When Talk Is Free : The Effect of Tariff Structure on Usage Under Two- and Three-Part Tariffs

Transcription

1 1 When Talk Is Free : The Effect of Tarff Structure on Usage Under Two- and Three-Part Tarffs Eva Ascarza, Anja Lambrecht, and Naufel Vlcassm Web Appendx In ths appendx, we present a detaled descrpton of the analyses performed to obtan certan results dscussed n the man manuscrpt. DESCRIPTIVE ANALYSES Analyss of tarff choce We analyze whether customers blls would have been lower on another than ther chosen tarff at the tme that three-part tarffs were ntroduced. Based on the three avalable usage perods pror to the three-part tarff ntroducton, we compute the ndvdual-level average usage and standard devaton. For smplcty, we exclude customers who have swtched more than once as well as the 1.1% of customers who swtched wthn these three months. To account for devatons from average usage due to random usage shocks, we then compute the bll for the usage level of [average usage +/ 1 standard devaton] under the current tarff, and the bll for the average usage under each of the remanng tarffs. We conclude that a customer would have had a lower bll on a dfferent tarff f the bll for ther average usage on a tarff other than the chosen tarff was below the lower bound of the bll-nterval that accounts for varaton n usage on the chosen tarff. Note that ths analyss focuses on potental savngs and does not account for the fact that customers may, on the same bll, be able to use more on a dfferent tarff. The next secton wll dscuss ths aspect n detal.

2 2 Table A1 llustrates that based on ther average usage and standard devaton of usage before the ntroducton of three-part tarffs, the large majorty of customers chose the tarff that mnmzes ther bll. In total, 26.2% of customers would pay less on a dfferent tarff. For customers that would pay less on a dfferent tarff average savngs were between MU 4.1 and MU 7.7. As a result, t would take customers more than one perod on average to amortze the swtchng fee of MU 10. We then exclude three-part tarffs from ths analyss and lmt the analyss to whether customers would have pad less on a dfferent two-part tarff. We fnd that only 10.9% of customers would have pad less on a dfferent two-part tarff. Ths further confrms that two-part tarff customers largely chose the bll-mnmzng tarff. Table A1: Potental savngs when three-part tarffs were ntroduced Tarff wth lowest bll (n %) Chosen tarff T_2_1 T_2_2 T_2_3 T_2_4 T_3_1 T_3_2 T_3_3 N Avg. savngs (n MU)* Tarff_2_ Tarff_2_ Tarff_2_ , Tarff_2_ , Excludng customers who swtched wthn the frst 3 months of our data and customers that n our data swtch more than once * Average savngs on tarff wth lowest bll, computed only for customers that would have had a lower bll on a dfferent tarff Detaled analyss of swtchng from two- to three-part tarffs The prevous secton focused on whether customers would have pad less on a dfferent tarff. We now focus on three-part tarffs and analyze n more detal whether customers would beneft from swtchng to a three-part tarff, accountng for both whether customers would have pad less on a dfferent tarff and whether they would have been able to use more for the same bll. Fgure A1 llustrates n whch stuatons a customer should or should not swtch to a threepart tarff. We abstract from swtchng costs and assume that a customer knows her optmal usage

