闡述生產(chǎn)者理論知識

TheFirm:ComparativeStaticsMicroEconomicsProductionOptimisationTheFirmandtheMarketComparativeStaticsComparativeStaticsOptimisationTheFirmandtheMarketProductionTheFirmOverview......wederivethefirm'sreactionstochangesinitsenvironment.Thesearetheresponsefunctions.Wewillexaminethreetypesofthem,treatingthefirmasa......BlackBox

thefirmMovingonfromtheoptimum...thefirmoutputlevel;inputdemandsmarketpricesHowitworks...

UsethefactthatthefirmisanoptimiserBehaviourcanbepredictedbynecessaryandsufficientconditionsforoptimumTheFOCcanbesolvedtoyieldbehaviouralresponsefunctions.TheirpropertiesderivefromthesolutionfunctionThefirmasa“blackbox”TheFirmProductionOutputSupplyOrdinaryInputDemandOptimisationComparativeStaticsTheFirmandtheMarketOutputSupplyConditionalInputDemandOrdinaryInputDemandblackboxproblemsChooseztominimiseS

wizi

m

i=1Q

￡

G(z)...subjecttotheproductionconstraint...z

3

0...andtheobviousnon-negativityconditionsThesolutiontothefirst-stageproblem...C(w,Q):=min?wizi

vectorofinputpricesSpecifiedoutputleveland,presumably...Yieldsminimisedcostasafunctionofexogenousvariables...oneforeachoftheminputsz1*=H1(w,Q)z2*=H2(w,Q)............zm*=Hm(w,Q)üyt...optimalinputdemandsasafunctionofexogenousvariablesdemandforinputi,conditionalonoutputQ

Weneedtoexaminethefirststageoftheoptimisationprocesszi*=Hi(w,Q)Afunctionofinputprices...andoutputlevelconditionalinputdemandfunction(ourfirstresponsefunction)ResultdependsonshapeofZz1z2z1z2z1z2z1z2z2z1Taketheconventionalcase...Maptheoptimuminto(z1,w1)-spacez2z1w1z1Startwithanarbitraryvalueofw1...Doitagainforalowervalueofw1......andagaintoget...H1(w,Q)theconditionaldemandcurveInthe"conventional"case......theconstraintsetisconvex,withasmoothboundaryWefindthesolutionisacontinuousmap......thatissinglevalued.PointstonoteResultdependsonshapeofZz1z2z1z2z1z2z1z2Z(Q)_2z2z1Whataboutthenon-convexcase...?againmaptheoptimuminto(z1,w1)-spaceZ(Q)_

2w1z2z1z1...nowtryaverylowvalueof

w1Butwhathappensinbetween?ademandcorrespondenceNonconvexZ:

jumpsin

z*w1z1nopriceyieldsasolutionheremultipleinputsatthispriceThedemandcorrespondencePointstonoteInthiscase......theconstraintsetisnonconvexWefindthesolutionisadiscontinuousmap…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

?___?wj

Nowlet'sdifferentiatethis...conditionalinputdemandfunctionWhichgivesus...Cij(w,Q)=Hji(w,Q)

secondderivativeCji(w,Q)=Cij(w,Q)

Why...??Andnowforasimpleresult:=

?2_____?wj?wi

?2_____?wi?wj

Secondderivativescommute...Theeffectofthepriceofinputj

onconditionaldemandforinputiHij(w,Q)=Hji(w,Q)

Theeffectofthepriceofinputi

onconditionaldemandforinputjTheeconomicmeaning...Nowforanevensimplerresult:Cii(w,Q)=We'veputj=i.SpecialcaseHii(w,

Q)

thismustbenegative...sothismustbenegativetoo...andso:Becausethecostfunctionisconcaveinprices:Considerthedemandforinput1conditionaldemandcurve

ziwiHi(w,Q)Hii(w,

Q)<0TheconditionaldemandcurveslopesdownwardsNonconvexZ

yieldsdiscontinuous

HCross-priceeffectsaresymmetricOwn-pricedemandslopesdownward.Fortheconditionaldemandfunction...TheFirmProductionOutputSupplyOrdinaryInputDemandOptimisationComparativeStaticsTheFirmandtheMarketConditionalInputDemandConditionalInputDemandOrdinaryInputDemandblackboxproblemsmaxPQ-C(w,Q)s.t.Q30YieldsoptimaloutputThesecond-stageproblemQ*=S(w,P)supplyofoutputWeneedtoexaminethesecondstageoftheoptimisationprocess(oursecondresponsefunction)PQForagivenPreadoffoptimalQQCQC/QPPQPQQPQPPPNowletPfall...NotewhathappensbelowAverageCost...PQ_

_Q=S(w,P)nopricegivessolutionhereSupplycurvez2Q0IRTShereProductionfunctionwithlocalIRTSz1SupplycurveslopesupwardNonconcaveGyieldsdiscontinuous

SIRTSmeansGisnonconcaveandso

S

isdiscontinuousForthesupplyfunction...OptimisationTheFirmProductionOutputSupplyOrdinaryInputDemandComparativeStaticsTheFirmandtheMarketConditionalInputDemandConditionalInputDemandOutputSupplyblackboxproblemsdemandforinputi,conditionalonoutputQ

zi*=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)DifferentiatefortheuncompensateddemandTotal=Substitutioneffect+OutputeffectUnconditionalDemandcanbedeterminedfromthecostfunctionFromShepherdslemmaAndSinceCansolveSj(w,P)intermsofC(w,P)Hotelling’sLemmaProveAssumingoneinputzTheProfitFunctionandit’sderivativesareButfromcostminimisationweknowConsiderthedemandforinput1Changeincostconditionaldemandcurvepricefall

z1*z1w1H1(w,Q)initialpricelevel...andallowthepriceofinput1tofallinput-pricefall:substitutioneffectNotionalincreaseinfactorinputifoutputtargetisheldconstantpricefallz1*z1w1Originalconditionaldemandcurveordinarydemandcurvez1**Conditionaldemandcurveatnewoutputlevelinput-pricefall:totaleffectNonconvexZmayyieldadiscontinuousD

Cross-priceeffectsaresymmetricOwn-pricedemandslopesdownward

Samebasicproperties

asforHfunctionTheordinarydemandfunction...ProductionOptimisationTheFirmComparativeStaticsTheFirmandtheMarketComparativeStaticsTheFirmandtheMarketProductionOverview...TheFirmProductionTheShortRunOptimisationComparativeStaticsTheFirmandtheMarketASpecialCaseThisisnotamomentintimebut…...isdefinedbyadditionalconstraintswithinyourmodelTheshortrun...Short-runproblembuildsonthestandardapproachChooseQ

andztomaximiseP:=P

Q-

S

wizi

m

i=1Q

￡

G(z)...subjecttotheproductionconstraint...Q

3

0z

30...andsomeprettyobviousconditions:zm

=`zmWithsidecondition(s)~{zm

=`zm}C(w,Q,`zm):=minSwizi

ThesolutionnowinvolvestheshortruncostfunctionSideconstraintinshortrunComparethistothesolutionwithoutthesideconstraint....C(w,Q):=minSwizi

￡

C(w,Q,`zm

)~Bydefinition...Sotherefore...~C(w