<p><strong>To add to my previous answer</strong> (and to reply to your comment), the set of equations for Pareto optimality include the MRS equations and the feasibility equations.</p>
<p>MRS equations:
<span class="math-container">\begin{align}
\text{MRS}^A_{x,y} = \text{MRS}^B_{x,y} = \text{MRS}^C_{x,y} &amp;\iff  \frac{\frac{\partial U_A}{\partial x}}{\frac{\partial U_A}{\partial y}} = \frac{\frac{\partial U_B}{\partial x}}{\frac{\partial U_B}{\partial y}} = \frac{\frac{\partial U_C}{\partial x}}{\frac{\partial U_C}{\partial y}} \\ &amp;\iff \frac{y_A z_A^2}{x_A z_A^2} = \frac{2x_By_Bz_B}{x_B^2z_B} = \frac{y_C^2z_C}{x_Cy_Cz_C} \tag{1} \\
\text{MRS}^A_{y,z} = \text{MRS}^B_{y,z} = \text{MRS}^C_{y,z} &amp;\iff  \frac{\frac{\partial U_A}{\partial y}}{\frac{\partial U_A}{\partial z}} = \frac{\frac{\partial U_B}{\partial y}}{\frac{\partial U_B}{\partial z}} = \frac{\frac{\partial U_C}{\partial y}}{\frac{\partial U_C}{\partial z}}\tag{2} \\
\text{MRS}^A_{z,x} = \text{MRS}^B_{z,x} = \text{MRS}^C_{z,x} &amp;\iff  \frac{\frac{\partial U_A}{\partial z}}{\frac{\partial U_A}{\partial x}} = \frac{\frac{\partial U_B}{\partial z}}{\frac{\partial U_B}{\partial x}} = \frac{\frac{\partial U_C}{\partial z}}{\frac{\partial U_C}{\partial x}} \tag{3}
\end{align}</span></p>
<p>The feasibility allocations are: <span class="math-container">$$x_A + y_A + z_A = 1 + 2 + 1 = 4 \tag{4}$$</span> <span class="math-container">$$x_B + y_B + z_B = 1 + 1 + 5 = 7 \tag{5}$$</span> <span class="math-container">$$x_C + y_C + z_C = 1 + 3 + 1 = 5 \tag{6}$$</span></p>
<p>The equations <span class="math-container">$1-6$</span> combined will give you the final Pareto Efficient allocations.</p>

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2022-10-16T23:03:12.073Z
2022-10-16T23:04:41.363Z
2022-10-16T23:04:41.363Z
Ricardo
<p><strong>Pareto Optimality:</strong></p>
<p>Since preferences are convex in your case, you can find Pareto optimality in the same way. You need to solve <span class="math-container">$\text{MRS}^{A}_{v,w} = \text{MRS}^{B}_{v,w} = \text{MRS}^{C}_{v,w}$</span> where <span class="math-container">$\{v,w : v \neq w\} \subset \{x,y,z\}$</span>.</p>
<p>The reason this works is because you can maximize <span class="math-container">$U_A$</span> subject to <span class="math-container">$U_B = \overline{u_B}$</span> and <span class="math-container">$U_C = \overline{u_C}$</span> and similarly for the other two agents. Since your utility functions are convex, when you set up the Lagrangian, you will find that the MRSs have to be equal to one another.</p>
<p><strong>Walrasian equilibrium:</strong></p>
<p>You have to solve the following (three) maximization problems: <span class="math-container">$$\max [U_P(x_P,y_P,z_P)] \text{ subject to } p_xx_P + p_y y_P + p_z z_P = p_x w_x^A + p_y w_y^A + p_z w_z^A$$</span></p>
<p>for each <span class="math-container">$P \in \{A, B, C\}$</span>. You will be able to find <span class="math-container">$p_x : p_y : p_z$</span> and the allocations.</p>
<p>Additionally, you can verify that these points lie in the Pareto optimal set due to the <a href="https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics" rel="nofollow noreferrer">first welfare theorem</a>.</p>
<hr />
<p>Later edit: Your set of equations for Pareto optimality will be:</p>
<ol>
<li><span class="math-container">$\begin{aligned}\text{MRS}^A_{x,y} = \text{MRS}^B_{x,y} = \text{MRS}^C_{x,y} &amp;\iff  \frac{\frac{\partial U_A}{\partial x}}{\frac{\partial U_A}{\partial y}} = \frac{\frac{\partial U_B}{\partial x}}{\frac{\partial U_B}{\partial y}} = \frac{\frac{\partial U_C}{\partial x}}{\frac{\partial U_C}{\partial y}} \\ &amp;\iff \frac{y_A z_A^2}{x_A z_A^2} = \frac{2x_By_Bz_B}{x_B^2z_B} = \frac{y_C^2z_C}{x_Cy_Cz_C}\end{aligned}$</span></li>
<li><span class="math-container">$\begin{aligned}\text{MRS}^A_{y,z} = \text{MRS}^B_{y,z} = \text{MRS}^C_{y,z} \iff  \frac{\frac{\partial U_A}{\partial y}}{\frac{\partial U_A}{\partial z}} = \frac{\frac{\partial U_B}{\partial y}}{\frac{\partial U_B}{\partial z}} = \frac{\frac{\partial U_C}{\partial y}}{\frac{\partial U_C}{\partial z}}\end{aligned}$</span></li>
<li><span class="math-container">$\begin{aligned}\text{MRS}^A_{z,x} = \text{MRS}^B_{z,x} = \text{MRS}^C_{z,x} \iff  \frac{\frac{\partial U_A}{\partial z}}{\frac{\partial U_A}{\partial x}} = \frac{\frac{\partial U_B}{\partial z}}{\frac{\partial U_B}{\partial x}} = \frac{\frac{\partial U_C}{\partial z}}{\frac{\partial U_C}{\partial x}}\end{aligned}$</span></li>
</ol>

