Hila Etzion. Min-Seok Pang

Transcription

1 RESERCH RTICLE COPLEENTRY ONLINE SERVICES IN COPETITIVE RKETS: INTINING PROFITILITY IN THE PRESENCE OF NETWORK EFFECTS Hla Etzon Department of Technology and Operatons, Stephen. Ross School of usness, Unversty of chgan, nn rbor, I 4810 U.S.. {etzonh@umch.edu} n-seok Pang Department of Informaton Systems and Operatons anagement, School of anagement, George ason Unversty, Farfax, V 00 U.S.. {mpang@gmu.edu} ppendx Dervatons of Equlbrum Prces and Profts per arket Confguraton Confguraton We derve the equlbrum n prces and demands gven the choces of the two frms n the frst stage of the game. We consder only cases n whch (1 each frm has postve demand for ts product, and ( market s covered. The requred condtons on the parameters values are gven n the followng assumpton. ssumpton 1. Condtons for Spatal Competton n Equlbrum ( t > (α + α, where α 0 when frm does not offer servce (1 ( c t + α I Δ < s < t α + c Δ (, j and, j ( ( t + (α + α Δ < s s j < t (α + α j Δ (v m m j < t (v t+ m+ m ( NS V > max, V, V, V ( (4 ssumpton 1-( ensures that an equlbrum n whch both frms have postve demand prevals when only one of the frms offers the servce (else, one frm would set a prce to undercut the other and capture the entre market. Smlarly, ssumpton 1-( ensures both frms have postve demand when both frms offer the servce, and ssumpton 1-(v ensures both frms have postve demand when both sell only the product. ssumpton 1-( s necessary for the ranges gven n ssumptons 1-( and ( to be none empty, and s thus mpled by the other two condtons. n dentcal assumpton s set n L and Chen (01 (where 1 and θ denote the degree of network effects, who state: If t < θ, the network effects domnate employees preferences over product s stand-alone value and employees wll always purchase form one sngle seller. s s shown below, gven condton (, all S.O.C are satsfed. IS Quarterly Vol. 8 No. 1 ppendces/arch 014 1

2 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Fnally, ssumpton 1-(v ensures that the nherent value of the product, V, s suffcently hgh so that the market for the product s covered by the two frms, whether both, nether, or only one frm offer the servce. 1. Case NN: oth Frms Sell Only Product When nether frm operates the servce, the surplus a consumer obtans when buyng the product sold by Frm and the surplus from buyng the product sold by Frm, are gven respectvely by u NN V tx p NN u NN V t(1 x p NN For spatal competton (the market s covered and the margnal customer has postve utlty, t must be that V that when ths condton holds, n equlbrum th product prce s * t+ m + m (,, j pnn j and j t m m > + + (5 (6. It s easy to show (7 The market share of Frm s ( t m+ mj 6t, and ts proft s gven by * ( y m mj + NN 18t (8 In ths paper, we lmt our attenton to cases of spatal competton; that s, we assume V t m m > + + (see ssumpton 1-(v.. Cases and NS: Only One Frm Offers a Servce Wthout loss of generalty, we assume that only Frm decded to offer a servce to ts customers. The soluton when only Frm offers the servce can be derved n a smlar manner. When expected network sze of frm s N, the consumer surplus when buyng from Frm, u, and when buyng from Frm, u are gven by u V tx + s N p u V t(1 x p (9 (10 The locaton of the customer who s ndfferent between the two frms, denoted by x, s thus ( t N s p p xn + α + + t (11 The demand for the product and servce of Frm, D, gven that consumers expect the number of servce users to be N, s gven by x (N. In the fulflled expectaton equlbrum, we requre that Solvng the above equaton for D, we get D x (D (1 ( + + t s p p D tα (1 Gven our assumpton that the market s covered, the demand for Product s gven by D D. Fnally, the proft functons of two frms are gven by D (p m c and N D (p m. IS Quarterly Vol. 8 No. 1 ppendces/arch 014

