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 11.4223) = (2%?) Is, while her daught er's utility is and (Id) = @, where I; and Id are nonnegative parts of the pizza consumed by her son and daughter, respectively. Of course, the total consumption cannot exceed the size of the pizza, is + Id 5 1. 1. Characterize all Paretoeicient allocations. Explain. (Hint: is (maa) = (.8, .2) Pareto efcient? How about ($3,221) = (.8, .1)?} 2. Derive the Utility Possibility Frontier. That is, consider a space with son's utility us on the xaxis and the daughter's utility \"d on the y axis. The UPF function will tell us what utility level the daughter can obtainJ for a given level of utility of the sonJ denoted \"1055); for any Pareto efcient allocation that you characterized in point 1 above. Illustrate graphically.1 3. Suppose that the mother wants to maximize the utilitarian welfare of her children what is the best division of the pizza? Show that point on the UPF.2 1You should get and = 1 u\"? (Ev/a. If you do not feel condent about your algebra: try to graph the UPF in a u,,ud space: using powers of Excel. Namely: you may check all possible divisions 3d = 000,031,032, etc. and see what utility levels the children get and graph them with u, on the horizontal axis and not on the vertical. 2Mathemati cal maximization will re quire simple calculus [rst order conditions involv ing derivatives]. If you are not familiar with this technique: use trialanderror method to approximate the answer. Fbr example: you can automatize this process in Excel; you may check all possible divisions 35:: = 0.00,U.01,0.D2, etc. and see when the utilitarian welfare is maximal. 4. Suppose that the mother wants to maximize the maximin welfare what is the best division of the pizza? Show that point on the UPF.3 5. What divisions of pizza are envyfree? What Pareto efcient divisions of pizza are envy-free? Show that point on the UPF. 6. Both maxmin welfare function and envyfreeness have very strong con cerns for equality. Why do they lead to different answers