Complex-Valued Modeling in Economics and Finance Sergey Svetunkov Complex-Valued Modeling in Economics and Finance SergeySvetunkov NationalMineralResources University-MiningUniversity St.Petersburg,Russia ISBN978-1-4614-5875-3 ISBN978-1-4614-5876-0(eBook) DOI10.1007/978-1-4614-5876-0 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012954285 #SpringerScience+BusinessMediaNewYork2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface As early as the mid-eighteenth century, mathematicians discovered a way to use modelsinvolvingcomplexvariables.Sincethen,workwithcomplexvariableshas been progressing, with the theory of complex variables emerging as a branch of mathematics. Nowadays this theory is widely implemented in all of the natural sciencesasworkwithcomplexvariablesmakesitpossibletodescribeadequately more complicated processes than do real variables. Economics as an object of scientific research and cognition is no less complicated than the natural sciences, which is why complex variables may be applied in economics to give a more precise description of the processes involved and to build even more complicated models than those that can be built using real variables. Complex variables are sometimesusedincertainbranchesofeconomicandmathematicalsimulation,but this study considers application ofmodels of economic andmathematical simula- tionexclusivelyintheformofmodelsofcomplexvariables. Models and mathematical methods of working with complex variables are consideredinthestudynotassomealternativetoreal-variablemodelsandmethods but as an instrument complementing and expanding the existing arsenal of eco- nomic and mathematical modeling. It is of principal importance that complex economics provides the economist with a new instrument of research, and the more flexible the instrument, the more diverse the tasks that can be solved by theresearcher The first chapter of this work presents basic principles of complex economics and certain data from the theory of functions of complex variables that are necessary for an understanding of further actions in the formation of complex economictheory. Very often, understanding a certain meaning of mathematical operations requires graphic methods of describing these operations. Since in the study in question complexeconomics involves functions ofcomplex variables, it is neces- sary to know the characteristics of these functions including their graphical v vi Preface representation. This task is solved in the second and third chapters of the study, whichconsiderconformalmappingsofbasicfunctionsofcomplexvariables.These chaptersdonotsimplystatetherespectivebranchesofthetheoryoffunctionsofa complex variable encountered in various textbooks. Textbooks on the theory of functionsofacomplexvariable,inthesectiondevotedtoconformalmappings,do not consider, for instance, exponential complex functions with a complex coeffi- cient;theneedforsuchtreatmenthasnotexisted.Textbooksconsiderexceptional conformalmappingsofthisexponentialcomplexfunctionwithrealexponent.For thepurposesofcomplexeconomics,thereisaneedtousecomplexexponentsnot onlyforexponentialfunctions. The fourth chapter presents an instrument of practical application in complex economics–complexeconometrics.Thechapterprovidesonlythebasicprinciples of complex econometrics because it is practically impossible for a group of scientists working in this field to develop or adapt to complex econometrics all the branches of real-variable econometrics. Moreover, it is simply impossible to presenttheentirescientificdisciplineofeconometricsinonechapter.Herewewill substantiateandadapttocomplexeconomicsthebasicsectionsofcorrelationand regressionanalysisofmathematicalstatistics–calculationofcomplexcoefficients of pair correlation, least-squares method for evaluating coefficients of complex models, method of construction of confidence limits for obtained statistical estimates, and new coefficients showing the adequacy of econometric constructions. The results obtained are sufficient for solving subsequent tasks of complexeconomicsanddevelopingcomplexeconometrics.Theideasofstatistical characteristics of complex random variables currently in use in mathematical statistics has led to a deadlock. This can be seen in calculations of complex coefficients of pair correlation – the obtained contradictory results that follow fromstandardsituationstestifytotheirerroneouscharacter.Thismadeitnecessary to devise other principles of statistics of complex random variables that underlie newandconsistentconclusionsandrecommendations. The fifth chapter contains the results of an investigation of one of the simplest typesofeconomicmodelsofcomplexvariables–productionfunctionsofcomplex arguments,whereactualproductionresultsdependonacomplexargument,thatis, production resources represented in the form of a complex variable. These functions possess some very important properties applicable in the successful solution of certain economic tasks. Here, we also demonstrate one remarkable feature of complex argument models – the sustainability of their assessments undermulticollinearity. Chapter 6 discusses production complex variable functions, more complicated models than complex argument ones. Here, the complex production result is represented in the form of dependence on a complex resource. Since functional relationshipsbetweentwocomplexvariablesmayhavevariousforms,thischapter