The short run approach to long run equilibrium in competitive markets

Lecture Notes in Economics and Mathematical Systems  684 Anthony Horsley Andrew J. Wrobel The Short-Run Approach to LongRun Equilibrium in Competitive Markets A General Theory with Application to Peak-Load Pricing with Storage Lecture Notes in Economics and Mathematical Systems Founding Editors: M Beckmann H.P Künzi Managing Editors: Prof Dr G Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Hagen, Germany Prof Dr W Trockel Murat Sertel Institute for Advanced Economic Research Istanbul Bilgi University Istanbul, Turkey and Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Bielefeld, Germany Editorial Board: H Dawid, D Dimitrov, A Gerber, C.-J Haake, C Hofmann, T Pfeiffer, R Slowi´nski, W.H.M Zijm 684 More information about this series at http://www.springer.com/series/300 Anthony Horsley • Andrew J Wrobel The Short-Run Approach to Long-Run Equilibrium in Competitive Markets A General Theory with Application to Peak-Load Pricing with Storage 123 Anthony Horsley (1939-2006) Watford, Hertfordshire, UK Andrew J Wrobel Warsaw, Poland Completed in August 2015, this book is a revised and restructured version of the STICERD Discussion Paper TE/05/490 “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” © Anthony Horsley and Andrew J Wrobel (London, LSE, 2005) ISSN 0075-8442 ISSN 2196-9957 (electronic) Lecture Notes in Economics and Mathematical Systems ISBN 978-3-319-33397-7 ISBN 978-3-319-33398-4 (eBook) DOI 10.1007/978-3-319-33398-4 Library of Congress Control Number: 2016939945 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface This book is dedicated to the memory of Anthony Horsley (1939–2006), nuclear physicist and mathematical economist, my friend and mentor Most of the book was Chap of my Ph.D Econ thesis “The formal theory of pricing and investment for electricity”, written at the London School of Economics under Tony’s supervision This part of the research was supported financially by Tilburg University’s Center for Economic Research (in 1989–1990) and by ESRC grant R000232822 (1991– 1993); their support is gratefully acknowledged The final manuscript was prepared at the Eastern Illinois University; I am grateful for the use of their premises, which sustained my conclusion I not think that I could have made this last effort without the moral support of my newly-wed wife Anita Shelton, professor of history at the EIU, who has encouraged me to return to academic work after a break of nearly a decade This work, which develops ideas of Boiteux and Koopmans, as well as a few new ones, is permeated by Horsley’s way of thinking about scientific problems His fundamental conviction, grounded in his training and research in elementary particle physics, was that new mathematical frameworks could offer opportunities for theories of greater verisimilitude with new insights and results I could not agree more Rigour is, of course, de rigueur these days, but it becomes rigor mortis if all it serves is a formal extension of existing knowledge I hope that this book will help to vindicate Tony’s stance Charleston, Illinois, USA August 2015 Andrew J Wrobel v Contents Introduction Peak-Load Pricing with Cross-Price Independent Demands: A Simple Illustration 2.1 Short-Run Approach to Simplest Peak-Load Pricing Problem 2.2 Reinterpreting Cost Recovery as a Valuation Condition 2.3 Equilibrium Prices for the Single-Consumer Case 15 15 17 18 Characterizations of Long-Run Producer Optimum 3.1 Cost and Profit as Values of Programmes with Quantity Decisions 3.2 Split SRP Optimization: A Primal-Dual System for the Short-Run Approach 3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price Decisions 3.4 SRP and SRC Optimization Systems 3.5 SRC/P Partial Differential System for the Short-Run Approach 3.6 Other Differential Systems 3.7 Transformations of Differential Systems by Using SSL or PIR 3.8 Summary of Systems Characterizing Long-Run Producer Optimum 3.9 Extended Wong-Viner Theorem and Other Transcriptions from SRP to LRC 3.10 