Taken from Chapter 17 of Mas Colell "Microeconomic Theory"

Consider an exchange economy with two commodities and two consumers. Both consumers have homothetic preferences of the constant elasticity variety. Morover, the elasticity of substitution is the same for both consumers and is small (i.e. goods are close to perfect complements). Specifically

$u_1(x_{11},x_{21})=(2x_{11}^{\rho}+x_{21}^{\rho})^{1/\rho}$ and $u_1(x_{12},x_{22})=(x_{12}^{\rho}+2x_{22}^{\rho})^{1/\rho}$

And $\rho=-4$, The endowments are $w_1=(1,0)$ and $w_2=(0,1)$. Compute the excess demand function of the economy and verify that there are multiple equilibria.

My attempt

After applying fist-condition and normalizing prices as $\frac{P_1}{P2}=p$, I have that the demand functions are:

For the first consumer: $x_1=\frac{p}{p+(p/2)^{\frac{1}{1-\rho}}}$ and $x_2=\frac{p(p/2)^{\frac{1}{1-\rho}}}{p+(p/2)^{\frac{1}{1-\rho}}}$

For the second consumer: $x_1=\frac{1}{p+(2p)^{\frac{1}{1-\rho}}}$ and $x_2=\frac{(2p)^{\frac{1}{1-\rho}}}{p+(2p)^{\frac{1}{1-\rho}}}$

So the excess demand function would be:

$\begin{pmatrix} z_1\\z_2 \end{pmatrix}=\begin{pmatrix} \frac{p}{p+(p/2)^{\frac{1}{1-\rho}}}+\frac{1}{p+(2p)^{\frac{1}{1-\rho}}}-1 \\ \frac{p(p/2)^{\frac{1}{1-\rho}}}{p+(p/2)^{\frac{1}{1-\rho}}}+\frac{(2p)^{\frac{1}{1-\rho}}}{p+(2p)^{\frac{1}{1-\rho}}}-1 \end{pmatrix}$

The thing is that this is not the answer stated in Mas Colell's Solutions. Here it is:

enter image description here

I guess the elasticity of substitution has something to do here? But I don't get it anyway because if we had some corner solution, the answers would be $(x_1,x_2)=(1,0)$ for this consumer and $(x_1,x_2)=(0,1)$ for second consumer.

Any idea?


1 Answer 1


Your answer is essentially the same as Mas-Colell's.

$z_{11}$ is excess demand for good $1$ by consumer $1$. $z_{12}$ is excess demand for good $1$ by consumer $2$. They sum up to zero at the price vector $(1,1)$, which means that the market for good $1$ clears. By Walras' law, that implies that the market for good $2$ clears as well, hence $(1,1)$ is an equilibrium price vector.


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