国际金融中微数学基础部分

1 Chapter2 THEMATHEMATICSOFOPTIMIZATION 2 TheMathematicsofOptimization ManyeconomictheoriesbeginwiththeassumptionthataneconomicagentisseekingtofindtheoptimalvalueofsomefunctionconsumersseektomaximizeutilityfirmsseektomaximizeprofitThischapterintroducesthemathematicscommontotheseproblems 3 Part1Unconstrainedoptimization 4 MaximizationofaFunctionofOneVariable Simpleexample Managerofafirmwishestomaximizeprofits f q Quantity Maximumprofitsof occuratq 5 MaximizationofaFunctionofOneVariable Themanagerwilllikelytrytovaryqtoseewherethemaximumprofitoccursanincreasefromq1toq2leadstoarisein f q Quantity q 1 q1 6 MaximizationofaFunctionofOneVariable Ifoutputisincreasedbeyondq profitwilldeclineanincreasefromq toq3leadstoadropin f q Quantity q 7 FirstOrderConditionforaMaximum Forafunctionofonevariabletoattainitsmaximumvalueatsomepoint thederivativeatthatpointmustbezero 8 SecondOrderConditions Thefirstordercondition d dq isanecessaryconditionforamaximum butitisnotasufficientcondition Quantity Iftheprofitfunctionwasu shaped thefirstorderconditionwouldresultinq beingchosenand wouldbeminimized 9 SecondOrderConditions Thismustmeanthat inorderforq tobetheoptimum and Therefore atq d dqmustbedecreasing 10 SecondDerivatives ThederivativeofaderivativeiscalledasecondderivativeThesecondderivativecanbedenotedby 11 SecondOrderCondition Thesecondorderconditiontorepresenta local maximumis 12 RulesforFindingDerivatives 13 RulesforFindingDerivatives aspecialcaseofthisruleisdex dx ex 14 RulesforFindingDerivatives Supposethatf x andg x aretwofunctionsofxandf x andg x existThen 15 RulesforFindingDerivatives 16 RulesforFindingDerivatives Ify f x andx g z andifbothf x andg x exist then Thisiscalledthechainrule Thechainruleallowsustostudyhowonevariable z affectsanothervariable y throughitsinfluenceonsomeintermediatevariable x 17 RulesforFindingDerivatives Someexamplesofthechainruleinclude 18 ExampleofProfitMaximization Supposethattherelationshipbetweenprofitandoutputis 1 000q 5q2Thefirstorderconditionforamaximumisd dq 1 000 10q 0q 100Sincethesecondderivativeisalways 10 q 100isaglobalmaximum 19 FunctionsofSeveralVariables Mostgoalsofeconomicagentsdependonseveralvariablestrade offsmustbemadeThedependenceofonevariable y onaseriesofothervariables x1 x2 xn isdenotedby 20 Thepartialderivativeofywithrespecttox1isdenotedby PartialDerivatives Itisunderstoodthatincalculatingthepartialderivative alloftheotherx sareheldconstant 21 CalculatingPartialDerivatives 22 CalculatingPartialDerivatives 23 PartialDerivatives Partialderivativesarethemathematicalexpressionoftheceterisparibusassumptionshowhowchangesinonevariableaffectsomeoutcomewhenotherinfluencesareheldconstant 24 Elasticity ElasticitiesmeasuretheproportionaleffectofachangeinonevariableonanotherunitfreeTheelasticityofywithrespecttoxis 25 ElasticityandFunctionalForm Supposethaty a bx othertermsInthiscase ey xisnotconstantitisimportanttonotethepointatwhichtheelasticityistobecomputed 26 ElasticityandFunctionalForm Supposethaty axbInthiscase 27 ElasticityandFunctionalForm Supposethatlny lna blnxInthiscase Elasticitiescanbecalculatedthroughlogarithmicdifferentiation 28 Second OrderPartialDerivatives Thepartialderivativeofapartialderivativeiscalledasecond orderpartialderivative 29 Young sTheorem Undergeneralconditions theorderinwhichpartialdifferentiationisconductedtoevaluatesecond orderpartialderivativesdoesnotmatter 30 UseofSecond OrderPartials Second