# Local nonsatiation

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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.[1]

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

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 either if the consumption set is unbounded (open) (in other words, it cannot be compact) or if x is on a section 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 does not hold.

## Notes

1. ^ Microeconomic Theory, by A. Mas-Colell, et al. ISBN 0-19-507340-1