# Why doesn't this exchange economy have a walrasian equilibrium?

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)?

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.
• 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 $x^*=y^*=1$. Oct 30, 2023 at 19:18