Huanxing Yang - Understanding Asymmetric Price Adjustments

7/28/2019 Huanxing Yang - Understanding Asymmetric Price Adjustments

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Search with Learning: Understanding Asymmetric Price AdjustmentsAuthor(s): Huanxing Yang and Lixin YeSource: The RAND Journal of Economics, Vol. 39, No. 2 (Summer, 2008), pp. 547-564Published by: Wiley on behalf of RAND CorporationStable URL: http://www.jstor.org/stable/25474382 .

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RAND Journal f EconomicsVol. 39,No. 2, Summer2008pp. 547-564

Search with learning: understandingasymmetric price adjustmentsHuanxing YangandLixin Ye*

In many retail markets, prices risefaster than they all. We develop a model of search withlearning to explain thisphenomenon of asymmetricprice adjustments. By extending our staticgame analysis to thedynamic setting,we demonstrate thatasymmetric price adjustments arisenaturally.When a positive cost shock occurs, all the searchers immediately learn the truestate;thesearch intensity,nd hence theprices, fully adjust in thenextperiod. When a negative costshock occurs, it takes longer for nonsearchers to learn the true state, and the search intensityincreases gradually, leading toslowfalling ofprices.

1. IntroductionFirms are quick to raise prices in response to their cost increases, but not so keen toreduce prices when theircosts fall. This widespread phenomenon isknown as asymmetric price

adjustment, or the rockets andfeathers. This pattern of asymmetric price adjustment has beenreported in a broad range of productmarkets. In fact, a growing empirical literaturedocumentsasymmetric price adjustment invariousmarkets, including gasoline (Bacon, 1991; Karrenbrock,1991;Duffy-Deno, 1996; Borensteinet al., 1997; Eckert, 2002; Deltas, 2004), fruit nd vegetables(PicketaL, 1991;Ward, 1982), beef and pork (Boyd andBrorsen, 1988;Goodwin andHolt, 1999;Goodwin andHarper, 2000), and banking (Hannan andBerger, 1991;Neumark and Sharpe, 1992;O'Brien, 2000).1 According toPeltzman (2000), asymmetric price adjustment is found inmorethan two of every threemarkets examined in a large sample with 77 consumer goods and 165producer goods.

Despite these extensive empirical studies confirming thegeneral patternof asymmetricpriceadjustment, there is little theoreticalwork examining thisphenomenon. In fact,asymmetric priceadjustment first appeared to be inconsistentwith conventional microeconomic theory,which"The Ohio State University; [email protected], [email protected] thankMatt Lewis, Howard Marvel, James Peck, and the seminar participants at the Third Summer Workshopon Industrial Organization and Management Strategy (2006, Beijing), theMidwest Economic Theory Meetings (Fall2006, Purdue), and the 5th Annual International Industrial Organization Conference (May 2007, Savannah) for valuablecomments and suggestions. We also benefited from insightful comments and suggestions from two anonymous referees.All remaining errors are our own.1It is found thatdeposit rates respond more quickly to an increase than to a decrease ofmoney market rates.

Copyright ? 2008, RAND. 547

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548 / THE RAND JOURNALOF ECONOMICSusually suggests thatan increase or decrease of input prices should affectmarginal costs, andhencemove prices up or down ina symmetric,rather thanasymmetric,way. As Peltzman (2000)puts it, the "stylized fact" of asymmetric price adjustment "poses a challenge to theory."Thisarticle attempts tohelp bridge such a gap in the literature.More specifically,we develop amodel of searchwith learning in a dynamic framework.We startwith a description of the static game. There are a continuum of consumers and acontinuum of firmswith capacity constraints. Firms have a common unit production cost (eitherhigh or low). Although known to thefirms, the cost is unknown to the consumers. There arethree types of consumers: the low search cost consumers who always search, thehigh searchcost consumerswho never search, and critical consumerswhose search cost is intermediate.Thedecision for a critical consumer to search or not depends on whether the expected benefit ofsearching outweighs her search cost, so the percentage of consumers who search (the searchintensity)will be endogenously determined.We adopt theprotocol of nonsequential search, thatis, consumers who search observe theprices charged by all firms, so searchers always shop atfirmswith the lowestprice available (unless theyare rationed due tofirms' capacity constraint,inwhich case theywill shop at thefirmswith thenext lowestprice, and so forth).On theotherhand, nonsearchers shop randomly and only observe one price.In the static game, we show that there is a unique equilibrium. Critical consumers holdheterogeneous beliefs regarding thefirms' production cost (the state), and theequilibrium searchintensity nly depends on critical consumers' distributionof initial beliefs. As more consumers'initialbeliefs about thehigh-cost state lie below some cutoff evel, theequilibrium search intensityincreases. This isbecause prices aremore dispersedwhen thecost is low due tocompetition amongfirms, leading to a higher expected gain from search. The equilibrium price distributiondependson the search intensity nd the actual cost state.Specifically, theequilibrium prices are increasingin the actual cost, and are decreasing in search intensity, ecause each firm's demand becomesmore elastic as more consumers search. Thus, the full adjustment of equilibrium prices requirestheadjustment of search intensity,hich solely depends on thecritical consumers' belief-updatingprocess.We then extend our static game analysis to a dynamic settingwhere the cost evolvesaccording to a Markov process with positive persistence. Because consumers never observe thecost realizations, each consumer updates her belief based on thehistory of prices she observed.Thus consumers have heterogeneous beliefs. In equilibrium, searchers and nonsearchers havedifferentbelief-updating processes. Searchers always correctly learn the

true state,because theyalways observe the lowestprice thatfullyreveals the true state.But nonsearchers do not alwayslearn the true state.

Asymmetric price adjustment thus arises naturally. In the event of positive cost shocks, allthe searchers among the critical consumers immediately learn the true state and stop searching.In the following period, no critical consumers search and the search intensity is the lowestpossible. Thus, the search intensityand hence the prices fully adjust within two periods. Inthe event of negative cost shocks, it takes longer for critical consumers who do not searchoriginally to learn the truestate and startsearching, thus the search intensity ncreases gradually,leading to slow falling of prices. To sumup, asymmetric price adjustment is caused by learningasymmetrybetween searchers and nonsearchers,which isclosely related to the evolution of searchintensity.More formally,we show that given the evolution of the underlying cost states, thereis a unique equilibrium in the dynamic game, with the evolution of the distribution ofbeliefs, the search intensity,and the prices uniquely determined. We demonstrate that aslong as the cost shocks are persistent, the pattern of asymmetric price adjustments emergesin statistical sense on the equilibrium path of the dynamic game. Moreover, as the costshocks become more persistent, the pattern of asymmetric price adjustments becomes moreprominent, because thedownward price adjustment on average spreads over longer periods oftime.?RAND 2008.



