379 

AdvancesinProductionEngineering&Management ISSN1854‐6250

Volume14|Number3|September2019|pp379–390 Journalhome:apem‐journal.org

https://doi.org/10.14743/apem2019.3.335 Originalscientificpaper

  

Optimal timing of price change with strategic customers under demand uncertainty: A real option approach

Lee, Y.a, Lee, J.P.b, Kim, S.b,*  aDepartment of Mechanical Engineering, University of Texas at San Antonio, San Antonio, United States of America (USA) bCollege of Business Administration, Hongik University, Seoul, South Korea   

A B S T R A C T   A R T I C L E   I N F O

Thispaperproposesamodeltodeterminetheoptimalmarkdowntimingforacompanywithstrategiccustomerpurchasingbehaviour. Since strategic cus‐tomers are aware of potentialmarkdownunder the posted pricing scheme,theymaychoosetowaitlongertomaximisetheirutilisationinsteadofbuyingaproductand fulfillingan instant surplus.On theotherhand, thesellercandelaythemarkdowndecisionuntilitisprovedtobeprofitableandhencehasanoptiontodeterminethetiming. Inestimatingthevalueof themarkdowndecision,theseller’soptionneedstobeestimated.However,thevalueoftheoptionishardtobecapturedbytheconventionalnetpresentvalueanalysis.Under market uncertainty where potential customer demand evolves overtime,theseller’srevenuefunctionisintheformofastochasticdynamicpro‐grammingmodel.Applyingarealoptionapproach,weinvestigatetheoptimalprice path and propose the optimal markdown threshold. Given the mark‐downcostsincurred,wefindthattheoptimaldiscounttimingforthefirmisdeterminedbyathresholdpolicy.Furthermore,ourresultsshowthatiffuturemarketbecomesmoreuncertain,thesellerneedstowaitlongerordelaythemarkdown decision. In addition, the optimal threshold of the markdowndecreasesexponentiallyinadecliningmarket,whichexplainstheearlymark‐downpolicyofsomeconsumerproductcompanies.

©2019CPE,UniversityofMaribor.Allrightsreserved.

  Keywords:Strategiccustomers;Pricechange;Postedpricing;Markdown;Demanduncertainty;Realoption

*Correspondingauthor:sbkim@hongik.ac.kr(Kim,S.)

Articlehistory:Received28February2019Revised10September2019Accepted12September2019

  

1. Introduction 

Demandmanagementbecomesthebasisforthedecisionmakingofthefirms;fromproductionplanningtoinventorymanagement[1,20].Pricingpoliciesarefrequentlyusedtoolswhenfirmsmanagetheirdemand[20].Thepricingpoliciesofafirmareoftencomplexanddiversedepend‐ingonthebusinessenvironmentinwhichthecompanylies[1,5,6,14,20].Inthefashionindus‐try, for instance,simplemarkdownpricing iswidelyusedtoselloutremainingstockaftertheregularsalesseason[21].Somecustomersmaychoosetowaitandpurchasetheproductlateratthemarkdownpriceratherthanbuyingitrightaway.Ontheotherhand,airlinecompaniescon‐tinuouslymarkupthepricesofticketsupondeparture.Whenlookingforanairlineticket,cus‐tomerscanexpectan increase inprices if theydelaytheirpurchase.Therefore,understandingcustomerpurchasingbehaviouriscriticalforthefirmtomakepricingdecisions.

Strategic customers in theoperationsmanagement literature aredefined as thosewhoareawareofthefirm'sdynamicpricingpoliciesandmakeinter‐temporalpurchasingdecisions[20].Sincesuchcustomersareconsciousofpotentialchangesinpricesatalaterpointintime,theyarebeingstrategicratherthanmyopic.Insteadofbuyingaproductandfulfillinganinstantsur‐

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plus, they strategicallywait for a future pricemarkdown and thereby seek tomaximise theirutilisation[5,6,18].Assuch,strategiccustomershavebecomeasubstituteformyopiccustom‐erswhosimplymakeabuyingdecisionifthepriceislowerthantheirvaluation[5,6,20].There‐fore, firmsmustcomprehendthestrategicbehaviourofcustomersandfindanoptimalpricingschemebasedonittomaximiserevenue.

Inresponsetothestrategicbehaviourofcustomers, the firm'sdecisionsaregenerallytwo‐fold:thetimingofpricechangesandtheavailabilityoftheproduct[3,20].Thefirmsellsaprod‐uctforadurationoftime,afterwhichitmaydecidetochangethepriceatacertainpointintime.Limited supply could also be used as a marketing strategy to increase the sense of urgencyamongcustomers.Therefore,customersinthemarketchooseeithertopurchaseaproductatitscurrentpriceor to revisit it after theprice goesdown, consideringnotonly the timingof themarkdownbutalsothepossibilityofsellouts.

