Consumer Basket Information, Supermarket Pricing and Loyalty Card Programs

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1 Consumer Basket Information, Supermarket Pricing and Loyalty Card Programs Nanda Kumar SM 3, School of Management The University of Texas at Dallas P.O. Box Richardson, TX Phone: ( Fax: ( Ram Rao SM 3, School of Management The University of Texas at Dallas P.O. Box Richardson, TX Phone: ( Fax: ( August 5, 3 We want to thank the seminar participants at the Summer Institute in Competitive Strategy (SICS at Berkeley for their valuable suggestions. The usual disclaimer applies.

2 Consumer Basket Information, Supermarket Pricing and Loyalty Card Programs ABSTRACT The central question we address here is: how are supermarket prices affected when competing stores implement loyalty card programs? We study a model of two competing supermarkets. We incorporate the novel feature that consumers are heterogeneous in their shopping basket, and focus on the information that loyalty card programs provide on size and composition of shopping baskets of consumers. Such information can lead to greater competition and so its effect on profits is not obvious. So a second question we address is: does the use of this information lead to higher profits? We find that using the information causes lower equilibrium prices for goods desired by large basket consumers and higher for other goods because large basket consumers are more valuable for stores. Second, use of the information leads to higher profits regardless of competitor s decision on the program. Indeed, if it is not too costly, implementing the program is a dominant strategy for both stores, and profits are maximized if both implement the program. Finally, by identifying and targeting large basket households with rewards, stores can increase their profits further although basket prices are reduced to the most valuable segment of large basket households. KEY WORDS: Loyalty cards, reward programs, supermarkets, competition, pricing, shopping basket, game theory SHORTABSTRACT: Yes, it is profitable to reward the large basket consumer.

3 Consumer Basket Information, Supermarket Pricing and Loyalty Card Programs. Introduction A major innovation that supermarkets have pursued for a little over five years now is the loyalty card. Despite its name, the motivation for the card has always been seen as a way to target individual shoppers with rewards, or more generally special offers, that take into account the preferences and buying patterns of individual households (See, for example, Schlumberger, 998. Consumer activists have been concerned with privacy issues surrounding these cards, as evidenced by the work of Consumers Against Supermarket Privacy Invasion and Numbering (C.A.S.P.I.A.N.. But of equal concern to them appears to be the effect of the loyalty card on prices, in particular posted prices. For example, in Albuquerque, New Mexico, one report had Albertsons Supermarkets increasing the posted price of a basket of 5 items by 6% over a period of three years while a local store without a loyalty card program saw an increase of % over the same period (Gordon,. Of course, the focus on posted price may not reveal all the forces at work. In practice, supermarkets often price specific items such that by using the loyalty card a consumer can obtain a lower price than the posted price on that item. Although this is in the nature of price discrimination, it is also the case that the overwhelming fraction of shoppers avail themselves of the lower price (for example, MPR,. It would seem, therefore, that in practice price discrimination is not a key reason for linking lower prices to the use of loyalty cards by supermarkets. To the extent that most shoppers, if not all, end up paying the discounted price rather than the posted price, the relationship of the loyalty cards to pricing is not obvious. In this paper, we clarify this by first focusing on the fact that the loyalty card can

4 provide information on purchase patterns of consumers and then asking how the use of such information might be used to formulate supermarket pricing. Thus, our focus is not so much on the reason that supermarkets implement loyalty card programs as it is on how the loyalty card programs can be put to good use for pricing decisions. A supermarket wanting to attract customers must provide value by way of discounts on a subset of the shopping basket relative to the customers willingness to pay for the entire basket at least to the extent that the discounts offset the cost of traveling to the store (Lal and Matutes (L- M, 994. In the L-M model and subsequent extensions of it, consumers are assumed to be identical with respect to their shopping baskets. In this paper we develop a model incorporating a novel feature by supposing the existence of shoppers who are heterogeneous in the basket of items they buy, which we term basket heterogeneity. This accords well with reality since it is well known that consumers differ in the baskets that they buy at supermarkets. Basket heterogeneity can be because of differences in the basket size, meaning some consumers have larger baskets than others (Bell and Lattin, 998. This is basket size heterogeneity. Another dimension of basket heterogeneity is on the items in a basket of a certain size. If items in a consumer basket differ, we have basket composition heterogeneity. A consequence of basket composition heterogeneity is that there is no single item in the baskets of all shoppers. And, further, some items appear in the baskets of relatively more shoppers than other items. Under these conditions, the stores problem is complicated and it is not obvious which products in a basket, if any, should be discounted. Loyalty cards can be helpful in identifying the items in the baskets of heterogeneous shoppers, possibly leading to better pricing decisions by the stores. In fact, a consumer can often get the lower price simply by telling the check-out clerk that he forgot to bring the card.

