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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.