On Cusomize Goos, Sanar Goos, an Compeiion Nilari. Syam C. T. auer College of usiness Universiy of Houson 85 Melcher Hall, Houson, TX 7704 Email: nbsyam@uh.eu Phone: (71 74 4568 Fax: (71 74 457 Nana Kumar School of Managemen The Universiy of Texas a Dallas Email: nkumar@uallas.eu Phone: (97 88 646 Fax: (97 88 677-1 -
ppenix Proof of Lemma 1: Opimizing ( an (4 wih respec o p S an p S respecively an simulaneously solving he necessary firs-orer coniions yiels he prices of he wo sanar proucs characerize in he lemma. Subsiuing hese prices in SS, x h >, an x S an aking he ifference we obain x h SS, > SS, > r h xh 1 r +. For 4 + incomplee coverage of he high-cos segmen we require his ifference o be posiive, which is he case if, > r + 9 r + r ( r (.1 Proof of Lemma : For any given, opimizing (5 wih respec o p S, pc an (6 QED wih respec o p S an simulaneously solving he necessary firs-orer coniions we obain he prices as a funcion of. Subsiuing he expression of hese prices in (5 we obain firm s secon sage profis as a funcion of. To fin he opimal we ake he erivaive of he profi wih respec o : (108 (76 15 6 ( (16 5 ( (48 5 (16 88( (1 + + I can be shown ha here is no soluion o he firs orer coniion for profi maximizaion w.r. in he inerval [0, 1]. Nex we show ha firm s profi ecreases in. The enominaor of he erivaive above is posiive an so he sign of his erivaive will epen on he sign of he numeraor. The numeraor is negaive if - -
(76 15 + 6 ( (16 5 + ( (48 + 5 (16 > 108 (. Since >1 an 1, we have 76-15 61, 16-5 11, an 16-1. The LHS is increasing in. Seing o 1 an making he above subsiuions an rearranging (. reuces o ( ( 48 + 8+ 7+ 4 > 0 which hols [0,1]. Therefore in he inerval [0, 1], s profi monoonically ecreases in. Thus will choose he smalles possible consisen wih all he coniions of our eman characerizaion. Specifically, we require ha a he opimum egree of cusomizaion be such ha: 1. 0 xl xl 1. 0 xh xh 1, S. The consumer locae a x > h mus erive non-negaive surplus from purchasing he cusomize prouc of firm. The hir coniion is necessary for complee coverage of he high-cos segmen. The, surplus U (, of he consumer a h x h, increases monoonically in [0, 1] as a funcion of. Moreover, i is negaive a 0 an is posiive a 1(see proof of Proposiion. Therefore he coniion ha he surplus of he marginal consumer be nonnegaive ses a lower boun on, an since will choose o be as small as possible, his coniion is he bining consrain. The firs wo coniions are auomaically saisfie if he hir is. Consequenly, firm chooses such ha U ( r ( 1 x p 0. We noe firm woul no eviae h from, S h C >. Since profis are ecreasing in he egree of cusomizaion we only nee o check for eviaions below egree of cusomizaion o, S >. Suppose firm unilaerally eviaes an ses he ε, ε > 0. The marginal consumer in he high cos - -
,, segmen locae a ( xh now erives negaive surplus from purchasing he cusomize prouc from firm. This will resul in incomplee coverage of he high cos segmen. To ensure sub-game perfecion, we solve for he prices of he sanar proucs an he cusomize proucs uner incomplee coverage for any given. We subsiue, hese prices in firm s firs sage profis an subsiue ε. Following he argumens in Lemma IV (in he Technical Supplemen we know ha profis wih incomplee coverage are increasing in he egree of cusomizaion. In urn his implies ha firm s profis from eviaing are ecreasing in ε. I is herefore in firm s bes ineres o se ε0 or no o eviae. QED Proof of Lemma : Similar o he proof of lemma, for any given, we opimize (7 wih respec o p S, pc an (8 wih respec o p S an p C. Solving he necessary firs-orer coniions simulaneously we obain he prices as a funcion of an. We hen subsiue hese prices in (7 an (8 o obain he secon sage profis of firms an respecively. Differeniaing he secon sage profi of (say wih respec o an hen invoking symmery (by seing we obain: > 1 4 1 + 48 ( 1 α + α 16 9 1 α 0 (. Solving (. for we obain: 1, 1, α 0,1 + α > > * ( 1 α ( - 4 -
