Bundling With Customer Self-Selection: A Simple Approach to Bundling Low-Marginal-Cost Goods

Transcription

1 Unversty of Pennsylvana ScholarlyCommons Operatons, Informaton and Decsons Papers Wharton Faculty Research Bundlng Wth Customer Self-Selecton: A Smple Approach to Bundlng Low-Margnal-Cost Goods Lorn. M. Htt Unversty of Pennsylvana Pe-yu Chen Follow ths and addtonal works at: Part of the Marketng Commons, Other Economcs Commons, and the Sales and Merchandsng Commons Recommended Ctaton Htt, L. M., & Chen, P. (2005). Bundlng Wth Customer Self-Selecton: A Smple Approach to Bundlng Low-Margnal-Cost Goods. Management Scence, 5 (0), Ths paper s posted at ScholarlyCommons. For more nformaton, please contact repostory@pobox.upenn.edu.

2 Bundlng Wth Customer Self-Selecton: A Smple Approach to Bundlng Low-Margnal-Cost Goods Abstract Wth declnng costs of dstrbutng dgtal products comes renewed nterest n strateges for prcng goods wth low margnal costs. In ths paper, we evaluate customzed bundlng, a prcng strategy that gves consumers the rght to choose up to a quantty M of goods drawn from a larger pool of dfferent goods for a fxed prce. We show that the complex mxed-bundle problem can be reduced to the customzed-bundle problem under some commonly used assumptons. We also show that, for a monopoly seller of low margnal cost goods, ths strategy outperforms ndvdual sellng (M = ) and pure bundlng (M = ) when goods have a postve margnal cost or when customers have heterogeneous preferences over goods. Comparatve statcs results also show that the optmal bundle sze for customzed bundlng decreases n both heterogenety of consumer preferences over dfferent goods and margnal costs of producton. We further explore how the customzed-bundle soluton s affected by factors such as the nature of dstrbuton functons n whch valuatons are drawn, the correlatons of values across goods, and the complementarty or substtutablty among products. Altogether, our results suggest that customzed bundlng has a number of advantages both n theory and practce over other bundlng strateges n many relevant settngs. Keywords nformaton goods, dgtal goods, prcng, bundlng, self-selecton, nternet Dscplnes Marketng Other Economcs Sales and Merchandsng Ths journal artcle s avalable at ScholarlyCommons:

3 Bundlng wth Customer Self-Selecton: A Smple Approach to Bundlng Low Margnal Cost Goods February, 2003 Lorn M. Htt Unversty of Pennsylvana, Wharton School 300 Stenberg Hall-Detrch Hall Phladelpha, PA 904 lhtt@wharton.upenn.edu Pe-Yu Chen Graduate School of Industral Admnstraton Carnege Mellon Unversty 5000 Forbes Ave Pttsburgh, PA 523 pychen@andrew.cmu.edu We would lke to thank Rav Aron, Erk Brynjolfsson, Erc Clemons, Davd Croson, Rachel Croson, Gerald Faulhaber, Steven Matthews, Mot Lev, El Snr, Raghu Srnvasan, Shny Wu, Denns Yao and partcpants of the 999 Workshop on Informaton Systems and Economcs for helpful comments on earler drafts of ths paper.

4 Bundlng wth Customer Self-Selecton: A Smple Approach to Bundlng Low Margnal Cost Goods Abstract The reducton n dstrbuton costs of dgtal products has renewed nterest n strateges for prcng goods wth low margnal costs. In ths paper, we evaluate the concept of customzed bundlng n whch consumers can choose up to a quantty M of goods drawn from a larger pool of dfferent goods (>M) for a fxed prce. We show that the complex mxed bundle problem can be reduced to the customzed bundle problem under some commonly used assumptons. We also show that, for a monopoly seller of low margnal cost goods, ths mechansm outperforms ndvdual sellng (M=) and pure bundlng (M=) when goods have a postve (even small) margnal cost, when users face costs of evaluatng goods n the bundle, or when customers have heterogeneous preferences over goods. Comparatve statcs results also show that the optmal bundle sze for customzed bundlng decreases n both heterogenety of consumer preferences over goods and margnal costs of producton. Altogether, our results suggest that customzed bundlng provdes an effcent and mathematcally tractable prcng scheme for sellng nformaton and other low-margnal cost goods, especally when consumers have heterogeneous preferences over goods or are budget or attenton constraned. 2

5 . Introducton The emergence of the Internet as a low-cost, mass dstrbuton medum has renewed nterest n prcng structures for nformaton and other dgtal goods (Shapro and Varan, 998; Cho, Stahl and Whnston, 998). One common settng, faced by publshers, software producers, musc dstrbutors, cable televson operators and a wde varety of other frms, s when a provder of a large number of nformaton goods seeks to sell to a group of heterogeneous consumers, who may place dfferent values on the ndvdual goods. Whle t s now possble to effcently sell ndvdual goods separately for even small payments (Metcalfe, 996), frms may be able to generate greater profts by engagng n bundlng, where large numbers of goods are sold as a unt. In theory, for goods frms could offer up to (2 -) possble bundles, each at a dfferent prce. However, ths problem s known to be computatonally ntractable and dffcult to solve n closed form except for small numbers of goods (Hanson and Martn, 990). Recent work (Bakos and Brynjolfsson, 999) has shown that when the margnal costs of goods are suffcently low and customers share a common probablty dstrbuton for valuaton of dfferent goods, pure bundlng (offerng all goods for a fxed prce) s optmal, greatly smplfyng the bundlng and prcng problem. However, less s known about stuatons where t may be optmal to bundle large numbers of goods, yet margnal costs and consumer preferences are such that pure bundlng s neffcent. These types of stuatons arse when goods have a small but non-neglgble margnal cost (e.g., dgtal vdeo dstrbuton on a congested network), when consumers value only a subset of all avalable goods (e.g., musc, moves, engneerng software modules, past news artcles), or when consumers face costs of evaluatng goods that ncrease n the number of goods offered n a bundle. In ths paper, we analyze a prcng approach whch generalzes exstng results on nformaton prcng whle preservng smplcty and analytcal tractablty, whch we term customzed bundlng. A customzed bundle s the rght for a consumer to buy her choce of up to M goods

