Question

18) Suppose a local swimming pool either charges $10 per person (per visit) OR charges a reduced price of $5 per person (per visit) if that person has an annual membership. The annual membership costs $50. Next, assume John preferences are formed such that he values trips to the swimming pool in the following manner: his first visit is valued at $20 and each subsequent visit is worth $1 less ($19 for visit #2, $18 for visit #3, #1 for the 20th visit, and the 21st visit brings him no value). If John does not buy the membership card, how many times should he visit the pool over the summer? What consumer surplus will he obtain under this scenario? If John purchases the membership, how many times should he go? What consumer surplus will he obtain now? Should John purchase the membership or not?

Answer 1

Solution:

John values each of the visit 1$ lesser than the previous visit. Thus make a table showing the decrease in willingness to pay from 20 to 1 as shown below. Consumer surplus is nothing but the difference between willingness to pay and Price. Calculae the same as below. Suppose he does not purchase the membership card and pays 10$ for every visit, his consumers surplus becomes zero on the 11th visit and he will not wish to go for swimming class after this visit as he is not willing to pay the price. In this scenario, he obtains consumers surplus of 55 which can be found by adding the consumer surplus for each visit from 1 to 11.

Visit	Willingness to pay	Price	Consumer surplus
1	20	10	10
2	19	10	9
3	18	10	8
4	17	10	7
5	16	10	6
6	15	10	5
7	14	10	4
8	13	10	3
9	12	10	2
10	11	10	1
11	10	10	0
12	9	10	-1
13	8	10	-2
14	7	10	-3
15	6	10	-4
16	5	10	-5
17	4	10	-6
18	3	10	-7
19	2	10	-8
20	1	10	-9
21	0	10	-10

Suppose John decides to purchase the membership, the price will be 5$ per visit and the table will be as follows. Here, he will be willing to go for 16 visits till his consumer surplus becomes zero and gains a total consumer surplus of 120.

Visit	Willingness to pay	Price	Consumer surplus
1	20	5	15
2	19	5	14
3	18	5	13
4	17	5	12
5	16	5	11
6	15	5	10
7	14	5	9
8	13	5	8
9	12	5	7
10	11	5	6
11	10	5	5
12	9	5	4
13	8	5	3
14	7	5	2
15	6	5	1
16	5	5	0
17	4	5	-1
18	3	5	-2
19	2	5	-3
20	1	5	-4
21	0	5	-5

John should consider purchasing the membership as it will give him incentive to visit the swimming pool 16 times compared to not purchasing the membership.