Dynamic pricing for inventories with reference price effects

Transcription

1 Discussion Paper No June 15, 2018 hp:// Please cie he corresponding Journal Aricle a hp:// Dynamic pricing for invenories wih reference price effecs Régis Chenavaz and Corina Paraschiv Absrac This aricle presens a dynamic pricing model of a reailer selling an invenory, accouning for consumer behavior. The auhors propose an opimal conrol model, maximizing he ineremporal profi wih consumers sensiive o he selling price and o a reference price. The opimal dynamic pricing policy is solved wih Ponryagin s maximum principle wih a srucural (general demand funcion. They obain an original pricing rule, which explicily accouns for he impac of price and invenory on fuure profis. The dynamics of price do no have o imiae he dynamics of he reference price. Insead, he dynamics of price are ied o opposing effecs linked o his reference price. The auhors also discuss managerial implicaions wih regards o behavioral pricing policies. JEL C61 D03 D40 M21 M37 Keywords Dynamic pricing; invenory; reference price; behavioral pricing; opimal conrol Auhors Régis Chenavaz, Kedge Business School, Aix-Marseille School of Economics, CNRS & EHESS, France, regis.chenavaz@kedgebs.com Corina Paraschiv, Paris Descares Universiy & Insiu Universiaire de France, Paris, France Ciaion Régis Chenavaz and Corina Paraschiv (2018. Dynamic pricing for invenories wih reference price effecs. Economics Discussion Papers, No , Kiel Insiue for he World Economy. hp:// Received May 12, 2018 Acceped as Economics Discussion Paper June 8, 2018 Published June 15, 2018 Auhor(s Licensed under he Creaive Commons License - Aribuion 4.0 Inernaional (CC BY 4.0

2 1 Inroducion Dynamic pricing is a common pracice where sellers adjus prices for producs or services based on curren demand. Developed iniially by he airline indusry wih yield managemen, dynamic pricing is now a wide-spread phenomenon (Den Boer, Recen echnological evoluions concerning real-ime access o demand-relaed daa (big daa, real-ime daa processing, pricing algorihm auomaizaion (allowing for real-ime compuaion of opimal prices, and new echnologies o implemen frequen price changes (smar labels conribue o dynamic pricing diffusion in he reailing indusry. A reailer is radiionally confroned wih invenory managemen issues, for which price is a main conrol (Goyal and Giri, 2001; Elmaghraby and Keskinocak, 2003; Bakker e al., New dynamic pricing models need o be developed ha also address invenory managemen issues ogeher wih acual consumer behavior. This poin moivaes our research. In his aricle, we sudy he dynamic pricing policy of a reailer selling an invenory when he reference price of consumers plays a role. In he modeling, we assume a fixed invenory wihou possibiliy of replenishmen. This assumpion describes well a wide range of reail producs: markdown producs, producs a he end of heir life cycle wih excess invenory, or producs wih a shor life cycle and long procuremen delays (Goyal and Giri, 2001; Elmaghraby and Keskinocak, 2003; Bakker e al., In such siuaions, he reailer needs o make pricing decisions given a fixed amoun of invenory and a limied ime horizon. Concerning consumer behavior, we posi a reference price effec. Consumer decisions o purchase a produc depend no only on he reail (or selling price, bu also on is reference price. The reference price is a psychological price used as a benchmark agains which consumers compare he reail price (Sorger, 1988; Mazumdar e al., A reail price below he reference price is perceived by consumers as a gain, which increases demand. Conversely, a reail price above he reference price is perceived as a loss, reducing demand (Kopalle e al., 1996; Popescu and Wu, 2007; Nasiry and Popescu, The lieraure on behavioral dynamic pricing and invenory managemen inform his research. The dynamic pricing research sream ha formally accouns for reference dependence was iniiaed decades ago by Sorger (1988 and Kopalle and Winer (1996. Building on his approach, Kopalle e al. (1996 and Fibich e al. (2003 consider asymmeric reference effecs. The join sudy of reference effecs wih invenories is more recen, wih Taudes and Rudloff (2012 proposing a wo-period model. Sochasic invenory is inroduced in Li e al. (2015 Dye and Yang (2016 and Xue e al. (2016 sudy he opimal pricing of deerioraing iems in he presence of a reference price. Demand depending on invenory is analyzed by Lu e al. (2016 and Hsieh and Dye (2017. Previous research uses parameric (specific demand funcions. By conras, we adop he approach of Popescu and Wu (2007, Chenavaz (2016, and Chenavaz (2017, using srucural (general demand funcions, which enables us o derive more general resuls. Building on he aforemenioned sudies, we propose a dynamic pricing policy of a non-perishable invenory wih a reference-dependen demand. To he bes of our knowledge, his is he firs aemp o inegrae a general demand funcion assuming lile resricion on consumer behavior in invenory managemen. Analyical resuls show how he price seing is ied o invenory and 2

