Competitive Equilibrium and Trading Networks : A Network Flow Approach

Under full substitutability of preferences, it is known that a competitive equilibrium exists in trading networks, and is equivalent to (chain) stable outcomes. In this paper, we formulate the problem of finding an efficient outcome as a generalized submodular flow problem on a suitable network. Existence of a competitive equilibrium and its equivalence with the seemingly weaker notion of stability follow directly from the optimality conditions of the flow problem. Our formulation enables us to perform comparative statics with respect to the number of buyers, sellers, and trades. In particular, we establish that if a new buyer is added to the economy, at an equilibrium the prices of all existing trades increase. In addition we give a polynomial time algorithm for finding competitive equilibria in trading networks and testing (chain) stability