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.

  • $\begingroup$ Your function is not always increasing in $p$. What happens at the borders? How is the equilibrium defined? $\endgroup$ – Bertrand Dec 11 '19 at 9:42
  • $\begingroup$ Thank you @Bertrand. The equilibrium is defined as Z=0 and p is strictly positive. $\endgroup$ – martian03 Dec 11 '19 at 9:57
  • $\begingroup$ Oh, p can be zero or infinity? $\endgroup$ – martian03 Dec 11 '19 at 10:22
  • $\begingroup$ In general $p$ can tend towards the boundaries. $\endgroup$ – Bertrand Dec 11 '19 at 11:28
  • $\begingroup$ 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. $\endgroup$ – martian03 Dec 11 '19 at 13:28

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