Hotelling Competition and Political Di erentiation with more than two Newspapers

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1 Hoelling Compeiion and Poliical Di ereniaion wih more han wo Newspapers Sefan Behringer Deparmen of Economics, Universiä Bonn Lapo Filisrucchi Deparmen of Economics, CenER and TILEC, Tilburg Universiy Deparmen of Economics, Universiy of Florence Ocober 8, 0 Absrac We analyse a newspaper marke where media rms compee for adverising as well as for readership. Firms rs choose he poliical posiion of heir newspaper, hen se cover prices and adverising ari s. We build on he duopoly work in wo-sided markes of Gabszewicz, Laussel, and Sonnac (00, 00) who show ha adverising nancing can lead o minimum poliical di ereniaion. We exend heir model o more han wo rms and show ha concerns for he emergence of a Pensée Unique as a resul of adverising nancing increase as he number of rms increases. In a simulaion exercise we derive equilibrium locaions and he welfare implicaions of an asymmeric shock as moivaed by he empirical ndings in Behringer and Filisrucchi (00b). Inroducion While he issue of endogenous produc posiioning in wo-sided markes is ineresing per se, concerns abou poliical pluralism imply ha produc di ereniaion (e.g. he choice of poliical conen covered) plays an even more crucial role in media markes han in sandard markes. This is he case because of he posiive exernaliy pluralism has in he poliical process. As a resul public concern abou he role and e ec of adverising on conen in media markes We are very graeful o Georg Göz for making his Mahemaica code available o us. We received many useful commens by paricipans of he TILEC Work in Progress Meeing in Tilburg, of he Deparmen of Economics Seminar in Siena and Bochum and of he "7h Media Economics Conference" in Siena. All remaining errors are ours. We also graefully acknowledge nancial suppor from he NET Insiue (

2 is much more pronounced and he wo-sided marke srucure of his indusry deserves special aenion. Previous heoreical work has modelled spacial compeiion for newspapers as aking place on he poliical line. The work on duopoly of Gabszewicz, Laussel, and Sonnac (00, 00) develops a model of oligopolisic compeiion among media rms who choose rs poliical posiion, hen cover prices and adverising ari s. In mos models he choice of conen is assumed o be exogenous. They show ha adverising nancing can lead o minimum produc di ereniaion (Pensée Unique). Our model feaures more han wo oligopolisic compeiors. Choosing wo rms has subsanial analyical advanages due o he implied complee symmery of he rms. Having more han wo rms on he poliical line implies ha some rms may be close o he cener of he poliical specrum wih wo immediae compeiors bu wo rms are he boundary only facing one. Hence wih more han wo rms individual rms are asymmeric in heir compeiive posiion prevening mos analyical shorcus ha could oherwise be applied. We nd ha despie his compeiive asymmery criical resuls of Gabszewicz, Laussel, and Sonnac and hence is policy implicaions can be generalized. We can conclude ha concerns for minimum produc di ereniaion as a resul of adverising nancing increase as he number of rms increases bu ha an alernaive form of condensed conen may appear in he four rm case where pairs of rms o er idenical conen. We calculae equilibrium oucomes for an asymmeric adverising shock and discuss he implied welfare consequences as moivaed by our work on he UK newspaper indusry in Behringer & Filisrucchi (009b). The Model In our model produc di ereniaion is assumed o be one dimensional and rms can charge di eren prices in uencing he posiion of marginal consumers drawn from some disribuion funcion over he characerisic space which we simplify o be he real line. The sandard ranspor cos parameerized by is hus a shared disuiliy ha occurs if a reader does no consume he newspaper ha exacly corresponds o her mos preferred variey. Mahemaically he model can become very complex once locaions of he rms on he characerisic space are no longer xed (e.g. see Anderson, (99), p.84). Also almos all heoreical models assume ha here are only wo rms and if an n> rm case is analyzed symmery assumpions abou subsiuion paerns (e.g. he logi model) or he use of a circular characerisic space (e.g.

3 he Salop model) erase many of he realisic properies of he original model for he newspaper indusry. An excepion o his endency is recen work by Brenner (005) who also relaxes he duopoly resricion. Following he possible non-exisence of Pure Sraegy Nash equilibria (PSNE) in price for given and close locaions noed by d Aspremon, Gabszewicz, & Thisse (979) we assume ha ranspor coss are quadraic. A full characerizaion of he heoreical seup wih more han wo rms, variable locaion, variable price, and endpoins in a wo-sided marke implies a non-rivial analyical challenge. In order o mee his challenge we have o pu some srucure on he order of he player s moves, on he shape of he demand funcion, and on he demand for adverising. We assume ha rms play a non-cooperaive wo-sage game in which in he rs sage hey simulaneously chose heir opimal locaions on he poliical line and in he second sage hey simulaneously chose boh prices and adverising raes. The soluion concep is pure sraegy subgame-perfec Nash Equilibrium (SPNE) in pure sraegies. To beer moivae he analysis assume ha he four rms are newspapers: he Guardian (G); he Independen (I); The Times (T ); and he Daily Telegraph (DT ) and are di ereniaed on he poliical uni-line from Lef o Righ according o common consensus. As oal demand for newspapers changes over he period under invesigaion we generalize he sandard Hoelling model allowing for elasic demand. In his we exend work on duopolisic seings by Böckem (994) who allows for individually disribued reservaion prices (independen of he disribuion of consumers) and Hinloopen and van Marrewijk (999) who look a common reservaion prices wihin he linear ranspor cos analysis of Economides (986).. The readers side Consumers have uiliy R p N from consuming he good where R 0 is a reservaion price. They also face a quadraic ranspor cos ha is proporional o he disance beween heir (poliical) locaion and ha of he rm l : so ha uiliies funcions of a consumer j who is locaed a x and may purchase a newspaper wih quaniy q j;i f0; g from rm i wih locaion l i are given by u ji = 0 if q j;i = 0 R j;i j;i p N i (x l i ) if q j;i = where j;i ( i ) gives a marginal disuiliy of price for consumer j. () 3