Derivation of Dual Programmes 3.11 Shephard-Hotelling Lemmas and Their Dual Counterparts 3.12 Duality for Linear Programmes with Nonstandard Parameters in Constraints Short-Run Profit Approach to Long-Run Market Equilibrium 4.1 Outline of the Short-Run Approach 4.2 Detailed Framework for Short-Run Profit Approach 21 21 25 26 38 40 42 43 45 47 52 53 62 73 73 80 vii viii Contents Short-Run Approach to Electricity Pricing in Continuous Time 91 5.1 Technologies for Electricity Generation and Energy Storage 91 5.2 Operation and Valuation of Electric Power Plants 97 5.3 Long-Run Equilibrium with Pumped Storage or Hydro Generation of Electricity 109 Existence of Optimal Quantities and Shadow Prices with No Duality Gap 6.1 Preclusion of Duality Gaps by Semicontinuity of Optimal Values 6.2 Semicontinuity of Cost and Profit in Quantity Variables Over Dual Banach Lattices 6.3 Solubility of Cost and Profit Programmes 6.4 Continuity of Profit and Cost in Quantities and Solubility of Shadow-Pricing Programmes Production Techniques with Conditionally Fixed Coefficients 7.1 Producer Optimum When Technical Coefficients Are Conditionally Fixed 7.2 Derivation of Dual Programmes and Kuhn-Tucker Conditions 7.3 Verification of Production Set Assumptions 7.4 Existence of Optimal Operation and Plant Valuation and Their Equality to Marginal Values 7.5 Linear Programming for Techniques with Conditionally Fixed Coefficients 119 119 122 131 133 137 137 142 148 150 152 Conclusions 155 A Example of Duality Gap Between SRP and FIV Programmes 157 B Convex Conjugacy and Subdifferential Calculus B.1 The semicontinuous Envelope B.2 The Convex Conjugate Function B.3 Subgradients and Subdifferentiability B.4 Continuity of Convex Functions B.5 Concave Functions and Supergradients B.6 Subgradients of Conjugates B.7 Subgradients of Partial Conjugates B.8 Complementability of Partial Subgradients to Joint Ones 161 161 162 164 166 167 168 171 176 C Notation List 183 References 193 List of Figures Fig 2.1 Fig 3.1 Fig 4.1 Short-run approach to long-run equilibrium of supply and (cross-price independent) demand for thermally generated electricity: (a) determination of the short-run equilibrium price and output for each instant t, given a capacity k; (b) and (d) trajectories of the short-run equilibrium price and output; (c) the short-run cost curve When k is such that the shaded area in (b) equals r, the short-run equilibrium is the long-run equilibrium 16 Decision variables and parameters for primal programmes (optimization of: long-run profit, short-run profit, long-run cost, short-run cost) and for dual programmes (price consistency check, optimization of: fixed-input value, output value, output value less fixed-input value) In each programme pair, the same prices and quantities— p; y/ for outputs, r; k/ for fixed inputs, and w; v/ for variable inputs—are differently partitioned into decision variables and data (which are subdivided into primal and dual parameters) Arrows lead from programmes to subprogrammes 32 Flow chart for an iterative implementation of the short-run profit approach to long-run market equilibrium For simplicity, all demand for the industry’s outputs is assumed to be consumer demand that is independent of profit income, and all input prices are fixed (in terms of the numeraire) Absence of duality gap and existence of the optima (Or, yO ) can be ensured by using the results of Sects 6.1 to 6.4 74 ix Appendix B 179 mere existence of such a complementary r, but also gives a method of calculating it This alternative proof is detailed next Alternative Proof of Theorem B.8.3 Take any y, any k at which C y; / is finite and continuous, and any p @y C y; k/ Introduce … WD C#1 , the convex-concave function on P K defined by (B.7.1) By the SSL (Lemma B.7.2), it suffices to show that @O k … p; k/ Ô ; Since p; / h p j yi C y; /, which is bounded from below on a neighbourhood of k, the concave function … p; / is continuous on all of intK dOom… p; /, which contains k (Sect B.4).16 Also, since p @y C y; k/, one has … p; k/ D h p j