Complementary Inputs and the Existence of Stable Outcomes in Large Trading Networks

This paper studies a model of large trading networks with bilateral contracts. Contracts capture exchange, production, and prices, as well as frictions such as complex taxes and the absence of transfers. In our setting, under standard continuity and convexity conditions, a stable outcome exists in any acyclic network, as long as all firms regard sales as substitutes and the market is large. Thus, complementarities between inputs do not preclude the existence of stable outcomes in large markets, unlike in discrete markets. Even when sales are not substitutable, tree stable outcomes exist in our setting. The model presented in this paper generalizes and unifies versions of general equilibrium models with divisible and indivisible goods, matching models with continuously divisible contracts, models of large (two-sided) matching with complementarities, and club formation models. Additional results provide intuition for the role of uni-directional substitutability conditions and acyclicity in the main existence results, and explain what kinds of equilibria are guaranteed to exist even when these conditions are relaxed. Unlike in two-sided large-market settings, the sufficient conditions described in this paper pin down maximal domains for the existence of equilibria.