Cellular Service Demand: Biased Beliefs, Learning, and Bill Shock. Online Appendix

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1 Cellular Servce Demand: Based Belefs, Learnng, and Bll Shock Mchael D. Grubb and Matthew Osborne Onlne Appendx May 2014 A Data Detals A.1 Unversy Prce Data Table 6: Popular Plan Prce Menu Plan 0 $14.99) Plan 1 $34.99) Plan 2 $44.99) Plan 3 $54.99) Date Q p OP Net Q p OP Net Q p OP Net Q p OP Net 8/02-10/ free not free not free not 10/02-12/ free free free not free not free not 12/02-1/ free free free not free not free not 1/03-2/ free free free not free not free not 2/03-3/ free free free not free not free not 3/03-9/ free free free not free not free not 9/03-1/ not free free not free not free not 1/04-4/ not free free not free free free not 4/04-5/ not free free not free free free not 5/04-7/ not free free not free free free not Entres descrbe the callng allowance Q), the overage rate p), whether off-peak callng s free or not OP), and whether n-network callng s free or not Net). Bold entres reflect prce changes that apply to new plan subscrbers. The Bold alcs entry reflects the one prce change whch also appled to exstng plan subscrbers. Some terms remaned constant: Plan 0 always offered Q = 0, p = 11, and free n-network. Plans 1-3 always offered free off-peak. Prces of the four popular plans are descrbed for all dates n Table 6. Ths prce seres was nferred from bllng data rather than drectly observed. For each plan and each date, we nfer the total number of ncluded free mnutes by observng the number of mnutes used pror to an overage n the call-level data. Ths calculaton s complcated by the fact that some plans offered free nnetwork calls, and our call-level data does not dentfy whether an ncomng call was n-network.) We were able to relably nfer ths prcng nformaton for popular plans from August 2002 to July We exclude other dates and the 11 percent of blls from unpopular plans natonal plans, free-long-dstance plans, and expensve local plans), groupng them wh the outsde opton n our 1

2 structural model. In fact, we treat swchng to an unpopular plan the same as qutng servce, hence we also drop all remanng blls once a customer swches to an unpopular plan, even f they eventually swch back to a popular plan. As stated n Secton II.A, an addonal 7 percent of ndvduals are excluded from our structural estmaton due to data problems. In partcular, we exclude ndvduals wh substantally negatve blls, ndcatng eher bllng errors or ex post renegotated refunds that are outsde our model. Also excluded are ndvduals who have nfeasble choces recorded plans outsde the choce set or negatve n-network callng) and 8 ndvduals for whom we could not fnd startng ponts nal parameter values from whch to begn maxmzng the lkelhood) wh posve lkelhood. A.2 Publc Prce Data Table 7 shows a subset of data obtaned from EconOne: publcly avalable local callng plans for October 2003 n the same geographc market as the unversy. Sprnt dd not offer local plans.) Table 7: Publcly Avalable Local Callng Plans - October 2003 AT&T Cngular Verzon Plan M Q p M Q p M Q p A.3 Addonal Evdence of Inattenton If consumers are attentve to the remanng balance of ncluded mnutes durng the bllng cycle they should use ths nformaton to contnually update ther belefs about the lkelhood of an overage and a hgh margnal prce ex post. Followng an optmal dynamc program, an attentve consumer should all else equal) reduce her usage later n the month followng unexpectedly hgh usage earler n the month. Ths predcton should be true for any consumers who are nally uncertan whether they wll have an overage n the current month. For these consumers, the hgh usage shock early n the month ncreases the lkelhood of an overage, thereby ncreasng ther expected ex post margnal prce, and causng them to be more selectve about calls. If callng opportunes arrved ndependently throughout the month, ths strategc behavor by the consumer would lead to negatve correlaton between early and late usage whn a bllng perod. However, lookng for 2

3 negatve correlaton n usage whn the bllng perod s a poor test for ths dynamc behavor because s lkely to be overwhelmed by posve seral correlaton n taste shocks. To test for dynamc behavor by consumers whn the bllng perod, we use our data set of ndvdual calls to construct both fortnghtly and weekly measures of peak usage. A smple regresson of usage on ndvdual fxed effects and lagged usage shows strong posve seral correlaton. However, we take advantage of the followng dfference: Posve seral correlaton between taste shocks n perods t and t 1) should be ndependent of whether perods t and t 1) are n the same or adjacent bllng cycles. However, followng unexpectedly hgh usage n perod t 1), consumers should cut back usage more n perod t f the two perods are n the same bllng cycle. Thus by ncludng an nteracton effect between lagged usage and an ndcator for the lag beng n the same bllng cycle as the current perod, we can separate strategc behavor whn the month from seral correlaton n taste shocks. Table 8 shows a regresson of log usage on lagged usage and the nteracton between lagged usage and an ndcator equal to 1 f perod t 1) s n the same bllng cycle as perod t. We also nclude tme and ndvdual fxed effects and correct for bas nduced by ncludng both ndvdual fxed effects and lags of the dependent varable n a wde but short panel Roodman, 2009). Reported analyss s for plan 1, the most popular three-part tarff. As expected, posve seral correlaton n demand shocks leads to a posve and sgnfcant coeffcent on lagged usage n the full sample column 1) and most subsamples columns 2-6). If consumers adjust ther behavor dynamcally whn the bllng cycle n response to usage shocks, then we expect the nteracton effect to be negatve. In the full sample column 1) the nteracton effect has a posve pont estmate, but s not sgnfcantly dfferent from zero. Ths result suggests that consumers are not attentve to past usage durng the course of the month. Table 8: Dynamc Usage Pattern at Fortnghtly Level. 1) 2) 3) 4) 5) 6) Overage Percentage 0-100% % 30-70% 71-99% 100% lnq t 1 ) 0.649*** 0.609*** 0.522*** 0.514*** *** ) ) ) ) 1.060) ) SameBll*lnq t 1 ) ) ) ) ) 1.174) 4.745) Observatons Number of ndvduals Dependent varable lnq t). Standard errors n parentheses. Tme and ndvdual fxed effects. Key: *** p<0.01, ** p<0.05, * p<0.1 3

