Horsley, Anthony and Wrobel, Andrew J. (2005) Characterizations of longrun producer optima and the shortrun approach to longrun market equilibrium: a general theory with applications to peakload pricing. TE (490). Suntory and Toyota International Centres for Economics and Related Disciplines, London, UK.

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Abstract
This is a new formal framework for the theory of competitive equilibrium and its applications. Our “shortrun approach” means the calculation of longrun producer optima and general equilibria from the shortrun solutions to the producer’s profit maximization 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 shortrun profit can be a nondifferentiable function of the fixed quantities, and the shortrun 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 generalizations of smoothcalculus results of microeconomics are given, including the key WongViner Envelope Theorem. This resolves longstanding discrepancies between “textbook theory” and industrial experience. The other tool employed to characterise longrun producer optima is a primaldual pair of programmes. Both marginalist and programming characterizations 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 shortrun approach is that using the shortrun profit programme and its dual. This programme pair is employed to set up a formal framework for longrun generalequilibrium pricing of a range of commodities with joint costs of production. This gives a practical method that finds the shortrun general equilibrium en route to the longrun equilibrium, exploiting the operating policies and plant valuations that must be determined anyway. These critical shortrun solutions have relatively simple forms that can greatly ease the fixedpoint problem of solving for equilibrium, as is shown on an electricity pricing example. Applicable criteria are given for the existence of the shortrun 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 fixedcoefficients production function to the case of multiple outputs. The shortrun approach is applied to the peakload pricing of electricity generated by thermal, hydro and pumpedstorage 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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