Optimal Pricing Scheme for Information Services

Transcription

1 Optml rcng Scheme for Informton Servces Shn-y Wu Opertons nd Informton Mngement The Whrton School Unversty of ennsylvn E-ml: e-yu (Shron) Chen Grdute School of Industrl Admnstrton Crnege Mellon Unversty E-ml: G. Anndlngm Decson nd Informton Technologes R. H. Smth School of Busness Unversty of Mrylnd nd Opertons nd Informton Mngement The Whrton School Unversty of ennsylvn E-ml: August 00

2 Optml rcng Scheme for Informton Servces Abstrct Ths pper exmnes whch, mong three commonly used prcng schemes: the flt-fee, pure usge-bsed nd the two-prt trff prcng, s optml for monopolst provdng nformton servces. Our nlyss suggests tht under zero mrgnl costs nd montorng costs, when customers re homogeneous or when customers hve heterogeneous mrgnl wllngness to py (whch corresponds to dfferent downwrd slopng demnd curves), flt-fee prcng nd two-prt trff prcng lwys cheve the sme proft level, nd domnte usge bsed prcng. However, when customers re chrcterzed by heterogeneous mxmum consumpton levels (or usge levels), the two prt trff prcng s the most proftble mong the three. We lso exmne how senstve the optml prcng scheme s to mrgnl costs nd montorng costs. Our nlyss shows tht when mrgnl cost s below certn vlue, the flt fee prcng s the optml scheme regrdless how lrge or how smll montorng cost s (s long s t s postve) when customers re homogeneous or hve heterogeneous mrgnl wllngness to py. But s montorng cost becomes zero, the two-prt trff wll lso become one of the optml prcng schemes.

3 . Introducton The dvnce of the Internet nd other telecommuncton networks hs mde mny new nformton servces vlble. For exmple, number of softwre pplctons servces re provded by onlne Applcton Servce rovders (AS) over the Web or prvte networks for busnesses or end users. Onlne dscounted brokerge s lso n exmple of nformton servces whch end users could get ccess to over the Web. Short Messgng Servces (SMS) nd WA servces re mong the most populr servces for moble users. Moreover, s crrers roll out ther.5g nd 3G networks, host of other new servces re beng offered-- from moble e-ml to multmed messgng servces (MMS)-- tht represent vluble new revenue opportuntes (Mer, 00). However, these nformton servce provders generlly fce one dffculty n provdng the servces, tht s, how to prce nd bll these nformton servces (Mer, 00), whch re chrcterzed by neglgble mrgnl, dstrbuton, nd montorng cost. In ths pper we exmne the ssue of prcng nformton servces. In prtculr, we re nterested n knowng whch mong the three most populr prcng schemes used n prctce: flt fee, usge-bsed nd two-prt trffs, s the best for monopolst provdng nformton servces. Whle some reserchers beleve tht reducton n montorng cost or dstrbuton cost mkes usge-bsed prcng reltvely more ttrctve opton (Cho, Sthl nd Whnston, 997; Metclfe, 997), some rgue tht neglgble mrgnl producton cost mkes flt-fee prcng more proftble (Fshburn, Odlyzko nd Sders, 997). There s so fr no cler gudelne bout when the nformton servce provders should dopt flt rte prcng nd when pure usge-bsed prcng (wthout subscrpton fee) or even two-prt trff (usge-bsed prcng plus subscrpton fee) s more proftble. Mny nformton servce provders hve struggled to fnd best wys to prce ther servces nd bll ther customers, nd ths s reflected from the non- 3

4 greed-upon prcng schemes offered by dfferent nformton servce provders. For exmple, Verzon Wreless, whch rolled out ts 3G Express Network n lte Jnury, 00 hs chosen flt fee prcng scheme, whle AT&T Wreless hs mplemented usge-bsed prcng scheme tht bll for the mount of dt customer uses. et nother scheme dopted by NTT DoCoMo's I-mode servce n Jpn chrges users $.50 monthly fee, plus 5 cents per dt pcket (one pcket s equvlent to 8 bytes of dt), s two-prt trff scheme. Outsourcng of IT servces uses both fxed-fee prcng n some cses, nd usge-bsed prcng wth or wthout subscrpton fees n some other cses (Gopl et l., 00). For exmple, the newly sgned contrcts between Amercn Express Co. nd IBM, nd convenence-store chn 7-Eleven Inc. nd EDS re both bsed on usge-bsed prcng wth some fxed fees,.e., the two-prt trff prcng (Greenemeer, 00). Whle there hs been ncresng nterest on how to prce nformton goods (Bkos nd Brynjolfsson, 999; Chung nd Srbu, 999; Vrn, 000), much of the work ether does not ddress nformton servces, or s only ndrectly pplcble to such cses. For exmple, the dvntges of pure bundlng n Bkos nd Brynjolfsson (999) result from reducton n the vrnce for customers vlutons for bundle of dfferent nformton goods. Ths modelng technque cnnot be ppled to nformton servces, snce ech unt of nformton servces s essentlly dentcl, nd therefore, we cnnot expect vrnce to be reduced through ggregton of dentcl unts. We use frmework n whch both buyers nd sellers of nformton goods optmze ther net vlues n order to determne whch prcng scheme works best under dfferent condtons. Recent works tht re relted to nformton servce prcng nclude Fshburn, Odlyzko nd Thus they wll hve perfect correlton cross unts n customers vlutons. 4