yi C y; k/, which is finite It follows that … p; / nowhere takes the value C1.17 Therefore, … p; / is superdifferentiable at k, i.e., @O k … p; k/ ¤ ; by Theorem B.3.1 A basic shortcoming of Theorem B.8.3 is its failure to apply at the boundary points of the function’s effective domain, dom C And indeed, at a boundary point, a partial subgradient may have no complement to a joint one—but it is useful to identify those cases in which such complements exist This is because the boundary points of dom C can be the points of greatest interest: for example, when C is the short-run cost as a function of the output bundle y and the fixed-input bundle k, all the efficient combinations of y and k lie on the boundary of dom C If, however, C has a finite convex extension CEx , defined on the whole space (or at least on a neighbourhood of dom C), and dom C can be represented as the sublevel set of another finite convex function CDm , then Theorem B.8.3 can be applied to both of these functions (CEx and CDm ) For the original function C, this yields a result that applies also to the boundary points of dom C Corollary B.8.4 (Complementary Subgradient at Boundary Point) Let CW Y K ! R [ fC1g be a (jointly) convex function, where Y and K are topological vector spaces paired with their continuous duals (P and R) Assume that: The effective domain of C has the form ˚ « dom C D y; k/ W CDm y; k/ Ä and k K0 (B.8.4) where K0 is a convex subset of K, and CDm W Y K ! R is a finite, continuous and convex function k K0 and CDm y; k/ Ä 0, i.e., y; k/ dom C There exists a yS Y with CDm yS ; k < Continuity of … p; / can also be proved more succinctly: the maximization programme that defines … by (B.7.1) satisfies, at k, Slater’s Condition for generalized perturbed CPs as formulated in [44, Theorem 18 (a)] So … p; / is continuous at k by [44, Theorem 18 (a)]; to apply this formally, the programme must of course be reoriented to minimization 17 If … p; / took the value C1 anywhere, then it would have to equal C1 everywhere on intK dOom… p; /—see the end of Sect B.1—but it is finite at k 16 180 Appendix B C (or, more precisely, its restriction to dom C) has a finite, continuous and convex extension CEx W Y K ! R Then for every p @y C y; k/ there exists an r such that p; r/ @y;k C y; k/ Proof Every p @y C y; k/ has the form p D p0 C ˛p00 for some p0 @y CEx y; k/, p00 @y CDm y; k/ and a scalar ˛ 0, with ˛ D if CDm y; k/ < This is because, since C D CEx C ı j dom C/, @y C y; k/ D @y CEx y; k/ C @y ı y; k j dom C/ ˚ « D @y CEx y; k/ C @ı y j y0 W CDm y0 ; k Ä D @y CEx y; k/ C cone @y CDm y; k/ (B.8.5) when CDm y; k/ D When CDm y; k/ < 0, the term @y ı—which is the outward normal cone to the sublevel set of CDm ; k/—equals f0g, in which case the term denoting the cone generated by @y CDm must be deleted from (B.8.5) For additivity of @ (with a similar application to a sum of the form C C ı), see, e.g., [42, 23.8 and proof of 28.3.1], [44, Theorem 20] or [48, 5.38 and 7.2] The relevant formula for the normal cone to a sublevel set is given in, e.g., [32, 4.3: Proposition 2], [42, 23.7.1] or [48, 7.8] Since CEx and CDm are finite and continuous (everywhere on Y K), Theorem B.8.3 applies to both; so there exist r0 and r00 with p0 ; r0 @y;k CEx y; k/ and p00 ; r00 @y;k CDm y; k/ (B.8.6) It now suffices to set r WD r0 C ˛r00 To see this, use again the formula for the normal cone and additivity of @ (this time for joint subdifferentials) to obtain from (B.8.6) that p; r/ D p0 C ˛p00 ; r0 ˛r00 « ˚ @y;k CEx y; k/ C @y;k ı y; k j y0 ; k0 W CDm y0 ; k0 Ä ˚ «  @y;k CEx y; k/ C @y;k ı y; k j CDm Ä C @y;k ı y; k j Y K0 / D @y;k CEx y; k/ C @y;k ı y; k j dom C/ D @y;k C y; k/ The penultimate equality follows from (B.8.4); by the way, on its l.h.s @y;k ı y; k j Y K0 / D f0g @k ı k j K0 / The Alternative Proof of Theorem B.8.3—the proof based on the SSL—has a counterpart that gives another proof of Corollary B.8.4 This is detailed next Alternative Proof of Corollary B.8.4 Let … WD C#1 (a saddle function on P K) By the SSL (Lemma B.7.2), it