CC BY-SA 4.0
2022-10-16T21:37:05.640Z
As for the endowment notation, $(w^A_x, w^A_y, w^A_z) = (1,1,1)$ and similarly for $B,C$.
Ricardo
CC BY-SA 4.0
2022-10-16T22:10:51.043Z
Thank you for your help. But, I am confused at the point of finding MRS for Pareto optimality. For example, $w={x,y}$ and $v=z$so $MRS_A=-2wv^2/2w^2v=-v/w$ and so on… can you please show how I find MRS explicitly? Thanks a lot.
studentp
CC BY-SA 4.0
2022-10-16T21:35:20.443Z
2022-11-04T17:31:29.357Z
2022-11-04T17:31:29.357Z
Ricardo
<p>We have 3 people and 3 goods</p>
<p><span class="math-container">$$U_A(x,y,z)=x_Ay_Az_A^2$$</span></p>
<p><span class="math-container">$$U_B(x,y,z)=x_B^2y_Bz_B$$</span></p>
<p><span class="math-container">$$U_C(x,y,z)=x_Cy_C^2z_C$$</span></p>
<p>Endowments are
<span class="math-container">$W_A= (1,1,1)$</span></p>
<p><span class="math-container">$W_B= (2,1,3)$</span></p>
<p><span class="math-container">$W_C= (1,5,1)$</span></p>
<p>I am confused due to the presence of three goods</p>
<p>In the case of two goods, I can find MRS=<span class="math-container">$\partial U_x/\partial U_y$</span></p>
<p>Now, how can find MRS to obtain Pareto optimal allocations and Walrasian equilibrium ?</p>
<p>Any help will be appreciated. Thank you.</p>

<microeconomics><self-study><pure-exchange-economy>
Finding Pareto optimal allocations and Walrasian equilibrium allocations in the case of 3 goods
studentp
# Finding Pareto optimal allocations and Walrasian equilibrium allocations in the case of 3 goods

(Asked by: username_0 on 2022-10-16)

We have 3 people and 3 goods

$$U\_A(x,y,z)=x\_Ay\_Az\_A^2$$

$$U\_B(x,y,z)=x\_B^2y\_Bz\_B$$

$$U\_C(x,y,z)=x\_Cy\_C^2z\_C$$

Endowments are
$W\_A= (1,1,1)$

$W\_B= (2,1,3)$

$W\_C= (1,5,1)$

I am confused due to the presence of three goods

In the case of two goods, I can find MRS=$\partial U\_x/\partial U\_y$

Now, how can find MRS to obtain Pareto optimal allocations and Walrasian equilibrium ?

Any help will be appreciated. Thank you.

**Tags:** microeconomics, self-study, pure-exchange-economy


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## 2 answers

### Answer by username_1 on 2022-10-16. Score: 4 | Accepted answer

**Pareto Optimality:**

Since preferences are convex in your case, you can find Pareto optimality in the same way. You need to solve $\text{MRS}^{A}\_{v,w} = \text{MRS}^{B}\_{v,w} = \text{MRS}^{C}\_{v,w}$ where $\{v,w : v \neq w\} \subset \{x,y,z\}$.

The reason this works is because you can maximize $U\_A$ subject to $U\_B = \overline{u\_B}$ and $U\_C = \overline{u\_C}$ and similarly for the other two agents. Since your utility functions are convex, when you set up the Lagrangian, you will find that the MRSs have to be equal to one another.

**Walrasian equilibrium:**

You have to solve the following (three) maximization problems: $$\max [U\_P(x\_P,y\_P,z\_P)] \text{ subject to } p\_xx\_P + p\_y y\_P + p\_z z\_P = p\_x w\_x^A + p\_y w\_y^A + p\_z w\_z^A$$

for each $P \in \{A, B, C\}$. You will be able to find $p\_x : p\_y : p\_z$ and the allocations.

