Fisher Meets Lindahl: A Unified Duality Framework for Market Equilibrium

The Fisher market equilibrium for private goods and the Lindahl equilibrium for public goods are classic and fundamental solution concepts for market equilibria. While Fisher market equilibria have been well-studied, the theoretical foundations for Lindahl equilibria remain substantially underdeveloped. In this work, we propose a unified duality framework for market equilibria. We show that Lindahl equilibria of a public goods market correspond to Fisher market equilibria in a dual Fisher market with dual utilities, and vice versa. The dual utility is based on the indirect utility, and the correspondence between the two equilibria works by exchanging the roles of allocations and prices. Using the duality framework, we address the gaps concerning the computation and dynamics for Lindahl equilibria and obtain new insights and developments for Fisher market equilibria. First, we leverage this duality to analyze welfare properties of Lindahl equilibria. For concave homogeneous utilities, we prove that a Lindahl equilibrium maximizes Nash Social Welfare (NSW). For concave non-homogeneous utilities, we show that a Lindahl equilibrium achieves $(1/e)^{1/e}$ approximation to the optimal NSW, and the approximation ratio is tight. Second, we apply the duality framework to market dynamics, including proportional response dynamics (PRD) and t\^atonnement. We obtain new market dynamics for the Lindahl equilibria from market dynamics in the dual Fisher market. We also use duality to extend PRD to markets with total complements utilities, the dual class of gross substitutes utilities. Finally, we apply the duality framework to markets with chores. We propose a program for private chores for general convex homogeneous disutilities that avoids the "poles" issue, whose KKT points correspond to Fisher market equilibria. We also initiate the study of the Lindahl equilibrium for public chores.