considersthebasicones. Preface vii The seventh chapter uses a case that has been insufficiently explored in the mathematics of complex variables; the existing theory operates with only one complex variable and is therefore called the theory of functions of a complex variable. Chapter 7 involves multifactor complex models, i.e., models of several complexvariables.Thedevelopmentofthetheoryoffunctionsofseveralcomplex variablesinmathematicsfindsverylittleapplicationineconomics.Thisiswhyin the sixth chapter we state for the first time the principles and approaches of the theoryofmultifactorfunctionsofcomplexvariables.Thiswasnecessitatedbyour wish to build complex production functions that are more applicable to real economic processes, and that could be done only by increasing the explanation factorsusedincomplexeconomicmodels.Thischapterpresentsthepropertiesand characteristics of simple multifactor complex models. The variety of possible applicationsofthemodelsandmethodsofcomplexeconomicsisnotlimitedonly to production function models. However, production functions provide a good example for seeing the advantages and disadvantages of complex models. In Chaps. 5 and 6 this is done by comparing such models with basic models of productionfunctionswithrealvariables. Another good example that demonstrates the advantages of complex variable modelsistheirapplicationintheanalysisofstockmarkets.Chapter8showshowto use complex indices of economic conditions and how to use the properties of complexnumberstoobtainphaseportraitsofstockmarketsthatcouldallowusto reveallawsthatremainhiddenwiththeuseofrealvariables.ThematerialsofChap. 9aimatshowingotherwaysofdevelopingcomplexeconomicsthanthosespecified inpreviouschapters. Inthisstudywemakereferencetotheliteratureusedintheformoffootnotes.At theendofthestudywegiveacompletelistofallthepublicationsbythosescientists workingontheformationofcomplexeconomics.Ifnecessary,thereadermayrefer tothesesources.Thelargevolumeofscientificresultsprovidedbythisstudycould not have been obtained without the support of the Russian Foundation of Funda- mental Research. The grants allocated by the foundation on a competitive basis from2006till2010renderedinvaluablefinancialandmoralassistance. The main ideas, hypotheses, and materials stated in this study belong to the author; however, these hypotheses would never have become a well-balanced theory without the active involvement of a group of scientists, chief among whom was Dr. Ivan S. Svetunkov. This collaboration laid the foundations of complex economics on which basis many other scientific results were obtained. We are extremely grateful to Prof. G.V. Savinov who reviewed my first works, written in cooperation with I.S. Svetunkov, and later formulated a number of interesting proposals published in various articles. A very important contribution in the creation of various sections of complex economics was made by Dr. T.V. Koretskaya, E.V. Sirotina, and A.F. Chanysheva. Their work and the results obtained by them are discussed in the corresponding sections of the book. Some partialconclusions,recommendations,andnewscientificresultsobtainedbyother youngscientistsarestatedinvariouspartsofthebook. viii Preface Insofarasthematerialspresentedinthisstudyarenewandthisisthefirsttime they have been systematized in this way and used for the proposed scientific purposes, the author understands that some points may be subject to debate or containinaccuracies.Itiseveryscientist’smission,havingobtainedanewscientific result,toencouragecomprehensivescientificdiscussionandgettothetruth.Thisis why any constructive criticism of the study is welcome. Comments and remarks maybemaileddirectlyviawww.sergey.svetunkov.ru. St.Petersburg,Russia SergeySvetunkov Contents 1 TheoreticalBasisofComplexEconomy. . . . . . . . . . . . . . . . . . . . . . 1 1.1 ComplexEconomiesasaNewBranchofEconomics andMathematicalSimulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 BasicConceptsoftheTFCV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 AxiomaticCoreoftheTheoryoftheComplexEconomy. . . . . . . 14 1.4 BasicModelofaComplexEconomy. . . . . . . . . . . . . . . . . . . . . 16 1.5 SomeDataonMinkowsky’sGeometry. . . . . . . . . . . . . . . . . . . . 21 1.6 LaplaceTransform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 PropertiesofComplexNumbersofaRealArgument andRealFunctionsofaComplexArgument. . . . . . . . . . . . . . . . . . 27 2.1 GeneralProblemofConformalMapping inComplex-ValuedEconomics. . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 ComplexFunctionsofaRealArgument. . . . . . . . . . . . . . . . . . . 28 2.3 FunctionsofaComplexArgument:LinearFunction. . . . . . . . . . 42 2.4 PowerFunctionofaComplexArgument withaRealExponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5 ExponentialFunctionofComplexArgument withImaginaryExponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6 PowerFunctionofComplexArgument withComplexExponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7 ExponentialFunctionofaComplexArgument. . . . . . . . . . . . . . . 55 2.8 LogarithmicFunctionofaComplexArgument. . . . . . . . . . . . . . 59 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3 ConformalMappingsofFunctionsofaComplexVariable. . . . . . . . 63 3.1 PowerFunctionsofaComplexVariable. . . . . . . . . . . . . . . . . . . 63 3.2 ExponentialFunctionsofComplexVariables. . . . . . . . . . . . . . . . 75 ix