The short run approach to long run equilibrium in competitive markets

2.1 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 ou

Lecture Notes in Economics and Mathematical Systems 684

A General Theory with Application to

Peak-Load Pricing with Storage

Lecture Notes in Economics

Murat Sertel Institute for Advanced Economic Research

Istanbul Bilgi University

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

Watford, Hertfordshire, UK 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).

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

This book is dedicated to the memory of Anthony Horsley (1939–2006), nuclearphysicist and mathematical economist, my friend and mentor Most of the book wasChap.5of my Ph.D Econ thesis “The formal theory of pricing and investment forelectricity”, written at the London School of Economics under Tony’s supervision.This part of the research was supported financially by Tilburg University’s Centerfor 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, whichsustained my conclusion I do not think that I could have made this last effort withoutthe moral support of my newly-wed wife Anita Shelton, professor of history at theEIU, who has encouraged me to return to academic work after a break of nearly adecade

This work, which develops ideas of Boiteux and Koopmans, as well as a fewnew ones, is permeated by Horsley’s way of thinking about scientific problems.His fundamental conviction, grounded in his training and research in elementaryparticle physics, was that new mathematical frameworks could offer opportunitiesfor theories of greater verisimilitude with new insights and results I could not agreemore Rigour is, of course,de rigueurthese days, but it becomesrigor mortisif all

it serves is a formal extension of existing knowledge I hope that this book will help

to vindicate Tony’s stance

August 2015

v

1 Introduction 1

2 Peak-Load Pricing with Cross-Price Independent Demands: A Simple Illustration 15

2.1 Short-Run Approach to Simplest Peak-Load Pricing Problem 15

2.2 Reinterpreting Cost Recovery as a Valuation Condition 17

2.3 Equilibrium Prices for the Single-Consumer Case 18

3 Characterizations of Long-Run Producer Optimum 21

3.1 Cost and Profit as Values of Programmes with Quantity Decisions 21

3.2 Split SRP Optimization: A Primal-Dual System for the Short-Run Approach 25

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price Decisions 26

3.4 SRP and SRC Optimization Systems 38

3.5 SRC/P Partial Differential System for the Short-Run Approach 40

3.6 Other Differential Systems 42

3.7 Transformations of Differential Systems by Using SSL or PIR 43

3.8 Summary of Systems Characterizing Long-Run Producer Optimum 45

3.9 Extended Wong-Viner Theorem and Other Transcriptions from SRP to LRC 47

3.10 Derivation of Dual Programmes 52

3.11 Shephard-Hotelling Lemmas and Their Dual Counterparts 53

3.12 Duality for Linear Programmes with Nonstandard Parameters in Constraints 62

4 Short-Run Profit Approach to Long-Run Market Equilibrium 73

4.1 Outline of the Short-Run Approach 73

4.2 Detailed Framework for Short-Run Profit Approach 80

vii

5 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

6 Existence of Optimal Quantities and Shadow Prices with No Duality Gap 119

6.1 Preclusion of Duality Gaps by Semicontinuity of Optimal Values 119

6.2 Semicontinuity of Cost and Profit in Quantity Variables Over Dual Banach Lattices 122

6.3 Solubility of Cost and Profit Programmes 131

6.4 Continuity of Profit and Cost in Quantities and Solubility of Shadow-Pricing Programmes 133

7 Production Techniques with Conditionally Fixed Coefficients 137

7.1 Producer Optimum When Technical Coefficients Are Conditionally Fixed 137

7.2 Derivation of Dual Programmes and Kuhn-Tucker Conditions 142

7.3 Verification of Production Set Assumptions 148

7.4 Existence of Optimal Operation and Plant Valuation and Their Equality to Marginal Values 150

7.5 Linear Programming for Techniques with Conditionally Fixed Coefficients 152

8 Conclusions 155

A Example of Duality Gap Between SRP and FIV Programmes 157

B Convex Conjugacy and Subdifferential Calculus 161

B.1 The semicontinuous Envelope 161

B.2 The Convex Conjugate Function 162

B.3 Subgradients and Subdifferentiability 164

B.4 Continuity of Convex Functions 166

B.5 Concave Functions and Supergradients 167

B.6 Subgradients of Conjugates 168

B.7 Subgradients of Partial Conjugates 171

B.8 Complementability of Partial Subgradients to Joint Ones 176

C Notation List 183

References 193

List of Figures

Fig 2.1 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

Fig 3.1 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

Fig 4.1 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, Oy) can be ensured by

using the results of Sects 6.1 to 6.4 74

ix

Fig 5.1 Trajectories of: (a) shadow price of stock O , and (b)

output of pumped-storage plant (optimum storage

policy) OyPSin Sect 5.2, and in Theorem 5.3.1 Unit rent

for storage capacity is VarCc

isRT

0 ˇˇˇp t/  O t/ˇˇˇ dt, the sum of grey areas By

definition, OPSD kSt=kCo 102

Fig A.1 The total capacity value (…SR) and the operating profit

(…SR) of a pumped-storage plant as functions of its

storage capacity kSt(for a fixed conversion capacity

kCo > 0 and a fixed TOU price, p 2 L1n L1, of the

storable good) When kSt> 0, Slater’s Condition is met

and so… D …, but a duality gap opens at kSt D 0,

where… is right-continuous but … drops to zero (Example A.1) 159

Chapter 1

Introduction

This is a new formal framework for the theory of perfectly competitive equilibriumand its industrial applications The “short-run approach” is a scheme for calculatinglong-run producer optimum and market equilibrium by building on short-runsolutions to the producer’s profit maximization problem, in which capital inputsand natural resources are treated as fixed These fixed inputs are valued at theirmarginal contributions to the operating profit and, where possible, their levels arethen adjusted accordingly.1 Since short-run profit is a concave but generally non-differentiable function of the fixed inputs, their marginal values are defined as thegenerally nonunique supergradient vectors Also, they usually have to be obtained

as solutions to the dual programme of fixed-input valuation because there is rarely

an explicit formula for the operating profit The key property of the dual solution

is therefore its marginal interpretation, but this requires the use of a generalized,multi-valued derivative of a convex function—viz., the subdifferential—because anoptimal-value function, such as cost or profit, is commonly nondifferentiable.Despite being essential for applications, differential calculus has been purgedfrom geometric and topological treatments of the Arrow-Debreu model, whichare limited to equilibrium existence and Pareto optimality results But the use ofsubgradients restores calculus as a rigorous method for equilibrium theory Themathematical tools employed here—convex programmes and subdifferentials—make it possible to reformulate some basic microeconomic results In addition tostatements of known subdifferential versions of the Shephard-Hotelling Lemmas,

a subdifferential version of the Wong-Viner Envelope Theorem is devised here forthe short-run approach especially (Sect.3.9) This facilitates economic analysis and

1 When carried out by iterations, the calculations might also be seen as modelling the real processes

of price and quantity adjustments.

© 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_1

1

resolves some long-standing discrepancies between textbook theory and industrialreality.2

These methods are used here to set up a framework for perfectly competitivegeneral-equilibrium pricing of multiple outputs with joint production costs Theterms “general equilibrium” and “market equilibrium” are used interchangeablyhere—i.e., the latter term refers to markets for all the commodities in the realeconomy being modelled The model focuses, however, on the differentiated goodsupplied by a particular industry, termed the Supply Industry (SI) All the othercommodities—except for the SI’s inputs and for the product of an industrial user

of the SI’s outputs—are aggregated into a homogeneous numeraire good Thisyields what is formally a closed model of general equilibrium, but it is a modelskewed towards partial equilibrium in the markets for the SI’s products—a generalequilibrium model with a “partial bent” (Sect.4.2)

This model is applied to the pricing, operation and investment problems of anElectricity Supply Industry (ESI) with a technology that can include hydroelectricgeneration and pumped storage of energy, in addition to thermal generation(Chap 5) This application draws on the much simpler case of purely thermalgeneration (Chap.2) and on the studies of operation and valuation of hydroelectricand pumped-storage plants in [21,23, 24, 27] and [30] Here, those results aresummarized and “fed into” the short-run approach

The short-run approach starts with fixing the producer’s capacities k and optimizing the variable quantities, viz., the outputs y and the variable inputsv

For a competitive, price-taking producer, the optimum quantities, Oy and Ov, depend

on their given prices, p and w, as well as on k.3 The primal solution (Oy and Ov)

is associated with the dual solution Or, which gives the imputed unit values of the fixed inputs (with Or  k as their total value); the optima are, for the moment, taken

to be unique for simplicity When the goal is limited to finding the producer’slong-run profit maximum (rather than the market equilibrium), it can be achieved

by part-inverting the short-run solution map of p; k; w/ to y; vI r/ so that the

prices p; r; w/ are mapped to the quantities y; k; v/ This is done by solving the equation Or p; k; w/ D r for k and substituting any solution for the k in Oy p; k; w/

and Ov p; k; w/ to complete a long-run profit-maximizing input-output bundle Such

a bundle may be unique, albeit only up to scale if the returns to scale are constant

(making Or p; k; w/ homogeneous of degree zero in k).

