T\^atonnement Dynamics for Fisher Markets with Chores

In this paper, we initiate the study of t\^atonnement dynamics in markets with chores. T\^atonnement is a fundamental market dynamics, capturing how prices evolve when they are adjusted in proportion of their excess demand. While its convergence to a competitive equilibrium (CE) is well understood in goods markets for broad classes of utilities, no analogous results are known for chore markets. Analyzing t\^atonnement in the chores market presents new challenges. Several elegant structural properties that facilitate convergence in goods markets-such as convexity of the equilibrium price set and monotonicity of excess demand under the t\^atonnement price updates-fail to hold in the chore setting. Consistent with these difficulties, we first show that na\"ive t\^atonnement diverges. To overcome this, we propose a modified process called relative t\^atonnement, where prices are updated according to normalized excess demand. We prove its convergence to a CE under suitable step-size choices for a broad class of disutility functions, namely continuous, convex, and 1-homogeneous (CCH) disutilities. This class includes many standard forms such as linear and convex CES disutilities. Our proof proceeds by introducing a nonsmooth, nonconvex yet regular objective function-a generalization of the objective in the Eisenberg-Gale-type dual program introduced by [CKMN24]. For convex CES disutilities, where disutility is the weighted $p$-norm of the individual chore disutilities for $p \in (1, \infty)$, we show that relative t\^atonnement converges to an $\varepsilon$-CE in $O(1/\varepsilon^2)$ iterations. This quadratic convergence rate is established by leveraging the polar gauge (or gauge dual) of the disutility function. Finally, following the framework of [AH58], we analyze the stability of CE and provide a complete characterization of local stability.