3 reference price consideraions. They also characerize how he dynamics of he selling price are ied o he dynamics of he reference price. A firm ignoring he psychological elemen would se an inadequae pricing policy, hus losing profi. The inegraion of consumer behavior provides a greaer undersanding of firm dynamic pricing sraegies. 2 Modeling Assumpions We sudy he opimal ineremporal pricing sraegy of a reailer in a monopoly siuaion. The reailer sells an invenory during he fixed and finie planning period T > 0. The ime [0, T ] is coninuous. 2.1 Consumer Reference Price To decide on a purchase decision a ime, a consumer compares he curren reail price p( 0 wih a reference price r( 0. The reference price represens a benchmark informing he cusomers abou he opporuniy of a ransacion (Sorger, 1988; Mazumdar e al., If p( > r(, he consumer perceives a loss; he is less likely o buy. Conversely, if p( r(, he consumer perceives a gain; he is more likely o buy. The reference price is usually formalized hrough a weighed average of pas produc prices (Sorger, 1988; Kopalle e al., 1996; Kopalle and Winer, 1996; Fibich e al., We define he reference price a ime, r( wih an exponenially decaying funcion: r( = e β (r 0 β 0 eβs p(sds where r 0 is he iniial reference price a ime = 0, and β 0 is he coninuous forgeing (or adjusmen speed parameer. Differeniae r( wih respec o ime : dr( = β(p( r(, wih r(0 = r 0. (1 Equaion (1 saes ha he variaion of he reference price increases wih he adjusmen speed β and wih he difference beween he reail price p( and he reference price r(. 2.2 General Reference-Dependen Demand The demand D 0 depends on he reail price p and he reference price r. We consider in his aricle a general reference-dependen demand funcion D = D(p(, r(. This general form offers he advanage of allowing for nonlineariies when capuring dynamics relaed o reference price variaions (Popescu and Wu, For breviy and in order o simplify equaions, funcion parameers will be furher omied, if here is no confusion. The demand funcion is wice coninuously differeniable, saisfying he following condiions D D < 0, > 0, 2 D 0. (2 The firs condiion capures a price effec ( D/ < 0 by imposing ha he demand decreases wih he reail price. The second condiion refers o a direc reference effec ( D/ > 0, where he demand increases wih he reference 3

