Hot and Spicy: Ups and Downs on the PriceFloor and Ceiling at Japanese Supermarkets ¤

Kenn Ariga Kenji MatsuiMakoto Watanabe

Institute of Economic Research, and,Graduate School of Economics, Kyoto University,

Yoshida Honmachi, Sakyo-ku Kyoto 6068501 Japan

September 18, 2000

Abstract

This paper develops a model of dynamic pricing with menu cost forgrocery stores with some local market power in retail markets. Our modelderives frequent price changes among a few focal prices as the optimalprice policy. The key assumption of the model is that customers di¤er intheir willingness to pay, depending upon they purchase the product for im-mediate consumption, or for the inventory to avoid stock-out. The latterprobability is important because the customers visit these stores occasion-ally and their shopping decisions are based upon overall consumption andshopping plan, not whether or not they need to purchase the particularitem of our interest.

The model is then applied for the estimation of the demand ‡uctua-tions and pricing policy for two brands of curry pastes sold at 18 di¤erentsupermarket stores in Japan. The empircal results strongly support thefollowing key predictions of the model: (1) stores tend to lower the pricewhen (a) the share of customers with inventory is lower, and when (b) theexpected shopping intensity is higher; (2) the demand exhibits negativedependence on price duration at lower sales price, whereas the demand ispositively dependent upon the price duration at high, regular price. Thispattern is consistent with accumulation (when the price is low) and decu-mulation (when the price is high) of the inventory at home as the drivingfactor of shortrun ‡uctuations in the demand.

¤Ariga’s research is partially funded by grant-in aid from Ministry of Educaiton and JapanSociety of Promotion of Science. We bene…tted greatly from interviews at purchasing divisionof the chain J. The chain J also provided us with the data on daily ‡ow of customers toindividual stores in our sample.

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1 IntroductionFigure 1 speaks better than any other form of introduction in clarifying theobjective of the paper. The …gure plots daily movements of two competingbrands of curry pastes1 sold at a store belonging to one of nationwide super-market chains in Japan. The …gure traces these prices for roughly 6 years duringwhich prices changed more than 300 times. Remarkably, more than 80% of therecorded prices are concentrated in only two speci…c levels in each brand. Thisstore was open for 1707 days during the observation period stretching 1889days2 . Among those days, this store priced House Vermont Curry paste at 253yen for 763 days, whereas S&B Golden Curry, another national brand, was soldat 272 yen for 566 days. Both brands were sold at 198 yen very frequently, 568and 768 days, respectively. Compared to these pairs of two focal prices, otherprices were observed very infrequently. There are 64 other prices observed forthe …rst brand but these observations constitute only 378 days, or 20% of thesample, whereas two focal prices comprise roughly 80% of price observations.There are 364 price changes in the …rst brand. Among 363 completed spells,more than 70% of them are less than 6 days, with one day spell comprising 40%of these spells. The longest uninterrupted spell is 43 days in the case of …rstbrand. Not surprisingly, frequent price changes among few focal prices inducecustomers to wait for the price mark-down to concentrate their purchases: forexample, among the 7 stores which belong to a national chain of supermarket,the price of the …rst brand was below 198 yen for 1,862 store-days, or 15% of12,032 store days of the total observation, during which the stores sold 130,394units, or, 45.6% of the total sales, 286,812.

What exactly are the underlying logic behind this type of pricing? Althoughit is clear that the customers do recognize such a pricing pattern and adjust theirpurchasing cycle to that of pricing, it is not at all obvious why these stores em-ploy such a pricing policy. This paper develops a model of sales by extendingthe Varian’s pioneering work [Varian(1980)] and then empirically estimates theoptimal pricing model with special emphasis on inventory accumulation by con-sumers.

A small but rapidly growing number of empirical researches use scanner dataat retail stores to investigate the underlying mechanism of price adjustments at

1 Curry paste is sold generally in a package containing half-solid bars which can be cookedwith vegetables and meats. Curry served with rice is extremely popular in Japan. The annualsales of the curry paste (or bar) is roughly 80 billion yen in recent years, which translates into1.5 billion curry rice dishes cooked and consumed in Japnese household in a typical year. Twobrands in our sample are chosen because these are by far the most well known and longtimebest sellers among the curry pastes. The market shares of these two brands are relativelysmall; although no o¢cial statistics exist, the share is likely to be less than 10% for bothbrands. The House with commanding 40-45% market shares of the curry pastes have 5 majorbrands and each brand comes in 3 to 4 di¤erent versions (versions are typically distinguishedby the spicyness). Our House sample is one version of the …ve major brands o¤ered by House.Curry paste in Japan is similar to Canbell’s soup cans in the United States: you are boundto …nd at least a few packages of curry pastes in any randomly picked Japnese house.

2 Missing observations due to no registered sales or store closing constitute 182 and 233days, respectively.

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individual store/ brand level. There are two important research issues that mustbe addressed in order to analyze the data.

First of all, it is not at all clear who exactly sets the price: Slades (1998,1999),for example, assumes that manufacturers make decisions which are transmit-ted (with noise) to individual retail stores. Agguireria (1999) and Pesendorfer(1998), on the other hand, analyze the pricing problem of supermarket chains.

Second issue concerns the nature of intertemporal pricing pattern. Onein‡uential approach is menu cost model. Lump sum cost of (nominal) price ad-justments is assumed and responsible for the irregular and sizable price changes.Slades (1998) estimates such a menu cost model combined with a model of ’cus-tomer capital’. There are two major testable implications in her model. First,the current demand is a decreasing function of the past prices because customercapital is accumulated (depleted) over time as the size of customers graduallyincreases (decreases) in response to the posted price in relation to the perceived’fair’ price. The pricing policy is driven by the level of customer capital; as thecustomer capital hits the upper (lower) critical level, the store raises (lowers)the price. Another implication of the model is that the frequency and the sizeof price change are cross-sectional negatively correlated. If the menu cost isrelatively large, price changes are infrequent but larger in size. The centralmessage shared by many menu cost models is that the observed irregular andsizable price changes are attributable to the interactions between exogenousshocks and costs of nominal price changes.

Varian’s in‡uential paper [Varian (1980)] gave rise to a variety of randompricing models as equilibria supported by mixed strategies in a variety of pric-ing games.3 Among the models in this strand, one common implication is thatthe current demand is increasing in its past price because maintaining highprice for extended periods is necessary to accumulate large size of shoppers, or, bargain hunters. Pesendorfer (1998) exploits this idea and develop a simplemodel of sales which is applied to explain pricing patterns of ketchup sold at alarge supermarket. Another common implication of these models is the factorthat determines the length of price cycle. In menu cost-cum-customer capitalmodel of Slade, relative size of menu cost determines the frequency of pricechanges, whereas the direction of price change is determined by the stochasticchanges in the customer capital. In shopper-sales model, the cycle necessarilyends with ’sales’ which is short lived and produces large amount of sales. Hencethe direction of price change is necessarily (gradual or probabilistic) price de-

3 In Varian’s original paper, and also in Sobel (1984), the game is played among oligopolistic…rms. The mixed strategy in price setting game arises because stochastic sales, if successful,generates supra normal pro…ts. To maintain the equilibria, it is necessary that success in salesoccur randomly so that the expected pro…t during stochastic sales period is compatible to thepro…t during the no sale period. The driving factor behind the stochastic sale is accumulationof ’shoppers’ who are willing to postpone consumption and wait for the sale. In Conlisk,J., Gerstener, E and Sobel, J.(1984), the game is played between a monopolist seller and itscustomer. The pricing pattern in his paper is such that the price is stable at high level forsome time before it begins to decline gradually until all the shoppers hit the store at the lowestprice period. The cycle then repeat itself. Gradual price decline is necessary to retain andaccumulate large number of shoppers.

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cline followed by a large price increase. Another common implication of thesemodels is that the length of the cycle depends upon the speed of accumulationof bargain hunter, in relation to the cost of postponing the sale. This suggeststhat exogenous changes in demand size predicts the occurrence of sales. Warrenand Barsky (1995) actually …nds supportive evidence showing that the largepredictable increase in customer signi…cantly raises the probability of sales.

Compared to menu cost models, models of sales generate endogenous pricecycles even in the absence of exogenous shocks. Large ‡uctuations in sales andoccasional price mark-downs are outcome of intertemporal price discrimination.This paper builds upon this intuitively appealing idea that the observed pricingpatterns are used to dynamically price discriminate consumers. We take indi-vidual retail stores as the basic decision unit of pricing, instead of the chainheadquarters, or the upstream …rms, wholesalers or manufacturers. Our choiceis based upon the following considerations. We learned through interviews thatthe only incidence in which manufacturers are involved in retail pricing is specialsales promotion4 . Even in the case of sales promotion, the primary role of themanufacturer in sales promotion is funding. We think it natural to treat indi-vidual stores as independent decision making unit because prices di¤er acrossdi¤erent stores belonging to the same chain.

One of the key driving force of the model is the inventory accumulation bycustomers. Customers di¤er in their reservation prices depending upon whetherthey purchase the product for immediate consumption or for the inventory athome. As a result, the reservation prices of two types of consumers di¤er, and,this is the source of price discrimination that retail stores exploit. Consumers tryto concentrate their purchases when stores occasionally mark down the price. Asa result, the sales pattern of such a store is characterized by extended periods ofrelatively stable and low level of sales, punctuated by a burst of sales for a shortperiod of lower price. As a matter of fact, the correlation in our data betweenthe current sales and the past price is positive and signi…cant, suggesting theplausibility of the basic idea. As we will see later, the evidence also suggestwhile the store charges high price, the number of customers waiting for thesale steadily increases, whereas the stock of these customers is rapidly depletedwhen the store sets lower price for the product. This is in sharp contrast to thenegative correlation found in the data that Slades (1998) uses. Moreover, highfrequency of price changes, especially those between two focal prices suggest thatthe underlying factor responsible for the price change also has relatively highfrequency. The ‡uctuation of customer capital seems somewhat implausible onthis ground.

High frequency of price changes is consistent also with the ‡uctuations of thestore inventory as the underlying factor, the avenue taken by Agguireria (1999).

4 From the viewpoint of manufacturers, frequent special sales can actually destroy its pro…tbase because large sales at special discount often cannibalize the sales at regular price inneighboring stores. On the other hand, the manufacturer can use special sales to steal bothcurrent and future sales of the competing brand because customers purchase the product tostore, as well as for immediate consumption. Due to the lack of suitable data, we do notanalyze this important issue.

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Circumstantial evidence suggests strongly against the (S-s) inventory dynamicsas the important factor responsible for the price changes in our data. The retailstores in our sample have free and unlimited access to the inventory of theirmerchandize through the network of warehouses and delivery system. Most ofthe items in each store can be ordered overnight in very small units; very oftenan order of even one unit is assembled without any signi…cant additional costs5 .

These considerations leave us with the choice over the source of consumerheterogeneity and their shopping behaviors as the major candidate for the factorresponsible for the pricing patterns described above. There is no doubt thatthe data suggests occasional low price and concurrent sales promotion activity.Most of the existing models of sales seem to …t to describe this type of low pricewell because the amount of sales during these short-lived periods are order ofmagnitude larger than the average sales at regular prices. For example, at 7stores in our sample which belong to one national chain of supermarket, theaverage daily sales volume is 70 when the price is below the regular low price(198 yen), compared to 5.7 when the price is at regular high price (253 yen).Arguably, huge amount of sales can be usefully approximated as the burst ofpent-up demand coming from those customers waiting for the sales. On theother hand, about one third of the observations are at regular low price (198yen), the price level signi…cantly below regular high price but above the specialsales prices. The average daily sales during these periods is 23, about 4 timeslarger than the average sales at the regular high price. The model thereforeneeds to incorporate not only occasional special low prices with sales promotionbut also the interim price as a part of optimal pricing policy. In more generalterms, we need to explicitly model the length and depth of the price markdownsas the key policy variable in shaping the pricing policy.

We develop a model that incorporate these features of the pricing policyin a uni…ed framework. One of the key assumptions of the model is that the‡ow of customers are exogenously given not only for the stores but also to thecustomers. We justify this assumption on the ground that the stores in oursample sell a large variety of grocery items and it is unlikely that customerstime their shopping solely on the basis of the price of any particular product.This implies that the pent-up demand cannot be disposed with within a singlesales period. This crucial property gives rise to shorter but still non-negligibleprice durations even at prices below regular price.

The second salient feature of the model is the source of heterogeneity ofcustomers. We allow each customer to store the product at home. The het-erogeneity arises because of the di¤erence between the demand for immediateconsumption and for the inventory. The cost associated with holding inven-

5 The logistics involved in this network can be summarized as follows. Both major whole-salers and national retail chain stores have regional warehouses and highly sophisticated trans-portation network connecting these warehouses. Individual retial stores prepares orders forthe next day (or the day after). The regional warehouse, upon receiving the orders, assemblesthe orders and stock them into tracks headed for the delivery. In so doing, hundreds of di¤er-ent kinds of merchandises are delivered to an individual store by a single trip. In a nutshell,the cost associated with replenishing inventory seems quite small and seems highly unlikelyto be the ma jor cause of price changes.

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tory comes from two distinct sources. First, grocery items purchased occupylimited storage space, an acute concern for the Japanese households. Anothercost stems from the strong preference for freshness, and concerns over the de-terioration of quality for foods stored for long time6 . We believe these costs aresubstantial, and arguably more important than the role played by time discountin the context of the pricing patterns in our sample.

