Cambridge Working Papers in Economics : 1813 A NETWORK APPROACH TO PUBLIC GOODS

Suppose agents can exert costly effort that creates nonrival, heterogeneous benefits for each other. At each possible outcome, a weighted, directed network describing marginal externalities is defined. We show that Pareto efficient outcomes are those at which the largest eigenvalue of the network is 1. An important set of efficient solutions—Lindahl outcomes—are characterized by contributions being proportional to agents’ eigenvector centralities in the network. The outcomes we focus on are motivated by negotiations. We apply the results to identify who is essential for Pareto improvements, how to efficiently subdivide negotiations, and whom to optimally add to a team. When economic agents produce public goods, mitigate public bads, or more generally create externalities, the incidence of the externalities is often heterogeneous across those affected. A nation’s economic policies—e.g., implementing a fiscal stimulus, legislating environmental regulations, or reducing trade barriers—benefit foreign economies differently. Investments by a firm in research yield different spillovers for various producers and consumers. Cities’ mitigation of pollution matters most for neighbors sharing the same environmental resources. And within a firm, an employee’s efforts (e.g., toward team production) will benefit other employees to different degrees. How does heterogeneity in the incidence of externalities translate into outcomes? Which agents contribute the most and least? Whose effort is particularly critical? An active research program addresses these questions by modeling agents playing a Nash equilibrium of a one-shot public goods game, in which they unilaterally choose how much effort to put forth; see, e.g., Ballester, Calvó-Armengol, and Zenou (2006), Bramoullé, Kranton, and D’Amours (2014), and Allouch (2015, 2013). These works model externalities via particular functional forms in which a network is a set of parameters. Links describe the pairs of players who directly affect each other’s payoffs or incentives, as when two people collaborate on a project. The main results then characterize equilibrium effort levels via certain network statistics. Since these statistics are major subjects of Elliott: University of Cambridge Golub: Harvard University We gratefully acknowledge the research support and hospitality of Microsoft Research New England. Elliott acknowledges financial support from NSF grant 1518941. Kevin He provided exceptional research assistance. The guidance of Nageeb Ali, Abhijit Banerjee, Matthew O. Jackson, Andy Skrzypacz, and Bob Wilson has been essential. For detailed comments on previous drafts, we thank Gabriel Carroll, Arun Chandrasekhar, Sylvain Chassang, Anil Jain, Juuso Toikka, Xiao Yu Wang, and Ariel Zucker. For helpful suggestions we thank Daron Acemoglu, Yann Bramoullé, Federico Echenique, Glenn Ellison, Maryam Farboodi, Alex Frankel, Drew Fudenberg, Andrea Galeotti, Jerry Green, Hari Govindan, Sanjeev Goyal, Johannes Hörner, Scott Kominers, David Kreps, John Ledyard, Jacob Leshno, Jon Levin, Mihai Manea, Eric Maskin, Vikram Manjunath, Muriel Niederle, Michael Ostrovsky, Antonio Penta, Michael Powell, Phil Reny, Larry Samuelson, Ilya Segal, Balazs Szentes, Harald Uhlig, Leeat Yariv, Alex Wolitzky, and many seminar participants. Finally, we thank four anonymous referees and the editor for extensive and constructive comments. 2 MATTHEW ELLIOTT AND BENJAMIN GOLUB study in their own right, the connection yields a rich set of intuitions, as well as analytical techniques for comparative statics, identifying “key players,” and various other policy analyses. We argue that it is valuable to also study different classes of solutions in a public goods economy—ones motivated by negotiations—and, paralleling the results above, to understand how network properties matter for these solutions. The static Nash equilibrium is a benchmark relevant in cases where decisions are unilateral, with limited scope for repetition or commitment. Under this solution concept, agents do not internalize the externalities of their effort. Indeed, in a public goods game, players free-ride on the contributions of others, leading to a classic “tragedy of the commons” problem. The resulting inefficiencies can be substantial; in the context of problems like climate change, some argue they are disastrous. In cases where large gains can be realized by improving on the unilateral benchmark, institutions arise precisely to foster multilateral cooperation. Global summits, the World Trade Organization, research consortia, and corporate team-building practices all aim to mitigate free-riding by facilitating commitment. Therefore, rather than working with the static Nash equilibrium, this paper focuses on the complementary benchmark of Pareto efficient public goods provision in the presence of nonuniform externalities. Our contribution is to show that taking a network perspective on the system of externalities sheds new light on efficient outcomes and the scope for efficient cooperation. First, we provide a new characterization of when Pareto improvements are possible, which relates such improvements to cycles of favor-trading, quantified in a suitable way. Second, we characterize certain efficient solutions—the Lindahl outcomes, which have microfoundations in terms of negotiation games. Our results describe agents’ contributions at these outcomes in terms of their positions in the network of externalities. The insights that the analysis generates can help address questions such as who should be given a seat at the negotiating table or admitted to a team. In contrast to the previous work mentioned above, our characterizations are non-parametric: A “network” representation of marginal externality incidence arises naturally from general utility functions. Finally, we provide new economic foundations and intuitions for statistics that are widely used to measure the centrality of agents in a network by relating these statistics to concepts such as Pareto weights and market prices. 1. Example and Roadmap We now present the essentials of the model in a simplified example. Section 2 defines all the primitives formally in the general case. Each agent has a one-dimensional effort/action choice, ai ≥ 0; it is costly for an agent to provide effort, which yields positive, non-rival externalities for (some) others. For a concrete example, suppose there are three towns: X, Y and Z, located as shown in Figure 1a, each generating air and water pollution during There are many empirical applications of these results. See, for example, Calvó-Armengol, Patacchini, and Zenou (2009) and Acemoglu, Garćıa-Jimeno, and Robinson (2014). Other theoretical papers that examine different issues related to the provision of public goods on networks include Bramoullé and Kranton (2007) and Galeotti and Goyal (2010). For example, it was at the Rio Earth Summit that the first international treaty on climate change was hammered out. There have been several other summits and associated climate change agreements since. A NETWORK APPROACH TO PUBLIC GOODS 3