Online Appendix. to Add-on Policies under Vertical Differentiation: Why Do Luxury Hotels Charge for Internet While Economy Hotels Do Not?

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1 Onine Appendix to Add-on Poicies under Vertica Differentiation: Wy Do Luxury Hotes Carge for Internet Wie Economy Hotes Do Not? Song Lin Department of Marketing, Hong Kong University of Science and Tecnoogy

2 Onine Appendix to Add-on Poicies under Vertica Differentiation: Wy Do Luxury Hotes Carge for Internet Wie Economy Hotes Do Not? List of Contents A. Proof of Proposition 1 B. Extension 1: Partia Market Coverage and Positive Base Cost C. Extension 2: Asymmetric Add-on D. Extension : Asymmetric Margina Cost of te Base Good E. Extension 4: Correated Tastes F. Proof of Proposition 2 and Two Additiona Resuts on Commitment G. Proof of Proposition H. Proof of Proposition 4

3 A Proof of Proposition 1 A-1 A Lemma for Firm L To prove te proposition, te foowing emma for firm L wi be usefu. Lemma A-1. For firm L, not seing te add-on is stricty dominated by seing it, if: (1) P > α wen firm H ses optiona add-on, or (2) P + standard add-on. > α( + w) wen firm H ses Te proof of te emma foows te simiar ogic of te proof of Lemma 1 in te main text: (1) assume tat firm L ses ony te base; (2) find te equivaent tota price and add-on price, () consider a sma deviation by owering te tota price and te add-on price; (4) sow tat te deviation is profitabe. In wat foows, I prove te first case wen H ses te add-on optionay. Te proof of te second case foows te same argument. Suppose tat firm L ses ony te base to consumers at price P. Ten te margina consumer indifferent between te two firms becomes θ = (P P )/. Firm L s profit is Π = (θ θ)p. Tis strategy is exacty equivaent to carging a bunde price P + add-on price p suc tat P + = P + p, and tat and an P P + w = P P = p w. (A-1) Consider a sma decrease of te bunde price to P + = P + ɛ wit ɛ > 0, and a sma decrease of te add-on price to p = p ɛ by te same amount. Te base price P remains uncanged. Tis resuts in a segmentation of consumers because Firm L s profit now becomes P P + w > P P > p w. Π (P +, p ) = ( P P + w θ)(p + c) ( p w θ)(p c). Evauating tis function around te origina prices (P +, p ) by Tayor series yieds: Π (P +, p ) = Π (P +, p ) ɛ Π P [ ɛ 2 Π 2 P + P + (P +, p ) ɛ Π p (P +, p ) (P +, p ) + 2ɛ 2 Π P + p (P +, p ) + ɛ 2 Π p p (P +, p ) ] + R 2

4 = Π (P +, p ) ɛ [ P P + w θ P + c w ] + ɛ [p w θ + p c w ] + ɛ2 2 [ 2 w 2 w ] = Π (P +, p ) + ɛ w (P ɛ 2 α ) w( w), }{{} M were te second equaity foows from te remainder term R 2 = 0 because te iger order derivatives are a zero, and te ast equaity foows from Equation (A-1). Terefore, M > 0 for sma positive ɛ wen P > α. A-2 Equiibrium Wen α 1 (2θ θ) To derive te equiibrium profie under tis case, note tat te indifferent consumer becomes θ = (P P )/. Te two firms profits are given by Π = (θ P P )P + (θ p w )(p c); Π = ( P P θ)p. It is straigtforward to find te foowing equiibrium prices using te first-order conditions: P = 1 (2θ θ), p = wθ ; P = 1 (θ 2θ), p > wθ, were te tresods are given by θ = 1(θ + θ) and θ = 1 (θ + α). Next I first estabis 2 tat tere is no profitabe deviation for eiter firm, and ten sow tat tis is te unique equiibrium in tis parameter range. A-2.1 No Profitabe Deviation for Firm L Tis is estabised by considering tree possibe non-oca deviations. Case (a): firm L does not se te add-on and a consumers of firm H buy te add-on. Te margina consumer indifferent between te two firms becomes θ = (P + P )/( +w). For tis case to arise, we need tat θ p /w so tat no consumer buys te base ony from H. But note tat in equiibrium te incentive constraint as to satisfy: θ < θ P P < p w + P P = + w < p w. (A-2) Firm L s deviation profit is given by Π = (θ θ)p. It turns out tat te optima profit under tis deviation case is acieved by te corner soution θ = p /w. To see tis, note 4

5 tat te first-order condition of te Lagrangian is given by P = (P + ( + w)θ λ)/2. Ten, θ p w = P + P + w p w < P P + w = 1 2( + w) ( p + wθ + λ), were te inequaity foows from Equation (A-2) and te ast equaity is obtained by substituting te best responses of firm L. Because p > wθ, to guarantee tat te constraint is nonnegative, we need λ > 0. Tis impies tat te constraint as to be binding by compementary sackness. Tis corner soution ten coincides wit te corner soution for te probem under te proposed equiibrium (wit L not seing te add-on and H seing it optionay) because: P + P + w = p w = P P Terefore, te deviation does not improve firm L s profit. = p w. Case (b): firm L ses te add-on and some consumers of firm H do not buy te add-on. First, et us consider te scenario wereby firm L ses te add-on optionay. Te market is divided into four consumer segments as in Figure 2 in te main text. Te tresods are given by θ = p /w, θ = (P P p )/( w), and θ = p /w. It wi ater become evident tat wen firm L ses te add-on to a of its consumers (i.e., θ < θ) te profit is not improved. Given firm H s equiibrium pricing, firm L maximizes its profit Π = (θ θ)(p + c) (θ θ)(p c) subject to te constraints: (1) θ θ, (2) θ θ, and () θ θ. Te objective function is concave and tus te necessary and sufficient condition for te optimization program is te Karus-Kun-Tucker conditions. Let λ 1, λ 2, and λ be te Lagrangian mutipiers for te tree constraints. Te first-order conditions ead to P + = (P ( w)θ + c λ 1 + λ )/2 and p = (wθ + c λ 1 + λ 2 )/2. Terefore, Constraint (1) can be rewritten as [ 1 θ θ = w ( 1 ] 2w( w) (2θ θ) α) λ 2( w) λ w +λ 1 0. }{{} M Because α > (2θ θ)/ and bot λ 2 and λ are nonnegative, te term M in te bracket is stricty negative (M < 0). To guarantee tat te constraint is nonnegative, we need λ 1 > 0. Tis impies tat Constraint (1) as to be binding. Terefore, te profit is no greater tan tat obtained from not seing te add-on (i.e., te equiibrium strategy). It remains to verify tat firm L wi not improve te deviation profit if it were to se te 5