Even within the confines of the producer problem, this approach saves effort

by building on the short-run solutions that have to be found anyway: the problems

of plant operation and plant valuation are of central practical interest and alwayshave to be tackled by producers But the short-run approach is even more useful

2 The theory of differentiable convex functions is, of course, included in subdifferential calculus as

a special case Furthermore, the subgradient concept can also be used to prove, rather than assume, that a convex function is differentiable—by showing that it has a unique subgradient This method

is used in [ 21 , 23 ], [ 27 , Section 9] and [ 30 , Section 9].

3 From Sect 3.2 on, short-run cost minimization is split off as a subprogramme, whose solution is denoted by Lv y; k; w/ In these terms, Ov p; k; w/ D Lv Oy p; k; w/ ; k; w/.

1 Introduction 3

as a practical method for calculating market equilibria For this, with the input

prices r and w taken as fixed for simplicity, the short-run profit-maximizing supply

Oy p; k; w/ is equated to demand for the products Ox p/ to determine the run equilibrium output prices p?SR.k; w/ The imputed capacity values Or p; k; w/, evaluated at p D p?SR.k; w/ together with the given k and w, are onlythenequated

short-to the given capacity prices r short-to determine the long-run equilibrium capacities

k?.r; w/—by solving for k the equation Orp?SR.k; w/ ; k; w D r Finally, the

long-run equilibrium output prices and quantities are determined by substituting

k?.r; w/ for k in the short-run equilibrium solution.4In other words, determination

of investment is postponed untilafter the equilibrium in the product markets hasbeen found: the producer’s long-run problem is split into two problems—that ofoperation and that of investment—and the short-run market equilibrium problem is

“inserted” in between Since the operating solutions usually have relatively simpleforms, doing things in this order can greatly ease the fixed-point problem of solvingfor equilibrium: indeed, the problem can even become elementary when approached

in this way (Chap 2) Furthermore, unlike the optimal investment of the pure

producer problem, the equilibrium investment k? has a definite scale (determined

by demand for the products) Put another way: Or

p?SR.k; w/ ; k; w, the value to

be equated to r, is not homogeneous of degree zero in k like Or p; k; w/ Thus

one can keep mostly to single-valued maps and avoid dealing with multi-valuedcorrespondences—even when the returns to scale are constant Last but not least,like the short-run producer optimum, the short-run general equilibrium is of interest

in itself

This exposition comes in six chapters (not counting the Introduction, sions, or Appendices), which can be divided into three parts The first and mainpart (Chaps 2 5) gives various characterizations of long-run producer optimum(Chap 3), but its final objective is a framework for the short-run approach tolong-run general-equilibrium pricing of a range of commodities with joint costs

Conclu-of production (Chap 4), which is applied to peak-load pricing of electricitygenerated by a variety of techniques (Chap 5) A much simplified version ofthe electricity pricing problem serves also as an introductory example (Chap.2).The characterizations of producer optimum (which are needed for the short-runapproach) are complemented by conditions for existence of the optimal quantitiesand shadow prices in the short-run profit maximization and cost minimizationproblems, and for equality of the total values of the variable quantities and of thefixed quantities—i.e., for absence of a gap between the primal and dual solutions.These results form the second part (Chap 6) The third and last part (Chap 7)introduces the concept of technologies with conditionally fixed coefficients, andspecializes the preceding general analysis to this class (which includes, e.g.,

4 A short-run approach to equilibrium might also be based on short-run cost minimization, in which

not only the capital inputs (k) but also the outputs (y) are kept fixed and are shadow-priced in the

dual problem, but such cost-based calculations are usually much more complicated than those using profit maximization: see Sect 4.1

thermal generation of electricity and pumped energy storage, but not hydroelectricgeneration) AppendixAgives a contextual example of a duality gap—a possible butrather exceptional mathematical complication in convex programming SectionsB.1

to B.7 of Appendix B give the required standard results of convex calculus—including one innovation, viz., LemmaB.7.2on subdifferential sections (the SSL),which underlies the Extended Wong-Viner Envelope Theorem (3.9.1) The typicalmathematical obstacle that necessitates the extension—viz., nonfactorability of jointsubdifferentials for nondifferentiable bivariate convex functions—is looked at inmore detail in Sect.B.8

First of all, for a simple but instructive introduction to the short-run approach tolong-run equilibrium, Boiteux’s treatment of the simplest peak-load pricing problem

is rehearsed: this is the problem of pricing the services of a homogeneous capacitythat produces a nonstorable good with cyclic demands (such as electricity) Adirect calculation of long-run equilibrium poses a fixed-point problem, but, withcross-price independent demands, short-run equilibrium can be determined by theelementary method of intersecting the supply and demand curves for each time

instant separately At each time t, the short-run equilibrium output price p?SR.t/ is the sum of the unit operating cost w and a capacity charge?

SR.t/  0 that is nonzero only at the times of full capacity utilization, i.e., when the output rate y?SR.t/ equals the given capacity k Finally, long-run equilibrium is found by adjusting the capacity

k so that its unit cost r equals its unit value defined as the unit operating profit, which

equals the total capacity charge over the cycle,RT

of electricity pricing with a diverse technology, including energy storage and hydro

as well as thermal generation Such a plant mix makes supply cross-price dependent,even in the short run (i.e., with the capacities fixed) Demand, too, is allowed to becross-price dependent

The setting up of the short-run approach to pricing and investment (Chap.4) isthe most novel part of this work Unlike the characterizations of producer optimum,and the existence results on it, this part of the study is not fully formalized intomathematical theorems: it is assumed, rather than proved, that the short-run equilib-rium is indeed unique, and as for its existence it is merely noted that this cannot be

5 Boiteux’s work is also presented by Drèze [ 15 , pp 10–16], but the short-run character of the approach is more evident from the original [ 9 , 3.2–3.3] because Boiteux discusses the short-run equilibrium first, before using it as part of the long-run equilibrium system When Drèze mentions short-run equilibrium on its own, it is only as an afterthought [ 15 , p 16].

p?SR.k; w/ ; k; w D r—as an equation for the investment k.8 But it is

shown that the SRP Programme-Based System, consisting of Conditions (4.2.12)–(4.2.16) together with (4.2.19)–(4.2.20), is a full characterization of long-run marketequilibrium And, as is seen already from the introductory example of Chap 2,the short-run approach can greatly simplify the problem of solving for long-runequilibrium (as well as finding the short-run equilibrium on the way) It seemsclear that the approach is worth applying not only to the case of electricity butalso to the supply of other time-differentiated commodities (such as water, naturalgas, telecommunications, and so on) The questions of uniqueness, stability anditerative computation of equilibria, although important, are not specific to the short-run approach; also, they have been much studied and are well understood (at leastfor finite-dimensional commodity spaces) The central and distinctive quantitativeelements of the approach are valuation and operation of plants; these problems havebeen fully solved for the various types of plant in the ESI (see Sect 5.2and itsreferences) The priorities in developing the short-run approach are: (i) to analyzethe valuation and operation problems for other technologies and industries, and(ii) to compute numerical solutions from real data by using, at least to start with, thestandard methods (viz., linear programming for producer optima and tâtonnementfor market equilibria) It would seem sensible to address the theoretical questions ofequilibrium uniqueness and stability in the light of future computational experience(in which more elaborate iterative methods could be employed if necessary) Thesequestions are potentially important for practice as well as for completing the theory,but they are not priorities for this study, and are left for further research

The bulk of Chap.3, between the introductory example and the setup for theshort-run approach, gives various characterizations of long-run producer optimum(Sects.3.1to3.11) Each of these is either anoptimization systemor adifferentialsystem, i.e., it is a set of conditions formulated in terms of either the marginaloptimal values or the optimal solutions to a primal-dual pair of programmes(although the two kinds of condition can also be mixed in one system) Thoughequivalent, the various systems are not equally usable, and the best choice of systemdepends on one’s purpose as well as on the available mathematical description

of the technology In the application to electricity pricing with non-thermal as

6 This is not an unacceptable condition, but some capacities can of course be zero in long-run equilibrium The long-run model meets the usual adequacy assumption, as does the short-run model with positive capacities, and so existence of an equilibrium follows from Bewley’s result [ 7 , Theorem 1], which is amplified in [ 31 , Section 3] and [ 29 ] by a proof based on continuity of demand in prices.

