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Characterizations of long-run producer optima and the short-run approach to long-run market equilibrium: a general theory with applications to peak-load pricing

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  • Horsley, Anthony
  • Wrobel, Andrew J.

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 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 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 generalizations 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 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 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, exploiting the operating policies and plant valuations that must be determined anyway. These critical short-run solutions have relatively simple forms that 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.

Suggested Citation

  • Horsley, Anthony & Wrobel, Andrew J., 2005. "Characterizations of long-run producer optima and the short-run approach to long-run market equilibrium: a general theory with applications to peak-load pricing," LSE Research Online Documents on Economics 19307, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:19307
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    File URL: http://eprints.lse.ac.uk/19307/
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    References listed on IDEAS

    as
    1. Anthony Horsley & Andrew J Wrobel, 1996. "Comparative Statics for a Partial Equilibrium Model of Investment with Wicksell-Complementary Capital Inputs," STICERD - Theoretical Economics Paper Series /1996/302, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    2. repec:cep:stitep:300 is not listed on IDEAS
    3. Horsley, Anthony & Wrobel, Andrew J., 2002. "Efficiency rents of pumped-storage plants and their uses for operation and investment decisions," Journal of Economic Dynamics and Control, Elsevier, vol. 27(1), pages 109-142, November.
    4. Takayama,Akira, 1985. "Mathematical Economics," Cambridge Books, Cambridge University Press, number 9780521314985, December.
    5. Anthony Horsley & Andrew Wrobel, 2005. "Continuity of the equilibrium price density and its uses in peak-load pricing," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(4), pages 839-866, November.
    6. repec:cep:stitep:301 is not listed on IDEAS
    7. Anthony Horsley & Andrew J. Wrobel, 2002. "Boiteux's solution to the shifting-peak problem and the equilibrium price density in continuous time," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 20(3), pages 503-537.
    8. Anthony Horsley & Andrew J Wrobel, 1996. "Efficiency Rents of Storage Plants in Peak-Load Pricing, I: Pumped Storage," STICERD - Theoretical Economics Paper Series /1996/301, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    9. Horsley, Anthony & Wrobel, A. J. & Van Zandt, Timothy, 1998. "Berge's maximum theorem with two topologies on the action set," Economics Letters, Elsevier, vol. 61(3), pages 285-291, December.
    10. repec:cep:stitep:302 is not listed on IDEAS
    11. Anthony Horsley & Andrew J Wrobel, 1996. "Uninterruptible Consumption, Concentrated Charges, and Equilibrium in the Commodity Space of Continuous Functions," STICERD - Theoretical Economics Paper Series /1996/300, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    12. Bewley, Truman F., 1972. "Existence of equilibria in economies with infinitely many commodities," Journal of Economic Theory, Elsevier, vol. 4(3), pages 514-540, June.
    13. Horsley, Anthony & Wrobel, Andrew J., 1999. "Efficiency rents of storage plants in peak-load pricing, ii: hydroelectricity," LSE Research Online Documents on Economics 19342, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Horsley, Anthony & Wrobel, Andrew J., 2007. "Profit-maximizing operation and valuation of hydroelectric plant: A new solution to the Koopmans problem," Journal of Economic Dynamics and Control, Elsevier, vol. 31(3), pages 938-970, March.

    More about this item

    Keywords

    general equilibrium; fixed-input valuation; nondifferentiable joint costs; Wong- Viner Envelope Theorem; public utility pricing;

    JEL classification:

    • D58 - Microeconomics - - General Equilibrium and Disequilibrium - - - Computable and Other Applied General Equilibrium Models
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory
    • L94 - Industrial Organization - - Industry Studies: Transportation and Utilities - - - Electric Utilities
    • D24 - Microeconomics - - Production and Organizations - - - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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