6 add-on to a of its consumers, in wic case te profit is Π = (θ θ)(p + c). Tis profit can be stricty improved if te firm sets te add-on price p to be anywere between [θw, c] so tat some consumers decide not to buy te add-on. Te incrementa benefit is equa to (θ θ)(p c) wic is stricty positive. Case (c): firm L ses te add-on and a consumers of firm H buy te add-on.. Te margina consumer indifferent between te two firms becomes θ = (P + P + )/. To ensure tat a consumers buy te add-on from H, it is necessary tat θ p /w. Consider tat firm L maximizes its profit Π = (θ θ)(p + c) (θ θ)(p c). Te profit differs from Case (b) ony in terms of te profit of seing te bunde, (θ θ)(p + c). Maximizing tis bunding profit under te constraint yieds P + = (P + θ + c λ)/2. Substituting tis best response back to te constraint gives θ p w = P + P + p w = 1 [ 2 1 (θ 2θ + α) + 1 (θ α)w }{{ 2 } M ] +λ 0. Note furter tat, M 1 (θ 2θ + 2θ θ) + 1 (θ (2θ θ))w 6 < (θ θ)w + 1 (θ + θ)w 6 < w 2, were te first inequaity foows from α (2θ θ)/, te second from > w, and te ast from θ > 2θ. Since M < 0, for te constraint to be nonnegative, it is necessary tat λ > 0. Tis impies tat te constraint as to be binding by compementary sackness. Terefore te profit is no greater tan tat obtained from te previous Case (b), wic is proved to be not profitabe. A-2.2 No Profitabe Deviation for Firm H Case (a): firm H ses te add-on as optiona and some consumers buy L +. Te margina consumer indifferent between te two firms becomes θ = (P P p )/( w). Consider firm H s maximization of profit, Π = (θ θ )P +(θ θ )(p c), subject to te constraint θ p /w. Notice tat te profit function is te same as te one of te proposed equiibrium except te tresod θ. It ten suffices to compare te profits from seing te base ony. Te first-order condition of te constrained probem is given by P = (P +p +( w)θ+λ)/2. 6

7 Terefore, θ p w = 1 2( w) [2 (θ + θ) θw p (2 w) + λ] w 1 2( w) [2 (θ + θ) θw 1 (θ + θ)(2 w) + λ] 1 = 2( w) ( 1 (2θ θ)w + λ), were te inequaity ods because p = 1 (θ+α)w (θ+θ)w/. Because (2θ θ) is positive, 2 to ensure tat te constraint is nonnegative it is necessary tat λ > 0. By compementary sackness te constraint must be binding, impying tat firm H soud eave no demand for L +. Terefore, te optima profit obtained in tis case is no greater tan te equiibrium profit. P + Case (b): firm H ses te add-on as standard and some consumers buy L +. Because > 0, by Lemma 1 tis strategy is stricty dominated by a oca deviation wereby firm H ses te add-on optionay by increasing te add-on price wie owering te base price. Tis deviation is exacty in te form of Case (a), wic yieds no greater profit tan te equiibrium one. Case (c): firm H ses te add-on as standard but no consumer buys L +. Because P > 0, by Lemma 1 tis strategy is stricty dominated by a oca deviation wereby firm H ses te add-on optionay by increasing te add-on price wie owering te base price. Tis oca deviation is exacty in te form of te maximization probem in te proposed equiibrium. Terefore, tere is no profitabe deviation. Case (d): firm H does not se te add-on at a. Tis deviation can take two forms. First, firm H can eave no demand for te bunde of L +. Te optima base price is ten te same as te equiibrium strategy P. However, te deviation forgoes te profit from seing te add-on, eading to reduced tota profit. Second, firm H can eave some demand for te bunde of L +. Te profit is ess tan tat in Case (a) because some positive saes of te add-on can improve te profit witout affecting te profit of seing te base good. Terefore, in eiter case, te deviation is not profitabe. 7

8 A-2. No Oter Equiibrium Exists Te ony possibe aternative equiibrium profie is suc tat firm L ses te add-on to some consumers, eading to te foowing pricing strategies: P = 1 (2θ θ)( w) + 1 c, p = wθ ; P + = 1 (θ 2θ)( w) + 2 c, and p = wθ if α > θ but p wθ if α θ. Te tresods are given by θ = (θ + θ)/ c/( w), θ = (θ + α)/2, and θ = (θ + α)/2. To verify tat tis equiibrium does not sustain, it suffices to notice tat θ < θ A- Equiibrium Wen α < 1 (2θ θ) wen α 1 (2θ θ). Reca tat te equiibrium profie were firm L ses te add-on optionay is summarized in Section 2. of te main text. Te next two subsections estabis te second and tird statements of te proposition by sowing tat neiter firm wi deviate wen > max{ 1, 2 }. Te ast subsection sows tat no pure strategy equiibrium exists wen max{ 1, 2 }. A-.1 No Profitabe Deviation for Firm L Case (a): firm L does not se te add-on and some consumers of firm H buy te add-on. By Lemma A-1 tis is not a profitabe deviation as ong as P > α. Notice tat θ θ = P P + w p w = 1 w [P 1 2 (P + c θ( w))] 1 (θ + α) 2 1 = 2( w) (P α ), were te second equaity resuts from substituting te best-response prices of firm L. Terefore, as ong as α < 1 (2θ θ) and > 1 (te conditions for θ > θ ), we ave P > α and tus te deviation is not profitabe. Case (b): firm L does not se te add-on and a consumers of firm H buy te add-on. By Lemma A-1 tis is not a profitabe deviation as ong as P + P + α( + w) = P α + p c > P α > 0, > α( + w). Notice tat were te first inequaity foows from p > c, and te second inequaity foows from te resut of Case (a). Hence, any deviation in tis case is not profitabe. Case (c): firm L ses te add-on and a consumers of firm H buy te add-on. Te margina consumer indifferent between te two firms now becomes θ = (P + P + )/. 8

