A tâtonnement process with fading memory, stabilization and optimal speed of convergence

The purpose of this work is to provide a way to improve stability and convergence rate of a price adjustment mechanism that converges to a Walrasian equilibrium. We focus on a discrete tâtonnement based on a two-agent, two-good exchange economy, and we introduce memory, assuming that the auctioneer adjusts prices not only using the current excess demand, but also making use of the past excess demand functions. In particular, we study the effect of computing a weighted average of the current and the previous excess demands (finite two level memory) and of all the previous excess demands (infinite memory). We show that suitable weights' distributions have a stabilizing effect, so that the resulting price adjustment process converge toward the competitive equilibrium in a wider range of situations than the process without memory. Finally, we investigate the convergence speed toward the equilibrium of the proposed mechanisms. In particular, we show that using infinite memory with fading weights approaches the competitive equilibrium faster than with a distribution of quasi-uniform weights.