Suppose that to be able to buy lunch at a University Faculty Club, a professor must pay a yearly membership fee of L dollars. Once a professor is a member, he/she can buy a standard lunch at the price of p (set by the Faculty Club). Assume that there are two types of professors: old professors, and young professors. The inverse demand function of a typical young professor is:p = 72 - q1 where q1 is the quantity of lunches (at the Faculty Club) demanded by a young professor per year, and p is the Faculty Clubs fixed price for each lunch.

The inverse demand function of a typical old professor is: p = 92 q2 where q2 is quantity of lunches (at the Faculty Club) demanded by an old professor per year, and p is the Faculty Clubs fixed price for each lunch. The university's regulation is that the Faculty Club cannot charge discriminatory lunch prices, and cannot charge different membership fees for different customers. Assume there are 10 young professors and 10 old professors. The Faculty Club wants to maximize its profit. The Faculty Clubs marginal cost of a lunch is $25 and it is required that the price p be not less than the marginal cost.

(i) Find the profit-maximizing solution p and L . Find the Faculty Club's total profit.

(ii) Suppose there is a change in preferences, such that a typical young professor's demand is now lower than before: p = 52 - q1. Under this preference scenario, what are the Faculty Club's optimal membership fee, L, and fixed price per lunch, p, for members? What is the Faculty Clubs total profit?