9 To ensure tat a consumers buy te add-on from H, firm L as to set te tota price P + sufficienty ow so tat θ p /w. But reca tat te margina consumer in te equiibrium profie satisfies: θ < θ P P + w < p w + P P + = < p w. (A-) Firm L s deviation profit is given by Π = (θ θ)(p + c) (θ θ)(p c), wic is different from te equiibrium profit ony in terms of te bunde component (θ θ)(p + c). Te first-order condition of te Lagrangian for te constrained maximization of tis bunde profit is given by P + = (P + θ + c λ)/2. Te constraint becomes θ θ = P + P + p w < P + P + = 1 2 ( p + wθ + λ), were te inequaity foows from Equation (A-) and te ast equaity is obtained by substituting te best responses of firm L. Because p > wθ, to guarantee tat te constraint is nonnegative, we need λ > 0. Tus, te constraint as to be binding. Te corner soution exacty coincides wit te one for te probem under te equiibrium because P + P + = p w = P P + w = p w. Hence, te deviation profit is no greater tan tat obtained from te equiibrium strategy. A-.2 No Profitabe Deviation for Firm H Case (a): firm H ses te add-on as optiona but no consumer buys L +. Te margina consumer indifferent between te two firms is θ = (P P )/. Firm H maximizes te profit Π = (θ θ )P + (θ θ )(p c) subject to te constraint, θ p /w, so tat tere is no demand for consumers wo buy L +. Note tat te profit is aso separabe into a component for te base good and a component for te add-on price. It suffices to examine te profit component of te base good wic is different from te equiibrium one. Te first-order condition for te constrained optimization probem of te base profit is P = (P + θ λ)/2. Tere are two cases to consider. First, if α > θ, ten te equiibrium add-on price is p = (θw + c)/2 and satisfies θ > θ. Te constraint becomes θ θ = p w P P < P P = 1 2 (p θw + λ), 9

10 were te inequaity foows from θ > θ, and te ast equaity is obtained by substituting te best responses of firm H. Because p < θw, for te constraint to be nonnegative we need λ > 0, impying tat te constraint must be binding. Te corner soution is te same as te one under te equiibrium probem because: P P = p w = P P + w = p w. Terefore te deviation profit is no greater tan te equiibrium profit. Second, if α θ, ten p θw. Tere is no demand for firm L at a. Te equiibrium strategy is optima. P + Case (b): firm H ses te add-on as standard and some consumers buy L +. Because > 0, by Lemma 1 tis strategy is stricty dominated by seing te add-on optionay by increasing te add-on price and owering te base price. Tis is exacty te equiibrium strategy. Terefore, deviation in tis case is not profitabe. Case (c): firm H ses te add-on as standard but no consumer buys L +. Firm H s profit becomes Π = (θ θ )(P c) subject to te constraint, θ p /w, were te indifferent consumer becomes θ = (P + P )/( + w). Te first-order condition for te constrained optimization probem of te base profit is P + = (P + c + ( + w)θ λ)/2. Tere are two cases to consider. First, if α > θ, ten te equiibrium add-on price is p = (θw + c)/2 and satisfies θ > θ. Te constraint under deviation becomes θ θ = p w P + P + w = 1 2( + w) [P (θ θ α)( + w) c +λ], }{{} It can be sown tat M < 0 using α < 1 (2θ θ). For te constraint to be nonnegative we need λ > 0, impying tat te constraint must be binding. Te corner soution is te same as te one under te deviation in Case (b) because: M P + P + w = p w = P + P + = p w, impying tat te deviation is not profitabe. Second, if α θ, ten p θw. Tere is no demand for firm L at a. Te equiibrium strategy is optima. Case (d): firm H does not se te add-on at a. Tis deviation can take two forms. First, firm H can eave no demand for buying te add-on from firm L. Te profit is smaer tan tat in Case (a) because a sma positive saes of te add-on can improve profit witout 10

11 affecting profit from te base good. Second, firm H can eave some demand for te owquaity bunde L +. Te optima base price is ten te same as te equiibrium strategy P. However, te deviation forgoes te profit from seing te add-on, resuting in a ower profit tan te equiibrium profit. Hence, in eiter case, te deviation is not profitabe. A-. No Oter Equiibrium Exists Wen > max{ 1, 2 } Te ony possibe aternative equiibrium is suc tat firm L does not se te add-on, resuting in a pricing profie described in Section A-2 in tis appendix. Firm H s base price is P = (2θ θ)/. Since α < (2θ θ)/, we ave P > α. Terefore, by Lemma A-1 firm L can profitaby deviate by seing te add-on to some consumers. A-.4 No Pure-Strategy Equiibrium Exists Wen max{ 1, 2 } Because in any possibe equiibrium firm H wi aways se te add-on as optiona by Observation 1 in te main text, tere are two possibe equiibrium outcomes depending on firm L s impementation: Case (a): firm L ses te add-on. Te equiibrium profies are te same as tose in te second and te tird statements of te proposition. Wen α > θ, bot firms se te add-on optionay. Te fraction of consumers wo buy te bunde from firm L is given by θ θ, wic is positive ony if > 1. 1 Wen α θ, firm L wi se te add-on as standard, and te fraction of consumers wo buy te bunde from te firm becomes θ θ wic is positive ony if > 2. Hence, no equiibrium is sustainabe in eiter case if max{ 1, 2 }. Case (b): firm L does not se te add-on. Foowing te same proof of A-., firm L can profitaby deviate by seing te add-on to some consumers. B Extension 1: Partia Market Coverage and Positive Base Cost Let C > 0 be te margina cost of te base good. Furter, te market is not fuy covered and assume tat θ > 2C/V to ensure tat te market can accommodate bot firms. Te first observation is tat firm H s incentive to price discriminate is reinforced under vertica differentiation, just as in Lemma 1 and in Observation 1 in te main text. Lemma B-1. For firm H, seing te add-on as standard is stricty dominated by seing it as optiona, if: (1) P C > α wen firm L does not se te add-on, or (2) P + C > α( w) wen firm L ses te add-on. Observation B-1. In any pure-strategy equiibrium, firm H ses te add-on as optiona. 1 Note tat 1 > 2 wen α > θ, and 2 > 1 wen α < θ. 11

