Bundling with Customer Self-Selection: A Simple Approach to Bundling Low Marginal Cost Goods On-Line Appendix

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1 Bundlng wh Cusomer Self-Selecon: A Smple Approach o Bundlng Low Margnal Cos Goods On-Lne Appendx Lorn M. H Unversy of Pennsylvana, Wharon School 57 Jon M. Hunsman Hall Phladelpha, PA 94 lh@wharon.upenn.edu Pe-yu Chen Tepper School of Busness Carnege Mellon Unversy 5 Forbes Ave Psburgh, PA 523 pychen@andrew.cmu.edu

2 Onlne Appendx Appendx A: Proof of Resul : Snce cusomzed bundlng s a consraned form of mxed bundlng, by defnon canno acheve a hgher prof level. Thus, f he mxed bundlng (MB) soluon s feasble under he cusomzed bundlng () problem, we know ha he opmal soluon wll yeld he same prof as he opmal MB soluon. Consder any wo bundle szes ha appear n he soluon o he mxed bundlng problem (hs ncludes he degenerae case where here s only one bundle sze le boh be he same): [ x, p( x )] and [ x, p( x )] where x g mand x g m. Choose prces for cusomzed bundles such ha [ m, p( m ) p( x )] and [ m, p( m ) p( x )] - hese prces, f hey sasfy he consrans of he cusomzed bundlng problem, acheve he same prof as he mxed bundlng problem. We mus now ensure ha hs soluon s feasble under cusomzed bundlng. All soluons o he mxed bundlng problem mus sasfy: IR : W ( x ) p( x ) IR : W ( x ) p( x ) IC : W ( x ) p( x ) W ( x ) p( x ) IC : W ( x ) p( x ) W ( x ) p( x ) Subsung n he consruced cusomzed bundle soluon, hs program becomes: IR : W ( x ) p( m ) IR : W ( x ) p( m ) IC : W ( x ) p( m ) W ( x ) p( m ) IC : W ( x ) p( m ) W ( x ) p( m ) We now seek condons on consumer preferences ha make hs program equvalen o he cusomzed bundlng program: -IR : w ( m ) p( m ) -IR : w ( m ) p( m ) -IC : w ( m ) p( m ) w ( m ) p( m ) -IC : w ( m ) p( m ) w ( m ) p( m )

3 We show ha can acheve he same prof level as MB under wo ses of preferences: (A) w ( m ) w ( m ) W ( x ) and (B) W ( x ) w ( m ). (A) Consder he condon w ( m ) w ( m ) W ( x ). Subsung he laer par of he condon ( ) ( w m W x ) for all x n he MB soluon mmedaely yelds an equvalence beween he IR condons n MB and he -IR condons n. ow we mus show ha he IC condons are also sasfed hs s grealy aded by he followng Lemma. Lemma. Gven w ( m ) w ( m ) W ( x ) he opmal soluon o he MB problem s of he form p( x ) W ( x ) x. Proof: Consder a canddae soluon of hs form f exss, yelds he hghes prof for hs se of x and IR s mmedaely sasfed. ow we show he condons mply IC s sasfed as well. Consder any IC consran n he problem beween any wo bundles W ( x ) p( x ) W ( x ) p( x ). Applyng he IR consrans for boh and reduces hs condon o W ( x ) W ( x ). In any problem here may be mulple IC consrans conanng bundles he same sze as x. If hs condon s sasfed when W ( x ) s he greaes, hen wll be sasfed for all ohers. By he defnon of w(m) hs greaes value s w ( m ). Thus, he wors case for sasfyng hs IC consran s w ( m ) W ( x ). Subsuon he laer par of Condon (A) hs becomes w ( m ) w ( m ) or w ( m ) w ( m ) whch s he frs par of he condon. QED. We now use he observaon ha all IRs are bndng n boh soluons o complee he proof for condon (A). Agan choose any IC consran, bu do so for he problem: w ( m ) p( m ) w ( m ) p( m ). Subsung n he IR consran for ype on he rgh and ype on he lef, we have w ( m ) w ( m ) whch s us he hypoheszed condon w ( m ) w ( m ). Therefore, we have shown ha he replcaon of he MB soluon sasfes boh -IC and -IR n he problem under Condon (A). (B) We now show ha when W ( x ) w ( m ), he soluon can replcae he MB soluon and yeld he same prof as he mxed bundlng soluon (.e., he MB soluon s feasble as a soluon). Consder prces for cusomzed bundles such ha [ m, p( m ) p( x )] and [ m, p( m ) p( x )]. The condon w ( m ) W ( x ) (mpled by condon B) for all choces of mmedaely guaranees ha all he -IR consrans n he cusomzed bundlng problem are sasfed snce hey are dencal afer hs condon s subsued for and from IR o -IR. I also mples ha all opmal bundles of he same sze mus have he same prce because hey have he same uly (oherwse, one could rase he prce of he lower prce bundle of he same sze whou volang any consrans). We now need o show ha he cusomzed bundlng -IC consrans also hold. The mxed bundlng soluon gves us a se of IC consrans of he form: IC : W ( x ) p( m ) W ( x ) p( m ) Subsuon for W ( x ) from he condon gves:

4 IC: w ( m ) p( m ) W ( x ) p( m ) Whle x s a mxed bundle of sze m, here may exs oher bundles of hs sze ha offer more uly o hose cusomers o whom hs bundle s no nended, makng he IC consran n he problem harder o sasfy. If we can guaranee ha no oher sze m bundles offered offer more uly o cusomer ype hen consumers wll no change her choce. The condon, W ( x ) w ( m ) guaranees ha consumers wll no ge hgher surplus from choosng a dfferen bundle sze (han from he sze ha s nended for her) when hey are allowed o choose freely. Thus we can wre IC: w ( m ) p( m ) W ( x ) p( m ) w ( m ) p( m ) Ths s us he -IC consran for he problem. The logc for all oher IC consrans s he same. Thus, we show ha all consrans of he cusomzed bundlng problem are sasfed by he replcaon. Tha s, an opmal soluon wll yeld he same prof as he MB soluon. QED. Proof of Resul 2: Proof (by consrucon). Le he h order sasc of he valuaon of goods gven by F( v) be denoed as X : (where he larges order sasc s gven by X : ). The wllngness o pay for any consumer s wm %( ) Xk :, a random varable. Ths funcon does k ( m) no depend on whch consumer s examned (equvalence). For any gven dsrbuon here exss a dsrbuon for each order sasc, and summaon s a Borel measurable funcon. Therefore, here exss a cdf for wm ( ) for each m (exsence). Fnally, we know E[ w% ( m ) ] < because E[ w % ( m) ] E[ X ] E[ X ] <. Across each consumer group, wm % ( ) s : : ( m) ndependen and dencally dsrbued wh fne mean (because Ewm [ %( )] E[ wm %( ) ] < ). Q Therefore, leng Q represen he number of cusomers, average valuaon wm % ( ) Ewm [ % ( )] as Q by he Srong Law of Large umbers. Proof of Resul 3: The maxmzaon program s gven by: n Q Max α P C( m ) IR : w ( m ) p DIC : w ( m ) p w ( m ) p for 2.. I, and < s.. UIC : w ( m ) p w ( m ) p for.. I, and < < I m [,/,2 /,...,] Assume nally ha all ypes can be profably served n hs marke by an orderng of bundles * 2* I* m m L m ). We can show ha only adacen DIC consrans can bnd. Defne he adacen ncenve compably consrans o be w ( m ) p w ( m ) p 2 ow we wll show ha f he adacen ncenve compably consrans hold, hen all he ncenve compably consrans hold. Consder any <-;

