<p>Hint: Prove by contradiction. That is, suppose <span class="math-container">$x_i\succsim_ix_i^*$</span> but <span class="math-container">$p\cdot x_i&lt;w_i$</span>, show that this would lead to a contradiction of <span class="math-container">$x_i^*$</span> being maximal for the feasible set.</p>

CC BY-SA 4.0
2020-09-23T06:44:18.447Z
2020-09-23T06:44:18.447Z
Herr K.
<blockquote>
<p>(MWG 16.C.2) Suppose that the preference relation <span class="math-container">$\succsim_i$</span> is locally nonsatiated and that <span class="math-container">$x_i^*$</span> is maximal for <span class="math-container">$\succsim_i$</span> in set <span class="math-container">$\{x_i \in X_i: p \cdot x_i \le w_i\}$</span>. Prove that the following property holds: &quot;If <span class="math-container">$x \succsim_i x_i^*$</span> then <span class="math-container">$p \cdot x_i\ge w_i.$</span>&quot;</p>
</blockquote>
<p>Locally nonsatiation means that for every <span class="math-container">$x_i \in X_i$</span> and <span class="math-container">$\epsilon &gt;0$</span>, there exists <span class="math-container">$x_i' \in X_i$</span> such that <span class="math-container">$||x_i - x_i'|| &lt; \epsilon$</span> and <span class="math-container">$x_i' \succ x_i$</span>.
I think that I need to prove that if <span class="math-container">$x_i \sim_i x^*_i $</span> for <span class="math-container">$x_i \not= x_i^*$</span>, then it is possible that <span class="math-container">$p\cdot x_i = w_i$</span> because I already know that if <span class="math-container">$x_i \succ x_i^*$</span>, then <span class="math-container">$p\cdot x_i &gt; w_i$</span>.</p>
<p>I am stuck and  cannot proceed. Can anyone give me some hint for this question?</p>

<microeconomics>
If $x \succsim_i x_i^*$ then $p \cdot x_i\ge w_i.$ (MWG 16.C.2)
shk910
# If $x \succsim_i x_i^*$ then $p \cdot x_i\ge w_i.$ (MWG 16.C.2)

(Asked by: username_0 on 2020-09-23)

> (MWG 16.C.2) Suppose that the preference relation $\succsim\_i$ is locally nonsatiated and that $x\_i^\*$ is maximal for $\succsim\_i$ in set $\{x\_i \in X\_i: p \cdot x\_i \le w\_i\}$. Prove that the following property holds: "If $x \succsim\_i x\_i^\*$ then $p \cdot x\_i\ge w\_i.$"

Locally nonsatiation means that for every $x\_i \in X\_i$ and $\epsilon >0$, there exists $x\_i' \in X\_i$ such that $||x\_i - x\_i'|| < \epsilon$ and $x\_i' \succ x\_i$.
I think that I need to prove that if $x\_i \sim\_i x^\*\_i $ for $x\_i \not= x\_i^\*$, then it is possible that $p\cdot x\_i = w\_i$ because I already know that if $x\_i \succ x\_i^\*$, then $p\cdot x\_i > w\_i$.

I am stuck and cannot proceed. Can anyone give me some hint for this question?

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2020-09-23. Score: 2

Hint: Prove by contradiction. That is, suppose $x\_i\succsim\_ix\_i^\*$ but $p\cdot x\_i<w\_i$, show that this would lead to a contradiction of $x\_i^\*$ being maximal for the feasible set.


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