Asymmetric Firms and their Willingness to Compete

Transcription

1 Asymmetrc Frms and ther Wllngness to Compete Prelmnary Verson Clemens Fedler Tlburg Unversty, CentER, TILEC February 18, 2016 Abstract In ths paper, we present a duopoly model of customer loyalty. Two frms compete n research and development efforts. Efforts ncrease the frms number of customers whle ncurrng a cost that depends partally on the number of customers. We show how the ths cost structure can generate a nverted u-shaped reacton functon of the frms efforts. We derve condtons under whch welfare s not rased by supportng the laggard and show how ths can help to understand the market for operatng systems. Keywords: Innovaton, Competton, Loyalty JEL Classfcaton: D21, L11 CentER, TILEC, Tlburg Unversty, P.O. Box 90153, 5000 LE Tlburg, The Netherlands; c.fedler@tlburgunversty.edu 1

2 1 Introducton Innovaton and the ncentves of frms to nvest n research and development are of fundamental nterest to socety. Innovaton s the engne that provdes ever new mprovements to the well-beng of customers and creates steppng stones for other ndustres to buld on and develop products never before consdered possble. Ths s true today just as t was n the last century when Wllg et al pondered the lnk of nnovaton and competton. 1 But how does market structure mpact the frms ncentve to create these nnovatons? Mght we be stallng nnovaton by regulatng markets gnorant of the consequences of our meddlng? The frms ncentves to nnovate are drven by two man aspects: Stealng market share from compettors and obtanng hgher profts from current customers. Ignorng dynamc consderaton, ths shows that a monopolst tends to nnovate less, as an nnovaton does not help to steal market share from compettors. Competng frms tend to nnovate more, as ganng the edge over compettors can ncrease profts by generatng addtonal revenue from each customer who s already purchasng from the frm and stealng market share from compettors. These aspects are even more mportant n the modern consumer electroncs market. Take the market for smartphone operatng systems. Here chefly two frms compete wth each other Google wth Androd and Apple wth OS. Only Apple derves profts from hardware sales, but both frms proft from app sales and advertsement. Accordng to documents leaked from an IP tral between Google and Oracle Inc, snce 2008 Google has obtaned profts of 22 bllon USD through both channels. 2 On the other hand Apple reported an Revenue of 32 bllon USD n the thrd quarter of 2015 n hardware IPhone sales. Whle t does not separately report the revenue from app sales total servce profts amount to 10 bllon USD. 3 Accordng to Wllg et al. 1991, an mportant paper on the US merger gudelnes, concentraton s seen as hghly suspcous. An mportant crteron to analyze the danger mposed by a merger s the Herfndahl-Hrschman Index HHI, whch s the sum of the squares of all market shares. The HHI s ncreasng n the degree of asymmetrc market shares. 4 Furthermore, mergers between small frms, resultng n small changes of the HHI, are treated more lenently. Smlarly, n duopoly models t s often the case that frms nnovate more f they are equal n strength and less f they are asymmetrc. We tend to consder nnovatve efforts as strategc substtutes - smlar to output n Cournot competton. If one frm ncreases ts research efforts, t erodes the other frm s ncentves. Even worse ths can lead to a monopolzaton n the long run based on some ntally small advantage of one frm. 1 [It] s a commonplace n today s economy that nnovaton s an mportant battleground for competton, and t seems evdent that market power and effcences based on asset holdngs play sgnfcant roles n shapng ts contours and vgor. Wllg et al. 1991, p Bloomberg accessed 22/01/ Quartz accessed 23/01/ HHI = N s2 = 1 N + N s 2 1 N 2 2