Inthispaper,weinvestigatethemarkdowndecisionofamonopolistwhowishestomaximiseexpectedrevenuesinthepresenceofstrategiccustomers.Ourmodelcapturesseveralimportantpropertiesofthemarketenvironmentforconsumerproducts.First,thesellercommitstoafixedpath of two prices: itmay sell a product for a duration of time, afterwhich itmay decide tochangetheprice.Themarkdowndecisioncanbemadenomorethanonceoverthesaleshorizonand is hence irreversible. Second, customers show strategic purchasing behaviour towardsfirms:evenifthevaluationoftheproductexceedsthepriceoftheproductduringthefirstpartofthesaleshorizon,customersmaynotsimplypurchaseit.Instead,theirdecisiontopurchaseisbasedonthevaluationthatexceedsacertainlevel,thusfollowingathresholdpolicy.Third,po‐tentialcustomerdemandisstochastic.Inparticular,themarketsizefollowsageometricBrown‐ianmotionthatevolvesdynamicallyovertime.

Weconsideraseller'sproblemondecidingtheoptimalpricepathandthetimingofamark‐down under demand uncertainty. Specifically,we present a stochastic dynamic programmingmodelwherethesellerhasasingleopportunity todiscountthepriceof theproductatasunkcost.Customersinthemarketareawareofapotentialmarkdownandthelikelihoodofasellout.Basedoncustomers'valuationinregardtothetwoprices,thevalueofthefirmisexpressedasastreamofexpectedrevenues.Solvingtheproblemusingarealoptionapproach,weshowthattheseller'soptimalmarkdowntimingdecision isbasedonthethresholdpolicy.Tothebestofourknowledge,thisisoneofthefirststudiesthatconsidersapostedpricingschemeundermar‐ketuncertainty.

Theremainderofthepaperisorganisedasfollows.Section2outlinespreviousrelatedworkstosummariseextantresearch.Section3proposesarevenuemaximisationmodel,andSection4continueswiththetopicbyanalysingthesolutionofthemodel.Finally,wediscussbroaderfind‐ings,conclusions,andpotentialfutureresearchopportunitiesinSection5.

2. Literature review 

Strategiccustomersandfirms'pricingpolicyproblemshavebecomeanincreasinglyproductiveresearch area.Amongothers, studies regarding customerpurchasingbehavior are thoroughlyreviewedbyShenandSu[20].Mostofthepapersintheliteratureconsidertwoimportantele‐ments inmodellingstrategiccustomerbehaviour.Thefirst isthearrivalprocessofcustomers.Whethercustomerspreexistedinthemarket[7,13,18]orsequentiallyarrivedinthemarket[3,14,21,23]isaquestionbasedonthispremise.Thesecondisthedecisionmakingofthecustom‐ersandhowthedecisionmakingultimatelyformsanequilibrium.Regardlessofthemarketsize,thedecisionmakingofanindividualcustomermakesanimpactonthedynamicsofthemarketto some extent. For instance, when many customers purchase goods in the early stages, theproductmayrunoutofstockforsomeofthosewhoinitiallydecidedtodelaythepurchase[13].Sometimescustomersmayhavetopurchasethegoodsatanevenhigherpriceifthefirmorsell‐erchoosestoadoptamarkuppricingpolicy[24].Astrategiccustomertriestomakeanoptimaldecision,foreseeingthesesituations,andthisprocessmay, inturn,compriseanequilibriumindecisionmaking.Theseller,ontheotherhand,setshisorherpricingpolicybasedonthisequi‐libriuminanefforttomaximiseprofit.Therefore,thisgame‐theoreticrelationshipwithconflict‐

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ing interestsbetweenthesellerandthestrategicbuyersnecessarily leadstoahighlycomplexmodelinmanystudies.

Thereare twotypesofsimplifications todealwith thecomplexity in themodelling.Firstly,thetimeofthepricechangeisoftenfixed.Aspecificnumberofperiodsarepresumed,andstaticpricingismaintainedforthedurationoftheperiods.Inotherwords,theanalysisofoptimalpric‐ingisbasedonthedefinitenumberofperiodsinwhichafixedpriceisoffered,ratherthanfind‐ingthepricechangingperiodonebyone[5‐7,13].Thesecondcaseisthesizeofthemarket:inapplyingagametheoryapproach,asmallmarketsizeisassumed.Inthissituation,acustomerpredicts the decisionmaking of other consumers tomake his or her optimal decision and anequilibriumofstrategicpurchases isachieved.AvivandPazgal[3] foundthata firm'sbenefitsfrompricedifferentiationmaydecreaseascustomersbecomemorestrategic,andhenceoptimalpricingpoliciesmayresultinpotentialrevenuelossesinthepresenceofstrategiccustomers.

