wit retail price p = 5, retailer olding cost H = 1, consumer olding cost = 0.5, and consumer risk aversion measure γ = 0.5. In Fig. 2, te two-dimensional area of (β, v) is divided into 3 regions. In te lower-left region, te retailer carries inventory at te end of Period 1 to cover all te demand in Period 2 and te realized fulfillment rate in Period 2 is 1. In te upper-left region, te retailer carries some inventory, but

16 16 Article submitted to Manufacturing & Service Operations Management; manuscript no. Figure 2 10 Te optimal inventory policy for te retailer facing risk averse consumers 9 Consumer Desirability Period 1 demand<q1<total demand Q1=total demand Q1=Period 1 demand Supply reliability does not cover all demand in Period 2, so te realized fulfillment rate in Period 2 is between β and 1. In te rigt region, te retailer does not carry any inventory at te end of Period 1 and te realized fulfillment rate in Period 2 is β. Te interpretation of tis result is as follows. Te retailer considers te trade-off between two types of costs wen making ordering decisions: te inventory olding cost and te stockout cost caused by supply disruption. If te retailer increases te inventory at te end of Period 1 by one unit, ten it incurs extra olding cost H but reduces stockout cost p wit probability (1 β). Furtermore, increasing te inventory at te end of Period 1 will also increase te fulfillment rate α in Period 2, wic decreases te number of consumers wo purcase two units of product in Period 1. Hence, increasing inventory may increase te retailer s olding cost by more tan H per unit. If te supplier reliability is ig, te reduced stockout cost by carrying one unit of inventory (1 β)p is less tan te increased olding cost. Tus, te retailer as no incentive to carry inventory at te end of Period 1. Now we consider low supply reliability and ig stockout cost. Wen consumer desirability for te product is low, not many consumers purcase two units of product in Period 1. Even if te retailer carries inventory to cover all demand in Period 2 (wic increases te fulfillment rate in Period 2 to 1), te demand of Period 1 does not significantly decrease. Te extra olding cost due to reducing consumer panic buying incentives and srinking te demand in Period 1 is relatively small. Hence, te retailer carries inventory at te end of Period 1 to cover all demand in Period 2. Wen consumer desirability for te product is ig, tere are a lot of consumers panic buying in Period 1. Te demand srinks more dramatically and te olding cost increases more significantly wen te inventory at te end of Period 1 is greater. Tere is a balance point of

17 Article submitted to Manufacturing & Service Operations Management; manuscript no. 17 inventory at te end of Period 1, were te reduced stockout cost is equal to te marginal olding cost. Terefore te retailer tends to carry less inventory to cover partial demand in Period 2 wen te consumer desirability for te product is ig. Comparing Teorems 2 and 3, we see tat consumers risk preferences influence te retailer ordering decisions. Interestingly, wen supply reliability is low and consumer desirability for te product is ig, te retailer does not carry any safety inventory if consumers are risk neutral but it does carry safety inventory if consumers are risk averse. 6. Cost of ignoring consumer beaviors In tis section we answer te following question: wen will ignoring consumer beaviors ave severe consequences for te retailer? To evaluate te cost of ignoring consumer beaviors, we first derive te optimal order quantity wen te retailer ignores consumer panic buying. Wit tis assumption, te retailer cooses Q I1 and Q I2 (te subscript I represents te case of ignoring consumer beavior) to maximize te expected profit Π I (Q I1, Q I2 ) = pq I1 H[Q I1 NF (p)] + βpq I2. (31) Clearly, for any given Q I1, it is optimal to order Q I2 = 2NF (p) Q I1. If we replace Q I2 by 2NF (p) Q I1, te objective function in Eqn. (31) is linear in Q I1. Tus te optimal initial order quantity is Q I1 = NF (p), if β > 1 H ; and p Q I1 = 2NF (p), if β 1 H. Te following teorem compares p te optimal order quantities for wen te retailer ignores consumer panic buying and wen it does not. Teorem 4. (a) Suppose γ = 1. (i) Q 1 = Q I1, if (β, v) Ω n 2 (ii) Q 1 Q I1, if (β, v) {(β, v) Ω n 1 v p + and (β, v) {(β, v) Ω n 1 v < p + 1 β }; wen β (0, 1 H p ), and decreasing in β wen β (1 H p, 1 v p ). (b) Suppose 0 < γ < 1. (i) Q 1 = Q I1, if (β, v) Ω v 2 and (β, v) {(β, v) Ω v 1 v < p + (ii) Q 1 Q I1, if (β, v) Ω v 3 and (β, v) {(β, v) Ω v 1 v p + }. Furtermore, 1 β Q 1 Q I1 is increasing in β 1 β γ 1 increasing in β wen β (β t, 1 H p ), and decreasing in β wen β (1 H p, (1 }; }. Furtermore, Q 1 Q I1 is v p )γ ). Teorem 4 sows tat te retailer can ignore consumer panic buying beavior wen te supply reliability is ig or te consumer desirability for te product is low. Oterwise, te retailer sould take consumer beavior into account. As in Figs. 1 (or 2) wen (β, v) lies in te area below te curve max{p(h + )/H, p + /(1 β)} (or, max{v t (β), p + /(1 β 1 γ )}), te retailer can ignore consumer beavior. Wen (β, v) lies in te area above tis curve, te retailer sould acknowledge

