Question

Dana's monthly demand curve for the gym is: p=10-q where p is the price and q is the quantity. The market price for a single visit is 3 RMB. Monthly membership (you can enter the gym for free if you have the membership) cost 22 RMB. Should Dana pay for the monthly membership? Prove.

Answer 1

Solution:

Demand curve for Dana is; Qd: p = 10-q

Given price for every single visit without any membership; 3RMB

Monthly Membership: 22RMB

Now, let us assume that the price for Gym service is 0, as in, it is free for Dana to go to the gym any number of times she wishes to.

0 = 10 - q

q = 10

Therefore, the maximum number of times Dana visits the gym even if it is free, is 10 Days in a month.

Now, let us assume that Dana does not pay for the membership, but prefers to pay every time she visits (3RMB)

3 = 10 - q

-7 = -q

q = 7

Therefore she visits the gym 7 days in a month if she were to pay every time she uses it and does not buy a membership.

Now, If Dana were to pay for a member ship, her demand function would be: 22 = 10 - q

q = -12 , which indicates that she would miss out the days for which she paid the membership for, which makes the monthly membership useless for Dana as she does not uses the Gym on the days she pays for.

This thus makes the monthly membership option unnecessary and Dana doesn't need to pay for one. Dana can thus opt to pay on singly visit basis and utilize the Gym on the days she actually pays for.