Online Appendix. t=1 (p t w)q t. Then the first order condition shows that

Transcription

1 Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate We drop the subscrpt for notatonal smplcty Followng backward nducton, frst consder the retaler s prcng decson Invertng the margnal customer s utlty, the retaler s demand n perod t, assumng the market s not fully covered, s Q t (m p t + αθ)/d, and ts proft s π R 2 t1 (p t w)q t Then the frst order condton shows that the optmal prce s p 1 p 2 (m + w + αθ)/2 and the total sales quantty s Q 1 + Q 2 (m w + αθ)/d The manufacturer maxmzes ts proft π M (w r)(q 1 + Q 2 ), whch gves the optmal wholesale prce w (m + r + αθ)/2 Fnally, the suppler maxmzes ts proft π S r(q 1 + Q 2 ) cθ 2 by frst choosng ts qualty θ and then ts prce r Smlarly, the frst order condtons lead to the optmal prce r (m + αθ)/2 and qualty θ mα, and thus the equlbrum decsons descrbed n ths proposton Here, we need 8cd α 2 m < dk (8 1) to avod full market coverage n perod 2, e, every customer purchases the product Ths condton s obtaned by requrng Q 2 < k, that s, the sales quantty n perod 2 does not exceed the market sze Note we assume m < dk dk (4 1) whch mmedately mples m < (8 1) Also, the unqueness of ths SPNE s straghtforward because frm profts are concave n every decson varable Smlarly, we derve the unque equlbrum decsons when the manufacturer backward or forward ntegrates These two cases yeld the same equlbrum decsons because the sequence of events are dentcal Here, we need m < dk (4 1) to avod full market coverage n perod 2, whch s assumed n Secton 41 Usng the SPNE decsons, we derve frm profts and the second result n the proposton s straghtforward by comparng frm profts across ntegraton scenaros Specfcally, let Π I, I {N, F, B}, be the manufacturer s equlbrum proft Then t can be shown that Π F Π N Π B Π N 2c 2 dm 2 (32c 2 d 2 α 4 )(12(cd) 2 +(6cd α 2 ) 2 ) (8cd α 2 ) 2 (4cd α 2 ) 2 > 0 Proof of Lemma 1 We present the proof for N N scenaro; the equlbrum decsons for other scenaros can be derved followng the same procedure Followng backward nducton, frst we derve the retalers equlbrum retal prces for each perod assumng products compete Retalers s sales quantty Q,t U(θ, p,t, Q,t ) U(θ j, p j,t, ρ t Q,t ) for j 3, whch yelds can be obtaned by solvng Q,t α(θ θ j ) p,t + p j,t + dρ t, (13) 2d where ρ 1 1 and ρ 2 k are the market sze n each perod Usng the sales quantty gven by (13), retalers determne ther retal prce p,t compettvely to maxmze ther profts It s straghtforward to show retaler proft n the NN scenaro π NN R s concave n p,1 and p,2 Snce nventory s not carred over to the second

2 Ln, Parlaktürk and Swamnathan 2 Vertcal Integraton under Competton: Forward, Backward, or No Integraton? perod, we can solve the retaler prcng problem separately for each perod usng the frst order condtons, and obtan the equlbrum prce and sales quantty: p,t dρ t + α(θ θ j ) + 2w + w j, (14) 3 Q,t ρ t 2 + α(θ θj) + (w j w ) (15) 6d Next we solve for the manufacturers game where they smultaneously choose a wholesale prce w to maxmze proft π NN M gven n (3) It s straghtforward to show π NN M s concave n w, and the equlbrum for the wholesale prce game can be derved by solvng π NN M / w 0 smultaneously for 1, 2, whch yelds: w 3d(1 + k) 2 + (2r + r j ) + α(θ θ j ) 3 Next we solve for the supplers problem where each of them determnes the materal prce r Each suppler sets r to maxmze ts proft π NN S gven n (4) Agan, t s straghtforward that the proft functon s concave n r Therefore