3 Ths s often seen as evdence that a hgher level of symmetry n the frms abltes to engage and proft from research and development maxmzes the ncentves to nnovate and that regulators should reduce the possblty of strong asymmetres. Even more mportantly, competton authortes are well known to am for symmetrc market shares and treat frms who compete wth smlar szed frms more lenently than f competng wth smaller frms. Contrary to that Bloom et al show emprcally that whle the technologcal spllover benefts amount to twce the prvate benefts [... ] smaller frms have sgnfcantly lower socal returns because they tend to operate n technologcal nches [... ]. Whch grants a reason to focus on encouragng market leaders. I wll argue n ths paper that the regulaton of asymmetrc markets mght be even more problematc than the lterature leads us to beleve. The efforts of frms can be complements and substtutes on the same market dependng on the relatve strength. I wll argue that the exstence of customer loyalty together wth not-perfectly scalng costs, requres a careful nterventon polcy. Supportng the weak frm can rase total efforts exerted f the weak frm s slghtly nferor to ts stronger compettor, but mght also reduce total efforts f the laggard s strongly nferor. Consequently, even when only consderng the short run mplcatons of a regulatory nterventon, amng for perfect symmetry n such markets mght reduce the ncentves to nnovate drectly. By creatng a smple model that nests both strategc substtutes and complements I can llustrate the dfference between Bertrand and Cournot competton. A deeper mechansm les underneath that leads to the dfference n frm behavor. I wll elaborate how ths can enter the decson makng of frms n a market that behaves more lke a Bertrand or more lke a Cournot market. Do frms harm each other s ncentve to nnovate or encourage each other? Furthermore, I wll dscuss how the underlyng parameters effect dfferent markets. What markets are more prone to have frms rase ther efforts or lower them n response to ther compettor s expanses. Especal the dfferent mpact that market condtons have on frms of dfferent szes s nterestng as t can help to better understand the ncentves of leaders and laggards, Fnally, I wll connect the model wth the smartphone operatng system market and llustrate how the unque stuaton there gves rase to non-monotonc reactons of the frms and how best to address ths ssue. 2 Lterature The model outlned here s smlar n sprt to Casell et al The authors dscuss conflcts between two countres and the effect an asymmetrc allocaton of resources has on the escalaton of conflcts. They come to the concluson that the exstence of natural resources and ther geographcal poston has a drastc mpact on the escalaton lkelhood. If only one country possesses natural resources, conflct becomes more lkely, the closer the resources are to the border. If both countres possess natural resources conflct becomes more lkely f the postonng relatve from the border s asymmetrc. I.e. war s more 3

4 lkely f one country has a resources n ts safe hnterland, whle the other country has ts resources close to the border. Ther model s smlar to ours, but nstead of natural resources and countres borders, we look at the number of customers and the separaton of market shares. The man dfference s that n ther model the outcome of the conflct s uncertan, whle n the model presented here, t s determned result of the efforts of both partes. Ishda et al show how and ncrease n competton mght have surprsng effects on frm profts and outcomes, when asymmetres are at play. They analyze a olgopolstc market wth hgh and low costs frms and shows how the entry of addtonal hgh costs frms mght stmulate R&D by low cost frms and reduce t by the hgh cost frms. Further more t rases the profts of the low cost frms. Closely related s Salant and Shaffer 1999, AER. The authors show two nterestng facts about the Cournot competton usng a two-stage model wth a R&D stage followed by a standard quantty competton: Frst, a change to the constant margnal costs of frms leaves aggregate output unchanged as long as the sum of all margnal costs remans constant and all frms reman n the market. Second, an ncrease n the varance of the costs wth the mean remanng constant leads to a shft of producton from hgh cost to low cost frms and decreases aggregate producton costs. Thus an ncrease n the cost spread that leaves the mean cost unchanged leads to hgher ndustry profts. If all frms reman n the market the consumer outcome remans unchanged and total welfare ncreases wth asymmetry. Bloom et al provde an emprc analyss of frms nnovatve efforts, ther prvate benefts and spll-over effects. They fnd that the externaltes of nnovaton are about twce the prvate benefts and undernvestment s lkely. Another possblty how asymmetry can mpact the competton ntensty n a market s the waterbed effect. Inderst and Vallett 2011 show that a powerful buyer mght negotate a better prce wth ts suppler. The latter s then forced to charge a hgher prce from other frms. Ths leads to a snowballng where ntal asymmetres are amplfed. 3 Model Consder the market for smartphones. Two frms {1, 2} compete over customers wth heterogeneous preferences. We are nterested n competton n R&D efforts, thus any prce competton s abstracted away from. Devces are sold at an exogenously gven prce such that frm derves a proft of γ from each customer. 5 Demand s gven by a contnuum of customers wth unt mass and a heterogeneous level of loyalty towards one of the brands. Customer k preference s gven by θ k Θ. θ k 0 mples that customer k has a strong preference for the devce sold by frm 2, θ k 0 that she has a strong reference for product 1 and θ k = 0 that she s ndfferent between the frms. Preferences are exogenously determned and dstrbuted accordng to the cdf G and the pdf g. 5 Thnk of prces beng determned by the retal market. 4

