Arrow debreu equilibrium or Radner equilibrium and spot prices

Question

Suppose there are 2 states, 2 goods and 2 consumers and consumers have identical expected utility function: $U^i (x)= \sum_{s=1,2} \pi_s (\ln x_{1s}+\ln x_{2s} )$ where $\pi=(1/3,2/3)$.
Endowments are $e_s=(e_1, e_2) = (12,12)$.
consumer 1 has everything in state 1 and consumer 2 has everything in state 2.

There is a Radner equilibrium which is $x^1_s=(4,4)$ and $x^2_s=(8,8)$.
What is the spot price $(p_{1s}, p_{2s})$? You can normalize $p_{1s}=1$.

I made lagrangian and got $p_{2s}=1$ from MRS=relative price condition.
But my answer is correct?
I can't understand why two goods have same price when probability of state is different.
That spot price is the same with ADE?

Answer 1

I find the notation a bit unusual; it might be better to actually write out the vector $$ x^i = (x_{11}^i, x_{21}^i,x_{12}^i, x_{22}^i), $$ where the first lower index denotes the good, the second the state, rather than relying on the fact that in this particular problem the equilibrium consumption of each agent is state independent. The same goes for the equilibrium price vector $(p_{11}, p_{21},p_{12}, p_{22})$.

The price ratios $p_{1s} / p_{2s}$ denote the exchange rate of two goods in a given state, the price ratio $p_{11} / p_{12}$ shows ex-ante how many state 2 units of wealth is worth a state 1 unit of wealth.

Given the above equilibrium this particular utility function does imply $$ 1 = \frac{4}{4} = |MRS^1_{x_{11}x_{21}}(4,4)| = \frac{p_{11} }{ p_{21}}. $$ Note that if the utility functions were $$U^i (x)= \sum_{s=1,2} \pi_s (\color{red}{2} \cdot \ln x_{1s}+\ln x_{2s} )$$ the same allocation would still be an equilibrium allocation but the above price ratio would change to $2$, so getting $1$ in this problem is merely an accident.

One can also get the price ratio $$ \frac{p_{11} }{ p_{12}} = |MRS^1_{x_{11}x_{\color{red}{12}}}(4,4)|. $$