Question

Consider a monopolist selling sweets to two types of consumers (assume sweets are

infinitely divisible, so that the monopolist can sell any nonnegative, real quantity).

The demand function of consumers of type 1’s is p1(q1) = 9 − 3q1.

The demand function of consumers of type 2’s is p2(q2) = 8 − 5q2.

The monopolist can produce sweets at no cost.

There are equal numbers of both types of consumers.

SOLVE by using calculus please

SHOW STEP-BY-STEP solution please

(a) Suppose the monopolist can distinguish between the two types of

consumers, but now, because of regulation, it must supply any amount of sweets

that the consumers choose (i.e., it cannot restrict consumers’ choice to a menu of

fixed packages). Thus, the monopolist will charge each type of consumer a different

constant price per kilo of sweets.

Furthermore, the monopolist is not allowed to charge consumers any fixed fees.

What price will the monopolist set for each type?

What quantities will be consumed?

What profit will the monopolist earn per consumer?

Answer 1

For a single price monopolist, p1 = p2 = P and market quantity (Q) = q1 + q2

For Type 1,

P = 9 - 3q1

3q1 = 9 - P

q1 = (9 - P)/3

For Type 2,

P = 8 - 5q2

5q2 = 8 - P

q2 = (8 - P)/5

Q = q1 + q2 = [(9 - P)/3] + [(8 - P)/5]

Q = (45 - 5P + 24 - 3P)/15

Q = (69 - 8P)/15

15Q = 69 - 8P

8P = 69 - 15Q

P = (69 - 15Q)/8

Profit is maximized when Marginal revenue (MR) = Marginal cost (MC) = 0.

Total revenue (TR) = P x Q = (69Q - 15Q²)/8

MR = dTR/dQ = (69 - 30Q)/8

Equating with MC,

(69 - 30Q)/8 = 0

69 - 30Q = 0

30Q = 69

Q = 2.3

P = [(69 - (15 x 2.3)] / 8 = (69 - 34.5) / 8 = 34.5 / 8 = 4.3125

Total Profit = TR (since cost is zero) = P x Q

Profit per consumer = TR/Q = (PQ)/Q = P = 4.3125