5. Envelope theorem, first- and second-order envelope properties, interpretation of Lagrange multipl

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1、Envelopetheorem,…rst-andsecond-orderproperties,interpretationofLagrangemultiplierInstructor:MinWang1.Considerunconstrainedorconstrainedoptimizationproblemsinwhichobjectiveand/orconstrainedfunc-tionsdependsonparticularparametera.maxF(x;a)w:r:t:xs:t:g(x;a)=b:whereF:

2、RnR!R.Viewoptimalvaluesofchoicevariablesasfunctionsofa:x(a)fori=1;2;:::;n;andde…nevaluefunctioniastheoptimalvalueoftheobjectivefunction:F(a)=F(x(a);a).Assumex(a)andF(a)areicontinuousanddi¤erentiable.Whatisthevalueofdx(a)i(comparativestatics)daanddF(a)(enve

3、lopetheorem)?da2.ComparativestaticsforthecaseofunconstrainedoptimizationproblemConsidertheproblemmaxF(x;a)w:r:t:xwhereF:RnR!Risdi¤erentiableandaisexogenouslygiven.Letxdenotealocalsolution.Thendxi(a)measureshowxvarieswithafori=1;2;:::;n.daiRecall…rstordernec

4、essaryconditions@F(x1;:::;xn;a)=0;@x1@F(x1;:::;xn;a)=0;@x2:::@F(x1;:::;xn;a)=0:@xnnequationsinnendogenousvariables,x;:::;x,and1exogenousvariablea.1nClaim:ifdetJacobian(x)6=0,thesystemimplicitlyde…nesdi¤erentiablefunctionsx:a!x(a)iifori=1;2;:::;n:2–TheJ

5、acobianofthe…rst-orderconditionsistheHessianofF(
),i.e.,@F(x;a).@x22Assumesecond-ordersu¢cientconditionsaresatis…ed,i.e.,@F(x;a)isnegativede…niteand@x2n@2F(1)det@x2(x;a)>0.Applyingimplicitfunctiontheorem,wehave01021dx1(a)@F(x;a)da@x1@a2Bdx(a)CB@2FC@FB2C

6、B(x;a)C(x;a)BdaC=B@x2@aC@x2@:::A@:::Adx(a)2n@F(x;a)da@xn@awherexistheregularsolution,asolutionatwhichsoscaresatis…ed.13.EnvelopetheoremforthecaseofunconstrainedoptimizationproblemConsidertheproblemmaxF(x;a)w:r:t:xwhereF:RnR!Risdi¤erentiableandaisexogenously

7、given.Letx(a)bealocalsolutionandassumex:R!Rdi¤erentiablefori=1;2;:::;n.De…neF(a)=iF(x(a);a).ThendF(a)@F=(x(a);a)da@aTheindirecte¤ectsarewashedoutatlocalsolution.4.ExampleProductionfunction:Q=f(L;K)=LKwhere;>0and+<1.Firmisapricetakerandthusthepricesofou

8、tputandinputsareexogenouslygivenfor…rms.Theoptimizationproblemfora…rmismax(L;K;p;w;r)=pLKwLrK:w:r:t:L;KFirst,weneedtowritedownthe…rst-ordernecessary