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An economy has two goods, x and y, and two individuals, 1 and 2. Let (x ):) be individual i's consumption bundle and u;: [0, +co)2 - (-00, too) be individual i's utility function, which is assumed to be strictly increasing and continuous. We are interested in making judgment about the living standard of the two individuals. Let > be a reflexive and transitive binary relation over [0, two) with the following interpretation: for any (x;, Vi) and (x;, y;), (xi, vi) > (xj, y;) means that individual i's living standard under (xi, );) is at least as high as individual j's living standard under (x,, y;), and (xi, vi) > (xjy;) means that individual i's living standard under (x;, yi) is higher than individual j's living standard under (x;, y; ). For the purpose of comparing living standards of the two individuals, consider the following two properties: (Pl) for any i E {1,2) and any (x;, y;), (z;, w;) = [0, +co)2, (x;, Vi) = (z;, wi) wi(xi Vi) > u;(Zi, Wi). (P2) for any i, j 6 {1,2] and any (x;, );). (x;, y;) = [0, two)2, if (xavi) > (x; y;) then ( xi, Vi ) > ( x jy;). Suppose u, (1,3) > uj (3,1) and u2 (1,3) 0, the following must be true: w, (1 - 6, 3 - 8) > u, (3,1) and u2 (1,3) (the living-standard ranking introduced earlier in this question) by using the information in (ii) and the two properties, (P1) and (P2). (iv) Explain whether there exists a reflexive and transitive binary relation > over [0, +co)2 introduced in this question that satisfies (P1) and (P2) simultaneously