Optimal Marketing Strategies for a Customer Data Intermediary. Technical Appendix

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1 Optal Marketng Strateges for a Custoer Data Interedary Techncal Appendx oseph Pancras Unversty of Connectcut School of Busness Marketng Departent 00 Hllsde Road, Unt 04 Storrs, CT oseph.pancras@busness.uconn.edu Phone: Fax: Sudhr Yale School of Manageent 35 Prospect St, PO Box 0800 ew Haven, CT Eal: k.sudhr@yale.edu Phone: Fax: Septeber 006

2 A. The Prcng Equatons Retaler Fro (5, the retaler s optzaton proble s as follows. r t,, r t Rt r w f S r D Mt = = ax Π = [ ]*[ ( ]* t (A For the purposes of the dervaton, we drop the superscrpts x and y ndcatng whether a anufacturer bought targetng servces and the subscrpt t that ndexes teperod for clarty. These can be ncluded approprately nto the fnal wholesale and retal argns. Hence the retaler obectve s: ax Π [ ]*[ ( ]*,, R = r w f S r D Mt r r = = Takng the dervatve of the obectve functon wth respect to the retal prces, the followng frst order condton for each product s: ( r D k [ r w] + f S( r D = 0 = = where w s the wholesale prce charged by anufacturer to the retaler for brand. (A Defne Θ R as the frst dervatves of all the (ndvdual consuers shares wth respect to all retal prces (retal prces are coon across consuers, wth eleent (, = ( r. The retaler frst order condtons can then be wrtten n atrx for as: = [ ] ΘR R + S = 0 where R s the vector of retal prces and s the vector of wholesale prces (whch are coon across all consuers and all the brands: S s the vector of shares for each consuer over

3 r w S R,, S r w S x x x The vector of retal argns [ R ] s obtaned by nvertng the above atrx equaton: [ R ] *[ ] = = R = Θ S Retal Margn (A3 where the shares are: S f S f S f S x and the ndvdual specfc share dervatve atrx wth respect to retaler prces s: p p p Θ R = = p p p x ] k f αs [ S ] f αs S f αs S k f αs S f αs S f αs[ S k = Therefore the retal prce s gven by [ R] *[ ] = = (A5 R = Θ S x (A4 Manufacturer A anufacturer offerng a subset wholesale prce ( D ℵ of brands n the arket sets the w (where ℵ and the coupon face values to ndvdual households so as to axze the anufacturer s profts. A anufacturer who has not been

4 sold the : arketng servce wll have coupon face values set to zero. The anufacturer takes nto account the knowledge that retaler prces ( r wll be set takng nto account the wholesale prces and the coupon face values that have been ssued to ndvdual households. t t [ w D c ]*[ S ( r ( w, D D ]* M t ℵ = Π = (A6 where c s the argnal cost of the anufacturer for brand n perod t, and S ( r ( w, D D s the probablty of household, buyng brand n perod t gven the decsons of anufacturers (denoted by x and (denoted by y to purchase the purchase hstory data, and M t s the total sze of the arket n perod t. e present the frst order condtons for the anufacturer droppng the x, y superscrpts and the t subscrpt and wrtng retal prce as r (not as r ( w, D for clarty. e wrte w = w D snce the anufacturer sets both the wholesale prce and the ndvdual coupon face values to axze proft. As dscussed earler, even though the anufacturer sets the wholesale prce and Catalna sets the coupon face value, analytcally t does not atter whether we ake ths dstncton. The frst order condton wth respect to w s: ds ( r D [ ]* + ( = 0 w c S r D = ℵ (A7 Defne Θ for each ndvdual consuer such that t contans the frst dervatves of all the (ndvdual consuers shares wth respect to all wholesale prces (wholesale prces are coon across consuers, wth eleent (, = S ( r D. To account w for the set of brands owned by the sae anufacturer, defne the anufacturer s ownershp atrx O such that eleent (, s equal to one f the anufacturer who sells brand also sells brand, and zero otherwse. The anufacturer s frst order condton can then be wrtten n atrx for as:

