MPRA Munch Personal RePEc Archve Second-Degree Prce Dscrmnaton on Two-Sded Markets Enrco Böhme 30. August 0 Onlne at http://mpra.ub.un-muenchen.de/4095/ MPRA Paper No. 4095, posted 30. August 0 09:0 UTC
Second-Degree Prce Dscrmnaton on Two-Sded Markets Enrco Böhme Johann Wolfgang Goethe-Unversty, Frankfurt August 0 Abstract The present paper provdes a descrptve analyss of the second-degree prce dscrmnaton problem on a monopolstc two-sded market. By mposng a smple two-sded framework wth two dstnct types of agents on one of ts market sdes, t wll be shown that under ncomplete nformaton, the extent of platform access for hgh-demand agents s strctly reduced below the benchmark level (complete nformaton). In addton, the paper s fndngs mply that t s feasble n the optmum to charge hgher payments from low-demand agents f the extent of nteracton wth agents from the opposte market sde s assumed to be bundlespecfc. Keywords: two-sded markets, second-degree prce dscrmnaton, monopoly JE Classfcaton: D4, D8,, 5 Char of Publc Fnance, Faculty of Economcs and Busness Admnstraton, Goethe Unversty Frankfurt, Grüneburgplatz, D-6033 Frankfurt am Man, Germany. E-mal: boehme@econ.un-frankfurt.de. I hghly apprecate the helpful comments and hnts of Chrstopher Müller. All remanng mstakes are my responsblty.
. Introducton Second-degree prce dscrmnaton s a well-known phenomenon n the feld of Industral Organzaton, snce t s present n many ndustres. For nstance, non-lnear prcng schemes are very common n the telecommuncatons ndustry, n nsurance markets, or n ralroad and arlne ndustres. The correspondng problem of a monopolstc frm seekng to maxmze proft by offerng type-specfc bundles that are voluntarly chosen by the approprate type of consumer has been wdely analyzed n the economc lterature. Semnal papers are Spence (977), Stgltz (977), Mussa and Rosen (978), Maskn and Rley (984), or Spulber (993). More recently, second-degree prce dscrmnaton has also been dscussed n the context of duopolstc competton, yeldng ambguous effects of prce dscrmnaton on profts. The most relevant paper dealng wth ths topc s Stole (995), whereas Armstrong (006a) and Stole (007) survey ths lterature. The economc lterature mentoned above refers to the case of tradtonal one-sded markets, whle many ndustres operate on two-sded markets,.e. markets where platforms enable nteracton between two dstnct groups of agents. Examples for two-sded networks are manfold: Real estate agences facltate nteracton between house buyers and sellers, credt card companes establsh a smple way of payments between consumers and merchants, whereas meda platforms allow advertsers to nteract wth meda consumers. The economcs of two-sded markets has been extensvely analyzed n the economc lterature. Whle Roche and Trole (003, 006), Callaud and Jullen (003), and Armstrong (006b) analyze monopolstc and duopolstc prce-settng behavour n more general two-sded frameworks, Anderson and Coate (005), Gabszewcz et al. (004), Gal-Or and Dukes (003), and Petz and Vallett (008)) specfcally focus on meda markets. So far, only lttle attenton has been gven to non-lnear prcng strateges on two-sded markets. Callaud and Jullen (003) and Armstrong (006b) analyze the case of groupspecfc prces,.e. thrd-degree prce dscrmnaton, whereas two-part tarffs are consdered by Roche and Trole (003), Armstrong (006b), and Resnger (00). In addton, u and Serfes (008) study frst-degree prce dscrmnaton on a two-sded duopolstc market. To our knowledge, second-degree prce dscrmnaton so far has not been analyzed n the context of two-sded networks. owever, offerng type-specfc bundles for dfferent types of agents s very common n two-sded ndustres. For nstance, move theatres offer mult-tcket bundles or even flat-rates, pay-tv platforms sell dfferent combnatons of content and prce,
whle onlne datng platforms allow men to subscrbe for, e.g., one month, sx months, or one year wth a decreasng prce per month. Recently, second-degree prce dscrmnaton became partcularly relevant n the newspaper ndustry as many newspaper companes started to addtonally offer ther content on the nternet. ere, dfferent strateges can be observed. Whle some companes smply offer dentcal content va pay-per-vew access, others publsh reduced content, e.g. shortened artcles or orgnal artcles wth a sgnfcant tme delay, wthout chargng any prce, but exposng readers to advertsng. Compared to the tradtonal prnted newspaper, ths may well be nterpreted as a dfferent bundle that contans qualtyreduced content for a lower prce. Surprsngly, t can also be observed that some newspapers, e.g. Germany s best-sellng newspaper Bld, regularly publsh exclusve prnt meda content (e.g. exclusve stores or soccer trade rumors) wthout any qualty reducton on the nternet free of charge, generatng revenues from advertsng only. It s the am of ths paper to make a frst step n analyzng second-degree prce dscrmnaton on monopolstc two-sded markets. Talored around the examples mentoned above, we wll develop a smple framework wth asymmetrc nformaton on one of ts two market sdes. Ths specfc sde of the market s supposed to consst of two dstnct types of agents wth dfferent valuatons regardng the ntrnsc utlty they obtan from jonng the platform. As per usual, the agents utlty on ether sde of the market s also affected from ndrect network externaltes. We wll show that many