Oligopoly

[Ameya]

Hi, I have a question from Oligopoly, Pindyck.
two firms facing demand curve P = 50 - 5Q, where Q= Q1 + Q2.
Cost functions, C1(Q1) = 20 + 10Q1
C2(Q2) = 10 + 12Q2

The MC in both cases are different which gives us 2 possible solutions. how do we maximise and how do we solve for a collusion with different marginal per unit cost?

[s]

Hi,

for a cartel solution the problem is :

profit = P(Q1+Q2).(Q1+Q2) - C1(Q1) - C2(Q2)

ie (50 - 5Q1 -5Q2)(Q1+Q2) -20 -10Q1-10-12Q2

now maximise this wrt Q1 and Q2 u'l get values for Q1* and Q2* then find total output Q* and then u can derive P*