清华大学中级微观经济学讲义(清华李稻葵).ppt

ChapterNineteen,Profit-Maximization,WhatDoWeDointhisChapter?,Afterworkingour“producersbudgetsets”(productionsets),Weareworkingon“producerschoices”Pleasepayattentiontothesimilarityanddifferencesbetween“producerschoices”and“consumerschoices”,EconomicProfit,Theeconomicprofitgeneratedby(x1,xm,y1,yn)isNotes:Forthetimebeing,werestricttothecaseofacompetitivefirm,whichisatinyrelativetothemarketsizeandtakespricesp1,pnw1,wmasgivenconstants;,Short-runEconomicProfit,Supposethefirmisinashort-runcircumstanceinwhichItsshort-runproductionfunctionisThefirmsfixedcostisanditsprofitfunctionis,Short-RunIso-ProfitLines,Aniso-profitlinecontainsalltheproductionplansthatyieldthesameprofitlevel.Theequationofaniso-profitlineisI.e.,Short-RunIso-ProfitLines,Increasingprofit,y,x1,Short-RunProfit-Maximization,x1,Increasingprofit,y,Short-RunProfit-Maximization,x1,y,Attheshort-runprofit-maximizingplan,theslopesoftheshort-runproductionfunctionandthemaximaliso-profitlineareequal.,Short-RunProfit-Maximization,isthemarginalrevenueproductofinput1,therateatwhichrevenueincreaseswiththeamountusedofinput1.Ifthenprofitincreaseswithx1.Ifthenprofitdecreaseswithx1.,AMathematicalApproachtoShort-RunProfit-Maximization,Mathematically,thefirmsshortrunproblemis:MaximizeSubjectto:Thisgivesus:pMP1=w1Important:WehaveassumedthatMP1isdecreasinginx1,ComparativeStaticsofShort-RunProfit-Maximization,Whathappenstotheshort-runprofit-maximizingproductionplanastheoutputpricepchanges?,ComparativeStaticsofShort-RunProfit-Maximization,Theequationofashort-runiso-profitlineis,soanincreaseinpcausesareductionintheslopeofthefamilyofiso-profitlines,ComparativeStaticsofShort-RunProfit-Maximization,x1,y,ComparativeStaticsofShort-RunProfit-Maximization,x1,y,AUsefulMath:TheEnvelopeTheorem,Supposex*maximizesg(x;t),wheretisaparameter;Thenx*varieswitht,i.e,x*=x*(t);Wehavegxx*(t),t=0;Letg*(t)=Maxg(x,t)=gx*(t),tThen,thedg*(t)/dt=gxx*(t),tx*(t)+gtx*(t),t=gtx*(t),tThisiscalledtheEnvelopeTheorem.,ApplyingtheEnvelopeTheorem,Where,wepluginLet*betheoptimalprofitlevel;Wegetd*/dp=y0;d*/dw1=-x10.,ComparativeStaticsofShort-RunProfit-Maximization,Whathappenstotheshort-runprofit-maximizingproductionplanasthevariableinputpricew1changes?,ComparativeStaticsofShort-RunProfit-Maximization,Theequationofashort-runiso-profitlineis,soanincreaseinw1causesanincreaseintheslope,and,ComparativeStaticsofShort-RunProfit-Maximization,x1,y,ComparativeStaticsofShort-RunProfit-Maximization,x1,y,ComparativeStaticsofShort-RunProfit-Maximization,Anincreaseinw1causesdecreasesinoptimalinputlevel;optimaloutput;optimizedprofit.,Long-RunProfit-Maximization,Nowallowthefirmtovarybothinputlevels.Sincenoinputlevelisfixed,therearenofixedcosts.,Long-RunProfit-Maximization,Bothx1andx2arevariable.FirmsproblemisMaximizepyw1x1w2x2Subjecttoy=f(x1,x2),Long-RunProfit-Maximization,Firmsproblemnowbecomes:Maximizepf(x1,x2)w1x1w2x2Theinputlevelsofthelong-runprofit-maximizingplansatisfyThatis,marginalrevenueequalstoinputpriceImportant:wehaveassumeddecreasingMarginalProducts!,Long-TermProfitintheCaseofDecreasingReturns-to-Scale,Inthiscase,thefirmhasasinglelong-runprofit-maximizingproductionplan;Seethegraphnextpage;Theoptimalprofitmightbe0.,Returns-toScaleandProfit-Maximization,x,y,y*,x*,Decreasingreturns-to-scale,Long-TermProfitintheCaseofIncreasingReturns-to-Scale,Inthiscase,thelong-termprofitisinfinite(seethegraphnextpage);But,thisisnotconsistentwithourperfectcompetitivefirmassumption,sincethefirmisinfiniteinsize.,Returns-toScaleandProfit-Maximization,x,y,y”,x,Increasingreturns-to-scale,y,x”,Increasingprofit,Long-TermProfitintheCaseofConstantReturns-to-Scale,Itslong-termprofitiseither0orinfinity-dependingonprices;Onlythe0profitcaseisconsistentwithourperfectcompetitivefirmassumption.,Returns-toScaleandProfit-Maximization,x,y,y”,x,Constantreturns-to-scale,y,x”,Increasingprofit,Returns-toScaleandProfit-Maximization,x,y,y”,x,Constantreturns-to-scale,y,x”,P=0,RevealedProfitability,Consideracompetitivefirmwithatechnologythatexhibitsdecreasingreturns-to-scale.Foravarietyofoutputandinputpricesweobservethefirmschoicesofproductionplans.Whatcanwelearnfromourobservations?,RevealedProfitability,Ifaproductionplan(x,y)ischosenatprices(w,p)wededucethattheplan(x,y)isrevealedtobeprofit-maximizingfortheprices(w,p).,RevealedProfitability,x,y,ischosenatpricessoisprofit-maximizingattheseprices.,RevealedProfitability,x,y,ischosenatpricessoisprofit-maximizingattheseprices.,wouldgivehigherprofits,sowhyisitnotchosen?Becauseitisnotafeasibleplan.,Sothefirmstechnologysetmustlieundertheiso-profitline.,RevealedProfitability,x,y,ischosenatpricessoisprofit-maximizingattheseprices.,Sothefirmstechnologysetmustlieundertheiso-profitline.,Thetechnologysetissomewhereinhere,RevealedProfitability,x,y,ischosenatpricessomaximizesprofitattheseprices.,Thetechnologysetisalsosomewhereinhere.,RevealedProfitability,x,y,Thefirmstechnologysetmustlieunderbothiso-profitlines,RevealedProfitability,x,y,Thefirmstechnologysetmustlieunderbothiso-profitlines,Thetechnologysetissomewhereinthisintersection,RevealedProfitability,x,y,Thefirmstechnologysetmustlieunderalltheiso-profitlines,RevealedProfitability,Whatelsecanbelearnedfromthefirmschoicesofprofit-maximizingproductionplans?,RevealedProfitability,x,y,Thefirmstechnologysetmustlieunderalltheiso-profitlines,ischosenatpricesso,ischosenatpricesso,RevealedProfitability,and,so,and,Addinggives,RevealedProfitability,so,Thatis,isanecessaryimplicationofprofit-maximization.,RevealedProfitability,isanecessaryimplicationofprofit-maximization.Supposetheinputpricedoesnotchange.ThenDw=0andprofit-maximizationimplies;i.e.,aco