Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2005-04

March 2005

On the continuity of the equilibrium price density and its uses

Anthony Horsley and Andrew J. Wrobel


With L-infinity as the commodity space, the equilibrium price density is shown to be a continuous function of the commodity characteristics. The result is based on symmetry ideas from the Hardy-Littlewood-Polya theory of rearrangements. It includes, but is not limited to, the case of symmetric (rearrangement-invariant) production costs and additively separable consumer utility. For example, in continuous-time peak-load pricing of electricity, it applies also when there is a storage technology and demands are cross-price dependent. In this context, a continuously varying price has two uses. First, it precludes demand jumps that would arise from discontinuous switches from one price rate to another. Second, in the problems of operating and valuing hydroelectric and pumped-storage plants (studied elsewhere), price continuity guarantees that their capacities (viz., the reservoir and the converter), the energy stocks, and in the case of hydro also the river flows, have well-defined marginal values.

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