4 The posiion of a marginal consumer (x :: ) beween wo rms i and i + can be deermined by he indi erence condiion R j;i i p N i (x i;i+ l i ) = R j;i+ i+ p N i+ (x i;i+ l i+ ) () of he consumer j locaed a x i;i+. As he reservaion price will no be speci c o consumpion of a paricular good we assume ha R j;i = R j;i+ 8j; i bu he reservaion price will deermine wheher a consumer purchases a newspaper a all. Hence a consumer locaed a x [l i ; l i+ ] solves max R j;i i p N i (x l i ) ; R j;i+ i+ p N i+ (x l i+ ) ; 0 (3) The marginal consumer, denoed x i;i+ who is jus indi eren of buying good i and i + is locaed a x i;i+ = i+ p N i+ + l i+ i p N i li (l i+ l i ) Clearly a posiive purchase decision from (3) of he marginal consumer x = x i;i+ implies a posiive purchase from any consumers in is neighbourhood. Symmerically he marginal consumer on his lef is locaed a (4) x i ;i = i p N i + li p N i i li (l i l i ) (5) Clearly R j;i = R j;i+ 8j; i ) R j;i = R j;i and hus boh indi erence locaions are independen of reservaion prices. As no consumers can be forced o purchase, individual raionaliy of he marginal consumer (u ji 08j; i) implies ha reservaion prices for posiive purchases have o saisfy R j;i = R j;i+ min i p N i + (x i;i+ l i ) ; i+ p N i+ + (x i;i+ l i+ ) 8j; i (6) from which i follows ha all locaions for a purchase from i need o saisfy Rj;i x l i + i p N i (7) and x l i Rj;i i p N i Similarly a purchase from i + needs consumer locaions o saisfy Rj;i+ i+ p N i+ l i+ x l i+ + Rj;i+ i+ p N i+ (8) (9) 4

5 Given R j;i ( R), i.e. a common reservaion price he marke share of some inerior newspaper i = I; T wih uniformly disribued consumers is ms i = z i;i+ z i ;i = min z NB ( R i p N ) i min x i;i+ ; l i + i;i+; z B i;i+ max zi NB ;i; zi B ;i (0) ( R i p N ) i max x i ;i ; l i Given ha reservaion prices R : are su cienly large so ha hey are no binding we nd ms i = z NB i;i+ p N i+ ( i+ i p N i l i+ l i z NB i ;i = x i;i+ x i ;i = () Oherwise, if boh reservaion prices bind we have R ms i = zi;i+ B zi B ;i = l i + p N i l i+ l i (l i+ l i ) (l i l i ) + i + (l i+ l i )) l i l i i p N i which is independen of he oher rms behaviour. l i R i p N i If only one reservaion price binds we eiher have ms i = z B i;i+ or R zi NB ;i = l i + i p N i ms i = zi;i+ NB zi B ;i = i+ p N i+ + l i+ i p N i li (l i+ l i )! R = i p N i () i p N i + li p N i i li (l i l i ) (3) l i R i p N i! (4) Noe ha, ceeris paribus, he propery ha own-price increases will decrease and oher-price increases will increase own marke shares will no depend on wheher reservaion prices are binding or no. Hence he quasiconcaviy of he pro funcion will be preserved. z i If we look a non-inerior rms, for he LHS rm G, seing R ip ;i = max 0; l N i i we have for su cienly large R ha ms G=i = z NB i;i+ z NB i ;i = x i;i+ = i+ p N i+ + l i+ i p N i li (l i+ l i ) = l G + l I + Ip N I G p N G (l I l G ) (5) If reservaion prices are binding for he non-inerior rms hey may also bind 5

6 on eiher side (see Böckem (994)). If only one reservaion price binds we may hus eiher have or ms G=i = z B i;i+ R zi NB ;i = l G + ms G=i = zi;i+ NB zi B ;i = I p N I + l I G p N G lg (l I l G ) G p N G 0 (6) l G R G p N G! (7) If boh are binding we again have he same resul () as wih inerior rms. R ip For he RHS rm DT, seing z i;i+ = min ; l i + N i we have for su cienly large R : ha ms DT =i = z i;i+ z i ;i = x i ;i = i p N i + li p N i i li = (l i l i ) lt + l DT + DT p N DT T p N T (8) (l DT l T ) again if one reservaion price binds we eiher have or R ms DT =i = zi;i+ NB zi B DT p N! DT ;i = x i ;i = l DT ms DT =i = z B i;i+ R zi NB ;i = l DT + DT p N DT (9) DT p N DT + l DT p N T T lt (l DT l T ) (0) and if are binding we again have he same resul () as wih inerior rms. If we furher assume i = 8i he non-binding sysem reduces o a demand for an inerior rm i = I; T of ms i = l i+ l i and ha of he rm G on he LHS as p N i+ p N i + (l i+ l i ) p N i p N i (l i l i ) () ms G = l G + l I p N I p N G + (l T l G ) () and ha for DT on he RHS as lt + l DT ms DT = p N DT p N T + (l DT l T ) (3) 6