7 As is well known, this process does not always converge, but there are other iterative methods.

8 In general, this is an inclusion rather than an equality: see ( 4.2.19 ).

well as thermal generation, the technology is described by production sets ratherthan by profit or cost functions (Sect.5.1)—and so the best tool for the short-runapproach is the system using the programme of maximizing the short-run profit(SRP), together with the dual programme of shadow-pricing the fixed inputs Foreachindividualplant type, the problem of minimizing the short-run cost (SRC) istypically easy (if it arises at all); therefore, it can be split off as a subprogramme

of profit maximization.9 The resulting Split SRP Optimization System serveshere as the preferred basis for the short-run approach to pricing and industrialinvestment (Chap 4) Because of its importance to applications, this system isintroduced as soon as possible, in Sect.3.2—not only before the differential systems(Sects.3.5,3.6and3.9), but also before the other optimization systems (Sects.3.4

and3.9), and even before the discussion of dual programmes in Sect.3.3

Of the differential systems, the first one to be presented formally, in Sect.3.5,

is that which generalizes Boiteux’s original set of conditions—limited though it is

to technologies that are simple enough to allow explicit formulae not only for theSRC function but also for the SRP function Another differential system, introducedinformally in Sect.2.2and formally in Sect.3.9, has the same mathematical form butuses the LRC instead of the SRP function (with the variables suitably swapped) Thetwo systems’ equivalence extends, to convex technologies with nondifferentiablecost functions, the Wong-Viner Envelope Theorem on the equality of SRMC andLRMC Stated in Formula (3.9.1), this is the result outlined earlier in Sect.2.2(in thecontext of Boiteux’s short-run approach to the simple peak-load pricing problem).The extension is made possible by using the subdifferential (a.k.a the subgradientset) as a generalized, multi-valued derivative This is necessary because the joint-cost functions may lose differentiability at the crucial points For example, in thesimplest peak-load pricing problem, the long-run cost is nondifferentiable at everyoutput bundle with multiple global peaks because, although the total capacity charge

is determinate (being equal to r, the given rental price of capacity), its distribution

over the peaks cannot be determined by cost calculations alone And, far from beingexceptional, multiple peaks forming an output plateau do arise in equilibrium as

a solution to the shifting-peak problem—as is shown in [26] under appropriateassumptions about demand.10 Short-run marginal costs are even less determinate:whenever the output rate reaches full capacity, an SRMC exceeds the unit operating

9 By contrast, SRC minimization for a system of plants can be difficult because it involves allocating the system’s given output among the plants Its complexity shows in, e.g., the case

of a hydro-thermal electricity-generating system studied by Koopmans [ 35 ] The decentralized approach taken here (Chaps 4 and 5 with their references) avoids having to deal directly with the formidable problem of minimizing the entire system’s cost: see the Comments with Formulae ( 4.1.3 ) and ( 4.1.4 ).

10 This shows how mistaken is the widespread but unexamined view that nondifferentiabilities of convex functions are of little consequence: the very points which, in a sense, are exceptional a priori turn out to be the rule rather than the exception in equilibrium Also, it is only on finite-dimensional spaces that convex functions are “generically smooth” or, more precisely, twice differentiable almost everywhere with respect to the Lebesgue measure (Alexandroff’s Theorem) On an infinite- dimensional space, a convex function can be nondifferentiable everywhere.

1 Introduction 7

cost w by an arbitrary amount —which makes capacity charges indeterminate

in their total as well as in their distribution This exemplifies a general inclusionbetween subdifferentials of the two costs, as functions of the output bundle: theset of SRMCs is larger than the set of LRMCs when the capital inputs are at anoptimum (i.e., minimize the total cost) It then takes a stronger condition than inputoptimality to ensure that a particular SRMC is actually an LRMC What is needed

is equality of the rental prices to theprofit-imputed values of the fixed inputs (i.e.,

to the fixed inputs’ marginal contributions to the operating profit) This equality isthe required generalization of Boiteux’s condition for long-run optimality, which,

for his one-station technology, equates the capacity price r to the unit operating

profitR

 dt D R p t/  w/ dt [9, 3.3, and Appendix: 12] The valuationsmust

be based on increments to the operating profit: it is generally ineffective to try

to value capacity increments by any reductions in the operating cost The station example shows just how futile such an attempt can be: excess capacitydoes not reduce the operating cost at all, but any shortage of capacity makes therequired output infeasible This leaves the capacity value completely undetermined

one-by SRC calculations—in contrast to the definite valueR

It seems practically out of the question that these costs should be equal; it is difficult to imagine, for instance, how the marginal cost of operating a thermal power station could become high enough to equal the development cost (including plant) of the thermal energy [its long-term marginal cost] The paradox is due to the fact that most industrial plants are

in reality very ‘rigid’ .

There is no question of equating the development cost to the cost of overloading the plant, since any such overloading is precluded by the assumption of rigidity The more usual types of plant have some slight flexibility in the region of their limit capacities but any

‘overloading’ which might be contemplated in practice would never be sufficient to equate its cost with the development cost; hence the paradox referred to above.

Its resolution starts with his

new notion which will play an essential part in ‘peak-load pricing’: for output equal to maximum, the differential cost [the SRMC] is indeterminate: it may be equal to, or less or greater than the development cost [the LRMC].

In the language of subdifferentials, Boiteux’s “new notion”—that the LRMC

is just one of many SRMCs—is a case of the afore-mentioned general inclusionbetween LRMCs and SRMCs, which is usually a strict one: @y CLR.y; r/  

@y CSR.y; k/ when r 2 @ k CSR.y; k/, i.e., when the bundle of capital inputs k minimizes the total cost of producing an output bundle y, given the capital-input prices r (and given also the variable-input prices w, which, being kept fixed, are

suppressed from the notation) For differentiable costs, the inclusion reduces to theWong-Viner equality of gradient vectors: ry CLRD ry CSR(when the capital inputsare at an optimum) But for nondifferentiable costs, all it shows is that each LRMC

is an SRMC—which is thereverseof what is required for the short-run approach.The way out of this difficulty is to bring in the SRP function,…SR, and require thatthe given prices for the capital inputs are equal to their profit-imputed values, i.e.,

that r D r k…SR p; k/ or, should the gradient not exist, that r 2 O@ k…SR (which

is the superdifferential a.k.a the supergradient set) This condition is stronger than

cost-optimality of the fixed inputs, when the output price system p is an SRMC: i.e., if p 2 @y CSR.y; k/ then O@ k…SR p; k/  @ k CSR.y; k/, and the inclusion is

generally strict (indeed, rk…SR can exist also when rk CSR does not, in whichcase rk…SR 2 @k CSR) But the new condition—that r 2 O@k…SR p; k/—is no

stronger than it need be: it is just strong enough to do the job and guarantee that if

p2 @y CSR.y; k/ then p 2 @ y CLR.y; r/.

The present analysis of the relationship between SRMC and LRMC bears out,amplifies and develops Boiteux’s ideas—which, at the time, he allowed, with ahint of exasperation, were “false in the theoretical general case, but more or lesstrue of ordinary industrial plant” Both cases are thus accommodated: the industrialreality of fixed coefficients and rigid capacities as well as the rather unrealistictextbook supposition of smooth costs The gap is bridged between the inadequateexisting theory and its intended applications, and an end is put to its disturbing andunnecessary divorce from reality This allows peak-load pricing to be put, for thefirst time, on a sound and rigorous theoretical basis (Chap.5)

From the new perspective, Boiteux’s condition for long-run optimality (r D

R

p t/  w/ dt) should be viewed as a special case, for the one-station technology,

of the equation r D r k…SR But staying within the confines of this particularexample, Boiteux interprets his condition merely as recovery of the total productioncost, including the capital cost [9, 3.4.2: (2) and Conclusions: 4] This is correct,but only in the case of a single capital input, and it cannot provide a basis fordealing with a production technique that uses a number of interdependent capitalinputs.11In such a case, the present generalization of Boiteux’s condition for long-run optimality is stronger than capital-cost recovery: under constant returns to scale,

if r 2 O@k…SR(or r D r k…SR) then r  k D…SR, but notvice versa if there are two

or more capital inputs (though also the converse is of course true when, with just

one capital input, k is a nonzero scalar) It is a dead end to think purely in terms of

11 Capital inputs are called independent if the SRP function (… SR ) is linear in the

capital-input bundle k D k1; k2 ; : : :/; an example is the multi-station technology of thermal electricity generation Such a technology in effect separates into a number of production techniques with

a single capital input each, and so Boiteux’s analysis applies readily: to ensure that a short-run equilibrium is a long-run equilibrium, it suffices to require cost recovery for each production technique  with k > 0, although one must also remember to check that any unutilized

production technique (one with k D 0) is unprofitable at the equilibrium prices (e.g., that

r  R

p t/  w/ dt for any unbuilt type  of thermal station, with unit capital cost r and

unit fuel cost w).

