生產技術應用和利潤最大化

生產技術應用和利潤最大化27.1技術（Technologies）7.1.1什么叫技術？技術是投入轉換為產出的過程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用產量面表示生產函數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1234567812345678R1X2:X1=8:1R2X2:X1=3:1R3X2:X1=1:1R4X2:X1=1:4現有6單位x1和3單位x22單位x1和2單位x2用于R3，生產產品50單位；4單位x1和1單位x2用于R4，生產產品50單位產量100單位的等產量線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:在邊際產出遞減的情況下，是否能存在規(guī)模報酬遞增現象？A:Yes.E.g.31Returns-to-Scalediminishesasx1increasesdiminishesasx1increases所以，即使邊際產出是遞減的，規(guī)模報酬也可能是遞增的。327.1.5TechnicalRate-of-SubstitutionAtwhatratecanafirmsubstituteoneinputforanotherwithoutchangingitsoutputlevel?33TechnicalRate-of-Substitutionx2x1yo100Theslopeofanisoquantisitstechnicalrate-of-substitution.34技術替代率的計算＝0357.1.6長期和短期Thelong-runisthecircumstanceinwhichafirmisunrestrictedinitschoiceofallinputlevels.Therearemanypossibleshort-runs.Ashort-runisacircumstanceinwhichafirmisrestrictedinsomewayinitschoiceofatleastoneinputlevel.36長期和短期isthelong-runproduction

function(bothx1andx2arevariable).Theshort-runproductionfunctionwhen

x2

o1is

Theshort-runproductionfunctionwhen

x2

o10is37x1y長期和短期Fourshort-runproductionfunctions.387.2Profit-Maximization7.2.1EconomicProfit廠商用投入j=1…,m生產產品i=1,…n.產出水平y(tǒng)1,…,yn，投入水平x1,…,xm.產品價格p1,…,pn，投入價格w1,…,wm.

廠商是價格接受者，即p1,…,pn

和w1,…,wm

給定。經濟利潤：39經濟利潤投入量、產出量及利潤均為流量；經濟成本407.2.2企業(yè)現值某廠商若干時期的經濟利潤為π0,π1,π2,…利率為r，廠商經濟利潤的現值為417.2.3不變要素和可變要素數量固定的生產要素稱為不變要素（固定要素）?？梢园床煌臄盗渴褂玫纳a要素稱為可變要素（變動要素）。長期、短期與不變要素、可變要素427.2.4短期利潤最大化假定要素2的投入水平保持不變，廠商的利潤最大化問題就可以表示為銷售收入變動成本固定成本43利潤最大化生產要素的邊際產品價值(themarginalrevenueproductofinput1)應該等于它的價格x1￥pMP1w1x*144幾何法等利潤線(Iso-ProfitLines)slopeverticalintercept45Iso-ProfitLinesIncreasing

profityx146Short-RunProfit-Maximizationx1y47Short-RunProfit-Maximization;ACobb-DouglasExample487.2.5比較靜態(tài)學產品價格p變動x1y低價格高價格49比較靜態(tài)學投入價格w1變動x1y高價格低價格50ComparativeStaticsofSh