* 0, > 1, α 0,1 9 1 ( ( α + α oh hese inerior soluions are inamissible since [ 0,1]. ( To characerize he equilibrium egree of cusomizaion we noe ha he erivaive of firm i s secon-sage profi wih respec o i, evaluae a i 1is negaive. { 48 + 15 + 8 + 4 } > 1 16( 1 ( 144 ( ( ( 1 α + α 4 ( 1 α 15 8( 16( 4 ( 4b ( 108 ( 108 5 + + (.4 Equaion (.4 evaluae a 1: > 1 1 (( 4( + + α ( 1 ( 47 + 4( 6 + 0 144 (.5 Thus will choose he smalles possible consisen wih all he coniions of our eman characerizaion. These consisency coniions are:,,, 1. 0 > > x > l xl xl 1,,,. 0 > > x > h xh xh 1,. The consumer locae a x > h mus erive non-negaive surplus from purchasing he cusomize prouc of eiher firm. The hir coniion is require for complee coverage of he high-cos segmen. The, surplus U > (, of he consumer a h x h, increases monoonically in [0, 1] as a funcion of, an as in Lemma, i is negaive a 0 an is posiive a 1(see proof of Proposiion. y a logic exacly similar o Lemma, firm chooses such - 5 -
ha, > U h ( r ( 1 x h p C 0. We noe ha neiher firm woul eviae from, nee o check for eviaions below he egree of cusomizaion o >. Since profis are ecreasing in he egree of cusomizaion we only, >. Suppose firm unilaerally eviaes an ses ε, ε > 0. Given firm s egree of cusomizaion,, > he marginal consumer in he high cos segmen locae a (, x > > h now erives negaive surplus from purchasing he cusomize prouc from firm. This will resul in incomplee coverage of he high cos segmen. Therefore, o ensure sub-game perfecion, we solve for he prices of he sanar proucs an he cusomize proucs uner incomplee coverage for any given an. We subsiue hese prices in he eviaing firm s firs sage profis an subsiue, > ε an, >. Following he argumens in Lemma IV an Lemma V (in he Technical Supplemen we know ha profis wih incomplee coverage are increasing in he egree of cusomizaion. In urn his implies ha he eviaing firm s profis are ecreasing in ε. This is illusrae in he following plo. pasc,sc> wih Incomplee Coverage 0.44 0.4 0.8 0.1 0. 0. 0.4 0.5 e - 6 -
The above figure illusraes ha i is in he eviaing firm s bes ineres o se ε o zero. In oher wors eviaing from, > is no opimal. QED Proof of Proposiion 1: The equilibrium prices an profis when boh segmens are covere wih only sanar proucs are obaine by sanar mehos.. The opimal prices are: p p /(1, an opimal profis are /( 1. + + Suppose eviaes from offering only sanar proucs o offering boh sanar an cusomize proucs. s profi from such a swich in sraegy is -k, where can be obaine from lemma an equals 576 84 5 4 (1 + S 88( >, ( + + (1 + (1 + (9 + 5 Subracing he above expression from /( 1+ gives {96 + 5 (1 + + 88( ( + (1 + (1 + (9 + 5} (.6 Clearly, he enominaor is posiive. Consier he hir an fourh erms insie he braces in he numeraor. Since + > 1+ we subsiue + by 1+ in he hir erm an collec erms o obain he inequaliy 96 + 5 (1 + + ( + (1 + (9 + 5 > 96 + 5 (1 + + ( (9 + 5(1 + (.7 Since 1, (9 + 5 > 18(1 +, an so he RHS of inequaliy (.7 is larger han 96 18(1 +. This quaniy is, in urn, larger han - 7 -
78 6 18 since 1. Finally, 78 6 18 > 0 if >.764. Since > 1, he numeraor of (.6 an hus he enire quaniy is posiive. QED Proof of Theorem 1: To esablish he proposiion we will show ha for inermeiae values of one firm, say firm will eviae from S, o, an for high values of firm will eviae from > o. In oher wors, for inermeiae values of he equilibrium is > an for high values of he equilibrium is. Define: (r : -k S, (.8 (r : -k (.9 We nee o show ha when ** ( r, *** ( r he equilibrium firs sage oucome is **, >. This will be accomplishe by showing ha here exiss ( r such ha if > ** *** ( r hen firm will eviae from S, o,, an ha here exiss ( r such *** ha, if > ( r hen firm will eviae from, > o,. Consier firs he propose eviaion by firm. s Deviaion from, > o, :, Such a eviaion will occur if (,,, > > (, > -k for all > *** ( r. The sraegy of proof is as follows: We will show numerically ha for small, (i >, (, >, -k > (,. - 8 -
