生产技术应用和利润最大化

生产技术应用和利润最大化27.1技术（Technologies）7.1.1什么叫技术？技术是投入转换为产出的过程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用产量面表示生产函数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有限种投入比的技术1234567812345678R1X2:X1=8:1R2X2:X1=3:1R3X2:X1=1:1R4X2:X1=1:4现有6单位x1和3单位x22单位x1和2单位x2用于R3，生产产品50单位；4单位x1和1单位x2用于R4，生产产品50单位产量100单位的等产量线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:在边际产出递减的情况下，是否能存在规模报酬递增现象？A:Yes.E.g.31Returns-to-Scalediminishesasx1increasesdiminishesasx1increases所以，即使边际产出是递减的，规模报酬也可能是递增的。327.1.5TechnicalRate-of-SubstitutionAtwhatratecanafirmsubstituteoneinputforanotherwithoutchangingitsoutputlevel?33TechnicalRate-of-Substitutionx2x1yº100Theslopeofanisoquantisitstechnicalrate-of-substitution.34技术替代率的计算＝0357.1.6长期和短期Thelong-runisthecircumstanceinwhichafirmisunrestrictedinitschoiceofallinputlevels.Therearemanypossibleshort-runs.Ashort-runisacircumstanceinwhichafirmisrestrictedinsomewayinitschoiceofatleastoneinputlevel.36长期和短期isthelong-runproduction

function(bothx1andx2arevariable).Theshort-runproductionfunctionwhen

x2

º1is

Theshort-runproductionfunctionwhen

x2

º10is37x1y长期和短期Fourshort-runproductionfunctions.387.2Profit-Maximization7.2.1EconomicProfit厂商用投入j=1…,m生产产品i=1,…n.产出水平y1,…,yn，投入水平x1,…,xm.产品价格p1,…,pn，投入价格w1,…,wm.

厂商是价格接受者，即p1,…,pn

和w1,…,wm

给定。经济利润：39经济利润投入量、产出量及利润均为流量；经济成本407.2.2企业现值某厂商若干时期的经济利润为π0,π1,π2,…利率为r，厂商经济利润的现值为417.2.3不变要素和可变要素数量固定的生产要素称为不变要素（固定要素）。可以按不同的数量使用的生产要素称为可变要素（变动要素）。长期、短期与不变要素、可变要素427.2.4短期利润最大化假定要素2的投入水平保持不变，厂商的利润最大化问题就可以表示为销售收入变动成本固定成本43利润最大化生产要素的边际产品价值(themarginalrevenueproductofinput1)应该等于它的价格x1￥pMP1w1x*144几何法等利润线(Iso-ProfitLines)slopeverticalintercept45Iso-ProfitLinesIncreasing

profityx146Short-RunProfit-Maximizationx1y47Short-RunProfit-Maximization;ACobb-DouglasExample487.2.5比较静态学产品价格p变动x1y低价格高价格49比较静态学投入价格w1变动x1y高价格低价格50ComparativeStaticsofSh