4 Consumers who eher never have an overage 43 percent of plan 1 subscrbers) or always have an overage 3 percent of plan 1 subscrbers) should be relatvely certan what ther ex post margnal prce wll be, and need not adjust callng behavor durng the month. For nstance, consumers who always make overages may only make calls worth more than the overage rate throughout the month. For such consumers we would expect to fnd no nteracton effect, and ths may drve the result when all consumers are pooled together as n our frst specfcaton. As a result, we dvde consumers nto groups by the fracton of tmes whn ther tenure that they have overages. We repeat our frst specfcaton for dfferent overage-rsk groups n Columns 2-6 of Table 8. The nteracton effect s ndstngushable from zero n all overage rsk groups. Moreover, n unreported analyss, more flexble specfcatons that nclude nonlnear terms 1 and a smlar analyss at the weekly rather than fortnghtly level all estmate an nteracton effect ndstngushable from zero. There s smply no evdence that we can fnd that consumers strategcally cut back usage at the end of the month followng unexpectedly hgh nal usage. We conclude that consumers are nattentve to ther remanng balance of ncluded mnutes durng the bllng cycle. A.4 Plan Swchng Calculatons We make two calculatons for each swch from an exstng plan j to an alternate plan j that cannot be explaned by a prce cut for plan j. Frst, we calculate how much the customer would have saved had they sgned up for the new plan j nally, holdng ther usage from the orgnal plan j fxed. By ths calculaton, average savngs are $10.87 to $15.24 per month and 60 to 61 percent of swches save consumers money. We calculate bounds because we cannot always dstngush n-network and out-of-network calls. Average savngs are statstcally greater than zero at the 99 percent level. The percent rates of swchng n the rght drecton are statstcally greater than 50 percent at the 95 percent level. Ths calculaton s based on 99 of the 136 swches whch cannot be explaned by prce decreases. The remanng 37 swches occur so soon after the customer jons that there s no usage data pror to the swch that s not from a pro-rated bll. Second, we calculate how much money the customer would have lost had they remaned on exstng plan j rather than swchng to the new plan j, now holdng usage from plan j fxed. By ths calculaton, average savngs are $13.80 to $24.56 per month, and 61 to 69 percent of swches save money. Ths calculaton s based on 132 of the 137 swches whch can not be explaned by 1 Average q t wll vary wh expected margnal prce, whch s proportonal to the probably of an overage. The probably of an overage n a bllng perod whch ncludes perods t and t 1) ncreases nonlnearly n q t 1. In one specfcaton, we frst f a prob on the lkelhood of an overage as a functon of the frst fortnghts usage, and then used the estmated coeffcents to generate overage probably estmates for all fortnghts. We then ncluded these lagged) values as explanatory varables. In an alternatve specfcaton we added polynomal terms of lagged q t 1. 4

5 prce decreases. The calculaton cannot be made for the remanng 5 swches snce there s no usage data followng the swch that s not from a pro-rated bll. Fgures are sgnfcant at the 95 percent or 99 percent confdence level. As explaned n Secton II.B, the frst and second calculatons descrbed above provde lower and upper bounds, respectvely, on the benefs of swchng plans. We therefore conclude that consumers expected benef from swchng s between $10.87 and $24.56 per month and 60 to 69 percent of swches save money. B Model Detals A Model Gude: Tables 9-12 provde a gude to model parameters. B.1 Dervaton of optmal callng threshold Defne qp, θ k ) ) ) arg max q V q, θ k pq to be a consumer s demand for category-k calls gven a constant margnal prce p. Ths s the quanty of category-k calls valued above p.) A consumer s nverse demand for category-k calls s V q q k, θk ) = 1 q k ) θk /β and thus: qp, θ k ) = θ k 1 βp) = θ k ˆqp). 12) Condonal on tarff choce j wh free off-peak callng, consumer chooses her perod t peak threshold v pk j to maxmze her expected utly condonal on her perod t nformaton I : v pk j [ = arg max E V v qv, θ pk ) ) ] ), θpk P j qv, θ pk ) I. Let F be the cumulatve dstrbuton of θ pk condon for the consumer s problem s as perceved by consumer at tme t. The frst-order θ θ ) V q qv, θ pk d θ ), θpk dv qv, θ pk )d F θ pk ) = θ j v ) p j d dv qv, θ pk )d F θ pk ), 13) where θ j v ) s the peak type whch consumes exactly Q j uns: qv, θ j v )) = Q j. Equaton 13) s smlar to Borensten s 2009) frst-order condon. Unlke Borensten 2009), we assume V q qv, θ k ), θ k ) s equal to v by defnon. Notce that substutng q v, θ k ) = θ k 1 βv ) 5

6 Table 9: Model Gude: Payments, Preferences, and Usage Parameter Descrpton Dstrbuton or equaton qj bllable bllable mnutes qj bllable = q pk + OP j q op OP j ndcator for costly off-peak plan j constant P j prce plan j P j q ) = M j + p j max{0, qj bllable Q j } M j monthly fee plan j constant Q j allowance plan j constant p j overage rate plan j constant V value functon V q k, ) θk = 1 β qk q k /θ k )) u j utly functon u j = k {pk,op} V q k, ) θk Pj q ) + η f U j expected utly equaton 5) β prce sensvy constant O outsde good utly constant P C plan consderaton Pr. constant η f log error d log θ k # callng opportunes θ k = max{0, θ k } θk latent taste shock θ = µ + ε v k j callng threshold equaton 4) vj callng threshold vector vj = vpk j, vop j ) ˆqv j k ) Frac. callng opp. value > vk j ˆqv j k ) = 1 βvk j ) q k mnutes of callng usage) q k = θk ˆqv k j ) Subscrpts and superscrpts denote customer, tme t, plan j, frm f, and callng category k {pk, op}. 6

7 Table 10: Model Gude: Tastes & Sgnals Parameter Descrpton Dstrbuton µ pk 1 µ pk µ op s ε pk ε op Pont Estmate µ pk Peak Type Off-Peak Type Sgnal Peak Shock Off-Peak Shock N µ pk 0 µ pk 0 µ op 0, 0 N 0, 0 σ pk µ ) 2 ρ µ,pk σ pk µ σ pk µ ρ µ,op σ op µ σ pk µ ρ µ,pk σ pk µ σ pk µ σ pk µ ) 2 ρ µ σ pk µ σ op µ ρ µ,op σ op µ σ pk µ ρ µ σ pk µ σ op µ σ op µ ) 2 1 ρ s,pk σ pk ε ρ s,pk σ pk ε ρ s,op σ op ε σ pk ε ) 2 ρ ε σ pk ε σ op ε ρ s,op σ op ε ρ ε σ pk ε σ op ε σ op ε ) 2 Parameter Descrpton Equaton δ µ pk σ t overconfdence Table 11: Model Gude: Belefs and Bases updated pont est. Bayes rule, equaton 17) SD[µ pk µ pk ] Bayes rule t > 1) equaton 18) pk SD[µ σ t µ pk θ σ θt µ pk ] [ ] Ẽ θ pk s [ ] SD θ pk s σ θt SD [ θ pk s ] σ t = δσ t equaton 6) equaton 7) σ θt = δσ θt b 1 aggregate mean bas b 1 = µ pk 0 µpk 0 b 2 condonal mean bas b 2 = 1 Covµpk V ar µ pk, µ pk 1 ) 1 ) pk σµ = 1 ρ µ,pk σ pk µ. equaton 8) Operators E and SD denote populaton moments, whle Ẽ and SD are wh respect to ndvdual s belefs. 7