the equlbrum satsfes π NN S / r 0 for 1, 2, whch yelds: r 27d(1 + k) + 2α(θ θ j ) 6 Fnally, we consder the suppler qualty game Each suppler determnes ts qualty θ to maxmze proft π NN S It can be shown that 2 πs NN < 0 c > α2, whch holds under assumpton A2 Thus, we solve for the θ 2 81d supplers equlbrum qualty decson followng the frst order condtons and obtan θ (1 + k)α 6c Then the results n Lemma 1 follow pluggng ths equlbrum qualty nto each frm s equlbrum decsons Pluggng the equlbrum retal prce and qualty nto the utlty functon of the margnal customer who s ndfferent between the products, t follows that we need assumpton A1, m > d( 3(5+4k) margnal customer generates postve utlty from the purchase 1+k 2 6 ), so that the We apply the same approach to derve equlbrum decsons for other scenaros Moreover, the equlbrum demand for product 2 n F N and BN scenaros, D F N 2,2 and D BN 2,2 respectvely, mply assumpton A2: 9(11k 1) 4k D F N 2,2 D BN 2,2 > (11 1 ) > 1 (16) k Fnally, t can be shown that under assumptons A1 and A2 the proft functons for frms are concave n decson varables n every scenaro and products compete Proof of Lemma 2 The proof proceeds by comparng manufacturer profts across ntegraton scenaros usng the equlbrum qualty and prce gven n Lemma 1 For part (a), t can be shown that Π F N Π NN 0 36(5 + 34k + 5k 2 ) 81( k + 23k 2 ) 2 4(1 + 6k + k 2 ) 0 whch has only one root k δ 1 that takes value n the parametrc space defned by assumptons A1 and A2 Smlarly, δ 2 s the only root for Π F I 2 Π NI 2 0 that takes value under assumptons A1 and A2 Part () proceeds by showng Π BI 2 > Π NI 2 under parametrc

3 Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 3 assumptons A1 and A2 Proof of Lemma 3 Parts () and () proceed by showng that Π BN > Π F N and Π F I 2 Π BI 2 > Π F N Π BN under parametrc assumptons A1 and A2 In part (), δ 3 s the only root for Π F I 2 Π BI 2 0 that takes value under assumptons A1 and A2 Proof of Proposton 2 Part () follows from Π BI 2 > Π NI 2 by Lemma 2 Part () follows from Π F F > Π NF for k < δ 2 and Π NN > Π F N for k > δ 1 by Lemma 2 Moreover, t can be shown that δ 1 δ 2 under assumptons A1 and A2 and Proposton 3 shows that NN Pareto domnates F F when both can be equlbrum outcomes The proof of part () proceeds by showng the manufacturers have no ncentve to devate Frst we show manufacturer 1 does not devate from backward ntegraton for k > δ 3 Ths result s establshed by two facts: (1) Lemma 2 shows devaton from BB to NB s unattractve, and (2) Lemma 3 states manufacturer 1 does not devate from BB to F B for k > δ 3 Now we show manufacturer 1 does not devate from forward ntegraton for k δ 3 Ths result s establshed by the followng two facts: (1) Lemma 3 shows manufacturer 1 does not devate from F F to BF for k δ 3, and (2) t can be shown that δ 3 < δ 2 and therefore manufacturer 1 does not devate from F F to NF for k δ 3 Smlarly, t can be shown that manufacturer 2 has no ncentve to devate from the equlbrum strategy followng the same procedure Proof of Proposton 3 The proof s straghtforward because Π NN Π F F d(1+6k+k2 ) > 0 and Π NN 4 Π BB (1+k)2 α 2 > 0 36c Proof of Corollary 1 In a monopolst settng, usng the equlbrum decsons descrbed n Proposton 1, t can be shown that Π B M > Π F M f and only f > 1 2 for a monopolst manufacturer In addton, Lemma 3 shows that a manufacturer backward ntegrates f and only f > that 1 2 > (1+k)2 9(1+6k+k 2 ) (1+k)2 9(1+6k+k 2 ) under duopoly supply chans It can be shown for 0 k 1, and therefore the manufacturer s more lkely to backward ntegrate than forward ntegrate under supply chan competton Proof of Proposton 4 Frst consder part () of ths proposton For I 2 {B, F }, we have Π NI 2 Π F I 2 Π NI 2 Π BI 2 (1+k) 2 (18 1)α(477 20) 16c(27 1) 2 > 0 For I 2 N, we have Π NI 2 Π F I 2 Π NI 2 Π BI 2 (1+k)2 α 2 ( ) 16c(27 1) 2 > 0 For part (), frst note Proposton 2 states that I 1I 2 NN, F F or BB dependng on the strateges that are consdered Then part () follows because Π NN Π F F Π NN Π BB 9 4 d(1 + k)2 > 0

