Romeo loves Juliet and Juliet loves Romeo. Besides love, they consume only one good, spaghetti. Romeo likes spaghetti, but he also likes Juliet to be happy and he knows that spaghetti makes her happy. Juliet likes spaghetti, but she also likes Romeo to be happy and
she knows that spaghetti makes Romeo happy. Romeo’s utility function is U_R(S_R, S_J) = S^a_RS^1−aJ and Juliet’s utility function is U_J(S_J, S_R) = S^a_jS_R^1−a, where S_J and S_R are the amount of spaghetti for Romeo and the amount of spaghetti for Juliet respectively. There is a total of 24 units of spaghetti to be divided between Romeo and Juliet.

(a) Suppose that a = 2/3. If Romeo got to allocate the 24 units of spaghetti exactly as he wanted to, how much would he give himself _________? How much would he give Juliet ________? 

(b) If Juliet got to allocate the spaghetti exactly as she wanted to, how much would she take for herself _______? How much would she give Romeo ________?

(c) What are the Pareto optimal allocations?

(d) Suppose that a = 1/3. If Romeo got to allocate the spaghetti, how much would he choose for himself _________. If Juliet got to allocate the spaghetti, how much would she choose for herself _____________? Label the Edgeworth line below, showing the two people’s favorite points and the locus of Pareto optimal points.

(e) When a = 1/3, at the Pareto optimal allocations what do Romeo and Juliet disagree about?