Customers'purchasingdecisionsdependontheinteractionamongthepricingpolicy,availa‐bility,customervaluations,remainingtime,andsoforth.Underthepostedpricing,forinstance,wherethesellerannouncesitspricepathinadvance,theavailabilityoftheproductorthepossi‐bilitytopurchaseitlaterintimewillbethemajorconcernforthecustomer[2,7,13].AsDasuandTong[7]specificallypointedout,theseller'sdynamicpricingdecisionismeaningfulonlyifcustomersareawareofthestock‐outpossibility,whiletheimpactoftheperceptiononstrategiccustomerbehaviourisdifferent inheterogeneouscustomervaluations[23]. Inmanystudies,atwo‐periodpostedpricingschemehasbeenusedduetoitssimplicityandapplicability,althoughthesellercanstillmakeapricechangeatanytime[4,7,13,15].DasuandTong[7],inparticular,foundthat theapproximationclose to themaximumrevenuecanbeachievedbytwoor threepricingchanges.Inthisstudy,ourmodelwillalsobebasedonthetwo‐periodpostedpricingincontinuoustimeperiodstofindtheoptimaltimingofpricechange,whiletheavailabilityoftheitemislimitedafterthemarkdown.

Asforthefirm'spointofview,ontheotherhand,marketsizeisthemainsourceofuncertain‐ty.Giventhepriceandthetimingofthepricechange,thefirm'srevenuemustbesignificantlydifferentdependingonchangesindemandatthemoment.Undermarketuncertainty,thesellercaneithermakean immediateprice changeor intentionallydelay thedecision toobserve theactualdemandmovement.Thissituationisverycommoninmanyoperationalpractices:compa‐nieshaveanopportunitytoinvestbuttheycanstillwaitfornewinformation.Inotherwords,afirmwith theability topostponeadecisionhas theoption,not theobligation, toexercise it –making it analogous to holding a financial call option. Since first proposed by Pindyck [19],McDonaldandSiegel[17],DixitandPindyck[8]andothers,thisrealoptionapproachhasbeenwidelyborrowedintheareasofmarketingandoperationsmanagementbecauseithelpsustobetterunderstandthetruevalueoftheinvestmentopportunity.

Adoptingtherealoptionconceptisnotcompletelynewinrevenuemanagementliterature.Innumerouspapers,thedynamicpricingdecisionisdeterminedbyconsideringtheoptionvalueofunsoldproducts[11,16].Sincethisoptionvaluedecreasestowardstheendofthetimehorizon,theoptimalpricepathalsodecreasesovertime.Inanotherpaper,GallegoandSahin[10]usedthe real option approach tomodel uncertain customer valuations. In this paper, however,weassumethatpotentialcustomerdemandevolvesovertimeandfollowsthegeometricBrownianmotion(GBM).Assumingtheknowndistributiononcustomervaluationsandthelevelofavaila‐bility,weexploretheoptimalmarkdowntimingproblembasedonthenetpresentvalueoftheseller'sexpectedrevenue.Tothebestofourknowledge,intheliteratureonstrategiccustomers,there areonly ahandful of studies thatdealwith theoptimal timingof price change, andyetfewerstillthatatthesametimeaddressoptimalpricingwithstrategiccustomersundermarketuncertainty.

3. Model description 

Inthispaper,weconsideramonopolisticfirmthatsellsasingleitemtopotentialcustomersovertwoperiods.Thefirmwantstomaximiseitsnetpresentvalueofexpectedrevenue.Below,weexplainfurtherassumptionsbeforebuildingourmodel.

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Assumption1.Themonopolistic firm followsatwo‐periodmarkdownpricingschemeandcom‐mitstothepricepathinbothphases.

Assumption2.Theoriginal( )andmarkdownprices( )arepre‐announcedandthe in‐stockprobability(π)inthesecondperiodisalsogiveninformation.

Assumption 3. Customersare strategic rather thanmyopicandareawareofmarkdownsandpossibilitiesofstock‐outs.

Assumption4.Thedistributionofcustomervaluations( ⋅ )isknown.

