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# Boiteux's solution to the shifting-peak problem and the equilibrium price density in continuous time

## Author

Listed:
• Anthony Horsley

() (Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK)

• Andrew J. Wrobel

() (Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK)

## Abstract

Bewley's condition on production sets, imposed to ensure the existence of an equilibrium price density when $L^{\infty}$ is the commodity space, is weakened to allow applications to continuous-time problems, and especially to peak-load pricing when the users' utility and production functions are Mackey continuous. A general form for production sets with the required property is identified, and examples are given of technologies which meet the weakened but not the original condition: these include industrial use and storage of cyclically priced goods. This gives a framework for settling Boiteux's conjecture on the shifting-peak problem. To make clear the restriction implicit in Mackey continuity, we interpret it as interruptibility of demand; and we point out that, without this assumption, the equilibrium can feature pointed peaks with singular, instantaneous capacity charges. The general equilibrium results are supplemented by results for prices supporting individual consumer or producer optima.

## Suggested Citation

• 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.
• Handle: RePEc:spr:joecth:v:20:y:2002:i:3:p:503-537
Note: Received: February 16, 2000; revised version: July 7, 2001
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## Citations

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Cited by:

1. 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.
2. 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.
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. Anthony Horsley & Andrew J Wrobel, 2005. "Characterizations of long-run producer optima and the short-runapproach to long-run market equilibrium: a general theory withapplications to peak-load pricing," STICERD - Theoretical Economics Paper Series 490, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
5. Tsitakis, D. & Xanthopoulos, S. & Yannacopoulos, A.N., 2006. "A closed-form solution for the price of cross-commodity electricity derivatives," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 371(2), pages 543-551.
6. Horsley, Anthony & Wrobel, Andrew J., 2005. "A practical short-run approach to market equilibrium," LSE Research Online Documents on Economics 19313, London School of Economics and Political Science, LSE Library.

## More about this item

### Keywords

Price density; Continuous-time peak-load pricing.;

### JEL classification:

• C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
• D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
• L94 - Industrial Organization - - Industry Studies: Transportation and Utilities - - - Electric Utilities

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