7. Consider a pure exchange economy with two commodities (called X and Y with quantities given by lower case letters x and y respectively) and two traders, denoted by subscripts 1 and 2, each having an initial endowment vector e1 = e2 = (1,1). Utility functions are given by u(x1,y1) = 1 and u2(x2,y2) = 2. Each trader's consumption set is the set of all nonnegative vectors in the plane or (equivalently) the closed rst quadrant. [85 points total] _ L (3) Find all competitive equilibria. Do not normalize prices in your answer. [20 points] (b) Do the preference preorders represented by these utility functions satisfy the following properties? Explain each of your answers briey. [35 points, 5 points each] (i) Transitivity. (ii) Continuity. (iii) Nonsatiation. (iv) Local nonsatiation (l.n.s.) (v) Weak convexity. (vi) Strict (or strong) convexity. (vii) Strict monotonicity. (c) Find all Pareto optimal allocations in this economy. [10 points] (d) Can we use the First Welfare Theorem to conclude that any competitive equilibrium must be Pareto optimal? Explain. [10 points] (e) Give an explicit numerical example of an allocation that is not Pareto optimal in this economy or explain why this is not possible. [10 points]