3 3 under a two- and a three-part tarff. We assume a utlty functon whch s quadratc n usage (bold curve; the Model secton of the man paper justfes the choce of utlty functon). The bll on a two-part tarff (straght lne) ncreases n the customer s usage. The bll on a three-part tarff (dashed lne) remans flat as long as usage remans wthn the allowance and then ncreases lnearly n usage. The maxmum dstance between the utlty functon and the bll ndcates a customer s surplus on that tarff. A ratonal customer should swtch to a three-part tarff f that entals a greater surplus than on a two-part tarff. The vertcal (horzontal) arrows ndcate how such a swtch would affect a customer s bll (usage). A customer should swtch to a three-part tarff f for the same optmal usage, she pays less on a three- than on a two-part tarff (II), for the same bll, her optmal usage s greater on a three- than on a two-part tarff (IV), or f she can ncrease her optmal usage and stll pay less on a three-part tarff (I). A customer should not swtch f for the same bll, her optmal usage on a three-part tarff would decrease (VI), for the same optmal usage her bll would ncrease (VIII) or f her bll would ncrease whle decreasng optmal usage (IX). She s ndfferent f the same optmal usage entals the same bll (V). If under a three-part tarff, both optmal usage and the bll would decrease (III) or ncrease (VII), swtchng may or may not be benefcal, dependng on the curvature of the utlty functon. To determne whch customers n our sample should or should not swtch to a three-part tarff, we compare actual usage and expendtures on a two-part tarff to (a) how much a customer could use under a three-part tarff for the same bll and (b) how much she would pay under a three-part tarff for the same usage (Fgure A1). To account for devatons from average usage due to random usage shocks, the nterval of [average usage +/ 1 standard devaton] and the nterval of the bll of [average usage +/ 1 standard devaton] serve as a reference pont. For

4 4 example, we classfy a customer as beng ndfferent between swtchng to a three-part tarff and stayng on a two-part tarff (Case V) f the same optmal usage entals a bll n the same nterval on a two- and a three-part tarff.

5 Bll ncreases Bll stays constant Bll decreases Bll Bll Usage Usage Usage Utlty VII VIII IX Better Off Worse Off Utlty Utlty 5 Fgure A1: Predcted swtchng from two- to three-part tarffs II III IV V VI Bll Utlty Usage Usage Better Off Worse Off Usage Usage IV I Bll Usage Utlty Bll Bll Usage Usage Usage ncreases Usage stays constant Usage decreases The maxmum dstance between the utlty functon and a tarff s bll ndcates maxmum surplus. The vertcal dotted lne represents the optmal level of usage on a two-part tarff. Note: Margnal utlty and the prce of the outsde good are set to 1, so utlty represents wllngness to pay. II Utlty Utlty Bll Bll Bll Usage Utlty Utlty

6 6 Table A2 summarzes the results of ths analyss. The frst four columns correspond to the results when the swtchng fee s not taken nto consderaton. They ndcate that customers who, accordng to our analyss, should swtch to a three-part tarff were far more lkely to swtch to a three-part tarff than customers who accordng to our analyss should not swtch to a three-part tarff. The next set of results accounts for the fee the customer has to pay for swtchng. Here we consder a swtch to be benefcal f savngs n the frst month would compensate for the swtchng fee. Snce the fee ncreases the bll, the share of customers classfed as unknown,.e., those for whom both optmal usage and the bll would ncrease on a three-part tarff, s larger than when abstractng from the swtchng fee. Table A2: Predcted and actual swtchng behavor Not consderng swtchng fee Consderng swtchng fee Category No. of customers % of sample % of customers n that group who swtched % of total swtchers belongng to category No. of customers % of sample % of customers n that group who swtched % of total swtchers belongng to category Should swtch 3, %0 8.95% 71.7% %0 13.2% %0 Should not swtch %0 4.71% 0.9% %0 6.4% %0 Indfferent (a) %0 7.19% 11.9% %0 10.4% %0 Unknown (b) %0 5.66% 15.6% %0 6.8% %0 (a) A customer s ndfferent f the same optmal usage entals the same bll on a two- and a three-part tarff. (b) If under a three-part tarff, both optmal usage and the bll would decrease or ncrease, swtchng may or may not be benefcal dependng on the curvature of the utlty functon. Persstence of three-part tarff usage over tme We next check whether the ncrease n three-part tarff usage perssts over tme. We focus on customers for whom we observe at least sx months of three-part tarff usage and plot the aggregate three-part tarff usage over tme. Fgure A2 llustrates that, apart from the holday seasons n months 5 and 7 after the ntroducton of the three-part tarffs, there are no clear trends of ncreasng or decreasng three-part tarff usage.

