MICro: cost curves for two plants

Posted by Mauli on Jun 05, 2013; 9:52am
URL: http://discussion-forum.276.s1.nabble.com/MICro-cost-curves-for-two-plants-tp7581776.html

MICRO: A monopoly faces the demand curve P=16-Q where Q is the total demand. and it has two plants and both have identical cost functions C(qi)=15+2qi^2 for qi>0 and zero otherwise..should monopoly produce in both the plants or only one plant..what are the profit maximizing levels of q1,q2.


guess answer
 min
  q1,q2            c(q1) +c(q2)
                   15+2q1^2 + 15+2q2^2
such that
q1+q2=Q

MIn  15+2q1^2 +15 +2(Q-q1)^2
q1    
        which gives us
4q1 -4(Q-q1)=0
q1= Q/2
since the question is symmetrical we get q2= Q/2.

if the monopolist uses only  one plant say plant 1.
then he max profits according to
(16-Q)Q-[15+2[(Q/2)^2].
i am getting Q= 16/3 =5.33


if we use both the plants ,
we max
(16-Q)Q-[ 15+2(Q/2)^2 + 15 +2(Q/2)^2]
  the profit in the first case is > than when both the plants are in operation.
also, if we see that though the marginal costs in both the plants are the same , the total cost faced by the monopolist rises if he uses both the plants due to the presence of TFC.
so he produces in plant 1.
hence the profit max levels are (q1,q2) = (5.33 , 0) or
q1,q2 = 0,5.33 .

please tell me , is it the correct way to do this problem.
PS: i was going through the old threads and this was a question asked during the time of ISI interview. i found it interesting and was curious to know the answer.