Assume quantities need not be integers. You are a profit-maximizing monopolist who operates a swimming pool and whose marginal cost equals $2. Each customer has demand MWTP(Q) = 23 – 0.5 x Q. You have chosen the profit-maximizing two-part tariff. What is the highest membership fee consumers would be willing to pay to join the swimming club?

Round to two decimal places. If your answer is 1.125, enter 1.13.

Question

Assume quantities need not be integers. You are a profit-maximizing monopolist who operates a swimming pool and whose marginal cost equals $2. Each customer has demand MWTP(Q) = 23 – 0.5 x Q. You have chosen the profit-maximizing two-part tariff. What is the highest membership fee consumers would be willing to pay to join the swimming club?

Round to two decimal places. If your answer is 1.125, enter 1.13.

BlackTom AI · Accepted Answer

To begin, restate the problem in my own words and outline the approach we’ll use.
We’re given a monopolist using a two-part tariff: a per-unit usage price and a fixed membership fee. The marginal cost is MC = 2, and the individual demand is defined by MWTP(Q) = 23 − 0.5Q, where Q is the quantity consumed.
First, in a two-part tariff, the monopolist typically sets the per-unit price equal to marginal cost to maximize usage and then extracts the entire consumer surplus through the fixed fee. So, we set the price charged for each unit, P, to P = MC = 2.
Next, determine the quantity demanded at this price using the demand relation: P = 23 − 0.5Q. Substituting P = 2 gives 2 = 23 − 0.5Q, which implies 0.5Q = 21 and thus Q = 42.
Now compute the consumer surplus at this quantity, which captures the maximum fixed fee a consumer would be willing to pay while still participating under the tariff. CS is found by integrating the demand function up to Q and then subtracting the expenditure P·Q: CS = ∫0^Q [23 − 0.5x] dx − P·Q.
Evaluate the integral: ∫0^Q (23 − 0.5x) dx = 23Q − 0.25Q^2. With Q = 42, this becomes 23(42) − 0.25(42^2).
Compute the numbers: 23 × 42 = 966, and 42^2 = 1764, so 0.25 × 1764 = 441. Therefore, the integral equals 966 − 441 = 525.
Subtract the expenditure at price P = 2: P·Q = 2 × 42 = 84. Finally, CS = 525 − 84 = 441.
Thus, the highest membership fee consumers would be willing to pay, i.e., the fixed fee, is 441.00 when rounded to two decimals.
If you want to double-check the arithmetic, you can re-run the steps: Q = (23 − 2)/0.5 = 21/0.5 = 42; CS = [23Q − 0.25Q^2] − 2Q = (23 × 42 − 0.25 × 42^2) − 84 = (966 − 441) − 84 = 525 − 84 = 441.
Overall, the two-part tariff yields a fixed fee equal to the consumer surplus at the usage price equal to MC, which in this setup is 441.00.

ECO101H1 S LEC0101 topic quiz: price discrimination and game-theory

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