Table Of ContentLecture Notes in Economics and Mathematical Systems 684
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
Lecture Notes in Economics
and Mathematical Systems 684
FoundingEditors:
M.Beckmann
H.P.Künzi
ManagingEditors:
Prof.Dr.G.Fandel
FachbereichWirtschaftswissenschaften
FernuniversitätHagen
Hagen,Germany
Prof.Dr.W.Trockel
MuratSertelInstituteforAdvancedEconomicResearch
IstanbulBilgiUniversity
Istanbul,Turkey
and
InstitutfürMathematischeWirtschaftsforschung(IMW)
UniversitätBielefeld
Bielefeld,Germany
EditorialBoard:
H.Dawid,D.Dimitrov,A.Gerber,C.-J.Haake,C.Hofmann,T.Pfeiffer,
R.Slowin´ski,W.H.M.Zijm
Moreinformationaboutthisseriesathttp://www.springer.com/series/300
Anthony Horsley (cid:129) 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
AnthonyHorsley(1939-2006) AndrewJ.Wrobel
Watford,Hertfordshire,UK Warsaw,Poland
CompletedinAugust2015,thisbookisarevisedandrestructuredversionoftheSTICERDDiscussion
PaperTE/05/490“Characterizationsoflong-runproduceroptimaandtheshort-runapproachtolong-run
marketequilibrium: ageneraltheorywithapplications topeak-load pricing”©AnthonyHorsleyand
AndrewJ.Wrobel(London,LSE,2005).
ISSN0075-8442 ISSN2196-9957(electronic)
LectureNotesinEconomicsandMathematicalSystems
ISBN978-3-319-33397-7 ISBN978-3-319-33398-4(eBook)
DOI10.1007/978-3-319-33398-4
LibraryofCongressControlNumber:2016939945
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland
Preface
This book is dedicated to the memory of Anthony Horsley (1939–2006),nuclear
physicistandmathematicaleconomist,myfriendandmentor.Mostofthebookwas
Chap.5ofmyPh.D.Econ.thesis“Theformaltheoryofpricingandinvestmentfor
electricity”,writtenattheLondonSchoolofEconomicsunderTony’ssupervision.
Thispartof the research was supportedfinanciallyby TilburgUniversity’sCenter
for Economic Research (in 1989–1990)and by ESRC grant R000232822(1991–
1993);theirsupportisgratefullyacknowledged.Thefinalmanuscriptwasprepared
attheEasternIllinoisUniversity;Iamgratefulfortheuseoftheirpremises,which
sustainedmyconclusion.IdonotthinkthatIcouldhavemadethislasteffortwithout
themoralsupportofmynewly-wedwifeAnitaShelton,professorofhistoryatthe
EIU,whohasencouragedmetoreturnto academicworkaftera breakofnearlya
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 mathematicalframeworkscouldofferopportunities
fortheoriesofgreaterverisimilitudewithnewinsightsandresults.Icouldnotagree
more.Rigouris,ofcourse,derigueurthesedays,butitbecomesrigormortisifall
itservesisaformalextensionofexistingknowledge.Ihopethatthisbookwillhelp
tovindicateTony’sstance.
