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 that the monopolist cannot distinguish between

the two consumers, but is allowed to offer fixed packages.

How can the monopolist achieve the highest profit (state quantities and fees for each package)?

What profit will the monopolist earn per consumer?

Answer 1

In this case, profit is maximized if for each group, price equals MC (= 0) and total profit equals entire consumer surplus (CS).

In group 1,

p1 = 9 - 3q1 = 0

3q1 = 9

q1 = 3

p1 = 0

From demand function, when q1 = 0, p1 = 9 (Vertical intercept of demand curve).

CS = Total Profit = Area between demand curve and price = (1/2) x (9 - 0) x 3 = 1.5 x 9 = 13.5

Profit per consumer = CS/q1 = 13.5/3 = 4.5

In group 2,

p2 = 8 - 5q2 = 0

5q2 = 8

q2 = 1.6

p2 = 0

From demand function, when q2 = 0, p2 = 8 (Vertical intercept of demand curve).

CS = Total Profit = Area between demand curve and price = (1/2) x (8 - 0) x 1.6 = 0.8 x 8 = 6.4

Profit per consumer = CS/q2 = 6.4/1.6 = 4