5 Sders (997), Esseger, Gupt nd Zhng (00) nd Sundrrjn (00). Our pper s complementry to these ppers. Sundrrjn (00) consders fxed-fee nd nonlner usgebsed prcng schemes. But the focus of ths study s dfferent from ours n ths pper. He shows tht n the presence of contrct dmnstrton costs, such s montorng cost for usge bsed prcng, monopolst cn mprove ts profts by offerng fxed-fee prcng n ddton to usgebsed contrct. However, whle ths study hs suggested tht frm could mprove ts profts by doptng mx of prcng schemes, the results re bsed on utlty functon tht hs to stsfy the Spence-Mrrlees sngle-crossng property, whch llows frm to possbly segment customers through ther self-selecton proftbly. When ths property does not hold, t s not cler whether doptng multple schemes wll stll be proft mprovng. In ddton, ths pper does not offer drect gudelne bout whch prcng scheme s most proftble when the frm could only opt for one prcng scheme, the mjor focus of our pper. There re severl stutons where frms my fvor to dopt one prcng scheme only. For exmple, when new nformton servce s just beng provded, the frm my prefer to dopt only one prcng scheme to keep the mrketng smple; eser dmnstrton nd mngement my lso mke the frm my prefer one prcng scheme only. Fshburn, Odlyzko nd Sders (997) compre the flt fee nd the usge-bsed prcng nd show tht flt fee s better thn metered rte for monopolst offerng nformton servces on the Internet. However, they hve smplfed the problem wth some very restrctve ssumptons. For exmple, they ssume tht consumers choose the quntty of servce to buy nd stck to t before exmnng the vlble prces. It s not cler whether ther results could be generlzed to more generl demnd functons,.e., downwrd slopng demnd functon. And fnlly, both of these works do not consder the two-prt trff prcng, whch s populr both n theory nd prctce; we exmne ths cse lso. Esseger, Gupt nd Zhng s (00) lso 5

6 consder the two-prt trff prcng together wth flt fee nd usge-bsed prcng. As n Fshburn, Odlyzko nd Sders (997), they ssume tht consumer usge s nelstc to prce chnge. Moreover, they ssume tht both hevy nd lght users hve the sme totl reservton prce for the servce, whch my be doubtful s users usully hve qute dfferent nd dmnshng mrgnl utlty for ech unt of servce they consume. It s lso questonble to ssume tht mrgnl cost s zero when servce provder hs cpcty constrnt (nother problem wth cpcty constrnt s the possble queung problems, whch s not dscussed n ther pper). Overll, our nlyss suggests tht under zero mrgnl nd montorng costs, when customers re homogeneous or when customers hve heterogeneous mrgnl wllngness to py (whch corresponds to dfferent downwrd slopng demnd curves), flt-fee prcng nd two-prt trff prcng lwys cheve the sme proft level, nd re strctly better thn the usge-bsed prcng. However, when customers re chrcterzed by heterogeneous mxmum consumpton levels, two-prt trff prcng s the most proftble mong the three. We lso exmne how senstve the optml prcng scheme s to mrgnl costs nd montorng costs when customers re homogeneous or when customers hve heterogeneous mrgnl wllngness to py. Our nlyss shows tht when mrgnl cost s below certn vlue, the flt fee prcng s the optml scheme regrdless how lrge or how smll montorng cost s (s long s t s postve) when customers re homogeneous or hve heterogeneous mrgnl wllngness to py. But s montorng cost becomes zero, the two-prt trff becomes one of the optml prcng schemes. The pper s orgnzed s follows: In secton, we provde the generl model for the mrket for nformton servces. Secton 3 reports on the nlyss of dfferent prcng schemes nd when ech of them s most proftble. Secton 4 outlnes some model extensons. We provde Although they try to extend ther model ner the end of the pper by usng the sme unt reservton prce for the two consumer segments, ths ssumpton stll fls to reflect the truth tht users usully hve qute dfferent nd dmnshng mrgnl utlty for ech unt of servce they consume. 6

