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Pasc al Courty 3

Universit at Pomp euFabra

And

Li, Hao 33

University of H ong Kong

Novemb er5, 1998

Abst rac t: We present a mo del of t iming of seasonal sales where st ores cho ose several

designs at the b e ginning of t he se ason without k nowing which one, if any, will b e

fash-ionable. Fashionable designs have a chance to fet ch high prices in f ashion markets while

non-fashionable one s must b e sold in a discount market . In the beginning of t he season,

store s charge high prices in the hop e of c apt uring their fashion market. As t he end of

the season approaches with goo ds st ill on the shelves, st ores adjust downward their

ex-p ect ations that t he yare c arry ing afashionable design,and may havesales t o capture the

discount market . H aving a greater numb er of designs induc es a st ore t o put one of t hem

on sales earlie r t o test the market. Moreover, price comp e tition in t he discount marke t

inducesstore s t ostart salesearlie r b e causeof agreate rp e rceivedrst-mover advantagein

capturingt he discountmarket. More comp e tition,p e rhaps dueto decreases inthe cost of

pro duct innovat ion, makes sales o ccur eve n earlie r. These result s are consistent with the

observat ion that the tre nd toward earlier sales since mid-1970's coincides with increasing

pro duct varieties in fashion go od markets andincre asing store comp etit ion.

Ac knowledgements: Wearegrate ful forthe comment smadebyPet erPashigian,St eve

Salant,theEditorandanony mousrefe ree,andse minarpart icipant satU niversit atPomp eu

Fabra and Hong KongUniversity of Scienceand Technology. Courty' work was supp orted

by grant DGES PB 96-0302 from the Spanish Ministry of Education and by grant 1997

SGR 00138 f rom the Generalit at de Catalunya, and Li's was supp ort ed by a grant from

the Research Grant s Council of the Hong Kong Sp ecial Administ rat ive Region, China

(Proj ect No. H KU 7148/98H) .

3 Department of Economics and Business, Ramon Trias Fargas, 25-27, 08005 Barcelona,

Spain ( courty@upf .e s) .

33 Scho ol of Economics and Finance, University of Hong Kong, Pokfulam Road, H ong

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A common phenomenon in retail pricing is sales. In recent ye ars, sales have b ecome an

imp ortant pricing strat egy f or seasonal goo ds (Pashigian, 1988) . An addit ional

observa-tion is that sales have started e arlier in the se ason in recent years. Whereas New Year's

Day t raditionally marke d the b eginning of sales in t he winte r season, sale s have recent ly

advancedt ob ef oreChrist masandevenaroundThanksgiving. Forexample,Pashigianand

Bowe n (1991)cite a rep ort by t he Nat ional Ret ail Merchants Asso ciationon t he monthly

dist ribut ionofyearlymarkdownsshowingthatforapparel,themarketshareoft otalannual

markdownstaken inJune (int he spring-summer season)and Decemb er(in thef all-winter

season) has b ee n increasing since t he 1970's. Pashigian and Bowen's own e mpirical work

alsoshowst hatwhenaveragemonthlypricesofwomen'sapparelarecomparedacrosst ime,

Decemb er prices have decreased relative to the average season price . What explains the

trend t oward early sales in t he clot hing industry? Should one ex p ect a similar t rend in

markets forother seasonal goo ds?

Around the same time as sales b egan to start earlier in t he se ason the numb er of

varietiesoe redhasb ec omemoreimp ort ant. Inc lot hingmarket s,f orexample,inaddit ion

to tradit ional formal and tailor-made suits and dresse s, there has b e en an increase in

sp ortswear, which gives producers more lat itude t o mix diere nt colors and fabric s and

create ne w styles. The market share of sp ortswe ar in women's apparel increase d from

around 42% in 1967, to 56% in 1972, 69% in 1977, and to 78% in 1982 ( Pashigian, 1988).

Do increasing pro duct variet ies play a role in the trend toward e arly sale s in market s f or

seasonal go ods?

Exist ingmodelsofsales(Varian1980,SalopandStiglitz1982,Lazear1986,Pashigian

1988) explain why a st ore may use sales t o maximiz e prots in the pre sence of demand

unce rt ainty. A bsentfromthelit erat ureisananalysisofthetimingofsalesinacomp e titive

cont ext . 1

Timingofsalesisanimp ort antst rate gicissueforseasonalgo o dsmarketsb ec ause

store s facea deadlineto sellt he capacity t hey orderedat theb eginning of theseason, and

they sue r great losses when they are forcedto carry theirgo ods t o t he next season.

We ex tend Lazear's mo del of sales to acc ount for timing of sales, pro duct variety,

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fashionable or not, that is, whet her it will command a market of fashionable consumers

or discount buyers. While fashionable consumers are impulsive buye rs who may or may

not buy at a list price, b ot h fashionable consumers and discount buyers buy at a lower

discount price. In deciding when t o start sale s, the store faces a trade-o b etween selling

the go ods early in the season and selling t hem cheap at the discount price . Sales occur

earlier, if t hereare fewerfashionable consumers inthe marke t asin re cessions,if thestore

learns faster ab out t he market as when it attracts more freq ue nt customer visit s, if the

store faces a great er cost of delaying sale s as when it has a premium lo c at ion wit h high

rent, or if the fashion premium decreases as whenpro duct labeling and bar coding reduce

the cost of price adjust ments and increase the net sales prots. Sales do not necessarily

o cc ur earlier in stores wit h more c omp et ent sales forces: these store s learn faste r ab out

consumer de mand, which te nds to induce e arly sales, butat t he same time they are more

likely t o sell t he ir goo dsat the list price, which t ends t o delay sales.

St ores oft en order mult iple designs of a pro duct at the beginning of the season. For

example, abrand nameof shirtscarriesslightlydiere ntiateddesignswithdierentcolors,

fabrics, orsty le s. Thereislit tlesubst it ut ionamongt he designs forf ashionableconsumers,

butfordiscountbuyerst hedesignscanb e p erfectsubstitute s. Inourmodel,t hemonop oly

store has incent ives to order several designs without knowing which one , if any, will b e

fashionable,evenifdoingsodoesnotincreaset hefashionpremiumorthesizeofthefashion

market. Themarginalb enetofanadditionalvarie ty comesf romthereduct ionoft he risk

in putting a \hot" de sign on sale, and as a result, having a great er numb er of designs

induces the store t o put one of them on sale earlier to test t he market . Thus, our mo del

of monop olyst oreimpliesa relationbetwee ninc reasingpro duct varietiesand e arliersale s.

When the cost of creating designs decreases, our mo del predicts b oth increasing pro duc t

varieties and e arlier sales.

Increasing pro duct varieties in recent ye ars in many markets have also intensied

comp etition among ret ail shops. Standard models of sales based on monop oly pric ing

cannotaccountfortheeec tsofincreasingcompetition. 2

Indeed,byapply inghismonop oly

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init ial prices and lower those prices more slowly." This c onclusion se ems to cont radic t

the ab ove-mentioned coincidenc e of t he trend t oward earlier sales and increasing pro duc t

varieties in the appare l market . The inadequacy of Lazear's prediction is due to the f ac t

that he did not take into account the role of price comp etition.

Ourmo del allows ust o e xamine the relat ion b etweenincreasingc omp et it ionin retail

businesses and t he timing of sales. At the beginning of the season, several st ores each

choose a design and do not know which one of t he designs, if any, will b e fashionable. As

in the case of monop oly st ore, the marke t c onsist s either of fashionable consumers who

are willing t o pay the list price fort he fashionable design only, or of discount buyers who

are willing t o pay at most the discount price and who buy from the st ore with the lowest

price. This st ructure of consumerdemandcaptures theintuit ionthat pricecomp etit ion is

not as imp ortant infashion markets as it is in t he disc ount market .

The main result here is that sales occur earlier under c omp et it ion. Stores advance

the dat e of sales in an atte mpt to capt ure the discount market b efore their comp e titors.

