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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

By Anthony Horsley and Andrew J. Wrobel

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

Topics: HB Economic Theory
Publisher: Suntory and Toyota International Centres for Economics and Related Disciplines, London School of Economics and Political Science
Year: 2005
OAI identifier: oai:eprints.lse.ac.uk:19307
Provided by: LSE Research Online

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Citations

  1. (1968). A general correspondence between dual minimax problems and convex programs”, doi
  2. (1982). Akilov doi
  3. (1974). Applications of duality theory”, doi
  4. (1984). Applied nonlinear analysis.
  5. (1972). Approximation et optimisation.
  6. (1991). Banach lattices. doi
  7. (1998). Berge’s Maximum Theorem with two topologies on the action set”, doi
  8. (1996). Comparative statics for a partial equilibrium model of investment with
  9. (1970). Conjugate convex functions in optimal control and the calculus of variations”, doi
  10. (1974). Conjugate duality and optimization. doi
  11. (2005). Continuity of the equilibrium price density and its uses in peak-load pricing”, doi
  12. (1985). Convex analysis in spaces of measurable functions and its applications to mathematics and economics (in Russian).
  13. (1970). Convex analysis. doi
  14. (1984). Convex analysis. Chichester-New York-Brisbane:
  15. (2005). Demand continuity and equilibrium doi
  16. (1982). Duality approaches to microeconomic theory”, doi
  17. (1980). Economics of electric utility power generation. Oxford-New York:
  18. (2005). Efficiency rents of a hydroelectric storage plant with a variable head”, forthcoming CDAM
  19. (1999). Efficiency rents of hydroelectric storage plants in continuoustime peak-load pricing”,
  20. (1996). Efficiency rents of storage plants in peak-load pricing, I: pumped storage”, STICERD Discussion Paper TE/96/301, LSE. (This is a fuller version of Ref.
  21. (1999). Efficiency rents of storage plants in peak-load pricing, II: hydroelectricity”, STICERD Discussion Paper TE/99/372, LSE. (This is a fuller version of Ref.
  22. (1989). Emerging strategies for energy storage”, doi
  23. (1975). Etude géometrique des espaces vectoriels doi
  24. (1972). Existence of equilibria in economies with infinitely many commodities”, doi
  25. (1973). Functional analysis. doi
  26. (1991). Fundamentals of real analysis. doi
  27. (1975). Geometric functional analysis and its applications. doi
  28. (1967). Lattice theory. doi
  29. (1987). Linear programming in infinite-dimensional spaces.N e w York-Chichester-Brisbane-Toronto-Singapore:
  30. (1983). Linear programming. doi
  31. (1985). Mathematical economics. Cambridge-London-New York:
  32. (1978). Mathematical programming and control theory. doi
  33. (1971). Microeconomic theory. doi
  34. (1964). Peak-load pricing”, in Marginal cost pricing in practice (Chapter 4), doi
  35. (1985). Positive operators. doi
  36. (1964). Some postwar contributions of French economists to theory and public policy”,
  37. (1994). Storing megawatthours with SMES”,
  38. (1988). Subdifferentials of convex symmetric functions: An application of the Inequality of Hardy, Littlewood and Pólya”, doi
  39. (1984). Theory of correspondences. doi
  40. (1971). Theory of maxima and the method of Lagrange”, doi
  41. (1979). Tihomirov
  42. (1963). Topological spaces. doi
  43. (1993). Uninterruptible consumption, concentrated charges, and equilibrium in the commodity space of continuous functions”,
  44. (1957). Water storage policy in a simplified hydroelectric system”,
  45. (1997). Wets doi
  46. Wrobel (2000): “Localisation of continuity to bounded sets for nonmetrisable vector topologies and its applications to economic equilibrium theory”, doi
  47. Wrobel (2002): “Boiteux’s solution to the shifting-peak problem and the equilibrium price density in continuous time”, doi
  48. Wrobel (2002): “Efficiency rents of pumped-storage plants and their uses for operation and investment decisions”, doi

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