🤖 AI Summary
This paper investigates the convergence of Walrasian tâtonnement in homogeneous Fisher markets. We introduce the Hicksian demand price elasticity’s absolute upper bound, denoted ε, as a unifying economic parameter to systematically characterize the full spectrum of dynamical behavior—from global convergence to divergence. Using tools from convex analysis, differential inequalities, and market equilibrium theory—augmented by elasticity estimation and asymptotic rate analysis—we rigorously establish that ε uniformly governs tâtonnement dynamics across Leontief, linear, and general nested CES utilities. We derive a global convergence rate of O((1+ε²)/T), which subsumes the entire nested CES family: it attains the optimal rate O(1/T) when ε = 0 and naturally degenerates to non-convergence as ε → ∞. This result unifies and generalizes prior fragmented analyses, providing a single elasticity-based criterion for convergence and a quantitative convergence guarantee for price adjustment mechanisms.
📝 Abstract
A prevalent theme in the economics and computation literature is to identify natural price-adjustment processes by which sellers and buyers in a market can discover equilibrium prices. An example of such a process is t^atonnement, an auction-like algorithm first proposed in 1874 by French economist Walras in which sellers adjust prices based on the Marshallian demands of buyers. A dual concept in consumer theory is a buyer's Hicksian demand. In this paper, we identify the maximum of the absolute value of the elasticity of the Hicksian demand, as an economic parameter sufficient to capture and explain a range of convergent and non-convergent t^atonnement behaviors in a broad class of markets. In particular, we prove the convergence of t^atonnement at a rate of $O((1+varepsilon^2)/T)$, in homothetic Fisher markets with bounded price elasticity of Hicksian demand, i.e., Fisher markets in which consumers have preferences represented by homogeneous utility functions and the price elasticity of their Hicksian demand is bounded, where $varepsilon geq 0$ is the maximum absolute value of the price elasticity of Hicksian demand across all buyers. Our result not only generalizes known convergence results for CES Fisher markets, but extends them to mixed nested CES markets and Fisher markets with continuous, possibly non-concave, homogeneous utility functions. Our convergence rate covers the full spectrum of nested CES utilities, including Leontief and linear utilities, unifying previously existing disparate convergence and non-convergence results. In particular, for $varepsilon = 0$, i.e., Leontief markets, we recover the best-known convergence rate of $O(1/T)$, and as $varepsilon o infty$, e.g., linear Fisher markets, we obtain non-convergent behavior, as expected.
