A Theory of Input-Output Architecture

Abstract

APPENDIX A: STABLE EQUILIBRIA A.1. Notation THIS SUBSECTION SHOWS that it is without loss of generality to associate each bilateral contract with a technique and restrict the arrangement so that there is one contract for each technique. A coalition is connected if, for any entrepreneurs i and j in the coalition, there is a sequence of entrepreneurs beginning with i and ending with j, all of whom are members of the coalition, such that for each consecutive pair there exists a technique in Φ for which one entrepreneur is the buyer and the other is the supplier. LEMMA 1: If there is a coalition of size/cardinality N or smaller with a dominating deviation , then there is a connected coalition of size/cardinality N or smaller with a dominating deviation. PROOF: Suppose that there is a coalition J with a dominating deviation that can be divided into two subsets that are not connected, J and J , so that J ∪ J = J and J ∩ J = ∅. The deviation would leave every member of J at least as well off and at least one member of J strictly better off. Without loss of generality, suppose that the member who is strictly better off is in J. Then J has a dominating deviation: Since no member of J is able to produce using intermediate inputs from members of J , J has a dominating deviation in which members of J drop all contracts for which there are positive payments to members of J and otherwise deviate according to the original deviation; every member of J is at least as well off as under the original deviation. Q.E.D. LEMMA 2: It is without loss of generality to use notation that associates each bilateral contract with a technique. PROOF: We first show that if j has no technique to use i's good as an input, then there is no equilibrium in which i provides goods to j or in which there is a payment between them. After that, we will show that if a coalition has a dominating deviation, then there is always an alternative dominating deviation in which each pairwise payment and trade of goods can be associated with a technique. Toward a contradiction, suppose first that there is an equilibrium in which entrepreneur i provides goods for entrepreneur j, but j has no technique that would use good i as an input. Since j cannot resell good i, if the payment to j is positive, then i would be strictly better off dropping the contract (setting the payment and the quantity of goods to zero). If the payment is weakly negative, i would be strictly better off dropping the contract. Thus this cannot be an equilibrium. Suppose second that there is an equilibrium in which Ezra Oberfield: edo@princeton.edu