经济学文档格式.docx
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经济学文档格式.docx
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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>
3.Decreasingreturnstoscaleiff(tz)<
∙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:
∙CostFunction.Thecostfunctionofafirmisthefunction
c(w,q)=minwz
s.t.zV(q)
definedforallw0,q0.
∙Ifthereisasingleoutputandtheproductiontechnologyisfullyrepresentedbytheproductionfunctionq=f(z),then
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:
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
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