學長只能幫你們到這里了ie hws

QUESTION1.Therearetwogoods,goodAandgoodB,thatareproduced+bothcapital(K)andlabor TheproductionfunctionsarefA:+

→R++fB:R2→R+.WritedownthecorrespondingproductionsetY+Answer.TheproductionsetY={(?K,?L,qA,qB)where:qA≤fA(KA,LA),qB≤fB(KB,LB),KA+KB≤K,LA+LB≤QUESTION2.ConsidertheCES(constanasticityofsubstitution)production ρtionf(x1,x2)=(x1+x2)ρ,whereρ≤1isaDoesthisproductionfunctionexhibitanytypeofreturnstoscale?Ifyes,whatComputetheTRS(technicalrateofsubstitution)andtheelasticityofsubstitutionforthisproductionfunction.ShowForρ=1weobtainthelinearproductionForρ=0weobtaintheCobb-DouglasproductionForρ=?∞weobtaintheLeontiefproductionDerivetheconditionalfactordemandx?(w,q)andcostfunctionc(w,Derivethe(unconditional)factordemandx?(p,w),outputsupplyq?(p,w)andprofitfunctionπ(p,w).[Becarefulhere.]Answer.Part(1).Itiseasytoverifythattheproductionfunctionischaracterizedbyconstantreturnstoscale(CTRS):foranyα>0wehave:f(αx1,αx2)=αf(x1,x2).Part(2).Thetechnicalrateofsubstitutionbetweeninputs1and2isdefined?fρ =? . Computingthepartialderivativesoffwe =(x?fxρ)1?1(x)ρ?1and

=(xρ+xρ)1?1(x)ρ?1,

2

2

(x1—x—2dTTheelasticityofsubstitutionbetweeninputs1and2isdefinedas:σ12=x1/x2dTtheexpressionforTRS12wefoundabove,aftersomealgebra,weobtain:σ12=?1Part(3a).Ifρ=1,weobtainf(x1,x2)=x1+x2,ie.theproductionfunctionisPart(3b).Ifρ=0,wecancomputetheproductionfunctionas 1ρ ρ

log(x1+x2

ρ(x1

lim(x1+x2) lim

=limex1

=e1(logx1+logx2)= 1x) 1)2(2 [AnotherwaytoanswerthisquestionistolookattheTRSforρ=0:TRS12ρ=0=x2whichcorrespondstoaCobb-Douglasproductionfunction. ρPart(3c).Ifρ=?∞,wewanttoshowthatlimρ→?∞(x1+x2)ρ=min{x1,Supposethatx1≤x2andletρ<0. 1≥0andx2≥0wehave:xρ+xρ≥ ρ

andsinceρ<0weobtain(x1+x2)ρ≤x1.Ontheotherhandx1≤x2andρ<0

ρ x1+x2≤x1+x1=2x1,or(x1+x2)ρ≥2ρx1.Thereforewe ρ2ρx1≤(x1+x2)ρ≤→ → ,weobtainlimρ→?∞(x1+x2)ρ=x1.Theothercaseof x1is[Therearemanyotherwaystoanswerthisquestion.Forexample,youcouldlook0,ifx2<theTRSwhenρ=?∞:TRS12|ρ=?∞=?∞,ifx2>x10,ifx2<totheLeontieftypeproductionPart(4).Thecostminimizationproblemc(w,q)=minw1x1+ ρs.t.(x1+x2)ρ=Thecostproblemcanbere-writtenc(w,q)=minw1x1+s.t.xρ+xρ= UsingthestandardLagrangianmethod,youshouldobtaintheconditionalfactordemand ρ 1—x?(w,q)=(wρ?1+wρ?1

ρwρ?1 ρ 1—x?(w,q)=(wρ?1+wρ?1

ρwρ?12andthecost

ρ

c(w,q)=(wρ?1+wρ?1)ρ NotethatsincetheproductionfunctionhasCRTS,thecostfunctionislinearinq.(seeLemma4inthe2Part(5).SincetheproductionfunctionfeaturesCRTS,theprofitsareeither0+∞.Toseethat,writethe izationproblem ρ

π(p,w)=maxpq?c(w,q)=(p?(wρ?1+wρ?1

ρThesolutiontothisproblemq(p,q(p,w)??