orderpartialsplayanimportantroleinmanyeconomictheoriesOneofthemostimportantisavariable sownsecond orderpartial fiishowshowthemarginalinfluenceofxiony y xi changesasthevalueofxiincreasesavalueoffii 0indicatesdiminishingmarginaleffectiveness 31 TotalDifferential Supposethaty f x1 x2 xn Ifallx sarevariedbyasmallamount thetotaleffectonywillbe 32 First OrderConditionforaMaximum orMinimum Anecessaryconditionforamaximum orminimum ofthefunctionf x1 x2 xn isthatdy 0foranycombinationofsmallchangesinthex sTheonlywayforthistobetrueisif Apointwherethisconditionholdsiscalledacriticalpoint 33 FindingaMaximum Supposethatyisafunctionofx1andx2y x1 1 2 x2 2 2 10y x12 2x1 x22 4x2 5First orderconditionsimplythat OR 34 TheEnvelopeTheorem TheenvelopetheoremconcernshowtheoptimalvalueforaparticularfunctionchangeswhenaparameterofthefunctionchangesThisiseasiesttoseebyusinganexample 35 TheEnvelopeTheorem Supposethatyisafunctionofxy x2 axFordifferentvaluesofa thisfunctionrepresentsafamilyofinvertedparabolasIfaisassignedaspecificvalue thenybecomesafunctionofxonlyandthevalueofxthatmaximizesycanbecalculated 36 TheEnvelopeTheorem Supposeweareinterestedinhowy changesasachangesTherearetwowayswecandothiscalculatetheslopeofydirectlyholdxconstantatitsoptimalvalueandcalculate y adirectly 37 TheEnvelopeTheorem Tocalculatetheslopeofthefunction wemustsolvefortheoptimalvalueofxforanyvalueofady dx 2x a 0 x a 2Substituting wegety x 2 a x a 2 2 a a 2 y a2 4 a2 2 a2 4 38 TheEnvelopeTheorem Therefore dy da 2a 4 a 2 x But wecansavetimebyusingtheenvelopetheoremforsmallchangesina dy dacanbecomputedbyholdingxatx andcalculating y adirectlyfromy 39 TheEnvelopeTheorem y a xHoldingx x y a x a 2Thisisthesameresultfoundearlier 40 TheEnvelopeTheorem Theenvelopetheoremstatesthatthechangeintheoptimalvalueofafunctionwithrespecttoaparameterofthatfunctioncanbefoundbypartiallydifferentiatingtheobjectivefunctionwhileholdingx orseveralx s atitsoptimalvalue 41 TheEnvelopeTheorem Theenvelopetheoremcanbeextendedtothecasewhereyisafunctionofseveralvariablesy f x1 xn a Findinganoptimalvalueforywouldconsistofsolvingnfirst orderequations y xi 0 i 1 n 42 TheEnvelopeTheorem Optimalvaluesforthesesx swouldbedeterminedthatareafunctionofax1 x1 a x2 x2 a 43 TheEnvelopeTheorem Substitutingintotheoriginalobjectivefunctionyieldsanexpressionfortheoptimalvalueofy y y f x1 a x2 a xn a a Differentiatingyields 44 TheEnvelopeTheorem Becauseoffirst orderconditions alltermsexcept f aareequaltozeroifthex sareattheiroptimalvaluesTherefore 45 Part2ConstrainedOptimization Veryimportantintheeconomic 46 ConstrainedMaximization Whatifallvaluesforthex sarenotfeasible thevaluesofxmayallhavetobepositiveaconsumer schoicesarelimitedbytheamountofpurchasingpoweravailableOnemethodusedtosolveconstrainedmaximizationproblemsistheLagrangianmultipliermethod 47 LagrangianMultiplierMethod Supposethatwewishtofindthevaluesofx1 x2 xnthatmaximizey f x1 x2 xn subjecttoaconstraintthatpermitsonlycertainvaluesofthex stobeusedg x1 x2 xn 0 48 LagrangianMultiplierMethod TheLagrangianmultipliermethodstartswithsettinguptheexpressionL f x1 x2 xn g x1 x2 xn where isanadditionalvariablecalledaLagrangianmultiplierWhentheconstraintholds L fbecauseg x1 x2 xn 0 49 LagrangianMultiplierMethod First OrderConditions L x1 f1 g1 0 L x2 f2 g2 0 L g x1 x2 xn 0 50 LagrangianMultiplierMethod Thefirst orderconditionscangenerallybesolvedforx1 x2 xnand Thesolutionwillhavetwoproperties thex swillobeytheconstraintthesex swillmakethevalueofL andthereforef aslargeaspossible 51 LagrangianMultiplierMethod