5 Assumpton 1 G s twce contnuously dfferentable on Θ such that g s contnuously dfferentable and the dstrbuton lacks atoms. Preferences are exogenously. They are non-marketable characterstc of the products such as past experences of the customer wth the frms. Ths can be used as the startng pont for a dynamc model. Frms compete n R&D efforts to ncrease the attractveness or qualty of ther own products whch steals customers from ther compettors. The profts derved from each customer γ x are weakly ncreasng n the efforts such that: γ x 0. R&D can lowers the manufacturng and dstrbuton costs or generates a hgher return from mproved thrd party servces. Furthermore, profts are concave γ x < 0. Consder a the development of a chp wth hgher transstor densty. The chp s faster, enablng better uses, but t also has lower varable costs. Customer k compares the utlty presented by both choces and purchases product 2 f and only f θ k x 1 x 2. By assumpton 1 tes occur wth measure 0. We defne the ndfferent customer as θ x 1 x 2. If frm 1 provdes a product superor to ts compettor s, only customers hghly loyal to frm 2 purchase from frm 2. The demand for frm 1 s q 1 = Gx 1 x 2. Example 1 Consder the sector for smartphone operatng systems and other platform servces. Here the supplers of operatng systems - chefly Apple, Google and Mcrosoft - provde a platform use by other frms to supply end users. Improvng the operatng system not only ncreases ther market share but also generates revenue from thrd party sales. As thrd partes develop better applcatons, the share n app sales that the OS supplers receve, grow. The unque characterstc of ths model are the costs of R&D: C x 1, x 2 = c q α x 1, x 2 x 2 /2 C x 1, x 2 = c q α α x2 q x + x x 2 q The costs depend on the efforts and the market share of frm. α [0, 1] measures how drectly costs scale wth the number of customers. If α = 0, costs are ndependent of the number of customers - e.g. an mprovement to the code can easly be mplemented on every devce. Conversely, for α = 1 R&D drectly rases the unt costs - e.g. lcensng a new chp from a thrd party for a fxed unt prce. In between, α 0, 1 and efforts partally scale wth quantty. 6 6 Thnk of Apple addng a pece of functonalty to OS. The ntal development s qute expensve, and adjustng the code for dfferent hardware requres further nvestment. As the number of devces compatble ncrease adjustment becomes cheaper. 5

6 Combnng everythng gves the proft functons: π x 1, x 2 = q x 1, x 2 γ x q α x 1, x 2 c x 2 /2 1 q 1 x 1, x 2 = Gx 1 x 2 q 2 x 1, x 2 = 1 Gx 1 x 2 Wthout loss of generalty we normalze the costs to c = 1 for {1, 2} by settng γ x γ x /c and c Optmal Frm Behavor Let x C be the optmal effort level under competton for frm and let θ C be the ndfferent customer under the optmal effort levels. The frst and second dervatves are gven as: π x 1, x 2 = q x 1, x 2 γ x αq α 1 x 2 /2 + q γ x x x q α x 2 π x 1, x 2 2 = 2 q x 1, x 2 x 2 γ x αq α 1 x 2 /2 x + q x 1, x 2 x 2γ x + α1 α 2 q α x 1, x 2 + q x 1, x 2 γ x For frm 1 ths expresson becomes: π 1 x 1, x 2 = gx 1 x 2 γ 1 αgx 1 x 2 α 1 x 2 x 1/2 1 Gx 1 x 2 α x 1 + Gx 1 x 2 γ 1x 1 q α 2 x 2 q x 1, x 2 2αq α 1 x x extensve margn ntensve margn An ncrease n frm s efforts helps t twce: It attracts customers from frm j extensve margn and t rases the return form customers already purchasng from t ntensve margn. The frst order condton gves an mplct soluton for the frms problem. For smplcty we assume that frms efforts are such that both frms reman n the market: q > 0 for {1, 2}. Lemma 1 Shared Market Let Θ = [ a 2, a 1 ] be the support of G wth a > 0 for {1, 2}. If lm ga γ 1 a 1 a2 1 a a 2 1 lm ga a a + 2 γ 2 a 2 a2 2 2 γ a a 0 + γ 1a 1 a γ 2a 2 a 2 0 both frms command a postve market share n equlbrum. 6