12 Te equiibrium outcomes ten rest on firm L s beaviors. Define α 1, α 2, α as foows: 2(V + )θ C 2(V + )( w)θ ( w)c α 1 =, α 2 = V + 4 (V + )( w) + ( w), (B-1) α = w [ (V + 2 w)θ (2V + 4 w) C ]. (B-2) (V + )w V It can be sown tat α 1 > α 2 > min{ C V, α }. Te foowing proposition summarizes a te pure-strategy equiibria of te game. Proposition B-1. (1) If α α 1, tere exists a unique equiibrium in wic firm H ses te add-on as optiona wereas firm L does not se it; (2) If C < α < α V 2, tere exists a unique equiibrium in wic bot firms ses te add-on as optiona; () If α min{ C, α V }, tere exists a unique equiibrium in wic firm H ses te add-on as optiona wereas firm L ses it as standard; (4) No pure-strategy equiibrium exists in oter parameter regions. Te proof foows exacty te same ine of proof in Onine Appendix A. First, a emma for firm L is obtained: Lemma B-2. For firm L, not seing te add-on is stricty dominated by seing it, if: (1) P C > α wen firm H ses optiona add-on, or (2) P + H ses standard add-on. C > α( + w) wen firm Next I derive te equiibria and some of te tedious proof tat no profitabe deviation exists wi be omitted as te proof is very muc simiar to tat in Onine Appendix A. B-1 Equiibrium Wen α α 1 Te best response prices in tis case are: P = 1 2 [P + θ + C]; P = Te resuting equiibrium profie becomes: P = P = V 2(V + ) P C. 1 V + 4 [2(V + ) θ + (V + )C], p = wθ; 1 [V θ + (V + 2 )C], V + 4 p > wθ, were te tresods are given by θ tat tere is no profitabe deviation for eiter firm under tis equiibrium. = (V +2 )θ+c V and θ = 1 (θ + α). It can be verified 2

13 Te ony possibe aternative equiibrium profie is were firm L ses te add-on to some consumers. Te equiibrium profie consists of te foowing pricing strategies: P = P + = V + V + 4 w [2( w)θ + C + c], p = wθ; 1 [(V + w)( w)θ + (V w)c + 2(V + )c], V + 4 w and p = wθ if α > θ but p wθ if α θ. To verify tat tis equiibrium does not sustain, it suffices to notice tat θ < θ wen α B-2 Equiibrium Wen α < α 2 2(V + ) V +4 w θ. Under tis equiibrium, firm H ses te add-on as optiona wie firm L ses it eiter as optiona or as standard. Firm L maximizes its profit: Π = (θ θ)(p + C + ) (θ θ)(p c) (θ 0 θ)(p + p C). Te first-order conditions are P + = 1[ V +w P 2 V + + C + ] and p = 1[ w P 2 V + + c]. Notice tat, regardess of firm H s prices, te demand of consumers wo buy te base witout te add-on from firm L is given by θ θ 0 = p w P V = 1 2 (α C V ). (B-) Terefore, if α > C V, firm L ses te add-on as optiona, but as standard if α C V. Now consider firm H s profit maximization probem: Te first-order condition for P is P = (P + ead to te foowing equiibrium strategies: P = P + = Π = (θ θ )(P C) + (θ θ )(p c). + ( w)θ + C)/2. Te best-response prices V + V + 4 w [2( w)θ + C + c], p = wθ; 1 [(V + w)( w)θ + (V w)c + 2(V + )c], V + 4 w and p = wθ if α > θ but p wθ if α θ. Te constraint θ > θ is equivaent to P C > α, wic ods if and ony if α < α 2. Wen α C, firm L ses te add-on V as standard. To guarantee equiibrium existence, te constraint becomes θ C/V, wic eads to α α. Hence, te equiibrium exists if and ony if α min{ C, α V }. It can be 1

14 verified tat tere is no profitabe deviation for eiter firm under tis equiibrium. To sow tat te above equiibrium is unique wen α [0, min{ C V, α }] ( C V, α 2), it suffices to consider te ony aternative equiibrium profie in wic firm L does not se te add-on. Te equiibrium profie is summarized in Section B-1. Firm H s base price is ten P = (2θ + C)(V + )/(V + 4 ). It foows tat P C > α if and ony if α < α 1. Hence, by Lemma B-2 firm L can profitaby deviate by seing te add-on. B- No Oter Pure-Strategy Equiibrium Exists Te ony parameter region eft to consider is α [α 2, α 1 ) and α (α, C V ) if α < C V. Because in equiibrium firm H wi aways se te add-on as optiona by Observation B-1, tere are two possibe equiibrium outcomes depending on firm L s impementation. Case (a): firm L ses te add-on. Te equiibrium profies are te same as tose in te second and te tird statements of te proposition. Wen α > C/V, bot firms se optiona add-on. Te fraction of consumers wo buy te bunde from firm L is given by θ θ wic is positive ony if α < α 2. Wen α C/V, firm L wi se te add-on as standard, and te fraction of consumers wo buy te bunde from firm L becomes θ θ wic is positive ony if α < α. In eiter case te equiibrium is not sustainabe. Case (b): firm L does not se te add-on at a. Foowing te same proof above, firm L can profitaby deviate by seing te add-on. C Extension 2: Asymmetric Add-on Consider te extension tat te cost and vaue of te add-on are different for te two firms: (c, w ) for firm H and (c, w ) for firm L. Ten te cost-to-vaue ratio aso varies across te firms: α = c /w and α = c /w. No restriction is paced on weter te cost and/or vaue soud be iger for firm H tan for firm L, and tus te cost-to-vaue ratio can be eiter iger or ower for firm L. It is, owever, assumed tat te quaity premium remains iger tan te maximum vaue of te add-ons, > max{w, w }. In addition, te assumption tat te cost of te add-on is not unreasonaby arge, α < θ, sti appies so tat firm H aways ses te add-on in equiibrium. Oter assumptions foow from te main anaysis. Wit tis specification, te concusions in te preceding anaysis are quaitativey uncanged. Let 1, = 2θ θ α w, and 2, = θ 2θ + α w. 2θ θ α θ 2θ Proposition C-1. (1) If α 1 (2θ θ), tere exists a unique equiibrium in wic H ses 14

