Increasing complexity in structurally stable models: An application to a pure exchange economy

A model M is defined (see Anderlini and Canning (2001) and Yu et al. (2009) ) as a quadruple M={Λ,X,F,R}, where Λ and X represent the parameter and actions spaces, respectively, F is a correspondence defining the feasible actions and R is a real-valued function which measures the degree of rationality of the feasible actions. We recall that structural stability means the continuity of the equilibrium set with respect to small perturbations of the parameters and that robustness to bounded rationality holds if small deviations from rationality imply small changes in the equilibrium set. In this paper we extend M to a model M̄={Λ̄,X̄,F̄,R̄}, where Λ̄ is defined as the set of all compact subsets of Λ, X̄=X, F̄ and R̄ are the feasibility and rationality correspondences which extend F and R, respectively. M̄ is more complex than M, since M is embedded into M̄ in a natural way. We show that the structural stability of M implies the structural stability of M̄ and that M̄ is robust to bounded rationality if R̄ is lower semi-continuous. This abstract characterization of complexity is important because it can be used to appraise the nontrivial issue of whether structural stability and robustness to bounded rationality are preserved when a structurally stable model M is extended to M̄. By applying this abstract construction to a pure exchange economy, the result by Loi and Matta (2010), concerning the stability of the equilibrium set with respect to perturbations of endowments along a given path, is extended to perturbations of paths under bounded rationality.