There is a negative externality among t he store s in the sense that they would b e b e tter

o if they could co ordinate todelay sales. Comp etitionreduc es t he stores' re ve nueat any

p oint in time and theref ore reduces the gains from delaying sales. Experimenting with

the listpric e b ecomes more cost ly under incre asing comp etit ion. Ournding is consistent

with theconsensus of ret ail industryoc ials thatincreasing st ore comp etition has played

an imp ortant role in theunraveling of t he sale date in recent seasons. 3

Pricecomp et itionfordisc ountbuyersimpliesthatsalesprice scanb equitevariable. A

monop olystorecho osesthesale spricetob et hereservationpriceofdiscountbuyers. Under

price c omp et it ion in the discount market, st ores must vary their sales prices randomly

in order not t o lose out in t he comp et ition. A lthough sales prices are random, expected

discount sincreaseasthese asonapproachest heend. Thisisconsistentwiththeobservat ion

that discounts are great er whe n sales o ccur later ( Pashigian, 1988). As in the c ase of

monop oly, an increase int he service quality of st ores has ambiguous eects on t he timing

of sales, because it increases the b ene t s from ex p erimenting with t he list price while

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fromex p erimentingwiththelist price decre aseas storec omp et it ionincre ases. Thus,more

comp etitive marke ts are more likely t o exhibit a relation b etween earlier sales and b e tter

serv ic es.

Pro duct varieties inamarket are closely re lat ed wit ht hedegree of store c omp et it ion.

If a store c an order a new design at some cost to comp et e against e xist ing designs, t hen

new varieties will b e creat ed until t he exp ect ed re ve nue from sharing the f ashion marke t

equals the cost of ordering a new de sign. The exp ec ted revenue from one share of the

fashion market do e s not dep endon t he disc ount pric e b e cause comp et itive stores ex trac t

no surplus from t he discount buyers. In contrast, the marginal b enet to a monop oly

store of acquiring an additional design arises fromt he reduction in t he riskof putting the

fashionable design on sale. The marginal b enet dep ends negat ively on t he willingness to

pay by discount buyers, b ecause the monop oly st ore ex trac ts all their surplus. It follows

that the equilibrium numb erof varieties under c omp et it ion is great er thanthe numb er of

varieties that a monop oly st ore will order.

Exist ing mo delsof sales c an explain somewell-do cume ntedwit hin-season and

across-season regularit ies in seasonal go ods market s. Lazear (1986) e xplains why price s f all as

theseasonpro ceeds. Pashigian(1988)andPashigianand Bowen(1991)arguethatgreater

demand uncertainty in themarket forwome n's apparel explains why p ercent age markups

andmarkdownsaregreaterf orwomen'sapparelthanformen'sapparel. Pashigian,Bowen,

andGould(1995)not ethatseasonalvariat ioninretailpriceshasincreasedinapparel

mar-ketbut decreased inautomobilemarket , and suggestt hatt he reasonlies int hedecreasing

cost of innovationforapparel and increasing cost forcars. By e xplicit ly c onsidering

prod-uct variety and price comp et ition, our mode l complement s the existing mo dels of sales

and explains also across-season changes in t he timing dime nsion of store sales st rate gie s.

In clothing marke ts, new pro duct ion t echnologie s t hat reduced t he cost of innovations

have made it easier b ot h for ex ist ing stores to order more designs and new ent rants to

establish their marke t share, prompt inge arlier sales in the season. Inc ontrast , increasing

inte rnationalcompetitionandincre asingcostof innovationint heautomobile markethave

opp osite eec tson t he t iming of sales, which may explain why seasonal pric e data in the

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of t iming of seasonal sales. Section 3 uses t he model t o study the dete rminant s of timing

of sales for a monop oly st ore, and section 4 studies comp etitive t iming of sale s. In each

of these sections, aft erderiv ing themain analyt icalresult s, we present t he ir empirical

im-plicat ions in separat e subsect ions. In section 5, we discuss how t o relax t he assumpt ion

of store and pro duc t symmetry, and ext endt he mo delto address issues suchas dierent

i-ated learning pace and pro duc t turnover rate . Section 6 concludes wit h a brief summary

and some remark s on re gulation p olicies. Pro of s of t he prop ositions can b e found in the

app endix.

2. A Mo del of Se asonal Sal es

The season begins with k 1 designs of a product in the market and lasts f or N selling

p erio ds. Stores cannot order new designs during the season due t o high ordering cost.

Let b e thestore's discount f actor b etween two adj ac ent se lling p erio ds. Small discount

fact orcan b e interpret edas high opp ortunity cost of ke epingt he designson t heshe lf . For

simplicity, we assume t hroughout the pap erthat at t he end of the N-th p erio d, t he store

has zero salvage value for unsold design. This assumption can b e easily relaxe d without

changing the basic result s of themo del.

We assume that the market is homogeneous wit h a xed size normaliz ed to one.

The market consists either of f ashionable consumers attracted to one of t he designs or of

discount buyers who do not c are ab out fashion. Fashion buyers are willing t o pay v

H f or

the design t he ylike and zero forany ot he r de signwhile discount buyers are willing topay

v L <v H f or any design. 4 We ref ert o v H

the\list pric e," v

L

the \discount pric e," and the

dierence \f ashion premium." At t he b eginning of the season, stores do not k now which

one of t he designs, if any, is fashionable. They b elieve t hat each design is fashionable

with probability =k and no design is fashionable wit h probability 10. St ret ching the

inte rpret ationof t hispriorb eliefabit ,wesaythatst oresestimatet hat thef ashionmarke t

is of size .

Since there are only two p ossible consumer valuat ions, timing of sale s can b e given

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L

list price v

H

. Whenever a store charges a discount price ( a price b elow v

L

), we say that

the st ore has \sales." St ores have no incentive to c ontinuously decre ase prices. 5

Throughout t he pap er, we maintain the assumption that consumer do not b ehave

strat egically. Thisassumptioniscommonlymadeinmo de lsofsales(Varian( 1980) ,Laze ar

(1986), Pashigian (1988)) . 6

Justications for the assumption inc lude limited supply of

fashion go o ds, high search cost, quick f ashion t urnover, and short selling season. These

condit ionsarelikely t ob esatised inmarke ts forhigh-endfashionclothesandac cessorie s,

Christmas cards, calendars, and souvenirs f or events such as N BA playos.

In each p erio d, consumers visit all stores and dec ide whether or not to purchase the

designsatt he pricesp ost ed. Int hissimplesett ing, storesoert hedesignsatt he listprice

v

H

in the rst p eriod if there is a go od chance of selling one of them. If consumers do

not buy any of t he designs at the list price, stores learn immediately t hat the designs are

not f ashionable and will charge disc ount price s (have sales). In re ality, stores are likely

to learn more slowly. We mo delt his by assuming that fashionable consumers are impulse

buyers who buy t he design they like only when t he y are in a buy ing mo od, which o ccurs

with probability q < 1. For simplicity, we assume that fashionable buyers buy the design

they like with probability q for any price b etwe en v

L

and v

H

, and 1 if the price is v

L or

lower. D iscount buyers buy t he cheap est design if the price does not exc eed v

L

. If several

designs are equally cheap, they are indierent and cho ose randomly. Finally, to simplif y

thederivationof t he main results,throughout thepap erwe maint aint he assumptionthat

qv

H >v

L :

As in t he existing models of sales (Varian, 1980, Lazear, 1986, and Pashigian, 1988),

the drivingforce in our mo del is c onsumer demand uncertainty and st ore learning, as

op-p osedtomostof thesales mode lsint he marketingliterature (e.g.,G allegoandvan Ry zin,

1994),whereinventory controlisthedrivingforce andlearningplaysnorole. Our

assump-tionofhomogeneousmarketallowsust of o cusontheissuesof timingofsalesandsupplyof

variety,whileignoringq uantitychoiceattheb eginningof t heseasonand invent orycont rol

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in t he ex ist ingsales models. The parameterq hast he interpretat ion of the service quality

of the store s: ast ore with more c omp et entsales force or more comfort able shopping env

i-ronmenthasagreaterprobabilityof sellingthedesignsatthelistprice. A not hernoveltyis

that product variety is taken int o ac count explicitly, which allows us to isolat e the eects

of demanduncertainty fromother fact ors ( e.g., fashionpremium) thatmay alsoaect the

imp ortance of f ashion to consumers. Moreover,we explicit ly mo delt iming of sales wit h a

simple de mand structure of fashionable consumers and discount consumers. The idea of

discount market is also cruc ial forour analysis of comp e titive timing of sale s.

