5. Consider three different pure exchange economies and for each nd (i) the set of all Pareto optimal allocations, (ii) the set of all competitive equilibria (prices and allocations), and (iii) for each Pareto optimal allocation, its corresponding set of all (normalized) competitive equilibrium prices which support the Pareto optimal allocation as a (no trade) competitive equilibrium allocation (after redistribution of commodities as in the second welfare theorem). Take prices to be nonnegative and normalize them to sum to one. Each economy has two commodities (called X and Y with quantities denoted by X and y respectively) and two traders. Each trader has an initial endowment vector of (1,1) in each economy. Also, in each economy both traders' consumption sets are the rst quadrant in the plane and both traders have the same utility functions or preferences, given as follows for each economy: Economy A: Cobb-Douglas utilities with parameters 1/2 and 1/2. Economy B: Perfect substitutes. Economy C: Left gloves and right gloves (with perfectly divisible goods), also known as the symmetric xed proportions case. Hint: It might help to draw an Edgeworth Box for each economy