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YANG AND YE / 549Several recentpapers (Lewis, 2005; Tappata, 2006; Cabral and Fishman, 2006) also attemptto explain asymmetric price adjustment based on searchmodels.2 Lewis (2005) develops areferenceprice searchmodel inwhich the expected distribution of prices is exogenously given,rather than endogenously determined; consumers have adaptive expectations and thus are notrational.On the otherhand, consumers are rational inourmodel, because theyformexpectationsof prices based on all the informationavailable.Cabral and Fishman (2006) develop a searchmodel inwhich thecost changes are positivelybut not perfectly correlated across firms.3They show thatconsumers have a greater incentive tosearch in the case of large price increases or small price decreases, but little incentive to searchwhen prices increase a littleor decrease by a lot.This implies thatfirmsare reluctant to changeprices when costs decrease by a little bit or increase by a lot,but quick to change prices ascosts increase by a littlebit or decrease by a lot. In otherwords, when the cost change is small,theprice adjustment exhibits downward rigidityand upward flexibility;when the cost change is

big, the asymmetry is reversed: prices exhibit downward flexibilityand upward rigidity.Theseimplications are quite differentfromours.The paper that ismost closely related to ours isTappata (2006). Our article differs fromTappata inan important spect in termsofmodelling. That is,although Tappata assumes thatthefirms'past costs are known to theconsumers,we do not impose thisassumption inour analysis.Because consumers know past costs, there is no learning inTappata. In contrast, the learningasymmetrybetween searchers and nonsearchers about theunderlying cost is thedriving force inour analysis. Thismodelling difference leads tovery differentempirical implications. InTappata'ssetting,because there isno learning, ittakes exactly twoperiods forprices to fully adjust toboththepositive and negative cost shocks.Moreover, inTappata, theasymmetry inprice adjustmentsis only present in the firstperiod after a cost shock occurs: themagnitude of price adjustmentin the firstperiod is bigger in the case of positive cost shocks than in the case of negative costshocks. In contrast,by endogenizing the timeperiods that re needed forprices to fully djust tocost shocks,we are able to show thatasymmetric price adjustment goes beyond the firstperiodafter a cost shock occurs: although it takes two periods for prices to fully adjust to positivecost shocks, it takesmuch longerperiods forprices to fully adjust inresponse tonegative costshocks.

Our article also contributes to the literatureon consumer search, in thatwe develop adynamic searchmodel with consumers, ina heterogeneous fashion, learningabout theunderlyingstates based on the personal histories of prices they observed.4 Several previous papers havestudied equilibrium search with learning (Benabou and Gertner, 1993; Dana, 1994; Fishman,1996). Benabou and Gertner (1993) studyhow the correlations among firms' cost shocks affectconsumers' incentive to search and theequilibrium prices. In a staticmodel, Dana (1994) showsthat f onsumers are uncertain about firms'costs, thentheresponse ofprices to cost shockswill belimited. In a dynamic framework,Fishman (1996) shows thatcost shocks have different hort-runand long-run effects on prices.5 None of these papers study asymmetric price adjustments.The article isorganized as follows. Section 2 presents the staticgame and characterizes theunique staticgame equilibrium. In Section 3,we extend the staticgame analysis to thedynamicsettingand show thatasymmetric price adjustment arises naturally on the equilibrium path. In

2Borenstein et al. (1997) propose an explanation for asymmetric price adjustments based on Rotemberg andSaloner's (1986) model of tacit collusion with stochastic shocks.3More specifically, they consider two firms. The costs of the firms either both increase or both decrease, althoughthemagnitude of the changes might be different.4For models based on nonsequential search, see, for example, Salop and Stiglitz (1977); Braverman (1980); andVarian (1980). For models based on sequential search, see, for example, Burdett and Judd (1983); Rob (1985); and Stahl(1989). 5More specifically, Fishman shows that in the case of a general cost (common to all firms) increase, consumerssearch toomuch so as to limit the extent towhich prices increase in the short run; however, in the case of an idiosyncraticcost (specific to only one firm) increase, consumers search too little, leading toprice overshooting.?RAND 2008.



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550 / THE RAND JOURNALOF ECONOMICSSection 4, we discuss the restrictions of our key assumptions and the robustness of our results.Section 5 concludes.

2. Static gameThe model. We consider amarket with a continuum of firmsproducing a homogeneousgood. The totalmeasure of firms is normalized to be 1.All thefirmshave the same cost c inproducing each unit of thegood (firmshave common cost shocks). Ex ante, c (i.e., the state oftheworld) can take value either cH or cL,where cL < cH. At thebeginning of theperiod, firmsobserve therealization of thecost and thencompete inprices.We also assume thateach firmhasa capacity constraint k (finite), that is,no firmcan sellmore thank units of thegood.6There is a continuum of consumerswith totalmeasure /3 1.The parameter ft can also beinterpreted s thenumber of consumers per firm in themarket. Each consumer has a unit demandwith a choke price of v > cH.We assume that t< k, that is, thenumber of consumers per firm inthemarket is less than each firm'scapacity constraint.7Consumers do not observe therealizationof c. Instead, consumers hold beliefs about the cost realization, which might be heterogeneousamong consumers. Let a denote a consumer's belief ifshe believes thattheprobability of c? cHis . The distribution ofbeliefs among consumerswill be specified later. efore observing prices,consumersmake decisions regardingwhether to search (become informed)or not to search (stayuninformed).We adopt theprotocol of nonsequential search. Informed consumers observe allthe realized prices and purchase from thefirms (stores) with the lowest price available.8 Eachuninformed consumer shops randomly at a firm (store) and only observes thatfirm'sprice.Each consumer's type is characterized by her search cost. The firsttypeof consumer (withproportion A.i) each has search cost sL = 0. These consumers are also called shoppers, who canbe interpreted s thosewho have obtained price informationwithout incurringnontrivial searchcost (e.g., fromTV or internet dvertisements, etc.). This typeof consumer always searches inequilibrium regardless of theirbeliefs about theunderlying state.The second typeof consumer(withproportion X2) each has search cost sH.We assume that H > v so that this typeof consumernever searches (regardless of theirbeliefs about theunderlying state). The restof the consumers(with proportion 1 ? Xx ? X2) each have intermediate search cost sM e (sL, sH) and they mayormay not search depending on theirbeliefs about theunderlying state.Because beliefs abouttheunderlying stateonlymatter for this typeof consumer, theyare henceforth referredtoas thecritical consumers. Let F(a) denote thecumulative distribution functionof thebeliefs among thecritical consumers. In otherwords, F(a) is thefraction of the critical consumers whose beliefsare lower than a.Because there is a capacity constraint for each firm, rationingmay occur: for a low-pricefirm, the number of consumers shopping at this firmmay be bigger than k. We adopt theproportional rationingrule: ifrationingoccurs at a firm,each consumer (nonsearcher or searcher)who shops at thatfirmwill be able topurchase a unit of thegood with the same probability. Ifanonsearcher isrationed, she shops randomly atother firmswithout incurring ny cost. Ifa searcherisrationed, she goes to thefirmwith the lowestprice among theremainingfirms. Should rationingalso occur there, the same search procedure applies until the searcher purchases a unit of thegood.