We have made a few simplifying assumptions ab ove, chie famong which is sy mmetry

regarding designs and stores: all designs have t he same list price even t hough there is

no substitut ion among t hem f or fashionable consume rs, and all st ores have t he same

ser-vice quality. Symmet ry regarding designs is assumed to fo cus our mo del on the eec ts of

pro duct variety, while symmet ry regarding st ores is t o highlight t he impac t of store

com-p et it ion. Insect ion 5, wediscuss theimplicationsofrelaxingtheseandotherassumptions,

and show how to enrich our model to address ot her issues re lat ed t o seasonal sales.

3. M onop ol i st ic Ti ming of Sal es

Imagine a single st ore wit h k designs at the b eginning of the season. The store's pric ing

problem can b e solved by backwardinduct ion. To underst and t he basic intuit ion, assume

fornow that k=1. Le t w m

n

( p) b e thestore's exp e ctedprot s whenthereare n p erio ds to

go ( sup ersc ript m stands f or \monop oly ")and t he estimatedf ashion market siz e is p. B y

the assumption of zero salvage value, w m

0

(p) = 0. For n 1, t he store chooses b etween

having sale (charging v

L

)and holding to the list price (charging v

H ): w m n ( p)=maxfv L ;pqv H +(10pq) w m n01 ( p 0 )g;

wherethe up dated estimat e of thesize of thefashion marketafter unsuccessfully charging

the list price is given by

p 0 = p(10q) 10pq :

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Since w

0

( p) =0, in the last p eriod the st ore charges v

L

(has sales) if and only if its

estimat ed size of t he fashion marke t is smaller than

t 1 = v L qv H :

Inallp erio dsb eforethelastone,t hestoreisindiere ntb etwe enhaving sale sinthisperio d

and hav ing sales in t he next p eriod aft er charging t he list price in t his p eriod, if t he size

of thefashion market p sat ises

v L =pqv H +(10pq) v L :

This implies that the t hreshold size of the fashion market under which the store is

indif-ferent b etween having sales in this p erio d andin t he next p erio dis

t= v L (10) q( v H 0v L ) :

Iftheestimatedf ashionmarke tsizeisgreate rthantwithn2p erio dstogo,thencharging

v

H

in t he current p erio d is optimal, b e cause if the de sign do es not sell at v

H

the store

can alway shave sales in the next p erio d, thatis, w m n01 ( p 0 )v L

. If the est imate d fashion

marketsize issmallerthan t,thenit 'soptimalf orthe storetohavesales rightaway: since

the up dated estimate is always smaller than the c urrent e st imate of the f ashion marke t

siz e, that is, p 0

< p, t he b est the st ore can do in t he next p erio d aft er unsuc cessfully

charging v

H

in t he current perio d is to have sales. 7

The ab ove analy sis can b e ext ended t o t he case of mult iple designs (k 2). This

ext ension is imp ortant b ecause it illust rates t he eect of pro duct variety on t iming of

sale s; it will also b e compared to the comp etitive mode l of sales to show t he ee ct of

store price comp etition. The monop oly store orders several de signs at the b eginning of

the season but do e s not know which one of the designs, if any, will b e fashionable. The

newc onsideration in thecase of mult iple de signsis how many designs the monop oly store

should choose t o put on sale when having sales is optimal. Since all designs are p e rfec t

substitutes for discount buye rs and since t he re is no substitution among the designs f or

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L H

theassumption qv

H >v

L

, inthe last p erio d it isnot optimalt o put moret han onedesign

on sale , and a fortiori, it is never optimal to have more than one design on sale in any

p erio d.

For the following prop osition, we dene

t m 1 = v L qv H +v L (k01) ; and t m = v L (10) qv H +v L ((10)( k01)0q) :

Proposition3.1. Inthelastperio d oftheseason,sales o ccurifand only ift hee st imated

p er-design size of the fashion market is smaller t han t m

1

. In any p erio d except t he last

one, sales o c cur if and only if t he store estimates t hat its f ashion market is smaller t han

t m

p e r-de sign.

The threshold f ashion market size t m

1

determining end-of-t he-season sales is not the

same as t he t hreshold t m

during t he season. Indeed, sales occur for dierent reasons in

the two cases. Salesatt he end oft he seasonmay b e int erpre tedas \clearancesales." The

store gets ze ro salvage value for any unsold design while it capt ures the whole discount

market by put ting one design on sale , so sale s o ccur if the e xp ect ed sales prot f rom a

singledesignisgreaterthant hee xp ect edprotof sellingitatthelist pric e. Thethreshold

t m

1

do es not de p end on the disc ount fact or. During the season (n 2) , t he monop oly

store 's choice isb etweenputt ing onedesign onsale in thecurre ntp erio d andhav ing sales

inthenext p eriod. Since thest orecanalwayshavesales inthenext p eriodif nec essary,as

opp osed t o getting zero salvage value in the last p erio d after unsuccessf ully charging the

listprice ,t he t hresholdt m

is smalle r andsalesareless likelyduringt heseason thanatthe

end of the season. Moreover, t he threshold t m

during theseason dep ends on thediscount

fact or. Indeed,from theexpression oft m

, we seethatsales do not o ccur during t heseason

whenitdoesnotc ostany thingtohold thego o dont heshe lf ( =1). Sinceour fo c us ison

learning and demand unce rtainty, unless ot herwise note d, by sales we mean sales during

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that determines t he timing of sales do e s not dep end on the numb er of p erio ds lef t in the

season. The reason f or this is already clear in the case of k = 1. In each p erio d, the

monop olyst orefaces achoiceof selling thesingledesignearly f orsureandselling itatthe

high list price withsome probability. Thet hreshold f ashion marketsiz e duringt he season

is theref ore determined by t he indiere nce c ondition b etwee n hav ing sales in this perio d

and having sales in t he nex t p e rio d aft er charging t he list price in t he current p erio d.

Sincethe t rade-o capt ured by this indiere nce condition do esnotdep endonthe numb er

of p eriods lef t in the season, thethreshold t m

is inde p endent of n.

AccordingtoProp osit ion3.1,priceb ehaviorduringtheseason issimple. Ifthestore 's

prior e st imate of the total size of its fashion market is great er t han kt m

, it st arts with

the list price v

H

for allit s designs. Ast he designs continue to st ay on theshelf, the store

adjusts its estimate downward. If t he designs are still unsold when, forthe rst t ime the

up dat ed estimat e falls b elow the threshold f ashion market size t m

, the st ore randomly

selects a design and puts it on sale at the discount price v

L

. Then, either the marke t

closes after purchases by discount consumers, or if nopurchase is madeat the sales price,

the st ore learns t hat consumers are fashionable and keeps the list price for the rest of the

season. 8

Although t he choice of design on sale is random, t he timing of sales, or the perio d

when sales start, if at all, is determinist ic in this monop oly mo de l. Since the threshold

fashion market size t m

does not dep end on t he numb er of p erio ds lef t, we can work out

the t iming of sales by reversing t he up dating rule, st art ing from t m and t m 1 . Le t b 1 =t m , and dene b n

re cursively for each n2 according tot he up dating rule:

b n01 = b n (10q) 10b n q : Similarly,let ^ b 1 =t m 1 ,anddene ^ b n

accordingt ot hesameup dat ingruleabove. Iftheprior

p er-designest imate=kex ceeds ^

b

N

,salesneve ro c curandt hest orechargesv

H t hroughout the se ason. If b N01 <=k < ^ b N

, sales occur, if atall, in thelast p eriodof the season. For

any i N 01, if t he prior satise s b

i0 1

< =k < b

i

, sale s o ccur in thei -t h p eriod of the

season, wit h probability (10)+( 10q) i01

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Store lo cat ion, bar coding, and business cycles. It c an b e shown direc tly from t he

ex-pression of t m

that the threshold fashion market size is higher for stores with a lower .

Thatis, stores thatdiscountfuture prots heaviertendt o have earliersales in theseason.

