MICROECONOMICS Principles and Analysis Frank Cowell Exercise 6.4 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

Ex 6.4: Question purpose: to construct and solve a simple model of profit-maximisation in a closed economy method: use model of Ex 2.14 to define transformation curve and then apply Lagrangean technique

Ex 6.4: Background from Ex 2.14 Production function is [ q1]2 + [ q2]2 + Aq3 ≤ 0 where qi is net output of good i good 3 is labour A is a positive constant The transformation curve is [ q1]2 + [ q2]2 = Aq3 because good 3is an input, q3 < 0 so that Aq3 > 0

Ex 6.4: Production possibilities Attainable set for given q3 q2 Transformation curve Increase the parameter A [ q1]2 + [ q2]2 = Aq3 Can consider A as a productivity parameter q1

Ex 6.4: Profits Profits are given by: If we use good 3 as numéraire P = p1 q1 + p2q2 + p3q3 If we use good 3 as numéraire p3 = 1 P = p1 q1 + p2q2 + q3 Solution must be on the transformation curve [ q1]2 + [ q2]2 = Aq3 So profits are [ q1]2 + [ q2]2 P = p1 q1 + p2q2   A

Ex 6.4: Profit maximisation The problem is to choose q1 and q2 to maximise profits [ q1]2 + [ q2]2 P = p1 q1 + p2q2   A First-order conditions for an interior maximum are : 2 q1 p1   = 0 2 q2 p2   = 0 Solve to get net output supply: q1 = ½Ap1 q2 = ½Ap2

Ex 6.4: Maximised profits Given the expression for profits [ q1]2 + [ q2]2 P = p1 q1 + p2q2   A Substitute in the optimised values of qi q1 = ½Ap1 q2 = ½Ap2 Therefore maximised profits are [ p1]2 + [ p2]2 P = A  4