7 Proof: Frst, for a = t s trval to show that ths s fulflled. If the nequalty s satsfed for a t s satsfed for all a > a as ga = 0 and γ < 0. If frm j provdes zero efforts and frm provdes enough effort that q = 1 the latter fnds t proftable to lower ts efforts. An ncrease n x j has the same affect as an ncrease n a 1, whch lowers the RHS of the nequalty by lm ga γ 1a 1 a 1 + γ a a 1 a Thus, the nequalty s enough to guarantee x < 1 for all x j. Lemma 2 Quasconcavty For g = 0 and γ x < 0 the proft functon s quas concave. Proof: C x, 1, x 2 s convex n x as t s the product of a convex functon wth an ncreasng functon. Thus, the proft functon can only be convex f q x, x j γ x s convex whch requres: 2gx x j γ x }{{} >0 > q x, x j γ x }{{} <0 As γ x s decreasng n x so s the LHS. The RHS s ncreasng n x. Thus, f the nequalty s volated for one x t s volated for any x > x. Furthermore as the nequalty s strct t also holds for γ = 0 + ε for some small ε. In-fact lemma 3.1 can be relaxed, as shown by fgure 1, showng the proft functon for dfferent values of γ x. 7 Theorem 1 Exstence If q x 1,x 2 x > 0, γ < 0 and f the requrements of Lemma 1 are satsfed, a par x 1, x 2 exsts n whch both frms set ther efforts to maxmze ther profts gven ther compettors choce wth x, q > 0 for {1, 2}. Proof: By desgn the efforts x 1, x 2 0. Furthermore as γ < 0 and q 1, q 2 [0, 1] a x exsts such that x > x : γ x < x and γ x < x 2 /2. Thus the best response for the players s bound between [0, x]. The proft functons are contnuously dfferentable for nteror solutons and the best response functons are contnuous. Thus, by Brouwer s Fxed Pont Theorem a Nash Equlbrum exsts. By Lemma 1 market shares are q 0, 1. Thus for x = 0, profts are ncreasng n x and for x = x profts are decreasng n x. The optmal efforts are gven by: x C = q 1 α q 1 γ αq α 1 x 2 /2 + γ x q x 2 7 The parameter values are x 2 = 0 and α = 0. dx j 7

8 γx =0.1 x +1.0 π 1 x 1,x γx =0.1 x γx = x 1 Fgure 1: Proft functon for dfferent γx Dependng on the shape of G the game mght feature multple equlbra. To avod havng to assume unqueness the followng dscusson focuses only on local changes. Efforts depend on three effects. Frst, efforts are the hgher, the more customers are under threat q 1 x q. Ths effect s more powerful for the laggard than for the leader who has less customers to gan relatve to ther share. Second, the hgher γ, the greater are the efforts of frm. A frm that derves a hgher proft from each customer has a hgher ncentve to exert effort. As γ x > 0 ths augments asymmetry. The efforts also depend on γ x. If the profts per customer react strongly to efforts, frms wll use hgher efforts to rase them. Thrd, f the costs per customer ncrease strongly wth the number of customers frms wll provde less efforts. For low market shares ths decreases n α whle t ncreases for hgh market shares. Consequently, a low α can boost an ntal asymmetry n value extracton. Fgure 2 shows the equlbrum efforts for a two dfferent levels of asymmetry n the profts per customers, such that γ 2 = l 2 γ 2 for l 2 > 1. In general efforts are ncreasng n α and the leader exerts more efforts than the laggards. However, for hgh values of α the market leader exerts less efforts aganst a weak opponent than aganst a only slghtly weaker opponent. The cost structure works aganst asymmetry. For low αs ths s the other way around. The drvng mechansm s the change to the costs as a functon of the marketshare. For a low α the costs of the leader change lttle wth changes to quantty. 8

9 x C Leader - hgh asymmetry Leader - low asymmetry Laggard - low asymmetry Laggard - hgh asymmetry 0.02 α Fgure 2: Efforts for hgh and low levels of asymmetres 4 Comparatve Analyss 4.1 Baselne: Unform Dstrbuton Frst, consder the most basc case. Customers are dstrbuted unformly on [ k, k] wth q x 1,x 2 x = gθ = 1 2k g and g = 0. Secondly, the qualty does not effect the value extracted per customers γ x = 0 and costs scale one-to-one wth quantty α = 1. For smplcty we defne γ x = l γx wth l 1 = 1. The optmal efforts are: 2 x C q = 2l γ + q 3 g g The mplcatons are standard. An ncrease n g q leads to more customers at the margn and ntensfes competton. An ncrease n the efforts of frm j reduces frm s market share, makng ts efforts cheaper and rasng x. Efforts are strategc complements, smlar to Bertrand competton. 8 The other extreme s α = 0 and l γ x > 0, whch gves a market that behaves lke a Cournot market. The optmal efforts of frm become: g x C = q l γ + γ 4 q γ 8 In Bertrand competton a decrease of the compettors prce leads to a reducton n the number of customers makes prce cuts less expensve and causes the frm to lower ts prce. 9

