Suppose that the gym's strategy is to offer personal training sessions at two price levels: all times during the day other than the 5pm-9pm slot will be priced at $50 per hour, while sessions between 5pm-9pm will be priced at some other level, which is to be determined.

Construct a linear demand model for the aggregate potential demand for personal training sessions between 5pm and 9pm from all member segments under the assumption that the prices for all other products is $50.

By aggregate, we mean the demand from all three member segments, i.e., students, members with fixed schedules, and members with flexible schedules. The way members decide which product to buy is as follows: each member calculates her net utility from a personal training session at a different time slot, which is given by her willingness-to-pay - the price to pay. She then selects the option that maximizes her net utility; if all options result in negative utility for a particular member, then that member does not buy at all. Fixing the price for all time periods except 5pm-9pm to $50, you can compute the net utilities for all these products for each member that participated in the survey. You can then vary the price for the 5pm-9pm slot and for each possible choice count the number (or even better the fraction) of members that would choose a session in the 5pm-9pm period. You should then try to approximate this demand function by a linear relation of the form d(p) = D - b*p, where d(p) is the fraction of members that purchase the 5pm-9pm session at price $p, D is the maximum potential demand when $p = $0, and b is the price sensitivity parameter that explain how the demand drops as the price increases. The goal is to find D and b.