# Challenging question in microeconomics - local nonsatiation

I'm studying advanced micro from the Mas-Colell book (exercise 16.C.1)

I was wondering if anyone can help me to solve the following exercise. I have no idea how to deal with it

Show that if a consumption set $$X_i \subset \mathbb{R}^{L}$$ is nonempty, closed, and bounded and the preference relation $$\succeq_i$$ on $$X_i$$ is continuous, then $$\succeq_i$$ cannot be locally nonsatiated. [Hint: Show that the continuous utility function representing $$\succeq_i$$ must have a maximum on $$X_i$$]

• Do you have a problem showing that a representing continuous utility function has a maximum? Or do you have a problem relating the existence of such a maximum to local nonsatiation? Feb 14 at 21:37

Continuous functions attain a minimum and a maximum value over a compact (closed and bounded set). This is a well known theorem.

Since the utility function $$u_i$$ representing $$\succeq_i$$ is continuous and $$X_i$$ is closed and bounded, then $$u_i$$ must attain a maximum value at some point $$x = (x_1,\dots,x_L)$$ in $$X_i$$.

Since $$u_i$$ has a maximum value at $$x$$, there is no $$y \in X_i : y \succ_i x$$.

Therefore, $$\succeq_i$$ is not locally non-satiable.

The definition of locally non-satiable I use is $$\forall x \in X \forall \epsilon > 0 \exists y \in X : ||y-x|| \leq \epsilon$$ and $$y \succ x$$.

Note this definition implies the non-existence of maximum values for the corresponding utility function, if the preferences are indeed representable by a utility function (which happens when the preferences are complete, transitive and continuous).

• I think that you are wrong, it should be, there is no $y$ in the consumption set preferred to $x$. Is the answer so simple? Feb 14 at 21:43
• Yes, it is a simple application of Weierestrass’s theorem. Since $u_i$ attains its maximum vale on $x$, there is no point $y$ strictly preferred to $x$. Feb 14 at 21:52
• Right, I was pointing out this just because you wrote at least as good as instead of strictly preferred. Thanks Feb 14 at 21:55
• @Maximilian I just corrected it as you pointed that out, it was actually a typo. Feb 14 at 21:57