No 144. Bundling and Joint Marketing by Rival Firms. Thomas D. Jeitschko, Yeonjei Jung, Jaesoo Kim

Transcription

1 No 144 Bundlng and Jont Marketng by Rval Frms Thomas D. Jetschko, Yeonje Jung, Jaesoo Km May 014

2 IMPRINT DICE DISCUSSION PAPER Publshed by düsseldorf unversty press (dup) on behalf of Henrch Hene Unverstät Düsseldorf, Faculty of Economcs, Düsseldorf Insttute for Competton Economcs (DICE), Unverstätsstraße 1, 405 Düsseldorf, Germany Edtor: Prof. Dr. Hans Theo Normann Düsseldorf Insttute for Competton Economcs (DICE) Phone: +49(0) , e mal: normann@dce.hhu.de DICE DISCUSSION PAPER All rghts reserved. Düsseldorf, Germany, 014 ISSN (onlne) ISBN The workng papers publshed n the Seres consttute work n progress crculated to stmulate dscusson and crtcal comments. Vews expressed represent exclusvely the authors own opnons and do not necessarly reflect those of the edtor.

3 Bundlng and Jont Marketng by Rval Frms Thomas D. Jetschko Yeonje Jung Jaesoo Km May 014 Abstract We study jont marketng arrangements by competng frms who engage n prce dscrmnaton between consumers who patronze only one frm (sngle purchasng) and those who purchase from both compettors (bundle purchasers). Two types of jont marketng are consdered. Frms ether commt to a component-prce that apples to bundle-purchasers and then frms set stand-alone prces for sngle purchasers; or frms commt to a rebate off ther stand alone prce that wll be appled to bundle-purchasers, and then frms set ther stand alone prces. Both methods allow frms to rase prces and earn hgher profts. However, the effect of prce dscrmnaton on socal welfare depends on how prces are chosen. The rebate jont marketng scheme ncreases jont purchasng, whereas bundle prcng dmnshes bundle purchases. If the margnal socal value of a bundle over a sngle purchase s large, the former ncreases total welfare. However, welfare can also ncrease wth bundle prcng compared to non-dscrmnatory prcng. Keywords: Bundlng, Jont Marketng, Prce Dscrmnaton JEL Classfcatons: D4, D8 We would lke to thank Mark Armstrong, Roger Calantone, Jay Pl Cho, Carl Davdson, Yoss Spegel, Zhyun Xu, Aleks Yankelevch, and partcpants at the Sprng 013 Mdwest Economc Theory Meetngs and the Sprng 014 IIOC, as well as semnar partcpants at IUPUI, Mchgan State Unversty and the U.S. Department of Justce for ther helpful comments and suggestons. All remanng errors are our responsblty. Department of Economcs, Mchgan State Unversty. E-mal: jetschk@msu.edu Department of Economcs, Mchgan State Unversty. E-mal: jungyeo1@msu.edu Department of Economcs, IUPUI. E-mal: jaeskm@upu.edu

4 1 Introducton Jont marketng arrangements nvolvng separate frms n whch customers are charged dfferentally when they patronze multple frms are not new. Thus, the unlateral cancellaton of a jont marketng agreement that nvolved dscounted prcng to skers who bought passes to sk resorts run by separate operators gave rse to clams of llegal refusal to deal n the famed Aspen Skng anttrust sut. 1 In recent years smlar jont marketng arrangements nvolvng separate frms have been on the rse. CtyPASS, for example, s a package that bundles multple tourst attractons n nne popular destnatons n North Amerca. The package for Chcago allows admsson to fve attractons: the Shedd Aquarum, the Feld Museum, Skydeck Chcago, ether of the Adler Planetarum or the Art Insttute of Chcago, and ether of the John Hancock Observatory or the Museum of Scence and Industry. The attractons compete aganst each other for tme-constraned travelers who cannot vst more than a few places. At the same tme, the partcpatng venues offer dscounted prcng by sellng the CtyPASS.,3 In ths paper we nvestgate the ncentves and welfare mplcatons of nstances n whch companes choose prcng strateges that target consumers who make jont purchases across frms. The frms offer horzontally dfferentated products and consumers vew the frms goods as mperfect substtutes. However, each frm s product has unque features and attrbutes that gve a consumer who has already purchased a unt an added utlty from buyng the competng product as well, and so consumers are endogenously dvded nto two groups. Whle some consumers purchase a sngle product from ether frm, others purchase both products. We show that frms are able to leverage ther jont marketng schemes nto hgher prces and hgher profts compared to both unform prcng and ndependent prce dscrmnaton. However, the mechansm through whch prces and profts are rased depend on the nature of the jont marketng scheme used. When frms market to jont purchasers by each settng a prce for ther contrbuton to the bundle (bundle prcng), frms commt to a hgh bundle prce, whch drves consumers nto sngle-purchasng. Ths enables the frms to capture more surplus from snglepurchasng customers. In contrast, f jont marketng takes the form of each frm settng a rebate 1 Aspen Skng Co. v. Aspen Hghlands Skng Corp., 47 U.S. 585, (1985). As of Aprl 014, the total prce for vstng fve attractons s $187.95, whle the dscounted CtyPASS prce s $ See for more nformaton. 3 Other examples cover ted promotons and other jont marketng efforts, ncludng entertanment venues, such as move theaters, restaurants, or museums, and tour-operators gvng recprocal dscounts nvolvng potental compettors; or newspapers jontly marketed under the Newspaper Preservaton Act of Also, a very smlar arrangement to the one n the Aspen case s currently n place for multple sk resorts across four countres that are ted together through jont marketng of the epc-pass. 1

