# Can the indifference curve (set) not be a curve at all?

Does the indifference set have to be in the form of a curve, or of a form that is well-known? If it is not necessary to be a curve, how would the set look like? Can I get some examples?

If we follow the usual definition of the indifference curve, I don't see any reason why it can not be a function that's not a curve. The Wikipedia article does not have a particular description of the function; it just gives a list of the possible forms.

• @Giskard It does not, because it assumes convexity. I am trying to figure out if an indifference set can have a form that is not well-known. I am not assuming monotonicity, or convexity. Let's assume the axioms and nothing more. Jun 22, 2021 at 21:24
• What are "the axioms"? Jun 22, 2021 at 22:12
• @Giskard The completeness axiom, the reflexivity axiom and the transitive axiom. Jun 22, 2021 at 22:15

You can make literally any set into an indifference curve of a "well-behaved" (complete, transitive, reflexive) preference relation.

Let $$H \subset \mathbb{R}^n$$.

Assume that the preference relation $$\preceq$$ is such that
(i) for all $$x,y \in H$$ we have $$x \sim y$$
(ii) for all $$x \in H$$, $$y \notin H$$, we have $$x \not\sim y$$ (meaning $$x \succ y$$ or $$x \prec y$$, the same for all $$y \notin H$$).
(iii) $$\preceq$$ is well-behaved on $$\mathbb{R}^n \setminus H$$.
A simple way to guarantee (iii) is to assume for all $$x,y \notin H$$ we have $$x \sim y$$.

Then $$H$$ (whatever shape it may have) is an indifference curve of $$\preceq$$.

Such a $$\preceq$$ may also be represented by a utility function $$u$$:
$$u(x) = 0, \ \forall x \in H$$
$$u(x) = 1, \ \forall x \notin H$$.

Note that this $$u$$ is not continuous, but you did not require that. For closed sets $$H$$ it can be made continuous, by setting
$$u(x) = \text{distance}(x, H)$$.

• Right, so if we have a consumption bundle of order $2$ (that is, two different goods) and we represent it in the 2D plane, then this can be an example of an indifference curve as well right? (If yes, then do such cases exist in reality or we tend to work with curves with more distinct properties?) Jun 22, 2021 at 23:03
• Well, since 2D is $\mathbb{R}^2$ and your curve is a subset of $\mathbb{R}^2$, my answer applies. Jun 22, 2021 at 23:34
• "do such cases exist in reality" I don't think anyone has ever observed an indifference curve in a non-laboratory setting, but you can post this as a new question. Jun 22, 2021 at 23:34