6 from a larger set, for a fxed prce p. ote that non-trval customzed bundlng s only relevant when margnal cost for each good s small and the goods share smlar cost structure otherwse, t wll be more proftable to sell ndvdual unts. When consumer demand can be characterzed n a specfc way, whch s consstent wth the assumptons used n some prevous work on nformaton goods prcng, we show that the mxed bundlng problem can be reduced to a smple problem of non-lnear prcng. Ths allows the applcaton of known results to solve otherwse very complcated bundlng problems such as when ndvdual sale can coexst wth pure bundlng, when more than one bundle wll be offered, how margnal cost affects the optmal sze of bundles, and the welfare effects of bundlng. In addton, because customzed bundlng contans unt sale and pure bundlng as extreme cases, our analytcal results can be compared to and extend prevous work on nformaton goods bundlng. 2. Prevous Lterature The lterature on bundlng has a long hstory begnnng wth the observaton by Stgler (963) that bundlng can ncrease sellers profts when consumers reservaton prces for two goods are negatvely correlated. In the two-goods case, offerng both a two-good bundle as well as the ndvdual tems (mxed bundlng) s typcally optmal (Adams and Yellen, 976; McAfee, McMllan and Whnston, 989). Ths s because bundlng reduces heterogenety n consumer valuatons, enablng a monopolst to better prce dscrmnate (Schmalensee, 984; Salnger, 995), whle stll capturng resdual demand through unt sale. Whle the nsght that bundlng reduces heterogenety n valuatons s qute general, other aspects of these solutons often do not generalze beyond the two-good case. Other work has extended the bundlng lterature to consder multple goods as well as multple types. Spence (980) generalzed the prncples of sngle product prcng problem to the case of several products usng a lnear programmng formulaton, and shows some cases where the Our defnton s almost dentcal to generalzed bundlng studed by MacKe-Mason, Rveros and Gazzale (999). Ths mechansm was also mentoned as a possble strategy for dscrmnatng between customers wth dfferent wllngness to pay (Bakos and Brynjolfsson, 997; Shapro and Varan, 998), but ths suggeston was not completely explored n prevous work. The earlest reference to ths approach of whch we are aware s Chen (997), who studed the propertes of customzed bundlng usng mxed nteger programmng, although varants of ths scheme have been used n practce n software and musc dstrbuton for many years. 2

7 problem can be solved n closed form. Other tractable analytcal solutons have been found for a varety of specal cases such as lnear utlty (McAfee and McMllan, 988) or when valuatons across dfferent consumers can be ordered n specfc ways or satsfy certan separablty condtons (Armstrong, 996; Sbley and Srnagesh, 997; Armstrong and Rochet, 999). These papers have found addtonal general results such as the observaton that t s usually optmal to leave some consumers unserved n order to extract more revenue from the other hgher value consumers (Armstrong, 996) and that t s sometmes optmal to nduce a degree of bunchng, so that consumers wth dfferent tastes are forced to choose the same bundle of products (Rochet and Chone, 998). These papers provde a general structure for solvng qute complcated bundlng problems n closed form, although the complexty ncreases dramatcally as more goods are consdered, makng t dffcult to extend them to large numbers bundlng problems. However, when margnal costs are very low, t s often optmal to bundle all goods together (Bakos and Brynjolfsson, 999) whch leads to a dramatc smplfcaton of the bundlng problem. These pure bundlng results are robust to any set of consumer preferences generated by a common dstrbuton functon for the value of each good across all consumers. However, when pure bundles are not optmal, such as when consumers are budget or attenton constraned or margnal costs are sgnfcant, ths mechansm provdes lttle gudance snce lmtng the sze of a bundle of ths form creates substantal deadweght loss when customers are heterogeneous. 2 There have been several studes that have consdered large numbers bundlng problems n specfc contexts related to nformaton goods prcng. These studes generally fnd that engagng n a form of mxed bundlng where a certan large bundle s offered along sde ndvdual sale domnates ether strategy alone (Chuang and Srbu, 999; Fshburn, Odlyzko and Sders, 997). In addton, these studes ntroduce the dea that allowng customers to self- 2 The nsght behnd ths shortcomng s straghtforward. Suppose there are a large number of consumers that have valuaton for each of 0 goods drawn from the same dstrbuton functon. It s clear that f we offer a pure bundle, all consumers wll obtan ther most preferred goods, although not all wll agree whch ones they are. However f we are constraned such that we can only sell, say, a 5 good bundle, there are now 252 possble bundles that a consumer mght want f they can only have 5 goods. If any sngle bundle among the 252 possble bundles s offered, as few as /252 of the customers wll receve ther hghest valued goods, creatng substantal deadweght loss. Only by offerng every possble combnaton that consumers desre would ths deadweght loss dsappear. 3

8 select the goods n the bundle (rather than havng them predesgnated) can often mprove outcomes whle mantanng smplcty n the prcng mechansm (Chen, 997; Chuang and Srbu, 999; Macke-Mason, Rveros and Gazzale, 999). 3. Model 3. Introducton We wll examne the optmal bundlng and prcng problem for a monopolst that dstrbutes dfferent goods to consumers. We are nterested n examnng the proftablty of customzed bundles for a monopolst, n whch a consumer s allowed to choose up to M goods ( M ) for a sngle prce p. In general, a monopolst may want to offer more than one customzed bundle when facng heterogeneous customers. For notatonal smplcty we wll use m [0,/,2 /,...,] to represent a fracton of the total number of goods avalable and let p( m) represent the prce for a customzed bundle of sze M = m. In addton, for a functon f ( m ) we defne the notaton f '( m ) as wth the dscrete nature of m. f m ( ) f( m ) for m > to be consstent 3.2. Multproduct onlnear Prcng for Heterogeneous Consumers We begn by defnng a structure for the standard bundlng problem n whch customers demand at most one unt of each good. Consumers purchase a bundle of goods x =< x,.., x,.., x > (where the elements of x are bnary varables, x {0,}, j =.. representng the purchase or non-purchase of a bundle component) over all goods avalable. Consumers derve benefts from these goods whch leads to a wllngness to pay (WTP) W ( x ), a weakly ncreasng functon n all components of x wth W(0)=0. In general there wll be more than one set of consumer preferences over these goods. Let there be I dstnct types of consumers ndexed [, 2,.. I] wth a unque wllngness to pay functon W ( x ). The proporton of each consumer type n the populaton s denoted by α (where α = ). If the prce of a set of goods s p( x ), I j = j 4