Store s at premium lo cat ions have lower b ec ause t hey face a greate r c ost of selling its

designs lat e due to higher re nt or higher opp ort unity cost of shelf space, and as a result

they tend t o have earlier sale s. St ores at premium locations are also likely to have more

frequent cust omer visit s. This means more opp ortunit ies fort he st oresto le arn ab out the

market,whichalsomakessalesapp earearlie rint heseason. 9

Wecanshowt hatt het

hresh-old fashion market size t m

increases as thediscountpric e v

L

increases,or as the list price

v

H

decreases, b ecause it is more costly t o de lay sales and less prot able t o keep the list

price. Recentinnovationsinproductlab elingandbarcodinghavereducedt hec ostofprice

adjustment s. B y increasing the ne t prot s fromsales, theseinnovat ions have equivalent ly

increased t he discount price v

L

, and as a result , sales te nd to occur earlier in t he season.

Timing of sales also depends on the busine ss cycles. Sales occur earlier during re cessions,

b ecause b oth thefashion premium v

H 0v

L

and thetotal size of t he fashionmarket are

likely t o decre ase in a recession. Ifbusiness downt urns are anticipatedb ef ore st ores order

their designs at t he st artof the se ason, then st ores willlikely resp ond by orde ring designs

with smaller f ashion premiums, which will f urther pre cipitat e sales in t he season.

Store service quality. Service quality q can have opp osite eect s on t he t iming of sale s.

On one hand, an increase in t he q uality of the serv ice makes it more likely for t he store

to sell t he designs to fashionable consumers, which tends t o delay the timing of sales by

increasingt hebenet sfromwaiting. This eectcanb eseenfromt hee xpressiont m

,which

decre ases as q incre ases. On the ot he r hand, wit h a b ette r service, the store also learns

faster( updat esfaster)whe theritsdesignsare fashionable,whichtendstodriveearlysale s.

This eectcan b e seen fromtheup dating rulep 0

=p( 10q) =(10pq): f ora givenestimat e

of the fashion market size, the up dated estimat e is smaller if q is greater. The overall

eectof changingstore se rvicequality de p endsonthecomparisonof these two eects. For

(13)

ont m

, forastore thatfaces amarketof smallfashionpremium orhasapre mium lo cat ion

withhigherrentor higheropp ort unitycostof shelf space ,improvedse rvice quality ismore

likely t o lead t o late r sales in the season.

Increasing imp ort ance of fashion. Pashigian ( 1988) observe s t hat since 1960's fashion

has b ecome more imp ortant in clothing markets. Increasing imp ort anc e of f ashion may

b e repre sented by increasing fashion premium: as fashion b e comes more imp ort ant to

consumers, they are willing to pay a higher premium for a fashionable design. As the

fashion premium inc reases, our model predicts late r sales b ecause exp erimenting with

the list price b ecome s more att ract ive t han switching t o t he discount price. Thus, our

mo de l seems t o suggest that asf ashion b ecomesmore imp ort ant t oc onsumers, sales start

later. H owever, the imp ortance of fashion cannot b e gauged solely from t he dierence

in willingne ss t o pay. Increasing imp ortance of fashion implies not only greate r demand

unce rt ainty as re presented by incre ased fashion pre mium, but also more pro duct variety.

The discussion of t he relation b etwee n t iming of sales and the imp ortanc e of fashion is

incompleteunless weaddresst heeects ofincreasingsupplyof varieties. Thiswillbedone

immediat elyb elow forthemonopoly c ase andint he nextsect ion fort hecomp etit ivecase.

Productvariety. Thee ectsof pro duct variety canb e readily ex aminedin ourmo del. Fix

thetotalsizeof thefashionmarketandconsiderwhathapp enst ot het imingofsalesasthe

fashionmarke tisse gment edbyagreaternumb erof designs. Thethre sholdfashionmarke t

siz et m

n

decre aseswithk,but kt m

n

increase s wit hk. Sincetheup dat ingis unae ctedby the

increaseint henumb erofdesigns, saleso cc urearlierwhent he monop olyb eginswithmore

designs on t he shelf. 10

In t he case with one go o d ( k = 1) , the monop oly store balances

the t rade-o b etween selling early and selling cheap. When t he re are more designs, the

monop oly c an use one design as a \t rial ballo on"|if consumers don't buy the design on

sale then they must have high valuation f or t he remaining it ems and the store will keep

the listpric e forthe re stof t he season. When t henumb erof designsis greate r, therisk of

put ting a \hot " design onsale is reduced,and therefore t he balance b etween se llingearly

and selling cheap is t ilt ed in favor of selling early. 11

(14)

so do es not increase t he f ashion premium or the size of t he fashion market. The b enets

of having more varieties can b e calculat ed by comparing t he ex p ecte d prots wit h k +1

and k designs at t he b eginning of the season, with t he prior est imate of the t otal fashion

market size xed at . Clearly, t he benet s are nil if is suciently large so t hat sales

never o c cur even wit h k+1 designs ( that is, ex ceeds ( k+1) ^ b N whe re ^ b N is c alculated

as ab ove, with k +1 designs): f rom t he derivation of Prop osit ion 3.1, since t here are no

sale s, the e xp ect ed prot is just

qv H (10(( 10q) ) N ) 10(10q) ;

indep e ndent of the numb er of designs. The di erence in the exp ect ed prots b etween k

and k+1 designs is t he largest, the marginal b enets of variety the gre at est, when is

so small t hat sales o ccur in t he rst p erio d of t he season e ve n with k designs ( that is,

is smallert han k t m

) . 12

From theexpressionof sales prots in thepro of of Prop osition3.1

(see the app endix ), the b enets t o the monopoly store of adding an additional design to

existing k designs are at most

qv H (10((10q)) N ) 10(10q) 0v L k ( k+1) :

Not e that the ab ove ex pression of marginal b ene t s of product variety is prop ort ional to

1=(k(k+1)). Theint uit ion issimple. Since thestore cho osesonly one design onsale ( and

since by assumption increasing the numb erof designs doesn't change the tot alsize of the

fashion market), having one more design reduces the probability of put ting a hot design

forsale. If the store cho ose s thehot design on sale, it getsv

L

inst ead of t he revenue from

sellingit at v

H

some time inthe se ason. Theprobability of choosing t he hotdesign is1=k ,

and so the marginal b ene t of an addit ional design is prop ort ional to 1=( k ( k+1)) .

4. Comp eti t i ve Ti mi ng of Sal e s

Nowimagine thattwo st ores,A and B, eachwitha de sign atthe b eginning of theseason,

(15)

that since there is no substitution b etween the two designs for fashionable consumers,

price competition is limit ed t o t he discount market where t he two designs are p e rfec t

substitutes. Furthermore, if b oth stores charge the list pric e v

H

and if consumers do not

purchase eit her design, the up dated estimat e of each store ab out it s f ashion marke t size

is p(10q)=( 102pq) . Thus, stores up date at the same pace under comp et itionand under

monop oly. The results we obtain b elow regarding earlier sale s under comp e tition do not

rely on any implicit assumpt ion that learning is fasterunder comp et ition.

Consider rst t he equilibrium in t he last p erio d. Since t he salvage value is zero f or

b oth st ores, each charging v

H

is a pure-strat egy Nashequilibrium if

pqv

H

(10p) v

L ;

where p < 1=2 denot es the common estimate of each store's fashion market size. In t his

case,twostoresareoptimisticab outt he irchanceattheirownfashionmarket,eventhough

eachofthemc ancapturethewholediscountmarketbyhavingsales. Iftheab ovecondit ion

is not satised, there is no pure-strat egy Nash e quilibrium, b ec ause the estimat ed size of

the fashion market is too small t o just if y giving up the discount market complet ely, but

since the two designs are p erfect substitutes in the disc ount market, price comp etit ion

te nds t o drive down the sales prot s b elow what each store c an get by sticking t o t he list

price. N ot e t hat neit her store charges any price b e tween v

L

and v

H

, b ecause such price

reduces the store's prots from it s fashion market wit hout attracting discount buyers.

Moreover, price comp e titionint he discount marketdo es not reducethe pricesand prots

to zero, b ecause each store can always hold on to its f ashion marke t, which provides a

lower b oundon its prot s and the amount of discount it is willing togiveaway.

