Online Appendix for "Customer Recognition in. Experience versus Inspection Good Markets"

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1 Online Appendix for "Cusomer Recogniion in Experience versus Inspecion Good Markes" Bing Jing Cheong Kong Graduae School of Business Beijing, 0078, People s Republic of China, bjing@ckgsbeducn November 5, 04 This echnical online Appendix conains he proofs of Proposiion and Lemma TA5 Lemmas TA-TA4 below are pivoal o he proof of Proposiion Lemma TA Suppose 0 p 5 5 0:55 and There is a SPE where each rm charges a D = ( ) + () and b R = + (6 5) () in period and b S = () o new and repea cusomers, respecively, in period In equilibrium, only some of he consumers realizing a low value from he rs purchase will swich brand, and hose realizing a high value never swich in period Proof We seek an equilibrium where only some of he consumers realizing a low value swich brands We rs look a he period- game Wihou loss of generaliy, suppose consumers in [0; D ] have purchased produc 0 and consumers in ( D ; ] have purchased produc in period Le b S i and b R i denoe rm i s (i = 0; ) prices for swichers and repea buyers, respecively Suppose a consumer x buying produc 0 in he rs period has realized a low value In period, she will repea buy produc 0 if L b R 0 x v b S ( x) or

2 equivalenly x x D 0, where x D 0 = bs b R 0 + () : Oherwise, she will swich o produc Suppose a consumer x buying produc in he rs period has realized a low value In period, she will repea buy produc if L b R ( x) v b S 0 x or equivalenly x x D, where x D = br b S () : She will swich o produc 0 oherwise The rms period- pro funcions are R D 0 = [ D + x D 0 ]b R 0 + (x D D )b S 0 and R D = [( D ) + ( x D )]b R + ( D x D 0 )b S : In he expression of R D 0, D is he consumers who purchased produc 0 previously and realized a high value, and x D 0 capures he consumers who purchased produc 0 previously, realized a low value bu are locaed su cienly close o produc 0 Boh groups will repea purchase produc 0 Those who purchased produc, realized a low value and are locaed su cienly disan from produc (hose in ( D ; x D )) will now swich o produc 0 The number of such swichers is (x D D ) Each erm in R D has an analogous inerpreaion For i = 0;, we can verify R R i ) R S i ) = < 0 R R i = 0 The Hessian marix of each rm s period- objecive is hus negaive de nie The period- equilibrium h i readily follows from he rs-order condiions: b S 0 = ( ( ) D ) + + (), b R 0 = h i h i h i (+) D + (), b S = ( ) D + (), and b R = ( (+) D ) ()

3 Subsiuing hese prices ino R D 0 and R D above and rearranging erms, we have rm i s (i = 0; ) period- equilibrium pro s R D i = [(br i ) +(b S i ) ] We now urn o period Observing he period- prices a 0 and a, each consumer decides which produc o buy Her objecive is o maximize expeced oal surplus over boh periods, anicipaing he period- price equilibrium To derive he period- consumer choice rule, i su ces o idenify he consumer indi eren beween producs 0 and in period, D From he discussions above, we see ha if consumer D purchases produc i (i = 0; ) in period and realizes a low value, she will swich o he oher produc in period Her expeced oal surplus of buying produc 0 in period is V 0 ( D ) = v a 0 D + H b R 0 D + v b S ( D ), and ha of buying produc in period is V ( D ) = v a ( D ) + H b R ( D ) + v b S 0 D Seing V 0 ( D ) = V ( D ) and subsiuing in b S 0, b R 0, b S, and b R yield D = (a 0 a ) 4( + : ) In period, rms 0 and choose prices a 0 and a o maximize heir oal expeced pro s D 0 = D a 0 + R D 0 and D = ( D )a + R D, respecively I is sraighforward o verify ha for i = 0;, d i da i = [7+8] 4(+ ) < 0, ie, i is sricly concave in a i The rms FOCs lead o a candidae SPE: a 0 = a = a D, where a D = ( ) () + ; and b S 0 = b S = b S and b R 0 = b R = b R, where b S = + () and b R = ()

