1 Anna and Bryan Annabelle and Bryan are room-mates. Anna has 10 apples and 5 bananas. When she consumes a A apples and bA bananas, her utility is uA (aA, bA) = (aA) 1/4 (bA) 1/2 Notice that this is a Cobb-Douglass function, with Marginal Utilities MU. = (1/4) (aA) /(bA) /2 and MUb = (1/2) (aA) /(bA) , and the Marginal Rate of Substitution (when apples are on the horizontal axis) is MRSanna = -bA/ (204). Bryan has 5 bananas and no apples. He treats these two fruit as perfect substitutes in 1:1 ratio; specifically, his utility function is up (aB, bB) = aB +bB, where (aB, bB) is the bundle of apples and bananas that Brian consumes. 1. Calculate Brian's Marginal Utilities MUa, MUb, and his Marginal Rate of Substitution, MRSBrian 2. Characterise all Pareto efficient allocations. Sketch the contract curve in the Edgeworth box. Note that we assume that fruit is perfectly divisible. (Hint: MRSAnna = MRSBrian) 3. Which allocation maximizes utilitarian welfare?1 4. Now, let's focus on equality or fairness. Is the initial endowment equal/fair in your opinion? Let's impose equality of outcomes (that is both room-mates get the same consumption of fruit). Is that allocation envy-free? Is it Pareto efficient? 5. If all your analysis so far is correct, you should be able to easily point to an allocation that is both Pareto efficient and envy-free (recall the equality of opportunity application from the lecture notes). Either indicate that allocation in your Edgeworth box, or calculate one. In any case, explain carefully