9 we could wrte the net utlty a consumer obtans from consumng ths bundle as: U ( x, p( x)) = W ( x) p( x ). 3 We denote the cost of provdng a vector of goods x as C( x ) whch s weakly ncreasng n all components of x. Usng ths notaton, the general bundlng problem the monopolst faces s the determnaton of the set of bundles offered { x} and a set of prces p( x ) solvng the well known mxed bundle prcng problem wth heterogeneous consumers (Spence, 980): I max α [ p( x ) C( x )] st.. = IR: W ( x ) p( x ) 0 j j IC: W ( x ) p( x ) W ( x ) p( x ), j () The frst set of constrants, ndvdual ratonalty (IR), guarantees that f a consumer chooses to purchase a bundle, t provdes non-negatve surplus. In other words, the monopolst cannot force a consumer to purchase the bundle. The second set of constrants, ncentve compatblty (IC), guarantees that a consumer segment receves at least as much surplus for purchasng the bundle ntended for them than they would for choosng another bundle. Implct n ths assumpton s that the monopolst cannot prce dscrmnate by group t must be n the consumer s self-nterest to purchase ther ntended bundle. Ths formulaton treats the problem as a drect revelaton mechansm where consumers reveal ther type through ther choce of product, whch wll yeld the proft maxmzng soluton for the monopolst, f such a soluton exsts (Myerson, 979). ote from ths formulaton that for I consumer groups and products, the monopolst must determne the optmal set of I bundle compostons and prces out of 2 possbltes Customzed Bundlng Our ntal nterest s n determnng the condtons under whch the full bundlng problem can be reduced to the much smpler customzed bundlng problem. Followng the lterature on nformaton goods prcng, we wll assume that the cost structure of provdng goods to 3 The assumptons on W guarantee that U obeys the normal propertes of utlty functons. 5

10 consumers depends only on the number and not on whch goods provded, thus C( x ) = C( m) where m = x (where denotes a vector dot product, and s a vector of all s). We further assume that C s weakly ncreasng, wth decreasng dfferences n m (that s, C ( m) 0 and C ( m) 0 ). ote also that C (0) = 0, consstent wth an addtonal assumpton that the monopolst has already sunk any fxed cost necessary to produce these goods. Gven ths cost structure, we now examne what types of consumer preferences (utlty or wllngness-to-pay) wll yeld an equvalence between the optmal bundlng problem and the optmal customzed bundlng problem. Before establshng these results, t s useful to ntroduce some addtonal notaton. Let w ( m ) represent the most a consumer of type s wllng to pay for m goods (formally, w ( m) = max x W ( x ) st.. x m ). ote that ths mples that w (0) = 0, w '( m) 0 and k = k w "( m) 0. Although there can exst as many as I such functons, 4 n general there wll be less than I because dfferent preferences W ( x) can yeld the same expresson for w(m). We can now formulate customzed bundlng problem as: I max α [ pm ( ) Cm ( )] st.. = IR: w ( m ) p( m ) 0 j IC: w ( m ) p( m ) w ( m ) p( m ), j (2) Ths problem s the well-known non-lnear quantty dscrmnaton prcng problem wth heterogeneous consumers (also known as second-degree prce dscrmnaton; see Trole, 988, p ). In addton to the mathematcal formulaton beng dentcal, customzed bundlng s also ntutvely smlar to second-degree prce dscrmnaton as t accomplshes dscrmnaton among dfferent groups through customer self-selecton from a menu of offerngs. The key dstncton s that the non-lnear prcng problem generally refers to dfferent quanttes of an dentcal good, whle customzed bundlng refers to heterogeneous goods wth 4 In the context of nformaton goods, t s reasonable to assume that I, number of consumer types, s much smaller than,.e., number of nformaton goods offered. 6

11 smlar valuatons. ote also that ths problem s sgnfcantly smpler than the full mxed bundlng problem (gven n ()) as t only requres a selecton of a maxmum of I prces from a total space of possble customzed bundles. An nterestng queston to explore s when we can reduce the complex mxed bundlng prcng problem () to the much smpler customzed bundlng prcng problem (2). Clearly, the two problems are equvalent f they yeld the same optmal prcng soluton. The condtons for ths to hold are presented n Result (all proofs are shown n the Appendx). Result : The optmal bundle prce schedule { x, p( x )}, s equvalent to a customzed j j bundlng soluton p( m) m ff for every par of optmal bundles x, x where x = x, j w ( m) = w ( m) Ths result shows for every optmal bundlng soluton there s an equvalent customzed bundlng soluton as long as the wllngness to pay functon over customzed bundles s the same for all customers whose optmal regular bundle s the same sze. Whle ths condton seems both abstract and rather restrctve, a surprsng number of assumptons on consumer preferences meet these condtons. Specfcally, ths wll hold f all preferences over regular bundles ( W ( x ) ) have a common customzed bundle representaton (w(m)). The smplest example of ths condton s when heterogeneous preferences map to a sngle wllngness to pay n customzed bundles. For nstance, f there are two consumer groups (={,2}) and three goods (=3) and the margnal cost per good s 0.25, the followng two sets of values for each of the three goods satsfy ths property: {0.,0.4,} for consumer and {,0.4,0.} for consumer 2. Ths yelds a WTP over customzed bundles for all consumers of {.5,.4,,0} for m={, 2/3, /3,0} respectvely, and the monopolst optmally offers a 2-good customzed bundle at a prce of.4, yeldng a proft of 0.9. Chuang and Srbu (999) and Fay and MacKe-Mason (200) consdered a more general form of ths relatonshp, representng consumer WTP by a lnear functon over ther rank ordered preferences that satsfes ths condton. We examne n detal a mnor generalzaton of ther formulaton n Secton

