Whats the difference between local non-satiation and monotonicity?

Question

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

Answer 1

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 $x_2=1$.