10 Leader - α = Leader - α = 1 x C Laggard - α = Laggard - α = l 2 Fgure 3: Effect of Asymmetry Agan the efforts are ncreasng n both, the share of customers under threat and the return generated from each customer. Efforts are also ncreasng n the market share as they rase the return per customer. The market behaves lke a Cournot market as efforts are strategc substtutes. 9 To sum up: γ x adds substtutonary submsson pressure, whle α adds complementary escalaton pressure. If a frm can ncrease the return from ts product by ncreasng ts qualty, any reducton of ts market share leads to a declne of efforts. If a frm has less to lose as t falls behnd ts compettor, efforts act lke strategc substtutes. α determnes f a frm becomes more aggressve α = 1 as t loses market share or f the frm becomes more complacent α = 0. For completeness, f α = 0 and γ x = 0 the efforts are x C = gγ. Frms stll compete but ther efforts are ndependent. 10 Fgure 3 llustrates how dfferent levels of α affect the outcome on an asymmetrc market. The γ was scaled so that the symmetrc outcome s the same. In both cases the efforts of the laggard are ncreasng n l 2 and are lower for α = 0. More nterestngly for α = 0 the efforts of the leader are decreasng n l 2, for α = 1 they are ncreasng. The two extreme cases help to understand the mechancs, but more nterestng are ntermedate cases wth α 0, 1 and γ x > 0 n whch case both pressures exst. The optmal efforts are: 9 In a Cournot market f the competng frm rases ts output t lowers the prce for the other frm whch leads to t reducng ts output. 10 Condtonal on the parameters of the model. 10

11 x C = g q q 1 α l γx α 2 x2 + q 1 α l γ x 5 Multple equlbra can exst. If frm exerts hgh efforts t rases the costs for frm j who s then lmted to exert less efforts. Thus, even for symmetrc condtons asymmetrc equlbra can exsts. For x C the RHS s negatve, for x C = 0 t s postve thus t must ntersect an odd number of tmes wth x C. The RHS can be ncreasng n x but not around the optmal choce. 2 π x 1, x 2 2 x = l 2γ x + q α α1 α 2 x q 2 2α x 1 + q l γ x q For α = 0 and α = 1 the optmal efforts become: g x C = q l γ + γ α = 0 6 q γ x C = g l γ 1 q 2 x2 + l γ α = 1 7 The reacton functon of frm s: dx C = α gxc q dx C dx C j l γ x + q α dx C + 1 ααq 2 x 2 /2 g dx C dx C j dx C dx C j + l γ x g q α l γ x + q α 1 dx C αq 1 x dx C dx C = B A + B dxc j A = 1 + α x l γ x q q α + l γ x q α 1 B = g α x 1 αα 2 x l γ x q 2 q q α 8 dx C dx C j dx C dx C j l γ x = g 1 2l γ x q l γ x x l γ x = g q + 2x 2l γ x q l γ x α = 0 9 α =

12 Reacton dx xj Laggard Leader dfferent sgned reacton 0.10 α Fgure 4: Reacton of frms For α = 0 and γ > 0 ths expresson s negatve and for α = 1 t s postve. 11 Increasng γ rases x and q. Ignorng ths effect, an ncrease of γ lowers the slope of the reacton functon. Furthermore, for any γ and x C 1, xc 2 an α exsts such that the reacton of frm 1 n equlbrum s negatve for smaller α. dx dx j x C,x C j { > 0 α > α < 0 α < α 11 Now magne an exogenous shock reducng the effcency of frm 1 n extractng value, such that γ s ncreased by a factor l such that γ x = γx l and γ x = γ x l. Ths drectly leads to an ncrease n the efforts of frm 1 and ts demand, whch n turn lowers the costs of frm 2 but reduces the ncentve to ncrease the value extracton from customers. If α > α the frst effect domnates and frm 2 exerts more efforts. If frm 2 loses market share t - paradoxcally - becomes more compettve as the costs are reduced encouragng t to exert more efforts. Such a market s characterzed by unt costs that drectly depend on R&D and do not depend on the quantty provded. In the other case frms suffer from hgh start-up costs of ther research and development programs. Increasng ts efforts s qute expensve f only a small number of customers are served, but gets ncreasngly cheaper as the customer base s expanded. Then the reacton of the frms becomes negatve and losng market shares rases the costs of research. Fgure 4 llustrates the reacton of the frms to an ncrease of ts compettors efforts for dfferent levels of α. For small α the reactons of both frms are negatve, for large 11 If x γ x was negatve a reducton n x would rase the proft per customer and thus the total proft. 12

