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ChapterEighteenTechnology1WhereAreWeintheCourse?Wehavestudiedconsumerbehavior;Wenowstudyproducerbehavior;Wewillthencombinethesetwotostudymarketequilibrium.WhereAreWeintheCourse?We2ThreeStepstoStudyofProducerBehavior1st:Weworkonproducer’s“budgetset”,whichisthetechnologyset;2nd:Westudyproducer’s“utilityfunction”,whichishisprofitfunction;3rd:Weexamineproducer’schoice,whichishisoptimalproductionplan.ThreeStepstoStudyofProduc3TechnologiesAtechnologyisaprocessbywhichinputsareconvertedtoanoutput.Usuallyseveraltechnologieswillproducethesameproduct.Howdowecomparetechnologies?TechnologiesAtechnologyisa4ProductionFunctions

Letydenotetheoutputlevel.Aninputbundleisavectoroftheinputlevels;(x1,x2,…,xn).Thetechnology’sproductionfunctionstatesthemaximumamountofoutputpossiblefromaninputbundle.ProductionFunctionsLetyden5ProductionFunctionsy=f(x)istheproductionfunction.x’xInputLevelOutputLevely’y’=f(x’)isthemaximaloutputlevelobtainablefromx’inputunits.Oneinput,oneoutputProductionFunctionsy=f(x)i6TechnologySetsAproductionplanisaninputbundleandanoutputlevel;(x1,…,xn,y).Aproductionplanisfeasibleif

Thecollectionofallfeasibleproductionplansisthetechnologyset.TechnologySetsAproductionpl7TechnologySetsy=f(x)istheproductionfunction.x’xInputLevelOutputLevely’y”y’=f(x’)isthemaximaloutputlevelobtainablefromx’inputunits.Oneinput,oneoutputy”=f(x’)isanoutputlevelthatisfeasiblefromx’inputunits.TechnologySetsy=f(x)isthe8TechnologySetsThetechnologysetisTechnologySetsThetechnology9TechnologySetsx’xInputLevelOutputLevely’Oneinput,oneoutputy”Thetechnology

setTechnically

inefficient

plansTechnically

efficientplansTechnologySetsx’xInputLevelO10TechnologieswithMultipleInputsWhatdoesatechnologylooklikewhenthereismorethanoneinput?Thetwoinputcase:Inputlevelsarex1andx2.Outputlevelisy.SupposetheproductionfunctionisTechnologieswithMultipleInp11TechnologieswithMultipleInputsTheyoutputunitisoquantisthesetofallinputbundlesthatyieldatmostthesameoutputlevely.Thecompletecollectionofisoquantsistheisoquantmap.TechnologieswithMultipleInp12IsoquantswithTwoVariableInputsyo8yo4x1x2yo6yo2IsoquantswithTwoVariableIn13Cobb-DouglasTechnologiesACobb-Douglasproductionfunctionisoftheform

E.g.

Cobb-DouglasTechnologiesACob14x2x1Allisoquantsarehyperbolic,

asymptotingto,butnever

touchinganyaxis.Cobb-DouglasTechnologiesx2x1Allisoquantsarehyperbol15Fixed-ProportionsTechnologiesAfixed-proportionsproductionfunctionisoftheform

E.g.

withFixed-ProportionsTechnologies16Fixed-ProportionsTechnologiesx2x1min{x1,2x2}=144814247min{x1,2x2}=8min{x1,2x2}=4x1=2x2Fixed-ProportionsTechnologies17Perfect-SubstitutesTechnologiesAperfect-substitutesproductionfunctionisoftheform

E.g.

Perfect-SubstitutesTechnologi18Perfect-SubstitutionTechnologies93186248x1x2x1+3x2=18x1+3x2=36x1+3x2=48AllarelinearandparallelPerfect-SubstitutionTechnolog19Marginal(Physical)ProductsThemarginalproductofinputiistherate-of-changeoftheoutputlevelasthelevelofinputichanges,holdingallotherinputlevelsfixed.Thatis,Marginal(Physical)ProductsTh20Marginal(Physical)ProductsE.g.ifthenthemarginalproductofinput1isMarginal(Physical)ProductsE.21Marginal(Physical)ProductsThemarginalproductofinputiisdiminishingifitbecomessmallerasthelevelofinputiincreases.Thatis,ifMarginal(Physical)ProductsTh22Marginal(Physical)ProductsandsoandBothmarginalproductsarediminishing.E.g.ifthenMarginal(Physical)Productsan23Returns-to-ScaleMarginalproductsdescribethechangeinoutputlevelasasingleinputlevelchanges.Returns-to-scaledescribeshowtheoutputlevelchangesasallinputlevelschangeindirectproportion(e.g.allinputlevelsdoubled,orhalved).Returns-to-ScaleMarginalprodu24Returns-to-ScaleIf,foranyinputbundle(x1,…,xn)andanyk,thenthetechnologydescribedbythe