5 Keeping our interests in mind, we ask if a loyalty card program would be profitable solely on the basis of whether it could improve pricing decisions. It would be natural to expect that a monopolist would always find it profitable to gain more information. It is not obvious that this would be the case under competition, since learning about consumers could possibly intensify competition, thus leading to lower profits. Certainly, promotions targeted at pricesensitive consumers often dissipate profits from that segment of consumers. Similarly, it has been observed that price discrimination under competition may not be profitable even though it may be profitable for a monopolist (Chen and Iyer,. An analogous result is obtained in the case of pricing in the presence of a learning curve: while a monopolist benefits from foresight, leading to lower prices, under competition myopic pricing by all firms would be preferable to competition with foresight (Rao and Bass, 985. However, recent research by Villas-Boas (3 demonstrates that even a monopolist could be worse off with additional information on its consumers. Our question in this paper is: what is the effect on profits of competing firms that use information obtained through loyalty card programs? To answer this, we study a market served by two supermarkets, each of which decides whether or not to implement a loyalty card program. We find that even under competition, in general, improved pricing decisions lead to higher profits, if the information on the baskets is available. In particular, when stores can use information from loyalty cards, price of items that are common to large basket shoppers decreases while that of other items increases. This is not inconsistent with the reported increase in posted prices at stores that implement loyalty card programs. The practice of tying discounts to the use of loyalty cards, then, merely reinforces the continued use of the card enabling stores to keep their information current. 3

6 A question of managerial interest is whether competing stores might differ in their decision to implement a loyalty card program. In other words, would one store find the loyalty card a profitable innovation while another operating in the same market would find it unprofitable? We find that, for stores that are symmetric, implementing a loyalty card program is a dominant strategy if the fixed cost of the program is not too high. This is an interesting finding, and may explain why, for example, after initially eschewing loyalty cards, Albertsons decided to follow competitors decision to implement the program (Sierra Times,. Indeed, after engaging in a No Card Needed promotional strategy, Albertsons claimed in its quarterly report in December : the Company bolstered its market leadership in Dallas/Ft.Worth by introducing a new loyalty card. Presumably this was in response to local conditions of competition and positioning. 3 The rest of the paper is organized as follows. In section, we present our model of retail competition and the corresponding game. In section 3, we analyze the Nash equilibrium of the game and establish that, even under competition, loyalty programs in our model are rendered profitable by the effect they have on prices of basket items. We also derive the implications for prices of individual goods and consumer surplus. In section 4, we discuss the effects of relaxing some of the assumptions of our model, and also issues pertaining to loyalty cards that allow pricing based on basket information and consumer identity. Section 5 presents our conclusions and direction for future research. The major competitors Kroger and Tom Thumb (part of Safeway had loyalty cards at the time. 3 The company has not introduced a loyalty card program in California, for example, as of January, 3. 4

7 . Model of Loyalty Cards We analyze retail competition by supposing that consumers can choose to shop at one of two supermarkets to buy one or more goods they are in the market for, and the two retailers compete to serve five segments of consumers. 4 We next elaborate on this by formally describing our model assumptions with respect to consumers, retailers and the game structure of retail competition.. Consumers ASSUMPTION : In our model there are five types, or segments, of consumers, indexed by t. Types, and 3 are in the market for one unit of good, and 3 respectively. Types 4 and 5 are in the market for one unit of two of the three goods. It is well known that consumers differ in the baskets that they buy at supermarkets. One basis of difference is the basket size, meaning some consumers have larger baskets than others. We capture this by including some consumers that are types 4 and 5, with large baskets, and others that are not. Another basis of consumer heterogeneity in shopping baskets is the items in the basket, meaning, for example, some baskets contain baby food while others contain hair coloring. We capture this by specifying multiple consumer types given each basket size. Finally, we should note that in practice consumer baskets contain more than one or two items, and so the goods in our model are best interpreted as sub-baskets. Also, in practice consumer heterogeneity in baskets is likely to be rather large. Through this assumption we capture heterogeneity among both small-basket and large-basket consumers. Within this general model of basket heterogeneity, we will also examine special cases: one in which there is only basket size heterogeneity, and another with only basket composition heterogeneity. 4 We will use the terms retailers, stores and supermarkets interchangeably throughout the paper. 5

8 ASSUMPTION : Consumers in our model are uniformly distributed along a straight line AB of unit length. Further, consumer location along this line is independent of the type of consumer. The total size of the market is normalized to unit. A consumer, denoted by x, x, is located at a distance x from A and (-x form B. Each consumer incurs a transportation cost of c/ per unit distance. In addition to basket heterogeneity captured by assumption, note that assumption implies that consumers are also heterogeneous by virtue of their location. Since consumers incur transportation cost at the rate of c/ per unit distance, depending on the location of stores along the line AB, consumers would differ in the convenience (inversely related to the transportation cost that each store affords them. The assumption that all consumers have the same unit transportation cost cuts down on an additional source of consumer heterogeneity that would make the model complicated without yielding any additional benefit or affecting our results. Later, we let a fraction of consumers have zero costs of transportation, and briefly examine the consequence of that. The uniform distribution of consumers along the line is a commonly made assumption that keeps the problem mathematically tractable. Note that the heterogeneity in basket size and composition is assumed to be independent of location. Thus, each type of consumer is uniformly distributed along the line. ASSUMPTION 3: There is a fraction α of each of types, and 3 consumers, α /3, and a fraction, (-3α/, of types 4 and 5 consumers. Assumption 3 says that the three small-basket type segments are equal in size, and the two large-basket type segments are equal in size. This can be relaxed, and we do so in section 4., but for now we impose this restriction to keep the mathematical developments simple. 6