(ii We will show ha, here exiss 0 large enough, such ha for > 0, > [, > (, (,, S ] 0. (iii The above wo saemens imply ha he ifference,, S, S [ (, > > (, > ] will sar posiive for small an, as increases i will become negaive for large enough. Thus here will exis *** ( r, efine by, > *** ** (, k (, * ***, such ha, for all > ( r, (, > -k > (,. Proof of Saemen (i: Recall ha * r an following lemma 1, ( r, for r. Recall ha is he inensiy of preference or he ransporaion cos parameer of he high cos segmen. ecause he ransporaion cos parameer of he low cos segmen is normalize o one, represens he inensiy of preference of he high cos segmen relaive o he low cos * segmen. In esablishing his saemen we subsiue ( r, S, S (, (, > > >, > + 0.001 in he expression so ha he ifference in firm s profis in sub-games,> an, is only a funcion r. We vary r in he inerval o 10 o obain a plo of, >,, (, (,, S S r (, > > > as a funcion of r (Figure 1. Figure 1 emonsraes ha for all r in he inerval o 10 he ifference in profis, S, S (, (, > > is posiive. lso noe ha for r 10, >, > * ( 10 16.4, which is acually a very high value of as i represens markes where he ransporaion cos parameer of consumers in he high cos segmen is a leas 16.4 imes ha of consumers in he low cos segmen. - 9 -
pbsc,sc> - 0.05 pbsc,s> 0.04 0.0 0.0 0.01 4 6 8 10 r Figure 1. Firm s Incenive o Deviae an Reservaion Price We conclue herefore ha for small, >,, (, (,, S S r (, > > > >. Proof of saemen (ii: (, (,, S ] ( [ >, >, + (, >, >,, (.10 Nex, we noe ha he profi of in he case ecreases in. Observe ha ( + (6 ( (0 + (1 + 8(1 + 7( (1 + 0, an 1. Similarly, > (1 + (1 + 0, an 1, an finally > 48 16 + (64 /( 9(1 + 88(1 + 16(1 + (7 1 > 0 Therefore - 10 -
>, >, (.11 Thus he profi of firm ecreases wr he egree of cusomizaion more seeply in he uopoly case han in he monopoly case. lso he profis of firm increase wr in boh he monopoly an uopoly case. > 1 (1 + > 0, an 16( (1 + 7( 4 >0, an 1. Comparing he wo slopes we fin ha he profi of firm increases wr more seeply in he monopoly cusomizaion case han in he uopoly cusomizaion case. Tha is >, > > 0 (.1 Lasly, we nee o sign / an /. We firs consier /. The opimal is obaine from he coniion ha he marginal high-cos consumer x (, h ha is inifferen beween firm s cusomize prouc, > an firm s cusomize prouc will receive zero surplus. Define U (, C as he uiliy of his marginal consumer inifferen beween he wo firms cusomize proucs. Making subsiuions for opimal quaniies we ge he surplus as a funcion of an such ha solves (, 0. Toally iffereniaing his w.r. gives U C U C U / C Evaluaing he parial erivaives on he RHS gives (1 ( + + 5 (1 + ( (5 + (1 + > 0. - 11 -
Using an analogous echnique i can be shown ha > 0 (see reamen of s eviaion below. I remains for us o eermine he relaive magniues of. an Claim: For any arbirary ε >0, here exiss 0 such ha for > 0,, >, ε. In oher wors, for large enough he rae of change in is arbirarily close o rae of change in. Proof of claim: Since increases in, consier 1 large enough such ha 1 lies wihin aε 1 -neighborhoo of 1 for arbirary ε 1 >0. Clearly any furher increase in o > + δ will sill keep ( + δ in ha neighborhoo since, 1. 1 1 Therefore, / ε1,. Similarly here exiss 1 an ε such ha / ε,. Le 0 max{ 1, }. y Schwarz inequaliy, we have, >, ε 1 + ε ε (say, 0. Le he quaniy on he LHS of (.10 be evaluae for > 0. Rewriing (.8 so as o reflec he signs of he various quaniies on he R.H.S yiels (, (,, S ] ( > [, >, - > ( ( ( (.1-1 -
