Hong Xu. School of Business and Management, Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, HONG KONG

Transcription

1 RESEARCH ARTICLE IDETITY MAAGEMET AD TRADABLE REPUTATIO Hong Xu Scoo of Business an Management, Hong Kong University of Science an Tecnoogy, Cearater Bay, Kooon, HOG KOG Jianqing Cen Jina Scoo of Management, Te University of Texas at Daas, 800 West Cambe Roa, Ricarson, TX U.S.A. Anre B. Winston McCombs Scoo of Business, Te University of Texas at Austin, 0 Seeay B6000, Austin, TX U.S.A. {ab@uts.cc.utexas.eu} Aenix A A. Proof of Proositions an We first anayze te incentive comatibiity conition for o tyes. otice tat te objective for a o-tye agent eviating to urcasing a goo reutation is formuate in Equation (9). Te first-orer conition of Equation (9) is α ' + β v 0, ic eas to te eviating effort as ĺ /. Te eviating rofit an equiibrium rofit for o tyes become ( ) [( ) ] π βfv + βfv + β v + v v g b g ( ) π βf v + βf v + βv v b b Incentive comatibiity for o tyes requires π ' # π, ic is equivaent to f f β v + ( f f) βv + ( β( ) ) v 0 Diviing bot sies by v, e can reorganize te conition as M βv # M, ere ( α) β( ) M β f f [( ) ( ) ] M β f f MIS Quartery Vo. 4 o. Aenix/June 08 A

2 Xu, Cen & Winston/Ientity Management an Traabe Reutation Simiary, e can estabis tat /. Te corresoning eviating rofit an equiibrium rofit for ig tyes are ( ) π βfv + βfv + βv v b b π β f v + β f v + β ( ) v + v v g b g Incentive comatibiity conition for ig tyes requires π # π, ic is equivaent to β ( α) β ( β ) f f / v + f f v + ( ) v 0 Furter simify te above by iviing bot sies it v, an e ave te conition as M βv M. Combining te to IC conitions eas to M Mβv M. We first so tat M > 0. To see tis, notice te equivaent conition is β < /[ + ( α)(f f )], ere te rigt-an sie is greater tan. Furtermore, te sign of M is consistent it f f. Terefore, if f f, or, equivaenty, if / < ( α)/[ ( α)], ten M < 0 an tus no searating equiibrium can be sustaine, ic roves te resut in Proosition (b). Part (a) is a secia case of Part (b) it α 0. We next focus on a strict reutation system it f > f suc tat bot M an M are ositive. We substitute in v λ /(βλ f + βλ f ) to reorganize te IC conitions as M M λ M f + f λ M Define λ an λ as te vaues of λ / λ, tat mae te to IC conitions bining; tat is, π π an π π. We can erive λ as in Equation () an λ as in te roosition. For any searating equiibrium tat satisfies te above conition, te corresoning roortion ratio λ / λ must be beteen λ an λ ; tat is, λ λ / λ λ. Furtermore, in orer to fin a λ tat satisfies te above conitions, it suffices to so tat ( M )/( M ) $ ( M )/( M ) an M /( M ) > f. Te first conition is aarenty true, an te secon conition can be simifie as f β > β ( f ) ( f ) ( α ) + ( + ) f + f f f In aition, because β 0 [0, ], a searating equiibrium requires reritten as / it as in Equation (0). f f f f f f 0, ic can be + Substituting f θ in Equation (7) an v in Equation (8) into Equations (5) an (6), e can erive θ as in Equations (3) an (4). A. Proof of Proosition 3 otice tat is ecreasing in α. We can verify tat te secon fraction in (i.e., ) is + 4 ( ) aso ecreasing in α. Terefore, is ecreasing in α. A MIS Quartery Vo. 4 o. Aenix/June 08

3 Xu, Cen & Winston/Ientity Management an Traabe Reutation A.3 Proof of Coroaries,, an 3 otice tat Equations (3) an (4) can be reorganize as λ( ) λ ( ) + λ ( ) an λ ( ) λ + λ Proof of Coroary : Because ( α)/( α) is ecreasing in a, it is easy to see tat v increases in α, increases in α, an ecreases in α. Proof of Coroary : It is straigtforar to verify tat increases in /. For, e tae erivative of te enominator it resect to / an obtains ( )/( ) λ /( λ). Wen / > ( ) λ / ( ) λ, te enominator ecreases an increases in / ; oterise, te enominator increases an ecreases in /. Proof of Coroary 3: ote tat λ / λ λ /( λ ) is increasing in λ. It foos irecty from te exressions of v,, an tat tey are a ecreasing in λ / λ an λ. A.4 Proof of Proosition 4 ( m) b Te equiibrium satisfies te fooing conitions: b λ + λ αb+ βv αb+ βv βv mλ, b,,, an Te first tree equations ea to ( ) ( ) ( ) λ m λ m + λ + λ m b (8) β Mv b M b β β v ( M) v ( M) (9) (30) (3) ere M [( m) λ + ( λ) ] /[ ( mλ) ]. Hig tyes at goo an ba reutations must be inifferent; tat is, π π b, ic eas to Substituting in Equations (9) to (3), v can be reresente as ( α) β + + v 0 b b MIS Quartery Vo. 4 o. Aenix/June 08 A3