5 offer that apples to the the stand-alone prce when a consumer makes a jont purchase, a generous rebate s offered. Ths draws consumers nto jont purchasng. The ncreased demand for the bundle s renforced by chargng hgh stand alone prces, whch yelds hgher profts because the fxed rebate then apples to a hgh prce. Welfare s affected dfferently n the two cases. If ncremental values assocated wth purchasng a second unt are hgh, then the rebate scheme whch nduces more jont purchasng ncreases welfare; whereas bundle prcng, whch reduces jont purchasng, tends to decrease welfare. However, there are also more subtle welfare mplcatons of jont marketng. When a frm lowers ts prce to bundle-purchasers then ths has both a demand-nducng effect n that some new customers are attracted to the frm; as well as a demand-shftng effect, as exstng customers of the frm obtan the second good n order to secure an effectve prce reducton on the frst good. An mplcaton of the latter effect s that consumpton decsons are nduced that do not lead to ncreases n welfare, so there are stuatons n whch jont marketng reduces total welfare regardless of whch scheme s used. There s an extensve lterature on bundlng for the purpose of prce dscrmnaton. 4 Early papers n ths lterature, however, restrct attenton to the case where bundle dscounts are offered by a mult-product monopolst (e.g., Adams and Yellen, 1976; McAfee et al., 1989; Armstrong, 1996; Rochet and Choné, 1998), rather than ndependent frms. Gans and Kng (006) are the frst to study the stuaton where bundle dscounts are offered by dfferent frms through jont marketng. In contrast to our settng, the two products sold are ndependent, yet all consumers must purchase both goods so that there are no sngle purchasers. They show that the unlateral bundlng by the par of frms aganst other frms s proftable, whereas bundle rebates by both pars of frms do not ncrease ther profts. At the same tme, mutual jont marketng dmnshes socal welfare substantally. In our settng, wth frms offerng partal substtutes, prce dscrmnaton through jont marketng s always proftable and ts mpact on socal welfare depends on the mechancs of the jont marketng. 5 The most closely related paper to ours s Armstrong (013), who studes ncentves to offer bundled dscounts by separate sellers n a very general setup. He shows that when product valuatons are negatvely correlated and/or sub-addtve (so that products are partal substtutes), frms beneft by offerng ndependent dscounts to consumers. He extends the analyss to show that co- 4 Armstrong (006) and Stole (007) provde excellent surveys. 5 Brto and Vasconcelos (Forthcomng) buld on the nvestgaton of Gans and Kng (006) by consderng frms that produce vertcally dfferentated goods. Ther man nsght s that bundled dscounts may nduce a decrease n consumer surplus and always nduce a reducton n total welfare.

6 ordnated dscounts mplemented by frms reduce competton by mtgatng the substtutablty of products and reduce total welfare. We also focus on sub-addtve and negatvely correlated values; but we consder asymmetrc frms and contrast dscounts wth a polcy of frms commttng to prces for ther contrbuton to the bundle. Compared to settng dscounts, frms commttng to bundle prces before settng stand-alone prces actually ncreases profts. The dscount polcy nduces more purchases, however, and total welfare s ted to total consumpton whenever goods are not too closely substtutable so that welfare can ncrease due to jont marketng. There s also a nascent lterature on the mpact of jont purchases to whch our paper contrbutes. Gabszewcz and Wauthy (003) analyze how jont purchases affect prce competton and show that varous types of equlbra arse dependng on the value of ncremental utltes from jont purchasng. Km and Serfes (006) and Anderson et al. (01) extend Gabszewcz and Wauthy (003) by nvestgatng jont purchasng n a horzontally dfferentated market. Km and Serfes (006) ask under what condtons the Prncple of Mnmum Dfferentaton s restored when frms choose ther locaton on the Hotellng lne; and Anderson et al. (01) fnd a nonmonotonc relatonshp between equlbrum prces and qualtes under jont purchasng. There, the addtonal gan by jont purchasng s valued more by closer consumers so frms have an ncentve to sacrfce some sales and set hgh prces to prevent jont purchases. In contrast to our work, these papers abstract from prce dscrmnaton as a motvaton for jont marketng. 6 Somewhat related to jont marketng arrangements are several other recent papers on jont prcng n the context of patents and patent poolng (e.g., Lerner and Trole, 004; Cheng and Nahm, 007; Cho, 010; Rey and Trole, 013; Jetschko and Zhang, 014). Whle ths work generally does not consder prce dscrmnaton, Lerner and Trole (004) and Rey and Trole (013) examne how ndvdually set royalty rates nteract wth prcng n patent pools. Our fndngs shed further lght on ths ssue and provde addtonal nsghts as n our settng we allow for the frms contrbutons to the bundle to asymmetrc and we consder alternatve jont marketng structures. The paper s organzed as follows. In Secton we ntroduce the base model and solve for equlbrum unform prces for both sngle and jont purchasng regmes and also study ndependent prce dscrmnaton. Secton 3 contans the two forms of jont marketng: bundle prcng and bundle rebates; and draws some ntal comparsons to unform prcng and jont prce dscrmnaton n terms of prces and consumpton patterns. Profts, consumer surplus, and total welfare are 6 Gabszewcz et al. (001) also consder competton between frms that produce complementary goods. There, too, multple equlbra emerge only for ntermedate degrees of complementarty. 3

7 presented n Secton 4, whch then also establshes the man fndngs between the dfferent welfare and consumpton mplcatons of the prcng schemes. The fnal secton concludes. Detaled dervatons and proofs are relegated to the Appendx. The Model and the Benchmarks We consder two frms that offer partally substtutable products. Regardless of whch frm s product s purchased, when consumers make a purchase, the base utlty from consumng the product s gven by V. However, frms are asymmetrc n that each frm s product has dosyncratc features that are only obtaned when purchasng that partcular product. We let v denote the consumer s gross added value from usng product s unque features. That s, a consumer who purchases product obtans the base value of havng purchased an tem (V) and then the added margnal value of product s features (v ) for a total (gross) utlty of V + v. In contrast, a consumer who purchases both products receves the base utlty and then each product s addtonal margnal value for a total (gross) utlty of V + v + v j. For example, PlayStaton offers free onlne games, whereas W s more famly frendly and has more games that are sutable for chldren famles who care about both of these attrbutes may purchase both systems, whereas others may be satsfed wth smply havng a hgh-end gamng system. We further assume that the two products are horzontally dfferentated, e.g., some have an ntrnsc preference for one product over the other (prefer the W over PlayStaton or vce versa), whch we capture through an extenson of the Hotellng (199) model. Thus, when a consumer located at x [0, 1] purchases product 1 only, her payoff s gven by U 1 (x; p 1 ) V + v 1 tx p 1, where t denotes the (lnear) transportaton costs. When she purchases product only, her payoff s gven by U (x; p ) V + v t(1 x) p. And because the common attrbutes of the product s captured by V, the payoff from a jont purchase s gven by U 1 (p 1 ) V + v 1 + v t p 1, where p 1 s the prce pad by a consumer who purchases products 1 and together. 7 Note that as the dosyncratc value from ether of the two products ncreases, more consumers undertake a jont purchase all else equal. Ths can lead to a corner soluton, n whch all consumers purchase both products. Also, f the ncremental values of the dosyncratc characterstcs are too small, then an equlbrum n whch some consumers purchase more than one unt my fal to exst. To rule out these trval cases we make the followng assumpton that holds throughout 7 We assume that V s suffcently large so that all consumers buy at least one product. Ths s not a partcularly restrctve assumpton; ndeed f the added values that each frm provdes are suffcently hgh then even wth V = 0 the market s covered. 4

