ASYMMETRIC SWITCHING COSTS CAN IMPROVE THE PREDICTIVE POWER OF SHY S MODEL

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1 Document de trvil / April 2012 ASYMMETRIC SWITCHIG COSTS CA IMPROVE THE PREDICTIVE POWER OF SHY S MODEL Evens Slies OFCE-Sciences-Po

2 Asymmetric switching costs cn improve the predictive power of Shy s model Evens Slies OFCE-Sciences-Po Astrct Economists Oz Shy introduced the definition of undercut-proof property ( PP ) prices in model of Bertrnd competition involving loyl consumers ( A quick-nd-esy method for estimting switching costs, Interntionl Journl of Industril Orgniztion, Vol. 20, pp , 2002). Shy s seminl pper llows pplied reserchers to mesure the switching costs fced y locked-in consumers. Although there is incresing interest in demonstrting consumer inerti in retil mrkets opened up to competition, Shy s pproch hs not received much ttention. The present pper shows tht the PP s lck of ppel in this context stems from strong ssumption of identicl switching costs in the theoreticl model, wheres rel dt re more likely to revel symmetric vlues for these costs. We revisit the PP y considering symmetric switching costs stright from the theoreticl model. Doing so enles us to show tht more rigorous conditions relting the vlues of switching costs to mrket shres re necessry in order for PP prices to e vlid predictions of these costs, which consequently increses the predictive power of Shy s model. This improvement is illustrted with two exmples orrowed from Shy s pper. JEL Codes: D43, D83. Keywords: Price competition, Switching costs, ndercut-proof Property. 2

3 Asymmetric switching costs cn improve the predictive power of Shy s model 1. Introduction In the present pper, I revisit the model of imperfect price competition, suggested in Morgn nd Shy (2000) then Shy (2002), which llows the mesurement of consumer switching costs s etween two or more rnds. In the cse of two firms, ech sets its price suject to the constrint tht the other will not find it profitle to undercut tht price nd gr ll its customers. Shy (2002, p. 75) suggests tht those prices stisfy property tht he clls the ndercut-proof Property ( PP ). At equilirium, PP prices re function of firms mrket shres nd unoservle switching costs. sing oservtions sed only on prices nd mrket shres, these vriles cn e mpped into two levels of switching costs (one for ech rnd). Though the method is quick nd esy s suggested y its inventor Shy, the PP hs not very received much ttention. In lortory experiment designed to investigte sptil price competition, Orzen nd Sefton (2008) nd Peters nd Strole (2009) find wek evidence in support of the PP s solution concept in fvour of other refinements of the Bertrnd-sh prdigm. There re t lest two resons for not fvouring one solution concept over nother. First, switching costs re neither restricted to eing equl nor to tking positive vlues; sitution tht is not llowed y the ove uthors who consider the originl PP solution concept. Second, Morgn nd Shy (2000) then Shy (2002, 2002) consider switching costs s prmeter tht is exogenous to the firm whilst endogenous to the modeller, wheres it is set t the eginning of the models tested in the forementioned lortory experiments. This endogenous nture of consumer switching costs in Shy s model is prt of wht seems to mke the PP so difficult for mny to ccept. 1 nlike trvel costs tht cn e clculted ex-nte the economic trnsction, the whole vlue of switching costs s modelled y Shy cn only e clculted ex-post, tht is to sy, given prices nd mrket shres. Rther thn ttempt to further situte Shy s model of price competition with loyl consumers reltive to other theoreticl solutions, I shll insted try to improve its predictive power. I revisit the PP in the two-firm cse y giving more precise conditions for the rnge of mrket shres nd switching costs under which prices stisfy the PP. nlike Morgn nd Shy (2000), nd Shy (2002), we ssume in the theoreticl model underlying the PP tht consumers my hve symmetric switching costs. One impliction is tht the theoreticl prediction mde y these uthors to the effect tht lrger firms chrge lower prices (Morgn nd Shy, 2000, p. 1) ecomes prticulr cse. Furthermore, our model ccommodtes negtive switching costs y interpreting them s net rther thn gross vlues s suggested 1. It is worth noting tht in the literture, the mening of endogenous switching costs differs from tht which we use here. Switching costs re considered s endogenous to firm when it cretes them (e.g., the impossiility of trnsferring one s locl phone numer from the incument to competing crrier). In the present pper, endogenous is used with respect to the modelling prctice. In Shy s model, switching costs re function of the firms prices nd mrket shres. In Orzen nd Sefton, prices nd mrket shres re functions of the switching cost. 3

