Clipping Coupons: Redemption of Offers with Forward-Looking Consumers

Transcription

1 Clipping Coupons: Redemption of Offers with Forward-Looking Consumers Kissan Joseph Oksana Loginova Marh 6, 2019 Abstrat Consumer redemption behavior pertaining to oupons, gift ertifiates, produt sampling, rebates, and the like, has been the fous of muh sholarly inquiry and the extant literature has doumented two noteworthy empirial regularities a bump in redemptions lose to offer expiry and greater redemption with shorter redemption windows. In the extant work, these phenomena have been explained by invoking myopi onsumers. Against this bakdrop, we ask a simple question: an these phenomena survive if we assume rational, forward-looking onsumers? Aordingly, we develop a model onsisting exlusively of forward-looking onsumers and inorporate two onstruts highlighted in the literature forgetting and stohasti redemption osts. We derive onsumers period-by-period redemption rule and subsequently illustrate the emergene of the two aforementioned empirial regularities. Keywords: onsumer redemption behavior, forward-looking onsumers, forgetting, stohasti osts JEL odes: D11, D81, D91 Shool of Business, University of Kansas, 2182 Capitol Federal Hall, Lawrene, KS 66045, USA; kjoseph@ku.edu; Phone: Department of Eonomis, University of Missouri, 118 Professional Bldg, Columbia, MO 65211, USA; loginovao@missouri.edu; Phone: (orresponding author). 1

2 1 Introdution Consumers often find themselves in a position where they an redeem marketing offers suh as oupons, gift ertifiates, invitations to partake in produt sampling, rebates, et. Not surprisingly, onsumer redemption behavior has reeived a fair amount of researh attention and the literature has doumented two empirial regularities a bump lose to offer expiry and greater redemption with shorter redemption windows. Speifially, with respet to the bump, Inman and MAlister (1994) inorporate regret theory to examine redemption behavior as the oupon approahes expiration. A key premise of their researh is that onsumers are myopi in that they do not antiipate or respond to the regret until they experiene it. In their empirial work, Inman and MAlister (1994) analyze oupon redemption patterns for various brands in the spaghetti saue ategory and find that they do indeed exhibit a bump lose to expiry. Similarly, Groupon redemptions also show a bump lose to expiry (Gupta, Weaver and Rood 2012). With respet to redemption windows, Shu and Gneezy (2010) hypothesize and empirially find that redemption rates are higher when onsumers are assigned to shorter redemption windows. In their oneptualization, Shu and Gneezy (2010) build on resoure slak theory (Zauberman and Lynh 2005) whih suggests greater disounting of time investments relative to money investments. Suh systemati underweighting of effort has also been disussed in Akerlof (1991). Another oneptualization utilized by Shu and Gneezy (2010) is temporal onstrual theory (Trope and Liberman 2003) whih examines how non-monetary osts and benefits are treated in the short run and in the long run. This theory suggests that individuals tend to fous on the desirability aspets of a task in the long run, but then swith to a fous on the feasibility aspets of the task as it atually approahes. Again, it is the onsumer s myopi pereption of his/her behavior in distant periods that leads to the empirial regularity of greater redemption assoiated with shorter redemption windows. While these are very interesting explanations, they nevertheless beg the question: how an suh phenomena persist over time as onsumers ome to learn about their myopia? Another way to ask the same question is whether suh phenomena will ever emerge with rational, forwardlooking onsumers. Aordingly, our primary ontribution to the literature is to develop a model that demonstrates how the two aforementioned empirial regularities an arise even in a world onsisting exlusively of forward-looking onsumers. In spirit, our work is losest to that of Gilpatri (2009) who examines the impat of the distribution of forward-looking onsumers on rebate redemptions. Taken together, these two works deepen our understanding of onsumer redemption behavior. In our model, we inlude two onstruts identified in the literature: forgetting and stohasti redemption osts. The notion of forgetting has a rih history in many ommuniation models 2