12 Interestngly, when there are a large number of consumers whose valuaton s drawn randomly from the same dstrbuton functon, as s common n marketng choce models (e.g,, McFadden, 974) and prevous work on nformaton good prcng (e.g., Bakos and Brynjolfsson, 999), the resultng dstrbuton of preferences over goods W ( x ) yelds a common dstrbuton of preferences over customzed bundles w(m). As shown n Result 2, the equvalence between regular bundlng and customzed bundlng holds n ths settng qute generally, only requrng that valuatons of goods can be descrbed by a common cumulatve dstrbuton functon where the expected absolute value of each good s fnte, a common assumpton n prevous work. Result 2: If each of a large number of ndvdual consumer s wllngness to pay for a vector of goods x [0,] s gven by a vector v R drawn ndependently from a common dstrbuton functon wth cdf F( v ) wth fnte expected absolute value for all goods, there exsts a wllngness to pay functon for customzed bundles common across consumers. Ths functon s gven by wm ( ) = E[ Xk : ] where X : s the th order statstc from F( v ) k= ( m) Result 2 shows that to calculate consumers wllngness to pay across customzed bundles for random dstrbutons, one need only calculate wm ( ) = E[ X : ]. The expresson nsde k= ( m) the expectaton, a lnear combnaton of order statstcs, s a specal case of a general class of functons called L-estmates (see a survey n Rychlk, 998). The fact wll prove useful n later results we derve for random valuatons. k 3.4 General Solutons We now consder the general soluton to the generalzed bundlng problem (2). To solve ths problem, t s common to mpose some addtonal structure on the varaton across consumers known as the Spence-Mrrlees sngle crossng property. Ths assumpton enables closed form solutons and straghtforward comparatve statcs results, and s commonly used n theoretcal work on non-lnear prcng. Let a and b represent bundle szes (dfferent values of m), let consumers be ndexed by and j, the sngle crossng property (SCP) holds f there exsts an orderng of consumers such that: 8

13 j w ( a) > w ( a) j j w ( a) w ( b) > w ( a) w ( b) a> b, > j. Ths mples that a hgher type consumer (meanng hgher value of n ths condton) places a greater value on any gven bundle than lower type consumers. For all subsequent dscusson assume that customer types are ordered to satsfy ths condton. In addton, gven any two bundles, there s a greater dfference n valuaton due to sze for hgher type consumers (or n other words, ncreasng dfferences n type and bundle sze). Whle ths appears to be restrctve, t s a common assumpton n most models of ths type, and smply rules out cases where the orderngs of consumer value change as the bundle sze changes. Moreover, t s necessary for any general characterzaton of the soluton to ths problem. * * Let { m, p } denote the optmal offerng of the monopolst when there are multple customer types, and { mˆ, pˆ } represent the bundle that would be offered to consumer type () f there were no ncentve compatblty constrants (that s, f they were the only consumer type beng served). Usng standard results and proof technques from the theory of non-lnear prcng we can show the followng Result. Result 3: A monopolst wll offer a set of customzed bundles that have the followng propertes: a) The lowest-type customer that s served s prced at ther wllngness to pay: p = w ( m ) * * b) The prces for all other bundles are determned to satsfy IC, and leave all consumers except the lowest type wth postve surplus (let mn be the lowest type that s served): * * * * * p p = + w ( m ) w ( m ) < w ( m ) > mn c) The hghest type customer s always served at the sze they would have receved f they were the only customer segment: m I* = mˆ I d) All other customers receve bundles smaller than the bundle sze they would have receved f they were the only customer segment. These szes are the greatest values of m [0,/,2 /,...,] that satsfy: 9

14 I I I j * j + * j * ( α ) '( ) ( α ) '( ) ( α ) '( ) w m w m C m < I j= j= + j= e) There, n general, may be a customer segment such that all customers below that segment are not served (that s, > = < ) * mn 0 st.. m 0 for mn f) The optmal sze of the customzed bundle s weakly decreasng n margnal cost Result 3 replcates some of the well known results on non-lnear prcng wth a small modfcaton to account for the dscrete nature of m. Frst, there s generally one optmal bundle per type of consumer f that segment s served at all. Second, not all consumers are served wth snce under sngle crossng t s more proftable to extract addtonal surplus form the hgher types than have them cannbalzed by bundles targeted to the lower types. Thrd, only the hghest type consumer s served at ther socally optmal bundle sze, the rest beng weakly lower to dscourage hgh types consumers from consumng bundles targeted at the lower types. Fourth, because the monopolst cannot perfectly prce dscrmnate, all consumers except the lowest types that are served earn some surplus, an nformaton rent due to ther hdden type. Fnally, a more subtle observaton s that wthout extremely restrctve assumptons on cost and preferences, prces wll not be lnear n bundle sze (wth or wthout a fxed fee component) suggestng that n general the optmal soluton wll outperform a two-part tarff (a formal proof of ths s avalable from the authors). All of these are a drect consequence of customzed bundlng beng a well-behaved nonlnear prcng problem under our assumptons. There are also some addtonal nsghts these results brng that are unque to the customzed bundlng problem. Frst, f cost and wllngness to pay are known for each customer segment (whether t s determnstc or the expectaton of a random valuaton), t s a smple calculaton of complexty O(I) to determne the optmal prce and bundle szes that wll be offered. Ths contrasts wth the ntractable mxed bundlng problem of a large number of goods. Second, n ths formulaton Result 3f provdes a smple but powerful result on the relatonshp between customzed bundlng, sngle good sellng and pure bundlng as the margnal cost per good ncreases, there s a monotonc shft between n optmal bundlng polcy from pure bundlng to customzed bundlng to unt sale. 0

15 Customzed bundlng becomes ncreasngly desrable relatve to pure bundlng when consumers bear addtonal margnal costs of consumng larger bundles, for nstance, when consumer attenton s scarce and larger bundles requre greater consumer attenton to evaluate and dentfy ther preferred goods. Let Z ( m ) represent the cost a consumer of type faces n evaluatng a bundle of sze m. Assume that Z ( m) s postve and weakly ncreasng n m wth weakly ncreasng dfferences ( Z '( m) 0 and Z "( m) 0). We fnd that under some mld assumptons, the optmal bundle szes are further reduced, makng t more lkely that customzed bundlng domnates pure bundlng as shown n Corollary below: Corollary : Under the assumptons above, f consumers face an addtonal (prvate) cost of evaluatng a bundle then bundle sze s weakly decreased and prces are strctly decreased compared to the stuaton of no evaluaton costs f + Z '( m) Z '( m) < ( + α I ) Z '( m) < I. Moreover, ths condton wll always be j α j=+ satsfed f Z ( m) = Z( m). In other words, as long as evaluaton costs are enough to matter n the optmal soluton, and are the same across consumers or at least do not ncrease too fast n consumer type, evaluaton costs wll tend to yeld a smaller optmal customzed bundle (the weak nequalty due to the dscrete nature of m requrng a certan level of evaluaton cost before bundle sze s affected). Overall, we are able to use our formulaton and slght modfcatons of standard results to derve some nterestng nsghts nto large numbers bundlng problems regardng the number of optmal bundles, the relatonshp between pure bundlng and customzed bundlng, the response to margnal cost changes, welfare effects, and the tractablty of the prcng algorthm. However, these general results do not say much how customzed bundlng s affected by the nature of consumer preferences over dfferent goods, the exstence of attenton lmts (a maxmum number of goods desred), or budget constrants. In the next two sectons, we make some specfc assumptons about cost and preferences to enable us to study these relatonshps.