18 18 Article submitted to Manufacturing & Service Operations Management; manuscript no. consumer beaviors. Teorem 4 also sows tat te deviation between ignoring consumer beaviors and acknowledging tem is te most significant wen supply reliability is close to 1 H p. Hence, we define a critical ratio, ˆβ, wic satisfies ˆβ = 1 H p. Fig. 3 illustrates tese results for parameters p = 5, H = 1, = 0.5 N = 100 and v = 10. We now consider te loss in profit from ignoring consumer beavior, and we consider ow tis loss canges wit model parameters. Define te percentage of profit loss (P L) caused by ignoring consumer beavior as P L = Π Π I Π, (32) were Π is te retailer s optimal profit wen considering consumer beavior and Π I is te retailer s profit wen it ignores consumer beavior (but consumer beavior still actually exists, i.e., te corresponding profit wen coosing Q I1). Wen te retailer s order quantity is less tan te demand in Period 1, tere will be lost sales. In tis case, te consumer arrival sequence affects te retailer s expected profit. We assume a uniform arrival sequence for te consumers (refer to te next section for te details of tis assumption). Wen stockout occurs in Period 1, te retailer s expected profit is calculated based on a uniform arrival sequence. We can sow tat te cost of ignoring consumer beavior is te igest wen te supply reliability approaces te critical ratio ˆβ. Te following numerical examples demonstrate ow P L canges wit supply reliability, consumers desirability for te product, and te degree of consumer risk aversion. Te parameter values p = 5, H = 1, and = 0.5 are used in te following examples. Table 2. Te percentage of profit loss caused by ignoring consumer beavior versus supply reliability and consumer desirability for te product; γ = 0.5. v β PL % 4.2% 5% 5.5% 15.8% 5.8% % 6.6% 7.1% 7.4% 22% 17.3% % 7.6% 7.9% 8.1% 24.1% 20.8% 10.9% % 8.1% 8.4% 8.5% 25.3% 22.5% 15.6% 0

