<p>I am assuming <span class="math-container">$x_{ij}$</span> denotes <span class="math-container">$i$</span>'s consumption of good <span class="math-container">$j$</span>. Let <span class="math-container">$\omega = (\omega_1,\omega_2)\gg 0$</span> be the endowment. Therefore, Set of feasible allocations is</p>
<p><span class="math-container">$\mathcal{F} = \{((x_{11},x_{12}),(x_{21},x_{22}))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_{11}+x_{21}=\omega_1 \ \wedge x_{12}+x_{22}=\omega_2\}$</span></p>
<p>Here consumer <span class="math-container">$1$</span>'s utility is affected by the amount of commodity <span class="math-container">$1$</span> consumed by consumer <span class="math-container">$2$</span> i.e. <span class="math-container">$x_{21}$</span>, and <span class="math-container">$u_1=x_{12}-x_{21}$</span>. Since we are interested in finding Pareto efficient allocations, we can consider the following adjustment in <span class="math-container">$u_1$</span> by taking into account the feasibility constraint.</p>
<p><span class="math-container">$u_1^{\text{ADJ}}=x_{12}-(\omega_1-x_{11}) =x_{11}+x_{12}-\omega_1$</span></p>
<p>and we know that</p>
<p><span class="math-container">$u_2 = x_{21}x_{22}$</span></p>
<p>Now we can apply the standard procedure to determine the Pareto efficient allocations. Interior Pareto efficient allocations will satisfy:</p>
<p><span class="math-container">$\dfrac{x_{22}}{x_{21}} = 1$</span>
i.e. <span class="math-container">$x_{22}=x_{21}$</span></p>
<p>Set of Pareto efficient allocations is given by
<span class="math-container">\begin{eqnarray*} \mathcal{P}=  \{((x_{11},x_{12}),(x_{21},x_{22}))\in\mathcal{F}|x_{22}=x_{21}\vee \min(x_{21},x_{22})=\min(\omega_1,\omega_2)\} \end{eqnarray*}</span></p>
<p><a href="https://i.sstatic.net/bM2wM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bM2wM.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-05-24T01:05:06.047Z
2023-05-24T01:05:06.047Z
Amit
<p>In the Edgeworth box we have <span class="math-container">$x_{21}=1-x_{11}$</span>, so, ignoring the constant, the first agent's utility function can simply be rewritten as <span class="math-container">$u_1=x_{11}+x_{12}$</span>. Thus, for agent 1 the two goods are perfect substitutes and her indifference curves are straight lines with slope <span class="math-container">$-1$</span>. Agent 2 has Cobb-Douglas indifference curves that are also symmetric between goods and therefore have slope <span class="math-container">$-1$</span> along the diagonal of the Edgeworth box. This diagonal is then also the set of Pareto efficient allocations.</p>