Using t he standard techniques ( see , e.g., Varian 1980), we can show t hat in the

random-st rat egy equilibrium when pqv

H

< (10p)v

L

, there is no probability mass p oint

in the randomizat ion supp ort of each store, except at v

H

. By symmetry, we can let the

common randomization supp ort b e [x ;v

L ][ fv

H

g. The lower b ound of t he supp ort x

(16)

H fashion market : ( 10p )x =pqv H ; which implies x= pqv H 10p :

The probability F(x) that each storecharge sa price b elowsome x2[x;v

L

] isdet ermined

by t he condit ion that theot her st ore is indierent b etween t he price x and v

H , ( 102p) (10F(x )) x+px=pqv H ; which implies F(x ) = ( 10p)x0pqv H (102p)x :

Not e thatF(x) is p osit ive f or all x2[x ;v

L

]. Moreover, the assumpt ion qv

H >v L implies thatF( v L

)<1,whichshowsthatt hereisaprobabilitymassatv

H

,thatis, theprobability

of each st ore having salesin strictly less than one. 13

Theab ove mo del of two comp etingstores c an b e easily e xtendedt o t hec ase of many

store s. For ex p ositional convenience, we rest rict to t he two-store model in t he main tex t,

although theresult and thepro of are statedfor thegeneral case of k st ores eachwit h one

design. For t he following proposition, we dene ( sup ersc ript cst ands for \c omp et it ive ")

t c n = v L qv H (10( ( 10q)) n ) =(10(10q))+v L ( k01) :

Proposition 4.1. Supp oset here aren p erio ds leftin the se ason. Then,sales o ccur with

p osit iveprobability if andonly if eachst ore's estimat eof its fashion marketsize issmaller

than t c

n .

From the derivation of Prop osition 4.1, when there are n p erio ds remaining in the

season, theex p ected prot of each store as af unct ion of its estimatedfashionmarket size

p is give n by: w c n (p)= pqv H ( 10( (10q)) n ) 10(10q) :

(17)

H

each p eriod. Obviously, t his myopic strategy cannot b e part of an equilibrium. Howe ver,

from an ex ant e p oint of view, comp e ting st ores cannot ex p ect to do any b ett er than by

following t his myopic strat egy. U nder comp et it ion, the store s receive no re nt from the

discount market. It is completely dissipat ed in storecomp etit ion.

The random-st rat egy equilibrium calls for a few comment s. In the monop oly store

mo de l of se ction3, wehavese ent hat thetimingof sales isdet erministic,but theselect ion

of designforsalesis random b ecause t he monopoly storehas nocluewhich designis likely

to b e fashionable. The random-strategy eq uilibrium in the comp et itive mo del here may

b e unde rsto od in a similar way. B ecause stores do not know whether they are carry ing

a fashionable design, only a random pric ing strategy can guarant ee t hem not to lose out

in the comp et it ion. Random strategies are not uncommon either in pract ic e or in the

literature. Existing mode ls of sales dist inguish two typ es of sales: \te mp oral" sales of

Varian (1980) wit h random prices, and \spatial" sales of Salop and St iglitz (1982) with

determinist icprice s. Te mp oral salescanb e interpret edas unadvertise d sales. 14

Compared

to adve rtised spatial sale s, temp oral sales are less v ulnerable to consumer arbit rage.

Timingof sales canb e derived fromProp osition4.1. Ifst ores' priorestimate of the

tot al size of t he fashion market is larger than 2t c

N

, b oth st ores st art the se ason wit h the

list price v

H

. As thedesigns continue to st ay on t he shelves, stores adjust t heir e st imates

downward, and at t he same t ime t he threshold fashion market size for the start of

sales-randomization increases ( t c

n

inc reases as n decreases). Le t i b e t he rst perio d such that

each store's e st imate of it s fashion market size is smaller than t c

N0i+1

. If the designs are

still unsold at theendof p eriodi01, which o c curs withprobability (10)+(10q) i01

,

the stores st art t o have sales in p e rio d i. From the derivation of Prop osition 4.1, the

probability that astore has sales, or charges a price b e low v

L , is F n (v L )= (10p)v L 0w c n (p) (102p)v L ;

where n=N 0i+1 is t he numb erof p erio ds remaining at the b eginning of p e rio d i .

Price b ehaviorafte r sales startin p erio di can b e easily derivedfromProp osit ion 4.1.

Ifb oth storeshavesale s inp e rio d i, whicho ccurswithprobability F 2

n (v

L

(18)

sale s, which occurs with probability 2F n ( v L )( 10F n (v L

) ), then consumers buy fromstore

A if t he y are discount buyers or if they like store A'sfashion, and they buy fromstore B

only iftheylikeB'sfashionandthey areina buyingmo o d. Ineithercase,theseasonends

afte rthepurchase . Ifconsumerslikestore B's fashionbutare notinabuy ing moo d, store

B infers that consumers like it s design with probability 1, and will charge v

H

fort he rest

of theseason. Finally, if b oth stores charge v

H

in p eriod i, which o ccurswith probability

10F 2

n (v

L

) , theseasonpro ce eds top erio d i+1ifconsumersare discountbuyers, orif they

like one design but are not in a shopping mo od. Inperio d i+1, t he up dat ed estimat e p 0

remains b elow t he threshold fashion market size t c N0i ( b ecause p 0 < p<t c N0i+1 < t c N0i ),

and the ab ove de script ion applies with i+1 replacing i and p 0

replacing p.

4.1. Impl i cati ons of the c omp e ti ti ve mo del

Within-season pricevariability. U nlikeint he monop oly case wherethesalespric eis eq ual

to the low valuation, price comp et ition for discount buyers implies t hat observed sales

prices can b e q uit e variable, b ec ause st ores must randomize their sale s prices in order

not t o lose to t heir comp etitors. However, as the season approaches the end, expected

discount s increase . Toseet his, not erst thataf terp erio di whensalesstart ,stores charge

the sales pric e only if b oth charged v

H

in all pre vious p e rio ds and either consumers are

discount buyers or t hey are fashionable but in a buying mo od. From the derivation of

Prop osition4.1, t he lower b ound of each st ore's randomizat ion supp ort x

n is given by x n = w c n (p) 10p :

Ast he numb erof p eriods nremainingdecreases andas thestore'se st imatepofits fashion

marketsizeis adj usteddown, theb oundx

n

decreases. Similarly,oncesales-randomizat ion

start , with n p erio ds remaining in t he season, the probability t hat each st ore charges a

price lower thanx in the randomizat ion supp ort is givenby:

F n (x )= (10p)x0w c n ( p) (102p)x :

(19)

n

lower the e st imates of their fashion market size (p dec reases). This result is consistent

witht he observation inPashigian(1988) thatstoreshaving sale searlier inthe seasongive

smaller discount s.

Although exp e cted discounts increase as t he season approache s t he end, observed

ave rage price ( ac ross designs) implied by the mo del may not always dec rease over the

season. When sales start in p erio d i, there is a p ositive probability 2F

n (v L ) (10F n (v L ) )

that only one store, say store A , has sales, and wit h probability p( 10q) consumers do

not buy from store B. In t his c ase, store B infe rs t hat consumers like it s design and

will charge v

H

for t he rest of the se ason. Thus, as in the case of monop oly store, the

observed average price may go down and t he n up instead of going down throughout the

wholeseason,alt houghunderb othmonop olyandcomp etition,t hepricef oragive ndesign

cannot go down and t hen up. This explains why somet imes stores are observed to have

sale s in t he b eginning of the season, which is me ntioned by Pashigian (1988) as a puzzle

to t he st andard theory of sale s. Our mo del is more successful in this resp ect because,

unlike in the st andard sales mo del where estimates of c onsumer willingne ss to pay are

always adj usted down as t he season proceeds, we capture t he idea thatit can go up after

unsuccessf ul sales by intro ducing the p ossibility t hat c onsumers are discount buye rs.

Serv ic equalityandotherde terminantsoft iming. Weconc ludedint hemonop oly casethat

increases in t he service quality of the store have ambiguous e ectson the timing of sale s,

b ecause b e netsfromexp erimentingwith thelistpriceare greaterwhilelearningisf ast er.

