<p>Let <span class="math-container">$z_j(p)$</span> be the excess demand function for good <span class="math-container">$j$</span>, where <span class="math-container">$p := \frac{p_2}{p_1}$</span> is the relative price between the two goods.</p>
<p>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 <span class="math-container">$0$</span> in prices, and hence, the excess demands are also homogeneous of degree <span class="math-container">$0$</span> in prices.</p>
<p>If <span class="math-container">$\lim_{p \to 0} z_2(p) = \infty$</span>, then there exists a relative price <span class="math-container">$c$</span> such that <span class="math-container">$z_2(c) &gt; 0$</span>.</p>
<p>On the other hand, if <span class="math-container">$\lim_{p \to \infty} z_2(p) = -1$</span>, then there exists a relative price <span class="math-container">$d$</span> such that <span class="math-container">$z_2(d) &lt; 0$</span>. (Note this <span class="math-container">$d$</span> can be taken greater than <span class="math-container">$c$</span> because of the directions of the limits.)</p>
<p>By the Intermediate Value Theorem, since the excess demand functions are continuous when both preferences are strictly convex,</p>
<p><span class="math-container">$\forall y \in (z_2(d), z_2(c)) \exists p \in (c,d) : z_2(p) = y$</span></p>
<p>In particular, there is a <span class="math-container">$p^\star : z_2(p^\star) = 0$</span></p>
<p>By Walras’s law, it must also hold that <span class="math-container">$z_1(p^\star) = 0$</span></p>
<p>Therefore, that <span class="math-container">$p^\star$</span> is a Walrasian equilibrium.</p>

CC BY-SA 4.0
2023-03-14T22:10:25.700Z
2023-03-14T22:21:27.860Z
2023-03-14T22:21:27.860Z
Nicolas Torres
<p>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).</p>
<p>Could anyone explain why these two conditions are sufficient to ensure this equilibrium exists?</p>

<microeconomics><general-equilibrium><walrasian><difference-equations><equilibrium>
Why is there a Walrasian Equilibrium if excess demand goes to infinity as price goes to 0?
aliosha karamazov
# Why is there a Walrasian Equilibrium if excess demand goes to infinity as price goes to 0?

(Asked by: username_0 on 2023-03-14)

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?

**Tags:** microeconomics, general-equilibrium, walrasian, difference-equations, equilibrium

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## 1 answers

### Answer by username_1 on 2023-03-14. Score: 3 | Accepted answer

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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