9 ASSUMPTION 4: All consumers are willing to pay $R per unit of any of the three goods. In practice, consumers differ in their willingness to pay for any good. Thus, our assumption must be seen as a way to keep the analysis simple. As we will see later, our results do not depend on the exact value of R relative to other parameters. It is easy to relax the assumption that all goods have the same reservation price. Relaxing this assumption will turn our attention to the relative profitability of goods, which is not the focus of our investigation. ASSUMPTION 5: Consumers know the prices of all goods before deciding on a store to shop at. Prior work by L-M and derivatives of it have established that consumers can make sensible shopping decisions based on advertised prices of a subset of the goods that they are in the market for by forming rational expectations of the prices of unadvertised goods. Given our focus on the profitability of loyalty programs rather than on advertising and randomized promotions, we have chosen to build on known results, and assume that consumers know the prices of a subset of the goods they are in the market for.. Retailers ASSUMPTION 6: There are two retailers (supermarkets, who compete with each other. Each retailer carries three goods that are neither complements nor substitutes, indexed by j, j =, and 3. Given our interest in supermarket pricing strategies, we assume that both retailers in our model carry multiple goods. For supermarkets the assumption that both stores carry the same goods that are neither complements nor substitutes is reasonable because it reflects the reality that supermarkets carry the same brands of goods such as shampoo and cake mix. The restriction to three goods simplifies the exposition considerably without affecting our results, and it is best to interpret the three goods as three baskets. 7

10 ASSUMPTION 7: The two stores are located at either end of the straight line AB, with store A at the left end and store B at the right end. We will index the stores by i, i = {A, B}. This assumption makes the two stores differentiated based on their location. In light of Assumption, consumers located closer to one store find it more convenient relative to the other store. In particular, a consumer x incurs a cost c x for a round-trip visit to store A and c (-x for a round-trip visit to store B. ASSUMPTION 8: Retailers know the size of all the segments. In other words, they know α. However, retailers do not know the composition of the baskets of large basket consumers. In light of assumption 3 on the equality of segment sizes within each basket size, knowing α is sufficient to learn the sizes of all segments. Turning to composition of the baskets, this has relevance only for large basket segments. While consumers know the composition of their basket retailers in our model do not know it. Note that what retailers know about types 4 and 5 consumers is that they are in the market for two goods. In other words, it is common knowledge among consumers and retailers that types 4 and 5 are in the market for two goods. The identity of the goods in their basket is private information held by types 4 and 5 consumers. Thus, retailers are ignorant of basket composition. This ignorance could be overcome by obtaining consumer information, which is elaborated on in the next assumption. ASSUMPTION 9: The two retailers can each choose to learn the basket composition by obtaining and analyzing appropriate information at a fixed cost of F, and a variable cost of v per customer. Without loss of generality we set v to be zero. In practice, stores may obtain basket information in a variety of ways. For example, stores could analyze aggregate sales of each good to infer the basket composition. In our model, given assumption and 3, a store could indeed do this. However if assumption 3, for example, 8

11 does not hold implying that stores do not know segment sizes, they would need information on more than just the aggregate sales. In such a situation, we can imagine stores analyzing cash register receipts to learn both segment size and basket composition. In practice, unlike in our model, consumer purchases vary over shopping trips and it may be necessary to aggregate information across shopping trips to learn the true nature of basket composition. This can be accomplished if purchases of individual households can be tracked and analyzed. Thus, although analyzing cash register receipts is sufficient for learning the parameters of our model, a more realistic interpretation would suggest the need for a sophisticated tracking mechanism. Loyalty, or reward, cards can be one such mechanism. Therefore, in this paper we view a store s decision to learn basket composition as one of implementing a loyalty card program. Let l i = (or, be an indicator variable denoting whether firm i implements (or does not implement a loyalty card program. The program collects information on each household that tells the retailer what items each household buys. It is important to note that in practice stores may have many reasons to implement a loyalty card program. We are interested in studying how the information that these programs make available can be used for pricing under competition. Specifically, a retailer that institutes a loyalty card program can learn which two goods types 4 and 5 consumers buy. Obviously such programs geared to generating information are expensive to set up and maintain. On the other hand, most of the attendant costs are fixed relative to the size of the customer base that is part of the program, and so the assumption that v is zero is not serious. Assumption 9 leaves open what a store does with the information. Thus, a store may undertake specific promotional campaigns based on the information that the loyalty program yields, by incurring additional costs. The fixed cost F does not take into account these additional 9