6 In the dynamic setting, this assumption implies thatno firm can sellmore than k units of the good per period.We believe that capacity constraint is prevalent inmany product markets. For example, sales of perishable goods suchas fruit,vegetables, beef, or pork are often constrained by the storage space for a given period; even the supply capacityof gasoline stationswithin a given period can be limited by things such as petroleum pipeline systems, which are oftenoperating at their full capacities. This being said,we impose the capacity constraintmainly for tractability of equilibriumanalysis, which is furtherdiscussed inSection 4.7As will become clear, this assumption makes competition among firms nontrivial.8The assumption that searchers observe all the realized prices follows Varian (1980). InBurdett and Judd (1983),searchers are only able to observe a subset of realized prices. This alternative setting is furtherdiscussed inSection 4.?RAND 2008.



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YANG AND YE / 551The timeline of thegame is as follows. First, theproduction cost (the state of theworld) isrealized and all the firmsobserve the common state.The firmsthen simultaneously setprices.

Finally, consumers decide whether to search and make purchases accordingly.9As is standard in the search literature, e focus on symmetricequilibria inwhich all firmsemploy the same pricing strategies. Let G be the price distribution and /xbe theproportionof informedconsumers, or the search intensity, hich is endogenously determined inourmodel(/^ A.i).Given thedistributionof critical consumers' beliefsF(a), a symmetricperfect Bayesianequilibrium is characterized by a pair (/x*,G* ( |c)), with the following properties: given theequilibrium search intensity /x*,firms' optimal pricing strategies yield the equilibrium pricedistribution G* ( | ); given G* ( |c) andF(a), consumers' optimal search decisions give rise totheequilibrium search intensity/x*.Analysis with fixed search intensity.We first erive theequilibrium price distributiongiven

the search intensity/x nd theproduction cost c (the state). Let/? be the lowestprice charged inequilibrium, and let theproportion of the firms thatcharge/?be r](p).Note that p firm'ssales aremin+(1-^*1.

The firsttermin thebracket is thedemand fora/? firm: t ttracts (1 ? /x)/3onsearchers, and getsY~ searchers. The quantity a/? firm ells is simply theminimum of itsdemand and capacity.Lemma 1. In any equilibrium, a firmthatcharges/?must sell k units of thegood.Proof. Suppose innegation, a firmcharging/? sells strictly ess thank units inequilibrium. Thenby undercuttingp by an arbitrarily small amount s, this firmcan attracta positive measure ofsearchers, thus increasing its sales to kwithout affecting theprofitmargin, which destroys theproposed equilibrium. This implies that inany equilibrium, -^ + (1 ? /x)/? k.Lemma 2. There isno equilibrium inwhich prices are continuously distributedon \p,/?],foranyp such that/? p < v.Proof. Consider a candidate equilibrium inwhich prices are continuously distributed on \p, ]for some p such that < p < v. A necessary condition for this to be an equilibrium is thatconsumers should be rationed at any p e (p, /?): ifconsumers are not rationed at such a/?, thenthere is no point to charge p -f s, because a firm can only attract nonsearchers in that case and itsdemand isgiven by (1 - /x)/*,hich is strictly ominated by charging v.But given that onsumersare rationed at any p e (/?, ?),which means thateach firmcharging any p e (p,p) sells k, apfirm can increase itsprofitmargin without affecting its sales by deviating to p e (/?, ?). Thisdestroys theproposed equilibrium.Lemma 3. In any equilibrium, consumers shopping at any p firm are not rationed. That is,^+(i-M)0 k. In thiscase, a positive measure of consumers(hence searchers) are rationed at/? firms.By Lemma 2, there is a positive number s such thatno firm charges at any price p (/?,/? s). Then a firmwho charges p can deviate to someprice p' (/?,/? ?). Under thisdeviation, this firm can still attractenough searchers (becausea positive measure of searchers are rationed atp firms), and thus sell k units with an increasedprofitmargin. This destroys theproposed equilibrium.Therefore,^ + (1 - /x)/3 k in equilibrium, if there is any. This implies that searchersare not rationed at/?firms.

9Because the first nd the second type consumers either always or never search, only the critical consumers havenontrivial decisions tomake.?RAND 2008.



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552 / THE RAND JOURNAL OF ECONOMICSLemma 1and Lemma 3 jointly implythat inequilibrium,we must have -^ + (1 - /x)/? k.That is,no rationing occurs at firms charging the lowest price. Next we will establish thatany

equilibriummust have a two-pointprice distribution.Lemma 4. Given /x nd c, in any candidate equilibrium each firmmust charge eitherp = v orp =p e (c, v)(p is tobe determined).Proof. Suppose there is an equilibrium price strictly etween/? and v. Then a firm thatchargesthisprice can increase itsprofitby deviating to charging v. This deviation leads toa higher profitmargin per unit of sales,with the demand unchanged (only the lowestprice can attract informedconsumers, given thattheyare not rationed at the lowestprice byLemma 3). So theonly possibleequilibrium is either all firmscharging the same price, or a two-pricedistribution on/? and v.First consider the candidate equilibrium inwhich firms charge the same price p =p > c.The equilibrium demand for each firm is /?.But thena firm an undercut/? a littlebit and sellk > ftwithout affecting theprofitmargin per unit of sales. Thus, this typeof equilibrium cannotexist.All firms chargingp = c cannot be an equilibrium either,because by deviating top ? v afirmcan get a positive profit by selling touninformed consumers (sH > vmeans thata measureof X2fi consumers are always uninformed).Now theonly candidate equilibrium left is thatfirms charge either v orp. Clearly, p < v.What remains tobe shown is that/? c. Because charging v yields a positive profit, charging/?should also yield a positive profit,which implies/?> c.

