Consumer theory problems from Microeconomics

1.6 EXERCISES 1.1 Let X = R4. Verify that X satisfies all five properties required of a consumption set in Assumption 1.1. 1.2 Let ~ be a preference relation. Prove the following: (a) UN= = (d) > n ~=0 1.3 Give a proof or convincing argument for each of the following claims made in the text. (a) Neither > nor ~ is complete. (b) For any x and x2 in X, only one of the following holds: x > x2, or x> > x', or x] ~ x2. 1.4 Prove that if > is a preference relation, then the relation > is transitive and the relation ~ is transitive. Also show that if x! ~ x2 > x3, then x! _ x3. 1.5 If > is a preference relation, prove the following: For any x0 E X, (a) ~ (x ) = =(x9) n = (x0 ) ( b ) = ( x 0 ) = ~ ( x 0 ) u > ( x 0 ) ( c ) ~ ( x9)n > (x 0 ) = 0 (d) ~ (xon - (x0) = 0