4 price. The hird condiion capures an indirec reference effec ( 2 D/ 0. Tha is, he decrease of he demand due o an increase in price is higher when he reference price is high. In line wih he general demand approach iniiaed by Popescu and Wu (2007 and followed by Chenavaz (2016, condiion (2 impose relaively weak resricions on he impac of reail price and reference price on demand. Also such assumpions cope wih he wo main cases of demand separabiliy. Addiively separable case D = f(p g(r, of which he classical linear demand funcion D = α δp γ(p r is an example. Such a funcion is widely used, for insance by Sorger (1988, Kopalle e al. (1996, Kopalle and Winer (1996, Dye and Yang (2016, Xue e al. (2016, and Hsieh and Dye (2017. The implicaion of addiive separabiliy is o impede any indirec reference effec as 2 D/ = 0. Such an implicaion simplifies he racabiliy of any model and hus is analysis, hough a a cos of lesser generaliy of he resuls. Muliplicaively separable case D = f(pg(r, for which he Cobb-Douglas (isoelasic demand funcion D = αp δ r γ represens an insance. Because of opposie monoonies, he indirec reference effec wries 2 D/ < 0, enriching he analysis wih a cross effec a he demand level. Popescu and Wu (2007 provide greaer deails on he properies of he demand funcions. 2.3 Invenory Firm invenory a wries I(. We exclude he possibiliy of replenishmen as in Taudes and Rudloff (2012, Dye and Yang (2016, Xue e al. (2016, and Hsieh and Dye (2017. The lifeime T of he invenory is fixed, ha is, here is a predeerminisic selling period (say several monhs or seasons. Invenory decrease depends on previous sales and iems do no perish over ime. This siuaion corresponds o class (1 in he grea survey by Bakker e al. (2012, p Formally, he invenory a ime, denoed I(, is given by I( = I 0 0 D(p(s, r(sds, wih I(0 = I 0 and I(T = 0. (3 The invenory a ime equals he iniial invenory minus he cumulaive demand during he period 0 o. The differeniaion of I( wih respec o ime yields di( = D(p, r. (4 Equaion (4 saes ha invenory variaion a ime decreases wih curren demand. Also invenory does no decrease auonomously as in he case of perishable iems. 4

5 2.4 Curren Profi The curren profi of he reailer π R corresponds o he difference beween revenues and coss. Revenues depend on price and demand. Coss depend on invenory level I and uniary invenory cos h. Thus, he profi of he reailer wries π(p(, r( = p(d(p(, r( hi(. (5 Subsiuing (3 in (5 gives ( π(p(, r( = p(d(p(, r( h I 0 0 D(p(s, r(sds. (6 Equaion (6 saes ha curren profi of he reailer increases wih price, curren demand, and also wih pas demand, which, by reducing curren invenory, reduces curren invenory coss. 3 Dynamic Pricing Policy The reailer, in a monopoly siuaion, is modeled wihin an opimal conrol framework. Table 1 provides he noaions. Table 1: Noaions T = fixed erminal ime of he planning horizon, ρ = ineres rae, p( = selling price a ime (conrol variable, r( = reference price a ime (sae variable, I( = invenory a ime (sae variable, D(p, r = curren demand, dr( = β(p r = reference poin dynamics a ime, I( = D(p, r = invenory dynamics a ime, λ r ( = curren-value adjoin variable for r a ime, λ I ( = curren-value adjoin variable for I a ime, h = invenory cos, π(p, r = pd hi = curren profi, H(p, a, r, λ = curren-value Hamilonian. 3.1 General Case The reailer maximizes he ineremporal profi by seing he opimal pricing, while accouning for reference price and invenory dynamics. Wih he discoun 5

6 rae ρ 0, he problem of he reailer wries max p( 0 0 subjec o e ρ π(, dr( = β(p( r(, wih r(0 = r 0, di( = D(p, r, wih I(0 = I 0. The maximum principle allows solving he dynamic opimizaion problem. Wih he shadow price of he reference price λ r ( and he shadow price of he invenory λ I (, he curren-value Hamilonian H is ( H(p, r, λ r, λ I = pd(p, r h I 0 D(p, rds λ r β(p r λ I D(p, r. (7 0 The inerpreaion of curren-value Hamilonian H is as follows: H sums (1 curren profi pd(p, r hi and (2 fuure profis due o reference poin variaion λ r β(p r and invenory variaion λ I D. H hus represens he insananeous oal (ineremporal profi a ime. The maximum principle imposes he dynamics of λ I o saisfy dλ I = ρλ I H I = ρλ I h, wih λ I (T = 0. (8 Inegraing (8 wih respec o ime gives 1 λ I ( = By compuing he inegral, (9 becomes e ρ(s hds. (9 λ I ( = (1 e ρ(t h ρ. (10 From (9 we derive λ I ( 0, meaning ha he shadow price of he invenory is negaive over he planning period. Thus, an invenory is always cosly for he reailer. Noe ha λ I ( is no reference dependen. Insead, i depends on he uniary invenory cos h, he ineres rae ρ, and he remaining selling period T. Moreover, he funcion increases over ime. The raional is he following: A he beginning of he selling period, he invenory is large, and he high invenory coss reduce he profi of he reailer. Bu, as ime passes, he invenory is lower, and he smaller invenory coss exer less influence on profi. The maximum principle dicaes he dynamics of λ r dλ r = ρλ r H = (ρ βλ r (p h λ I D, wih λ r(t = 0. (11 Inegraing (11 wih respec o ime gives 2 λ r ( = e (ρβ(s (p h λ I D ds. (12 1 The proof of (9 is in Appendix A.1 2 The proof of (9 is in Appendix A.2 6

