A Network Approach to Public Goods

An economy can be thought of as a network in which the nodes are agents and links among them represent heterogeneous opportunities for exchange or cooperation. This paper argues that studying properties of such a network -- how dense it is, how "central" various agents are in it -- yields insights about issues such as the efficiency and fragility of an economic system, as well as its market outcomes. We develop this conceptual point in a model of a public goods economy: one in which each agent can incur a private cost to take an action -- e.g., reducing pollution -- that creates nonrival but heterogeneous benefits for others. The network we study is a directed, weighted graph in which an edge from agent i to agent j captures the marginal benefits i can provide to j, at the current action profile, as i increases his public good provision. We find that when the largest eigenvalue of this network differs from one there are Pareto inefficiencies. The largest eigenvalue can be interpreted in terms of cycles in the benefits network (e.g., X can help Y, who can help Z, who can help X). These are critical to finding a Pareto improvement on a given outcome, and the network's largest eigenvalue quantifies the marginal returns available from exploiting such cycles. Building on the eigenvalue result, we propose a simple algorithm to find the players who are essential to a negotiation -- in the sense that without their participation, there is no Pareto improvement on the status quo. They are the ones whose removal causes a sufficiently large disruption of cycles in the benefits network, as measured by the decrease in its largest eigenvalue. Since Wicksell [1896] and Lindahl [1919] it has been recognized that a system of personalized taxes and subsidies can incentivize efficient public good provision. Lindahl outcomes are analogues of Walrasian market allocations in which the taxes and subsidies internalize all the externalities. Our main result characterizes the Lindahl outcomes in terms of the same marginal benefits matrix we use to diagnose inefficiencies. A nonzero action profile is a Lindahl outcome if and only if it is an eigenvector centrality action profile for the marginal benefits matrix. At such an action profile, the agents contribute in proportion to how much they value the efforts of those who help them. We deduce two main interpretable consequences from this characterization. First, it is the benefits an agent receives, rather than those he can confer, that determine his level of effort. Second, the players contributing the most are those who are most "central" in the benefits network, in the sense that they receive strong direct and indirect benefit flows from others. As Samuelson [1954] discusses, Lindahl taxes and subsidies often cannot be directly implemented in practice because they require a planner to know agents' utility functions. Nevertheless, Lindahl outcomes turn out to play a distinguished role in several strategic settings. First: any implementable, continuous social choice correspondence that is Pareto efficient and individually rational must contain all the Lindahl outcomes Hurwicz [1979a, 1979b], and Hurwicz, Maskin, and Postlewaite [1995]). Second the Lindahl outcomes are the equilibrium outcomes of a natural non-cooperative bargaining game among the n agents (Dávila, Eeckhout, and Martinelli [2009] and Penta [2011]). Finally, the Lindahl outcomes are robust to coalitional deviations (Aumann [1959], Shapley and Shubik [1969], Foley [1970]). Ballester et al [2006] as well as Bramoullé et al [2011] have found relationships between the Nash equilibria of games and the spectral properties of an interaction network under the assumption of linear best responses. In contrast, we study efficiency and market outcomes and do not require functional form assumptions.