Welfare targeting in networks

Abstract

This paper investigates welfare targeting for public goods in networks. First, we show that a tax/subsidy scheme (not necessarily budget-balanced) affects each consumer only insofar as it affects his neighbourhood. Second, we show that either a Pareto-improving income redistribution can be found or there exist Negishi weights, which we relate to the network structure. Third, in the case of Cobb–Douglas preferences, we show that a Law of Welfare Targeting holds and links two well-known notions of the comparative statics of policy interventions: neutrality and welfare paradoxical effects. Collectively, our findings uncover the importance of the $-1$ eigenvalue to economic and social policy: it is an indication of how consumers absorb the impact of income redistribution.

Introduction

Many major challenges facing modern societies (essential infrastructure, information acquisition, emerging infectious disease) relate to enhancing public good provision across different consumers. While market outcomes often provide scope for policy intervention, central planners seldom have the luxury to completely change the state of the economy and implement optimal outcomes. Instead, social planners typically aim to design welfare-improving reforms — involving small changes as in the policy reform literature in the tradition of Dixit (1975) and Guesnerie (1977), and with important policy implications as in Ahmad and Stern (1984).

This paper explores welfare targeting with small tax/subsidy schemes, not necessarily budget-balanced, which are traditionally viewed as a benchmark for broader policy interventions, for the private provision of public goods in networks. In a key contribution, Bramoullé and Kranton (2007) showed that the network context, where local influences are heterogeneous among consumers, is a natural setting to examine the private provision of public goods. Bramoullé et al. (2014) investigated the whole range of strategic substitution and identified a threshold of impact related to the lowest eigenvalue of the network. Below the threshold, the uniqueness and stability of a Nash equilibrium hold. Beyond it, multiple Nash equilibria will in general exist, and stability holds only for corner equilibria. Allouch (2015) extended this model to the non-linear case, with a condition on the normality of the public good which follows the classical (Bergstrom et al., 1986) (henceforth BBV) approach, and showed that their neutrality no longer holds for income redistribution in general networks. Galeotti et al. (2020) analyse optimal policy interventions informed by the eigenvalues of the underlying network of spillovers. Other recent and relevant contributions to the network literature include those by: Galeotti et al., 2010, Ghiglino and Goyal, 2010, Acemoglu et al., 2016, Bourlès et al., 2017, Kinateder and Merlino, 2017, López-Pintado, 2017, Chen et al., 2018, Belhaj and Deroïan, 2019, Elliott and Golub, 2019, Allouch, 2017, Allouch and King, 2019, Akbarpour et al., 2020, Ushchev and Zenou, 2020, Li et al., 2021, Sun et al., 2021, Faias et al., 2015, and Herskovic and Ramos (2020).

First, we establish a property that is key to understanding the impact of welfare targeting in networks. More specifically, we show that a tax/subsidy scheme affects each consumer only insofar as it affects the consumer’s neighbourhood, formed by himself and his neighbours. That is, it is the neighbourhood scheme, rather than the tax/subsidy scheme that affects consumption. Hence the policy implications can be derived by focusing just on the neighbourhood schemes. In particular, neutral tax/subsidy schemes (budget-balanced or not) in general networks are those with null neighbourhood schemes, or equivalently those that are the eigenvectors to the $-1$ eigenvalue.

Secondly, to inform/guide welfare targeting, we use an approach based on a weighted utilitarian social welfare function. More specifically, the social planner maximises a weighted sum of consumers’ utility functions. We find two mutually exclusive cases — either there is a Pareto-improving income redistribution or, if not, there exist Negishi welfare weights. These Negishi weights provide the implicit welfare weights at the initial equilibrium. In particular, we find that a Pareto-improving income redistribution always exists in the class of networks that has a neutral and budget-unbalanced tax/subsidy scheme, since Negishi weights cannot exist. This finding is very much in the spirit of the BBV neutrality result as it holds for any profile of preferences of consumers. Additionally, in the case of Cobb–Douglas preferences, we show that the feasibility of Pareto-improving reform turns out to be readily interpreted and easily checked from the network structure. As a consequence, our analysis leads to a useful characterisation of welfare targeting.

Thirdly, we provide a link between key, but seemingly unrelated, notions of comparative statics: neutrality and paradoxical welfare effects. In fact, by focusing on tax/subsidy schemes that are also eigenvectors, our analysis shows that in the case of Cobb–Douglas preferences, neutrality (or, equivalently, the $-1$ eigenvalue) corresponds to the point of policy switch between tax/subsidy schemes where the utility levels of the donors and the recipients move in the same direction as the scheme (normal welfare impact) and tax/subsidy schemes where the utility levels of the donors and the recipients move in the opposite direction (paradoxical welfare impact). In addition, we show that a Law of Welfare Targeting holds for more general tax/subsidy schemes determined by the $-1$ eigenvalue.

In different settings, our results highlight the importance of the $-1$ eigenvalue to social and economic outcomes, since our findings identify it as a condition for neutral tax/subsidy schemes, Pareto improvement, and the policy switch. In interpretation, the $-1$ eigenvalue is an indication of how consumers, via their neighbourhood, absorb the impact of tax/subsidy schemes, and hence of the welfare implications. Despite the frequency with which the $-$1 eigenvalue appears for many (but not all) networks, as far as we know the $-1$ eigenvalue is not used as a common measure of network analysis in any other fields, including sociology, computer science, and physics. Given that the $-1$ eigenvalue provides a key to social and economic outcomes, perhaps its relationship to the underlying network structure could usefully be studied alongside classic network statistics such as the highest, the second, and lately the lowest eigenvalues.

The paper is structured as follows. Section 2 sets out the model and Section 3 investigates welfare targeting. Section 4 looks at Pareto improvement and Negishi weights and Section 5 provides a new perspective on neutrality and paradoxical welfare effects. Section 6 concludes the paper. An Appendix provides proofs of the propositions and corollaries.