QI. A small island, Whiteknights, has two residents, Alex and Sarah. They consume only two goods, cake x and coffee y and in total 120 units are available for each good. Initially Alex consumes two-third of the cake and one-third of coffee and Sarah consumes the rest of the cake and coffee. Alex and Sarah have different preferences, which are represented by the utility function. Alex's utility function is = X 2/5)' 3/5 and Sarah's one is = x 1/4)' 3/4 . [30 marks] a) [5 marks] Is the initial consumption efficient in terms of consumption egalitarianism o r welfare egalitarianism? (We assume that utilities are interpersonally comparable.) b) [10 marks] Derive Alex's and Sarah's marginal rate of substitution of cake for coffee. Define the contract curve and draw it on the Edgeworth box. Is the initial allocation Pa reto efficient? Justify your answer. c) [5 marks] Characterize the set of allocations of cake and coffee which Pareto improve the initial allocation. d) [10 marks] Now there is a market price for each good and Alex and Sarah are price-ta kers. The price of cake is Px=l and the price of coffee is Py=2. Are these prices in equi librium? Justify your answer.