Question

Consider a monopolist facing two types of consumers. Normalise the total population to 1. Type one consumers are in proportion 1/2, and type two are in proportion 1/2. The monopolist has marginal cost of production c = 1/2. The two types have demand curves

q₁ =1-p

q₂ =1-(p/2).

If the monopolist can identify the two types and can charge different two-part tariffs to different types: {A1, p1} and {A2, p2}. [All type one consumers are identical and have the q1 demand curve, All type 2 consumers are identical and have the q2 demand curve. When one consumer shows up, the firm knows exactly his type and can discriminate directly.]

1) Consider the situation that the monopolist cannot distinguish between the two types of consumers and wishes to serve both types with one two part tariff. That is, the monopolist knows that there are two types of consumers with the above mentioned demand curves and also is aware of the proportions of each type of consumers. But the monopolist cannot distinguish the two types of consumers and cannot price discriminate directly. The monopolist now offers one two part tariff {A, p} to serve both types. The optimal p is equal to?

2) In the two-part tariff in Q1, the optimal fixed fee A is equal to?

3) If the monopolist now offers two pricing plans {T₁,q₁} and {T₂,q₂}. That is, the consumers have two options: paying T1 for q1 (fixed quantity) of goods or paying T2 for a fixed q2 quantity of the good. For example {T1 = 10, q1 = 8} refers to the pricing plan such that the consumer pays $10 for 8 units of the good.

Take your answer from Q1 and Q2 (the one where the monopolist offers one two part tariff to both types of consumers). Set T1 to be the total expenditure from type 1 consumer under that tariff, that is T₁=A+pq₁ where q1 is the quantity demanded by type 1 under the previous tariff.

With {T₁,q₁} given as the above, we now want to work out the optimal  {T₂,q₂}. The optimal T2 is?

4) And the resulting q2 is?

Answer 1