By the above lemmas, there is only one possible equilibrium, with equilibrium pricescharacterized by a two-point distribution.Denote n(v) and n(p) as the profits of a firm thatcharges v and/?, respectively. Explicitly,

7t(v) = (l-n)l3(v-c)tt(p) = k(p- c).

By Lemma 3, a firmcharging v can only attractnonsearchers. Thus itsdemand and sales are (1 ?/ji)P, nd itsprofitmargin is v ? c.10By Lemma 1,a firmcharging/? sells k,with a profitmarginp ? c. The equilibrium is characterized by thefollowing two conditions:n(v) = n(p) (1)

^+(1-M)? = ?. (2)Condition (1) says thata firm should be indifferent etween charging v and/?, and condition (2)says that the demand fora/?firmexactly equals itscapacity k.Proposition 1. Given /xnd c, there isa unique equilibrium: a proportion of 1 rj(p)firmschargeprice v, and a proportion of rj(p)firms charge price/? G (c, v),where r](p) and/? are determinedby conditions (1) and (2).More explicitly,

n(p) = -? (3)

p?c+Q-Miv-cy (4)kProof. We only need to show theprice distribution specified above is an equilibrium. Note thatfrom condition (2), searchers are not rationed at/?.We show that irmshave no incentive todeviate.First consider a firmcharging v.Deviating to anyp e (p,v) would lead to a lowerprofitmarginwithout increasing the sales (because searchers are not rationed at /?),hence such a deviation is

10Note that (1 - /x)? < k because ? < k.?RAND 2008.



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YANG AND YE / 553not profitable.Deviating to/?yields the same profitby condition (1). Deviating to/? p is strictlydominated by charging /?,because a firm cannot sellmore thank. Thus a firm charging v hasno incentive todeviate. Next consider a firmcharging /?.By a similar argument, thefirmhas noincentive todeviate top < p. If thefirmdeviates to /? (/?, ], itonly attractsnonsearchers andhence sells (1 ? /x)/Jnly, because searchers are not rationed at/?.11Thus themost profitabledeviation is to set theprice at v,which, by condition (1) yields the same profitas no deviation.Thus, a firmcharging/?has no incentive todeviate either.Solving (1) and (2) yields expressions(3) and 4).Note thatinthisunique equilibrium, no consumer isrationed: thedemand fora/?firmexactlyequals itscapacity k, and the demand fora vfirm is strictly ess thank. From (3),we can see thatr)(p) is increasing in the search intensity/x. ntuitively, s xt increases the demand becomes moreelastic, andmore firmscharge lowerprice.Moreover, ij(p) does not depend on c directly.By (4),/? s increasing inc and decreasing in /x. s demand becomes more elastic (/x ncreases), chargingthe lowerprice becomes relativelymore profitable,other things equal. To restore the indifferencecondition, the lowerprice must decrease to reduce theprofitmargin for the lower-pricefirms.Note that thepresence of the capacity constraintkeeps theprofitmargin ofp firmspositive, byrestrictingthecompetition among /? irms.

Equilibrium with endogenously determined search intensity.Now we analyze theequilibrium of the game, with search intensity /xendogenously determined.We assume thatfirmsknow thedistribution of beliefs F(a), while consumersmay not know F(a).n Recall thatthe cost c can only take twopossible values cH and cL,which is the same among all the firms.Proposition 1 characterizes firms' equilibrium price distribution given any search intensity/x.The remaining task is toderive theequilibrium search intensity/x*. e firstcompute theexpectedgain from searching given theequilibrium price distribution, /x, nd a consumer's belief, a:

E[p -p\a]= a[(\ - r](pH))v+ rj(pH)pH -pH] + (1 - a)[(l - r\{pj)v + r](pL)p_L/?J= (1 - i/(?.))[a(v ~PH) + (1 - a)(v -pj]=

k~=^- {v- [otcH (1 - a)cL]}, (5)

where d denotesp(cH) and/? denotesp(cL). Note thatby (4),/? > /? given /x.H ? ?L ? ?H ?LFrom (5) we can see thatE[p ? p \] does not depend on /x. hus, search does not exhibitcomplementarity.This isbecause an increase in /x as two countervailing effects.The firsteffectis that /? decreases as /x increases, which increases the return of search. On the other hand, anincrease in /x auses more firms to chargep(r](p) increases),which reduces theaverage price andraises thepayoffof a nonsearcher. These two effectsexactly offset each other,as shown by (5).From (5),we see that theexpected gain from search isdecreasing in forcritical consumers.Thus, there is a cutoffbelief a such that all critical consumers with beliefs below a search andthose above a do not search. If sM > ~^- (v ? cL), then even the consumer with themost optimisticbelief (a = 0) cannot afford the search, so a = 0; on theotherhand, ifsM < ^(v ? cH), theneven the consumer with themost pessimistic belief (a = 1) can afford the search, so5= 1. Inwhat followswe will focus on themost interesting ase inwhich

11A single firmdeviation would not affect the totalmeasure of thefirms who set price p (due to our assumption ofcontinuum of firms). Thus, condition (2) implies that the following condition continues tohold even afterone single firmdeviates:IjlP

Measure of firmssettingpwhich in turn implies that searchers will not be rationed, even though one single firmdeviates by setting a higher price(all before consumers move, as specified inour timeline).12These properties will be justified in thedynamic model.?RAND 2008.



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554 / THE RAND JOURNALOF ECONOMICSk-p k-fi? (v - cH) < sM < ?? (v - cL). (6)

In thiscase, a lies in (0,1),which isdetermined by thefollowing indifferencecondition:sM= E[p -/? | ] = ?j? {v- [acH + (1 - a)cL]}. (7)

Solving (7) we have^ = (v ~ cL)(k -P)- ksM(cH - cL)(k - P) ' U

Then theequilibrium search intensity/x* an be computed as follows:/x* *i + (1 - Xx X2)F(a). (9)

