Frank Cowell Microeconomics November 2006 Exercise 6 4
Frank Cowell: Microeconomics November 2006 Exercise 6. 4 MICROECONOMICS Principles and Analysis Frank Cowell
Ex 6. 4: Question Frank Cowell: Microeconomics n n 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 Frank Cowell: Microeconomics n Production function is u u n [ q 1]2 + [ q 2]2 + Aq 3 ≤ 0 where qi is net output of good i good 3 is labour A is a positive constant The transformation curve is u u u [ q 1]2 + [ q 2]2 = Aq 3 because good 3 is an input, q 3 < 0 so that Aq 3 > 0
Ex 6. 4: Production possibilities Frank Cowell: Microeconomics §Attainable set for given q 3 q 2 §Transformation curve §Increase the parameter A § [ q 1 ] 2 + [ q 2 ] 2 = A q 3 § Can consider A as a productivity parameter q 1
Ex 6. 4: Profits Frank Cowell: Microeconomics n Profits are given by: u n If we use good 3 as numéraire u u n p 3 = 1 P = p 1 q 1 + p 2 q 2 + q 3 Solution must be on the transformation curve u n P = p 1 q 1 + p 2 q 2 + p 3 q 3 [ q 1 ]2 + [ q 2 ]2 = Aq 3 So profits are: u [ q 1]2 + [ q 2]2 P = p 1 q 1 + p 2 q 2 ���� A
Ex 6. 4: Profit maximisation Frank Cowell: Microeconomics n The problem is to choose q 1 and q 2 to maximise profits u n First-order conditions for an interior maximum are: u u n [ q 1]2 + [ q 2]2 P = p 1 q 1 + p 2 q 2 ���� A 2 q 1 p 1 ���= 0 A 2 q 2 p 2 ���= 0 A Solve to get net output supply: u u q 1 = ½Ap 1 q 2 = ½Ap 2
Ex 6. 4: Maximised profits Frank Cowell: Microeconomics n Given the expression for profits u n Substitute in the optimised values of qi u u n [ q 1]2 + [ q 2]2 P = p 1 q 1 + p 2 q 2 ���� A q 1 = ½Ap 1 q 2 = ½Ap 2 Therefore maximised profits are u [ p 1 ]2 + [ p 2 ]2 P = A ����� 4
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