Problem 3 Sam is a personal trainer working for private clients. He just moved to a new city and he is building his new client base. A colleague tells him that in this city a personal trainer typically faces an inverse demand P = 360 lUQ. That is when working with Q clients, a personal trainer can nd one more client willing to pay P = 300 lUQ for a weekly training session. Sam's opportunity cost of a two-hour training session is MC = ZDQ. We make the heroic assumption that clients are innitely divisible so that we can use calculus to solve Sam's prot maximizing problems. At rst, Sam doesn't know his clients that well and he can't tell who is willing to pay more money for a training sessions. Hence, he charges all clients the same price. a) In part naively in part to get to know more people, Sam decides to sign up any customer willing to pay a price at least as high as his marginal cost of time. That is, Sam behaves like a pticc~taken How many clients does he signals? How much does he charge for a training session? What is Sam's producer surplus? As time goes by, Sam realizes that he can be more discerning when signing up clients. In economic parlance, he now behaves like a one-price monopolist. b) How many clients does he keep working with? How much does he charge them per session? What is his producer surplus? In a year or so. Sam has his client based gured out and starts charging each client on a sliding scale. In economic parlance he behaves like a perfectly price discriminating monopolist. c) How many clients does Sam work with each week? What is the highest price he charges? What is the lowest price he charges? What is Sam's prot