4 Ln, Parlaktürk and Swamnathan 4 Vertcal Integraton under Competton: Forward, Backward, or No Integraton? Proof of Proposton 5 The proof proceeds by comparng the equlbrum prce and sales quantty gven n Lemma 1 across scenaros n the parameter space specfed by assumptons A1 and A2 Proof of Proposton 6 The proof proceeds by comparng manufacturer profts across scenaros usng the equlbrum qualty and prce descrbed n Lemma 4 In the followng, we demonstrate the dervaton of τ N 1 and the dervaton of τ F 1, τ B 1, τ I 2 2 and τ 3 can be obtaned followng the same procedure Here, we use β to denote β F for ease of notaton It can be shown that Π F N Π NN d 4υ 3 (υ 2 υ 1 υ 3 ), where υ 1 3(1 + k) 2, υ 2 81c 2 d 2 ( k + 85k 2 )β 2 36cdα 2 β(3 + 10β + k(8β 6) + k 2 (3 + 10β)) + α 4 (1 + 2β + 5β 2 + k 2 (1 + 2β + 5β 2 ) k(2 + 4β 6β 2 ), υ 3 β(54cdβ α 2 (1+β)) 2 Thus solvng Π F N Π NN 0 s equvalent to solvng υ 2 υ 1 υ 3 0 whch has only one real root for 0 < β < 1 Then τ N 1 s gven by ths root because υ 2 υ 1 υ 3 < 0 for β 1 and υ 2 υ 1 υ 3 > 0 for β 0 Proof of Proposton 7 NN cannot be an equlbrum, because Lemma 2 () states Π BN > Π NN, showng manufacturer 1 has ncentve to devate by choosng backward ntegraton Proof of Proposton 8 Let β denote β F for ease of notaton The proof proceeds by comparng the equlbrum qualty, prcng decsons and sales quantty gven n Lemma 4 For part (a), solvng p F I 2 1,t p NI 2 1,t 0 reveals that p F I 2 1,t > p NI 2 1,t f and only f β < σ I 2, where σ I 2 s the larger root of χ I 2 { 0, I2 {F, N, B}, where α 2 (k 1 β kβ + 14β kβ 2 ) 9cdβ(84β 55 + k(72β 43)), for t 1 χ F α 2 (1 β + 12β 2 + k(14β 2 1 β)) 9cdβ(72β 43 + k(84β 55)), for t 2 { 18cdβ(28β 15 + k(22β 9)) + α 2 (5 16β β 2 + k(3 16β + β 2 )), for t 1 χ N 18cdβ(22β 9 + k(28β 15)) + α 2 (3 16β + β 2 + k(5 16β β 2 )), for t 2 α 4 (1 k 2β 2kβ 11β 2 9kβ 2 ) + 243c 2 d 2 β(15 28β + k(9 22β)) 9cdα 2 (3 + 14β 61β 2 k(3 8β + 49β 2 )), for t 1 χ B α 4 (k(1 2β 11β 2 ) 1 2β 9β 2 ) 243c 2 d 2 β(22β 9 + k(28β 15)) +9cdα 2 (3 8β + 49β 2 k(3 + 14β 61β 2 )), for t 2 Proof of Proposton 9 Let β denote β F for ease of notaton Frst, we obtan frms equlbrum decsons gven n Appendx C followng the procedure descrbed n the proof of Lemma 1 Then the results n ths proposton proceed by comparng equlbrum qualty and sales quantty across scenaros, whch leads to the followng threshold values ξ θ 1 α2 (4β 2 + β 1 (3β 2 5) + β 2 1(5 3β 2 ) 2 16β 1 β β 2 2) 54dβ 1 β 2,

5 Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 5 ξ θ 2 α2 (15 11β 2 2β β 2 + β β β 1 β 2 + 4β 2 1) 54d(6 5β 2 )β 1, ξ Q 1 ξ Q 2 ξ θ 1 Part () of ths proposton follows by replacng β F 1 β F 2 and the fact that dξθ d Proof of Proposton 10 Q dξ > 0 and > 0 d In ths extenson, the manufacturers can set the wholesale prce they charge to retalers dynamcally n each perod Frst we derve the SPNE decsons under every ntegraton scenaro (as n Lemma 1 for the base model) Then we compare frm profts across scenaros to characterze the manufacturers equlbrum ntegraton strategy (as n Proposton 2 for