Collusion, competition and piracy serie - Ivie

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where p i represents the price of the original information good i =1, 2. By comparing the levels of utility obtained from each consumer’s strategy we find the demand function for information good i =1, 2: 2 ⎧ ⎪⎨ D i (p i ,p j )= ⎪⎩ 1 θ−θ 1 θ−θ ³ ³ ´ θ − p i−K s θ − p i+p j −K 1 θ−θ 2s ¡ θ − p i s ¢ ´ if p j + K ≤ p i if p j − K ≤ p i
multiproduct monopoly. From Belleflamme and Picard (2007), we have that firms price p C and obtain the profit π C in this case, where p C = 2θs + K 4 ¡ ¢ 2 2θs + K and π C = . (4) 16s The punishment profits are the Bertrand-Nash profits corresponding to the duopoly equilibrium. Belleflamme and Picard (2007) show that, in duopoly, there exists a unique symmetric equilibrium in which both firms focus on switchers and price p B−N = ¡ 2θs + K ¢ /3 if and only if K > b K = ¡ 3 √ 2 − 4 ¢ θs/2. Otherwise, there is no equilibrium in pure strategies. In duopoly, the profit ofeach firm is ¡ ¢ 2 2θs + K π B−N = . (5) 18s Hence, we assume that K> b K, because we focus on pure strategy equilibria given the multiplicity and complexity of mixed-strategy equilibria. 3 according to its best-response function: 4 ⎧ If a firm deviates from the collusive agreement, it prices ¡ θs + K ¢ /2 if pj ≤ ¡ θs − K ¢ /2, p BR i (p j )= ⎪⎨ ⎪⎩ p j + K if ¡ θs − K ¢ /2 ≤ p j ≤ ¡ 2θs − K ¢ /3, θs +(K − p j ) /2 if ¡ 2θs − K ¢ /3 ≤ p j ≤ ¡ 2 − √ 2 ¢ θs + K, θs/2 if p j ≥ ¡ 2 − √ 2 ¢ θs + K. (6) Therefore, the optimal deviation price and profit are ( 2θs+5K p D 4 if bK
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