3.1 Valuation of a strategic customer 

Supposethatthereisamonopolistwhohasasufficientlylargenumberofanitem.Untiltime ,theitemisinitiallysoldatprice ,andafter theitemissoldatthemarkdownprice .Thetwoprices, and ,arepre‐announced.CustomerdemandfollowsageometricBrownianmotion(GBM),andeachcustomerissupposedtopurchaseonlyoneunitoftheitem.Whentheselleroffersamarkdownprice,weassumethatthesellercancontrolthelevelofproductavaila‐bility, ,toinducescarcity.Inotherwords,inthesecondperiod,thein‐stockprobabilitydecidedbythesellerwillbesetto 1.Controllingtheavailabilityofservicesoritemsofdifferentclas‐sesisprevalentinrevenuemanagement[24]andinducingalevelofscarcityisalsooneofthemostcommonstrategiesinmarketing[9,22].

Let denotethecustomer'ssurplus.Thentheutilisationofthecustomerwhopurchasestheitemrightnowisasfollows:

(1)

where isthesurplusofthecustomerand isthecurrentpriceoftheproduct.Similarly,theutilisationofthecustomerwhodecidestowaitforthediscountisasfollows:

(2)

where is the customer'svaluationof theproduct, is the currentpriceof theproduct, isthe future price of the product, and is the service level of the product at the lowerprice .Thus, the stock‐out probability is1 . stands for the customer's preference for risk; 0indicates risk‐averse, 0risk‐taking, and 0risk‐neutral attitude. Furthermore, we as‐sumethatthecustomersareeitherrisk‐averseorrisk‐neutral,whichisaprevalentassumptionmadebymanyresearchers[13,15].

Inthissetting,thestrategiccustomersdecidetopurchaseinthefirstperiodiftheirvaluationisgreaterthanorequaltothethresholdvalue.Thepurchasingdecisionofthestrategiccustomerisdeterminedbythefollowinglemma.

Lemma1.Thethresholdofastrategiccustomer'svaluationisgivenby:

1 (3)

Proof.Thetwochoices,purchasingrightnoworwaitingforadiscount,generatethesamesur‐pluswhen .Solvingtheequation,astrategiccustomerwillhavethefollowingthreshold.Thatis,

. (4)Thiswouldfinishtheproof.

If 1,nocustomerswouldbuyinthefirstperiod.Furthermore,weassumethatcustomersdonotpurchaseiftheutilisationislessthanzerowithoutlossofgenerality.Thatmeans, isnotlessthan .Therefore,itissufficienttoconsideronlythecasewherethethresholdisbetweenthefirstperiod'spricingandone.Thatis,

1 (5)

Thisalsodecidestheupperandthelowerboundsof accordingly.Followingtheliterature,potentialcustomerdemandisassumedtobeamultiplicationofthe

customervaluefunctionandthetime‐varyingpotentialdemand.Thatis,thedemandfunctionisgivenby

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1 (6)

where is a knowndistribution function of the product at customer valuation , and isthemultiplicativedemandshockprocess.Thismaybethoughtasdemandinwhichtheproducthasaunitprice.

In the firstperiod,a strategiccustomerwouldpurchase theproduct, if and 0.Therefore,fromtheconditions,

 and  (7)

wehave

. (8)

Since forthecustomerspurchasinginthefirstperiod,thecurrentdemandfunctionis

1 11

. (9)

Ontheotherhand,aproportionofcustomerswouldwaitandpurchaselateratalowerprice,if and 0.Thevaluationofsuchcustomersisasfollows:

. (10)

Hence,thedemandfunctionofthecustomerswhocomebacklaterinthesecondperiodtopur‐chasewillbe:

1. (11)

Finally,whentheproductstartsbeingsoldatamarkdownprice ,anycustomerwhoseval‐uationisatleastgreaterthanthepricewouldpurchaseit.Namely,thedemandfunctionwillbe:

1 . (12)

Without loss of generality, let the valuation of customers, , be uniformly distributed over[0,1].Thenthedemandforeachcaseisgivenasfollows:

1 11

1 (13)

1 1 1 (14)

1 . (15)

3.2 Customer demand 

Inthispaper,weuseageometricBrownianmotion(GBM)toformulatethemulti‐plicativede‐mandshock attime .Thatmeanstherelativechangeindemand, / , withinashorttimeinterval, , ,canvarywithtime .Thedynamicsofdemandarerepresentedbythefollow‐ingformula:

, (16)

where isthegrowthrateordriftrateindemand, isthevolatilityoftheprocess,and isastandardWienerprocess. If 0,market size is increasingover time. If 0,market size isdecreasing.

