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 tâtonnement. 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.

翻译：私人商品的费雪市场均衡与公共商品的林达尔均衡是市场均衡的经典且基本解概念。尽管费雪市场均衡已得到充分研究，但林达尔均衡的理论基础仍显著不足。本文提出一个市场均衡的统一对偶框架。我们证明，公共商品市场的林达尔均衡对应于具有对偶效用的对偶费雪市场中的费雪市场均衡，反之亦然。对偶效用基于间接效用，两种均衡间的对应关系通过交换分配与价格的角色实现。利用该对偶框架，我们填补了林达尔均衡在计算与动态性方面的空白，并为费雪市场均衡提供了新的见解与进展。首先，我们运用此对偶性分析林达尔均衡的福利性质。对于凹齐次效用函数，我们证明林达尔均衡最大化纳什社会福利（NSW）。对于凹非齐次效用函数，我们表明林达尔均衡对最优NSW达到$(1/e)^{1/e}$近似比，且该近似比是紧的。其次，我们将对偶框架应用于市场动态，包括比例响应动态（PRD）与试探过程。我们从对偶费雪市场的动态中推导出林达尔均衡的新市场动态。我们还利用对偶性将PRD扩展至具有完全互补效用的市场，即总替代效用的对偶类。最后，我们将对偶框架应用于含厌恶品的市场。针对一般凸齐次负效用函数，我们提出一个避免“极点”问题的私人厌恶品规划，其KKT点对应于费雪市场均衡。同时，我们首次对公共厌恶品的林达尔均衡展开研究。