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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 $X_i$]

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  • $\begingroup$ 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? $\endgroup$ Feb 14, 2023 at 21:37

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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).

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  • $\begingroup$ 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? $\endgroup$
    – Maximilian
    Feb 14, 2023 at 21:43
  • $\begingroup$ 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 $x$. $\endgroup$ Feb 14, 2023 at 21:52
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    $\begingroup$ Right, I was pointing out this just because you wrote at least as good as instead of strictly preferred. Thanks $\endgroup$
    – Maximilian
    Feb 14, 2023 at 21:55
  • $\begingroup$ @Maximilian I just corrected it as you pointed that out, it was actually a typo. $\endgroup$ Feb 14, 2023 at 21:57

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