The same forces e xist under comp etition as we ll. H owever, while t he eect of increases

in t he service quality on t he sp eedof learning do e snot change when t here are c omp et ing

store s, the b e nets from exp erimenting with the list price de crease as store comp etit ion

increases. Thus, a comp etit ive market is more likely to exhibit a correlation b etween

earlier sales and b ett er service s. As in t he case of monop oly st ore, other determinants

of earlier sales are small fashion premium, great opp ortunity cost of holding the goo d on

the shelf and small size of t he fashion market. We also c onclude d in t he monop oly case

that decreasing cost of adj usting prices drives earlier sale s and inc reases st ore prot s by

(20)

obtain no prots from t he discount market .

Storecomp e tition. Themainresultint hissect ionisthatcomp et itiondrivesearlysales. A

monop olyst ore startssales when itsestimat eof p er-designf ashionmarketsize falls below

the threshold size t m

, which does not dep end on t he numb er of p eriods le ft. With two

comp etingstores,salesstartwithnp erio dstogoif t hestore s'estimat esof theirresp e ctive

fashion market size falls b elow t c

n

. U nder t he assumption that the prior estimates of the

tot al fashion market siz e are the same under monop oly and under comp et ition, up dat ing

of est imate so ccursat t he samepace. Therefore,salesstarte arlierunder competitionwith

n p erio ds t o go if t c n > t m . 15 Recall t hat t c n

decreases wit h n. We have (in the c ase of

k =2) t c 1 = lim n!1 t c n = v L ( 10( 10q)) qv H +v L (10(10q)) :

It can b e veried wit h a lit tle algebra that t c

1 > t

m

, and there fore t c

n >t

m

forall n.

The reason that price c omp et it ion in the discount market drives early sale s can b e

underst o o dasfollows. A monop olystore balanc es thetrade-ob etweent akingthechance

at the f ashion marke t and capturing t he discount market for sure . In t he comp e titive

equilibrium, the two st ores f ac e the same trade -o. H owever, while a monop oly store

can always have sales in the next p eriod after unsuccessf ully charging v

H

in the c urrent

p erio d,suchsecond-chanceisnottakenforgrantedundercomp et it ion. Instead,eachstore

p erceives arst -move r advant agein capt uring t hediscountmarket . Thecont rast b etween

the se cond-chance under monop oly and t he rst-mover advantage under competition can

b eseenbycomparingthenumerat orint heexpressionoft m

tothatof t c

n

: undermonop oly,

the b enet of havingsales in t he c urrent p erio d instead of in the ne xt p erio d, represe nted

byv

L

(10 ) ,isext ract ingthemaximumsurplusv

L

fromdiscountbuyersonep eriodearlier,

whereasunder comp e titionthe b enetof havingsales int hec urrentp erio d isv

L

,whichis

larger t han under monop oly. The refore, under comp etition t he trade-o is tilte d in favor

of capturing the disc ount market, imply ing t hat sales are more likely under c omp et it ion.

The paradox , of course, is t hat in equilibrium the rst -move r advantage do e s not exist,

(21)

b e mode le d by an increase in the numb erof stores while maint aining t he t ot al size of the

fashionmarket andtheassumpt ionthate achstore hasonedesign. Itis straightforwardto

showt hatkt c

n

increase swithk. Thus,morecomp e titiondrivese arliersales. Tounderst and

the int uit ion b ehind this result , not e thatas the numb erof st ores inc reases wit ht he total

siz e of f ashion marke t xed, e ach store commands an increasingly smaller share. Since

the prot f rom each store 's fashion market decreases as it s share shrink s, the perceived

advant age f rom being the rst to capture t he whole discount market b ec omes great er.

Theref ore, sales occur earlierduring t he season as comp et itioninte nsie s.

Product variety. The comp etit ive model can b e used t o consider the implication of store

comp etition t o pro duct variety. Supp ose that there are already k stores in the market,

each carryingadesign, andc onsider t heinc entives of a newentrant. Ift he ent rantcarries

a design that gets 1=(k + 1) share of the fashion market , then from the derivat ion of

Prop osition4.1, t he expected prot t o t he store at the b eginning of the season is

qv H ( 10( (10q)) N ) 10( 10q) k+1 ;

where is thepriort hat c onsumersarefashionable. The ab oveexpression canb e thought

ofast heaverageb enetofvarie tyundercomp etition,whicharisesfromsharingthefashion

market. N ote t hat the maximum discount price v

L

do e s not ent er t he expression, as the

store sdo notext ractsurplusf romdiscountbuyers. Att heendof sect ion3,wederivedthe

marginal b enet of variety unde r monop oly, which arises from there duction int he risk of

put tingthef ashionabledesignon sale andt heref oredep endsne gat ive ly onv

L

. Comparing

the average b enet under comp etition t o t he marginal b ene t under monop oly, we nd

thattheave rageb enetis great er,implyingt hatc omp et it ivemarket s oermore variet ie s.

If thereis free ent ry intot he ret ailing business in thatany store can order a di erent

designatsomecostt ocomp et eagainstthedesignscarriedbyexistingst ores, newvarieties

willb e c reate duntiltheexpectedprot fromprice comp etitionequals thecostof ordering

a newdesign. If the costof orderinga new de sign falls, t herewill b e more varietiesin the

market, and henc e greater st ore comp etit ion. As a result, sales will o c cur earlier in the

(22)

an explanation of t he empirical relat ion b etween the t rend of earlier sale s and increasing

pro duct variety in seasonalgo ods marke ts may lie in b oth mo dels.

5. Exte nsi ons

The stylized model presented in prev ious sections allows us t o fo cus on the main issues

of sales t iming, pro duct variety, and price c omp et it ion. H owever, several feat ures of our

results are atodds wit hcommonly observedprac ticesinre tail market s f or se asonalgoo ds.

In this section, we ext end our mode l t o address some of t hese inconsistencies. These

ext ensions are not meant tob e comprehe nsive; rat her, they are partly chosen toillustrat e

how our mo del can b e deve lop ed furt her t o address other issues regarding seasonal sales.

A st raightf orward ex tension is t o combine t he monop oly mo del of sect ion 3 and the

comp etitive mo del of sect ion 4 by allowing comp e ting st ores to have multiple designs.

Supp osethatt he reare k st ores, each wit h l designst hat sharethe fashionmarket eq ually.

As in the monop oly mo del,the assumption qv

H >v

L

implies thatstore s randomly se lec t

a single design when it is opt imal to have sales. Following t he pro ofs of Prop ositions 3.1

and 4.1, we c an show that t he p e r-de sign threshold fashion market size with n p erio ds to

go is given by v L qv H ( 10((10q)) n )=(10(10q) )+v L (kl01) :

Thus, the comp etit ive t iming of sales dep ends only on t he numb e r of designs that share

thef ashionmarket , notdire ctly ont henumb erof competingst ores (aslongasitisgreater

than one) . An implicat ion is t hat the assumption in section 4 t hat each comp eting store

has just one design is without loss of generality.

Inourmo delwhenastorewithmultiplede signsof apro ducthassale s,asingledesign

ischosenrandomly,b ec auseputt ingmorethanonedesignonsaleonlysacricesthechance

ofcapt uringthefashionpremiumandb ecausealldesignsare exanteidenticaltothest ore.

Howe ver, one some times observes stores putting several designs on sale at t he same t ime,

andt he choiceof designs f or sale sis systematic rat her t hanrandom. Thisobservat ioncan

b e accommo date din our mo del by relaxing the assumpt ion of qv

H > v

L

(23)

qv

H <v

L

, the price b e haviorandsales t imingstrat egy in themonop oly mode l are similar

asdescrib edinProp osit ion 3.1. Inparticular,asingleitemwillb e chosenwhensalesoccur

fort he rstt ime, so thatgre at er product variety hasthe same eectof pre cipitat ing sales

as in the case of qv

H > v

L

. However, unlike in the case of qv

H > v

L

, when the \trial"

sale s t urn out to b e unsuc cessful and t he st ore infers that it is f ac ing a market of fashion

buyers,itwill not chargev

H

untilt heendof t heseason. Inst ead,at somep ointitwillnd

it opt imal toput all remaining designs ona \c le arance" sale. A morerealist ic assumpt ion

is t hatdesigns die rin fashion premiumso that qv

H >v

L

forsome de signs but qv

H < v

L

for the ot hers. Then, a few designs wit h small premium will b e chosen rst when sales

o cc ur later in the season. 16

The result that greater product varieties precipitate sales in

the se ason should carry t hrough to this case as well.

