Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200502February 2005 
The shortrun approach to longrun equilibrium: a general theory with applications
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 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 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 generalisations 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 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 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, and exploits the operating policies and plant valuations that must be determined anyway. These critical shortrun solutions have relatively simple forms, which 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.
A PDF file (2.2 MB) with the full contents of this report can be downloaded by clicking here.
Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSECDAM200502, together with your name and postal address to:
CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K. 

Phone: +44(0)207955 7732. Fax: +44(0)207955 6877. Email: info@maths.lse.ac.uk 
Introduction to the CDAM Research Report Series.  
CDAM Homepage. 
Last modified: Mon 21st February 2005