4 Xu, Cen & Winston/Ientity Management an Traabe Reutation βv α ( ) M ( ) ( ( )) ( ) ( ) β α α M (3) Te equiibrium eves of v an m can be sove troug (8) an (3). For o tyes, enote teir eviating effort an rofit as ' an π '. If tey eviate to a goo reutation, teir incentive can be escribe troug te first-orer conition as α t ' + β v 0, ic eas to ' /. For o tyes to refer ba reutations to goo reutations, e nee π < π, or equivaenty ( β ) + + v < ( ) ( α)( b) ( ) 0 ote tat ' / an b /, an / <. Te above IC conition os as ong as ' >, ic can be simifie to te conition on / in Proosition 4. We aso nee to ensure tat m 0 [0. ] in equiibrium. First, notice tat bot M an βv are ecreasing in m. We ten rearrange Equation (8) as ( ) ( ) + ( ) λ m λ m λ b ote tat te eft-an sie (LHS) of tis equation is increasing in m ie te rigt-an sie (RHS) of te equation is ecreasing in m. In orer for m to be beteen 0 an, it suffices to so tat () en m 0, LHS < RHS in Equation (8); an () en m, LHS > RHS in Equation (8). Wen m 0, e ave M λ / + ( λ)/, LHS 0, an RHS βv ( λ + ( λ ) )/[ ( M)]. Te conition tat RHS > LHS is equivaent to βv > 0, or β < β, ere β is efine as in Equation (8). Simiary, en m, e ave M /, LHS λ βv /( ), an RHS ( λ ) βv / ( ). Te conition LHS > RHS is equivaent to βv < λ( )( )/[ λ( ) + ( λ) ( )], or β > β, ere β is efine as in Equation (7). It is easy to so tat β < β because α > 0 λ λ λ + In aition, e aso nee to ensure β <, ic eas to conition λ / λ < λ ere λ is efine as in Equation (9). A.5 Proof to Proosition 5 λ + λ ng αg + βv αg + βv βv Te equiibrium satisfies te fooing conitions: g,, g,, an λ + ( λ) n Te first tree equations ea to ( λ )( ) λ ( ) + ( λ ) ( g) n n (33) β v g (34) A4 MIS Quartery Vo. 4 o. Aenix/June 08

5 Xu, Cen & Winston/Ientity Management an Traabe Reutation βv ( ) v g β ( ) (35) (36) ere λ + λ n ( λ + λ n) [ ]/[ ]. Lo tyes at goo an ba reutations must be inifferent; tat is, π π g, ic eas to ( g ) + ( g ) + ( β( ) ) v 0 We can sove for v by substituting in Equations (34) to (36) as βv ( ) ( α )( ) ( ) ( ( ) ) β + + (37) Te equiibrium eve of v an n are etermine by (33) an (37). For ig tyes, enote teir eviating effort an rofit as ' an π '. If tey eviate to a ba reutation, teir incentive can be escribe troug te first-orer conition as α ' + β v 0, ic eas to ' /. For ig tyes to refer goo reutations to ba reutations, e nee π ' < π, or equivaenty + v < ( ) ( α)( g) ( β ) 0 ote tat ' / an g /, an / >. Te above IC conition os as ong as ' >, ic can be simifie to te conition on / in Proosition 5. We aso nee to ensure tat in equiibrium n 0 [0, ]. First, notice tat bot an βv are ecreasing in n, ic can be verifie it sime agebra. We ten rearrange Equation (33) as ( λ )( n ) λ ( ) + ( λ ) n( ) g (38) otice tat te eft-an sie of tis equation is ecreasing in n ie te rigt-an sie is increasing in n. For te equiibrium n to be beteen 0 an, it must satisfy to conitions: () en n 0, Equation (38) becomes an inequaity it te eft-an sie (LHS) greater tan te rigt-an sie (RHS); an () en n, Equation (38) becomes an inequaity it te eft-an sie ess tan te rigt-an sie. At n 0, /( ), LHS ( λ) βv /( ), an RHS λ( βv)/( ). Te equivaent conition of LHS > RHS is β ( ) ( α)( ) ( α ) β α α λ v + + > ( λ) λ + Simiary, en n, ten λ + λ, LHS 0, an RHS > 0 is equivaent to MIS Quartery Vo. 4 o. Aenix/June 08 A5