7 The resuling elasiciy marix given he 4 newspapers is " G;G " G;I " G;T " G;DT " I;G " I;I " I;T " I;DT " T;G " T;I " T;T " T;DT = (4) " DT;G " DT;I " DT;T " DT;DT p N G p N I (l I l G ) ms G (l I l G ) ms G 0 0 p N G l T l G p N I p N T (l I l G ) ms I (l T l I )(l I l G ) ms I (l T l I ) ms I 0 p 0 N I l DT l I p N T p N DT (l T l I ) ms T (l DT l T )(l T l I ) ms T (l DT l T ) ms T p 0 0 N T p N DT (l DT l T ) ms DT (l DT l T ) ms DT Noe ha hese elasiciies depend on locaions, i.e. he newspaper s opimal choices in he rs sage of he game. Locaions l; newspaper prices p N ; and adverising raes p A are deermined in a non-cooperaive supply side game. Thus we deermine equilibrium prices p (l) and r given he locaion vecor in sage II and hen he SPNE locaion vecor l = (l G ; l I ; l T ; l DT )0 in sage I. 7

8 . The adverising side Pro also depend on he oher side of he marke, i.e. on adverising revenue. Hence rms also simulaneously chose opimal adverising raes p A i. As in he model of Rysman (004), adverising uiliy will depend posiively on newspapers share of he readers marke. Gabszewicz, Laussel, and Sonnac (00) look a he poliical di ereniaion of wo newspapers in a sage game model and make an exension o he classical model of verical produc di ereniaion (as in Gabszewicz & Thisse (979) or Shaked & Suon (983)) by allowing for muliple purchases of adverisers. We exend heir analysis of newspaper adverising o I > wo rms. An adveriser k has a bene from adverising in newspaper i given by u k (i) = ms N i p A i (5) where [0; ] gives he adveriser s inensiy (or abiliy) of preference of buying he adver. Assume ha he are also disribued uniformly on he uni line wih densiy 4 > 0: If he adveriser purchases from any m > newspapers hen his uiliy is simply Pm u k (m): The adveriser will hus buy an adver from newspaper i raher han none (paricipaion condiion) if u k (i) 0, pa i ms N i (6) so ha only he per-reader adverising rae maers. Wihou loss of generaliy order all newspapers by heir reader marke shares as ms N > ::: > ms N i > ms N i > ms N i+ > ::: > ms N I (7) Now k will prefer adverising wih h f; :::; i ; i; i + ; :::I g o any oher h 0 fj + ; :::Ig if and only if his ype sais es u k (h) u k (h 0 ), pa h p A h 0 ms N h ms N h 0 (8) Noe ha by he ordering denominaors are sricly posiive so ha p A h > 0 pa h implies ha his condiion holds for all [0; ] and numeraor and denominaors canno be negaive a he same ime. If (8) is sricly binding his is he indi erence condiion of he classical model. 8

9 An imporan propery ha follows from he addiive uiliy srucure of muliple purchases for any m > newspapers is ha mx uk (m) + u k (i) mx uk (m), u k (i) 0 (9) so ha he addiional purchase is bene cial if and only if he paricipaion condiion (a degenerae indi erence condiion) for he addiional adverising holds. Whence one has o decide wheher a saisfacion of his condiion on he uni line implies ha only his adverising is purchased or wheher i is purchased wih oher adverising in a muliple purchase. The possibiliy of muliple purchases implies ha he indi erence condiion of he classical model has no in uence on marke shares in his model ha will be deermined by he paricipaion consrains only bu i will in uence he ordering of hese paricipaion condiions. We can show he following: Lemma The verical di ereniaion model wih muliple purchases of Gabszewicz, Laussel, and Sonnac (00) exends o I > rms and leads o adverising demand as ms A i = ms A i (p A ; ms N (p N (l))) = 4 p A i ms N i (pn (l)) wih he Nash equilibrium in adverising raes necessarily saisfying (30) p A i = msn i (pn (l)) 8 i f; :::; Ig (3) Proof: As emphasized by Gabszewicz, Laussel, and Sonnac (00) wih muliple purchases he indi erence condiion (given by he criical ype for which (8) is sricly binding) is imporan o deermine which adverising is purchased as D h;h 0 pa h p A h 0 ms N h ms N h 0 pa i ms N i and conversely D h;h 0 pa h p A h 0 ms N h ms N h 0 pa i ms N i for any i = h; h 0, for any i = h; h 0, pa h ms N h pa h ms N h pa h 0 ms N h 0 (3) pa h 0 ms N h 0 (33) These condiions imply ha an indi erence condiion for any wo adverisemens canno be in beween he wo respecive paricipaion condiions. In our seing wih I rms and muliple purchases here are (I I)= indi erence condiions ha, for any given prices p A i R +, may fall anywhere 9