Charleston,Illinois,USA AndrewJ.Wrobel
August2015
v
Contents
1 Introduction .................................................................. 1
2 Peak-Load Pricing with Cross-Price Independent
Demands:ASimpleIllustration............................................ 15
2.1 Short-RunApproachtoSimplestPeak-LoadPricingProblem...... 15
2.2 ReinterpretingCostRecoveryasaValuationCondition............. 17
2.3 EquilibriumPricesfortheSingle-ConsumerCase................... 18
3 CharacterizationsofLong-RunProducerOptimum .................... 21
3.1 CostandProfitasValuesofProgrammeswithQuantity
Decisions............................................................... 21
3.2 Split SRP Optimization: A Primal-Dual System
fortheShort-RunApproach........................................... 25
3.3 Duality:Cost andProfit asValuesof Programmes
withShadow-PriceDecisions......................................... 26
3.4 SRPandSRCOptimizationSystems................................. 38
3.5 SRC/PPartialDifferentialSystemfortheShort-RunApproach.... 40
3.6 OtherDifferentialSystems............................................ 42
3.7 TransformationsofDifferentialSystemsbyUsingSSLorPIR..... 43
3.8 Summary of Systems Characterizing Long-Run
ProducerOptimum .................................................... 45
3.9 Extended Wong-Viner Theorem and Other
TranscriptionsfromSRPtoLRC ..................................... 47
3.10 DerivationofDualProgrammes ...................................... 52
3.11 Shephard-HotellingLemmasandTheirDualCounterparts......... 53
3.12 Dualityfor LinearProgrammeswith Nonstandard
ParametersinConstraints ............................................. 62
4 Short-RunProfitApproachtoLong-RunMarketEquilibrium ........ 73
4.1 OutlineoftheShort-RunApproach................................... 73
4.2 DetailedFrameworkforShort-RunProfitApproach ................ 80
vii
viii Contents
5 Short-RunApproachtoElectricityPricinginContinuousTime....... 91
5.1 TechnologiesforElectricityGenerationandEnergyStorage ....... 91
5.2 OperationandValuationofElectricPowerPlants................... 97
5.3 Long-RunEquilibriumwithPumpedStorageorHydro
GenerationofElectricity .............................................. 109
6 Existence of Optimal Quantities and Shadow Prices
withNoDualityGap......................................................... 119
6.1 Preclusion of Duality Gaps by Semicontinuity
ofOptimalValues...................................................... 119
6.2 SemicontinuityofCostandProfitinQuantityVariables
OverDualBanachLattices............................................ 122
6.3 SolubilityofCostandProfitProgrammes............................ 131
6.4 Continuity of Profit and Cost in Quantities
andSolubilityofShadow-PricingProgrammes...................... 133
7 ProductionTechniqueswithConditionallyFixedCoefficients.......... 137
7.1 ProducerOptimumWhenTechnicalCoefficientsAre
ConditionallyFixed.................................................... 137
7.2 DerivationofDualProgrammesandKuhn-TuckerConditions ..... 142
7.3 VerificationofProductionSetAssumptions.......................... 148
7.4 ExistenceofOptimalOperationandPlantValuation
andTheirEqualitytoMarginalValues ............................... 150
7.5 LinearProgrammingforTechniqueswithConditionally
FixedCoefficients ..................................................... 152
8 Conclusions................................................................... 155
A ExampleofDualityGapBetweenSRPandFIVProgrammes.......... 157
B ConvexConjugacyandSubdifferentialCalculus ........................ 161
B.1 ThesemicontinuousEnvelope ........................................ 161
B.2 TheConvexConjugateFunction...................................... 162
B.3 SubgradientsandSubdifferentiability................................ 164
B.4 ContinuityofConvexFunctions...................................... 166
B.5 ConcaveFunctionsandSupergradients............................... 167
B.6 SubgradientsofConjugates........................................... 168
B.7 SubgradientsofPartialConjugates ................................... 171
B.8 ComplementabilityofPartialSubgradientstoJointOnes........... 176
C NotationList.................................................................. 183
References......................................................................... 193
List of Figures
Fig.2.1 Short-runapproachtolong-runequilibriumofsupply
and(cross-priceindependent)demandforthermally
generatedelectricity:(a)determinationoftheshort-run
equilibriumpriceandoutputforeachinstantt,given
acapacityk;(b)and(d)trajectoriesoftheshort-run
equilibriumpriceandoutput;(c)theshort-runcost
curve.Whenkissuchthattheshadedareain(b)equals
r,theshort-runequilibriumisthelong-runequilibrium............. 16
Fig.3.1 Decision variables and parameters for primal
programmes(optimizationof:long-runprofit,short-run
profit, long-runcost, short-runcost) and for dual
programmes(priceconsistencycheck,optimization
of:fixed-inputvalue,outputvalue,outputvalueless
fixed-inputvalue).Ineachprogrammepair,thesame
pricesandquantities—.p;y/foroutputs,.r;k/forfixed
inputs,and.w;v/forvariableinputs—aredifferently
partitionedintodecisionvariablesanddata(whichare
subdividedintoprimalanddualparameters).Arrows
leadfromprogrammestosubprogrammes........................... 32
Fig.4.1 Flow chart for an iterative implementation of
the short-runprofit approachto long-runmarket
equilibrium. For simplicity, all demand for the
industry’soutputsisassumedtobeconsumerdemand
thatisindependentofprofitincome,andallinputprices
arefixed(intermsofthenumeraire).Absenceofduality
gapandexistenceoftheoptima(rO,yO)canbeensuredby
usingtheresultsofSects.6.1to6.4 .................................. 74
ix