Alternatively, the occasional price mark-down can be modeled as an dynamicprice discrimination in the presence of heterogenous consumers (shoppers andnon-shoppers). Although popular, we do not employ this assumption for thefollowing reasons. First of all, bargain hunters do buy a large amounts to storeup at home, the very behavior that we focus upon in our speci…cation. Ourmodel incorporates this crucial feature without the heterogeneity assumption.The distinction between bargain hunters and non-hunters are also problematicwhen we apply the model to our data: the crucial di¤erence between the two isthat the bargain hunters are more patient (low time discount) than non-hunters.Highly frequent price mark-downs that we explained above and the fact thattypical grocery items purchased are bought regularly by households (e.g., atleast once every month) makes this types of speci…cation unsuitable.

The sequel of the paper is organized as follows. In the next section, wedevelop a model of dynamic price discrimination in the spirit of Varian (1980)and Sobel (1984). In section 3, we o¤er a preliminary analysis of the data andmotivates our empirical model speci…cation. The econometric model of pricingdecision is developed and estimated in section 4. Section 5 concludes.

2 A Model of Dynamic Price DiscriminationThis section develops a model of dynamic price discrimination for a retail …rm.In 2.1, we explain the basic set up. In 2.2, we start the analysis with simple staticpricing and show that the retail store chooses from the two alternatives: statichigh price at which customers purchase only for the immediate consumption;and static low price at which the customers are marginally induced to buy theproduct also to store. In 2.3, we explain the intuition behind the dynamic pricediscrimination. In 2.5 we formally develop and analyze the model. In 2.6, weconsider the impact of temporal variations in the shopping intensity and …ndthat the store prefer to time low price period to heavy shopping days. Theformal analysis is supplemented by numerical examples in 2.7.

2.1 Basic SetupWe consider a local monopolist retail store catering to a stable population ofregular customers. The total size of customers is constant and normalized to

6 According to a PR o¢cer of a major manufacturer of the curry paste, most of the currypastes purchased will be consumed within two months from the purchase and those leftoversbeyond two months are more likely to be disposed of , rather than consumed. He also noted,however, that the product sorted in room temperature will last at least one year withoutdeterioration in quality.

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unity. In anticipation of the sample stores used in the empirical analysis, weassume that the store sells a large variety of grocery items and its customerstypically visit store on regular basis and purchase a variety of goods. Ouranalysis in this section focuses on the pricing of a particular brand of a good,holding as given all the other aspects of the store activities. We assume thateach customer visits the store with probability s for any given day. In otherwords, shopping cost is very large for the rest of customers on that particularday. Thus s is also the number of shoppers every day. As we explained inIntroudction, the assumption on shopping behavior is motivated by the fact thecustomers visit the store to purchase a large variety of grocery items and theshopping behavior is not dictated by the price of any particular item. Assumingexogenous probability of shopping is a convenient shortcut to characterize theenvironment in which the …rm sets the price of an individual merchandize.

The customers are far-sighted and may purchase the good today for futureconsumption if the price is right. We assume a constant cost " of holding oneunit of inventory per unit of time. For simplicity, we assume that the storagecost for more than one unit is prohibitively large so that each customer has atmost one unit of the good stored.7

The customers’ preference for the good is random: on each day, with proba-bility c, consumption of one unit of the good generates utility (in money terms)u: Otherwise she gets no utility from consumption. At the beginning of eachday, each customer observes her own preference and shopping cost, and thenmakes her shopping decision8 .

The store that sells the good faces constant marginal cost ! per unit. Denoteby ± the (menu) cost of price change. Another simplifying assumption is zerodiscount which can be justi…ed in this framework as the relevant time horizonfor storage and price adjustment is likely to be short relative to any reasonablemagnitude of time discounting9.

2.2 Static PricingWe start the analysis with a simple benchmark case wherein the store is exoge-nously constrained to o¤er time invariant price. Since the store cannot recognizethe type of each customer, the only choice is the constant level of the price forthe good1 0 .

7 Allowing multiple units of inventory substantially complicate the analysis without sub-stantive additional results.

8 Again, stochastic consumption pattern is used as a conveninet shortcut to represent thedi¢culty of perfectly alligning consumption and shopping patterns.

9 The model can be easily exntended to incorporate positive time discount witout changingthe substance of the analysis.

10 The store can in principle use non-linear price (volume) discount to price discriminate thecustomers. We assume away this alternative for the following reasons. First of all, throughinterview, we learned that the volume discount is rarely used. Although scanner technologysubstantially reduce the cost of price changes, the volume discount is singularly unsuitableand costly under the scanner technology. Price data is matched to individual item through barcode printed on the merchandize. To use volume discount, the store must prepare di¤erentpackaging for multiple units purchase and register distinct code numbers.

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The pricing problem is further simpli…ed by noting the fact that in e¤ectonly two types of customers exist in this case: those who are potentially willingto buy to store and those buying for immediate consumption. Hence if thestore decides not to cater to the former type of customers, the optimal priceis obviously the maximum that it can charge without loosing the …rst typecustomers. Hence we have

pH = u (1)

In this case the demand for the good is given by

DH = sc (2)

because only those who visit the store and plan to consume the good immedi-ately will buy the product. Therefore the pro…t per day under this policy isgiven by

¦H = (u ¡ !)sc (3)

On the other hand, suppose that the store lowers the price enough to inducethe former customer to buy. Denote by ¼ the share of customers without oneunit of inventory. On each day, s customers visit the shop, among which sc¼of them purchase two units for immediate consumption and to store, sc(1 ¡ ¼)purchase one unit for consumption, s(1¡c)¼ also purchase one unit for inventory,and s(1 ¡ c)(1 ¡ ¼) do not purchase any. Summing up these, we get

DL = s(c + ¼) (4)

Denote by pL the optimal price in this case (yet to be determined). We have

¦L = (pL ¡ !)DL = (pL ¡ !)s(c + ¼) (5)

By comparing (3) and (5) , we obtain

¦H Q ¦L $ pL ¡ !

u ¡ !R c

c + ¼(6)

If p = pL forever, far-sighted customer without inventory always purchasethe good to store whenever she visits the store, whereas the inventory will be

Another reason to assume away discount is that it is imperfect measure: although consumerspurchasing multiple units are necessarily those without inventory, the reverse is not true. Somecustomers buy only one unit to store, not for immediate consumption.

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consumed with probability c which will not be replenished if she does not shopfor the day, which occurs with probability (1-s). Thus we get

¢¼ ´ ¼t ¡ ¼t¡1 = c(1 ¡ s)(1 ¡ ¼t¡1) ¡ s¼t¡1

Hence the steady state level of ¼is given by

¼¤ =c(1 ¡ s)

c(1 ¡ s) + s(7)

Therefore the condition (6) is rewritten as

¦H Q ¦L $ pL ¡ !

u ¡ !R c(1 ¡ s) + s

1 + c(1 ¡ s)(8)

Let us now consider the determination of pL: Denote by Vi(i = 0; 1) theexpected present value of net utility stream for a customer with i (=0,1) unitsof inventory of the good at home. We have

rV0 = sc(eu + eÀ) + s(1 ¡ c)eÀrV1 = ¡" + sceu + c(1 ¡ s)feu ¡ eÀg

eu ´ u ¡ pL

eÀ ´ V1 ¡ V0 ¡ pL

Solving for eÀ; we get

eÀ =c(1 ¡ s)eu ¡ " ¡ rpL

r + s + c(1 ¡ s)

Setting r = 0, we get

eÀ =c(1 ¡ s)eu ¡ "

s + c(1 ¡ s)

Since the customer without inventory is willing to purchase to store thegood if and only if doing so yields net gain in the expected utility. Hence themaximum that the store can charge is given by

c(1 ¡ s)eu ¡ " = 0

or,

pL = u ¡ "

c(1 ¡ s)

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Therefore, the previous condition can be written as

¦H Q ¦L $ 1 ¡ "

c(1 ¡ s)(u ¡ !)R c(1 ¡ s) + s

1 + c(1 ¡ s)(9)

2.3 Optimality of Hi-Lo pricing policy: IntuitionIn a static setting, the store cannot e¤ectively price discriminate customersunless the store knows the type of each customer. The only choice left for thestore is whether or not to price the good low enough to induce customers tobuy for the inventory at home. Inspection of (6) suggests that this strategyis chosen if the share of customers without inventory is large. We also knowthat the customer will not purchase the good to store if the price is above thereservation level, eÀ:

Once we allow the …rm to price discriminate by changing the price fromtime to time, the …rm can exploit the above property that the lower (high)price increases pro…t when the average inventory at customers’ homes is low(high). In what follows, we demonstrate through several steps that a type ofHi-Lo pricing policy emerges as the optimal intertemporal price discriminationwith menu cost. To proceed, we …rst describe intuitively the idea before weformally prove the optimality of such a policy.

Suppose the store employs a Hi-Lo pricing policy in the following manner:high price is posted for [1 ¿ 1] ´ TH ; and low price is posted for

[¿1 + 1; ¿ 1 + ¿2] ´ TL: Then the process is repeated. First of all, we noticethat during the high price period, the customer will not buy the good to storeup since the customer gets zero net utility even if she immediately consumesthe good upon purchase. This implies that during the high price period, theaverage level of inventory at home will be depleted gradually so that the shareof customers without inventory increases. The low price must be such that thecustomers are marginally induced to buy the good to store. The crucial questionhere is whether or not such a reservation price is constant. Intuition suggeststhat such a price will not be constant. The point here is the fact that towardsthe end of the low price period, customers are willing to pay more to store upthan they are at the beginning of the low price period. This is the case becausecustomers get strictly positive net utility from buying and consuming the goodat the low price level. The fact that such a valuable opportunity will soon endacts as an inducement to buy during the low price period. These arguments inturn suggest that the customer’s willingness to pay for the inventory should bedeclining during the high price period in anticipation of the low price periodsahead, although such a shadow price is immaterial during the high price period.

The pricing pattern that emerges from the argument above is the following.A typical cycle starts (arbitrarily) with high price period during which no in-ventory accumulation occur and the inventory will be gradually depleted as thecustomers consume them. The actual demand during the high price period iscon…ned to those without inventory who happens to shop on the day and also

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happens to get u from consumption. At ¿1, the price is reduced to the lowestlevel. The price then gradually rises during the low price period, mirroring therising willingness to pay to store up the inventory. The cycle ends with thesudden jump at ¿1+ ¿2 back to the high price. Then the cycle is repeated.

Now consider the role of menu cost. It is clear that at high enough menu cost,the cyclical pricing policy described above will not be optimal. Instead eitherpermanently high or low price policy will be taken. As we gradually lower themenu cost, the …rst pattern that will emerge is the one with minimum numberof price change: i.e., a simple Hi-Lo pricing policy in which the ’low’ price iskept constant until the price is returned to the high level. As we further lowerthe menu cost, the number of price adjustments during the low price periodincreases, more and more closely mirroring the price path without menu cost.As we will see shortly, however, without menu cost, the optimal policy calls forin…nitely frequent alternations between high and low prices because doing somaximizes the average size of ¼ consistent with low pricing.

2.4 Optimal Price Policy: Formal AnalysisIn this sub-section, we present and analyze the dynamic pricing model. We …rstcharacterize the optimal purchase policy of the customers, assuming that thestore employs stable Hi-Lo pricing policy described above. We then demonstrateindeed that the store optimally choose such a High-Lo pricing policy given theoptimal response of the customers. In Appendix, we prove that the optimal pol-icy starting from arbitrary level of ¼ indeed converges to the optimal stationarypolicy analyzed in this section.

2.4.1 Customer’s Optimal Policy

Let us start with customers. As we speci…ed above, the relevant choice for eachof the customers is either to buy for the good as inventory or not. Since pricenever exceeds the reservation utility level, u, customers always purchase thegood for immediate consumption.

We denote, as above, by V it (i = 0; 1 : t = 1; ::t1 + t2) the asset value of

discounted consumption stream net of purchasing and inventory holding costsfor customers with i units of inventory in period t during the cycle. We have

V 1t = ¡" + c(1 ¡ s)M ax[0; u ¡ Wt+1] (10)

+csMax[u ¡ pt+1; u ¡ Wt+1] + V 1t+1

V 0t = s(1 ¡ c)Max[0; Wt+1 ¡ pt+1]

+csfu ¡ pt+1 + Max[Wt+1 ¡ pt+1; 0]g + V 0t+1 (11)

Wt ´ V 1t ¡ V 0

t

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where it is understood t + 1 = 1 if t = t1 + t2: These equations canbe understood as follows. First, for those with one unit of inventory alreadyat hand, she has to pay " per period for inventory holding. She will get toconsume one unit of inventory with probability c. If she does not shop thatday, she depletes the inventory by doing so and su¤er capital loss given by Wt :By construction, postponing consumption is never optimal and u ¡ Wt is non-negative. If she happens to shop that day, she can replenish the inventory (inwhich case she gets u ¡ pt) or simply depletes the inventory (in which case shegets u ¡Wt): Those without inventory consumes the good only if she also shopson the day, upon which she gets u ¡ pt . Whether or not she purchases onemore unit for inventory depends upon the sign of Wt ¡ pt . Now recall that byassumption,

pt = u for t 2 TH (12)

pt 5 Wt for t 2 TL

The latter property is obtained because Wt is the maximum price that con-sumers are willing to pay for the good to store. We can use these pricing policyto rewrite (10) and (11). The results are

V 1t = ¡" + c(u ¡ Wt+1) + V 1

t+1 for t 2 TH (13)

V 1t = ¡" + c(1 ¡ s)(u ¡ Wt+1) + cs(u ¡ pt+1) + V 1

t+1 for t 2 TL

V 0t = V 0

t+1 for t 2 TH (14)

V 0t = s(1 ¡ c)(Wt+1 ¡ pt+1) + cs(u ¡ pt+1 + Wt+1 ¡ pt+1)

+V 0t+1 for t 2 TL

These two pairs of equations summarize the intertemporal pricing constraintfor the store, assuming that the store follows the Hi-Lo pricing policy.

