Electronic Companion to Supply Chain Dynamics and Channel Efficiency in Durable Product Pricing and Distribution
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1 i Eleconic Copanion o Supply Chain Dynaics and Channel Efficiency in Duable Poduc Picing and Disibuion Wei-yu Kevin Chiang College of Business Ciy Univesiy of Hong Kong wchiang@ciyueduh I Poof of Poposiion (The Opial Picing Saegy Based on (8 we obain he following opialiy condiions Hp ( α( N p + c αλ 0 p (S Hp ( λ ( λ λ( α + + α( p c (S ( α Np (S3 o (S we have p ( N + c λ / which when subsiued ino (S and (S3 gives wo diffeenial equaions in es of and λ : ( ( a a( N c + ( A λ( b whee A and α+ b (S4 α The wo eigenvalues of A ae ( α ( α vaiables u ( and v ( as linea cobinaions of ( and λ ( : + / and + + / Define wo new u ( ( v ( H λ( whee α+ α+ H α α (S5 Noe ha each colun in H is an eigenveco of A Then we can ansfo (S4 ino a diagonal syse consising of single-endogenous-vaiable diffeenial equaions: u ( ( ( ( u ( v ( H ( H A λ( H b H HΛH λ( H b Λ v ( H b (S6 whee Λ is he diagonal ai whose diagonal eleens ae he wo eigenvalues of A I is saighfowad o obain he following geneal soluion fo he ansfoed syse in (S6: u ( e 0 v ( 0 e H b (S7
2 ii whee and ae abiay consans o be deeined Subsiuing in (S5 we conve he soluion bac ino he oiginal vaiables ( and λ ( Tha is ( u( e 0 e 0 H H H H b H A b λ( v( 0 e Λ 0 e α+ α+ + α e 0 α( Nc α α α 0 e α Nc α+ α+ e e N c α α + e e 0 (S8 λ The bounday condiions (0 0 and li e ( ( 0 iply α( Nc in (S8 i follows ha ( ( N c( e γ and ( / Subsiuing in (S yields he opial pice pah p ( and ( α+ 0 Subsiuing λ N c γ α e γ whee γ II Poof of Poposiion 4 (yopic Equilibiu Plugging ( ino ( yields α ( / ( w iplies w ( N + c / w w c N N + w The fis ode condiion of which afe subsiuing ino (6 yields ( α/4( N c Solving he diffeenial equaion wih (0 0 yields (4 The esul in (3 follows iediaely afe plugging (4 ino w above and hen ino ( III Poof of Poposiion 5 (Benefi fo yopic Picing Wih (0 and (5 i can be veified ha α α α + + ( N c 4α 4( α + N c N c α( α + α( α + α + + ( α + ( + α 3 3 α + α + + α α < 0 Siila ly we can veify < 0 < 0 and < 0 The esul hen follows Wih ( and (5 he condiion α 4 can be deived by equaing + o and hen solving fo α IV Poof of Poposiion 6 (Saegic Decenalizaion o (5 we have + ( N c ( α + 3α 8 and fo (3 we now α( N c 4( α + Equaing o + and hen solving fo α esul in α which concludes + > if α >
3 iii V Poof of Poposiion 7 (Disineediaion Condiions When he fowad-looing anufacue sells diecly o cusoes i acs as a onopolis; hus accoding o ( is ne discouned pofi wih α is given by ( α α + + ( N c /( α (S9 On he ohe hand when selling hough a fowad-looing eaile wih he ial α based on Table (a he fowad-looing anufacue will obain he following pofi α + α + ( N c/(4 α (S0 4α By equaing (S9 and (S0 and hen solving fo α we obain ( Siilaly 5 + α + 3 α + we can obain he ohe hesholds in he case of open-loop equilibiu: α ( + + α α + and ( ( α / In he sae vain wih ( and Table (b he following hesholds in he case of feedbac equilibiu can be deived: ( and 3α α + 4 4α ( ( 6 α ( ( + + ( + ( + 6α α 3α α α 3 α α 3α α ( + ( ( + ( + ( α + The esul ( < α / can be veified by showing and li ( ( α + + α ( + α α α α ( + 4α ( + α + 3 α α + α 4α α li To veify 5 + α / + 3 α / + ( > ( since ( ( ( ( α α α α 3 α i suffices o show ( α α + ( + ( + α α + ( α α + α α > > (S The diffeence beween he lef hand side and he igh hand side of (S afe squaing he ies on boh 4 sides is 6 + α 3 + 4α 4 + α 6 α + α + which is posiive The es of he esuls can be veified wih he sae appoach