of the well-known results from second-degree prce dscrmnaton on one-sded markets stll preval n our two-sded framework. owever, n contrast to the no-dstorton-at-the-top result from one-sded markets, we fnd that due to the twosdedness of the market, the proft-maxmzng quantty for hgh-demand agents s strctly reduced under ncomplete nformaton. In addton, our fndngs ndcate that f the nteracton wth agents from the opposte market sde depends on the chosen bundle, t s a feasble optmal soluton that the bundle for low-demand agents s more expensve than the bundle for hgh-demand agents. The paper s organzed as follows: In Secton, we wll develop the analytcal framework, whereas Secton 3 analyzes the prce settng behavour of a monopolstc platform operator. In ths context we wll dscuss the benchmark case of complete nformaton and compare our results to the case of ncomplete nformaton. Secton 4 follows the same structure, but mposes bundle-specfc nteracton,.e. the extent of nteracton wll depend on the chosen 3
bundle. In Secton 5, we wll summarze our fndngs and the contrbuton of our paper. In addton, we wll suggest drectons for further research.. Analytcal Framework In the followng secton, we wll develop a benchmark model of a monopolstc platform operator that operates on a two-sded market. Our theoretcal framework consders two market sdes k =,, where market sde conssts of two dstnct groups of agents, labelled and. Agents of both groups dffer n ther ntrnsc valuaton for jonng the platform. Whle the agents know ther ndvdual type, the platform operator s not able to dstngush agents wth respect to ther type. ence, except for the benchmark case, ths stuaton s characterzed by asymmetrc nformaton. The agents utlty s supposed to consst of two elements: an ntrnsc utlty that depends on each agent s access to the platform, denoted by n, as well as an ndrect network effect from the presence of market sde agents and the total payment t. The correspondng utlty functon s assumed to be addtve-separable and can be descrbed by ( ) θ ( ) ( ) U n, n, t = u n n. t =,, where u(.) represents the utlty from jonng the platform, θ reflects the type-dependent valuaton, and denotes the ndrect network externalty resultng from the presence of n market sde agents. For, the network externalty s negatve, whle < 0 mples an addtonal beneft for agents on market sde from nteractng wth agents from the opposte market sde. In addton, we assume u (.), u (.) < 0 and θ > θ, whereas the absolute number of agents n each group s set to one. The latter assumpton mples that our results wll be ndependent from the type dstrbuton of agents, whch allows for an analyss that s strctly focused on the effects resultng from extendng the problem from one-sded markets to two-sded markets. Market sde s characterzed by a tradtonal downward slopng demand functon n ( n, p), where p denotes the prce that all agents on ths market sde have to pay, n order to jon the platform. Snce we assumed that the absolute number of agents on market sde s fxed, agents on market sde do not care about the total number of agents from market sde, but about the extent of ther access to the platform. Ths mples that a potental beneft from 4
nteracton s assumed to depend on the extent of the nteracton process. If the externalty s postve, we have, whle a negatve ndrect network effect mples < 0. As the demand functon was supposed to be downward slopng n prces, we assume that < 0. Addtonal assumptons are =, and the exstence of a unque nteror soluton,.e. t s supposed that both types of agents on market sde are served n the optmum. Reservaton utlty of market sde agents s supposed to be U = 0. The cost functon of the monopolstc platform operator s assumed to consst of constant margnal cost, c, for each unt of access to market sde. In order to smplfy the analyss, total costs on market sde are assumed to be equal to zero. The resultng cost functon s gven by (, ) C n n = c n + c n. Our theoretcal framework refers to the extent of platform access that s sold to agents on market sde, whch may well be nterpreted as beng equvalent to sellng dfferent quanttes of a consumpton good to dfferent types of consumers. Ths allows for a comparson of our results to the well-known second-degree prce dscrmnaton outcome from tradtonal onesded markets. 3. Model Analyss The benchmark case of complete nformaton Under complete nformaton, the monopolst maxmzes profts by sellng a type-specfc bundle of platform access and payment to each type of agents on market sde. It must be taken nto account that both types must be wllng to accept ther offer (partcpaton constrants). ence, the maxmzaton problem can be descrbed by () max (, Π= t + +, t c n c n p n n n p ) t, t, n, n, p s.t. (a) ( ) ( ) θ u n n n, n, p t 0, θ u n n n, n, p t 0. (b) ( ) ( ) Obvously, the partcpaton constrants have to be bndng n the optmum as the platform operator s able to explot the entre consumer surplus. Therefore, t and t n () can be substtuted by (a) and (b), whch leads to the agrangan For nstance, on meda markets ths assumpton corresponds to the well-known concept of persuasve advertsng. 5