2.4.2 Optimal Pricing Policy without Menu Cost

We now move on to consider the store’s optimal policy. As we said above,we limit our attention to the choice among the set of stationary policy rules.Hence the available choice for the …rm is simply to choose either constant priceor Hi-Lo pricing. The question on the choice over the two will be discussedlater. For the time being, we assume that the …rm pursue a variant of Hi-Lopricing policy. Without loss of generality, we assume that the cycle begins withthe high price period, followed by the low price period.

As it turns out, the analysis without menu cost is not only simpler butalso the nature of optimal pricing policy can be seen far more clearly in thisbenchmark case. Without menu cost, the …rm can costlessly change prices

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provided that during the ’low’ price periods, pt � Wt is satis…ed. Obviously thestore sets price such that:

pt = Wt for t 2 TL

Subsituting the equality for pt in (13) and (14), and then subtracting (13)from (14) for both sides, we obtain:

Wt+1 = (1 ¡ c)¡1fWt + " ¡ cug for t 2 TH (15)

Wt+1 = f1 ¡ c(1 ¡ s))g¡1fWt + " ¡ c(1 ¡ s)ug for t 2 TL

Corresponding changes in ¼t for each period is given by

¼t = c + (1 ¡ c)¼t¡1 for t 2 TH (16)

¼t = c(1 ¡ s) + f1 ¡ c(1 ¡ s) ¡ sg¼t¡1 for t 2 TL

Denote by W and ¼, respectively the initial level of each variable at t = 0:We can solve respective di¤erence equations to obtain:

Wt = WH ¡ (1

1 ¡ c)t(WH ¡ W ) for t 2 TH (17)

Wt = WL + ´t¡t1fWH ¡ WL ¡ (1

1 ¡ c)t1(WH ¡ W )g

for t 2 TL

WH ´ cu ¡ "

c; WL ´ c(1 ¡ s)u ¡ "

c(1 ¡ s); ´ ´ 1

1 ¡ c(1 ¡ s)

¼t = 1 ¡ (1 ¡ c)t(1 ¡ ¼) for t 2 TH (18)

¼t = ¼¤ + f1 ¡ ¼¤ ¡ (1 ¡ c)t1(1 ¡ ¼)gf1 ¡ c(1 ¡ s) ¡ sgt¡t1 for t 2 TL

Since the cycle repeats itself after (t1 + t2) periods, we require:

¼ t1+t2 = ¼ (19)

Wt1+t2 = W

Using these terminal conditions, we obtain:

¼ = 1 ¡ (1 ¡ AB)¡1(1 ¡ B)(1 ¡ ¼¤) (20)

A ´ (1 ¡ c)t1 ; B ´ f1 ¡ c(1 ¡ s) ¡ sgt2

13

W = (DE ¡ 1)¡1fE(D ¡ 1)WH + (E ¡ 1)WLg (21)

D ´ (1

1 ¡ c)t1 = A¡1; E ´ ´t2 = f 1

1 ¡ c(1 ¡ s)gt2

Using these expressions, equations in (17) are re-written as follows:

Wt = WH ¡ W¢ (DE ¡ 1)¡1(E ¡ 1)(1 + c)t for t 2 TH (22)

Wt = WL + ´t¡t1W¢ (DE ¡ 1)¡1(D ¡ 1) for t 2 TL

W¢ ´ WH ¡ WL ´ s"

c(1 ¡ s)> 0

Notice that Wt decreases during the high price period in re‡ection of theexpected price decline in the low price period. By the same token, Wt increasesduring the high price period anticipating the price increase in the high priceperiod.

¼t = 1 ¡ (1 ¡ c)t(1 ¡ AB)¡1(1 ¡ B)(1 ¡ ¼¤) for t 2 TH (23)

¼t = ¼¤ + f1 ¡ c(1 ¡ s) ¡ sgt¡t1(1 ¡ AB)¡1(1 ¡ A)(1 ¡ ¼¤) for t 2 TL

Let us now compute the store’s net pro…t for a representative cycle. Thisvalue is given by

M = (u ¡ !)sc

t1X

t=1

¼t¡1 + s

t2X

j=1

(Wt1+j ¡ !)(c + ¼t1+j¡1) (24)

We can substitute (22) and (23) for Wt and ¼t in (24). After tedious butstraight-forward computations and re-arranging terms, we obtain:

M

s= (u ¡ !)ct1 + (WL ¡ !)(c + ¼¤ )t2

+[(WL ¡ !) ¡ fc(1 ¡ s) + sg(u ¡ !)] £ (1 ¡ ¼¤ )(1 ¡ A)(1 ¡ B)

fc(1 ¡ s) + sg(1 ¡ AB)

+W¢ (D ¡ 1)(E ¡ 1)(c + ¼¤)

(DE ¡ 1)c(1 ¡ s)

+W¢ (D ¡ 1)(1 ¡ ¼¤)(1 ¡ A)(1 ¡ F )

(DE ¡ 1)(1 ¡ AB)s] (25)

F ´ BE = [f1 ¡ c(1 ¡ s) ¡ sg

1 ¡ c(1 ¡ s)]t2(< 1)

14

The optimal solution is given by

ft¤1; t

¤2g ´ argmax[

M

t1 + t2] (26)

As it turns out, without menu cost, the optimal policy calls for in…nitelyfrequent price changes in order to minimize the deviation of ¼ around the steadystate. On the other hand, the ratio of t¤1to t¤

2 converges to a positive …nite value.To establish the claim, we employ the following Lemma.

Lemma 1 Consider a function:

à (x; a; b; ®; ¯) ´ (1 ¡ a®x)(1 ¡ b¯x)

x(1 ¡ a®xb¯x); 0 < a; b < 1; ®; ¯ > 0

Ã(x) is monotonically decreasing in x in (0,1] and

Ã0(a; b; ®; ¯) = limx!+0

Ã(x) =®¯log(a)log(b)

¡(®log(a) + ¯log(b))

Proof. The result is immediate by using the L’Hopital’s rule twice: i.e.,di¤erentiate the denominator and the numerator twice and evaluate them atx = 0:

Lemma 2 Proof.

Lemma 3 Consider a function:

Á(x; a; b; ®; ¯) ´ (a®x ¡ 1)(b¯x ¡ 1)

x(a®xb¯x ¡ 1); a; b > 1; ®; ¯ > 0

Ã(x) is monotonically decreasing in x in (0,1] and

Á0(a; b; ®; ¯) = limx!+0

Á(x) =®¯log(a)log(b)

(®log(a) + ¯log(b))

Proof. Apply the same argument as in Lemma 1.

Lemma 4 Consider a function:

»(x; a; b; d; ²; ®; ¯ ) ´ (1 ¡ a®x)(d®x ¡ 1)f1 ¡ (b²)¯xgx(1 ¡ a®xb¯x)(d®x²¯x ¡ 1)

0 < a; b < 1 < d; ²; ®; ¯ > 0; b² < 1

Ã(x) is monotonically decreasing in x in (0,1] and

»0(a; b;d; ²; ®; ¯) = limx!+0

»(x) =®2¯flog(a) £ log(d)gflog(b) + log(²)g

¡f®log(a) + ¯log(b)gf®log(d) + ¯log(²)g

15

Proof. Apply the L’Hopital’s rule thrice.We now rewrite the maximand using these lemmas.

fM ´ M

sT= (u ¡ !)cµ + (WL ¡ !)(c + ¼¤)(1 ¡ µ)

+[(WL ¡ !) ¡ fc(1 ¡ s) + sg(u ¡ !)](1 ¡ ¼¤)

fc(1 ¡ s) + sg £

à [T ; 1 ¡ c; 1 ¡ c(1 ¡ s) ¡ s; µ; 1 ¡ µ]

+W¢(c + ¼¤ )

c(1 ¡ s)Á[T ; (

1

1 ¡ c);

1

1 ¡ c(1 ¡ s); µ; 1 ¡ µ]

+W¢(1 ¡ ¼¤)

s£

» [T ; 1 ¡ c; 1 ¡ c(1 ¡ s) ¡ s; (1

1 ¡ c);

1

1 ¡ c(1 ¡ s); µ; 1 ¡ µ] (27)

where the following new variables are introduced:

T ´ t1 + t2 (28)

µ ´ t1T

Then we obtain the following result.

Proposition 5 The optimal policy is obtained by setting T ! 0, and choosingµ that maximizes

fM(µ) = (u ¡ !)cµ + (WL ¡ !)(c + ¼¤)(1 ¡ µ)

+[(WL ¡ !) ¡ fc(1 ¡ s) + sg(u ¡ !)](1 ¡ ¼¤)

fc(1 ¡ s) + sg £

Ã0[1 ¡ c; 1 ¡ c(1 ¡ s) ¡ s; µ; 1 ¡ µ]

+W¢ (c + ¼¤)

c(1 ¡ s)Á0[

1

1 ¡ c;

1

1 ¡ c(1 ¡ s); µ; 1 ¡ µ]

+W¢ (1 ¡ ¼¤)

s£

»0[1 ¡ c; 1 ¡ c(1 ¡ s) ¡ s;1

1 ¡ c;

1

1 ¡ c(1 ¡ s); µ; 1 ¡ µ] (29)

Proof. Immediate from Lemma 1 through 3 since the …rst two terms ofthe maximand is independent from T , whereas the remaining three terms aremaximized by setting T ! 0:

Having converted the maximization problem into the choice of µ; we nextestablish the following:

16

Lemma 6 The optimal policy entails the following choices:

Static High Price Policy µ¤ = 1

Dynamic Hi ¡ Lo P olicy 0 < µ¤ < 1

Static Low Price Policy µ¤ = 0

The value function M is applicable to all of these cases. In particular, we have

fM (1) = (u ¡ !)c

fM (0) = (WL ¡ !)(c + ¼¤ )

Moreover, if the parameters (c,s) are su¢ciently small positive numbers11 , thefollowing conditions characterize the partitions of the parameter space into eachof the e three types of the optimal policy.

µ¤ = 0 iff eR 5 c(1 ¡ s)2

fc(1 ¡ s) + sg2 + c(1 ¡ s)2(30)

0 < µ¤ < 1 iffc(1 ¡ s)2

fc(1 ¡ s) + sg2 + c(1 ¡ s)2< eR <

1

(1 + c)

µ¤ = 1 iff eR = 1

(1 + c)

where eR ´ "

(u ¡ !)c(1 ¡ s)

Proof. By di¤erentiations, we obtain:

dà 0

dµ=

log(a) £ log(b) £ [µ2log(a) ¡ (1 ¡ µ)2log(b)]

[µlog(a) + (1 ¡ µ)log(b)]2(31)

dÁ0

dµ=

log(d) £ log(e) £ [µ2log(d) ¡ (1 ¡ µ)2log(e)]

[µlog(d) + (1 ¡ µ)log(e)]2(32)

d»0

dµ=

flog(a) £ log(d)gflog(b) + log(²)g[fµlog(a) + (1 ¡ µ)log(b)gfµlog(d) + (1 ¡ µ)log(²)g]2 £ (33)

µ[µ3log(a) £ log(d) ¡ µ(1 ¡ µ)2log(a) £ log(e)

¡µ(1 ¡ µ)2log(b) £ log(d) ¡ (2 ¡ 3µ)(1 ¡ µ)2log(b) £ log(e)]

where, a = 1 ¡ c; b = 1 ¡ c(1 ¡ s) ¡ s;d = (1 ¡ c)¡1; e = f1 ¡ c(1 ¡ s)g¡1:Using these expressions, the total di¤erentiation of the maximand is given

by

11 Since the choice of the time unit is arbitrary, we can choose the unit small (but still …nite)enough so that paramters c; s; and " are all small positive numbers.

