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External Effects, Separability, and Resource Allocation



External Effects, Separability, and Resource Allocation by Ernst Baltensperger* In a well-known article, Ronald Coase [3] argued, in the context of a two-firm externality situation, that a socially optimal resource allocation will be reached through a negotiation process, regardless of whether the externality-causing firm is liable for the damage it causes to the other firm or not1. In a recent paper, James Marchand and Keith Rüssel [5] reexamined the Coase argument, and came to the conclusion that Coase's result regarding the neutrality of liability arrangements on resource allocation is not generally true, but holds only in the restricted case of additively separable cost functions. Marchand and Rüssel formalize the Coase argument, as suggested by Coase, by assuming that both firms A (the one causing the externality) and B (the damaged firm) include as part of their objective func- tion the compensation which B would pay to A for a reduction of the latter's out- put, and that this compensation is equal to the cost saving B would experience as a result of the reduction in A's rate of output. That is, the two firms' objective functions are ITA = Piqi - A(qO + C and n B = P2q2 - B(qi, q2) - C, with C = B(qi, q2) — B(qi, q2), where ITA and ITB denote the two firms' profits, qi and q2 their outputs, Pi and P2 the (market determined) prices of these outputs, A(qi) and B(qi, q2) the total cost of firms A and B, respectively, and qi is the quantity of output firm A would produce in the absence of any compensation (the quantity qi satisfying the condition Pi = Ai(qi)). Marchand and Rüssel show that under these conditions the optimal output for firm A (denoted qï) exceeds the socially optimal quantity (denoted q?), while firm B's optimal output (q2) is less than the socially optimal quantity (q|), unless B's cost function B(qi, q2) is separable into two independent parts B^qi) and B2(q2). Although this model does in fact seem to be an accurate formalization of Coase's verba

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