Local nonsatiation

The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it. What this means is that a consumer always either prefers more of an item or less of an item, never a particular amount of a good. An additional requirement is that there is some good that consumer does prefer more of.

Formally if X is the consumption set, then for any $x\in X$ and every $\varepsilon >0$, there exists a $y\in X$ such that $\|y-x\|\leq \varepsilon$ and $y$ is preferred to $x$.

Several things to note are:

1. Local nonsatiation is implied by monotonicity of preferences. Because the converse isn't true, local nonsatiation is a weaker condition.

2. There is no requirement that the preferred bundle y contain more of any good - hence, some goods can be "bads" and preferences can be non-monotone.

3. It rules out the extreme case where all goods are "bads", since the point x = 0 would then be a bliss point.

4. Local nonsatiation can only occur if the consumption set is either unbounded (open) (in other words, it cannot be compact) or on sections of a bounded consumption set sufficiently far away from the ends. Near the ends of a bounded set, there would necessarily be a bliss point where local nonsatiation doesn't hold.