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

## Author Info

• Anthony Horsley

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• Andrew Wrobel
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## 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

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## Bibliographic Info

Article provided by Springer in its journal Economic Theory.

Volume (Year): 26 (2005)
Issue (Month): 4 (November)
Pages: 839-866

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Handle: RePEc:spr:joecth:v:26:y:2005:i:4:p:839-866

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## Related research

Keywords: Price continuity; Equilibrium existence; Efficiency rents.;

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## References

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1. 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.
2. Anthony Horsley & Andrew J Wrobel, 2000. "The Short-Run Approach to LRMC Pricing for Multiple Outputs with Nondifferentiable Costs," STICERD - Theoretical Economics Paper Series 393, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
3. Larry E. Jones, 1982. "A Competitive Model of Commodity Differentiation," Discussion Papers 526, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
4. Anthony Horsley & Andrew J Wrobel, 1996. "Uninterruptible Consumption, Concentrated Charges, and Equilibrium in the Commodity Space of Continuous Functions," STICERD - Theoretical Economics Paper Series 300, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
5. 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.
6. Anthony Horsley & Andrew J Wrobel, 1996. "Efficiency Rents of Storage Plants in Peak-Load Pricing, I: Pumped Storage," STICERD - Theoretical Economics Paper Series 301, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
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## Citations

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Cited by:
1. Anthony Horsley & Andrew J Wrobel, 2000. "Efficiency Rents of Pumped-Storage Plants and their Uses for Operation and Investment Decisions," STICERD - Theoretical Economics Paper Series 405, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
2. Anthony Horsley & Andrew J Wrobel, 2005. "The Wong-Viner Envelope Theorem for subdifferentiable functions," STICERD - Theoretical Economics Paper Series /2005/489, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
3. 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 /2005/490, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
4. Anthony Horsley & Andrew J Wrobel, 2005. "A Practical Short-run Approach to Market Equilibrium," STICERD - Theoretical Economics Paper Series /2005/488, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
5. 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.

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