19 Article submitted to Manufacturing & Service Operations Management; manuscript no. 19 Table 3. Te percentage of profit loss caused by ignoring consumer beavior versus degree of consumer risk aversion; v = 10. γ β PL % 8.5% 8.1% 7.8% 7.4% 6.9% 6.5% 6.1% 5.7% % % 23.3% 22% 20.6% 19.1% 17.4% 15.7% Wen supply reliability is eiter ig or low, te percentage of profit loss caused by ignoring consumer beavior is relatively low. Neverteless, wen supply reliability is moderate (around ˆβ), te percentage of profit loss can be very ig (it can be as large as 25% as sown in Table 2). We interpret tis observation as follows. Wen supply reliability is ig, consumer panic buying beavior is not significant and ignoring consumer beavior will not incur muc loss. Wen supply reliability is low, many consumers tend to purcase two units of product in Period 1. If te retailer accounts for consumer beavior, te retailer will place a large order for Period 1 to satisfy te demand, or te retailer will put a lot of product into inventory to edge against te risk of supply disruption in Period 2. Coincidentally, if te retailer ignores consumer beavior ten it is also optimal for te retailer to make an order before Period 1 to satisfy demand in bot periods. Terefore, wen supply reliability is low, te cost of ignoring consumer beavior is not very ig. Wen supply reliability is moderate, ignoring consumer beavior makes te retailer incur iger olding costs (if β < ˆβ) or iger stockout costs (β > ˆβ). As we can see, te stockout cost p is muc iger tan te olding cost H. It follows tat te percentage of profit loss is muc iger wen β approaces ˆβ from above tan from below. Te percentage of profit loss caused by ignoring consumer beavior is increasing wit respect to consumer desirability for te product. Wen consumer desirability for te product increases, consumer beavior becomes more significant to te retailer. Tus, te percentage of profit loss caused by ignoring consumer beavior is larger in tis case. Finally, te percentage of profit loss caused by ignoring consumer beavior is iger wen te degree of consumer risk aversion is iger. 7. Capacity constraint and Quota Policy In tis section, we consider te case were te retailer ave limited capacity and examine te resultant canges to te retailer s inventory decision. More importantly, we study te mecanism of quota policy (i.e., limiting te maximum number of units tat eac consumer can purcase) and evaluate its effectiveness for dealing wit consumer panic buying beavior under limited-capacity case.

20 20 Article submitted to Manufacturing & Service Operations Management; manuscript no. Tere are various practical reasons to consider a capacity constraint, e.g., te vendor as limited product supply or te retailer as limited storage space. Wen tere is supply sortage and consumer panic buying beavior, quota policy as been often observed in practice. For example, during te milk powder crisis in Hong Kong, eac consumer from mainland Cina can only purcase a maximum of two tins of milk powder; e/se would be eavily penalized if carrying more tan two tins troug te border between Hong Kong and mainland. 7.1 Impact of Capacity Constraint on Retailer s Inventory Decision We suppose tat te retailer s order cannot exceed K units during eac period. To focus on te non-trivial cases, we assume tat te capacity K is greater tan or equal to te one-period demand witout consumer panic buying, i.e., K NF (p). ( First consider te case were K N F (p) + F ( p + ) ). In tis case, te retailer can satisfy all consumer demand in Period 1. Te retailer must decide weter or not to carry inventory to increase te fulfillment rate in Period 2 and ow muc inventory to carry. We find tat te managerial insigts ere are similar to tose in Section 5, so we omit te discussion ere. ( Next, consider te case were NF (p) K < N F (p) + F ( ) ) p +. Te retailer cannot meet all consumer demand in Period 1 so te retailer as lost sales in Period 1. We assume tat consumers wo were not able to purcase in Period 1 will still come to te retailer to purcase in Period 2. From te retailer s point of view, tere are two types of consumers: consumers purcasing one unit of product in Period 1 (we refer to tese consumers as Type-1 consumers), and consumers purcasing two units of product in Period 1 (Type-2 consumers). Under Constraint (33), te retailer cannot serve all consumers in Period 1. Altoug te retailer always sells K units of product in Period 1, te arrival sequence of consumers does affect te demand in Period 2. If all te Type-2 consumers arrive before te Type-1 consumers, ten te demand in Period 2 will be lower tan it would be if all te Type-1 consumers arrived before te Type-2 consumers. To simplify our exposition, denote K = δnf (p). Hereafter we focus on te case were F ( ) p + 1 β γ 1 1 δ < 1 +. (33) F (p) We assume a uniform arrival distribution for consumers as follows. Suppose te proportion of Type-2 consumers among te wole consumer population is ξ and so far x consumers ave arrived. Among tose x consumers, ξx are Type-2 consumers and (1 ξ)x are Type-1. Under a uniform arrival pattern, for a large population of consumers (i.e., large N), te number of Type-2 consumers served in Period 1 is K ( NF (p) + NF p + ) NF (p) ( F p + F (p) ) ( F p + = K ( F (p) + F p + ) ). (34)