10 on he -uni line hus deermining he relaive posiion of he wo paricipaion condiions concerned. A unique nal ordering of paricipaion condiions does no require knowledge of all indi erence condiions however. For example if I = 4 hen u k () > u k (), > D ; pa p A ms N ms N (34) To ransform he ordering of marke shares o ha of he paricipaion condiions on he -uni line we need o have D ; < pa ms N < pa ms N ^ D ;3 < pa ms N < pa 3 ms N 3 ^ D ;4 < pa ms N < pa 4 ms N 4 (35) so ha we need D ;: < pa ms N < Min( pa h=;3;4 ms N ) (36) h=;3;4 By he same reasoning in order o order he paricipaion condiions for rm we need ha or D ;3 < pa ms N Evenually for rm 3 we need < pa 3 ms N 3 D ;: < pa ms N ^ D ;4 < pa ms N < pa 4 ms N 4 (37) < Min( pa h=3;4 ms N ) (38) h=3;4 D 3;4 < pa 3 ms N 3 < pa 4 ms N 4 (39) Noe ha hese condiions unravel backwards so ha we only need hree condiions, D 3;4 ; D ;3 ; and D ; o deermine a full ordering of adveriser ypes. If we also have ha D ; < D ;3 < D 3;4 (and he oher D :: locaed such ha ransiiviy is sais ed for any [0; ]) his implies ha adverisers will purchase from f; (); (; ); (; ; 3); (; ; 3; 4)g whenever he respecive paricipaion condiion holds. In he classical model wih wo rms and a single purchase, D ; < pa ms N no sales. If D ; > pa ms N < pa ms N > pa ms N implies ha his se is f; ()g so ha rm has his se is f; (); ()g and marke shares are deermined by paricipaion and he indi erence condiion. Hence here is no general demand form. 0

11 Wih muliple purchases he nal locaions of D :;: will no a ec marke shares and hence demand for adverising can always be wrien as ms A i = ms A i (p A ; ms N (p N p A i (l))) = 4 ms N (40) i (pn (l)) The decision o adverise wih one rm will be independen of raes and circulaions of he oher newspapers so ha pro maximizaion of he adverising pro yields : p A i = msn i (pn (l)) 8 i f; :::; Ig (4).3 Solving sage II Pro s wih di ereniaed producs in sage II are II i (p A i ; p N i ) = ms N i (p N (l))(p N i (l) c N i ) + ms A i (p A ; ms N (p N (l)))p A i (4) As he newspaper indusry is a prima facie case of a wo-sided marke here is a second side o he rm s overall pro ha resuls from sales of adverising slos o adverisers. Adverising demand for a paricular newspaper will increase in he newspaper s reach and hence is reader marke share. Lemma The Nash equilibrium in prices necessarily sais es 0 0 p N i (l) cn i p N i (l) + pa i " ii p N N " ii + A i i ms N N " hi ms N AA h h6=i h (43) where " ii i p N i (l) i (l) ms N i are respecively ownand cross-price elasiciies. (pn (l)) and " hi h p N i (l) i (l) ms N h (pn (l)) Proof: Take derivaive. Given he Hoelling demand and elasiciy srucure he markup 0 0 p N i (l) cn i p N i (l) + pa i " ii p N N " ii + A i i ms N N " hi ms N AA h h6=i h (44)

12 for example for he Independen, using " II ; " GI ; and " T I simpli es o p N I (l) c N I = msn I (l T l I ) (l I l G ) (l T l G ) 0 + pa I ms A N A N I (l I l G ) (l T l G ) (l T l I )(l I l G ) N T + (l T l I ) AA (45) Solving for he equilibrium prices of he second sage closed-form soluions are possible..3. Equilibrium prices wih non-binding reservaion consrains Given he marke shares from (0) and some simpli caions we nd ha equilibrium prices for he Independen are p N I (l) = cn I + (l T l I ) (l I l G ) (l T l G ) where A I p A I A I : By symmery for he Times we N I p N T (l) = cn T + (l DT l T ) (l T l I ) (l DT l I ) ldt For he Guardian, using " GG (44) simpli es o l G + pn T (l) (l T l I ) + pn G (l) (l I l G ) A I (46) l I + pn DT (l) (l DT l T ) + pn I (l) (l T l I ) A T (47) p N G (l) c N G = (l I l G ) ms G A i = (l lg + l I I l G ) + pn I (l) pn G (l) (l I l G ) A G (48) which we can solve for prices as p N G (l) = cn G + (l I l G ) (l G + l I ) + pn I (l) For he Daily Telegraph, using " DT;DT we have A G (49) p N DT (l) c N DT = (l DT l T ) ms DT A DT = (50) (l lt + l DT DT l T ) + (pn DT (l) pn T (l)) A DT (l DT l T ) which can be solved as

13 p N DT (l) = cn DT + l DT ( l DT ) l T ( l T ) + pn T (l) A DT (5) We hus have as sysem of reacion funcions of he price game consising of 4 equaions in 4 unknowns as p N G = max 0; cn G + (l I l G ) (l G + l I ) + pn I A G (5) p N I = max 0; cn I + (l T l I ) (l I l G ) lt (l T l G ) l G p N T + (l T l I ) + p N G (l I l G ) A I (53) p N T = max 0; cn T + (l DT l T ) (l T l I ) ldt (l DT l I ) l I + p N DT (l DT l T ) + p N I (l T l I ) (54a) A T p N DT = max 0; cn DT + l DT ( l DT ) l T ( l T ) + pn T A DT (55) For all prices o be 0 simulaneously we need o be in a region where (by adding up he 4 inequaliy consrains ha resul if he RHS of he brackes above are less han zero for zero prices): A DT c N DT + A T c N T + A I c N I + A G c N G + ((l DT l T + l I ) l DT + lg + (l T l I ) l G + l T ( l I )) > 0 (56) i.e. if adverising is very imporan for pro s. Noe ha in equilibrium A i = p A A N i = msn i (pn (l)) 4 (ms N i (pn (l)) pa i = (57) In marix form de ne his sysem of equaions as A = pn (l) (58) 3