8 the paper. Assumpton 1. v ( t v j, t ),, j = 1, and = j..1 Unform Prcng and Jont Purchasng We begn by assumng unform prcng, so consumers who purchase from both frms pay a prce of p 1 = p 1 + p. Lettng x denote a consumer who s ndfferent between buyng from frm only and buyng from both frms the ndfference condton U ( x ; p ) = U 1 (p 1 ) provdes x 1 = 1 (v p )/t and x = (v 1 p 1 )/t. Ths mples that the mass of consumers who buy good only s n = 1 v j p j, = 1, ; t and, assumng a postve measure of jont purchasers, the mass of consumers who buy both goods s gven by n 1 = x x 1 = v 1 + v p 1 p 1. t Frm 1 Frm Ths gves each frm s demand functon as 0 x J 1 x J 1 1 q (p, p j ) = + 1 t (v v j + p j p ) f p v + v j t p j, Good 1 only Jont purchase Good only 1 t (v p ) f p v + v j t p j. Fgure 1: The locaton of margnal consumers (1) p v + v j t p j 1 t (t v j + p j ) q Fgure Fgure 1: Frm : Frm s nverse s demand demand functon functon Note that demand has an nward knk at v + v j t p j (see Fgure 1). Above the knk prces p are so hgh that there are no jont purchasers, yeldng the standard Hotellng model. Below the p = v 1 + v t p knk prces are low enough to have jont purchasers. Here 1 consumers are more prce senstve (demand s flatter), because the value-added of a second unt les below the value of purchasng v 1 + v t the frst unt. Note also that the other frm s prce s rrelevant on ths segment and each frm 5 S J

9 behaves as a monopolst wth regard to provdng ther dosyncratc value v ths s because when prcng to the jont-purchasers on the margn, there s no busness stealng from the rval. Gven demand, frm s proft functon s π (p, p j ) = p q (p, p j ); the Appendx shows that ths yelds best responses of 1 φ (p j ) = (t + v v j + p j ) f p ˆp j = ( 1)v + v j t, 1 v f p ˆp j = ( 1)v + v j t. The key mplcaton of the knk n demand s that each frm s margnal revenue s non-monotonc, and therefore the frst-order condton may be satsfed twce possbly permttng two dfferent prcng strateges: prcng hgh to compete for sngle purchasers wth less elastc demand; or prcng low to attract jont purchasers whose demand s more elastc. As a result, frms best response correspondences are not contnuous. Fgure depcts the three possble confguratons. p BR 1 p p BR 1 v 1 v ˆp 45 (p US 1, pus ) BR p 1 ˆp v 1 v 45 BR 1 BR (p UJ 1, puj ) p 1 v ˆp 1 v 45 (p US 1, pus ) (p UJ 1, puj ) BR p 1 ˆp 1 1 v 1 v 1 (a) when (v 1, v ) Φ S \ Φ J (a) Only Sngle Purchasng 1 v 1 v 1 ˆp 1 (c) when (v 1, v ) Φ J \ Φ S (b) Some Jont Purchasng 1 v 1 v 1 (b) when (v 1, v ) Φ S Φ J (c) Both Eq. Confguratons Fgure : Best response correspondences (here: v 1 > v ) The followng proposton gves the equlbrum confguratons, the proof s n the Appendx. { Proposton 1. Let Φ S := (v 1, v ) (3 )v 1 + v 3 t and (3 )v + v 1 3 } t, { and Φ J := (v 1, v ) ( + 1)(t 1 v ) v 1 t and ( } + 1)(t 1 v 1) v t ; then f () (v 1, v ) Φ S, there s a sngle-purchasng regme n whch p U S = 1 3 (3t + v v j ), n U S = 1 6t (3t + v v j ) and n U S 1 = 0, π U S = 1 18t (3t + v v j ) ; 1 6 3

10 () (v 1, v ) Φ J, there s a regme wth jont purchases n whch p U J = 1 v and p U J 1 = 1 (v + v j ) n U J = 1 t (t v j) and n U J 1 = 1 t (v 1 + v t), π U J = v 4t. As shown n Fgure 3, when the addtonal gan from a second purchase s small,.e., (v 1, v ) Φ S \ Φ J, there s a unque equlbrum n whch frms charge hgh prces to sngle purchasers. As dosyncratc values ncrease, (v, v j ) Φ S Φ J, frms may prce low to attract jont-purchasers. However, the reducton n proft from sngle purchasers due to the lower prces s greater than the gan from addtonal sales so, assumng coordnaton on the preferred equlbrum, frms keep prces hgh and only target sngle-purchasers. Lastly, when a jont purchase adds a large addtonal gan, (v 1, v ) Φ J \ Φ S, there s agan a unque equlbrum as secondary customers are an attractve prospect and frms lower ther prces to capture these consumers. t v t Φ J t Φ S Φ J Φ S t t t v 1 Fgure 3: Sngle- Fgure and Jont-Purchasng 1: Equlbrum pattern Confguratons n the benchmark under Unform Prcng. Jont Purchasng and Independent Prce Dscrmnaton Customers who are located close to a frm purchase from that frm; and those far removed from ths frm wll not purchase from the frm. Thus, the purchase decson reveals somethng about the consumer s locaton on the lne. Frms can use ths nformaton to segment the market and prce dscrmnate even when they do not engage n jont marketng efforts. Ths scenaro yelds 7