4 Each rm makes oal pro s D = ( ) () + +R D, where R D is is period- pro s R D = [ ()] + [ + ()] : 8 We can check ha x D 0 < < xd always holds, ha x D 0 0, x D, H b R v b S, For, ha ( ) (ie, hose realizing a high value do no swich) < ( ), we need p Nex, we sill need o ensure ha neiher rm has an incenive o unilaerally deviae from he period- equilibrium above Due o symmery, i su ces o rule ou rm 0 s such incenive We see from above ha each rm s period- price for swichers (b S ) is independen of ha for repea buyers (b R ), as he wo prices serve separae marke segmens Below, we derive he condiions for rm 0 no o deviae from eiher price Firs, suppose rm 0 wishes o deviae o a di eren price for repea cusomers, b RD 0 From he above deducion, i is clear ha rm 0 will no deviae o any b RD 0 ha aracs all of is own H cusomers Therefore, suppose b RD 0 is high enough such ha rm 0 loses some of is H cusomers o rm, ie, H b RD 0 < H + L b S, b RD 0 b S > () Le y 0 denoe rm 0 s H cusomer who is indi eren o repea buying produc 0 and swiching o produc, ie, H y 0 b RD 0 = H + L ( y 0 ) b S, which leads o y 0 = bs D RD b RD 0 ++() = y 0 + x 0 = bs b RD Firm 0 s period- demand from repea buyers is hen, where x 0 is as given above By deviaing, rm 0 wishes o maximize b RD 0 D RD 0, which leads o he opimal deviaing price, b RD 0 = bs + p, we have ha b RD 0 b S > (), ( ), <, which always holds Therefore, when will no unilaerally deviae from b R > (6 5) and ha When (6 5) > ( ), rm 0 Second, suppose rm 0 wishes o deviae o a di eren price for swichers, 0 From he above deducion, we see ha rm 0 will no deviae o any 0 ha does no arac any of rm s H cusomers Therefore, suppose 0 is low enough such ha i also aracs 4

5 some of rm s H cusomers, ie, H + L 0 > H b R, 0 b R () Le y denoe rm s H cusomer who is indi eren o repea buying produc and swiching o produc 0, ie, H ( y ) b R = H + L y 0, which leads o y = br D SD 0 = (y ) + (x 0 + () ) = br Firm 0 s period- demand from swichers is 0 +, where x is as given above Firm 0 wishes o maximize 0 D SD 0, which leads o he opimal deviaing price, p, we have ha b SD 0 b R (), (6 5) < ( ) always holds Therefore, when 0 will no unilaerally deviae from b S Noe ha which holds when Lemma TA When p 5 5 QED period and b R = and bs = 0 = br When (6 5) and ha (6 5), rm (6 5), > 0, > ( ), here is a SPE where each rm charges a D = 4 in o repea cusomers and swichers, respecively, in period, making oal pro s of D = 7 In equilibrium, some of he consumers realizing a low 8 or high value swich brands in period Proof We now seek an equilibrium where some of he consumers realizing a low or high value swich brands in he second period We sar wih analyzing he period- compeiion Again, wihou loss suppose consumers in [0; D ] have purchased produc 0 and consumers in ( D ; ] have purchased produc in period Consider consumer x who has purchased produc 0 previously and realized value L In period, she will repea purchase produc 0 if and only if x x D 0 Similarly, if consumer x has purchased produc and realized value L in period, in period she will repea purchase produc if and only if x x D Here x D 0 and x D are as given in he proof of Lemma TA above If consumer x has purchased produc 0 and realized value H, in period she will repea purchase produc 0 if H b R 0 x v b S ( x) or equivalenly x y D 0, where y0 D = bs b R () : 5

6 Oherwise, she will swich o produc If consumer x has purchased produc and realized value H, in period she will repea purchase produc if H b R ( x) v b S 0 x or equivalenly x y D, where y D = br b S 0 + () : She will swich o produc 0 oherwise Therefore, rms 0 and s period- pro funcions are R D 0 = [y D 0 + x D 0 ]b R 0 + (y D D ) + (x D D ) b S 0 and R D = [( y D ) + ( x D )]b R + ( D y D 0 ) + ( D x D 0 ) b S : R R i ) R S i ) = R R i = 0, each rm i s period- Hessian marix is negaive de nie The unique period- price equilibrium hen follows from he FOCs: b S 0 = 4 D, br 0 = +D, b S = +4D, and b R = D We hen obain he period- pro s: R D 0 = (br 0 ) +(b S 0 ) and R D = (br ) +(b S ) We now urn o period For given period- prices a 0 and a, we rs characerize he consumer indi eren o producs 0 and, D Noice, here he consumers in [y D 0 ; y D ] always swich brands, even afer realizing a high value from heir rs purchase Therefore, consumer D s oal expeced surplus of buying produc 0 in period is V 0 ( D ) = v a 0 D + v b S ( D ), and ha of buying produc in period is V ( D ) = v a ( D ) + (v b S 0 D ) Seing V 0 ( D ) = V ( D ) and subsiuing in b S 0 and b S yield D = (a 0 a ) In period, rms 0 and se prices a 8 0 and a o maximize heir oal expeced pro s D 0 = D a 0 + R D 0 and D = ( D )a + R D, respecively Since d D i da i = 7 6 < 0 for i = 0;, a candidae SPE hen follows from he FOCs: a 0 = a = 6