0,ifp≤(0,ifp≤(wρ?1+wρ?1 ρ+∞,ifp>(wρ?1+wρ?1)sotheprofitfunctionπ(π(p,w)

0,ifp≤(0,ifp≤(wρ?1+wρ?1 ρ12+∞,ifp>(wρ?1+wρ?1)12QUESTION3.Supposethatafirmhastwonts,withcostfunctionsc1()andc2().Computethecostofproducingsometyqif:c1(q1)=3(q1)2andc2(q2)=c1(q1)=4√q1andc2(q2)=Answer.Theproblemofthefirmistominimizethetotalcost:c1(q1)+c2(q2)subjecttoconstraintq1+q2=q.Part1.Forthisfunctionalformofthetwocostfunctions,theproblemc(q)=

3(q1)2+s.t.q1+q2=4Solvingthisproblem,eitherusingtheLagrangian,orsubstitutingforq2fromthecon-straintintotheobjectivefunction,youshouldobtainc(q)=3q2.4Part2.Forthesecondcase,thefirmc(q)=

4√q1+s.t.q1+q2=Youhavetobecarefulherebecausetheobjectivefunction4√q1+2√q2isconcave.Takingfirstorderconditionswillyieldaum,notaminimum.Inthiscaseitisenoughtocheckthetwocornercases:(q1=0,q2=q)and(q1=q,q2=0),andchoosetheonegiveslowestcost.Asitturnsout,thesolutionis(q1=0,q2=q),whichimpliesthatthecostfunctionisc(q)=2√q.QUESTION4.Suppo ?isasetofinputsthat izesprofitswheninputpricesarew.Letq?=f(x?)betheoutputsupply.Showthatx?minimizesthecostof3outputq?.Inotherwords,anycombinationofinputsthat productioncost.Answer.Letx?andq?solvethe ization max w s.t.q≤fbutsupposethatx?doesnotsolvethecostminimization·minw·xs.t.f(x)= whenq=q?.Thatis,thereexistsanothercombinationofinputsx0suchthatf(x0)=q?andwx0<wx?.Computingtheprofitsassociatedwithx0,weobtainpf(x0)wx0=pq?wx0>pq?wx?.Ofcourse,thiscontradictsthefactthe(x?, QUESTION5.Aprice-takingfirmproducesoutputqfrominputsx1andx2accordingtoadifferentiableconcaveproductionfunctionf(x1,x2).Thepriceofitsoutputisp>0,andthepricesofitsinputsare(w1,w2)0.However,therearetwounusualthingsaboutthisfirm.First,ratherthan izingprofit,thefirm managerwantsherfirmtohavebiggerdollarsalesthananyother).Second,thefirmiscashconstrained.Inparticular,ithasonlyCdollarsonhandbeforeproductionand,asaresult,itstotalexpenditureoninputscannotexceedC.Supposeoneofyoureconometricianfriendslsyouthatshehasusedrepeatedobser-vationsofthefirm’srevenuesundervariousoutputprices,inputprices,andlevelsofthefinancialconstraintandhasdeterminedthatthefirm’srevenuelevelRcanbeexpressedasthefollowingfunctionofthevariables(p,w1,w2,C):R(p,w1,w2,C)=p[logC?αlogw1?(1?α)logw2]whereαisascalarwhosevalueshelsyou.Whatisthefirm’suseofinputsx1andx2whenpricesare(p,w1,w2)andithasdollarsofcashonWhatistheminimumcostandthecorrespondinginputsdemandifthefirmwantstoproducequnitsofoutput?Whatisfirm’sproductionAnswer.(1)Theproblemofthefirm

pf(x1,s.t.w1x1+w2x2≤4Thisissimilartotheutilityizationproblemfromconsumertheorywithf()corre-spondingtotheutilityu()andw1x1+w2x2≤Ccorrespondingtothebudgetconstraint.SincewearegiventhevaluefunctionR(p,w1,w2,C),whichcorrespondstotheindirectutility,wecanuseRoy’sidentitytocomputetheinputdemand: 1x(p,w,w)=??w1=??αw1=

1 1wx(p,w,w)=??w2=??(1?α)w

=(1? 1 ToobtaintheminimumcostwesimplyinverttherevenuefunctionR()andFinallywecanrecovertheproductionfunctionbyf(x1,x2)=

