nicholson微观经济理论（答案+ppt+习题库）-20-03-033593-0

THEMATHEMATICSOFOPTIMIZATIONMANYECONOMICMODELSSTARTWITHTHEASSUMPTIONTHATANAGENTISSEEKINGTONDTHEOPTIMALVALUEOFSOMEFUNCTIONFORCONSUMERSTHATFUNCTIONMEASURESTHEUTILITYPROVIDEDBYTHEIRPURCHASESFORRMSITMEASURESTHEIRPROTSBUTINBOTHCASESTHEFORMAL,MATHEMATICALASPECTSOFTHESOLUTIONAREIDENTICALINTHISCHAPTERWEEXAMINETHEMATHEMATICSCOMMONTOALLSUCHPROBLEMSFORTHOSEFAMILIARWITHMULTIVARIABLECALCULUS,THISCHAPTERWILLBELARGELYINTHENATUREOFAREVIEWFORTHOSEWHOAREFAMILIARONLYWITHSOMECONCEPTSFROMBASICCALCULUS,THISCHAPTERSHOULDPROVIDEENOUGHBACKGROUNDTOSTARTLOOKINGATTHEWAYSINWHICHCALCULUSISUSEDTOCONSTRUCTMICROECONOMICMODELSMOREGENERALLY,THECHAPTERISINTENDEDTOPROVIDEAREFERENCETHATMAYBEUSEFULASTHESEVARIOUSMATHEMATICALCONCEPTSAREENCOUNTEREDLATERINTHETEXT2CHAPTERMAXIMIZATIONOFAFUNCTIONOFONEVARIABLELETSSTARTWITHASIMPLEEXAMPLESUPPOSETHATAMANAGEROFARMDESIRESTOMAXIMIZE1THEPROTSRECEIVEDFROMSELLINGAPARTICULARGOODSUPPOSEALSOTHATTHEPROFITSH9266RECEIVEDDEPENDONLYONTHEQUANTITYQOFTHEGOODSOLDMATHEMATICALLY,H9266H11005FQ21FIGURE21SHOWSAPOSSIBLERELATIONSHIPBETWEENH9266ANDQCLEARLY,TOACHIEVEMAXIMUMPROTS,THEMANAGERSHOULDPRODUCEOUTPUTQ,WHICHYIELDSPROTSH9266IFAGRAPHSUCHASTHATOFFIGURE21WEREAVAILABLE,THISWOULDSEEMTOBEASIMPLEMATTERTOBEACCOMPLISHEDWITHARULERSUPPOSE,HOWEVER,ASISMORELIKELY,THEMANAGERDOESNOTHAVESUCHANACCURATEPICTUREOFTHEMARKETHEORSHEMAYTHENTRYVARYINGQTOSEEWHEREAMAXIMUMPROTISOBTAINEDFOREXAMPLE,BYSTARTINGATQ1,PROTSFROMSALESWOULDBEH92661NEXT,THEMANAGERMAYTRYOUTPUTQ2,OBSERVINGTHATPROTSHAVEINCREASEDTOH92662THECOMMONSENSEIDEATHATPROTSHAVEINCREASEDINRESPONSETOANINCREASEINQCANBESTATEDFORMALLYASH110220ORH110220,22WHERETHEH9004NOTATIONISUSEDTOMEAN“THECHANGEIN”H9266ORQASLONGASH9004H9266/H9004QISPOSITIVE,PROTSAREINCREASINGANDTHEMANAGERWILLCONTINUETOINCREASEOUTPUTFORINCREASESINOUTPUTTOTHERIGHTOFQ,HOWEVER,H9004H9266/H9004QWILLBENEGATIVE,ANDTHEMANAGERWILLREALIZETHATAMISTAKEHASBEENMADEIFHEORSHECONTINUESTOEXPANDQH9004H9266H5007H9004QH92662H11002H92661H5007Q2H11002Q122PART1INTRODUCTION1INTHISCHAPTERWEWILLGENERALLYEXPLOREMAXIMIZATIONPROBLEMSAVIRTUALLYIDENTICALAPPROACHWOULDBETAKENTOSTUDYMINIMIZATIONPROBLEMSHYPOTHETICALRELATIONSHIPBETWEENQUANTITYPRODUCEDANDPROTSIFAMANAGERWISHESTOPRODUCETHELEVELOFOUTPUTTHATMAXIMIZESPROTS,QSHOULDBEPRODUCEDNOTICETHATATQ,DH9266/DQH110050H92661H92663H92662H9266H11005FQH9266H9266Q3Q2Q1QQUANTITYFIGURE21DERIVATIVESASYOUPROBABLYKNOW,THELIMITOFH9004H9266/H9004QFORVERYSMALLCHANGESINQISCALLEDTHEDERIVATIVEOFTHEFUNCTION,H9266H11005FQ,ANDISDENOTEDBYDH9266/DQORDF/DQORFH11032QMOREFORMALLY,THEDERIVATIVEOFAFUNCTIONH9266H11005FQATTHEPOINTQ1ISDENEDASH11005H11005LIMH0023NOTICETHATTHEVALUEOFTHISRATIOOBVIOUSLYDEPENDSONTHEPOINTQ1THATISCHOSENVALUEOFTHEDERIVATIVEATAPOINTANOTATIONALCONVENTIONSHOULDBEMENTIONEDSOMETIMESONEWISHESTONOTEEXPLICITLYTHEPOINTATWHICHTHEDERIVATIVEISTOBEEVALUATEDFOREXAMPLE,THEEVALUATIONOFTHEDERIVATIVEATTHEPOINTQH11005Q1COULDBEDENOTEDBYH5007DDH9266QH5007H20841QH11005Q124ATOTHERTIMESONEISINTERESTEDINTHEVALUEOFDH9266/DQFORALLPOSSIBLEVALUESOFQ,ANDNOEXPLICITMENTIONOFAPARTICULARPOINTOFEVALUATIONISMADEINTHEEXAMPLEOFFIGURE21,H20841QH11005Q1H110220,WHEREASH20841QH11005Q3H110210WHATISTHEVALUEOFDH9266/DQATQITWOULDSEEMTOBE0,SINCETHEVALUEISPOSITIVEFORVALUESOFQLESSTHANQANDNEGATIVEFORVALUESGREATERTHANQTHEDERIVATIVEISTHESLOPEOFTHECURVEINQUESTIONTHISSLOPEISPOSITIVETOTHELEFTOFQANDNEGATIVETOTHERIGHTOFQATTHEPOINTQ,THESLOPEOFFQIS0FIRSTORDERCONDITIONFORAMAXIMUMTHISRESULTISQUITEGENERALFORAFUNCTIONOFONEVARIABLETOATTAINITSMAXIMUMVALUEATSOMEPOINT,THEDERIVATIVEATTHATPOINTIFITEXISTSMUSTBE0HENCE,IFAMANAGERCOULDESTIMATETHEFUNCTIONFQFROMSOMESORTOFREALWORLDDATA,ITWOULDBETHEORETICALLYPOSSIBLETONDTHEPOINTWHEREDF/DQH110050ATTHISOPTIMALPOINTSAYQ,ITWOULDBETHECASETHATH5007DDQFH5007H20841QH11005QH11005025SECONDORDERCONDITIONSANUNSUSPECTINGMANAGERCOULDBETRICKED,HOWEVER,BYANAVEAPPLICATIONOFTHISRULEALONEFOREXAMPLE,SUPPOSETHATTHEPROTFUNCTIONLOOKSLIKETHATSHOWNINDH9266H5007DQDH9266H5007DQFQ1H11001HH11002FQ1H5007H5007HDFH5007DQDH9266H5007DQCHAPTER2THEMATHEMATICSOFOPTIMIZATION23EITHERFIGURE22AOR22BIFTHEPROTFUNCTIONISTHATSHOWNINFIGURE22A,THEMANAGER,BYPRODUCINGWHEREDH9266/DQH110050,WILLCHOOSEPOINTQATHISPOINTINFACTYIELDSMINIMUM,NOTMAXIMUM,PROTSFORTHEMANAGERSIMILARLY,IFTHEPROTFUNCTIONISTHATSHOWNINFIGURE22B,THEMANAGERWILLCHOOSEPOINTQB,WHICH,ALTHOUGHITYIELDSAPROTGREATERTHANTHATFORANYOUTPUTLOWERTHANQB,ISCERTAINLYINFERIORTOANYOUTPUTGREATERTHANQBTHESESITUATIONSPOINTUPTHEMATHEMATICALFACTTHATDH9266/DQH110050ISANECESSARYCONDITIONFORAMAXIMUM,BUTNOTASUFCIENTCONDITIONTOENSURETHATTHECHOSENPOINTISINDEEDAMAXIMUMPOINT,ASECONDCONDITIONMUSTBEIMPOSEDINTUITIVELY,THISADDITIONALCONDITIONISCLEARTHEPROTAVAILABLEBYPRODUCINGEITHERABITMOREORABITLESSTHANQMUSTBESMALLERTHANTHATAVAILABLEFROMQIFTHISISNOTTRUE,THEMANAGERCANDOBETTERTHANQMATHEMATICALLY,THISMEANSTHATDH9266/DQMUSTBEGREATERTHAN0FORQH11021QANDMUSTBELESSTHAN0FORQH11022QTHEREFORE,ATQ,DH9266/DQMUSTBEDECREASINGANOTHERWAYOFSAYINGTHISISTHATTHEDERIVATIVEOFDH9266/DQMUSTBENEGATIVEATQSECONDDERIVATIVESTHEDERIVATIVEOFADERIVATIVEISCALLEDASECONDDERIVATIVEANDISDENOTEDBYORH5007DDQ2F2H5007ORFH11033QD2H9266H5007DQ224PART1INTRODUCTIONTWOPROTFUNCTIONSTHATGIVEMISLEADINGRESULTSIFTHEFIRSTDERIVATIVERULEISAPPLIEDUNCRITICALLYINATHEAPPLICATIONOFTHERSTDERIVATIVERULEWOULDRESULTINPOINTQABEINGCHOSENTHISPOINTISINFACTAPOINTOFMINIMUMPROTSSIMILARLY,INBOUTPUTLEVELQBWOULDBERECOMMENDEDBYTHERSTDERIVATIVERULE,BUTTHISPOINTISINFERIORTOALLOUTPUTSGREATERTHANQBTHISDEMONSTRATESGRAPHICALLYTHATNDINGAPOINTATWHICHTHEDERIVATIVEISEQUALTO0ISANECESSARY,BUTNOTASUFCIENT,CONDITIONFORAFUNCTIONTOATTAINITSMAXIMUMVALUEH9266AH9266QAQUANTITYABH9266BH9266QBQUANTITYFIGURE22THEADDITIONALCONDITIONFORQTOREPRESENTALOCALMAXIMUMISTHEREFOREH20841QH11005QH11005FH11033QH20841QH11005QH110210,26WHERETHENOTATIONISAGAINAREMINDERTHATTHISSECONDDERIVATIVEISTOBEEVALUATEDATQHENCE,ALTHOUGHEQUATION25DH9266/DQH110050ISANECESSARYCONDITIONFORAMAXIMUM,THATEQUATIONMUSTBECOMBINEDWITHEQUATION26D2H9266/DQ2H110210TOENSURETHATTHEPOINTISALOCALMAXIMUMFORTHEFUNCTIONEQUATIONS25AND26TOGETHERARETHEREFORESUFCIENTCONDITIONSFORSUCHAMAXIMUMOFCOURSE,ITISPOSSIBLETHATBYASERIESOFTRIALSTHEMANAGERMAYBEABLETODECIDEONQBYRELYINGONMARKETINFORMATIONRATHERTHANONMATHEMATICALREASONINGREMEMBERFRIEDMANSPOOLPLAYERANALOGYINTHISBOOKWESHALLBELESSINTERESTEDINHOWTHEPOINTISDISCOVEREDTHANINITSPROPERTIESANDHOWTHEPOINTCHANGESWHENCONDITIONSCHANGEAMATHEMATICALDEVELOPMENTWILLBEVERYHELPFULINANSWERINGTHESEQUESTIONSRULESFORFINDINGDERIVATIVESHEREAREAFEWFAMILIARRULESFORTAKINGDERIVATIVESWEWILLUSETHESEATMANYPLACESINTHISBOOK1IFBISACONSTANT,THENH5007DDBXH5007H1100502IFAANDBARECONSTANTSANDBH110140,THENH11005BAXBH1100213H11005WHERELNSIGNIESTHELOGARITHMTOTHEBASEEH110052718284H11005AXLNAFORANYCONSTANTAAPARTICULARCASEOFTHISRULEISDEX/DXH11005EXNOWSUPPOSETHATFXANDGXARETWOFUNCTIONSOFXANDTHATFH11032XANDGH11032XEXISTTHEN5H11005FH11032XH11001GH11032X6H11005FXGH11032XH11001FH11032XGX7H11005,PROVIDEDTHATGXH110140FH11032XGXH11002FXGH11032XH5007H5007H5007GX2DH20898H5007GFXXH5007H20899H5007DXDFXH11080GXH5007H5007DXDFXH11001GXH5007H5007DXDAXH5007DX1H5007XDLNXH5007DXDAXBH5007DXD2H9266H5007DQ2CHAPTER2THEMATHEMATICSOFOPTIMIZATION25FINALLY,IFYH11005FXANDXH11005GZANDIFBOTHFH11032XANDGH11032ZEXIST,THEN8H5007DDYZH5007H11005H5007DDXYH5007H11080H5007DDXZH5007H11005H5007DDXFH5007H11080H5007DDGZH5007THISRESULTISCALLEDTHECHAINRULEITPROVIDESACONVENIENTWAYTOSTUDYHOWONEVARIABLEZAFFECTSANOTHERVARIABLEYSOLELYTHROUGHITSINUENCEONSOMEINTERMEDIATEVARIABLEXEXAMPLE21PROTMAXIMIZATIONSUPPOSETHATTHERELATIONSHIPBETWEENPROTSH9266ANDQUANTITYPRODUCEDQISGIVENBYH9266H110051,000QH110025Q227AGRAPHOFTHISFUNCTIONWOULDRESEMBLETHEPARABOLASHOWNINFIGURE21THEVALUEOFQTHATMAXIMIZESPROTSCANBEFOUNDBYAPPLYINGRULE2FORNDINGDERIVATIVESH110051,000H1100210QH11005028SOQH1100510029ATQH11005100,EQUATION27SHOWSTHATPROTSARE50,000THELARGESTVALUEPOSSIBLEIF,FOREXAMPLE,THERMOPTEDTOPRODUCEQH1100550,PROTSWOULDBE37,500ATQH11005200,PROTSAREPRECISELYZEROTHATQH11005100ISA“GLOBAL”MAXIMUMCANBESHOWNBYNOTINGTHATTHESECONDDERIVATIVEOFTHEPROTFUNCTIONISH1100210SEEEQUATION28HENCE,THERATEOFINCREASEINPROTSISALWAYSDECREASINGUPTOQH11005100THISRATEOFINCREASEISSTILLPOSITIVE,BUTBEYONDTHATPOINTITBECOMESNEGATIVEINTHISEXAMPLE,QH11005100ISTHEONLYLOCALMAXIMUMVALUEFORTHEFUNCTIONH9266WITHMORECOMPLEXFUNCTIONS,HOWEVER,THEREMAYBESEVERALSUCHMAXIMAQUERYSUPPOSETHERMSOUTPUT,Q,DEPENDEDONLYONLABORINPUT,L,ACCORDINGTOQH110052H20857LH33526WHATWOULDBETHEPROTMAXIMIZINGLEVELOFLABORINPUTDOESTHISAGREEWITHTHEPREVIOUSSOLUTIONHINTYOUMAYWISHTOSOLVETHISPROBLEMDIRECTLYBYSUBSTITUTIONORBYUSINGTHECHAINRULEFUNCTIONSOFSEVERALVARIABLESECONOMICPROBLEMSSELDOMINVOLVEFUNCTIONSOFASINGLEVARIABLEONLYMOSTGOALSOFINTERESTTOECONOMICAGENTSDEPENDONSEVERALVARIABLES,ANDTRADEOFFSMUSTBEMADEAMONGTHESEVARIABLESFOREXAMPLE,THEUTILITYANINDIVIDUALRECEIVESFROMACTIVITIESASACONSUMERDEPENDSONTHEAMOUNTOFEACHGOODCONSUMEDFORARMSPRODUCTIONFUNCTION,THEAMOUNTPRODUCEDDEPENDSONTHEQUANTITYOFLABOR,CAPITAL,DH9266H5007DQ26PART1INTRODUCTIONANDLANDDEVOTEDTOPRODUCTIONINTHESECIRCUMSTANCESTHISDEPENDENCEOFONEVARIABLEYONASERIESOFOTHERVARIABLESX1,X2,XNISDENOTEDBYYH11005FX1,X2,XN210PARTIALDERIVATIVESWEAREINTERESTEDINTHEPOINTATWHICHYREACHESAMAXIMUMANDINTHETRADEOFFSTHATMUSTBEMADETOREACHTHATPOINTITISAGAINCONVENIENTTOPICTURETHEAGENTASCHANGINGTHEVARIABLESATHISORHERDISPOSALTHEXSINORDERTOLOCATEAMAXIMUMUNFORTUNATELY,FORAFUNCTIONOFSEVERALVARIABLES,THEIDEAOFTHEDERIVATIVEISNOTWELLDENEDJUSTASINCLIMBINGAMOUNTAINTHESTEEPNESSOFASCENTDEPENDSONWHICHDIRECTIONYOUGO,SODOESTHESLOPEORDERIVATIVEOFTHEFUNCTIONDEPENDONTHEDIRECTIONINWHICHITISTAKENUSUALLY,THEONLYDIRECTIONALSLOPESOFINTERESTARETHOSETHATAREOBTAINEDBYINCREASINGONEOFTHEXSWHILEHOLDINGALLTHEOTHERVARIABLESCONSTANTTHEANALOGYOFMOUNTAINCLIMBINGMIGHTBETOMEASURESLOPESONLYINANORTHSOUTHOREASTWESTDIRECTIONTHESEDIRECTIONALSLOPESARECALLEDPARTIALDERIVATIVESTHEPARTIALDERIVATIVEOFYWITHRESPECTTOTHATIS,INTHEDIRECTIONOFX1ISDENOTEDBYORORFX1ORF1ITISUNDERSTOODTHATINCALCULATINGTHISDERIVATIVEALLOFTHEOTHERXSAREHELDCONSTANTAGAINITSHOULDBEEMPHASIZEDTHATTHENUMERICALVALUEOFTHISSLOPEDEPENDSONTHEVALUEOFX1ANDONTHEPREASSIGNEDVALUESOFX2,XNASOMEWHATMOREFORMALDENITIONOFTHEPARTIALDERIVATIVEISH20841XH335262,XH33526NH11005LIMH00,211WHERETHENOTATIONISINTENDEDTOINDICATETHATX2,XNAREALLHELDCONSTANTATTHEPREASSIGNEDVALUESXH335262,XH33526NSOTHEEFFECTOFCHANGINGX1ONLYCANBESTUDIEDPARTIALDERIVATIVESWITHRESPECTTOTHEOTHERVARIABLESX2,XNWOULDBECALCULATEDINASIMILARWAYPARTIALDERIVATIVESANDTHECETERISPARIBUSASSUMPTIONINCHAPTER1WEDESCRIBEDTHEWAYINWHICHECONOMISTSUSETHECETERISPARIBUSASSUMPTIONINTHEIRMODELSTOHOLDCONSTANTAVARIETYOFOUTSIDEINUENCESSOTHEPARTICULARRELATIONSHIPBEINGSTUDIEDCANBEEXPLOREDINASIMPLIEDSETTINGPARTIALDERIVATIVESAREAPRECISEMATHEMATICALWAYOFREPRESENTINGTHISAPPROACHTHATIS,THEYSHOWHOWCHANGESINONEVARIABLEAFFECTSOMEOUTCOMEWHENOTHERINUENCESAREHELDCONSTANTEXACTLYWHATECONOMISTSNEEDFORTHEIRMODELSFOREXAMPLE,MARSHALLSDEMANDCURVESHOWSTHERELATIONSHIPBETWEENPRICEPANDQUANTITYQDEMANDEDWHENOTHERFACTORSAREHELDCONSTANTUSINGPARTIALDERIVATIVES,WECOULDREPRESENTTHESLOPEOFTHISCURVEBYH11128Q/H11128PTOINDICATETHECETERISPARIBUSASSUMPTIONSTHATAREINEFFECTTHEFUNDAMENTALLAWOFDEMANDTHATPRICEANDQUANTITYMOVEINOPPOSITEDIRECTIONSWHENOTHERFACTORSDONOTCHANGEISTHEREFOREREECTEDBYTHEMATHEMATICALSTATEMENT“H11128Q/H11128PH110210”AGAIN,THEUSEOFAFX1H11001H,XH335262,XH33526NH11002FX1,XH335262,XH33526NH5007H5007H5007H5007H5007HH11128FH5007H11128X1H11128FH5007H11128X1H11128YH5007H11128X1CHAPTER2THEMATHEMATICSOFOPTIMIZATION27PARTIALDERIVATIVESERVESASAREMINDEROFTHECETERISPARIBUSASSUMPTIONSTHATSURROUNDTHELAWOFDEMANDCALCULATINGPARTIALDERIVATIVESITISEASYTOCALCULATEPARTIALDERIVATIVESTHECALCULATIONPROCEEDSASFORTHEUSUALDERIVATIVEBYTREATINGX2,XNASCONSTANTSWHICHINDEEDTHEYAREINTHEDENITIONOFAPARTIALDERIVATIVECONSIDERTHEFOLLOWINGEXAMPLES1IFYH11005FX1,X2H11005AX21H11001BX1X2H11001CX22,THENH11005F1H110052AX1H11001BX2ANDH11005F2H11005BX1H110012CX2NOTICETHATH11128F/H11128X1ISINGENERALAFUNCTIONOFBOTHX1ANDX2ANDTHEREFOREITSVALUEWILLDEPENDONTHEPARTICULARVALUESASSIGNEDTOTHESEVARIABLESITALSODEPENDSONTHEPARAMETERSA,B,ANDC,WHICHDONOTCHANGEASX1ANDX2CHANGE2IFYH11005FX1,X2H11005EAX1H11001BX2,THENH11005F1H11005AEAX1H11001BX2ANDH11005F2H11005BEAX1H11001BX23IFYH11005FX1,X2H11005ALNX1H11001BLNX2,THENH11005F1H11005ANDH11005F2H11005NOTICEHERETHATTHETREATMENTOFX2ASACONSTANTINTHEDERIVATIONOFH11128F/H11128X1CAUSESTHETERMBLNX2TODISAPPEARUPONDIFFERENTIATIONBECAUSEITDOESNOTCHANGEWHENX1CHANGESINTHISCASE,UNLIKEOURPREVIOUSEXAMPLES,THESIZEOFTHEEFFECTOFX1ONYISINDEPENDENTOFTHEVALUEOFX2INOTHERCASESTHEEFFECTOFX1ONYWILLDEPENDONTHELEVELOFX2BH5007X2H11128FH5007H11128X2AH5007X1H11128FH5007H11128X1H11128FH5007H11128X2H11128FH5007H11128X1H11128FH5007H11128X2H11128FH5007H11128X128PART1INTRODUCTIONSECONDORDERPARTIALDERIVATIVESTHEPARTIALDERIVATIVEOFAPARTIALDERIVATIVEISDIRECTLYANALOGOUSTOTHESECONDDERIVATIVEOFAFUNCTIONOFONEVARIABLEANDISCALLEDASECONDORDERPARTIALDERIVATIVETHISMAYBEWRITTENASORMORESIMPLYASH11005FIJ212FORTHEEXAMPLESABOVE1H11005F11H110052AF12H11005BF21H11005BF22H110052C2F11H11005A2EAX1H11001BX2F12H11005ABEAX1H11001BX2F21H11005ABEAX1H11001BX2F22H11005B2EAX1H11001BX23F11H11005H5007H11002X21AH5007F12H110050F21H110050F22H11005H5007H11002X22BH5007YOUNGSTHEOREMTHESEEXAMPLESILLUSTRATETHEMATHEMATICALRESULTTHAT,UNDERQUITEGENERALCONDITIONS,THEORDERINWHICHPARTIALDIFFERENTIATIONISCONDUCTEDTOEVALUATESECONDORDERPARTIALDERIVATIVESDOESNOTMATTERTHATIS,FIJH11005FJI213FORANYPAIROFVARIABLESXI,XJTHISRESULTISSOMETIMESCALLED“YOUNGSTHEOREM”FORANINTUITIVEEXPLANATIONOFTHETHEOREM,WECANRETURNTOOURMOUNTAINCLIMBINGANALOGYINTHISEXAMPLETHETHEOREMSTATESTHATTHEGAININELEVATIONAHIKEREXPERIENCESDEPENDSONTHEDIRECTIONSANDDISTANCESTRAVELED,BUTNOTONTHEORDERINWHICHTHESEOCCURTHATIS,THEGAININALTITUDEISINDEPENDENTOFTHEACTUALPATHTAKENASLONGASTHEHIKERPROCEEDSFROMONESETOFMAPCOORDINATESTOANOTHERHEORSHEMAY,FOREXAMPLE,GOONEMILENORTH,THENONEMILEEASTORPROCEEDINTHEH111282FH5007H11128X1H11128X1H111282FH5007H11128XJH11128XIH11128H11128F/H11128XIH5007H5007H11128XJCHAPTER2THEMATHEMATICSOFOPTIMIZATION29OPPOSITEORDERBYGOINGONEMILEEASTRST,THENAMILENORTHINEITHERCASE,THEGAININELEVATIONISTHESAMESINCEINBOTHCASESTHEHIKERISMOVINGFROMONESPECICPLACETOANOTHERINLATERCHAPTERSWEWILLMAKEQUITEABITOFUSEOFTHISRESULTBECAUSEITPROVIDESAVERYCONVENIENTWAYOFSHOWINGSOMEOFTHEPREDICTIONSTHATECONOMICMODELSMAKEABOUTBEHAVIOR2MAXIMIZATIONOFFUNCTIONSOFSEVERALVARIABLESUSINGPARTIALDERIVATIVES,WECANNOWDISCUSSTHEMAXIMIZATIONOFFUNCTIONSOFSEVERALVARIABLESTOUNDERSTANDTHEMATHEMATICSUSEDINSOLVINGTHISPROBLEM,ANANALOGYTOTHEONEVARIABLECASEISHELPFULINTHISONEVARIABLECASE,WECANPICTUREANAGENTVARYINGXBYASMALLAMOUNT,DX,ANDOBSERVINGTHECHANGEINYCALLTHISDYTHISCHANGEISGIVENBYDYH11005FH11032XDX214THEIDENTITYINEQUATION214THENRECORDSTHEFACTTHATTHECHANGEINYISEQUALTOTHECHANGEINXTIMESTHESLOPEOFTHEFUNCTIONTHISFORMULAISEQUIVALENTTOTHEPOINTSLOPEFORMULAUSEDFORLINEAREQUATIONSINBASICALGEBRAASBEFORE,THENECESSARYCONDITIONFORAMAXIMUMISTHATDYH110050FORSMALLCHANGESINXAROUNDTHEOPTIMALPOINTOTHERWISE,YCOULDBEINCREASEDBYSUITABLECHANGESINXBUTSINCEDXDOESNOTNECESSARILYEQUAL0INEQUATION214,DYH110050MUSTIMPLYTHATATTHEDESIREDPOINT,FH11032XH110050THISISANOTHERWAYOFOBTAININGTHERSTORDERCONDITIONFORAMAXIMUMTHATWEALREADYDERIVEDUSINGTHISANALOGY,LETSLOOKATTHEDECISIONSMADEBYANECONOMICAGENTWHOMUSTCHOOSETHELEVELSOFSEVERALVARIABLESSUPPOSETHATTHISAGENTWISHESTONDASETOFXSTHATWILLMAXIMIZETHEVALUEOFYH11005FX1,X2,XNTHEAGENTMIGHTCONSIDERCHANGINGONLYONEOFTHEXS,SAYX1,WHILEHOLDINGALLTHEOTHERSCONSTANTTHECHANGEINYTHATIS,DYTHATWOULDRESULTFROMTHISCHANGEINX1ISGIVENBYDYH11005DX1H11005F1DX1THISSAYSTHATTHECHANGEINYISEQUALTOTHECHANGEINX1TIMESTHESLOPEMEASUREDINTHEX1DIRECTIONUSINGTHEMOUNTAINANALOGYAGAIN,THISWOULDSAYTHATTHEGAININALTITUDEACLIMBERHEADINGNORTHWOULDACHIEVEISGIVENBYTHEDISTANCENORTHWARDTRAVELEDTIMESTHESLOPEOFTHEMOUNTAINMEASUREDINANORTHWARDDIRECTIONTOTALDIFFERENTIALIFALLTHEXSAREVARIEDBYASMALLAMOUNT,THETOTALEFFECTONYWILLBETHESUMOFEFFECTSSUCHASTHATSHOWNABOVETHEREFORETHETOTALCHANGEINYISDENEDTOBEDYH11005DX1H11001DX2H11001H11001DXN215H11005F1DX1H11001F2DX2H11001H11001FNDXNH11128FH5007H11128XNH11128FH5007H11128X2H11128FH5007H11128X1H11128FH5007H11128X130PART1INTRODUCTION2YOUNGSTHEOREMIMPLIESTHATTHEMATRIXOFTHESECONDORDERPARTIALDERIVATIVESOFAFUNCTIONISSYMMETRICTHISSYMMETRYOFFERSANUMBEROFECONOMICINSIGHTSFORABRIEFINTRODUCTIONTOTHEMATRIXCONCEPTSUSEDINECONOMICSSEETHEEXTENSIONTOTHISCHAPTERTHISEXPRESSIONISCALLEDTHETOTALDIFFERENTIALOFFANDISDIRECTLYANALOGOUSTOTHEEXPRESSIONFORTHESINGLEVARIABLECASEGIVENINEQUATION214THEEQUATIONISINTUITIVELYSENSIBLETHETOTALCHANGEINYISTHESUMOFCHANGESBROUGHTABOUTBYVARYINGEACHOFTHEXS3FIRSTORDERCONDITIONFORAMAXIMUMANECESSARYCONDITIONFORAMAXIMUMORAMINIMUMOFTHEFUNCTIONFX1,X2,XNISTHATDYH110050FORANYCOMBINATIONOFSMALLCHANGESINTHEXSTHEONLYWAYTHISCANHAPPENISIFATTHEPOINTBEINGCONSIDEREDF1H11005F2H11005H11005FNH110050216APOINTWHEREEQUATIONS216HOLDISCALLEDACRITICALPOINTEQUATIONS216ARETHENECESSARYCONDITIONSFORALOCALMAXIMUMTOSEETHISINTUITIVELY,NOTETHATIFONEOFTHEPARTIALSSAY,FIWEREGREATERORLESSTHAN0,THENYCOULDBEINCREASEDBYINCREASINGORDECREASINGXIANECONOMICAGENTTHENCOULDNDTHISMAXIMALPOINTBYNDINGTHESPOTWHEREYDOESNOTRESPONDTOVERYSMALLMOVEMENTSINANYOFTHEXSTHISISANEXTREMELYIMPORTANTRESULTFORECONOMICANALYSISITSAYSTHATANYACTIVITYTHATIS,THEXSSHOULDBEPUSHEDTOTHEPOINTWHEREITS“MARGINAL”CONTRIBUTIONTOTHEOBJECTIVETHATIS,YIS0TOSTOPSHORTOFTHATPOINTWOULDFAILTOMAXIMIZEYSECONDORDERCONDITIONSAGAIN,HOWEVER,THECONDITIONSOFEQUATIONS216ARENOTSUFCIENTTOENSUREAMAXIMUMTHISCANBEILLUSTRATEDBYRETURNINGTOANALREADYOVERWORKEDANALOGYALLHILLTOPSAREMOREORLESSAT,BUTNOTEVERYATPLACEISAHILLTOPASECONDORDERCONDITIONSIMILARTOEQUATION26ISNEEDEDTOENSURETHATTHEPOINTFOUNDBYAPPLYINGEQUATIONS216ISALOCALMAXIMUMINTUITIVELY,FORALOCALMAXIMUM,YSHOULDBEDECREASINGFORANYSMALLCHANGESINTHEXSAWAYFROMTHECRITICALPOINTASINTHESINGLEVARIABLECASE,THISNECESSARILYINVOLVESLOOKINGATTHESECONDORDERPARTIALDERIVATIVESOFTHEFUNCTIONFTHESESECONDORDERPARTIALSMUSTOBEYCERTAINRESTRICTIONSANALOGOUSTOTHERESTRICTIONTHATWASDERIVEDINTHESINGLEVARIABLECASEIFTHECRITICALPOINTFOUNDBYAPPLYINGEQUATIONS216ISTOBEALOCALMAXIMUMLATERINTHISCHAPTERWEWILLLOOKATTHESERESTRICTIONSCHAPTER2THEMATHEMATICSOFOPTIMIZ