# Whats the difference between local non-satiation and monotonicity?

Is there a practical difference between local non-satiation and montonicity? Can one exist in a utility function without the other?

Monotonicity of preferences is a stronger condition than local nonsatiation. Monotonicity implies local nonsatiation, but not the other way around.

To see this:

Claim: Let $$\succsim$$ be a monotonic preference relation over $$\mathbb{R}^n_{+}$$. Therefore, $$\succsim$$ is locally nonsatiated.

Proof: Fix some $$\varepsilon > 0$$. Let there be an arbitrary $$x \in \mathbb{R}^n_{+}$$ and let $$\mathbf{1} \in \mathbb{R}^n_{+}$$ be the unit vector. For any $$\lambda > 0$$, we also have $$x + \lambda \mathbf{1} \in \mathbb{R}^n_{+}$$. Since clearly $$x + \lambda \mathbf{1} \gg x,$$ $$x + \lambda \mathbf{1} \succ x$$ by monotonicity. Consider the following metric over $$\mathbb{R}^n_{+}$$:

$$d(x+\lambda \mathbf{1}, x) = ||x+\lambda \mathbf{1} -x|| = \lambda||\mathbf{1}|| = \lambda \sqrt{n}.$$

Thus for $$\lambda < \frac{\varepsilon}{\sqrt{n}}$$, $$d(x+\lambda \mathbf{1}, x) < \varepsilon$$ yet $$x + \lambda \mathbf{1} \succ x$$. Since $$x$$ was arbitrary, the existence of such a point implies that $$\succsim$$ is locally nonsatiated. $$\blacksquare$$

To show that locally nonsatiated preferences do not imply monotonic preferences, you can come up with a utility function $$u(\cdot)$$ over various goods that strictly increases with respect to some of the goods but reaches a satiation point for at least one of the others. For example, in $$\mathbb{R}^2_{+}$$:

$$u(x_1,x_2) = x_1 - |1-x_2|.$$

A consumer with such preferences is satiated with respect to $$x_2$$ at $$x_2=1$$.