Harmful Upward Line Extensions: Can the Launch of Premium Products Result in Competitive Disadvantages? Web Appendix

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1 Harmful Upward Line Extensions: Can te Launc of Premium Products Result in Competitive Disadvantages? Faio Caldieraro Ling-Jing Kao and Marcus Cuna Jr We Appendix Markov Cain Monte Carlo Estimation e Markov cain Monte Carlo algoritm for estimating te empirical data is provided in tis appendix Let n e te total numer of respondents Let denote respondents Let * xt denote te optimal product quantity of rand at te respondent s te sopping trip t and x kt denote te oserved quantity coice x k of rand at te respondent s sopping trip t Let te superscripts and ) represent te pre- data and te post- data respectively ere are ntrip pre-launc sopping trip and ntrip ) post-launc sopping trips Let nvar denote te numer of pre independent variales including lnx kt ) Let nvar ) denote te numer of post independent variales including lnx ) kt ) Since te new item rand G + is only availale at te post sopping trip nvar ) nvar + z α ) is an nvar -) vector and ) ) z α ) ) is an nvar ) -) vector e - term is necessary ecause te coefficients of lnx kt ) and lnx ) kt ) are fixed to e for identification purpose e term ) G + denotes excluding te intercept of rand G + ere are nsku ) SKUs in te pre study and nsku ) SKUs in te post study As descried in te Empirical Model section te likelioods of te pre and post data are

2 { }) a e * * kt ) kt t ) ) t u x ) z x ) p x ) e e e var ) ~ ); Pr x Pr ln u x x > max ln u x ln + ln + ln + kt kt z kt kt t e N I t t t n t n; k nsku ; nrd ; t ntrip ; i e distriutions of pre- and post-study consumer eterogeneity are: ) ~ N V ~ N σ 3) ) i ) nvar ) + i ) ) ~ N µ V 5I V ~ IW f nvar ) 5 G f I nvar Estimation is carried out y sequentially generating draws from te following distriutions: Draw { u xkt ) n; k nsku ; nrd ; t ntrip ; i } and retain { εt n nrd t ntrip } e algoritm of generating pre and post latent utilities and teir corresponding errors are illustrated as follows First of all determine te utility maximizing quantity of rand y searcing over all availale packsize of rand : 4) { a ) ε } { zktz xkt ) a p xkt ))} x arg max z + ln x + ln p x + * t kt z kt kt kt k arg max + ln + ln k e error e t vanises ecause it is assumed to e te same for all packsizes wit te same rand name Let denote rand not purcased en generate te latent utility of cosen SKU from te univariate normal wit truncation region:

3 3 5) { }) * * kt kt ) ) kt t ) t ) * * * * kt t ) t z + ln t ) + a ln t ) u x ) ~ N u x I u x x > max u x u x x z x p x Let * x t e te optimal quantity of rand tat is not purcased y respondent at te sopping trip t e latent utility for any oter rand not purcased at te sopping trip t can e generated from 6) * * * t t ) ) kt ) > t ) * * * * t ) + ln t ) + α ln t ) u x ) ~ N u x I u x u x u x z x p x Because of te assumption tat all items wit te same rand name ave te same errors ε t te latent utilities of tose items not in te optimal set can e constructed y adding sared errors to te deterministic part of utilities Specifically after generating te latent utilities for all items in te optimal set te error realizations for eac rand can e otained y 7) ε u x ) * * * * i i i i i i t t z ln t ) α ln t ) u x * * t t t u x z + x + p x en te utilities of items not included in te optimal set can e computed y adding te error term ε i otained in Eq7) to te determinist part of latent utility: t 8) u x ) u x + ε kt kt t ln z + ln x + α p x + ε kt z kt kt t Draw and for i and n Let e te previous draw of i) p) and te next draw e given y i) n) i n i p + ς

4 4 wereς is a draw from a candidate generating density Normal 5) Let i) p) e te previ- ous draw of i and te next draw i ) n) is given y i n i p + ς wereς is a draw from a candidate generating density Normal 5) e coice for parameters of tis density ensures 3% acceptance rate p) Let ln t ) u x e te latent utility otained from Eq Given i ) n) and i ) n) te new value of latent utility ln n i t ) us te proaility of move i and and ) i) p) i) p) ) p p u x can e computed y adding up te error realizations ε i is given y i n) i n) exp V i n) exp ) σ i ) ntrip n) * i i) I y { u x ) k nrd }) i p) i p) exp V i p) exp ) σ i ) ntrip p) * i i) I yt { u xkt ) k nrd }) t i n i n t kt t e indicator function κ t suggests a procedure to ceck and see if te candidate draws of i ) n) and i ) n) can form an optimal set in wic te co- i) n) i) n) ) ) p p i) p) i) n) i) p) i) n) ) min i p i p i ) ntrip t n) * i t { kt ) }) I y u x k nrd

5 5 sen SKU as te greatest latent utility e candidate draws will e accepted if te cosen SKU is included in te new optimal set 3 Draw n V V m ~ N mean Var were ) Var n V + V n i mean Var V ) + V ) m 4 Draw V n V V V f G n ) ) exp / V ) V i ) i ) exp tr V ) G f + n var ) + ) V ~ IW n + f n i) + G Policy Analysis Estimation e algoritm for estimating te policy analysis estimation is provided in tis appendix In tis policy simulation prices are set according to te prices in te real data te estimated udget constraints and following principle: - Post price of rand R pre price of rand R; - Post price of rand G pre price of rand G;

6 6 - Price of rand G + > price of rand R> price of rand G; - Prices of rand S are as in te real data Note tat prices cannot e set aove certain tresold owing to te estimated udget constraints us in tis simulation study te maximum increase of usage-unit price is 8 Let te superscript i) denote te stage of study e pre study is indicated y i and te post study is indicated y i e term i represent te simulated coice of te respondent at t te sopping trip t U i is an nsku i) y vector wic represent te t respondent s latent utili- ties of all SKUs at te sopping trip t i are model parameters wic represent consumer preferences to rands product attriutes and outside goods y t denotes te udget constraint of te respondent e is an nrd i) y vector of random components of latent utilities t According to te empirical model consumer coices are simulated y te following procedure: Step : Simulate pre and post prices according to te price assumption Step : Generate e ε ε ε i ) ) ~NI) t t t nrd t Step 3: Given te posterior draws of and e t compute te latent utility of all SKUs Note tat te SKUs wit te same trademark ave te same random error ε t Step 4: Given te latent utilities determine te rand coice Repeat Step to Step 4 for all posterior draws After otaining all simulated rand coices te expected rand coice can e computed y te expression:

7 7 Pr y y U U e d d de t t t t t t maxiter max iter ˆ ˆ ˆ ˆ y ˆ t Ut Ut et iter Measuring te impact of price cange on utility Let pw k ) e te price of te K t SKU of rand wit te packsize w e log latent utility of purcasing w k is defined as: ln u w ) z ln w ) ln p w )) ε k k z k k k o measure te impact of price cange on utility take te first derivative of utility wit respect to price pw k ): k ) p w ) u w p w k ln k p w k p wk )) Since > and -pw k ) > wen te price of SKU w k increases its latent utility will decrease y p w k