Formally if X is the consumption set, then for any and every , there exists a such that and is preferred to .
Several things to note are:
- Local nonsatiation is implied by monotonicity of preferences. Because the converse isn't true, local nonsatiation is a weaker condition.
- 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.
- It rules out the extreme case where all goods are "bads", since the point x = 0 would then be a bliss point.
- 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.
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