mathematics for microeconomics

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1、MathematicsforCHAPTERTWOMicroeconomicsMicroeconomicmodelsareconstructedusingawidevarietyofmathematicaltechniques.Inthischapterweprovideabriefsummaryofsomeofthemostimportanttechniquesthatyouwillencounterinthisbook.Amajorportionofthechapterconcernsmathemati-calproceduresforfindi

2、ngtheoptimalvalueofsomefunction.Becausewewillfrequentlyadopttheassumptionthataneconomicactorseekstomaximizeorminimizesomefunc-tion,wewillencountertheseprocedures(mostofwhicharebasedoncalculus)manytimes.Afterourdetaileddiscussionofthecalculusofoptimization,weturntofourtopicsth

3、atarecoveredmorebriefly.First,welookatafewspecialtypesoffunctionsthatariseineconomics.Knowledgeofpropertiesofthesefunctionscanoftenbehelpfulinsolvingproblems.Next,weprovideabriefsummaryofintegralcalculus.Althoughintegrationisusedinthisbookfarlessfrequentlythanisdifferentiation

4、,wewillneverthelessencountersituationswherewewillwanttouseintegralstomeasureareasthatareimportanttoeco-nomictheoryortoaddupoutcomesthatoccurovertimeoracrossmanyindividuals.Oneparticularuseofintegrationistoexamineproblemsinwhichtheobjectiveistomax-imizeastreamofoutcomesovertim

5、e.Ourthirdaddedtopicfocusesontechniquestobeusedforsuchproblemsindynamicoptimization.Finally,Chapter2concludeswithabriefsummaryofmathematicalstatistics,whichwillbeparticularlyusefulinourstudyofeconomicbehaviorinuncertainsituations.MaximizationofaFunctionofOneVariableWecanmotiv

6、ateourstudyofoptimizationwithasimpleexample.Supposethataman-1agerofafirmdesirestomaximizetheprofitsreceivedfromsellingaparticulargood.Supposealsothattheprofits(p)receiveddependonlyonthequantity(q)ofthegoodsold.Mathematically,p¼fðqÞ:(2:1)Figure2.1showsapossiblerelationshipbetween

7、pandq.Clearly,toachievemaximumprofits,themanagershouldproduceoutputq,whichyieldsprofitsp.IfagraphsuchasthatofFigure2.1wereavailable,thiswouldseemtobeasimplemattertobeaccom-plishedwitharuler.1Herewewillgenerallyexploremaximizationproblems.Avirtuallyidenticalapproachwouldbetake

8、ntostudyminimizationproblemsbecausemaximizationoff(x)isequivalenttom