Optimization第二讲VIP免费

FacultyofEconomicsOptimizationLecture2MarcoHaanFebruary21,2005第一页，共三十九页。Lastweek•Optimizingafunctionofmorethan1variable.•Determininglocalminimaandlocalmaxima.•Firstandsecond-orderconditions.•Determiningglobalextremawithdirectrestrictionsonvariables.Thisweek•Constrainedproblems.•TheLagrangeMethod.•InterpretationoftheLagrangemultiplier.•Second-orderconditions.•Existence,uniqueness,andcharacterizationofsolutions.2第二页，共三十九页。Supposethatwewanttomaximizesomefunctionf(x1,x2)subjecttosomeconstraintg(x1,x2)=0.Example:AconsumerwantstomaximizeutilityU(x1,x2)=x1½x2½subjecttobudgetconstraint2x1+3x2=10.Inthiscase:f(x1,x2)=x1½x2½andg(x1,x2)=10–2x1–3x2.3第三页，共三十九页。*****11112111'()(,())(,())'()0.xfxxfxxxSupposethat,fromg(x1,x2)=0wecanwritex2=γ(x1).Takethetotaldifferential:dx2=γ’(x1)dx1Also:g1(x1,x2)dx1+g2(x1,x2)dx2=0211111211(,())'()(,())dxgxxxdxgxx12111max(,)max(,())().fxxfxxxWewanttomaximizef(x1,x2)subjecttog(x1,x2)=0.Hence:Wecannowwritetheobjectivefunctionas:****112112****212212(,)(,)(,)(,)fxxgxxfxxgxxWe’veseenthisinMicro1!4第四页，共三十九页。Theorem13.1If(x1*,x2*)isatangencysolutiontotheconstrainedmaximizationproblem1212max(,)s.t.(,)0fxxgxxthenwehavethatx1*andx2*satisfy****112112****212212**12(,)(,)(,)(,)(,)0.fxxgxxfxxgxxgxx5第五页，共三十九页。Backtotheexamplef(x1,x2)=x1½x2½andg(x1,x2)=10–2x1–3x2.Weneed****112112****212212(,)(,)(,)(,)fxxgxxfxxgxx112211221122112223xxxxSo2213xx122310xx551223;.xxWithHenceThisyieldsNote:thisonlysaysthatthisisalocaloptimum.6第六页，共三十九页。*****1121121*****2122122**12(,)+(,)0(,)+(,)0(,)0fxxgxxxfxxgxxxgxxLLLLagrangeMethodAgain,wewantto1212max(,)s.t.(,)0fxxgxxConsiderthefunction121212(,,)(,)+(,)xxfxxgxxLThefirsttwoequalitiesimply****112112****212212(,)(,)(,)(,)fxxgxxfxxgxxLet’smaximizethis:Hence,wegetexactlytheconditionsweneed!7第七页，共三十九页。Definition13.2TheLagrangemethodoffindingasolution(x1*,x2*)totheproblem1212max(,)s.t.(,)0fxxgxxconsistsofderivingthefollowingfirst-orderconditionstofindthecriticalpoint(s)oftheLagrangefunction121212(,,)(,)+(,)xxfxxgxxLwhichare*****1121121*****2122122**12(,)+(,)0(,)+(,)0(,)0fxxgxxxfxxgxxxgxxLLL8第八页，共三十九页。Backtotheexamplef(x1,x2)=x1½x2½andg(x1,x2)=10–2x1–3x2.1122121212(,,)(1023)xxxxxxL11221122112211122212203010230xxxxxxxxLLL2213xx122310xx***551223;;0.204.xx112211221122112223xxxxAgain,thisonlysaysthatthisisalocaloptimum.9第九页，共三十九页。Themethodalsoworksforfindingminima...(Definition13.2)TheLagrangemethodoffindingasolution(x1*,x2*)totheproblem1212min(,)s.t.(,)0fxxgxxconsistsofderivingthefollowingfirst-orderconditionstofindthecriticalpoint(s)oftheLagrangefunction121212(,,)(,)+(,)xxfxxgxxLwhichare*****1121121*****2122122**12(,)+(,)0(,)+(,)0(,)0fxxgxxxfxxgxxxgxxLLL10第十页，共三十九页。Theinterpretationofλ*•λ*istheshadowpriceoftheconstraint.•Ittellsyoubyhowmuchyourobjectivefunctionwillincreaseatthemarginasthetheconstraintisrelaxedby1unit.•Later,wegointomoredetailsastowhythisisthecase.•Intheconsumptionexample,wehadincome10andλ*=0.204.•Thistellsusthatasincomeincreasesby1unit,utilityincreasesby...