Marginal Analysis and Bewley Equilibria: The Use of Subgradients
Anthony Horsley and Andrew J Wrobel
Using subdifferentials we develop a marginal-analytical approach to equilibrium and use it in the commodity space L? to derive new results on the representation of prices by a density. Apart from conferring advantages when the intepretation of prices as marginal costs is required, the calculus approach gives scope for improved techniques. For example, in production economies Bewley (1972) shows only that any price singularities can be removed from any equilibrium price, but the method of 'containment of singularities' that we introduce can be used to establish, in equilibrium, the absence of price singularity from the outset. This is achieved in the marginal analysis by identifying concentration sets for singularities of subgradients of cost and uility/production functions. By keeping to bounded subsets of L?, we also simplify Bewley's Mackey continuity condition on preferences to the transparent condition of continuity for convergence in measure, which, unlike Mackey continuity, can be fully accounted for in terms of comprehensive economic properties. We apply our methods to examples which illustrate the cardinal points.