() max ( ) (). ( ) (. = θ u n n + θ u n n ) c n c n + p n (. ) n, n, p wth the frst-order condtons u = θ + c= 0, (3) ( p ) u = θ + p c= 0, (4) ( ) (5) () ( ) = n. + p = 0. Snce (). 0 > (!) n and < 0, we can mmedately conclude from equaton (5) that an nteror soluton requres ( ) < 0 p. Then, respectng the assumptons specfed n Secton, * * * equatons (3) to (5) mplctly defne the unqe nteror soluton ( n, n p ),. As per usual, the monopolst s proft s maxmzed where margnal proft s equal to zero. owever, t s not surprsng that our results are more complex than the analog outcome on one-sded markets as the margnal proft also accounts for the arsng network externaltes. Snce we know that ( ) p, t s obvous that n case of, =,, a margnal ncrease n n generates addtonal proft, whereas for < 0 each addtonal unt of n has a negatve mpact on margnal proft. Comparng equatons (3) and (4), whle takng nto account that = and θ > θ, t s easy to verfy that the margnal proft from an addtonal unt of access for the -type, n, strctly exceeds the margnal proft from an ncrease n n. ence, we know that n the * * optmum n > n must hold, whch s n lne wth the correspondng result from one-sded markets. As we know that (a) as well as (b) are bndng n the optmum, proft-maxmzng tarffs are gven by Therefore, respectng that θ ( ) (,, ) t = θ u n n n n p, * * * * * ( ) (,, ) t = θ u n n n n p. * * * * * > θ and n > n, we can conclude that * * ( ) (,, ) ( ) (,, ) t = θ u n n n n p > t = θ u n n n n p. * * * * * * * * * * Obvously, under complete nformaton there s no qualtatve dfference n the results of our two-sded markets model when compared to the correspondng outcome of the second-degree prce dscrmnaton problem on one-sded markets: The type-specfc bundle for the -type contans more platform access and a hgher payment than the one for the -type. 6
The case of ncomplete nformaton In the case of asymmetrc nformaton, the monopolstc platform operator s not able to dstngush the type of market sde agents. ence, t must be taken nto account that each type of agent on market sde must be wllng to voluntarly choose ts desgnated bundle (ncentve constrants). Wth ncentve and partcpaton constrants, the correspondng optmzaton problem s gven by (6) max (, Π= t + +, t c n c n p n n n p ),,,, t t n n p s.t. θ u n n n, n, p t 0, (6a) ( ) ( ) θ u n n n, n, p t 0, (6b) ( ) ( ) (6c) ( ) (,, ) ( ) (,, ) θ u n n n n p t θ u n n n n p t, (6d) ( ) (,, ) ( ) (,, ) θ u n n n n p t θ u n n n n p t. Usng equatons (6a) and (6d) as well as θ > θ, we fnd that ( ) (,, ) ( ) (,, ) θ u n n n n p t θ u n n n n p t ( ) ( ) ( ) ( ) > θ u n n n, n, p t 0 θ u n n n, n, p t, whch mples that the partcpaton constrant for the -type s never bndng. ence, ths restrcton can be gnored wth respect to the optmzaton process, so that the resultng Kuhn- Tucker problem s formally descrbed by (7) t, t, n, n, p, λ, λ, λ3 ( ) max = t + t c n c n + p n n, n, p ( ) (,, ) ( ) ( ) + λ θ u n n n n p t λ θ u n t θ u n t + + ( ) ( ) + λ3 θ u n t θ u n t +, leadng to the frst-order condtons (8) = λ λ + λ 3 = 0, t (9) = + λ λ 3 = 0, t u u = λ + λ θ λ θ + p λ c= 0, (0) ( ) ( ) 3 7
u u = λ θ λ θ + p λ c= 0, () ( ) 3 = n. + p λ = 0. () () ( ) Snce the Kuhn-Tucker condtons requre λm 0, m =,,3, we fnd from equatons (8) and (9) that a soluton s characterzed by λ =, λ = 0, and λ 3 =. Therefore, the remanng frst-order condtons become u u (3) = θ θ + ( p ) c= 0, u = θ + p c= 0, (4) ( ) (5) () ( ) = n. + p = 0. (!) Equaton (5) mmedately mples that an nteror soluton stll requres ( ) already the case under complete nformaton. Assumng that ( ) nteror soluton ( ** ** **,, ) < 0 p as was p holds, the unque n n p of the maxmzaton problem under ncomplete nformaton s mplctly descrbed by the equaton system (3) to (5). Snce θ > θ, t s easy to show that u u u u θ θ = θ θ < θ. (6) ( ) As we have addtonally assumed that =, we know from comparng equatons (3) and (4) that > n n. Therefore, we can conclude that n > n must hold n the optmum, so ** ** that the monopolst stll offers a hgher extent of platform access to the hgh-demand agents. Interpretng the Kuhn-Tucker condtons shows that the partcpaton constrant for the -type and the ncentve constrant for the -type are bndng n the optmum. Usng (6a) and (6d), the optmal payments are therefore gven by whch mmedately mples that ( ) (,, ) t = θ u n n n n p ** ** ** ** ** ( ) θ ( ) t = θ u n u n + t, ** ** ** ** ( ) ( ) ( ) ( ) ( ) t ** ** ** ** ** ** t = θ u n θ u n = θ u n u n. 8
Obvously, under ncomplete nformaton we stll fnd that the bundle for the -type contans more platform access and a hgher payment compared to the bundle for the -type. In addton, we can conclude from the Kuhn-Tucker condtons that the -type s consumer surplus n the optmum s equal to zero, whle the -type enjoys a strctly postve consumer surplus. Snce (6d) s bndng n the optmum, we know that the -type s ndfferent between buyng the bundles offered. The results obtaned so far are entrely consstent wth the correspondng fndngs on one-sded markets. Comparng the outcomes The fnal step n analyzng the model s to compare the results under ncomplete nformaton to the benchmark case of complete nformaton. On one-sded markets ths comparson produces the well-known no-dstorton-at-the-top rule,.e. one fnds that n case of assymetrc nformaton, the bundle for low-demand consumers contans less quantty than under complete nformaton, whle hgh-demand consumers are provded wth the effcent quantty. The correspondng fndngs of our two-sded market model are summarzed n Proposton : Proposton : Under ncomplete nformaton, the proft-maxmzng amount of platform