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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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