经济学.docx

经济学.docx

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经济学

XianJiada

Topic5:

Production&Cost

PrimaryReadings:

黄有光/张定胜,chapter5;DL–Chapter2&JR–Chapter5

Inthislecture,wewillpresentageneralframeworkofproductiontechnology.Wewillfocusonwhatchoicescouldbemade;andtheissueofwhatchoiceswouldbemadewillbedeferredtothenextlecturewhenwelookintothefirm’sbehaviour.

Thefirstpartwilldescribeproductionpossibilitiesinphysicalterms;whilethesecondpartwillrecastthisdescriptionintoacostfunctionframework.

Thetreatmentinthislectureisabitabstractandquitegeneral.Youarerequiredtounderstandtherelevanceofthisabstractframeworkintermsofparticulartechnologicalprocesses.

5.1ProductionPossibilitySets

Therearemanywaystodescribethetechnologyofafirm,suchas,productionfunctions,graphs,orsystemsofinequalities.Butinmathematicalterm,theserepresentationscanallbeexpressedasaset.

∙Thefirmusesandproducesatotalofmcommodities.

∙AparticularproductionplanisyinRm:

∙yi>0impliesthatanetamountyiofi-thcommodityisproduced;

∙yj<0impliesthatanetamount–yjofj-thcommodityisused;

∙yiscalledanetputvector.

∙ProductionpossibilitysetofafirmisasubsetYRm.AfirmmayselectanyvectoryYasitsproductionplan.

PropertiesofProductionPossibilitySet

∙Closed:

IfthelimitofanyconvergingsequenceofvectorsinYisinY.

∙Freedisposal:

IfyYimpliesthaty’Yforally’y.

∙Meaningthat:

commodities（inputsoroutputs）canbethrownaway.

InputRequirementSet:

V（q）={z:

（-z,q）Y}

z2

V（q）

Q（q）

z1

Isoquant:

Q（q）={z:

（-z,q）Y,（-z,q’）Yq’q,q’q}

∙TheisoquantQ（q）isusuallytheboundaryclosesttotheoriginofV（q）.

Proposition:

IfYisconvex,soisV（q）.

∙Wenormallydonotrequirethattheproductionpossibilitysetisconvex.Ifso,itwillruleout"start-upcosts"andothersortsofreturnstoscale.（Doyouseewhy?

）

5.2ProductionFunctions

TransformationFunctionoftheProductionPossibilitySet

Formostproductionpossibilitysets,itispossibletodescribetheminitemofsingleinequalityoftheformT（y）0.Thatis,

Y={y:

T（y）0}

AfunctionTthatdescribesYthiswayiscalledatransformationfunction.

EfficientProduction

∙AproductionpointyYisefficientisthereisnoy’Y,y’y,withy’y.

∙Anefficientproductionimpliesthatitisnotpossibletoeitherunilaterallyincreasetheoutput（s）orunilaterallydecreasetheinput（s）whilestillremaininginY.

ProductionFunctions（JoanRobinson）

∙Forthosetechnologiesthathaveasingleoutputcanbedescribedbyaproductionfunction,whichhasboththetheoreticalandempiricalappeal.

∙Thenetputvectorhastheform:

（-z,q）,whereqistheoutput.

∙IfthetechnologyhasatransformationfunctionT,i.e.,Y={（-z,q）:

T（-z,q）0},thenundercertainregularityconditions,wecansolveT（-z,q）=0forallq,whichleadstoanotherfunction:

q=f（z）.Thisfunctionfistheproductionfunction.

∙Thespecificationofq=f（z）involvesthenotionofefficiencysinceitrepresentsthemaximumoutputlevelthatcanbeachievedwiththeinput,i.e.,

f（z）=max{q’:

T（-z,q’）0}.

∙Withasingleoutput,theinputrequirementsetV（q）isconvexifandonlyifthecorrespondingproductionfunctionf（z）isaquasiconcavefunction.

MRTSandSeparableProductionFunctions

∙Withagivenproductionfunctionq=f（z）,themarginalrateoftechnicalsubstitution（MRTS）betweentwoinputsiandjisdefinedasfollows:

∙Normally,MRTSijdependsonthespecificationofallinputs.WecanuseMRTStodefineseparableproductionfunctions,whichinvolvesregroupingtheinputsintoseveralmutuallyexclusiveandexhaustivesubsets.Fordetails,refertop.221ofJehle&Reny.

ElasticityofSubstitutions

Foraproductionfunctionf（z）,theelasticityofsubstitutionbetweeninputsiandjatthepointzisdefinedas

wherefiandfjarethemarginalproductsofinputsiandj,andd（.）isthetotaldifferentiation.

∙MRTSisalocalmeasureofsubstitutabilitybetweentwoinputsinproducingagivenlevelofoutput.MRTSisnotindependentoftheunitsofmeasurement.

∙Theelasticityofsubstitutionisdefinedasthepercentagechangeintheinputproportion（zj/zi）associatedwitha1percentchangeintheMRTSbetweenthetwoinputs.Theelasticityofsubstitutionisunitless.

∙Ingeneral,theclosertheelasticityofsubstitutionistozero,themoredifficultsubstitutionbetweentheinputs;thelargeritis,theeasiersubstitutionbetweenthem.

CESProductionFunction

Theconstantelasticityofsubstitution（CES）productionfunctionhasthefollowingform:

∙Itcanbeshownthatfortheaboveproductionfunction,

SpecialCasesofCESProductionFunction

∙LinearHomogeneousCobb-DouglasProductionFunction:

∙Correspondtothecasewhen0.（ij1）

∙Thebasicfunctionalformis

∙Aproof（forthecaseofn=2）isintheAppendixofthisnote.

