<p>Given the pure-exchange economy:</p>
<ul>
<li>Utility functions: <span class="math-container">$u_A=\max(5,\min(x_A,y_A))$</span>, <span class="math-container">$u_B=\min(x_B,\frac{3}{2}y_B)$</span></li>
<li>Endowments: <span class="math-container">$\omega_A=(20,0)$</span>, <span class="math-container">$\omega_B=(0,15)$</span></li>
</ul>
<p>Set of Feasible allocations is given by:</p>
<p><span class="math-container">$\mathcal{F}=\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=20 \ \wedge \ y_A+y_B=15\}$</span></p>
<p>Set of Pareto optimal Allocations is given by:</p>
<p><span class="math-container">$\mathcal{PE}=\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|x_A=0 \ \wedge 0\leq y_A\leq \frac{5}{3}\} \ \cup \ \{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|5&lt;\frac{5}{3}+\frac{2}{3}x_A\leq y_A \leq x_A\} $</span></p>
<p>Set of competitive equilibrium allocations is given by</p>
<p><span class="math-container">$\mathcal{CE}=\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|y_B=\frac{2}{3}x_B\geq 10\}$</span></p>
<p>Each allocation in the set above corresponds to exactly one equilibrium price ratio <span class="math-container">$\frac{p_X}{p_Y}$</span> in the set <span class="math-container">$[\frac{1}{12},\frac{1}{3}]$</span>. To be specific, for each <span class="math-container">$x_B\in[15,20]$</span>, the equilibrium allocation is <span class="math-container">$y_B=\frac{2}{3}x_B$</span>, <span class="math-container">$x_A=20-x_B$</span>, <span class="math-container">$y_A=15-\frac{2}{3}x_B$</span> and the corresponding price ratio is <span class="math-container">$\frac{p_X}{p_Y}=\frac{45-2x_B}{3x_B}$</span>.</p>
<p>There are no equilibria with either <span class="math-container">$p_X=0$</span> or <span class="math-container">$p_Y=0$</span> because at <span class="math-container">$p_X=0$</span>, consumer 2 will demand at least <span class="math-container">$22.5$</span> units of <span class="math-container">$X$</span> leading to excess demand for it, and at <span class="math-container">$p_Y=0$</span>, consumer 1 will demand at least <span class="math-container">$20$</span> units of <span class="math-container">$Y$</span> causing exceed demand for <span class="math-container">$Y$</span>.</p>
<p><a href="https://i.sstatic.net/LGmpR6dr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LGmpR6dr.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-09-28T18:58:26.043Z
Thank you so much Amit, you are a smart guy!
Nicolas Torres
CC BY-SA 4.0
2024-09-28T19:53:07.380Z
Thank you @NicolasTorres
Amit
CC BY-SA 4.0
2024-09-28T04:16:58.397Z
2024-09-28T04:52:45.543Z
2024-09-28T04:52:45.543Z
Amit
<p>I'm trying to solve this General Equilibrium excercise which I find quite challenging as both agents have funky utility functions.</p>
<p>Find the Pareto Efficient allocations and Competitive Equilibra for this pure exchange economy:</p>
<p><span class="math-container">$U_A(x,y)=\max(5,\min(x,y))$</span></p>
<p><span class="math-container">$U_B(x,y)=\min(x,\frac{3}{2}y)$</span></p>
<p><span class="math-container">$\omega_A = (20,0)$</span></p>
<p><span class="math-container">$\omega_B = (0,15)$</span></p>
<p>Here I have my Edgeworth box where</p>
<p>Red: Indifference curve shape for A (L shape above and to the right of the kink)</p>
<p>Blue: Indifference curve shape for B (L shape below and to the left of the kink)</p>
<p>Green: Budget line for <span class="math-container">$\frac{p_1}{p_2}=\frac{1}{3}$</span></p>
<p>Orange: Leontief expansion path for A: <span class="math-container">$x=y&gt;5$</span></p>
<p>Pink: Expansion path for B: <span class="math-container">$y=\frac{2}{3}x$</span></p>
<p><a href="https://i.sstatic.net/BO9h79Kz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/BO9h79Kz.png" alt="Edgeworth box" /></a></p>
<p>Is the Pareto efficient set the portion of the blue indifference curve that intersects the y axis?</p>
<p>Now for the competitive equilibria: The set of optimal bundles for B is the entire pink curve.</p>
<p>For A, the optimal bundles are: the set below the budget line for <span class="math-container">$p=\frac{1}{3}$</span>, which is supported by prices <span class="math-container">$\frac{p_1}{p_2}\leq \frac{1}{3}$</span> union the orange curve which is supported by prices <span class="math-container">$\frac{p_1}{p_2} &gt; \frac{1}{3}$</span>.</p>
<p>So would the competitive equilibria allocations be the interesction of the optimal points for A and B, which would be the portion of the pink segment below the budget line for <span class="math-container">$\frac{p_1}{p_2} = \frac{1}{3}$</span> ?</p>
<p>If this is correct, how would I write the Competitive Equilibria set in the form <span class="math-container">$(p,x)$</span> where <span class="math-container">$p$</span> is the price vector and <span class="math-container">$x$</span> is the allocation vector, i.e. the quantities each consumer gets of each good.</p>
<p>Also, are there any equilibria for <span class="math-container">$p_1 = 0$</span> or <span class="math-container">$p_2 = 0$</span> ?</p>