R(p,w1,w2,s.t.w1x1+w2x2=UsingthestandardLagrangianmethod,weobtainthewell-knownCobb-Douglastech-nologyf(x1,x2)=(x1)α(x2)1?α.51.AB，(K)(L)為fA:R2+→R+和fB:R2+→R+。寫下相應的生產(chǎn)集Y。Y={(.K,.L,qA,qB)其中：qA≤fA(KA,LA),qB≤fB(KB,LB),KA+KB≤K,LA+LB≤o2.CES（恒定替代彈性）生產(chǎn)函數(shù)f(x1,x2)=(x1ρ+x2ρ)1，其中ρ≤1計算該生產(chǎn)函數(shù)的TRS（技術替代率）ρ1當ρ=0時，我們得到生產(chǎn)函數(shù)當ρ=.∞時，我們得到Leontief生產(chǎn)函數(shù)。導出條件因子需求x.(w,q)和成本函數(shù)c(w,q)要。]回答。第1部分）。很容易驗證生產(chǎn)函數(shù)的特點是規(guī)模不變（CTRS）：對于任何α>0，我們有：f(αx1,αx2)=αf(x1,x2)2）12.x1.fTRS12=..f(x1,x2)。計算f的偏導數(shù)，我們有：.x1=(x1x2ρ.1.fρ.1(x2)ρ.1x2)1(x1)ρ.1且.x2=(x1+x2)1x1TRS12=d(x1/x2)投入1和投入2之間的替代彈性定義為：σ12=dTRS12。使用TTRS12：σ12.11ρ3b)部分。如果ρ=0，我們可以計算生產(chǎn)函數(shù)如下log(xρ+xρ)1ρ12ρρ(xlog(x1)+(x2)ρρρ1lim(x1+x2)=lime=limexρ→0ρ→01.e=(x1)2(x2)2回答這個問題的另法是查看ρ=0的TRS：TRS12|ρ=0=.x2它對應于生產(chǎn)函數(shù)。參見Varian(1992)，第19-20頁。]ρ3c)ρlimρ→.Infini(xρ1xρ2)min{x1,x2}。ρρx1x2ρ<0x10x20，我們有：x1x2x1，ρ由于ρ<0，我們得到(xρ1+xρ2)≤x1。另一方面，x1≤x2且ρ<0ρρρρρρρx1+x≤x1+x=2x1，或(x1+x≥2ρx1)21ρ1ρx1≤(xρ1+xρ2)1≤ρ取ρ→.∞，我們得到limρ→.∞(x1ρ+x2ρ)1=x1。x2≤x1還有很多其他方法可以回答這個問題。例如，您可以查看是否x2當ρ=.∞時的TRS：TRS12|ρ=.∞0，x2<x1，Leontief(4)c(w,q)minw1x1+w2x2ρs.t.(x1ρx2ρ)1qc(w,q)minw1x1w2x2英石。x1ρ+x2ρ使用標準日方法，您應該獲得條件因子需求為ρ1。ρ.1ρ.1)。1ρ.1x(w,q)=(w+wρ1.ρ1。ρ.1ρ.1)。x(w,q)=(w+wρ2.ρ.1ρ.1c(w,q)=(w1+w2)q請注意，由于生產(chǎn)函數(shù)具有CRTS，因此成本函數(shù)對于輸出q（4）+∞pro.tρρ.1ρ.1ρ.1π(p,w)pqc(w,q)=(p(w1w2))qq≥0ρ.10，如果p≤(w+wρρρ.1q.(p,w)=12ρρ1p>(w1w2所以pro.t函數(shù)是ρ.1ρ.10，如果p≤(w1+w2ρρρ.1ρ.1π(p,w)=ρ.1如果p>(w1+w2問題3.假設.rm有兩個工廠，成本函數(shù)為c1()和c2()。計算生產(chǎn)一定數(shù)量q：c1(q1)=3(q1)2且c2(q2)=(q2)2c1(q1)=4q1且c2(q2)=2q2.rm：c1(q1)+c2(q2)q1q2q。第1部分。對于兩個成本函數(shù)的函數(shù)形式，問題是：c(q)=最小值3(q1)2英石。q1+q2=q解決這個問題，無論是使用日函數(shù)，還是將約束中的q2替換到目標函數(shù)中，您應該得到c(q)=43q2。第2部分.對于第二種情況，.rm求解c(q)=min4q1+2英石。q1+q2=q這里必須，因為目標函數(shù)4q1+2q2是凹的。采用.rst訂單條件將產(chǎn)生最大值，而不是最小值。在這種情況下，檢查兩個情況就足夠了：(q1=0,q2=q)和(q1q,q20)，(q1=0,q2q)，這意味著成本函數(shù)為c(q)=2q。問題