7 Mnutes Mnutes 7 Fgure A2: Monthly average usage after swtchng to a three-part tarff Months after three-part tarff ntroducton Second, we compare average usage before and after the three-part tarff ntroducton, as we do n the Descrptve Analyss secton of the man manuscrpt, but now analyze dfferences by cohorts (.e., groups of customers who swtched to a three-part tarff n the same month). Fgure A3 shows, for each cohort, the average usage before the three-part tarffs were ntroduced and the average usage n the last perod of our data and compares t to customers who dd not swtch to a three-part tarff. We observe a consstent ncrement n usage among three-part tarff swtchers, regardless of how long customers have been on a three-part tarff. Fgure A3: Average usage before and after the ntroducton of three-part tarffs, by cohorts 550 Before Last perod Non-swtchers 5 months 6 months 7 months 8 months # perods on a three-part tarff

8 Fracton.1.15 Fracton Fracton.1 Fracton Further detals on three-part tarff usage behavor We summarze nformaton on customers usage behavor on three-part tarffs. Frst, we analyze the dstrbuton of usage as percentage of the allowance (Fgure A4). Across all three-part tarffs, we observe a mass pont of usage observatons when usage s approxmately equal to the allowance. Ths mass pont results from the type of budget constrant mposed by a three-part tarff that mples bunchng of usage observatons at 100% of the allowance (see equaton (4) n the man manuscrpt). It s reassurng that we ndeed fnd such a mass pont n our data snce t provdes addtonal evdence that customers are aware of ther usage behavor. Fgure A4 also llustrates that many customers use more than ther usage allowance. Ths s n lne wth the behavoral motvaton that leads to greater three-part tarff usage as dscussed n the man paper. It outlnes that the postve effect from a three-part tarff should persst when consumers have exceeded ther allowance. Fgure A4: Usage as a percent of allowance All tarffs Tarff_3_ Usage/Allowance rato Usage/Allowance rato Tarff_3_2 Tarff_3_ Usage/Allowance rato Usage/Allowance rato

9 9 Second, we analyze whether three-part tarff customers had chosen the ex-post bllmnmzng tarff based on ther frst three months of three-part tarff usage. As n the frst secton of ths web appendx, we rely on the bll for the usage level of [average usage +/- 1 standard devaton] under the current tarff, and the bll for the average usage under each avalable tarff. Table A3 llustrates that overall 86.8% of customers chose the three-part tarff that mnmzes ther bll based on ther ex post usage. Snce the dfferences between access prces and allowances between the three-part tarffs are large, even customers that use more than ther allowance are largely n the bll-mnmzng tarff. Table A3: Optmalty of chosen three-part tarff (based on frst three perods on a three-part tarff) Tarff wth lowest bll (n %) Chosen tarff Two-part tarff T_3_1 T_3_2 T_3_3 N Tarff_3_ Tarff_3_ Tarff_3_ Includes all customers wth at least three perods on a three-part tarff, excludes customers who swtched agan n ther frst three perods on a three-part tarff DEMAND ESTIMATION Lnear demand estmaton of three-part tarff usage We compare actual usage on two- and three-part tarffs to predcted usage for the last month n our data. We estmate a lnear demand functon for two-part tarff usage, qjt dt bp j, where q jt denotes the number of mnutes that ndvdual consumes on tarff j at tme t, d t denotes the sataton level, or demand ntercept, b refers to the prce coeffcent and p j s the usage prce of tarff j. Snce we have lttle wthn-customer varaton of the usage prce, the prce coeffcent s assumed to be homogenous across customers. We ncorporate an ndvdual-level preference,, and a multplcatve shock, t, nto the demand ntercept, d t te. We assume that follows

10 0 jt 1 2 r 2 a normal dstrbuton wth mean and varance, and that t s dstrbuted lognormal wth 2 2 parameters 0.5,, such that E( t ) 1. We use MCMC methods to estmate the model. We choose dffuse hyperprors for b,,,and 10. We burn-n 90,000 teratons and use the next 10,000 to sample from the posteror dstrbutons of the parameters of nterest and to predct consumpton n the last perod of data. The parameters estmates are shown n Table A4. Table A4: Estmaton Results (Homogeneous prce coeffcent) Mean 95% Interval b For customers who remaned on a two-part tarff, we predct consumpton n the last perod of the data as: q * jt 0 f djt bp j djt bp j f djt bp j. For customers who have swtched to a three-part tarff, we predct consumpton n the last perod of the data as q * jt djt f djt q j Max( q j, djt bp j ) f djt q j Fgure A5 llustrates that the model accurately predcts usage for customers who reman on a two-part tarff whle notably underpredctng consumpton for customers who swtch to a three-part tarff. In other words, the model does not capture the ncrement n usage observed for three-part tarff customers.

