# Is non-monotonic local non-satiation supported by consumer theory in economics?

In general (Walrasian) equilibrium, local non-satiation is one of the assumptions that guarantee existence of equilibrium. Question is, is non-monotonic local non-satiation preference supported by rational consumer theory in economics?

• Do you mean is the theory supported by empirical evidence? – Pburg Jan 6 '15 at 13:36

Suppose we're talking about some bundle of consumables, $x$. If preferences are monotone, either weakly or strongly, the general idea is that more is better. With local non-satiation the idea is that, no matter what $x$ you have, there is always some small change in $x$ that would make you better off.

Is your question, does there exist rational preferences where:

1. More is better does not hold.
2. You can always make a tiny change that leads to a preferable bundle.

Imagine a consumption set on the surface of an infinite dimensional unit sphere, where the dimensions are indexed by the counting numbers and represent different varieties of goods. The consumer prefers goods found in higher dimensions. So the 1 unit of good on the second dimension is preferred to the 1 unit of good on the first dimension.

Thus, the consumer is never satiated because they would always prefer one unit of a good found in a higher dimension, and there are infinitely many dimensions. But these preferences do not satisfy monotonicity, because you must always give up a little bit of at least one good to acquire more of another.

Hence, these preferences are rational, because you can rank all pairs on the surface of the sphere, and transitivity holds trivially.

• Also, note I did not mention general equilibrium in this answer--I simply focused on the second part of your question. – Bryce Jan 6 '15 at 17:00
• Just wondering: Do these preferences correspond to any kind of realistic or empirically documented behavior? Is this a known formulation or does it just happen to demonstrate the principle? – shadowtalker Jan 10 '15 at 22:31
• -1 The unit sphere is not a natural consumption set in any context. Lns makes perfect sense in finite dimensional contexts and for good reasons, general equilibrium theory with infinitely dimensional commodity spaces uses generally stronger assumptions. Most importantly, on a purely technical level, lns is a notion that involves the topology and it is not clear what topology you have in mind. For such a fancy example, you should be a bit more explicit. – Michael Greinecker Jan 11 '15 at 11:02

The simplest examples of preferences which are lns but not monotonic come from allowing satiation for some commodities, but greed for others. Here is a simple example given by a utility function on $\mathbb{R}^2_+$: $$u(x,y)=x-|7-y|.$$ The consumer always wants more of the first good but is satiated with respect to the second good at having $7$ units of it.