the base model) Next, we llustrate n the N N scenaro how the dervaton of equlbrum decsons dffers from the base model The dervaton for SPNE decsons n the NN scenaro s dentcal to the proof of Lemma 1 untl (15) Next when consderng the manufacturers choce of wholesale prces, manufacturer maxmzes ts proft π M 2 t1 (w,t r )Q,t where Q,t s gven by (15) It s straghtforward to show π M s concave n w,t, and the equlbrum for the wholesale prce game can be derved by solvng π M / w,t 0 smultaneously for 1, 2 and t 1, 2, whch yelds: w,t 9dρ t + 2r + r j + α(θ θ j ) 3 Gven the manufacturers response for the wholesale prces n each perod, next we characterze the supplers choce of materal prces Each suppler sets ts materal prce r to maxmze ts proft gven n (4) Agan, t s straghtforward that the proft functon s concave n r Therefore the equlbrum satsfes π S / r 0 for 1, 2, whch yelds: r 27d(1 + k) + 2α(θ θ j ) 6 Fnally, we consder the supplers qualty nvestment Each suppler determnes ts qualty θ to maxmze proft π S It can be shown that 2 π S < 0 c > α2, whch s satsfed under the revsed assumpton A2 θ 2 81d Next, we solve for the supplers equlbrum qualty decson followng the frst order condtons and obtan: θ (1 + k)α 6c Then the SPNE prce and sales quantty can be derved usng ths equlbrum qualty Pluggng the equlbrum decsons nto the utlty functon, t can be shown that the margnal customer who s ndfferent between the products generates postve utlty from the purchase when the revsed assumpton A1 holds Followng the same procedure we derve the SPNE decsons n every possble vertcal ntegraton scenaro It can be shown that D 2,2 45(5k 1)+(1 9k) > 0 for the BN and F N scenaros leads to the revsed assumpton A2 Under the revsed assumptons A1 and A2, the proft functons for frms are concave n ther decsons varables and products compete Fnally, we compare the manufacturers profts across ntegraton scenaros to characterze ther equlbrum ntegraton strategy It can be shown that π BN > π NN and π BB > π BN and thus no ntegraton at any

6 Ln, Parlaktürk and Swamnathan 6 Vertcal Integraton under Competton: Forward, Backward, or No Integraton? manufacturer cannot be an equlbrum The threshold value n part () can be derved by solvng π F I π BI for I {F, B} Proof of Corollary 2 The proof proceeds by showng 1 8(18 1)+ 12 (18 1) dynamc wholesale prcng BB regon becomes larger n the parameter space Proof of Proposton 11 δ 3 wth equalty holds for k 1 That s, wth Ths extenson dffers from the base model n that each retaler also chooses a qualty nvestment level mmedately after the supplers determne ther qualty nvestments Agan, frst we derve the SPNE decsons under every ntegraton scenaro (as n Lemma 1 for the base model) Then we compare profts across scenaros to characterze the manufacturers equlbrum choce of ntegraton strategy (as n Proposton 2 for the base model) Next, we derve the SPNE decsons for the NN scenaro n ths extenson Followng the proof of Lemma 1, we obtan suppler s equlbrum materal prce: r 27d(1 + k) + 2(α(θ θ j ) + b(θ r θ jr )) (17) 6 We next derve the retalers choce of qualty nvestment by pluggng ths prce nto retaler s proft functon: Here, we focus on c > b2 729d π N R 2 (p,t w )Q,t cθ 2 (18) t1 so that retaler proft s concave n θ r We solve the frst order condtons for both retalers whch yelds the followng equlbrum retaler qualty: θ r b(27c(27d(1 + k) + 2α(θ θ j )) 2b 2 (1 + k))) (19) 54c(729cd 2b 2 ) Pluggng ths retaler qualty nto the supplers proft functons, t reveals that we need c 8b 2 +9α 2 so that each