<p>Ok I think I might understand now why local nonsatiation is important for tending toward a pareto optimal market allocation. Consider the following picture, where all the circles represent possible allocations, and their position on the graph represents the utility received by each person in a simple two-person market:</p>

<p><a href="https://i.sstatic.net/QdBWL.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QdBWL.png" alt="Satiated pareto efficiency graph"></a></p>

<p>In this case, X, Y, Z, and D all give person 1 the same utility. In such a situation, X, Y, and Z are all possible equilibria given complete markets and price taking behavior even though they're not pareto-optimal. </p>

<p>In a situation with local nonsatiation, this situation couldn't exist, and thus a pareto optimal equilibrium is ensured.</p>

<p>Weak pareto optimality doesn't require local non-satiation.</p>

CC BY-SA 3.0
2016-03-27T06:20:48.453Z
2016-03-27T07:00:16.603Z
2016-03-27T07:00:16.603Z
B T
<p>Local non-satiation assumption is required to prove the first welfare theorem.</p>
<p>Consider the following pure exchange economy consisting of two consumers - <span class="math-container">$1$</span> and <span class="math-container">$2$</span>, and two Goods - <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>:</p>
<ul>
<li>Preferences of <span class="math-container">$1$</span> and <span class="math-container">$2$</span> are given by the following utility functions: <span class="math-container">$u_1(x_1, y_1) = 0$</span> and <span class="math-container">$u_2(x_2, y_2) = x_2+y_2$</span></li>
<li>Endowments of <span class="math-container">$1$</span> and <span class="math-container">$2$</span> are given by the following: <span class="math-container">$\omega_1 = (10,0)$</span> and <span class="math-container">$\omega_2 = (0,20)$</span></li>
</ul>
<p>Observe that the endowment allocation is also the competitive equilibrium of this economy i.e. <span class="math-container">$(x_1^*, y_1^*) = (10, 0)$</span> and <span class="math-container">$(x_2^*, y_2^*) = (0, 20)$</span> supported by any pair of prices <span class="math-container">$p = (p_X, p_Y)$</span> satisfying the condition <span class="math-container">$0 &lt; p_Y \leq p_X$</span>. However, this allocation is not Pareto efficient. This is because an allocation where <span class="math-container">$2$</span> consumes all of both the goods is Pareto superior to it.</p>
<p>In this example, it is clear that <span class="math-container">$1$</span>'s preferences does not satisfy LNS. Also we can see that this condition fails to hold:</p>
<p><span class="math-container">$(x_1, y_1) \succsim (x_1^*, y_1^*) = (10,0)$</span> implies <span class="math-container">$p_Xx_1+p_Yy_1 \geq p\cdot\omega_1 = 10p_X$</span></p>
<p>To see this, consider <span class="math-container">$(x_1, y_1) = (0,0)$</span>.</p>
<p>When preferences satisfy LNS, we can prove that the following proposition is true:</p>
<p><strong>Proposition</strong>. If <span class="math-container">$x_i^*$</span> is the demand of consumer <span class="math-container">$i$</span> when price is <span class="math-container">$p$</span> and endowment is <span class="math-container">$\omega$</span>, then
<span class="math-container">$x_i\succsim x_i^* \ \Rightarrow \ p\cdot x_i \geq p \cdot \omega_i $</span>.</p>
<p><strong>Proof</strong>. Suppose <span class="math-container">$x_i\succsim x_i^*$</span> but <span class="math-container">$p\cdot x_i &lt; p \cdot \omega_i$</span>. Then by LNS, there exists a bundle <span class="math-container">$x_i'$</span> sufficiently close to <span class="math-container">$x_i$</span>, so that <span class="math-container">$x_i' \succ x_i$</span> and <span class="math-container">$p\cdot x_i'&lt;p \cdot \omega_i$</span> holds. This implies that there exists a bundle <span class="math-container">$x_i'\succ x_i^*$</span> which is also affordable, contradicting that <span class="math-container">$x_i^*$</span> is the demand.</p>

CC BY-SA 4.0
2022-07-20T07:41:49.613Z
2022-07-20T07:58:19.653Z
2022-07-20T07:58:19.653Z
Amit
<p>The profit maximization assumption implies 
$$\text{if } x_i \succ x_i^* \text{ then } p_ix_i &gt; p_i w_i$$</p>

<p>Okay so this just says if the agent is utility maximizing / rational, then if he doesn't choose a bundle strictly preferable to his bundle then it must not be affordable. </p>

<p>Why is the local non-satiation assumption needed to then say </p>

<p>$$\text{if } x_i \succeq x_i^* \text{ then } p_ix_i \geq p_i w_i$$</p>

<p>Why isn't this just automatic from the profit maximization assumption? If we know $x_i \succ x_i^* \implies p_ix_i &gt; p_i w_i$, isn't it obvious that $x_i = x_i^*  \implies p_ix_i =  p_i w_i$ and so $$\text{if } x_i \succeq x_i^* \text{ then } p_ix_i \geq p_i w_i$$</p>

CC BY-SA 4.0
2022-07-20T15:15:01.810Z
The consumer problem is not about profit maximization.
Michael Greinecker
<microeconomics><preferences><welfare-economics>
What is the purpose of the local non-satiation assumption in the first welfare theorem?
Stan Shunpike
# What is the purpose of the local non-satiation assumption in the first welfare theorem?

