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 teams) = (2J5) 13,, while her daughter's utility is ad (ma) = fwd, where 3:5 and and are non-negative parts of the pizza consumed by her son and daughter, respectively. Of course, the total consumption cannot exceed the size of the pizza, $5 + 113,; g 1. 1. Characterize all Pareto efcient allocations. Explain. (Hint: is ($5,:cd) = (.8, .2) Pareto efcient? How about ($3,513d) = (8,1)?) 2. Derive the Utility Possibility Frontier. That is, consider a space with son's utility as on the xaxis and the daughter's utility ad 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 ud (us), 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 no; = 1 usf (2m. If you do not feel condent about your algebra, try to graph the UPF in a umud space, using powers of Excel. Namely, you may check all possible divisions and = 0.00,0.01,0.02, etc. and see what utility levels the children get and graph them with us on the horizontal axis and ad on the vertical. 2Mathematical maximization will require simple calculus (rst order conditions involv ing derivatives). If you are not familiar with this technique, use trialanderror method to approximate the answer. For example, you can automatize this process in Excel; you may check all possible divisions 37:: = 0.00,0.01,0.02, 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 envy-free? What Pareto efficient divisions of pizza are envy-free? Show that point on the UPF. 6. Both maxmin welfare function and envy-freeness have very strong con- cerns for equality. Why do they lead to different answers