Thiscontinuousrandomvariable issaidtohavelognormaldistributionbecausetheinte‐gralofEq.16givesthefollowingdemandfunction(seeAppendixAforthederivation):

⁄ , (17)

where istheinitialdemand.Whilethebell‐shapedpatternofdemandisexpectedbyEq.17,therealisationofdemandwillsubstantiallydeviatefromit,dependingonthemarketvolatility,asshowninFig.1.

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Fig.1 SamplepathofpotentialdemandfromthegeometricBrownianmotioninadecreasingmarket.

Note that the threesamplepaths inFig.1aredrawn fromEq.16withameandrift rateof

0.1and three standard deviations of 0.05, 0.1, and 0.2. As shown in the figure, thesamplepathwithalargerstandarddeviationtendstofluctuatesignificantly,whileallthreetra‐jectorieshaveadecreasingtrendincommonduetothenegativemeandriftrate.

3.3 Optimal timing of price discount 

Next,weconsidertheoptimaltimingproblemconditionedonthecustomer'spurchasingstrat‐egy.Wedevelopamodelforanoptimaldiscounttimingdecisionusingarealoptionmodel.Inpractice,thecompanyhasan"option"todelaythediscountandhenceneedstodeterminewhenthepriceshouldbediscounted.Aftermarkdown,thecompanywouldmakerevenue ,withanirreversiblesunkcost beingincurredfromsalespromotion,inventorymanagement,andsoforth.

Hereinweformulatethevaluefunctionofthefirmwithanopportunityofthediscounttiming.Whentheproductissoldattheoriginalprice ,aproportionofcustomers, ,whosevalua‐tionisfarhigherthan ,orgreaterthan ,willdecidetopurchasetheitem.Agroupofstrategiccustomers, , whose valuation is between and would like to wait and see if the price ismarkeddown.Once the firmdecides todiscount theoriginalprice to themarkdownprice, ,theycomebacktopurchasetheproductbutonly ofthemwillbeabletogetone.Weassumethat suchdemand is instantaneous,meaning that customerdemandaccumulatedup to time willberealisedattime [3].Fromtime ,anycustomerswhosevaluationisatleasthigherthanwouldliketopurchasetheproductbut,again,only ofthemwouldgetone.Webeginwith thevalue functionof the firm for theoptimaldiscount timingproblem.The

valueofthefirm, ,isthestreamofrevenue,whichconsistsofthreecasesstatedearlier.Weusedynamicprogramming,stipulatinganexogenousdiscountrate .Then istheexpectedpresentvalue

max 1 (18)

where

1 . (19)

Sincethedemandofstrategiccustomerswillberealisedattime ,werearrangetheformulasothattherevenueisincludedintheterminalpayoff .Then

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max 1 (20)

where

1 (21)

1 (22)

1 (23)

Since 1 / (See Appendix A) and ⁄ , we

finallyhave:

1 1 (24)

Substituting inEq.3intotheformula,thevaluefunctionofthefirmissummarisedasfollows.

Proposition1.Thevaluefunctionofthefirmforoptimaldiscounttiming isgivenby:

max1

1 (25)

where

11 1 (26)

Bysolvingthisstochasticdynamicprogrammingproblem,wecanobtaintheoptimaltimingforamarkdown.Theoption‐likeapproachshownin[17]and[19]isusedtosolvethedynamicstochasticproblem.Asthepotentialdemand evolvesstochastically,theoptimalstrategyistoexercise(markdown)sothatthevalueisatleastgreaterthanthecriticalvalue ∗.Afirm’sop‐timalmarkdowntimingsolutionisrepresentedinthefollowingproposition.Proposition2.Acompanyconsideringmarkdownoftheretailpricewillhaveavaluefunctionasfollows:

1 ∗

1 1 ∗ (27)

where∗

1⋅

1 11 (28)

1 1⋅

∗

(29)

and12

12

2 (30)

Proof.Let denotethetimingatwhichthefirmdiscountstheoriginalpriceoftheitem.Asde‐scribedearlier,thefirmmakesrevenueflowsof 1 beforethemarkdown.Attime ,thefirmwouldmakerevenueflow withtheirreversiblesunkcost .Therefore,asshownin[8],theBellmanequationinthecontinuationregion,wherevaluesof arenotoptimaltomark‐down,isgivenby:

1 , (31)

whichimpliesthatoveratimeinterval ,thetotalexpectedreturnonthemarkdownopportu‐nityisequaltoitsexpectedrateofcapitalappreciation.

