<p>The most common version of Pareto dominance says that an allocation is Pareto-dominated if there is another feasible allocation in which at least one agent is better off and everyone else is at least as well off. In particular, the latter allocation can include consumers that are just as well off as before.</p>
<p>It follows from the very definition of a competitive equilibrium that (1) holds. That a consumer chooses the best bundle they can afford means that every better bundle must be unaffordable and, therefore, more expensive.</p>
<p>Now, the sum in (3) is not just over those agents that are strictly better off, it is also over those who are equally well off. For those, one only gets the weak inequality (2) using local-nonsation.</p>
<p>Here is an example where local-nonsation is violated and in which a competitive equilibrium is not Pareto-efficient, showing the importance of the assumption of local non-satiation for the first welfare theorem:</p>
<blockquote>
<p>There are two goods and two consumers. Both consumers have the initial
endowment <span class="math-container">$\omega_1=\omega_2=(1,1)$</span>. For consumer 1, both goods are
perfect substitutes; consumer 1 wants the sum of the amounts of both
commodities they consume as large as possible. Consumer 2, does not
care what they get, they are indifferent among all consumption
bundles. You can check that there is a competitive equilibrium in
which both consumers simply receive their endowment and both goods
have the same price, <span class="math-container">$p_1=p_2&gt;0$</span>. But the unique Pareto-efficient
allocation gives everything to consumer 1, they receive the bundle
<span class="math-container">$(2,2)$</span> and consumer 2 the bundle <span class="math-container">$(0,0)$</span>. Every allocation that
Pareto-dominates the proposed equilibrium allocation has consumer 1
being better off and therefore receiving a more expensive bundle (for
the equilibrium prices), but consumer 2 receiving a cheaper bundle
that is still as good as before.</p>
</blockquote>
<p>There is also a weak notion of Pareto-efficiency: An allocation is weakly Pareto-efficient if there is no feasible allocation in which everyone is strictly better off. One can prove that every competitive equilibrium is weakly Preto-efficient even without local non-satiation: If everyone is better off, (1) implies that everyone receives a more expensive bundle and then we can directly go to (3).</p>

CC BY-SA 4.0
2021-02-14T02:12:09.397Z
2021-02-14T02:12:09.397Z
Michael Greinecker
<p>Local non-satiation means that you always want a little bit more. There are no sweet spots where you are content and wouldn't accept any more of <span class="math-container">$x_i$</span>.</p>
<p>What this means in practice is that if you are optimising your allocation of goods <span class="math-container">$x_i$</span>, you will exhaust all of your resources (<span class="math-container">$w$</span>).</p>
<p>If you don't have (2), it's possible that your optimal bundle will involve some combination of <span class="math-container">$x_i$</span> and <span class="math-container">$w$</span>.</p>
<p>With (2), your optimal bundle will only contain <span class="math-container">$x_i$</span>.</p>

CC BY-SA 4.0
2021-02-15T21:43:32.860Z
2021-02-15T21:43:32.860Z
Jamzy
<p>Can somebody tell me why part (2) of this proof is of any relevance?</p>
<p>(1) Assume that <span class="math-container">$x$</span> is Pareto-dominated by <span class="math-container">$y$</span>. Since <span class="math-container">$x_i$</span> is optimal for consumer <span class="math-container">$i$</span>:
if <span class="math-container">$y_i\succ x_i$</span> then <span class="math-container">$p\cdot y_i &gt; p\cdot x_i$</span></p>
<p>(2) Local non-satiation even implies</p>
<p>if <span class="math-container">$y_i\succeq x_i$</span> then <span class="math-container">$p\cdot y_i\geq p\cdot x_i$</span></p>
<p>(3) Adding up: <span class="math-container">$p\cdot\sum y_i &gt; p\cdot\sum x_i= p\cdot w$</span>, in violation of feasibility. (<span class="math-container">$w$</span>= initial endowment)</p>

CC BY-SA 4.0
2021-02-12T14:45:36.053Z
Are you asking why one needs (2) in order to get (3)?
Michael Greinecker
CC BY-SA 4.0
2021-02-12T17:00:04.353Z
May I ask what is the full original question?
High GPA
<microeconomics><welfare-economics>
Microeconomics: first fundamental theorem of welfare
lzudems
# Microeconomics: first fundamental theorem of welfare

(Asked by: username_0 on 2021-02-12)

Can somebody tell me why part (2) of this proof is of any relevance?

(1) Assume that $x$ is Pareto-dominated by $y$. Since $x\_i$ is optimal for consumer $i$:
if $y\_i\succ x\_i$ then $p\cdot y\_i > p\cdot x\_i$

(2) Local non-satiation even implies

if $y\_i\succeq x\_i$ then $p\cdot y\_i\geq p\cdot x\_i$

(3) Adding up: $p\cdot\sum y\_i > p\cdot\sum x\_i= p\cdot w$, in violation of feasibility. ($w$= initial endowment)

**Tags:** microeconomics, welfare-economics

> **Comments:**
> **username_1**: Are you asking why one needs (2) in order to get (3)?  
> **username_2**: May I ask what is the full original question?  

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## 2 answers

### Answer by username_1 on 2021-02-14. Score: 4

The most common version of Pareto dominance says that an allocation is Pareto-dominated if there is another feasible allocation in which at least one agent is better off and everyone else is at least as well off. In particular, the latter allocation can include consumers that are just as well off as before.

It follows from the very definition of a competitive equilibrium that (1) holds. That a consumer chooses the best bundle they can afford means that every better bundle must be unaffordable and, therefore, more expensive.

Now, the sum in (3) is not just over those agents that are strictly better off, it is also over those who are equally well off. For those, one only gets the weak inequality (2) using local-nonsation.

Here is an example where local-nonsation is violated and in which a competitive equilibrium is not Pareto-efficient, showing the importance of the assumption of local non-satiation for the first welfare theorem:

> There are two goods and two consumers. Both consumers have the initial
> endowment $\omega\_1=\omega\_2=(1,1)$. For consumer 1, both goods are
> perfect substitutes; consumer 1 wants the sum of the amounts of both
> commodities they consume as large as possible. Consumer 2, does not
> care what they get, they are indifferent among all consumption
> bundles. You can check that there is a competitive equilibrium in
> which both consumers simply receive their endowment and both goods
> have the same price, $p\_1=p\_2>0$. But the unique Pareto-efficient
> allocation gives everything to consumer 1, they receive the bundle
> $(2,2)$ and consumer 2 the bundle $(0,0)$. Every allocation that
> Pareto-dominates the proposed equilibrium allocation has consumer 1
> being better off and therefore receiving a more expensive bundle (for
> the equilibrium prices), but consumer 2 receiving a cheaper bundle
> that is still as good as before.

There is also a weak notion of Pareto-efficiency: An allocation is weakly Pareto-efficient if there is no feasible allocation in which everyone is strictly better off. One can prove that every competitive equilibrium is weakly Preto-efficient even without local non-satiation: If everyone is better off, (1) implies that everyone receives a more expensive bundle and then we can directly go to (3).


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### Answer by username_3 on 2021-02-15. Score: 3

Local non-satiation means that you always want a little bit more. There are no sweet spots where you are content and wouldn't accept any more of $x\_i$.

What this means in practice is that if you are optimising your allocation of goods $x\_i$, you will exhaust all of your resources ($w$).

If you don't have (2), it's possible that your optimal bundle will involve some combination of $x\_i$ and $w$.

With (2), your optimal bundle will only contain $x\_i$.


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