生產(chǎn)技術(shù)應(yīng)用和利潤(rùn)最大化

生產(chǎn)技術(shù)應(yīng)用和利潤(rùn)最大化27.1技術(shù)（Technologies）7.1.1什么叫技術(shù)？技術(shù)是投入轉(zhuǎn)換為產(chǎn)出的過(guò)程E.g.labor,acomputer,aprojector,electricity,andsoftwarearebeingcombinedtoproducethislecture.Usuallyseveraltechnologieswillproducethesameproduct.ablackboardandchalkcanbeusedinsteadofacomputerandaprojector.Whichtechnologyis“best”?Howdowecomparetechnologies?37.1.2ProductionFunctionsydenotestheoutputlevel.Thetechnology’sproductionfunctionstatesthemaximumamountofoutputpossiblefromaninputbundle.4ProductionFunctions

-Oneinput,oneoutputy=f(x)x1InputLevelxOutputLevely1y1=f(x1)isthemaximaloutputlevelobtainablefromx1inputunits.5TechnologieswithMultipleInputsSupposetheproductionfunctionis(x1,x2)=(1,8)(x1,x2)=(8,8)6TechnologieswithMultipleInputsOutput,yx1x2(8,1)(8,8)7IsoquantswithTwoVariableInputsyo8yo4x1x28用產(chǎn)量面表示生產(chǎn)函數(shù)LOKTPK1L1L2K2AA’BB’CC’DD’Q1Q2E1F1G1E2F2G2E1’E2’G1’G2’9IsoquantswithTwoVariableInputsOutput,yx1x2yo8yo410Cobb-DouglasTechnologiesACobb-Douglasproductionfunctionisoftheform11Fixed-ProportionsTechnologiesAfixed-proportionsproductionfunctionisoftheform12Fixed-ProportionsTechnologiesx2x1min{x1,2x2}=144814247min{x1,2x2}=8min{x1,2x2}=4x1=2x213Perfect-SubstitutesTechnologiesAperfect-substitutesproductionfunctionisoftheform14Perfect-SubstitutionTechnologies93186248x1x2x1+3x2=9x1+3x2=18x1+3x2=2415有限種投入比的技術(shù)1234567812345678R1X2:X1=8:1R2X2:X1=3:1R3X2:X1=1:1R4X2:X1=1:4現(xiàn)有6單位x1和3單位x22單位x1和2單位x2用于R3，生產(chǎn)產(chǎn)品50單位；4單位x1和1單位x2用于R4，生產(chǎn)產(chǎn)品50單位產(chǎn)量100單位的等產(chǎn)量線167.1.3Marginal(Physical)ProductsThemarginalproductofinputiistherate-of-changeoftheoutputlevelasthelevelofinputichanges,holdingallotherinputlevelsfixed.Thatis,17Marginal(Physical)Products187.1.4Returns-to-ScaleMarginalproductsdescribethechangeinoutputlevelasasingleinputlevelchanges.Returns-to-scaledescribeshowtheoutputlevelchangesasallinputlevelschangeindirectproportion(e.g.allinputlevelsdoubled,orhalved).19Constantreturns-to-scaleIf,foranyinputbundle(x1,…,xn),thenthetechnologydescribedbythe

productionfunctionfexhibitsconstant

returns-to-scale.

.(k=2)doublingallinputlevels

doublestheoutputlevel.20Constantreturns-to-scaley=f(x)x’xInputLevelOutputLevely’2x’2y’Constant

returns-to-scale21Diminishingreturns-to-scaleIf,foranyinputbundle(x1,…,xn),thenthetechnologyexhibitsdiminishingreturns-to-scale.

.(k=2)doublingallinputlevelslessthandoublestheoutputlevel.22Diminishingreturns-to-scaley=f(x)x’xInputLevelOutputLevelf(x’)2x’f(2x’)2f(x’)Decreasing

returns-to-scale23Increasingreturns-to-scaleIf,foranyinputbundle(x1,…,xn),thenthetechnologyexhibitsincreasing

returns-to-scale.

