Choice theory: assumptions

October 26, 2008 - Leave a Response

Problem 3 doesn’t assume X is transitive. Can we assume that X is NOT necessarily transitive? Thanks.

Sorry, the question I asked just now should be rephrased as: can we assume that the preference operator on X is NOT necessarily transitive in problem 3 and 4?

On question 3, there is not actually a preference relation, there is only a (revealed preference) choice rule. That is to say, all you should be considering is a mapping from scriptB to subsets of X.

Similarly, in question 4, the question is about showing there can always be a rational preference relation that is consistent with any (revealed preference) choice rule.

I hope this clears things up some, but I’m not sure it will, so please let me know if not. I’m happy to try to be clearer.

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Monotonicity and non-satiation

October 26, 2008 - Leave a Response

Does Monotonicity o preferences imply their local non-satiation?
I guess yes, because if I draw a small ball around any point, no matter how small the radius is, as long as preferences are monotone, I can always find a point that’s more preferred because it has more of at least one good. Hence local non-satiation.
Is this reasoning correct?

Strict monotonicity implies local nonsatiation (on e.g., R^n).

Monotonicity does not imply local nonsatiation (think of being indifferent between all points). However, the MWG definition of monotone *does* imply local non-satiation (again on e.g., R^n).

The space does actually matter; suppose I have preference on [0,1]^2 represented by u(x,y) = x+y. These preferences are strictly monotone. However, they are not locally non-satiated: (1,1) is a point of satiation.

Continuous preferences

October 26, 2008 - Leave a Response

I’m going thorugh your slides and the notes for HW3, Q7. I don’t understand the notation for continuous preference relationships.

What are xsubn (in your slides) or xsupn (in the notes)? Do these subscripts/superscripts refer to the different dimensions x and y? Do they refer to different values which x and y may take on? Do they refer to something else entirely?

They refer to elements of a sequence of elements in X. That is, x_1 , x_2 , … are each elements of X, as are y_1 , y_2 , … .
Continuity obtains iff for all such sequences where
(1) x_i is preferred to y_i for all i,
(2) lim i -> infty x_i exists, and
(3) lim i -> infty y_i exists
Then lim i-> infty x_i is preferred to lim i-> infty y_i

OK, my intuition is that a lexicographic preference relationship is not continuous, but I don’t know how  I’d prove that.

Eg,

x_i = (.5 + 1/2i , .25)
Lim x_i = (.5, .25)

y_i = (.5 – 1/2i , .75)
Lim y_i = (.5, .75)

For all i, x_i preferred to y_i, but lim y preferred to lim x.

Sorry for terse, but sent from phone…

Welcome

October 26, 2008 - Leave a Response

So I get a bunch of questions, and I often think the answers could be relevant to folks other than she who sent it. My new idea is to post questions here. Let’s see how it goes.

Welcome!