# How can I find a Radner equilibrium?

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?