ECON UN3211, Problem Set #2
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ECON UN3211, Problem Set #2
Due date: Wednesday, February 1, 2023
1. Let X = Z+ , 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.
(a) The relation =.
(b) The relation 
.
(c) The relation YES where xYESy for every x and y .
(d) The relation NopE where we don’t have xNopEy for any x and y .
(e) The relation Y, where x Y y if and only if either x = 0 or x > y > 1.
2. Let X be a set, and let Y be a binary relation on X .
(a) Define the binary relation 5 on X by x 5 y 年÷ y Y x. Show that:
i. 5 is transitive if Y is transitive;
ii. 5 is complete if Y is complete;
iii. 5 is a preference relation if Y is a preference relation.
[The relation 5 can be thought of as the “opposite” of Y. For example, if Ann and Bob are playing heads-up poker, and Y describes Ann’s pref- erences over which starting hands they each have, then it’s natural that Bob’s preferences would be 5.]
(b) Suppose Y is transitive. Suppose a, b, c e X . Show that:
i. If a > b and b Y c, then a > c.
ii. If a Y b and b > c, then a > c.
(c) Suppose Y is transitive. Explain why the relation ~ (given by x ~ y 年÷ both x Y y and y Y x) is transitive.
3. Let X be a set, and let Y1 and Y2 be two binary relations on X . Define the lexicographic binary relation YL as follows: for any x, y e X, we have x YL if and only if at least one of the following two conditions holds:1 .
● x >1 y
● x ~1 y and x Y2 y .
[If Y1 and Y2 capture how a decision maker feels about different aspects of a decision, then YL 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 x, y e X have x YL y, then x Y1 y .
(b) YL is complete if both Y1 and Y2 are complete.
(c) YL is transitive if both Y1 and Y2 are transitive. [Hint: part (a) of this question and parts (b,c) of the previous question may be helpful.]
(d) YL is a preference relation if both Y1 and Y2 are preference relations. (e) If Y2 is exactly Y1 , then YL is exactly Y1 .
(f) If Y2 is exactly 51 (see question 2(a)), then YL is exactly Y1 .
4. [In addition to giving you some practice thinking about utility representations, this question will give us a bit more intuition for the idea that utility is an “ordinal” concept. It will also show how some of our intuitions about utility might apply when X is finite but not when X is infinite.]
Let X be a set, let Y be a preference relation on X, and let u be a utility representation for Y.
(a) Suppose X is finite. Explain why some (large enough) number M > 0 exists such that .M < u(x) < M for every x e X . That is, the utility is bounded.
(b) Show that, even if X is infinite, some alternative utility representation 
and some (large enough) number M > 0 exists such that .M < 
(x) < M for every x e X . [Hint: The function ϕ : R → R given by ϕ(t) = 
is strictly increasing.]
(c) Suppose X is finite. Show that some (small enough) number m > 0 exists such that u(x) . u(y) > m for any x > y .
(d) Suppose X = 勿+ , and Y is the preference relation defined in question 1(e).
i. Explain why any whole number k > 3 has u(1) < u(2) < · · · < u(k) < u(0).
ii. Explain why no number m > 0 exists such that u(x) . u(y) > m for any x > y .
5. Let X be a set, and let Y1 and Y2 be two preference relations on X, and let YL be the lexicographic preference relation as defined in question 3. Suppose u1 : X → R is a utility representation for Y1 , and u2 : X → R is a utility representation for Y2 . For every number ∈ > 0, define the function u5 : X → R by letting u5 (x) = u1 (x) + ∈u2 (x).
[We might say YL puts “much more weight” on Y1 than on Y2 . This question gives a sense in which that interpretation is valid, but also shows that this interpretation can be a bit misleading.]
(a) Consider the special case in which X is finite. Show that, for some ∈ > 0, the function u5 is a utility representation for YL . [Hint: parts (a,c) of the previous question might be helpful.]
(b) [Hard, optional question. You don’t need to know this, but you can think about it if you would find it fun.] Suppose X is countably infinite.2 Must there exist some ∈ > 0 such that u5 is a utility representation for YL ?
2023-02-03