# Continuity of the equilibrium price density and its uses in peak-load pricing

## Author

Listed:
• Anthony Horsley

()

• Andrew Wrobel

## Abstract

With $L^{\infty}$ as the commodity space, the equilibrium price density is shown to be a continuous function of the commodity characteristics. The result is based on symmetry ideas from the Hardy-Littlewood-Pólya theory of rearrangements. It includes, but is not limited to, the case of symmetric (rearrangement-invariant) production costs and additively separable consumer utility. Examples arise in continuous-time utility pricing, e.g., electricity pricing. In this context, a continuously varying price has two uses. First, it precludes demand jumps that would arise from discontinuous switches from one price rate to another. Second, in the problems of operating and valuing hydroelectric and pumped-storage plants (studied elsewhere), price continuity guarantees that their capacities (viz., the reservoir and the converter), the energy stocks, and in the case of hydro also the river flows, have well-defined marginal values. Copyright Springer-Verlag Berlin/Heidelberg 2005

## Suggested Citation

• 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.
• Handle: RePEc:spr:joecth:v:26:y:2005:i:4:p:839-866 DOI: 10.1007/s00199-004-0568-3
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File URL: http://hdl.handle.net/10.1007/s00199-004-0568-3

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## References listed on IDEAS

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1. Horsley, Anthony & Wrobel, Andrew J., 2000. "The short-run approach to LRMC pricing for multiple outputs with nondifferentiable costs," LSE Research Online Documents on Economics 19338, London School of Economics and Political Science, LSE Library.
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, pages 109-142.
4. Jones, Larry E, 1984. "A Competitive Model of Commodity Differentiation," Econometrica, Econometric Society, vol. 52(2), pages 507-530, March.
5. repec:cep:stitep:301 is not listed on IDEAS
6. Richard, Scott F., 1989. "A new approach to production equilibria in vector lattices," Journal of Mathematical Economics, Elsevier, vol. 18(1), pages 41-56, February.
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), 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. 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.
10. 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.
Full references (including those not matched with items on IDEAS)

## Citations

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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.
2. 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.
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, pages 109-142.
4. Anthony Horsley & Andrew J Wrobel, 2005. "The Wong-Viner Envelope Theorem for subdifferentiable functions," STICERD - Theoretical Economics Paper Series 489, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
5. 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.
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.

### Keywords

Price continuity; Equilibrium existence; Efficiency rents.;

### JEL classification:

• D58 - Microeconomics - - General Equilibrium and Disequilibrium - - - Computable and Other Applied General Equilibrium Models
• L94 - Industrial Organization - - Industry Studies: Transportation and Utilities - - - Electric Utilities
• D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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