17

dfM(µ)

dµ= f¹2(1 ¡ s)g¡1[¡Ceu(µ)eu + Ce"(µ)e"]

Ceu(µ) ´ (1 ¡ s)f¹c(1 ¡ s) ¡ s(1 ¡ ¹)dÃ0

dµg

Ce"(µ) ´ ¹c(1 ¡ s)f1 + c(1 ¡ s)g ¡ s(1 ¡ s)dà 0

dµ

+¹sf1 + c(1 ¡ s)gdÁ0

dµ+ ¹s(1 ¡ s)

d»0

dµ¹ ´ c(1 ¡ s) + s;

eu ´ u ¡ !;

e" ´ "

c(1 ¡ s):

Now the following properties are immediate from (31) through (33):(1) dÃ0

dµand dÁ0

dµare both monotonically decreasing in µ, strictly positive at

µ = 0;and negative at µ = 1:(2) d»0

dµis strictly convex in µ; is zero at µ = 0; reaches a strictly positive

value at bµ(0 < bµ < 1);and negative at µ = 1:Using these properties, we …nd:(3) Ceu(µ) is monotonically increasing in µ and strictly positive in [0,1](4) Ce"(µ) is monotonically decreasing in µ and strictly positive in [0,1].Consequently, the interior solution to the maximization problem, if it exists,

isgiven by

"

(u ¡ !)c(1 ¡ s)´ eR =

Ceu(µ¤)

Ce"(µ¤)

Now evaluate Ceu(µ¤) and Ce"(µ¤) at µ = 0: We have

Ceu(0)

Ce"(0)=

c(1 ¡ s)2

fc(1 ¡ s) + sg2 + c(1 ¡ s)2

where we have used the approximation for small positive x:

¡log(1 ¡ x) + x

Similarly, at µ = 0 :

Ceu(1)

Ce"(1)=

1

1 + c

18

Then conditions given in (30) follow immediately.This conditions can be interpreted as follows. Notice that "

c(1¡s) can beconsidered as the normalized holding cost. The term c(1 ¡ s) is the probabilitythat a customer wants to consume the product on a day that she will not visitthe store. Hence this is the probability that the inventory will be used forconsumption. Thus its inverse is the mean duration of the inventory. Then theratio eR signi…es the relative importance of the normalized inventory cost to thesize of the surplus. The lemma above states that this ratio must lie within theinterval given above in order for the store to choose Hi-Lo pricing policy. If theratio is too small, the store optimally set the price low permanently, whereasthe ratio exceeds the upper limit, the store always choose Hi price.

2.4.3 Optimal Pricing Policy with Menu Cost

We now introduce the cost of changing prices. Then, each of Hi-Lo pricingpolicy is characterized by (k + 1) ¡ tuple of positive integers denoted by

T ´ ft1; t12; t

22:::t

k2g; k ¸ 1

where the …rm adopts the following price policy:

pt = u for t 2 TH (34)

pt = pjL for t 2 T j

L

TH = [1; t1]

T jL = [t1 +

j¡1X

n=1

tn2 + 1; t1 +

jX

n=1

tn2 ]; j = 1; 2::k

Viz., the …rm changes the price (k + 1) times during the cycle and incurs(k + 1)± of menu costs. Since the cycle repeats itself, k + 1 ¸ 2: For the …rstt1 periods, the price is set at the reservation utility level, u. At t = t1 + 1, priceis reduced and the low price periods begins. For the …rst t12 periods, the priceis set at p1

L. Then the price is changed to p2L which lasts t22 periods, and so on,

until the price is …nally increased back to u after t1 +Pk

n=1 tn2 periods.

In order to compute the willingness to pay for the good to store, we use thefollowing notations:

Wt = WHt for t 2 TH

Wt = Wjt for t 2 T

jL

Since the di¤erence equation (15) is still applicable to WHt ; we obtain:

Wt = WH ¡ (1

1 ¡ c)t(WH ¡ W ) for t 2 TH

19

Hence we have:

Wt1 = WH ¡ D(WH ¡ W )

where D ´ (1

1 ¡ c)t1 ;

as before. The di¤erence equation for Wjt is given by

(1 + r)Wt = ¡" + c(1 ¡ s)(u ¡ Wt+1) ¡ s(Wt+1 ¡ pjL) + Wt+1

for t + 1 2 TjL

Which can be solved to obtain:

Wt = W j¡1L + ¸¿ j

t (W¿

j0

¡ W j¡1L )

¸ ´ 1

1 ¡ c(1 ¡ s) ¡ s> 1 (35)

¿jt = t ¡ ft1 +

j¡1X

n=1

tn2g

¿j0 = t1 +

j¡1X

n=1

tn2

WjL ´

c(1 ¡ s)u + spjL ¡ "

c(1 ¡ s) + s(36)

Notice that, as in the case without menu cost, Wt increases during the lowprice period and decreases in the high price period. Therefore, in order to meetthe constraint (12) during each sub-periods in which price is …xed, pt � Wt

must hold with equality at the beginning of each sub-period. Viz., we have

pjL = W¿ j

0+1 for 8j; j = 1; 2::k (37)

In order to solve the system, we …rst solve for pjL and W

jL :

WjL =

¹ + s¸W¿

j0

c(1 ¡ s) + s¸; (38)

pjL =

(¸ ¡ 1)¹ + ¸fc(1 ¡ s) + sgW¿j0

c(1 ¡ s) + s¸(39)

¹ ´ c(1 ¡ s)u ¡ "

Moreover we have:

20

W¿10

´ Wt1 = WH ¡ D(WH ¡ W )

W¿j0

= Wj¡1L + Ej¡1(W¿

j¡10

¡ Wj¡1L ) for j = 2; 3; ::k:

Ej ´ ¸tj2 (40)

Subsituting the above expressions for W jL and re-arranging, we get:

W¿ j0

= µ0j¡1W¿ j¡10

+ µ1j¡1¹

µ0j ´ Ej ¡ 1

c(1 ¡ s) + s¸;

µ1j ´ c(1 ¡ s)Ej + s¸

c(1 ¡ s) + s¸:

which has a general solution in the form given by

W¿ j0

= µ1j¡1µ1j¡2 ¢ ¢µ11Wt1

+fµ1j¡1µ1j¡2 ¢ ¢µ12µ01 + µ1j¡1µ1j¡2 ¢ ¢µ13µ02 + ¢ ¢ +µ1j¡1µ0j¡2 + µ0j¡1g¹

The terminal condition is

W¿k+10

= W

which can be solved to obtain:

W = (£1kD ¡ 1)¡1f£1k(D ¡ 1)WH ¡ £0k¹g

£1j ´jY

i=1

µ1i; j = 1; 2; ::::k

£0j ´jX

m=1

µ0m

jY

i=m+1

µ1i; j = 1; 2; :::k (41)

Then this solution can be substituted back into (40) to obtain W¿ j0:

W¿ j0

= (£1kD ¡ 1)¡1[£1j¡1(D ¡ 1)WH + f£0j¡1(£1kD ¡ 1) ¡ £1j¡1D£0kg¹]

(42)

This solution can be substituted back into (38) and (39) to obtain the solu-tions for pj

L and W jL:

21

For the time path of ¼t , the equations derived for the case of no-menu costare still applicable: they are reproduced here for convenience:

¼t = 1 ¡ (1 ¡ c)t(1 ¡ AB)¡1(1 ¡ B)(1 ¡ ¼¤) for t 2 TH (43)

¼t = ¼¤ + f1 ¡ c(1 ¡ s) ¡ sgt¡t1(1 ¡ AB)¡1(1 ¡ A)(1 ¡ ¼¤) for t 2 TL

where it should be understood that B is de…ned as

B ´ f1 ¡ c(1 ¡ s) ¡ sgt2 ´ B1B2B3:::Bk

Bj ´ f1 ¡ c(1 ¡ s) ¡ sgtj2

The maximand with menu cost is given by

M = ¡(k + 1)± + (u ¡ !)sc

t1X

t=1

¼t¡1 + s

kX

j=1

¿ j+10X

m=¿j0+1

(pjL ¡ !)(c + ¼m¡1) (44)

The …rst and second terms of the maximand are rewritten as

¡(k + 1)± + (u ¡ !)s[ct1 ¡ (1 ¡ ¼¤)(1 ¡ A)(1 ¡ B)

(1 ¡ AB)]

We use

¿ j+10X

m=¿ j0+1

¼m¡1 = tj2¼

¤ +B1B2B3:::Bj¡1(1 ¡ Bj)(1 ¡ A)(1 ¡ ¼¤)

(1 ¡ AB)fc(1 ¡ s) + sg

to rewrite the last term as

skX

j=1

¿j+10X

m=¿j0+1

(pjL ¡ !)(c + ¼m¡1)

= s

kX

j=1

(pjL ¡ !)(c + ¼¤)tj2

+skX

j=1

(pjL ¡ !)

B1B2B3:::Bj¡1(1 ¡ Bj)(1 ¡ A)(1 ¡ ¼¤)

(1 ¡ AB)fc(1 ¡ s) + sg

Therefore the maximand is given by

22

M = ¡(k + 1)± + (u ¡ !)s[ct1 ¡ (1 ¡ ¼¤ )(1 ¡ A)(1 ¡ B)

(1 ¡ AB)]

+skX

j=1

(pjL ¡ !)(c + ¼¤)tj

2

+s

kX

j=1

(pjL ¡ !)

B1B2B3:::Bj¡1(1 ¡ Bj)(1 ¡ A)(1 ¡ ¼¤)

(1 ¡ AB)fc(1 ¡ s) + sg

= ¡(k + 1)± + (u ¡ !)sct1 + skX

j=1

(pjL ¡ !)(c + ¼¤)tj

2

+(1 ¡ ¼¤)(1 ¡ A)(1 ¡ B)

(1 ¡ AB)fc(1 ¡ s) + sg £

[s

kX

j=1

(pjL ¡ !)(1 ¡ B)¡1B1B2B3:::Bj¡1(1 ¡ Bj)

¡sfc(1 ¡ s) + sg(u ¡ !)] (45)

The optimal solution is given by

fT¤; kg ´ ft¤1; t

1¤2 ; t2¤

2 ; :::tk¤2 ; k¤g = argmax[

M

t1 +Pk

j=1 tj2

]

Unfortunately, it is practically impossible to characterize the optimal solu-tion because not only the choice set is non-convex (k can take only positiveinteger values), but also the maximand is not concave (this we will demonstratein the simulation results below).

2.5 Christmas Bargain E¤ects, Sales Promotion, and Hi-Lo Price Cycle

In the base model, we assumed that the daily size of customers visiting the storeis constant. As we show in the next section, however, the data shows a largevariation in the number of visitors within a week, a month, or a year. In thissection we analyze the e¤ect of deterministic variations of s.

2.5.1 Christmas Bargain and Sales Promotion E¤ects: Intuition

The concentrations of price mark-down and sales during the heavy sales peri-ods is noted by Warner and Barsky (1995) who collects a variety of retail salesand pricing data in the United States and …nd that ’frequent markdowns in theintensive shopping period prior to Christmas, and a tendency for such sales tooccur in weekends’ (p.322). Our model can be modi…ed to incorporate deter-ministic ‡uctuations in the shopping probability. Such seasonality is common in

23

retail stores: weekends are heavy shopping periods, so are the Christmas-NewYear periods. The business is generally slow in mid-Winter and Summer days,etc. Faced with such patterns, a store can time its price cycle to increase pro…t.Intuitions suggest that the cycle should be timed in such a way that the priceshould be low in shopping days. The underlying mechanism is again that ofprice discrimination. We noted in the beginning that the Hi-Lo pricing in ourmodel is a compromised form of price discrimination: instead of perfectly pricediscriminate between bargain hunters and the other shoppers, retail stores lowerthe price when the share of the bargain hunters is large. In our model bargainhunters are those who buy the good to store rather than for immediate consump-tion. Timing the low price period to heavy shopping days help contribute toincrease the share of purchase by the bargain hunters during the period becauseas the low price period continues the share of bargain hunters decrease gradually(decrease in ¼). By timing the low price period to heavy shopping days, thestore can accelerate the sales to the targeted bargain hunters within a shorterperiod. As a result, the sales at low price to non-bargain hunters (those buyingfor immediate consumption) is reduced, thus enhancing the e¤ectiveness of thedynamic price discrimination. Compared to this strategy, setting price high inheavy shopping days is less pro…table because the size of potential customers isreduced.

The observation that the store bene…ts from timing the low price periodto heavy shopping days suggest that the store has an incentive to arti…ciallygenerate ‡uctuations in the shopping intensity12 . If the sales volumes duringthe low price periods can be increased by sales promotion activities, the storecan reduce the length of low price period and hence reduce the sales at low priceto those customers purchasing for the immediate consumption.

2.5.2 Synchronization of Sales Promotion and Price Reduction

In this subsection, we employ a particularly simple example to demonstrate thebene…t of timing the sales promotion to the low price period. Suppose thata retail store can pinpoint the sales promotion and advertizing activities toincrease the shopping intensity of a particular day during the price cycle froms to (1 + °)s (° > 0). Denote by (*) the optimal policy corresponding to aparticular con…guration of parameters:

P ¤ = argmax[P ¤

M (P : £0)

t¤1 + t¤2

]

£0 ´ f!0; "0; u0; c0; s0; ±0g

Consider:12 This …nding also justi…es the comment in interview by a director in charge of pricing at a

national chain of supermarket: manufacturer-sponsored special sales are always accompaniedby corresponding sales promotions including newspaper advertizement, in-house demonstra-tions, special display, etc.

24

¢ =M (P¤ : £1)

t¤1 + t¤

2

¡ M(P : £0)

t¤1 + t¤

2

£1 ´ f!0; "0; u0;c0; ±0; fstggfstg = s0 if t =2 tS

tS = t0S + j (t¤1 + t¤

2); j = 0; 1; 2::

fstg = (1 + °)s if t 2 tS ; ° > 0

The increase in pro…t measured by ¢ depends upon the timing of the shockwithin the cycle. Hence we denote:

¢ = ¢(bt)

where it is understood that bt denotes the timing within the cycle at whichsbt = (1 + °)s:

Lemma 7 Suppose t¤1; t

¤2 > 0: Viz, the optimal policy calls for Hi-Lo pricing.

