The Robust Stability of Every Equilibrium in Economic Models of Exchange Even under Relaxed Standard Conditions

In an economic model of exchange of goods, the preference structure can be specified by utility functions. Under utility conditions identified here more broadly than usual, except for concavity in place of quasi-concavity, every equilibrium will be stable in a doubly local sense with respect to shifts in the agent’s holdings and Walrasian tâtonnement. This result, fully allowing the boundary of the goods orthant to come into play, is obtained by paying attention not only to prices but also to the closeness of initial holdings to equilibrium holdings. The utility conditions are classically standard for stability investigations, in that they invoke properties coming from second derivatives, but are significantly relaxed in not forcing all goods to be held in positive amounts. Agents can be more than consumers, and the goods are viewed very generally, not just as commodities and not only for immediate disposal. For a given agent some goods are allowed to have no effect at all on utility, while others, although insatiably interesting, may anyway end up at zero in equilibrium. Recent advances in variational analysis provide the support needed for working in that context, which requires in particular a convenient “ample” survivability condition for existence to replace the usual assumption that agents start with at least a little bit of every good. The stability results also point the way toward further developments in which an equilibrium might evolve in response, say, to incremental consumption or inputs in the agents’ holdings as stockpiles or other sources of benefit.