2 Inverse demand function for a good supplied by a monopolis

2. Inverse demand function for a good supplied by a monopolist is P 64. This 9 monopolist has three plants whose marginal cost functions are given as MC,(Q) = 2 + 202 MC,(Q,)-6+0 Find the monopolists profit maximizing price and output at each plant.

Solution

Let us sum up Q = Q1 + Q2 + Q3
Q1= (MC)/4 = 0.25MC
Q2= (MC-2)/2 = 0.5MC-1
Q3= MC-6,
Q =0.25MC+0.5MC-1+MC-6
which can be rearranged as
MC = (4/7)*Q+4
MR=differentiation of TR = 64*Q-Q^2/7
= 64-(2/7)*Q
Setting MR = MC yields
64-(2/7)*Q = (4/7)*Q+4
Q = 70
And P =54
At this output level, MC = (4/7)*Q+4= (4/7)*70+4 = 44, implying that
Q1 = 44/4= 11,
Q2 = 44*0.5-1 = 21, and
Q3 = 44-6 = 38