Of ten,somestoreslearnf ast erthanothersab outthedemandfortheirpro ductsdueto

b et terse rvice quality, andstores can also dierint heir fashionmarket share. A sy mmetry

amongstores inourmo del aect sb ot ht iming andpricingstrat egiesof sales. Toillustrat e

this p oint, consider the situation in t he last p e rio d where one of the two st ores, say store

A, has an edge ove r its comp etit or B in t he fashion market, eit her b e cause its service

qualityis higher(q A

>q B

) , orb ecauseit hasagreatershare offashionmarket( p A

>p B

).

Then, sale s can b e more likely in store B than in store A. More pre cisely, supp ose that

p A (10p A ) q A > p B (10p B )q B . Then, if p A q A v H >( 10p B )v L and p B q B v H (10p A )v L ,

store A charges the list price v

H

and store B has sales (charges v

L

), but t he opp osit e

sit uat ion can not o ccur; if p A q A v H (10p B ) v L and p B q B v H (10p A ) v L , each store charges v H

and sale s do not o c cur in e it he r st ore; nally, if p A q A v H < ( 10p B )v L and p B q B v H < (10p A ) v L

,st oreA'sedgeinthef ashionmarketisto osmalltojust ifygiv ingup

the disc ount marketc omple tely, so there isno pure-st rategy equilibrium. Int he last case,

b oth stores can have sale s, but only st ore A has a p osit ive probability of charging the list

price while store B concentrates on the discount marke t. The prec ise characte rization of

the eq uilibrium and the pro of are in the app endix. This analy sis can also b e extended to

during theseason. The n, if ast ore has an edgein the fashion market due tolarge marke t

(24)

store has an edge in the fashion market due to it s high serv ice q uality, there will b e two

opp osite f orces aecting timing of sales: t he st ore has a go o d chance of selling its designs

at the list price, which t ends t o delay sales, but at t he same time it learns fast er ab out

the demand, whichtends t o induceearly sales. The overall e ects ontiming are t he refore

ambiguous.

Inour mo del,fashion pre miumdo es not changeforthe whole durat ion of t he season.

That is, even at the e nd of season, fashionable consumers are willing to pay t he same

list price for the design they like. However, in a seasonal go o ds market with fast fashion

turnover, the fashion premium is likely tob e de clining t owardt he end of t he season. This

can b e mode le d by an exogenous pac e of declining list pric e as t he season pro ceeds. An

implication is t hat as fashion t urnover b e comes fast er (t he list price de clines fast er) , the

eect ive lengthof t he season b ecomes short er, and salesst art earlier. Such mode l withan

exogenouspac e of declining list price is also more realist ic b ec ause st ore prices b ef ore the

startof sales follow agradual de cline, as opp osed t oa one-timedropf romthe list priceto

some discount prices in the stylized mo del of previous sec tions.

6. C onc ludi ng Remarks

Since t he 1970's, sales of seasonal designs such as apparel have st arte d earlier in the

season while product variety has incre ased in these markets. This pap er discove rs two

link s b etwe en these two trends: one through a monop oly store's opt imal pricing, and the

other t hrough st ore competition. We consider a simple model where retail stores start

the season without k nowing which of the designs t hey have, if any, will b e fashionable.

Store s initially charge a fashion premium in hop es of capturing t heir fashion market , but

as the end of t he se ason approaches with designs st ill on t he shelves, they adj ust t heir

exp e ctations downward. At some p oint in t he season, it b ec omes more protable t o have

sale stocapt ure thediscountmarket. Havingagreat ernumb erofdesignsinducesthestore

to put one of t he m on sales e arlier t o t est the market. Price c omp etit ion in t he discount

(25)

delaying sale s. Ina marketwith fre e ent ry, afall int he cost of pro ductinnovat ion results

in more pro duct varieties, gre ater store comp etit ion, and e arlier sales in the season.

Pashigian's ( 1988) work is most close ly related t o ours. He use s a monop oly mo del

to show t hat as fashion b ecomes more imp ortant , there is more within-season price

vari-ation. Increasing imp ort ance of fashion is mo dele d by a me an-preserv ing spread of the

dist ribut ion of consumer valuat ion. In Pashigian's mo del, great er pro duct variety means

more uncertainty facing t he monop oly st ore and greater het erogeneity among consumers

at t he same t ime. By c ontrast , in our mode l increasing imp ortance fashion implies more

unce rt ainty facing c omp eting st ores only. Although inc reasing demand for

individual-ism has result ed in greate r heteroge neity among consume rs, a large fraction of consumers

strives toownthe design thatis \in." A mo deldrivenentirely by great erunc ert ainty and

comp etitionsuchasours capt ures t he essentialherdingaspectof t hefashionphenomenon.

Our result s indicat e a c lose relat ion b etween the timing of sales, store s prots and

equilibrium pro duct variety. This observation has at least two imp ort ant implications f or

the retailing indust ry. First, price comp e tition in the discount market imp ose s negative

ext ernalityonthecomp etingstore sanddrivesearlysales. Thus,thereiscollect iveincentive

for the stores t o lobby re gulators to imp ose rest rict ions on the timing of sales and on the

amount of discounts. Second, industry regulators must take int o acc ount the eect on

pro duct variety of regulat ion p olicies restric ting the amount of discount or t he t iming of

sale s. When consumers value pro duct variety, abinding restriction ont he timing of sale s,

toget he rwithp olic ie st hate nsurefreeentryint heretailbusiness,mayresultinanincrease

(26)

A .1. P ro of of Proposi ti on 3.1

Proposition3.1. Inthelastp eriod of theseason,sales o ccur if andonly if thee st imated

p er-design size of the fashion market is smaller t han t m

1

. In any p erio d except t he last

one, sales o c cur if and only if t he store estimates t hat its f ashion market is smaller t han

t m

p e r-de sign.

Proof. Suppose that t here are n p erio ds remaining in t he season. With probability p

consumers like a give n design, and wit h probability 10kp t he y are discount buye rs. The

optimal protof the monop oly st ore w m

n

(p)satises thef ollowing equation:

w m n ( p)=maxfkpqv H +(10k pq) w n01 ( p 0 );s n (p) g; where p 0 =p (10q)=(10kpq), and s n (p)=( 10kp+p)v L +( k01)pqv H +(k01)p(10q) qv H 10( ( 10q)) n01 10( 10q)

is t he ex p ected prots f rom putt ing one design on sale. The rst term in t he sales prots

s

n

( p),( 10kp+p)v

L

,is theex p ectedsale s protsint he curre nt p erio d|consumers buyif

either t he yare discount buyers ( wit h probability kp, or t hey like the de sign on sale (with

probability p) . The second t erm, ( k01)pqv

H

gives the ex p ected prots from selling one

of k01 designs not on sale ( with probability (k01) pq). The third t erm gives t he prots

(discounted by ) af ter an unsuccessf ul sale (with probability (k 01)p( 10q)) when the

monop oly store learns t hat consumers like one of k 01 remaining designs f or sure and

charges v

H

f or e ach of the remaining n01 p erio ds. The rst part in the max expression

ab ove gives the ex p ecte d prots of sticking to v

H

for all designs for one more p erio d.

The rst term, k pqv

H

, gives the ex p ecte d prot s of selling one of k de signs at t he fashion

price v

H

. The se condt erm, ( 10kpq)w

n0 1 (p

0

) , givest he optimal discountedprot s after

unsuccessf ully charging v

H

for all designs, with an up dated e st imate p 0

and numb er of

p erio ds n01.

Since t he salvage value is zero, we have w m

0

(p) = 0 f or all p. Supp ose n = 1. The

following solut ion is easily obtained:

w m 0 (p)= kpqv H , if p>v L =(qv H +( k01)v L ) ( 10kp+p)v L +( k01)pqv H , if otherwise. Thus, t m 1

is thet hreshold f ashion market size when n=1.

For n 2, since t he up dat ed est imate p 0

of the f ashion market size is alway s smaller

(27)

thatthet hresholdfashionmarketsizet

n

isdeterminedbythec onditionthatt hemonop oly

store isindierent b etween having sale s inthec urrent p eriodand having sale s inthe nex t

p erio d aft ercharging v

H

fort he current perio d. That is, t he t hreshold t m n sat ises: s n (p)=kpqv H +(10k pq) s n0 1 (p 0 ) ;

which implies that t m

n =t

m

as given in the stat ement of t he prop osition. Q.E.D.

