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1、ChapterNineteenProfit-MaximizationWhatDoWeDointhisChapter?Afterworkingour“producer’sbudgetsets”(productionsets),Weareworkingon“producer’schoices”Pleasepayattentiontothesimilarityanddifferencesbetween“producer’schoices”and“consumer’schoices”EconomicProfitTheeconomicprofitgenerat
2、edby(x1,…,xm,y1,…,yn)isNotes:Forthetimebeing,werestricttothecaseofacompetitivefirm,whichisatinyrelativetothemarketsizeandtakespricesp1,…,pnw1,…,wmasgivenconstants;Short-runEconomicProfitSupposethefirmisinashort-runcircumstanceinwhichItsshort-runproductionfunctionisThefirm’sfixe
3、dcostis anditsprofitfunctionisShort-RunIso-ProfitLinesAniso-profitlinecontainsalltheproductionplansthatyieldthesameprofitlevel.Theequationofaniso-profitlineisI.e.Short-RunIso-ProfitLinesIncreasing profityx1Short-RunProfit-Maximizationx1Increasing profityShort-RunProfit-Maximiza
4、tionx1yAttheshort-runprofit-maximizingplan,theslopesoftheshort-runproductionfunctionandthemaximal iso-profitlineare equal.Short-RunProfit-Maximizationisthemarginalrevenueproductof input1,therateatwhichrevenueincreases withtheamountusedofinput1.Ifthenprofitincreaseswithx1.Ifthen
5、profitdecreaseswithx1.AMathematicalApproachtoShort-RunProfit-MaximizationMathematically,thefirm’sshortrunproblemis:MaximizeSubjectto:Thisgivesus:pMP1=w1Important:WehaveassumedthatMP1isdecreasinginx1ComparativeStaticsofShort-RunProfit-MaximizationWhathappenstotheshort-runprofit-
6、maximizingproductionplanastheoutputpricepchanges?ComparativeStaticsofShort-RunProfit-MaximizationTheequationofashort-runiso-profitline issoanincreaseinpcauses areductionintheslopeofthefamilyofiso-profitlinesComparativeStaticsofShort-RunProfit-Maximizationx1yComparativeStaticsof
7、Short-RunProfit-Maximizationx1yAUsefulMath:TheEnvelopeTheoremSupposex*maximizesg(x;t),wheretisaparameter;Thenx*varieswitht,i.e,x*=x*(t);Wehavegx[x*(t),t]=0;Letg*(t)=Maxg(x,t)=g[x*(t),t]Then,thedg*(t)/dt=gx[x*(t),t]x*’(t)+gt[x*(t),t]=gt[x*(t),t]ThisiscalledtheEnvelopeTheorem.App
8、lyingtheEnvelopeTheoremWhere,wepluginLet*betheoptimal
