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1、The Firm: Optimisation,MicroEconomics 2001/2,The Firm,Overview,Objectives,Constraints,Method,-Profit maximisation?,- Technology; other,- 2-stage optimisation,Lets make a quick list of the components of a standard optimisation problem.,Lets also look ahead to the way we will do it for the firm.,The o
2、ptimisation problem,.and quantities,Use the information on prices.,To set up.,the objective of the firm,-,=,+,P Q,The objective of the firm,A simple definition of profits,Choose Q and z to maximise,Q G(z),.subject to the production constraint.,Q 0,z 0,.and some pretty obvious conditions:,Optimisatio
3、n: the standard approach,You cant have negative inputs,You cant have negative output,Set up a Lagrangean to take care of the constraints,Bang out the First Order Conditions (FOC),Check out second-order conditions,sufficiency,necessity,Using standard methods.,FOC: the First-Order Conditions,L (. ), z
4、,First order conditions are used over and over in optimisation problems as we will see.,L (. )=0,The Lagrangean,Differentiate,The Firm,Production,Optimisation,Comparative Statics,The Firm and the Market,Two stages of optimisation,Pick a target output level Q,Take as given the market prices of inputs
5、 w (and of output P),Maximise profits.,.by minimising costs,Stage 1 optimisation,For 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 cost,We int
6、roduce the isocost,increasing cost,Iso-cost lines,z2,z1,warning - infeasible!,z*,Cost-minimisation,input MRTS = price ratio,If both i and j are in use and MRTS is defined.,implicit price = market price,z2,z1,What would happen if the input-requirement set looked like this?,Any z-value in this set of
7、points is cost-minimising,z2,z1,z*,or this?,There could be multiple solutions.,z*,But note that theres no solution point between z* and z*.,z2,z1,z*,or this?,MRTS21 is undefined exactly at z*. But z* is cost-minimising point for given Q and all finite values of w1/w2,z2,z1,z*,And what happens here?,
8、Here MRTS21 w1/w2 at the solution. Input 2 is “too expensive” and so isnt used at all: z2*=0.,THE LAGRANGEAN IS,The first order condition for input two is,Or if its not used,A set of m+1 conditions, ,lG1(z) = w1,lG2(z) = w2,lGm(z) = wm, .,Q = G(z),And if isoquants touch the axes.,You can get “” if o
9、ptimal value of this input is 0,we get a cost-minimising value for each input. .for the Lagrange multiplier. .and, of course, for the minimised value of cost itself. Each of these can be written as a function of the given prices (w) and given output (Q).,The solution.,C(w, Q) :=,vector of input pric
10、es,Specified output level,min S wi zi,G(z) Q,The solution is the cost function,Because it is a solution function.,.it automatically has very nice properties.,These are true for all production functions G.,And they carry over to applications other than the firm,The cost function is an amazingly usefu
11、l concept,Draw cost as a function of w1,C(w, Q),w1,C,Cost is non-decreasing in wi,w1,C,C(w, Q),Cost increases with output,w1,D,A,B,C,Cost is concave in prices,z2,z1,z*,Cost-minimising inputs for w, given Q,z*,Cost-minimising inputs for tw, given Q,C(tw,Q) = t Siwi zi* = tC(w,Q),What happens to cost
12、if w changes to tw?,w1,C,Shephards lemma,slope = z1*,Non-decreasing in every input price (Increasing in at least one input price) Increasing in output Concave Homogeneous of degree 1 Shephards Lemma,The Cost Function: Five things to remember,The Firm,Production,Cost minimisation,Profit maximisation,
13、Optimisation,Comparative Statics,The Firm and the Market,Two stages of optimisation,Q,Q,C/Q,Shape of AC depends on RTS,average cost and output.,Q,Q,P,C/Q,CQ,P,marginal cost,P,Q,Revenue,P,Q,C/Q,Revenue and Costs,C/Q,profits vary with Q .,P,CQ,P,Maximum Profits,P,C/Q,CQ,P,Q,Q* = 0,What happens if pric
14、e is low.,First-order conditions if Q* 0:,Price equals marginal cost,Price covers average cost,. and in general:,PQ* C(w, Q*),To Finish this Lecture,Again check the “Quickie” example Youve got enough now to do the first class assignment. Start today. Review the key concepts,Profit maximisation can be viewed in two stages,Stage 1: choose inputs to minimise cost,Stage 2: choose output to maximise profit,Use these to predict firms reactions,Summary,
