<p>Set of feasible allocations in this economy is
<span class="math-container">$\mathcal{F} = \left\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=1 \ \wedge \ y_A+y_B=20\right\}$</span></p>
<p>Set of Pareto efficient allocations in this economy is given by</p>
<p><span class="math-container">$\mathcal{P} = \left\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|(x_A=\frac{1}{2}) \ \vee \ (x_A&lt;\frac{1}{2} \ \wedge y_A= 0) \ \vee (x_A&gt;\frac{1}{2} \ \wedge y_A= 20) \right\}$</span></p>
<p>Here is the graphical presentation of the set of efficient allocations:</p>
<p><a href="https://i.sstatic.net/BHQ6hJlz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BHQ6hJlz.png" alt="enter image description here" /></a></p>
<p>Observe that there are three types of efficient allocations.</p>
<ul>
<li>For an efficient allocation <span class="math-container">$((\frac{1}{2},y_A),(\frac{1}{2},y_B))\in\mathcal{P}$</span>, consider <span class="math-container">$((\frac{1}{2},y_A),(\frac{1}{2},y_B))$</span> as the endowment, the resulting competitive equilibrium will be the allocation <span class="math-container">$((\frac{1}{2},y_A),(\frac{1}{2},y_B))$</span> itself supported by the price ratio <span class="math-container">$\frac{p_X}{p_Y}=v'(\frac{1}{2})$</span>.</li>
<li>For an efficient allocation <span class="math-container">$((x_A,0),(x_B,20))\in\mathcal{P}$</span>, consider <span class="math-container">$((x_A,0),(x_B,20))$</span> as the endowment, the resulting competitive equilibrium will be the allocation <span class="math-container">$((x_A,0),(x_B,20))$</span> itself supported by the price ratio <span class="math-container">$\frac{p_X}{p_Y}=v'(x_B)$</span>.</li>
<li>For an efficient allocation <span class="math-container">$((x_A,20),(x_B,0))\in\mathcal{P}$</span>, consider <span class="math-container">$((x_A,20),(x_B,0))$</span> as the endowment, the resulting competitive equilibrium will be the allocation <span class="math-container">$((x_A,20),(x_B,0))$</span> itself supported by the price ratio <span class="math-container">$\frac{p_X}{p_Y}=v'(x_A)$</span>.</li>
</ul>
<p>We can also write these three points together in the following way: for any allocation <span class="math-container">$((x_A,y_A),(x_B,y_B))\in\mathcal{P}$</span>, consider the same allocation <span class="math-container">$((x_A,y_A),(x_B,y_B))$</span> as the endowment, the resulting competitive equilibrium will be that allocation <span class="math-container">$((x_A,y_A),(x_B,y_B))$</span> itself supported by the price ratio <span class="math-container">$\frac{p_X}{p_Y}=\min(v'(x_A),v'(x_B))$</span>.</p>

CC BY-SA 4.0
2024-08-24T11:50:29.837Z
2024-08-24T14:13:24.527Z
2024-08-24T14:13:24.527Z
Amit
<p>There is a two-person exchange economy</p>
<p>Each agent has the following utility <span class="math-container">$u_i(x_i,y_i)=v(x_i)+y_i$</span> for agent <span class="math-container">$i=\{A,B\}$</span></p>
<p>Assume that <span class="math-container">$v$</span> is strictly concave and increasing function that has a continuous first derivative. <span class="math-container">$v(0)=0$</span> and <span class="math-container">$v(x)&lt;1$</span>.</p>
<p>Agent A has the endowment <span class="math-container">$(1,10)$</span>. And agent B has the endowment <span class="math-container">$(0,10)$</span>.</p>
<blockquote>
<p>For each Pareto efficient allocation, suggest how we might change the endowments so that the Pareto efficient allocation in the question is a walrasian equilibrium.</p>
</blockquote>
<p>I found the Pareto optimal allocation set as</p>
<p><span class="math-container">$$v’(x_A)=v’(1-x_B)$$</span>
<span class="math-container">$$y_A+y_B=20$$</span></p>
<p>I also found the Walrasian equilibrium set as <span class="math-container">$\{(x^*_A, y^*_A)=(1/2, 10+\frac{P_x}{2P_y}), (x^*_B, y^*_B)=(1/2, 10-\frac{P_x}{2P_y})\}$</span> with the Walrasian equilibrium price ratio</p>
<p><span class="math-container">$\frac{P_x}{P_y}= min\{v’(x_A),v’(x_B)\}$</span></p>
<p>If <span class="math-container">$y^*_A&gt;0$</span> Then <span class="math-container">$\frac{P_x}{P_y}= v’(x_A)$</span></p>
<p>If <span class="math-container">$y^*_A=0$</span> Then <span class="math-container">$\frac{P_x}{P_y}&gt; v’(x_A)$</span> so,  <span class="math-container">$x^*_A&gt; x^*_A$</span></p>
<p>I could only found Walrasian and Pareto optimal allocations.  But I am not sure. And I don’t understand the questions. How can I show this question. All helps will be appreciated. Thanks a lot.</p>
<p>*duplicated question</p>