<microeconomics><general-equilibrium><pareto-efficiency><competitive-equilibrium><edgeworth-box>
Find the Pareto Efficient allocations and Competitive Equilibrium when both agents have funky functions
Nicolas Torres
# Find the Pareto Efficient allocations and Competitive Equilibrium when both agents have funky functions

(Asked by: username_0 on 2024-09-28)

I'm trying to solve this General Equilibrium excercise which I find quite challenging as both agents have funky utility functions.

Find the Pareto Efficient allocations and Competitive Equilibra for this pure exchange economy:

$U\_A(x,y)=\max(5,\min(x,y))$

$U\_B(x,y)=\min(x,\frac{3}{2}y)$

$\omega\_A = (20,0)$

$\omega\_B = (0,15)$

Here I have my Edgeworth box where

Red: Indifference curve shape for A (L shape above and to the right of the kink)

Blue: Indifference curve shape for B (L shape below and to the left of the kink)

Green: Budget line for $\frac{p\_1}{p\_2}=\frac{1}{3}$

Orange: Leontief expansion path for A: $x=y>5$

Pink: Expansion path for B: $y=\frac{2}{3}x$

[![Edgeworth box](https://i.sstatic.net/BO9h79Kz.png)](https://i.sstatic.net/BO9h79Kz.png)

Is the Pareto efficient set the portion of the blue indifference curve that intersects the y axis?

Now for the competitive equilibria: The set of optimal bundles for B is the entire pink curve.

For A, the optimal bundles are: the set below the budget line for $p=\frac{1}{3}$, which is supported by prices $\frac{p\_1}{p\_2}\leq \frac{1}{3}$ union the orange curve which is supported by prices $\frac{p\_1}{p\_2} > \frac{1}{3}$.

So would the competitive equilibria allocations be the interesction of the optimal points for A and B, which would be the portion of the pink segment below the budget line for $\frac{p\_1}{p\_2} = \frac{1}{3}$ ?

If this is correct, how would I write the Competitive Equilibria set in the form $(p,x)$ where $p$ is the price vector and $x$ is the allocation vector, i.e. the quantities each consumer gets of each good.

Also, are there any equilibria for $p\_1 = 0$ or $p\_2 = 0$ ?

**Tags:** microeconomics, general-equilibrium, pareto-efficiency, competitive-equilibrium, edgeworth-box

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## 1 answers

### Answer by username_1 on 2024-09-28. Score: 2 | Accepted answer

Given the pure-exchange economy:

* Utility functions: $u\_A=\max(5,\min(x\_A,y\_A))$, $u\_B=\min(x\_B,\frac{3}{2}y\_B)$
* Endowments: $\omega\_A=(20,0)$, $\omega\_B=(0,15)$

Set of Feasible allocations is given by:

$\mathcal{F}=\{((x\_A,y\_A),(x\_B,y\_B))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_A+x\_B=20 \ \wedge \ y\_A+y\_B=15\}$

Set of Pareto optimal Allocations is given by:

$\mathcal{PE}=\{((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}|x\_A=0 \ \wedge 0\leq y\_A\leq \frac{5}{3}\} \ \cup \ \{((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}|5<\frac{5}{3}+\frac{2}{3}x\_A\leq y\_A \leq x\_A\} $

Set of competitive equilibrium allocations is given by

$\mathcal{CE}=\{((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}|y\_B=\frac{2}{3}x\_B\geq 10\}$

Each allocation in the set above corresponds to exactly one equilibrium price ratio $\frac{p\_X}{p\_Y}$ in the set $[\frac{1}{12},\frac{1}{3}]$. To be specific, for each $x\_B\in[15,20]$, the equilibrium allocation is $y\_B=\frac{2}{3}x\_B$, $x\_A=20-x\_B$, $y\_A=15-\frac{2}{3}x\_B$ and the corresponding price ratio is $\frac{p\_X}{p\_Y}=\frac{45-2x\_B}{3x\_B}$.

There are no equilibria with either $p\_X=0$ or $p\_Y=0$ because at $p\_X=0$, consumer 2 will demand at least $22.5$ units of $X$ leading to excess demand for it, and at $p\_Y=0$, consumer 1 will demand at least $20$ units of $Y$ causing exceed demand for $Y$.

[![enter image description here](https://i.sstatic.net/LGmpR6dr.png)](https://i.sstatic.net/LGmpR6dr.png)

> **Comments:**
> **username_0**: Thank you so much Amit, you are a smart guy!  
> **username_1**: Thank you @NicolasTorres  

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