7 Because λ I ( 0, hen (p h λ I > 0. Thus and recalling D 0, we conclude λ r 0. A any ime, an increase in he reference price augmens he ineremporal profi. Tha is, a higher reference price is associaed wih larger ineremporal profis of he reailer (consumers are more likely o purchase. We confine our ineres o an inerior soluion for he price, provided i exiss. The necessary and sufficien firs- and second-order condiions for p impose for all (0, : H 2 H 2 = 0 = (1 hd p D βλ r D λ I = 0. < 0 = (2 h D (13a (p λ I 2 D > 0. (13b 2 The firs-order condiion on price (13a yields a firs resul. 3 Proposiion 1. A any ime, price seing is such ha D p D = 1 h 1 ( βλ r D D λ I, for all in [0, T ] wih λ r 0 and λ I 0 given by (10 and (12. Proof. Divide (13a by D and rearrange. Proposiion 1 shows how he pricing sraegy of a reailer inegraing invenory and reference price consideraions differs from he pricing of a sandard saic monopoly, for which he price elasiciy of demand is uniary ( ( D//(p/D = 1. When he price elasiciy of demand is greaer han one, he demand is called elasic, and he reailer loses marke power. Alernaively, when he price elasiciy of demand is lower han one, he demand is called inelasic. According o Proposiion 1, he price elasiciy of reailer demand relies on hree compeing effecs: one ied o curren profi and wo ied o he dynamics of fuure profis. The presence of invenory cos h has a direc posiive effec on he price elasiciy of he demand. The demand becomes elasic, reflecing a loss of marke power for he reailer because of he reducion of he curren profi due o increased curren coss. A greaer reference effec also weakens he marke power of he firm. The pas prices se by he reailer have more influence on he curren reail price. Tha is, he reailer has less freedom o se prices, and he demand becomes more elasic. Formally, he price elasiciy of demand increases when he shadow price of he reference price λ r increases. A dynamic effec of he invenory, ied o λ I, is also observed. Indeed, due o he fuure coss of he invenory, he reailer has an incenive o sell more in he presen. This sraegy enables diminishing fuure invenory, reducing he associaed coss, and increasing fuure profis. Thus, he shadow price of he invenory exers a negaive impac, reducing he price elasiciy of he demand. Evenually, he larger he demand D, he lower is he impac of he shadow prices of reference price and invenory on price seing. Depending on he relaive srengh of hese compeing effecs, he pricing policy is such ha he demand is inelasic, uniary, or elasic. 3 The proof of (13a is obvious and he proof of (13b appears in Appendix A.3. 7