Note thatcondition (7) pins down a unique a e (0, 1), and hence /x* s also unique. Moreover,(8) shows that is independentof all theendogenous variables, and thus is common knowledge.It is easily seen that,given F(a), /x* nd G* ( | /x*, ) described by (3) and (4) constitutetheunique equilibrium of the staticgame. Given F(a) and G* ( | /x*, ), the optimal decisionrules about search give rise to /x*.Because firms know F(a), they can correctly anticipate/x* y (9). And given /x*, hefirms' optimal pricing strategies result in the equilibrium pricedistribution G* ( | xx*, ). Note that rationing does not occur in equilibrium. This is the casefor two reasons. First,firmscan correctlyanticipate theequilibrium search intensity/x*. econd,althoughpH > PL, ^iPfj) ? ^O^Xmat *s'me proportion of thefirms charging the lowerpricedoes not depend on thecost realization.The equilibrium price distribution is determined by the cost realization and consumers'beliefs F(a). The average price is lower under state cL than under state cH.Moreover, thepricedistribution ismore dispersed under state cL (the gap between the average price and the lowestprice is larger),which leads to a higher expected return to search. Thus, as more consumers'beliefs liebelow a (orF(a) increases), theequilibrium search intensity/x* ncreases.As a result,bothp and p are decreasing inF(a), and rj(p )= r\(p ) are increasing inF(a). Define theaverage equilibrium price as p(c(, /x*):

p(ci9 /x*) rj(p.)p.+ [1- rjipfiv= k-p+M [c+?r-(v ~ c)\+ k-p+?*pv- wIt is easily seen from(10) thatp(ct, /x*) s increasing inct and decreasing in /x*. hus, theaverageprice is also decreasing inF(a). The following proposition summarizes these results.Proposition 2. There is a unique equilibrium with the equilibrium search intensitygiven by (8)and (9).Moreover, /x* s increasing inF(a); both/? and/? are decreasing inF(S), and bothn(p ) and r\(p ) are increasing inF(a); p(ct, /x*) sdecreasing in /x*ndF(a).H ?L

According to the previous analysis, changes in equilibrium price distribution can bedecomposed into two components. The first component is the change resulting from changesincost realization, and the second component is thechange resultingfromchanges inconsumers'search intensity, hich isgoverned by thedistribution of consumers' beliefs.3. Dynamic model

We now extend our analysis to the dynamic setting, and endogenize critical consumers'beliefs. Time t is discrete and t= 1,2,_In each period the staticgame isplayed.We assumethat the common cost evolves according to a Markov process, with p being the persistenceparameter. That is,?RAND 2008.



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YANG AND YE / 555Pr(c,+1 = cH\ct = cH) = Pr(c,+1 = cL\ct = cL) = p,

where p > 1/2.At t= 1, the two cost states are equally likely.This Markov structureof costevolution is common knowledge. Although firms always observe the cost state of the currentperiod, consumers never observe past or current cost realizations. Instead, each consumer updatesher belief about the cost based on theprice history she observes. Tappata (2006) assumes thatlastperiod cost realization is observable toconsumers, and thusprices adjust fully to cost shocksin twoperiods.13From the staticmodel, we see that the equilibrium price distribution in a particular perioddepends on consumers' beliefs and the cost realization. The key inour dynamic game analysis isto trace thebelief-updating process among critical consumers. From the staticmodel, the lowerprice/? is responsive to cost realizations: /? > p forany given /x. o simplifyour analysis, wemake assumptions about theparameter values such thatthe lowerbound ofp (under thehighestpossible /x) is greater than theupper bound ofp (under the lowest possible /x).Note that thehighest possible /x s 1? X2 (all critical consumers search), and the lowestpossible /x sXx (nocritical consumers searches). So in effect, we assume

cH-cL P(\-Xx-X2) - > -, (11)v - cL k- p\2which implies/? (/x 1 X2) > p (/x Xx), that is, there isno overlap between the supportsof/? and /? inequilibria of thedynamic game (hence (11) can also be termed thenonoverlappingH ?Lcondition).Given thisnonoverlapping condition, a consumerwho observes/? can correctly infer he truecost of thatperiod. As a result,her initialbelief next period is eitherp or 1- p. On the otherhand, thehigh price (v) isnot responsive to cost realizations. Thus, if consumer observes a pricev, her belief about the truecost in the currentperiod (af, superscript/?denotes theposterior) isupdated as follows (suppose her initialbelief is thatPr(c, = cH) = at):

p_ttrO~

V(PH))_ _"' "


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556 / THE RAND JOURNALOF ECONOMICSprice v. In thiscase, her posterior isnot updated. If she observes the lowerprice, she updates herposterior correctly.As we will see, thedifference inbelief-updating processes between searchersand nonsearchers is thekey tounderstanding asymmetric price adjustments.

Equilibrium. By equation (8), a does not depend on /x* the equilibrium search intensityinperiod t) or the cost state inperiod t.Therefore, thedistribution of beliefs inperiod /can besummarized by Ft(a). Hence in each period, the state of the economy can be summarized byFt(a) and ct, and the equilibrium in thedynamic game can be characterized by the sequence of{c?F,(a),/x,*,G*(. |/x,*,c,)}.

Proposition 3. The equilibrium in thedynamic game exists and is unique for any evolution of

Proof. Suppose firms correctly anticipate Ft(a). Then theequilibrium /x* sdetermined bytf = Xx+(\-Xx-X2)Ft(a). (14)

From the analysis of the static game, the equilibrium price distribution inperiod t,G* ( | /x*,ct), isuniquely determined. The existence of equilibrium in thedynamic game boils down to thefollowing condition: firms are able to anticipate Ft(a) correctly.The uniqueness of equilibriumisguaranteed ifthe evolution ofFt(a) isunique given {ct}. Below we will show that these twoproperties hold.First,we show thatfirms can inferFx(a). In period t= 1 all critical consumers hold thesame initial belief 1/2 (because there is no prior history of prices). Given a < 1/2,Fx(a) = 0,and no critical consumers search in period t= 1. Because it is common knowledge that allcritical consumers hold the same initial belief 1/2 inperiod 1,firms can correctly infer thatFx(a) = 0.Second, we show thata critical consumer's beliefwill never be in (a, 1/2). To see this,firstnote that each critical consumer's initial belief inperiod 1 is 1/2. Ifa critical consumer observesa sequence of higher prices v, her initial belief remains 1/2by (12). Her beliefwill be differentonly ifshe observes the lowerprice in the lastperiod. In thatcase, her initial beliefwill either bep or 1? p. Ifher belief is 1? p, then shewill search and observe the lowerprice, and her nextperiod belief will be eitherp or 1? p. Ifher belief is p hence she does not search, then itwilleither converge to 1/2from above ifshe keeps observing thehigh price, or be revised top or 1? p if she happens to observe the lowerprice in some period. Therefore, in all cases a criticalconsumer's belief is outside therange of (a, 1/2).Third,we show that ifthefirms know Ft(a), thencombining thiswith the information boutct, they can correctly infer Ft+X(a). We discuss two cases in order.

