Get Strong is the only gym in the neighborhood. It offers different memberships, which differ in price, and in the times a month the members can access the gym facilities. In the same neighborhood there are two types of consumers: Fitness enthusiasts and regular gym-goers.

The customers have a utility function Ui(s,F) = Wi(s)F, where F is the fee, i E,R is the client type, and s is the number of sessions per months. Wi(s) is the willingness to pay of each consumer type i for a package with a given number of sessions s. The willingness to pay for gym Enthusiasts is WE(s) = 30s (s^2)/2 , and for regular gym-goers WR(s) = 15s (s^2)/2 . The utility of not going to the gym at all, and paying nothing, is normalized to zero. The marginal cost of a having a client accessing the gym per day is zero.

1. Suppose the Get Strong gym is able to observe the type of consumer and can offer two different memberships with number of sessions si at fixed monthly fee Fi. What quantities and fees maximize the firm's profits?
2. Hint: take into account that you do not need to compute the price for one time of gym access, but a fee to access the gym si days in a month.
3. Now suppose that the gym cannot observe the consumer types, and so it cannot force them to pick a specific package. If the same packages you found in item 1 were offered, which of the two would each of the consumer types choose? Hint: you can apply the utility function to the different packages.
4. Now suppose that a local law forces them to offer only two packages, one with s=10 and an- other with s=20. They can choose the fees for each package as they wish, but prices can only vary by package and not by consumer type (as in item 2 above). A market research by Get Strong told them that, in their neighborhood, there are 500 potential enthusiast gym-goers and 500 potential regular gym-goers. Find the fees that maximize profits for Get Strong.Hint: start by making a table with the willingness to pay for each package, for each consumer type.