12 costs, if any. Alternatively, as we will see in our analysis, a store could use the information to make more profitable pricing decisions. Following assumption, types 4 and 5 consumers have two goods in their basket, and the baskets differ. Note that one of the three goods would be present in both baskets. We will label this as the common good. We label any item other than the common good as a unique good. Denote s, s =,, 3 respectively to be the state corresponding to good s being the common good. Denote µ i, µ i and µ i 3 to be retailer i s probability assessment that s =, and 3 respectively, and µ i to be the corresponding (row vector, with µ i +µ i +µ i 3 =. Denote θ s to be this probability vector corresponding to the true state of s. If a retailer institutes the loyalty program, then he can discover the true state of s. 5 Thus, (µ i l i = = θ s ( where θ = [ ], θ = [ ] and θ 3 = [ ]. It is necessary to specify µ i corresponding to l i =. We do this in the next assumption. i ASSUMPTION : A retailer who does not institute a loyalty card program has a diffuse prior µ on s. In other words, µ i = (µ i l i = = [/3 /3 /3]. Furthermore, it is known to both retailers that a retailer with a loyalty card program knows s. This is common knowledge. Since there is no other source of information on consumers in our model, it makes sense to assume that stores have a diffuse prior on s in the absence of a loyalty card program. 6 The assumption of common knowledge follows simply from the recognition by each retailer that the other has no potential source of information on consumers other than a loyalty card program. Thus, Assumption implies that depending on whether or not each store implements a loyalty card program, we have the probability assessments as in Table. 5 The timing of the availability of information is spelt out later in Assumption.

13 Stores decisions (l A, l B µ A µ B (, [/3 /3 /3] [/3 /3 /3] (, [/3 /3 /3] θ s (, θ s [/3 /3 /3] (, θ s θ s Table : Probability Assessments Conditional on Stores Decision on Loyalty Cards ASSUMPTION : The store competition is modeled as a two-period game. In the first period each store decides whether or not to implement a loyalty card program. In addition each chooses prices of the three goods using its prior information. In the first period both stores make pricing decisions based on µ. In the second period, each store chooses prices of the three goods based on the probability assessments in table. We assume that firms discount future profits by a discount factor δ. Assumption implies that in the first period stores pricing decisions are based on the probability assessments corresponding to (,. This captures the idea that there is a time lapse between implementing a loyalty card program and the availability of information on basket composition. In the second period, a store that implemented a loyalty card program in the first period has the basket composition information. By the same token, if it did not implement a loyalty card program in period, its probability assessment continues to be µ. We may wonder if a store can learn the basket composition by observing its rival s pricing. Such a possibility has no force in our model since it is a two-period game and first period prices can contain no information. Now the question arises how our model would apply in practice where firms compete for more than two periods. However, in this case there would also be other forces at 6 The assumption of a diffuse prior is not crucial to the development of our model.

14 work. For one, supermarkets carry thousands of goods with multiple brands. Moreover, shopping baskets would contain many goods, and several of them may be common goods. Further, heterogeneity of basket composition among large basket shoppers would also be considerable. Thus, it would be reasonable to assume that a store that does not have a loyalty card program would remain substantially ignorant of basket composition if it relied only on its rival s prices. In this way our assumption can be seen as a reasonable model of reality. 3. Analysis of Loyalty Card Programs Let is i p jn and p jn denote the price of item j {,,3} in state s, chosen by retailer i {A,B} i i in period n, when l = and l = respectively. Note that the price would depend on l i in the second period because by implementing a loyalty card program in the first period a store can get is information on the common good for use in the second period pricing decisions. Let P i i denote the column vector of the prices chosen by store i in period when l = and l = and P respectively. However, since stores do not have this information in the first period, their prices in the first period do not depend on l i. Therefore, first period prices are independent of the true i state, s. We let P denote the column vector of prices chosen by store i in period, the superscript emphasizing the fact that in period state s is not known. 3. Characterization of Retail Demand and Profits Each consumer chooses which store to buy from based on his basket, the stores prices and his location x. These consumer choices determine the share of each type of consumer that the two stores attract. We first compute demand of the small basket consumers at store i as a function of its prices and those of its competitor, store k, and denote it by i ( i, k t The consumer located at x, purchases good j from store A iff: g P P, t=,, 3. i

15 B { j } A R cx p max R c( x p, j The LHS of the above inequality denotes the net utility from buying good j at store A, and the RHS that from buying at store B, or choosing to not buy at all, whichever is higher. Similarly, if the net utility from buying at store B is higher then the consumer would buy from store B. Given B A p j pj + c this store choice rule a consumer in segment t = {,, 3}, located at is indifferent to c buying at either store jt, {,,3}. 7 A A B The demand, gt ( P, P B A ( p j pj c α + =. c We now turn to type 4 and 5 consumers. Recall that the common good in the basket of type 4 and 5 consumers is one of goods, or 3 with the corresponding states being s =, or 3. For example when s =, type 4 consumers buy {, } and type 5 consumers buy {, 3}, or vice versa. 8 We next compute demand of the large basket consumers in state s at store i as a function of its prices and those of its competitor, store k, and denote it by i ( i, k ts h P P, t=4, 5. Suppose s=. Then, we can obtain h by solving the problem of store choice of consumers in segments 4 and 5. A consumer in segment 4 located at x, will purchase the basket comprising goods and from store A iff: { ( } R cx p p max R c x p p, R x p p, R cx p p, A A B B A B B A The LHS of the above inequality denotes the net utility of a type 4 consumer located at x from buying the basket at store A, and the RHS that from buying the basket at store B, or buying one good at store A and the other at store B or choosing to not buy at all, whichever is highest. Similarly, if the net utility from buying at store B is higher then the consumer would buy form 7 We are implicitly assuming that shopping at either store will yield positive net utility to the marginal consumers. This turns out to be the case in equilibrium. 3