8 Table 12: Model Gude: Near-9pm Callng Parameters Parameter Descrpton Dstrbuton or equaton r k,9 share of k-callng near 9pm r k,9 censored on [0, 1] r k,9 latent shock r k,9 = α k,9 + e k,9 α pk,9 pk-9pm type α pk,9 Nµ pk,9 α, σ pk,9 α ) 2 ) e k,9 pk-9pm shock e k,9 N0, σ k,9 e ) 2 ), d across, t, and k. α op,9 op-9pm type Defned mplcly by equaton 9) Subscrpts and superscrpts denote customer, tme t, and callng category k {pk, op}. from equaton 12) nto V q q k, θk ) yelds v ). Thus equaton 13) reduces to: v = p j θ θ j v ) θ θ d dv qv, θ pk )d F θ pk ) d dv qv, θ pk )d F. θ pk ) Wh multplcatve separably equaton 12)), θ jv ) = Q j /ˆq v ) and d dv qv, θ pk ) = θpk d dv ˆq v ), d so we can factor out and cancel dv ˆq v ). Ths smplfcaton yelds equaton 4). It s apparent by nspecton that equaton 4) has a unque soluton. Equaton 4) may seem counter-ntuve, because the optmal v pk j s greater than the expected margnal prce, p j Prqv pk j, θpk ) > Q j I ). Ths s because the reducton n consumpton from rasng v pk j s proportonal to θpk. Rasng vpk j cuts back on calls valued at vpk j more heavly n hgh demand states when they cost p j and less heavly n low demand states when they cost 0. Fgure 9 plots the optmal callng threshold v pk j as a functon of consumer belefs and the correspondng plan choce. The dstnct regons vsble on the contour plot correspond to the plan choce regons dentfed n Fgure 7. Fgure 9 shows that, on three-part tarffs, v pk j ncreases wh the perceved mean and varance of callng opportunes because both ncrease the upper tal of usage and hence the lkelhood of payng overage fees. Fgure 9 also shows that consumers who choose a three-part tarff use callng thresholds that are small relatve to the overage rates of $0.35 or $0.45 per mnute. Ths s because f the perceved lkelhood of an overage were large the consumer would have chosen a larger plan. B.2 Optmal callng threshold under bll-shock regulaton For smplcy we temporarly drop the peak superscrpt as well as customer, date, and plan subscrpts. Gven bll-shock regulaton, a consumer wll begn the bllng cycle consumng all peak 8

9 σ ~ θ µ~ θ1 Fgure 9: Callng threshold v pk j as a functon of nal belefs { µ θ1, σ θ1 } and the correspondng plan choce shown n Fgure 7) mpled by the model evaluated at fall 2002 prces gven β = 2. calls above some threshold v but rase ths callng threshold to the overage rate p upon recevng a bll-shock alert that the allowance Q has been reached. Let θ be the peak type whch consumes exactly Q uns gven nal callng threshold v : qv, θ ) = Q. If θ < θ then the customer has a total of θ callng opportunes, makes θˆq v ) < Q calls wh value V q v, θ), θ) and never receves a bll-shock alert. If θ > θ then a bll-shock alert s receved and Fgure 10 llustrates demand curves and callng thresholds before and after the alert. Pror to recevng a bll-shock alert, the customer has θ callng opportunes. Of these, fracton ˆq v ) are worth more than v so that pror to the alert the consumer makes θ ˆq v ) = Q calls wh total value V Q, θ ) and declnes θ 1 ˆq v )) callng opportunes. After recevng a bll-shock warnng, the customer has θ a = θ θ addonal callng opportunes. Of these, fracton ˆq p) are worth more than the overage rate p so that after the alert the consumer makes an addonal θ aˆq p) calls wh total value V q p, θ a ), θ a ). $ 1 β p v * Q pre-alert calls worth more than v* * θ a q p) post-alert calls worth more than p q pre-alert calls declned post-alert calls declned Fgure 10: Inverse demand curves and callng thresholds before and after a bll-shock alert. 9

10 Notce that calls wh the hghest value 1/β are made both before and after the alert. Ths s because the demand curves orderng of calls from hgh to low value s not chronologcal. Both low and hgh value calls arrve throughout the month. We now characterze the optmal callng threshold v wh bll-shock alerts. chooses v to maxmze θ U = M + V q v, θ), θ) f θ) dθ + 0 θ The consumer V Q, θ ) + V q p, θ a ), θ a ) pq p, θ a )) f θ) dθ, where θ = Q/ˆq v ) and θ a = θ θ. After cancelng terms recognzng that q v, θ ) = Q, q p, 0) = 0, and V 0, θ) = 0) and substutng v = V q q v, θ), θ) and dθ a /dv = dθ /dv, the frst dervatve s du θ dv = 0 v d dv q v, θ) f θ) dθ + dθ dv θ V θ Q, θ ) V θ q p, θ a ), θ a )) f θ) dθ. The multplcatve demand assumpton mples d dv q v, θ) = θ d dv ˆq v ) and dθ /dv = Q/ˆq 2 v )) d dv ˆq v ). Therefore the d dv ˆq v ) terms cancel and the frst-order condon s θ v θf θ) dθ = 0 Q ˆq 2 v ) θ V θ Q, θ ) V θ q p, θ a ), θ a )) f θ) dθ. Notce that because V θ q, θ) = q 2 / 2βθ 2) and q p, θ) = θˆq p) holds that V θ q p, θ), θ) = ˆq 2 p) /2β s ndependent of θ. Substutng ths expresson n yelds θ v θf θ) dθ = Q 1 2β Rearrangng terms, ths expresson s equvalent to 0 v = Q 2β 1 ) ) ˆq p) 2 ˆq v 1 F θ )). ) whch characterzes v gven θ = Q/ˆq v ) and ˆq v ) = 1 βv. ) ) ˆq p) 2 1 F θ )) ˆq v ) θ 0 θf θ) dθ, 14) B.3 Bayesan Updatng In ths appendx we use operators E and V to denote mean and varance n the populaton or gven objectve probables. We use operators Ẽ and Ṽ to denote mean and varance gven an ndvdual consumers subjectve belefs. 10

11 Updatng from sgnal s At the begnnng of each perod, each consumer receves a sgnal s that s nformatve about the taste nnovaton ε. Gven the jont dstrbuton of ε and s and the restrcton ρ s,op = ρ s,pk ρ ε, condonal on the sgnal s : ε s normally dstrbuted wh mean [ E ε pk ] ) [ ε op s = ρ s,pk σ pk ε s ρ s,pk ρ ε σ op ε s ], and varance V [ ε pk ] ) ε op s = σpk ) ε ) 1 2 ρ 2 s,pk ) ρ ε σ pk ε σ op ε 1 ρ 2 s,pk ) ρ ε σ pk ε σ op ε 1 ρ 2 s,pk σ op ε ) 2 1 ρ 2 s,pk ρ2 ε). However, as consumers underestmate the uncondonal standard devatons of s and ε pk factor δ, consumers perceve the condonal dstrbuton to be normal wh mean [ Ẽ ε pk by a ] ) [ ] ε op s = ρ s,pk σ pk 1 ε s δ ρ s,pkρ ε σ op ε s, 15) and varance Ṽ [ ε pk ] ) ε op s = ) σpk ε ) 1 2 ρ 2 s,pk ) ρ ε σ pk ε σ op ε 1 ρ 2 s,pk ) ρ ε σ pk ε σ op ε 1 ρ 2 s,pk σ op ε ) 2 1 ρ 2 s,pk ρ2 ε), 16) where σ pk ε = δσ pk ε. 2 The precedng belefs about the taste nnovaton ε correspond to the belef that θ s normally dstrbuted wh mean Ẽ θpk θ op s = µpk + ρ s,pkσ pk ε s µ op + ρ s,op σ op ε s, and varance Ṽ θpk θ op s = σ2 t + 1 ρ 2 s,pk ) σpk ε ) 2 1 ρ 2 s,pk )ρ ε σ pk ε σ op ε 1 ρ 2 s,pk )ρ ε σ pk ε σ op ε 1 ρ 2 s,pk ρ2 ε) σ op ε ) 2. 2 As belefs about off-peak tastes are not dentfed by our data, we assume that consumers understand the uncondonal varance of off-peak tastes σ op ε ) 2. Ths assumpton mples that consumers also correctly understand the condonal varance σ op ε s )2 = σ op ε ) 2 1 ρ 2 s,pk ρ2 ε ). However, consumers are overly sensve to the sgnal s when formng expectatons about off-peak tastes. Whle the true condonal expectaton s E[ε op s] = ρ s,pk ρ ε σop ε s, consumers msperceve to be 1 ρ δ s,pk ρ ε σop ε s. An alternate assumpton s that consumers also underestmate the off-peak uncondonal standard devaton σ op ε by δ. In ths alternatve, consumers are equally based about the varance of peak and off-peak tastes but have correct condonal expectatons for both peak and off-peak tastes. Ths alternatve specfcaton yelds smlar results. 11