suffices to show that @O k p; k/ Ô ; for the given k and any p @y C y; k/ For this, note first that … D …0 ı j P K0 /, where Appendix B 181 …0 p; kQ denotes, for each kQ K, the supremum of h p j yi CEx y; kQ over y subject to CDm y; kQ Ä Next, since CDm yS ; k < and CDm˝ is continuous, on a neighbourhood of k one ˛ has CDm yS ; < and so …0 p; / p j yS CEx yS ; , which is bounded from below on a neighbourhood of k So the concave function …0 p; / is continuous on all of intK dOom…0 p; /, which contains k (Sect B.4).18 Also, since k K0 and p @y C y; k/, one has …0 p; k/ D … p; k/ D h p j yi C y; k/, which is finite It follows that … p; / nowhere takes the value C1.19 Therefore, …0 p; / is superdifferentiable at k, i.e., @O k …0 p; k/ Ô ; by Theorem B.3.1 Finally, since p; / is continuous, @O k … p; k/ D @O k …0 p; k/ @ı k j K0 / Ô ; since both terms are nonempty sets (@ı is a cone); for additivity of @, see, e.g., [44, Theorem 20 (i) under (a)] or [48, 5.38 (b)] Continuity of …0 p; / can also be proved more succinctly: by the third assumption of Corollary B.8.4, the maximization programme that defines …0 satisfies, at kQ D k, Slater’s Condition as generalized to infinite-dimensional inequality constraints in [44, (8.12)] So …0 p; / is continuous at k by [44, Theorem 18 (a)] 19 If …0 p; / took the value C1 anywhere, then it would have to equal C1 everywhere on intK dOom…0 p; /—see the end of Sect B.1—but it is finite at k 18 Appendix C Notation List Notation is grouped below in several categories See also Table 5.1 for correspondence between the general duality scheme (Sects 3.3 and 3.12) and its application to electricity supply (Sects 5.2 and 5.3).1 Profit and Cost Optimization and Shadow-Pricing Programmes: Parameters and Decision Variables, Solutions, Optimal Values and Marginal Values y Y an output bundle, in a vector space Y k K a fixed-input bundle, in a vector space K v V a variable-input bundle, in a vector space V p P an output price system, in a vector space P r R a fixed-input price system, in a vector space R w W a variable-input price system, in a vector space W y, k, etc increments to y, k, etc ( differs from the upright ) Y a production set (in the commodity space Y K V) A, B and C matrices or linear operations, esp such that y; k; v/ Y if and only if Ay Bk Cv Ä AT the transpose of a matrix A ı j Y/ the 0-1 indicator function of a set Y (equal to on Y, and to C1 outside) Yı the polar cone of Y (a cone in P R W when Y is a cone in Y K V) Yıp;w the polar cone’s section through p; w/ Note two unrelated meanings of the symbols s and : in the general duality scheme of Sects 3.3 and 3.12, these mean the standard parameters (s) paired with the standard dual variables ( ), but in the description of energy storage techniques in Sect 5.1 they mean the energy stock (s) and spillage ( ) Also, the n, nSt and nTu of Sect 5.2 mean lower constraint parameters (whose original, unperturbed values are zeros), whereas in the short-run approach to equilibrium and its application to electricity pricing, in Sects 4.2 and 5.3, n means an input of the numeraire © Springer International Publishing Switzerland 2016 A Horsley, A.J Wrobel, The Short-Run Approach to Long-Run Equilibrium in Competitive Markets, Lecture Notes in Economics and Mathematical Systems 684, DOI 10.1007/978-3-319-33398-4 183 184 Appendix C G and G 00 respectively, the sets of generators and of spanning vectors of Yı when Y is a polyhedral cone in a finite-dimensional space projY Y/ projection on Y of a subset, Y, of Y K V YSR k/ short-run production set (the section of Y through k) ILR y/ long-run input requirement set (the negative of the section of Y through y) ISR y; k/ short-run input requirement set (the negative of the section of Y through y; k/) vmax Z and vmin Z sets of all the maximal and of all the minimal points of a subset, Z, of an ordered vector space (used with YSR k/, ILR y/ or ISR y; k/ as Z) …LR the maximum long-run profit, a function of p; r; w/ …SR the maximum short-run a.k.a operating profit, a function of p; k; w/ CLR