suppler s proft functon s concave n ts qualty choce θ Ths requrement 4b 2 +9α 2 +3α 1458d s satsfed under the revsed assumpton A2 We solve the frst order condtons for both supplers whch yelds the followng equlbrum suppler qualty: θ 243d(1 + k)α 1458cd 4b 2 (20) The SPNE retaler qualty, prces and sales quantty are then derved usng ths equlbrum suppler qualty Pluggng the equlbrum prces and qualtes nto the utlty functon of the margnal customer n perod 1 who s ndfferent between the products, t follows that we need the revsed assumpton A1 stated n Secton 53 so that the margnal customer generates postve utlty from the purchase and products compete Followng the same procedure, we derve the equlbrum outcome for other scenaros It can be shown that D 2,2 27b2 cd(1 23k)+2b 4 k+486cd(9cd(11k 1) 4kα 2 ) 4(b 4 324b 2 cd+972cd(27cd α 2 )) > 0 n the F N and BN scenaros mples the revsed assumpton A2 Under the revsed assumptons A1 and A2, the proft functons for frms are concave n ther decson varables and products compete

7 Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 7 Next we plug the equlbrum decsons nto proft functons to derve the manufacturers profts under every ntegraton scenaro It can be shown that π BN > π NN and π BB > π BN and thus no ntegraton at any manufacturer cannot be an equlbrum In other words, the manufacturers choose to vertcally ntegrate n equlbrum Then the threshold value n part () can be derved by solvng π F I π BI for I {F, B} Proof of Proposton 12 In ths extenson supply chan 2 has a lower cost for qualty mprovement Frst we derve the SPNE decsons under every ntegraton scenaro (as n Lemma 1 for the base model) Then we compare manufacturer profts across scenaros to characterze manufacturers equlbrum choce of ntegraton strategy (as n Proposton 2 for the base model) For ease of notaton, we drop the subscrpt 1 for c 1 n ths proof The dervaton of SPNE decsons for each scenaro essentally follows the same procedure descrbed n the proof of Lemma 1 The only dfference s that the cost for qualty mprovement n supply chan 2 becomes vcθ 2, where v (0, 1), nstead of cθ 2 as n the base model Once the SPNE decsons are derved, we plug them nto proft functons to derve the manufacturers equlbrum profts under every ntegraton scenaro The result of ths proposton s then derved by comparng manufacturer profts across ntegraton scenaros and shown that no manufacturer can acheve hgher proft by unlaterally devatng from the equlbrum strategy descrbed n the proposton Here, δ 4 s the soluton to π NB s the soluton to π F B π NB Proof of Proposton 13 π BB, δ 5 s the soluton to π F B π F F, δ 6 s the soluton to π F B π BB, δ 7 Here we analyze the manufacturers equlbrum vertcal ntegraton strategy when the qualty mprovement level for both products s exogenously determned at θ Frst we derve the SPNE decsons under every ntegraton scenaro followng the procedure descrbed n the proof of Lemma 1 Usng the equlbrum decsons n the NN scenaro, t can be shown that the margnal customer who s ndfferent between buyng the product from ether supply chan generates postve utlty for m > 3d(5+4k) 2 αθ Ths condton also ensures that margnal customers n every ntegraton scenaro generate postve utlty from ther purchase Then t can be shown that π BN > π NN > π F N Thus no ntegraton cannot be an equlbrum outcome Fnally, the result for ths proposton proceeds as π BI > π F I for I {F, B} Proof of Lemma 4 The dervaton for the equlbrum decsons n ths lemma follows the procedure descrbed n the proof of Lemma 1 wth β F < 1