Abstract
This illustrates the short-run approach to long-run equilibrium with Boiteux’s treatment of the simplest peak-load pricing problem—the problem of pricing the services of a homogeneous capacity which produces a nonstorable good with cyclic demands (such as electricity). A direct calculation of long-run equilibrium poses a fixed-point problem, but, with cross-price independent demands, short-run equilibrium can be determined instant-by-instant by the elementary method of intersecting the supply and demand curves. Long-run equilibrium can then be found by adjusting the capacity so that its unit cost equals its unit value defined as the unit operating profit.
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Notes
- 1.
In terms of the subdifferential, ∂ C, of the long-run cost (2.2.1) as a function of output, the fixed-point problem is to find a function p such that \(p \in \partial C_{\mathrm{LR}}\left (D\left (\,p\right )\right )\), where \(D\left (\,p\right )\left (t\right ) = D_{t}\left (\,p\left (t\right )\right )\) if demands are cross-price independent.
- 2.
The SRMC and the short-run supply correspondences are inverse to each other, i.e., they have the same graph: in Fig. 2.1a, the broken line is both the supply curve and the SRMC curve.
- 3.
This condition (\(r = \nabla _{k}\Pi _{\mathrm{SR}}\)) is stronger than cost-optimality of the fixed inputs (when p is an SRMC).
References
Horsley A, Wrobel AJ (2002) Boiteux’s solution to the shifting-peak problem and the equilibrium price density in continuous time. Econ Theory 20:503–537. DOI:10.1007/s001990100226
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Horsley, A., Wrobel, A.J. (2016). Peak-Load Pricing with Cross-Price Independent Demands: A Simple Illustration. In: The Short-Run Approach to Long-Run Equilibrium in Competitive Markets. Lecture Notes in Economics and Mathematical Systems, vol 684. Springer, Cham. https://doi.org/10.1007/978-3-319-33398-4_2
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DOI: https://doi.org/10.1007/978-3-319-33398-4_2
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