I don't think so but my TA was saying no. Is it correct?
Of course they're not continuous, but strictly monotone.
Yes, vacuously.
All distinct bundles can be strictly ranked with lexicographic preferences. So the definition of convexity starts with, "for any bundles a and b, where a ~= b and a is indifferent to b...." You've now defined an empty set. So any conclusion can follow.
Okay, I figured it out. They are in fact strictly convex. Thanks!
Yes, vacuously.
All distinct bundles can be strictly ranked with lexicographic preferences. So the definition of convexity starts with, "for any bundles a and b, where a ~= b and a is indifferent to b...." You've now defined an empty set. So any conclusion can follow.
Yes, vacuously.
All distinct bundles can be strictly ranked with lexicographic preferences. So the definition of convexity starts with, "for any bundles a and b, where a ~= b and a is indifferent to b...." You've now defined an empty set. So any conclusion can follow.
Do you even MWG bro?
Wow, what a pile of nonsense here. The indifference curves are not singletons. They are line segments. Convexity is not what 3e31 says it is. It is that if x and y in set A, then tx + (1-t)y is also in A, that is the line segment between any two points in A is also in A.
As it is, indeed the upper contour sets for lex prefs are convex. What it violates is continuity. The upper contour sets are not closed, so continuity it violated. They are also not open. Anyway, it is the violation of continuity that makes it the case that one cannot form a utility function from lexicographic preferences.
BTW, this is just garden variety grad micro stuff. Those making the above comments should not be hired by anybody. This is basic incompetence.
Wow, what a pile of nonsense here. The indifference curves are not singletons. They are line segments. Convexity is not what 3e31 says it is. It is that if x and y in set A, then tx + (1-t)y is also in A, that is the line segment between any two points in A is also in A.
As it is, indeed the upper contour sets for lex prefs are convex. What it violates is continuity. The upper contour sets are not closed, so continuity it violated. They are also not open. Anyway, it is the violation of continuity that makes it the case that one cannot form a utility function from lexicographic preferences.
BTW, this is just garden variety grad micro stuff. Those making the above comments should not be hired by anybody. This is basic incompetence.
Give me an example where preferences are lexicographic, and two distinct bundles are indifferent.
The problem I had is about the idcs for lexicographic preferences being dots. I was thinking in terms of taking two points and drawing a line by joining them. But the thing is there are many dots so the line can be filled with an infinite number of dots, which I think will satisfy the convexity proposition. Not sure how to think about the strictly convexity proposition.
I haven't seen a formal proof of lexicographic preferences being strictly convex so was just wondering.
They are
Lexicographic preferences have no utility representation because there is no countable order dense subset, not because lexicographic preferences aren't continuous. Continuity implies existence of a continuous utility function, but isn't necessary for existence.
The problem I had is about the idcs for lexicographic preferences being dots. I was thinking in terms of taking two points and drawing a line by joining them. But the thing is there are many dots so the line can be filled with an infinite number of dots, which I think will satisfy the convexity proposition. Not sure how to think about the strictly convexity proposition.
I haven't seen a formal proof of lexicographic preferences being strictly convex so was just wondering.They are
There you go (proof is for two-dimensional consumption space but you can extend easily to general dimensions):
Preferences are strictly convex if for all bundles x, and all bundles y and z that are not the same and that are both weakly preferred to x, the bundle ty + (1-t)z is strictly preferred to x for all 0<t<1.
Take any x in R^2. Take y in R^2 and z in R^2 both weakly preferred to x. Since indifference sets are singletons under lexicographic preferences and z and y are not the same, at least one of y and z must be strictly preferred to x. Wlog, let this be y, which means that either y_1 > x_1, or y_1 = x_1 and y_2 > x_2. As for z, either z_1 > x_1, or z_1 = x_1 and z_2 > x_2, or z=x.
Now set w = ty + (1-t)z, for some 0<t<1. Given the above, either one the following must be true: w_1 > x_1, or w_1 = x_1 and w_2 > z_2. This means that w is strictly preferred to x; hence lexicographic preferences are strictly convex. QED.
Now please gtfo and go back to your problem set.
Wow, what a pile of nonsense here. The indifference curves are not singletons. They are line segments. Convexity is not what 3e31 says it is. It is that if x and y in set A, then tx + (1-t)y is also in A, that is the line segment between any two points in A is also in A.
As it is, indeed the upper contour sets for lex prefs are convex. What it violates is continuity. The upper contour sets are not closed, so continuity it violated. They are also not open. Anyway, it is the violation of continuity that makes it the case that one cannot form a utility function from lexicographic preferences.
BTW, this is just garden variety grad micro stuff. Those making the above comments should not be hired by anybody. This is basic incompetence.
I know plenty of APs that have forgotten all of this stuff and couldn't give you an answer off the top of their head. Many work at top 50 Ph.D programs and have Ph.Ds from top 10 places.