14 where we wrie equaions (46), (47), (49), and (5) as: c N G (l I l G ) (l G + l I ) + A G c N I (l T l I ) (l I l G ) + A I c N T (l DT l T ) (l T l I ) + A T = c N DT l DT ( l DT ) + l T ( l T ) + A DT 0 0 l T l I l (l T l G ) I l G p N G (l T l G ) 0 l 0 DT l T l (l DT l I ) T l I x p N I (l DT l I ) p N T 0 0 p N DT (59) which can be solved analyically for he equilibrium price vecor p N (l) 0 = (p N G (l); pn I (l); p N T (l); pn DT (l))0 by invering he sysem as. Lemma 3 In he case of a non-binding reservaion consrain Proof: See N i = where A i = A 8i = G; I; T; DT Hence as in Gabszewicz a. al. wih wo rms a "full pass-hrough" effec prevails and an increase in he per-reader advering revenue decreases he equilibrium newspaper prices one-o-one..3. Equilibrium prices wih binding reservaion consrains If reservaion prices bind on boh sides of he rms we have ha for inerior and non-inerior rms R ms N i = zi;i+ B zi B ;i = i p N i (60) which is independen of he oher rms and increasing in he reservaion price. Hence i is a dominan sraegy for all rms o se p N i = R + 3 i (cn i A i ) 8i = G; I; T; DT (6) s ms i = p 3 R c N i + A i 8i = G; I; T; DT (6) 3 i 4

15 and equilibrium pro s from selling o readers given by i = 4p r 3 R c N i 9 i i + A i i (R c N i i + A i i ) 8i = G; I; T; DT (63) i which is increasing in he reservaion price (which generaes more sales and increases he equilibrium price) bu decreasing in and ;he (income-) sensiiviy of a price change on uiliy. Lemma 4 In he case of a binding reservaion N i = r p 3 R + i (A i c N i 3 i > 0 8i = G; I; T; DT Proof: Take derivaive. Hence unlike in he case wih non-binding reservaion consrain here is no "full-pass hrough e ec" and an increase in he advering demand increases he equilibrium price..4 Solving sage I Using he resuls from sage II of he game we can rs show he following: Proposiion 5 Given he srucure of he adverising side, he problem a he rs sage can be ransformed ino a Hoelling problem wih pro s depending only on locaion as 0 (p A i ; p N i ) = (p i (l) c0 i )ms i(p (l)). Proof: See Appendix. Toal pro s in sage I given he equilibrium price vecor p N (l) 0 wih generic elemen p N i and p A i from sage II can hen be wrien as 0 i (p A i ; p N B i (l)) N i (l) c N C i + A i A ms N i (p N (l)) (64) {z } c 0 i so ha sage I equilibrium pro s are linear in he reader marke shares ha are given by he Hoelling speci caion, i.e. for inerior rms see (0). 5

16 3 Guaraneeing pluralism One major nding in he work of Gabszewicz, Laussel, & Sonnac (00) is ha in order for he sandard maximum di ereniaion resul in duopoly of d Aspremon, Gabszewicz, and Thisse (979) o hold adverising demand canno be oo pronounced. Inversely a siuaion exiss where he adverising demand is so imporan ha sharing he marke for readers becomes he primal objecive and minimum di ereniaion resuls. Boh cases can overlap. We assume an equal sharing rule in case of minimum di ereniaion. version of heir respecive lemma saes A Lemma 6 There exiss a Nash equilibrium wih full minimum di ereniaion of (l L ; l R ) in he cener in he duopoly game if A c > Proof: Noe ha heir l R is l R in our noaion. Bes response funcions in he duopoly case are p L = max 0; (c A + p R + l R + lr; ll) p R = max 0; (c A + p I + l L + ll; lr) Adding up LHS condiions a zero prices we nd A > c + ( l R l L ) The game is hen a pure locaion game a zero equilibrium prices for given locaions. A su cien condiion for his o hold for any locaion (and hence any possible deviaion) is A > c + ( 0 0) so ha here canno be any deviaion ha leads o posiive prices and hence o any oher equilibrium bu he PSNE in locaions l L = l R =. 6

17 In our 4- rm case we nd: Lemma 7 There exiss a Nash equilibrium wih full minimum di ereniaion of (l G ; l I ; l T ; l DT ) in he cener in he quadropoly game if A c > 8 Proof: The equivalen zero-price condiion was found in (56) as A DT c N DT + A T c N T + A I c N I + A G c N G + ((l DT l T + l I ) l DT + lg + (l T l I ) l G + l T ( l I )) > 0 Noe for a symmeric coss and ads revenues we nd symmeric locaions as l G = l DT and l I = l T he condiion simpli es o where 4(A c) + ( (l T l DT + ) (l T l T D )) > 0 (l T l DT + ) (l T l DT ) < 0 To make his as small as possible we aim o max (l T l DT + ) (l T l DT ) s.. l DT l T l T ;l DT The rs order condiions yield he condiion l T + = l DT Using symmery and he ordering l T l I he unique soluion is lt = ; l DT = The bes one can do in order o guard pluralism is hen o selec locaions lg = 0; li = lt = ; l DT = for which we nd he all-zero price condiion (56) (normalize = ) as A > c + 8 If his condiion holds hen for any oher symmeric locaion he zero price consrain binds and hus l G ; l I ; l T ; l DT = is a Nash equilibrium given he sharing rule hold. 7