8 The managerial implicaions are sraighforward. A firm considering only he curren profi, because i ignores lasing reference price and invenory effecs, ses a price such ha he demand is elasic (1 h. By conras, a firm inegraing curren and fuure profis, because i accouns for lasing effecs, may charge a price for which he demand is inelasic (if he lasing invenory effec ouweigh he lasing reference effec and he invenory cos effec. Proposiion 1, informing abou he price a any ime offers preliminary insighs. As such, i characerizes he opimal relaionship beween he price level on he one side and invenory cos and dynamics effecs on he oher side. Ye, i says nohing abou he condiions under which he opimal price goes up or down afer a change in he invenory cos and dynamics elemens. Tha is, Proposiion 1 mainains silence abou any causal relaionship. Proposiion 1 has o hold during he whole selling period. To maximize he ineremporal profi, marginal revenue variaions mus balance marginal cos variaions. The relaionship beween he dynamics of price and he ineremporal elemens is made explici by he ime differeniaion of he firs-order condiions (13a. This ime decomposiion mehod has been used o show explici causaliy (see, for example, Kalish (1983 and Chenavaz and Jasimuddin (2017. The firs-order condiion on price (13a mus hold a any ime [0, T ]. Is decomposiion wih respec o ime suppors he following resul. Proposiion 2. Over ime, price dynamics are such ha dp (2 h D (p λ I 2 D }{{ 2 = dr (1 h D (p λ I 2 D }}{{} }{{} β (ρ βλ r }{{} (p h λ I D } {{ } D ρλ I h, }{{} for all in (0, T wih λ r 0 and λ I 0 given by (10 and (12. Proof. Differeniae (13a wih respec o ime. The deailed proof is in Appendix A.4. Proposiion 2 capures he evoluion of he opimal price over ime. The second order condiion (13b guaranees ha, on he lef side of equaion (2, he second facor is posiive. On he righ-side of equaion (2 appear hree addiively separable erms. The firs erm relaes o reference price dynamics, he second o he adjusmen speed of he reference price, and he hird o he price sensiiviy of demand. For he firs wo erms he impac is ambiguous as hey resul from wo compeing effecs. The hird erm has a posiive impac, 4 4 The proof is sraighforward. Subsiuing λ I from (10 yields ρλ I h = h(1 e ρ(t h = he ρ(t, which is posiive. Tha is, ρλ I h > 0. Furher, demand condiions (2 impose D D (ρλ I h > 0. < 0, enabling derivaion of he posiiviy of he hird facor: 8

9 informing ha, counerinuiively, he price increases wih he price sensiiviy of consumers. The lesson is ha, for a general demand, he dynamics of price relae o he reference price level and adjusmen and also o he price sensiiviy of consumers, hough, he oal effec is ambiguous. To obain more conclusive resuls, we consider more specific cases, rading off generaliy for srengh of he resuls. 3.2 Special Cases Absence of Invenory Cos When here is no invenory cos (h = 0, he shadow price of invenory equals zero (subsiuing h = 0 in (10 yields λ I = 0. Subsiuing h = 0 and λ I = 0 in Proposiions 1 and 2 yields D p D = 1 βλ r D, dp D p 2 }{{ 2 = dr } 2 D D }{{} p 2 D β (ρ βλ r }{{}}{{} p D } {{ } This case rules ou he opposing effecs due o invenory. Only remains he reference price effec, which is sricly posiive. Consequenly, ( he reailer ses a price such ha he price elasiciy of demand is elasic D p D. > 1 Tha is, because of he lasing effec of pas prices, he reailer enjoys less marke power in price seing. Also, he selling price may exhibi a decreasing or increasing paern, or a mix of boh. Noe ha he opimal pricing rules above mach he rules in Chenavaz (2016, Proposiions 1 and 2, which also characerize a dynamic pricing policy wih a reference-dependen demand. More precisely, our case wih a fixed invenory and no invenory cos corresponds o heir case, which does no consider producion cos and invenory issues Absence of Reference Adjusmen When he reference price does no adjus over ime (β = 0, i is consan over ime (dr/ = 0 and equals o he reference price a ime 0: r( = r 0. Subsiuing β = 0 in Proposiions 1 and 2 yields. D p D = 1 h D λ I D, dp (2 h D (p λ I 2 D }{{ 2 = D (ρλ I h. }}{{} The firs equaion informs ha when he reference price does no adjus, he price elasiciy of demand decreases compared o he general case reaed 9

