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Multimarket Contact, Bundling and Collusive Behavior - Cemfi

Multimarket Contact, Bundling and Collusive Behavior - Cemfi

that the deviation is

that the deviation is more attractive since there will be fewer one-stop shoppers thatunder linear pricing. One may see the introduction of a premium as an increase inproduct differentiation. The proposition below presents results for two extreme cities =0( =1)and =12 ( = −1). These results illustrate cleanly the mechanismat play and so they provide the basis for the discussion that follows covering the othercases.Proposition 5 In cities with either perfectly positive correlation ( =0) or perfectlynegative correlation ( =12), it is more difficult for the two conglomerates to sustainthe collusive-optima bundling agreement of Proposition 3 than linear monopoly prices.Proof. Consider first the case of a collusive-bundling agreement in city =12: =2 and = + 2 for = or . Note that the bundling discount doesnot need to be equal to for the result to hold. The critical discount factor for suchcollusive agreement is given by () = () − () () − ()where () = + 1 Φ(; ) are per-period profits along the collusive-bundling path2and () are (optimal) deviation profits in the period of deviation for a given bundlingdiscount . From the proof of Proposition 3 we have that () = + − 2 . To obtain the optimal deviation, note firstthatwhen → 12 themostdifficultconsumers to attract for conglomerate with a reduction in the price of the bundle arethe two-stop shoppers. Since a two-stop shopper that decides to buy the bundle (fromeither conglomerate for that matter) enjoys the bundle discount , the reduction in theprice of the bundle just enough to attract the "most distant" two-stop shopper is equalto 2 − . Thus, deviation profits can be written as () = ()+, where ()are profits when optimally deviating from linear monopoly prices (see above). Therefore, () () ⇐⇒ () − + − + 2 () − ()+⇐⇒ 1 − + () − () − () () − () − ()(27)(28)The left-hand side of (28) is greater than 1/2 for =12 and 0; hence, it suffices toshow that the right-hand side of (28) is smaller than 1/2 for all 0, whichisthecaseas indicated by the term on the left-hand side of (25).Consider now the case of a collusive-bundling agreement in city =0: =2 +and = for = or (again does not need to be equal to for the resultto hold). Proceeding as before, we have from Proposition 3 that collusive profits aregiven by ( ) = + 1Ψ( ) = 2 +( 1 − ) − 2 2 . To obtain theoptimal deviation ( ), notefirst that when → 0 themostdifficult consumers for24

conglomerate to attract with a reduction in the price of the bundle are −’s one-stopshoppers. As seen in Proposition 1, it is required to reduce the price of the bundle by∆ ≡ (1−2). Since everyone now ends up paying the premium, deviation profits become ( ) = ()+. Therefore, ( ) () ⇐⇒ () − + − ( 1 − ) + 2 2 () − ()+⇐⇒ 1 2 + + () − () − () () − () − ()(29)(30)As above, the left-hand side of (30) is greater than 1/2 for =0and 0, whichconcludes the proof.The two cases presented in the proposition share the exact same outcome: the additionaldeviation incentives created by collusive-bundling are far greater than the extraprofit. The reason for this, however, is different. In the case of =12 the deviationincentives are large because the bundling discount hasmadeiteasierfor to stealconsumers from −; infact,−’s one-stop shoppers can be attracted at almost no cost.Conversely, in the case of =0thedeviationincentivesarelargenotbecauseithasbecome easier for to steal consumers (actually it remains equally difficult) but becausethe bundling premium has made it increasingly attractive to attend more consumers.Understanding these two polar cases gives a hint on how things change as we moveaway from them. Consider first a departure from =0(towards lower correlations).There will be a city 14 for which it will be equally difficult for deviator to attract−’s one stop-shoppers, with a reduction in the price of the bundle of ∆ =(1− 2), andthe most distant of the two-stop shoppers, with a reduction of 2+ (recall that suchtwo-stop shopper must be compensated not only for the extra travel, 2, butalsoforthe premium, ). Solving (1 − 2) = 2+ we obtain (note that =14 for =0) = 1 4 − 2City is plot in Figure 7a that, according to Proposition 3, reduces to = 16when = (). More importantly, for cities ∈ (0 ] expression (30) in Proposition 5continues holding because the extra deviation incentives brought by the premium remainsthe same, that is, ( ) = () +. However, for cities ∈ ( 14], these extraincentives start falling because the price reduction is not longer (1 − 2) but 2+,and at =14 these extra incentives completely disappear, that is, ( )| =14 = ()| =14 =2 − 2 − 2. Thus, there exists a city ∗ ∈ ( 14) for which ( ) = (). Between ∗ and 1/4, the premium makes it easier to sustain collusion, i.e., theextra collusive-profit effect of the premium dominates its deviation effect.*** figure 7 here or below ***25

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