12 Note that ths expresson reles on our restrcton ρ s,op = ρ ε ρ s,pk.) Ẽ Note that, n the text, we focus on the margnal dstrbuton of θ pk [ ] θ pk s n equaton 6) and σ 2 θt = Ṽ gven σ pk ε = δσ pk ε and σ t = δσ t. and defne both µ pk θ = [ θ pk s ] n equaton 7), where we have factored out δ 2 Learnng from past usage At the end of bllng perod t, consumer learns z = θ pk ρ s,pk σ pk ε s, whch she beleves has dstrbuton Nµ pk, 1 ρ 2 ) s,pk σ pk ε ) 2 ). Defne z = 1 t t τ=1 z τ. Then by Bayes rule DeGroot, 1970), updated tme t + 1 belefs about µ pk are µ pk I,t+1 N µ pk,t+1, σ2 t+1 ) where substutng σ 1 = δσ 1, σ 1 = σ pk µ 1 ρ2 µ,pk, σpk ε = δσ pk ε, and factorng out δ) µ pk,t+1 = 1 1 ρ 2 µ,pk 1 ρ 2 µ,pk µ pk ) 1 ) 1 σ pk µ ) 2 + t z 1 ρ 2 s,pk σ pk ε ) 2 ) 1 σ pk µ ) 2 + t 1 ρ 2 s,pk ) 1 σ pk ε ) 2, 17) σ 2 t+1 = 1 ρ 2 µ,pk ) 1 σ pk µ ) 2 + t 1 ρ s,pk ) 1 σ pk ε ) 2) 1, 18) and σ t+1 = δσ t+1. Note that σ t+1 s always underestmated by a factor δ n each perod, but that µ pk,t+1 approprately because σ pk ε s underestmated by the same factor δ. s updated C Complete Model The complete model dffers from the llustratve verson presented n the man text by accountng for pecewse-lnear demand, nactve accounts, graduaton, and free n-network callng. C.1 Pecewse-lnear Demand We estmate β = 2.7. For any callng threshold above 1/β = $0.37, the value functon defned n equaton 2) leads to zero callng ndependent of θ because the maxmum call value s V q 0, θ) = 1/β. 12

13 Ths predcton s not only unrealstc but also complcates estmaton as mples a $0.37 upper bound on v and hence an upper bound on s for all observatons wh posve usage. As a result, n the model we estmate, we adjust the value functon to ensure that the frst 1 percent of callng opportunes have an average value of at least $0.50. In partcular, we assume that the nverse demand curve for callng mnutes, V q q k, θk ), s: V q q k, θ k ) { 1 = max 1 ) q k β 1 β /θ k, 1 1 )} q k 1 β /θ k, 2 where β 1 = β and β 2 = If we estmate β 2 the estmate does not move from the startng value β 2 = 0.01 and has a large standard error.) Ths nverse demand curve s llustrated n Fgure 11. As before, demand s qv k j, θk ) = θ k ˆqv k j ). Now, however, ˆqvk j ) s { ) )} ˆqv j) k = max 1 β 1 vj k, β 2 1 vj k. $ 1 1 β V q q, θ 0.01 q Fgure 11: Pecewse-Lnear Inverse Demand Curve The correspondng value functon s somewhat more cumbersome to express. It s V ) q, k θ k = V II q k, ) θk V I q k, ) θk + V II q, θ k ) V I q, θ k )), q k q θ k ) q k q θ k ), where V I q k, θk ) = q k 1 θ k 1 1/β 1 1/β 2 1/β. 1 β 1 1 2β q k 1 /θ k ) ), V II q k, ) θk = q k 1 1 2β q k 2 /θ k ) ), and q θ k ) = C.2 Inactve Accounts and Graduaton Consumers have the ably to put ther accounts nto nactve status, durng whch phones are dsabled and fees are zero, and many do so over summer vacatons. To account for these nactve perods we assume that no learnng occurs durng nactve perods and no taste shocks are observed. By lettng tme t ndex the number of actve perods to date, formulas for belefs based on Bayes 13

14 rule n Appendx B.3 reman correct. A substantal fracton of students termnate servce at the end of the sprng quarter. We assume that these termnatons are exogenous because students were requred to transfer servce from the unversy to the frm upon graduaton. C.3 Free n-network callng Callng plans dstngush between n-network and out-of-network calls as well as between peak and off-peak calls. 3 Thus the usage vector has four dmensons rather than two, q = q pk,out, q pk,n, q op,out, q op,n ), where the superscrpt notaton s: 1) pk for peak calls, 2) op for off-peak calls, 3) out for out-of-network calls, and 4) n for n-network calls. Popular prcng plans are the same functon of total bllable mnutes as before, but bllable mnutes now depend on an ndcator for whether the plan charges for n-network calls NET j ): q bllable j = q pk,out + NET j q pk,n + OP j q op,out + NET j OP j q op,n. The peak and off-peak taste shock vector θ follows the same process as n the llustratve model. In addon, we ncorporate the taste shock r = r pk, rop ) [0, 1]2 whch captures the share of peak and off-peak demand that s for n-network callng rather than out-of-network callng. Together, θ and r determne category specfc taste shocks x : x = x pk,out x pk,n x op,out x op,n = 1 r pk )θpk r pk θpk 1 r op )θop r op θop We contnue to assume that calls from dfferent categores are neher substutes nor complements. Consumer s utly from choosng plan j and consumng q n perod t s thus. u j = k V q k, x k ) P j q ) + η j, k {pk-n, pk-out, op-n, op-out}. 3 We matched area codes and exchanges of outgong calls to carrers to determne n-network status usng Telco- Data.us data TelcoData.us, 2005). 14