the minimum long-run cost, a function of y; r; w/ CSR the minimum short-run cost, a function of y; k; w/ @C the subdifferential of a convex function C O @… the superdifferential of a concave function … r… the (Gâteaux) gradient vector of a function … @=@k partial differentiation with respect to a scalar variable k VL y; k; v/ the set of all variable-input bundles that minimize the short-run cost vL y; k; v/ the variable-input bundle—if unique—that minimizes the short-run cost YO p; k; w/ the set of all output bundles that maximize the short-run profit, i.e., maximize the function h p j i CSR ; k; w/ yO p; k; w/ the output bundle—if unique—that maximizes the short-run profit, i.e., maximizes the function h p j i CSR ; k; w/ KO p; r; w/ the set of all fixed-input bundles that maximize the long-run profit kO p; r; w/ the fixed-input bundle—if unique—that maximizes the long-run profit (under decreasing returns to scale) C SR y; k; w/ the maximum, over shadow prices, of output value less fixed-input value (and less …LR when Y is not a cone) C LR y; r; w/ the maximum, over shadow prices, of output value (less …LR when Y is not a cone) …SR p; k; w/ the minimum, over shadow prices, of total fixed-input value (plus …LR when Y is not a cone) RO p; k; w/ the set of all fixed-input price systems that minimize the total fixedinput value (plus …LR when Y is not a cone) rO p; k; w/ the fixed-input price system—if unique—that minimizes the total fixed-input value PL y; k; w/ the set of all output price systems that maximize the output value less fixed-input value, h j yi …SR ; k; w/, less …LR when Y is not a cone pL y; k; w/ the output price system—if unique—that maximizes h j yi …SR ; k; w/ s vector of the standard primal parameters for a convex or linear programme (paired to its equality and inequality constraints) Appendix C 185 vector of the standard dual variables (Lagrange multipliers of the constraints) for a convex or linear programme †O p; s/ the set of all the standard dual solutions (Lagrange multiplier systems) when the primal is a linear programme with s as its primal parameters and h p j i as its objective function O p; s/ the standard dual solution (a.k.a Lagrange multiplier system)—if unique—when the primal is a linear programme with s as its primal parameters and h p j i as its objective function L the Lagrangian (i.e., the Lagrange function of the primal and dual variables and parameters) Characteristics of the Supply Industry  a production technique of the Supply Industry ˆÂ the set of fixed inputs of production technique  „ the set of variable inputs of production technique  Y the production set of technique  , a cone in Y RˆÂ R„ a variable input, with a price w a fixed input, with a price r ˆF the set of fixed inputs with given prices rF ˆE the set of fixed inputs with prices rE to be determined in long-run equilibrium G the supply cost of an equilibrium-priced input ˆE , a function of the supplied quantity q (or k ) Characteristics of Consumer and Factor Demands (From Industrial User) F production function of the Industrial User—a function of inputs: n of the numeraire and z of the differentiated good (e.g., electricity) Uh consumer h’s utility, a function of consumptions: ' of the Industrial User’s product, m of the numeraire and x of the differentiated good (e.g., electricity) u t; x/ the consumer’s instantaneous utility from the consumption rate x at time t (when U is additively separable) mEn consumer h’s initial endowment of the numeraire h & h consumer h’s share of profit … from the supply of input ˆE & h IU consumer h’s share in the Industrial User’s profit, …IU $ h consumer h’s share in the operating profit from production technique  of the Supply Industry B p; %; M/ consumer’s budget set when his income is M, the differentiated good (electricity) price is p and the Industrial User’s product price is % MO SR h pI rE ; rF I w; % j k consumer’s income in the short run MO LR h p; rE ; % consumer’s income in the long run (Supply Industry’s pure profit is zero) xO h p; %I M/ consumer