7 concludng remrks n secton 5.. Mrket Model for Informton Servces We exmne the optml prcng scheme for n nformton servce provder who sells one knd of nformton servce (such s voce communcton servce or dt trnsmsson servce) to consumers. We consder three prcng schemes: pure flt fee, pure usge-bsed nd two-prt trff prcng. The nformton servce provder chooses whch prcng scheme to dopt nd the prce(s) to offer. Consumers then mke decsons bout whether to jon the pln, nd how much to consume gven the prcng scheme nd prces set by the nformton servce provder. Snce nformton servces usully experence some pek-hours nd some non-pek hours, we ssume tht consumers my hve dfferent utlty functons n pek hours nd non-pek hours. As result, nformton servce provders my chrge dfferent prces for the two tme segments when usng usge-bsed prcng. In ddton, gven lmted tme nd ttenton, we ssume tht consumers fce certn upper bounds n consumng the servces. For exmple, gven tht there re only 4 hours n dy, consumers cn t consume the servce for more thn 4 hours dy... Consumers Optmzton roblem Gven the prcng scheme (flt rte, usge-bsed, or two-prt trff) nd prce(s) set by the nformton servce provder, consumer wll decde whether or not she wnts to jon the servce progrm nd her consumpton level of the servce n both pek hours nd non-pek hours to mxmze her totl net utlty. Gven rmeters: 7

8 : the subscrpton fee for the consumer to jon the progrm : the unt prce of the servce set by the provder n pek hours : the unt prce of the servce set by the provder n non-pek hours U(, ): the utlty functon of consumer t the consumpton level of n pek hours nd n non-pek hours : consumer 's mxmum consumpton level of the servce n pek hours : consumer 's mxmum consumpton level of the servce n non-pek hours Decson Vrbles: : consumer 's consumpton level of the servce n pek hours : consumer 's consumpton level of the servce n non-pek hours Z : the decson vrble whch s f consumer chooses to jon the progrm nd 0 otherwse Consumers Optmzton roblem: Mx U, Z (),, Z ( ) s.t. Z () Z (3) (, ) U Z 0 (4) (the Indvdul Rtonlty constrnts) Z = 0or (5) The objectve functon () s to mxmze the consumer surplus gven the prce(s) set up by the nformton servce provder. In our model, we do not consder the ntlzton cost for the consumer to jon the progrm, such s the purchse of 3G moble devces n the 3G wreless servce scenros for two resons. Frst, when we consder the long-run reltonshp between the suppler nd consumers, ths knd of one-tme-expense my not be s mportnt s the monthly usge fee nd the subscrpton fee. Further, ths one tme fee does not ffect the optmzton problem, nd t cn be bsorbed by U (, ). Note lso tht there s no prmeter n ths model 8

9 tht ndctes the prcng scheme dopted by the nformton servce provder. Rther thn usng ddtonl prmeter to ndcte the prcng mechnsm, the prcng scheme chosen ctully s reflected by the vlues of,, nd. For exmple, when nd re both zero nd s postve, t s the pure flt rte prcng; when nd re postve nd s zero, t s the pure usge-bsed prcng; nd when,, nd re ll postve, t s the two-prt trff prcng. Addtonlly, n ths pper, we mjorly consder the smple nd most commonly dopted usgebsed nd two-prt trff prcng n whch the unt prce of the servce s constnt nd doesn t chnge wth the consumer s consumpton level. For exmple, lmost ll, f not ll, resdentl long dstnce voce communcton servce (wth or wthout monthly fee) nd wreless dt trnsmsson servce hve constnt unt prce. Gven,, nd, consumer wll decde f she wnts to jon the progrm. If she decdes not to jon by choosng Z = 0, constrnt () nd (3) wll enforce her consumpton level nd to be zero, nd her totl utlty nd cost re both zero. On the other hnd, f she decdes to jon the progrm by choosng =, she then hs to decde her optml consumpton level nd Z, whch cnnot exceed her upper bounds nd, s enforced by constrnt () nd (3). Also note tht the consumpton level nd here could be the consumpton tme, such s n the voce communcton servce, the trffc volume, such s n the dt trnsmsson servce, or number of uses/ccesses, such s n the pplcton servces, or number of messges sent n SMS/MMS servces... The Suppler s Optmzton roblem Gven the optmzton problem fced by the consumers, the nformton servce provder 9

10 wll decde wht prcng scheme to dopt so s to mxmze ts totl proft. We ssume tht mrgnl cost,.e., mrgnl producton cost for provder one more unt of the servce to the customer, nd montorng cost,.e., mrgnl dmnstrton cost or montorng cost for one unt of the servce n usge-bsed prcng, 3 re both neglgble,.e., zero. We ll dscuss ths ssumpton n Secton 4. Gven rmeters: * = (,, ) : consumer 's consumpton level of the servce n pek hours * = (,, ) : consumer 's consumpton level of the servce n non-pek hours Z* = Z(,, ) : consumer 's decson vrble regrdng prtcpton U(, ) : the utlty functon of consumer t the consumpton level of n pek hours nd n non-pek hours : consumer 's mxmum consumpton level of the servce n pek hours : consumer 's mxmum consumpton level of the servce n non-pek hours Decson Vrbles: : the subscrpton fee for the consumer to jon the progrm : the unt prce of the servce set by the provder n pek hours : the unt prce of the servce set by the provder n non-pek hours The Suppler s Optmzton roblem: Mx,, ( * + * + Z* ) (6) where ( *, *, * = rgmx U Z ( ) ), Z s.t. Z Z ( ) U, Z 0 3 Snce there s no need to montor customer usge level n flt-fee prcng, we ssume tht flt-fee prcng does not ncur montorng cost n the nlyses throughout. 0

