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In one exercise, we have to argue that a Walrasian Equilibrium exists and the solution says that if we can see that excess demand goes to infinity as price goes to 0, and as price goes to infinity, excess demand tends to -1, then we could argue there exists a p* such that excess demand equals to 0 (meaning there is a Walrasian Equilibrium).

Could anyone explain why these two conditions are sufficient to ensure this equilibrium exists?

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Let $z_j(p)$ be the excess demand function for good $j$, where $p := \frac{p_2}{p_1}$ is the relative price between the two goods.

Note it is possible to express the excess demand functions as single variable functions of the relative price because the Walrasian demands are homogeneous of degree $0$ in prices, and hence, the excess demands are also homogeneous of degree $0$ in prices.

If $\lim_{p \to 0} z_2(p) = \infty$, then there exists a relative price $c$ such that $z_2(c) > 0$.

On the other hand, if $\lim_{p \to \infty} z_2(p) = -1$, then there exists a relative price $d$ such that $z_2(d) < 0$. (Note this $d$ can be taken greater than $c$ because of the directions of the limits.)

By the Intermediate Value Theorem, since the excess demand functions are continuous when both preferences are strictly convex,

$\forall y \in (z_2(d), z_2(c)) \exists p \in (c,d) : z_2(p) = y$

In particular, there is a $p^\star : z_2(p^\star) = 0$

By Walras’s law, it must also hold that $z_1(p^\star) = 0$

Therefore, that $p^\star$ is a Walrasian equilibrium.

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