h’s demand for the differentiated good (e.g., electricity) when its price system is p, the Industrial User’s product price is %, and the consumer’s income is M 186 Appendix C 'O h p; %I M/ consumer h’s demand for the Industrial User’s product when its price is %, the differentiated good’s (e.g., electricity) price system is p, and the consumer’s income is M zO p; %/ the Industrial User’s factor demand for the differentiated good (e.g., electricity) when its price system is p and the User’s product price is % nO p; %/ the Industrial User’s factor demand for the numeraire when the User’s product price is % and the differentiated good’s (e.g., electricity) price system is p Short-Run General-Equilibrium Prices and Quantities p?SR , %?SR prices for the differentiated good (electricity) and for the IU’s product y?SR  output of the differentiated good (electricity) by production technique  ? vSR variable input into production technique   ? xSR h , z?SR consumer demand and factor demand for the differentiated good (electricity) m?SR h , n?SR consumer demand and factor demand for the numeraire ' ?SR the Industrial User’s output Long-Run General-Equilibrium Prices and Quantities w the given prices of the Supply Industry’s variable inputs rF the given rental prices of the Supply Industry’s fixed-priced capital inputs rE rental prices of the Supply Industry’s equilibrium-priced capital inputs—to be determined in long-run equilibrium r? the equilibrium prices of the equilibrium-priced inputs (i.e., the equilibrium value of rE ) kÂ? equilibrium capacities of producer  in the Supply Industry p?LR , y?LR  , etc equilibrium prices and quantities—as above, but for the longrun equilibrium Electricity Generation (All Techniques) p t/ electricity price at time t (in $/kWh), i.e., p is a time-of-use (TOU) tariff y t/ rate of electricity output from a plant, at time t (in kW) Dt p/ cross-price independent demand for electricity (in kW) at time t, if the current price is p (in $/kWh) Thermal Generation S p/ in the short run, the cross-price independent rate of supply (in kW) of thermally generated electricity, if the current price is p (in $/kWh) cSR y/ the instantaneous short-run thermal cost per unit time (in $/kWh), if the current output rate is y (in kW); the common graph of the correspondences S and @cSR is the thermal SRMC curve  a type of thermal plant fuel type used by plant type   v fuel input of a thermal plant (in kWh of heat) Á technical efficiency of a thermal plant, i.e., 1=Á is the heat rate Appendix C 187 w unit running cost of a thermal plant (in $ per kWh of electricity output), equal to the price of fuel (in $ per kWh of heat) times the heat rate k a thermal generating capacity (in kW) Ä unit value of a thermal generating capacity at time t, per unit time (in $/kWh) RT r D Ä t/ dt unit value of a thermal generating capacity in total for the cycle (in $/kW) RT t/ D Ä t/ = Ä t/ dt density, at time t, of the distribution of capacity charges over the cycle, i.e., a function representing a subgradient of the convex functional EssSup on L1 Œ0; T (more generally, a subgradient of any capacity requirement function) rF the given rental price of a thermal generating capacity (in $/kW) t/ unit value of nonnegativity constraint on the output of a thermal plant at time t, per unit time (in $/kWh) YO Th p; k; w/ the set of all the electricity output bundles that maximize the operating profit of a thermal plant of capacity k with a unit running cost w when p is the TOU electricity tariff yO Th p; k; w/ the electricity output bundle—if unique—that maximizes the operating profit of a thermal plant of capacity k with a unit running cost w when p is the TOU electricity tariff y? t/ the general-equilibrium rate of electricity output from the thermal plant of type  at time t (in kW) Pumped Storage kSt the plant’s storage a.k.a reservoir capacity (in kWh) Ä St dt/ unit value of storage capacity on a time interval of length dt (in $/kWh) RT rSt D Ä St dt/ unit value of storage capacity in total for the cycle (in $/kWh) ? rSt the (long-run) equilibrium rental price of storage capacity (in $/kWh) G kSt / the supply cost of kSt of storage capacity St dt/ unit value of nonnegativity constraint on energy stock on an interval of length dt (in $/kWh) kCo the plant’s conversion capacity (in kW) Ä Pu t/ unit value of converter’s pump capacity at time t, per unit time (in $/kWh) Ä Tu t/ unit value of converter’s turbine capacity at time t, per unit time (in $/kWh) Ä Co t/ D Ä Pu t/ C Ä Tu t/ unit value of converter’s capacity at time t, per unit time (inR $/kWh) T rCo D Ä Co t/ dt unit value of conversion capacity in total for the cycle (in $/kW) F rCo the given rental price of conversion capacity (in $/kW) O Y PS pI kSt ; kCo / the set of all the electricity output bundles that maximize the operating profit of a pumped-storage plant with capacities kSt and kCo when p is the TOU electricity tariff 188 Appendix C yO PS pI kSt ; kCo / the electricity output bundle—if unique—that maximizes the operating profit of a pumped-storage plant with capacities kSt and kCo when p is the TOU electricity tariff y?PS t/ the general-equilibrium rate of electricity output from the pumpedstorage plant at time t (in kW) s0 energy stock at time and T (in kWh) unit value of energy stock at time and T (in $/kWh) s t/ energy stock at time t (in kWh) & h St household h’s share of profit from supplying the storage capacity (i.e., share of the rent for the storage site) t/ unit value of energy stock at time t (in $/kWh) O PS pI kSt ; kCo / the set of all TOU shadow prices for energy stock (profit‰ imputed time-of-use unit values of energy stock) in a pumped-storage plant with capacities kSt and kCo when the TOU electricity tariff is p O PS pI kSt ; kCo / the TOU shadow price for energy stock (profit-imputed timeof-use unit value of energy stock) in a pumped-storage plant with capacities kSt and kCo when the TOU electricity tariff is p—if the shadow price is indeed unique (as a function of time) Hydro kSt the plant’s storage a.k.a reservoir capacity (in kWh) Ä St dt/ unit value of storage capacity on a time interval of length dt (in $/kWh) RT rSt D Ä St dt/ unit value of storage capacity in total for the cycle (in $/kWh) ? rSt the (long-run) equilibrium rental price of storage capacity (in $/kWh) G kSt / the supply cost of reservoir of capacity kSt St dt/ unit value of nonnegativity constraint on water stock on an interval of length dt (in $/kWh) kTu the plant’s turbine-generator capacity (in kW) Ä Tu t/ unit value of turbine capacity at time t, per unit time (in $/kWh) RT rTu D Ä Tu t/ dt unit value of turbine capacity in total for the cycle (in $/kW) F rTu the given rental price of turbine capacity (in $/kW) Tu t/ unit value of nonnegativity constraint on turbine’s output at time t, per unit time (in $/kWh) e t/ rate of river flow at time t (in kW) YO H pI kSt ; kTu I e/ the set of all the electricity output bundles that maximize the operating profit of a hydro plant with capacities kSt and kTu and with river inflow function e when the TOU electricity tariff is p yO H pI kSt ; kTu I e/ the electricity output bundle—if unique—that maximizes the operating profit of a hydro plant with capacities kSt and kTu and with river inflow function e when the TOU electricity tariff is p y?H t/ the general-equilibrium rate of electricity output from the hydro plant at time t (in kW) t/ rate of spillage from the reservoir at time t (in kW) s0 water stock at time and T (in kWh) unit value of water stock at time and T (in $/kWh) Appendix C 189 s t/ water stock at time t (in kWh) t/ unit value of water stock at time t (in $/kWh) O H pI kSt ; kTu I e/ the set of all TOU shadow prices of water (profit-imputed ‰ time-of-use unit value of water) in a hydro plant with capacities kSt and kTu and with river inflow function e when the TOU electricity tariff is p O H pI kSt ; kTu I e/ the imputed TOU shadow price of water (profit-imputed time-of-use unit value of water) in a hydro plant with capacities kSt and kTu and with river inflow function e when the TOU electricity