3 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Solvng the frst-order condtons smultaneously (S.O.C requres t > α, whch s satsfed due to ssumpton 1-(, we fnd that n equlbrum prces and profts are as follows: * t s c m m p α * t s c m m, p α (14 ( α ( tα * t c s m m * ( t+ cα s+ m+ m, 9 ( tα (15 t the above prces, the condton for both frms to have postve demand (.e., 0 < D < for, s c t + α m + m < s < t α + c m + m (16 To ensure spatal competton at the above prces, we need to fnd the surplus of the customer ndfferent between the two products and requre t to be postve. Dong so we get the followng condton: ( ( t t c s m m m V > V α + α + α 6tα (17. Case : oth Frms Offer a Servce When both frms offer the servce, the utlty functons are gven by The locaton of ndfferent customer x u V tx + s N p u V t(1 x + s + α N p s found by solvng u u and s gven by ( N xn, t s s N N p p + + α α + t (18 (19 (0 The demand for the product and servce of Frm, D, gven consumers expectatons regardng network szes, s x (N, N, and the demand for the product and servce of Frm, D, gven the assumpton that the market s covered s D. In the fulflled expectaton equlbrum, we requre that Solvng the above two equatons smultaneously for D and D, we get D x (D, D and D (1 x (D, D (1 D ( α + +, D ( α + + t s s p p N t N s s p p t ( α + α t ( α + α ( The proft functons of the two frms are gven by D (p m c ( and ( Solvng the frst order condtons smultaneously (second order condton requres t > (α + α, whch s satsfed accordng to ssumpton 1-(, we fnd the equlbrum prces p 1 (s s + c + t (α + α + m + m p 1 (s s + c + t (α + α + m + m (4 The profts at the optmal prces are gven by IS Quarterly Vol. 8 No. 1 ppendces/arch 014

4 Etzon & Pang/Complementary Onlne Servces n Compettve arkets ( α + α ( ( α + α ( α + α ( t ( α + α t ( s s m m t ( s s m m, t (5 The condton for both frms to have postve demand (.e., the margnal customer s locaton s nteror s whch also requres that t + (α + α + m m < s s < t (α + α + m m (6 t > (α + α or else above range for s s values s empty. Fnally, wth the above prces, there s spatal completon f and only f ( 5 ( α α 1 V > V C s + s m m t α + α + ( t ( s s t m m t ( α + α (7 (8 ppendx Proofs Proof of Proposton 1 Havng obtaned the equlbrum prces and profts n ppendx (see also Tables and n the paper, we now derve the condtons for each possble market confguraton to be an equlbrum. The condtons are derved as follows: ( ( oth frms offer the servce n equlbrum f and only f > NS and >. oth frms sell only product n equlbrum f and only f NN > and NN > NS. ( Only Frm offers a servce n equlbrum f and only f > NN and >. (v Only Frm offers a servce n equlbrum f and only f NS > and NS > NN. Equlbrum n Whch oth Frms Sell the Servce In order for both Frm and Frm to offer the servce n equlbrum, t must be that > NS and >, so that nether frm has ncentve to devate and not sell the servce. These two condtons are gven by t where X 1 ( α + α tα j s > X s j + Y for (, j and for (, j ( ( and Y ( t α j+ c+ mmj t ( α + α t + + m + m α. t α α α j j j 4 IS Quarterly Vol. 8 No. 1 ppendces/arch 014

5 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Equlbrum n Whch Nether Frm Sells the Servce n equlbrum n whch nether frm provdes the servce exsts f and only f NN > and NN > NS, so that nether frm has ncentve to devate and offer the servce. From the proft expressons n Table, we fnd that NN > f and only f ( t m+ mj ( tα s < t + α + c+ m m t (9 We denote ths upper bound by s G. s G can be derved n a smlar manner. Equlbrum n Whch Only Frm Sells the Servce The condtons under whch there s an equlbrum n whch only Frm offers the servce are ( > and ( > NN. Condton ( mples that Frm does not have an ncentve to devate and offer the servce. Condton ( ndcates that Frm does not have an ncentve to devate and not offer the servce. Condton ( and ( translate to s < X s + Y and s > s G, respectvely. The condtons under whch an equlbrum n whch only Frm sells the servce s feasble can be derved n a smlar manner. Proof of Proposton We derve the condton for < NN. ( ( α+ α j + j + j 9 ( t ( α+ α j t s s m m ( ( α+ α j + j + j 9 ( t ( α+ α j t s s m m, and gven ssumpton 1-(, we have t (α + α j + s s j + m m j > 0. Thus ( + j t m m NN 18t < If and only f ( t m mj ( t ( j < + α + α ( t ( α α s sj m mj t (0 Rearrangng terms, we get ( t m mj ( t ( j s s < + + s s + m m + + α + α ( α α j j j j t Proof of Proposton ( We examne the dervatve of the proft of Frm, when both frms offer the servce, wth respect to the α. α ( ( α + α ( t ( α + α t s s m m α D ( α + + ( ( α + α ( t ( α + α t s s m m t s s m m ( α + + ( t ( α + α t s s m m (1 The above s postve f and only f condton stated n Proposton -(. ( α + + ( t ( α + α t s s m m s negatve, whch s equvalent to s > s + t α + m m. Ths s the IS Quarterly Vol. 8 No. 1 ppendces/arch 014 5