CC BY-SA 4.0
2021-06-18T08:19:13.847Z
2021-06-18T08:19:13.847Z
VARulle
<p>It seems like any allocation where <span class="math-container">$X_{11}=X_{12}$</span> is Pareto efficient.</p>
<p>Recall from Edgeworth box, <span class="math-container">$1 = X_{11} + X_{21}$</span> and <span class="math-container">$1 = X_{12} + X_{22}$</span>
Start with the utility function:
<span class="math-container">$$U_1 = X_{12} - X_{21} $$</span>
Use the first Edgeworth box identity to substitute out <span class="math-container">$X_{21}$</span>.
<span class="math-container">$$\Rightarrow U_1 = X_{12} + (X_{11} - 1) $$</span>
<span class="math-container">$$\Rightarrow MU_{11} = \frac{\partial U_1}{X_{11}} = 1 = \frac{\partial U_1}{X_{12}} = MU_{12}  = 1 $$</span>
At interior bundle utility maximization:
<span class="math-container">$$\Rightarrow \frac{MU_{11}}{P_1} = \frac{1}{P_1} = \frac{MU_{12}}{P_2} = \frac{1}{P_2} \rightarrow P_1 = P_2$$</span>
So the first guy is happy as long as the two goods have the same price.
<span class="math-container">$$U_2 = X_{21} \cdot X_{22} $$</span>
<span class="math-container">$$\Rightarrow MU_{21} = \frac{\partial U_2}{X_{21}} = X_{22} $$</span>
<span class="math-container">$$\Rightarrow MU_{22} = \frac{\partial U_2}{X_{22}} = X_{21} $$</span>
<span class="math-container">$$\Rightarrow \frac{MU_{21}}{P_1} = \frac{X_{22}}{P_1} = \frac{MU_{22}}{P_2} = \frac{X_{21}}{P_2} \rightarrow X_{21} = X_{22}$$</span>
But, again because of the Edgeworth both, this implies
<span class="math-container">$$ \Rightarrow 1-  X_{11} = X_{21} = X_{22} = 1-  X_{12} $$</span>
<span class="math-container">$$ \Rightarrow X_{11} = X_{12} $$</span></p>
<p>Which, when I check the result, manually seems to work. Consider <span class="math-container">$$\Rightarrow X_{11} = X_{12} = 0.25 \Rightarrow X_{21} = X_{22} = 0.75$$</span>
How many units of <span class="math-container">$X_{12}$</span> does person 1 want to give up 0.01 units of <span class="math-container">$X_{11}$</span>? He wants .02 units:
<span class="math-container">$$U_{1,old} = X_{12} + (X_{11} - 1) = 0.25 + 0.25 - 1 = -0.5$$</span>
<span class="math-container">$$U_{1,new} = X_{12} + (X_{11} - 1) = 0.24 + 0.26 - 1 = -0.5$$</span>
But that trade leaves the second person worse off:
<span class="math-container">$$U_{2,old} = X_{21} \cdot X_{22} = 0.75^2 = 9 / 16 = 0.5625 $$</span>
<span class="math-container">$$U_{2,new} = X_{21} \cdot X_{22} = 0.76 \cdot 0.74 = 0.562400365 &lt; U_{2,old}$$</span>
So there is no trade to do.</p>
<p>We should also check that the corner solutions aren't better.</p>
<ol>
<li>{1,0} is worse for person two</li>
<li>{0,1} is worse for person two</li>
<li>{1,1} is worse for person two</li>
<li>{0,0} is worse for person one</li>
</ol>
<p>More generally:</p>
<p>If <span class="math-container">$X_{11}=0 \rightarrow X_{21}=1$</span> and then <span class="math-container">$U_2 \geq 0.5625 \rightarrow  X_{22} \geq 0.5625$</span> this in turn implies that <span class="math-container">$X_{12} \leq 0.4375 \rightarrow U_1 \leq 0.4375 - 1 = -0.5625 &lt; -0.5$</span> which is not a pareto improvement.</p>
<p>If <span class="math-container">$X_{11}=1 \rightarrow X_{21}=0$</span> and then <span class="math-container">$U_2 \geq 0.5625$</span> has no valid solutions.</p>
<p>Similarly, if <span class="math-container">$X_{12}=1 \rightarrow X_{22}=0$</span> and then <span class="math-container">$U_2 \geq 0.5625$</span> has no valid solutions.</p>
<p>If <span class="math-container">$X_{12}=0 \rightarrow X_{22}=1$</span> and then <span class="math-container">$U_2 \geq 0.5625 \rightarrow  X_{21} \geq 0.5625$</span> this in turn implies that <span class="math-container">$X_{11} \leq 0.4375 \rightarrow U_1 \leq 0.4375 - 1 = -0.5625 &lt; -0.5$</span> which is not a pareto improvement.</p>
<p>So none of the corner solutions are pareto improvements. We could generalize this corner solution checking for arbitrary <span class="math-container">$X_{11}=X_{12}$</span>, but I'll leave that as an exercise to the reader.</p>

CC BY-SA 4.0
2021-06-17T20:25:08.957Z
2021-06-17T21:01:50.710Z
2021-06-17T21:01:50.710Z
BKay
<p>The pareto set is given by: boundary points <span class="math-container">$[(1,1), (0,0)]$</span> and <span class="math-container">$[(1,0), (0,1)]$</span>, and then all points on the far left, bottom and right boundaries of the edgeworth box.</p>
<p>I can provide working if anyone would like.</p>