access for -type agents s strctly smaller than under complete nformaton, whle the optmal level of platform access for -type agents s also strctly below the benchmark level, whch contradcts the fndngs from one-sded markets. Ths result s ndependent of the sgn of the network externalty exerted on market sde agents. Proof: Frst, we consder the case of postve ndrect network externaltes on market sde,.e. =. Usng (6) as well as equatons (3) and (3), we fnd that u u u + p c> + p c, (7) θ ( ) θ θ ( ) whch mples that the optmal amount of platform access for low-demand agents on market sde s ceters parbus smaller under ncomplete nformaton than under complete nformaton. Ths effect has an addtonal mpact on the demand of market sde agents, whch n turn nfluences the proft-maxmzng prce that s charged on market sde. The optmal prces p * and p ** were mplctly determned by equatons (5) and (5). n Dfferentatng (5) or (5) wth respect to n and addtonally assumng = 0 yelds (.) n = +, pn pn ( p ) = 0 9 pn
whch allows us to conclude that p ** < p *, so that the optmal prce on market sde s strctly smaller than n the benchmark case of complete nformaton. Snce p smultaneously enters equaton (3), t follows that the margnal proft from an ncrease n n s addtonally ** * reduced under ncomplete nformaton. ence, we fnd that n < n. At the same tme p also enters equaton (4), whch leads to a decrease n the margnal proft generated from an ** * addtonal unt n. Therefore, we obtan n < n. If we consder the case of = < 0, we fnd that (7) stll holds, whch agan mples that n s ceters parbus reduced under ncomplete nformaton. Due to the negatve network externalty on market sde, ths effect leads to an ncrease n the demand of market sde n agents. Assumng ** = 0, the correspondng mpact on p s descrbed by pn whch obvously mples that soluton requres ( p ) (q.e.d.) (.) n = + < 0, pn pn ( p ) < 0 = 0 p > p. Usng (3) and (4), and respectng that an nteror ** *, t follows that n < n as well as n ** * < n. ** * The economc ntuton behnd our fndngs s straghtforward: The monopolstc platform operator s maxmzng her proft by nducng a self-selecton process among both types of agents on market sde. Therefore, the bundle of platform access and payment that s sold to low-demand agents ceters parbus contans less platform access than under complete nformaton as ths allows the platform operator to extract addtonal consumer surplus from hgh-demand agents. Ths result s well-known from second-degree prce dscrmnaton on one-sded markets. owever, the reducton of n nduces an addtonal effect on the opposte market sde as the demand of market sde agents was supposed to depend on n and n. For nstance, assume a postve network externalty exerted on market sde. Then, a reducton of n shfts the demand functon ( n n, n, p ) nwards, whch reduces the optmal ** prce p that s charged on market sde. Ths n turn makes a margnal ncrease n n even less proftable, so that the proft-maxmzng extent of platform access for low-demand agents ** s strctly smaller under ncomplete nformaton. In addton, the reducton of p also affects ** n as each margnal unt of platform access for hgh-demand agents generates less margnal ** proft than under complete nformaton. Ths effect strctly reduces n below the benchmark level, whch contradcts the correspondng results from one-sded markets, where the quantty for hgh-demand consumers was not affected under ncomplete nformaton. 0
The outcome of the second-degree prce dscrmnaton problem on one-sded markets may also serve as an alternatve benchmark for a comparson wth our results under ncomplete nformaton. The case of one-sded markets s equvalent to the specal case of our model where = 0 and = = 0. In ths specal case, equatons (3) and (4) become u u u = θ θ + ( p 0) 0 c= 0 θ > c, u u = θ + ( p 0 ) 0 c= 0 θ = c, whch reproduces the no-dstorton-at-the-top outcome from one-sded markets. We wll denote ths specfc soluton by ( nˆ ˆ, n ). If we consder the case of 0 and =, whle addtonally respectng ( p ), we fnd by usng (3) and (4) that u u u = θ θ + ( p ) c= 0 θ + ( p ) > c, u u = θ + ( p ) c= 0 θ < c, ** whch mples that ˆ ** n > n as well as ˆ n > n. It s easy to verfy that for 0 and ** = < 0, we fnd the opposte result, that s ˆ ** n < n and ˆ n < n. The economc ntuton behnd our results s rather smple: In case of postve network externaltes on market sde,.e. =, each addtonal unt of n and n generates addtonal margnal proft. ence, the monopolst strctly ncreases the amount of platform access n both bundles compared to the soluton under asymmetrc nformaton on one-sded ** markets. In the optmum, we fnd that the -type s margnal wllngness to pay for n s strctly below margnal cost, whch mples that hgh-demand agents on market sde are subsdzed n any case. Under specfc parameter sets, the same even holds for low-demand agents. Consderng the case of = < 0, we fnd the opposte: A margnal ncrease n n and n leads to addtonal margnal cost, thus nducng the monopolstc platform operator to strctly reduce the extent of platform access sold to each type of agents on market sde. 4. A Model wth bundle-specfc Interacton So far, we have assumed that both types of agents on market sde are equvalently affected by the presence of market sde agents. In ths secton, we wll extend the analyss by
