Where I live, most people do not have swimming pools despite the fact that it gets very hot in the summers. Thus, families, especially those with children, try to find swimming pools in the area. Our local swimming pool offers two ways in which we can get by the entrance guard: We can either purchase a “family pass” for the whole season, or we can pay an entrance fee for the family every time we want to go swimming.
A: Suppose we have $1,000 to spend on activities to amuse ourselves during the summer, and suppose that there are exactly 100 days during the summerwhen the swimming pool is open and usable. The family passes costs $750 while the daily passes cost $10 each (for the whole family).
(a) With “days swimming” on the horizontal axis and “dollars spent on other amusements” on the vertical, illustrate our budget constraint if we choose not to buy the season pass.
(b) On the same graph, illustrate the budget constraint we face if we choose to purchase the season pass.
(c) After careful consideration, we decided that we really did not prefer one option over the other —so we flipped a coin with “heads” leading to the season pass and “tails” to no season pass. The coin came up “tails”—so we did not buy the season pass. Would we have gone swimming more or less had the coin come up “heads” instead? Illustrate your answer on your graph.
(d) My brother bought the season pass. After the summer passed by, my mother said: “I just can’t understand how two kids can turn out so differently. One of them spends all his time during the summer at the swimming pool, while the other barely went at all.” One possible explanation for my mother’s observation is certainly that I am very different than my brother. The other is that we simply faced different circumstances but are actually quite alike. Could the latter be true without large substitution effects?
(e) On a separate graph, illustrate the compensated (Hicksian) demand curve that corresponds to the utility level u∗ that my family reached during the summer. Given that we paid $10 per day at the pool, illustrate the consumer surplus we came away with from the summer experience at the pool.
(f) Since we would have had to pay no entrance fee had we bought the season pass, can you identify the consumer surplus we would have gotten? (Hint: Keep in mind that, once you have the season pass, the price for going to the pool on any day is zero. The cost of the season pass is therefore not relevant for your answer to this part.)
(g) Can you identify an area in the graph that represents how much the season pass was?
B: Suppose that my tastes can be represented by the utility function u(x1, x2) = x1a x2(1−α) with x1 denoting days of swimming and x2 denoting dollars spent on other amusements.
(a) In the absence of the possibility of a season pass, what would be the optimal number of days for my family to go swimming in the summer? (Your answer should be in terms of α.)
(b) Derive my indirect utility as a function of α.
(c) Suppose α = 0.5. How much utility do I get out of my $1,000 of amusement funds? How often do I go to the swimming pool?
(d) Now suppose I had bought the season pass instead (for $750). How much utility would I have received from my $1,000 amusement funds?
(e) What is my marginal willingness to pay for days at the pool if I am going 100 times?
(f) On a graph with the compensated demand curve corresponding to my utility this summer, label the horizontal and vertical components of the points that correspond to me taking the season pass and the ones corresponding tome paying a per-use fee.
(g) Derive the expenditure function for this problem in terms of p1, p2 and u (with α = 0.5).
(h) In (g) of part A, you identified the area in the MWTP graph that represents the cost of the season pass. Can you now verify mathematically that this area is indeed equal to $750? (Hint: If you have drawn and labeled your graph correctly, the season pass fee is equal to an area composed of two parts: a rectangle equal to 2.5 times 100, and an area to the left of the compensated demand curve between 2.5 and 10 on the vertical axis. The latter is equal to the difference between the expenditure function E (p1, p2, u) evaluated at p1 = 2.5 and p1 = 10 (with p2 = 1 and u equal to the correct utility value associated with the indifference curve in your earlier graph.))