In the first case, suppose ct = cH. For critical consumers whose at < a, they will search inperiod tand learn the true statect= cH; hence their initialbelief inperiod t 1 isp. For criticalconsumers whose at e [1/2,p], theywill not search inperiod t.Among those consumers, anrjt(p ) portion observe p and learn the true state cH. Thus their initial belief inperiod t+ 1becomes p. The remaining 1 rjt(p ) portion of consumers observe v hence no belief updatingoccurs. Their initial beliefs in period t+ 1 remainwithin [1/2, p]. Aggregating over all thecritical consumers,we can see that t+X(a) = 0. Thus, we have thefollowing transitionequationfor thedistributionof beliefs:

ifct= cH:Ft+x(a) = 0 given anyFt(a). (15)In the second case, suppose ct = cL. For critical consumers whose at < a, they will searchinperiod tand learn the true statect=cL; hence their initialbelief inperiod t 1 is 1? p. Forcritical consumerswhose at e [1/2, p], theywill not search inperiod t.Among those consumers,

mr]t(p ) portion observe/? and learn the true state cL. Thus their initialbelief inperiod t+ 1becomes 1? p. The remaining 1 rjt(p ) portion of consumers observe vwith no information?RAND 2008.



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YANG AND YE / 557updated. Their initialbeliefs inperiod t+ 1remain in [1/2, p]. Aggregating over all the criticalconsumers,we obtain another transitionequation for the distribution of beliefs,

ifc, = cL:Ft+x(a) = Ft(a) + [1- Ft(a)]r)t(p_L)= Ft(S) ff [1 Ft(a)]. 16)k- P + tfpFrom (15) and (16), we can see thatgiven ct and Ft(a), Ft+X(a) is uniquely determined.Moreover, firms can correctly anticipate Ft+X(a) based on their information about ct,Ft(a), andTherefore, given the evolution of {ct}, there is a unique equilibrium characterized by thesequence of {ct,Ft(a), /x*, *(- | /x*, t)} in thedynamic game.Note that except for period 1, critical consumers' beliefs are heterogeneous due to the

differentprice histories they experienced. Also note thatwe do not need to trace the evolution oftheexact distributionof beliefs Ft(a), which would be cumbersome todescribe. Instead,we onlyneed to trace the evolution of Ft(a) to determine the equilibrium search intensity/x*. notherinterestingproperty is that,whereas firms can correctly inferFt(a), /x*, nd G* ( | /x*, t),consumers inperiod tdo not need tohold correctbeliefs about Ft(a), xx*,nd G* ( | /x*,t). Themain reason is thatconsumers do not observe thehistory {ct}; instead, theyupdate theirbeliefsabout {c t} and G* ( | /x*,t) based on theirpersonal histories of theprices theyencountered.

Asymmetric price adjustments. According to the analysis of the staticgame, changes inequilibrium price distribution can be decomposed into two components, one due to the changeincost realization and the other due to thechange inconsumers' search intensity. n thedynamicgame, the highest average price across periods (and the highest possible lower price acrossperiods) arises when ct = cH and /x* /x k\ (no critical consumers search). Note that thiscorresponds to the case where ct = cH and consumers have full information about ct. On theotherhand, the lowestpossible average price (and the lowestpossible lowerprice) arises when c= cL and /x* /x 1 X2 (all critical consumers search). Similarly, thiscorresponds to thecasewhere c= cL and consumers have full information bout ct.Therefore, in statecHwe sayprice isfullyadjusted if/x* /x Xx,and in statecLwe sayprice is fullyadjusted if/x* /x 1 X2.In thedynamic game, because consumers do not observe past cost realizations, their eliefs,hence the search intensity /x*, o not adjust as quickly as the underlying cost state changes.More importantly, he speed of adjustments for thebeliefs and search intensity s different nderpositive cost shocks and negative cost shocks. Because the average price and the lower price movein the same direction, forbrevityof exposition inwhat follows theyare often simply referredtoas theprices. The following propositions identifytheasymmetry inprice adjustments ifthecoststate persists after a shock occurs.Proposition 4. Suppose a positive cost shock occurs in period t + 1. Then regardless ofFt+X(a), Ft+2(a) = 0 and /x*+2 jx= Xx.Regardless of previous history, ifcost statesLHH arerealized inperiods t, t+ 1,and t+ 2, then theprices fully adjust to thehighest level inperiodt+ 2.Proof Suppose L andH are the realized cost states forperiods tand t+ 1, respectively.Thenby the transition equation (14), Ft+2(a) = 0 irrespective of Ft+X(a). By (15), /x*+2 Xx - /x.Therefore, ifct+2= cH, theprices reach thehighest level, and hence are fullyadjusted inperiodt 2.

Proposition 4 implies that tate is theabsorbing state in termsof critical consumers' searchbehavior: regardless of theprevious history, ifa positive shock occurs in the currentperiod, thesearch intensity ill fully djust downward inthenextperiod.Another implicationofProposition 4is that theprices fully djust upward intwoperiods when a positive cost shock occurs and thehighcost persists inthenextperiod.More specifically, inthefirstperiod a positive shock occurs, prices?RAND 2008.



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558 / THE RAND JOURNAL OF ECONOMICSonly adjust upward partially because the search intensity isnot fullyadjusted. However, in thesecond period the search intensity s fully djusted downward, leading to fullupward adjustmentof prices.The key to this result is that,when a positive shock occurs, the critical consumers whosearch in thecurrentperiod immediately learn thecost statehas switched to , and thus theystopsearching in thenext period. For thenonsearchers, theyeitherobserve the lowerprice, learn thetrue stateH and do not search in thenext period, or observe thehigh price, do not update theirbeliefs, and remain nonsearchers in thenext period.Proposition 5. Suppose a negative cost shock occurs inperiod t+ 1. Then

F


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YANG AND YE / 559state (N > 2 because p > 1/2).We thus consider the following expected evolution path of thestates: L...LH...HL...L.... That is,N periods of stateL followed byN periods of stateH, whichare in turnfollowed byN periods of stateL, and so on.

Along this expected evolution path, the lowestprices will occur in the h period of theLstate,and thehighest prices emerge in the2nd throughthe h periods of the state.The patternof asymmetric price adjustment is quite striking.When a positive cost shock occurs, thepricesadjust from the lowest (theMh period of theL state) to thehighestwithin twoperiods (the 2ndperiod of theH state), and then theprices stay at the same level until the h period of theHstate.On the otherhand,when a negative cost shock occurs, ittakesN >2 periods for thepricesto adjust from thehighest (theMh period of theH state) to the lowest (theMh period of theL state), before the state switches toH. During theN periods of theL state, theprice decreasesgradually to the lowest level.We summarize the resultbelow.Proposition 6. Assume p > 1/2.Along an expected evolution path, price adjustments exhibitan asymmetric pattern inequilibrium.Whereas theprices always fully adjust upward within twoperiods after a positive cost shock occurs, it takes a longer time for theprices to fully adjustdownward aftera negative cost shock occurs.