11 the second benchmark. Specfcally, suppose that frms provde a rebate ρ off ther prce to consumers who buy the rval s good.e., who purchase both goods. The jont purchaser s prce s p 1 = (p 1 ρ 1 ) + (p ρ ), yeldng a net payoff of U 1 (p 1 ) = V + v 1 + v t (p 1 ρ 1 ) (p ρ ). The locatons of the margnal consumers follow from U (x ; p ) = U 1 (p 1 ), mplyng that n = 1 (v j p j + ρ + ρ j )/t and n 1 = (v 1 + v p 1 p + ρ 1 + ρ )/t 1. Frm s proft maxmzaton problem s gven by max p, ρ π = p n + (p ρ )n 1. () Whle the problem entals a more complex prcng strategy than unform prcng, the soluton s n some sense easer n that the frst order condtons are necessary and suffcent and there s a unque equlbrum. Proposton. Under ndependent prce dscrmnaton, p IPD = 1 3 (3t + v v j ), ρ IPD = 1 3 (t v j) and p IPD 1 = 1 3 (t + v 1 + v ) n IPD = 1 3t (t v j) and n IPD 1 = 1 3t (v 1 + v t), π IPD = 1 9t (v + v j + tv 4tv j + 5t ). Note that frm s choce of the bundle rebate s ndependent of v and s decreasng n v j, mplyng that as frm j s unque features are more valuable, frm can reduce the bundle rebate, as the rval product s attractveness serves to ncrease the demand. Compared to unform prcng, the stand alone prce under ndependent prce dscrmnaton s dentcal to the equlbrum prce n the sngle purchasng regme. That s, the prce charged to the relatvely nelastc (.e., the captured ) consumers s the same as when only these are targeted. Ths follows readly, snce the margn on whch ths prce operates s the same across the two cases. On the other hand, the prce pad by jont-purchasers under unform prcng s lower than the prce pad by jont-purchasers under ndependent prce dscrmnaton,.e., p U J = v / < (t + v 1 + v )/3 = p IPD 1. Ths s because when the frm prce dscrmnates and lowers ts prce to jont purchasers, t cannbalzes ts hgh-proft sales to nelastc consumers who otherwse snglepurchase ths lmts the amount the frm s wllng to lower the prce to jont purchasers compared to the case of unformly low prces (that s, n s decreasng n the rebate ρ ). An mplcaton of consumers beng able to obtan rebates upon purchasng the second product s that despte the bundle prce beng hgher than the unform prce under jont purchasng, the 8

12 mass of consumers who jont-purchase ncreases when the frms prce dscrmnate. Ths can be seen by comparng n IPD 1 = (v 1 + v t)/3t to n U J 1 = (v 1 + v t)/t and n U S 1 = 0. That s, the hgh stand alone prce pushes people nto makng a jont purchase n order to obtan the prce break. Thus, ndependent prce dscrmnaton leads to (some) jont purchasng and ncreased profts when otherwse there would only be sngle purchasng under unform prcng. Moreover, ndependent prce dscrmnaton also rases profts when under unform prcng there are some jont purchasers, because both sngle purchasers and jont-purchasers now face hgher prces, and there are more purchases n total as some consumers shft from sngle to jont purchasng to avod the hgher non-dscounted prce that sngle purchasers face. Ths result s smlar to one n Armstrong (013, Proposton 3). He shows that f there are separate sellers, then each seller has an unlateral ncentve to offer bundle dscounts for consumers who buy goods from both sellers whenever the demand for the bundle product s more elastc than total demand for one frm s product. 3 Bundlng and Jont Marketng Consder now jont marketng. Frms act non-cooperatvely n ther prcng decsons so jont marketng does not take the form of prce colluson. Jont marketng s done n two stages, wth the frms frst commttng to ther prcng strateges vs-á-vs jont purchasers, and then settng ther stand alone prces n lght of the offer that s made to jont purchasers. We consder two types of jont marketng schemes. Frst we suppose that frms non-cooperatvely set the prce for ther product n the bundle, p, and then choose stand-alone prces p n lght of the bundle prce of p1 P p 1 + p. The second scheme we consder s a rebate scheme n whch frms frst announce a rebate offer ρ, and then choose the stand alone prce n lght of the bundle rebate of ρ := ρ + ρ j. 8 In prncple ether scheme mght be mplemented by the frms, possbly through a thrd party facltator, or, for nstance, through marketng on the nternet. In practce, however, t could be that for logstcal reasons ether one or the other scheme s more readly nsttuted. 8 Camnal and Matutes (1990) make a smlar dstncton between prces and rebates. In ther model, however, customers are dstngushed by purchasng across perods. 9

13 3.1 Bundle Prce Marketng Wth bundle prcng, each frm frst chooses the prce p for ts contrbuton to the bundle. Ths yelds the bundle prce p P 1 p 1 + p that wll be marketed to consumers. The frms stand-alone prces p are then set n lght of ths bundle prce. Gven the choce of prces, consumers make ther purchasng decsons, wth a frm recevng p for each bundle that s sold, and obtanng p from consumers that purchase only frm s product. Facng a bundle prce of p P 1 = p 1 + p, a jont purchaser s net payoff s U 1 (p P 1 ) V + v 1 + v t p1 P, resultng n np = 1 (v j p1 P + p )/t and n1 P = (v 1 + v + p 1 + p p1 P )/t 1. We assume that n1 P 0, whch yelds the canddate prcng structure. We then confrm the condtons on the canddate prcng structure that entals a postve measure of jont purchasers. Gven the bundle prce p P 1, each frm chooses p to maxmze max p π P = p n P + p n P 1. From the frst-order condtons, each frm s stand-alone prce s: p ( p, p j ) = ( ) t vj + p + p j. Note that p s ncreasng n p and p j. Thus, commtment to the component prce leads to rasng the stand-alone prce. Moreover, p rses wth t and falls wth v j, because an ncrease n t and a decrease n v j make demand for frm s product less elastc. In antcpaton of the stand alone prces as a functon of the bundle prce, the frms prce ther ndvdual contrbuton to the bundle: max π P = t v j + p + p j t v j + p j p t + p v + v j p p j. t Jont purchases take place provded that the dosyncratc values of the two frms are suffcently hgh, yeldng the followng equlbrum. Proposton 3. When v + v j t, there exts a jont marketng equlbrum n whch frms set bundle 10