5 k+ k k+ k w ( m ) w ( m ) [ w ( m ) w ( m )] [ p p ] p p k k or equvalenly, w ( m ) p w ( m ) p As a resul, DIC can be replaced by he adacen ncenve compably consrans. Afer hese smplfcaons, he orgnal problem s equvalen o he followng problem, gnorng UIC: n Max α P C( m ) w ( m ) p s.. w ( m ) p w ( m ) p m [,/,2 /,...,] By subsung n all he IC consrans (whch generaes a recursve equaon ha gves all prces n erms of wllngness o pay) and collecng he erms ogeher for each m we can see ha he IC consrans nclude a erm for + w ( m ) for all prces n sequence above p and w ( m ) for all prces n he sequence of prces above p +. Takng dervaves for each m yelds * + * * α w '( m ) α+ w '( m ) α C'( m ) < I where I (C) α α I I I* I* and a I, α [ w '( m ) C'( m )], because here s no group above hem wh an IC consran. Therefore, he hghes ype purchases he effcen opmal bundle sze. Two fnal ssues arse wha s he lowes group served and wha happens when (C) canno be sasfed for a posve m. One approach s o compue he enre sequence of m ha arses from he program above. Wherever monooncy s volaed n he m x* x *, say m < m, a more profable sraegy s o pool ype x and x* ( x )* ype x-, ha s, se m m. oe ha hs acon does no aler any oher consrans we have, namely IR and IC consrans. In parcular, he relevan IC consran o h group, * * * * * * * * p p + w ( m ) w ( m ) p + w ( m ) w ( m ) p, s sll sasfed. Consder he followng modfcaon process (*). Inspec he sequence of opmal szes o he problem and denfy he relavely unprofable groups (hose wh opmal sze smaller han ha of her adacen lower ypes). Pool hese relavely unprofable ypes o her adacen lower ypes, relabel he segmens and adus he sze of each segmen. oe ha by dong hs he number of bundles (prces and szes) we have o deermne s reduced. Calculae he soluon o hs modfed problem. If monooncy s sasfed, hen he soluon s an opmal one; f no, we run (*) agan unl monooncy s sasfed. Proof of Corollary : Consder a prcng scheme (T) gven by T(m)E+Pm. Ths s a menu (connuum) of bundles {T,m} locaed on a sragh lne whch need no pass hrough he orgn. For smplcy, we assume here are only wo ypes of consumers (ype and ype 2) wh a proporon of ype cusomers α. Assume ha ype 2 consumers have a hgher demand han ype consumers. In hs case, he opmal wo-par arff soluon sasfes he followng obecve funcon: 2 2 ( Pm,, m) arg max E+ ( P c ) [ αm+ ( α) m] 2 Pm,, m 2 2 s. w '( m ) w '( m ) P E S ( P) w ( m ) m P The profs derved from hese wo ypes of cusomers are:

6 w ( m) mc w ( m) mp + m P mc 2 Gven hs soluon, we can show ha he equvalen consumpon level m, m can be mplemened n 2 cusomzed bundles. Le < ( p, m),( p2, m ) > be he menu offered by cusomzed bundlng and le: p w ( m ) p2 w ( m) + w ( m ) w ( m) Gven hs, he assocaed profs derved from each ype s: w ( m ) m c w ( m ) + w ( m ) w ( m ) m c The concavy of w(.) funcon yelds: w ( m ) w ( m) 2 2 w '( m ) 2 P m m w ( m ) w ( m) ( m m) P And as a resul, we have: w ( m) mp + m P mc < w ( m) + w ( m ) w ( m) mc whle. QED. oe also ha he same argumens generalze o more han wo cusomer ypes. Proof of Resul 4. Usng Rychlk (999, p. 8, eq. ) we have ha he rmmed mean of a se of random varables has ghes bounds (for a general dsrbuon): k/ n k k n QF( z) dz E X : n QF( z) dz n k+ n+. Subsung ( )/ n n, m +, k yelds Q ( z) dz E X Q ( z) dz. Mulplyng F : n F m m m m boh szes by m and applyng he defnon of expeced WTP yelds he resul. Proof of Resul 5: For..d random varables, he h (hghes) order sasc from a sample of sze. If c : F : E c X Q ( z) c ( z) dz where X : < m m s hen he frs erm n he expresson becomes E cx: E[ w( m)] and he rgh hand erm becomes he expresson shown m above. Proof of Resul 6: Denoe he soluon o he monopolss problem max ( m) max w( m) C( m) as m *. From Resul 4, m m F :. The range of possble values (,, ) m wm ( ) [ a+ bq ( z;,)] ( zdz ) abm forms a lace 2 ( [,/...,] ). Therefore comparave sacs on he parameers can be examned usng Topks