Additionally, you can verify that these points lie in the Pareto optimal set due to the [first welfare theorem](https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics).

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Later edit: Your set of equations for Pareto optimality will be:

1. $\begin{aligned}\text{MRS}^A\_{x,y} = \text{MRS}^B\_{x,y} = \text{MRS}^C\_{x,y} &\iff \frac{\frac{\partial U\_A}{\partial x}}{\frac{\partial U\_A}{\partial y}} = \frac{\frac{\partial U\_B}{\partial x}}{\frac{\partial U\_B}{\partial y}} = \frac{\frac{\partial U\_C}{\partial x}}{\frac{\partial U\_C}{\partial y}} \\ &\iff \frac{y\_A z\_A^2}{x\_A z\_A^2} = \frac{2x\_By\_Bz\_B}{x\_B^2z\_B} = \frac{y\_C^2z\_C}{x\_Cy\_Cz\_C}\end{aligned}$
2. $\begin{aligned}\text{MRS}^A\_{y,z} = \text{MRS}^B\_{y,z} = \text{MRS}^C\_{y,z} \iff \frac{\frac{\partial U\_A}{\partial y}}{\frac{\partial U\_A}{\partial z}} = \frac{\frac{\partial U\_B}{\partial y}}{\frac{\partial U\_B}{\partial z}} = \frac{\frac{\partial U\_C}{\partial y}}{\frac{\partial U\_C}{\partial z}}\end{aligned}$
3. $\begin{aligned}\text{MRS}^A\_{z,x} = \text{MRS}^B\_{z,x} = \text{MRS}^C\_{z,x} \iff \frac{\frac{\partial U\_A}{\partial z}}{\frac{\partial U\_A}{\partial x}} = \frac{\frac{\partial U\_B}{\partial z}}{\frac{\partial U\_B}{\partial x}} = \frac{\frac{\partial U\_C}{\partial z}}{\frac{\partial U\_C}{\partial x}}\end{aligned}$

> **Comments:**
> **username_1**: As for the endowment notation, $(w^A_x, w^A_y, w^A_z) = (1,1,1)$ and similarly for $B,C$.  
> **username_0**: Thank you for your help. But, I am confused at the point of finding MRS for Pareto optimality. For example, $w={x,y}$ and $v=z$so $MRS_A=-2wv^2/2w^2v=-v/w$ and so on… can you please show how I find MRS explicitly? Thanks a lot.  

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### Answer by username_1 on 2022-10-16. Score: 2

**To add to my previous answer** (and to reply to your comment), the set of equations for Pareto optimality include the MRS equations and the feasibility equations.

MRS equations:
\begin{align}
\text{MRS}^A\_{x,y} = \text{MRS}^B\_{x,y} = \text{MRS}^C\_{x,y} &\iff \frac{\frac{\partial U\_A}{\partial x}}{\frac{\partial U\_A}{\partial y}} = \frac{\frac{\partial U\_B}{\partial x}}{\frac{\partial U\_B}{\partial y}} = \frac{\frac{\partial U\_C}{\partial x}}{\frac{\partial U\_C}{\partial y}} \\ &\iff \frac{y\_A z\_A^2}{x\_A z\_A^2} = \frac{2x\_By\_Bz\_B}{x\_B^2z\_B} = \frac{y\_C^2z\_C}{x\_Cy\_Cz\_C} \tag{1} \\
\text{MRS}^A\_{y,z} = \text{MRS}^B\_{y,z} = \text{MRS}^C\_{y,z} &\iff \frac{\frac{\partial U\_A}{\partial y}}{\frac{\partial U\_A}{\partial z}} = \frac{\frac{\partial U\_B}{\partial y}}{\frac{\partial U\_B}{\partial z}} = \frac{\frac{\partial U\_C}{\partial y}}{\frac{\partial U\_C}{\partial z}}\tag{2} \\
\text{MRS}^A\_{z,x} = \text{MRS}^B\_{z,x} = \text{MRS}^C\_{z,x} &\iff \frac{\frac{\partial U\_A}{\partial z}}{\frac{\partial U\_A}{\partial x}} = \frac{\frac{\partial U\_B}{\partial z}}{\frac{\partial U\_B}{\partial x}} = \frac{\frac{\partial U\_C}{\partial z}}{\frac{\partial U\_C}{\partial x}} \tag{3}
\end{align}

The feasibility allocations are: $$x\_A + y\_A + z\_A = 1 + 2 + 1 = 4 \tag{4}$$ $$x\_B + y\_B + z\_B = 1 + 1 + 5 = 7 \tag{5}$$ $$x\_C + y\_C + z\_C = 1 + 3 + 1 = 5 \tag{6}$$

The equations $1-6$ combined will give you the final Pareto Efficient allocations.


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