tariff is p—if the shadow price is indeed unique (as a function of time) & h St household h’s share of profit from supplying the reservoir capacity (i.e., share of the rent for the hydro site) Specific Vector Spaces, Norms and Functionals meas the Lebesgue measure, on an interval Œ0; T of the real line R L1 Œ0; T the space of meas-integrable real-valued functions on Œ0; T L1 Œ0; T the space of essentially bounded real-valued functions on Œ0; T EssSup y/ D ess supt2Œ0;T y t/ the essential supremum of a y L1 Œ0; T kyk1 WD EssSup jyj the supremum norm on L1 C Œ0; T the space of continuous real-valued functions on Œ0; T M R Œ0; T the space of Borel measures on Œ0; T Œ0;T s t/ dt/ the integral of a continuous function s with respect to a measure "t the Dirac measure at t (i.e., a unit mass concentrated at the single point t) BV 0; T/ the space of functions of bounded variation on 0; T/ VarC / the total positive variation (upper variation) of a BV 0; T/ C VarC T//C the cyclic positive variation of c / WD Var / C 0/ Norms and Topologies on Vector Spaces, Dual Spaces, Order and Nonnegativity, Scalar Product Y the norm-dual of a Banach space Y; k k/ k k the dual norm on Y Y the Banach predual of Y; k k/, when Y is a dual Banach space k k0 the predual norm on Y 0 YC , YC and YC the nonnegative cones in Y, Y and Y (when these are Banach lattices): e.g., LC and L1C are the nonnegative cones in L1 and L1 yC and y the nonnegative and nonpositive parts of a y Y (when Y is a vector lattice) k means that k is a strictly positive vector (in a lattice paired with another one); here, used only with a finite-dimensional k h j i a bilinear form (scalar product) on the Cartesian product, P Y, of two vector spaces (if P D Rn D Y then p y is an alternative notation for the scalar product h p j yi WD pT y, where y is a column vector and pT is a row of the same, finite dimension n) w Y; P/ the weak topology on a vector space Y for its pairing with another vector space P (e.g., with Y or Y when Y is a dual Banach space) 190 Appendix C m Y; P/ the Mackey topology on Y for its pairing with P (e.g., with P D Y or with P D Y when Y is a dual Banach space) w and m abbreviations for w P ; P/ and m P ; P/, the weak* and the Mackey topologies on the norm-dual of a Banach space P bw the bounded weak* topology (on a dual Banach space) clY;T Z the closure of a set Z relative to a (larger) set Y with a topology T intY;T Z the interior of a set Z relative to a (larger) set Y with a topology T Y a the algebraic dual of a vector space Y TSLC D m Y; Y a / the strongest locally convex topology on a vector space Y Sets Derived from a Set in a Vector Space cone Z the cone generated by a subset, Z, of a vector space (i.e., the smallest cone containing Z) conv Z the convex hull of a subset, Z, of a vector space (i.e., the smallest convex set containing Z) cor Z the core of a subset, Z, of a vector space ext Z the set of all the extreme points of a subset, Z, of a vector space span Z the linear span of a subset, Z, of a vector space N y j Z/ D @ı y j Z/ the outward normal cone to a convex set Z at a point y Z (a cone in the dual space) Na y j Z/ D @a ı y j Z/ the algebraic normal cone to Z at y (a cone in the algebraic dual space); @a is the algebraic subdifferential Sets and Functions Derived from Functions or Operations on a Vector Space argmaxZ f means the set of all maximum points of an extended-real-valued function f on a set Z dom C the effective domain of a convex extended-real-valued function C dOom… the effective domain of a concave extended-real-valued function … epi C the epigraph of a convex extended-real-valued function C (on a vector space) ker A the kernel of a linear operation, A lsc C the lower semicontinuous envelope of C (the greatest l.s.c minorant of C) usc … the upper semicontinuous 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the problem of solving for long -run equilibrium (as well as finding the short- run equilibrium on the way) It seems clear that the approach is worth applying not only to the case of