7 heorem (Topks, 978). Denoe m m. From Topks heorem, we know ha m * ncreases ( m) ( m ) n ζ f. Wrng wm ( ) wm ( ) [ a bqf( z;,)] m: ( zdz ) ζ ζ +. For ( m) ( m ) comparave sacs on a we have [ wm ( ) wm ( ) c] m: ( zdz ) > a a a (he Bernsen polynomals are all posve) so m * ncreases n a. For comparave sacs on b, he dervave of he dfference reduces o: [ a + bqf( z;,)] m: ( z) dz QF( z;,) m: ( z) dz b. Ths s us an m order sasc from a sandardzed dsrbuon. Defne M as he lowes order sasc of he sandard dsrbuon wh non-negave expeced value. The comparave sacs w.r.. b depend on wheher he opmum M * s greaer han or less han hs sze. If he opmum bundle sze s less han he M order sasc wll be posve hs s guaraneed f EX [ M : ] c(noe ha for symmerc dsrbuons wh an odd number of goods, hs condon smplfes o a c ). By he same argumen, one can guaranee he oppose sgn for he order sasc a opmum when EX [ M+ : ] c, so M * decreases n b under hs condon. Par c follows drecly from he observaon ha he value of he margnal (lowes valued) good n he bundle mus, a opmum, be greaer han margnal cos, and so conrbues posvely o prof. Thus, opmal bundles ha are larger mus have greaer profs han smaller ones snce prof for all goods above he margnal good are no less, and he prof conrbuon from he margnal good s non-negave. Proof of Resul 7: Le R : represen he h order sasc from repeaed samplng of he sandard normal. Le S : be he order sascs from samplng from a equcorrelaed ( ρ ) sandard /2 mulvarae normal. Owen and Seck (962) showed ha ES [ ] ( ρ) ER [ ]. Usng hs relaon and argumen from he proof of Resul 6 we have : : /2 ρ F :. m wm ( ) ( ) Q ( z) ( zdz ) Applyng he same argumen as we dd for he scale parameer n he proof of Resul 6, we have ha /2 m* ncreases f sgn ( ρ) and µ < c. Calculang he dervave yelds <. /2 ρ 2( ρ) Thus, he sgn of he relaonshp beween opmal bundle sze and correlaon s ha he opmal bundle sze decreases wh correlaon f µ < c and ncreases wh correlaon oherwse. Proof of Resul 8: When goods are complemens,.e. α>, we have EW ( x ) M α µ > M µ µ, where M x. Ths corresponds o an ncrease n he mean (or he locaon parameer consdered n Resul 6) from µ o M α µ for he valuaon of a gven bundle sze, whch mples wm ( α > ) wm ( α ) m. Wh w(m) ncreases wh α bu C(m) remans unchanged, he opmal bundle sze ncreases n α when we solve for m arg max w( m) C( m). Appendx B: Relaxng he Sngle-Crossng Assumpon (Two group case) Mos of he resuls n he paper follow drecly from he ably o rank order cusomers on her valuaon curve, whch s essenally he sngle-crossng propery (SCP) assumpon. When hs does no hold, here are some smlares o he soluon n Resul 3, bu he srucure of he soluon depends heavly on he exac condons of he problem. Consder a suaon n whch here are only m

8 wo consumer groups where SCP s volaed. Le he pon where he wo consumers uly curves cross for he second me be denfed as m c. Consder he problem where here are wo consumer segmens (I2) and he sngle-crossng propery s volaed. Then, here are hree possble oucomes: 2 a) Case : he crossng pon s beween he opmal pons for each marke mˆ < mc < mˆ. In hs case, he markes are fully separable he opmal soluon s he same as f each marke exsed ndependenly. 2 b) Case 2: he crossng pon s below he opmal pons for boh cusomers: mc < mˆ < mˆ, hen he * * 2* 2 2* 2 2* resul s eher he soluon n Resul or { m mc, p w ( mc); m mˆ, p w ( m )} 2 c) Case 3: he crossng pon s beyond he opmal quany for boh consumers: mc < mˆ < mˆ. In hs case, he problem has he same srucure as n Resul, bu he roles of he hgh-ype and low-ype consumers are reversed. Proof Skech: In he frs case, no IC consrans are bndng, herefore he opmzaon problem s separable for each group. In he second case, a he crossng pon, he IC consran s sasfed he monopols eher chooses ha pon or chooses he soluon o he orgnal problem. In he hrd case, because cusomer ype now has hgher value han cusomer ype 2, he roles are swched (full proof avalable from auhors). Wha s clear from he above dscusson s ha no much can be sad for he general problem n some cases he orgnal soluon holds, n ohers holds a dfferen pons, and n some condons he role of hgh and low valuaon consumers s reversed. Ths complexy ncreases furher f more han wo consumers are nvolved and any of he I(I-) IC consrans mgh be bndng. Therefore, whou some orderng over consumer segmens n demand, he problem mus be solved as a full lnear programmng problem (e.g., Spence, 98). Appendx C: Dervaons of soluons under he wo-parameer preference funcon: Dervaons for ndvdual sellng: Gven a fxed un prce for each good, consumers wll choose o consume addonal goods unl her margnal uly (wllngness o pay per good) s equaed wh he prce of he good. Therefore: P arg max p ( P c) m s.. w'( M) P When he soluon s neror (ha s, <m*<k) he opmum s gven by: b( + a) + ck b + a c PIS ( ) + 2k k 2 2 kb( + a) ck mis a 4b b ( + a) c k pis PIS mis 8ab 2 [ b( + a) ck] IS 8ab From hs, we know ha opmal prce for each good n he ndvdual sale scenaro s deermned by b he average wllngness o pay for all goods he consumers posvely value ( w ). Prce k ncreases wh wllngness o pay, skewness of valuaon (a) and margnal cos. Under our assumpons, he uly curves necessarly cross a m, and hey wll cross a mos one addonal me.

