# What is the assumption behind "indifference curve does not cross"?

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"?

• 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. Mar 7, 2019 at 22:52
• 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. Mar 7, 2019 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.

• Thanks for your answer! So what is the assumption needed for drawing indifference curves? Is weak-order + local non-satiation enough? Mar 7, 2019 at 23:15
• @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. Mar 7, 2019 at 23:21
• 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" Mar 7, 2019 at 23:24
• @HighGPA: By monotonicity I mean if $x\ge x'$ (element by element), then $x\succsim x'$, and if $x>x'$ then $x\succ x'$. Mar 7, 2019 at 23:31
• 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 Jun 9, 2019 at 17:00

Consider any utility function $$u: \mathbb{R}^2_+ \rightarrow\mathbb{R}$$. Indifference curve for satisfaction level $$\mu$$ is defined as: $$\text{IC}(\mu) = \{(x, y)\in\mathbb{R}^2_+|u(x, y) = \mu\}$$

I am assuming that by ICs cannot cross, you mean that ICs for two different utility levels don't have a non-empty intersection i.e.
$$(\forall (\mu_1, \mu_2)\in\mathbb{R}^2) (\mu_1 \neq \mu_2 \Rightarrow \text{IC}(\mu_1) \cap \text{IC}(\mu_2) = \emptyset)$$.

This is obviously true because of the fact that $$u$$ is a function.

Related Question: Can an IC look like a cross? https://qr.ae/pG0AoL

• Regarding the link: if we assume either strong monotonicity or strict convexity then the example cannot be correct. May 5, 2022 at 19:06
• Yes, that's right.
– Amit
May 6, 2022 at 2:09
• In fact, AMit, the original motivation of this question is: "if an IC cannot be looked like a cross, then what are some assumptions behind that?". However, later I found that I had a misconception about ICs: I thought that, if the ICs look like a cross, then they can be different ICs. In fact, they can only be the same IC. May 6, 2022 at 11:21