TheLagrangianmultiplier hasanimportanteconomicinterpretationThefirst orderconditionsimplythatf1 g1 f2 g2 fn gn thenumeratorsabovemeasurethemarginalbenefitthatonemoreunitofxiwillhaveforthefunctionfthedenominatorsreflecttheaddedburdenontheconstraintofusingmorexi 52 LagrangianMultiplierMethod Attheoptimalchoicesforthex s theratioofthemarginalbenefitofincreasingxitothemarginalcostofincreasingxishouldbethesameforeveryx isthecommoncost benefitratioforallofthex s 53 LagrangianMultiplierMethod Iftheconstraintwasrelaxedslightly itwouldnotmatterwhichxischangedTheLagrangianmultiplierprovidesameasureofhowtherelaxationintheconstraintwillaffectthevalueofy providesa shadowprice totheconstraint 54 LagrangianMultiplierMethod Ahighvalueof indicatesthatycouldbeincreasedsubstantiallybyrelaxingtheconstrainteachxhasahighcost benefitratioAlowvalueof indicatesthatthereisnotmuchtobegainedbyrelaxingtheconstraint 0impliesthattheconstraintisnotbinding 55 Duality Anyconstrainedmaximizationproblemhasassociatedwithitadualprobleminconstrainedminimizationthatfocusesattentionontheconstraintsintheoriginalproblem 56 Duality Individualsmaximizeutilitysubjecttoabudgetconstraintdualproblem individualsminimizetheexpenditureneededtoachieveagivenlevelofutilityFirmsminimizethecostofinputstoproduceagivenlevelofoutputdualproblem firmsmaximizeoutputforagivencostofinputspurchased 57 ConstrainedMaximization Supposeafarmerhadacertainlengthoffence P andwishedtoenclosethelargestpossiblerectangularshapeLetxbethelengthofonesideLetybethelengthoftheothersideProblem choosexandysoastomaximizethearea A x y subjecttotheconstraintthattheperimeterisfixedatP 2x 2y 58 ConstrainedMaximization SettinguptheLagrangianmultiplierL x y P 2x 2y Thefirst orderconditionsforamaximumare L x y 2 0 L y x 2 0 L P 2x 2y 0 59 ConstrainedMaximization Sincey 2 x 2 xmustbeequaltoythefieldshouldbesquarexandyshouldbechosensothattheratioofmarginalbenefitstomarginalcostsshouldbethesameSincex yandy 2 wecanusetheconstrainttoshowthatx y P 4 P 8 60 ConstrainedMaximization InterpretationoftheLagrangianmultiplierifthefarmerwasinterestedinknowinghowmuchmorefieldcouldbefencedbyaddinganextrayardoffence suggeststhathecouldfindoutbydividingthepresentperimeter P by8thus theLagrangianmultiplierprovidesinformationabouttheimplicitvalueoftheconstraint 61 ConstrainedMaximization Dualproblem choosexandytominimizetheamountoffencerequiredtosurroundthefieldminimizeP 2x 2ysubjecttoA x ySettinguptheLagrangian LD 2x 2y D A x y 62 ConstrainedMaximization First orderconditions LD x 2 D y 0 LD y 2 D x 0 LD D A x y 0Solving wegetx y A1 2TheLagrangianmultiplier D 2A 1 2 63 EnvelopeTheorem ConstrainedMaximization Supposethatwewanttomaximizey f x1 xn a subjecttotheconstraintg x1 xn a 0OnewaytosolvewouldbetosetuptheLagrangianexpressionandsolvethefirst orderconditions 64 EnvelopeTheorem ConstrainedMaximization Alternatively itcanbeshownthatdy da L a x1 xn a ThechangeinthemaximalvalueofythatresultswhenachangescanbefoundbypartiallydifferentiatingLandevaluatingthepartialderivativeattheoptimalpoint 65 SecondOrderConditions FunctionsofOneVariable Lety f x Anecessaryconditionforamaximumisthatdy dx f x 0Toensurethatthepointisamaximum ymustbedecreasingformovementsawayfromit 66 SecondOrderConditions FunctionsofOneVariable Thetotaldifferentialmeasuresthechangeinydy f x dxTobeatamaximum dymustbedecreasingforsmallincreasesinxToseethechangesindy wemustusethesecondderivativeofy 67 SecondOrderConditions FunctionsofOneVariable Notethatd2y 