For the question below, use these symbols correspondingly:

Preference relation: ?

Strictly preferred to: ?

Indifference: ~

Is a subset of: ? i.e. A ? B means A is a subset of B

Please bold things like R (real numbers), R+means positive real numbers, vice versa.

Using these questions as context:

3. Let X be a set, and let :1 and :2 be two binary relations on X. Dene the lexicographic binary relation :L as follows: for any 2:, y E X, we have a: :1, if and only if at least one of the following two conditions holdszl. O :17 >1 y o xmlyandrzy. [If :1 and :2 capture how a decision maker feels about different aspects of a decision, then :L is one way of combining them: use aspect 1 to make decisions, but if aspect 1 does not make the decision easy, then use aspect 2 to break ties] Explain why each of the following is true: (a) If 56,3; 6 X have s :L y, then 3: 2:1 y. (b) :L is complete if both :1 and :2 are complete. (c) 2:1,; is transitive if both :31 and :32 are transitive. [Hintz part (a) of this question and parts (b,c) of the previous question may be helpful] (d) KL is a preference relation if both 31 and :2 are preference relations. N (e) If 2:2 is exactly :1, then 3;; is exactly 2:1. (f) If :2 is exactly jl (see question 2(a)), then :L is exactly fa. 1. Let X = 2+, the set of nonnegative integers. In each of the following parts, we will name a binary relation on X. For each one, tell me (i) whether or not it is complete, (ii) whether or not it is transitive, (iii) whether or not it is a preference relation, and if so, (iv) whether or not it admits a utility representation. Justify your answers. The relation :. The relation 7A a) ) ) The relation YES where :I;YESy for every :13 and y. l l ( (b (c (d (e The relation :3 where m z y if and only if either a: : O or :2: 2 y 2 1. The relation NDPE where we don't have wNOPEy for any a: and y. The notion of convex preferences is very important in economics, capturing the idea that a consumer likes "balanced" bundles of goods. This question gives you some practice reasoning about convex preferences. For each of the preferences relations YES, 21, 12, L defined in the previous question, is the relation convex? Justify your answer. (a) YES? (b) z1? (c) -2? (d) zz?3. [Upper/lower contour sets and indifference curves are an important part of our toolkit for reasoning about preferences. This example gives you some practice deriving them from preferences. Let X = R2, and let a denote the vector (3,3) E X. In each part of this question, we will name a preference relation (you should convince yourself the relations we name are in fact preference relations, but you don't need to prove it), and ask you to draw a picture of various sets associated with it. (a) Consider the preference relation YES defined on the previous problem set. What does the "weakly better than" set ByEs () look like? What does the "weakly worse than" set WyES () look like? (b) Consider the preference relation ~1 given by x zly it2 zyity2. What does B (x) look like? What does the "strictly better than" set By, (x) look like? What does the indifference curve I, (x) look like? (c) Consider the preference relation _2 given by x Zly + 2 2 42. What does B, (x) look like? What does By, (x) look like? What does Wy, (x) look like? (d) Consider the preference relation Zz, the lexicographic preference that first uses ~1, and then uses _2 to refine _1-indifference. [See the previous problem set for a formal definition.] What does B, (x) look like? What does W, () look like? What does It, (x) look like