<p>Given the economy:</p>
<ul>
<li>Utility functions: <span class="math-container">$u_A(x_A,y_A) = x_A - |y_A-\alpha_A|$</span>, <span class="math-container">$u_B(x_B,y_B) = x_B - |y_B-\alpha_B|$</span>, where <span class="math-container">$\alpha_A \geq 0$</span>, and <span class="math-container">$\alpha_B\geq 0$</span> are given.</li>
<li><span class="math-container">$\omega = (\omega_X, \omega_Y)$</span>, where <span class="math-container">$\omega_X&gt;0$</span>, and <span class="math-container">$\omega_Y&gt;0$</span></li>
</ul>
<p>Set of feasible allocations is given by</p>
<p><span class="math-container">$\mathcal{F} =\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=\omega_X \ \wedge \ y_A+y_B=\omega_Y \} $</span></p>
<p>Set of Pareto efficient Allocations is given by</p>
<p><span class="math-container">$\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|\min(\alpha_A,\omega_Y-\alpha_B)\leq y_A \leq \max(\alpha_A,\omega_Y-\alpha_B)\}$</span></p>
<p><a href="https://i.sstatic.net/3Etl3.png" rel="noreferrer"><img src="https://i.sstatic.net/3Etl3.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-05-23T04:01:26.083Z
2023-05-23T12:01:51.143Z
2023-05-23T12:01:51.143Z
Amit
<p>Let’s rename the variables <span class="math-container">$x := x_1$</span>, <span class="math-container">$y:= x_2$</span>.</p>
<p>First let’s consider the set of feasible allocations such that <span class="math-container">$y_i \neq \alpha_i$</span> for both <span class="math-container">$i = A,B$</span>.</p>
<p>In this set, both utilities are completely differentiable.</p>
<p>The optimization problem we want to solve is</p>
<p><span class="math-container">$\max x_A - |y_A - \alpha_A|$</span></p>
<p><span class="math-container">$s.t. x_B - |y_B - \alpha_B| = \overline{U}$</span></p>
<p>and the endowment constraints</p>
<p><span class="math-container">$x_A + x_B = \omega_x$</span></p>
<p><span class="math-container">$y_A + y_B = \omega_y$</span></p>
<p>where <span class="math-container">$(\omega_x,\omega_y)$</span> are the total endowments for <span class="math-container">$(x,y)$</span>.</p>
<p>Since both utilities are completely differentiable in this region, the efficiency condition is</p>
<p><span class="math-container">$MRS_A = MRS_B$</span></p>
<p><span class="math-container">$\frac{\frac{\partial U_A}{\partial x_A}}{\frac{\partial U_A}{\partial y_A}} = \frac{\frac{\partial U_B}{\partial x_B}}{\frac{\partial U_B}{\partial y_B}} $</span></p>
<p>Note <span class="math-container">$\frac{d}{dx} |x| = sign(x)$</span> for <span class="math-container">$x \neq 0$</span>.</p>
<p><span class="math-container">$\frac{1}{- sign(y_A - \alpha_A)} = \frac{1}{- sign(y_B - \alpha_B)}$</span></p>
<p><span class="math-container">$sign(y_A - \alpha_A) = sign(y_B - \alpha_B)$</span></p>
<p>This equation holds exactly when <span class="math-container">$\{y_A - \alpha_A, y_B - \alpha_B \}$</span> are either both positive or both negative.</p>
<p>Now for the points where for some agent <span class="math-container">$i$</span> it happens that <span class="math-container">$y_i = \alpha_i$</span>, let’s do a contour plot of indifference curves.</p>
<p>For the contour plot, note from the <span class="math-container">$MRS_i$</span> that the indifference curves are two connected line segments, each with slope <span class="math-container">$-MRS_i = sign(y_i - \alpha_i)$</span>.</p>
<p>Let’s hold an agent’s indifference curve constant. Pick a point where <span class="math-container">$y_i = \alpha_i$</span> for either <span class="math-container">$i$</span>. You can check that moving along the indifference curve, you can only move into the same or a worse indifference curve for the other agent.</p>
<p>This implies the points where <span class="math-container">$y_i = \alpha_i$</span> for either <span class="math-container">$i$</span> are efficient as well.</p>
<p>Therefore, the contract curve/efficient set <span class="math-container">$CC$</span> is given by</p>
<p><span class="math-container">$CC = \{ ((x_A,y_A),(x_B,y_B)) \in \mathcal{F} : [y_A \geq \alpha_A \text{ and } y_B \geq \alpha_B] \text{ or } [y_A \leq \alpha_A \text{ and } y_B \leq \alpha_B] \} $</span></p>
<p>where <span class="math-container">$\mathcal{F}$</span> is the set of feasible allocations (where the endowment constraints hold), i.e. points in the Edgeworth box.</p>
<p>Here I’ll show all the <span class="math-container">$11$</span> possible cases graphically depending on where in the box are the <span class="math-container">$\alpha_i$</span> relative to each other and the endowment constraints (the signs I write on the right of the box are the corresponding signs of the <span class="math-container">$y_i - \alpha_i$</span>):</p>
<p>Note: Whatever is in red corresponds to agent <span class="math-container">$A$</span>, and whatever is in black corresponds to agent <span class="math-container">$B$</span>.</p>
<p><a href="https://i.sstatic.net/vbLwz.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vbLwz.jpg" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/uHENe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uHENe.png" alt="enter image description here" /></a></p>
<ol start="4">
<li><span class="math-container">$CC$</span> is the below region.</li>
<li><span class="math-container">$CC$</span> is the above region.</li>
<li><span class="math-container">$CC$</span> is the entire box.</li>
</ol>
<p><a href="https://i.sstatic.net/rTlo5.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/rTlo5.jpg" alt="enter image description here" /></a></p>
<ol start="7">
<li><span class="math-container">$CC$</span> is the entire box.</li>
</ol>
<p><a href="https://i.sstatic.net/MG4Yi.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/MG4Yi.jpg" alt="enter image description here" /></a></p>
<ol start="8">
<li><span class="math-container">$CC$</span> is the below region.</li>
<li><span class="math-container">$CC$</span> is the above region.</li>
</ol>
<p><a href="https://i.sstatic.net/gD3GU.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gD3GU.jpg" alt="enter image description here" /></a></p>
<ol start="10">
<li><span class="math-container">$CC$</span> is the below border.</li>
<li><span class="math-container">$CC$</span> is the above border.</li>
</ol>
<p>Note for 10) and 11), my algebraic description of <span class="math-container">$CC$</span> yields no solution, so this means you have to graphically search for frontier solutions.</p>