16 3.5. Bundlng Under a Two-Parameter Preference Functon For ths secton, we buld on results by Chuang and Srbu (999) (CS) by consderng a problem where dfferent consumers can be descrbed by a wllngness to pay functon that depends on two parameters: an overall budget constrant or total wllngness to pay (b) and the number of goods they value postvely (K). We denote k as the fracton of goods that consumers value K postvely (that s k = ). Consumers are assumed to have smlar utlty functons over a rank orderng of goods, whch n the CS model s assumed to be lnear n the rank order of ther m preferences. We generalze ths case to allow other relatonshps by assumng wm ( ) = by ( ), k where y(.) captures customers relatve valuatons, or degree of preference, for dfferent goods. 5 Therefore we can wrte: m by( ) pm f m k ump (, m) = k b pm f m> k (3) where y s such that y(0) = 0, y() =, y' > 0, y'' 0 over the doman [0,] (recall that ths s consstent wth w'( m) 0 and w"( m) 0 gven our defnton of w(m) and that y refers to a dfference, not a dervatve, to account for the dscrete nature of m). For any common functon y( ) and any set of parameters { b, k, α } characterzng consumer preferences that satsfes sngle crossng we can drectly apply Result 3 to obtan the optmal bundlng soluton. To make ths analyss concrete, however, we focus on a sngle consumer type and make some functonal form assumptons for preferences and costs (the mult-type case s also easly solved under SCP, but adds lttle nsght over the general sngle type case). Specfcally, assume that 5 One can thnk y(t) as the proporton or fracton of total budget that a customer s wllng to spend on the top t percent of the goods she postvely values. Intutvely, y s an ncreasng and concave functon of t. 2

17 there s a constant margnal cost (c) for all goods, Cm ( ) = cm, and that relatve valuaton (y) across goods can be descrbed by a quadratc functon wth a parameter (a) capturng customers preferences across dfferent nformaton goods. The quadratc formulaton s, perhaps, the smplest functonal form assumpton that enables us to examne the dfferences between relatvely unform preferences over goods and preferences skewed toward a few hgh value goods. Moreover, gven the propertes of the y functon ( y(0) = 0, y() =, y' > 0, y'' 0 ), quadratc functons provde a reasonably good local (and often global) approxmaton to arbtrary functonal forms for y( ) whle mantanng computatonal tractablty. Thus we have: 2 m y() t = ( + a) t at where t = [0,] and a [0,] (4) k By varyng a, we can examne dfferent condtons of valuaton for a gven consumer or representatve consumer across dfferent goods. If a= then we have the CS assumptons (lnearly decreasng value n rank order). If a=0, the consumer values all goods equally. Under ths formulaton we can compare the effcency (ablty to maxmze socal welfare) as well as the proftablty of dfferent bundlng schemes. In ths example, pure bundlng s a trval soluton as long as the pure bundle s proftable (that s, b C() > 0). The monopolst sets prce to total value (p=b) and extracts all surplus, although not at mnmum cost when k < so t s not effcent (the monopolst ncurs margnal cost to offer goods that are not consumed). The optmal prce per good for ndvdual sale ( P IS ) s found by maxmzng profts subject to a constrant that the margnal utlty of customers ganed by purchasng addtonal unts of the good s equated wth the prces pad: P = arg max Pm C( m) st.. w'( m) = P IS P The optmal customzed bundlng soluton s just a specal case of Result 3 where there s only one type of customer. Because there s only one type, we can gnore ncentve compatblty and focus entrely on ndvdual ratonalty. Thus the prce for a customzed bundle ( gven by: p ) s 3

18 p = argmax p C( m) st.. w( m) p 0 p ote that the constrant s always bndng at the optmum, so we can rewrte the objectve functon as: m = arg max w( m) cm m Ths equaton s dentcal to the problem of maxmzng socal welfare and, thus, there s no deadweght loss, so the customzed bundlng soluton s effcent. We summarze the solutons and results to ndvdual sale ( m, π ), pure bundlng ( m, π ) and customzed bundlng ( m, π ) n the followng two graphs (detaled dervatons appear n b the Appendx). For ease of comparson, we defne w to be the average wllngness to pay k for the goods that have postve values. Fgures a and b characterzes proft and bundle sze for varous regons of margnal cost per good (c) and customer preference parameter over goods (a). ote that snce customzed bundlng contans both pure bundlng and ndvdual sale as extreme cases, t wll always weakly domnate. However, the degree of dfference depends on margnal cost and the dsperson of values across goods. As shown n Fgure a, only when margnal cost s zero s customzed bundlng and pure bundlng equvalent n profts. Ths contnues up untl margnal costs are equal to kw when pure bundlng s no longer feasble whle customzed bundlng s stll proftable. Fnally at w( + a) customzed bundlng s no longer feasble. Altogether, these results suggest that the proftable regon of customzed bundlng expands as a ncreases (.e., when there s ncreasng dfference n good valuatons). IS IS PB PB 4

