Compute the optimal profit maximising menu under complete

Question

Question: Electricity is provided by a monopoly There are two types of consumers, Θ_H and Θ_L, with utility functions over quantity-price bundles (q, t) given by

u_H = Θ_H ln(q + 1) - t, u_L = Θ_L ln(q + 1) - t.

The marginal cost of producing electricity is c > 0 (let's ignore fixed costs). The monopolist believes both types of consumers occur with the same probability.

(a) Compute the optimal (profit maximising) menu under complete information. Determine the monopolist's profits when Θ_H = 6, Θ_L = 4, and c = 1.

(b) Assume the monopolist continues to offer the package designed for the eu-consumer in your solution to part (a). What is the profit maximising bundle offered to the 0H-consumer, assuming that the monopolist cannot sort out consumers (i.e., assuming there is asymmetric information). Determine the monopolist's associated profits when Θ_H = 6, Θ_L = 4, and c = 1. Hint: with the bundle for the Θ_L type fixed, write the monopolist profit maximisation problem for the eH type and impose the IR and IC conditions for the high type.

(c) Find the optimal (profit maximising) menu under asymmetric information. De- termine the monopolist's associated profits when Θ_H = 6, Θ_L = 4, and c = 1.

(d) Rank the three profit levels found in (a), (b) and (c).