Then, t¤1 + 1 = argmax(¢(bt)):

Proof. We have

¢(bt) = °(u ¡ !)sc¼bt¡1 if bt 2 TH

¢(bt) = °(Wbt ¡ !)s(c + ¼bt¡1) if bt 2 TL

Note that ¢(bt) is increasing in bt if bt 2 TH because ¼bt¡1 increases as thecustomers gradually deplete their inventory during the high price period. Onthe other hand, we can con…rm easily that ¢(bt) is decreasing in bt if bt 2 TL:Hence to prove the claim, it is su¢cient to show :¢(t1) < ¢(t1 + 1): We have

¢(t1) = °(u ¡ !)sc¼t1¡1

¢(t1 + 1) = °(Wt1+1 ¡ !)s(c + ¼t1)

We use (42) and (43) to get

¼t1¡1 =¼t1 ¡ c

1 ¡ c

and

Wt1+1 ¡ ! > WL ¡ !

= (1 ¡ eR)(u ¡ !)

25

Thus we have

¢(t1) = °(u ¡ !)sc¼t1 ¡ c

1 ¡ c

¢(t1 + 1) > ¢1 ´ °(1 ¡ eR)(u ¡ !)s(c + ¼t1)

Then we have

¢1 > ¢(t1) iff

(1 ¡ eR)(c + ¼t1) >¼t1 ¡ c

1 ¡ c

Rewriting the above, the condition is

eR <¼t1(1 ¡ 2c) + c

(c + ¼t1)(1 ¡ c)

Since the RHS is decreasing in ¼t1, we have

¼t1(1 ¡ 2c) + c

(c + ¼t1)(1 ¡ c)¸ 1

(1 + c)

On the other hand, from Lemma *, we know

0 < µ¤ < 1 iff

c(1 ¡ s)2

fc(1 ¡ s) + sg2 + c(1 ¡ s)2< eR ´ "

(u ¡ !)c(1 ¡ s)<

1

(1 + c)

if the menu cost is zero. With menu cost a fortiori we have

eR ´ "

(u ¡ !)c(1 ¡ s)<

1

(1 + c)

Thus we obtain

¢(t1 + 1) > ¢1 > ¢(t1)

26

2.6 Numerical Examples of Optimal Pricing PolicyThe optimal pricing policy analyzed above is highly non-linear and conventionalbattery of comparative analysis are powerless to examine the characteristics ofthe optimal pricing policy. We therefore provide a variety of numerical exam-ples of these policies in this subsection to illustrate the impact of changes inparameters on the pricing policy1 3 .

We start with the description of the benchmark equilibrium as shown inTable 2. In this benchmark case, we set: u = 10; ! = 5:11; s = :35; c = :19;" = :35; and ¾ = :05: The optimal policy14 is a price cycle of 6.76 days, roughlyone week, in which the high price is set for the …rst .67 of the cycle, or, 4.53 days,then the price is cut to 7.3 from the high price, 10. This …rst low price periodlasts 1.15 days, then the price is increased slightly to (7.47) , which last for1.08 days, and the cycle ends by returning to the high price level. I.e., k¤ = 2The average discount during the low price period is 26% ( i.e., the averageprice during the low price period is about 7.4). The maximand (per unit oftime) in this case is 1.03977. Compared to this value of the maximand at theoptimal policy, it is, respectively .33%, .52%, 1.23%, 2.07%, and 10.64% lowerif k = 1; 3; 4; 5 and 0 (static pricing policy). On the other hand, this maximandis 8.52% smaller than the maximum value which we would obtain if the menucost, ¾; is zero.

Table 1 shows the comparative statics results which is obtained by varyingone of the parameters and keeping the rest of the parameters at the benchmarkcase.

The impacts of changes in s are shown in the second column of the Table 1and they are quite intuitive. Both cycle length and average price duration areincreasing in s: As the customer shops more frequently, the bene…t of holdinginventory declines. Hence to induce customers to store inventory, the shopsneed to discount more during the low price period and also shorten the lowprice period relative to the length of the cycle. Evetually, beyond the thresholdvalue of s, it becomes optimal to sell always at the high price. Similarly, with sbelow the threshold , it is optimal to set the price always at the low thresholdlevel so that customers without inventory always purchase to store.

Similar logic can be applied for the impact of a change in c. An increase inc increases the probability of stock-out and hence raises the reservation price ofthe inventory. As a result, low price period increases but the average discountduring the low price period becomes smaller. Evevtually, beyond the thresholdvalue, the optimal policy calls for static low price policy.

An increase in the marginal cost (!) obviously raises the average price notonly by shortening the relative share of the low price period but also by raisingthe average price during the period.

The impact of " is quantitatively large: the impact is through its directe¤ect on the cost of holding inventory at household. As a result, the store must

13 The computer program for numerical examples shown in this subsection is written byMATLAB (v.5.3) and available upon request from the authors.

14 We ignore the integer constraints except for the choice of k.

27

set lower price during the low price period, whereas the consequent decline inthe pro…t from low price sales reduces the share of the low price period. Theimpacts on price level and discounts are relatively small and it is unclear if theresults shown in the Table 1 are robust.

In Table 2 we show the impact of a change in the menu cost on the optimalchoice of k: To begin with, the optimal policy calls for longer cycle length andlarger value of k as we decrease ¾ (bottom row): Starting with the benchmarkvalue of .05, the optimal policy changes from k = 2 to 1 at ¾ = :091 or larger:Reducing ¾ further, optimal policy changes to 3 at or around ¾ = :0074; andthen to 4 at ¾ = :00059: Correspondingly, the cycle length (not shown in Table 3)gets shorter and shorter as we decrease ¾: For example, at ¾ = :2 (the maximumvalue in our numerical example), the cycle length is 16.6 days, whereas it is lessthan 2.5days when optimal k …rst becomes 3. Notice ¾ = :05 correspondsto .5% of the high price. The simulations results suggest that thrice or morefrequent price changes during the low price period can be found only in extremecon…guration of parameter values and optimal policy, e.g., extremely small menucosts (less than .1% of the price tag) and extremely short cycle length (less than3days). The results are hence consistent with our data in that most of pricechanges are between regular high price and regular low price, i.e., HiLo pricingpolicy. The rest of the simulation results shown in Table 2 adds more evidencepointing the optimality of HiLo type pricing policy. Aside from the changes inmenu cost, we did not …nd any parameter con…gurations in which optimal policycalls for more than three ( k > 3)price changes. Frequent price changes duringa cycle is typically dominated by either a static low or static high price policy.

3 Data Explorations and Preliminary EmpiricalAnalysis

In this section we introduce our data taken from Nikkei Data Base. This sectionalso explores the data, especially to ascertain the plausibility of alternativetheories advanced in the existing literature in this …eld.

3.1 DataThe data used in this paper consists of daily observations on actual price andquantities sold of the items listed below at 18 di¤erent stores which belong to oneof the six national and regional supermarket chains in Japan. Unfortunately,none of these stores are in close proximity against the other stores in the data toexplicitly analyze the impact of strategic interactions. The items in the sampleare : two brands of curry pastes, two brands of ketchup bottle, two brands ofinstant co¤ee, and two brands of box of paper tissues. The sample period isJuly 1st 1991 to August 31, 1996. During the entire period, retail prices inJapan remained stable and there was virtually no discernible e¤ect of ongoingor future in‡ation. ( CPI rose only by 3.6% between 1991 and 1996).

28

Figure 1 through 8 depict the daily prices and frequency distribution of twobrands of curry pastes at two stores. Unmistakable patterns emerge from these…gures. First of all, in spite of highly frequent price changes, prices actuallyposted are highly concentrated in a few particular levels. As we explained inIntroduction, in the case of Stores of Chain 1, the focal two prices comprise morethan 70% of the observations in 6 of the 7 stores which belong to this nationalchain. The patterns di¤er considerably in other stores: all other stores belong tosmaller regional chains and the average shop size is also smaller. Figures 3 and4 show the pricing patterns of such a store: in this case the regular ’high’ priceis easily identi…ed (268 yen) that comprises 87% of the observations, whereasthere is no obvious candidate for regular ’low’ price.

3.2 Price Changes: Frequencies, Duration Dependence,and Store Characteristics

Table 3 provides several key indicators which we use to characterize the pricingpolicy of the stores. We notice immediately that the sample data exhibit ex-tremely high frequency of price changes. Psendorfer (1998) uses similar scannersales data on two brands of ketchup sold at US supermarket stores. The du-rations of prices are shorter for lower prices, characteristics shared in our data.The durations in his data are much longer, however. In our sample, averageprice durations are less than 10 days in the …rst chains and none of the samplestores have durations longer than 60 days even at the regular high price level.In Slade (1998,1999), she reports that 80% of weekly price observations are zeroprice changes, which implies unconditional mean duration of price is larger thanone month.

Although both Pesendorfer and Slade indicate the existence of focal pricelevels, our data shows not only the existence but that they are actually domi-nant: typically, only two focal prices constitute a majority of daily price obser-vations. In the subsequent empirical analysis, we exploit this …nding and usethe following representation of the price patterns:

pranget = 1 if pt < pregLow

pranget = 2 if pt = pregLow

pranget = 3 if pregLow < pt < pregHigh

pranget = 4 if pt = pregHigh

We investigated if there exists any simple time pattern of price changes inthe data. We thoroughly checked the relative frequency of price changes in bothdirections by the day of a week, day of a month, by month, on holidays, all ofthem for each store. We found a pair of statistically signi…cant regularities.Among the seven stores that belong to a same national chain of supermarket(chain 1), we found that the prices of both brands are reduced on they day 20thof a month, only to be increased again on 21st. As it turns out, the day 20th.

29

of each month is a regular monthly store sales day, common to all the storesof this chain. Except for this pattern, we failed to detect any simple regularityin price changes, although we do …nd mild seasonality in the over all frequencyof price changes over a year: In House Brand, overall price increase frequencyis around 8.6%, and 8.9% for the decrease. The frequency of price changes ishighest in February, at 10 to 11% of the observations for price increases anddecreases, whereas these …gures are lowest in July around 7.5%. For S&B brand,we found February registers highest frequency of price decrease but April …gureis the highest for the increase. Both frequencies are the lowest in September15 .Somewhat unexpectedly, pricing policies of the two competing brands exhibitsigni…cant positive correlations. Table 4 shows relative frequencies of pricesconditional upon the price of the competing brand. Table 4 shows the tendencythat the probability of higher price is signi…cantly larger when the price levelof the competing brand is also higher. Correlation is seen most clearly in thediagonal elements of the Table 4. These are all signi…cantly higher than theunconditional frequencies.

These …ndings are consistent with our modeling speci…cations in several di-mensions. First of all, they suggest strongly that the retailers set these pricesunder their initiative and with information they have. Otherwise, it is extremelyunlikely that upstream suppliers coincidentally adjust respective prices, giventhe extreme high frequency of the price changes. Secondly, the evidence in-dicates that prices of the two brands re‡ect common factors relevant to theoptimal pricing policy. The time pattern we found on the day 20th is a primeexample, as the store sales day of a month is likely to be accompanied by sig-ni…cant sales promotion and advertizing activities, which in turn signi…cantlyincreases the size of the customers visiting the store. On the other hand, it is un-likely that the changes in the wholesale price are the driving force behind thesepricing policies. As we will see later, our results indicate that large day-to-day‡uctuations in the number of customers is the most likely candidate.

3.3 ’Stylized’ Facts and Alternative ModelsAs we sketched in Introduction, frequent price changes without any apparenttime trend or seasonality is observed regularly in many grocery items sold typi-cally in supermarket stores. The lack of apparent seasonality distinguishes thistype of pricing from that of many apparel merchandises. Strong seasonality,uncertainty, or, importance of fashion in demand are relatively unimportant ingrocery items as they are purchased on regular basis, and, except for some freshfoods, consumption patterns are smooth and no strong seasonality exist.

Excluding the impact of seasonality, recurrent price increases and decreases

15 Weeekly ‡uctuations in the relative frequency of price changes are equally mild: for Housebrand, Tuesdays have the highest frequency in both directions (11.2 and 9.7%, respectively),and at 6.3%, Friday is the lowest for the price increase,whereas Thursday …gure is the lowestat 6.8% for the decrease. S&B brand exhibits similar pattern: Tuesdays register the highestfrequency in both directions (8.4 and 6.2%),. with Fridays being lowest for the increase (4.4%)and Thursdays lowest (4.6%) for the decrease.

30

without accompanying changes in costs can be an optimal pricing policy only ifsome type of the inter-temporal linkage in demand is important. As pointed outcorrectly in Slade (1998), many candidate explanations imply gradually decay-ing negative impact of the past price on current demand. Think, for example, acustomer capital model. Lower than a ’fair’ price gradually accumulates good-will or customer capital so that the (current) demand size also increases. Similarpatterns can be predicted if consumers over time grow tastes or habits in con-sumption of a good: lower price attracts …rst time customers and the stores canbuild up the customer base. Information imperfection is yet another possibilityconsistent with such a pattern. Suppose customers collect price informationonly occasionally so that, at any point of time, customers di¤er in the timeelapsed since they collected price information for the last time. In this case,lower prices set in the past induce some of the customers to shop, resultingagain in gradually decaying negative e¤ect of the past price on current demand.

