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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)?

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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.

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  • $\begingroup$ 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 $x^*=y^*=1$. $\endgroup$ Commented Oct 30, 2023 at 19:18
  • $\begingroup$ 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: youtu.be/eFe25HVJSWg?feature=shared $\endgroup$
    – Amit
    Commented Oct 31, 2023 at 0:09

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