3 where onsumers forget messages delivered to them. (See, for example, Keller (1987), who highlights the endemi problem of onsumers memory performane in the ontext of advertising, and Tellis (1998), who disusses the notion of less than perfet arry-over for advertising investments.) We also believe that stohasti redemption osts are an integral aspet of onsumer behavior, an idea highlighted by O Donoghue and Rabin (1999). (See also Chen, Moorthy and Zhang (2005), who utilize a stohasti speifiation for onsumers marginal utility for money.) 2 Model Setup We onsider a ohort of onsumers who in period 1 reeive a marketing offer (oupon, gift ertifiate, invitation to partake in produt sampling, rebate) of value x that expires in period N. We make the following key assumptions. A1: Forgetting. A onsumer in possession of the offer faes the possibility that he/she forgets, misplaes, or loses the offer as he/she rosses from one period to the next. We assume that the probability that the onsumer will remember the offer is ρ [0, 1]; onversely, the probability that he/she will forget the offer is 1 ρ. The forgotten offer annot be realled at a future time. A2: Stohasti Redemption Costs. Redeeming the offer is ostly. We assume that this ost varies stohastially aross the periods. In every period, eah onsumer learns his/her redemption ost that is drawn from a ontinuous distribution F ( ) with support [, ]. A3: Forward-Looking Consumers. All the parameters of the model, x, N, ρ and F ( ) are known to the onsumers. We assume that the offer is suh that x. When a onsumer redeems the offer, his/her utility is the differene between x and the realized redemption ost. Thus, the onsumer may strategially postpone redeeming the offer to take advantage of the possibility of a lower ost realization in the future. For simpliity, we assume there is no disounting between periods. 3 Analysis Consumer deision-making an be haraterized by a set of thresholds 1, 2,... N. When a onsumer arrives in period n and the offer has not yet been redeemed or forgotten, the onsumer redeems the offer if and only if his/her realized redemption ost does not exeed n. We begin our analysis with the last period. In period N, the onsumer redeems the offer as 3

4 long as the realized ost does not exeed x: N = min{x, }. The onsumer s expeted utility at the beginning of period N (period N value funtion) is, therefore, V N = N (x ) df (). In the next to last period, period N 1, the onsumer ompares x with ρv N and redeems the offer if and only if x ρv N, or x ρv N. Hene, N 1 = min{x ρv N, }. We then alulate period N 1 value funtion, whih inorporates the onsumer s optimal redemption rule in periods N 1 and N: V N 1 = = N 1 N 1 (x ) df () + ρv N df () N 1 (x ) df () + (1 F ( N 1 ))ρv N. Continuing with bakward indution, we obtain the sets of thresholds { n } and value funtions {V n }, n = 1, 2,... N. The results are summarized in Proposition 1. Proposition 1. Among onsumers in possession of the offer in period n, only those with the ost realizations below n will redeem the offer in that period. The set of thresholds { n } an be alulated reursively, with the initial ondition N = min{x, }, V N = N (x ) df (). and the formula n 1 = min{x ρv n, }, V n 1 = n 1 (x ) df () + (1 F ( n 1 ))ρv n. Corollary 1. Consumers beome less stringent as the expiration date of the offer approahes: N. For the proof of Corollary 1, onsider a onsumer in possession of the offer in period n. 4

5 This onsumer an do at least as well as a onsumer in possession of the offer in period n + 1 by behaving as if the offer expires one period earlier. This simple observation immediately implies that the value funtion is higher in period n than in period n + 1, V n V n+1. Sine n 1 = min{x ρv n, } and n = min{x ρv n+1, }, we have n 1 n for all n. Intuitively, onsumers are more demanding early on beause there are still ample opportunities to realize a low redemption ost. However, as periods go by, the hanes to draw a good redemption ost diminish, thereby ausing a onsumer to be less likely to delay redemption. We now alulate the number of redemptions in eah period n, whih we denote by p n, n = 1, 2,... N. In the first period, the number of redemptions is p 1 = 1 1 df () = F ( 1 ). The number of onsumers who find themselves in possession of the offer in period 2 is ρ 1 1 df (). Sine fration 2 1 df () of them will redeem the offer, the number of redemptions in period 2 equals 2 p 2 = ρ 1 df () 1 df () = ρ(1 F ( 1 ))F ( 2 ). 1 In period 3 the number of onsumers in possession of the offer is ρ df () 2 1 df (); fration 3 1 df () of them will redeem the offer in that period. Thus, p 3 = ρ df () 2 1 df () We have the following proposition. 3 1 df () = ρ 2 (1 F ( 1 ))(1 F ( 2 ))F ( 3 ). Proposition 2. The number of redemptions in period n is given by F ( 1 ), if n = 1, p n = ρ n 1 F ( n ) n 1 k=1 (1 F ( k)), if n 2. In the remainder of the paper, we use the preeding analysis to investigate onsumer redemption behavior. Our primary objetive in this endeavor is to demonstrate that it is possible to obtain the two aforementioned redemption patterns a bump at expiry and greater redemption with shorter redemption windows exlusively with forward-looking onsumers. Another objetive is to provide some intuition for the emergene of these phenomena. To failitate meeting these objetives, we restrit our attention to x. 5