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ApplyingIto'slemma,wehave

12

, (32)

where ⁄ and ⁄ .SubstitutingEq.16anddividingthroughby ,wehavethefollowingBellmanequation(see

AppendixBforproof):

12

1 0(33)

Toensuretheexistenceoftheoptimalsolution,weassumethat .Thedifferentialequa‐tion mustsatisfythefollowingthreeboundaryconditions:

0 0 (34)

∗ 1 1∗

(35)

∗ 1 (36)

Eq.34holdsbasedon theobservation that itwill stayzero if thestochasticprocess goes tozero.Theothertwoequationsaretoimposecontinuityandsmoothnessatthecriticalpoint ∗,thepotentialdemandatwhichitisoptimaltodiscount.Eq.35isthevalue‐matchingcondition,indicatingtherevenuethefirmmakesuponmarkdown.Eq.36isthesmooth‐pastingconditionatthepoint.

Therefore,thesolutionofthedifferentialEq.33musttaketheform

1, (37)

where isaconstanttobedeterminedand isoneofthesolutionsofthefollowingquadratureequation:

12

1 0. (38)

Solvingtheequationandtake

12

12

21 (39)

toensuretheboundarycondition.Fromthesmoothpastingandthevalue‐matchingconditions,wehave

∗ ∗ 1 ∗ 1 1∗

(40)

and

∗ ∗ 1 1. (41)

Solvingtheseequationsresultsin:

∗1

⋅1 1

1 (42)

1 1⋅

∗

(43)

and12

12

2. (44)

Herein, only if 1 / and 1 1 ,we have a positive

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threshold ∗ 0.ThisresultleadstoProposition3.Again,thethreshold ∗determinestheoptimalmarkdowntimingforafirm.Whentheactual

customerdemandofthefirmattime islowerthanthethreshold ∗,itisbeneficialtoselltheproductattheoriginalretailprice ,makingtherevenuestreamof 1 / aswellasgiving the flexibility that the firmcanhold for thepricemarkdown,measuredbyαX .On theotherhand,whentheactualdemandisgreaterthanthethresholdX∗,thefirmwilldecidetodis‐

countthepriceandtakethebenefitofmarkdown 1 byspend‐

inginvestmentcost .

4. Analysis and discussion 

Thissectionexplainssomeof the importantcharacteristics foroptimalmarkdownapproachessuggestedearlier.First,thefollowingpropositionillustratesthatthereexistsapositivethresh‐oldforthefirmatanygiventime underspecificconditionsfor and .

Proposition3. Let ∗denotetheoptimaltimingofmarkdowntomaximisethe firmvalue.Thentheoptimalmarkdowntimeisfinite ∗ ∞[12],andthefirstepochthatdemandexceedsthethre‐sholdisestimatedatthefollowingtime:

∗ inf 0 | ∗ , (45)

where there exists a positive threshold ∗ ⋅ 1

attime if 1 / and 1 1 / .Proof.ByProposition2.

Notethatweassumeadecreasingmarketsize( 0).Astime increases,therefore,wecanobserve that the threshold ∗decreases exponentially,while theminimumvalue for the fixedcost thatisrequiredforthisapproachtobefeasibleincreasesexponentiallybeforehittingthelowerbound asshowninthefollowingproposition.

Proposition 4. As → ∞, we have a threshold ∗ → 0and the lower bound of the fixed cost→ / .Again,as the threshold ∗for themarkdowndecreasesexponentially,a firmis likely tode‐

cideonapricediscountintherelativelyearlystages.Italsoindicatesthatnosignificantrevenueis expected after a certain amountof timebecauseof the reduction in customerdemand, andhencethefirmnolongerneedstoinvestmoreinlaterstagesunderthisapproach.

Astheoptimaltiming ∗isrepresentedbysomeexogenousfactors,weexploittheimpactoftheparametersonthethreshold.

Proposition5.Theoptimaltimingthresholdincreaseswithrespecttodemandvolatility.Thatis,

∗

0 (46)

Proof.Notingthefractionalvalue 0,wetakethederivativeof fromEq.16withrespecttothedemandvolatility, .Weknow 11isoneofthesolutionstothefollowingquadraticfunction

0(theothersolutionis 0),where12

1 .(47)

Takingaderivativeoftheequation,wehave

0. (48)

Since ⁄ 0and ⁄ 0,wehave

0. (49)

Furthermore,

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111

0. (50)

Finally,thederivativeoftheoptimalthreshold ∗withrespecttoσis,∗

1 1 11 0 (51)

Thispropositionindicatesthattheoptimalmarkdownthresholdincreasesasthevariationindemandincreases.Simplyput,itismorebeneficialforthefirmtowaitanddelaythemarkdown,therebyavoidingtheriskofmakingtheinstantaneousdecisionwhenthemarketishighlyuncer‐tain. The firm iswilling tomake themarkdown decision, onlywhen excessive revenue is ex‐pectedwheretheamountofuncertaintyregardingfuturedemandislarger.

