Question

I have question about answer ? please see every ***** I noted below:

question :A monopolist is deciding how to allocate output between two
geographically separated markets (East Coast and West Coast). Demand for thetwo markets are:

P1 = 10 – 0.25Q1 and P2 = 15 – Q2

The corresponding aggregate demand curve is given as P = 11 – 0.2Q, where Q =Q1+Q2.
The monopolist’s marginal cost is fixed at $5 and there are no fixed costs. What are
price, output, and profits

(i) if the monopolist can price discriminate?

(ii) if the lawprohibits charging different prices in the two regions?

Answer:

Marginal revenue:
P1 =10 -0.25Q1 implies MR1 =10 -0.5Q1 *******(why is become 0.5Q1?)
P2 =15 -Q2 implies MR2 =10 -2Q2 *******(why here MR2= 10- 2Q2 how do you get 2Q2????????)

Choose quantity in each market such that marginal revenue is equal to marginal cost. The
marginal cost is equal to 5. The profit-maximizing quantities in the two markets are:
10 -0.5Q1 = 5, or Q1 10 in Market 1,

and
15 -2Q2 5, or Q2 5 in Market 2.

Substituting into the respective demand equations, prices for the two markets are:
P1= 10- 0.25(10)= $7.5, and P2 15- 5= $10.
Noting that the total quantity produced is 11.5, profit is
=$7.50(10)+ $10(5)- 5(15)= $50.

(ii) Without price discrimination the monopolist must charge a single price for the entire
market. To maximize profit, find the quantity such that marginal revenue is equal to
marginal cost. Adding demand equations,

P = 11-0.2Q ******(where get this ?)
This implies marginal revenue equations of
MR = 11 -0.4Q ******(where get this ?)

With marginal cost equal to 3, equate marginal revenue and marginal cost:
11- 0.4Q= 5, or Q 15.
Substituting the profit-maximizing quantity into the demand equation to determine
price:
P =11- (0.2)(15)= $8.
Profit is
$8(15) -5(15) =$45.

Answer 1

P1 = 10 – 0.25Q1

Total Revenue: TR = P1*Q1 = (10 - 0.25Q1)*Q1 = 10Q1 - 0.25Q1²

Marginal Revenue can be derived by differentiating TR with respect to Q1

MR = dTR/dQ1 = 10 - 0.5Q1

Similarly,

P2 = 15 – Q2

Total Revenue: TR = P2*Q2 = (15 - Q2)*Q2 = 15Q2 - Q2²

Marginal Revenue can be derived by differentiating TR with respect to Q2

MR = dTR/dQ2 = 15 - 2Q2

ii)

Without price discrimination, the monopolist will have to charge a single price and thus will optimize on the aggregate demand function:  P = 11 – 0.2Q

Total Revenue: TR = P*Q = (11 - 0.2Q)*Q = 11Q - 0.2Q²

Marginal Revenue can be derived by differentiating TR with respect to Q

MR = dTR/dQ = 11 - 0.4Q

I hope this was helpful.