1 Introduction 9

marginal costs and cost recovery: with multiple capital inputs, cost recovery isnot

sufficient to guarantee that a short-run equilibrium is also a long-run equilibrium or,equivalently, that an SRMC tariff is also an LRMC tariff The SRP function with its

marginals (derivatives w.r.t k), or the SRP programme with the dual solution, must

be brought into the short-run approach This is done here for the first time

In mathematical terms, the Extended Wong-Viner Theorem (3.9.1) comes fromwhat is named the Subdifferential Sections Lemma (SSL), which gives the jointsubdifferential of a bivariate convex function,@y ;k C, in terms of one of its partial

subdifferentials,@y C, and of a partial superdifferential, O@k … p; k/, of the relevant partial conjugate of C (denoted by …, it is a saddle function)—see (3.7.3), andLemmaB.7.2in AppendixB For the extension (3.9.1), the SSL is applied twice:

to either the SRP or the LRC as a saddle function obtained by partial conjugacy

from the SRC, which is a jointly convex function (C) of the output bundle y and the fixed-input bundle k, with the variable-input prices w kept fixed (Sect.3.9) Inthe wider context of convex calculus and its applications, the SSL can be usefullyregarded as a direct precursor of the Partial Inversion Rule (PIR), a well-knownresult that relates the partial sub/super-differentials of a saddle function (@p… andO@k…) to the joint subdifferential of its bivariate convex “parent” function (@y ;k C):

see Lemmas B.7.3 and B.7.5 (whose proofs do derive the PIR from the SSL).One well-known application of this fundamental principle is the equivalence oftwo conditions for optimality, viz., the parametric version of Fermat’s rule andthe Kuhn-Tucker characterization of primal and dual optima as a saddle point ofthe Lagrange function: see, e.g., [45, 11.39 (d) and 11.50] Another well-knownuse of the PIR establishes the equivalence of Hamiltonian and Lagrangian systems

in convex variational calculus; when the Lagrange integrand is nondifferentiable,this usefully splits the Euler-Lagrange differential inclusion (a generalized equationsystem) into the pair of Hamiltonian differential inclusions, and it may eventransform the inclusion into ordinary equations because the Hamiltonian can bedifferentiable also when the Lagrangian is not: see, e.g., [44, (10.38) and (10.40)],[43, Theorem 6] or [4, 4.8.2].12 The present application of the PIR or the SSLrelates the marginal optimal values for a programme to those of a subprogramme,

to put it in general terms In the specific context of extending the Wong-VinerTheorem, SRC minimization is a subprogramme both of SRP maximization and of

LRC minimization (their optimal values are CSR.y; k/, …SR p; k/ and CLR.y; r/,

respectively) This is a new use of what is, in Rockafellar’s words, “a strikingrelationship: : :at the heart of programming theory” [41, p 604]

One half of this argument—the application of the SSL to the saddle function

…SR as a partial conjugate of the bivariate convex function CSRto prove the firstequivalence in (3.9.1)—is given already in Sect 3.7 It comes along with otherapplications of the PIR and the SSL that establish the equivalence of the partial

12 To distinguish the two quite different meanings of the word “Lagrangian”, it shall be occasionally expanded into either “Lagrange function” (in the multiplier method of optimization) or “Lagrange integrand” (in the calculus of variations only).

differential systems to the saddle differential systems of Sect.3.6(which use jointsubdifferentials).

Like all optimization, economic theory has to deal with the nondifferentiability

of optimal values that commonly arises even when the programmes’ objective andconstraint functions are all smooth This has led to the eschewing of marginalconcepts in rigorous equilibrium analysis, but any need for this disappeared withthe advent of nonsmooth calculus Of course, in using generalized derivativessuch as the subdifferential, one cannot expect to transcribe familiar theorems fromthe smooth to the subdifferentiable case simply by replacing the ordinary singlegradients with multi-valued subdifferentials—proper subdifferential calculus must

be applied This not only extends the scope for marginal analysis, but also leads to

a rethinking and reinterpretation that can give a new economic content to knownresults The Wong-Viner Theorem is a case in point: a useful extension depends onrecasting its fixed-input optimality assumption in terms of profit-based valuations(i.e., on restating optimality of the fixed inputs as equality of their rental prices

to their marginal contributions to the operating profit) After this reformulation ofoptimality in terms of marginal SRP—but not before—the “smooth” version of thetheorem can be transcribed to the case of subdifferentiable costs (by replacing each

r with a @) Without this preparatory step, all extension attempts are doomed: adirect transcription of the original Wong-Viner equality of SRMC and LRMC tothe subdifferentiable case is plainly false, and although it can be changed to a trueinclusion without bringing in the SRP function, that kind of result fails to attain thegoal of identifying an SRMC as an LRMC.13

One well-known condition for optimality is, perhaps, conspicuous by its absence from the main part of this analysis The Lagrangian Saddle-Point Condi-tions of Kuhn and Tucker are central to the duality theory of convex programmes(CPs)—and they are used in the studies of hydro and energy storage [21,23,27] and[30], which feed the application of the short-run framework to electricity supply inSects 5.1to 5.3—but here the Kuhn-Tucker Conditions are not used before thestudy of technologies with conditionally fixed coefficients (in Chap 7), althoughthey do appear earlier on the margin (in Comments in Sects.3.3and4.1) Instead

near-of the Kuhn-Tucker Conditions, for a general analysis with an abstract productioncone it is preferred here to use the Complementarity Conditions (3.1.5) on theprice system and the input-output bundle This system is a case of what will becalled the FFE Conditions, which consist of primal feasibility, dual feasibility andequality of the primal and dual objectives (at the feasible points in question) TheFFE Conditions form an effective system whenever the dual programme can beworked out from the primal explicitly This is always so, in principle at least, withthe profit and cost problems because they become linear programmes (LPs) once the

13 Without involving … SR , the inclusion ( @y CLR  @y CSR) can be improved only by making it more precise but no more useful: @y CSR.y; k/ can be shown to equal the union of @ y CLR.y; r/ over

r 2 @k CSR.y; k/, i.e., over all those input price systems r for which k is an optimal input bundle for the output bundle y (given also the suppressed variable-input price system w):

fixed-see ( 3.9.11 ).

1 Introduction 11

production cone has been represented by linear inequalities For an LP, the system

of FFE Conditions is linear in the primal and dual variables jointly—unlike thesystem of Kuhn-Tucker Conditions (which is nonlinear because of the quadraticterm in the Complementary Slackness Condition): compare (3.3.3) with (3.3.2) And

a linear system (i.e., a system of linear equalities and inequalities) is much simpler

to deal with; in particular, it can be solved numerically by the simplex method (oranother LP algorithm) The problem’s size is smaller, though, when the method

is applied directly to the relevant LP (or to its dual), rather than to its system ofFFE Conditions.14Either way, there is no need for the Kuhn-Tucker Conditions insolving the SRP programmes with their fixed-input valuation duals—although theyare instrumental in proving uniqueness of solutions, as in [21,23,27] and [30]

In the LP formulation of a profit or cost programme, the fixed quantities areprimal parameters but need not be the same as the standard “right-hand side”parameters—and so their shadow prices, which are the dual variables, need not beidentical to the standard dual variables Yet the usual theory of linear programmingworks with the standard parameterization, and it is the standard dual solutionthat the simplex method provides along with the primal solution But, as isshown in Sect 3.12, this is not much of a complication because any other dualvariables can be expressed in terms of the standard dual variables a.k.a the usualLagrange multipliers of the constraints This is used in valuing the fixed inputs forelectricity generation (Sect.5.2) The principle has also a counterpart beyond thelinear or convex duality framework: it is the Generalized Envelope Theorem forsmooth optimization, whereby the marginal values of all parameters, including anynonstandard ones, are equal to the corresponding partial derivatives of the ordinaryLagrangian—and are thus expressed in terms of the constraints’ multipliers See [1,(10.8)] or [47, 1.F.b]

The exposition of producer optimum pauses for “stock-taking” in Sect.3.8 Inparticular, Tables3.1and3.2summarize the various characterizations of long-runoptimum, though not their “mirror images” which result from a formal substitution

of the LRC for the SRP These tables record also the methods employed to transformthese systems into one another This shows a unity: the same methods are applied

to systems of the same type, even though this exposition gives special places tothe two systems of most importance for the application of the short-run approach

to the ESI (in Chap 5)—viz., the Split SRP Optimization System of Sect 3.2

and the SRC/P Partial Differential System of Sect.3.5 The latter system’s “mirrorimage”, the L/SRC Partial Differential System of Sect.3.9, is also directly involved

in applications when its conditions of LRMC pricing and LRC minimization serve

as the definition of long-run optimum—as is often the case in public utility pricing,including Boiteux’s work and the account thereof in Chap.2 The other fourteensystems are not applied here, but any of them may be the best tool (for the short-runapproach as for other purposes) when the technology is described most simply in

14 For a count of variables and constraints, see the last Comment in Sect 3.12 before mula ( 3.12.15 ).

For-the system’s own terms; see also For-the Comments at For-the end of Sect.4.1 In particular,one should not be trapped by the language into thinking that a system using theLRC programme or function is somehow inherently unsuitable for the short-runapproach.