5. Conclusion 

Inthispaper,theoptimalpricingpolicyofamonopolisticfirmisinvestigatedwithstrategiccus‐tomerbehaviour.Whencustomersstrategicallywait foradiscount, themonopolisthasanop‐tiontoofferamarkdowntomaximiseitsrevenue.Assumingthattheunderlyingcustomerde‐mandisstochastic,evolvingdynamicallyovertime,wedevelopavaluefunctionforthefirmtofindtheoptimaltimeforthediscount.Usingarealoptionapproach,thestochasticdynamicpro‐grammingmodelissolved.Giventhefixedcostofthemarkdown,servicelevel,andaknowndis‐countedprice,theoptimalpolicyforthefirmistofollowthethresholdpolicy.Thesellermax‐imisesitsrevenuebydiscountingthepriceoftheproductwhenthepotentialcustomerdemandisgreaterthanthethresholdvalue.

Thecontributionofthispaperisasfollows:Consideringtheoptimalmarkdowndecisionforamonopolistic sellerwith strategic customers,we address the gap in other literature on thesecustomerswithproblemsundermarketuncertainty.Astochasticdynamicoptimisationmodelisproposedtofindtheoptimalmarkdownstrategyoftheseller.Arealoptionapproachisappliedtoobtainaclosed‐formsolutionofthefirm’sdemandthreshold.Theanalysisoftheoptimaltim‐ingrevealstherelationshipbetweenthedegreeofmarketuncertaintyandthemarkdowndeci‐sion‐making.

Althoughtheoptimalthresholdpolicyisfound,carefulinterpretationsoftheresultareneed‐ed.First,customersareawareofpotentialmarkdownswhilethediscountedpriceisknown.Theseller may not exercise the option to markdown if the potential demand never exceeds thethreshold. Second,we found that there is an exponential decrease in the threshold value in adecliningmarket,whichjustifiestheearlymarkdowninsomeindustries.Ontheotherhand,theoptimalmarkdownthresholdincreasesasthevariationindemandincreases.Thisindicatesthata firmneeds to avoid the riskof committingmarkdownpricing too earlywhen themarket ishighlyuncertain.Thecompany’smanufacturingandproductionplanningmustbealignedwiththisstrategicdecisiononthemarkdowntiming.

There aremany challenges involved in the proposed study for future research. Discussionoverpotentialdemandisrecommended.Furtherinvestigationonthepostedpricingschemeofdemanddiffusioncanbedevelopedwherethenewproductgetsadoptedinthepopulationovertime.Anotherpotentialareaofresearchwouldbethepredictionofstrategiccustomerdemandbyapplyingdata‐drivenapproaches,suchasmeta‐heuristicsandmachinelearningalgorithms.Finally,aninterestingextensionwouldbetoimplementtheproposedframeworkonreal‐worldproblemstodemonstratethepracticalimplicationsofourmodel.

Conflict of interests 

Theauthors thank theeditorand tworeviewers for their constructivecomments,whichhelpedus to improve thispaper.Theauthorsdeclarethatthereisnoconflictofinterestsregardingthepublicationofthispaper.

   

Optimal timing of price change with strategic customers under demand uncertainty: A real option approach

Acknowledgement This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2016S1A5A8019542).

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Advances in Production Engineering & Management 14(3) 2019 389

Lee, Lee, Kim

Appendix A: Proof of Proposition 1

Since 𝐸𝐸 �∫ 𝑋𝑋𝑡𝑡𝑇𝑇0 𝑑𝑑𝑑𝑑� = ∫ 𝐸𝐸 𝑇𝑇

0 [𝑋𝑋𝑡𝑡]𝑑𝑑𝑑𝑑 by Fubini's Theorem, we first apply Ito's lemma to 𝑑𝑑 ln𝑋𝑋𝑡𝑡 , to find 𝐸𝐸[𝑋𝑋𝑡𝑡]:

𝑑𝑑 ln𝑋𝑋𝑡𝑡 =1𝑋𝑋𝑡𝑡𝑑𝑑𝑋𝑋𝑡𝑡 −

12

1𝑋𝑋𝑡𝑡2

(𝑑𝑑𝑋𝑋𝑡𝑡)2 (52)

=1𝑋𝑋𝑡𝑡

(𝜇𝜇𝑋𝑋𝑡𝑡 + 𝜎𝜎𝑋𝑋𝑡𝑡𝑑𝑑𝑑𝑑)−12

1𝑋𝑋𝑡𝑡2

(𝑋𝑋𝑡𝑡2𝜎𝜎2𝑑𝑑2) (53)

