生产者理论3

TheFirm:ComparativeStaticsMicroEconomicsProductionOptimisationTheFirmandtheMarketComparativeStaticsComparativeStaticsOptimisationTheFirmandtheMarketProductionTheFirmOverview......wederivethefirm'sreactionstochangesinitsenvironment.Thesearetheresponsefunctions.Wewillexaminethreetypesofthem,treatingthefirmasa......BlackBoxthefirmMovingonfromtheoptimum...thefirmoutputlevel;inputdemandsmarketpricesHowitworks...UsethefactthatthefirmisanoptimiserBehaviourcanbepredictedbynecessaryandsufficientconditionsforoptimumTheFOCcanbesolvedtoyieldbehaviouralresponsefunctions.TheirpropertiesderivefromthesolutionfunctionThefirmasa“blackbox”TheFirmProductionOutputSupplyOrdinaryInputDemandOptimisationComparativeStaticsTheFirmandtheMarketOutputSupplyConditionalInputDemandOrdinaryInputDemandblackboxproblemsChooseztominimiseSwizimi=1QG(z)...subjecttotheproductionconstraint...z0...andtheobviousnon-negativityconditionsThesolutiontothefirst-stageproblem...C(w,Q):=minwizivectorofinputpricesSpecifiedoutputleveland,presumably...Yieldsminimisedcostasafunctionofexogenousvariables...oneforeachoftheminputsz1*=H1(w,Q)z2*=H2(w,Q)............zm*=Hm(w,Q)...optimalinputdemandsasafunctionofexogenousvariablesdemandforinputi,conditionalonoutputQWeneedtoexaminethefirststageoftheoptimisationprocesszi*=Hi(w,Q)Afunctionofinputprices...andoutputlevelconditionalinputdemandfunction(ourfirstresponsefunction)ResultdependsonshapeofZz1z2z1z2z1z2z1z2z2z1Taketheconventionalcase...Maptheoptimuminto(z1,w1)-spacez2z1w1z1Startwithanarbitraryvalueofw1...Doitagainforalowervalueofw1......andagaintoget...H1(w,Q)theconditionaldemandcurveIntheconventionalcase......theconstraintsetisconvex,withasmoothboundaryWefindthesolutionisacontinuousmap......thatissinglevalued.PointstonoteResultdependsonshapeofZz1z2z1z2z1z2z1z2Z(Q)_2z2z1Whataboutthenon-convexcase...?againmaptheoptimuminto(z1,w1)-spaceZ(Q)_2w1z2z1z1...nowtryaverylowvalueofw1Butwhathappensinbetween?ademandcorrespondenceNonconvexZ:jumpsinz*w1z1nopriceyieldsasolutionheremultipleinputsatthispriceThedemandcorrespondencePointstonoteInthiscase......theconstraintsetisnonconvexWefindthesolutionisadiscontinuousmap…Themapismultivaluedatthediscontinuity.Letussetthisdifficultyaside...Let’stakeitforgrantedthatsingle-valuedinput-demandfunctionsexist.Howaretheyrelatedtothecostfunction?Whataretheirproperties?Howaretheyrelatedtopropertiesofthecostfunction?Doyourememberthese...?AssumetheexistenceofaconditionalinputdemandfunctionC(w,Q)wi=zi*_______Rememberthis...?slopeofthecostfunctionoptimaldemandforinputiCi(w,Q)=zi*...yes,it'sShephard'slemmaAndso...Ci(w,Q)=Hi(w,Q)i___wjNowlet'sdifferentiatethis...conditionalinputdemandfunctionWhichgivesus...Cij(w,Q)=Hji(w,Q)secondderivativeCji(w,Q)=Cij(w,Q)Why...??Andnowforasimpleresult:=2_____wjwi2_____wiwjSecondderivativescommute...TheeffectofthepriceofinputjonconditionaldemandforinputiHij(w,Q)=Hji(w,Q)TheeffectofthepriceofinputionconditionaldemandforinputjTheeconomicmeaning...Nowforanevensimplerresult:Cii(w,Q)=We'veputj=i.SpecialcaseHii(w,Q)thismustbenegative...sothismustbenegativetoo...andso:Becausethecostfunctionisconcaveinprices:Considerthedemandforinput1conditionaldemandcurveziwiHi(w,Q)Hii(w,Q)0TheconditionaldemandcurveslopesdownwardsNonconvexZyieldsdiscontinuousHCross-priceeffectsaresymmetricOwn-pricedemandslopesdownward.Fortheconditionaldemandfunction...TheFirmProductionOutputSupplyOrdinaryInputDemandOptimisationComparativeStaticsTheFirmandtheMarketConditionalInputDemandConditionalInputDemandOrdinaryInputDemandblackboxproblemsmaxPQ-C(w,Q)s.t.Q0YieldsoptimaloutputThesecond-stageproblemQ*=S(w,P)supplyofoutputWeneedtoexaminethesecondstageoftheoptimisationprocess(oursecondresponsefunction)PQForagivenPreadoffoptimalQQCQC/QPPQPQQPQPPPNowletPfall...NotewhathappensbelowAverageCost...PQ__Q=S(w,P)nopricegivessolutionhereSupplycurvez2Q0IRTShereProductionfunctionwithlocalIRTSSupplycurveslopesupwardNonconcaveGyieldsdiscontinuousSIRTSmeansGisnonconcaveandsoSisdiscontinuousForthesupplyfunction...OptimisationTheFirmProductionOutputSupplyOrdinaryInputDemandComparativeStaticsTheFirmandtheMarketConditionalInputDemandConditionalInputDemandOutputSupplyblackboxproblemsdemandforinputi,conditionalonoutputQzi*=Hi(w,Q)Q*=S(w,P)supplyofoutputzi*=Hi(w,Q)Q*=S(w,P)NowputtogetherthetwostagesoftheoptimisationprocessBysubstitution:Hi(w,S(w,P))Di(w,P)demandforinputi(unconditional)=:(ourthirdresponsefunction)Differentiatefortheuncompensateddemand(.)(.)(.)(.)jiQijijSHHD(.)(.)(.)(.)jiQijijSHHDTotal=Substitutioneffect+OutputeffectUnconditionalDemandcanbedeterminedfromthecostfunctionFromShepherdslemma(.)),(ijijHQCw(.)),(ijijHQCwAnd(.)),(iQiQHQCwSincepQCQ),(wCansolveSj(w,P)intermsofC(w,P)Hotelling’sLemmaQPSPp),(),(wwiiizPDwp),(),(wwProveAssumingoneinputzTheProfitFunctionandit’sderivativesare],[]),[(),(pwwzpwzpGwpzzwpGwzzzpGwzwwzw)(GzwpGwzzpGGpzppzp)(ButfromcostminimisationweknowwpGzConsiderthedemandforinput1Changeincostconditionaldemandcurvepricefallz1*z1