Discreteness or Convexity of the State Space: Implications for Nonlinear Economics

Mohammed H. I Dore, Brock University, St. Catharines

**Abstract: **
This paper investigates the generality of equilibrium in economics. It contrasts the discrete methods of social choice with the continuous (convex) methods of general equilibrium, and argues that if the same discrete methods were applied to general equilibrium, then the generality of the invisible hand story becomes doubtful, and the existence of the general equilibrium becomes questionable. On the other hand the choice of the same continuous methods in social choice would make the Arrow difficulties disappear. Next, if the structural stability of general equilibrium is investigated, it can be shown that very restrictive assumptions are required about the number of traders and about the nature of preferences. These restrictive preferences are essentially the same as those required for an invariant measure of value, namely the Chipman and Moore conditions which can also be interpreted in terms of the constancy of the marginal utility of income. The applicability or otherwise of convex methods requires that attention be given to the nature of the state space (or Manifold) before proposing the equations of motions (vector fields) that govern economic variables. If the vector fields governing economic time series are not known or are not knowable, then the nonlinear economics must begin with an investigation of the geometric dimension of a given nonlinear time series, along lines of research in chaotic dynamics.

*Keywords: *general equilibrium, social choice, discrete methods, convexity, invariant measure of value, nonlinearity, geometric dimension, chaotic dynamics