4 iv VI Opialiy Condiions fo he Nueical Sudy in Secion 73 (i The Opial Picing The poble is o aiize (7 subjec o (6 (7 (9 and (30 Accodingly he cuen-value Lagangian is given by Lp ( λ λ u ( p ( c (( + λ(( + λ(( + uk ( ( whee λ ( and λ ( ae he shadow pices associaed wih and especively and he scala u > 0 is he Lagange uliplie The opial picing can be obained by solving he following opialiy condiions: u ( c0 + ce L N + Ω λ λ κ 0 p + (S p ( + Ω ( + Ω ( α + β / N ( ( α + β/ N N + Ω + ( + Ω λ u ( c + ce / λ κ/ (S3 0 κ N ( + Ω λκ ( λ u ( c0 ce (S4 ( +Ω ( +Ω ( α + β / N λκ N +Ω λ + λκλce N + Ω + ( + Ω ( λ u ( c0 + ce + β α + β/ N N( + Ω λ u ( c0 ce βλκ α β / N (S β + Λ ce ( α + β/ N N N( +Ω ( α + β/ N +Ω ( α β / N N λκ + Ω + Ω ( + κ λ λ u ( c0 ce (S6 +Ω ( +Ω ( α + β / N ( ( α β ( ( λ 0 λκ u K + / N N + Ω + ( + Ω u ( c + ce / + / 0 (S7 (ii yopic Picing in he Decenalized Supply Chain When he anufacue and he eaile ae yopic hey aiize hei especive cuen-e pofis ( w c and ( p w subjec o (6 (7 (9 and (30 Given he wholesale pice w he bes pice eacion fo he eaile is 0 N + Ω p + w (S8 p ( + Ω which afe plugging ino (9 and (30 yields he following sales ae and efeence pice ae: ( ( α + β/ N N + Ω ( + Ω w / (S9 N w ( κ + Ω + ( + Ω (S0 Subjec o (S9 (S0 and (7 he Lagangian fo he anufacue s opiizaion poble is given by Lwu ( ( w ( c (( + uk ( ( whee he scala u > 0 is he Lagange uliplie Accoding-
5 v ly he yopic equilibiu picing coesponds o he soluion of he following opialiy condiions: 0 N + Ω p + w (S p ( + Ω ( 0 ( ( α + β/ N N + Ω ( + Ω u ( + Ω ( c + ce /4 (S ( κ ( 3 ( N (4 + Ω + ( + Ω ( u+ ( c0 + ce (S3 4( +Ω ( α β ( 0 u K ( + / N N + Ω ( + Ω u ( + Ω ( c + ce /4 0 (S4 VII Copuaional Resul of he Nueical Sudy in Secion 7 Cos Leaning Effec: Absen (Λ0 ai (Λ005 High (Λ00 Iiaion Effec Absen (β0 ai (β0 High (β0 Absen (β0 ai (β0 High (β0 Absen (β0 ai (β0 High (β0 Discoun Refeence Pice Effec Refeence Pice Effec Refeence Pice Effec Absen ai High Absen ai High Absen ai High (Ω0 (Ω05 (Ω050 (Ω0 (Ω05 (Ω050 (Ω0 (Ω05 (Ω050 Rae No Capaciy Consain (K Low ( % 9966% 9997% 965% 9883% 9976% 9690% 988% 9973% ai (00 939% 967% 9803% 9003% 9367% 9593% 904% 9406% 965% High (05 898% 9305% 955% 8593% 8963% 905% 8600% 8983% 90% Low ( % 998% 9996% 9685% 9896% 9974% 9706% 98% 9938% ai ( % 965% 9804% 896% 9348% 9578% 9009% 9386% 96% High ( % 96% 9486% 8467% 8873% 99% 8494% 8896% 96% Low ( % 9983% 9984% 975% 9899% 9958% 9744% 990% 9958% ai ( % 9649% 987% 8973% 9359% 9588% 904% 940% 963% High (05 888% 953% 9504% 845% 889% 9099% 8446% 8864% 936% ai Capaciy Consain (K4 Low ( % 9969% 9998% 9653% 988% 9976% 9673% 988% 9973% ai (00 93% 9635% 9806% 9008% 9373% 9595% 904% 9393% 96% High ( % 93% 953% 8598% 8869% 90% 8666% 898% 96% Low ( % 998% 9996% 975% 9896% 9974% 970% 9909% 997% ai (00 936% 963% 9806% 8966% 9345% 9577% 8996% 9386% 9608% High (05 89% 993% 9489% 8478% 8867% 900% 8487% 8877% 969% Low ( % 9983% 998% 975% 990% 996% 978% 990% 9959% ai (00 933% 9649% 989% 897% 9338% 9578% 907% 9389% 9646% High ( % 960% 9478% 849% 8853% 904% 8437% 8930% 93% High Capaciy Consain (K3 Low ( % 9968% 9996% 9653% 988% 9976% 9673% 9895% 9977% ai ( % 9634% 9805% 8978% 9367% 9607% 9036% 944% 9676% High ( % 96% 9480% 8606% 9048% 937% 8677% 934% 9488% Low ( % 998% 9995% 9686% 9896% 9956% 9686% 9890% 9993% ai (00 953% 968% 9775% 9005% 9458% 9776% 904% 956% 9883% High ( % 950% 9504% 8640% 998% 9637% 8763% 9344% 9785% Low ( % 9983% 998% 9698% 993% 9989% 978% 9946% 9984% ai (00 973% 964% 984% 977% 9703% 988% 9305% 9773% 99% High ( % 938% 9637% 8848% 9509% 9795% 900% 90% 9843% Noe ha he shaded aea in he uppe lef cone of he able coesponds o he analyical esuls in Secion 5 whee all addiional effecs ae absen
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