productionfunctionfexhibitsconstant

returns-to-scale.

Returns-to-ScaleIf,foranyin25Returns-to-Scaley=f(x)x’xInputLevelOutputLevely’Oneinput,oneoutput2x’2y’Constant

returns-to-scaleReturns-to-Scaley=f(x)x’xInp26Returns-to-ScaleIf,foranyinputbundle(x1,…,xn)andanyk>1,thenthetechnologyexhibitsdiminishing

returns-to-scale.

Returns-to-ScaleIf,foranyin27Returns-to-Scaley=f(x)x’xInputLevelOutputLevelf(x’)Oneinput,oneoutput2x’f(2x’)2f(x’)Decreasing

returns-to-scaleReturns-to-Scaley=f(x)x’xInp28Returns-to-ScaleIf,foranyinputbundle(x1,…,xn)andanyk>1,thenthetechnologyexhibitsincreasing

returns-to-scale.Returns-to-ScaleIf,foranyin29Returns-to-Scaley=f(x)x’xInputLevelOutputLevelf(x’)Oneinput,oneoutput2x’f(2x’)2f(x’)Increasing

returns-to-scaleReturns-to-Scaley=f(x)x’xInp30LocalScaleofReturns-to-ScaleWecanmodifyourdefinitionabovetodescribesituationsof‘local’returns-to-scale.LocalScaleofReturns-to-Scal31Returns-to-Scaley=f(x)xInputLevelOutputLevelOneinput,oneoutputDecreasing

returns-to-scaleIncreasing

returns-to-scaleReturns-to-Scaley=f(x)xInput32ExamplesofReturns-to-ScaleTheperfect-substitutesproduction

functionisWehave:Theperfect-substitutesproduction

functionexhibitsconstantreturns-to-scale.ExamplesofReturns-to-ScaleTh33ExamplesofReturns-to-ScaleTheperfect-complementsproduction

functionisWehave:Theperfect-complementsproduction

functionexhibitsconstantreturns-to-scale.ExamplesofReturns-to-ScaleTh34ExamplesofReturns-to-ScaleTheCobb-DouglasproductionfunctionisWehaveExamplesofReturns-to-ScaleTh35ExamplesofReturns-to-ScaleTheCobb-DouglasproductionfunctionisTheCobb-Douglastechnology’sreturns-

to-scaleis

constantifa1+…+an=1

increasingifa1+…+an>1

decreasingifa1+…+an<1.ExamplesofReturns-to-ScaleTh36ComparingReturns-to-ScalewithMarginalProductQ:Canatechnologyexhibitincreasingreturns-to-scaleevenifallofitsmarginalproductsarediminishing?A:Yes.E.g.

ComparingReturns-to-Scalewit37ComparingReturns-to-ScalewithMarginalProduct(Cont.)sothistechnologyexhibits

increasingreturns-to-scale.Butdiminishesasx1increasesanddiminishesasx1increases.ComparingReturns-to-Scalewit38Returns-to-ScaleAmarginalproductistherate-of-changeofoutputasoneinputlevelincreases,holdingallotherinputlevelsfixed.Marginalproductdiminishesbecausetheotherinputlevelsarefixed,sotheincreasinginput’sunitshaveeachlessandlessofotherinputswithwhichtowork.Returns-to-ScaleAmarginalpro39TechnicalRate-of-SubstitutionAtwhatratecanafirmsubstituteoneinputforanotherwithoutchangingitsoutputlevel?TechnicalRate-of-Substitution40TechnicalRate-of-Substitutionx2x1yo100Theslopeistherateatwhichinput2mustbegivenupasinput1’slevelisincreasedsoasnottochangetheoutputlevel.Theslopeofanisoquantisitstechnicalrate-of-substitution.TechnicalRate-of-Substitution41TechnicalRate-of-SubstitutionHowisatechnicalrate-of-substitutioncomputed?TheproductionfunctionisAsmallchange(dx1,dx2)intheinputbundlecausesachangetotheoutputlevelofTechnicalRate-of-Substitution42TechnicalRate-of-SubstitutionButdy=0sincethereistobenochange