(Asked by: username_0 on 2015-10-22)

The profit maximization assumption implies
$$\text{if } x\_i \succ x\_i^\* \text{ then } p\_ix\_i > p\_i w\_i$$

Okay so this just says if the agent is utility maximizing / rational, then if he doesn't choose a bundle strictly preferable to his bundle then it must not be affordable.

Why is the local non-satiation assumption needed to then say

$$\text{if } x\_i \succeq x\_i^\* \text{ then } p\_ix\_i \geq p\_i w\_i$$

Why isn't this just automatic from the profit maximization assumption? If we know $x\_i \succ x\_i^\* \implies p\_ix\_i > p\_i w\_i$, isn't it obvious that $x\_i = x\_i^\* \implies p\_ix\_i = p\_i w\_i$ and so $$\text{if } x\_i \succeq x\_i^\* \text{ then } p\_ix\_i \geq p\_i w\_i$$

**Tags:** microeconomics, preferences, welfare-economics
> **Comments:**
> **username_1**: The consumer problem is not about profit maximization.  

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## 2 answers

### Answer by username_2 on 2016-03-27. Score: 1

Ok I think I might understand now why local nonsatiation is important for tending toward a pareto optimal market allocation. Consider the following picture, where all the circles represent possible allocations, and their position on the graph represents the utility received by each person in a simple two-person market:

[![Satiated pareto efficiency graph](https://i.sstatic.net/QdBWL.png)](https://i.sstatic.net/QdBWL.png)

In this case, X, Y, Z, and D all give person 1 the same utility. In such a situation, X, Y, and Z are all possible equilibria given complete markets and price taking behavior even though they're not pareto-optimal.

In a situation with local nonsatiation, this situation couldn't exist, and thus a pareto optimal equilibrium is ensured.

Weak pareto optimality doesn't require local non-satiation.


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### Answer by username_3 on 2022-07-20. Score: 1

Local non-satiation assumption is required to prove the first welfare theorem.

Consider the following pure exchange economy consisting of two consumers - $1$ and $2$, and two Goods - $X$ and $Y$:

* Preferences of $1$ and $2$ are given by the following utility functions: $u\_1(x\_1, y\_1) = 0$ and $u\_2(x\_2, y\_2) = x\_2+y\_2$
* Endowments of $1$ and $2$ are given by the following: $\omega\_1 = (10,0)$ and $\omega\_2 = (0,20)$

Observe that the endowment allocation is also the competitive equilibrium of this economy i.e. $(x\_1^\*, y\_1^\*) = (10, 0)$ and $(x\_2^\*, y\_2^\*) = (0, 20)$ supported by any pair of prices $p = (p\_X, p\_Y)$ satisfying the condition $0 < p\_Y \leq p\_X$. However, this allocation is not Pareto efficient. This is because an allocation where $2$ consumes all of both the goods is Pareto superior to it.

In this example, it is clear that $1$'s preferences does not satisfy LNS. Also we can see that this condition fails to hold:

$(x\_1, y\_1) \succsim (x\_1^\*, y\_1^\*) = (10,0)$ implies $p\_Xx\_1+p\_Yy\_1 \geq p\cdot\omega\_1 = 10p\_X$

To see this, consider $(x\_1, y\_1) = (0,0)$.

When preferences satisfy LNS, we can prove that the following proposition is true:

**Proposition**. If $x\_i^\*$ is the demand of consumer $i$ when price is $p$ and endowment is $\omega$, then
$x\_i\succsim x\_i^\* \ \Rightarrow \ p\cdot x\_i \geq p \cdot \omega\_i $.

**Proof**. Suppose $x\_i\succsim x\_i^\*$ but $p\cdot x\_i < p \cdot \omega\_i$. Then by LNS, there exists a bundle $x\_i'$ sufficiently close to $x\_i$, so that $x\_i' \succ x\_i$ and $p\cdot x\_i'

holds. This implies that there exists a bundle $x\_i'\succ x\_i^\*$ which is also affordable, contradicting that $x\_i^\*$ is the demand.


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