∙LeontiefProductionFunction:

∙Correspondtothecasewhen-.（ij0）

∙Thefunctionalformis

q=f（z）=min{1z1,…,mzm}.

∙TheeasiestwayofprovingthisresultistocheckthecorrespondingMRTSijofCESproductionfunctionas-,whichleadtospecificisoquantsthatareuniquetoLeontieftechnology.

∙AnotherfunctionformfortheLeontiefproductionfunctionisasfollows:

∙ItisclearfromthefunctionspecificationthataLeontieftechnologyusesinputsinfixedproportion,whichimpliesthatthereisasinglefixedformulaforproduction.

ReturnstoScale

∙Aproductionfunctionf（z）hasthepropertyof（globally）

1.Constantreturnstoscaleiff（tz）=tf（z）forallt>1andallz.

2.Increasingreturnstoscaleiff（tz）>tf（z）forallt>1andallz.

3.Decreasingreturnstoscaleiff（tz）1andallz.

∙Themostnaturalcaseofdecreasingreturnsto“scale”isthecasewhereweareunabletoreplicatesomeinputs.Infact,itcanalwaysbeassumedthatdecreasingreturnstoscaleisduetothepresenceofsomefixedinput.

∙Toseethis,letf（z）beaproductionfunctionwithdecreasingreturnstoscale.Supposethatweintroduceanother"newinput"andmeasuredbyz0.Nowdefineanewproductionfunction:

F（z0,z）=z0f（z/z0）.

ItiseasytoseethatFexhibitsconstantreturnstoscale.Inthissense,theoriginaldecreasingreturnstechnologyf（z）canbethoughtasarestrictionoftheconstantreturnstechnologyF（z0,z）thatresultsfromsettingz0=1.

ElasticityofScale

∙Theelasticityofscaleisalocalmeasureofreturnstoscale.It,definedatapoint,specifiestheinstantaneouspercentagechangeinoutputasaresultof1percentincreaseinallinputs:

∙Wesaythatreturnstoscalearelocallyconstant,increasing,ordecreasingwhen（x）isequalto,greaterthan,orlessthanone.

ConstantReturnstoScaleandtheMarginalProductivityTheoryofDistribution

Fromthedefinitionofahomogenousproductionfunction,differentiationwithrespecttok,evaluatedatk=1,wehavesy=Σxifiwherefi≡δf/δxi.

Aproductionfunctionhomogenousofdegrees,themarginalproductofeachfactorishomogenousofdegrees-1.Toshowthis,differentiatewithrespecttoxi.

5.3TheCostFunction

BasicSettings:

∙outputvector:

qR+n;;inputvector:

zR+m;

∙inputfactorpricevector:

wR+m;

∙Recallthatforthegivenoutputvectorq,theinputrequirementsetisdefinedas

V（q）={z:

（-z,q）Y}

∙CostFunction.Thecostfunctionofafirmisthefunction

c（w,q）=minwz

s.t.zV（q）

definedforallw0,q0.

∙Ifthereisasingleoutputandtheproductiontechnologyisfullyrepresentedbytheproductionfunctionq=f（z）,then

c（w,q）=minwz

s.t.f（z）q

Ifz（w,q）solvesthisminimizationproblem,then

c（w,q）=wz（w,q）

∙Thesolutionz（w,q）isreferredtoasthefirm'sconditionalinputdemandfunctions（alsoknownasconditionalfactordemandfunctions）,sinceitisconditionalonthelevelofoutputq,whichatthispointisarbitraryandsomayormaynotbeprofit-maximizing.

∙Theinequalityconstraintcanusuallybereplacedbytheequality.

CalculusAnalysisofCostMinimization

Considerthefollowingcost-minimizationproblem:

c（w,q）=minwz

s.t.f（z）=q

ThenthecorrespondingLagrangefunctionis

L（z,）=wz-（f（z）-q）

Factor2（z2）

C=w1z1+w2z2（Isocost）

f（z1,z2）=q（Isoquant）

Factor1（z1）

whichleadstothegeographicalillustrationofthecostminimization（tangencycondition）indicatedasbelow.

Theabovefigureindicatesthatthereisalsoasecond-orderconditionthatmustbechecked,namely,theisoquantmustlieabovetheisocostline.This,forthecaseoftwoinputs,leadstothattheborderedHessianmatrixoftheLagrangian,

hasanegativedeterminant.

Examples:

∙CostfunctionfortheCobb-Douglastechnology:

q=K1/2L1/2,whereKisthecapital（withaunitpriceofw1-rental）andListhelabor（withaunitpriceofw2-wage）.Thenthecorrespondingcostfunctionis

∙ForthegeneralCobb-Douglasproductionfunction:

Thecorrespondingcostfunctionisgivenby:

∙CostfunctionforCESTechnology:

q=（az1+bz2）1/,byusingthefirst-orderLagrangianconditions,wecanderivethecostfunctiongivenby:

∙CostfunctionforLeontiefTechnology:

Itscostfunctionisgivenby:

c（w,q）=qwa.

GeneralPropertiesofCostFunctions

∙Iftheproductionfunctionfiscontinuousandstrictlyincreasing,thenc（w,q）is

1.Zerowhenq=0.

2.Continuousonitsdomain

3.Foranyallw>0,strictlyincreasingandunboundedaboveiny.

4.Increasinginw.

5.Homogenousof