6 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Next, we examne the dervatve of the proft of Frm, when both frms offer the servce, wth respect to the degree of network effects of Frm. α ( t ( α α s s m m α 9 ( t ( α + α D ( 5 ( α + α + + ( ( α + α ( t ( α + α ( 5 ( α + α + + ( t ( α + α t s s m m t s s m m t s s m m ( 5t ( α + α s+ s+ mm The above s negatve f and only f. y ssumpton 1-, t > (α + α. Thus, ( t ( > 0 α + α f and only f s s < 5t (α + α + m m. In addton, due to ssumpton 1-( we have α ( < 0 (5t (α + α (t (α + α t (α + α > 0 nd due to ssumpton 1-( we have s s < t (α + α + m m, whch leads to s s < 5t (α +α + m m. Therefore, α s always negatve. ( Suppose that n equlbrum Frm offers the servce and Frm does not. Then, the dervatve of Frm s proft wth respect to α s α ( α ( tα t c s m m α D ( + α + + ( α ( tα ( t+ α c+ s m+ m ( ( tα t c s m m t c s m m Gven our assumpton that both frms have postve product demands, whch also requres t > α, we see that f s > t α + c + m m. α ( s postve f and only Next we examne the dervatve of the proft of Frm : α ( t+ cα s+ mm α 9 ( tα D ( 5 + α + + ( + α + 9 ( tα ( 5t+ α + c s+ mm ( ( tα t c s m m t c s m m (4 We see that s negatve f and only f ( 5t+ α + c s+ mm α. Gven that t > α, we fnd that Frm s proft s ( t < 0 α decreasng n α f and only f s > 5t + α + c + m m. Furthermore, ( ( t m+ mj tα 5 + < 0 t ( t α c m m s ( t α 6 IS Quarterly Vol. 8 No. 1 ppendces/arch 014

7 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Thus, s G > 5t + α + c + m m. We conclude that when Frm offers the servce n equlbrum (whch mples s > s G accordng to Proposton 1, t must be that s > 5t + α + c + m m, and thus < 0. ( We examne the dervatve of the proft of Frm, when both frms offer the servce, wth respect to the common degree of network effects: α ( ( α + ( + ( tα ( + ( 9 ( tα t s s m m s s m m α α In equlbrum we have 1 ( s s m + m and 1 ( s s + m m. Under our assumpton that Frm has ( D 6 ( + tα D 6 ( + tα postve demand D s s m + m > 0, t must be that <. Thus, tα ( ss m+ m α ( ( tα 18 9 < 0 (5 (6 Smlarly, α s negatve when both frms have postve product demand. Proof of Proposton 4 ( We examne the dervatve of Frm s proft, when both offer the servce, wth respect to. ( ( α + α ( t ( α + α ( + + ( α + α ( ( ( α α α α t + t s s m m t s s m m t D (7 bove s negatve f and only f t( ss m+ m+ t ( α + α ( t ( α α α + α s negatve, whch s equvalent to ( α α ( α + α ( α + α s s < + t + m m t (8 The RHS of 8 can be ether negatve or postve. ( Suppose only Frm offers the servce. ( α ( tα ( α ( + α + ( tα t s c m m D t c t s m m (9 The above s postve f and only f equvalent to ( α t( c t+ α s+ mm ( tα s postve, whch, gven the assumpton that t > α, s α s > c t+ α + m m t (40 IS Quarterly Vol. 8 No. 1 ppendces/arch 014 7