4 Evens Slies in Green s 2000 working pper (see Krfft nd Slies, 2008 with respect to Green s contriution). Shy s definition (2002, pges ) of the undercut-proof property solution concept ssumes symmetric costs, however these costs ecome symmetric vlues when he pplies the PP for mesuring switching costs from rel dt. This improvement in the predictive power of Shy s model should contriute to mking the PP more le to explin the dt in future lortory experiments. 2. PP with rel nd symmetric switching costs Two firms nd sell homogenous product to consumers. There is shre α > 0 of rnd oriented consumers nd 1 α > 0 of rnd oriented consumers. Firm s customers perceive net cost s of switching to firm while firm s customers perceive net cost s of switching to. The utility functions of ech type of consumer re: u T T s if stying with rnd if switching to rnd (1) T u T s if stying with rnd if switching to rnd (1) where i denotes gross utility for ech type of consumer. T is firm s price nd T is firm s price. egtive switching costs my e defined s follows: let s e the gross level of switching costs common to oth types of consumers, nd denote s the vlue tht firm s customers ttch to uying from firm. Let v, then s nd s re s v nd s v, respectively where v v. Let n nd n 1 n denote the shres of customers uying rnd nd, respectively. These numers depend on prices nd switching ehviours of oth types of customers: 0 if T > T + s nα α if T s T T + s 1 if T < T s (2) 0 if T > T + s n 1 α if T s T T + s 1 if T < T s (2) The following definition generlises Shy s definition (2002, p. 75) of PP prices nd mrket shres to the sitution where the two types of consumers hve symmetric net switching costs. 4

5 Asymmetric switching costs cn improve the predictive power of Shy s model Proposition 1. Prices T s + n s 1 n n α, T n 1 α stisfy the PP if { α s } s + n s α 1 nαn min 0, (1 ) < s, where s IR., nd mrket shres n α α, Assume the lrger firm is ( α 1 α ). If, α () s >0 nd (1 α)s <s < s then chrges the lower price. 1 α α () s >0 nd s > s then chrges the higher price. 1 α (c) s <0 nd s 0 then chrges the higher price. Assume firms shre the mrket eqully. If, (d) s > s then chrges the higher price. Proof: see Appendix. ote tht if s < 0 then s must e strictly greter thn zero for PP prices to exist. In ll cses where prices stisfy the PP, switching costs re s T (1 α) T nd s T αt. Prices T nd T ecome simplified so s to effectively ssimilte to those in Shy (2002, p. 75) when s s. In this cse (under equl switching costs) Shy s model (2002) predicts tht the lrger firm chrges the lower price. Indeed it is possile to write sgn( T T ) sgn(1 2 α) [see (iii) in the proof of Proposition 1 in the Appendix]. We drw the reder s ttention to the fct tht our ssertion is different to nd stronger thn tht ctully stted y the uthor: under the PP, the firm with the higher mrket shre chrges lower price (Shy, 2002, p. 76, footnote 4). In fct, Shy s ssertion is true under stronger conditions relting mrket shres to switching costs of which one is tht switching costs re identicl. Proposition 1 demonstrtes tht the lrger firm my lso chrge the higher price thus showing tht the direction of the reltion etween price differentils nd mrket shres depends on the reltive levels of switching costs. Accordingly, the ssertion tht equilirium prices of rnds monotoniclly diverge when the rnds ecome more differentited (Morgn nd Shy, 2000, p. 1) my e restted s: holding s (respectively s ) constnt, equilirium prices of rnds monotoniclly diverge when s increses (respectively s decreses). More criticlly, Proposition 1 shows tht more rigorous conditions relting the vlues of switching costs to mrket shres re necessry in order for PP prices to e vlid predictions of these costs, 5

6 Evens Slies given ctul dt on prices nd mrket shres. This contrsts with Morgn nd Shy (2000, p. 2) who found tht n PE exists for ny vlue of the switching cost. It is lso now cler tht these uthors ssertion tht the lrger firm chrges the lower price is true not only under equl switching costs ut lso in lrger region of the vlues for s nd s. Our refinement of Shy s model (2002, 2002) mkes it of greter interest to studies of firms pricing ehviour in mrkets where consumers fce switching costs. In fct, rel dt tend to show positive correltion coefficient etween prices nd mrket shres nd negtive net cost of switching from smll firms (see e.g. Krfft nd Slies, 2008 nd the pplictions in Shy, 2002). Our generlistion widens its ppliction to include situtions such s where the lrger firm chrges the higher price nd the smller firm serves customers with the higher switching costs. The region of the vlues for s nd s comptile with PP prices is given in Figure 1 s well s the su-regions where T > T nd those where T < T. Any vlues for s nd s elow the thick line re not comptile with the existence of PP prices. Figure 1: PP prices given the region of the vlues for s nd s. s s /α T > T T > T α 1 α s T > T T > T T < T T < T s ( 1 α )s 6

