This paper studies when competitive markets work well in economies where people’s consumption affects others through multiple kinds of social ties. The authors model an economy with several goods and with externalities—effects that one person’s consumption has on others—organized by a multiplex network. A multiplex network means there are several layers of connections; each layer corresponds to one good and can have a different pattern of who influences whom. The main finding is that competitive markets can still be efficient despite these externalities, but only under clear structural conditions on the networks that generate those spillovers.

The authors build a general equilibrium model (an Arrow–Debreu economy) in which each good is tied to a network layer and people are heterogeneous in their initial endowments and in where they sit in each layer. Consumers choose how much of each good to buy subject to a budget. Markets set prices and determine who ends up with what. The paper shows that a market equilibrium exists under mild restrictions (in particular when negative externalities are not too strong) and can be unique under stronger assumptions. The authors also give a precise description of equilibrium: an agent’s consumption of a good is proportional to a network centrality measure (Katz–Bonacich centrality, which roughly counts a person’s direct and indirect influence in a network) weighted by an “effective endowment” that bundles an agent’s initial wealth and the pattern of externalities she gives and receives.

Why does the network structure matter for efficiency? The paper revisits the two classical welfare theorems. The First Welfare Theorem says a competitive equilibrium is Pareto efficient (no one can be made better off without making someone else worse off). The Second says any efficient allocation can be supported as a competitive equilibrium after a suitable redistribution of endowments. Here those theorems hold if and only if a condition called “centrality parallelism” is satisfied: for every pair of goods, the vectors of unweighted network centralities across people must be proportional. Two practical sufficient cases are identified where this condition holds: either every network layer is regular (each person has the same number of links in that layer) or every good shares the same network of links. If the condition fails, the competitive equilibrium is not Pareto efficient and the authors construct explicit Pareto-improving reallocations.