16 B B A A p + p p p + c store B. Given this choice rule a type 4 consumer located at c is indifferent to buying the basket at either store. 9 Hence, the demand from consumers in segment 4 in state, at store A: h4 ( P, P B B A A ( 3α ( + + p p p p c A A B =. Similar analysis of type 5 4c consumers store choice problem yields the demand from consumers in segment 5 in state, at store A: h5 ( P, P B B A A ( 3α ( p p p p c A A B =. Using the same procedure, we can 4c i i k compute demand from these two segments in the other states, ts ( h P, P, t = {4,5}, s= {,3}. Period Demand and Pricing Strategies If retailer i implements a loyalty card program in period, the identity of the common good in the basket of type 4 and 5 consumers (or the true state, s is known to i at the beginning of period. In that case i ( i, k t and i ( i, k ts g P P represents the demand of retailer i in segment t, t = {,, 3} h P P represents its demand in segment t, t = {4, 5} when the true state is s. However, when store i decides not to implement a loyalty card program in period it remains ignorant about the common good in the basket of type 4 and 5 consumers. In this case, we derive the expected demand. First, consider the demand in each of the segments through 3. Denote m i t to be the expected demand at store i in segment t, t =,, 3. The expected demand is a function of prices charged by store i and the prices of its competitor, store k, the latter depending on l k. Then, for any choice P i by store i we have 8 Notice that the case when type 4 consumers buy {, 3} and type 5 consumers buy {, } results in the same state. This would not be the case if the size of type 4 and 5 segments were different. The symmetry in size of these two segments helps limit the state space. 9 This would be true if c is sufficiently large. 4

17 3 i i i k k µ sgt ( P, P, if l = i i k s= mt ( P ; l =, t = {,,3}, i, k { A, B} 3 i i i ks k µ sgt ( P, P, if l = s= ( For the expected demand in segments 4 and 5, let i ( i ; k ts M P l denote the demand in segment t, t = {4, 5} that retailer i obtains given the competitor s first period decision l k and state s. Then, we have ( ( i i k k,, if i i k hts P P l = M ts ( P ; l =, t = {4,5}, s= {,,3}, i, k { A, B} i i ks k hts P, P, if l = (3 Consider two cases, in turn. First, when l i i = the expected profits of retailer i in period, π, are given by π 3 ( ; = t ( ; P l m P l p i i k i i k i t t= ( M ( P ; l ( p p M ( P ; l ( p p M ( P ; l ( p p M ( P ; l ( p p M ( P ; l ( p + p + M ( P ; l ( p + p i i i k i i i i k i i µ i i i k i i i i k i i + µ ( i i i k i i i i k i i µ 3 ( (4 In (4 the first term corresponds to profits from small-basket shoppers in segments, and 3. The profit in each segment is the price of the good that segment buys times the expected demand in that segment. The latter, as specified in (, depends on retailer i s price, the price of that good chosen by the competitor k given l k and i s assessment µ i. The second term is the profit from segments 4 and 5 given by the sum of the prices of the two goods that consumers in each segment buys times retailer i s expected demand in each state, s, weighted by firm i s beliefs. The demand, as specified in (3, depends on retailer i s prices, the prices chosen by the 5

18 competitor k given l k, and i s assessment µ i. Second, when l i =, retailer i s profits in period, π i s, are given by: π 3 ( ; (, P l = g P P p + i is k i is k is s t t t= ( ( ( ( ( ( ( ( ( 3 ( 3 3 ( 3 ( 3 3 i i k i i i i k i i M4 P ; l p + p + M5 P ; l p + p3, if s= i i k i i i i k i i M4 P ; l p + p + M5 P ; l p + p3, if s = i i k i i i i k i i M43 P ; l p + p3 + M53 P ; l p + p3, if s= 3 (5 As in (4, the first term in (5 corresponds to profits from segments -3 and the second term to profits from segments 4 and 5 when retailer i knows the state to be s. Note that in contrast to (4 we do not take expectations with respect to state in (5 because the state is known. Furthermore, the prices in (5 are state contingent. Corresponding to each outcome of period, the retailers must determine their prices that maximize their profits in (4 or (5 given their probability assessments. There are four price vectors for each retailer that we must characterize. For firm i, if l i =, the equilibrium choice of prices in period, * i P is given by: ( * P i = arg max π i P i ; l k, i, k = { A, B} (6 For firm i, if l i =, the equilibrium choice of i, P is* is given by: ( * is i i k P = arg max π s P; l, i, k = { A, B}, s= {,,3} (7 The Nash equilibrium prices in (6 and (7 maximize the expected profits of each store. Period Demand and Pricing Strategies Since stores are ignorant about the common good in period, the pricing decisions of both stores in period are independent of the state s, and so for any P i = A B the expected i, {, } demand in segments t = {,, 3}: 6