<microeconomics><self-study><general-equilibrium>
Walrasian equilibrium with quasi linear function
studentp
# Walrasian equilibrium with quasi linear function

(Asked by: username_0 on 2022-10-14)

There is a two-person exchange economy

Each agent has the following utility $u\_i(x\_i,y\_i)=v(x\_i)+y\_i$ for agent $i=\{A,B\}$

Assume that $v$ is strictly concave and increasing function that has a continuous first derivative. $v(0)=0$ and $v(x)<1$.

Agent A has the endowment $(1,10)$. And agent B has the endowment $(0,10)$.

> For each Pareto efficient allocation, suggest how we might change the endowments so that the Pareto efficient allocation in the question is a walrasian equilibrium.

I found the Pareto optimal allocation set as

$$v’(x\_A)=v’(1-x\_B)$$
$$y\_A+y\_B=20$$

I also found the Walrasian equilibrium set as $\{(x^\*\_A, y^\*\_A)=(1/2, 10+\frac{P\_x}{2P\_y}), (x^\*\_B, y^\*\_B)=(1/2, 10-\frac{P\_x}{2P\_y})\}$ with the Walrasian equilibrium price ratio

$\frac{P\_x}{P\_y}= min\{v’(x\_A),v’(x\_B)\}$

If $y^\*\_A>0$ Then $\frac{P\_x}{P\_y}= v’(x\_A)$

If $y^\*\_A=0$ Then $\frac{P\_x}{P\_y}> v’(x\_A)$ so, $x^\*\_A> x^\*\_A$

I could only found Walrasian and Pareto optimal allocations. But I am not sure. And I don’t understand the questions. How can I show this question. All helps will be appreciated. Thanks a lot.

\*duplicated question

**Tags:** microeconomics, self-study, general-equilibrium

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## 1 answers

### Answer by username_1 on 2024-08-24. Score: 1 | Accepted answer

Set of feasible allocations in this economy is
$\mathcal{F} = \left\{((x\_A,y\_A),(x\_B,y\_B))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_A+x\_B=1 \ \wedge \ y\_A+y\_B=20\right\}$

Set of Pareto efficient allocations in this economy is given by

$\mathcal{P} = \left\{((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}|(x\_A=\frac{1}{2}) \ \vee \ (x\_A<\frac{1}{2} \ \wedge y\_A= 0) \ \vee (x\_A>\frac{1}{2} \ \wedge y\_A= 20) \right\}$

Here is the graphical presentation of the set of efficient allocations:

[![enter image description here](https://i.sstatic.net/BHQ6hJlz.png)](https://i.sstatic.net/BHQ6hJlz.png)

Observe that there are three types of efficient allocations.

* For an efficient allocation $((\frac{1}{2},y\_A),(\frac{1}{2},y\_B))\in\mathcal{P}$, consider $((\frac{1}{2},y\_A),(\frac{1}{2},y\_B))$ as the endowment, the resulting competitive equilibrium will be the allocation $((\frac{1}{2},y\_A),(\frac{1}{2},y\_B))$ itself supported by the price ratio $\frac{p\_X}{p\_Y}=v'(\frac{1}{2})$.
* For an efficient allocation $((x\_A,0),(x\_B,20))\in\mathcal{P}$, consider $((x\_A,0),(x\_B,20))$ as the endowment, the resulting competitive equilibrium will be the allocation $((x\_A,0),(x\_B,20))$ itself supported by the price ratio $\frac{p\_X}{p\_Y}=v'(x\_B)$.
* For an efficient allocation $((x\_A,20),(x\_B,0))\in\mathcal{P}$, consider $((x\_A,20),(x\_B,0))$ as the endowment, the resulting competitive equilibrium will be the allocation $((x\_A,20),(x\_B,0))$ itself supported by the price ratio $\frac{p\_X}{p\_Y}=v'(x\_A)$.

We can also write these three points together in the following way: for any allocation $((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{P}$, consider the same allocation $((x\_A,y\_A),(x\_B,y\_B))$ as the endowment, the resulting competitive equilibrium will be that allocation $((x\_A,y\_A),(x\_B,y\_B))$ itself supported by the price ratio $\frac{p\_X}{p\_Y}=\min(v'(x\_A),v'(x\_B))$.


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