ncludng bundle-specfc nteracton,.e. the extent of the ndrect externalty depends on the chosen bundle. For nstance, consder an onlne datng platform that sells dfferent pars of platform access and payments to two dstnct types of market sde agents (men) wth dfferent valuatons, whle market sde agents (women) are generally allowed to jon the platform for free. Then, t could be the case that the -type bundle only allows market sde agents to contact a lmted number of agents from the opposte market sde, whereas the bundle for hgh-demand agents contans unlmted access. We wll account for ths by mposng a dfferent utlty functon for agents on market sde. Ths functon s stll supposed to be addtve separable and takes the form ( ) θ ( ) ( ) U n, n, t = u n n n. t =,, whle the notaton ntroduced n Secton remans the same. In addton, all assumptons made so far are stll vald. The motvaton behnd ths model specfcaton stems from the move theatre example mentoned n Secton. Assume that a cnema operator offers two bundles: The -type bundle contans a sngle tcket for a specfc prce, whle the -type bundle ncludes 5 tckets for a total payment that mples a per tcket prce below the prce of the sngle tcket. If both types of consumers buy ther desgnated bundles, the -type s only once exposed to the cnema-specfc amount of advertsng, whle the -type watches the same amount of advertsng fve tmes. Proft maxmzaton under complete nformaton In case of symmetrc nformaton, the monopolstc platform operator s agan able to perfectly dscrmnate between both types of agents on market sde by sellng type-specfc bundles ( n, n, t ), as take-t-or-leave-t offers. Respectng the partcpaton constrants, the correspondng optmzaton problem s gven by (8) max (, Π= t, + t c n c n + p n n n p) t, t, n, n, p (8a) θ ( ) ( ) s.t. u n n n n, n, p t 0, (8b) θ ( ) ( ) u n n n n, n, p t 0. As we have already mentoned n Secton, perfect prce dscrmnaton mples that the consumer surplus for market sde agents s equal to zero. ence, we know that the partcpaton constrants are bndng n the optmum, so that the maxmzaton problem smplfes to a agrangan of the form
(9) max ( ) (. = θ u n n n ) + θ u( n ) n n (). n, n, p yeldng the frst-order condtons ( ) c n c n + p n,. ( ) u = n. + p n + n c= 0, (0) θ () ( ) ( ) u = n. + p n + n c= 0, () θ () ( ) ( ) () = n (). + p ( n n + ) = 0. 0 > (!) n Snce all assumptons from the prevous sectons are stll holdng, we can conclude from equaton () that a feasble soluton requres p ( n n ) < 0 ( ) + >. Assumng that an 0 nteror soluton exsts, ths soluton s mplctly characterzed by equatons (0) () and. wll be denoted by ( * n * *, n, p ) Gven our extenson of bundle-specfc nteracton, t s not surprsng that the effects resultng from the presence of a second market sde are more complex than n the benchmark case of Secton 3. Analysng equatons (0) and () reveals that the frst-order condtons reflect three dfferent effects (apart from the tradtonal mpact on the margnal wllngness to pay): Frst, a margnal change of n drectly affects the demand of market sde agents, whch n turn has an ndrect mpact on the margnal proft generated on ths market sde. In addton, a dfferent number of market sde agents nfluences the utlty of both types of market sde agents and hence ther wllngness to pay. These two (ndrect) effects are reflected by the ( ) term ( ) p n + n n and are n lne wth our fndngs from the benchmark case of Secton 3. owever, n case of bundle-specfc nteracton there s an addtonal effect from a margnal change of n as type s utlty on market sde s drectly affected, because the extent of nteracton wth agents from the opposte market sde changes. Ths addtonal effect s reflected by the expresson n (. ). Snce we assumed that θ > θ and =, t s easy to see by comparng equatons (0) and () that the margnal proft from an addtonal unt n stll exceeds the margnal proft from an ncrease n n,.e. >. Therefore, we know that n the optmum t stll holds * * that n > n, whch corresponds to our fndngs from the prevous secton. In addton, we know that the partcpaton constrants are bndng n the optmum. ence, usng equatons (8a) and (8b), the proft-maxmzng tarffs for the type-specfc bundles are gven by 3
( ) (,, ) t = θ u n n n n n p, * * * * * * ( ) (,, ) t = θ u n n n n n p. * * * * * * Therefore, the dfference of payments s descrbed by (3) t * * ( * ) * ( * * * ) ( * ) * ( * * *,,, t = θ u n n n n n p θ u n + n n n n, p ) ( ) ( ) (,, ) ( ) t t = θ u n θ u n + n n n p n n. * * * * * * * * * So far, our results are not surprsng as they are consstent wth the results from Secton 3. owever, analyzng the relaton of the bundle-specfc payments yelds an nterestng result, denoted n Proposton. Proposton : Under complete nformaton, the proft-maxmzng tarff for the -type bundle strctly exceeds the optmal prce-level for the -type bundle f the ndrect network externalty exerted on market sde agents s postve or absent,.e. 0. In case of a negatve network externalty on market sde,.e., t s a feasble proft-maxmzng soluton that the bundle for low-demand agents s more expensve than the one for hghdemand agents, even though the latter contans a strctly hgher extent of platform access. * * Proof: Suppose < 0. As we know that θ > θ as well as n > n, we can mmedately conclude from equaton (3) that ( ) ( ) ( ) ( ) t t = θ u n θ u n + n n, n, p n n. * * * * * * * * * < 0 < 0 For = 0, equaton (3) becomes ( ) θ ( ) * * * * t t = θ u n u n, whch closes the proof for the frst part of Proposton. In case of a negatve ndrect network effect,.e., we fnd by usng (3) that ( ) ( ) (,, ) ( ) t t = θ u n θ u n + n n n p n n, * * * * * * * * * < 0 * * whch obvously mples that t t can be ether postve, negatve or equal to zero. In order * * to prove that t t < 0 s feasble n the optmum, we have to show that * * * * * * * ( ) ( ) ( ) ( ) θ u n θ u n + n n, n, p n n < 0 < 0 4