11 11 Fgure A5: Usage predctons usng lnear model (all customers) 450 Actual Predcted Non_swtchers Swtchers Persstence of over-usage over tme We next check whether the unpredcted ncrease n three-part tarff usage perssts over tme. We use the estmates obtaned n the analyss presented n the prevous secton but now analyze threepart tarff customers n cohorts of customers who swtched to a three-part tarff n the same month. For each cohort, we predct usage n the last month of the data and compare t wth actual usage n that month. The model under-predcts three-part tarff usage regardless of how long customers have been on a three-part tarff. Specfcally, we under-predct usage by 22.1% for the fve-month cohort, by 12.7% for the sx-month cohort, by 19.8% for the seven-month cohort, and by 12.1% for the eght-month cohort. Robustness to non-lnear demand specfcatons If customers usage followed a convex demand functon, our lnear demand model n the prevous secton would predct demand accurately n the area of usage prces smlar to those of the twopart tarffs,.e., MU, but would possbly underpredct usage at a zero prce. As a consequence, the over-usage we fnd n the descrptve analyss presented n the man manuscrpt

12 could smply be due to the specfcaton of the demand functon. We rule out ths possblty by estmatng two addtonal demand specfcatons. Frst, we use a polynomal specfcaton (as a Taylor approxmaton to the true demand functon) to estmate demand. We buld on the demand functon presented n the prevous secton, qjt dt bp j, and nclude a quadratc term, 2 bp, 2 j and a cubc term 12 3 bp. 3 j We estmate demand as q d b p b p b p. If the quadratc and cubc terms do not sgnfcantly dffer from 2 3 jt t 1 j 2 j 3 j zero, that would support the choce of a lnear demand functon. We replcate the analyss presented n the man manuscrpt. The results show that the quadratc and cubc terms of the demand functon are not sgnfcantly dfferent from zero (Table A5). We next use the parameter estmates to predct usage n the last perod. Fgure A6 dsplays the results. Smlarly to our man specfcaton, predcted usage of customers who swtched to a three-part tarff s only 86.4% of ther actual usage whle the model predcts 98.9% of actual usage for customers who reman on a two-part tarff. Ths provdes evdence that the ncrease n usage s not due to the specfc form of the demand functon. Table A5: Estmaton results (quadratc and cubc terms) Mean 95% Posteror Interval b b b

13 13 Fgure A6: Usage predctons usng quadratc and cubc terms Actual Predcted Non_swtchers Swtchers Second, we estmate an addtonal model specfcaton that allows for convex demand: q jt e p t j. Ths demand specfcaton s obtaned by maxmzng the utlty functon U ( q, q ) log( q ) q q jt jt 0t jt jt 0t, wth, 0 and 0. The term q 0 t denotes the outsde good, when ts prce s beng normalzed to 1. To emprcally dsentangle and, the data needs to have ndvdual-level varaton of the usage prce. However, n our data there s lttle tarff swtchng before the three-part tarffs were ntroduced. An alternatve s to fx the value of at a reasonable level and estmate the remanng parameters based on the frst two perods and predct usage for the last perod. We proceed n three steps: 1. To avod havng to arbtrarly set, we estmate the demand model usng all observatons from the frst sx perods of data. We obtan an estmate of (-0.049). 1 1 We conduct two sets of robustness checks to our estmate of. Frst, we estmate based on a dfferent number of perods (4 and 6 perods). Second, we estmate based on a random subsample of 50% of the customers n our dataset. We fnd that our estmate of s robust to these alternatve specfcatons.