13 values they are postve. However, for a small range of values the reacton s postve for the laggard, but negatve for the leader. The reason for ths s that the reacton depends on the market shares. For a large market share losng customers only effects the costs of R&D by a small amount, whle the other channels do not mpact frms dfferently. Thus for a small α the market leader uses efforts substtutonary whle the laggard acts complementary. The sgn of the reacton functon depends on B as A + B s proportonal to the second order condton and thus has a postve sgn. From equaton 5 we have that: α 2 x 2 = 1 x l γx + q g q q 1 α + l γ x q α 1 x l γx < g q q 1 α + l γ x q α Whch substtuted nto the reacton functon 9 gves: B1 = α + 1 g 1 α x 1 α B1 = α x q α 2 x q q 2 l γ x q α l γx q 1 α 2 α l γ x q α For x = 0 ths expresson s negatve, for x t s negatve. The market shares mpact the result for any α [0, 1] and a larger market share makes the expresson smaller and more lkely to be negatve. Keepng efforts, quantty and the base return per customer γ fxed, the sgn depends on l γ as t determnes the mportance of the ntensves margn for efforts. Any loss of customers erodes the effort ncentves the stronger, the greater l s. γ An ncrease n α has a more unpredctable consequence. l x q s ncreasng n α, α whle α x q α 2 x q 2 s ncreasng f postve and decreasng f negatve. Consequently, for a negatve reacton an ncrease n α makes t more negatve, for a postve reacton the change can be negatve or postve. For small efforts an ncrease n x makes the reacton more postve, for large efforts an ncrease n x makes the reacton more negatve. A hgh effort level means steep effort costs, thus frms are more wllng to cut costs f possble. Fgure 5 shows efforts and reactons for dfferent values of γ, l 2 and g for l 1 = 1. Not surprsngly the efforts of frm 2 are ncreasng n l 2 x-axs and n g y-axs. However, the contour lnes of the efforts are smlar for the values of α. The efforts of frm 1 are ncreasng n g and ether ncreasng α = 0 or decreasng α = 1 n l 2. Ths s seen one to one n the reacton of frm 1, that s negatve for small values of α and postve for large values. 13

14 g α = α = l 2 a Efforts α = g α = α = l 2 b Efforts α = α =0.0 α =0.08 α = α =0.0 α =0.08 α = g g l 2 c Reacton l 2 d Reacton 2 Fgure 5: Comparson 14

15 The reacton of frm one s negatve for small a α and postve for a large α. In between the pattern s more complcated, nterestngly for ntermedate values of α the reacton functon changes non-monotone n l 2 and reaches ts maxmum for an nteror value of l 2. For α = 0.08 the reacton functon of frm 2 s domnated by the rato of x 2 q 2, whch leads to a nverse-u shaped reacton. If the efforts of frm 2 are hgh an ncrease n x 1 reduces the market share. 4.2 Loyalty: Double Peaked Dstrbuton As dscussed the market for smartphones exhbts a hgh decree of customer loyalty. Accordng to a 2013 survey by WDS 76% of all IPhone customers contnue to buy a new IPhone. Ths percentage s 58% for Samsung - the manufacturer wth the greatest sales among Androd phones. 12 Ths shows that both customers stayng wth a frm and swtchng are relevant n numbers gθ θ Fgure 6 Loyalty s modeled wth a double peaked dstrbuton of g. Half of the customers exhbt a preference for each of the brands. Customers loyal to frm 1 are drawn from a dstrbuton wth Eθ < 0. More generally ths mples that 2 q = 2 q x 2 x j x j > 0. Fgure 6 llustrates one example for such a gθ. The peaks represent two groups of loyal customers. The frst order condton reman unchanged and the second dervatve wth respect to x j s: 12 WDS Accessed 24/01/

16 π x 1, x 2 = q x 1, x 2 γ x αq α 1 x 2 /2 q α x + q γ x x x π x 1, x 2 2 = 2 q x 1, x 2 γ x αq α 1 x 2 /2 x x j x x j + q x 1, x 2 α1 αq α 2 x 2 /2 x q x 1, x 2 αq α 1 x γ x x j π x 1, x 2 2 γ x αq α 1 x 2 /2 x 2 = 2 q x 1, x 2 x 2 The reacton functon s gven as: + q x 1, x 2 α1 αq α 2 x 2 /2 x q x 1, x 2 αq α 1 x γ x x + q x 1, x 2 γ x x αq α 1 x q α + q γ x dx = π2 x 1 1, x 2 dx j x 2 }{{} >0 π 2 x 1, x 2 x x j Usng that q x 1,x 2 x = q x 1,x 2 x j and 2 q x 1,x 2 x x j = 2 q x 1,x 2 x 2 2 π x 1, x 2 = q x 1, x 2 x x j x 2 π x 1, x 2 x 2 2 q x 1, x 2 x 2 α1 αq α 2 γ x αq α 1 x 2 /2 the equatons smplfy to: x 2 /2 + αq α 1 x γ x = 2 π x 1, x 2 x x j q x 1, x 2 γ x x + αq α 1 x + q α + q γ x The only new aspect of the reacton functon s 2 q x 1, x 2 x 2 γ x αq α 1 x 2 /2 = π q whch gves the change to revenue per customer n response to a change n the number of customers, gnorng frm reactons. 16