6 A4: Offer Attrativeness. The offer is suffiiently attrative in that x. For redemption osts, we onsider three distributions from the beta family: B(0.4, 0.2), B(0.2, 0.4), and B(1, 1). Their densities are depited in Figure 1. Note that B(1, 1) orresponds to the uniform distribution, while the other two have more mass at the extremes. We inlude the uniform distribution for ompleteness but expet the distribution of redemption osts to be bi-modal and asymmetri. The distribution B(0.4, 0.2) depits a world where the likelihood of extremely large redemption osts is more pronouned than the likelihood of extremely low redemption osts. The distribution B(0.2, 0.4) depits the reverse. Our expetation for this last distribution is generally onsistent with the distribution found for other traits suh as intelligene, wherein there is a seond mass of highly gifted individuals (Burt 1963). density B(0.4,0.2) B(0.2,0.4) B(1,1) Figure 1: Probability Density Funtions for B(0.4, 0.2), B(0.2, 0.4), B(1, 1) Suppose N = 4, ρ = 0.95 and x = 1.2. For eah of the three distributions we alulate { n } and {p n }, n = 1, 2,... N. We also alulate the total number of redemptions p = N n=1 p n. The results are reorded in Table 1. We see that in all three ases { n } is an inreasing sequene, as it should be (Corollary 1). Under assumption A4, the onsumer in possession of the offer in period N redeems it irrespetive of the realized ost, N =. This explains 4 = 1 in the table. As to the redemption pattern {p n }, it is non-monotoni. In the ase of B(0.4, 0.2), there is a 6

7 bump in the last period: p 4 = > p 3 = (shaded dark gray) p 1 p 2 p 3 p 4 p B(0.4, 0.2) B(0.2, 0.4) B(1, 1) Table 1: N = 4, ρ = 0.95, x = 1.2 If we derease the probability of remembering the offer, the ending period bump under B(0.4, 0.2) will get smaller. In Table 2, ρ = We see that p 4 = is only slightly above p 3 = (shaded dark gray). The bump will ompletely disappear if we push ρ below p 1 p 2 p 3 p 4 p B(0.4, 0.2) B(0.2, 0.4) B(1, 1) Table 2: N = 4, ρ = 0.85, x = 1.2 Thus, we onlude that the ending period bump is present in the redemption pattern when: (i) high values of reeive more weight under F ( ), and (ii) the probability of remembering the offer is lose to one. Intuitively, more weight on high values of and a higher probability of remembering the offer inrease the likelihood that the onsumer arrives into the last period without redeeming the offer. This leads to the ending period bump, as the onsumer is fored to redeem. If we drop assumption A4 (i.e., assume x instead of x ), there will be fewer redemptions in the final period beause redemption may not provide positive value. This will obviate the bump. It will also redue the total number of redemptions, and therefore sale it to be more in line with real-world redemption rates without impating the essential alulus of our model dynamis for those onsumers desribed by x. Overall, we note that our findings relate very well to offers haraterized by our analysis, e.g., Groupon offers. Groupon offers are very attrative (x ), exhibit high redemption rates (total redemptions about 45%), and are haraterized by a substantial ending period bump (approximately 15%) please see Figure B in Gupta, Weaver, and Rood (2012). Next, we want to study the effet of N on the total number of redemptions, p. Table 3 differs from Table 1 in that the number of periods, N, equals 3. Comparing the last olumn in Table 1 with the last olumn in Table 3, we obtain < 0.948, < 0.970, and < (shaded light gray). In fat, the result that the total number of redemptions is higher under 7

8 shorter redemption windows holds for any F ( ) and ρ p 1 p 2 p 3 p B(0.4, 0.2) B(0.2, 0.4) B(1, 1) Table 3: N = 3, ρ = 0.95, x = 1.2 Proposition 3. The total number of redemptions is higher under shorter redemption windows. The proof of Proposition 3 is illustrated by inspeting Tables 1 and 3. For all three distributions, the redemption ost thresholds in the N = 3 period world (Table 3) are unsurprisingly idential to the redemption ost thresholds in the last three periods of the N = 4 period world (Table 1). Moreover, from Corollary 1, it follows that the threshold in the first period of the N = 4 period world is even lower. Thus, as onsumers journey aross the first three periods of the N = 3 period world, they fae higher (less stringent) redemption thresholds than their ounterparts journeying aross the first three periods of the N = 4 period world. This fat, oupled with the fats that all surviving offers are redeemed in the last period and the N = 4 period world has one more hane for forgetting, yields the finding that the total number of redemptions is greater in the N = 3 period world. The essential logi in this line of reasoning is invariant to the hoie of F ( ) and ρ; moreover, it an be repliated for any two redemption windows. As suh, it proves the general laim inherent in Proposition 3. Of ourse, offering a longer redemption window is good for onsumers and enhanes their welfare. However, a longer redemption window results in a lower total number of redemptions whih may be at variane with the objetives of the entity making the offer (e.g., a firm attempting to obtain initial produt sampling). 4 Conlusion Previous researh on onsumer redemption behavior has doumented two noteworthy empirial regularities a bump lose to offer expiry and greater redemption with shorter redemption windows. In the extant work, these phenomena have been explained by invoking myopi onsumers. In ontrast, we develop a model onsisting exlusively of forward-looking onsumers and demonstrate the emergene of these phenomena for plausible parameter values. Our work thus reveals how these phenomena may arise, and persist, even in the absene of onsumer myopia. 8

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