19 ( i i k g P, P, i, k = { A, B} (8 t and the demands in segments 4 and 5 in state s: ( i i k h P, P, t = {4,5}, i, k = { A, B}, s= {,,3} (9 ts i Given, (8 and (9 period profits of store i, π : π 3 ( P ; l = gt ( P, P pt t= i i i k i i i i k i i µ h4( P, P ( p p h5( P, P ( p p3 i i i k i i i i k i i µ h4( P ; P ( p p h5( P ; P ( p p3 i i i k i i i i k i i µ h43( P ; P ( p p3 h53( P ; P ( p p3 i i k i i k i ( ( ( i lf ( Note that first period gross profits do not depend on the decision to implement a loyalty card program. But the net profits do because of the fixed cost F. Thus, for firm i, the equilibrium choice of prices in period, * i P is obtained by solving: ( i * π i i k P = arg max[ P ; l ] ( Given these pricing strategies, retailers must determine whether or not to implement a loyalty card program in period. Denote i * ( l A, l B given that the outcome in stage of period is (l A, l B, where: Π to be the equilibrium profits of retailer i ( l, l i* A B Π = ( * δπ ( * ( * δπ ( * ( * 3 s ( * i i k i i k i k π P ; l + P ; l, l =, l = i i k i i k i k π P ; l + P ; l, l =, l = i i k i i is k i k π P ; l + δ µ π P ; l, l =, l = s= 3 i i* k i i is* k i k π ( P ; l + δ µ πs ( P ; l, l, l = = s= ( Note that in ( the second term in the first two entries differ from the second term in the third and fourth entries. This is because at the time when the loyalty card decision is made firm i 7

20 does not know which state will be realized in period. Consequently, we need to weight the second period profit in a given state with the probability that that state is realized to compute the ex-ante profit in period. The first stage decision of period is then a Nash equilibrium to the loyalty card game displayed in Table. Retailer B, l B Retailer A, l A A* B* A* B* Π (,, Π (, Π (,, Π (, Π A* (,, Π B* (, Π A* (,, Π B* (, Table : Loyalty Card Game Pay-Offs In table the entries represent expected profits of the retailers corresponding to their choice of l. We next turn to exploring the properties of this equilibrium. In particular, we investigate the profitability of loyalty card programs. 3. Profitability of Loyalty Card Programs In this section we demonstrate that loyalty card programs can help stores make better pricing decisions, and thereby increase profits even when both implement the programs. This highlights the informational role of loyalty card programs and the fact that pricing based on such information does not increase competition overall. We proceed by first characterizing the equilibrium prices in period depending on first period outcomes. The period decisions on A B loyalty cards induce four sub-games in period corresponding to l = l =, denoted (, ; l A B A B A B = l =, denoted (, ; l =, l =, denoted (, ; and l =, l =, denoted (,. These four sub-games correspond respectively to the cases in which loyalty card programs are 8

21 implemented by neither store, both stores, by store B but not store A, and finally by store A but not store B. Because firms in our model are symmetric it is sufficient to analyze only the first three sub-games. Equilibrium prices and profits for these three sub-games are characterized in Lemmas -3. We then work backwards to characterize equilibrium strategies of stores in both the pricing and loyalty card game in period. This is done in proposition. 3.. Neither Retailer Has a Loyalty Card Program First consider the case in which neither store implements a loyalty card program in the first period. The equilibrium pricing strategies in period and the corresponding retail profits in this sub-game are summarized in the following Lemma. Lemma. When neither store implements a loyalty card program (l A = l B = then i* j ( 3α ( 4 9α c p =, j = {,, 3}, i = { A, B} * and, π ( ( 3α ( α P ; l =, i, k = { A, B}, i k 4 9 i i k c Proof. See Appendix Note that in this sub-game stores do not know the state s (identity of the common good, and so they set prices of all three items to be the same. To get an insight into the equilibrium pricing strategies, first consider the case when there are no large basket shoppers (α=/3. The equilibrium prices of each good turn out to be c. This is identical to the price of the shopping basket in L-M. Since there is no overlap in consumer baskets if α=/3, we find that each good is priced at c. In other words, there is complete segmentation of the market. Next, if the size of both large and small basket shopper segments are greater than zero (<α</3, the equilibrium price Proofs all lemmas and propositions are in the appendix. 9

22 of each good, and therefore of each small basket, is less than c. However, note that in this case the price of the large basket is p * i j c( α ( 4 9α 3 = > c. Thus, compared to the basket price in Lal and Matutes small basket shoppers pay a lower price while large basket shoppers pay a higher price. In other words, the large basket shoppers exert a positive externality on small basket shoppers or said differently small basket shoppers exert a negative externality on large basket shoppers. The intuition here is that stores would like to compete aggressively for large basket shoppers by offering lower prices on the items in their basket. Since s is unknown, retailers set the price of all three goods to be equal. This leads to giving small basket shoppers higher consumer surplus than they would get in the absence of large basket shoppers. This is because the market is not completely segmented. Finally, if there are no small basket shoppers (α= we find that the equilibrium price of each good is c/. Although in this case, the price of the individual good is less than the price when α=/3, the basket price is c. Presence of basket heterogeneity causes equilibrium basket prices to be different from c. We will elaborate on the role of basket heterogeneity in section Both Retailers Have Loyalty Card Programs Now consider the case when both stores implement a loyalty card program, and learn the items desired by large basket consumers. In contrast to the earlier sub-game, stores can now price so as to compete aggressively for these consumers and reduce subsidy for small basket consumers. The following Lemma summarizes the equilibrium pricing strategies and retail profits in this sub-game.