s, at least under specfc restrctons, n lne wth the frst-order condtons. Suppose that θ θ. Then we have n the optmum that n = n ε, where ε s assumed to be very small. In ths case, t approxmately holds that * * u u θ u n θ u n = θ n n + θ n n * * * * * * ( ) ( ) ( ) * * n n u u u u = θ ε + θ ε = ε θ + θ Snce an optmal soluton requres we fnd by usng (3) that t * * * * n n n n * * u θ = θ u * * n n t can be expressed as * * u u t t θ n(). ( ) n(). n ε ε ε θ * n = + =. * n n ence, we have that u u t t < 0 θ n. < 0 θ < n..,. () () * * * * n n By analyzng the frst-order condtons, n partcular equaton (), we can conclude that ths nequalty s satsfed n the optmum, f and only f t s true that * * * ( p n n ) c (4) ( ) u + > θ, whch obvously only holds for (q.e.d.). * 0 * n > n (!) The results of Proposton are surprsng, snce they contradct the standard result from onesded markets where the monopolst n any case charges a hgher tarff for the -type bundle. owever, n case of bundle-specfc nteracton on a two-sded market, the optmzaton problem s more complex: Consderng the case of θ θ, we can conclude from (4), that * * * * * t t < 0 requres a specfc relaton between the term ( p ( n + n )) and the margnal cost c as ther dfference must be suffcently large. For, the expresson * * * ( p ( n + n )) covers the (net) margnal proft that arses from the ndrect mpact of an ncrease n n on the demand of market sde agents. In case that ths expresson s very large, t mples that an addtonal unt n s partcularly proftable due to the presence 5
of the opposte market sde. If, on the other hand, c s very small, there s a strong ncentve for the monopolst to choose n (as well as n ) as large as possble. owever, the platform operator s facng a dffcult trade-off: For, any ncrease of n and n reduces the utlty of market sde agents, but ther partcpaton s crucal n order to explot the hgh margnal profts from market sde. Therefore, the platform operator has to compensate both types of agents for ther utlty losses by reducng the bundle-specfc payment. As we have * * found that n > n, we know that hgh-demand agents on market sde are facng a larger extent of utlty reducton n the optmum, snce the nteracton wth market sde agents s assumed to depend on the chosen bundle. ence, n order to respect the partcpaton constrants, the prce reducton for the -type bundle wll exceed the one for the -type bundle. In case that (4) holds, ths process leads to the surprsng result t * * < t. The economc ntuton s as follows: As per usual, hgh- demand agents obtan more utlty from. At the same tme they are facng more dsutlty from the presence of market sde agents,.e. we have that * * jonng the platform,.e. t holds that θ u( n ) > θ u( n ) (,, ) (,, ) ( * ) ( * ) * ( * * * ) * ( * * * u n u n < n n n, n, p n n n, n, p ) n n n n p > n n n n p. If * * * * * * * * θ θ, whch holds f (4) s satsfed, the platform operator has to compensate hgh-demand agents * * by choosng t < t. Optmzaton under ncomplete nformaton As was already dscussed n Secton 3, the monopolst has to respect the ncentve constrants and the partcpaton constrants when maxmzng her proft under ncomplete nformaton. ence, the correspondng (Kuhn-Tucker) optmzaton problem s descrbed by (5) max (, Π= t, + t c n c n + p n n n p) t, t, n, n, p (5a) θ ( ) ( ) s.t. u n n n n, n, p t 0, (5b) θ ( ) ( ) u n n n n, n, p t 0, (5c) θ ( ) (,, ) θ ( ) (,, ) u n n n n n p t u n n n n n p t, (5d) θ ( ) (,, ) θ ( ) (,, ) u n n n n n p t u n n n n n p t. Respectng θ > θ, we fnd by usng equatons (5a) and (5d) that 6
( ) (,, ) ( ) (,, ) θ u n n n n n p t θ u n n n n n p t ( ) ( ) ( ) ( ) > θ u n n n n, n, p t 0 θ u n n n n, n, p t. ence, the partcpaton constrant for hgh-demand agents can be dsregarded as t s never bndng. The resultng Kuhn-Tucker optmzaton problem s therefore gven by (6) max (, = t, + t c n c n + p n n n p) t, t, n, n, p, λ, λ, λ3 ( ) (,, ) +λ u n n n n n p t θ ( ) (,, ) ( ) (,, ) +λ θ u n n n n n p t θ u n + n n n n p + t ( ) (,, ) ( ) (,, ) +λ3 u n n n n n p t u n n n n n p t θ θ + + wth the frst-order condtons (7) = λ λ +λ 3 = 0, t (8) = +λ λ 3 = 0, t u u = λ + λ θ λ θ n. λ + λ λ +Ψ c= 0, (9) ( ) () ( ) 3 3 u u = λ θ λ θ n. λ λ +Ψ c= 0, (30) () ( ) 3 3 (3) (). = n +Ψ = 0, ( ( 3 3 )) where p n ( λ λ λ ) n ( λ λ ) Ψ= + +. Takng nto account that the Kuhn-Tucker condtons requre λm 0, m =,,3, equatons (7) and (8) mply that n the optmum λ =, λ = 0, and λ 3 = must hold. ence, equatons (9) to (3) become ( ) u u = n. + p n + n c= 0, (3) θ θ () ( ) < θ u ( ) u = n. + p n + n c= 0, (33) θ () ( ) ( ) (34) = n (). + p ( n + n ) = 0. 0 > (!) < 0 7