15 Moreover, we contnue to assume that consumers choose separate callng thresholds for each type of call, v j = v pk,out j, v pk,n, v op,out, v op,n j j j ), and, at the end of the month, realzed usage n category k s q k = xkˆqvk j ). Implcly ths assumes that consumers can dstngush n-network and out-of-network numbers when choosng to make a call. In realy, consumers lkely can t always do so but they lkely can for partes they call n hgh volume. Fnally, customer s perceved expected utly from choosng plan j at date t s U j = E k {pk-n,pk-out,op-n,op-out} V ) q k vj, k x k ), x k P j qv j, x ) ) I + η j. 19) In general, the frst-order condons for threshold choce are analogous to the base model: [ ] ) E x k vj k = p j Pr qj total qtotal j > Q j > Q j E [ ] x k. Gven the structure of the taste shocks, n-network and out-of-network thresholds only dffer when n-network calls are free. There are four classes of tarff to consder. Frst, plan 0 pror to fall 2003 when both n-network and off-peak were free: vj =.11, 0, 0, 0). Second, plan 0 n fall 2003 or later when only n-network was free: vj =.11, 0,.11, 0). Thrd, three-part tarffs wh free n-network callng, such as plan 2 n January 2004: vj = vpk,out j, 0, 0, 0) and v pk,out j [ ) E x pk,out = p j Pr x pk,out Q j /ˆqv pk,out j ) x pk,out Fourth, standard three-part tarffs whout free n-network callng: v = v pk E ] Q j /ˆqv pk,out j ); I [ ]. 20) x pk,out I, vpk, 0, 0) and [ ] ) E θ pk v pk j = p j Pr θ pk Q j /ˆqv pk j ) θ pk Q j /ˆqv pk j ); I [ ]. 21) E θ pk I As n the man text Secton IV), we break out callng demand for weekday outgong calls to landlnes mmedately before and after 9pm to help dentfy the prce sensvy parameter. Now, however, such calls are a subset of out-of-network callng. Thus the shock r 9pm = r pk,9, r op,9 ) [0, 1] 2 captures the share of peak and off-peak out-of-network callng demand that s whn one 15

16 hour of 9pm on a weekday and s for an outgong call to a landlne: x 9pm = xpk,9 x op,9 = rpk,9 r op,9 x pk,out x op,out Our dentfyng assumpton that consumer s expected outgong callng demand to landlnes on weekdays s the same between 8:00pm and 9:00pm as s between 9:00pm and 10:00pm s unchanged, but ths now corresponds to a revsed restrcton: [ E r pk,9 ] [ E 1 r pk,n ] [ E θ pk ] [ = E r op,9 ]. [ E 1 r op,n ] E [θ op ], 22) n place of equaton 9). Equaton 22) mplcly defnes α op,9 parameters. as a functon of α pk,9 and other We model all callng share shocks r k for k {pk-n,op-n,pk-9,op-9} n the same manner as rk for k {pk-9,op-9} n the llustratve model. We assume the dstrbuton s a censored normal, r k = α k + e k 0 f r k 0 r k = r k f 0 < r k < 1, 1 f r k 1 where α k s unobserved heterogeney and e k s a mean-zero shock normally dstrbuted wh varance σ k e) 2 ndependent across, t, and k. For k {pk-n,op-n,pk-9} we assume that α k are normally dstrbuted n the populaton ndependently across and k) wh mean µ k α and varance σ k α) 2. Belefs about µ and θ are the same as n the llustratve model. We assume that there s no learnng about the share of demand that s for n-network callng. Consumers know α k,n and the dstrbuton of e k,n for k {pk,op} up to the fact that they underestmate the share of ther callng opportunes that are n-network by a factor δ r [0, 1]. δ r = 1 corresponds to no bas.) Specfcally, consumers beleve that n-network callng shares have the dstrbuton of δ r r k,n. We ncorporate ths addonal bas to help explan consumers choce of plan 1 over plan 0, whch offered free n-network callng. Modelng n-network callng adds seven addonal parameters to be estmated. These estmates are reported n Table 13. The parameters µ pk α and µ op α govern the average shares of θ that can be apportoned to peak and off-peak n-network usage, respectvely, whle the next four govern the standard devaton of n-network usage shares between and whn ndvduals. Snce our estmate of δ r s close to zero, consumers beleve that almost all usage s out 16

17 of network. Table 13: In-Network Callng Parameter Estmates Parameter Descrpton Est. Std. Err Parameter Descrpton Est. Std. Err µ pk α µ op α σ pk α σ op α E[α pk,n ] ) σ pk e E[α op,n ] ) σ op e SD[α pk,n SD[e pk,n ] ) SD[e op,n ] ) ] ) δ r In-network bas ) SD[α op,n ] ) C.4 Identfcaton of Complete Model Identfyng belefs nvolves one complcaton relatve to the llustratve model: plan choce depends on belefs about n-network callng shares as well as peak usage. A consumer s plan choce depends on her expected n-network peak-callng share δ r E[r pk,n depend on the populaton dstrbuton of δ r E[r pk,n α pk,n ]. Thus nal plan-choce shares α pk,n ]. Frst consder a restrcted model n whch consumers are unbased about n-network callng shares δ r = 1). Then the populaton dstrbuton of δ r E[r pk,n α pk,n ] s dentfed whout knowng belefs usng data pror to fall Durng ths perod, all plans offered free nghts-and-weekends so that r op,n = q op,n /q op. Moreover, only plan 0 offered free n-network callng. Thus for plans 1-3, peak callng-thresholds are the same for n-network and out-of-network callng and r pk,n = q pk,n /q pk callng-thresholds are 0 cents n-network and 11 cents out-of-network and hence r pk,n = q pk,n q pk,n pk,n + q pk,out /ˆq 0.11) = q q pk,n β) q pk,out,. For plan 0, peak where 1/ˆq 0.11) = β). Observng r k,n for k {pk, op} a censorng of r k,n = α k,n +e k,n ) [ ] dentfes E α k,n, V arα k,n ), and V are k,n ). A potental complcaton s that q pk n s only observed precsely for plan 0 subscrbers and q op n s only observed precsely for fall 2003 and later subscrbers to plan 0. For other plans we only observe bounds and a nosy estmate of q pk n because we can only dstngush n-network and out-of-network for outgong calls. Ths measurement error problem s solvable because only apples to a subset of the data. Ths comment also apples to r op,9 as: r op,9 = q op,9 /q op,out and r pk,9 = q pk,9 /q pk,out. and r pk,9 whch are now computed In an unreported robustness check, we fnd that a specfcaton estmated wh δ r restrcted to 17