CC BY-SA 4.0
2021-06-17T19:18:05.463Z
Showing your work would greatly improve this answer.
BKay
CC BY-SA 4.0
2021-06-17T19:27:32.133Z
I haven't the time. Shall do so shortly.
Charles
CC BY-SA 4.0
2021-06-17T19:16:46.130Z
2021-06-17T19:16:46.130Z
Charles
<p>I have a two-agent exchange economy where utility functions are given by: <span class="math-container">$u_{1}=x_{12}-x_{21}$</span> and <span class="math-container">$u_2=x_{21}x_{22}$</span>. I am looking to find the set of pareto efficient allocations.</p>
<p>Since utility functions are differentiable, then at any interior pareto efficient allocation, we know that the MRS of either consumer must be equal. So we need that: <span class="math-container">$\frac{0}{1}=\frac{x_{22}}{x_{21}}$</span>. Thus, in any interior pareto efficient allocation, <span class="math-container">$x_{22}=0$</span>. But of course any such allocation is along the boundary of the edgeworth box (and so is not interior). Does this mean that there are no interior pareto efficient allocations?</p>
<p>Thank you.</p>

<microeconomics><general-equilibrium>
Finding the set of pareto efficient allocations with externalities
Charles
# Finding the set of pareto efficient allocations with externalities

(Asked by: username_0 on 2021-06-17)

I have a two-agent exchange economy where utility functions are given by: $u\_{1}=x\_{12}-x\_{21}$ and $u\_2=x\_{21}x\_{22}$. I am looking to find the set of pareto efficient allocations.

Since utility functions are differentiable, then at any interior pareto efficient allocation, we know that the MRS of either consumer must be equal. So we need that: $\frac{0}{1}=\frac{x\_{22}}{x\_{21}}$. Thus, in any interior pareto efficient allocation, $x\_{22}=0$. But of course any such allocation is along the boundary of the edgeworth box (and so is not interior). Does this mean that there are no interior pareto efficient allocations?

Thank you.

**Tags:** microeconomics, general-equilibrium

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## 4 answers

### Answer by username_1 on 2023-05-24. Score: 3

I am assuming $x\_{ij}$ denotes $i$'s consumption of good $j$. Let $\omega = (\omega\_1,\omega\_2)\gg 0$ be the endowment. Therefore, Set of feasible allocations is

$\mathcal{F} = \{((x\_{11},x\_{12}),(x\_{21},x\_{22}))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_{11}+x\_{21}=\omega\_1 \ \wedge x\_{12}+x\_{22}=\omega\_2\}$

Here consumer $1$'s utility is affected by the amount of commodity $1$ consumed by consumer $2$ i.e. $x\_{21}$, and $u\_1=x\_{12}-x\_{21}$. Since we are interested in finding Pareto efficient allocations, we can consider the following adjustment in $u\_1$ by taking into account the feasibility constraint.

$u\_1^{\text{ADJ}}=x\_{12}-(\omega\_1-x\_{11}) =x\_{11}+x\_{12}-\omega\_1$

and we know that

$u\_2 = x\_{21}x\_{22}$

Now we can apply the standard procedure to determine the Pareto efficient allocations. Interior Pareto efficient allocations will satisfy:

$\dfrac{x\_{22}}{x\_{21}} = 1$
i.e. $x\_{22}=x\_{21}$

Set of Pareto efficient allocations is given by
\begin{eqnarray\*} \mathcal{P}= \{((x\_{11},x\_{12}),(x\_{21},x\_{22}))\in\mathcal{F}|x\_{22}=x\_{21}\vee \min(x\_{21},x\_{22})=\min(\omega\_1,\omega\_2)\} \end{eqnarray\*}

[![enter image description here](https://i.sstatic.net/bM2wM.png)](https://i.sstatic.net/bM2wM.png)


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### Answer by username_2 on 2021-06-18. Score: 1

In the Edgeworth box we have $x\_{21}=1-x\_{11}$, so, ignoring the constant, the first agent's utility function can simply be rewritten as $u\_1=x\_{11}+x\_{12}$. Thus, for agent 1 the two goods are perfect substitutes and her indifference curves are straight lines with slope $-1$. Agent 2 has Cobb-Douglas indifference curves that are also symmetric between goods and therefore have slope $-1$ along the diagonal of the Edgeworth box. This diagonal is then also the set of Pareto efficient allocations.


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### Answer by username_3 on 2021-06-17. Score: 1

It seems like any allocation where $X\_{11}=X\_{12}$ is Pareto efficient.

