Consider the problem of the sophisticate who has a membership at the gym. Let us limit the problem to four periods. Exercising in any period yields an immediate cost of c and a reward in all subsequent periods of v. Suppose that the member must pay k0 each week as part of the gym membership and that he cannot easily rescind his membership.
a. Suppose that the gym offers two potential gym memberships, one that results in a weekly cost of k0, and another that adds k1 to the weekly cost. However, for this additional fee, the gym agrees to fine the gym member k2 after any week in which he does not use the gym. Solve for the maximum amount k1 and minimum amount k2 such that the sophisticate will commit for periods 2, 3 and 4.
b. Now consider the problem of the entrepreneur who, for a small fee, will pose as the gym member, allowing him to shirk his commitment without penalty. Solve for the maximum fee k3 that entrepreneur can charge given the amounts k1 and k2 in part a.