18 1 sgnfcantly over-predcts the share of plan 0 after the plan stops offerng free off-peak callng. Hence we allow consumers to underestmate ther n-network callng share. To separately dentfy δ r from overconfdence δ) s mportant to use both pre and post fall-2003 plan-choce-shares. Reducng δ r or δ both make plans 1-3 more favorable relatve to plan 0. However, only δ r has a dfferentally larger effect post fall 2003 when plan 0 stopped offerng free nghts-and-weekends. Thus the larger the drop n share of plan 0 between fall 2002 and fall 2003, the more fall 2002 plan 1 choces should be explaned by low δ r rather than overconfdence. D Estmaton Detals D.1 Model F As shown n Table 14, our model does a good job of ftng plan choce shares. We also f the monthly rate of swchng well, whch we predct to be 4.5 percent and s observed to be 3.7 percent. We note that our model s not flexble enough to capture the entre shape of the usage dstrbuton. In Fgure 12, we plot observed and predcted usage denses for both peak and off peak usage. The model tres to f the shape of the usage dstrbuton as closely as can, but the censored normal specfcaton we use produces a hump near zero that s not replcated n the data. A more flexble usage specfcaton, such as a mxture of normals, mght f the observed data better. Table 14: Plan Shares for Inal Choces percent) October 2002 October 2003 Plan 0 Plan 1 Plan 2 Plan 3 Plan 0 Plan 1 Plan 2 Plan 3 Observed Predcted D.2 Smplfed Lkelhood Formulaton We begn ths secton by descrbng the structure of the lkelhood functon whch arses from our model. To smplfy the exposon, we nally make several smplfyng assumptons. In partcular, we gnore for the moment n-network and 8:00 pm to 10:00 pm usage shares, censorng of θ, and the fact that the set of frms consdered by F ) s unobserved. Addonal detals about the lkelhood functon, ncludng censorng of θ k, a treatment of n-network callng, 8:00 pm to 10:00pm callng, and qutng, are n Appendx D.3. As dscussed below, the lkelhood functon for our model does not have a closed form expresson due to the presence of unobserved heterogeney. We therefore 18

19 Peak Off Peak Densy Observed Predcted Densy Observed Predcted Mnutes Mnutes Fgure 12: Observed and Predcted Usage Denses Kernel Densy Fs) turn to Smulated Maxmum Lkelhood to approxmate the lkelhood functon Goureroux and Monfort, 1993). Smulaton detals are gven n Appendx D.4. An observaton n our model s a plan-choce and usage par for a consumer at a gven date: {j, q }. For ndvdual at tme perod t, we observe a plan choce j as well as a vector of usage, q, where q = {q pk, qop }. We denote the sequence of observatons for up to tme t by {jt, qt }, all observatons for an ndvdual by {j T, qt } and the full data set by {jt, q T }. We construct the lkelhood functon from the dstrbutonal assumptons on our model s unobservables. To faclate the exposon, we dvde the unobservables nto ndvdual type ω, sgnals s, and taste shocks η and ε. By ndvdual type ω, we mean the vector of unobservables that are ndependent across ndvduals but constant across tme: µ pk 1, µpk, µ op. The taste shocks are ndependent across ndvduals and tme, and nclude the vector of frm-specfc log errors η ) and the shocks governng total peak and off-peak usage θ ). For each observaton {j, q }, we frst evaluate the jont lkelhood of observed plan choce and usage condonal on 1) consumer type ω, 2) past observatons and sgnals { j t 1, q t 1, s t 1 }, and 3) the current sgnal s. Ths lkelhood s denoted by L j, q ω, s t, jt 1, q t 1 ), where we suppress the dependence on Θ n our notaton. The lkelhood for an ndvdual s then constructed by takng the product of perod t lkelhoods and ntegratng over ω and s T : L j T, q T ) = ω s T T t=1 L j, q ω, s t, j t 1, q t 1 ) ) ) f s s T fω ω ) ds T dω. 23) 19

20 As has no closed form soluton, we approxmate the ntegral over ω and s T n equaton 23) usng Monte Carlo Smulaton. For each ndvdual, we take N S draws on the random effects from f ω ω ) and f s s T ) and approxmate the lkelhood usng ˆL Θ) = 1 N S N S T b=1 t=1 L j, q ω b, s t b, jt 1, q t 1 ) ). 24) The smulated model log-lkelhood s the sum of the logarhms of the ndvdual smulated lkelhoods: ˆ LLΘ) = I logˆl Θ)). 25) =1 We use N S = 400 randomly shuffled) Sobol smulaton draws to calculate the smulated loglkelhood LLΘ) ˆ see Appendx D.4 for detals). D.2.1 Condonal perod t lkelhood We now construct the perod t lkelhood that enters equaton 23). For convenence, we reexpress ths lkelhood condonal on the ndvdual s peak type µ pk ncludes past choces j t 1 and her nformaton set I, whch and q t 1, sgnals to date s t, and all elements of type ω excludng µ pk. Our constructon follows naturally from the dstrbutonal assumptons on the taste shocks η and θ. For a customer who qus, we only observe that j J \J,unversy and thus we sum: Prqu µ pk, I ) = j J \J,unversy P j I ). Otherwse, condonal on {µ pk, I }, the lkelhood of an observaton {j, q } s the product of the plan choce probably and the lkelhood of observed usage condonal on plan choce. In partcular, condonal on s, we wll nfer θ from observed q and wre the lkelhood as: L j, q ω, s t, j t 1, q t 1 ) = L j, q µ pk, I ) = Prj I )f q q j, µ pk, I ). 26) Usage lkelhood: Frst, we consder the lkelhood of the observed usage q. We know that for k {pk, op}: q k = θk ˆqv k j, I )). Therefore, condonal on v k j, I ), we can nfer θ k from usage: θ k q k, vk j, I )) = q k /ˆqvk j, I )). Then, the jont normal assumpton on ε and s leads drectly to the dstrbuton of θ condonal on s : θ N µ k + E [ε s ], V [ε s ] ) See Appendx B.3). Assumng no censorng so that θ = θ ) ths yelds the usage lkelhood: f q q j, µ pk, I ) = f θ s θ q, v j, I )) µ, s ) θ q, v j, I ) q, 27) 20

21 where the term θ q, v j, I ) / q s the Jacoban determnant of the transformaton between θ and q condonal on v j, I )). The usage lkelhood must be adjusted whenever an element of θ s censored at zero, and n the complete model there are addonal terms to ncorporate data on n-network callng and 8-10pm callng Appendx D.3). Plan Choce Probably: Next we consder plan choce condonal on I and an actve choce, whch s denoted by condonng on C. Let F denote the set of frms consdered by the consumer, J f) denote the set of plans avalable from frm f F, and f j ) denote the frm whch offers plan j. Note that J f) depends on j,t 1 because consumers can always keep ther exstng plan.) Then the probably of choosng plan j equals the probably of choosng frm f j ) multpled by the probably of choosng plan j condonal on choosng frm f j ): Pr j C; I, F ) = Pr f j ) C; I, F ) Pr j C; I, f j )) Condonal on an actve choce, our assumpton of frm-specfc log errors gves rse to the followng frm choce probably: Prf C; I, F ) = expmax k J f) {U k I )}) g F expmax k J g) {U k I )}). Condonal on an actve choce of frm f j ), the probably of choosng plan j s smply equal to an ndcator functon specfyng whether or not plan j offered the hghest expected utly to consumer of any plan offered by frm f j ): { } Pr j C; I, f j )) = 1 U j I ) = max {U ki )}. 28) k J fj )) Typcally we characterze Pr j C; I, f j )) by numercally computng bounds s and s such that s equal to 1 f an only f s [s, s ].) In perod 1, every consumer makes an actve choce from the set of unversy plans, so Pr j 1 I 1 ) = Pr j 1 C; I 1, f j 1 )). In every later perod, however, consumers keep ther exstng plan j,t 1 wh probably 1 P C ) and otherwse choose between the unversy, the outsde opton, and one of three outsde frms. Thus, uncondonal on an actve choce, the probables that an exstng customer swches to plan j j,t 1 n perod t where j could be the outsde good) or keeps the exstng plan j = j,t 1 are P C Pr j C; I, F ) and P C Pr j C; I, F )+1 P C ) respectvely: P C Pr j C; I, F ) f j j,t 1 Pr j I, F ) =. 29) P C Pr j C; I, F ) + 1 P C ) f j = j,t 1 21