CC BY-SA 4.0
2023-05-23T07:16:12.830Z
(+1) Good Answer.
Amit
CC BY-SA 4.0
2023-05-22T22:20:19.440Z
2023-05-23T14:14:04.490Z
2023-05-23T14:14:04.490Z
Nicolas Torres
<p>Suppose we have an pure exchange economy with 2 consumers, and 2 goods <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>. Fix some <span class="math-container">$\alpha_1$</span> and <span class="math-container">$\alpha_2$</span>. The utility for consumer <span class="math-container">$i$</span> is defined by:
<span class="math-container">$$u_i(x_{1i},x_{2i}) = x_{1i} - |x_{2i} - \alpha_{i}|.$$</span></p>
<p>How to find all the Pareto efficient allocations for this economy?</p>

CC BY-SA 4.0
2023-05-23T14:21:30.330Z
@Stef The first index is for the goods and the second index $i$ is for consumer $i$. I feel uncomfortable with that notation, so I renamed the goods in my answer as $x,y$ rather than $x_1,x_2$.
Nicolas Torres
<general-equilibrium><pareto-efficiency>
Pareto efficient allocations for non-monotonic, quasi-linear utility function
Mondayisgood
# Pareto efficient allocations for non-monotonic, quasi-linear utility function

(Asked by: username_0 on 2023-05-22)

Suppose we have an pure exchange economy with 2 consumers, and 2 goods $x\_1$ and $x\_2$. Fix some $\alpha\_1$ and $\alpha\_2$. The utility for consumer $i$ is defined by:
$$u\_i(x\_{1i},x\_{2i}) = x\_{1i} - |x\_{2i} - \alpha\_{i}|.$$

How to find all the Pareto efficient allocations for this economy?

**Tags:** general-equilibrium, pareto-efficiency

> **Comments:**
> **username_1**: @Stef The first index is for the goods and the second index $i$ is for consumer $i$. I feel uncomfortable with that notation, so I renamed the goods in my answer as $x,y$ rather than $x_1,x_2$.  