11 Z = 0or The objectve functon (6) s to mxmze the totl proft gven the optmzton problems fced by the consumers. Note tht we do not consder the ntlzton fxed cost of provdng the servce to ech consumer s t s not s mportnt f we consder the long-run reltonshp between the suppler nd consumers. In ddton, we ssume the servce provder hs enough cpcty, so tht the mrgnl cost of provdng the servce s zero. 4 Bsed on ths model, we cn fnd the most proftble prcng scheme nd prce(s) to chrge the consumers gven the consumers t fces. 3. Anlyss 3.. The Bse Cse: Homogeneous Consumers As the frst cse, we consder homogeneous consumers n the mrket wth the sme utlty functon nd the sme upper bounds nd on the consumpton level n pek hours nd nonpek hours, respectvely. For nlytcl convenence, we dopt the frequently used Cobb- ( ) Dougls utlty functon, U, = log + blog, wth one mnor modfcton. 5 U(, ) = log( + ) + blog( + ) (7) Wth ths modfcton, when the consumpton level s zero, consumers wll get zero utlty rther thn negtve nfnte utlty. Note tht ths utlty functon s ncresng nd strctly concve n consumpton level nd tht nd re substtutes n tht one could substtute ech 4 Gven ny cpcty, we cn ssume tht the mrgnl cost wthn cpcty s zero, frm only fces lrge mrgnl cost when t needs to ncrese cpcty, but ths s ctully nother nvestment decson tht needs to be mde by frm, rther thn mrgnl cost of the servce, becuse wth new lrger cpcty, mrgnl cost goes to zero gn. 5 Log denotes nturl log here.

12 other to get the sme utlty. We dopt ths specfc utlty functon for two resons: t not only gretly smplfes our devtons but lso llow us to explore how the homogenety (n ths secton) nd heterogenety (n Secton 3.) of consumer utlty functons (wth dmshng mrgnl untlty property) ffect frm s choce of prcng structure. Workng on the generl form of utlty functon U (, ) would hve mde our nlyss much less trctble nd trnsprent wthout ny pprent promse for new nsghts. Wth ths specfc utlty functon, ech consumer wll then fce the followng optmzton problem: Consumers Optmzton roblem: Mx Z,, s.t. ( ) log + + blog( + ) Z (8) Z (9) Z (0) ( ) log + + blog( + ) Z 0 () Z = 0or () Gven consumers optmzton problem bove, the nformton servce provder tres to solve the followng optmzton problem: The Suppler s Optmzton roblem: Mx,, ( * + * + Z* ) (3) where ( * = rgmx, *, *) Z ( ) log + + blog( + ) Z s.t. Z

13 Z ( ) log + + blog( + ) Z 0 Z = 0or roposton When ll consumers n the mrket re homogeneous nd hve utlty functon gven by (7), the pure flt rte prcng nd the two-prt trff prcng yeld the sme proft, whch s strctly hgher thn the pure usge-bsed prcng. roof: We frst mke the followng observtons: Frst, snce ll consumers re ssumed homogeneous, ll consumers wll mke the sme jon-or-not decson nd the servce provder ether serve ll of them or serve none of them. In order to mxmze the proft, the servce provder wll mke sure tht ll consumers wnt to jon the progrm. Second, snce the mjor prcng mechnsms we re studyng n ths pper re pure flt rte, pure usge-bsed, nd the two-prt trff prcng, we cn do the nlyss seprtely nd see wht s the best proft the servce provder cn get by ech prcng pln. ) If the servce provder uses the pure flt rte prcng by settng = 0, = 0, nd > 0: It s cler tht gven ths prcng pln, the consumers wll fully utlze the servce by choosng the consumpton level = nd = wth the mxmum utlty the consumers cn get log( +) + b log( +). It s then obvous tht the mxmum flt rte the servce provder cn chrge s log( +) + b log( +), wth mxmum proft : [ log( +) + b log( +)]. ) If the servce provder uses the pure usge-bsed prcng by settng > 0, > 0, nd = 0: 3

14 Tkng frst-order condtons for optmlty of consumer s optmzton problem yeld: = * + b = * + => * = b => * = Supplers Optmzton roblem becomes: Mx ( * + *) = Mx ( - + b - ) It s cler tht to mxmze the equton bove, the suppler wll hve to mnmze nd. From FOC bove, we know tht s nd decrese, * nd * wll ncrese. But snce nd re bounded, * nd * wll eventully become nd. In other words, the best nd wll be = + nd b =, wth mxmum proft: ( b - b + ) = [( - + ) + b( - + )]. 3) If the servce provder uses the two-prt trff prcng by settng > 0, > 0, nd > 0: Agn, the frst-order condtons for optmlty of consumer s optmzton problem re: = * + b = * + => * = b => * = Supplers Optmzton roblem becomes: Mx ( * + * + ) = Mx ( - + b - + ) Lkewse, t s cler tht to mxmze the equton bove, the suppler wll hve to mnmze nd. From FOC bove, we know tht s nd decrese, * nd * wll 4