0impliesthatf x dx2 0Sincedx2mustbepositive f x 0Thismeansthatthefunctionfmusthaveaconcaveshapeatthecriticalpoint 68 SecondOrderConditions FunctionsofTwoVariables Supposethaty f x1 x2 Firstorderconditionsforamaximumare y x1 f1 0 y x2 f2 0Toensurethatthepointisamaximum ymustdiminishformovementsinanydirectionawayfromthecriticalpoint 69 SecondOrderConditions FunctionsofTwoVariables Theslopeinthex1direction f1 mustbediminishingatthecriticalpointTheslopeinthex2direction f2 mustbediminishingatthecriticalpointBut conditionsmustalsobeplacedonthecross partialderivative f12 f21 toensurethatdyisdecreasingforallmovementsthroughthecriticalpoint 70 SecondOrderConditions FunctionsofTwoVariables Thetotaldifferentialofyisgivenbydy f1dx1 f2dx2Thedifferentialofthatfunctionisd2y f11dx1 f12dx2 dx1 f21dx1 f22dx2 dx2d2y f11dx12 f12dx2dx1 f21dx1dx2 f22dx22ByYoung stheorem f12 f21andd2y f11dx12 2f12dx1dx2 f22dx22 71 SecondOrderConditions FunctionsofTwoVariables d2y f11dx12 2f12dx1dx2 f22dx22Forthisequationtobeunambiguouslynegativeforanychangeinthex s f11andf22mustbenegativeIfdx2 0 thend2y f11dx12ford2y 0 f11 0Ifdx1 0 thend2y f22dx22ford2y 0 f22 0 72 SecondOrderConditions FunctionsofTwoVariables d2y f11dx12 2f12dx1dx2 f22dx22Ifneitherdx1nordx2iszero thend2ywillbeunambiguouslynegativeonlyiff11f22 f122 0thesecondpartialderivatives f11andf22 mustbesufficientlynegativesothattheyoutweighanypossibleperverseeffectsfromthecross partialderivatives f12 f21 73 SecondOrderConditionsfortheConstrainedMaximization f11f22 2f12f1f2 f22f12 0Thisistosayunderlinearconstrainedconditions theobjectfunctionneedstobequasi concavetomakethefirstorderconditionsbethesufficientconditions 74 ConcaveandQuasi ConcaveFunctions Thedifferencesbetweenconcaveandquasi concavefunctionscanbeillustratedwiththefunctiony f x1 x2 x1 x2 kwherethex stakeononlypositivevaluesandkcantakeonavarietyofpositivevalues 75 ConcaveandQuasi ConcaveFunctions Nomatterwhatvaluektakes thisfunctionisquasi concaveWhetherornotthefunctionisconcavedependsonthevalueofkifk0 5 thefunctionisconvex 76 HomogeneousFunctions Afunctionf x1 x2 xn issaidtobehomogeneousofdegreekiff tx1 tx2 txn tkf x1 x2 xn whenafunctionishomogeneousofdegreeone adoublingofallofitsargumentsdoublesthevalueofthefunctionitselfwhenafunctionishomogeneousofdegreezero adoublingofallofitsargumentsleavesthevalueofthefunctionunchanged 77 HomogeneousFunctions Ifafunctionishomogeneousofdegreek thepartialderivativesofthefunctionwillbehomogeneousofdegreek 1 78 Euler sTheorem Ifwedifferentiatethedefinitionforhomogeneitywithrespecttotheproportionalityfactort wegetktk 1f x1 xn x1f1 tx1 txn xnfn tx1 txn ThisrelationshipiscalledEuler stheorem 79 Euler sTheorem Euler stheoremshowsthat forhomogeneousfunctions thereisadefiniterelationshipbetweenthevaluesofthefunctionandthevaluesofitspartialderivatives 80 HomotheticFunctions Ahomotheticfunctionisonethatisformedbytakingamonotonictransformationofahomogeneousfunctiontheydonotpossessthehomogeneitypropertiesoftheirunderlyingfunctions 81 HomotheticFunctions Forbothhomogeneousandhomotheticfunctions theimplicittrade offsamongthevariablesinthefunctiondependonlyontheratiosofthosevariables notontheirabsolutevalues 82 HomotheticFunctions Supposeweareexaminingthesimple twovariableimplicitfunctionf x y 0Theimplicittrade offbetweenxandyforatwo variablefunctionisdy dx fx fyIfweassumefishomogeneousofdegreek itspartialderivativeswillbehomogeneousofdegreek 1 83 HomotheticFunctions Theimplicittrade offbetweenxandyis Ift