14 2. We then set and estmate the remanng parameters, ncludng, usng two-part tarff usage observatons pror to the three-part tarff ntroducton. 3. We then use the set of estmated parameters to predct usage n the last perod of our data. Fgure A7 llustrates predcted versus actual usage. Consstent wth the results obtaned n the prevous secton, we under-predct three-part tarff usage by 19.2% whle predctng two-part tarff usage very accurately (under-predcton of only 0.9%). Ths provdes further evdence that the assumpton of lnear demand does not lead us to artfcally under-predct three-part tarff usage. 14 Fgure A7: Usage predcton convex demand functon actual predcted Non_swtchers Swtchers Robustness to non-homogeneous prce senstvty It s possble that customers who swtch to a three-part tarff dffer n ther usage prce senstvty from customers who reman on a two-part tarff. Gven the lmted wthn-customer prce varaton n our data, we cannot estmate a model wth an ndvdual-level prce coeffcent, b. Nevertheless, we conduct an ad hoc analyss n whch we allow for a dfferent set of parameters for swtchers to a three-part tarff compared to all other customers. We then test whether ths specfcaton stll under-predcts three-part tarff usage.

15 15 As n the Descrptve Analyss secton n the man manuscrpt, we estmate a demand model usng the two-part tarff perods pror to the three-part tarff ntroducton and then predct usage n the last perod of our data. We now estmate two sets of coeffcents, one for customers who remaned on a two-part tarff and one for customers who swtched to a three-part tarff. The same dffuse prors were chosen for both sets of parameters. Table A6 summarzes the posteror dstrbutons and Fgure A8 shows the model predctons. The model wth heterogenety n prce senstvty under-predcts three-part tarff usage by 9.8% whle two-part tarff usage s predcted very accurately. We conclude that whle heterogenety n usage prce senstvty may possbly contrbute to greater three-part tarff usage, t does not explan the large ncrease n usage we observe n the data. To capture some degree of heterogenety n usage prce senstvty, our full model specfcaton (the Model secton of the man manuscrpt) ncorporates observed heterogenety n the usage prce senstvty. Table A6: Estmaton results (heterogeneous prce coeffcent) Customers who do not swtch to a three-part tarff Customers who swtch to a three-part tarff Mean 95% Interval Mean 95% Interval b

16 16 Fgure A8: Usage predctons for heterogenety n prce senstvty (lnear model) 450 Actual Predcted Non_swtchers Swtchers Analyss of autocorrelaton n the usage process leadng to self-selecton As dscussed n the man manuscrpt, autocorrelaton n the usage process could be a possble explanaton for the usage ncrease we observe. If usage followed an autoregressve process and customers swtched to a three-part tarff after havng receved a postve usage shock, then we would expect that customers ncrease ther consumpton after swtchng to a three-part tarff. However, we fnd that ths pattern of behavor s not consstent wth our data. We frst nvestgate the level of autocorrelaton among the usage shocks. Gven that our demand s specfed wth multplcatve usage shocks n the demand coeffcent, shocks do not enter n a lnear way. Hence, we cannot run smple autocorrelaton tests usng usage observatons. To solate the usage shocks, one would need to take logs of the quantty q jt bp j, whch s not feasble snce b s one of the parameters to be estmated. To overcome ths ssue, we consder sub-samples of customers for whch p j does not vary, reducng the term pb j to a constant, and then estmate the degree of autocorrelaton n each sub-sample. We do so by successvely lmtng

