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E-raamat: Short-Run Approach to Long-Run Equilibrium in Competitive Markets: A General Theory with Application to Peak-Load Pricing with Storage

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Short-Run Approach to Long-Run Equilibrium in Competitive Markets: A General Theory with Application to Peak-Load Pricing with StorageSuurem pilt
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The authors present a new formal framework for finding the long-run competitive market equilibrium through short-run equilibria by exploiting the operating policies and plant valuations. This "short-run approach" develops ideas of Boiteux and Koopmans. Applied to the peak-load pricing of electricity generated by thermal, hydro and pumped-storage plants, it gives a sound and practical method of valuing the fixed assets-in this case, the river flows and the geological sites suitable for reservoirs. Its main mathematical basis is the producer"s short-run profit maximization programme and its dual; their solutions have relatively simple forms that can greatly ease the fixed-point problem of solving for the general equilibrium. Since the optimal values (profit and cost functions) are usually nondifferentiable-this is so when there are joint costs of production such as capacity constraints-nonsmooth calculus is employed to resolve long-standing discrepancies between textbook theory and

industrial reality by giving subdifferential extensions of basic results of microeconomics, including the Wong-Viner Envelope Theorem.

Introduction.- Peak-load pricing with cross-price independent demands: a simple illustration.- Characterizations of long-run producer optimum.- Short-run profit approach to long-run market equilibrium.- Short-run approach to electricity pricing in continuous time.- Existence of optimal quantities and shadow prices with no duality gap.- Production techniques with conditionally fixed coefficients.- Conclusions.
1 Introduction
1(14)
2 Peak-Load Pricing with Cross-Price Independent Demands: A Simple Illustration
15(6)
2.1 Short-Run Approach to Simplest Peak-Load Pricing Problem
15(2)
2.2 Reinterpreting Cost Recovery as a Valuation Condition
17(1)
2.3 Equilibrium Prices for the Single-Consumer Case
18(3)
3 Characterizations of Long-Run Producer Optimum
21(52)
3.1 Cost and Profit as Values of Programmes with Quantity Decisions
21(4)
3.2 Split SRP Optimization: A Primal-Dual System for the Short-Run Approach
25(1)
3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price Decisions
26(12)
3.4 SRP and SRC Optimization Systems
38(2)
3.5 SRC/P Partial Differential System for the Short-Run Approach
40(2)
3.6 Other Differential Systems
42(1)
3.7 Transformations of Differential Systems by Using SSL or PIR
43(2)
3.8 Summary of Systems Characterizing Long-Run Producer Optimum
45(2)
3.9 Extended Wong-Viner Theorem and Other Transcriptions from SRP to LRC
47(5)
3.10 Derivation of Dual Programmes
52(1)
3.11 Shephard-Hotelling Lemmas and Their Dual Counterparts
53(9)
3.12 Duality for Linear Programmes with Nonstandard Parameters in Constraints
62(11)
4 Short-Run Profit Approach to Long-Run Market Equilibrium
73(18)
4.1 Outline of the Short-Run Approach
73(7)
4.2 Detailed Framework for Short-Run Profit Approach
80(11)
5 Short-Run Approach to Electricity Pricing in Continuous Time
91(28)
5.1 Technologies for Electricity Generation and Energy Storage
91(6)
5.2 Operation and Valuation of Electric Power Plants
97(12)
5.3 Long-Run Equilibrium with Pumped Storage or Hydro Generation of Electricity
109(10)
6 Existence of Optimal Quantities and Shadow Prices with No Duality Gap
119(18)
6.1 Preclusion of Duality Gaps by Semicontinuity of Optimal Values
119(3)
6.2 Semicontinuity of Cost and Profit in Quantity Variables Over Dual Banach Lattices
122(9)
6.3 Solubility of Cost and Profit Programmes
131(2)
6.4 Continuity of Profit and Cost in Quantities and Solubility of Shadow-Pricing Programmes
133(4)
7 Production Techniques with Conditionally Fixed Coefficients
137(18)
7.1 Producer Optimum When Technical Coefficients Are Conditionally Fixed
137(5)
7.2 Derivation of Dual Programmes and Kuhn-Tucker Conditions
142(6)
7.3 Verification of Production Set Assumptions
148(2)
7.4 Existence of Optimal Operation and Plant Valuation and Their Equality to Marginal Values
150(2)
7.5 Linear Programming for Techniques with Conditionally Fixed Coefficients
152(3)
8 Conclusions
155(2)
A Example of Duality Gap Between SRP and FIV Programmes
157(4)
B Convex Conjugacy and Subdifferential Calculus
161(22)
B.1 The semicontinuous Envelope
161(1)
B.2 The Convex Conjugate Function
162(2)
B.3 Subgradients and Subdifferentiability
164(2)
B.4 Continuity of Convex Functions
166(1)
B.5 Concave Functions and Supergradients
167(1)
B.6 Subgradients of Conjugates
168(3)
B.7 Subgradients of Partial Conjugates
171(5)
B.8 Complementability of Partial Subgradients to Joint Ones
176(7)
C Notation List
183(10)
References 193
Anthony Horsley (1939--2006) was a British mathematical economist and a nuclear physicist who in the 1960s worked for the U.K. Atomic Energy Authority (becoming then its youngest ever Senior Scientific Officer) and for the Council for Scientific Policy. Later, devoting himself to academic research and teaching, he worked on the theory and applications of competitive equilibrium at the University of Sussex, Oxford University and, from 1979 until retirement and untimely death, the London School of Economics. A Renaissance mind, he also had a keen interest in literature, history and politics. He received a Ph. D. in Mathematical Physics from the University of Birmingham and a D. Phil. in Economics from the University of Oxford.





Andrew J. Wrobel (b. 1955) is a Polish-born mathematical economist. He has worked at the Institute of Computer Science in Warsaw and, as a Senior Research Fellow, at the London School of Economics and the Catholic University of Brabant in Tilburg. His main research, joint with Anthony Horsley, is on the theory of competitive equilibrium and its applications to the electricity supply industry. Educated in Warsaw, and in Bonn and London on postgraduate grants from the Studienstiftung des deutschen Volkes and the LSE, he holds an M.A. in Mathematics from the University of Warsaw and a Ph.D. in Economics from the University of London.
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