14 component prces wth p P = 1 4 (5t + v v j ), p = 1 (t + v ) and p P 1 = 1 (t + v 1 + v ), n P = 1 4t (3t v j) and n1 P = 1 4t (v 1 + v t) π P = 1 ) (11t + (v 3v j )t + (v + v j 16t ). Compared to the case of unform prcng wth jont purchasers the prce pad by jont purchasers s now hgher,.e., p1 P > pu J 1 ; and so s the prce pad by sngle purchasers when compared to the case of unform prcng wth only sngle purchasng, p P > p U S, (whch, of course, s hgher than the unform jont purchasng prce,.e., p U S > p U J ). That s, when jontly marketng the bundle prce, both prces (those appled to sngle purchasers and those appled to jont purchasers) are above the the unform prces despte all prces beng set non-cooperatvely. The fact that the bundle prce s hgher under jont marketng suggests that fewer consumers purchase the bundle. However, the fact that the stand alone prce s also hgh, makes the bundle attractve n that t leads to a prce dscount. The former effect domnates the latter so that fewer jont purchases take place when compared to the case of unform prcng wth jont purchases,.e., n P 1 = (v 1 + v t)/4t < n U J 1 = (v 1 + v t)/t, gven v 1 + v t; (naturally, more jont purchases take place when compared to unform prcng wth no jont purchases). Intutvely, by commttng to hgh bundle component prces the frms push demand towards sngle purchasng and use ths ncreased demand to charge hgher stand alone prces. Overall then, profts are greater compared to ether of the unform prcng cases,.e., π P > π U S, π U J. Consder now the comparson to ndependent prce dscrmnaton. Wrtng p = p ρ, the ndependent prce dscrmnaton optmzaton problem for frm, Equaton (), s: max π IPD (p, p j, p, p j ). p,p The frst-order condton wth respect to the bundle component prce s π IPD p n = p + n 1 + p IPD p n 1 p = 0, (3) where p IPD = p IPD ρ IPD. In contrast, wth the jont marketng scheme, frm s choce of ts bundle component prce s mpled by max π P p (p P ( p, p j ), p P j ( p, p j ), p, p j ). 11

15 From the envelope theorem, the effect of changes n the stand alone prce p on π P s of second order ( π / p = 0), so the optmal bundle prce component of frm, p, satsfes the followng frst-order condton: dπ = π 0 π + p d p p p P + π p P j p p j p [ ] 0 n = n p + n n p + p n p p p } {{ } + 1 p P j p j p j p =Drect Effect } {{ } =Indrect Effect = 0. (4) The frst term captures the drect effect that frm s component prce has on profts. It mrrors the frst order condton that the frm has n ndependent prce dscrmnaton, namely Equaton (3). The second term s the ndrect effect that frm s component prce has on profts by affectng frm j s stand alone prce that s set n lght of the bundle prces. The ndrect effect s postve, because commttng to a bundle component prce leads to the rval rasng ts stand-alone prce. Evaluatng the frst-order condton (4) at p = p IPD whch mples that p P > p IPD. dπ d p p = p IPD yelds = p n 1 p j p P j p p = p IPD Thus, the frms charge a hgher bundle prce when they commt to the bundle component prces pror to the stand-alone prces. As a consequence, the mass of jont purchaser s smaller n the bundle prce commtment case than n the case of ndependent prce dscrmnaton,.e., n P 1 < n1 IPD. Ths makes sngle-purchasng relatvely more attractve and allows the frms to rase the stand alone prce as well, p P > p IPD, and so profts on each unt sold exceed those of ndependent prce dscrmnaton, leadng to an overall ncrease n proft, π P > π IPD. > 0, 3. Rebate Marketng In the case of non-cooperatve jont rebate marketng, each frm determnes a dscount or rebate of ρ that wll be appled to ther stand alone prce to any consumer who purchases both goods. Together the dscounts consttute a rebate offer of ρ := ρ + ρ j that wll be jontly marketed to consumers. The frms stand alone prces are then set n lght of the rebate offer. Jont purchasers pay a prce of p1 R p + p j ρ so that frm receves a net prce of p ρ for ts own product n the bundle, and obtans p from sngle purchasers. 1

16 Gven the jont marketng rebate of ρ, a jont purchaser s net payoff s U 1 (p1 R ) V + v 1 + v t (p 1 + p ρ) mplyng n R = 1 (v j p j + ρ)/t and n1 R = (v 1 + v p 1 p + ρ)/t 1, wth jont purchasng ncreasng as the rebate offer ρ ncreases. In lght of the jont marketng rebate ρ, frm solves max p π R = p n R + (p ρ ) n R 1, where ρ can be thought of as the cost of havng a product n the bundle. The frst-order condton yelds the ndvdual frm s stand-alone prce p (ρ, ρ j ) = v + ρ + ρ j. To compare wth the benchmark case of unform prcng wth jont purchasng, the prce can be wrtten as p (ρ, ρ j ) = p U J + ρ + ρ j /. Ths shows the advantage of the jont marketng rebate scheme: each frm rases the stand-alone prce to consumers by ts rebate and by half the rval s rebate as well. That s, snce the frms choose the rebate before choosng ther stand-alone prces, ther choce of rebate serves as a commtment to rase the stand-alone prces. Gven the optmal stand-alone prces as a functon of the jont rebate, each frm selects ther partal rebate non-cooperatvely: max π ρ R = v + ρ + ρ j t v j ρ + v + ρ j t v + v j + ρ + ρ j t. t The second-order condtons are satsfed, and the problem has the unque soluton. Proposton 4. When frms undertake a jont marketng arrangement n whch a rebate s gven to bundle purchasers, p R = 1 4 (6t + v v j ), ρ R = 1 (t v j) and p R 1 = 1 4 (4t + v 1 + v ), n R = 1 4t (t v j) and n1 R = 1 4t (v 1 + v ), π R = 1 ) (1t + 4(v v j )t + (v + v j 16t ). Compared to the unform prcng regme wth jont purchasng, jont marketng agan leads to an ncrease n the prce for jont purchasers. That s, jont purchasers pay p R 1 = (4t + v 1 + v )/4 > (v 1 + v )/ = p U J 1. And, as was also the case for jont marketng wth bundle component prcng, wth a rebate beng jontly marketed, the stand alone prces are hgher than they are when a unform prce s charged to sngle purchasers, p R 13 > p U S. In contrast to the case of jont marketng