17 17 the sample to customers who are on the same tarff and do not swtch to a dfferent tarff. Then we run a fxed effect lnear regresson for the whole hstory of each set of customers, usng log jt q as dependent varable and ts lagged value as ndependent varable. 2 For each of the subset of customers, we fnd no evdence of strong autocorrelaton among the usage shocks (ρ ranges from 0.16 to 0.35 across all tarffs). We then perform further analyses to ensure that the weak seral correlaton we fnd does not bas our model estmates. We frst smulate tarff choce and usage behavor for a synthetc panel of customers where we use the estmated parameters from our man model as the data generatng process. We ncorporate weak autocorrelaton (values of 0.2, 0.3 and 0.4) nto the usage process through autocorrelated usage shocks. We estmate all parameters usng our man model. We fnd that n all cases the smulated values lay wthn the posteror nterval of the estmated parameters. Ths provdes further confrmaton that our results are not affected by a possble weak autocorrelaton. Second, we nvestgate whether past usage shocks affect swtchng behavor. We estmate a logstc regresson wth swtchng to a three-part tarff as dependent varable. 3 As ndependent varables, we use past usage, dummy varables for the current two-part tarff, and the rato of usage n the last perod to usage n the perod before last. The latter varable serves as a proxy for the usage shock receved n the prevous perod. If past usage shocks affected swtchng to threepart tarffs, then the shock varable should be sgnfcant. We fnd that ths s not the case. Table A7 summarzes the results of three dfferent specfcatons. In the frst specfcaton, we nclude the usage shock n the last perod as a predctor for swtchng behavor, controllng for 2 We use the method proposed by Blundell and Bond (1998) to correct for the Nckell bas nduced by the fxed effect.

18 18 the chosen tarff. In the second specfcaton, we also control for the average usage level prevous to the three-part tarff ntroducton, and n the thrd specfcaton, we add a quadratc term for average usage. 4 In all specfcatons, the proxy for a past usage shock s not sgnfcant. We therefore conclude that autocorrelaton does not explan the over-usage we observe n the data. Table A7: Logstc regresson results for swtchng to three-part tarffs Varable Coeffcent p-value Coeffcent p-value Coeffcent p-value Constant Prevous usage (avg.) Prevous usage (avg.) ^ Past usage shock Dummy for prevous tarff 2_ Dummy for prevous tarff 2_ Dummy for prevous tarff 2_ MODEL Asymptotc propertes of the learnng model We show that for any value of the ntal parameters 0, 0, the expected value of the belef converges to the true value,, and ts varance goes to zero as the consumer gets more experence on a three-part tarff (.e., the number of perods on a three part tarff goes to nfnty). We compute the lmt of the mean and the varance of the belefs, as shown n equaton (22), when n goes to nfnty: 3 We estmate tarff choce n the fourth month of data. As a robustness check we also estmate the same model usng months 5, 6, etc. and n all cases, obtan qualtatvely the same results. 4 We perform the same analyss usng (1) current usage, and (2) lagged usage. We obtan the same qualtatve results.

19 19 (A1) 0 nr lm E( ) lm n n n n s 0 t1 0 r lm n. n n 0 1 s t n n t t1 We know from equaton (19), that s s gamma-dstrbuted wth shape and scale parameters t r, r / e. Thus, as n, we know that n 1 lm s e. n t n t1 r Therefore, substtutng ths result nto (A1), we obtan that lm E( ) n n e. 0 nr lm Var( ) lm n n n n 0 s t1 t 2 (A2) 0 r lm n n n 0 1 n s n n t1 t 2 0. Posteror dstrbutons for the full model The model s estmated usng a Bayesan framework. We obtan estmates of all model parameters by drawng from the margnal posteror dstrbutons. Gven the nonlneartes of our lkelhood functon and the complexty of the herarchy n the parameters, most condtonal dstrbutons do not have conjugate posterors. We use the Metropols-Hastng (MH) algorthm to draw from these condtonal posteror dstrbutons. We use a data augmentaton approach to nclude the unobserved ndvdual-level parameters as well as the tme-varant belefs.

