Managing Capacity Through Reward Programs. on-line companion page. Byung-Do Kim Seoul National University College of Business Administration

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1 Managng Caacty Through eward Programs on-lne comanon age Byung-Do Km Seoul Natonal Unversty College of Busness Admnstraton Mengze Sh Unversty of Toronto otman School of Management Toronto ON M5S E6 Canada Emal: Kannan Srnvasan Carnege Mellon Unversty Graduate School of Industral Admnstraton Pttsburgh PA 5 E-mal: annans@rewcmuedu

2 Aendx Analyzng the basc model The analyss follows the rncle of bacward nducton Prooston on second erod rce rofts If the caacty allocated for rewards then the second-erod avalable caacty Further f max{ } there s a ure strategy equlbrum where both frms sell u to ther caactes Otherwse frms follow the mxed strategy equlbrum The resultng roft functons are derved n Kres Schenman 98 ; ; } mn{ f f f A In A condton } mn{ s equvalent to max{ } Prooston follows the result that when dem s low rce rofts ncrease as the sze of caacty decreases by hng out caacty rewards Prooston on the sze of caacty allocated for rewards The caacty allocaton roblem s defned n P P argmax where s gven by equaton A Calculate the frst-order dervatve } mn{ f ; f 0 ; f A

3 In A frst 0 when but becomes negatve hence decreases wth when Second 0 Based on these two results we can conclude wth the followng equlbrum allocatons resultng rofts If then The equlbrum rofts are: A If then s solved by 0 A To see the relaton between we tae dervatve aganst A wth resect to at both sdes of A We obtan the followng: 0 A5 Snce 0 at but 0 at The roft A6 If or but s suffcently large then the equlbrum allocatons are solved by 0 A7 From Equaton A7 for both frms we can solve the equlbrum allocaton A8 Smlar wth A5 we can obtan from A8 the relaton between

4 0 Fnally the roft A9 Prooston on frst-erod rce P argmax d δ where d s gven by Equaton are solved from P We frst show that If frm lowers below d remans at due to caacty constrant As a result reman the same but s lower wth a smaller Thus frm s total roft would decrease wth the rce decrease Assumng that we consder a small ncrease of from Then d We frst calculate the mact on the frm s frst-erod roft - 0 A0 We now calculate the mact of rce ncrease on second-erod roft followng A~A9 for dfferent values of Snce d d the reward amount the condton s equvalent to If from A we have Then If d from A we have 0 Then d 0

5 Therefore f we have 0 Together wth A0 δ 0 A Therefore If then s gven by A6 Alyng the Enveloe Theory d 0 The mact of the ncrease of on frm s total roft δ δ A If then deends on only through The mact of the ncrease of on frm s total roft δ δ A From A5 we now that 0 Therefore n both A A when the maret dem s close to rewards are large an ncrease of above could lead to an ncrease of total rofts or In ths case δ -δ A From roft functon A9 t s clear that 0 Therefore as n A A can be ostve when rewards are large s close to

6 Prooston on reward decsons Our roof for the result range roceeds n two stes Frst we show that wthn the the equlbrum reward amount s Second we show that no frm can do better by unlaterally ncreasng rewards from We begn wth the rewards From equaton A we now that n the frst erod From A we now that n the second erod We can then wrte frms reward decson roblem as follows: PA argmax Snce reaches maxmum at f f we can conclude that gven the equlbrum rewards Under ths reward amount n the second erod the allocated caacty for rewards r For both frms 9 9 Now we show that nether frm can be better off by unlaterally ncreasng rewards To demonstrate we let fxed consder a small ncrease of above Frst as we have shown earler n the second erod the caacty allocated for rewards wll be The avalable caacty n the second erod wll be hence the second-erod roft 9 Second based on Equaton A when s 5

7 suffcently small frm s roft Otherwse Overall Combnng the above dscusson on frm s roft n both erods we can conclude that 9 when frm unlaterally ncreases above Therefore no frm wll devate from reward amount In the equlbrum r r 9 A5 Prooston 5 on ntal caacty sze We now extend the model formulate the frms decsons on the ntal caacty PA argmax δ Caacty wthout reward rograms Wthout reward rograms are ndeendent Snce frms decde the equlbrum caacty to maxmze δ the equlbrum caacty should be wthn the nterval Q c Q c where Q c s the equlbrum caacty for only Q c for only Caacty should be closer to Q c wth smaller δ 0 Caacty wth reward rograms 0 Followng the same argument that leads to A5 we have 9 r a or b A6 Substtute A6 nto PA we fnd that frm decdes to maxmze only Followng Kres Schenman 98 the caacty should be at the Cournot level Q c a b 0 Q c 0 A7 Substtute A7 nto A6 we have reward amount r Q c 0 A8 6

8 Aendx Analyzng the generalzed model In ths aendx we analyze the generalzed model dscussed n Secton Consstent wth the results obtaned n the basc model we assume Prooston 6 on allocatng caacty for rewards when The roblem formulaton PA s smlar wth P PA argmax [ - ] - The frst-order dervatve s [ ] ab A The otmal can be ether zero or ostve If 0 [- ] a b 0 e a b A then 0 for both frms Substtute 0 a b nto PA we have - d A If A does not hold we solve the frst-order condton of A 0 ab A Solvng A smultaneously for both frms we can obtan ab A5 We can obtan the equlbrum rofts by substtutng A5 nto frms roft functons Prooston 7 on frst-erod rce cometton P6 argmax δ[ ρ ρ ] d 7

9 where d s gven by Equaton are solved from PA PA resectvely ere we let of reward rograms on frst-erod rces nvestgate the equlbrum mact Prooston 7 gh dem n the frst erod Ths art of analyss s smlar wth that of Aendx Frst followng the same logc we can argue that Second to study f a frm may rce hgher than we let consder a small ncrease of above The frst-erod dem becomes d The mact of rce ncrease on - When The mact of an ncrease of 0 A6 accordng to Equaton A we have on When we have shown n A~A5 that can be zero or ostve 0 If as n A then 0 In ths case - [ ] A7 The mact of an ncrease of on frm s total roft δ ρ ρ δ ρ ρ 0 Therefore δρ 0 A8 If then s gven by A5 Substtutng nto equaton A5 we have d Then 8

10 [ d d d d ] -[ d d ] A9 Now we can calculate the overall roft effect of the rce ncrease: δ ρ ρ - δρ [ ] 0 - {-δρ [ ]} δρ [ ] A0 The above exresson A0 can be ostve when s small Prooston 7 ow dem n the frst erod We frst characterze the mxed strategy equlbrum wth rewards Φ We then comare the rce dstrbuton wth that wthout rewards When r 0 accordng to A Uer bound - d If a frm charges the frm wll be undersold by ts comettor wth robablty equal to one The frm s exected roft at the uer bound s E [ ] 9