15 ncrese. But snce nd re bounded, * nd * wll eventully become nd. In other words, the best nd wll be = + nd b =. The mxmum subscrpton fee + the suppler cn chrge s then the dfference between the mxmum utlty the consumers cn get, log( +) + b log( +), nd the pyment for ther usge, ( b - b + ). Therefore, the mxmum proft chevble by the servce provder s [ log( +) + b log( +)], the sme s n the cse when the servce provder dopts the flt rte prcng mechnsm. Note tht snce log( +) > ( - + ) nd log( +)> ( - hve log( +) + b log( +)> [( - + ) + b( )] for ll >, 0, we )]. Tht s, the pure flt rte prcng nd the two-prt trff prcng re strctly better thn the pure usge-bsed prcng from the servce provder s proft mxmzton pont of vew. QED. 3.. Heterogeneous Customers In prevous nlyss, we hve shown tht flt fee nd two-prt trff s more proftble thn the pure usge-bsed prcng. However, the ssumpton of homogeneous consumers my be somewht restrctve, so we relx ths ssumpton by consderng dfferent types of heterogeneous customers. Followng Jn et l. (999), we exmne two-sets of customer segmentton: hgh-end nd low-end n terms of wllngness-to-py (Secton 3..), nd hevy nd lght n terms of level of usge (Secton 3..). We further ssume tht t s the frm s nterest to serve both segments n ech cse; otherwse, the problem s reduced to tht 5

16 consdered n Secton 3.. We further ssume tht the nformton servce provder cnnot dscrmnte between these two consumer segments. Ths ssumpton s resonble snce t s usully hrd for the servce provder to tell whch segment the consumers belong to. Note tht f the nformton servce provder cn dscrmnte these two types of consumers, the problem gn becomes tht consdered n Secton 3., nd the nformton servce provder cn smply offer dfferent flt rte prcng or two-prt trff prcng to dfferent consumer segments. 3.. Heterogeneous customers: the hgh-end customers nd the low-end customers For smplcty, we wll cll the hgh-end busness consumers, nd the low-end personl consumers. We suppose there re m busness consumers (=) nd n personl consumers (=). To study how heterogeneous wllngness to py ffects frm s prcng scheme, we ssume ech consumer n both segments hs the sme upper bounds nd n pek hours nd non-pek hours, nd, b > b. > Consumers Optmzton roblem: Mx Z,, ( ) log + + blog( + ) Z (4) s.t. Z (5) Z (6) ( ) log + + b log( + ) Z 0 (7) Z = 0or (8) The Suppler s Optmzton roblem: 6

17 Mx,, m( * + * + Z *) + n( * + * + Z *) (9) where ( *, *, * = r Z gmx ( ) ) log + + blog( + ) Z s.t. Z Z ( ) log + + b log( + ) Z 0 Z = 0or m+ n m+ n Lemm : when < nd b < b, 6 f the servce provder uses the pure flt rte, m m the prce chrged wll be ( ) ( be: ( m+ n) log ( + ) + blog ( ) roof : log + + b log + ), nd the mxmum proft chevble wll +. It s cler tht gven = 0, = 0, nd > 0, f consumer chooses to jon the progrm, she wll fully utlze the servce by choosng the consumpton level =, = or =, =. Gven ths, t s obvous tht the servce provder cn chrge ech busness consumer no more thn log ( ) lo ( ( ) ( + ) + + b g + ), nd ech personl consumer no more thn m+ n m+ n log + + b log. It cn be esly shown tht f < nd b < b, the m m 6 These condtons correspond to the cse tht t s more proftble for the frm to serve both segments. 7

18 servce provder wll chrge ( ) ( log + + b log + ) nd serve both busness nd personl consumers wth the mxmum proft chevble ( + ) ( + ) + ( m n log b log +. QED. ) Lemm : If the servce provder uses the pure usge-bsed prcng, when m > n, 7 the optml prce n the pek hours s = ; when m > n, the optml non-pek hour prce s + b = +. The mxmum proft s: b b m( + ) + n( + b ) ; otherwse, the optml prces re gven by = + nd b = + wth proft : b ( m+ n) ( + ). + + roof : When > 0, > 0, nd = 0, the frst-order condtons for optmlty of busness/personl consumer optmzton problem yeld: = * = * + (0) b b = * = * + () = * = * + () b b = * = * + (3) The Suppler s Optmzton roblem becomes: 7 If we normlze to be, ths condton mens m > n. 8

