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Consider an economy with 2 consumers and 1 good. There are 2 dates, date 0 and date 1. At date 1 there are 3 states of nature. The utility functions for the 2 consumers are the same: $$u_i(x_i)=\frac{1}{2}\sqrt{x_{i1}}+\frac{1}{3}\sqrt{x_{i2}}+\frac{1}{6}\sqrt{x_{i3}}$$ Consumer 1's endowments are $w_1=(2,0,0)$ and those for consumer 2 are $w_2=(0,2,2)$.

Suppose there are 2 assets with return vectors $r_1^T=(1,0,0),r_2^T=(0,1,1)$. How can I find a Radner equilibrium?

Right now I'm trying to solve for Walrasian equilibrium first. I got $x_1^*=(1,1,1)=x_2^*$, and the prices at each state are $(3,2,1)$. Then I try to find the arbitrage free asset prices, and got $q=(5,4,3)$. However, I'm skeptical as the rank of $R$ is 2, which is not equal to the number of states, so a Walrasian equilibrium and a Radner equilibrium are not equivalent. How do I proceed?

In addition, how do I argue that if $r_1^T=(1,1,0),r_2^T=(0,0,1)$, then a Radner equilibrium doesn't exist?

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