7 a D, where a D = 4 We can furher verify ha in period each rm charges price bs = o swichers and b R = o repea cusomers Each rm makes oal pro s D = 7 8 In equilibrium, x D 0 = (), x D = () +, y D 0 = () +, y D = ha 0 < x D 0 < < xd <, > and ha 0 < yd 0 < < yd <, () Noe > Nex, we need o ensure ha neiher rm has an incenive o unilaerally deviae from he period- equilibrium above Due o symmery, we focus on rm 0 We see from above ha each rm s period- prices for swichers and for repea buyers serve separae marke segmens Below, we derive he condiions for rm 0 no o deviae from eiher price Firs, suppose rm 0 wishes o deviae o a di eren price for repea cusomers, b RD 0 From he above deducion, i is clear ha rm 0 will no deviae o any b RD 0 ha induces some of is H cusomers o swich o rm Therefore, suppose b RD 0 is low enough such ha i reains all of is H cusomers, ie, H b RD 0 > H+L b S, b RD 0 b S < () Firm 0 s demand from is own repea cusomers is D0 RD = + [bs b RD 0 + ()] Maximizing b RD 0 D0 RD leads o he opimal deviaing price b RD 0 = bs () + However, we have b RD 0 b S < (), )g Therefore, when di eren price for repea cusomers < ( ) +, which never holds when > maxf ; ( > maxf ; g, rm 0 will no unilaerally deviae o a Second, suppose rm 0 wishes o deviae o a di eren price for swichers, 0 From he above deducion, we see ha rm 0 will no deviae o any 0 ha aracs any of rm s H cusomers Therefore, suppose 0 is high enough such ha i aracs only some of rm s L cusomers, ie, H + L 0 < H b R, 0 > b R ( )(H L) Firm 0 s period- demand from swichers is hen D0 SD = x = Maximizing 0 D0 SD leads o he opimal deviaing price, b R 0 ++() 0 = br +() We can check ha 0 > b R (), ha ( ) > maxf ; g Therefore, when deviae o a di eren price for swichers QED < ( ) Noe > ( ), rm 0 will no unilaerally Lemma TA When > p 5 0:68 and [(+) p p + ] ( ) p (+) p (+), + 7

8 here is a SPE where each rm charges price a D = [( ) ()] in period and b R = (+)+() o repea cusomers and b S = ( ) () o swichers in period In equilibrium, all consumers realizing a low value ( L) and some of he consumers realizing a high value ( H) swich brands Proof We seek an equilibrium where all consumers realizing a low value swich brands and where some of he consumers realizing a high value swich brands Because he deducion of he curren equilibrium closely parallels ha of Lemma TA, here we will be brief We rs derive he period- price equilibrium Suppose consumers in [0; D ] have purchased produc 0 and hose in [ D ; ] have purchased produc in period Then, rms 0 and s period- pro funcions are R D 0 = y D 0 b R 0 + (y D D ) + ( D ) b S 0 and R D = ( y D )b R + ( D y D 0 ) + D b S ; where y D 0 and y D are as given in he proof of Lemma TA above For i = 0;, we have = 0 The second-order condiions are hus sais ed From he FOCs, we obain a unique period- price equilibrium: b R 0 = b S = (+)+(), b S 0 R R i ) R S i ) = R R i (4 4) (), b R = (4 ) () Each rm i s period- pro s are R i = (+ )+() h b R i + b S i i, and We now urn o period Given period- prices a 0 and a, we can easily check ha he consumer indi eren o producs 0 and is D = (a 0 a ) Firms 0 and s period- 8 objecives are D 0 = D a 0 +R D 0 and D = ( D )a +R D, respecively Since d D i da i for i = 0;, he second-order condiions are sais ed = 7 6 < 0 The SPE hen follows from he FOCs: a 0 = a = a D, where a D = [( ) ()] In period, each rm charges b R = swichers We have y D 0 = (+)+() o repea cusomers and b S = (+)+() 6 and y D = ( ) () o ( +5) () We can easily 6 8