10 in Proposiion 1 (he erm βλ r was posiive and is now null. This suggess D ha, when he reference price is consan, he firm benefis from more marke power. Indeed, pas prices do no affec he reference price, and hus hey do no affec he consumer purchase decision for he curren price. The second equaion saes ha, in he absence of reference dynamics, dp is always posiive (recall ha he second facor on he lef-side is posiive see foonoe 4. The opimal pricing policy of he invenory is such ha he price increases over ime; he firm adops a peneraion sraegy. Proposing a low price in he firs period enables reducing invenory coss hrough he enire ime period by selling more producs a he beginning of he selling period. This low iniial reail price is possible because i has no impac on he consumer reference price Absence of Invenory Cos and Reference Adjusmen When here is no invenory cos (h = 0 and no adjusmen of he reference price (β = 0, hen he shadow price of he invenory is null (λ I = 0 and he reference price remains consan (dr/ = 0. In his siuaion, Proposiions 1 and 2 wrie D p D = 1, dp = 0. These resuls claim ha in a dynamic seing when he firm is no consrained by reference and invenory consideraions, he pricing policy is ha of a sandard monopoly. Indeed, he price elasiciy of demand is uniary. Also, he price is consan over ime because all dynamic effecs disappear. In a nushell, his special case conforms o he pricing of a monopoly in a saic siuaion. 4 Conclusion This aricle sudied he dynamic pricing of an invenory, of a monopoly operaing in a marke where consumers are subjec o reference-dependence. The modeling in an opimal conrol framework conribues o exising lieraure on behavioral dynamic pricing hrough he following aspecs: The pricing rules explicily consider he impac of curren reference prices and invenory levels of he fuure profis. Finally, i discusses he managerial implicaions of behavioral pricing sraegies of profi-maximizing firms. By poining o he influence of he psychological elemen, his aricle enhances he comprehension of invenory pricing. Firms and reailers would benefi from adaping heir pricing policies o consumer behavior, hereby becoming more profiable. Furher profi-maximizing opporuniies require fuure research on he opimal relaionships among direcly conrollable variables (such as he selling price, markeing expenses, or promoion aciviies and indirecly conrollable variables (such as he reference price, reference qualiy, or produc goodwill. 10

11 A Appendix A.1 Proof of Equaion (9 Recall ha he dynamic of λ I wries in (8 dλ I = ρλ I H I = ρλ I h, wih λ I (T = 0. Consider he inegraing facor e ρ, such ha dλ I (e ρ ( = e ρ dλi ( ρλ I (. Because dλ I( d(λ(e ρ = he ρ, d(λ(se ρs = ρλ I ( = h, hen λ(t e ρt λ(e ρ = e ρs hds, e ρs hds. Subsiuing he ransversaliy condiion λ I (T = 0 yields λ I ( = e ρ(s hds, which complees he proof. A.2 Proof of Equaion (12 Recall ha he dynamic of λ r wries in (11 dλ r = ρλ r H = (ρ βλ r (p h λ I D, wih λ r(t = 0. Consider he inegraing facor e (ρβ, such ha (ρ βλ r (. d(λ r (e (ρβ Because dλ r( d(λ r (e (ρβ = e (ρβ ( dλr ( (ρ βλ r ( = (p h λ I D, hen d(λ r (se (ρβs = [ = e (ρβ (p h λ I D λ r (T e (ρβt λ r (e (ρβ = ], ( e (ρβs (p h λ I D ds, e (ρβs ( (p h λ I D ds. 11

12 Subsiuing he ransversaliy condiion λ r (T = 0 yields λ r ( = which complees he proof. e (ρβ(s (p h λ I D ds, A.3 Proof of Equaion (13b Recall he firs-order condiion wih respec o p (13a H = 0 = D p D hd βλ r D λ I = 0, = (1 hd p D βλ r D λ I = 0. The second-order condiion wih respec o p (13b saes 2 H 2 < 0 = (1 h D D D p D 2 λ I < 0, = (2 h D 2 D 2 (p λ I < 0, = (2 h D ( λ I p 2 D 2 > 0, which complees he proof. A.4 Proof of Proposiion 2 Recall he firs-order condiion (13a D(p(, r( (1 hd(p(, r( p( βλ r ( D(p(, r( λ I ( = 0. Differeniae he firs order condiion (13a wih respec o gives ( D dp (1 h D dr dp ( D 2 p D dp 2 2 D dr β dλ (( r 2 D dp 2 2 D dr λ I D dλ I = 0, giving afer rearrangemen dp ( (2 h D 2 D 2 (p λ I β dλ r D dλ I. Subsiuing (8 and (11 above provides ( dp (2 h D 2 D 2 (p λ I = dr which complees he proof. = dr ( (1 h D 2 D (p λ I β ( (1 h D 2 D (p λ I [ (ρ βλ r (p h λ I D ] D [ρλ I h], 12