23 Lemma. When both stores implement a loyalty card program (l A = l B = then is* j c( α ( 3 7α p =, j = s, i= { A, B} is* j c( α ( 3 7α p =, j s, i= { A, B} i is* k and, π s( ( α( α 3 ( 7α 3c P ; l =, i k, i, k = { A, B}, s= {,,3} We can obtain several interesting insights by comparing the prices and profits in this sub-game (, with that in the sub-game (,. First, in any state s, for the common good j, j=s, we have is* i* j j For unique goods, j s, we find ( α + α c p p = <, α / 3, i= { A, B} is* i* j j ( 3 7α( 4 9α ( α + 5α c p p = >, α / 3, i= { A, B} ( 3 7α( 4 9α These two inequalities imply that when both stores have loyalty cards they charge a lower price for the common good in the large basket relative to the case when neither has a loyalty card and they charge a higher price for the unique goods. Further, the basket price is also lower when both stores have loyalty cards. Thus, information on the large basket shoppers has resulted, as expected, in greater competition for them. In turn, they and the small basket shoppers who buy the common good obtain lower prices and higher consumer surplus. In contrast, consumers who buy the unique good face higher prices and get lower consumer surplus. In this way, we can see is* is* i* i* cα( 3α s j s j ( 3 7α( 4 9α p + p ( p + p = <, j s, α /3, i= { A, B}

24 that by offering loyalty cards both stores are able to compete effectively, and intensely, for the large basket shoppers and reduce the subsidy to certain small basket shoppers. The next interesting question is the impact of the information generated by the loyalty card programs on profits. Comparing the equilibrium profits in the two sub-games, we find that * * ( ( i is k i i k c α( 3α ( α( α π P ; l = π P ; l = = >, α /3, i = { A, B} We can see that period profits are higher if both stores implement a loyalty card program. This happens even when more consumers (fraction -α ½ if α /4 benefit from lower prices and fewer consumers (fraction α / face higher prices in the sub-game (,. It is natural to ask if this is due to the assumed symmetry in the size of the three small basket segments. It turns out that lack of symmetry in segment sizes does not overturn this result, as we will see in section Only one Retailer Has a Loyalty Card Program If only one store implemented a loyalty card program would that lead to higher profits to either or both stores? A related question is: should a store implement a loyalty card program if its competitor does so? To answer these questions, assume that retailer A does not implement a loyalty card program but retailer B does. In this case, B will know the identity of the common good while A will not, and will therefore believe that items -3 are equally likely to be the common good. Consequently, it will set equal prices for all three items. Store B, on the other hand, will set prices depending on the state. The following Lemma summarizes the equilibrium pricing strategies and retail profits in this sub-game.

25 Lemma 3. When one store implements a loyalty card program and the other does not, (l B = and l A = then p * A j ( 3α ( 4 9α c =, j = {,, 3} * Bs j 5c 3cα( 6 5α ( 3 7α( 4 9α p =, j = s c cα( α ( α( α * Bs 4 57 p j =, j s π * ( P ; l A A B c = B Bs* A and, π s( P ; l ( 3α ( α 4 9 ( α( α( α 83 ( 7α( 4 9α 3c = As in sub-game (,, retailer A, who does not have a loyalty card program charges the same price for all items. In contrast, retailer B, who implements a loyalty card program charges a lower price than A for the common good but a higher price for the unique goods. Next, what is the effect on profits to retailer B who implements a loyalty card program? As might be expected, its profits are higher in this case than in sub-game (, but, interestingly, its profits are lower than what it would obtain in sub-game (,. This is because its competitor s prices of the unique goods are too low. As we saw in lemma, the higher profits in the sub-game (, resulted from the higher prices on unique goods. Since prices are strategic complements in this model, store B which knows the identity of the common good can raise its price on the unique goods further if its competitor A also had the same information. 3

26 3..4 The Loyalty Card Decision In period, stores decide on prices and on implementing the loyalty card program, taking into account period (sub-game equilibria characterized in lemmas -3. We now solve for the sub-game perfect equilibrium of this two-period game. First, note that in period, stores do not know the state s (identity of the common good, and so their equilibrium pricing strategies, and resulting gross profits in period, are identical to those in lemma. Net profits, however, will depend on the cost F and on the store s decision on loyalty cards. We now invoke the results from lemmas -3 to characterize the equilibrium outcome in the loyalty card game. ( ( ( * δ c 3α α Proposition. Assume F < F =. Then implementing a loyalty card program 83 7α 4 9α is a dominant strategy, and so in equilibrium both stores implement loyalty card programs. Proposition says that regardless of what its competitor does, it is best for a store to implement a loyalty card program if the cost F is not too large. Note that the threshold value of F increases with the discount factor δ, implying that ceteris paribus less myopic firms are more likely to implement a loyalty card program. Also, note that if there are no large basket shoppers (α=/3, F*=, and similarly if there are no small basket shoppers (α=, once again F*=. Thus, loyalty card programs can increase profits only if there is basket size heterogeneity. The key intuition from our analysis is that by implementing a loyalty card program retailers have information that enables them to price more effectively in the presence of basket heterogeneity. In particular, the loyalty card program allows retailers to compete for large basket shoppers without subsidizing the small basket shoppers too much. Interestingly, a retailer s profits from a loyalty card program do not decrease as a result of increased competition if its rival also offers a loyalty card program. In fact, profits increase if the rival also implements a loyalty card program. 4