On the other hand, the current demand depends positively on the past pricesin the case of bargain hunter models. The longer the time elapsed since the lastsales, the larger the size of the pent-up demand waiting for the sales.

Table 5 shows the results of the regressions on the current demand on pastprices. There is no doubt that in both brands the current demand is positivelycorrelated with the past prices ( in one, two, and four weeks). It might also benoted here that regressions indicate signi…cant and sizable e¤ects of the currentprice of the competing brand.

3.4 Dependence of the current sales on the current Priceand its Duration

Although the positive dependence of sales volume on the past prices is a char-acteristic shared by many (including our own) variants of the model on sales,our model di¤er sharply form the others in the prediction on the e¤ect of theduration of the high price on the current demand during the high price period.The dependence of the current sales on the level of ¼ implies that the salesmonotonically decline during the low price period as the customers continue tobuild up the inventory at home, whereas the sales monotonical ly increases overtime during the high price period as the customer gradually depletes the inven-tory. In a nutshell, the sales depends negatively on the price duration duringthe low price, whereas the sales depends positively on the duration during thehigh price period.

Table 6 shows the panel regression on the sales, incorporating di¤erential im-pacts of the price durations on the sales. As the model in this paper predicts,the duration has generally signi…cant negative impact on the sales, with onlyexception on price range 3, House brand case (the coe¢cient is insigni…cant),whereas at the regular high price ( price range 4), the durations have positiveimpact on the sales. It might be noted again that the price of the competingbrand has signi…cant positive impact on sales, as we found in preliminary regres-sions in Table 5. Recall the …nding in Table 4 that prices of the two competingbrands are highly correlated. Taken together, these facts suggest that the prices

31

of the two brands do respond to a common demand shock which is store speci…cand highly volatile.

In summary, we found that neither simple time-dependent rule or expla-nations based upon information imperfection, consumer habit formation, orcustomer capital model can account for the statistical regularities documentedabove. On the other hand, these facts are consistent with our model of sales,in particular, our model is the only one that predicts di¤erential impacts of thepast prices on the current sales .

4 Empirical AnalysisThe base model developed in section 2 is deterministic, giving rise to in…nitesequence of price cycles of identical length and pattern. As is clear from Figure1, the observed pricing patterns are highly irregular, and in particular, we …ndlarge variations in the length and magnitude of price cycles.

As we will show below, the two most important temporal variations whichwe believe responsible for the ‡uctuations are (a) changes in the number ofcustomers per period, (b) occasional but sharp changes (declines) in wholesaleprice of the curry paste.

In order to take account of these impacts, we use regression result for eachstore and brand, whose speci…cation is identical to those panel regressions inTable 6. Since the regressions in Table 6 estimate the daily sales, not the sizeof the daily visitors to the store, we need to …lter out the temporal variationsof the demand, given the size of the visiting customers to a store. For thispurpose, we use the …tted value of the sales and subtract the estimated e¤ectsof (1) those due to the duration as they re‡ect the gradual changes in ¼, and(2) the e¤ect of price range dummy variables as they represent the averagesize of ¼; and (3) the e¤ect of cproxy, as this is the proxy for the temporalvariations in c: Although we have no data on the wholesale price, we learnedthrough interview that (1) at regular low price (prange2), stores intentionallyavoid advertizing the lower price level and these discounts are unaccompaniedby corresponding wholesale price changes, (2) occasional sales lasting for a shortperiods (at most 3 days) at prices below regular low price (prange1) is alwaysaccompanied not only by corresponding reductions in wholesale price but alsoby promotion and advertizing activities conducted jointly with the store andwholesalers. Based upon these …ndings we treat the impacts of these occasionalsales as those re‡ecting the increase in the shopping intensity, s: Thus weinclude the estimated e¤ect of the dummy variable for the lowest price rangebecause we expect that the dummy primarily represents the increased customerawareness and increase in s.

To sum up, we used the following procedure. First, we ran the followingregression for each brand-store observations (indexed by i ).

32

log(salesit) = ®i

0 +X

k

®i1k £ dummyi

kt + ®i2cproxyt +

4X

j=1

[®i3jprangeji

t + ®i4jprangej i

t £ durationjjt

(+®i5jprangeji

t £ fdurationjjt g2)] + ui

t

Denoting byb the estimated coe¢cients, we obtain the shopping intensitymeasure as follows:

bsit = b®i

0 +X

k

b®i1k £ dummyi

kt + ®i1jprange1i

t (46)

+

4X

j=1

[®i4jprangeji

t £ durationj jt (+®i

5jprangej it £ fdurationjj

t g2)]

We obtain two intensity measures for each case, depending upon whether ornot the terms in the parenthesis in the equation above is included: henceforthwe denote by type 1 [type 2] the measure without [with] the parenthesized term.

In view of the theoretical model in this paper, the amount of sales ‡uctuatesover time as (1) ¼ changes over time, (2) c ‡uctuates over time, (3) s changes overtime for exogenous reasons (such as heavy shopping in weekends, lower shoppingrate in February), and (4) advertizement and sales promotion increases s: The

variable bsit is obtained by subtracting from the …tted value of sales, \log(salesi

t),the e¤ects due to (1) and (2) in the sales regressions and incorporate the e¤ectsof (3) and (4) as measured in the sales regressions. Finally, we normalized thevariable by taking the anti-log and then adjusting the average to unity.

4.1 Empirical Analysis of the Demand ShiftsOne of the key characteristics of the model in our paper is that the compositionand the size of the demand systematically evolves depending upon the pricepolicy chosen by a store. To be speci…c, the demand during the low price periodis given by

QLt = st(ct + ¼t¡1) (47)

This can be solved to

¼t¡1 = QLt=st ¡ ct

Using the equation below

33

¢¼ ´ ¼t ¡ ¼t¡1 = ct(1 ¡ st)(1 ¡ ¼t¡1) ¡ st¼t¡1

we can eliminate ¼ from the system to get

Qt+1 =st+1

stf1 ¡ ct(1 ¡ st) ¡ stgQt + st+1fctg2(1 ¡ st) (48)

Similarly, in the high price periods, we have

Qt+1 =st+1ct+1

stct(1 ¡ ct)Qt + st+1ct+1ct (49)

These pair of equations can be estimated using one of non-linear estimationmethods once we have data on Qt ; st ; and ct : In the subsection above we haveobtained the estimate bsi

t except that the average of the sales is unknown. simi-larly, our data on consumption, cproxyt needs the estimate of the average level.Finally, our data on gross sales needs to be re-adjusted to incorporate unknownsize of the customer (which is normalized to unity in the theoretical model insection 2). Hence we have

sikt = ¸ik

s bsikt ´ ¸ik

s esikt

cikt = ¸ik

c cproxyt ´ ¸ikc ect

Qikt = Q

ik

t =X ik

In the last equation, Qikt is the observed sales volume, and X ik is the un-

known size of the regular customer for store i; brand k: We substitute theseexpressions in (47). The corresponding regression form is:

Qt+1 = a11est+1

estQt ¡ a12

est+1

estectQt + a13

est+1

estectestQt ¡ a14

est+1

estestQt

+a15Xest+1fectg2 ¡ a16Xest+1fectg2est

where we omitted subscripts and the following non-linear restrictions on thecoe¢cients must be imposed

a11 = 1

a12 = ¸c

a13 = ¸c¸s

a14 = ¸s

a15 = ¸sf¸cg2

a16 = f¸sg2f¸cg2

34

which can be summarized as

a11 = 1

a12 = ¸c

a13 = a12a14

a14 = ¸s

a15 = a14fa12g2

a16 = fa14g2fa12g2

Similarly, the regression form for the high price periods is

Qt+1 = a21est+1ect+1

estectQt ¡ a22

est+1ect+1

estectectQt + a23Xest+1ect+1ect

a21 = 1

a22 = ¸c

a23 = ¸sf¸cg2

Combining everything together we have:

Qt+1 = rt [a11est+1

estQt ¡ a12

est+1

estectQt + a13

est+1

estectestQt

¡a14est+1

estestQt + a15Xest+1fectg2 ¡ a16X est+1fectg2est]

+(1 ¡ r t)[a21est+1ect+1

estectQt ¡ a22

est+1ect+1

estectectQt + a23Xest+1ect+1ect ]

where r is unity during the low price period and is zero otherwise. The setof restrictions are:

a11 = 1 (50)

a12 = ¸c

a13 = a12a14

a14 = ¸s

a15 = a14fa12g2

a16 = fa14g2fa12g2

a21 = 1

a22 = a12

a23 = a14fa12g2

35

The entire system can be estimated by maximum likelihood estimation ofnon-linear regressions. Speci…cally, we employed the following procedure. Wehad obtained an estimate of the average size of shoppers to each store in thesample. We used this as our proxy for X ik : This leaves us with the two un-knowns, ¸c and ¸s which we estimated as using maximum likelihood non-linearestimation: The results are shown in Table 7. Except for stores 8, 9, 13 and16 for House and store 16 for S&B brands, estimated coe¢cients are positiveand less than unity. Most of them are also highly signi…cant. Since estimationrequires positive sales …gures for the current and the previous period, we losemany observations in some store/brand combinations and the regressions per-formed relatively poorly in those cases. Overall, these estimate provide us withintuitively reasonable magnitudes for the two key parameters, ¸c and ¸s : theaverage consumption probability ranges .2 to 1% per day, whereas the averageshopping probability ranges between .1 to .3.

4.2 Ordered Probit Model of Price ChangesThe second important prediction of the model is that the pricing policy de-pends crucially upon the level of ¼: The probability of price decline should bemonotonically increasing in ¼ , whereas the price increase probability decreasesmonotonically with ¼: This prediction can be posited as a ordered probit model:

¢p ´ pt ¡ pt¡1

¢p > 0 if ª > P U

¢p = 0 if P D < ª < PU

¢p < 0 if ª < P D

ª ´ ®1¼ + ®2¼Low + ®3Low + ®4se + dummies

®1 < 0; ®2 > 0; ®3 7 0; ®4 < 0

wherein the ª equation above, Low is a dummy variable which is unitywhenever the price is below regular high price (p < prange4), and se is theexpected shopping intensity for the next two weeks. One of the two unknownthreshold values, PU ; PD ;can be set at arbitrarily at, say, zero and the remainingunknown together with other paramters should be estimated. We expect ®1 tobe negative as the store is more likely to lower the price when the share ofcustomers without inventory is higher. This impact is smaller, however, if theprice is already below the regular price (®2 > 0). We cannot sign ®3 because thelower price today reduces the probability of price decline whereas the currentprice may predict further decline in price to the extent that the current pricepredicts future increase in shoppers. We expect that the expected increase inshoppers should induce the store to reduce price, hence we expect ®4 < 0: Weuse the …tted value of the regression on shopping intensity used to construct ¼series16 .

16 In view of the RHS variables used to forecast shopping intensity, it is reasonable to assume

36

To test these predictions of the model in the data, we …rst construct the timeseries for the unobservable variable, ¼t : We use the estimates of the two keyparameters s and c obtained above in order to dynamically construct ¼ seriesusing :

st = csbst (51)

ct = bccproxyt

b¼ t = ct + (1 ¡ ct)b¼t¡1 for t 2 TH

b¼ t = ct(1 ¡ st) + fct(1 ¡ st) + stgb¼t¡1 for t 2 TL

The equations above generate the time series for a given initial condition,i.e., the value of ¼ in the initial period. In order to eliminate the e¤ect ofinitial condition on the generated series, we used two alternatives. First, wearbitrarily pick an initial value and we can compute the empirical density of¼. Then we choose the next initial condition by drawing from this probabil-ity distribution. The process is continued until we obtain convergence of theprobability distribution. Alternatively, we can simply eliminate some portionof the early observations of ¼. The results of the probit analysis hardly di¤eredirrespective of the methods used. Figures 9 and 10 show the estimated seriesfor two brands at the …rst seven stores belonging to the Chain 1. It is evidentthat the estimated series ‡uctuate widely at high frequency concurrently withcorresponding price changes. In comparison with House, S&B brand prices arechanged less frequently and the corresponding ¼ series exhibit lower frequency‡uctuations. It should be also noted that the estimated series across these sevenstores are highly correlated for the second brand, whereas for the …rst brandthe correlations are positive but smaller17 .

Table 9 reports the estimations based upon the …rst method. The resultsshown are obtained by pooling all the observations for the stores for which weobtained theoretically correct estimates of s and c; i.e., they are all positive andless than unity. The results are highly robust and the estimated coe¢cientsare of correct signs and signi…cant at 1% level18 . In particular, the constructedseries of ¼ is highly signi…cant and negative.

4.3 Cross Section Variations in the Estimated Coe¢cientsThe model presented in this paper has …ve free parameters (!, "; ¾; s; and c) andthe sixth parameter, u; can be used as a numeraire. We obtained estimates of

that the store can perfectly predict those RHS variables in the near future.17 Among the 7 stores, the correlation ranges between just about zero to .56 with mean .23

for House, whereas for S&B, the correlations ranges between .56 to .85 with mean close to.7. For the rest of 11 stores in the sample, within or across chain correlations are small inboth brands : none of them are larger than .2 in absolute value and the mean is close to zero.

18 Although we cannot sign ®3, the comparison of the regression results without se againstthsoe with se shows that ®3 are always smaller in the absolute value for the latter cases. Thisis consistent with our expectation because the part of the predictive power of the current priceon future sales is taken away by inclusion of se .