totheoutputlevel,sothechangesdx1

anddx2totheinputlevelsmustsatisfyTechnicalRate-of-Substitution43TechnicalRate-of-Substitutionistherateatwhichinput2mustbegiven

upasinput1increasessoastokeep

theoutputlevelconstant.Itistheslope

oftheisoquant.TechnicalRate-of-Substitution44Well-BehavedTechnologiesAwell-behavedtechnologyismonotonic,andconvex.Well-BehavedTechnologiesAwel45Well-BehavedTechnologies-MonotonicityMonotonicity:Moreofanyinputgeneratesmoreoutput.yxyxmonotonic

not

monotonicWell-BehavedTechnologies-Mo46Well-BehavedTechnologies-ConvexityConvexity:Iftheinputbundlesx’andx”bothprovideatleastyunitsofoutputthenthemixturetx’+(1-t)x”providesatleastyunitsofoutput,forany0<t<1.Well-BehavedTechnologies-Co47Well-BehavedTechnologies-Convexityx2x1yo100yo120Well-BehavedTechnologies-Co48Well-BehavedTechnologies-Convexityx2x1ConvexityimpliesthattheTRS

increases(becomesless

negative)asx1increases.Well-BehavedTechnologies-Co49Well-BehavedTechnologiesx2x1yo100yo50yo200higheroutputWell-BehavedTechnologiesx2x1y50TheLong-RunandtheShort-RunsThelong-runisthecircumstanceinwhichafirmisunrestrictedinitschoiceofallinputlevels.Therearemanypossibleshort-runs.Ashort-runisacircumstanceinwhichafirmisrestrictedinsomewayinitschoiceofatleastoneinputlevel.TheLong-RunandtheShort-Run51TheLong-RunandtheShort-RunsAusefulwaytothinkofthelong-runisthatthefirmcanchooseasitpleasesinwhichshort-runcircumstancetobe.TheLong-RunandtheShort-Run52TheLong-RunandtheShort-RunsWhatdoshort-runrestrictionsimplyforafirm’stechnology?Supposetheshort-runrestrictionisfixingthelevelofinput2.Input2isthusafixedinputintheshort-run.Input1remainsvariable.TheLong-RunandtheShort-Run53TheLong-RunandtheShort-Runs

isthelong-runproduction

function(bothx1andx2arevariable).Theshort-runproductionfunctionwhen

x2

o1is

Theshort-runproductionfunctionwhen

x2

o10is

TheLong-RunandtheShort-Run54TheLong-RunandtheShort-Runsx1yFourshort-runproductionfunctions.TheLong-RunandtheShort-Run55Summary:KeyConceptsProductionfunctionandisoquantsMarginalproductandreturns-to-scaleTechnicalrateofsubstitutionLong-runandshort-runproductionfunctionsSummary:KeyConceptsProductio56/

ChapterEighteenTechnology57WhereAreWeintheCourse?Wehavestudiedconsumerbehavior;Wenowstudyproducerbehavior;Wewillthencombinethesetwotostudymarketequilibrium.WhereAreWeintheCourse?We58ThreeStepstoStudyofProducerBehavior1st:Weworkonproducer’s“budgetset”,whichisthetechnologyset;2nd:Westudyproducer’s“utilityfunction”,whichishisprofitfunction;3rd:Weexamineproducer’schoice,whichishisoptimalproductionplan.ThreeStepstoStudyofProduc59TechnologiesAtechnologyisaprocessbywhichinputsareconvertedtoanoutput.Usuallyseveraltechnologieswillproducethesameproduct.Howdowecomparetechnologies?TechnologiesAtechnologyisa60ProductionFunctions