17 The sgn of π q must be postve. Although t can be locally optmal to exert efforts such that π q < 0 f the γ s suffcently large, ths would mply that π x C, xc j < 0 < π 0, x C j and volate the partcpaton constrant. Theorem 2 Let x C be the optmal efforts of the orgnal game. Increasng the splt between both customer groups such that more customers become loyal to ther respectve brands whle the market shares reman unchanged such that q x 1,x 2 x x remans un- =x C changed and q2 x 1,x 2 x ncreases, leaves the equlbrum unchanged but reduces the 2 x =x C reacton functons of frm. Proof: The second dervatve of the demand does not enter the FOCs of the frms. Thus the optmal efforts reman optmal. Addtonally, the sgn of π q must be postve. Otherwse, π x C, xc j < 0 < π 0, x C j and the frm would prefer to exert no efforts. Consequently, a postve term s subtracted from the numerator and denomnator. Thus, the reacton depends on the sze of the orgnal reacton. 5 Dscusson Based on the prevous analyss we can now dscuss three typcal markets. Frst, a hardware market exhbts nnovaton costs that scale nearly perfect wth market shares α = 1. Frms add more hardware nto ther devces whch rases manufacturng costs. The mplcaton s smple. A lower market share lowers the cost of competton and encourages the frm. Second, a software market exhbts nnovaton costs that are ndependent from the market share α = 0. Improvng the underlyng code has a fxed costs rrespectve of the number of users. Agan the mplcaton s smple. A lower market share lowers the return to nnovaton and dscourages the frm. Thrd, a platform market features frms competng wth ther platforms for developers to use. Costs only partally scale wth market share. Developng a platform for a small number of customer s expensve as the costs need to be splt up on fewer people. On the other hand a hgher market share also means that more efforts need to be exerted to guarantee compatblty and debuggng servces to platform thrd party companes. Consder Apple addng an addtonal feature to OS that allows applcatons to better connect wth the operatng system. The ntal development s costly and barely depends on the number of customer usng OS. However, wth a large user base also come more complants about compatblty wth specfc programs, thus rasng the costs for debuggng. In ths case a very small market share blocks the frm from competng effcently as t faces very steep start up costs. However f the frms are equal n sze losng a few customers lowers the costs of the frm, whch helps t to compete more fercely. Thus efforts exerted are maxmal for an ntermedate value of asymmetry. 17

18 5.1 Thrd Party Now we consder a thrd party that can ncrease the profts l of one frm, whle extractng a lump sum payment of l γx C from the market partcpants. If costs are ndependent of the quantty sold, nterventon on the sde of one frm wll crowed out efforts of the other frm. If the costs scale perfectly wth the demand, encouragng one frm rases both efforts. However, for ntermedate levels of α, effort can be strategc complements for the leader and substtutes for the laggard. In whch case encouragng the laggard rases the efforts of both frms. The drver of ths s the dfferent change caused to the costs of R&D to the leader and the laggard. A change n the quantty sold, barely affects the leader, whle t strongly affects the laggards. Thus, the costs channel that adds complementary pressure to the frms reacton s stronger for the laggard than the leader. The second consequence s that small changes to the underlyng parameters can have bg consequences to the symmetry wthn the market. A postve shock to the market leader lowers ther costs of development and rases the costs for the laggard, whch leads to an even larger gap n research. The mplcatons are very useful for the consumer electroncs market: Here mprovng the qualty of the product has a hgh fxed proporton and hgh economes of scale. A plant that produces 100,000 chps s not much more expensve than a plant for 10, Welfare Consumer welfare n ths model s gven as: W x 1, x 2 = = θ θgθdθ + x 1 Gθ + x 2 1 Gθ x 1 x 2 θgθdθ + x 1 x 2 Gθ + x 2 An ncrease n x rases the welfare of the customers of frm and attracts customers from frm j to, who are ndfferent between frm and j. Addtonally, frm j reacts. dw x 1, x 2 dx = q x 1, x q x 1, x 2 dx j dx If efforts are strategc complements a small nterventon always ncrease customer welfare. If the efforts are strategc substtutes an nterventon rases welfare f: q x 1, x 2 q j x 1, x 2 > dx j dx 12 If frm 1 s the market leader wth x 1 > x 2 and q 1 > q 2 an nterventon n the leader tends to be more lkely to ncrease total welfare. 18