The summarizing Sect.3.8ends by noting that some of the systems—includingthe two “special” ones—can be partitioned into a short-run subsystem (whichcharacterizes SRP maxima) and a valuation condition that generalizes Boiteux’scondition for long-run optimality and requires that investment be at a profitmaximum

A complete formalization of all the duality-based systems is deferred toSects.3.10and3.11, in which the programmes’ duality and the systems’ equivalenceare cast as rigorous results with proofs To this end, Sect 3.11restates formallythe subdifferential versions of the Shephard-Hotelling Lemmas (some of which areannounced earlier in Sect 3.6) As has long been known [14, pp 555 and 583],these are cases of the derivative property of optimal value, which transcribes to thesubdifferentiable case directly (by replacing r with@)

The characterizations of long-run producer optimum are complemented byresults on solubility of the primal and dual programmes and on equality of theirvalues (absence of a duality gap) Such an analysis is given in Sects.6.1to6.4; ityields sufficient conditions for existence of a pair of solutions with equal values.First, it is recalled from the general theory of CPs that absence of a duality gap isequivalent to semicontinuity of either optimal value, and this is spelt out for theprofit and cost programmes (Sect.6.1) To make this criterion applicable, Sect.6.2

gives some sufficient conditions for the required semicontinuity of SRP as well

as of LRC and SRC, as functions of the programmes’ quantity data (the fixedquantities) When the commodity space for either the fixed or the variable quantities(the programme’s quantity data or its decision variables) is infinite-dimensional,these criteria use its weak* topology as well as its vector order The commodityspaces are therefore taken to be dual Banach lattices (i.e., the duals of completely

normed vector lattices) One example is L1Œ0; T, which serves here as the output

space in the application to peak-load pricing With this or any other nonreflexivecommodity space, these semicontinuity criteria for profit or cost (as a function

of the fixed quantities) apply only when the given price system (for the variablequantities) lies not just in the Banach dual of the commodity space but actually

in the smaller predual space Such a criterion is therefore adequate for the run approach to general equilibrium (and other analysis thereof) only when theequilibrium price system is known to lie in the predual space—as is the case for the

short-commodity space L1and its predual L1under Bewley’s assumptions [7], which areweakened in [26] to make his density representation of the price system apply to atleast some continuous-time problems Unavoidably, even the weakened assumption

is restrictive: it requires that brief interruptions of a consumption or input flow causeonly small losses of utility or output (i.e., interruptibility of consumer demand and

of input demand) When this is not so and the programme’s given price system

cannot be taken to lie in L1Œ0; T—or in whatever price space is the predual of

the commodity space in some other economic context—a duality gap can still be

in the predual of the commodity space When it does not, the programme can still

be soluble in some, though not all, cases (it must be soluble perforce in generalequilibrium, also when the equilibrium price system does not lie in the predualspace).15

Both thermal generation and pumped storage of electricity are examples ofproduction techniques with conditionally fixed coefficients (c.f.c.)—a conceptwhich extends that of the fixed-coefficients production function to the case of amulti-dimensional output bundle It is introduced in Sect.7.1, which also spells out:the convex programme of SRP maximization (profit-maximizing plant operation)for a c.f.c technique, the dual programme of fixed-input valuation (plant valuation),and the Kuhn-Tucker Conditions—although their fully formalized statements andproofs are deferred to Sect 7.2 In Sect 7.3, the assumptions of Sects 6.2 to

6.4are verified for c.f.c techniques Therefore, the solubility and no-gap results

of Sects.6.2,6.3and6.4can be applied to the profit and cost programmes withsuch a technology, and this is done for the SRP programme (with its dual) inSect 7.4 Finally, Sect 7.5gives a general method of handling c.f.c techniques

by linear programming (formulated in terms of input requirement functions, theseLPs are, however, different from those which come from another description ofthe production sets—such as their original definitions in the case of electricitygeneration and storage in Sect.5.1)

Notation is explained when first used, but it is also gathered at the end, inAppendix C In the main text, Table 5.1 shows the correspondence of notationbetween the general duality scheme (Sects 3.3 and 3.12) and its application toelectricity supply (Sects.5.2and5.3)

15 See [ 21 ] and [ 23] for examples of an SRP programme in which the output space is L1Œ0; T and

a “singular” price term places the price system outside the predual L1Œ0; T, but it is the timing of

the singularity, and not just its presence, that determines whether the programme is soluble or not.

Peak-Load Pricing with Cross-Price

Independent Demands: A Simple Illustration

Problem

The short-run approach to solving for long-run market equilibrium is next illustratedwith the example of pricing, over the demand cycle, the services of a homogeneous

productive capacity with a unit capital cost r and a unit running cost w The

technology can be interpreted as, e.g., electricity generation from a single type of

thermal station with a fuel cost w (in $/kWh) and a capacity cost r (in $/kW) per

period The cycle is represented by a continuous time intervalŒ0; T Demand for the time-differentiated, nonstorable product, D t.p/, is assumed to depend only on the

time t and on the current price p (a scalar) As a result, the short-run equilibrium can

be found separately at each instant t, by intersecting the demand and supply curves

in the price-quantity plane This is because, with this technology, short-run supply

is cross-price independent: given a capacity k, the supply is

k for p> w

(2.1.1)

where p is the current price That is, given a time-of-use (TOU) tariff p—i.e., given

a price p t/ as a function of time t 2 Œ0; T—the set of all the profit-maximizing

output trajectories, OY p; k; w/, consists of selections from the correspondence t 7!

S p t/ ; k; w/ When D t w/ > k, the short-run equilibrium TOU price, p?

SR.t; k; w/, exceeds w by whatever is required to bring the demand down to k (Fig.2.1a) Thetotal of this excess, or “capacity premium”, over the cycle is the unit operating profit,

which in the long run should equal the unit capacity cost r That is, the long-run

© 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_2

15

16 2 Peak-Load Pricing with Cross-Price Independent Demands: A Simple Illustration

Fig 2.1 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

equilibrium capacity, k?.r; w/, can be determined by solving for k the equation

whereC D max f; 0g is the nonnegative part of —i.e., by equating to r the

shaded area in Fig.2.1b The equilibrium capacity can then be put into the short-runequilibrium price function to give the long-run equilibrium price

p?LR.tI r; w/ D p?

SR.t; k?.r; w/ ; w/ (2.1.3)

An obvious advantage of this method is that the short-run equilibrium is ofinterest in itself Also, the short-run calculations can be very simple—as in thisexample For comparison, to calculate the long-run equilibrium directly requirestiming the capacity charges so that they are borne entirely by the resulting demandpeaks—i.e., it requires finding a function  0 such that

Z T

0  t/ dt D 1 and if  t/ > 0 then y t/ D sup

where: y t/ D D p t// and p t/ D w C r t/

This poses a fixed-point problem that, unlike the short-run approach, is not muchsimplified by cross-price independence of demands.1

Since the operating profit is…SR p; k; w/ D kRT

0 p t/  w/Cdt, the break-even

condition (2.1.2) can be rewritten as r D@…SR=@k, i.e., it can be viewed as equating

the capital input’s price to its profit-imputed marginal value And this is indeed,with any convex technology, the first-order necessary and sufficient condition for a

profit-maximizing choice of investment k Together with a choice of output y that maximizes the short-run profit (SRP), such a choice of k maximizes the long-run

profit (LRP), thus turning short-run equilibrium into long-run equilibrium

Furthermore—with any technology and any number of capital inputs—r D

rk…SRif and only if r is the unique solution to the dual of the SRP maximization

programme (and there is no duality gap): this is the derivative property of theoptimal value…SRas a function of the primal parameter k This identity of marginal

values and dual solutions is useful when, with a more complex technology, the SRPprogramme has to be solved by a duality method, i.e., solved together with its dual

It means that the dual solution Or p; k; w/, evaluated at the short-run equilibrium output price system p?SR.k; w/, can be equated to the capital inputs’givenprices r

to determine their long-run equilibrium quantities k?

When the producer is a public utility, competitive profit maximization usually

takes the form of marginal-cost pricing In this context, the equality r D@…SR=@k,

or r D r k…SR when there is more than one type of capacity, guarantees that

an SRMC price system is actually an LRMC price system The result applies toany convex technology—even when the cost functions are nondifferentiable, andmarginal cost has to be defined by using the subdifferential as a generalized, multi-valued derivative This is so in the above example of capacity pricing, since thelong-run cost

respect to y (w.r.t y) And multiple peaks are more the rule than the exception in

equilibrium—note the peak output plateau in Fig.2.1d here, and see [26] for anextension to the case of cross-price dependent demands Similarly, the short-run

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 2 @CLR.D p//, where D p/ t/ D D t p t// if

demands are cross-price independent.