20 20 We denote as all parameters n our model, ncludng the populaton parameters b,,,, a, a,exp( r), the ndvdual-level parameters,, mxng parameters,,,,,, the, and the ndvdual specfc tme-varant belefs t. The full jont posteror dstrbuton can be wrtten as: f ( Data) L( Data ) f ( ) I T 1 t1 f ( q k,,,, Z, X ) t t t j f ( k,,,,, Z, X ) t t t j f ( t,, 0, Zt ) f (, ) f (, ) f (, ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ) f ( ). where f ( q k,,,, Z, X ), f ( k,,,,, Z, X ), and f(,, Z ) are the t t t j t t t j t t expressons derved n the appendx, (App-1), (App-2), and equaton (21) n the man paper. Expressons (, ), (, ), and (, ) correspond to the mxng f f f dstrbuton for the populaton parameters, as specfed n the Model secton. We choose dffuse pror dstrbutons for all populaton parameters. We use a normal dstrbuton wth mean and standard devaton (0,100) for,,, and nverse-gamma wth shape and scale parameters (1, 10 ) for,,. We assume that b,,,, a, a,exp( r) follows a multvarate normal dstrbuton wth parameters and dag( ) 100 I,1 n 1,3, where n s the dmenson of, n 1 s a 1n vector of zeros, and I n s the dentty matrx of dmensons n1

21 n n. (The values of and were chosen to ensure unnformatve prors n the transformed space.) We draw recursvely from the followng posteror dstrbutons: 1. (Gbbs) Parameters,,,,, are obtaned by samplng from the followng dstrbutons: 21 f 2 1 (, ) Normal, I I 1 I I 1 f (, ) Inverse Gamma 1, I 2. We proceed smlarly for parameters,,,. 2. (MH) Draws for are obtaned by samplng from (,,, t,data) exp.5 P(data, t, ) 1 f 3. (MH) Draws for are obtaned by samplng from: 2 f(,,,,, t,data) exp.5 P(data,, ) 2 t We proceed smlarly for. 4. (MH) Draws for t are obtaned by samplng from: (,,,,data), (data,,, ), n f t t g t r0 nr 0 s P t t t1 n where g t r0 nr, 0 s t s the gamma pdf as derved n (21). t1

22 Snce there s no closed-form expresson for the posteror dstrbutons of and, we use a Gaussan random-walk Metropols-Hastng algorthm to draw from these dstrbutons. Followng the Metropols-Hastng procedure proposed by Atchade (2006), for each teraton, s, we draw a ( s) proposal vector of parameters (ether for and ): ~ Normal,, ( l ) ( l 1) ( l 1) ( l 1) and then accept the vector usng the Metropols-Hastngs acceptance rato. The tunng parameters 22 ( l 1) ( l 1) and are adapted n each teraton to get an acceptance rate of approxmately 20%. We ran the smulaton for 30,000 teratons. The frst 20,000 teratons were used as a ``burn-n'' perod, and the last 10,000 teratons were used to estmate the condtonal posteror dstrbutons. Fgure A9 and Fgure A10 show the posteror draws obtaned n the smulaton. Fgure A9: Posteror draws for the populaton parameters (MH steps) b x 10 4 rho x 10 4 beta x r x 10 4 rho x 10 4

23 23 Fgure A10: Posteror draws for mxng ndvdual-level parameters (Gbbs) mu eta sgma eta x x 10 4 mu delta sgma delta x x 10 4 mu lambda sgma lambda x x 10 4 FURTHER ROBUSTNESS CHECKS Senstvty analyss for the effect of swtchng costs on counterfactual analyses Our econometrc model assumes that customers choce decsons are based on the next perod only. Ths assumpton does not affect the estmates of our man varable of nterest,, but could potentally lead us to overestmate consumers senstvty to the swtchng fee, 1. If ths were the case, the effect of lowerng the swtchng fee on provder revenues could be lower than what our results about recommendatons to the frm suggest. We run a senstvty analyss to measure whether the effect of reducng the swtchng fee, as presented n n the man manuscrp, s robust to lower levels of 1. We reduce the estmate of 1 by 5%, 10%, and 20%. Fgure A11, Fgure A12 and Fgure A13 replcate the results obtaned n the man manuscrpt (see Fgure 3 of the man manuscrpt) for lower levels of 1. We fnd that the revenue mpact from lowerng the swtchng fee s very robust to lower levels of 1. In an addtonal analyss, we smlarly vary the level of the senstvty to cost of swtchng to a dfferent provder,

