<p>The first foundamental theorem of welfare economics states that under locally-non satiated preferences, and if <span class="math-container">$(x^{\ast},y^{\ast},p)$</span> is a price equilibrium with transfers, then, the allocation <span class="math-container">$(x^{\ast},y^{\ast})$</span> is Pareto optimal. Note that a Walrasian equilibrium is a special case of a price equilibrium with  transfers.</p>
<p>Few remarks:</p>
<p><span class="math-container">$i)$</span> <span class="math-container">$x^{\ast}=(x^{\ast}_1,~x_2^{\ast},...,~x_i^{\ast},...,x_I^{\ast})$</span> is a list of optimal FEASIBLE consumption bundles (those maximizing utility for all <span class="math-container">$i=1,2,...,I$</span> consumers). Each <span class="math-container">$x_i^{\ast}$</span> optimal consumption bundle describes the set of optimal <span class="math-container">$L$</span> commodities consumed, that is, <span class="math-container">$~x_i^{\ast}=(x_{1i},x_{2i},...,x_{Li})$</span>. We basically have <span class="math-container">$L$</span> commodities.
Also, <span class="math-container">$y^{\ast}=(y^{\ast}_1,~y_2^{\ast},...,y_j^{\ast},...,y_J^{\ast})$</span> is a list of optimal FEASIBLE production plans maximizing profits for all <span class="math-container">$j=1,2,...,J$</span> firms. Each <span class="math-container">$y^{\ast}_j$</span> optimal production plan is of size <span class="math-container">$L$</span>, i.e., <span class="math-container">$y^{\ast}_j(y_{1j},y_{2j},...,y_{Lj})$</span>. The idea is that each commodity can be either consumed or used as input.
Finally, <span class="math-container">$p$</span> is the price vector of the <span class="math-container">$L$</span> commodities.</p>
<p><span class="math-container">$ii)$</span>A price equilibrium with transfers is an allocation <span class="math-container">$(x^{\ast},y^{\ast})$</span> and a price vector <span class="math-container">$p=(p_1,p_2,...,p_L)$</span> with an assignment of wealth levels <span class="math-container">$(w_1,w_2,...,w_I)$</span>, one for each consumer, such that all firms maximize profits, all consumers maximize their well-being and all markets clear.</p>
<p>Why the theorem you asked requires local non-satiation?
Because, if <span class="math-container">$(x^{\ast},y^{\ast})$</span> is Pareto optimal, it cannot be any other feasible allocation that Pareto dominates it.
Indeed, <span class="math-container">$x^{\ast}$</span> contains a list of the best consumption bundles that consumers can afford. It must that, if there is an <span class="math-container">$x_i \succ_i x_i^{\ast}$</span>, then <span class="math-container">$p \cdot x_i &gt; w_i$</span>, and so <span class="math-container">$x_i$</span> must be non affordable.
Also, if <span class="math-container">$x_i \succsim_i x_i^{\ast} $</span>, then it must be <span class="math-container">$p \cdot x_i \ge w_i$</span>. Suppose that <span class="math-container">$p \cdot x_i \ge w_i$</span> is false, and so there is an <span class="math-container">$x_i \succsim
_i x_i^{\ast}$</span> such that <span class="math-container">$p \cdot x_i &lt; w_i$</span>. If local nonsatiation holds, there must be another bundle, say <span class="math-container">$x_i^{'}$</span> that is extremely closed to <span class="math-container">$x_i$</span> such that <span class="math-container">$x_i^{'} \succ x_i$</span> and still <span class="math-container">$p \cdot x_i^{'}&lt; w_i$</span>. By transitivity, it must be that <span class="math-container">$x_i^{'} \succ x_i^{\ast}$</span>. But this is a contraddiction beacuse it cannot be another bundle strictly preferred to <span class="math-container">$x_i^{\ast}$</span>. So, <span class="math-container">$x_i^{'} \succ x_i^{\ast}$</span> is NOT POSSIBLE, and local non satiation rules out this possibility.</p>

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2023-02-04T14:50:48.350Z
2023-02-04T14:50:48.350Z
Tony
<p>Why the First foundamental theorem of welfare economics requires local nonsatiation ?</p>

<microeconomics>
First foundamental theorem of welfare economics
John M.
# First foundamental theorem of welfare economics

(Asked by: username_0 on 2023-02-04)

Why the First foundamental theorem of welfare economics requires local nonsatiation ?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2023-02-04. Score: 1 | Accepted answer

The first foundamental theorem of welfare economics states that under locally-non satiated preferences, and if $(x^{\ast},y^{\ast},p)$ is a price equilibrium with transfers, then, the allocation $(x^{\ast},y^{\ast})$ is Pareto optimal. Note that a Walrasian equilibrium is a special case of a price equilibrium with transfers.

Few remarks:

$i)$ $x^{\ast}=(x^{\ast}\_1,~x\_2^{\ast},...,~x\_i^{\ast},...,x\_I^{\ast})$ is a list of optimal FEASIBLE consumption bundles (those maximizing utility for all $i=1,2,...,I$ consumers). Each $x\_i^{\ast}$ optimal consumption bundle describes the set of optimal $L$ commodities consumed, that is, $~x\_i^{\ast}=(x\_{1i},x\_{2i},...,x\_{Li})$. We basically have $L$ commodities.
Also, $y^{\ast}=(y^{\ast}\_1,~y\_2^{\ast},...,y\_j^{\ast},...,y\_J^{\ast})$ is a list of optimal FEASIBLE production plans maximizing profits for all $j=1,2,...,J$ firms. Each $y^{\ast}\_j$ optimal production plan is of size $L$, i.e., $y^{\ast}\_j(y\_{1j},y\_{2j},...,y\_{Lj})$. The idea is that each commodity can be either consumed or used as input.
Finally, $p$ is the price vector of the $L$ commodities.

$ii)$A price equilibrium with transfers is an allocation $(x^{\ast},y^{\ast})$ and a price vector $p=(p\_1,p\_2,...,p\_L)$ with an assignment of wealth levels $(w\_1,w\_2,...,w\_I)$, one for each consumer, such that all firms maximize profits, all consumers maximize their well-being and all markets clear.

Why the theorem you asked requires local non-satiation?
Because, if $(x^{\ast},y^{\ast})$ is Pareto optimal, it cannot be any other feasible allocation that Pareto dominates it.
Indeed, $x^{\ast}$ contains a list of the best consumption bundles that consumers can afford. It must that, if there is an $x\_i \succ\_i x\_i^{\ast}$, then $p \cdot x\_i > w\_i$, and so $x\_i$ must be non affordable.
Also, if $x\_i \succsim\_i x\_i^{\ast} $, then it must be $p \cdot x\_i \ge w\_i$. Suppose that $p \cdot x\_i \ge w\_i$ is false, and so there is an $x\_i \succsim
\_i x\_i^{\ast}$ such that $p \cdot x\_i < w\_i$. If local nonsatiation holds, there must be another bundle, say $x\_i^{'}$ that is extremely closed to $x\_i$ such that $x\_i^{'} \succ x\_i$ and still $p \cdot x\_i^{'}< w\_i$. By transitivity, it must be that $x\_i^{'} \succ x\_i^{\ast}$. But this is a contraddiction beacuse it cannot be another bundle strictly preferred to $x\_i^{\ast}$. So, $x\_i^{'} \succ x\_i^{\ast}$ is NOT POSSIBLE, and local non satiation rules out this possibility.


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