In a small town called Lake Gift all the residents live around a lake as illustrated in Figure 10.7. For over 200 years, the town has the tradition 1 2 3 ✾ ✠ ✎ η η − 1 ■  ❥ ✒ >?>? >?>? >?>? >?>? Figure 10.7: Lake Gift. of giving gifts every Christmas in a counterclockwise pattern. That is, resident 1 gives a gift to resident 2, resident 2 gives a gift to resident 3, and so on. Resident η gives a gift to resident 1. Assume that there are λ products to buy in Lake Gift. Suppose that each resident of Lake Gift has a utility function of receiving a gift given by (10.10), and a utility function of giving a gift given by (10.12). Answer the following questions.

(a) Calculate the total number of gifts exchanged every Christmas in Lake Gift.

(b) Calculate eachindividual’s expected utility from receiving a gift.

(c) Calculate the social loss function of Lake Gift, and demonstrate how the social loss varies with an in increase in the population size (i.e., an increase in η).