# Centre for Discrete and Applicable Mathematics

## CDAM Research Report, LSE-CDAM-2005-02

February 2005

The short-run approach to long-run equilibrium: a general theory with applications

Anthony Horsley and Andrew J. Wrobel

Abstract

This is a new formal framework for the theory of competitive equilibrium and its applications. Our "short-run approach" means the calculation of long-run producer optima and general equilibria from the short-run solutions to the producer's profit maximisation programme and its dual. The marginal interpretation of the dual solution means that it can be used to value the capital and other fixed inputs, whose levels are then adjusted accordingly (where possible). But short-run profit can be a nondifferentiable function of the fixed quantities, and the short-run cost is nondifferentiable whenever there is a rigid capacity constraint. Nondifferentiability of the optimal value requires the introduction of nonsmooth calculus into equilibrium analysis, and subdifferential generalisations of smooth-calculus results of microeconomics are given, including the key Wong-Viner Envelope Theorem. This resolves long-standing discrepancies between "textbook theory" and industrial experience.

The other tool employed to characterise long-run producer optima is a primal-dual pair of programmes. Both marginalist and programming characterisations of producer optima are given in a taxonomy of seventeen equivalent systems of conditions. When the technology is described by production sets, the most useful system for the short-run approach is that using the short-run profit programme and its dual. This programme pair is employed to set up a formal framework for long-run general-equilibrium pricing of a range of commodities with joint costs of production. This gives a practical method that finds the short-run general equilibrium en route to the long-run equilibrium, and exploits the operating policies and plant valuations that must be determined anyway. These critical short-run solutions have relatively simple forms, which can greatly ease the fixed-point problem of solving for equilibrium, as is shown on an electricity pricing example. Applicable criteria are given for the existence of the short-run solutions and for the absence of a duality gap. The general analysis is spelt out for technologies with conditionally fixed coefficients, a concept extending that of the fixed-coefficients production function to the case of multiple outputs. The short-run approach is applied to the peak-load pricing of electricity generated by thermal, hydro and pumped-storage plants. This gives, for the first time, a sound method of valuing the fixed assets - in this case, river flows and the sites suitable for reservoirs.

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 Introduction to the CDAM Research Report Series. CDAM Homepage.