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## 2 answers

### Answer by username_2 on 2023-05-23. Score: 5

Given the economy:

* Utility functions: $u\_A(x\_A,y\_A) = x\_A - |y\_A-\alpha\_A|$, $u\_B(x\_B,y\_B) = x\_B - |y\_B-\alpha\_B|$, where $\alpha\_A \geq 0$, and $\alpha\_B\geq 0$ are given.
* $\omega = (\omega\_X, \omega\_Y)$, where $\omega\_X>0$, and $\omega\_Y>0$

Set of feasible allocations is given by

$\mathcal{F} =\{((x\_A,y\_A),(x\_B,y\_B))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_A+x\_B=\omega\_X \ \wedge \ y\_A+y\_B=\omega\_Y \} $

Set of Pareto efficient Allocations is given by

$\{((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}|\min(\alpha\_A,\omega\_Y-\alpha\_B)\leq y\_A \leq \max(\alpha\_A,\omega\_Y-\alpha\_B)\}$

[![enter image description here](https://i.sstatic.net/3Etl3.png)](https://i.sstatic.net/3Etl3.png)


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### Answer by username_1 on 2023-05-22. Score: 3

Let’s rename the variables $x := x\_1$, $y:= x\_2$.

First let’s consider the set of feasible allocations such that $y\_i \neq \alpha\_i$ for both $i = A,B$.

In this set, both utilities are completely differentiable.

The optimization problem we want to solve is

$\max x\_A - |y\_A - \alpha\_A|$

$s.t. x\_B - |y\_B - \alpha\_B| = \overline{U}$

and the endowment constraints

$x\_A + x\_B = \omega\_x$

$y\_A + y\_B = \omega\_y$

where $(\omega\_x,\omega\_y)$ are the total endowments for $(x,y)$.

Since both utilities are completely differentiable in this region, the efficiency condition is

$MRS\_A = MRS\_B$

$\frac{\frac{\partial U\_A}{\partial x\_A}}{\frac{\partial U\_A}{\partial y\_A}} = \frac{\frac{\partial U\_B}{\partial x\_B}}{\frac{\partial U\_B}{\partial y\_B}} $

Note $\frac{d}{dx} |x| = sign(x)$ for $x \neq 0$.

$\frac{1}{- sign(y\_A - \alpha\_A)} = \frac{1}{- sign(y\_B - \alpha\_B)}$

$sign(y\_A - \alpha\_A) = sign(y\_B - \alpha\_B)$

This equation holds exactly when $\{y\_A - \alpha\_A, y\_B - \alpha\_B \}$ are either both positive or both negative.

Now for the points where for some agent $i$ it happens that $y\_i = \alpha\_i$, let’s do a contour plot of indifference curves.

For the contour plot, note from the $MRS\_i$ that the indifference curves are two connected line segments, each with slope $-MRS\_i = sign(y\_i - \alpha\_i)$.

Let’s hold an agent’s indifference curve constant. Pick a point where $y\_i = \alpha\_i$ for either $i$. You can check that moving along the indifference curve, you can only move into the same or a worse indifference curve for the other agent.

This implies the points where $y\_i = \alpha\_i$ for either $i$ are efficient as well.

Therefore, the contract curve/efficient set $CC$ is given by

$CC = \{ ((x\_A,y\_A),(x\_B,y\_B)) \in \mathcal{F} : [y\_A \geq \alpha\_A \text{ and } y\_B \geq \alpha\_B] \text{ or } [y\_A \leq \alpha\_A \text{ and } y\_B \leq \alpha\_B] \} $

where $\mathcal{F}$ is the set of feasible allocations (where the endowment constraints hold), i.e. points in the Edgeworth box.

Here I’ll show all the $11$ possible cases graphically depending on where in the box are the $\alpha\_i$ relative to each other and the endowment constraints (the signs I write on the right of the box are the corresponding signs of the $y\_i - \alpha\_i$):

Note: Whatever is in red corresponds to agent $A$, and whatever is in black corresponds to agent $B$.

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

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

4. $CC$ is the below region.
5. $CC$ is the above region.
6. $CC$ is the entire box.

[![enter image description here](https://i.sstatic.net/rTlo5.jpg)](https://i.sstatic.net/rTlo5.jpg)

7. $CC$ is the entire box.

[![enter image description here](https://i.sstatic.net/MG4Yi.jpg)](https://i.sstatic.net/MG4Yi.jpg)

8. $CC$ is the below region.
9. $CC$ is the above region.

[![enter image description here](https://i.sstatic.net/gD3GU.jpg)](https://i.sstatic.net/gD3GU.jpg)

10. $CC$ is the below border.
11. $CC$ is the above border.

Note for 10) and 11), my algebraic description of $CC$ yields no solution, so this means you have to graphically search for frontier solutions.

> **Comments:**
> **username_2**: (+1) Good Answer.  

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