倫敦經(jīng)濟(jì)學(xué)院高微講義企業(yè)理論課件

1、ProductionOptimisationThe FirmComparative StaticsThe Firm and the MarketComparative StaticsThe Firm and the MarketProductionOverviewObjectivesConstraints Method-Profit maximisation?- Technology; other- 2-stage optimisation Lets make a quick list of the components of a standard optimisation problem.

2、Lets also look ahead to the way we will do it for the firm.The optimisation problemUse the information on prices.and quantitieswiPprice of input iprice of outputziQamount of input iamount of outputP Q - S wi zi m i=1Profits:P QRevenue:S wi zi m i=1Cost of inputs:So we get.the objective of the firmTo

3、 set up.P Q - S wi zi m i=1-=+S wi zi m i=1P QThe objective of the firmChoose Q and z to maximiseP :=P Q - 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 outputSet up a Lagrangean to take ca

4、re of the constraintsBang out the First Order Conditions (FOC)Check 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 (. )=0DifferentiateThe FirmProdu

5、ctionCost minimisationOptimisationComparative StaticsThe Firm and the MarketProfit maximisationProfit maximisationTwo stages of optimisation S wi zi m i=1Pick a target output level QTake as given the market prices of inputs w (and of output P)Maximise profits.by minimising costsStage 1 optimisationF

6、or a given set of input prices w.this is the set of points z in input space.that yield a given level of factor cost.These form a hyperplane (straight line).because of the simple expression for factor costWe introduce the isocost S wi zi m i=1 S wi zi m i=1z2z1increasing costw1z1 + w2z2 = c (constant

7、)w1z1 + w2z2 = cw1z1 + w2z2 = cIso-cost lines z2z1warning - infeasible!z*What condition holds at exactly this point? Cost-minimisation _ _ =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 = mar

8、ket pricez2z1What would happen if the input-requirement set looked like this?z2z1z*or this? z* But note that theres no solution point between z* and z*.z2z1z*or this? 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 h

9、ere? Lets look at the formal result from the FOC. THE LAGRANGEAN ISThe first order condition for input two isOr if its not usedAnd if isoquants touch the axes.A set of m+1 conditionslG1(z) = w1lG2(z) = w2lGm(z) = wm .Q = G(z)+ outputconstraintone for each inputThe result?You can get “ 1/2 Cost at A

10、+ Cost at BCCost is concave in pricesz2z1 z*Cost-minimising inputs for w, given Q 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 lemmaNon-decreasing in every input price (Increasing in at least one input p

11、rice)Increasing in output Concave Homogeneous of degree 1 Shephards LemmaThe Cost Function:Five things to rememberThe FirmProductionCost minimisationProfit maximisationOptimisationComparative StaticsThe Firm and the MarketCost minimisationTwo stages of optimisationQQdecreasing returnsto scaleincreas

12、ing returnsto scaleC/QShape of AC depends on RTSaverage cost and output.QQPC/QCQPcuts average cost at its minimummarginal costPQPQRevenuePQC/QRevenue and CostsC/QMaximum Profitsprofits vary with Q . QQQQQQQPCQPprice =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*=0PQ* C(w, Q*) To Finish this Lecture Again check the “Quickie” example Youve got enough now to do the first class assignment. Star