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This paper makes three advances: 1) It fixes the empty-core problem of the Coase theorem; 2) it provides the smallest upper bound of transaction costs below which the optimal or efficient outcomes can be achieved; and 3) it establishes two mathematical theorems that capture the main insights and major aspects of the Coase theorem. A simpler version of the theorems says that in a coalitional production economy without transaction costs, the maximal payoff will be produced by the optimal firmsdoi:10.22574/jmid.2018.12.005 fatcat:eam6i7zqwjhnxc4x6isfcd3msy