18 Noe ha minimum di ereniaion is no he only symmeric pure locaions equilibrium in he Hoelling 4- rm game. The alernaive pure locaion equilibrium candidae ha does no require a sharing rule o hold is (=4; =4; 3=4; 3=4) for which he zero-price condiion similarly reduces o A > c; i.e. he same condiion ha has o hold for he minimum di ereniaion equilibrium. Hence even if he zero-price oucome is more probable in he 4- rm case his need no imply ha here necessarily is he endency for minimum di ereniaion of all rms. Mos generally we nd for he n- rm case (for n 4) ha: Proposiion 8 There exiss a Nash equilibrium wih full minimum di ereniaion of (l ; l ; :::; l n ; l n ) in he cener in he n- rm oligopoly game if A c > n Proof: See Appendix. Noe ha in order o make he occurrence of minimum di ereniaion leas likely he locaions for he rms will always be l = 0; l = ::: = l n = l n = ; l n = : Here he (n ) rms locaed in he middle have a zero price and only he wo exreme rms have a sricly posiive one. Thus in order o bes guaranee a full minimum di ereniaion resul one chooses a parial minimum di ereniaion of (n ) rms and les he remaining rms cover he exreme posiions. For compleeness we repor he condiion for n = 3 which reads A c > 4 : This does no ino he above general proposiion as he rm in he cener reains a posiive price. The is no SPNE in he 3 rm pure locaion game. 8

19 4 Simulaion Given he model oulined above, we can simulae changes of he exogenous variables and derive equilibrium comparaive saics resuls. In paricular Proposiion 5 implies ha our wo-sided marke seup can be reduced o a Hoelling problem wih four rms, simulaneous choices of locaion and prices and adverising raes. A similar problem has been analysed in Göz, (005) who exends work on he Hoelling model by Neven (987) and Economides (993). We hus modify his Mahemaica algorihm o solve for he equilibrium of he rs sage locaions explicily. This exercise has been underaken also in Brenner (005) for up o nine rms. The algorihm is based on a Newon-Raphson approximaion of he equilibrium rs sage locaion wih saring values l 0 =(0;:3; :6; ) 0 : The algorihm proceeds by evaluaing he angen on he original funcion a he saring value, nds is inercep wih he abscissa which is hen used o nd he funcional value and angen again recursively unil he poin of angency and he inercep wih he abscissa coincide. The algorihm converges and su ciency of he rs order necessary condiions for opimaliy can be checked by looking a he pro funcion for local deviaions. 4. The symmeric siuaion We assume ha = = 0 and ha he exended cos is c 0 i = :5 for all rms. The NR-algorihm hen yields equilibrium locaion on he poliical line in he rs sage of he game as l = (l G; l I ; l T ; l DT ) 0 = (:4; :396; :604; :876) 0 (65) Noe ha despie he symmery of he siuaion here remains a di erence beween he inerior and he non-inerior newspapers. This implies ha only he locaions of inerior newspapers I and T and hose of he non-inerior newspapers G and DT are mirror images, i.e. l G = l DT and l I = l T. These equilibrium locaions imply equilibrium prices in he second sage as p N (l ) = (p G(l ); p I(l ); p T (l ); p DT (l )) 0 = (:566; :6; :6; :566) 0 (66) where again we observe he symmery beween inerior and non-inerior rms. The equilibrium marke share vecor of he readers marke is ms N (p N (l )) = (:96; :304; :304; :96) 0 which corresponds o he marke share vecor of he adverising marke by (30). Equilibrium pro s are I (p A ; p N (l )) = (:09; :8; :8; :09) 0 (67) 9

20 Noe ha marke share are sill asymmeric wih regard o own locaion of he inerior and non-inerior rms as ms qin R = ms G l G = :96 :4 = :07 (68) ms qin L = ms G ms qin R = :96 :07 = :4 ms in R = l I = :396 = :04 ms in L = ms I ms in R = :304 :04 = : Welfare coss o consumers in his case can be calculaed as W C S = (69) R msqinl x dx + R! ms qinr x dx+ 0R 0 msinl x dx + R msqin ms inr + (p x pin ; p in ) = dx ms 0 0 in R 0:4 x dx + R! 0:07 x dx+ 0R 0 :96 0: x dx + R 0:04 x + (:566; :6) = dx : : : 353 for = 0 ( = ) his is W C S = : 49 : 0

21 4. An asymmeric adverising shock Moivaion from he UK newspaper indusry: As described in deail in Behringer & Filisrucchi (00b) as a consequence of he economies of scale and scope wihin media marke a possible consequence of Ruper Murdoch s acquisiion of he Times in 98 is an increased asymmery beween The Times and is compeiors. To explore he e ec of such asymmeric changes in he compeiive environmen we assume ha saring from a symmeric siuaion he marginal cos of producion of he newspaper Times c N i decreases and/or he per-capia adverising revenue A i increases (reducing he exended cos c 0 i ). This may be due o a subsanial prining cos advanage resuling from more e cien producion mehods a Wapping or due o some exogenous increase of he demand for adverising for The Times only (boh are consisen wih he Times joining a media conglomerae such as Ruper Murdoch s). We invesigae he e ec of his change on he equilibrium magniudes of he model. Assuming ha he exended cos of The Times T exogenously drops o c 0 i = : he equilibrium magniudes change o l = (l G; l I ; l T ; l DT ) 0 = (:03; :353; :577; :887) 0 (70) Thus he equilibrium locaion on he poliical line of all newspapers bu he DT shif o he Lef. This nding is akin o resuls in he non-sraegic rm seing of Behringer (007) based on marke daa only. The implied equilibrium prices in he second sage are p N (l ) = (p G(l ); p I(l ); p T (l ); p DT (l )) 0 = (:4; :58; :8; :647) 0 (7) so ha all equilibrium prices bu ha of he DT go down. The equilibrium marke share vecor of he readers marke is now ms N (p N (l )) = (:8; :79; :357; :85) 0 (7) so ha marke shares of all newspapers bu ha of he T go down. Finally equilibrium pro s are now I (p A ; p N (l )) = (:6; :83; :33; :) 0 (73) so ha equilibrium pro s of he G and he I go down bu ha of he T and he DT go up. Thus by moving furher o he poliical Righ, he DT is able o reduce he compeiive pressure (and even increase is pro despie is decreasing marke share) ha resuls from he T s unilaerally lower exended coss. The I on he oher hand, being an inerior rm does no have his opion as a move o he poliical Lef will auomaically increase he compeiive pressure from