.(k=2)doublingallinputlevels

morethandoublestheoutputlevel.24Increasingreturns-to-scaley=f(x)x’xInputLevelOutputLevelf(x’)2x’f(2x’)2f(x’)Increasing

returns-to-scale25Returns-to-Scaley=f(x)xInputLevelOutputLevelDecreasing

returns-to-scaleIncreasing

returns-to-scale26ExamplesofReturns-to-ScaleTheperfect-substitutesproduction

functionisTheperfect-substitutesproduction

functionexhibitsconstantreturns-to-scale.27ExamplesofReturns-to-ScaleTheperfect-complementsproduction

functionisTheperfect-complementsproduction

functionexhibitsconstantreturns-to-scale.28ExamplesofReturns-to-ScaleTheCobb-Douglasproductionfunctionis29ExamplesofReturns-to-ScaleTheCobb-Douglastechnology’sreturns-to-scaleisconstantifa1+…+an=1increasingifa1+…+an>1decreasingifa1+…+an<130QuestionandanswerQ:在邊際產(chǎn)出遞減的情況下，是否能存在規(guī)模報(bào)酬遞增現(xiàn)象？A:Yes.E.g.31Returns-to-Scalediminishesasx1increasesdiminishesasx1increases所以，即使邊際產(chǎn)出是遞減的，規(guī)模報(bào)酬也可能是遞增的。327.1.5TechnicalRate-of-SubstitutionAtwhatratecanafirmsubstituteoneinputforanotherwithoutchangingitsoutputlevel?33TechnicalRate-of-Substitutionx2x1yo100Theslopeofanisoquantisitstechnicalrate-of-substitution.34技術(shù)替代率的計(jì)算＝0357.1.6長(zhǎng)期和短期Thelong-runisthecircumstanceinwhichafirmisunrestrictedinitschoiceofallinputlevels.Therearemanypossibleshort-runs.Ashort-runisacircumstanceinwhichafirmisrestrictedinsomewayinitschoiceofatleastoneinputlevel.36長(zhǎng)期和短期isthelong-runproduction

function(bothx1andx2arevariable).Theshort-runproductionfunctionwhen

x2

o1is

Theshort-runproductionfunctionwhen

x2

o10is37x1y長(zhǎng)期和短期Fourshort-runproductionfunctions.387.2Profit-Maximization7.2.1EconomicProfit廠商用投入j=1…,m生產(chǎn)產(chǎn)品i=1,…n.產(chǎn)出水平y(tǒng)1,…,yn，投入水平x1,…,xm.產(chǎn)品價(jià)格p1,…,pn，投入價(jià)格w1,…,wm.

廠商是價(jià)格接受者，即p1,…,pn

和w1,…,wm

給定。經(jīng)濟(jì)利潤(rùn)：39經(jīng)濟(jì)利潤(rùn)投入量、產(chǎn)出量及利潤(rùn)均為流量；經(jīng)濟(jì)成本407.2.2企業(yè)現(xiàn)值某廠商若干時(shí)期的經(jīng)濟(jì)利潤(rùn)為π0,π1,π2,…利率為r，廠商經(jīng)濟(jì)利潤(rùn)的現(xiàn)值為417.2.3不變要素和可變要素?cái)?shù)量固定的生產(chǎn)要素稱為不變要素（固定要素）。可以按不同的數(shù)量使用的生產(chǎn)要素稱為可變要素（變動(dòng)要素）。長(zhǎng)期、短期與不變要素、可變要素427.2.4短期利潤(rùn)最大化假定要素2的投入水平保持不變，廠商的利潤(rùn)最大化問(wèn)題就可以表示為銷售收入變動(dòng)成本固定成本43利潤(rùn)最大化生產(chǎn)要素的邊際產(chǎn)品價(jià)值(themarginalrevenueproductofinput1)應(yīng)該等于它的價(jià)格x1￥pMP1w1x*144幾何法等利潤(rùn)線(Iso-ProfitLines)slopeverticalintercept45Iso-ProfitLinesIncreasing

profityx146Short-RunProfit-Maximizationx1y47Short-RunProfit-Maximization;ACobb-DouglasExample487.2.5比較靜態(tài)學(xué)產(chǎn)品價(jià)格p變動(dòng)x1y低價(jià)格高價(jià)格49比較靜態(tài)學(xué)投入價(jià)格w1變動(dòng)x1y高價(jià)格低價(jià)格50ComparativeStaticsofSh