13 References Bakker, M., J. Riezebos, and R. H. Teuner (2012. Review of invenory sysems wih deerioraion since European Journal of Operaional Research 221 (2, Chenavaz, R. (2016. Dynamic pricing wih reference price dependence. Economics: The Open-Access, Open-Assessmen E-Journal 10 ( , Chenavaz, R. (2017. Dynamic qualiy policies wih reference qualiy effecs. Applied Economics 49 (32, Chenavaz, R. and S. Jasimuddin (2017. An analyical model of he relaionship beween produc qualiy and adverising. European Journal of Operaional Research 263, Den Boer, A. V. (2015. Dynamic pricing and learning: Hisorical origins, curren research, and new direcions. Surveys in Operaions Research and Managemen Science 20 (1, Dye, C.-Y. and C.-T. Yang (2016. Opimal dynamic pricing and preservaion echnology invesmen for deerioraing producs wih reference price effecs. Omega 62, Elmaghraby, W. and P. Keskinocak (2003. Dynamic pricing in he presence of invenory consideraions: Research overview, curren pracices, and fuure direcions. Managemen Science 49 (10, Fibich, G., A. Gavious, and O. Lowengar (2003. Explici soluions of opimizaion models and differenial games wih nonsmooh (asymmeric referenceprice effecs. Operaions Research 51 (5, Goyal, S. and B. C. Giri (2001. Recen rends in modeling of deerioraing invenory. European Journal of operaional research 134 (1, Hsieh, T.-P. and C.-Y. Dye (2017. Opimal dynamic pricing for deerioraing iems wih reference price effecs when invenories simulae demand. European Journal of Operaional Research 262 (1, Kalish, S. (1983. Monopolis pricing wih dynamic demand and producion cos. Markeing Science 2 (2, Kopalle, P. K., A. G. Rao, and J. L. Assuncao (1996. Asymmeric reference price effecs and dynamic pricing policies. Markeing Science 15 (1, Kopalle, P. K. and R. S. Winer (1996. A dynamic model of reference price and expeced qualiy. Markeing Leers 7 (1, Li, S., J. Zhang, and W. Tang (2015. Join dynamic pricing and invenory conrol policy for a sochasic invenory sysem wih perishable producs. Inernaional Journal of Producion Research 53 (10, Lu, L., J. Zhang, and W. Tang (2016. Opimal dynamic pricing and replenishmen policy for perishable iems wih invenory-level-dependen demand. Inernaional Journal of Sysems Science 47 (6,

14 Mazumdar, T., S. Raj, and I. Sinha (2005. Reference price research: Review and proposiions. Journal of Markeing 69 (4, Nasiry, J. and I. Popescu (2011. Dynamic pricing wih loss-averse consumers and peak-end anchoring. Operaions research 59 (6, Popescu, I. and Y. Wu (2007. Dynamic pricing sraegies wih reference effecs. Operaions Research 55 (3, Sorger, G. (1988. Reference price formaion and opimal pricing sraegies. In G. Feichinger (Ed., Opimal Conrol Theory and Economic Analysis 3, pp Elsevier. Taudes, A. and C. Rudloff (2012. Inegraing invenory conrol and a price change in he presence of reference price effecs: a wo-period model. Mahemaical Mehods of Operaions Research 75 (1, Xue, M., W. Tang, and J. Zhang (2016. Opimal dynamic pricing for deerioraing iems wih reference-price effecs. Inernaional Journal of Sysems Science 47 (9,

15 Please noe: You are mos sincerely encouraged o paricipae in he open assessmen of his discussion paper. You can do so by eiher recommending he paper or by posing your commens. Please go o: hp:// The Edior Auhor(s Licensed under he Creaive Commons License - Aribuion 4.0 Inernaional (CC BY 4.0.