In the 1970s, Disneyland charged an entrance fee to get into the park and then required customers to separately buy tickets for each ride once they were in the park. In the 1980s, Disneyland switched to a different pricing system that continues to this day. Now, customers simply pay an entrance fee and then all rides in the park are free.
A: Suppose you own an amusement park with many rides (and assume, for the sake of simplicity that all rides cost the same to operate.) Suppose further that the maximum number of rides a customer can take on any given day (given how long rides take and how long the average wait times are) is 25. Your typical vacationing customer has some exogenous daily vacation budget I to allocate between rides at your park and other forms of entertainment (that are, for purposes of this problem) bought from vendors other than you. Finally, suppose tastes are quasilinear in amusement park rides.
(a) Draw a demand curve for rides in your park. Suppose you charge no entrance fee and only charge your customers per ride. Indicate the maximum price per ride you could charge while insuring that your consumer will in fact spend all her day riding rides (i.e. ride 25 times).
(b) On your graph, indicate the total amount that the consumer will spend.
(c) Now suppose that you decide you want to keep the price per ride you have been using but you’d also like to charge a separate entrance fee to the park. What is the most you can charge your customer?
(d) Suppose you decide that it is just too much trouble to collect fees for each ride—so you eliminate the price per ride and switch to a system where you only charge an entrance fee to the park. How high an entrance fee can you charge?
(e) How would your analysis change if x1, amusement part rides, is a normal good rather than being quasilinear?
B: Consider a consumer on vacation who visits your amusement park for the day. Suppose her tastes can be summarized by the utility function u(x1,x2) = 10x10.5 + x2 where x1 represents daily rides in the amusement park and x2 represents dollars of other entertainment spending. Suppose further that her exogenous daily budget for entertainment is $100.
(a) Derive the uncompensated and compensated demand functions for x1 and x2.
(b) Suppose again there is only enough time for a customer to ride 25 rides a day in your amusement park and suppose that congestion and wear-and-tear on equipment in the park is not a problem. Suppose then that you’d like your customer to ride as much as possible so he can spread the word on how great your rides are. What price will you set per ride?
(c) How much utility will your consumer attain under your pricing?
(d) Suppose you can also charge an entrance fee to your park—in addition to charging the price per ride you calculated above. How high an entrance fee would you charge? (Hint: You should be evaluating an integral, which draws on some of the material from the appendix.)
(e) Now suppose you decide to make all rides free (knowing that the most rides the consumer can squeeze into a day is 25) and you simply charge an entrance fee to your park. How high an entrance fee will you now charge to your park? (Note: This part is not computationally difficult—it is designated with ** only because you have to use information from the previous part.)
(f) How does your analysis change if the consumer’s tastes instead were given by u(x1, x2) = (3−0.5) x10.5 + x20.5?