Doesnt convexity prevent thick indifference curves aswell?

Question

It is standard in many micro textbooks when analyzing the relationship between preference axioms and the shape of the utility function (and consequently the shape of indifference curves), to attributed the "non-thickness" of indifference curves to "local non-satiation". While it is easily seen (and proven) that LNS preferences don't admit thick IC, my question is the following:

Thick IC violate as well strict convexity (notice that weak convexity appears to "survive"), so, isn't convexity alone another property that prevents thickness? Put it differently, can we find non monotonic, convex preferences that don't admit thick IC?

If any proof or reference to relevant literature would be very useful.

Answer 1

My approach would be to define thick indifference curves in terms of a stronger form of local non-satiation:

Definition (thick indifference curves) Preferences are said to have thick indifference curves if there exists at least one bundle $A\in\mathbb{R}^l$ and an open ball $\mathscr{B}(A)$ around $A$ such that $A'\sim A$ for every $A'\in\mathscr{B}(A)$.

Let's set up a definition for strict convexity to be sure we are on the same page:

Definition (strict convexity) Preferences obey strict convexity if

$$A\sim B\implies \lambda A+(1-\lambda) B\succ A\sim B$$

for $A,B\in\mathbb{R}^l$ and any $\lambda\in(0,1)$.

Now we can state the desired result:

Proposition If $\succsim$ is strictly convex then $\succsim$ does not have thick indifference curves.

Proof Suppose that $\succsim$ has thick indifference curves. Then there exists an $A$ such that $A'\sim A$ for all $A'\in\mathscr{B}(A)$. Thus, fix two bundles $A$ and $B$ such that $A\sim B$ and $B\in\mathscr{B}(A)$. We know that $$\lambda A+(1-\lambda) B \in \mathscr{B}(A).$$ Thus, if the indifference curve is thick around point$A$ we must have $\lambda A+(1-\lambda) B\sim A\sim B$. But this contradicts the strict convexity hypothesis. QED

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That should do the job for strict convexity. As frage_man already pointed out, the proposition fails to hold generally under weak convexity. Indifference everywhere is a very elegant example why.

Answer 2

Great question, never thought about this:

So the main thing that LNS preferences get rid of is a satiation point, preventing the budget from binding in the consumer's problem. So there are tons of preferences that are non-monotonic, strictly convex, with a satiation point.

Now a simple example of weakly convex with thick IC is indfference everywhere. So $\forall x,y\in X, x\sim y$. In this case the IC is all of $\mathbb{R}$.

I agree with you on that strict convexity eliminates thick IC. I'm working on a proof but defining thick IC in mathmatical terms is tricky. However, just recall non-thick IC is not enough to guarantee LNS, so that's why strict convexity is not a substitute for LNS in major theorems like the 2nd Welfare Theorem.