22 he G. Hence his move is punished wih a subsanially lower pro, bu also he pro of he G declines. Welfare coss o consumers in his asymmeric case follow from ms G R = ms G l G = :8 :03 = :077 (74) ms G L = ms G ms G R = :8 :077 = :03 ms I R = (ms G + ms I ) l I = :06 ms I L = ms I ms I R = :79 :06 = :73 ms T R = (ms G + ms I + ms T ) l T = :39 ms T L = ms T ms T R = :8 ms DT R = l DT = :887 = :3 ms DT L = ms DT ms DT R = :85 :3 = :07 as W C AC = ( Z 0:8 0 x dx + Z 0:077 0 Z 0:3 x dx + 8: : x dx + Z 0:03 0 Z 0:07 x dx + Z 0:06 0 x dx + again wih = 0 ( = ) his becomes W C AC = : 365 9: 0 Z 0:73 0 x dx + Z 0:39 0 x dx + x dx) + p N (l ) 0 ms N (p N (l )) = (75) Comparing wih he symmeric siuaion where W C S = : 49 we noe ha he cos drop of he imes leads o higher price welfare of consumers bu he new and more asymmeric locaions o slighly higher locaion welfare coss. Hence wheher he cos drop is advanageous o consumers in general will depend on he level cos saving and he level of : Calculaions for he sysem where reservaion consrains are all binding are subsanially simpler. Wih binding consrains, locaions decisions (and hence he rs sage of he game) does no maer for pro consideraions for neiher inerior nor non-inerior rms. A small locaion change of a paper will lead o a loss of cusomers on one side ha is exacly compensaed for by he gain of cusomers on he oher no a ecing a neighbours marke share. This move will be pro neural as price choices are (from (6)) dominan sraegies. Parameers (and in paricular he reservaion price R) have o be chosen such ha he uni-line consrain o (symmeric) marke shares remains sais ed. Given his consrain we hen know ha equilibrium pro s will also be symmeric and decreasing in and a i :

23 5 Conclusion We propose a heoreical model of he newspaper marke wih (more han) 4 rms encompassing demand for di ereniaed producs on boh sides of he marke and pro maximizaion by compeing oligopolisic publishers. These publishers recognize he exisence of indirec nework e ecs beween he wo sides of he marke as hey choose rs he poliical posiion, hen simulaneously he cover prices and he adverising price We nd ha he form of he adverising side as proposed in model of Gabszewicz a. al. (00, 00) exends o quadropoly and also ha he "full pass hrough" resul of he adverising revenue on equilibrium prices holds. We show ha in general he concern for a Pensée Unique as a resul of adverising - nancing is increasing. Also, he bes guardian agains such a Pensée Unique i.e. full minimum di ereniaion is a parial (n-) rm minimum di ereniaion wih he remaining wo rms covering he exremes of he poliical specrum. For he paricular case of quadropoly we nd ha despie he fac ha for a large adverising revenue componen equilibrium reader prices for he newspapers will fall o zero, unlike in duopoly he endency o produce overly similar papers is bounded by he fac ha anoher pure sraegy Nash Equilibrium exiss. The implicaions for consumer welfare of a drasic bu symmeric adverising demand increases are hus more comforing. Evenually we derive he locaion equilibrium by simulaion for an asymmeric adverising shock for The Times only o he symmeric siuaion as moivaed by occurences in he alleged newspaper price war in he UK in he 990s (see Behringer & Filisrucchi (00b) for deails). Findings reveal surprising equilibrium reacions of The Times compeiors and explain asymmeric prices, circulaion, adverising volumes, and locaions. From a welfare perspecive he equilibrium price drops for all of he papers bu he Daily Telegraph and is advanageous o consumers. On he oher hand he asymmery produces a less diverse coverage han under symmery which bene s he readers wih exreme poliical opinions bu, on average, resuls in higher welfare coss. The ne e ec of hese will hus depend on he size of he producion cos savings (adverising increase) relaive o he ranspor cos parameer puing a measure of poliical diversiy in newspapers. 3