19 Fgure a: Pure bundlng vs. customzed bundlng for dfferent margnal cost and customer preference parameter c 2w π = π = 0 PB c= w( + a) w kw c= kw π π > π = 0 π PB PB 0 π 0 0 = π PB = b /3 2/3 a Fgure b: Indvdual sellng vs. customzed bundlng for dfferent margnal cost and customer preference parameter c 2w m π IS IS = m = 0 = π = 0 c= w( + a) w 0 < m < m < k IS 0 < π < π = 2π IS IS m = m = k IS π π = b ck IS 0 0 c= w( 3 a) c= w( a) 0 < m < m = k IS 0 < π < π = b ck IS /3 2/3 a In Fgure b, at very low margnal cost and low dsperson of valuaton across goods when C' = c< w( 3 a), ndvdual sellng and customzed bundlng are equvalent n number of goods sold, whch s the effcent soluton. They also acheve the same proft level when a=0. But as a departs from zero but s smaller than w( 3 a), ndvdual sellng s effcent but not 5

20 proft maxmzng. ote also that market demand (goods that are postvely valued) can be fully satsfed usng customzed bundlng for a values three tmes as large as the case of ndvdual sellng for the same level of margnal cost. Fnally, as margnal costs ncrease, the sze of customzed bundle decreases untl margnal cost s so hgh that bundlng s nfeasble. The key nsght from ths analyss s found from an examnaton of the normal case wth nonzero margnal costs and consumers placng dfferent values over dfferent goods (a>0). In ths case, customzed bundlng domnates the alternatve approaches, even when there s only a sngle customer segment (except n the boundary cases where they are equvalent, m = k 6 for ndvdual sale, and m= for pure bundlng). In other words, as margnal costs (c) ncrease or customer heterogenety across goods (a) becomes larger, t s ncreasngly attractve to consder customzed bundlng over the alternatves of pure bundlng and ndvdual sale. Whle b pure bundlng s only feasble when average cost drops below average budget ( c = kw), customzed expands the range of feasble bundlng to nclude kw c w( + a), and customzed bundlng s strctly better whenever there s heterogenety n valuatons (a>0) or consumers do not value all goods (0<k<). These analyses are llustrated n Fgures 2a-2c (see remanng fgures at the end) showng the proftablty of the varous approaches for varyng levels of the parameters (c,a,b,k). All of these results hold for a sngle group of customers. The contrast wll only ncrease f we allow multple customer segments snce customzed bundlng can offer talored bundles to each segment, a strategy not possble wth ndvdual sale or pure bundlng wthout some segmentaton mechansm Customzed Bundlng Under Random Valuaton 6 m represents total number of goods sold to each customer n the case of ndvdual sellng. 6

21 We earler showed that smple customzed bundlng solutons may exst when consumers have valuatons for multple goods drawn from a sngle valuaton dstrbuton. 7 Snce a number of prevous papers n nformaton goods bundlng have used such dstrbutonal assumptons (especally Bakos and Brynjolfsson, 999, referred to hereafter as BB), t may be useful to compare the customzed bundlng soluton to the pure bundlng alternatve under random valuaton drawn from a certan dstrbuton. Unlke BB who consdered the effects of changng the number of goods n the populaton, we wll consder a smplfed structure compared to BB n whch s fxed at the largest possble number of goods. It should be noted that once s fxed, pure bundlng s a specal case of customzed bundlng. We wll retan the BB assumptons of dentcally dstrbuted v (the value of the th good) and ther assumpton of constant margnal cost per good (whch may be zero). In addton, all dstrbutons consdered here are assumed to meet the condtons descrbed n Result 2 (essentally fnte expected absolute value). For the followng results, t s useful to defne the Quantle or Inverse Dstrbuton Functon of a dstrbuton functon F(t) as Q ( z) = sup( t: F( t) z). F The most general result we can show for arbtrary dstrbuton functons, ncludng those where valuaton s dependent, s that customzed bundlng value s bounded below by mean valuaton, and above by an ntegral expresson nvolvng the quantle functon: Result 4: If the valuaton for any ndvdual good s drawn from a common but possbly dependent dstrbuton F( ) wth fnte mean ( µ ) then m Q ( z) dz w( m) mµ F There are two nterestng nsghts from Result 4. Frst, the upper bound can sometmes serve as a reasonable approxmaton for the value of customzed bundlng, as t can be nterpreted as an approxmate average value of the porton of the dstrbuton that exceeds the m th percentle. Smulaton results on many common dstrbutons (unform, normal, logstc and exponental) suggest ths approxmaton s good when the valuaton of dfferent goods s not too dependent. Second, the result shows that the expected value of a customzed bundle always (weakly) 7 Ths can be extended to multple dstrbutons as long as they generate wllngness to pay functons that obey the sngle crossng condton. However, ths formulaton yelds few ncremental nsghts beyond the sngle type case dscussed here and the general formulaton n Result 3. 7

22 exceeds the mean value of the same number of goods, mplyng that average valuaton per good of customzed bundles under most crcumstances exceeds the average value of per good of pure bundles. The strct lower bound only holds when m= or valuatons of goods are perfectly correlated. Wth addton dstrbutonal assumptons we can apply the theory of L-estmates to obtan a number of addtonal general results. The most straghtforward exact results can be obtaned when we further assume ndependence of good valuatons, that s assumpton yelds an explct expresson for wm ( ). F( v ) = F( v ). Ths = Result 5: If the valuaton of ndvdual goods s ndependently and dentcally dstrbuted wth quantle functon Q ( z ) then (the Bernsten Polynomals). F F : where : 0 = m wm ( ) = Q ( z) ( zdz ) ( z) = z ( z) Ths expresson can be used to numercally calculate the values of the consumer wllngness to pay for arbtrary dstrbuton functons and may be solvable n closed form for some dstrbutons such as the unform and the exponental. It can also be appled to make comparatve statcs predctons regardng dstrbuton parameters such as mean or varance and the sze of the optmal customzed bundle. Because the customzed bundlng proft functon ( π ( m) = w( m) C( m) ) has a drect relatonshp wth the quantle functon, a number of useful results can be obtaned restrctng our attenton to dstrbutons n the locaton-scale famly, whch ncludes most common dstrbutons assumed n pror work such as the exponental, normal and unform. Locatonscale dstrbutons are such that for a dstrbuton wth two parameters ( ab, ) known, we can wrte the quantle functon as Q ( z; a, b) = a+ bq ( z;0,) where a s referred to as the locaton F and b as the scale. Usng ths defnton and the usual assumpton of a constant margnal cost for each good (c), t s now possble to derve a relatonshp between locaton (proportonal to F 8

