This paper studies cross-community risk sharing. There is now a large body of theoretical and empirical work on informal insurance, where people mitigate risk by sharing income. A consistent empirical finding is that risk-sharing is not complete within villages, often the observed sets of individuals.2 One reason, researchers suspect, is that risk-sharing does not take place at the village level, but between individuals and families.3 We build a theoretical model where risk-sharing takes place between pairs of agents. There are idiosyncratic shocks to individual income and community-level shocks. We consider how the opportunity for cross-community links affects the shape and efficiency of risk sharing arrangements. We find that when links across villages form, there can be less risk sharing within a village. Welfare is higher for those directly or indirectly connected across villages, but lower for those with no path connecting them to the other village. Overall, welfare can be higher. Thus, empirical findings that insurance within a village is not complete is not necessarily evidence of an inefficient pattern of risk-sharing relations. Rather, the finding is consistent with risk-sharing patterns that involve cross-community relations, and such patterns may yield higher aggregate welfare despite incomplete insurance within a village.

Geography and social links shape economic interactions. In industries, schools, and markets, the entire network determines outcomes. This paper analyzes a large class of games and obtains a striking result. Equilibria depend on a single network measure: the lowest eigenvalue. This paper is the first to uncover the importance of the lowest eigenvalue to economic and social outcomes. It captures how much the network amplifies agents' actions. The paper combines new tools—potential games, optimization, and spectral graph theory—to solve for all Nash and stable equilibria and applies the results to R&D, crime, and the econometrics of peer effects.

We provide the first analysis of altruism in networks. Agents are connected through a fixed, weighted network and care about the well-being of their network neigh-bors. Given some initial distribution of incomes, agents may provide financial support to their poorer friends. We characterize the Nash equilibria of this transfer game for general networks and utility functions. We show that equilibria solve a well-behaved maximization program, related to classical problems of optimal transportation on networks. We build on this reformulation and establish existence, uniqueness in consumptions and generic uniqueness in transfers. We show that transfers are affected by the geometry of the altru-istic network. They flow through shortest paths and chains of transfers emerge when the network is not transitive. We analyze the effects of changes in incomes and in the network. When an agent suffers a negative income shock, the equilibrium consumption of every agent decreases weakly. We characterize the impact of small redistributions and show that decreasing income inequality may increase consumption inequality. We also characterize the impact of a small increase in altruism. While altruistic networks reduce inequality, more altruism may lead to more inequality.