8 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Proof of Proposton 5 We start by dervng consumer surplus under each of the four possble market confguratons (, NN, NS, and. Defne x ndf as the locaton of the consumer ndfferent between buyng the product from Frm and buyng from Frm. Then, when both frms offer the servce n equlbrum x ( α α ( t ( α + α t s s m m ndf (41 When only Frm sells the servce, n equlbrum x t c s m m ndf α + + 6tα j (4 Consumer surplus when Frm sells the servce and Frm does not s gven by 1 ( α ndf ( ( 1 xndf t+ c α + s m+ m ( α ndf t+ cα s+ mm ( 1 ( ndf CS V tx p + s + x dx + N V t x p dx x x ( ( ndf V tx + s + x dx V t x dx xndf ( 5tα ( scmm V 4 ( α + scmm ( t( s c+ m m + α ( 7tα + 6( tα Smlarly, consumer surplus when only Frm sells the servce s gven by: CS V + ( 5 α ( t s c m m 4 ( α + ( ( + + α ( 7 α 6( tα s c m m t s c m m t Consumer surplus when both frms offer the servce s gven by xndf t ( α + α + s s+ c m+ m ( ( α ndf 0 CS V tx + s + x dx + 1 t ( α + α + s s+ c+ mm ( ( 1 ( + + α ( 1 ndf V t x s x dx xndf ( 6( ( α + α 1 6 ( α ( α + s+ mm 18 18( t ( α + α s s c t m m t t s V + + ( α ( t m( α + α t t s s m m Fnally, consumer surplus when nether frm offers the servce s gven by: + 8 IS Quarterly Vol. 8 No. 1 ppendces/arch 014

9 Etzon & Pang/Complementary Onlne Servces n Compettve arkets t+ m+ m 1 t+ m+ m ( t m + m ( ( 1 6t (( 18 ( t CS V tx dx + V t x dx NN t m+ m 0 m m t m m t V + 6t We denote the socal welfare when both frms offer servce, + + CS, by SW, the socal welfare when nether frm offers servce NN + NN + CS NN, by SW NN,, and the socal welfare when only Frm offers servce by SW. The proft expressons are gven n Table, and were derved n ppendx. F (s, s j s defned as the dfference between socal welfare when both frms offer servce to socal welfare when only Frm offers servce, specfcally: ( F s, s SW SW j ( α α ( t( t α+ sj s+ mmj 9 7s + sj j 8c7 m m + + ( t ( α+ α j ( 4t 5α+ ( sj s+ mmj ( tα s+ sj+ mmj t ( α+ α j ( t+ scm mj( t+ s+ c m+ mj t( c+ mmjst tα ( tα Thus, when F (s, s j < 0, socal welfare when only Frm offers the servce exceeds socal welfare when both frms offer the servce. Settng m m j, F (s, s j becomes ( j F s, s 9 t( t α j+ sjs ( 4t 5α+ ( sjs ( tα s+ sj 7s + sj + ( 5α + αj 8c+ + ( t t ( α ( + α α j + α j t+ s c t+ s+ c t cs t ( ( ( tα ( t α c s N > α When m m, gven the condtons on s specfed n ssumpton 1, we can show that SW > SW NN ff. Smlarly, SW > SW NN ff s > α cα N. In addton, t s easy to show that < s. Thus, as long as n equlbrum at least one frm offers the servce (.e., at s > α s > α least one s s larger than s G, we know that NN s not socally optmal. s long as or (or both, socal welfare when one frm offers servce exceeds socal welfare when nether offers, and thus socal welfare s maxmzed when both offer servce f and only f F (s, s > 0and F (s, s > 0. s < α s < α Fnally, when and, socal welfare when nether frm offers servce s larger than socal welfare when only Frm s < s < or only Frm offers the servce. In addton, when α, α N, and c > α, we fnd that SW < SW NN. Fnally, when s < α and s < α cα N, n equlbrum, nether frm offers servce (as < s. Thus the equlbrum s NN, whch s also socally optmal. The rest s trval based on the results from Proposton 1. (4 IS Quarterly Vol. 8 No. 1 ppendces/arch 014 9

10 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Proof of Proposton 6 In the case n whch frms choose the drect servce qualty (s endogenously, to ensure that the second-order condtons are met, the market s covered, and the two frms have postve demands, the followng parameter assumptons are needed. ssumpton. ( t > α ( and c > ( ( ( and 18 t α c > + ( ( ( and t α c (v c < t α ( and c < o (v 9 ( ( and t α ( In Case, α ( 9 ( ( 9c( tα c c t s c < c > Ths s postve f and only f ( c > t and or ( c < t and. In the latter case, 9 ( c t 9 ( c t ( c t and thus, when c < t, we have α s > 0 for all postve c. < 0 9 ( In Case, ( ( 9 ( α ( + 9 ( ( α + α c c c c t α s ( c c c c t y ssumpton -(v, 9c (t α + ( 9c (t α. Therefore, α f and only f. s > 0 9c( t α c c < α ( ( + 9 ( α ( + 9 ( ( α + α s c c c c t ( c c c c t c c < + Ths s postve f and only f (. 9c t α ( When α α α, the optmal drect value s whch s postve f any only f c < c. ( 9c ( tα s ( c+ c 18cc( tα. Then ( ( ( c+ c 18cc( tα c c c α s 10 IS Quarterly Vol. 8 No. 1 ppendces/arch 014