6 Xu, Cen & Winston/Ientity Management an Traabe Reutation βv ( ) ( α) + + β α α < λ λ α ( α ) ( α ) ( α ) + Te corresoning conition in terms of β becomes β 3 < β < β 4, ere β 3 an β 4 are efine as in Equations (0) an (). We can verify tat β 3 < β 4. In aition, e aso nee to ensure β 3 < an β 4 > 0, for ic it suffices to so tat / β 3 >, or equivaenty, λ / λ > λ ere λ is efine as in Equation (). A.6 Proof of Proosition 6 Consier an equiibrium ere () o tyes on reutation j 0, an ig tyes ave a te rest; () te reutation vaue ifference is suc tat v v 0 α, an v j v j b for j $. Te corresoning equiibrium efforts an rofits are a 0 π + ( α) a+ b π + a b j π + b ( α) j j j Te searating equiibrium as to satisfy te fooing five conitions. (39) Conition : π π j $, ic eas to a+ 3b ( )( ) (40) Conition : te roortion of reutation 0 is λ 0 λ. Conition 3: te roortion of a te oter reutations are suc tat λ ( ) λ0 / [( )( ) ( a+ b)], λ ( a+ b) λ/( b), an λj 3 bλj /( b). Te aggregate of tese roortions must be λ ; tat is, + a 3b λ ( ) ( a+ b) 4b λ (4) Conition 4 (ICH): ig tyes refer j $ to 0; tat is, b b a a + b + (4) Tis is equivaent to π π /. Conition 5 (ICL): o tyes refer 0 to j $ ; tat is, A6 MIS Quartery Vo. 4 o. Aenix/June 08

7 Xu, Cen & Winston/Ientity Management an Traabe Reutation a a b + ( α) + b b (43) ic is equivaent to π π / b( )/. We rove te existence of suc an equiibrium in to stes. Ste : We ant to so tat Equation (4) simy moves te equiibrium aong te ine escribe by Equation (40). In oter ors, as λ / ( λ ) varies from 0 to 4, te corresoning variations of a an b cover every singe oint in Equation (40). First, substitute Equation (40) into Equation (4) an rerite te atter as a function of b ony: λ M 3b+ M 6b λ λ 4b M + b ere M ( α)( ) /, an α. otice tat 3 3 λ 60 b + (36 0M + 6 M ) b + (7M 4M 3 4 M ) b ( 4 b) ( M+ b) an 60 > 0, 36 ² 0M + 6M ² > 0, an 7M ² 4M ² 3 ³ 4M ³ < 0. Hence, λ ecreases in b en b is sma, an increases in b en b is arge. We aso nee to cec eter Equation (4) itsef imoses any restrictions on te variations of b. Te ony conition tat as to be satisfie is λ $ 0. ote tat, α (a + b) > 0 is equivaent to b > ( α)( )/, ic is aays true because < 0. Simiary, α 3b + α $ 0 eas to a $ ( α)( )/, ic is aso true for a a $ 0. Finay, α 4b > 0 eas to b < ( α)/4, ic is equivaent to a $ ( α)(5 4)/(4 ) because b ( α)( )/(3 ) a/3. ote tat, if # 4/5, te inequaity aays os. Oterise, Equation (4) imoses te aitiona conition tat b > ( α)/4. So, en # 4/5, λ first ecreases an ten increases in b, as b goes u. In tis case, every oint on te ine in (40) can be reace at a certain λ, an e ony nee to fin one oint on (40) to emonstrate te existence of searation. Wen > 4/5, λ first ecreases an ten increases in b, an goes to infinity as b aroaces ( α)/4. In tis case, Equation (4) covers ony art of te ine in (40); tat is, 0 # b < ( α)/4. We nee to fin a oint itin tis range tat satisfies te ICs to so te existence of searation. Ste : Reorganizing te ICH an ICL by substituting in a M 3b, e ave We enote te mie exression as T(b). otice tat ( M 3 ) b b b + ( α) b ( M 3 ) b b T 3 8b + 6( M 3b) + ( ) > 0 b It is easy to see T(0) < 0, an T(M/3) > 0. Hence, as ong as T(b M/3) $ b en # 4/5 or T(b ( α)/4) > b en > 4/5, a b b searation must exist. Wen # 4/5, substituting in b M/3, e nee ( + ) b 0, ic eas to $ 3α/. In orer for 0 # # 4/5, α as to satisfy tat /3 # α #. Terefore, en /3 # α # an 3α/ # # 4/5, tere exists a searating equiibrium. MIS Quartery Vo. 4 o. Aenix/June 08 A7