17 through bundle prcng, n ths case the hgher stand alone prce drves purchasers to buy the bundle n order to obtan the rebate despte the effectve bundle prce beng hgher than before, so n1 R > nu J 1. Once agan, jont marketng leads to ncreased profts compared to unform prcng, that s π R > π U S, π U J. Turnng to the comparson wth ndependent prce dscrmnaton, wthout jont marketng of the rebate, frm solves max π ρ,p IPD (p, p j, ρ, ρ j ). The frst-order condton wth respect to the ndvdual rebate n ths case s π IPD ρ = p n ρ + (p ρ ) n 1 ρ n 1 = 0. (5) In contrast, when jontly marketng the bundle rebate, the optmal ndvdual rebate s derved by solvng max π ρ R wth the followng frst-order-condton dπ dρ = π 0 π + p ρ p R + π p R j ρ p j ρ [ n = p n 1 + (p ρ ) n ] 1 ρ ρ } {{ } =Drect Effect (p R (ρ, ρ j ), p R j (ρ, ρ j ), ρ, ρ j ), [ n + p + (p ρ ) n ] 1 p R j p j p j ρ } {{ } =Indrect Effect = 0. (6) Agan, the drect effect reflects the condton obtaned n ndependent prce dscrmnaton, Equaton (5); and the ndrect effect captures the effect that one s own rebate has on the stand-alone prcng decson of the rval. Evaluatng the frst order condton (5) at ρ = ρ IPD mplyng that ρ R dπ dρ > ρ IPD. ρ =ρ IPD = [ n p + (p ρ ) n ] 1 p R j p j p j ρ ρ =ρ IPD > 0, yelds That s, frm ncreases ts rebate when t commts to jontly marketng the bundle rebate pror to settng the stand-alone prce. The effect of ths s to ncrease the attractveness of the bundle relatve to the stand alone prce. For ths reason, there are more jont purchasers when rebates are jontly marketed compared to the case of ndependent prce dscrmnaton,.e., n R 1 = (v 1 + v )/4t > (v 1 + v t)/3t = n IPD 1. 14

18 However, because frms set stand alone prces only after the rebates have been fxed, ths commtment leads to the ablty to ncrease the stand alone prces, dp R /dρ j > 0. Ths commtment s suffcently strong so that jont purchasers end up payng more wth jont marketng then wth ndependent prce dscrmnaton, despte the former gettng hgher rebates,.e., p R 1 = (4t + v 1 + v )/4 > (t + v 1 + v )/3 = p IPD 1. In sum, when the frms jontly market the bundle rebate they gve generous rebates, as ths allows very hgh stand alone prces that drve consumers to jont purchasng, who despte large rebates are very proftable, because the rebate s taken off very hgh ntal (.e., stand-alone) prces. 4 Welfare Analyss We now consder the welfare effects of the varous prcng schemes derved. We begn wth profts, then move to consumer surplus, and fnally compare total welfare for the cases analyzed. 4.1 Proft Comparsons Profts are derved n the prevous secton and are gven n the propostons coverng the equlbrum confguratons. Here, we dscuss ther relaton across the dfferent prcng schemes. Frms profts are ncreasng n ther ablty to engage n and coordnate prce dscrmnaton. 9 Thus, profts are lowest under unform prcng and frms do better when they ndependently prce dscrmnate. And regardless of whether the frms choose a bundle prce or a rebate scheme when they launch a jont marketng strategy, ther profts ncrease over and above what ndependent prce dscrmnaton acheves. Ths occurs despte the non-cooperatve nature of the agreement, because jont marketng commts the frms to the bundle strategy a commtment that s leveraged nto hgher prces and profts. Gven that proft comparsons are qute ntutve when t comes to the degree of commtment and coordnaton, the ntrgung queston s whether jont marketng through bundle prcng or through rebates s preferred. Whle t may be the case that for logstcal reasons one method or the other may not be practcal or avalable, whenever frms can engage n ether scheme t s worth knowng whch yelds the greater proft. Comparng Proposton 3 and Proposton 4 shows that the answer to whch scheme domnates 9 Ths s n contrast to Thsse and Vves (1988) who fnd that prce dscrmnaton n the sngle-purchasng Hotellng model decreases profts. The reason for the dfference s that prce dscrmnaton wthout jont purchasng leads to fercer Bertrand competton, whereas wth the possblty of jont purchasng there sn t head-to-head competton and busness stealng. 15

19 s not unambguously clear. In partcular, when frms use a jont marketng scheme n whch the bundle s prced, then the effectve prce for both the bundle and the stand alone products are hgher when compared to the case of a rebate scheme, that s, p P 1 = pp 1 + pp = (v 1 + v + t)/ > (v 1 + v + 4t)/4 = p R 1 + pr ρ = pr 1, and pp = (5t + v v j )/4 > (6t + v v j )/4 = p R. On the flp sde, n the case of rebates the mass of jont purchasers s larger when compared to the bundle prce marketng arrangement, n1 R = (v 1 + v )/4t > (v 1 + v t)/4t = n1 P, so the rebate scheme leads to a greater sales volume when compared to the bundle prcng regme. Whch of the two arrangements s more proftable depends on the sze of the dosyncratc values of the two products. Specfcally, note from Proposton 3 and Proposton 4 that π R π P = ((v j t) v )/16t, from whch t follows that f the dosyncratc values are smlar to oneanother (close n sze), then the bundle prcng s preferred whenever the dosyncratc values are suffcently large. When the dosyncratc values dffer (are asymmetrc), constellatons can arse n whch the frm wth the smaller v would prefer the rebatng scheme whereas the other frm would prefer to nsttute a bundle prce. Ths suggests that frms that are very dfferent may fnd t dffcult to decde on a jont marketng scheme. However, even n these cases t turns out that the overall proft rankng s unambguous n favor of bundle prcng. Thus, when dosyncratc values are large, jont profts are hgher under bundle prcng, even f frms ndvdual dosyncratc values dffer substantally. As a consequence, f sde payments are permssble, then even under noncooperatve jont marketng, frms can agree on whch scheme to employ. In sum, lettng Π denote ndustry profts, we have the followng theorem. Theorem 1 (Profts under Prce Dscrmnaton and Jont Marketng). Frms profts are ncreasng n the extent of prce dscrmnaton n that ndependent prce dscrmnaton ncreases profts above unform prcng, and prce dscrmnaton through jont marketng rases profts even more: Π U S, Π U J Π IPD < Π R, Π P. Moreover, the bundle prcng scheme s preferred to the rebate scheme whenever the dosyncratc values are large enough to mplement the bundle prce; otherwse the rebate scheme s chosen: Π P > Π R v 1 + v t. 16