5 [ O Θ][ C] + S = 0 (A8 = where [ O Θ ] s the eleent by eleent ultplcaton of the two atrces, s the vector of wholesale prces less the ndvdual coupon values, C s the vector of argnal costs of the anufacturer (C s coon across all consuers, and shares for each consuer : w D c S =, C, S = = w D c S X X X S s the vector of Fro the anufacturer frst order condtons, we can wrte the anufacturer argn fro a partcular household [ C] as follows: [ C] [ O ] *[ S ] = Θ (A9 The share dervatves wth respect to wholesale atrx Θ need to be calculated. As entoned earler, the anufacturer response atrx has the eleents (, = ( r D. Defne the acoban atrx of dervatves of all retal prces to all w wholesale prces (for consuer as be re-wrtten as: rw dr ( x, wth the eleent(,x =. Then Θ can Θ = rwθ R (A0 In the Manufacturer Stackelberg gae, anufacturers antcpate how the retaler wll respond to changes n wholesale prces and use these reactons when settng wholesale dr ( x prces. e can solve for the retal reactons by takng the total dervatve of the retaler s frst order condton wth respect to the retal prce r and the wholesale prce w : Vllas Boas and Hellersten (006 dscuss two condtons to assure the nvertblty of the acoban atrx: ( addtve separablty of costs across products for retaler and anufacturer and ( no nteracton between anufacturer and retaler costs. Snce these assuptons are antaned n our paper, the acoban s nvertble.

6 dr Ψ * dr where dr dr dr dr x = x Ψ S S + [ r -w] + + [ r ] -w = = = S S + + [ r -w] [ r -w] + = = x (A The second dervatves are obtaned for these relatonshps of a,b,c (where there s an equalty sgn, the ndex a wll be preferred to c or b f a s n the equalty, and b wll be preferred to c f b s n the equalty. : α *S *[ S ]*[ *S ] a = c = b *α *S *S *S a c b a a a a b c Sa a b a c b a c a a b b = α *S *S *[*S ] a = c b α *S *S *[*S ] a = b c α *S *S *[*S ] a c = b (A rtng the total dervatve of the retaler s frst order condton n atrx for: Ψ *[ ] =Θ T r R T where[ ] s the transpose of the atrx r T r. Therefore r s obtaned as: r = [ Ψ ] * Θ R (A3 The wholesale prce to the retaler s gven by specfc dscount s gven by D = w w. w = ax w and the ndvdual B. Endogenety Correcton e correct for prce endogenety usng the control functon approach developed n Petrn and Tran (004. The control functon approach has slartes to Rvers and

7 Vuong (988 and Vllas Boas and ner (999. The control functon approach (Hausan 978 uses extra varables to control for the part of the unobserved coponent of deand that s correlated wth prce. In prncple, the control functons are constructed usng as arguents the dfferences between observed prces and the predcted prces whch are arrved at usng all the relevant deand and supply varables observed by the econoetrcan. Consder the utlty equaton: u = X β r α + ξ + ε (B and rewrte t ncorporatng the control functon as: ( ( u = X β r α + f µ ; ω + [ ξ f µ ; ω ] + ε (B where ( µ ;ω f s the functon that controls for the correlaton of the unobserved coponent ξ wth the prce r, µ are control varables used n such a correcton, and ω are the coeffcents for = [ f ( ; ]. If the functon ( µ ;ω η ξ µ ω µ. Let the redefned unobserved coponent be f could be constructed and added to the utlty functon, t s clear fro equaton (B that the resultng rando coponent η + ε would no longer be correlated wth prce (by constructon, and the estates obtaned would be corrected for prce endogenety. Petrn and Tran (004 show that (under a wde range of condtons the control functon ( µ ;ω f s lnear n the prce resduals of a regresson of prce on ts prtves. In our context, we estate a regresson of prces aganst factor costs as follows: r = κ + ς * + µ where B t Β t are the factor prces, fro ths regresson. Thus ( κ are brand specfc ntercepts and f µ ; ω = ωµ, and we wrte equaton (B as : µ are the resduals u = X β r α + ωµ + η + ε (B3 Ths utlty equaton (B3 s used n estatng the latent class odel rather than equaton ( of the text to perfor the endogenety correcton. Dfferent specfcatons can be used for ω (Petrn and Tran 004 pages 5-6, and we present the results where ω s ] = segent-specfc,.e., [ ω.