7 Asymmetric switching costs cn improve the predictive power of Shy s model 3. Comment on Shy s results (2002) in the nking nd cellulr phone mrkets Once the pplied resercher hs clculted switching costs for customers of two competing firms (note tht the model cn esily e pplied for mesuring consumer switching costs etween sets of more thn two competing firms; see Krfft nd Slies, 2008; Shy, 2002), it is recommended tht he or she identify whether these vlues re comptile with one of three situtions () (c) descried in Proposition 1. 2 For exmple, in the cse of the Isreli cellulr phone mrket, Shy (2002, Tle 1) finds tht the lrger firm hs consumers with lower switching costs nd who lso py the lower price. Since the existence of PP prices in Shy s model is demonstrted where switching costs re identicl, this cse is ctully not predicted y the model. Point () of Proposition 1 in the present pper remedies tht prolem since it shows tht the lrger firm my even hve consumers with negtive switching costs. All mesures of switching costs given y Shy (2002) when pplying his model to the Finnish nking industry in 1997 re lso supported y our proposition. In prticulr, the mesure of the cost of switching from nk 3 to the smllest nk ($464) is no longer n exception in the present pper. Consider nk 3 s firm nd the smllest nk (nk 4) s firm. From Tle 2 of Shy (2002, p. 80) we cn clculte the switching cost in the opposite direction (from nk to ). We otin Since s 464 is greter thn ( α/(1 α))s ( / ) This cse is predicted y () in our proposition, while the negtive vlue of switching costs for consumers willing to switch to the lrgest nk (nk 1 in the Tle) is predicted y point (c). 4. Conclusion nd discussion This pper increses the predictive power of the undercut-proof property (PP) introduced y Shy (2002). We show tht more rigorous conditions relting the vlues of switching costs to mrket shres re necessry in order for PP prices to e vlid predictions of these costs. The sitution where the lrger firm chrges higher price is tken into ccount, mking the model more relistic. A negtive consumer cost for switch to the smllest competitor is lso predicted y the model, sitution which is common in empiricl pplictions. This improvement in the predictive power of Shy s model should contriute to mking the PP more le to explin the dt in future lortory experiments. We lso expect tht it will motivte more pplied reserchers to use the PP to the mesurement of switching costs in mrkets with pprent consumer inerti. A next nd quite importnt step in this reserch could consist of providing proilistic frmework for testing hypothesis on the vlues found for switching costs where some of these vlues pper very similr. Acknowledgements I m grteful to Adm Cutforth for ssistnce in finlizing the text of this pper. Any remining errors re my entire responsiility. 2. Sitution (d) of equl mrket shres is not very likely to occur. 7

8 Evens Slies References Green, R. 2000, Cn Competition Replce Regultion for Smll tility Customers? Center for Economic Policy Reserch s discussion pper, o Krfft, J. nd Slies, E. 2008, The Cost of Switching Internet Providers in the French Brodnd Industry, or Why ADSL hs Diffused Fster thn other Innovtive Technologies, Reserch Policy, vol. 37, pp Morgn, P.B. nd Shy, O. 2000, ndercut-proof Equiliriums, Mimeo, niversity of Hif, ccessed Mrch 22 nd, Orzen, H. nd Sefton, M. 2008, An Experiment on Sptil Price Competition, Interntionl Journl of Industril Orgniztion, vol. 26, pp Peeters, R. nd Stroel, M. 2009, Pricing Behviour in Asymmetric Mrkets with Differentited Products, Interntionl Journl of Industril Orgniztion, vol. 27, pp Shy, O. 2002, The Economics of etwork Industries, Cmridge niversity Press, Cmridge MA. Shy, O. 2002, A Quick-nd-Esy Method for Estimting Switching Costs, Interntionl Journl of Industril Orgniztion, vol. 20, pp

9 Asymmetric switching costs cn improve the predictive power of Shy s model Appendix Proof of Proposition 1: for given T nd n, firm chooses the highest price T suject to π nt T ( s ) (A1) Simultneously, for given T nd n α, firm chooses the highest price T suject to π nt T ( s ) (A2) α Let s find the PP prices nd ssocited mrket shres. It cn e shown tht (A1) nd (A2) hold with equlity. This system hs solution in prices if 2 nαn > 0. We otin, ( s+ n s T 2 nαn ) ( s+ nαs ), T 2 nαn (A3) 0, ; (ii): As consumers spred cross firms, there re three possile cses for ( n α, n ). (i): ( ) (,0) ; (iii): (, ) α. Sutrct s price from s ( T not equiliriums. From (2), (2), T T s > is ssocited with ( n, n ) ( ns nαs ) T ). Cses (i) nd (ii) re 2 nαn α ( ) 0, wheres price differentil t equilirium from (A3) is T T s. Similrly, T T < s is ssocited with ( n, n ) α (,0) wheres price differentil is T T s. In cse (iii), price differentil is T T ( s s 2 α α ). The reltive vlues of s nd s determine severl equilirium. Denote δ s ( α )/. nder cse (iii), prices stisfy the PP if s T T s, i.e. 9

10 Evens Slies s ( s s 2 α α ) s (A4) This doule inequlity imposes some restrictions on the reltive vlues of mrkets shres nd switching costs. Replce s with 1 2 s ( α) s ± then price differentil is: s α( αs+ s 2 α ) (A5) Similrly, if we replce α s with 1 2 αs ( α) s ± we otin s ( s+ s + 2 α ) (A6) If switching costs re positive, (A4) is esy to verify. Switching costs cn however e negtive, which requires the constrints tht the second terms in oth (A5) nd (A6) e simultneously greter or equl to zero. This leds us to the reltionship etween mrket shres nd switching costs stted in Proposition 1. We finlly mention two prticulr cses which re esy to demonstrte: s ( 1+ δ) s nd δ 0 (firms shre the mrket eqully). In the first cse, firms chrge the sme price. In the second cse, 2 T T ( s s ). 3 10