Question

A monopolist has two specific demanders with demand equations: qA = 10 – p and qB = 10 – 2p. This monopolist implements an optimal two-part tariff pricing scheme, under which demanders pay a fixed fee a for the right to consume the good and a uniform price p for each unit consumed. The monopolist chooses a and p to maximize profits. This monopolist produces at constant average and marginal costs of AC = MC = 2. The monopolist’s profits are ___________ and the average price paid by demander B is ____________.

Answer 1

Inverse demand functions are PA = 10 - QA and PB = 5 - 0.5QB. Under a two part tariff, fixed fee is the consumer surplus of smaller demand which is PB. Hence the profit is given by

Profit function = 2 x F + (P - MC)*(QA + QB)

= 2 x 0.5*(5 - P)*(10 - 2P) + (P - 2)*(20 - 3P)

= 50 + 2P^2 - 20P + 20P - 40 - 3P^2 + 6P

= 10 - P^2 + 6P

Profit is maximum when

2P = 6

P* = 3

Hence the price per unit is 3 and is same for A and B. Fees amount is F = 0.5*(5 - 3)*(10 - 2*3) = $4

Profit = 4*2 + (3 - 2)*(20 - 3*3) = $19