19 Mx m ( * + *) + n ( * + *) = Mx m ( + b ) + n ( + b ) To mxmze the equton bove, the suppler wll hve to mnmze nd. From (0)- (3), we know tht s nd decrese, *, *, * nd * wll ncrese. But snce,, nd re bounded (constrnts (5) nd (6)), *, *, * nd * cnnot exceed nd respectvely, nd ths suggests tht s prce goes down further, no ncrese n demnd cn be expected. To derve the optml prces, we consder the pek-hour problem frst. The pek-hour demnd curves of the busness nd personl consumers (constrnts (0) nd ()) re shown n Fgure ( nd D ). D + + D D * Fgure. The pek-hour demnd curves of the busness nd personl consumers The suppler s optmzton problem s: Mx m ( *) + n ( *) = Mx m ( ) + n ( ). To mxmze ths equton, the suppler wll hve to mnmze nd therefore the best prce cnnot be lrger thn. On the other hnd, f the suppler sets the prce below +, the proft s not optml snce * nd * cnnot exceed + nd user demnd won t 9

20 ncrese s the prce decreses. Hence, the best prce must be somewhere between + nd. When the prce s n ths ntervl, the demnd of the busness consumer s fxed t + whle the demnd of the personl consumer keeps on ncresng s the prce goes down. Thus, we hve: Mx m ( *) + n ( *) = Mx m ( ) + n( ) = Mx n + ( m n ). When m > n, the best prce n ths ntervl s therefore +, otherwse, = +. Smlr nlyss cn be done on the non-pek-hour problem nd we cn get the best b prce = + when m n, or b = otherwse. QED. + > m+ n Lemm 3: If the servce provder uses the two-prt trff prcng, when < nd m b m+ n b, optml m < cn be set nywhere between 0 nd, cn be set nywhere + between 0 nd b + log +, nd = log( + ) b ( + ) wth the mxmum proft chevble: ( m+ n)[ log( + ) + blog( + )]. roof : When > 0, > 0, nd > 0, the frst-order condtons for the busness/personl consumer optmzton problem yeld (0)-(3). Lkewse, we use the dvde-nd-conquer technque to do the nlyss. As before, we consder only the pek-hour problem here nd the jont problem wth non-pek-hour consderton cn be solved n smlr wy. Equtons (0) nd () re the pek-hour demnd 0

21 curves of the busness nd personl consumers ( nd D n Fgure..). D + + D D * Fgure.. The best subscrpton fee s equl to the consumer surplus of the personl consumers Frst, we mke the followng observton: no mtter wht usge prce the suppler sets for the servce, the best subscrpton fee t cn chrge the consumers s the consumer surplus of the personl consumers (the trngle re under D nd bove ). 8 Any subscrpton fee more thn ths wll let the suppler lose ll of the personl sonsumers. Note tht gven these demnd functons, f the suppler set the usge prce +, the suppler s optmzton problem wll be: = Mx m ( *) n ( *) ( m n ) Mx m ( ) n ( ) ( m n) d = m + m+ n Mx ( ) ( ) log( ) To mxmze ths equton the suppler wll hve to mnmze nd therefore the best prce 8 Due to our ssumpton tht t s more proftble for the frm to serve both mrket segments, nd the condtons for m+ n m+ n ths to be true re: < nd b < b. m m

22 n ths ntervl s + wth the mxmum proft chevble m + m+ n +). ( ) ( ) log ( Second, f the suppler set the usge prce, the suppler s optmzton + + problem wll be: Mx m ( *) ( *) ( ) Mx ( ) ( ) ( ) + n + m+ n= m + n + m+ n d 0 + = Mx m ( + ) m + ( m+ n ) log( ) The best prce n ths ntervl s + wth the mxmum proft chevble ( m+ n) log( + ). Note tht ths proft s lrger thn m ( ) + ( m+ n ) log ( +) (the proft we cn get from nother boundry pont + ). Thrd, f the suppler set the usge prce 0, the suppler s optmzton problem + wll be: Mx m ( *) n ( *) ( m n ) Mx m ( ) n ( ) ( m n) d = Mx ( m+ n) log( + ) = It s cler tht to mxmze ths equton the suppler cn set the prce nywhere between 0

23 nd + wth the mxmum proft chevble ( m+ n) log( + ). Gven the bove nlyss, we know tht the suppler cn set the optml prce nywhere between 0 nd, nd nywhere between 0 nd + b, nd subscrpton fee equl to + the consumer surplus of the personl consumers, wth the mxmum proft chevble ( m+ n)[ log( + ) + b log( + )]. QED. roposton When there re two types of consumers chrcterzed by heterogeneous wllngness to py n the mrket, the flt-fee prcng nd the two-prt trff prcng yeld the sme proft, whch s hgher thn the pure usge-bsed prcng. roof : Drcetly from Lemm -3, we lredy get the mxmum profts chevble when the servce provder dopts ech of the prcng mechnsm: pure flt rte, pure usge-bsed nd the two-prt trff prcng. It s not hrd to show tht for ll 0 nd 0, m[ log( +) + b log( +)] + n[ log( +) + b log( +)] > m( - + b + b - + ) + n( b - b ). Therefore we cn sy the pure flt rte prcng nd the two-prt trff prcng re strctly + better thn the pure usge-bsed prcng from the servce provder s proft mxmzton pont of vew. QED. Note tht the conclusons from ths subsecton s exctly the sme s tht derved n 3