18 2 Peak-Load Pricing with Cross-Price Independent Demands: A Simple Illustration

graph of the instantaneous cost function

cSR.y/ D



wy if0  y  k

in terms of which CSR.y/ equalsRT

0 cSR.y t// dt, so that a TOU price p is an SRMC

at y if and only if p t/ is an instantaneous SRMC at y t/ for each t With this technology, CSR is therefore nondifferentiable whenever k is the cost-minimizing capital input for the required output y: here, cost-optimality of k means merely that

k provides just enough capacity, i.e., that k D Sup y/ Since this condition does not even involve the capital-input price r, it obviously cannot ensure that an SRMC price system p is an LRMC To guarantee this, one must strengthen it to the condition that

r D RT

0 p t/  w/Cdt in this example, or, generally, that r D r k…SR (or that r

belongs to the supergradient set O@k…SR p; k; w/, should …SRbe nondifferentiable

in k).3The capital’s cost-optimality would suffice for the SRMC to be the LRMC ifthe costs were differentiable; this is the usual Wong-Viner Envelope Theorem Thepreceding remarks indicate how to reformulate it to free it from the differentiabilityassumption; this is detailed in Sect.3.9

Cross-price independent demand arises from price-taking optimization by sumers and industrial users with additively separable utility and production func-tions In this case, the short-run equilibrium prices can readily be given in terms ofthe marginal utility of the differentiated good and its productivity in industrial uses.For the simplest illustration, all demand is assumed to come from a single household

con-2 The SRMC and the short-run supply correspondences are inverse to each other, i.e., they have the same graph: in Fig 2.1 a, the broken line is both the supply curve and the SRMC curve.

3This condition (r D r k… SR) is stronger than cost-optimality of the fixed inputs (when p is an

SRMC).

that maximizes the utility function

U x / ; m/ D m C

Z T

0 u t; x t// dt over x /  0 and m  0, the consumptions of the nonstorable good and of the

numeraire, subject to the budget constraint

mC

Z T

0 p t/ x t/ dt  M where M is the income and p./ is a TOU price in terms of the numeraire

(which represents all the other goods and thus closes the model) For each t, the instantaneous utility u t; x/ is taken to be a strictly concave, increasing and

differentiable function of the consumption rate x 2RC, with.@u=@x/ t; 0/ > w (to ensure that, in a short-run equilibrium with k> 0, consumption is positive at every

t) The household’s income consists of an endowment of the numeraire (mEn) andthe pure profit from electricity sales, i.e.,

In other words, D t p/ D @u=@x/ t; //1.p/ When w < @u=@x/ t; k/, this value

of@u=@x is the price needed to equate demand to k So the short-run equilibrium

By (2.1.2) and (2.1.3), the long-run equilibrium capacity k?.r; w/ is determined

from the equation

Chapter 3

Characterizations of Long-Run Producer

Optimum

with Quantity Decisions

Costs and profits of a price-taking producer are, by definition, the optimal values ofprogrammes with quantities as decision variables With several decision variables,

it can be easier to solve the mathematical problem in stages, by fixing some ofthe variables and dealing with the resulting subproblem first The subproblem mayalso be of independent interest, especially if it corresponds to a stage in a practicalimplementation of the complete solution In the context of production, the decision

on how to operate a plant with a fixed equipment corresponds to short-run profitmaximization as a subproblem of long-run profit maximization: although plantoperation is usually planned along with the investment, the producer is still free

to make operating decisions after constructing the plant In other words, his final

choices of the outputs y and the variable inputsv are made only after fixing the

capital inputs k Such a multi-stage problem can be solved in the reverse order: the

decisions to be implemented last are determined first, but are made contingent onthe decisions to be implemented earlier, and the complete solution is put together by

back substitution For the producer, this means first choosing y andv to maximizethe short-run profit, given anarbitrary k as well as the prices, p and w, for the

variable commodities Even within the confines of the purely periodic (or static)problems considered here, this approach has a couple of analytical advantages First,

in addition to being of independent interest, the short-run equilibrium (with a fixed

k) may be easier to find than the long-run equilibrium—as in Chap.2 Second, whenthere is a number of technologies, the short-run equilibrium is usually easier tofind by solving the profit maximization programmes to determine the total short-runsupply (and then equate it to demand) than by solving the duals of cost minimizationprogrammes to determine the SRMCs (which would then have to be equated both

to one another and to inverse demand) Thisprofit approachis simpler than the cost

© 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_3

21

approach in two ways: it gives unique solutions to the producer problem and its dual,and it reduces the number of unknowns in the subsequent equilibrium problem (seeSect.4.1).

A third advantage of the short-run approach emerges when the framework—unlike this one—takes account of non-periodic demand and price uncertainty Pricesfor the variable commodities p; w/, or their probability distribution in a stochastic

model, will change in unforeseen ways between the planning and the completion ofplants, and will keep shifting thereafter also As a result, both the plant mix and thedesign of individual plants will become suboptimal But whether a plant is optimal

or not, it should be operated optimally, and a solution to this problem is a part of theshort-run approach

The above considerations are what makes short-run profit maximization thesubproblem of central interest to us It, too, may be solved in two stages, though

this time the order in which the decision variables (y andv) are determined is only amatter of convenience—it is usually best to start with the simpler subproblem Here,

it is assumed that short-run cost minimization (findingv given k and y) is easier than revenue maximization (finding y given k andv) The solution sequence (first v, then

y and finally k) corresponds to a chain of three problems: (i) the “small” one of

short-run cost minimization (with k and y as data,v as a decision variable), (ii) an

“intermediate” problem of short-run profit maximization (with k as a datum, y andv

as decision variables), and (iii) the “large” problem of long-run profit maximization

(with k, y andv as decision variables)

A fourth problem—another intermediate one—is that of long-run cost

minimiza-tion (with y as a datum, k andv as decision variables) It is in terms of this problemand its value function that public utilities usually formulate their welfare-promotingprinciples of meeting the demand at a minimum operating cost, optimizing theircapital stocks, and pricing their outputs at LRMC Together, these policies result inlong-run profit maximization and competitive equilibrium in the products’ markets.Although the separate aims are usually stated in terms of long-run costs, as LRMCpricing and LRC minimization, their combination is best achieved through short-runcalculations—for the reasons outlined above and detailed in Sect.4.1

Each of the four problems, when formulated as one of optimization constrained

by a convex (and nonempty)production setY, has a linear objective function.1Thishas several implications One is that each problem (SRC or LRC minimization,

or SRP or LRP maximization) can be formulated as a linear programme (LP), byrepresenting Y as the intersection of a finite or infinite set of half-spaces This

is discussed further in Sect 3.12 What matters for now is that in passing to a

subproblem, once a decision variable has become a datum (like k in passing from the long to the short run), the corresponding term of the linear optimand (r  k)

1 Even if the objective were nonlinear, it could always be replaced by a linear one with an extra scalar variable, subject to an extra nonlinear constraint: as is noted in [ 12 , p 48], minimization of

f y/ over y is equivalent to minimization of % over y and % subject to: %  f y/, in addition to any original constraints on y.

3.1 Cost and Profit as Values of Programmes with Quantity Decisions 23

can be dropped, since it is fixed Its coefficient (r) can then be removed from the subproblem’s data (which include k).2

Thecommodity spacesfor outputs, fixed inputs and variable inputs are denoted

by Y, K and V, respectively These are paired with price spaces P, R and W

by bilinear forms (a.k.a scalar products) denoted by h p j yi, etc.; the alternative notation p  y is employed to mean pTy when both P and Y are equal to the finite-

dimensional spaceRn (where pTis the row vector obtained by transposing a column

p) Unless specified, the range of a decision variable (say y) is the whole space (Y).

With p, r and w denoting the prices for outputs, fixed inputs and variable inputs (y, k andv, respectively), thelong-run profit maximizationprogramme is:

Given p; r; w/ , maximize h p j yi  hr j ki  hw j vi over y; k; v/ (3.1.1)

Its optimal value, the maximum LRP as a function of the data, is denoted by

…LR p; r; w/ By definition, y; k; v/ solves (3.1.1)–(3.1.2) if and only if

.y; k; v/ 2 Y and h p; r; w j y; k; vi D …LR p; r; w/ (3.1.3)

In the central case of constant returns to scale (c.r.t.s.), the production setY is acone, and…LRis the0-1 indicator of thepolar cone

YıD f p; r; w/ 2 P  R  W W 8 y; k; v/ 2 Y h p j yi  hr j ki  hw j vi  0g

(3.1.4)i.e.,…LR p; r; w/ is 0 if p; r; w/ 2 Yı, and it is C1 otherwise Condition (3.1.3)

is then equivalent to the conjunction of: technological feasibility, price consistencyand breaking even, which together make up theComplementarity Conditions

.y; k; v/ 2 Y; p; r; w/ 2 Yı

and h p ; r; w j y; k; vi D 0. (3.1.5)One subprogramme of LRP maximization (3.1.1)–(3.1.2) is short-run profitmaximization, i.e., the programme

Given p; k; w/ , maximize h p j yi  hw j vi over y; v/ (3.1.6)

Its optimal value is…SR p; k; w/, the maximum SRP.

2 More generally, this is so whenever the optimand separates into a function of.r; k/ plus terms independent of r and k.

Another subprogramme of (3.1.1)–(3.1.2) islong-run cost minimization, i.e., theprogramme

Given .y; r; w/ , minimize hr j ki C hw j vi over k; v/ (3.1.8)

Its optimal value is CLR.y; r; w/, the minimum LRC.

The common subprogramme of all these isshort-run cost minimization, i.e., theprogramme

Given .y; k; w/ , minimize hw j vi over v (3.1.10)

Its optimal value is CSR.y; k; w/, the minimum SRC.