In ligh of inequaliies (.11 an (.1, an he fac ha / is arbirarily close o /, he quaniy on he RHS of (.1 is negaive. This esablishes saemen (ii above. s Deviaion from S, o, : Now, consier he propose eviaion by firm. Such a eviaion will occur if S, ** ** -k > ( for all > ( r. We nee o show ha such a ( r, (, will inee exis. The sraegy of proof is exacly he same as ha for s eviaion. Consier [ (,,, C> S, ( ] +, - S, (.14 We have alreay esablishe ha S, 0. Furher, (, 64( (1 + 88(,, S (18 > > 0, an 48 (16 (, 1 (108 (76 15 + 0 88 ( 64( (1 + ( (1 + The las inequaliy hols because he firs wo erms ominae he hir an fourh erms. Finally, we have o sign /. Recall ha he opimal is obaine from he coniion ha he marginal consumer, S x h > ha is inifferen beween firm s cusomize prouc an firm s sanar prouc will receive zero surplus. Defining - 1 -
U C (, as he surplus of his marginal consumer an making subsiuions for opimal quaniies we ge he surplus as a funcion of an such ha solves U C (, 0. Toally iffereniaing his w.r.. gives U C U / C Evaluaing he parial erivaives on he RHS gives 4( + ( (1 + ( + (1+. The numeraor is clearly (1 + + 6(5 + 4( + 11 + (9 + 17 posiive an he enominaor is ecreasing in. If we evaluae he enominaor a 1 (he lowes possible value of he enominaor, we ge a posiive quaniy, an so he enominaor is always posiive. Thus, / > 0, an moreover by repeae applicaions of L Hospial s rule i can be shown ha lim 0. In oher wors, for large enough / is negligibly small, an hus, he RHS of (.14 is ** S, posiive. Hence, here exiss a criical ( r such ha (, ( >k ** for > ( r. This consiues a necessary an sufficien coniion for firms o offer cusomize proucs an will eviae from S, o for his range of parameers. Sai ifferenly, for large he irec effec of on ominaes he inirec effec hrough. Since he irec effec of increasing increases firm will fin i opimal o eviae o., - 14 -
** Finally, we nee o pu some resricions on k. Noe ha ( r is increasing in k, an *** ( r is ecreasing in k. Therefore we nee k o be small enough such ha *** > **. Le ** k be such ha *** ** when k k **. We nee k k **. Lasly, we can show ha *** > ** for a leas some values of r. To o so, i is enough o show he exisence of 1 an, wih > 1 such ha when 1 hen > sricly ominaes for firm, an when hen sricly ominaes >. This we o by consrucion. Le r, is minimum value. The minimum value of * require for incomplee coverage of he sanar prouc marke is (. We nee 1 * > (, so le 1.5. Wih hese values of r an, an wih k0.01, he opimal profis wih sanar proucs are S, S, 0.45. If firm offers a cusomize prouc hen he opimal quaniies are, 0.511, 0.618 an 0.456. So firm will eviae from S, o. If firm respons wih is own cusomize prouc he opimal quaniies are >, 0.666, 0.54. Thus we have > in equilibrium. Le 10. We hen have S, S, 0.,, 0.949, 0.644 an 0.5, so ha will offer is cusomize prouc. However, >, 0.748, an 0., so ha will no offer is *** cusomize prouc. Thus we have in equilibrium. So for r, 1 > **, esablishing ha *** > **. QED - 15 -
Proof of Proposiion : The opimum an in he monopoly an uopoly cusomizaion cases are boh obaine from he coniion ha he marginal consumer inifferen beween he wo firms proucs receives zero surplus. In he firs case he marginal consumer x h is he one ha is inifferen beween firm s cusomize an firm s sanar prouc an, in he secon case, he marginal consumer x h is he one ha is inifferen beween firm s cusomize prouc an firm s cusomize prouc. In equilibrium, he consumer a x h erives a surplus of,,, U h ( r (1 x (, (, h p where we wrie he surplus as a funcion of cusomizaion,. Making he appropriae subsiuions, he opimal egree of, solves C (1 + ( + 11 6 { r(1 + (5 + } 6{ r(1 + + (5 + } U h ( 6( (1 + 0, where, is resrice o lie beween 0 an 1. > Similarly, in uopoly cusomizaion, solves (1 + (5 + (5 + + r(1 + U h ( 0, where (1 + resrice o lie beween 0 an 1. >, is, Now consier U, ( an U > ( as funcions of. We firs noe ha U (0 h h h, > U S (0 h (5 + + r(1 + (1 + 0, he las inequaliy following from he fac ha > r., > Seconly, we noe ha U S (1 > 0, U (1 > 0, an, > U (1 - U S (1 (7 + > 0. (1 + h h h h - 16 -
