<p>I tried to solve by myself the exercises of MWG(Mas Colell). However, I think the exercise has an error at 17.D.1.</p>

<p>The 17.D.1 asks us that "verify that there are multiple equilibria". However, The excess demand function 
<span class="math-container">$Z_1(p,1)=\frac{p}{p+(\frac{p}{2})^\frac{1}{1-\rho}}+ \frac{1}{p+(2p)^\frac{1}{1-\rho}}-1$</span></p>

<p>And since <span class="math-container">$\rho=-4$</span>,</p>

<p><span class="math-container">$Z_1(p,1)=\frac{p}{p+(\frac{p}{2})^\frac{1}{5}}+ \frac{1}{p+(2p)^\frac{1}{5}}-1$</span></p>

<p>At equilibrium, <span class="math-container">$Z_1=0$</span>, and I think this has only one solution at <span class="math-container">$p=1$</span>  <del>since Z is increasing</del>.</p>

<p>How can we find multiple equilibria?</p>

<blockquote>
  <p>Original problem is following: An exchange economy with two commodities and two consumers. Both consumershave homothetic preferences of the constant elasticity variety. Moreover, the elasticity of substitution is the same for both consumers and is small (i.e., goods are close to perfect complements).<span class="math-container">$U_1(x_{11},x_{21}) = (2x^ρ_{11} +x^ρ_{21})^{\frac{1}{ρ}}$</span>
  and <span class="math-container">$U_2(x_{12},x_{22}) = (x^ρ_{12} +2x^ρ_{22})^{\frac{1}{ρ}}$</span>
  and
   <span class="math-container">$ρ=−4$</span>.</p>
  
  <p>The endowments are <span class="math-container">$ω_1= (1,0)$</span>  <span class="math-container">$ω_2= (0,1)$</span>.
  Compute the excess demand function of this economy and verify that there are multiple equilibria. </p>
</blockquote>

CC BY-SA 4.0
2019-12-11T09:42:03.790Z
Your function is not always increasing in $p$. What happens at the borders? How is the equilibrium defined?
Bertrand
CC BY-SA 4.0
2019-12-11T09:57:27.290Z
Thank you @Bertrand. The equilibrium is defined as Z=0 and p is strictly positive.
martian03
CC BY-SA 4.0
2019-12-11T10:22:03.667Z
Oh, p can be zero or infinity?
martian03
CC BY-SA 4.0
2019-12-11T11:28:49.473Z
In general $p$ can tend towards the boundaries.
Bertrand
CC BY-SA 4.0
2019-12-11T13:28:12.093Z
For example, let p=(0,1), then $X_{21}$*=0 and $X_{22}$*=1 and $Z_2(0,1)=0$ but $Z_1(0,1)$ goes infinity.
martian03
<microeconomics><general-equilibrium><mwg>
Question about MWG 17.D.1
martian03
# Question about MWG 17.D.1

(Asked by: username_0 on 2019-12-11)

I tried to solve by myself the exercises of MWG(Mas Colell). However, I think the exercise has an error at 17.D.1.

The 17.D.1 asks us that "verify that there are multiple equilibria". However, The excess demand function
$Z\_1(p,1)=\frac{p}{p+(\frac{p}{2})^\frac{1}{1-\rho}}+ \frac{1}{p+(2p)^\frac{1}{1-\rho}}-1$

And since $\rho=-4$,

$Z\_1(p,1)=\frac{p}{p+(\frac{p}{2})^\frac{1}{5}}+ \frac{1}{p+(2p)^\frac{1}{5}}-1$

At equilibrium, $Z\_1=0$, and I think this has only one solution at $p=1$ ~~since Z is increasing~~.

How can we find multiple equilibria?

> Original problem is following: An exchange economy with two commodities and two consumers. Both consumershave homothetic preferences of the constant elasticity variety. Moreover, the elasticity of substitution is the same for both consumers and is small (i.e., goods are close to perfect complements).$U\_1(x\_{11},x\_{21}) = (2x^ρ\_{11} +x^ρ\_{21})^{\frac{1}{ρ}}$
> and $U\_2(x\_{12},x\_{22}) = (x^ρ\_{12} +2x^ρ\_{22})^{\frac{1}{ρ}}$
> and
> $ρ=−4$.
>
> The endowments are $ω\_1= (1,0)$ $ω\_2= (0,1)$.
> Compute the excess demand function of this economy and verify that there are multiple equilibria.

**Tags:** microeconomics, general-equilibrium, mwg

> **Comments:**
> **username_1**: Your function is not always increasing in $p$. What happens at the borders? How is the equilibrium defined?  
> **username_0**: Thank you @username_1. The equilibrium is defined as Z=0 and p is strictly positive.  
> **username_0**: Oh, p can be zero or infinity?  
> **username_1**: In general $p$ can tend towards the boundaries.  
> **username_0**: For example, let p=(0,1), then $X_{21}$*=0 and $X_{22}$*=1 and $Z_2(0,1)=0$ but $Z_1(0,1)$ goes infinity.  

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## 0 answers