15 te add-on as optiona wereas L does not se it; (2) If θ < α < 1 (2θ θ), tere exists a unique equiibrium wen > 1,, in wic bot firms se te add-on as optiona; () If α θ, tere exists a unique equiibrium wen > 2,, in wic H ses te add-on as optiona wereas L ses it as standard; (4) No pure-strategy equiibrium exists in oter parameter regions. Te proof of te proposition argey foows te one for Proposition 1 of te main mode. Tus, I wi just derive te equiibrium pricing in wat foows and omit te tedious proof tat no deviation can improve equiibrium profit. C-1 Equiibrium Wen te Add-on is Too Costy (α 1 (2θ θ)) In tis case, te indifferent consumer becomes θ = (P P )/. Te two firms profits are given by Π = (θ P P )P + (θ p )(p c ); w Π = ( P P θ)p. It is straigtforward to find te foowing equiibrium prices using te first-order conditions: P = 1 (2θ θ), p = w θ ; P = 1 (θ 2θ), p > w θ, were te tresods are given by θ = 1 (θ + θ) and θ = 1 2 (θ + α ). C-2 Equiibrium Wen te Add-on is Not Too Costy (α < 1 (2θ θ)) In tis case, te indifferent consumers become θ = (P P + )/( w ). Te two firms profits are given by Π = (θ P P + )P +(θ p )(p c ); Π = ( P P + w w θ)(p + w It is straigtforward to derive te soution using first-order conditions: c ) ( p w θ)(p c ). P = 1 (2θ θ)( w ) + 1 c, p = w θ, P + = 1 (θ 2θ)( w ) + 2 c = w, p θ if α > θ w θ if α θ were te tresods are given by θ = (θ + θ)/ c /( w ), θ = (θ + α )/2, and θ = (θ + α )/2. 15

16 D Extension : Asymmetric Margina Cost of te Base Good Consider next te extension tat te margina cost for firm H is iger. Te margina cost of firm L remains zero by normaization, but te cost for firm H now becomes C 0. Define te cost-to-vaue ratio for te base good as A = C/, wic measures te unit cost of te quaity premium. Let A = (θ 2θ + α) (θ 2θ + α)w. 2 Tis is te bound above wic it is costy for firm H to serve additiona consumers wo buy ony te base by owering te base price. Terefore we sa focus on te case A < A. 2 Oter assumptions foow from te main anaysis. Let 1 = max{w, 2θ θ α 2A + 2θ θ α w}, and 2 = max{w, θ 2θ + α A + θ 2θ w}. Proposition D-1. (1) If α 1(2θ θ) + 2 A, tere exists a unique equiibrium in wic H ses te add-on as optiona wereas L does not se it; (2) If θ < α < 1(2θ θ) + 2 A, tere exists a unique equiibrium wen > 1, in wic bot firms se te add-on as optiona; () If α θ, tere exists a unique equiibrium wen > 2, in wic H ses te add-on as optiona wereas L ses it as standard; (4) No pure-strategy equiibrium exists in oter parameter regions. Again, te proof argey foows te proof of Proposition 1 of te main mode. equiibrium pricing wi be summarized beow wit te detaied proof omitted. Te D-1 Equiibrium Wen te Add-on is Too Costy (α 1 (2θ θ) + 2 A) To derive te equiibrium profie under tis case, note tat te indifferent consumer becomes θ = (P P )/. Te two firms profits are given by Π = (θ P P )(P C) + (θ p w )(p c); Π = ( P P θ)p, Te first-order conditions ead to te foowing equiibrium profie: P = 1 (2θ θ) + 2 C, p = wθ, (D-1) 2 Wen A A tere exists equiibrium were firm H ses te add-on as standard. 16

17 P = 1 (θ 2θ) + 1 C, p > wθ, (D-2) were te tresods are given by θ = (θ + θ + A)/ and θ = (θ + α)/2. D-2 Equiibrium Wen te Add-on is Not Too Costy (α < 1 (2θ θ) + 2 A) In tis case, te indifferent consumer becomes θ = (P P )/. Te two firms profits are given by Π = (θ P P + w )(P C) + (θ p w )(p c); Π = ( P P + w θ)p ( p w θ)(p c), Te equiibrium profie now becomes: P = 1 (2θ θ)( w) + 1 c + 2 C, p = wθ, P + = 1 (θ 2θ)( w) + 2 c + 1 C, = wθ p if α > θ wθ if α θ were te tresods are given by E θ = 1 (θ + θ) c C ( w), θ = 1 2 (θ + α), θ = 1 (θ + α). 2 Extension 4: Imperfecty Correated Tastes In tis extension, I reax te assumption tat te tastes for te base and te add-on are perfecty correated. For tractabiity, I focus on te case were tere is at east some correation between te two tastes. Suppose te taste for te base good remains to be θ, but te taste for te add-on is given by λ = βθ + e, were te constant β > 0 and e U[e, e]. Witout oss of generaity, assume tat β = 1. Terefore, eac consumer is summarized by a pair of taste parameters (θ, e). Te utiity of buying from firm j for type-(θ, e) consumer is ten θv j P j U j = θv j + (θ + e)w P j p j if ony te base good is purcased; if bot te base good and te add-on are purcased. Let e = e e. To reduce te compexity of te soution, et us assume tat e > α θ and e < 1 (2α 2θ + e). Te impication is tat type-(θ, e) consumer does not buy te add-on in 17

