(a) if the monopolist discriminates
Examine the price structures:
more sales mean lower prices
=> there is an upper limit
p1=15-q1 and p2=25-2q2 =>
(q1<15, q2<12.5)
Examine the cost structure:
c=5+3(q1+q2) =>
there is no discrimination of cost for each region.
Assuming the objective is to maximize profit, then we can formulate
total revenue,
tr(q1,q2)=q1*p1(q1)+q2*p2(q2)
=-q1²+15q1+25q2-2q2²
and total profit
tp(q1,q2)=tr(q1,q2)-c(q1,q2)
=tr(q1,q2)-(5+3(q1,q2)
=-q1²+12q1+22q2-2q2²-5
To maximize tp(q1,q2) which contains two variables q1 and q2, we need multi-variable calculus. Luckily, p1 is a simple function of q1, and p2 is a function of q2, and c does not discriminate between q1 and q2, so maximizing independently q1 and q2 works... in this case.
d(tp)/d(q1)=12-2q1=0 => q1=6
d(tp)/d(q2)=22-4q2=0 => q2=5.5
So q1=6, q2=5.5 for maximum total profit where
tp(6,5.5)=91.5
[check by varying q1 and q2 slightly to confirm that the solution gives the maximum profit]
To find marginal revenues,
mr1(q1,q2)=d(tr)/d(q1)=15-2q1
=> mr1(6,5.5)=15-2(6)=3
mr2(q1,q2)=d(tr)/d(q2)=25-4q2
=> mr2(6,5.5)=25-4(5.5)=3
This confirms the solution is optimal since
mr1=mr2 at the given production level.
It would be instructive if you read up on monopolist with price discrimination in your school textbook. If you do not have a text book, try reading articles from the Internet, for example,
http://www.economicsdiscussion.net/monopoly/price-discriminating-monopoly-economics/25670
focus on discrimination of the third degree.
A monopolist is deciding how to allocate output between two market that are separated geography.demands for the two markets are p1=15-q1 and p2=25-2q2.the monopolist TC is c=5+3(q1+q2).what are price,output,profits,and mr if:a)the monopolist discriminate?
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