Given the random nature of theunderlying state switches, the actual evolution pathwouldnot exactly follow an expected evolution path, and theduration of a given statewould also berandom. However, the expected evolution path can be regarded as the average overall possibleevolution paths. Thus, Proposition 6 implies thatthepattern of asymmetricprice adjustment canemerge on the actual evolution path ina statistical sense.The underlying reason for asymmetric adjustments lies in the asymmetric belief updating,which resultsfrom consumers' searchbehavior.Although searchers always learn thetrue ost stateimmediately,nonsearchers do not learn thetruecost stateunless theyhappen toobserve the lowerprice.When there is a positive cost shock, the searchers among critical consumers immediatelylearn thatthecost has gone up. As a result, those consumers stop searching in thenext period andprices are fully adjusted upward. On theotherhand,when thecost goes down, those consumerswho observe the high price do not learn the true state and remain as nonsearchers, whereas onlythosewho observe the lowerprice learn the true stateand begin to search.As a result, thebeliefsof the consumers are gradually adjusted downward as more andmore nonsearchers observe thelowerprices,which leads togradual downward price adjustments.

Comparative statics. In this subsection, we study how changes in some exogenousparameters affect thepattern of price adjustments. First,we provide a measure to evaluate thedegree of asymmetry inprice adjustments. Denote themagnitude of upward (downward) priceadjustment within two periods after a positive (negative) cost shock asMAP (MAN), and wedefine the adjustment ratioAR = MAN/MAP. AR measures the degree of asymmetry of priceadjustments: the smallerAR, the smaller themagnitude of downward price adjustment withintwoperiods relative to thatof upward price adjustmentwithin twoperiods, and hence themoreasymmetric theprice adjustments.We startwith the changes in thepersistence parameter p. Consider twoMarkov processes,1 and 2,with p2 > px > 1/2.That is,Markov process 2 ismore persistent thanprocess 1. In theprevious analysis about theexpected evolution path,we see that increases as p increases. Thuswe have N2 > Nx. Note that /x Xxdoes not depend on p. Hence thehighest prices on the twopaths are the same. However, because N2 > Nx, the lowestprices under process 2 (occurring inperiodN2 in stateL) are lower than thoseunder process 1 (occurring inperiodNx in stateL). Asa result,MAP < MAP. On the otherhand,MAN does not depend on p, which is evident from theequation

^ m? xxpF^a) =rr^p = k-v + - (17)?RAND 2008.



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560 / THE RAND JOURNAL OF ECONOMICSThusM4f = MA%. As a result,ARX > AR2. That is, theasymmetric price adjustments aremoreprominentwhen thecost evolution ismore persistent.So as theMarkov process becomes more persistent, themagnitude of fullprice adjustmentsbecomes larger. In thecase of positive shocks, this largermagnitude of upward price adjustmentis still completed within twoperiods. However, in the case of negative shocks, the downwardprice adjustment spreads over longerperiods. This makes theprice adjustmentmore asymmetric.The following proposition summarizes the result,which is also a testable implication.Proposition 7. As theMarkov process becomes more persistent (p increases), the asymmetricpattern ofprice adjustments becomes more prominent.

Next we consider the impact of Xx (the proportion of shoppers) on the pattern of priceadjustment. Let X[ < Xx .We show that the downward price adjustment inresponse toa negativeshock is slower under X[ thanunder X x.Proposition 8. As theproportion of shoppers (Xx) decreases, theprices adjust downward moreslowlywhen a negative cost shock occurs.Proof Suppose a negative shock occurs inperiod t 1,and theL statepersists in the subsequentperiods. Our goal is to show that theprices inanyperiod t j (j > 2) are strictly ower under X xthan under X[. Because prices are decreasing in the search intensity, t is sufficientto show that/x*|y /x*+yorall j > 2 (superscriptprime isused todistinguish variables under X[ from thoseunder Xx). According to equation (17), F't+2(a) < Ft+2(a). Now we proceed with induction. Inthe firststepwe show that ifF/+y(a) < Ft+j(a), then /x*|y /x*+yorj > 2 . In the second stepwe prove that ifF;+j(a) < Ft+J(a) and /x^- < fi*+j9 henF/+y+1(c?)< Ft+j+x(a).

Step 1.By equation (14),tf+j - aC, = (*i - *i) + 0-*i- i2)Ft+J(&) - (i - a; -W+y(2)= (Xx X\)[l - F;+j(a)] + (l-Xx- X2)[Ft+J(a) - F/+,(8)]

>0.

Step 2. First, define A = W,+J k-fi + rt+jP'At+j > A't+j because /x*+7- /x*+y,nd both of them are strictlybetween 0 and 1.Now by thetransitionequation (16),

Ft+j+x(a) - F;+J+l?2) = [Ft+J(a) - F;+J(&)] + At+j[\ - Ft+j(a)] - A't+j[\ - Ft+j(a)]= (At+j - A't+j)[l - Ft+J(S)] + [Ft+j(a) - F;+j(a)](l - A't+J)>0.

Thus we obtain the desired result.Intuitively, hen theproportion of shoppers is smaller, in the case of a negative cost shock

fewer consumers search initially.As a result, a smaller proportion of firms sets the lowerprice.This reduces the speed of learning for the critical consumers, leading to slower adjustments ofthe search intensity nd theprices.Although a smaller Xx slows down theprocess of downward price adjustment, theupwardprice adjustment isnot affected: it is always completedwithin twoperiods. Thus a decrease inA.2makes thepatternof asymmetric price adjustmentsmore prominent.

By a similar argument,we can show thatan increase ink/P leads to a slower downwardprice adjustment. The intuition is thatan increase ink/P, other things equal, results in fewerfirms setting the lowerprice. This slows down the critical consumers' learningwhen a negativecost shock occurs, leading to a slower downward price adjustment.?RAND 2008.



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YANG AND YE / 5614. Discussion

Our model is stylized,because we make a number of simplifying ssumptions tofacilitateouranalysis. Here we discuss the restrictions f these assumptions,whywe impose these assumptions,andwhether theyaffect thegeneral insightsof the article.First, unlike the standard search literaturewhich considers a finite number of firms,wework with an infinitenumber of firms.This assumption is adopted for technical convenience.

