<p>The economy you described does have a Walrasian Equilibrium when <span class="math-container">$n$</span> is even. The Equilibrium price ratio is <span class="math-container">$\dfrac{p_X}{p_Y}=1$</span>. The corresponding equilibrium allocation is any allocation in which <span class="math-container">$\frac{n}{2}$</span> agents consumes the bundles <span class="math-container">$(x,y)=(2,0)$</span> each, and the remaining <span class="math-container">$\frac{n}{2}$</span> agents consumes the bundles <span class="math-container">$(x,y)=(0,2)$</span> each. Check that this allocation is Pareto efficient. So, the first welfare theorem holds.</p>
<p>However, when <span class="math-container">$n$</span> is odd, the equilibrium does not exist. Since there is no competitive equilibrium in this case, first welfare theorem holds vacuously.</p>

CC BY-SA 4.0
2023-10-30T19:18:10.783Z
Thank you so much Amit! May I ask why we can't use the Lagrangian to solve the problem in this case? Because if I continue from my result (included in the question), I got $x^*=y^*=1$.
Ludwig Gershwin
CC BY-SA 4.0
2023-10-31T00:09:54.493Z
This is because of the shape of the ICs. But you can use Kuhn-Tucker conditions to find demand if you wish. Here is the procedure: https://youtu.be/eFe25HVJSWg?feature=shared
Amit
CC BY-SA 4.0
2023-10-30T16:27:35.623Z
2023-10-30T16:32:44.427Z
2023-10-30T16:32:44.427Z
Amit
<p>Consider an exchange economy with <span class="math-container">$n$</span> individuals and 2 goods <span class="math-container">$x,y$</span>. All individuals have the same utility function <span class="math-container">$u(x,y)=x^2+y^2$</span> and the same endowment <span class="math-container">$w=(1,1)$</span>.</p>
<p>I tried to solve the problem with Lagrangian:
<span class="math-container">\begin{align*}
x^2+y^2\\
p_x x+p_y y=p_x+p_y\\
\implies L=x^2+y^2+\lambda (p_x+p_y-p_x x+p_y y)\\
[x]2x=\lambda p_x\\
[y]2y=\lambda p_y\\
\implies \frac{x}{y}=\frac{p_x}{p_y}\\
\implies x=y\frac{p_x}{p_y}\\
\implies \frac{p_x^2}{p_y}y+p_y y=p_x+p_y\\
\implies x^*=\frac{p_y(p_x+p_y)}{p_x^2+p_y^2}
\end{align*}</span>
then by market clearing, <span class="math-container">$n\frac{p_y(p_x+p_y)}{p_x^2+p_y^2}=n\implies \frac{p_y(p_x+p_y)}{p_x^2+p_y^2}=1\implies p_x=p_y$</span>, which seems like a perfectly fine equilibrium price to me. This implies that every individual is autarky. But this economy doesn't have a Walrasian equilibrium. Why is that? And does the first welfare theorem hold in this economy (every walrasian equilibrium is pareto optimal)?</p>

<pure-exchange-economy>
Why doesn't this exchange economy have a walrasian equilibrium?
Ludwig Gershwin
# Why doesn't this exchange economy have a walrasian equilibrium?

(Asked by: username_0 on 2023-10-30)

Consider an exchange economy with $n$ individuals and 2 goods $x,y$. All individuals have the same utility function $u(x,y)=x^2+y^2$ and the same endowment $w=(1,1)$.

I tried to solve the problem with Lagrangian:
\begin{align\*}
x^2+y^2\\
p\_x x+p\_y y=p\_x+p\_y\\
\implies L=x^2+y^2+\lambda (p\_x+p\_y-p\_x x+p\_y y)\\
[x]2x=\lambda p\_x\\
[y]2y=\lambda p\_y\\
\implies \frac{x}{y}=\frac{p\_x}{p\_y}\\
\implies x=y\frac{p\_x}{p\_y}\\
\implies \frac{p\_x^2}{p\_y}y+p\_y y=p\_x+p\_y\\
\implies x^\*=\frac{p\_y(p\_x+p\_y)}{p\_x^2+p\_y^2}
\end{align\*}
then by market clearing, $n\frac{p\_y(p\_x+p\_y)}{p\_x^2+p\_y^2}=n\implies \frac{p\_y(p\_x+p\_y)}{p\_x^2+p\_y^2}=1\implies p\_x=p\_y$, which seems like a perfectly fine equilibrium price to me. This implies that every individual is autarky. But this economy doesn't have a Walrasian equilibrium. Why is that? And does the first welfare theorem hold in this economy (every walrasian equilibrium is pareto optimal)?

**Tags:** pure-exchange-economy

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## 1 answers

### Answer by username_1 on 2023-10-30. Score: 2 | Accepted answer

The economy you described does have a Walrasian Equilibrium when $n$ is even. The Equilibrium price ratio is $\dfrac{p\_X}{p\_Y}=1$. The corresponding equilibrium allocation is any allocation in which $\frac{n}{2}$ agents consumes the bundles $(x,y)=(2,0)$ each, and the remaining $\frac{n}{2}$ agents consumes the bundles $(x,y)=(0,2)$ each. Check that this allocation is Pareto efficient. So, the first welfare theorem holds.

However, when $n$ is odd, the equilibrium does not exist. Since there is no competitive equilibrium in this case, first welfare theorem holds vacuously.

> **Comments:**
> **username_0**: Thank you so much Amit! May I ask why we can't use the Lagrangian to solve the problem in this case? Because if I continue from my result (included in the question), I got $x^*=y^*=1$.  
> **username_1**: This is because of the shape of the ICs. But you can use Kuhn-Tucker conditions to find demand if you wish. Here is the procedure: https://youtu.be/eFe25HVJSWg?feature=shared  

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