20 4. Consumer Surplus Let CS X be the level of consumer surplus for X {U S, U J, IPD, P, R}, defned by CS X = = x1 X 0 X x U 1 (x; p1 X )dx + [ V + v 1 n X 1 + (v 1 + v ) n X 1 + v n X [ ] p1 X nx 1 + p1 X nx 1 + p X n X. 1 U 1(x; p X 1)dx X + U (x; p X X )dx x 1 x [ ] n X 1 t xdx + n1 X + 0 n X 0 xdx ] The frst term s the gross utlty from consumpton, the second term s the transportaton costs, and the fnal term s the total expendtures. 10 It s natural to conjecture that consumer surplus s drectly related to total consumpton, because consumers only purchase the second product when the added utlty exceeds ther transportaton costs. In that case consumer surplus would be drectly proportonal to n X 1. Indeed, ths logc apples when comparng unform prcng wth sngle purchasng to ndependent prce dscrmnaton. The stand alone prce s the same for both cases, but under ndependent prce dscrmnaton some obtan the lower prce and more consumers are drawn nto purchasng each frm s product. Thus, n IPD 1 > n U S 1 and CSIPD > CS U S. However, comparng unform prcng wth jont purchasng to ndependent prce dscrmnaton reveals that even though there are more purchases under prce dscrmnaton, n IPD 1 > n U J 1, consumer surplus s actually lower, CS IPD < CS U J. In ths case some consumers make the second purchase even f the addtonal margnal value from consumng the second unt s smaller than the added transportaton costs, smply because the second purchase allows them to take advantage of the lower prces for both the frst and the second purchase. And because the stand alone prce s hgher wth prce dscrmnaton, consumer surplus s lower, despte more total purchases. Ths dynamc, that consumers base ther decson on purchasng a second unt n part on how ths determnes the effectve prce pad on the frst unt, s mportant n understandng the mpact of jont marketng on consumer surplus. In partcular, as was shown n Secton 3, frms use ther commtment to the jont marketng scheme to ncrease overall prces. An unambguous result of ths s that consumer surplus s always lower under a jont marketng scheme compared to ndependent prcng schemes wth or wthout prce dscrmnaton. That s, CS U S, CS U J, CS IPD > CS P, CS R. And yet, because a jont marketng rebate scheme ncreases jont purchases, total con- 10 Note that x 1 U S = x U S so n U S 1 = 0. 17

21 sumpton s larger under ths scheme compared to any other: n1 R = max X n1 X. Fnally, the postve relatonshp between consumpton volume and consumer surplus holds agan when comparng the two jont marketng schemes. In partcular, consumer surplus s always larger under a rebate scheme compared to the bundle prcng arrangement, CS R > CS P, and so s total consumpton: n1 R > np 1. The man fndngs concernng consumer surplus are recapped n the followng theorem: Theorem (Consumer Surplus under Prce Dscrmnaton and Jont Marketng). All forms of prce dscrmnaton harm consumers, unless frms engage n ndependent prce dscrmnaton and thereby nduce jont purchases, when otherwse consumers only purchase a sngle good gven unform prcng, CS IPD > CS U S. Jont marketng rases prces compared to any of the other settngs and as a result leads to lower consumer surplus even when more s purchased, CS U S, CS U J, CS IPD > CS P, CS R. Fnally, consumers are better off under a rebate jont marketng scheme compared to the bundle prcng scheme, CS R > CS P. 4.3 Total Welfare We take the sum of consumer surplus and ndustry profts as a measure of socal welfare as usual. Because frms costs are normalzed to zero, consumer expendtures are equal to ndustry profts, so t s mmedate that socal welfare s the dfference between gross utlty and transportaton costs: TW X = agan wth X {U S, U J, IPD, P, R}. [ [ ] n X V + v 1 n1 X + (v 1 + v ) n1 X + v n X 1 t 0 n X xdx + n1 X + 0 xdx ], (7) There are two opposng effects of jont purchases on socal welfare. Frst, more jont purchases ncreases socal welfare because extra surplus s realzed from the consumpton of addtonal features. In the case of the standard Hotellng model wth nelastc consumer demand, there s no demand creaton effect wth prcng, as long as the market s covered. Here, however, the demand for a frm s product ncrease when t charges a relatvely lower prce for jont purchasers. The frst term n (7) captures the base surplus assocated wth consumng ether good, as well as the added gan assocated wth the dosyncratc features of the two goods. The second effect of jont purchasng s that t ncreases total transportaton costs, whch s captured n the second term n (7). Recall from the dscusson on consumer surplus that added surplus through addtonal consumpton need not be enough to offset the added transportaton costs, because the consumpton decson s based on the prce dfference between jont and sngle 18

22 purchasng due to prce dscrmnaton. As a result, the net effect on socal welfare depends on the relatve sze of the surplus creaton and transportaton cost effects. Whle the nterests of frms are often drectly opposed to those of consumers, assessng overall welfare requres closer scrutny n all but one case: compared to unform prcng n whch consumers only purchase one good, ndependent prce dscrmnaton ncreases consumer surplus whle also rasng frm profts. So total welfare s clearly greater under ndependent prce dscrmnaton compared to unform prcng when consumers do not engage n jont purchasng, TW IPD > TW U S. In contrast, ndependent prce dscrmnaton lowers consumer surplus when compared to unform prcng wth jont purchases. The reason for ths s that the ncreased surplus from added consumpton s more than offset by the hgher prces pad to frms by those who only purchased a sngle good. Ths latter effect, however, s a welfare-neutral prce transfer from consumers to frms. Therefore total welfare s also greater under ndependent prce dscrmnaton compared to unform prcng wth jont purchases, TW IPD > TW U J. Ths result s somewhat remnscent of welfare effects under thrd degree prce dscrmnaton, n whch welfare ncreases when consumpton (output) ncreases (see Varan, 1985), whch s the case here: n IPD 1 > n U J 1 > nu S 1 = 0. However, note that there s a crtcal dfference n our analyss compared to thrd degree prce dscrmnaton: n our model t s the prces that segment the markets nto sngle and jont purchasers, and consumers self-select nto whether they are sngle or bundle purchasers. Hence the comparson to prcng across dstnct markets wthout an arbtrage possblty s not apt n our settng. Nevertheless, the assocaton between greater consumpton and hgher surplus also holds across jont marketng schemes. Thus, total consumpton under a rebate scheme s greater than total consumpton under bundle prcng, n1 R > np 1, and total welfare s also greater n the former case, TW R > TW P. A postve assocaton between consumpton and total welfare holds more generally, provded that the products dosyncratc values are suffcently large and not too asymmetrc. Thus, Theorem 3 (Total Consumpton and Total Welfare). Whenever the dosyncratc values are not too small (3v 1 + 3v 90tv 1 90tv + 54t + 80v 1 v > 0) and not too asymmetrc, ((v 1 t /17) + (v t /17) 88 /89), then ncreases n total equlbrum consumpton mply ncreases n total welfare: n X 1 > n Y 1 TW X > TW Y, wth TW R > TW IPD > TW U J > TW P > TW U S. 19