27 3.3 Discussion of Results Asymmetry in F and Zero-cost Consumers: Case of Cherry Picking We have assumed that firms are symmetric in their costs. Suppose the cost of implementing loyalty card programs is different for the two stores. What might be some of the consequences of this? For the sake of exposition, suppose that, F A =, and F B =. Then, the A B equilibrium to the loyalty card game consists of l = andl =. Suppose also that there is a very small segment (a fraction ε of consumers, in the large basket segment, that have zero transportation costs. We know, from lemma 3, that in the sub-game (,, for sufficiently small ε, the common good would be lower priced at store B while the unique goods would be lower priced at store A. This would cause cherry picking by the zero-cost consumers that would buy the common good at store B and the unique good in their basket at store A. Such cherry picking occurs in our model only when one store implements a loyalty card program. Were both stores to implement a loyalty card program, there would be no variation in prices of goods across stores and cherry picking would disappear. The idea that loyalty card programs, under some conditions, could reduce cherry picking is also present in Bell and Lal (3. While they find that cherry picking could be eliminated even if one store were to implement the loyalty card program, we find that it occurs if only one store implements such a program but disappears if both stores implement it. 3 Role of Basket Heterogeneity We can understand the role of basket heterogeneity better by examining special cases of our model. The first is the simplest in which there is no basket size heterogeneity but only This mathematical simplification can be replaced by assuming F A to be sufficiently large and F B to be sufficiently small. 5

28 basket composition heterogeneity. So, the market consists of only small basket shoppers or only large basket shoppers. Suppose there are only small basket shoppers. Then the equilibrium price for all items, following L-M is c. And, this does not require the use of information from loyalty card programs. Similarly, if there are only large basket shoppers, the equilibrium price of each item is c/, so that both baskets are priced at c. Once again, information on basket composition is not useful. Thus, we see that information on basket composition is useful only if there is basket size heterogeneity. Next, we can ask if basket size heterogeneity is sufficient to make the information useful, or is it also necessary to have basket composition heterogeneity given (within a basket size? Now consider the case in which there is no basket composition heterogeneity within a basket size. In other words, there is a segment of small basket shoppers all of whom desire the same product, and a segment of large basket shoppers all of whom desire the same basket. In this case, the good purchased by small basket shoppers can be either common or unique. If it is the unique good, its equilibrium price is c, while that of the other two goods is c/. If it is the common good, its equilibrium price is c, and the other two goods are priced at. Thus, we see that the information on basket composition affects prices, and it can be shown that it also increases profits. Note that in contrast to our model with both basket and composition heterogeneity, in this case there is no subsidy to small basket shoppers because of the presence of large basket shoppers. Indeed, basket information leads to complete segmentation of the consumer types. In fact, in the case in which there is composition heterogeneity within the large, but not the small, basket segment, basket composition information once again leads to complete segmentation of the market. On the other hand, if composition heterogeneity exists within the 3 In comparing our results to theirs it is necessary to keep in mind that cherry picking occurs in their model under exogenously specified prices. In our model cherry picking may occur even when prices are endogenous. 6

29 small, but not the large, basket segment, basket composition information is not sufficient to completely segment the market. In reality, both large and small basket shoppers exhibit composition heterogeneity, making it difficult to completely segment the market based on composition information. A desirable model should have this feature of stores inability to completely segment the market even with loyalty card information. A parsimonious model of heterogeneity that maintains this desirable feature is one in which there is composition heterogeneity in the small basket segment but not in the large basket segment. In the next section we examine extensions of our model with this kind of heterogeneity. 4. Model Extensions In our model we have two assumptions that could be thought to drive our results. In particular, we have assumed that the size of small basket shopper segments are equal, a certain kind of symmetry. In reality, this need not be the case and the segments that buy the unique goods could be large or small relative to the segments that buy the common good. Another assumption that we have made is that firms know the size of the segments but not the composition of the baskets. In practice, stores may be better informed about basket composition than segment sizes. We now relax these assumptions, in turn so that we can speak better to the possibility of more profitable pricing from the use of information from loyalty card programs. Finally, in this section, we ask if household level information on consumers, that could be used for targeting rewards, would help increase profits further. As mentioned earlier, all three extensions use a parsimonious version of our base model. Specifically, we examine extensions of the case with three small basket segments and one large basket segment (say, type 4. 7

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