Final answer:
To find the profit-maximizing two-part tariff for Elvis Land's admission, we set marginal cost equal to marginal revenue to determine the optimal quantity of rides, which also leads to the per-ride price. The admission fee is then the consumer surplus per person, resulting from the difference between the price per ride and the marginal cost, over the total number of rides. For 1,000 interested individuals, Elvis Land's admission fee is calculated to be $28.125 per person.
Step-by-step explanation:
To determine the profit-maximizing two-part tariff for admission to Elvis Land, we need to calculate the price per ride to maximize revenue and the corresponding admission fee that would maximize profit.
The inverse demand function given is ply) = 50 - y, where y is the number of rides. To find the price per ride that maximizes revenue, we first identify the price that would make demand unitary elastic. We know that the quantity of rides demanded will be maximized when the per-ride price is set equal to the marginal cost of the rides. The marginal cost from the cost function cly) = 20y + y^2 is found by differentiating the cost function with respect to y, which is MC = 20 + 2y. To find the optimal quantity y* , set MC equal to the marginal revenue that is derived from the inverse demand function, which yields:
- MR = 50 - 2y
- Set 20 + 2y = 50 - 2y
- Solve for y, yielding y* = 7.5.
Once we have the optimal quantity, the price per ride is found using the inverse demand function: p* = 50 - 7.5 = $42.50.
To calculate the admission fee, we subtract the marginal cost at y* from the price per ride: 42.50 - (20 + (2 × 7.5)) = $42.50 - $35 = $7.50. The consumer's surplus is the area of the triangle above the price per ride and below the demand curve: (1/2 × (50-42.50) × 7.5 = $(28.125). This is the maximum admission fee for profit maximization. Hence, the admission fee for Elvis Land is $28.125 multiplied by the 1,000 people interested, resulting in a total admission revenue of $28,125.