Obvously, we fnd from equaton (34) that an nteror soluton stll strctly requres that ( p ( n n )) + >. Addtonally assumng that both types of agents on market sde are 0 served n the optmum, equatons (3) - (34) determne the unque proft-maxmzng soluton ( ** n ** **, n, p ). Snce θ > θ and =, we fnd by analyzng equatons (3) and (33) that > whch mples that the optmal level of platform access for hgh-demand agents stll exceeds the one for low-demand agents,.e. we have n > n. From the Kuhn-Tucker constrants ** ** we can conclude that there s stll no consumer surplus for the -type, whle the -type s ndfferent between both bundles and enjoys a strctly postve consumer surplus. These results are consstent wth our fndngs from the prevous secton.. As we know that (5a) as well as (5d) are bndng n the optmum, we fnd that the optmal payments are gven by ( ) (,, ) t = θ u n n n n n p ** ** ** ** ** ** ( ) (,, ) ( ) ( ) t = θ u n n n n n p θ u n + θ u n. ** ** ** ** ** ** ** ** ence, the prce dfferental s expressed by. (35) ** ** ( ** ) ( ** ) ( ** ** ** ) ( ** t ** t = θ u n θ u n + n n, n, p n n ), In the benchmark case of complete nformaton we found that under specfc restrctons the -type bundle was more expensve than the -type bundle. owever, n case of asymmetrc nformaton, the monopolstc platform operator s not able to sell type-specfc take-t-orleave-t offers to agents on market sde, whch potentally nfluences our results. The correspondng fndngs under ncomplete nformaton are summarzed by Proposton 3: Proposton 3: In case of ncomplete nformaton, t holds n the optmum that the proftmaxmzng tarff for the -type bundle s strctly larger than the payment for the -type bundle f the presence of market sde agents exerts a nonnegatve externalty on both types of agents on market sde,.e. for 0. For, the results are ambguous, whch mples that, under specfc condtons, the prce level of the -type bundle exceeds the payment for the -type bundle. Proof: Frst, we consder the case of postve network effects on market sde,.e. < 0. ** ** Then, respectng n > n, we fnd by usng equaton (35) that ( ) ( ) ( ) ( ) t t = θ u n θ u n + n n, n, p n n. ** ** ** ** ** ** ** ** ** < 0 < 0 8
For = 0, equaton (35) smplfes to ( ) θ ( ) ** ** ** ** t t = θ u n u n, whch proves the frst statement that s contaned n Proposton 3. Now, suppose that, whch mples the presence of a negatve ndrect network externalty exterted on market sde agents. In ths case, we fnd from (35) that ( ) ( ) (,, ) ( ) t t = θ u n θ u n + n n n p n n, ** ** ** ** ** ** ** ** ** < 0 yeldng ambguous results,.e. ** ** t t can have any sgn or s equal to zero. Showng that ** ** t t < 0 s feasble n the optmum, requres to verfy that ** ** ** ** ** ** ** ( ) ( ) ( ) ( ) θ u n θ u n + n n, n, p n n < 0 < 0 s covered by the frst-order condtons. We start the proof by assumng that θ θ. ence, ** ** we have n the optmum that n = n ε wth ε beng very small. By approxmaton, t holds that θ u u u n θ u n θ n n θ = = ε. ** ** ** ** ( ) ( ) ( ) ** ** n n Then, usng equaton (35) we fnd that t t can be descrbed by ** ** t t = θ u ε + n ε ε = θ u n whch allows us to conclude that (). ( ) (). * * ** ** n n u u t t < 0 θ n. < 0 θ < n.., () () * * ** ** n n By analyzng (33) we fnd that ths nequalty s satsfed n the optmum f t holds that ** ** ** ( p n n ) c (36) ( ) +, whch strctly requres postve ndrect network effects on market sde,.e. =. (q.e.d.) (!) ** n Proposton 3 shows that our results from the benchmark case of complete nformaton preval under ncomplete nformaton: It s stll a feasble soluton n the optmum that the -type bundle s more expensve than the bundle for hgh-demand agents. Ths s not surprsng as 9
we know that the -type enjoys a strctly postve consumer surplus under ncomplete nformaton, whle the monopolst extracts the entre consumer surplus n case of perfect prce dscrmnaton. ence, n order to make hgh-demand agents better off than under complete nformaton, the monopolstc platform operator ceters parbus has an ncentve to ncrease the -type s utlty n the optmum by choosng t ** below the level of the benchmark case. Therefore, as we already know that t * * t < 0 s feasble under complete nformaton, t s not surprsng that ths outcome prevals under ncomplete nformaton. Comparson wth the benchmark case We wll close the analyss by comparng our fndngs from the case of asymmetrc nformaton to the benchmark case of complete nformaton. Proposton 4 contans the correspondng results. Proposton 4: Restrctng the analyss to < < for the case of postve network effects on market sde,.e., =,, and n n n n < < for < 0, we fnd that under ncomplete nformaton, the optmal extent of platform access for low-demand agents s ** * strctly smaller than the benchmark level,.e. t holds that n < n. In addton, the proftmaxmzng amount of platform access for hgh-demand agents s also negatvely affected ** * under ncomplete nformaton. ence, we fnd that n < n. Gven the restrctons above, ths result s robust wth respect to the sgn of the network effect on market sde. Proof: We start by consderng the case of. Snce nequalty (6) s stll satsfed, t s easy to show by usng equatons (0) and (3) that the monopolstc platform operator under ncomplete nformaton ceters parbus chooses a smaller amount of platform access for the -type bundle. Ths effect nduces a change n the demand of market sde agents, whch n turn nfluences the proft-maxmzng prce p ** that s mplctly gven by equaton (34). Dfferentatng (34) wth respect to n, whle respectng that < < and n = 0, yelds n ( p ( n n )) n = + +, pn pn = 0 whch mmedately mples that p ** < p *. Ths reducton has an addtonal effect on the optmal amount of platform access for -type agents. By solvng (34) for n () we obtan ( ) (). ( ) n = p n + n, p 0.