23 the mean) and scale (proportonal to varance) for an arbtrary..d. dstrbuton n ths famly and the optmal customzed bundle sze ( m *). Result 6: Let the valuaton for any ndvdual good be drawn..d. from a dstrbuton F( x ) wth mean ( µ ), n the locaton-scale famly wth locaton a and scale b. Then: a) m* s weakly ncreasng n a. b) For general dstrbutons m * s weakly ncreasng n b f E[ XM+ : ] < cwhere M s the lowest order statstc of the standard dstrbuton for F( x ) (a=0, b=) wth nonnegatve expected value. m * s weakly decreasng n b f EX [ : ] > c. c) Profts are always ncreasng n m * at optmum M For any fxed margnal cost, an ncrease n locaton smply shfts the valuaton curve outward n margnal value-sze space, ncreasng optmal bundle sze (unless the optmal bundle s already the pure bundle). The ntuton behnd the scale s somewhat more complex. ote that as scale (varance) ncreases, the dstrbuton spreads out. The hghest order statstcs become larger and the lowest order statstcs become lower. If the optmum les n a regon where the order statstcs are ncreasng n varance (.e., when c> E[ X M + : ], that s, when c s relatvely large and the optmal bundle only ncludes the very hghest valued goods), then ncreasng scale rases the sze of the bundle. If the optmum les n a regon where they are decreasng n scale (that s, when c s relatvely small), the optmal bundle sze s decreasng n scale. The condtons n 6b guarantee the locaton of ths optmum and that ths optmum doesn t move to the opposte sde of the curve as scale s changed. Ths result shows an nterestng relatonshp between the pure bundlng and customzed bundlng. When t s feasble to have a pure bundlng soluton (the average value greater than margnal cost), then greater varance wll decrease the performance of pure bundlng relatve to customzed bundlng because t means that the lowest valued goods n the bundle become even worse wth ncreasng varance and thus wll make customzed bundlng more attractve. Ths provdes an addtonal reason why greater ex-ante uncertanty about consumer valuatons makes pure bundlng less undesrable. The BB explanaton s that t slows convergence of valuaton to 9

24 the mean n fnte samples whch leaves consumers wth more surplus. However, we add an addtonal explanaton that the goods beng bundled on the margn under hgher varance get ncreasngly worse. We also show that ncreasng varance wll decrease the sze of customzed bundle when margnal cost s relatvely small. Interestngly, however, n the regon where pure bundlng s nfeasble but there are stll feasble customzed bundles ( µ < c< E[ X : ]) varance actually leads to larger customzed bundles and greater bundlng profts n contrast to the BB results for low margnal costs. If we are wllng to make stronger dstrbuton assumptons, t s possble to relax the ndependence assumpton slghtly. Whle n general t s dffcult to calculate order statstcs from non-ndependent dstrbutons, smple expressons exst for the multvarate normal wth common correlaton ( ρ ). These results are gven n Result 7, usng the same notaton ntroduced n Result 6b: Result 7: The optmal bundle sze and total bundle profts are decreasng n the correlaton among goods when EX [ + : ] < c, and ncreasng n correlaton when EX [ : ] > cwhere M s the medan. M M Ths Result ndcates that negatve correlaton acts smlarly to varance, wth negatve correlatons rasng the value of the hghest valued goods, but also decreasng the value of the lower valued goods. In the regon where pure bundlng s effcent we have another contrastng result wth BB whle negatve correlaton mproves prce dscrmnaton ablty of the monopolst, t s offset by the fact that the lowest value goods are worth even less, reducng the prce dscrmnaton gans of pure bundlng and favorng customzed bundles. These analyses can be straghtforwardly expanded to multple consumer types provded that the mpled customzed bundle valuatons satsfy sngle crossng usng Result 3 (or can be solved by mxed nteger programmng when that s not true). However, ths analyss does not yeld any addtonal nsghts beyond the ndvdual contrbuton of Result 3 and the sngle-type results n ths secton. 20

25 4. Summary and Concluson We have analyzed an alternatve bundlng mechansm for low margnal cost goods that allows a consumer to choose up to M of ther preferred goods from a larger set for a fxed prce p. Comparng to tradtonal second-degree prce dscrmnaton, n whch frms try to solve p(x) over 2 - possbltes, customzed bundlng s much smpler n mplementaton when consumer preferences are such that the optmal full bundlng soluton has an equvalent customzed bundlng representaton. We further show that these requrements are satsfed by assumptons used n pror bundlng work, especally when consumer valuatons are drawn from an dentcal valuaton dstrbuton across goods. Customzed bundlng yelds a natural form of prce dscrmnaton for heterogeneous consumers by offerng a prce-bundle sze schedule that ncludes ndvdual sale and pure bundlng as specal cases. Snce customzed bundlng s smply an applcaton of a well-behaved non-lnear prcng problem, t retans the propertes of these problems ncludng one type of offerng per customer segment, the no dstorton at the top result that the hghest valuaton consumers receve ther optmal bundle, nformaton rents to all but the lowest type consumers, and the possblty that some of the lowest types are not served. In addton, customzed bundles become optmal when margnal costs are non-zero but not so large that the soluton s ndvdual sellng. We also demonstrate usng specfc representatons of consumer preferences that customzed bundlng can be advantageous when consumer preferences are concentrated on a few goods, consumers have lmted attenton for evaluatng goods n a bundle, or they have budget constrants (ether attenton or fnancal). In addton, for the case when consumer valuatons are generated by common dstrbutons we also show that uncertanty about consumers valuatons (varance) makes customzed smaller bundles more attractve when margnal costs are low, but the optmal customzed bundle sze ncreases n varance when margnal costs are hgh (n contrast to Bakos and Brynjolfsson, 999). However, unless margnal costs are zero and customers value all goods, customzed bundlng wll strctly domnate pure bundlng. 2