Working with a finitenumber of firmswould involve two technical difficulties.The firstdifficultyis that theequilibrium of the staticgame (with endogenously determined search intensity) annotbe analytically derived, as shown inTappata (2006) with more than threefirms.14The seconddifficulty s that the inferenceabout theunderlying statewould be too complicated inthedynamicsetting.With a finitenumber of firms, each firm randomly chooses a price according to somedistribution (usually some continuous distribution over a compact support). Thus theBayesianupdating based on different realized prices can easily get very involved.Moreover, assuminga finite number of firms adds randomness in the realized prices in each period, which furthercomplicates the analysis of belief updating.With a continuum of firms (along with capacityconstraints) inourmodel, we pin down a simple two-pointdistribution inequilibrium.15Moreover,theevolution of consumers' beliefs (hence equilibrium search intensity) isdeterministic given theevolution of theunderlying states, although individual firmsmight playmixed strategyin settingprices. This greatly simplifies theanalysis and makes theBayesian inferences tractable.We believe that the general insights of ourmodel carry over to the settingwith a finitenumber of firms. In the static game, consumers have stronger incentives to search when theybelieve thatthe low-cost state ismore likely, ecause ina low-cost stateprices aremore disperseddue to competition among firms; in thedynamic setting,asymmetry in learning continues tobepresent. Searchers observe all theprices setby thefirms,whereas each nonsearcher only observesone price and, consequently, the searchers learn the true state more quickly than nonsearchers.All these features do not seem tobe restrictedto the settingwith an infinite umber offirms. Thusour assumption of an infinite umber of firms ismainly for technical convenience, which helpssimplifyconsumers' Bayesian inferences and enables our equilibrium analysis tobe tractable.16Note that nother crucial assumption forour equilibrium analysis is thecapacity constraint,without which price dispersion simply does not arise inour equilibrium.17 Although capacityconstraints are prevalent inmany retailmarkets, our specific assumption that all firms have thesame capacity may appear restrictive.Again, we maintain this assumption mainly for technicalconvenience, which should not affect the robustness of our main results. To see this, we consideran alternative setting inwhich a proportion y of firms each has capacity k2, and the rest ofthefirms each have capacity kx,where k2 > kx > p. We assume thatyk2 < P, so that lowcapacity firmsare viable. In this setting,the staticequilibriumwith fixed search intensity/x s stillcharacterized by a two-point distribution.18 Depending on parameter values, we may have twotypesof equilibria. In thefirst-type quilibrium, firmswith thebigger capacity k2 charge the lowprice/?with probability 1,and firmswith the smaller capacity kxmix between/? and the chokeprice v. Let r]x(p)(rj2(p))denote theprobabilitywith which a firmwith capacity kx(k2) charges/?.Then theequilibrium isdetermined by thefollowing twoequations that re similar to (1) and (2):

7TX(V) 7Tx(p) (1-

fl)P(V ~C) = kx{p~

C) 18)14The main reason is that the expected gain of search cannot be integrated out explicitly.15The two-point price distribution derived inourmodel is somewhat special; nevertheless, we believe that itcapturesthe essence of real-world product market price dispersions while enabling our analysis tobe tractable.16Note that our approach is not without a cost?an undesirable feature of our equilibrium is that it lacks anappropriate continuity (in the number of firms); infact, our current equilibrium construction does not seem towork withany finite number of firms.17Recall that the capacity constraint keeps theprofitmargin of/? firmspositive by restricting the competition among(infinitelymany)/? firms.18The arguments are similar toLemmas \~A.Firms charging the lowest price have sales equal to their capacities.Searchers are not rationed at the lowest price.?RAND 2008.



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562 / THE RAND JOURNALOF ECONOMICStip+ (1 - n)P[y + (1 - y)r]x(p)] = kx(\- y)r]x(p) + k2y. (19)

Equation (18) says thata firmwith capacity kx is indifferent etween charging v and charging/?, nd equation (19) says that the totaldemand for the firmscharging/? equals the total capacityof those firms.Note thatfirmswith capacity k2 have no incentive to charge v, because n2(p) =k2(p? c) > 7Tx(p) 7^(1;) = n2(v). This is theunique equilibrium if \xP (1 - fx)Py > k2y(the search intensity sbig enough).On theotherhand, if/x/J (1 - pC)Py


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YANG AND YE / 563adjustment. Obviously, the asymmetry in price adjustment will be most prominentwhen thesearchers can observe all theprices. In this sense, our assumption can be regarded as the limitingcase when n goes to infinity.

Finally, in thedynamic game, we make thenonoverlapping assumption (11) such that theequilibrium ranges of/? and/? do not overlap. As a result,whenever a nonsearcher observes thelowerprice, she learns the truecost state immediately. In amore generalmodel, we should allowfor thecase that/? and/? may overlap inequilibrium. Under such a setting,we believe thatthegeneral insightof thearticle still holds. To see this,note that searchers can always infer the truecost state immediately.They have twopieces of information,the lowerprice/? and theproportionof firms charging the lowerprice r](p).Thus, from two equations (3) and (4), the searchers cancorrectly inferthe twounknowns, \x nd c.However, the belief updating fornonsearchers wouldbe much more involved. If a nonsearcher observes a lowerprice which lies in theoverlappingrange, she cannot immediately infer the true state.Although it is difficulttopin down the exactbelief-updating process in thiscase,20 it is clear thatnonsearchers' belief updating is slower thanthatof the searchers.Given this,asymmetric price adjustmentwill emerge aswell.

5. ConclusionIn thisarticle,we build a simple searchmodel with learning todemonstrate how asymmetric

price adjustments can arise as firms' optimal responses to cost shocks. Although theupwardprice adjustment is always completed within twoperiods aftera positive cost shock occurs, thedownward price adjustment takesmuch longer to complete when a negative shock occurs.The underlying reason forasymmetric price adjustments is thatsearchers and nonsearchershave differentbelief-updating processes. Because searchers observe thewhole spectrumof pricedistributionwhereas nonsearchers only observe one singleprice, searchers learn the truecost statea lotquicker thannonsearchers do. This learningasymmetrynaturally leads to asymmetricpriceadjustments.When a positive cost shock occurs, searchers quickly learn the true state and stopsearching. Thus thequick downward adjustment of search intensity leads to thequick upwardprice adjustment. On theother hand,when a negative cost shock occurs, it takes amuch longerperiod of time for nonsearchers to learn the true state and start searching. This slow upwardadjustment of search intensity eads to slow downward price adjustment.Thus, our article provides an explanation for thewidespread phenomenon of asymmetricprice adjustments. Although ourmodel is simple, we believe that it captures the essence ofreal-world productmarkets with imperfectlyinformedconsumers. Our model also predicts thatasymmetric price adjustments are more prominent inmarkets where the cost shocks are morepersistent or where more critical consumers are present.

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Huanxing Yang - Understanding Asymmetric Price Adjustments