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A \ " Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/optimalcollusionOOathe .,r•_-_..- 2 f\-f\\ working paper department economics of Optimal Collusion With Private Information Susan Athey Kyle Bagwell No. 99-17 October 1999 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 WORKING PAPER DEPARTMENT OF ECONOMICS Optimal Collusion With Private Information Susan Athey Kyle Bagwell No. 99-17 October 1999 MASSACHUSETTS OF INSTITUTE TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASS. 02142 99-17 ABSTRACT We consider an infinitely-repeated Bertrand game, in which prices are perfectly observed and each firm receives a privately-observed, i.i.d. cost shock in each period. In our base model, side-payments are prohibited. In order for a cartel to achieve productive efficiency, firms that receive a high cost draw in the current period must be induced to give up market share. To provide this incentive, firms may need to reduce current prices, make compensating adjustments in future market shares or allow for future price wars. In the most profitable collusive schemes, firms implement productive efficiency, but they do not resortto low prices or future price wars; instead, firms with bad cost draws may be favored with higher expected market share in future periods. Iftypes are discrete, there exists a discount factor strictly less than one above which first-best profits can be attained purely through history-dependent reallocation ofmarket share between equally-efficientfirms. We provide further characterizations of equilibrium play as well as several computational examples. We next considerthe role ofthree characteristics ofthe institutional environment. First, we examine the costs and benefits to the cartel ofexplicitcommunication about cost types in a given period. We show thatfirms may find it beneficial to communicate after some histories but notothers. Second, we show that iffirm identities cannot be tracked over time (so thatfirm-specific future market-share favors are unavailable), the best collusive scheme sacrifices all productive efficiency. Third, we examine the role ofexplicit side- payments, which may entail inefficiencies ifthey are illegal and bear the risk of detection. Unless side-payments are perfectly efficient, optimal collusive equilibria are non-stationary and thus involve the use offuture market-share favors. _-— HMsTffUTE trrepff