22 Fnally, the set of frms consdered by consumers s unobserved. Each exstng consumers consders unversy plans, the outsde opton, and one of the three outsde frms offerng local plans AT&T, Cngular, or Verzon). Dependng on the denty of the outsde frm, there are three possble consderaton sets: F { F 1, F 2, F 3 }, each of whch s equally lkely. Thus the fnal plan choce probably s: Pr j I ) = 1 3 D.3 Complete Lkelhood Formulaton 3 k=1 Pr j I, F k ). 30) In ths secton we revse the lkelhood functon descrbed n Appendx D.2 to fully account for 1) censorng of taste shocks θ, 2) free n-network callng and n-network callng data, and 3) near 9pm callng data. There are three changes that affect the ndvdual lkelhood, for whch equaton 23) gves the smplfed verson. Frst, the type vector ω s expanded to nclude parameters governng s dstrbuton of near 9pm and n-network usage: ω = µ pk 1, µpk, µ op, α pk,9, α pk,n, α op,n ). Second, the usage vector q s expanded to nclude n-network and near 9pm callng data: q = {q pk,n, q pk,out, q pk,9, q op,n, q op,out, q op,9 }. Thrd, censorng of the latent taste shocks means that there s an addonal vector of unobservable varables. Let θ be the vector of all latent taste shocks for ndvdual that are negatve and hence unobserved each correspondng to an observed θ k = 0). Let θ,t be the subset of θ realzed at or before tme t. As before, we wll construct the jont lkelhood {j, q } by condonng on {µ pk, I }. Whout censorng, ths was equvalent to condonng on { ω, j t 1, q t 1, s t }. Now, however, requres also condonng on θ,t 1 because consumer s nformaton set I ncludes all elements of θ realzed before tme t but these cannot be nferred from q t 1. Thus we must now ntegrate out over ths addonal unobserved heterogeney: L j T, q T ) = f θ ω s T θ T t=1 L j, q ω, s t, θ,t 1, j t 1 θ s T, ω, θ ) ) 0 f s s T fω ω ) d θ ds T dω. ) ), q t 1 31) The expresson for the lkelhood L j, q µ pk, I ) gven by equaton 26) must also be 22

23 revsed. In partcular, we replace the densy of q, f q, wh the lkelhood of q, l q : L j, q µ pk, I ) = P j I )l q q j, µ pk, I ). 32) The remander of ths secton s devoted to fully specfyng the lkelhood of observed usage, l q q j, µ pk, I ). The usage vector q s a functon of the random varables θ and r = {r pk,n, r op,n, r pk,9, r op,9 }. To compute the lkelhood l q q j, µ pk, I ), we compute the lkelhood of θ, l θ θ j, µ pk, I ) and the lkelhood of r, l r r j, I ), put them together and then do the change of varables from these varables to q : ) l q q j, µ pk, I ) = l θ θ j, µ pk, I )l r r j, I ) det θ, r ). Below we 1) outlne the constructon of l θ θ j, µ pk, I ) whch accounts for censorng, 2) outlne the constructon of l r r j, I ) by ncorporatng a) n-network callng data, and b) near 9-pm callng data, and 3) fnsh by outlnng det θ, r ) / q ) for the change of varables. q D.3.1 Censorng Condonal on µ and s, the taste shock θ follows a bvarate normal dstrbuton: θ N ) µ k + E [ε s ], V [ε s ] where E [ε s ] and V [ε s ] follow from Bayes rule n Appendx B.3 equatons 15)-16). We denote the jont densy by f θ s θ µ, s ), and the margnal densy of θ k for k {pk, op} by f k θ s θ k µ k, s ). As descrbed n Secton V, θ k can be nferred from observed usage. If θ k > 0, then ths gves the value of the latent varable: θ k = θ k. If θ k = 0, however, we can only nfer that θ k 0. When θ s not censored, s lkelhood s smply f θ s θ q, v j, I )). Otherwse, the lkelhood of θ q, v j, I ) must be adjusted by substutng the probably that θ k s censored for f θ s : l θ θ j, µ pk, I ) = f θ s θ µ, s ) Pr θ pk f θ pk f θ pk > 0, θ op > 0 0 µ pk, s, θ op )f k θ s θ op µop, s ) = 0, θ op > 0 Pr θ op 0 µ op, s, θ pk )f k θ s θ pk µpk, s ) f θ pk > 0, θ op = 0 Pr θ pk 0, θ op 0 µ, s ) f θ pk = 0, θ op = 0, 33) 23

24 where θ s dependence on q and v j, I ) s suppressed from the notaton. We then use the change of varables formula to arrve at the lkelhood of q rather than θ ) We leave ths step, however, untl after consderng n-network and near-9pm callng. D.3.2 In-network and Near-9pm Callng By ncludng data on whether calls are n or out of network and whether calls are whn 60 mnutes of 9pm, the usage vector becomes q = {q pk,n, q pk,out, q pk,9, q op,n, q op,out, q op,9 }, where n, out, and 9 sgnfy n-network, out-of-network, and near-9pm callng respectvely. before, q pk = q pk,n + q pk,out and q op = q op,n + q op,out As denote total peak and total off-peak callng respectvely. In ths secton we dscuss three ssues related to handlng n-network callng: 1) a data lmaton, 2) the nference of θ, and 3) an addonal term n the lkelhood. 1) Data Lmaton: We always observe total peak and off-peak callng because we observe the tme and date of all calls. An mportant data lmaton, however, s that call logs only drectly dentfy outgong calls as n-network or out-of-network. Ths nformaton provdes lower and upper bounds on n-network callng: q k,n and q k,n for k {pk, op}. The lower bound on total n-network usage s smply the total outgong n-network mnutes we observe. The upper bound s outgong n-network mnutes plus all ncomng mnutes. q k,out and q k,out Analogous bounds on out-of-network usage are = q k qk,n = q k qk,n for k {pk, op}. Fortunately, the network status of plan 0 peak calls and off-peak calls for plan 0 that dd not nclude free off-peak) can be nferred from whether they were charged 11 cents or 0 cents per mnute. Thus, precsely when n-network calls are dfferentally prced, we can nfer q pk,n and q op,n exactly. 2) Inferrng θ : The fact that n-network and out-of-network callng may be prced dfferently complcates our nference of θ from usage data. For k {pk, op}, θ k s calculated by equaton 34) f category k calls are not prced dfferentally by network status or by equaton 35) f category k calls are prced dfferentally by network status: θ k = q k /ˆqv k ), 34) θ k = q k,n /ˆqv k,n ) + q k,out /ˆqv k,out ). 35) There s always suffcent nformaton to nfer θ from usage condonal on the threshold vector v j, I ) because q k,n and q k,out are observed precsely when equaton 35) apples. 3) Lkelhood Functon: Next we turn to the lkelhood of n-network and near 9pm callng 24

