3 Parenting Suppose that a mother contemplates how to divide pizza of size 1 between her two children. She may divide it in any way; she may even throw some away. She knows that her son's utility is tr, (1:3) = (Sh/S) 11,, while her daughter's utility is try, (17,1) = ,g'zg, where 11,, and 11,; are nonnegative parts of the pizza consumed by her sen and daughter, respectively. Of course, the total consumption cannot exceed the size of the pizza, I; + 93,; E. 1. 1. Characterize all Pareto efficient allocations. Explain. (Hint: is (Imfd) = (.8, .2) Pareto efficient? How about ($173,175) = (3,1)?) 2. Derive the Utility Possibility Frontier. That is, consider a space with son's utility it,l on the x-axis and the daughter's utility try on the y- axis. The UPF function will tell us what utility level the daughter can obtain, for a given level of utility of the son, denoted rides), for any Pareto efcient allocation that you characterized in point 1 above. Illustrate graphically.l 3. Suppose that the mother wants to maximize the utilitarian welfare of her children what is the best division of the pizza'il Show that point on the UF'F.2 \"You should get ad = 131 at,\"z {EVE}. If you do not feel condent about your algebra, try to graph the UPF in a nun\": space. using powers of Exoel. Namely. you may check all possible divisions me = .l],.l,.2, etc. and see what utility levels the children get and graph them with u. on the horizontal axis and up on the vertical. 2Mathematical maximization will require simple calculus [first order conditions involv- ing derivatives]. If you are not familiar with this technique, use trial-and-error method to approximate the answer. For exampleI you can automatize this process in Exoel; you may check all possible divisions my = ...l..02, etc. and see when the utilitarian welfare is maximal