26 The advantages of customzed bundlng are lkely to be especally relevant when monopolsts are sellng large numbers of hgh value goods (e.g., moves) snce t s very lkely that budget constrants are bndng and dstrbuton costs are sgnfcant, at least usng current technologes or when customers have heterogeneous valuatons over dfferent goods and do not postvely value all goods. Other markets that have smlar characterstcs are hgh-qualty dgtal musc or modular packaged software (Offce sutes, e-commerce platform software, or the SAP R/3 system, for example). For smaller numbers or lower value goods, t s lkely that pure bundlng wll preval, at least f consumers are homogeneous n the ways consdered n our model and those of our predecessors. However, even n these cases, customer heterogenety s lkely to create opportuntes for offerng dfferent sze of customzed bundles, as t provdes a greater ablty to target dfferent segments than ndvdual prcng or pure bundlng, whch must rely on thrd-degree prce dscrmnaton to deal wth resdual customer dfferences. Gven these advantages whle mantanng a relatvely smple prcng structure, t s somewhat surprsng that these mechansms are not already wdespread. However, there s ncreasng evdence that these types of mechansms are beng mplemented or could be favourably employed. In a feld experment MacKe-Mason, Rveros and Gazzale (999) found that lbrarans shfted toward purchasng journals through customzed bundlng (or generalzed bundlng n ther termnology) when ths opton was offered along wth other more tradtonal prcng schemes. Ther consumpton of customzed bundles ncreased over tme relatve to other prcng approaches, even when t was lkely that preferences over journal artcles was largely unchanged. We have also dentfed a number of other examples used n practce. Many frms offerng engneerng drawng software a moderately expensve and modular software package offer modules n customzed bundles (e.g., purchase a 0-pack for $2000 or a 5-pack for $250 where the consumer chooses from all avalable modules). There s also the well known 0 CDs for a $ promotons by frms such as Columba House, whch represent the purchase of a customzed bundle of around 4 CDs for approxmately $75 (once contractual requrements are met). At least one onlne move rental club (netflx.com) currently uses a customzed bundlng scheme ther prcng scheme allows users to choose dfferent plans that enable them to smultaneously borrow vdeos for p() dollars per month where multple values of are allowed (currently 2, 3,4, 5, and 8). The ew York Tmes has 22

27 expermented wth a bundle prcng scheme for access to ther artcle archves wth four prcng optons, wth whch the artcles purchased are chosen by the consumer: 25 artcles for $25.95, 0 artcles for $5.95, four artcles for $7.95 or a sngle artcle for $ Ths suggests that frms are explorng the use of these mechansms, although t s lkely that many of the domans n whch ths s most relevant have not yet seen substantal expermentaton n prcng structure. Whle we have specfcally focused our attenton to nformaton goods, customzed bundlng technque s not unque to nformaton goods, and t s especally attractve when number of goods offered s large but margnal cost and valuatons are such that t s optmal for consumers to purchase more than one good. For example, some restaurants have tradtonally bundled a selecton of two or three sde dshes wth a meal where consumers choose from a specfed set. More recently, McDonalds changed ther menu from tradtonal second-degree prce dscrmnaton wth fxed bundles ( value meals wth a sandwch, french fres and drnk) to an arrangement where customers choose two sde dshes form a lst of tems. In addton to the potental for practcal use, our customzed bundlng analyss provdes another smplfcaton to the general problem of mxed bundlng that may be approprate n some crcumstances. Gven the complexty of the general problem, there has been tremendous nterest n the marketng, management, computer scence, and economcs communtes for approaches that yeld tractable analytc bundlng solutons

28 Appendx (Proofs): Proof of Result (suffcency). Startng the program descrbed by (), we show that t can be converted to the problem stated n (2). Consder any collecton of consumers {} where the optmal soluton to () s { x } s.t. x = m for all members of ths set. For every other value of m for ths set of consumers, there must exst some x, wth x= m x for ths set of x. By the suffcent such that W ( x) s maxmzed, thus w ( m ) and W ( ) = w ( m) condton, all consumers n ths set must be such that w ( m) w( m) =. We now establsh an equvalence p( x ) = pm ( ) for all consumers n ths set. If there s only one such element, there s a - mappng between x m and thus p( x ) = pm ( ) for that value of m. If there are two or more, then choose any two arbtrarly, l x and x. For both of these bundles, ether IR or IC must bnd, otherwse p() s unbounded. and labelng them k Changng varables n the wllngness to pay functons from x to m and applyng the assumed condton we can k l x and rewrte the constrants as: wm ( ) p( x ) 0, wm ( ) p( ) 0 k j j l j j wm ( ) p( x ) w( x / ) p( x ) j and wm ( ) p( x ) w( x / ) p( x ) j. There are 4 possble cases, both IR are bndng, both IC, and two cases where IR and IC bnds. If IR s bndng for both j l x x. If both IC bnd, then t must bnd at some value j for then t mmedately follows that: p( ) = p( ) = p( m) whch ( j j w x / ) p( x ) s maxmzed. Ths value (call t Q) does not depend on k or l. Thus f both IC l k constrants hold, wm ( ) p( x ) = wm ( ) p( x ) = Q or ( j l p x ) = p( x ) = p( m). The case where IC bnds and IR bnds cannot be an optmum snce t mples that Q must be both greater than and less than zero. Ths establshes that, x = m, p( x ) = pm ( ). Substtutng for wm ( ), Cm ( ), pm ( ) for C( x ), p( x ) and deletng redundant constrants we obtan the customzed bundlng problem shown n (2) (necessty). Gven an optmal customzed bundlng schedule p( m) m, suppose that there exsts two bundles k l x and x for some m where j l x = m where p( x ) > p( x ). From the same type of argument shown above, f at least constrant (IR, IC) must be bndng for each, ths mples that W ( x j ) > W ( x l ) whch contradcts the condton W k ( x k ) = W l ( x l ) = w( m). Proof of Result 2: (by constructon) Proof (by constructon). Let the th order statstc of the valuaton of goods gven by F( v) be denoted as X : (where the largest order statstc s gven by X : ). The random wllngness to pay for any consumer s wm ( ) X : = k= m k. Ths functon does not depend on the consumer examned (equvalence). For any gven dstrbuton there exsts a dstrbuton for each order statstc and summaton s a Borel measurable functon. Therefore, there exsts a cdf for wm ( ) for each m (exstence). Fnally, we know snce E w m E X : E X : E[ w( m ) ] < [ ( ) ] = [ ] [ ] <. Across each consumer, wm ( ) = m = ndependent and dentcally dstrbuted wth fnte mean (snce Ewm [ ( )] E[ wm ( ) ] < ). Therefore, lettng Q Q represent the number of customers, average valuaton wm ( ) Ewm [ ( )] as Q by the Strong Law of Large umbers. Q = s 24