25 shares. 4 For k {pk, op}, f n-network usage s observed exactly then we can calculate the exact share of category k callng opportunes that are n-network as r k,n = q k,n /ˆqv k,n q k,n /ˆqv k,n ) + q k,out ) /ˆqv k,out ), and the exact share of category k callng opportunes that are near-9pm as r k,9 = q k,9 /q k,out. For k {pk-n,pk-9,op-n,op-9}, as r k follows a censored-normal dstrbuton, where the underlyng normal dstrbuton s defned by α k + ek, we can wre the lkelhood of rk as: Φ α k /σk e) f r k = 0 f k r α k k ) = φr k αk )/σk e)/σ k e f r k 0, 1). 1 Φ1 α k )/σk e) f r k = 1 For k {pk, op}, f we only observe bounds on q k,n for r k,n and r k,9 : Note that q k,9 r k,n r k,9 and q k,out [ ] q k,n k,n θ k ˆqv k,n ), q [ θ k ˆqv k,n = ) [ ] q k,9 q k,9 [ ] q k,out, q k,out = r k,9, rk,9. q k,out, so q k,9 > 0 mples the bounds on r k,9 then we can only calculate bounds r k,n ], r k,n, are whn [0, 1]. For k {pk, op}, f the upper bound on the category k n-network usage share s below one r k,n lkelhood for the bounds on r k,n and r k,9 are as follows. l k,n r k,n, r k,n α k ) = f k r k,n Φ r k,n Φ r k,n α k,n ) f r k,n ) ) ) ) α k,n /σ k,n e Φ r k,n α k,n /σ k,n e ) ) α k,n /σ k,n e < 1) then the = r k,n f 0 < r k,n f 0 = r k,n = r k,n < r k,n < 1 < r k,n < 1 4 Note that, for k {pk, op}, when q k = 0 we have no nformaton about the share of n-network usage r k,n when q k,out = 0 and we have no nformaton about the share of near-9pm r k,9. and 25

26 l k,9 r k,9, rk,9 α k,9 ) = f k,9 r k,9 αk,9 ) f r k,9 = r k,9 = r k,9 ) ) ) ) Φ r k,9 α k,9 /σ k,9 e Φ r k,9 α k,9 /σ k,9 e f 0 < r k,9 < r k,9 < 1 ) ) Φ r k,9 α k,9 /σ k,9 e f 0 = r k,9 < r k,9 < 1 ) ) 1 Φ r k,9 α k,9 /σ k,9 e f 0 r k,9 < r k,9 = 1 Moreover, n the case r k,n < 1, the lkelhoods of r k,n and r k,9 are ndependent and we can wre the jont lkelhood as the product of l k,n r k, rk αk ) and lk,9 r k,9, rk,9 α k,9 ). When r k,n = 1, constructng the lkelhood of r k,n and r k,9 s more complcated. The complcaton stems from the way that bounds on n-network and out-of-network calls are constructed. The lower bound on out-of-network calls s the total number of outgong calls to out-of-network numbers, plus f n-network callng s free) the total number of non-free mnutes used after an overage occurs. If ths lower bound s zero, then total outgong-calls to landlnes were also zero. Total landlne calls near 9pm could be zero for two reasons: r k,n = 1, or r k,9 = 0. If the upper bound on r k,n bnds, then r k,9 could take any value. However, f the upper bound on r k,n does not bnd, then r k,9 r k,9 [0, 1] s = must be zero. Followng ths logc, the jont lkelhood of r k,n ) l r,k r k,n, r k,n = 1, r k,9 [0, 1] α k,n, α k,9 1 Φ1 α k,n )/σ k,n e ) + Φ α k,9 /σ k,9 e ) Φ1 αk,n Φr k,n /σ k,9 e 1 Φ1 α k,n )/σ k,n e ) + Φ α k,9 )/σ k,n e ) α k,n )/σ k,n e ) )Φ1 α k,n )/σ k,n r k,n f r k,n > 0 e ) f r k,n = 0. 1 and Thus the jont lkelhood of n-network and near-9pm callng opportuny shares for category k {pk, op} s l r,k r k,n, r k,n, r k,9, rk,9 α k,n, α k,9 ) = l r,k r k,n, r k,n α k,n ) l k,9 r k,9, rk,9 α k,9 ) f r k,n < 1 l r,k r k,n, r k,n = 1, r k,9 [0, 1] α k,n, α k,9 ) f r k,n = 1. Fnally, l r r j, I ) = Π k {pk,op} l r,k r k,n, r k,n, r k,9, rk,9 α k,n, α k,9 ). In the followng secton we perform the change of varables transformaton to wre the lkelhood as a functon of q rather than r. 26

27 D.3.3 Change of Varables The jont lkelhood of θ and r wll be the product of the l θ and l r. Ths s not the lkelhood of the observed usage vector q, however, because q s a functon of θ and r and we need to make the change of varables between them. Summarzng what we ve outlned above, the transformaton from the data to θ k, r k,n, r k,9 for k {pk, op} s θ k = qk,out ˆqv k,out ) + qk,n ˆqv k,n ) ; rk,n = q k,out q k,out + q k,n ˆqvk,out ) ˆqv k,n ) ; r k,9 = qk,9 q k,out. We need to take the Jacoban determnant of ths transformaton and multply by each lkelhood observaton. We note that the Jacoban we take depends on what we observe. Below, we descrbe the case where all the sx dfferent q s are observed. The Jacoban s smpler f less data s observed. For example, f only θ k were observed, and we could not compute the r s, we would only take the Jacoban of the transformaton between θ k and q k for k {pk, op}. Defnng y = q pk,out, q pk,n, q pk,9, q op,out, q op,n, q op,9 ) and x = θ pk, r pk,n, r pk,9, θ op, r op,n, r op,9 ), ) fy) = fx) x det. y For k {pk, op}, the dervatves we need are: θ k / q k,out = 1/ˆqv k,out ), θ k / q k,n = 1/ˆqv k,n ), r k,9 / q k,out = q k,9 q k,out ) 2, r k,9 / q k,9 = 1/q k,out, θ k rk,n = qk,9 q k,9 = rk,9 = 0, qk,n and r k,n q k,out = q k,n ˆqvk,out ) ˆqv k,n q k,out + q k,n ˆqvk,out ) ) ˆqv k,n ) r k,n q k,n = q k,out ˆqvk,out ) ˆqv k,n ) ) 2 q k,out + q k,n ˆqvk,out ) ˆqv k,n ) ) 2. 27