37

s and c in the nonlinear estimation model reported above. This leaves us withthree remaining parameters. Since we have no strong reasons to believe thatcustomers across stores and /or brands di¤er in their costs of holding inventoryat home, we treat " as an unknown but common constant across brands andstores.

Next, given the scanner technology and inventory and logistic system com-mon to supermarket stores in our sample, we assume that the cost of changingprice tags, " does not vary across brands or stores within a chain. Moreover,the menu cost should be independent from the sales size19 . Thus we postulatethe menu cost per unit of customer is given by:

¾ijk = ¾k=Q

i

jk (52)

wherein ¾k is chain speci…c menu cost and Qijk is the average customer size

of brand i at store j, which we normalized to unity in the theoretical model.In section 2.6, we obtained the following comparative statics results using

numerical simulations.

logfE[pL]ijkg = ®0 + ®1log(sijk ) + ®2log(ci

jk) + ®3log(!ijk)

¡®4log(¾ijk) ¡ ®5log(") (53)

logfE[T

k + 1]ijkg = ¯0 + ¯1log(si

jk ) ¡ ¯ 2log(cijk) + ¯3log(!i

jk )

+¯ 4log(¾ijk ) + ¯5log(") (54)

logfE [TH

T]ijkg = °0 + °1log(si

jk ) ¡ °2log(cijk ) + °3log(!i

jk)

+°4log(¾ijk) + °5log(") (55)

wherein all the parameters (®, ¯, and°) are positive. We use (51) above tosubstitute for ¾i

jk to get

logfE[pL]ijkg = ®0 + ®1log(sijk) + ®2log(ci

jk ) + ®3log(!ijk ) ¡ ®4log(¾k )

+®4log(Qijk ) ¡ ®5log(") (56)

logfE [T

k + 1]ijkg = ¯0 + ¯1log(si

jk) ¡ ¯2log(cijk ) + ¯3log(!i

jk) + ¯4log(¾k)

¡¯4log(Qi

jk ) + ¯5log(") (57)

logfE[TH

T]ijkg = °0 + °1log(si

jk) ¡ °2log(cijk) + °3log(!i

jk ) + °4log(¾k)

¡°4log(Qijk) + °5log(") (58)

19 There is no requirement that the price tag be attached to the individual merchandize. Allof the sample stores in our data use scanners and the system is linked to the price tags on theshelves of each store. As shown in whether or not the stores are required to attach the pricetag on individual merchandize makes signi…cant di¤erence in the cost of changing prices.

38

Denote by ¢ log-di¤erencing across stores and brands within a chain. Theequations above can be written to get ’within chain’ regressions.

¢logfE [pL]ijkg = ®k0 + ®1¢ log(si

jk ) + ®2¢log(cijk )

+®4¢ log(Qi

jk) + u1 (59)

¢logfE [T

k + 1]ijkg = ¯k

0 + ¯1¢log(sijk) ¡ ¯2¢log(ci

jk )

¡¯4¢log(Qi

jk ) + u2 (60)

¢ logfE[TH

T]ijkg = °k

0 + °1¢log(sijk ) ¡ °2¢log(ci

jk )

¡°4¢log(Qijk) + u3 (61)

wherein chain speci…c constants and error terms ( u0s ) represent the impactof the unobservable, ¢!i

jk :Table 9 shows the ’within chain’ estimates of these parameters. The …rst

column is within chain panel OLS and the second set of regression results areSUR (seemingly unrelated regressions) run for joint estimations of ®, ¯, and°:

The results are mixed: most of the estimated coe¢cients for s and c are notstatistically signi…cant and some of them are wrong signs. On the other hand,the coe¢cients on log(Q

i

j ) are statistically signi…cant and signs are correct inall of the three regressions.

5 ConclusionIn this paper, we developed a dynamic model of sales and applied it to thepricing policy of Japanese supermarkets, with the results consistent and sup-port the main thrust of the theoretical model. In particular, we found thatthe customers respond to occasional price markdown by accumulating the lowpriced merchandize as inventory at home, which in turn are gradually consumedduring the high price period. Such an behavior is con…rmed by the regressionresults demonstrating negative (positive) price duration dependence of the salesduring the low (high) price period. We also estimated the determinants of theprice changes and found that the share of customers without inventory havestatistically signi…cant impact on price changes.

These facts are not easily reconcilable with competing models of sales. Forexample, customer capital model is at variance with the data because the modelpredicts negative dependence of the sales on the past prices, which as we sawis strongly contradicted by the data. Variants of models of sales developed inthe past also have di¢culty in one way or the other once confronted with thedata. Frist of all, the duration of sales (low price) is typically shorter than theprice durations at higher price level but in no way approximated by the burst ofsales completed within a single period. The theory is thus silent on exactly howlong sales periods last, not to mention when they start. Random pricing models

39

of sales are again at variance with the fact: there is no indication whatsoeversupportive of such pricing.

Our model scored better than any of these candidate models but we stillleave many important questions left unanswered.

One of the major unexplored issues is the interactions between brands withina store. As we noted in section 3, statistical evidence indicates signi…cant cor-relations between the pricing of the two brands within a store20 . It seems quitelikely that customers can easily switch between brands in response to occasionalsales of one of the brands. It is not clear at this moment how the model shouldbe altered once we incorporate brand substitutions into the model.

Another issue of interest is the linkage between pricing and sales promotionactivities. We found evidence in support of the close coordination between pric-ing and sales promotion but our results fall short of identifying the mechanismof the interactions.

In more general terms, this paper can be considered as a …rst step towardscomprehensive analytic of the retail store pricing and sales promotion activities.Our results indicate that supermarket stores do induce ‡uctuations in shoppingintensity and purchase patterns by employing sophisticated dynamic pricing andsales promotion.

20 Lach and Tsiddon (1996) also found signi…cant within store price synchronization in winesbut not in meat products. Theri …ndings are consistent with our model of sales.

40

References[1] Aguirregabiria, V., ’The Dynamics of Markups and Inventories in Retailing

Firms,’ Review of Economic Studies, 66 (1999): 275-308

[2] Benabou, R., ’Search Price Setting and In‡ation,’ Review of EconomicStudies, LV (1988): 353-376

[3] Conlisk, J., Gerstener, E and Sobel, J.,”Cyclical Pricing by A DurableGood Monopolist,” Quarterly Journal of Economics XCIX,August, 1984:489-505

[4] Ferstman, C., and F. Fishman, ’Price Cycles and Booms: Dynamic SearchEquilibrium,’ American Economic Review (1992): 1221-1233

[5] Kashyap, A., ’Sticky Prices: New Evidence from Retial Cata-logues,’Quarterly Journal of Economics 110 (1995): 245-274

[6] Lach, S. and D. Tsiddon, ’Staggering and Synchronization in Price Set-ting: Evidence from Multiproduct Firms,’ American Economic Review86(5)(1996): 1175-96

[7] Levy, D., Bergen, M., Dutta, S., and Venable, R., ’ The Magnitude of MenuCosts: Direct Evidence from Large U.S. Supermarket Chains,’ QuarterlyJournal of Economics 113 (1997): 791-825

[8] Pashigian, P. and B. Browen, ’Why are Products Sold on Sale?: Ex-planations of Pricing Regularities,’Quarterly Journal of Economics 106(1991):1015-38

[9] Pesendorfer, M., ’Retail Sales. A Study of Pricing Behavior in Supermar-kets,’ mimeo., 1998

[10] Slade, M.E., ’Optimal Pricing with Costly Adjustment and Persistent Ef-fects: Empirical Evidence, Quarterly Journal of Economics XCXI,

[11] Slade, M.E., ’Sticky Prices in a Dynamic Oligopoly: An Investigationof (s,S) Theresholds’, International Journal of Industrial Organization17(1999): 477-511

[12] Sobel, J.,”The Timing of Sales” Review of Economic Studies LI,(1984)353-368

[13] Sobel, J.,’Durable Good Monopoly with Entry of New Customers,’ Econo-metrica 59(5)(1991): 1455-1485

[14] Varian, H.R., ’A Model of Sales,’ American Economic Review,70 (1980):651-59

[15] Warren, E. and R. Barsky, ’The Timing and Magnitude of Retail StoreMarkdowns: Evidence from Weekedns and Holidays,” Quarterly Journalof Economics 110 (1995):321-52

41

6 Appendix Covergence to the Stationary Hi-Lo Pricing

Consider a sequence of Hi-Lo pricing cycles, each of which consists of high priceperiods follow by the low price periods. Index by n (= 1; 2; 3; :::) each of thesecycles. The optimal policy for each cycle is denoted by

P ¤ = argmax[P ¤

M(P : ¼; £)

t¤1 + t¤

2

]

P ´ (k;t1; ftj2gj=1;2:::k )

£ ´ f!; ";u; c; s; ±gsubject to ¼(0) = ¼

The optimal policy can be written as

P ¤ = ª(¼; £)

Using (18), we obtain:

¼t¤1+t¤

2= [1 ¡ ¼¤ ¡ A(t¤

1(¼))]B(t¤2(¼)) + ¼¤ + A(t¤

1(¼))B(t¤2 (¼))¼

Since ¼t¤1+t¤

2is the initial value of ¼ for the next Hi-Lo price cycle, we obtain:

¼n+1 = [1 ¡ ¼¤ ¡ A(t¤1(¼n ))]B(t¤2 (¼n)) + ¼¤ + A(t¤

1(¼n))B(t¤2 (¼n))¼n

Rewritng the above we have

e¼n+1 = ©(e¼n )e¼n;

0 < © < 1

© ´ A(t¤1(e¼n + ¼))B(t¤2 (e¼n + ¼))

e¼n ´ ¼n ¡ ¼

¼ ´ 1 ¡ [1 ¡ AB ]¡1(1 ¡ B)(1 ¡ ¼¤)

The mapping © given above is contraction and we can apply conventionalargument to con…rm that the {¼ng converges monotonically towards the …xedpoint, ¼.

42

7 Tables and FiguresTable 1 E¤ects of Changes in Parameters

T Tk+1

TH

TM E(p) E(pL jt 2 TLj)

cycle lengthprice

durationshare of highprice periods

maximandmean price

(time average)mean price

in TL

s +2:59 +3:87 +1:05 +0:65 +0:18 +0:09c ¡0:60 ¡0:80 ¡0:36 +1:29 ¡0:22 +0:08

! +3:16 +4:90 +1:49 ¡1:44 +0:24 +0:10" +4:59 +6:31 +1:32 ¡0:45 +0:18 ¡0:16

¾ +1:13 +1:39 +0:12 ¡0:05 +0:04 ¡0:04

notes: the …gure in each cell shows the average % change of an endogenous variable(column) when one of the parameters (row) is increased by 1% when the rest of the parametersare set at the benchmark values. In computing the …gures, only the results from interiorequilibria are used ( we exclude equilibria wherein static pricing policies are optimal).

Table 2 Optimal number of price changes during a cycleparameter range static low k = 2 k = 1 k > 2 static high

s [:122 .999] <:296 [:296 .423] [:423 .521] n:a: >.521c [.036 .999] <.141 [.166 .237] [.141 .166] n:a: >.237! [2:26 9:65] < 3:74 [3:74 5:63] [5:63 6:11] n:a: > 5:63

" [:122 .999] < :284 [.284 .396] [:396 :460] n:a: > :460parameter range static policy k = 1 k = 2 k = 3 k ¸ 4+

¾ [.00013 high price if ¾ > :24 [.088 .24] [.0073 .088] [.00063 .0073 <.00063+we programmed simulation model up to k=5. Obviously, as we reduce ¾ further, the

optimal number of price changes increases inde…nitely. We …nd that k=5 overtakes k=4 at.00013.