Letydenotetheoutputlevel.Aninputbundleisavectoroftheinputlevels;(x1,x2,…,xn).Thetechnology’sproductionfunctionstatesthemaximumamountofoutputpossiblefromaninputbundle.ProductionFunctionsLetyden61ProductionFunctionsy=f(x)istheproductionfunction.x’xInputLevelOutputLevely’y’=f(x’)isthemaximaloutputlevelobtainablefromx’inputunits.Oneinput,oneoutputProductionFunctionsy=f(x)i62TechnologySetsAproductionplanisaninputbundleandanoutputlevel;(x1,…,xn,y).Aproductionplanisfeasibleif

Thecollectionofallfeasibleproductionplansisthetechnologyset.TechnologySetsAproductionpl63TechnologySetsy=f(x)istheproductionfunction.x’xInputLevelOutputLevely’y”y’=f(x’)isthemaximaloutputlevelobtainablefromx’inputunits.Oneinput,oneoutputy”=f(x’)isanoutputlevelthatisfeasiblefromx’inputunits.TechnologySetsy=f(x)isthe64TechnologySetsThetechnologysetisTechnologySetsThetechnology65TechnologySetsx’xInputLevelOutputLevely’Oneinput,oneoutputy”Thetechnology

setTechnically

inefficient

plansTechnically

efficientplansTechnologySetsx’xInputLevelO66TechnologieswithMultipleInputsWhatdoesatechnologylooklikewhenthereismorethanoneinput?Thetwoinputcase:Inputlevelsarex1andx2.Outputlevelisy.SupposetheproductionfunctionisTechnologieswithMultipleInp67TechnologieswithMultipleInputsTheyoutputunitisoquantisthesetofallinputbundlesthatyieldatmostthesameoutputlevely.Thecompletecollectionofisoquantsistheisoquantmap.TechnologieswithMultipleInp68IsoquantswithTwoVariableInputsyo8yo4x1x2yo6yo2IsoquantswithTwoVariableIn69Cobb-DouglasTechnologiesACobb-Douglasproductionfunctionisoftheform

E.g.

Cobb-DouglasTechnologiesACob70x2x1Allisoquantsarehyperbolic,

asymptotingto,butnever

touchinganyaxis.Cobb-DouglasTechnologiesx2x1Allisoquantsarehyperbol71Fixed-ProportionsTechnologiesAfixed-proportionsproductionfunctionisoftheform

E.g.

withFixed-ProportionsTechnologies72Fixed-ProportionsTechnologiesx2x1min{x1,2x2}=144814247min{x1,2x2}=8min{x1,2x2}=4x1=2x2Fixed-ProportionsTechnologies73Perfect-SubstitutesTechnologiesAperfect-substitutesproductionfunctionisoftheform

E.g.

Perfect-SubstitutesTechnologi74Perfect-SubstitutionTechnologies93186248x1x2x1+3x2=18x1+3x2=36x1+3x2=48AllarelinearandparallelPerfect-SubstitutionTechnolog75Marginal(Physical)ProductsThemarginalproductofinputiistherate-of-changeoftheoutputlevelasthelevelofinputichanges,holdingallotherinputlevelsfixed.Thatis,Marginal(Physical)ProductsTh76Marginal(Physical)ProductsE.g.ifthenthemarginalproductofinput1isMarginal(Physical)ProductsE.77Marginal(Physical)ProductsThemarginalproductofinputiisdiminishingifitbecomessmallerasthelevelofinputiincreases.Thatis,ifMarginal(Physical)ProductsTh78Marginal(Physical)ProductsandsoandBothmarginalproductsarediminishing.E.g.ifthenMarginal(Physical)Productsan79Returns-to-ScaleMarginalproductsdescribethechangeinoutputlevelasasingleinputlevelchanges.Returns-to-scaledescribeshowtheoutputlevelchangesasallinputlevelschangeindirectproportion(e.g.allinputlevelsdoubled,orhalved).Returns-to-ScaleMarginalprodu80Returns-to-ScaleIf,foranyinputbundle(x1,…,xn)andanyk,thenthetechnologydescribedbythe

productionfunctionfexhibitsconstant

returns-to-scale.

Returns-to-ScaleIf,foranyin81Returns-to-Scaley=f(x)x’xInputLevelOutputLevely’Oneinput,oneoutput2x’2y’Constant

returns-to-scaleReturns-to-Scaley=f(x)x’xInp82Returns-to-ScaleIf,foranyinputbundle(x1,…,xn)andanyk>1,thenthetechnologyexhibitsdiminishing

returns-to-scale.