19 θ = x C 1 x C 2 = q1 gθ 1 α 1q1 q 1 α 2 γx 1 αq α 1 1 x 2 1/2 + γ x 1 gθ 1q2 l 2 γx 2 αq α 1 2 x 2 2/2 + l 2 γ x 2 For α = 0, gθ = g and q 0, 0 = 1/2 we have: 1 θ = g γx 1 l 2 γx gθ γ x 1 gθ = g γx 1 l 2 γx γ x 1 l 2 γ x 2 1/ g γ x 1 + l 2 γ x 2 By usng a talor expanson at θ = 0, we get: q x 1, x 2 q j x 1, x 2 1 gθ 1 2 gθ l 2 γ x 2 Comparng ths wth equaton 9 allows us to make some nferences. Startng form parameter values such that q 1 q 2 = dx j dx and ncrease n g makes an nterventon welfare enhancng. Consder the case of strategc substtutes wth 1 beng the leader wth x 1 > x 2. An nterventon on x 1 drectly, rases welfare of the larger group of people, whle the crowdng out effects only a relatve small number. As the reacton s less than unty. Intervenng on the sde of the leader wll always rase welfare. For the laggard the stuaton s slghtly dfferent. Intervenng on ts efforts rases welfare, however f the reacton of frm 1 s suffcent strong and ts market share suffcently large total welfare wll decrease. Ths mplcaton changes slghtly, once we consder the costs of efforts as socal costs. Total costs are gven as: Cx 1, x 2 = q 1 x 1, x 2 α x 2 1/2 + 1 q 1 x 1, x 2 α x 2 2/2 dcx 1, x 2 dx 1 = α qα 1 1 x q 1 α 1 x 2 2 dq 1 + q1 α x q 1 α dx 2 x 2 2 dx 1 dx 1 If development costs do not depend on the demand α = 0 the total costs are equal to the sum of both efforts. If both efforts ncrease cost go up. However, as the costs are hgher for the leader an ncrease of the laggard that leads to a reducton of the costs of the leader can reduce total costs. 6 Concluson I presented a model to llustrate the dfference between markets that can behave more lke a Cournot, where efforts are strategc substtutes, or lke a Bertrand market, wth efforts beng strategc complements. 19

20 Frst, f the costs of efforts closely depend on the market share of the frms α 1 efforts for both frms are strategc complements. Consequently, any nterventon encouragng any party rases efforts. A competton authorty need not pay much attenton to the dentty of the frms t encourages. In ths hardware case the thrd party should could red tape, offer subsdes and relax legal barrers for all frms. Second, f the costs of efforts do not depend on the market shares α 0 efforts are strategc substtutes and every postve nterventon leads to crowdng out. In general t wll stll lead to a postve gan, but wth two caveats. If the regulator decdes to encourage the laggard the loss of efforts of the leader mght lead to a total decrease of welfare, especally f the market s splt unequal. Ths mples that nterventon n markets where a hgher qualty does not rase unt costs should target a leader or should be avoded. Ths s the case for software systems, where addng a new feature has fxed costs but hardly varable costs. On the other hand to avod snowballng the contnuous ncrease n asymmetry t can support the laggard, by havng laws that treat smaller busness more lenent or by puttng strcter requrements on monopolsts. In between les the platform market, the specal case occuped by the market for operatng systems. Two frms compete here, both wth a low but postve α and hgh start up costs of nnovaton. Encouragng the leader n terms of market share, rases total efforts n the market, whle encouragng the laggard does not do so. Ths ntuton can be appled to the smartphone market. We can nfer that an ncrease n the efforts of Google would motvate Apple to exert more efforts, whereas an ncrease n the efforts of Apple would reduce the efforts by Google. The model shows that regulators should be more reluctant to support drectly or ndrectly frms on markets wth varable costs. Wthn platform markets they should treat the leader more lenent than the laggards. The model used can also be appled to ncumbent-entrant markets. A heavly entrenched ncumbent has a hgher market shares and any shock to ts market share leads to an ncrease n ts efforts f costs are varable. If costs are manly fxed, the entry and ncrease n the efforts of a compettor reduces the ncentves to nnovate by the leader. Ths also mpacts the decson of an entrant who has to decde f he wants to enter wth a large market share or wth a small market share. Because of the strategc substtutes the latter s more attractve n case R&D costs are fxed. Of course a polcy supportng the market leader or ncumbent can have severe problems n the long run as t creates a natural monopoly. However, the mpact that supportng frms n an asymmetrc stuaton has should be better taken nto account by regulators to perform a more dfferentated polcy. Ths paper llustrated the underlyng mechansms behnd these effects and helps regulators to understand when encouragng the ncumbent mght be advantageous for competton. By dong so t also emphaszed the mportance of asymmetrc markets n whch frms compete on dfferent level wth each other. 20

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