24 Revenue change Revenue change 2. We fnd that whle a lower senstvty to cost of leavng the provder affects the level of provder revenues, t does not change the optmal level of the swtchng fee. Hence, we are confdent that the assumpton that customers make tarff choce decsons takng nto account ther usage n the next perod only does not sgnfcantly bas our polcy smulatons. 24 Fgure A11: Change n revenue due to reducton of the swtchng fee f 1 s reduced by 5% 6% 4% 2% 0% -2% -4% -6% (a) Posteror mean for revenue change Delta set to zero Model Swtchng fee (n MUs) 6% 4% 2% 0% -2% -4% -6% 6% 4% 2% 0% -2% -4% -6% (b) Posteror nterval for Model (c) Posteror nterval for Delta set to zero Swtchng fee (n MUs) Fgure A12: Change n revenue due to reducton of the swtchng fee f 1 s reduced by 10% (a) Posteror mean for revenue change (b) Posteror nterval for Model 3 6% 4% 2% 6% 4% 2% 0% -2% -4% -6% 0% (c) Posteror nterval for Delta set to zero -2% -4% -6% Delta set to zero Model 3 6% 4% 2% 0% -2% -4% -6% Swtchng fee (n MUs) Swtchng fee (n MUs)

25 Revenue change 25 Fgure A13: Change n revenue due to reducton of the swtchng fee f 1 s reduced by 20% (a) Posteror mean for revenue change (b) Posteror nterval for Model 3 6% 4% 2% 0% 6% 4% 2% 0% -2% -4% -6% (c) Posteror nterval for Delta set to zero -2% -4% -6% Delta set to zero Model Swtchng fee (n MUs) 6% 4% 2% 0% -2% -4% -6% Swtchng fee (n MUs) Alternatve model specfcaton: Addtonal value for free mnutes only An alternatve way to buld our model would be to assume that three-part tarff customers assgn greater value only to mnutes strctly below the allowance, and not to all three-part tarff mnutes. In our data, three-part tarff usage mostly les beyond the allowance: 72% of three-part tarff observatons exceed the usage allowance, by an average of 88.4%. As a consequence, a behavoral theory that lmts the effect of free mnutes to usage below the allowance seems, n prncple, unable to explan the pattern n our data. To further confrm ths clam, we re-estmate Model 2 as presented n the man manuscrpt but allow the effect of free mnutes,, to apply to mnutes wthn the allowance only. We fnd that such a model does not reflect the phenomenon we observe well. Frst, the ft s worse than that of Models 2 and 3 (Model secton of the man manuscrpt) that assume that the addtonal valuaton apples to all three-part tarff mnutes. The MSE of the alternatve model s versus a MSE of n Model 2 and n Model 3. In the alternatve model we obtan a MAPE of versus a MAPE of 72.4 n Model 2 and of n Model 3. Second, we obtan a negatve

26 posteror mean of the varable relatng to the value of free mnutes. Ths estmate s negatve because n our sample customers generally consume above the allowance. As a consequence, a model that only estmates from mnutes wthn the allowance would overestmate the sataton 26 level for customers who swtch to a three-part tarff and often consume above the allowance (.e., the majorty of our three-part tarff customers). Then, n the perods n whch these customers consume wthn the allowance, needs to be negatve to compensate for the overestmaton of ther sataton level. A negatve delta cannot explan the usage ncrease observed n the data and documented n the man manuscrpt, and t s not consstent wth prevous lterature ndcatng that free would lead to ncreased valuaton of the good. We conclude that ths model specfcaton s not a good representaton of the phenomenon we observe. Table A8: Posteror dstrbuton of parameter estmates for model where apples to free mnutes wthn the allowance only Model 2 free mnutes only Mean 95% Interval Demand ntercept Mean, µ η Std. dev., σ η Demand slope, b Varance of usage shock, 1/r Valuaton of free unts Mean, µ δ Std. dev., σ δ Preferences n tarff choce, ζ jt SC bw. tarffs, ρ SC to other provder, ρ Preference for the three-part tarff Mean, µ λ Std. dev., σ λ Log Margnal Densty MSE ( 000) MAPE N=5,831 customers, 63,449 usage and 63,616 choce observatons Demographc shfters of the demand slope ncluded but not reported for readablty.

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