24 roposton, tht s, the flt-fee prcng nd the two-prt trff lwys yeld the sme proft, nd tht the usge-bsed prcng s strctly domnted. Whle these results re estblshed under one or two segments of customers, they could be generlzed to contnuous type of customers (s shown s roposton 5 n Appendx). 3.. Heterogeneous customers: the hgh-demnd customers nd the low-demnd customers In ths subsecton, we consder how heterogeneous mxmum consumpton level mght ffect frm s prcng choce. Agn, we ssume two types of customers, the hgh-demnd customers (type ) wth mxmum consumpton level t nd nd the low-demnd customers (type ) wth mxmum consumpton level t nd, where > nd >. As before, there re m type customers nd n type customers wth = = nd b = b =. b roposton 3 When there re two types of consumers chrcterzed by heterogeneous mxmum consumpton levels, the two-prt trff lwys domntes the flt-fee prcng nd the usge-bsed prcng. roof Sketch: The optml prce(s) nd mxmum proft under ech scheme re chrcterzed n the followng: Under the flt fee scheme : = log ( + ) + log ( ) b +, wth mxmum proft: ( ) ( ) ( m+ n)[ log + + blog + ]. Under the pure usge-bsed prcng: 4

25 b b when n m, = nd =, wth proft=( m + n)( + ) b b b when n < m, = nd =, wth proft= m( + ) + n( + ) Under the two-prt trff prcng: = +, b = + nd the subscrpton fee: log ( ) blog ( ) = b ( + ). Therefore, the mxmum proft chevble by the servce provder + + ( + b ) + ( m+ n)( log( + ) + blog( + )), whch s greter thn tht cn be + + s m cheved n the flt fee prcng nd the usge-bsed prcng. QED. 4. Model Extenson Mrgnl Cost nd Montorng Cost In prevous nlyses, we ssume tht mrgnl cost (denoted by c) nd montorng cost (denoted by t) re both zero. In ths secton, we relx ths ssumpton nd exmne how mrgnl cost nd montorng cost ffect optml prcng scheme. In generl, postve mrgnl cost s expected to mke the flt fee prcng less ttrctve nd fvor two prt trff prcng nd usge bsed prcng, whle postve montorng cost tend to mke the two-prt trff prcng or usgebsed prcng less desrble thn the flt fee prcng snce there s no need to ncur montorng expenses n the flt-fee prcng. The optml prcng scheme thus depends on the trdeoff between these two costs. However, under the cse where customers re homogeneous or re chrcterzed by heterogeneous wllngness to py, the optml scheme becomes nsenstve to these two costs when c s below certn vlue, n prtculr, holdng c constnt, reducng 5

26 montorng cost does not mke the two-prt trff prcng better choce thn the flt fee prcng. roposton 4 Under the cse where customers re homogeneous or re chrcterzed by b heterogeneous wllngness to py, when c mn{, }, the flt fee prcng ( + ) ( + ) domntes the two-prt trff prcng nd the usge bsed prcng. roof sketch: The proof here s bsed on homogeneous customers for smplcty. However, the sme nlyss cn be done on heterogeneous customers cse, the results remn the sme, but only the condton on c chnges. As before, we nlyze the pek hour problem, nd the jont problem wth non-pek hour consderton cn be solved n the sme mnner. Wth postve mrgnl cost nd montorng cost, the proft from ech customer (denoted by π ) for the frm under ech prcng scheme s gven by: Under the flt-fee prcng: π= c, nd to mxmze the proft, wll be set t log( + ), wth proft chevble: log( + ) c. Under the usge-bsed prcng: π = ( c t) ( ), where ( ) s the demnd functon chrcterzed by ( ) = from the frst order condton n customer s optmzton problem. The proft-mxmzton per-use prce cn be shown to be: π * = mx{, ( c+ t)} from the frst order condton = 0, nd the mxmum proft cn + be shown to be: ( * c t) ( *) = ( *) * ( *) ( c+ t). Under the two-prt trff prcng: π = ( c t) ( ) +, gn ( ) s the demnd 6

27 functon chrcterzed by ( ) = from the frst order condton n customer s π optmzton problem. Accordng to the frst order condton = 0, we cn derve the proftmxmzton per-use prce: * = mx{, ( c+ t)}, nd, the subscrpton fee, wll be set + t: log( ( *) + ) ( *) *,.e., to fully extrct customer surplus. The mxmum proft s thus equl to: log( ( *) + ) ( *) ( c+ t). We cn show tht when c ( + ), flt fee prcng lwys domntes the two-prt trff prcng nd the usge bsed prcng no mtter how lrge or how smll the montorng cost s (s long s t s postve), nd when montorng cost, t, goes down to zero, the two-prt trff prcng wll derve the sme proft s the flt fee prcng. Smlrly, we cn show tht n non-pek hours, when b c ( + ), flt fee prcng lwys domntes the two-prt trff prcng nd the usge bsed prcng no mtter how lrge or how smll the montorng cost s (s long s t s postve). QED. Ths result s very nterestng becuse t suggests tht s mrgnl cost goes down (but not necessrly zero), flt fee prcng becomes the optml scheme even though tht montorng cost my goes down t lrger scle. A drect mplcton of ths roposton s tht s mrgnl costs nd montorng costs re both lowered wth the dvnce of nformton technology or wth evoluton of electronc mrkets, flt-fee prcng wll become more ttrctve for nformton servce provders. However, when montorng cost becomes neglgble, two-prt trff prcng wll become s ttrctve. 7