23 The area dentfed n the Theorem 3 s gven n Fgure 4. v t TW P TW U S > 0 t t t TW R TW IPD < 0 t Fgure 4: Socal Welfare t v 1 The reason for the exstence of a threshold that values must exceed s ntutve. Frms are balancng ther prcng decsons across two margns, namely the margn at whch they capture new customers who otherwse only purchase from the rval, and the margn at whch ther own customers decde to become jont purchasers or reman sngle purchasers. As a result, the frms prcng decsons nduce transportaton costs that need not be offset by margnal ncreases n consumer surplus ted to consumpton of the second product. However, greater consumpton leads to greater surplus provded that the value from added consumpton s suffcently hgh to offset added transportaton costs that are ncurred by consumers whose second purchase s partly drven by the desre to lower ther expendture on ther frst purchase. Two aspects of the theorem are partcularly noteworthy. Frst, Theorem 3 demonstrates that total welfare s greatest when frms engage n jont marketng that sets a rebate for bundle purchasers compared to any of the other schemes, so jont marketng can ncrease welfare above what frms can do ndependently. Whle consumer surplus s lower under ths scheme than when compared to ndependent prce dscrmnaton, the ncrease n profts more than offsets the reducton n consumer surplus. However, there s an mportant caveat to note here. From the second part 1 of Theorem 1 we know that when dosyncratc values are hgh, frms actually prefer the bundle prcng scheme over the rebate scheme. Nonetheless, the fndng ponts n the drecton of whch type of jont marketng arrangements should be vewed as benefcal especally as the case of bundle prcng leads to the lowest mass of jont purchasers, and the lowest welfare of any equlbrum confguraton n whch there s jont 0

24 purchasng. Because of ths, the comparson between surplus through added consumpton and losses through addtonal transportaton s relatvely easy for the case of jont marketng: f a jont marketng agreement leads to greater consumpton then t lkely also ncreases total welfare. The second noteworthy fndng n Theorem 3 s that compared to unform prcng wth (only) sngle purchasng, (even) jont marketng through bundle prcng can rase total welfare. And, thus, the welfare mplcatons of jont marketng are not clear ex ante, but whenever dosyncratc values are hgh and not too asymmetrc, there s a chance that ether type of jont marketng may ncrease total welfare. There s, of course, an mmedate mportant corollary to Theorem 3. If the dosyncratc values of the products are not that large, then equlbrum consumpton need not be postvely correlated wth equlbrum welfare. The ntuton for ths s straghtforward: f the added value from consumng a second good s not that large, then prcng schemes that nduce the added purchase may generate less addtonal surplus than the added transportaton that the second purchase entals, thus reducng total welfare. In fact, when the dosyncratc values are very small, then no prcng scheme generates more total welfare than smple unform prcng wth sngle purchasng. Related to ths s the case where dosyncratc values are large yet very asymmetrc. Adverse welfare effects occur on the margn between jont and sngle purchasng when t comes to the relatvely lower-valued good when swtchng from ndependent prce dscrmnaton to bundle jont marketng. That s, bundle jont marketng wll nduce addtonal purchases of the relatvely lower valued good for whch the added transportaton costs are not offset by the added consumpton value. 5 Concluson We have studed the effect of prce dscrmnaton through jont marketng by frms producng mperfectly substtutable goods. The man fndng s that the mpact on overall purchasng and welfare depend on whether frms offer a bundle prce or offer rebates off ndvdual prces to consumers who buy from both frms. Frms may want to ncrease jont purchasng by settng bundle rebates before determnng stand-alone prces. Alternatvely, they may want to mtgate prce competton by prcng the bundle components before settng the ndvdual prces. Loosely speakng, when a second purchase adds lttle n terms of ncremental utlty, frms prefer to offer dscounts off ther prces. Ths has the effect of ncreasng the mass of customers to both frms, whch s benefcal because rent-extracton s lmted by the added value of the second unt. In contrast, f consumers place a hgh value on each of the goods and therefore the second 1

25 unt s relatvely valuable, frms prefer to drectly prce and market the bundle, as ths allows them to ncrease ther stand-alone prces and extract more rents from ther captured customers. However, we show that the frms nterests need not be algned n ths respect f ther ncremental values to the consumers vary greatly. Wth respect to welfare we fnd that f second purchases add a lot of addtonal utlty, then prce dscrmnaton that nduces added purchases rases total welfare. However, as frms fnd bundle prcng more proftable compared to the rebate scheme when ncremental values are hgh, total purchasng can actually decrease wth hgh addtonal values when both jont marketng optons are avalable to the frms. Nevertheless, there are parameter values for whch total welfare ncreases due to jont marketng compared to unform prcng regardless of whch jont marketng scheme s used. Indeed, whenever both dosyncratc values are suffcently hgh jont marketng rases total welfare; otherwse ndependent prce dscrmnaton s better from a total welfare perspectve. Appendx A Dervaton of best response correspondences We wll derve frm 1 s best response correspondence frst. Frm s problem wll be solved n a smlar manner. Gven the demand functon (1), the proft functon of frm 1 s gven by, ( ) 1 π 1 = + v 1 v +p p 1 t p 1 f p 1 v 1 + v t p, 1 t (v 1 p 1 )p 1 f p 1 v 1 + v t p. Consder frst the case n whch p 1 v 1 + v t p. Usng the frst order necessary condton, we dentfy the canddate canddate best reply b 1 (p ) = 1 (t + v 1 v + p ). It yelds a payoff π1 S(p ) = (t + v 1 v + p ) /8t. Notce that ths case s vald only when p 1 v 1 + v t p, so we need to have the condton: p 1 v 1 + v t p p 1 3 (v 1 + 3v 3t). For the case n whch p 1 v 1 + v t p, frm 1 sets the prce at p J 1 = v 1/, yeldng a payoff π J 1 = v 1 /4t. Ths payoff s vald only when p 1 v 1 + v t p, requrng the followng condton: p 1 v 1 + v t p p v 1 + v t. Thus, for p [ 1 3 (v 1 + 3v 3t), 1 v 1 + v t ] both choces satsfy the requrements. In order