Question

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

Visit number	Value of visit, $
1	20
2	19
3	18
4	17
5	16
6	15
7	14
8	13
9	12
10	11
11	10
12	9
13	8
14	7
15	6
16	5
17	4
18	3
19	2
20	1
21	0

a)

If John does not buy the membership,

He will increase the number of visits as long as the value of visit is higher than or equal to price of visit i.e. $10. We can see that value of 11th visit is equal to $10. So, optimal number of visits is 11.

Consumer surplus=(20-10)+(19-10)+(18-10)+(17-10)+(16-10)+(15-10)+(14-10)+(13-10)+(12-10)+(11-10)+(10-10)=$55

b)

If John buys the membership,

He will increase the number of visits as long as the value of visit is higher than or equal to price of visit i.e. $5. We can see that value of 16th visit is equal to $5. So, optimal number of visits is 16.

Consumer surplus=(20-5)+(19-5)+(18-5)+(17-5)+(16-5)+(15-5)+(14-5)+(13-5)+(12-5)+(11-5)+(10-5)+(9-5)+(8-5)+(7-5)+(6-5)+(5-5)-50=$70

Consumer Surplus is higher if he goes for annual membership. He should buy the membership