Advanced Price Theory高级价格理论讲义

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1、Lecture0MathematicalEssentialsLecture0.MathematicalEssentialsThroughoutthiscoursewewillbedealingwithmodelsinvolvingconstrainedoptimization.Inthissectionweoutlinearatherstraightforwardmethodforfindingsolutionstotheseproblemsandanalyzingthecomparativestaticsofthesesolutions.Theformalsolutiontothes

2、eproblemsismosteasilyunderstoodintermsofthemethodoflagrangemultipliers.Ingeneralwewillconsideraproblemoftheform(0.1)MaxF(X1...Xk,B)s.tG(X1...Xk,B)£MxwhereXrepresentstheindividual'sdecisionvariables(howmuchofeachgoodtoconsumeforthegenericconsumerproblemorhowmuchofeachtypeofproducttoproduce)andBre

3、presenttheparametersinfluencingtheindividual'sdecisionsuchasthepricesofindividualproductsorthemarketratesofreturnonalternativeinvestments.ThefunctionFrepresentstheindividual'sobjectivewhilehischoiceoftheX'sisrestrictedtocombinationsthatsatisfyG(X,B)£M.Mgivesthelevelofthisindividual'sconstraint.T

4、hesolutiontothisproblemcanbefoundbysettinguptheLagrangianfunction(0.2)L=F(X1...Xk,B)+l(M-G(X1...Xk,B))ThesolutionfortheoptimallevelsofXwillbeacriticalpointofL(i.e.apointwherethederivativesoftheLagrangianwithrespecttoXandlaresimultaneouslyzero.Hencetheseconditionsare(0.3)dL/dX1=dF/dX1-ldG/dX1=0=>

5、dF/dX1=ldG/dX1dL/dX2=dF/dX2-ldG/dX2=0=>dF/dX2=ldG/dX2...dL/dXk=dF/dXk-ldG/dXk=0=>dF/dXk=ldG/dXkdL/dl=M-G(X,B)=0=>M=G(X,B)ThefirstkequationstellusthattheoptimalsolutionrequirestheimpactofeachoftheXvariablesontheobjectivefunction,dF/dXj,tobeproportionatetoitseffectontheconstraintfunction,dG/dXj(wi

6、thlbeingthefactorofproportionality).Thenecessityofthisconditionfollowsfromthefollowingargument:ifweincreaseXibysomesmallamountdXithenthevalueofourconstraintfunctionwouldfallbedG/dXi*dXiwhichwouldthenallowustochangeXjbyanamount-dG/dXi/dG/dXjdXi(tokeepthevalueofG(X,B)=M).Theneteffectofthisontheobj

7、ectivefunctionwouldbe1Lecture0MathematicalEssentials(0.4)dF=dF/dXidXi-dF/dXjdG/dXi/dG/dXjdXi.AttheoptimumdFmustbezerowhichimpliesthat(0.5)dF/dXi/dFdXj=dG/dXi/dG/dXjorthattheeffectsofthexvariablesontheobjectivemustbeproportio