Returns-to-ScaleIf,foranyin83Returns-to-Scaley=f(x)x’xInputLevelOutputLevelf(x’)Oneinput,oneoutput2x’f(2x’)2f(x’)Decreasing

returns-to-scaleReturns-to-Scaley=f(x)x’xInp84Returns-to-ScaleIf,foranyinputbundle(x1,…,xn)andanyk>1,thenthetechnologyexhibitsincreasing

returns-to-scale.Returns-to-ScaleIf,foranyin85Returns-to-Scaley=f(x)x’xInputLevelOutputLevelf(x’)Oneinput,oneoutput2x’f(2x’)2f(x’)Increasing

returns-to-scaleReturns-to-Scaley=f(x)x’xInp86LocalScaleofReturns-to-ScaleWecanmodifyourdefinitionabovetodescribesituationsof‘local’returns-to-scale.LocalScaleofReturns-to-Scal87Returns-to-Scaley=f(x)xInputLevelOutputLevelOneinput,oneoutputDecreasing

returns-to-scaleIncreasing

returns-to-scaleReturns-to-Scaley=f(x)xInput88ExamplesofReturns-to-ScaleTheperfect-substitutesproduction

functionisWehave:Theperfect-substitutesproduction

functionexhibitsconstantreturns-to-scale.ExamplesofReturns-to-ScaleTh89ExamplesofReturns-to-ScaleTheperfect-complementsproduction

functionisWehave:Theperfect-complementsproduction

functionexhibitsconstantreturns-to-scale.ExamplesofReturns-to-ScaleTh90ExamplesofReturns-to-ScaleTheCobb-DouglasproductionfunctionisWehaveExamplesofReturns-to-ScaleTh91ExamplesofReturns-to-ScaleTheCobb-DouglasproductionfunctionisTheCobb-Douglastechnology’sreturns-

to-scaleis

constantifa1+…+an=1

increasingifa1+…+an>1

decreasingifa1+…+an<1.ExamplesofReturns-to-ScaleTh92ComparingReturns-to-ScalewithMarginalProductQ:Canatechnologyexhibitincreasingreturns-to-scaleevenifallofitsmarginalproductsarediminishing?A:Yes.E.g.

ComparingReturns-to-Scalewit93ComparingReturns-to-ScalewithMarginalProduct(Cont.)sothistechnologyexhibits

increasingreturns-to-scale.Butdiminishesasx1increasesanddiminishesasx1increases.ComparingReturns-to-Scalewit94Returns-to-ScaleAmarginalproductistherate-of-changeofoutputasoneinputlevelincreases,holdingallotherinputlevelsfixed.Marginalproductdiminishesbecausetheotherinputlevelsarefixed,sotheincreasinginput’sunitshaveeachlessandlessofotherinputswithwhichtowork.Returns-to-ScaleAmarginalpro95TechnicalRate-of-SubstitutionAtwhatratecanafirmsubstituteoneinputforanotherwithoutchangingitsoutputlevel?TechnicalRate-of-Substitution96TechnicalRate-of-Substitutionx2x1yo100Theslopeistherateatwhichinput2mustbegivenupasinput1’slevelisincreasedsoasnottochangetheoutputlevel.Theslopeofanisoquantisitstechnicalrate-of-substitution.TechnicalRate-of-Substitution97TechnicalRate-of-SubstitutionHowisatechnicalrate-of-substitutioncomputed?TheproductionfunctionisAsmallchange(dx1,dx2)intheinputbundlecausesachangetotheoutputlevelofTechnicalRate-of-Substitution98TechnicalRate-of-SubstitutionButdy=0sincethereistobenochange

totheoutputlevel,sothechangesdx1

anddx2totheinputlevelsmustsatisfyTechnicalRate-of-Substitution99TechnicalRate-of-Substitutionistherateatwhichinput2mustbegiven

upasinput1increasessoastokeep

theoutputlevelconstant.Itistheslope

oftheisoquant.TechnicalRate-of-Substitution100Well-BehavedTechnologiesAwell-behavedtechnologyismonotonic,andconvex.Well-BehavedTechnologiesAwel101Well-BehavedTechnologies-Monotonicity