18 equiibrium. Define te foowing quantities 1 = max{ θ α e θ 2α + 2e w, α + e θ 2α 2e θ w}, θ 2θ + α e 2 = max{ θ 2θ w, α + e θ 2α 2e θ w} Proposition E-1. (1) If α 1 (2θ θ) + e, tere exists a unique equiibrium in wic H ses te add-on as optiona wereas L does not se it; (2) If θ + e < α < 1 θ + e, tere exists 2 a unique equiibrium wen > 1, in wic bot firms se te add-on as optiona; () If α θ + e, tere exists a unique equiibrium wen > 2, in wic H ses te add-on as optiona wereas L ses it as standard; (4) No pure-strategy equiibrium exists in oter parameter regions. E-1 Equiibrium Wen te Add-on is Too Costy (α 1 (2θ θ) + e) To derive te equiibrium profie under tis case, note tat te indifferent consumers between te two firms are defined by θ = (P P )/. Te intra-margina consumers for firm H are defined by e + θ = p /w, wic intersect wit te upper bounds of e and θ at e = p θ w and θ = ( p e). Te two firms profits are ten w Π = e (θ P P )P + π (p ); Π = ( P P θ)p, were π (p ) = 1 2 (θ θ )(e e )(p c) = 1 2 (θ p w + e)2 (p c). Soving for te equiibrium prices yied P = 1 (2θ θ), p = w (θ + 2α + e); P = 1 (θ 2θ), p > wθ, were te tresods are given by θ = 1(θ + θ) and θ = 1 (θ + 2α 2e). E-2 Equiibrium Wen te Add-on is Not Too Costy (α < 1 (2θ θ) + e) In tis case, te indifferent consumers are defined by te condition θ( w) ew = P P +. To determine te market sares, it is sufficient to know te upper and ower bounds of te margina consumers θ,u = ew + P P + w, and θ, = ew + P P + w. 18

19 Terefore, firm H s profit is Π = e [ θ 1 ] 2 (θ,u + θ, ) P (θ θ )(e e )(p c), (E-1) }{{} π (p ) were θ = ( p e) and e w = p θ. Simiar to te case of perfect correation, te profit w maximization probem reduces to maximizing two profit components separatey using te base price P and te add-on price p. Firm L s profit is [ 1 ] Π = e 2 (θ,u + θ, ) θ (P + c) 1 2 (θ θ)(e e)(p c). (E-2) }{{} π (p ) were θ = ( p e) and e w = p θ. Simiar to te case of perfect correation, maximizing w profit reduces to maximizing te two components of te profit separatey using te bunde price P + and te add-on price p. Te price competition between te two firms pins down te equiibrium prices P = 1 ( w)(2θ θ)+ 1 c+ 1 + (e+e)w, and P = 1 2 ( w)(θ 2θ)+ 2 c+ 1 2 (e+e)w, Independent from te strategic interaction tat determines equiibrium market sares, te firms set te optima add-on prices. Add-on prices p and p are cosen to maximize π (p ) and π (p ) in Equations (E-1) and (E-2) respectivey, wic ead to p = 1 (θ + e + 2α)w, and p = 1 (θ + e + 2α)w. F Proof of Proposition 2 and Two Additiona Resuts on Commitment Before proving Proposition 2, I first anayze two bencmark cases assuming partia commitment abiity: (1) bot firms can ony make commitment to te no-add-on poicy, and (2) bot can ony make commitment to te standard-add-on poicy. Buiding on tese two resuts, in te ast subsection, I provide te proof of Proposition 2 assuming bot firms can commit to eiter poicy (fu commitment abiity). 19

20 F-1 Firms Can Ony Commit to Not Seing te Add-on Proposition F-1. Suppose tat firms can commit to te no-add-on poicy. In any purestrategy equiibrium, (a) firm H does not commit; (b) firm L commits if α < 1 (2θ θ), and is indifferent between committing and not committing if oterwise. First observe tat firm H wi never commit to te no-add-on poicy. By committing to not seing te add-on, firm H aways uses te base good to compete wit firm L in te second stage. Breaking te commitment aows it to se te add-on to a fraction of consumers wo ave te igest taste for quaity. Tis does not affect te profit from seing te base good but can generate additiona profit from seing te add-on. Te tota profit is ten stricty improved. Given tis observation, firm H wi aways ses te add-on as optiona in any equiibrium, appying Observation 1. It remains to consider firm L s strategies. If firm L commits to not seing te add-on, ten its equiibrium profit is given by Π (opt,no) = 1 9 (θ 2θ)2. If firm L does not commit, ten te equiibrium prices and profits are exacty te same as in Proposition 1. Weter or not it is optima to commit depends on te size of α. Wen α > 1 (2θ θ), it wi not se te add-on in te second stage and tus obtains te same profit Π (opt,no) as in te commitment case. Hence, committing or not does not cange te equiibrium profit. Wen θ < α < 1 (2θ θ), te firm cooses to commit if and ony if Π (opt,no) Π (opt,opt) = w [ (θ 2θ) 2 + 2α(θ 2θ) 9 (θ α)2 9 }{{ 4 } Y ] α2 w > 0. (F-1) w It can be sown tat Y > 0 wen θ < α < 1 (2θ θ). Terefore, commitment is optima if > (1 + α 2 /Y )w. If <, ten te firm does not commit, eading to te possibiity of an optiona add-on in equiibrium as in Proposition 1. However, tis equiibrium exists ony wen > 1. It can be sown tat < 1, so tis equiibrium is not sustainabe. Wen α θ, te firm cooses to commit if and ony if Π (opt,no) Π (opt,std) = w [ (θ 2θ) 2 + 2α(θ 2θ) 9 }{{} Y ] α 2 w > 0. w Since Y > 0, committing not to offer te add-on is optima if > (1 + α 2 /Y )w. If <, ten te firm does not commit, eading to te possibiity of a standard add-on in 20

