<blockquote>
<p>A counterexample would work as well.</p>
</blockquote>
<p>Consider the utility function <span class="math-container">$U(x) = 0$</span>, and the budget constraint <span class="math-container">$1 \cdot x \leq 1$</span>. The solution <span class="math-container">$x^* = 1$</span> is feasible and maximizes utility, but it does not minimize expenditure for this utility level (0) for this utility function, <span class="math-container">$x' = 0$</span> would achieve a lower expenditure with the same utility.</p>

CC BY-SA 4.0
2023-11-27T17:31:07.100Z
2023-11-27T17:31:07.100Z
Giskard
<p>I have been trying to use the contradiction method to prove this, but it does not seem to be working.</p>
<p>Suppose <span class="math-container">$x^*$</span> is optimal in both EMP and UMP. Then <span class="math-container">$u(x^*) \geq u(x')$</span> for all <span class="math-container">$x'$</span> in <span class="math-container">$B_pw$</span>.</p>
<p>And <span class="math-container">$p.x^*$</span> &lt; <span class="math-container">$p.x'$</span> <span class="math-container">$\leq w$</span></p>
<p>Also <span class="math-container">$x^*$</span> is optimal in UMP. So <span class="math-container">$u(x^*)$</span> &gt; <span class="math-container">$u(x')$</span> for all <span class="math-container">$x'$</span>.</p>
<p>I can't find a way to reach a contradiction here. Is it that the demand correspondence lies on the budget plane so there is no way LNS can hold because it would lie outside Bp,w?</p>
<p>This is basically MWG proposition 3.E.1 without the LNS assumption.</p>
<p><em>Show that if <span class="math-container">$x^*$</span> is optimal in the utility maximization problem when <span class="math-container">$w &gt; 0$</span>, then <span class="math-container">$x^*$</span> is optimal in the expenditure minimization problem when the constraint is to attain a utility level <span class="math-container">$u(x^*)$</span>.</em></p>

CC BY-SA 4.0
2023-11-27T13:53:38.463Z
I think that contrary to your title. you don't want a contradictory example (which is easy to give), but a proof by contradiction. Please consider editing the title.
Giskard
CC BY-SA 4.0
2023-11-27T14:55:29.917Z
A counterexample would work as well. I was not able to come up with a counterexample, so just thought that maybe proving by contradiction could be the easier approach (as I could tie it with MWG's explanation), but apparently that's not the case either.
pacmanscuriousbloob
CC BY-SA 4.0
2023-11-27T15:00:27.750Z
I just realized if a person has the greatest integer function as his utility function, it could work as a valid counterexample. However, I would appreciate it someone could provide an answer with a more formal approach to my question.
pacmanscuriousbloob
<microeconomics><utility><consumer-theory><demand>
Showing UMP and EMP do not exhibit duality if the assumption of local non-satiation is absent
pacmanscuriousbloob
# Showing UMP and EMP do not exhibit duality if the assumption of local non-satiation is absent

(Asked by: username_0 on 2023-11-27)

I have been trying to use the contradiction method to prove this, but it does not seem to be working.

Suppose $x^\*$ is optimal in both EMP and UMP. Then $u(x^\*) \geq u(x')$ for all $x'$ in $B\_pw$.

And $p.x^\*$ < $p.x'$ $\leq w$

Also $x^\*$ is optimal in UMP. So $u(x^\*)$ > $u(x')$ for all $x'$.

I can't find a way to reach a contradiction here. Is it that the demand correspondence lies on the budget plane so there is no way LNS can hold because it would lie outside Bp,w?

This is basically MWG proposition 3.E.1 without the LNS assumption.

*Show that if $x^\*$ is optimal in the utility maximization problem when $w > 0$, then $x^\*$ is optimal in the expenditure minimization problem when the constraint is to attain a utility level $u(x^\*)$.*

**Tags:** microeconomics, utility, consumer-theory, demand
> **Comments:**
> **username_1**: I think that contrary to your title. you don't want a contradictory example (which is easy to give), but a proof by contradiction. Please consider editing the title.  
> **username_0**: A counterexample would work as well. I was not able to come up with a counterexample, so just thought that maybe proving by contradiction could be the easier approach (as I could tie it with MWG's explanation), but apparently that's not the case either.  
> **username_0**: I just realized if a person has the greatest integer function as his utility function, it could work as a valid counterexample. However, I would appreciate it someone could provide an answer with a more formal approach to my question.  

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## 1 answers

### Answer by username_1 on 2023-11-27. Score: 2

> A counterexample would work as well.

Consider the utility function $U(x) = 0$, and the budget constraint $1 \cdot x \leq 1$. The solution $x^\* = 1$ is feasible and maximizes utility, but it does not minimize expenditure for this utility level (0) for this utility function, $x' = 0$ would achieve a lower expenditure with the same utility.


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