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The purpose of this problem is to make sure that you fully understand the basic concepts of utility representation Prove or disprove the following statement....

(20 points) Let X = R. Let be a preference relation on X. In Lecture notes-2 (see Lecture 2, page 7, Definition 9) we have encountered the definition of additivity property of a preference relation. (a) (10 points) Suppose that the preference relation is represented by the utility function u(21, C2) where, u(x1, x2) = ax1 + B x2 where a, > 0. Verify that the preference relation satisfies addivity and strict monotonicity. 2 (b) (10 points) Consider now the lexicographic preference EL on X = R4. Prove or disprove the following statement : "The Lexicographic preference relation El also satisfies additivity and strict monotonicity. (20 points) Let X = R. Let be a preference relation on X. In Lecture notes-2 (see Lecture 2, page 7, Definition 9) we have encountered the definition of additivity property of a preference relation. (a) (10 points) Suppose that the preference relation is represented by the utility function u(21, C2) where, u(x1, x2) = ax1 + B x2 where a, > 0. Verify that the preference relation satisfies addivity and strict monotonicity. 2 (b) (10 points) Consider now the lexicographic preference EL on X = R4. Prove or disprove the following statement : "The Lexicographic preference relation El also satisfies additivity and strict monotonicity

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