Table 3 Summary Statistics of Price Data: House Brandstore# price (yen) prange shares (%) price duration (days) at prange=[chain #] mean min. max. 1 2 3 4 all 1 2 3 41 [1] 221 128 253 13.5 32.9 8.7 45.0 5.29 2.67 6.79 2.05 9.232 [1] 199 145 253 22.3 49.3 21.6 6.8 5.14 2.76 6.82 6.70 6.893 [1] 215 130 253 11.4 47.8 8.8 32.1 5.23 2.21 8.08 2.31 7.534 [1] 218 146 253 12.3 42.9 6.9 37.9 5.16 2.34 7.09 1.81 9.255 [1] 221 149 253 13.8 34.3 9.0 42.8 5.14 2.82 6.96 2.16 7.876 [1] 217 138 258 12.6 32.5 20.0 34.8 5.03 1.99 6.65 4.41 8.627 [1] 213 95 253 22.5 3.1 46.0 28.4 7.10 5.12 6.33 7.64 8.908 [2] 199 135 228 4.3 87.2 0.3 8.2 11.30 1.29 21.71 2.00 8.259 [2] 199 126 228 6.6 81.1 0.2 12.1 8.32 1.25 15.92 1.00 7.6010 [3] 198 98 253 47.3 35.4 8.5 8.8 9.34 9.75 7.75 10.19 18.7811 [4] 202 135 358 17.9 48.4 7.3 26.3 13.24 6.34 23.21 6.23 18.6112 [4] 212 128 248 10.1 45.0 2.4 42.5 17.81 4.48 28.97 4.55 33.7513 [5] 223 128 268 15.8 15.8 21.3 47.2 10.22 4.42 12.04 6.95 21.0914 [6] 220 148 255 10.5 45.6 4.5 39.4 5.46 2.06 8.29 1.60 10.0115 [6] 219 127 255 11.0 45.2 4.1 39.7 5.54 2.22 8.88 1.66 9.3116 [7] 241 100 248 2.4 15.5 0.4 81.6 20.29 1.24 17.13 1.17 40.5617 [7] 238 100 248 4.4 16.9 5.9 72.8 14.30 2.34 9.76 5.89 28.8218 [7] 242 100 248 3.0 9.0 2.0 85.9 23.02 1.60 12.77 4.25 52.90

Table 3 (cont’d) Summary Statistics of Price Data: S&B Brandstore # price (yen) prange shares (%) price duration (days) at prange=[chain #] mean min. max. 1 2 3 4 all 1 2 3 41 [1] 225 136 272 8.5 46.9 9.8 34.8 7.90 2.28 13.08 3.47 12.962 [1] 227 138 272 4.7 42.0 17.5 35.8 12.93 1.90 18.10 10.58 22.563 [1] 227 128 272 5.8 48.9 7.9 37.4 10.67 2.21 15.95 4.29 19.134 [1] 227 145 272 6.5 48.0 7.7 37.8 10.29 2.65 15.57 4.38 16.865 [1] 226 138 272 8.2 45.7 9.2 36.8 9.70 3.16 15.23 3.67 16.576 [1] 226 99 272 7.9 43.6 12.3 36.2 13.15 3.19 19.76 7.61 23.757 [1] 226 148 272 7.0 41.8 17.9 33.3 13.11 3.13 20.49 11.77 19.368 [2] 238 138 278 4.2 36.1 12.6 47.1 14.66 1.90 13.91 12.92 24.679 [2] 241 121 278 5.9 29.3 16.4 48.3 13.66 2.64 14.54 16.58 21.8310 [3] 213 100 368 30.4 26.3 23.5 19.8 15.86 11.65 11.98 27.50 27.3611 [4] 197 100 228 21.3 46.7 3.2 28.8 13.67 7.65 21.08 3.17 21.0812 [4] 202 128 228 13.6 42.1 2.9 41.3 15.93 5.68 23.03 4.50 29.2313 [5] 245 128 283 17.9 2.2 16.2 63.6 11.39 6.18 1.50 6.00 21.6514 [6] 203 148 268 12.3 77.3 1.3 9.1 5.78 1.84 9.95 1.38 8.2015 [6] 205 134 268 14.5 70.2 1.6 13.8 5.84 2.40 9.59 1.60 8.8916 [7] 261 100 268 0.5 3.4 6.8 89.3 35.29 1.0 6.50 8.61 64.0817 [7] 249 100 268 20.1 14.2 2.1 63.6 22.67 11.00 20.50 11.33 33.5518 [7] 243 100 268 20.9 18.9 6.3 53.8 25.22 8.75 26.70 5.22 49.06

Table 4 Correlations in Pricing of the Competeing BrandsS&B Brand Price House Brand Price

overall range=4 range=3 range=2 range=1price range=4 .428 .524 (1.24) .334 (.78) .377 (.88) .363 (.85)price range=3 .095 .072 (.76) .240 (2.53) .071 (.75) .122 (1.28)price range=2 .362 .318 (.88) .264 (.73) .464 (1.28) .279 (.77)

price range=1 .115 .085 (.74) .162 (1.41) .089 (.77) .236 (2.06)

Table 5 Panel Regression of the Sales on the Current and Past Pricesyprice of average price House Brand Price S&B Brand Pricerival brand in the last current past rival current past rivalnot included 1 week -.030¤¤¤z .974¢10¡3¤¤¤ - -.022¤¤¤ .914¢10¡3¤¤¤ -

2 weeks -.030¤¤¤ .326¢10¡2¤¤¤ - -.022¤¤¤ .186¢10¡2¤¤¤ -4 weeks -.031¤¤¤ .551¢10¡2¤¤¤ - -.022¤¤¤ .327¢10¡2¤¤¤ -

included 1 week -.030¤¤¤ .954¢10¡3¤¤¤ .141¢10¡2¤¤¤ -.022¤¤¤ .926¢10¡3¤¤¤ .130¢10¡2¤¤¤

2 weeks -.030¤¤¤ .321¢10¡2¤¤¤ .104¢10¡2¤¤¤ -.022¤¤¤ .188¢10¡2¤¤¤ .516¢10¡3¤

4 weeks -.031¤¤¤ .542¢10¡2¤¤¤ .691¢10¡3¤¤ -.022¤¤¤ .327¢10¡2¤¤¤ .184¢10¡3

yAll regressions are …xed e¤ect panel regressions and aside from those shown in the table,they include the following variables; year, month, day of the week, holidays, the day beforeshops are closed, and the day after shops are closed (all of these dummy variables), andcproxy, representing seasonal ‡uctuations in the consumption. z***indicates signi…cant at1%, ** and * correspond to 5% and 10%, respectively.

Table 6 Panel Regressions of the Sales on Price Range and Durationsy

House Brand S&B Brandregression 1 regression 2 regression 1 regression 2

coeff : p ¡ value coeff : p ¡ value coeff : p ¡ value coeff: p ¡ valueprange1 80.99 [.000] 87.34 [.000] 45.14 [.000] 46.73 [.000]prange2 21.84 [.000] 25.64 [.000] 15.13 [.000] 15.45 [.000]prange3 11.94 [.000] 14.50 [.000] 8.24 [.000] 9.20 [.000]

duration ¤ p1 -1.81 [.000] -1.70 [.000] -.43 [.000] -.42 [.000]duration ¤ p2 -.090 [.021] -0.22 [.000] -.024 [.030] -.031 [.027]duration ¤ p3 .097 [.433] .032 [.026] -.010 [.186] -.012 [.122]duration ¤ p4 .046 [.071] .062 [.007] .083 [.000] .080 [.000]rivalprange1z - - -6.43 [.000] - - -3.98 [.000]rivalprange2 - - -3.51 [.001] - - -1.49 [.083]rivalprange3 - - -4.86 [.000] - - -2.38 [.001]

R2

.357 .377 .244 .258model …xed e¤ect panel regressions

yBoth regressions include the following variables; year, month, day of the week, holidays,the day before shops are closed, and the day after shops are closed (all of these dummyvariables), and cproxy, representing seasonal ‡uctuations in the consumption.zThe variablelabeled 1-3 indicate price range of the competing brand, i.e., in the case of House (S&B)brand, rivalprangex is a dummy variable which is unity if the price of S&B (House) curryis in the price range x (=1,2,or 3)

Table 7 Non-Linear Regressions on SalesChain # Store # House Brand

type 1 type 2 N bXb

cb

sb

cb

s

1 1 .00921 .102 .0151 .121 1087 45791 2 .0116 .336 .0118 .344 1093 43361 3 .00637 .213 .00670 .156 1101 46071 4 .00878 .146 .00884 .137 1168 37431 5 .0109 .118 .0130 .127 1213 44471 6 .0113 .201 .00942 .185 1139 24901 7 .00825 .195 .00839 .204 1048 34742 8 -.00104££ .533 -.00092££ .546 1139 6153 9 -.0240££ .140 .00525££ .210 1012 20653 10 .00841 .374 .00836 .378 1152 24034 11 .00620 .262 .00656 .298 844 25484 12 .0137 .116 .0150 .149 837 34555 13 1.437 .00144 1.427 .00155 480 24166 14 .00760 .147 .00840 .171 1277 36046 15 .00977 .103 .00959 .113 946 25497 16 -.00961££ .194££ -.0169££ .280££ 802 8077 17 .00153££ .128 .00875££ .157 710 8817 18 .628 601£10¡4££ -.0244££ .071££ 241 728

Table 7 Non-Linear Regressions on Sales (cont’d.)Chain # Store # S&B Brand

type 1 type 2 N bXb

cb

sb

cb

s

1 1 .00518 .141 .00529 .147 954 45791 2 .00271 .142 .00281 .137£ 708 43361 3 .00299 .168 .00297 .173 973 46071 4 .00501 .156 .00506 .154 1021 37431 5 .00899 .0791 .00890 .0748 1085 44471 6 .00632 .180 .00660 .146 817 24901 7 .00372 .0968 .00395 .0863 837 34742 8 .00818 .113£ .00814 .107 201 6153 9 .00293 .209 .00285 .194 562 20653 10 .00429 .298 .00428 .297 825 24034 11 .00979 .305 .00949 .370 837 25484 12 .0204 .118 .0185 .149 883 34555 13 .0539 .00258 .0205 .0813 282 24166 14 .00463 .321 .00549 .227 1352 36046 15 .00569 .125 .00667 .135 1186 25497 16 -.00679££ .0644£ -.0160££ .0308££ 290 8077 17 .00670 .130 .00696 .141 362 8817 18 .00814 .171 .00875 .172 146 728

All the regressions are estimated by non-linear maximum likelihood estimation and theyinclude …rst-order serial correlation corrections. Based upon standard error computed from

heteroscedastic consistent matrix,all the coe¢cients are signi…cant at 1% unless otherwise noted: £not signi…cant at 1% but

signi…cant at 5%, ££not signi…cant at 5%.

Table 8 Ordered Probit Model of Price ChangesHouse Brand S&B Brand

shopping intensityy type bs2 bs1 bs2

b¼ -2.05 -2.06 -2.38 -2.37 -2.29 -2.30 -2.45 -2.46low -0.67 -0.64 -0.69 -0.66 -0.81 -0.76 -0.82 -0.77

low £ b¼ 0.16 0.12 0.17 0.13 0.30 0.25 0.34 0.29shopping intensityez - -0.048 - -0.042 - -0.069 - -0.077storesale -1.04 -1.04 -1.04 -1.04 -1.25 -1.25 -1.25 -1.25storesale + 1 0.95 0.95 0.96 0.95 1.19 1.19 1.19 1.19

R2

.085 .086 .088 .088 .092 .093 .091 .092

# of samples 21958 21872 21958 21872 25918 25820 25918 25820Log Likelihood -14529 -14467 -14505 -14444 -11890 -11830 -11900 -11840

Notes: (*)the RHS is an index variable; 0 for price decrease, 1 for no change, and 1 forprice increase, respectively, from the previous day. All the estimated coe¢cients are correctsigns and signi…cant at 1% signi…cance level. (y) We use the estimated values of bs shown inTable. (z) see the explanations in the main text.

Table 9 Cross Section Regressions on Estimated Coe¢cientss = bs1 s = bs2

s c Q R2

s c Q R2

logfE [pL]g -2.951 -2.057yes -1.856¤¤yes .459 -4.186 -17.627yes -1.676¤¤

yes .480[SUR ] -2.952 -2.055yes -1.856¤¤

yes .632 -4.186 -17.627yes -1.677¤¤yes .647

logfE [ Tk+1

]g -.0230 .0486yes .0416¤¤yes .666 .0412yes -.614 .0400¤¤

yes .653[SUR ] -.0228 .0489yes .0416¤¤

yes .773 .0412yes -.611 .0400¤¤yes .764

logfE [TH

T]g -.793 .0738 -.107¤

yes .769 1.005¤yes -1.829yes -.0814¤

yes .809[SUR] -.792 .0738 -.107¤¤

yes .843 1.000¤yes -1.829yes -.0814¤¤

yes .871

All the regressions above include dummy variables for each chain (1 through 7). [SUR]indicates the results for seeming unrelated regressions in which three equations are

simultaneously estimated incorporating possible covariations in the error terms.yes indicatescorrect signs and ¤¤(¤) indicates 1%(5%) signi…cance level.

t1 1889

128

253

Figure 1 Daily Price of House Brand Curry Paste at Store 1 of Chain 1

t1 1889

136

272

Figure 2 Daily Price of S&B Brand Curry Paste at Store 1 of Chain 1

t1 1889

100

248

Figure 3 Daily Price of House Brand Curry Paste at Store 3 of Chain 7

t1 1889

100

268

Figure 4 Daily Price of S&B Curry Paste at Store 3 of Chain 7

54

Fra

ctio

n

ph128 253

0

.461043

Figure 5 Frequency Distribution of House Brand Curry Paste at Store 1 of Chain 1

Fra

ctio

n

ps136 272

0

.474638

Figure 6 Frequncy Distribution of S&B Brand Curry Paste at Store 1 of Chain1

Fra

ctio

n

ph100 248

0

.859398

Figure 7 Frequency Distribution of House Brand Curry Paste at Store 3 of Chain 7

Fra

ctio

n

ps100 268

0

.518784

Figure 8 Frequency Distribution of S&B Brand Curry Paste at Store 3 of Chain 7

55

phe1

t1 1889

0

.562179

phe1

t1 1889

0

.2579

phe1

t1 1889

.002151

.206074

phe1

t1 1889

.002692

.338126

phe1

t1 1889

.005631

.380012

phe1

t1 1889

0

.361933

phe1

t1 1889

.003685

.295116

Figure 9 Estimated Series of ¼ for House Brand at Stores 1 -7 in Chain 1

56

pse1

t1 1889

.000179

.333287

pse1

t1 1889

.006708

.347625

pse1

t1 1889

.002569

.250643

pse1

t1 1889

.003077

.328486

pse1

t1 1889

.018636

.557619

pse1

t1 1889

.004846

.408184

pse1

t1 1889

.011477

.328536

Figure 10 Estimated Series of ¼ for S&B Brand at Store 1-7 of Chain 1

57