<p>I find the notation a bit unusual; it might be better to actually write out the vector
<span class="math-container">$$
x^i = (x_{11}^i, x_{21}^i,x_{12}^i, x_{22}^i),
$$</span>
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
<span class="math-container">$(p_{11}, p_{21},p_{12}, p_{22})$</span>.</p>
<p>The price ratios <span class="math-container">$p_{1s} / p_{2s}$</span> denote the exchange rate of two goods in a given state, the price ratio <span class="math-container">$p_{11} / p_{12}$</span> shows ex-ante how many state 2 units of wealth is worth a state 1 unit of wealth.</p>
<p>Given the above equilibrium this particular utility function does imply
<span class="math-container">$$
1 = \frac{4}{4} = |MRS^1_{x_{11}x_{21}}(4,4)| = \frac{p_{11} }{ p_{21}}.
$$</span>
Note that if the utility functions were
<span class="math-container">$$U^i (x)= \sum_{s=1,2} \pi_s (\color{red}{2} \cdot \ln x_{1s}+\ln x_{2s} )$$</span>
the same allocation would still be an equilibrium allocation but the above price ratio would change to <span class="math-container">$2$</span>, so getting <span class="math-container">$1$</span> in this problem is merely an accident.</p>
<p>One can also get the price ratio
<span class="math-container">$$
\frac{p_{11} }{ p_{12}} = |MRS^1_{x_{11}x_{\color{red}{12}}}(4,4)|.
$$</span></p>

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2022-02-05T14:40:17.953Z
2022-02-05T18:01:15.193Z
2022-02-05T18:01:15.193Z
Giskard
<p>Suppose there are 2 states, 2 goods and 2 consumers and consumers have identical expected utility function:
<span class="math-container">$U^i (x)= \sum_{s=1,2} \pi_s (\ln x_{1s}+\ln x_{2s} )$</span> where <span class="math-container">$\pi=(1/3,2/3)$</span>.<br />
Endowments are <span class="math-container">$e_s=(e_1, e_2) = (12,12)$</span>.<br />
consumer 1 has everything in state 1 and consumer 2 has everything in state 2.</p>
<p>There is a Radner equilibrium which is <span class="math-container">$x^1_s=(4,4)$</span> and <span class="math-container">$x^2_s=(8,8)$</span>.<br />
What is the spot price <span class="math-container">$(p_{1s}, p_{2s})$</span>? You can normalize <span class="math-container">$p_{1s}=1$</span>.</p>
<p>I made lagrangian and got <span class="math-container">$p_{2s}=1$</span> from MRS=relative price condition.<br />
But my answer is correct?<br />
I can't understand why two goods have same price when probability of state is different.<br />
That spot price is the same with ADE?</p>

<microeconomics><competitive-equilibrium><uncertainty><radner-equilibrium>
Arrow debreu equilibrium or Radner equilibrium and spot prices
guest
# Arrow debreu equilibrium or Radner equilibrium and spot prices

(Asked by: username_0 on 2022-02-05)

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?

**Tags:** microeconomics, competitive-equilibrium, uncertainty, radner-equilibrium


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## 1 answers

### Answer by username_1 on 2022-02-05. Score: 1 | Accepted answer

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
$$
\frac{p\_{11} }{ p\_{12}} = |MRS^1\_{x\_{11}x\_{\color{red}{12}}}(4,4)|.
$$


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