so that equaton (3) can be reformulated as ( ( )) u u = θ θ + p n + n + c= 0. Takng nto account that < <, ths allows us to conclude that Snce For (q.e.d.) p ** p smultaneously enters equaton (33), t can be analogously shown that n n n n < 0, the proof follows the same logc, but requres < <. n < n. ** * n < n. ** * Proposton 4 confrms our fndngs from Secton 3 and shows that our results are stll holdng n case of bundle-specfc nteracton. The logc behnd our fndngs remans the same as n Secton 3: Under ncomplete nformaton, the monopolst ceters parbus has an ncentve to reduce the optmal extent of platform access for low-demand agents below the benchmark level, n order to make ths bundle less attractve for hgh-demand agents. Thus, the frm s able to extract addtonal consumer surplus from the -type on market sde. owever, for the reducton of n has a negatve nfluence on the demand of market sde agents, whch addtonally reduces the margnal proft from an addtonal unt of n. ence, there s a second effect that reduces the optmal amount of platform access for low-demand agents on market sde. At the same tme ths effect has an mpact on the margnal proft that s generated from an addtonal unt n. Snce each unt n becomes less proftable due to the reduced demand on market sde, the proft-maxmzng extent of platform access that s provded to the -type s also strcly below the benchmark level. 5. Conclusons and Implcatons The present paper provdes a postve analyss of second-degree prce dscrmnaton on a monopolstc two-sded market. We found that many of the results from the equvalent problem on one-sded markets are stll vald n our two-sded settng: The extent of platform access (whch may well be nterpreted as qualty) for low-demand agents s strctly reduced under ncomplete nformaton, n order to nduce the well-known self-selecton process among agents on market sde, whle allowng the monopolst to extract addtonal consumer surplus from hgh-demand agents. In addton, n the optmum agents wth low valuaton are stll left wthout any consumer surplus, whereas hgh-demand agents are n any case ndfferent between the two bundles, whle enjoyng an nformaton rent.
owever, the paper contrbutes to the exstng lterature by revealng some mportant dfferences n relaton to the second-degree prce dscrmnaton problem on one-sded markets: In Secton 3, we found that the famous no-dstorton-at-the-top -rule from onesded markets does not preval n our two-sded framework, snce our analyss yelded that ** * n < n,.e. under ncomplete nformaton, the extent of platform access for the -type s strctly below the level under complete nformaton. In the subsequent analyss of Secton 4 t was also shown that ths result s robust wth respect to the model specfcaton of bundlespecfc nteracton. Ths result mples that the monopolstc optmzaton problem s more complex n the presence of network externaltes as the ceters parbus proft-enhancng reducton of n has an mpact on the demand of market sde agents, whch n turn (negatvely) affects the margnal proft generated from an addtonal unt n. The analyss of Secton 4 mpled another mportant result: In case of bundle-specfc nteracton, t s a feasble proft-maxmzng soluton that the bundle for the -type s more expensve than the one for the -type, whch contradcts the fndngs from one-sded markets. Ths surprsng result holds for the case of complete nformaton and prevals under ncomplete nformaton. Interpretng these fndngs sheds some new lght on the examples provded n Secton. For nstance, recall the case of German newspaper Bld that offers exclusve prnt meda content on the nternet free of charge, but n combnaton wth advertsng. Snce the onlne content s dentcal to the content of the prnted newspaper, there s no qualty reducton, but a sgnfcant amount of advertsng consstng of numerous banner ads and lnks. Intutvely, one would expect that offerng the onlne content for free targets low-demand agents, snce they do not have to pay for beng able to read the same content that s offered n the prnt medum. owever, respectng the results of Propostons and 3, t may well be the case that the combnaton of free onlne access and hgher advertsng levels s a desgnated bundle for hgh-demand agents. Ths s supported by the followng consderaton: If we assume that hgh-demand agents on a newspaper market have a strong preference for recevng exclusve news as soon as possble, the results of our model would strctly requre publshng any nformaton on the nternet at frst. Ths can ndeed be observed as the artcles of Bld are usually publshed on the nternet the nght before the prnted newspaper s sold. Fnally, t has to be mentoned that the focus of our analyss s exclusvely postve, so that we are not able to draw any normatve conclusons wth respect to the welfare aspects of the second-degree prce dscrmnaton problem on a two-sded market. Extendng the analyss by
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