生产者理论1

李永波TheFirm:ProductionAdvancedMicroEconomicsInthislecturewesetoutsomeoftheelementsneededforananalysisofthefirmTechnicalefficiencyReturnstoscaleConvexitySubstitutabilityMarginalproducts...and(fornexttime)assumingacompetitiveenvironment.Wedoitwithinthecontextofasingle-outputfirm...Butfirstweneedthebuildingblocksofamodel...Thebasicsofproduction...ziamountofinputiQamountofoutputThebasicsofproduction...inputvectorz:=(z1,z2,...,zm)wipriceofinputiw:=(w1,w2,...,wm)inputpricevectorPpriceofoutputNotation:PricesQG(z1,z2,....,zm)Thesingle-outputproductionfunctionWrittenmorecompactlyQG(z)technologyoutputYes,butwhynot=signhere?inputsThemeaningofthefunctionthemaximumamountofoutputthatcanbeproducedfromthislistofinputsUsethisrelationtodistinguishtwocases...ComponentsoftherelationshipQG(z)Q=G(z)21ThecasewhereproductionistechnicallyefficientThecasewhereproductionis(technically)inefficientTechnicalefficiencyG(z)zi____iG(z):=wheredifferentiableSomehandynotation...z2Q0outputinput2QG(z,z)12Feasible,butinefficientpointsQG(z,z)12infeasiblepointsQ=G(z,z)12technicallyefficientpointsNowlet’sslicethissetuptogetsomeusefultoolsThefullproductionfunctionPickaparticularoutputlevelQZ(Q):={z:G(z)Q}FindafeasibleinputvectorzG(z)QRepeattofindallsuchinputvectorsInputrequirementsetsZ(Q):={z:G(z)Q}thesetofinputvectorsthatmeetthetechnicalfeasibilityconditionforoutputlevelQ......butwhatwouldZlooklike??Thisdependsontheassumptionswemakeaboutproduction...First,a“standard”case...?Whatisthisthing...?z1z2G(z,z)=`Q12infeasiblepoints_Z(Q)technicallyefficientpointsfeasible,butinefficientpointsG(z,z)`Q12G(z,z)`Q12Theinputrequirementset:z1z2zzPicktwoboundarypointsDrawthelinebetweenthemIntermediatepointsmustlieintheinteriorofZmeaning:acombinationoftwotechniquesmayproducemoreoutputG(z)=`QG(z)=`QG(z)`QButwhatifwechangedsomeoftheassumptionshere?_Z(Q)Case1:Zissmoothandstrictlyconvexz1z2_Z(Q)zzPickanytwopointsinZDrawthelinebetweenthemIntermediatepointsmustlieinZmeaning:acombinationoffeasibletechniquesisalsofeasibleCase2:Zconvexbutnotstrictlyconvexz1z2_Z(Q)Thisregioncausesaproblemmeaning:inthisregionthereisanindivisibilityCase3:ZissmoothbutnotconvexAnExample...LondonNewYork3131z1z2slopeundefinedatthispointtheonlyefficientpointforQ=`Q_Z(Q)Case4:Zisconvexbutnotsmoothz1z2z1z2z1z2z1z2Standardcase,butstrongassumptionsaboutdivisibilityandsmoothnessalmostconventionalcase:mixturesmaybejustasgoodassingletechniquesPresentsproblems:thedentrepresentsanindivisibilityunusualcase:onlyoneefficientpointandnotsmooth.Butnotperverse.Summary:4possibilitiesfortheinputrequirementsetZ{z:G(z)=Q}Thisistheisoquant.Let'slookatitsshape...PickanoutputlevelQFindtheinputrequirementsetZ(Q)DrawtheboundaryofthissetIsoquants{z:G(z)=Q}Az1inputsrequiredtoproduceatAz2IsoquantatQTheinputratiodescribestheparticulartechniqueTheisoquantistheboundaryofZTheisoquantthroughA(Q)MarginalrateoftechnicalsubstitutionTheslopeoftheisoquantisthemarginalrateofsubstitutionatA.Itmeasurestheimplicit“price”ofinput1intermsofinput2.Thehigheristhis“price”,thesmalleristherelativeusageofinput1z1z2AA'G1(z)/G2(z)TheresponsivenessoftheinputratiototheMRTSisgivenbytheelasticityofsubstitution-log(z1/z2)log(G1/G2)Canbeseenastheisoquant’s“curvature”Nowforaspecialcase...Aconstantelasticityofsubstitution:Increasetheelasticityofsubstitution...z1z2Nowlookatthestructureofthecontourmap...z1z2Homotheticcontoursz1z2Qtz2tztQr1G(tz)=tG(z)rContoursofahomogeneousfunctionTheisoquantsformacontourmap.Ifwelookedatthe“parent”diagram,whatwouldwesee?Let'sdothisfor2inputs,oneoutput.Let'srebuildfromtheisoquantsz2Qexpansionray0G(tz)=tG(z)constantreturnstoscaleProportionalincreaseinallinputsz2Q0t1G(tz)tG(z)IncreasingreturnstoscaleProportionalincreaseinallinputsz2Q0t1G(tz)tG(z)DecreasingreturnstoscaleProportionalincreaseinallinputsz2QisoquantQ=`Q0Takeahorizontalsection...togettheisoquantagainz2Q0…thisgivesusournextconceptNowtakeaverticalsection...PickatechnicallyefficientinputvectorKeepallbutoneinputconstantQ=G(z)Measurechangeinoutputw.r.t.thisinputG(z)zi____MPi=Gi(z)=Marginalproductsz1QG(z)z1QG(z)z1QG(z)possiblerelationshipsbetweenoutputandoneinputz1QG(z)Let'staketheconventionalcase…feasiblesetG(z)Qz1input1isessentialSetoftechnicallyefficientpointsTaketherelationshipbetweenoutputandinput1...Ifz1=0thenQ=0z1QG(z)G1fallswithz1ifGisconcaveMarginalproductslope=G1(z)TechnicalefficiencyReturnstoscaleConvexityMRTSMarginalproductKeyconcepts