If only weak-ordering and continuity is assumed, "ICs" can definitely intersect.

If we assume Monotonicity or convexity in addition to weak-ordering, then we can get "no cross of IC".

But those two assumptions are argueably too strong. In some real world scenarios they are violated.

Are there any weaker assumptions for the "IC cannot cross"?

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    $\begingroup$ 1. Convexity is no problem. In case of $U(x,y) = 0$ the IC is the whole plane. This is a convex preference relation, and no "ICs cross" as there is only one IC. $\endgroup$ – Giskard Mar 7 '19 at 22:52
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    $\begingroup$ 2. The problem with your question is that "weaker" is not a mathematical concept in this context. Would you accept transitivity as a weaker condition? Please edit your question to clarify. $\endgroup$ – Giskard Mar 7 '19 at 22:52

If only weak-ordering and continuity is assumed, ICs can definitely intersect.

This is not true. First, if you're speaking of indifference curves, you'd already be assuming either local non-satiation or monotonicity.

Let's speak of indifference sets instead. The analogue of two sets, $I_1$ and $I_2$, "crossing" each other can be formalized as $I_1\ne I_2$ and $I_1\cap I_2\ne \varnothing$.

Take two alternatives $x_1,x_2\in X$ and define two indifference sets as follows: \begin{align} I_1:=\{x\in X:x\sim x_1\}\quad\text{and}\quad I_2:=\{x\in X:x\sim x_2\}. \end{align} WLOG, assume that $x_1\succ x_2$, so that $I_1\ne I_2$. If we allow $I_1$ to "cross" $I_2$, then $I_1\cap I_2$ must be non-empty. Let $\bar x\in I_1\cap I_2$ be an element in the intersection. By the transitivity of the indifference relation $\sim$, we have $x_1\sim \bar x$ and $\bar x\sim x_2$, implying $x_1\sim x_2$. But this is contradictory to our assumption that $x_1\succ x_2$. The contradiction therefore suggests that transitivity, a property of the weak ordering, is violated.

  • $\begingroup$ Thanks for your answer! So what is the assumption needed for drawing indifference curves? Is weak-order + local non-satiation enough? $\endgroup$ – High GPA Mar 7 '19 at 23:15
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    $\begingroup$ @HighGPA: You're welcome. To have indifference curves, you'd need completeness, transitivity, continuity, and non-satiation. Non-satiation can be substituted with monotonicity if ICs are required to be downward sloping. $\endgroup$ – Herr K. Mar 7 '19 at 23:21
  • $\begingroup$ I guess your monotonicity here means strict monotonicity? Without non-satiation, Weak-order + Cont + Convexity or Weak-order + cont + Mono (not strict) seem to imply something like "indifference manifold" $\endgroup$ – High GPA Mar 7 '19 at 23:24
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    $\begingroup$ @HighGPA: By monotonicity I mean if $x\ge x'$ (element by element), then $x\succsim x'$, and if $x>x'$ then $x\succ x'$. $\endgroup$ – Herr K. Mar 7 '19 at 23:31
  • $\begingroup$ Hi Herr, I just learned how to prove that non-saitation (along with completeness, transitivity and continuity) implies that the indifference set must have empty interior. However I am still not sure how to show that the indifference set can only contain curves and cannot contain a cross $\endgroup$ – High GPA Jun 9 '19 at 17:00

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