伦敦经济学院高微讲义企业理论.ppt

The Firm:OptimisationMicroEconomics 2001/2ProductionOptimisationThe FirmComparative StaticsThe Firm and the MarketComparative StaticsThe Firm and the MarketProductionOverviewnObjectivesnConstraints nMethod-Profit maximisation?-Technology;other-2-stage optimisation lLets make a quick list of the components of a standard optimisation problem.lLets also look ahead to the way we will do it for the firm.The optimisation problem.and quantitieswiPprice of input iprice of outputziQamount of input iamount of outputUse the information on prices.P Q -S S wi zi m i=1Profits:P QRevenue:S S wi zi m i=1Cost of inputs:So we get.To set up.the objective of the firmP Q -S S wi zi m i=1-=+S S wi zi m i=1P QThe objective of the firmA simple definition of profitsChoose Q and z to maximiseP:=P Q -S S wi zi m i=1Q G(z).subject to the production constraint.Q 0z 0.and some pretty obvious conditions:Optimisation:the standard approachYou cant have negative inputsYou cant have negative outputqSet up a Lagrangean to take care of the constraintsqBang out the First Order Conditions(FOC)qCheck out second-order conditionssufficiencynecessityUsing standard methods.FOC:the First-Order Conditions L(.)zFirst order conditions are used over and over in optimisation problems as we will see.L(.)=0The LagrangeanDifferentiateThe FirmProductionCost minimisationOptimisationComparative StaticsThe Firm and the MarketProfit maximisationProfit maximisationTwo stages of optimisation S S wi zi m i=1nPick a target output level QnTake as given the market prices of inputs w(and of output P)nMaximise profits.n.by minimising costsStage 1 optimisationlFor a given set of input prices w.l.this is the set of points z in input space.l.that yield a given level of factor cost.lThese form a hyperplane(straight line).l.because of the simple expression for factor costWe introduce the isocost S S wi zi m i=1 S S wi zi m i=1z2z1increasing costw1z1+w2z2=c(constant)w1z1+w2z2=cw1z1+w2z2=cIso-cost lines z2z1warning-infeasible!z*What condition holds at exactly this point?Cost-minimisation l _ _=Gi(z)wiGj(z)wj input MRTS =price ratioIf both i and j are in use and MRTS is defined.What if we changed the assumptions about technology?implicit price =market pricez2z1What would happen if the input-requirement set looked like this?Any z-value in this set of points is cost-minimisingz2z1z*or this?There could be multiple solutions.z*l l But note that theres no solution point between z*and z*.z2z1z*or this?l MRTS21 is undefined exactly at z*.But z*is cost-minimising point for given Q and all finite values of w1/w2z2z1z*And what happens here?l Here MRTS21 w1/w2 at the solution.Input 2 is“too expensive”and so isnt used at all:z2*=0.Lets look at the formal result from the FOC.THE LAGRANGEAN ISThe first order condition for input two isOr if its not usedA set of m+1 conditionslG1(z)=w1lG2(z)=w2lGm(z)=wm .Q=G(z)+outputconstraintone for each inputThe result?And if isoquants touch the axes.You can get“1/2 Cost at A+Cost at BCCost is concave in pricesz2z1l z*Cost-minimising inputs for w,given Ql z*Cost-minimising inputs for tw,given QC(tw,Q)=t Siwi zi*=tC(w,Q)What happens to cost if w changes to tw?C(w,Q)wi=zi*_w1CShephards lemmaslope=z1*oNon-decreasing in every input price(Increasing in at least one input price)oIncreasing in output oConcave oHomogeneous of degree 1 oShephards LemmaThe Cost Function:Five things to rememberThe FirmProductionCost minimisationProfit maximisationOptimisationComparative StaticsThe Firm and the MarketCost minimisationTwo stages of optimisationQQdecreasing returnsto scaleincreasing returnsto scaleC/QShape of AC depends on RTSaverage cost and output.QQPC/QCQPcuts average cost at its minimummarginal costPQPQRevenuePQC/QRevenue and CostsC/Qprofits vary with Q.QQQQQQQP PCQPMaximum Profitsprice =marginal costWhat condition holds at exactly this point?PC/QCQprice 0:Price equals marginal costPrice covers average costP=CQ(w,Q*)P C(w,Q*)Q*_covers both the cases:Q*0 and Q*=0.and in general:PQ*C(w,Q*)To Finish this Lecturel Again check the“Quickie”example lYouve got enough now to do the first class assignment.Start today.lReview the key conceptsnProfit maximisation can be viewed in two stagesnStage 1:choose inputs to minimise costnStage 2:choose output to maximise profitUse these to Use these to predictpredictfirms reactionsfirms reactionsSummary此课件下载可自行编辑修改，供参考！感谢您的支持，我们努力做得更好！