In a remote town, Mr. Knightley opened an amusement park where various kinds of mechanical rides are provided. There are N identical residents living in the town, and they are the only people who could possibly visit the amusement park. With the assistance of an economic consultant, Mr. Knightley has figured out that each local resident has a monetary income M and a utility function U = C + R^2, where C is the quantity of consumption goods and R is the number of rides enjoyed in the amusement park. Consumption goods can be bought on the market at a price equal to 1.
To operate the amusement park, Mr. Knightley has to pay a fixed cost F and a constant marginal cost k. That is, to provide Q rides, his total costs are F + kQ.
(a) Suppose Mr. Knightley charges a price P for each ride, find a resident's demand for rides and for consumption goods, R* and C*. Given each resident's R*, what is Mr. Knightley's optimal price in order to maximize profit?
The consultant suggested that Mr. Knightley should consider an alternative pricing scheme: Charge each person a one-time entrance fee E for entering the amusement park, and then charge a price P for each ride the person enjoys in the park.
(b) Under this alternative pricing scheme, find a resident's demand for rides and for consumption goods, R** and C**.
Knowing each person's demand for rides under the alternative pricing scheme, Mr. Knightley immediately realized that given the price P for each ride, profit will increase with the entrance fee E. However, his consultant reminded him that he cannot increase the entrance fee E too high because after a certain point, people would choose not to enter the park at all.
(c) For an arbitrary price P of a ride, find the maximum entrance fee Mr. Knightley could charge so that a resident will still visit his amusement park.
(d) Given the result in part (c), find the optimal E and P Mr. Knightley will charge in order to maximize profit.
(e) Under which pricing scheme can Mr. Knightley make more profit?