, In oher wors, boh U, ( an U > ( sar from he same negaive value a 0, h h, an finally a 1 we fin ha U > ( ens up a a higher posiive value han U h (. h Lasly, i can be shown ha U ( > h 0 U > ( h 0,, an ha boh U ( an h U h ( are monoonically increasing in [0, 1]., Therefore he graph of U >, ( sars higher han he graph of U ( a 0, an h, remains higher hroughou he inerval [0, 1]. This implies ha, boh U ( an U U h h ( sar from he same negaive quaniy a 0 an, as increases,, ( his zero before U ( (i.e. a a smaller value of. Therefore, in equilibrium, h >. h h QED Proof of Proposiion : We sar by noing ha, following a logic similar o heorem 1 we can show ha for >r, firm will eviae from S, o,. lso, following similar seps as in heorem 1, we can show ha >0, an, > >0. Moreover,,, > since ( > ( for all values of, he maximum amissible, enoe by max,, is such ha ( max 1. For any value of > max, he surplus of he marginal consumer a x h will be negaive. Recall ha his consumer receives a surplus of U h (, (, r (1 x (, h pc (, when her preference inensiy is an when firm offers a cusomize prouc wih egree of - 17 -
, cusomizaion (. Noe he epenence of he consumer surplus on, boh explicily, an implicily hrough he egree of cusomizaion. Now, U (, ( ecreases in an so he maximum allowable value of is obaine by solving U h ( max,1 0, which yiels max ( r 1 /. s in heorem 1, we can show ha for h large enough, 0. In oher wors, if [ (, (, ], can be an equilibrium oucome, i can only be when is large enough. Conrarily, we will show ha firm will eviae from, o, > even a he larges possible, an so, canno be an equilibrium. Firm s profi in he, subgame a 1 an ( r 1 / is r /(6r. In he, >, > subgame firm will choose he egree of cusomizaion ( such ha U (, ( 0. ( r 1 /, h > ( 9r + r(81r 94 / 4. Wih hese values of > an, he profi of firm wihou he fixe cos is (9r + r(81r 94 + (r 1 (7 9r + 51(r 1 + r(81r 94. I can be checke ha > for r > 0.675. Since r, excees by a large margin an he inequaliy remains rue for small k. Thus, -k > as long as k is small, an will eviae o, >. For > max here will be incomplee coverage of he marke even wih cusomize proucs. Wih incomplee coverage, he opimal profis in he various subgames are > S, S i r 4 ; ( 4r( 16(1 + 4r, r ; 4-18 -
i ( 4r( 16(1 i + 4r i i. I is easily checke ha for, 1, > > S, S, an for >, 1, i >. Thus, for >r he only Nash equilibrium is, >. QED Proof of Proposiion 4: We will aop he viewpoin of firm. Le us assume ha firm commis o offering only is sanar prouc. If offers only is cusomize prouc is profi is ( ( 7( C, C, C, C, (6 + (1 + C,. If i offers boh is sanar an cusomize proucs, is profi can be obaine from lemma an equals 576 84 4 5 (1 + ( + + 88( (1 + (1 + (9 + 5. Since C,, 1 (say, hese profis can be irecly compare an i can be shown ha - C, 1 (1 + >0. Suppose ha commis o offering only is cusomize prouc. If oes likewise is profi is C, C > C, (1 1+ C >, an if i offers boh he sanar an cusomize proucs, is profi is, C >, C> (1 + (1 +, C> (1 +. Since C C >, C >, (say, he profis can be irecly compare, an, C > - C, C > (1 + >0. Finally, if commis o offering boh is cusomize an sanar proucs, hen s (1 profi from offering only is cusomize prouc is 1+, C>. Is profi from offering - 19 -
boh a sanar an a cusomize prouc is obaine from lemma. gain, will make greaer profi by offering boh proucs. QED - 0 -