21 equiibrium as in Proposition 1. However, tis equiibrium exists ony wen > 1. It can be sown tat < 2, so tis equiibrium is not sustainabe. F-2 Firms Can Ony Commit to Standard Add-on Proposition F-2. Suppose tat firms can commit to te standard-add-on poicy (bunding). Tere exists a tresod ˆα > 1 (2θ θ) suc tat, in any pure-strategy equiibrium: (a) firm H commits if α < ˆα, but does not (and ses te add-on as optiona) if oterwise; (b) firm L does not commit if α > θ, and is indifferent between committing and not if oterwise. Case (a): α > 1 (2θ θ). First it is easy to sow tat firm L as no incentive to se te add-on in te second stage, regardess of H s commitment coices, because of its ig cost. Terefore, it remains to consider firm H s incentive to commit, given firm L doesn t se te add-on. Te equiibrium profit of committing to bunding is given by Π (std,no) = 1 [ 2. (2θ θ)( + w) c] 9( + w) Te equiibrium profit of not committing is given in Tabe 1 in te main text. Terefore, firm H commits if and ony if Π (std,no) Π (opt,no) = w [ 9 (2θ θ) 2 2α(2θ θ) 9 (θ α)2 }{{ 4 } X ] + α2 w > 0. (F-2) + w Tere exists a tresod ˆα > 1 (2θ θ) suc tat wen α < ˆα, te above inequaity ods. Wen α > ˆα, it is optima for H to not commit and se te add-on optionay. Case (b): θ < α < 1 (2θ θ). First note tat firm L as no incentive to commit to bunding, because not committing aows te firm to carge some consumers for a fee to recover some cost. Terefore, it remains to consider firm H s incentive to commit, given tat firm L ses te add-on to some consumers. Te equiibrium profit of committing to bunding is given by Π (std,opt) = 1(2θ 9 θ)2. Te equiibrium profit of not committing is given in Tabe 1 in te main text. Terefore, firm H commits if and ony if Π (std,opt) Π (opt,opt) = w [ (2θ θ) 2 2α(2θ θ) 9 ] 9 4 (θ α)2 α2 w > 0. w Tere exists a tresod ˆα suc tat wen α < ˆα, te above inequaity ods. Wen α > ˆα, it is optima for te firm to not commit. However, for suc equiibrium to exist, 21

22 we need α < (2θ θ)( w)/( w) ˆα 1. Wit some agebra, one can sow tat ˆα > ˆα 1. Terefore, not committing does not constitute an equiibrium in tis case. Case (c): α < θ. First note tat firm L as no strict incentive to commit to bunding because, even if it doesn t commit, it wi se te add-on to a consumers anyway. It ten remains to consider firm H s incentive to commit, wic is exacty te same condition as in Case (b): it wi commit if and ony if Π (std,opt) Π (opt,opt) > 0. Tis is because firm L s commitment coice does not affect firm H s probem as ong as L ses te add-on to some consumers. Te condition ods wen α < ˆα. Wen α > ˆα, it is optima for te firm to not commit. However, for suc equiibrium to exist, we need α < (θ 2θ)( w)/w ˆα 2. 4 Wit some agebra, one can sow tat ˆα > ˆα 2. Terefore, not committing does not constitute an equiibrium in tis case. F- Fu Commitment Abiity (Proposition 2) Case (a): α > 1 (2θ θ). Foowing te same ine of proof of te proposition in Section F-2, we ave: wen α < ˆα, tere exists an equiibrium in wic firm H commits to bunding; Wen α > ˆα, tere exists an equiibrium in wic firm H does not commit and ses te add-on as optiona. In bot cases, firm L does not se te add-on (and is indifferent between committing to no-add-on and not). Case (b): θ < α < 1 (2θ θ). First note tat if firm L commits to te no-add-on poicy, ten Π (std,no) > Π (opt,no) given tat X > 0 (defined in Equation F-2). If firm H commits to bunding, ten Π (std,no) Π (std,opt) = w [ 9 (θ 2θ) 2 + 2α(θ 2θ) 9 (θ α)2 }{{ 4 } Y ] + α2 w > 0, + w aways ods given tat Y > 0. Terefore, te profie tat firm L commits to te no-add-on poicy wereas firm H commits to te standard-add-on poicy constitutes an equiibrium. Te oter possibe equiibrium woud be tat neiter firm commit and tus tey bot se it as optiona in te second-stage game. However, as sown in te proof of Proposition F-1, tis equiibrium is not sustainabe because < 1. Case (c): α < θ. Foowing te same argument of Case (b), te profie tat firm L commits to te no-add-on poicy wereas firm H commits to te standard-add-on poicy constitutes an equiibrium. It remains to verify tat tis is te ony possibe equiibrium Tis condition foows from > w(2θ θ α)/(2θ θ α) 1. 4 Tis condition foows from > w(θ 2θ + α)/(θ 2θ) 2. 22

