Suppose that you are also helping the book store with understanding the customers' preferences for three different scientific fiction novels by the author: "The Three Body Problem", "The Wandering Earth", and "Ball Lightening". The market prices for the three books are respectively 150 HKD, 35 HKD, and 15 HKD. Through collaboration with other data scientists, you find out that customer' expected willingness-to-pays are respectively, 170 HKD, 60 HKD, and 20 HKD. You decide to use the multinomial logit model to describe the customers' choices. Assume for simplicity that the weight for the "No Purchase" option is normalized to 1. Furthermore, assume that customers are similar, and that their expected surplus is simply the difference of the willingness-to-pay and the price. The dispersion parameter is estimated to be 0.04.

(a). What are the customers' probabilities of purchasing each book (i.e., the demand function)?

(b). Suppose that you want to measure customers' relative preference of the "Three Body Problem" to that of the "The Wandering Earth" by the ratio of the probabilities of purchasing the books. For example, if the customers' probability of choosing "The Three Body Problem" is 0.1, and that of choosing "The Wandering Earth" is 0.05, then the customers' relative preference of "The Three Body Problem" to that of the "The Wandering Earth"the should be 0.1/0.05 = 2. What is the customers' relative preference of "The Three Body Problem" to that of the "The Wandering Earth" in this question?

(c). Suppose that the third book "Ball Lightening" is EXCLUDED from customers' choice list (i.e., the customers can only select among the remaining two books and the "No Purchase" option). What are the customers' new probability of choosing "The Three Body Problem" and that of choosing "The Wandering Earth"? What is the customers' relative preference of "The Three Body Problem" to that of the "The Wandering Earth" now?

(d). Following part (a), you are also helping the store with setting the price for "The Three Body Problem". Suppose that the objective is to maximize the total expected revenue from each customer. For example, if the probabilities of purchasing each book are respectively 0.3, 0.2 and 0.1, then the expected revenue would be 150*0.3 + 35*0.2 + 15*0.1 = 53.5 HKD. Holding the rest of the parameters the same, you know that the price range for "The Three Body Problem" should be somewhere between 70 HKD and 150 HKD. Without changing other parameters, what should be the optimal price for "The Three Body Problem"? Hint: you can use any software tools (e.g. Excel) to help you with this problem. Round the solution to the nearest integer.