Fisher Meets Lindahl: A Unified Duality Framework for Market Equilibrium

The Fisher market equilibrium for private goods markets and the Lindahl equilibrium for public goods markets are classic and fundamental solution concepts for market equilibrium. While the Fisher market equilibrium has been well-studied, the theoretical foundations for the Lindahl equilibrium-including characterizations, computation, and dynamics-remain substantially underdeveloped.

In this work, we propose a unified duality framework for market equilibria in private goods and public goods markets. We show that every Lindahl equilibrium of a public goods market corresponds to a Fisher market equilibrium in a dual private goods 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. This duality framework enables us to transfer insights and results between the two settings. The framework also extends to markets with chores.

Using the duality framework, we address the gaps concerning the computation and dynamics for the Lindahl equilibrium and obtain new insights and developments for the Fisher market equilibrium. 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 tâtonnement. We obtain new market dynamics for the Lindahl equilibria from market dynamics in the dual Fisher market, significantly extending existing results for linear utilities. Moreover, the duality framework also introduces new insights into market dynamics. We show that the recently proposed PRD in gross substitutes Fisher markets is a best-response expenditure procedure in the dual Lindahl setting. Using this observation, we 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, and every KKT point of the program corresponds to a Fisher market equilibrium. We also initiate the study of the Lindahl equilibrium for public chores using duality to the private chores setting.