Recall from Edgeworth box, $1 = X\_{11} + X\_{21}$ and $1 = X\_{12} + X\_{22}$
Start with the utility function:
$$U\_1 = X\_{12} - X\_{21} $$
Use the first Edgeworth box identity to substitute out $X\_{21}$.
$$\Rightarrow U\_1 = X\_{12} + (X\_{11} - 1) $$
$$\Rightarrow MU\_{11} = \frac{\partial U\_1}{X\_{11}} = 1 = \frac{\partial U\_1}{X\_{12}} = MU\_{12} = 1 $$
At interior bundle utility maximization:
$$\Rightarrow \frac{MU\_{11}}{P\_1} = \frac{1}{P\_1} = \frac{MU\_{12}}{P\_2} = \frac{1}{P\_2} \rightarrow P\_1 = P\_2$$
So the first guy is happy as long as the two goods have the same price.
$$U\_2 = X\_{21} \cdot X\_{22} $$
$$\Rightarrow MU\_{21} = \frac{\partial U\_2}{X\_{21}} = X\_{22} $$
$$\Rightarrow MU\_{22} = \frac{\partial U\_2}{X\_{22}} = X\_{21} $$
$$\Rightarrow \frac{MU\_{21}}{P\_1} = \frac{X\_{22}}{P\_1} = \frac{MU\_{22}}{P\_2} = \frac{X\_{21}}{P\_2} \rightarrow X\_{21} = X\_{22}$$
But, again because of the Edgeworth both, this implies
$$ \Rightarrow 1- X\_{11} = X\_{21} = X\_{22} = 1- X\_{12} $$
$$ \Rightarrow X\_{11} = X\_{12} $$

Which, when I check the result, manually seems to work. Consider $$\Rightarrow X\_{11} = X\_{12} = 0.25 \Rightarrow X\_{21} = X\_{22} = 0.75$$
How many units of $X\_{12}$ does person 1 want to give up 0.01 units of $X\_{11}$? He wants .02 units:
$$U\_{1,old} = X\_{12} + (X\_{11} - 1) = 0.25 + 0.25 - 1 = -0.5$$
$$U\_{1,new} = X\_{12} + (X\_{11} - 1) = 0.24 + 0.26 - 1 = -0.5$$
But that trade leaves the second person worse off:
$$U\_{2,old} = X\_{21} \cdot X\_{22} = 0.75^2 = 9 / 16 = 0.5625 $$
$$U\_{2,new} = X\_{21} \cdot X\_{22} = 0.76 \cdot 0.74 = 0.562400365 < U\_{2,old}$$
So there is no trade to do.

We should also check that the corner solutions aren't better.

1. {1,0} is worse for person two
2. {0,1} is worse for person two
3. {1,1} is worse for person two
4. {0,0} is worse for person one

More generally:

If $X\_{11}=0 \rightarrow X\_{21}=1$ and then $U\_2 \geq 0.5625 \rightarrow X\_{22} \geq 0.5625$ this in turn implies that $X\_{12} \leq 0.4375 \rightarrow U\_1 \leq 0.4375 - 1 = -0.5625 < -0.5$ which is not a pareto improvement.

If $X\_{11}=1 \rightarrow X\_{21}=0$ and then $U\_2 \geq 0.5625$ has no valid solutions.

Similarly, if $X\_{12}=1 \rightarrow X\_{22}=0$ and then $U\_2 \geq 0.5625$ has no valid solutions.

If $X\_{12}=0 \rightarrow X\_{22}=1$ and then $U\_2 \geq 0.5625 \rightarrow X\_{21} \geq 0.5625$ this in turn implies that $X\_{11} \leq 0.4375 \rightarrow U\_1 \leq 0.4375 - 1 = -0.5625 < -0.5$ which is not a pareto improvement.

So none of the corner solutions are pareto improvements. We could generalize this corner solution checking for arbitrary $X\_{11}=X\_{12}$, but I'll leave that as an exercise to the reader.


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### Answer by username_0 on 2021-06-17. Score: -2

The pareto set is given by: boundary points $[(1,1), (0,0)]$ and $[(1,0), (0,1)]$, and then all points on the far left, bottom and right boundaries of the edgeworth box.

I can provide working if anyone would like.

> **Comments:**
> **username_3**: Showing your work would greatly improve this answer.  
> **username_0**: I haven't the time. Shall do so shortly.  

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