Pricing Infinite Horizon Programs *

This paper continues the earlier studies [ 17, 181 whose object was to develop an abstract framework for the analysis of problems of resource allocation in continuous time over an infinite horizon. These papers gave conditions on the preferences and technology sufficient to guarantee the existence of an optimal program. In this paper I shall provide an answer to the following question. What strengthening of these conditions guarantees the existence of supporting prices? I pose the problem of resource allocation as a convex programming problem in a suitable infinite dimensional space (Section 2). A convex programming problem is characterised by a family of preferred sets induced by a utility function U and a technologically feasible set 3. Such a framework is not confined to the analysis of an economy with a single representative agent. If the economy consists of a finite number of consumers, with preference orderings representable by concave increasing utility functions (Cr, ,..., cl,), and a finite number of producers, with technology sets (-5 ,..., 3,), then we may let CT = Et=, a, Vi, where aj 2 0, xi"=, aj= 1, and F= Cy!, .-T. Suppose now that the existence of an optimal program and an associated supporting price is established for each parameter value a in the simplex. Under certain conditions (see e.g., Section 5, Theorem 5.15) such a pair leads to an equilibrium with transfer payments. A fixed point argument involving the parameter a and the transfer payments can then be added to ensure the existence of a competitive equilibrium [6, 16, 201. In this way an equilibrium (allocation-price) emerges as an optimal program and an associated supporting price for a particular parameter value in a family of convex programming problems. When the problem of resource allocation ceases to be finite dimensional, an essential step in the analysis is the choice of an appropriate program (commodity) space. In [ 171 I considered a weighted version of the space of