A .2. P ro of of Proposi ti on 4.1

Proposition 4.1. Supp osethere are n p e rio dslef t in the se ason. The n, sales occur with

p osit iveprobability if andonly if eachst ore's estimat eof its fashion marketsize issmaller

than t c

n .

Proof. Thepro of is by induct ion. Supp ose thatn=1. Since consume rs have notmade

thepurchase,st ores have symmet rice st imates of t he ir fashion marketsize. Let thestore s'

estimat eb e p. Then,if p>t c

1

, each store charging v

H

is apure-st rate gy N ash eq uilibrium

b ecause pqv H >v L ( 10kp+p): If p<t c 1

, t here is no pure-strat egy Nashe quilibrium. Each st ore randomly select s a price

fromt hesupp ort[x

1 ;v

L ][fv

H

g. Thelowerb oundof t he supp ortx

1

isdete rmined by the

condit ionthateach storeisindie rentb e tween chargingt hispriceand chargingv

H , which implies that x 1 = pqv H 10(k01)p ; The probability F 1

(x ) that each store charges a price b elow a price x 2 [x

1 ;v

L

] is det

er-mined by thec ondition that eachstore is indie rent betwee n the price x and v

H : (10k p) (10F 1 ( x)) k 01 x+px=pqv H :

Since either it is opt imal f or each st ore to charge v

H

or each store is indierent b etween

v

H

and a price b elow v

L

, the ex p ected prot of each store as a funct ion of its e st imated

siz e of fashion marke t is given by w c

1

(p)=pqv

H

. The prop osition holds for n=1.

Supp osen=2,andeachst oreest imatesthatitsdesignisfashionablewit hprobability

p. Against a price of v

H

by otherstores, charging v

H is optimal if pqv H +( 10kpq)w c 1 (p 0 )( 10( k01)p )v L ; where p 0

= p( 10q)=(10kpq). Theref ore, when n =2, charging v

H is optimal against a price of v H by t he other store if pt c 2 = v L qv H (1+(10q))+v L (k01) :

(28)

If p<t

2

,each st ore charging v

H

cannot b e an equilibrium. As inthe case of n=1, t here

is nopure strategy eq uilibrium, and b othstores must randomize withp ositiveprobability

ofchargingv

H

andhavingsales. Notethatregardlessof thepric eschargedbyotherstore s,

charging v

H

givesa storet he sameexp ect ed protsof pqv

H

+p( 10q)qv

H

: wehaveseen

that this is the payo if all ot her st ores charge v

H

; if l stores charge discount prices (v

L

or lower), the payo of charging v

H is pqv H +( p(10 q)+ (k0 l 01) p)w c 1 (p 0 ) , where p 0

=p (10q)=(p( 10q)+(k0l01)p ),re sulting in t he same exp ec ted payo. Since eit her

it's opt imal for a store to charge v

H

or the store is indierent b etween charging v

H and

a discount price f rom the randomization supp ort,the ex p ected prot as a f unction of the

estimat ed fashion market size p is:

w c 2 (p)=pqv H +p(10q) qv H :

The lower b ound of t he supp ort x

2

is dete rmined by t he c ondition t hat each store is

indierent b etween charging t his pric e and charging v

H ,implying x 2 = w c 2 (p) 10(k01)p : The probability F 2

(x ) that each store charges a price b elow a price x 2 [x

2 ;v

L

] is det

er-mined by thec ondition that eachstore is indie rent betwee n the price x and v

H : ( 10kp)( 10F 2 (x )) k 0 1 x+px=w c 2 ( p):

The proposition holds for n=2.

Theargumentis similarforn=3: given thefunction w c

2

(p ) andsymmet rice st imates

of fashion market sizes, all stores charge v

H

if t he estimate p t c

3

, while t hey randomize

with p osit ive probability of charging v

H

and having sales if p<t c

3

. By induct ion, forany

n, the exp ec ted prot to each store, as a funct ion of the estimat e p of it s fashion marke t

siz e, is w c n (p)= pqv H ( 10( (10q)) n ) 10(10q) ;

and the threshold fashion market size t c n satises w c n (t c n )=(10t c n (k01))v L ;

which gives the ex pressionf or t c

n

stat ed in t he prop osition. Q.E.D.

A.3. The Asymmetri c C ase

Proposition A.1. Supp ose n = 1, p A q A v H < (10p B ) v L and p B q B v H < (10p A )v L but p A (1 0 p A ) q A > p B (1 0 p B )q B

. Then, in t he random-strategy eq uilibrium, store

A's probability of having sale s is p ositive and less than 1, while st ore B has sales with

(29)

Proof. First, we show that store B charges v

H

wit h zero probability. Let x b e the

gre at estlowerb oundof price st hatstore A charges in equilibrium. StoreA alwayshasthe

option of charging v

H

and getting prot s of p A q A v H . B y charging x A

, st ore A can earn

prots at most eq ual to (10p B

)x A

. The refore, we have x A p A q A v H =( 10p B ). Now

store B can alway s undercut store A by charging a price just b elow x A

and get prots

equal t o (10p A ) x A . Since x A p A q A v H =( 10p B

) , store B's prot is at least as great as

(10p A )p A q A v H =( 10p B

). By assumpt ion, this prot is greater than p B q B v H , which is

what st ore B gets by charging v

H

. There fore, st ore B never charges v

H .

Next , we show that st ore A charges v

H

with p ositive probability. Supp ose not. Le t

x A

and x B

be the smallest upp er b ound of equilibrium prices of st ore A and store B

resp ec tively. We already know x A ;x B v L . In eq uilibrium x A = x B = x , otherwise

the st ore wit h the greate r upp er b ound of randomization supp ort has no incent ive s to

charge the pric es b etwee n t he two upp er b ounds. Supp ose that t here is no mass p oint

at x f or store B. Since st ore B never charges v

H

, by charging x store A gets zero share

of t he discount market . St ore A 's equilibrium prot is theref ore arbit rarily close to p A

x,

which is smaller than p A

q A

v

H

, contradict ing the assumpt ion t hat store A ne ve r charges

v

H

. Thus, storeB assigns ap ositive probability tox. Similarly,sinceby assumptionstore

A nevercharges v

H

,andwealready know st oreB neve r charges v

H

,store Amust assigna

p osit ive probability to x. B ut t hen each st ore could get greate r prots by reassigning the

probability mass fromx to just b elowit , a contradict ion.

Next , we show t hat the upp er b ound of t he price supp ort for store B is v

L

. By the

ab ove argument, since store A charges v

H

wit h p ositive probability and store B never

charges v

H

, t here is a mass point at x in store B's price supp ort, and t here is no mass

p oint at x in store A's pric e supp ort. It f ollows t hat by charging x , st ore B get s all the

discountmarketwhenstoreAchoosesv

H

andzeroshare oft hemarketwhenstoreAisnot

charging v

H

. Ifx<v

L

,store B c ouldobt aingreater protsby reassigning theprobability

mass at x t o a price b etwee n x and v

L

, a contradic tion.

Cle arly, in equilibrium the lower b ound of t he price supp ort for the two st ores must

b e equal, ot he rwise the st ore wit h smaller lower b ound of randomiz at ion supp ort has

no incent ives to charge the price s b etween the two lower b ounds. Denote the common

lower b ound as x . Since store A is indierent b etween charging x and v

H , x is equal to p A q A v H =(10p B

). Forst oreA tob eindierentb etweenchargingx andchargingany price

in the inte rval [x;v

L

], the pric e dist ribution F B

( x) function of st ore B sat ises

(10p A 0p B )(10F B ( x) )x+p A x =p A q A v H ; which gives F B ( x)= (10p B ) x0p A q A v H ( 10p A 0p B )x :

Similarly, the price dist ribut ion func tion F A

( x) of store A must satisfy

F A (x )= ( 10p A )( x0x ) ( 10p A 0p B )x :

It can b e veried that F A

( v

L

) <1, so t hat st ore A charges v

H

with p ositive probability,

and F B

(v

L

) <1, so thatst ore B puts a probability mass at v

L

References

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