General equilibrium and welfare modeling in spatial continuum: a practical framework for land use planning

The application of continuous distributions from statistics in spatial modeling makes it possible to represent discrete choices in a spatial continuum, and to obtain efficiency results and competitive equilibrium prices where aggregate or discretized models fail. Along these lines, and combining principles established by Aumann and Hildenbrand in the sixties with recent results from stochastic optimization, the paper develops a practical modeling framework for land use planning and presents the associated stochastic algorithms for numerical implementation. We consider groups of consumers and producers whose activities are distributed over space, and who have to make decisions, say, about where to live, which marketplace to visit, and which infrastructure facilities to invest in. After presenting a general equilibrium model in which all consumers meet their own budget with given transfers, we focus on the case in which transfers among consumer groups adjust to support the maximization of a given social welfare criterion. It appears that this optimization problem becomes more tractable if it is treated as the minimization of a dual welfare function, that solely depends on prices but is evaluated after integration over space. Next, we apply the dual welfare function to represent (non-rival) demand that simultaneously benefits several agents, reflecting a general informational infrastructure as well as investments with uncertain outcomes. This leads to a minimax problem, in which the dual welfare function is to be minimized with respect to prices and maximized with respect to non-rival demand. Finally, we endogenize welfare weights jointly with prices to model, for example, a land consolidation process whereby none of the participants should lose relative to the initial situation, and the gains could be shared according to agreed principles. This gives rise to a bargaining problem whose solution can be found by jointly minimizing the dual welfare function over prices and welfare weights, subject to constraints

Global convergence of the stochastic tatonnement process

The paper introduces stochastic elements into the Walrasian tâtonnement process, both to make it more realistic and to ensure its global convergence, with probability 1. It is assumed that the aggregate excess demand satisfies standard assumptions but is subject to measurement error. We distinguish two cases. First, the true aggregate excess demand is assumed to satisfy the Weak Axiom of Revealed Preference. This condition will be met if the underlying economy has a single consumer, several consumers with identical homothetic utility functions, or if it maximizes a social welfare function. We prove that after two minor modifications, under a fairly general specification of the measurement error and by imposing a certain consistency property on the estimator of excess demand, the tâtonnement process converges with probability 1 to an equilibrium. Second, we consider the case that the Weak Axiom only holds around some of the equilibria. The procedure proposed imposes a random shock at time intervals of increasing duration. We also discuss how the procedure could be extended to determine all equilibria, to deal with jumps in excess demand including those that do not satisfy the Weak Axiom, and to represent agents who only gradually learn how to find an optimum