9 check ha y D 0 > 0 and y D < always hold, and ha y D 0 yd, L b R v b S, H have (+) +, p 5 When (+), hose realizing a low value always swich We also L + Nex, we need o ensure ha neiher rm has an incenive o unilaerally deviae from he period- equilibrium above Due o symmery, we focus on rm 0 We see from above ha each rm s period- prices for swichers (b S ) and for repea buyers (b R ) serve separae marke segmens Below, we derive he condiions for rm 0 no o deviae from eiher price Firs, suppose rm 0 wishes o deviae o a di eren price for repea cusomers, b RD 0 Again, rm 0 will no deviae o any b RD 0 ha induces some of is H cusomers o swich o rm Therefore, suppose b RD 0 is low enough such ha i reains all of is H cusomers, ie, b RD 0 b S < () We can easily derive he opimal deviaing price, b RD 0 = b S () + When > p 5, we have 6+ < 0 and b RD 0 b S < (), < (5 ) ( 6+) When > p 5 and (5 ) ( 6+) < when > p 5 and when >, he las inequaliy never holds since, > 0 Therefore, rm 0 will no unilaerally deviae from b R (+) + Second, suppose rm 0 wishes o deviae o a di eren price for swichers, 0 Again, suppose 0 is high enough such ha i aracs only some of rm s L cusomers, ie, b R 0 > b R () Firm 0 s period- demand from swichers is hen D0 SD = ( ) Maximizing 0 D0 SD leads o he opimal deviaing price, 0 = b R +() 0 ++(), and he associaed pro from swichers, [br +()] We can check ha 8 0 > b R (), since (+) + < (5 ) <, which always holds when (+), + + (5 ) From he above deducion process, rm 0 s pro from swichers in he + candidae equilibrium is (bs ) When > p 5, we have ha ( ) p (+) p > 0 and ha [br +()] 8 < (bs ) is equivalen o > p 5, we can furher verify ha > [(+) p p + ] < [(+) p p + ] ( ) p (+) p ( ) p (+) p When is equivalen o ( ) p < p, which always holds Therefore, when [(+)p + p ] ( ) p (+) p (+) +, rm 0 will no unilaerally deviae from bs QED 9

10 Lemma TA4 () There is no pure-sraegy SPE where all consumers realizing a low value swich and no consumer realizing a high value swiches in period () There is no pure-sraegy SPE where all consumers swich brands in period Proof () Suppose here is a pure-sraegy SPE where all consumers realizing L swich brands and no consumer realizing H swiches brands in period Suppose ha in equilibrium consumers in [0; ] purchase produc 0 and hose in [; ] purchase produc in period, and ha rm i (i = 0; ) charges b R i and b S i o repea cusomers and swichers, respecively, in period The rms period- pro s are hen R 0 = b R 0 + b S 0 and R = b R + b S, respecively Afer realizing L, consumer 0 swiches o produc i L b R 0 v b S Afer purchasing produc 0 and realizing H, consumer will repea buy produc 0 i H b R 0 v b S We herefore have () b R 0 b S () + However, if b R 0 b S > (), rm bene s from slighly raising b S If b R 0 b S = (), rm 0 bene s from slighly raising b R 0 A conradicion () Suppose here is a pure-sraegy SPE where all consumers swich brands in period Suppose ha in equilibrium consumers in [0; ] ([; ]) purchase produc 0 () in period Then, he rms period- pro s are R 0 = b S 0 and R = b S Afer realizing H, consumer 0 swiches o i H b R 0 v b S, b R 0 b S + + () However, by unilaerally deviaing o a price b R 0 < b S + + () rm 0 can also sell o some of he consumers who realize H wih is produc while sill selling o all consumers in [; ] a price b S 0 This sricly increases rm 0 s period- pro s A conradicion QED Proof of Proposiion We noe ha here are only ve possible brand-swiching paerns in period : () only some of he L consumers swich, () some of he L and H consumers swich, () some of he H consumers and all L consumers swich, (4) all L consumers swich bu no H consumer swichers, and (5) all consumers (L and H) swich Noe ha he condiions in Lemmas TA- do no overlap The saemens of he Proposiion hen direcly follow from Lemmas TA-4 above QED 0