9 There are wo possble boundary condons o hs problem ha can be found by examnng he behavor of m*. Frs, f coss are suffcenly hgh, hen no goods are sold. Ths occurs when c w( + a). A second boundary soluon occurs when he cos-benef radeoff s such ha all goods c ha are posvely valued are sold. Ths holds when c w( 3 a) or a ( ). For example, f 3 w coss are below average value and all goods have he same valuaon (a), he soluon s on he b boundary: p IS b (or P IS k ), and m ISk, whle IS b- ck. As a depars from zero, hs boundary condon becomes p IS w (k) k (or P IS w (k)/), m IS k, and IS w (k) k- ck, and here exss posve consumer surplus. Ths mples ha f we have a boundary soluon, ndvdual sale s effcen bu no prof maxmzng. If margnal coss are zero, hen we can eher have an neror soluon or a boundary condon f preferences are suffcenly unform (a</3). If he soluon s neror, here s quany resrcon due o he consumers equang prce wh her margnal raher han oal benef here s generally a non-zero consumer surplus as well as some deadwegh loss. Dervaons for cusomzed bundlng: Under cusomzed bundlng, he frm s opmzaon problem s he same, bu he consumer ndvdual raonaly (IR) consran s changed. Whereas prce for ndvdual sellng s deermned by he margnal good, prce under cusomzed bundlng s he oal bundle prce (whch s average un prce mulpled by quany) and s deermned by overall wllngness o pay. Thus he frm s problem becomes: p arg max p p cm s.. ( IR) w( m) p IR s always bndng o acheve prof maxmzaon, so we can rewre he obecve funcon as: m arg max m w( m) cm Ths equaon s dencal o he problem of maxmzng socal welfare and, hus, here s no deadwegh loss. The opmal soluon of hs program when m s neror s: k b( + a) kc m a 2b 2 [ b( + a) kc] 4ab ( + a) b c k p 4ab As saed before, here s no deadwegh loss, bu because he monopols can perfecly prce dscrmnae, here s no consumer surplus eher. Agan, s neresng o explore he boundary condons for hs soluon. Posve quanes are sold whenever: c w( + a), whch s he same condon as wh ndvdual sale. Ths s nuve snce he smalles cusomzed bundle s an ndvdual un. However, cusomzed bundlng aans s upper bound a a much hgher level of he cos and preference heerogeney parameers han ndvdual sellng. The opmal cusomzed bundle s he larges necessary o serve all consumers fully (m k) f c c w( a) or a ( ). In oher words, he enre marke demand can be fully sasfed usng w cusomzed bundlng for a values hree mes as large as he case of ndvdual sellng. And even when all goods ha are posvely valued are sold n boh sraeges (cusomzed bundlng and ndvdual sale), cusomzed bundlng wll yeld hgher profs as long as a s greaer han zero. Ths mples ha cusomzed bundlng becomes more effecve as a sraegy f consumers have more heerogeney of valuaons of goods. Summary of resuls:

10 Quany (m) Toal sales or bundle prce (p) Pure Bundlng 2 Indvdual Sellng Cusomzed Bundlng f kb( + a) ck kb( + a) ck b c m IS mn(, k ) m mn(, k ) a 4b a 2b f c w( + a), oherwse f c w( + a), oherwse b f f nonzero quany hen f nonzero quany hen b c ( + a) b ( ck ) ( + a) b ( ck ) p IS p 2 p IS 8ab 4ab f c w( 3 a), f c w( a) b oherwse b oherwse Profs b-c f b c f < m IS <k hen 2 [ b( + a) ck ] IS 8ab oherwse b-ck f m IS >k f < m <k hen 2 [ b( + a) ck ] 2 4ab oherwse b-ck f m >k IS 2 We do no consder suaons where a random pure bundle of sze smaller han s offered. Calculaon of he value of hs ype of bundle s a complex combnaoral problem and depends srongly on assumpons regardng he dsrbuon of values over each poenal good combnaon. We know from he example gven n foonoe 3 however, ha hs ype of bundle wll always creae deadwegh loss unless consumer preferences over each good are dencal.