Norton and Ralph have a utility possibility frontier that is given by the following equation, UR + (UN)^2 = 100.

(a) Plot the utilities possibilities frontier on a graph with UR on the vertical axis and UN on the horizontal axis.

(b) Derive an equation for the slope of the utilities possibilities curve.

(c) Both Ralph and Norton believe that the ideal allocation is given by maximizing an appropriate social welfare function. Ralph thinks that UR = 75, UN = 5 is the best distribution of welfare, and presents the maximization solution to a weighted-sum-of-the-utilities social welfare function that confirms this observation. What was Ralph’s social welfare function? (Hint: What is the slope of Ralph’s social welfare function?)

(d) Norton, on the other hand, believes that UR = 19, UN = 9 is the best distribution of welfare. What is the social welfare function that Norton has in mind?