Question

Suppose a company has a monopoly on yogurt, and sells it to two
types of consumers. The company is unable to identify which type of consumer is
which. One type of consumers really enjoys yogurt, and has an individual demand
for servings of yogurt of Q_h= 20 − 2P. The other type likes yogurt less, and has
an individual demand for servings of Q_l=16−2P. The marginal cost of a serving
of yogurt is 4. The company decides to adopt a menu pricing strategy, meaning it
packages a number of servings of yogurt together and sells the entire package at
some price. It may choose to produce packages of multiple sizes.

1.What is the optimal package size for the company to sell to “low-demand” customers? How much should it charge for this package?

2.If the “high-demand” (type-h) customers purchased the package specified in question 1 above, what would their surplus be?

3.What is the optimal package size to sell to high-demand customers?
What is the optimal price to charge for this “high-demand” package?

4. What is the consumer surplus of each type of consumer?

5. How much profit does the company make for each low-type package
it sells? How much profit does the company make for each high-type package
it sells?

Answer 1

Marginal Cost = 4

High Demand : Qh = 20 -2P

=>P = 10 - 0.5Qh

TR = P*Qh = 10Qh - 0.5 Qh²

MR = 10 -1Qh

Low Demand: Ql = 16 - 2P

=>P = 8 - 0.5Ql

TR =P*Ql = 8Ql -0.5Ql²

MR = 8 -1Ql

1. For monopolist, Equilibrium is at MR = MC

For Optimal Low-demand

8 -1Ql = 4

Ql* = 8 - 4 = 4 units

P* = 8 - 0.5Ql = 8 - 0.5(4) = 8 - 2 =6

2. If customer purchases the package from question 1 , then P = 10 - 0.5(4) = 10 - 2 = 8 units.

Surplus is (8 - 4)= 4 units

3.

10 - 1Qh = 4

=> Qh* = 6 units

P* = 10 - 0.5Qh = 10 - 0.5(6) = 10 - 3 = 7

4.

Consumer surplus in High demand = 1/2*3*6 = 9

Consumer surplus in Low demand = 1/2*2*4 =4

5.

Profit for Low demand = TR - TC

= 8Ql -0.5Ql² - 4Ql

=4Ql - 0.5Ql²

= 4(4) - 0.5(16)

=16 - 8

= 8

Profit for High demand = TR - TC

= 10Qh - 0.5 Qh² - 4Qh

= 6Qh - 0.5 Qh²

=6(6) - 0.5(36)

= 36 - 18

=18