Lihini is a CEO of an opera house called the ‘Lihini’s Opera House”. It is the only opera house in the region, giving Lihini’s Opera House a monopoly position in the market. Each customer’s demand curve for shows per year is given by the following: P = 132 – 6Q. The marginal cost of an extra visitor in a show (i.e., of an extra ticket sold) is $20. Lihini is considering a new pricing scheme for her business: a membership fee that allows people to purchase tickets to the shows after which they can purchase as many opera tickets as they wish for $20 each. If Lihini wants to maximize profits, what is the optimal membership fee that she should charge?

[Round your final answer to 2 decimal points when necessary]

Question

Lihini is a CEO of an opera house called the ‘Lihini’s Opera House”. It is the only opera house in the region, giving Lihini’s Opera House a monopoly position in the market. Each customer’s demand curve for shows per year is given by the following: P = 132 – 6Q. The marginal cost of an extra visitor in a show (i.e., of an extra ticket sold) is $20. Lihini is considering a new pricing scheme for her business: a membership fee that allows people to purchase tickets to the shows after which they can purchase as many opera tickets as they wish for $20 each. If Lihini wants to maximize profits, what is the optimal membership fee that she should charge?

[Round your final answer to 2 decimal points when necessary]

BlackTom AI · Accepted Answer

We’re evaluating a two-part tariff (membership fee plus per-ticket price) for a monopoly with demand P = 132 − 6Q and constant marginal cost MC = 20 per ticket.

First, set the per-ticket price equal to MC to maximize welfare per unit and allow the fixed membership fee to capture the entire consumer surplus. So choose p = MC = 20.

With p = 20, solve for the quantity demanded Q from the inverse demand P = 132 − 6Q:
20 = 132 − 6Q → 6Q = 112 → Q = 112/6 = 18.666... (approximately 18.67 tickets per member).

Consumer surplus (CS) per member at this price is the area under the demand curve above the price of 20, up to Q = 18.666.... For a linear demand, CS = 0.5 × (maximum price intercept − current price) × Q. The maximum price intercept when Q = 0 is 132, so:
CS = 0.5 × (132 − 20) × Q = 0.5 × 112 × (112/6) = 56 × (112/6) = 6272/6 = 1045.333...

In a two-part tariff, the monopolist should set the fixed membership fee equal to the consumer surplus to extract all surplus while keeping the per-ticket price at MC. Therefore, the optimal membership fee is CS = 1045.333..., which rounds to 1,045.33 when stated to two decimal places.

Note: This analysis assumes a single market, no membership constraints beyond the fixed fee, and that every member buys up to the available demand at the marginal cost price; in other words, all potential buyers participate and pay the fixed fee to enjoy the per-ticket price of 20. The key steps are (1) set p = MC, (2) compute Q from P = 132 − 6Q at p = 20, (3) compute CS at that Q, and (4) set the membership fee equal to CS.

BUSS1040 (ND) Quiz 3: Topics 5-7

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