24 6 Appendix Proof of Lemma 3: The invered sysem wries as p N (l) = A Firs nd he deerminan of as de() = (6l T l G )(l T D l I ) + 6l T l T D 3l T 3l I 6(l T l G )(l T D l I ) > 0 hen he inverse can be wrien as 0 0 l T l I l (l T l G ) I l G (l T l G ) 0 l 0 DT l T l (l DT l I ) T l I (l DT l I ) 0 0 3l T l T D +l T D (l T l I ) 3l G (l T D l I ) 4(l T l G )(l T D l I ) l T l I l T (l T l I )(4l T D l T 3l I ) 8(l T l G )(l T D l I ) (l T l I )(l T D l T ) 4(l T l G )(l T D l I ) (l T l I )(l T D l T ) 8(l T l G )(l T D l I ) 4l T D l T 3l I 8(l T D l I ) 4l T D l T 3l I 4(l T D l I ) l T D l T (l T D l I ) l T D l T 4(l T D l I ) de() adj() = de() ( l I l G 4(l T l G ) l I l G (l T l G ) 3l T +l I 4l G 4(l T l G ) 3l T +l I 4l G 8(l T l G ) ) (l I l G )(l T l I ) 8(l T l G )(l T D l I ) (l I l G )(l T l I ) 4(l T l G )(l T D l I ) (l T l I )(3l T +l I 4l G ) 8(l T l G )(l T D l I ) 3l T D (l T l G ) l G l T +4l G l I l T l I 4(l T l G )(l T D l I ) l I Summing each of he rows in he adjoin Marix adj() yields he same value, namely X row adj() = ( 3 3 (l T l I ) 8 (l T l G ) (l T D l I ) ) col and by premuliplying wih he inverse of he deerminan de() yields de() X adj() = ( col ) 8l G ; l I ; l T ; l DT Wih p N (l) = A and all rows of he inverse of summing o noe ha each elemen of he vecor A is linear in A i and premuliplied by he scalar =: For a common adverising demand densiy A i = A 8 i = G; I; T; DT herefore equilibrium price sais N i (l)=@a = : 4

25 Proof of Proposiion 8: For an even number of rms n we nd equilibrium prices as p N (l) = cn + (l l ) (l + l ) + pn (l) A p N (l) = cn + (l 3 l ) (l l ) (l 3 l ) l3 l + pn 3 (l) (l 3 l ) + pn (l) (l l ) A p N 3 (l) = cn 3 + (l 4 l 3 ) (l 3 l ) (l 4 l ) l4 l + pn 4 (l) (l 4 l 3 ) + pn (l) (l 3 l ) A 3 p N (n ) (l) = cn (n ) + l (n ) l (n ) l(n ) l (n 3) (l (n ) l (n 3) ) l (n ) l (n 3) p N (n ) + (l) p N (n 3) + (l)! l (n ) l (n ) l (n ) l (n 3) A (n ) p N (n ) (l) = cn (n ) + l n l (n ) l(n ) l (n ) (l n l (n ) ) l n l (n ) + pn (n ) (l) l n l (n ) + pn (n ) (l) l n l (n )! A (n ) p N n (l) = cn n + l n ( l n) l (n ) ( l (n )) + pn (n ) (l) A n Adding up he locaion erms analogous o (56) and using symmery l n = ( l ); l n = ( l )::: we can derive he condiion (56) for n rms as! nx (A i c i ) = (n(a c)) > i= + l n l n (l l ) (l + l ) + (l 3 l ) (l l ) + (l 4 l 3 ) (l 3 l ) + ::: l n l n 3 + l n l n l n l n + l n + l n where we have added symmery in adverising revenue and demands. Also by symmery we have l n + = l n 5 l n l n

26 so he las line reads :::+ l n l n l n l n 3 + l n l n l n l n as n is even and rms symmeric around =. Maximizing ordering we nd he rs = l + l l 3 = l + l 3 l 4 = = l j3 + l j l j + l j+ l j+ = n n n + l n = l n 4 + l n 3 l n + l n l n = 0 = l n + 3l n = 0 = l n + 3l n 4l n + = 0 subjec o he Noe ha all of hese =@l n are sricly concave. If l n = 0 we have =@l n ha l n 3 = l n 4 + l n + l n which conradics he ordering. If l n = he ordering implies ha l n = oo. Now we have ha l n l n l n l n = l n and he las par of cancels. The remainder is now ::: + l n l n l n l n 3 + l n l n l n l n so ha =@l n is no longer concave. The enire problem unravels from he back and we evenually nd =@l implies ha l = 0 The opimal value ha he condiion above akes a hese locaions is A c > n for any n 4: The proof for a general odd number of rms is similar. 6

27 7 Bibliography Anderson, S. (99): Discree Choice Theory of Produc Di ereniaion, MIT Press. Behringer, S. & Filisrucchi, L. (00b): "Price Wars in Two-Sided Markes: The case of he UK Qualiy Newspapers", Universiä Bonn, mimeo. Böckem, S. (994): A Generalized Model of Horizonal Produc Di ereniaion, The Journal of Indusrial Economics, Vol. XLII, No. 3, p Brenner, S. (005): Hoelling Games wih hree, four, and more Players, Journal of Regional Science, Vol. 45, No.4, p d Aspremon, C., Gabszewicz, J.J., & Thisse, J.-F. (979): On Hoelling s Sabiliy in Compeiion, Economerica, Vol. 47, No. 5, p Economides, N. (993): Hoelling s Main Sree Wih More Than Two Compeiors, Journal of Regional Science, vol. 33, no. 3, p Gabszewicz, J.J., Laussel D., Sonnac N. (00): Press adverising and he ascen of he Pensée Unique, European Economic Review, 45, p Gabszewicz, J.J., Laussel D., Sonnac N. (00): Press Adverising and he Poliical Di ereniaion of Newspapers, Journal of Public Economic Theory 4(3), p Gabszewicz, J.J. & Thisse, J. (979): Price compeiion, qualiy, and income dispariies, Journal of Economic Theory, 0, p Göz, G. (005): Endogenous sequenial enry in a spaial model revisied, Inernaional Journal of Indusrial Organizaion, 3, p Hinloopen, J. & van Marrewijk, C. (999): On he limis and possibiliies of he principle of minimum di ereniaion, Inernaional Journal of Indusrial Organizaion, 7, p Hoelling, H. (99): Sabiliy in Compeiion, The Economic Journal, 39, p