Stochastic Non-T\^atonnement Processes and the Attraction Principle

I introduce stochastic non-t\^atonnement processes and argue that they are a natural outcome of general equilibrium theory. To this end, I revisit classical demand theory to define two diffeomorphisms: the normalized Walrasian demand and a map that flattens indifference hypersurfaces. These diffeomorphisms are applied to the three canonical manifolds in the consumption domain (i.e., the indifference and the offer hypersurfaces, and the trade hyperplane) to analyze their images in the normalized and flat domains, and their relations to the set of Pareto optimal allocations are discussed. Then, I derive, for arbitrary non-t\^atonnement processes, an attraction principle based on the dynamics of marginal rates of substitution seen on the floor of the flat domain. This principle motivates the definition of stochastic non-t\^atonnement processes and leads to a stochastic version of the First Welfare Theorem. When utility functions are attractive and sharp, a class of stochastic non-t\^atonnement processes, called common density processes, is particularly well suited for Monte Carlo simulations, which can be used to compute the distribution of trade outcomes on the contract curve and to model price stickiness.