11 Lemma TA5 Suppose ex ane consumers observe heir values of boh producs Wihou BPD, here is a unique symmeric SPE where each rm prices a in boh periods and makes oal pro s of Wih BPD, here is a unique symmeric SPE where each rm prices a 4 in period and charges prices o repea cusomers and of 7 8 o swichers, making oal pro s Proof () In he inspecion good duopoly, ex ane consumers observe heir values of boh producs Wihou BPD each consumer s period- choice only depends on he rms period- prices, and no on her period- purchase hisory The equilibrium hus reduces o replicaions of he saic equilibrium, which we now derive Le p 0 and p denoe he prices of rms 0 and, respecively, in he saic model There are + ( ) consumers (called he Type- consumers) who value boh goods equally (a L or H), ( ) consumers who value good 0 a H and good a L (called he Type- consumers), and ( ) consumers who value good 0 a L and good a H (he Type- consumers) Le k (k = ; ; ) denoe he Type-k consumer indi eren beween he wo goods, ie, v p 0 = v ( ) p, H p 0 = L ( ) p, L p 0 = H ( ) p We hen have = p p 0 +, = p p 0 ++(), and = p p 0 + () Firm 0 s demand is D 0 = [ + ] + ( + ) = p p 0 + Firms 0 and s per-period pro funcions are (p p 0 +)p 0 and p p 0 + p, respecively The unique equilibrium is hen p 0 = p = Each rm hus makes oal pro s of () Each rm i (i = 0; ) charges a single price a i in period due o he lack of consumer purchase hisory In period, rm i charges prices b R i o is repea buyers and b S i o swichers We sar wih analyzing he period- compeiion I is simple o verify ha in period, if a Type-k consumer x purchases produc 0, hen he Type-k consumers in [0; x) will also purchase produc 0 Similarly, if a Type-k consumer x purchases produc, hen he Type-k consumers in (x; ] will also purchase produc Therefore, wihou loss of generaliy le k

12 (k = ; ; ) denoe he Type-k consumer indi eren beween he wo goods in period We now idenify each Type-k consumer s period- choice If a Type- consumer x purchases produc 0 in period, in period she will repea purchase 0 if and only if v x b R 0 v ( x) b S or x x A bs b R 0 + Here v equals eiher H or L If a Type- consumer x purchases produc in period, she will repea purchase if and only if v ( x) b R v x b S 0 or x x B br b S 0 + If a Type- consumer x purchases produc 0 in period, she will repea purchase 0 if and only if H x b R 0 L ( x) b S or x x A bs b R 0 ++() If a Type- consumer x purchases produc in period, she will repea purchase if and only if L ( x) b R H x b S 0 or x x B br b S 0 ++() Similarly, if a Type- consumer x purchases produc 0 in period, she will reeap purchase 0 if and only if L x b R 0 H ( x) b S or x x A bs b R 0 + () If a Type- consumer x purchases produc in period, she will repea purchase if and only if H ( x) b R L x b S 0 or x x B br b S 0 + () In period, rm 0 s demand from repea cusomers is D R 0 = [ + ]x A + (x A + x A ); and is demand from swichers is D S 0 = [ + ](x B ) + [(x B ) + (x B )] : Firm s demand from repea cusomers is D R = [ + ]( x B ) + [( x B ) + ( x B )] ; and is demand from swichers is D S = [ + ]( x A ) + [( x A ) + ( x A )] :

13 Each rm i s period- pro funcion is R i = D R i b R i + D S i b S i Le [ + ( ) ] + ( + ) The rms period- rs-order condiions R S 0 = br b S 0 + R check R S 0 = S = b R b S 0 + = S = b S b R 0 + = bs b R 0 + = 0, = 0 We can = 0 joinly lead o b R 0 = (+) and b S = ( +4), and ha ( 4) = 0 = 0 joinly lead o b R 0 = and b R = h edious algebra hen veri es R i = b R i i + b S i ( ) Sraighforward bu Nex, we urn o period Recall, by consrucion k (k = ; ; ) represens he Typek consumer indi eren o he wo goods in period For example, consider Buying produc 0 in period yields her oal surplus V 0 ( ) = (H a 0 ) + L b S ( ) Buying produc yields her oal surplus V ( ) = [L a ( )] + H b S 0 Tha V 0 ( ) = V ( ) leads o b S 0 b S = a 0 a : We can easily check ha his relaion is also obained by analyzing he period- choices of and Subsiuing b S 0 and b S ino he above equaion and rearranging, we have = (a 0 a ) : 8 Noe ha by consrucion sands for rm 0 s period- demand Firms 0 and s period- objecive funcions are 0 = (a 0 a ) a 0 + R 0 8 and = + (a 0 a ) a + R ; 8 respecively The rms period- rs-order condiions are d 0 da 0 = + 7a 0+a 6 d da = + a 0 7a 6 = 0 and = 0 These wo equaions joinly lead o a 0 = a = 4 and = We hen readily obain b R 0 = b R =, bs 0 = b S = 7 Each rm makes oal pro s of QED 8