Competitive economic systems: Stability, decomposition, and aggregation

The purpose of this work is to formulate, analyze, and resolve stability of competitive equilibrium under structural perturbations in the Hicks-Metzler algebraic setting. Nonstationary models are introduced for multiple markets of commodities or services, to consider a reduction in the number of commodities, nonlinear saturation phenomena, changing of a commodity from a substitute to a complement to another commodity, time-varying shifts in excess demand, etc. By using the modern mathematical methods of the comparison principle and the vector Lyapunov function, a decomposition-aggregation approach is proposed to treat large market systems. A commodity can be split up into a number of subcommodities, or several commodities can be combined into one composite commodity. Then, a low-order linear aggregate market model can be formed which represents the price adjustment mechanism for the composite commodities. Stability of the aggregate model implies stability of the original nonlinear and nonstationary market, and stability is connective. That is, stability of the aggregate model implies stability of each subset of markets for the composite commodities. This is remarkable in that it shows a wide tolerance of stable market models to nonlinear nonstationary phenomena and, therefore, inherent robustness of competitive equilibrium in economic systems.