Question 3 (Problem 3, page 349 of the coursebook) Five civic-minded patrons of a public library contemplate donations of time to its annual fundraiser. Each individual i bases his or her decision of how much time to donate, xi, on the following utility function: Ui(zi) = q:25 -.25xi where q = =1*; (the total amount of time given by all library patrons) and .25x; is the cost of losing x of one's leisure time. 1. What is the socially optimal amount of time donated? Determine this by summing the utility functions of five individuals, and finding the q that maximizes this function. 2. Now consider the case of an individual who assumes that everyone else will contribute no time. How much time will this individual donate, and is it at the socially optimal level? 3. Now assume that each individual assumes that the other four members will donate .8 units of time. Does this individual donate more or less time than before? Why is this? 4. Why is a socially suboptimal amount of time donated in parts 2 and 3? Consider the balance between internalized costs and internalized benefits. 5. [Bonus] Repeat the same exercise for the case of ten individuals. What proportion of the socially optimal amount is donated in an equilibrium now? What does this suggest about the difficulties of supplying quasi-public goods as the number of people increases