The partial conjugacy relationships between these value functions (…LR,…SR,

CLR, CSR) are summarized in the following diagram:

For example, the arrow from the y next to CSRto the p next to…SRindicates that

…SR is, as a function of p, the Fenchel-Legendre convex conjugate of CSR as a

function of y, with k; w/ fixed—i.e., by definition,

and C or… Details such as the signs and convexity or concavity are omitted

3.2 Split SRP Optimization: A Primal-Dual System for the Short-Run Approach 25

As is spelt out next, those y and k which yield the suprema in (3.1.13) and (3.1.14)are parts of an input-output bundle that maximizes the long-run profit

for the Short-Run Approach

A joint programme—one with two or more decision variables—can be split byoptimizing in stages: first over a subset of the variables (keeping the rest fixed), thenover the other variables to obtain the complete solution by back substitution Themethod can be applied to solve the LRP maximization programme (3.1.1)–(3.1.2)for.y; k; v/ by:

1 first minimizinghw j vi over v (subject to y; k; v/ 2 Y) to find the solution

set LV y; k; w/, orthesolution Lv y; k; w/ if it is indeed unique, and the minimum value CSR.y; k; w/, which is hw j Lvi;

2 then maximizingh p j yi  CSR.y; k; w/ over y to find the solution set OY p; k; w/,

orthe solution Oy p; k; w/ if it is unique, and the maximum value …SR p; k; w/,

which ish p j Oyi  CSR.Oy/;

3 and finally maximizing …SR p; k; w/  hr j ki over k to find the solution set

O

K p; r; w/, orthe solution Ok p; r; w/, should it be unique (which it obviously

cannot be if returns to scale are constant, in the long run)

Every complete solution to (3.1.1)–(3.1.2) can then be given (in terms of p, r and w) as a triple y; k; v/ such that: k 2 OK p; r; w/, y 2 OY p; k; w/ and v 2

LV y; k; w/ With decreasing returns to scale, if the solution is unique, then it is the triple: Ok p; r; w/, Oyp ; Ok p; r; w/ ; wand LvOy

p ; Ok p; r; w/ ; w; Ok p; r; w/ ; w

In other words, the LRP programme (3.1.1)–(3.1.2) for.y; k; v/ can be reduced to

an investment programme, for k alone, by first solving the SRP programme (3.1.6)–(3.1.7) for.y; v/ and substituting its maximum value (…SR) for the yet-unmaximizedterm h p j yi  hw jvi in (3.1.1) The SRP programme for .y; v/ can, in turn, be reduced to a programme for y alone by solving the SRC programme (3.1.10)–(3.1.11) and substituting its value (CSR) for the termhw jvi in (3.1.6)

So an input-output bundle .y; k; v/ maximizes long-run profit at prices p; r; w/ if and only if both

k maximizes…SR p; ; w/  hr j i on K (given p; r and w) (3.2.1)and the bundle.y; v/ maximizes short-run profit (given k) at prices p; w/ or,

equivalently,

y maximizes h p j i  CSR.; k; w/ on Y (given p; k and w) (3.2.2)

v minimizes hw j i on fv 2 V W y; k; v/ 2 Yg (given y; k and w). (3.2.3)

The system (3.2.1)–(3.2.3) shall be called the Split LRP Optimization System Its

SRC subprogramme forv in (3.2.3) is taken to be readily soluble By contrast, the

reduced SRP programme for y in (3.2.2) may require the duality approach Thisconsists in pricing the constraining parameters and solving the dual programme ofvaluation together with the primal programme of operation (when there is no duality

gap) For the SRP programme as the primal, this means valuing the fixed inputs k: a dual solution (with no duality gap) is a shadow-price system r such that

r minimizes h j ki C…LR p; ; w/ on R (given p; k and w) (3.2.4)and the minimum value, hr j ki C…LR p; r; w/ , equals …SR p; k; w/ (3.2.5)Under c.r.t.s., Conditions (3.2.4) and (3.2.5) become

r minimizes h j ki on fr 2 R W p; r; w/ 2 Yıg (given p; k and w) (3.2.6)and the minimum value, hr j ki , equals…SR p; k; w/ (3.2.7)The duality scheme that produces the programme in (3.2.4) or (3.2.6) as the dual ofSRP maximization is set out in detail in Sect.3.3

As well as helping to solve the operation problem in (3.2.2), the dual solutioncan be used to check the investment for optimality, i.e., (3.2.1) is equivalent

to (3.2.4)–(3.2.5).3 The system (3.2.2)–(3.2.5) is therefore equivalent to (3.2.1)–(3.2.3), and hence also to LRP maximization (3.1.3), and to ComplementarityConditions (3.1.5) under c.r.t.s It is, however, put entirely in terms of solutions

to the SRP programme for.y; v/ and its dual programme for r, with the primal split

into the SRC programme (forv) and the reduced SRP programme (for y) Therefore,

(3.2.2)–(3.2.5) shall be called the Split SRP Optimization System It is likely to

be the best basis for the short-run approach when the technology is specified bymeans of a production set (as is usual in an engineer’s description of a technology)and, in addition, the SRC is readily calculable Alternative systems are presented inSects.3.4to3.6,3.8and3.9

with Shadow-Price Decisions

Unless there are duality gaps, short-run and long-run cost and profit are also theoptimal values of programmes that are dual to those of Sect 3.1 The schemeproducing the duals is an application of the usual duality framework for convex

3 Formally, this follows from the definitional conjugacy relationship ( 3.1.14 ) between … SR and … LR

(as functions of k and r, respectively) by using the first-order condition (B.5.5 ) and the Inversion Rule ( B.6.2 ) of Appendix B

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price 27

programmes (CPs), expounded in, e.g., [44] and [36, Chapter 7] The present scheme

is, however, a little different in that it starts not from a single programme, yet to beperturbed, but from a family of programmes that depend on a set of data, whoseparticular values complete a programme’s specification So, one way to perturb

a programme is simply to add an increment to its data point, thus “shifting” itwithin the given family Some, possibly all, of the scheme’s primal perturbationsare therefore increments to some—though typically not all—of the data The samegoes for dual perturbations

Before the duality scheme is applied to the profit and cost programmes, it isbriefly discussed and illustrated in the framework of linear programming A centralidea is that the dual programme depends on the choice of perturbations of the primalprogramme: different perturbation schemes produce different duals Theoreticalexpositions of duality usually start from a programme without any data variables

whose increments might serve as primal perturbations: say, f y/ is to be maximized over y subject to a number of inequalities G1.y/  0, G2.y/  0; : : :, abbreviated

to G y/  0 In such a case, any perturbations must first be introduced, and the

standard choice is to add  D 1; 2; : : :/ to the zeros on the right-hand sides

(r.h.s.’s)—thus perturbing the original constraints G y/  0 to G y/   The original programme has no data other than the functions f and G themselves,

and the increments f and G (which change the programme to maximization

of f C f / y/ over y subject to G C G/ y/  0) can never serve as primal perturbations—not even if they were taken to be linear, i.e., if f and G were a vector and a matrix of coefficients of the primal variables y D y1; y2; : : :/ This

is because the perturbed constrained maximand must be jointly concave in thedecision variablesandthe perturbations,4but the bilinear form f y is neither concave nor convex in f and y jointly.5

But in applications, the primal programme usually comes with a set of data that itdepends on, and increments to some of the programme’s data can commonly serve

as primal perturbations Such data shall be called theintrinsic primal parameters;some or all of the other data will turn out to be dual parameters For example, inSRP maximization (3.1.6)–(3.1.7), the fixed-input bundle k is a primal parameter

because, since the production set Y is convex, the constrained maximand is aconcave function of.y; k; v/: it is

5 A linear change of variables makes it a saddle function:4f y D f C y/ f C y/ f  y/ f  y/

is convex in f C y and concave in f  y.

primal parameter (i.e., its increment p cannot be a primal perturbation) because

the bilinear formh p j yi is not jointly concave in p and y For these reasons, all

of the quantity data, but no price data, are primal parameters for the profit or costoptimization programmes of Sect.3.1 As for the production set, it cannot itselfserve as a parameter because convex sets do not form a vector space to begin with.However, once the technological constraint.y; k; v/ 2 Y has been represented

in the form Ay  Bk  Cv  0 (under c.r.t.s.), the matrices or, more generally, the

linear operations A, B and Carevectorial data But none can be a primal parameter,

for lack of joint convexity of Ay in A and y, etc Nor can A, B or C be a dual

parameter (for a similar reason) Such data variables—which are neither primal nordual parameters, and hence play no role in the duality scheme—shall be calledtertial

parameters

It can be analytically useful, or indeed necessary, to introduce other primalperturbations, i.e., perturbations that are not increments to any of the data (whichare listed after the “Given” in the original programme) This amounts to introducingadditional parameters, which shall be calledextrinsic; their original, unperturbedvalues can be set at zeros, as in [44] When the constraint set is represented by

a system of inequalities and equalities, the standard “right-hand side” parametersare always available for this purpose (unless they are all intrinsic, but this is soonly when the r.h.s of each constraint is a separate datum of the programme andcan therefore be varied independently of the other r.h sides) Section3.12showshow to relate the marginal effects of any “nonstandard” perturbations to those of thestandard ones—i.e., how to express any “nonstandard” dual variables in terms of theusual Lagrange multipliers of the constraints This is useful in the problems of plantoperation and valuation, including those that arise in peak-load pricing (Sect.5.2).6Once a primal perturbation scheme has been fully defined, the duality framework

is completed automatically (except for the choice of the topologies and thecontinuous-dual spaces in the infinite-dimensional case): dual decision variables areintroduced and paired to the specified primal perturbations (both the intrinsic andany extrinsic ones) To re-derive the primal programme as its dual’s dual, the dualperturbations are introduced so as to be paired with the primal variables (i.e., thismatch is set up “in reverse”) The perturbed dual minimand—a function of the dualvariables, the dual perturbations and the data of the original, primal programme—isdefined in the usual way (as in [44, (4.17)] but with the primal problem reoriented