23 under tis parameter region. Te aternative equiibrium woud be tat neiter firm commits and tus H ses te add-on as optiona wereas L ses it to a consumers. However, as sown in te proof of Proposition F-1, tis equiibrium is not sustainabe because < 2. G Proof of Proposition G-1 Wen α < (θ + θ)/ For te equiibrium in wic firm L ses te add-on as optiona to exist, te foowing incentive constraints ave to be satisfied: (a) θ > θ ; (b) θ > θ θ > 2θ α. For te equiibrium in wic firm L ses te add-on as standard to exist, it requires tat (c) θ > θ; (d) θ < θ. Note tat (a) and (b) impy (c). Define te foowing quantities: (u) 0 = θ 2θ + α w, (u) 1 = θ 2θ 2θ + θ α 2(θ + θ α) w, and (u) 2 = 2θ 9θ + 7α 2(θ 5θ + α) w. Constraint (a) is equivaent to θ > α since θ = 1 2 (θ + c). Note tat θ α is positive as ong as > (u) 1. 5 Constraint (b) is equivaent to θ > 2θ α since θ = 1 2 (θ + α). Note tat θ (2θ α) is positive as ong as > (u) 2. Note furter tat (u) 1 > (u) 2 is equivaent to 2(α θ)(θ 2θ + 6α) > 0, wic ods if and ony if α > θ. Terefore, if firm L ses optiona add-on in equiibrium, ten eiter one of te foowing must od: (1) α > θ and > (u) 1, or (2) α < θ and > (u) 2. Finay, if firm L ses standard add-on in equiibrium, Constraint (c) gives θ (u) θ = 2[(θ 2θ) (θ 2θ + α)w] 6 5w > 0 > (u) 0. To satisfy Constraint (d), it suffices tat < (u) 2. Note tat (u) 2 > (u) 0 is equivaent to (θ α)(θ 2θ + 6α) > 0, wic aways ods wen α < θ. It remains to verify tat tere is no profitabe deviation from eiter firm. G-1.1 No Profitabe Deviation for Firm L Te ony non-oca deviation is tat firm L does not eave demand for H. Fixing firm H s equiibrium prices, (P, pe + ), and given firm L s tota price, P, te consumers wit θ [θ, θ ] coose to pay te base price to visit firm L at time t = 1. Te margina consumer is given by θ = (P + P + )/. At time t = 2, tese consumers observe p and decide weter to buy it or not. Te firm maximizes its ex post profit, π = (θ θ )(p e c), wit 5 Note aso tat > (u) 1 occurs ony if α < (θ + θ)/. If, owever, α > (θ + θ)/, ten < (u) 1. 2

24 θ = p e /w. Again, te optima strategy is to set θ as a function of te margina consumer θ : θ (θ ) = (θ + α)/2. Te firm s probem at time t = 0 is ten finding te optima θ tat maximizes te tota profit taking into account te second-period probem max(θ θ)(p + θ c) (θ (θ ) θ)(w θ (θ ) c). θ }{{} =P +p e Te (unconstrained) optima coice of te margina consumer is θ d + = [2P 2c + (2 + w)θ]/(4 + w). Tis deviation requires tat no consumer woud buy ony te base good from firm H, tat is, θ d θ. However, notice tat θ d θ = 4(θ 2θ + α) 2 (θ 2θ 5α)w + 7 (θ 2θ + α)w2 2, (4 + w)(6 5w) wic is negative because > w. Terefore, te optima soution is binding suc tat θ = θ. Hence, firm L as no incentive to deviate. G-1.2 No Profitabe Deviation for Firm H Te ony non-oca deviation tat coud be profitabe is tat firm H impements te standardadd-on poicy. Since it is aways optima for firm H to set te add-on price, p e = (θw +c)/2, tat maximizes ex post profit, te deviation is acieved by increasing te base price so tat no consumer wi buy te base good ony. In te first case, te firm deviates by carging te tota price P +d = P d + pe (P + deviation is P +d = 1 2 te equiibrium profit because Π Π d = In te second case, te firm carges P +d at t = 0 wie eaving some demand for L+. Te optima + θ + c). Te profit from tis deviation, Π d, is no greater tan [ w [ P + + θ( w) ] ] 2 ( w)(θw c) 4( w)w deviation is P +d tan te deviation profit in te first case, because 0 suc tat tere is no demand for L +. Te optima = (P + (θ +w)+c)/2. Te profit from tis deviation, Π d Π d Π d = 1 [ ] P [ + θ c P 4 }{{} 4( + w) + θ( + w) c }{{} =A+B =A is no greater ] 2 24

25 = B2 ( + w) + 2AB( + w) + A 2 w 4( + w) given tat A + B > 0 and B < 0. Terefore, tere is no profitabe deviation for firm H. G-2 Wen α (θ + θ)/ In tis case, te equiibrium prices are exacty te same as case (1) in Proposition 1 (see subsection A-2), except tat firm H s add-on price is unobserved but correcty expected by rationa consumers. Terefore, as ong as α (2θ θ)/, tere is no profitabe deviation from eiter firm. However, wen α ((θ + θ)/, (2θ θ)/), firm L can profitaby deviate by seing te add-on to some consumers. H Proof of Proposition 4 Let superscripts o and u denote te observed-price and te unobserved-price cases, and Note tat B > 0 if and ony if > (u) 0. H-1 Market Sares 0 [ ] B = (θ 2θ) (θ 2θ + α)w w ( w)(6 5w). Te difference in te margina consumers indifferent between te two firms is θ (u) θ (o) = B < 0, were te inequaity foows from B > 0. Te difference in te margina consumers indifference between buying te add-on and not buying is θ (u) θ (o) were te inequaity foows from B > 0. H-2 Prices = 1 2 (θ (u) θ) = Te difference in te base prices of firm H is given by: ( w) B > 0, w P (u) P (o) = w( w)(θ 2θ + 6α) (6 5w) > 0. Te difference in te add-on prices for firm L is given by: p (u) p (o) = 1 2 (θ (u) θ)w = ( w)b > 0, 25

26 were te inequaity foows from B > 0. Note tat under eiter unobserved-price or observed-price equiibrium, te indifference condition and firm H s best response satisfy: P + = P θ ( w) P = 1 + (P 2 + θ( w)) P + = ( w)(θ 2θ ). (H-1) Wit some agebras, one can sow tat te difference in te tota prices is positive: P + (u) P + (o) = 2( w)(θ (o) θ (u) ) > 0. Te difference in te base price is given by P (u) P (o) = 2( w)(θ (o) θ (u) ) 1 2 (θ (u) θ)w = ( w)b, wic is negative given tat B > 0. H- Profits Firm H s profit is ceary increased because it ses te base to more consumers at a iger price wie keeping te add-on profit uncanged. For firm L, te profit difference is Π = (θ (u) Note tat θ (u) θ)(p + (u) c) (θ (u) θ)(p (u) c) (θ (o) θ)(p + (o) c)+(θ (o) θ)(p (o) c). = θ (o) B, θ (u) = θ (o) +B( w)/w, P + (u) = P + (o) +2( w)b, +( w)b. Substituting tese quantities into te profit difference yieds and p (u) = p (o) Π = ( 2w)( w)b 2 /w < 0, were te inequaity foows from > 2w. 26