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Consider an $n$ person economy with only 1 good and 2 states of nature $r,s$. Consumer $i$'s utility function is $$u_i(x_{ir},x_{is})=\pi_i \ln x_{ir}+(1-\pi_i)\ln x_{is}$$ where $\pi_i\in (0,1)$ is the "subjective probability of state $r$". Suppose all of them are endowed with $w_i=(1,1)$.

How do I find a Walrasian equilibrium of this economy? And how do I argue that in this equilibrium, prices aggregate beliefs?

Here's what I've done. Consumer $i$'s problem is \begin{align} \max \pi_i \ln x_{ir}+(1-\pi_i)\ln x_{is}\\ s.t. p_r x_{ir}+p_s x_{is}=p_r+p_s \end{align} By Lagrangian and FOC, I can get $$\frac{\pi_i x_{is}}{(1-\pi_i)x_{ir}}=\frac{p_r}{p_s}$$ then $x_{ir}=\frac{\pi_i}{1-\pi_i}\frac{p_s x_{is}}{p_r}$. Plug this into the budget constraint, I could get $p_s x_{is}=1-\pi_i$. I'm so close! How do I proceed?

Intuition tells me that at equilibrium, $x_i=(1,1)$. Doesn't it mean that this economy is autarky?

In addition, suppose that there are no contingent markets at date 0. Instead, 2 assets can be traded at date 0. Asset #1 pays 1 unit of good in $r$ and nothing in $s$; asset #2 pays 1 unit of good in $r$ and $s$. How do I find a Radner equilibrium?

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The equilibrium wont necessarily be $(1,1)$. Why? Well imagine an economy with just you and I. Your beliefs are (.99,.01) and my beliefs are (.01,.99). Why would we both consume (1,1)? Wouldnt we both be better off consuming more in the state we believe to be more likely?

First things, lets just assume $p_r=1$, we can do that. First order condition (dropping the i subscript for a second),

$$x_sp_s=\frac{1-\pi}{\pi}x_r$$

Toss that in the budget constraint to get

$$x_r+\frac{1-\pi}{\pi}x_r=1+p_s$$

$$\implies x_r^i=(1+p_s)\pi^i$$

Now, market clearing says $\sum_ix^i_r=N$ which implies $$\sum_i\frac{\pi_i}{N}=\frac{1}{1+p_s}$$

The lefthand side is the average belief, hence, prices are a function of the average belief. Let $\Pi=\sum_i\frac{\pi_i}{N}$, then we have

$$p_s=\frac{1-\Pi}{\Pi}$$

Recall that we scaled $p_r=1$, hence $p_s$ is really just the relative price of good $s$. So we really have

$$\frac{p_s}{p_r}=\frac{1-\Pi}{\Pi}$$

The ratio of prices is the same as the ratio of average beliefs.

For the Radner, your asset structure is complete. Does that help?

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