28 5. Dscusson nd Conclusons The mn objectve of ths pper s to offer gudelne for nformton servce provders bout wht prcng scheme s most proftble nd wht prce(s) t should chrge. Ths ssue bout how to prce nformton servces hs become ncresngly more mportnt s mrgnl producton nd montorng costs for nformton servces re beng reduced wth the dvnce of modern nformton technology, nd more nd more nformton servces re beng offered wth the evoluton of the Internet. Mny reserchers hve suggested tht zero mrgnl cost wll fvor the flt-fee prcng scheme. Overll, our nlyss shows tht when customers re homogeneous or hve heterogeneous mrgnl wllngness to py, f mrgnl cost s below certn vlue, flt-fee prcng s ndeed more ttrctve. However, s processng power keeps ncresng nd montorng costs become neglgble, the two-prt trff prcng becomes s ttrctve. Our nlyss lso shows tht under zero mrgnl cost nd zero montorng cost, when customers re frly homogeneous, the usge-bsed prcng s strctly domnted by the flt fee nd two-prt trff prcng schemes, wth the ltter two lwys chevng the sme proft level. The sme results sustn when customers hve heterogeneous mrgnl wllngness to py (whch corresponds to dfferent downwrd slopng demnd curves). However, when customers re chrcterzed by heterogeneous mxmum consumpton levels, the two-prt trff prcng domntes both the flt-fee prcng nd the usge-bsed prcng. Appendx: roposton 5 When consumers re chrcterzed by heterogeneous wllngness to py n the 8

29 mrket, the flt-fee prcng nd the two-prt trff prcng lwys yeld the sme proft, whch s hgher thn the pure usge-bsed prcng. roof sketch: Assume customer hs type: nd b, whch follow the unform dstrbuton,.e.,, b ~ U [0,]. As before, we solve the pek hour problem, nd the jont problem wth non-pek hour consderton cn be solved wth the smlr procedure. Under the flt-fee prcng: Gven ny flt fee,, chrged by the frm, we cn determne the mrgnl customer, 0, who wll sgn up by 0 log( + ) =, wth proft = ( ). To mxmze the proft, the frm wll 0 set = log( + ), wth 0 =,.e., customers wth type hgher thn wll jon the servce. The mxmum proft under ths scheme s thus ( 0 ) = ( ) log( + ) = log( + ). 4 Under the pure usge-bsed prcng: As before, the demnd functon s chrcterzed by: =, tht s, when, we hve, nd when <, we hve + + demnd functon, customer wth type hgher thn ( + ) wll consume. =. And gven ths The frm wll choose tht mxmzes ts proft: ( + ) d + ( ( + )). 0 Solvng ths optmzton problem, we get =, wth proft: ( + ) ( + ), whch cn be shown to be lower thn log( + ), the proft n the flt-fee prcng. 4 9

30 Under the two-prt trff prcng: The frm wll choose nd the subscrpton fee,, to mxmze ts proft. Note tht gven ny nd, the mrgnl customer,, who wll sgn up the servce s determned by: 0 0 0log( ( 0) + ) ( 0) = 0, whch cn be smplfed to = 0log( ) ( 0 ). The proft gven nd s : 0 ( + ) d + ( (+ )) + ( ). To mxmze the 0 proft, the frm wll set = nd ( + ) = log( + ) +, wth mxmum proft: lo g( ) 4 + QED., whch s exctly the sme s wht cn be cheved n the flt-fee prcng scheme. References. Bkos,. nd E. Brynjolfsson. Bundlng Informton Goods: rcng, rofts nd Effcency. Mngement Scence 45, Cho, S., Dle O., nd A. B. Whnston. The Economcs of Electronc Commerce. Mcmlln Techncl ublshng, Indnpols, Indn, Chung, C. I. nd Srbu, M. A. Optml Bundlng Strtegy for Dgtl Informton Goods: Network Delvery of Artcles nd Subscrptons. In Informton Economcs nd olcy, Esseger, S., S. Gupt nd Z. J. Zhng. rcng Access Servces. Mrketng Scence (), Fshburn,.C., A.M. Odlyzko nd R.C. Sders. Fxed fee versus unt prcng for nformton goods: competton, equlbr, nd prce wrs. roceedngs of the Conference on Internet 30

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