= 𝜇𝜇 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑑𝑑𝑑𝑑 −12𝜎𝜎2 𝑑𝑑𝑑𝑑 (54)

After integrating and applying the fundamental theorem of calculus, we obtain:

ln𝑋𝑋𝑡𝑡 − ln𝑋𝑋0 = �𝜇𝜇 −12𝜎𝜎2� 𝑑𝑑 + 𝜎𝜎𝑊𝑊𝑡𝑡 (55)

𝑋𝑋𝑡𝑡 = 𝑋𝑋0𝑒𝑒�𝜇𝜇−12𝜎𝜎

2�𝑡𝑡+𝜎𝜎𝑊𝑊𝑡𝑡 (56)

The general form of expectation for Gaussian random variable is 𝐸𝐸[𝑒𝑒𝑋𝑋] = 𝐸𝐸 �𝑒𝑒𝜇𝜇+12𝜎𝜎

2�, where 𝑋𝑋

has the law of a normal random variable with mean 𝜇𝜇 and variance 𝜎𝜎2. Since we know the stand-ard Brownian motion 𝑊𝑊𝑡𝑡~𝑁𝑁(0, 𝑑𝑑), taking expectation on both sides yields the following [9]:

𝐸𝐸[𝑋𝑋𝑡𝑡] = 𝑋𝑋0𝑒𝑒�𝜇𝜇−12𝜎𝜎

2�𝑡𝑡𝐸𝐸[𝑒𝑒𝜎𝜎𝑊𝑊𝑡𝑡] (57)

= 𝑋𝑋0𝑒𝑒�𝜇𝜇−12𝜎𝜎

2�𝑡𝑡𝑒𝑒0+12𝜎𝜎

2𝑡𝑡 (58)

= 𝑋𝑋0𝑒𝑒𝜇𝜇𝑡𝑡 (59) Finally taking integral produces the following results:

� 𝐸𝐸 𝑇𝑇

0[𝑋𝑋𝑑𝑑]𝑑𝑑𝑑𝑑 = � 𝑋𝑋0𝑒𝑒𝜇𝜇𝑡𝑡𝑑𝑑𝑑𝑑

𝑇𝑇

0=𝑋𝑋0𝜇𝜇

(𝑒𝑒𝜇𝜇𝑇𝑇 − 1) (60)

Appendix B: Proof of Theorem 1 Substituting Eq. 16 into Eq. 32, we have the following equation:

𝑑𝑑𝑑𝑑 = 𝑑𝑑′(𝜇𝜇𝑋𝑋 𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑋𝑋 𝑑𝑑𝑑𝑑) +12𝑑𝑑′′(𝜇𝜇𝑋𝑋𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑋𝑋 𝑑𝑑𝑊𝑊)2 (61)

= 𝜇𝜇𝑋𝑋𝑑𝑑′𝑑𝑑𝑑𝑑 + 𝜎𝜎𝑋𝑋 𝑑𝑑′𝑑𝑑𝑊𝑊 +12𝜇𝜇2𝑋𝑋2𝑑𝑑′′(𝑑𝑑𝑑𝑑)2 + 𝜇𝜇𝜎𝜎𝑋𝑋2𝑑𝑑′′(𝑑𝑑𝑑𝑑)(𝑑𝑑𝑊𝑊) +

12𝜎𝜎2𝑋𝑋2𝑑𝑑′′(𝑑𝑑𝑊𝑊)2 (62)

Taking expectations on both sides to apply some properties of GBM and discarding all terms involving dt to a power higher than 1, we have

E[𝑑𝑑𝑑𝑑] = �𝜇𝜇𝑋𝑋𝑑𝑑′ +12𝑑𝑑′′𝜎𝜎2𝑋𝑋2�dt = [𝑟𝑟𝑑𝑑 − 𝑝𝑝𝑜𝑜(1 − 𝜏𝜏)]𝑑𝑑𝑑𝑑. (63)

Note that the term (𝑑𝑑𝑑𝑑)(𝑑𝑑𝑊𝑊) has magnitude (𝑑𝑑𝑑𝑑)3/2, 𝐸𝐸[𝑑𝑑𝑊𝑊] = 0, 𝐸𝐸[(𝑑𝑑𝑊𝑊)2] = 𝑑𝑑𝑑𝑑, and 𝐸𝐸[𝑑𝑑𝑑𝑑] = 0. After dividing through by 𝑑𝑑𝑑𝑑, we have the Bellman Eq. 33.

390 Advances in Production Engineering & Management 14(3) 2019