经济学Word文档格式.docx

经济学Word文档格式.docx

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LetusreturntothegeneralframeworkwhereafirmisdescribedbyaproductionpossibilitysetYRm.LetyYbeanetputvectorandptheassociatedpricevector.Here,pcontainscomponentpricesforallnetputs,inputs,outputs,andquantitiesthatcanbeeitherinputoroutput.

ProfitFunction.LetYbeaproductionpossibilityset.Thenthecorrespondingprofitfunctionis

Agraphicalillustrationoftheprofitfunctionisasfollows.

y2

（p）=p.y

Isoprofit

p

Y

y1

Ifthefirmhasasingleoutput,theprofitfunctionbecomes:

（p,w）=maxpf（z）–w.z

whereq=f（z）istheproductionfunctionofthefirm.Thenthefirst-orderconditionsforthisspecialcaseare（interiorsolutionsonly）:

Thatis,thevalueofthemarginalproductofeachfactormustbeequaltothefactor'

sprice.（DoyouseethatthisisaspecialcaseofMR=MC?

）

ThediagrambelowillustratestheaboveFOCforsingleinputcase.

output

q=/p+（w/p）z

slope=w/p

q=f（z）

/p

input

∙Thesecondorderconditionisasusual:

theHessianmatrixoffisnegativesemidefiniteattheoptimalpoint.

PropertiesofProfitFunctions:

Theabovedefinedprofitfunction（p）is

1.non-decreasinginoutputprices,non-increasingininputprices;

2.homogeneousofdegree1inp;

3.convexinp;

4.continuousinp.

∙Propertiesoftheprofitfunctionhaveseveraluses.Inparticular,thesepropertiesoffersomeobservableimplicationsofprofit-maximizingbehavior:

oWheneversomepropertyisnottrue,wecanclaimthatthefirmisnotaprofit-maximizer.

Example

∙（Cobb-DouglasProduction,decreasingreturnstoscale）

Letw1andw2bethepricesofthetwoinputsandpthepriceoftheoutput.Then

whichleadsto

（AderivationisplacedintheTechnicalAppendixattheendofthenote.）

Note:

∙Assumethattheproductiontechnologyshowstheconstantreturnstoscale,thentheprofitfunctionisdegenerate,namely,takingavalueofeither0orinfinite.

oWhatwillbetheprofitfunctionforCEStechnologies?

Forexample,Leontiefproductionfunction?

6.1.2NetSupplyFunctionsandHotelling'

sLemma

NetSupplyFunctions-InputDemand&

OutputSupplyFunctions

Thesolutionoftheprofitmaximizationproblem:

isdenotedbyy=y（p）,whichiscommonlycallednetsupplyfunctionofthefirm.Clearly,

（p）=py（p）.

Inparticular,

∙ifyiisaninput,thenthecorrespondingfunctionyi（p）iscalledtheinputdemandfunction,alsoknownasfactordemandfunction.

∙similarly,ifyiisanoutput,thenthecorrespondingfunctionyi（p）iscalledtheoutputsupplyfunction,orsimplysupplyfunction.

Hotelling'

Ifyouknowtheprofitfunction,thenaccordingtothefollowingwell-knownlemma,Hotelling'

sLemma,itiseasytofindthenetsupplyfunction:

justdifferentiatetheprofitfunction.

sLemma.Letyi（p）bethefirm'

snetsupplyfunctionforgoodi（i=1,…,m）.Then,

assumingthatthederivativeexistsandthatpi>

0.

Proof:

Supposey*isaprofitmaximizingnetputvectoratpricesp*.Thendefinethefunction:

g（p）=（p）-py*.

Clearly,theprofit-maximizingproductionplanatpricespwillalwaysbeatleastasprofitableastheproductionplany*.But,theplany*willbeaprofit-maximizingplanatpricesp*,sothefunctiongobtainsaminimumvalueof0atp*,whichisaninteriorsolutionaccordingtothe（positivity）assumptionsontheprices.Wecanthenusethefirst-orderconditionong:

Sincethisistrueforallchoicesofp*,theproofiscompleted.

∙Ofcourse,wecandirectlyapplytheEnvelopeTheorem.

∙Trytogainsomeintuitionbylookingintothesimplecase:

（p,w）=maxz（pf（z）-wz）

Ageometricalintuitionisasfollows:

（bothcurveincreaseatthesameratey*atthetangencypoint）

Profits

（p）

py*=（p）=py*-wz*

（p*）

p*OutputPrices（p）

6.2ComparativeStaticsAnalysis

∙Economistsrefertotheanalysisofhowaneconomicvariablerespondstochangesinitsenvironmentascomparativestatics.

∙Theterm“comparative”referstocomparinga“before”anda“after”solution.

∙Theterm“statics”referstotheideathatthecomparisonismadeafteralladjustmentshavebeen“workedout;

”thatis,wemustcompareoneequilibriumsituationtoanother.

∙Inoptimization,itisknownassensitivityanalysis.