11 Etzon & Pang/Complementary Onlne Servces n Compettve arkets Proof of Proposton 7 ( In Case, D α ( α c ( tα ( 9c( t α+ c α ( 9c( t α ( 9c( tα ( c( t α + c 9 ( c( tα c t c D We can show that s postve by ssumpton and 9c (t α > 0 by ssumpton -(. Thus, > 0 f and only f c+ t 9c (t α + c > 0, whch s equvalent to >. α 9c α ( α ( ( ( α + α ( + ( α + α 9 ( c+ c 9cc( t ( α + α cc 9c t 9c c 6 ( ( + ( α + α ( ( + ( ( α + α 9( ( c+ c 9cc( t ( α + α c 9ct c 6 c c 9cc 7t 4 The frst term of α s negatve because 9c (t (α + α > 0 and (c + c 9c c (t (α + α > 0 by ssumpton -( and (v. lso by ssumpton -(v, 9c + c (α + 6α > 0. Thus, the second term s negatve f (c + c 9c c (7t (α + 4α > 0 ( ( c c 1 6t whch s equvalent to α > 6α +. 7 ( When α α α and both frms offer the servce, ( 18 ( α ( 9 ( α 9( ( c+ c 18cc( tα ( ( 9 ( α + ( 18 ( α + 7 ( α D 9( ( c+ c 18cc( tα c c t c t α ( c c t c t c c t c ( ( ( ( ( α > 0 ( Therefore f and only f c 9c t α + 18c t α + 7c c t α c > 0. s the c α > 0 coeffcent of s negatve by ssumpton -( and (v, f and only f < c < c c 7c t + 81c t + 6c t 4 7c t 81c t + 6c t 4 ( α ( α ( α ( α ( α ( α c ( ( ( ( ( However, when α (the lower bound of c gven by ssumpton -, 18( t α α 9c t D 1458 c + c 18c c tα > 0 by ssumpton -(v. lso, ( ( ( c+ c cc( tα f c c then D 4c 9c( tα <. 0 α α > Therefore, f and only f c < c < c. 7c( tα 81c( t α + 6c( tαn 4 Proof of Proposton 8 ( For example, n Case, IS Quarterly Vol. 8 No. 1 ppendces/arch

12 Etzon & Pang/Complementary Onlne Servces n Compettve arkets ( tα c 9( tα( tαc ( tα c s c ( α D c t ( c ( t α ( ( α < c c t 9 c ( t α c ( t α c c c ( c ( t < 9 α 0 ( 9c( tα c ( t α ( c( t α c + c ( 9c( tα 6 ( tα( tαc ( c( t α+ c ( t α( t α c D 6 ( c ( t 9 α ( 9c( tα In, t α c > 0 by ssumpton -(v. Thus, > 0. c ( When both frms offer the servce, c c s c ( 9ct+ c( α + 6α ( + ( ( α + α c c 7c c t ( ( ( 9 ( α 6α 9 ( ( α α ( c+ c 9cc( t ( α + α c t+ c + c t + y ssumpton -( and (v, the numerator of s s postve. Thus, s > 0. c ( ( ( c c c ( 9 ( ( α + α ( 9 + ( α + 6α 9 ( c+ c 9cc( t ( α + α c c t ct c ( ( 9 + ( α + 6α ( + ( 9 ( ( α + α 9( ( c+ c 9cc( t ( α + α c t c c c c t c + c 9c t α + α s postve by ssumpton -(. Thus, > 0. c c s ( 9 + ( 6α + α ( + ( ( α + α c t c ( c c c c t 9 The numerator of s s postve by ssumpton -(v. Thus, s < 0. c c c ( ( ( ( α α ( ( α α ( ( α α 9 ( c+ c 9cc( t ( α + α c 9ct+ c 6 + 9ct+ c + 6 9c t + Smlarly, by ssumpton -( and (v, the numerator and denomnator are postve. Thus, < 0. c Reference L, X., and Chen. Y. 01. Corporate IT Standardzaton: Product Compatblty, Exclusve Purchase Commtment, and Competton Effects, Informaton Systems Research (:4, pp IS Quarterly Vol. 8 No. 1 ppendces/arch 014