6 In this as in other contexts, it can be convenient to think of extrinsic perturbations either as (i) complementing the intrinsic perturbations (which are increments to the fixed inputs) by varying some aspects of the technology (such as nonnegativity constraints), or as (ii) replacing the intrinsic perturbations with finer, more varied increments (to the fixed inputs) For example, the time-

constant capacity k in ( 5.2.3 ) is an intrinsic primal parameter The corresponding perturbation

is a constant increment k, which can be refined to a time-varying increment k / The perturbation k./ is complemented by the increment n / to the zero floor for the output rate

y / in ( 5.2.3 ) The same goes for all the occurrences of k and n in the context of pumped

storage and hydro (where

constraint ( 5.2.15 ) or ( 5.2.35 )).

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price 29

to maximization) When all the primal perturbations are intrinsic, the resulting dualprogramme is called theintrinsic dual

Some or possibly all of the dual perturbations may turn out to perturb the dualprogramme just like increments to some of the data—which are thus identified astheintrinsic dual parameters Any other dual perturbations are called extrinsic, andthese can be thought of as increments toextrinsic dual parameters(whose original,unperturbed values are set at zeros) However, in the profit or cost programmes, allthe dual parameters are price data (and are therefore intrinsic)

In thereduced formulations of the profit or cost problems, some of the pricedata arenotdual parameters because the corresponding quantities have been solvedfor in the reduction process, and have thus ceased to be decision variables: e.g.,

the variable-input price w isnot a dual parameter of the reduced SRP programme

in (3.2.2) because the corresponding input bundle v has been found in SRCminimization (and so it is no longer a decision variable) But in thefull(not reduced)formulations, all the price data are dual parameters, and thus the programme’s data(other than the technology itself) are partitioned into the primal parameters (thequantity data) and dual parameters (the price data)

The primal and dual optimal values can differ at some “degenerate” parameterpoints (see AppendixA), but such duality gaps are exceptional, and they do notoccur when the primal or dual value is semicontinuous in, respectively, the primal ordual parameters (Sect.6.1) Note that both optimal values, primal and dual, depend

on the data, which are the same for both programmes So, in this scheme, either ofthe optimal values (primal or dual) is a function of both primal and dual parameters,and so it can have two types of continuity and of derivatives (marginal values):

• continuity/derivative of Type One is that of theprimal value with respect to the

primal parameters, or of thedualvalue w.r.t thedualparameters;

• continuity/derivative of Type Two is that of the dual value w.r.t the primal

parameters, or of theprimalvalue w.r.t thedualparameters

This useful distinction cannot be articulated when, as in [44] and [36], the primaland dual values are considered only as functions of either the primal or the dualparameters, respectively

Comments (Parameters and Their Marginal Values, Dual Programme and the FFE Conditions, the Lagrangian and the Kuhn-Tucker Conditions for LPs)

• Let the primal linear programme be: Given any p 2 Rn and s 2 Rm, and an

m  n matrix A, maximize p  y over y 2 R n subject to Ay  s Here, the only intrinsic primal parameter is the standard parameter s There is no obviously

useful candidate for an extrinsic primal parameter, and if none is introduced,then the dual is thestandard dualLP: Given p and s (and A), minimize  s over

2 Rm subject to AT D p and  0, where AT is the transpose of A.7The

only dual parameter is p.

• If both programmes have unique solutions, Oy s; p; A/ and O s; p; A/, with equal

valuesV s; p; A/ WD p  Oy D O  s DW V s; p; A/, then the marginal values of all

the parameters, including the tertial (non-primal, non-dual) parameter A, exist as

ordinary derivatives Namely: (i) rs V D r s V D O, (ii) r p V D r p V D Oy, and

(iii) rA V D r A V D  O ˝ Oy D  O OyT (the matrix product of a column and arow, in this order, i.e., the tensor product), where rAis arranged in a matrix like

A (i.e., @V=@ A ij D  Oi Oy j for each i and j) The first two formulae (for r s V and

rp V) are cases of a general derivative property of the optimal value in convex

programming: see, e.g., [44, Theorem 16: (b) and (a)] or [32, 7.3: Theorem 1’].The third formula follows heuristically from either of the first two by comparing

the marginal effect of A with the marginal effect of either s or p on the primal

or dual constraints, respectively It can also be proved formally by applying theGeneralized Envelope Theorem for smooth optimization [1, (10.8)],8 wherebyeach marginal value (rs V, r p V and r A V) is equal to the corresponding partial

derivative of the Lagrangian, which is here

Condition, is nonlinear in the decision variables (y and jointly)

• By contrast, the FFE Conditions—primal feasibility, dual feasibility and equality

of the primal and dual objectives—form the equivalent system10

Ay  s;  0; pTD TA and p  y D  s (3.3.3)

7The dual constraint AT D p must be changed to AT  p if y  0 is adjoined as another primal

constraint (in which case the primal LP can be interpreted as, e.g., revenue maximization—given

a resource bundle s, an output-price system p and a Leontief technology defined by an coefficient matrix A).

input-8 Without a proof of value differentiability, the Generalized Envelope Theorem is given also in, e.g., [ 47 , 1.F.b].

9 These are the Lagrangian Saddle-Point Conditions ( 0 2 @ Land 0 2 O@y L) for the present LP case.

10 In this case, equivalence of the Kuhn-Tucker Conditions and the FFE Conditions can be seen directly, but it holds always (since, by the general theory of CPs, each system is equivalent to the conjunction of primal and dual optimality together with absence of a duality gap).

3.3 Duality: Cost and Profit as Values of Programmes with Shadow-Price 31

whichislinear in.y; / This makes it simpler to solve than the system of

Kuhn-Tucker Conditions (3.3.2) The FFE system (3.3.3) is so effective because, inlinear programming, the dual programme can be worked out from the primalexplicitly

• But the dual of a general CP cannot be given explicitly (i.e., without leaving

an unevaluated extremum in the formula for the dual constrained objectivefunction in terms of the Lagrangian).11That is why, as a general solution methodfor convex programming, the Kuhn-Tucker Conditions are better than the FFEConditions, although the latter system is simpler in some important specificcases (such as linear programming) Whereas using the FFE Conditions requiresforming the dual from the primal to start with, using the Kuhn-Tucker Conditionsrequires only the Lagrangian Thus the latter Kuhn-Tucker Conditions offer aworkable general method of solving the primal-dual programme pair, and thismatters more than an explicit expression for the dual programme The FFEConditions can, however, be simpler in the case of a specific CP that, like an

LP, has an explicit dual

The duality scheme is next applied to all four of the profit and cost programmes

of Sect.3.1; the one of most importance in the context of a decentralized industry(such as the ESI of Sects.5.1to5.3) is the programme of SRP maximization Theduals are shown to consist in shadow-pricing the given quantities—and so theirsubprogramme relationship is the reverse of that between the primals: the morequantities that are fixed, the more commodities there are to shadow-price (In otherwords, the fewer primal variables, the more primal parameters, and hence moredual variables.) For this reason, the duals are listed, below, in the order reverse tothat in which the primals are listed in Sect 3.1 See also Fig.3.1, in which thelarge single arrows point from primal programmes to their subprogrammes, and thedouble arrows point from the dual programmes to their subprogrammes Each ofthe four middle boxes gives the data for the pair of programmes represented by thetwo adjacent boxes (the outer box for the primal and the inner box for the dual); thedata are partitioned into the primal parameters (the given quantities) and the dualparameters (the given prices) There are no other parameters in this scheme (i.e., ithas no extrinsic parameters)

11The standard dual to the ordinary CP of maximizing a concave function f y/ over y subject

to G y/  s (where G1, G2, etc., are convex functions) is to minimize supy L y; / WD

supy f y/ C  s  G y/// over  0 (the standard dual variables, which are the Lagrange

multipliers for the primal constraints): see, e.g., [ 44 , (5.1)] And supy L (the Lagrangian’s supremum over the primal variables) cannot be evaluated without assuming a specific form for

f and G (the primal objective and constraint functions).