# What properties must a utility function have such that we can define level sets and thus indifference curves?

Based on what I learned, it seems like we usually assume a level set $L(f)$ and its function $f$ defined on $\Bbb{R}^n$ have the following properties:

1. $f$ is continuous
2. Thus, the curves in the level set are closed curves. (not sure why)
3. All closed curves are "converging" to a point. (not sure why)
4. This point will be a maximum for the function: the derivative in the point will be zero, since if it was not then the point would be a 1-manifold, which is absurd.
5. $\nabla f \neq 0$ on the rest of the points

I don't understand how we know a preference relation on a consumption bundle has a utility function representation such that these properties hold. According to my professor, we use level sets to describe indifference curves.

Therefore, if indifference curves are described by a level set, that level set must have the properties listed above. Note, for economics, we are considering a function $U(\mathbf{x})$, where $\mathbf{x}=(x_1,...,x_n)$, and the level set $L(U)$. Also, note $U: \Bbb{R}^n\rightarrow \Bbb{R}$.

My Question:

In economic terms, what assumption do we have to make to ensure, $U$, $L(U)$ have these properties to create a level set used for indifference curves?

Ideally, it would be great to have an enumerated list of the assumptions (ie 1,2,3).

Look take a rational preference $\succeq$ defined on X, endow X with some metric (thus a assume it is a metric space). Assume also X is separable (e.g. $\mathbb{R}^n$ satisfies this conditions but is more general). Now we let $\succeq$ be (i) rational (complete, transitive), (ii) continuous (it means that if $x^n\rightarrow x$ , $y^n\rightarrow y$ and $x^n \succeq y^n$ $\forall n$, then $x \succeq y$. Under this assumptions, then it can be represented by a continuous utility function. 1. u is continuous by the above conditions. 2. Curves in the level set are closed. It follows from the fact that the indifference relation is a closed set and the continuous utility representation: take $x^n \sim y\quad \forall n$ by continuity of the preferences $x \sim y$, now this implies that $u(x^n)$ is a sequence on the level curve $u(x^n)=u(y)$ for fixed y and by above $u(x)\rightarrow u(y)$ thus making it closed. The other properties are deeper in the sense that one needs more assumptions, for instance to have the gradient of u it has to be differentiable, uniqueness of the extremum needs some sort of convexity of the level curves and so on.
Level sets are always well defined. No property of utility functions has to be assumed. For any utility level $\bar u$ just define the level set to be $\{x\in \mathbb{R}^n|U(x)=\bar u\}$. No property of $U$ must be assumed. This is well-defined regardless of the properties of $U$. Properties of $U$ are useful if one wants to derive specific properties of the level sets, but are not needed to define them.
• What if we aren't using $\Bbb{R}^n$? Or is that always assumed? – Stan Shunpike Apr 6 '15 at 17:15
• Even then, the above definition should is fine. Just replace in the above definition $\mathbb{R}^n$ by whatever set $X$ the agent chooses from. – TMB Apr 6 '15 at 17:19
As stated by TMB, levels sets of points at which the same utility is achieved ($\{x\in \mathbb{R}^n|U(x)=\bar u\}$). These depend on pretty much nothing, except that the ensemble has an equality relation - not much to ask really.
1 and 2. The curves are defined as $f^{-1} (\bar u)$ and $\{\bar u\}$ is closed, so the antecedent by a continuous function of $\{\bar u\}$ is closed too.
3,4 and 5 sounds like bullshit to me, maybe you can give us the reference of your course or a course refereeing to the same assumption? How do you define a maximum in $\Bbb{R}^n$?