6.2.1ComparativeStaticsofInputDemandFunctions

Case1:

Singleinput:

maxzpf（z）-wz

Assumethatfisdifferentiable.Letz=z（p,w）betheinputdemandfunction.Thenthefirst-orderandthesecond-orderconditionsare:

∙AstheFOCholdsforallp,wedifferentiateitw.r.t.wandget

∙Keyimplicationofthisresult:

∙Inputdemandcurveslopesdownwardinitsownprice

∙Similarly,wecanalsodifferentiatetheFOCw.r.t.p:

∙Keyimplications:

∙Inputdemandcurveshiftsupwardwhenoutputpriceincreases

∙（Subjecttono-shuttingdown）,aonepercentageincrease/decreaseinoutputpricehasthesame（positive/negative）effectonoutputasaonepercentagedecrease/increaseininputprice

Now,iflookattheissuefromtheprofitfunction:

（p,w）=pf（z（p,w））-wz（p,w）,

then

ThisjustverifiestheHotelling'

slemma.

Case2:

Twoinputs:

maxpf（z1,z2）-（w1z1+w2z2）

Denotetheinputdemandfunctionsasz1（w1,w2）andz2（w1,w2）（wedeliberatelydropofftheoutputpriceargumentforeaseofdiscussion.）TheFOCsareasfollow:

Differentiatingw.r.t.w1,wehave

Differentiatingw.r.t.w2,wehave

Therefore,weget

Thematrixonleft-handsideofthelastequationisknownasasubstitutionmatrixasitspecifieshowthefirmsubstitutesoneinputforanotherastheinputpriceschange.

Thesecond-orderconditionfor（strict）profitmaximizationistheHessianmatrixHisasymmetricnegativedefinitematrix.FromLinearAlgebra,weknowthatH-1mustalsobeasymmetric,negativedefinitematrix.Thisresultleadstothefollowingimportantpropertiesoftheinputdemandfunctions:

1.zi/wi<

0fori=1,2,sincethediagonalentriesofanegativedefinitematrixmustbenegative;

2.zi/wj=zj/wi,bythesymmetryofthematrix.

Case3:

Generalcase-multipleinputs

Wecannormalizep=1.TheFOCis

f（z（w））=w

Differentiateitw.r.t.wleadsto

2f（z（w））z（w）=IHz（w）=I

Solvingthisforthesubstitutionmatrix,wehave

z（w）=[H（f（z））-1|z=z（w）

Fromthisidentity,wewillhavesimilarresultsasforthecaseoftwoinputs.

6.2.2ComparativeStaticsUsingtheProfitFunction

ImplicationsofPropertiesoftheProfitFunction

Wenowgetbacktothekeypropertiesoftheprofitfunction:

∙monotonicity,homogeneousofdegree1andconvexity

∙Monotonicityimpliesthatpartialderivativeof（p）withrespecttothepriceofgoodiisnegativeifthegoodiisaninputandpositiveifthegoodiisanoutput.

∙Homogenousofdegree1impliesthat

∙Scalingallpricesbyacommonpositivefactorwillnotchangetheoptimalchoiceofthefirm,i.e.,thenetsupplyfunctionsarehomogenousofdegree0.

6.2.3TheLeChatelierPrinciple

∙Wearemotivatedtolookintotheshort-runresponseofsfirm'

ssupplybehaviorascomparedtothelong-runresponse.

∙Sincethereisnofixedinputinthelongrun,wewouldexpectthatthefirmwillrespondmore（netsupply）toapricechangeinthelongrunthanintheshortrun.（Tryingtoplacetheissueinthecontextoflaborandcapital.）

∙Thisistheso-calledLeChatelierPrinciple.

Letusconsiderthecaseofsingleoutputandtwoinputswithaproductionfunction

q=f（z1,z2）

Assumethatz2=z20isanfixedinput.Theprofitfunctionthenbecomes

ThentheFOCissimply

pf1（z1,z20）=w1

andthe（sufficient）second-orderconditionispf11<

0.Thecorrespondinginputdemandfunction（solvingz1fromtheFOC）is

z1=z1S（w1,p,z20）

（Notethatw2doesnotenterthisdemandcurve）.Nowdifferentiatingthefollowingequality（bypluggingz1SintotheFOC）:

pf1（z1S,z20）=w1

w.r.t.w1,wehave

Recallthatforthelong-runcase（bothinputsarevariableinputs）,wehavethefollowingresult（withoutassumingp=1）:

Fromlinearalgebra,wewillhavethefollowing:

Rememberthatthesecond-orderconditionsare:

f11<

0,f22<

0andf11f22-f122>

0.Therefore,

Sinceboth

arenegative,theaboveinequalitysaysthatthechangeinz1,duetoachangeinitspriceislarger,inabsolutevalue,whenz2isvariable（thelong-run）thanz2isfixed（short-run）.

∙Theaboveconclusionissometimesreferredtoasthe"

secondlawofdemand"

.

6.3DualityinProduction

Inourdiscussionofthecostfunction,wearemoreorlessrelyingonthespecifi