Here's how gym membership works for a potential customer: If the customer decides to join the gym, she will pay J onday 0. Then, she can use the gym starting the next day for 5 days (1; 2; 3; 4; 5), paying an additional F on each day shevisits. If she goes to the gym on any given day, it costs her (10+ F) that day, but beneÖts her 30 the next day. In otherwords, her "costs" are a sum of her psychic costs (10) and actual Onancial costs (F). Assume that, when indi§erent, theindividual will go to the gym.(a) Suppose the potential customer is a standard exponential discounter with a discount factor of = 12. If gymmembership was free (i.e. J = 0 and F = 0) would she join the gym? If so, how many days would she actually go?(b) Suppose the gym did not charge a joining fee. How high could it set the usage fee? What would its revenues from thiscustomer be?(c) Suppose instead the gym decided not to charge a usage fee. How high could it set the joining fee? [Hint: To Ögure outthe maximum the customer is willing to pay on day 0, think about her discounted utility from future gym usage.] Based onyour answer, determine how the gym should set its fees.(d) Now imagine the individual is a hyperbolic discounter with = 12 (same as before) and = 12. If the gym membershipwas totally free, how many days would she like to go to the gym (from her period-0 perspective)? How many days will sheactually go to the gym?(e) Suppose she is a naive hyperbolic discounter. How should the gym set its prices? [Hint: Is there any point to charging ausage fee? If not, what is the highest joining fee the individual will be willing to pay?](f) Suppose she is a sophisticated hyperbolic discounter. Now, is there any combination of J and F that will allow the gymto make money from her? What if the gym was to set a negative value of F? [Donít solve, just think about it and explain inwords.]

Question

Here's how gym membership works for a potential customer: If the customer decides to join the gym, she will pay J onday 0. Then, she can use the gym starting the next day for 5 days (1; 2; 3; 4; 5), paying an additional F on each day shevisits. If she goes to the gym on any given day, it costs her (10+ F) that day, but beneÖts her 30 the next day. In otherwords, her &#34;costs&#34; are a sum of her psychic costs (10) and actual Onancial costs (F). Assume that, when indi§erent, theindividual will go to the gym.(a) Suppose the potential customer is a standard exponential discounter with a discount factor of = 12. If gymmembership was free (i.e. J = 0 and F = 0) would she join the gym? If so, how many days would she actually go?(b) Suppose the gym did not charge a joining fee. How high could it set the usage fee? What would its revenues from thiscustomer be?(c) Suppose instead the gym decided not to charge a usage fee. How high could it set the joining fee? [Hint: To Ögure outthe maximum the customer is willing to pay on day 0, think about her discounted utility from future gym usage.] Based onyour answer, determine how the gym should set its fees.(d) Now imagine the individual is a hyperbolic discounter with = 12 (same as before) and = 12. If the gym membershipwas totally free, how many days would she like to go to the gym (from her period-0 perspective)? How many days will sheactually go to the gym?(e) Suppose she is a naive hyperbolic discounter. How should the gym set its prices? [Hint: Is there any point to charging ausage fee? If not, what is the highest joining fee the individual will be willing to pay?](f) Suppose she is a sophisticated hyperbolic discounter. Now, is there any combination of J and F that will allow the gymto make money from her? What if the gym was to set a negative value of F? [Donít solve, just think about it and explain inwords.]

Accepted Answer

Given that the customer is a standard exponential discounter with a discount factor of Δ = 1/2, we…

we hyperbolic discounter. H