<p>I've been given the following question and would really appreciate any help on part a. I've looked over all of my resources for this course and we have always been given the probability of the realisation of each state, so any hints on where to start would be great. I understand that they are not the same as the risk neutral probabilities, but that's about all we've been given on the topic! </p>

<p><img src="https://i.sstatic.net/cvHkN.png" alt="enter image description here"></p>

CC BY-SA 3.0
2015-05-04T21:09:31.437Z
I believe that the asset prices are equilibrium prices. In equilibrium both agents make optimal choices given the prices and their budget constraints. Also, in equilibrium (with insatiable preferences) the aggregate demand in any state will match the aggregate supply. From this, you get several equations (six to be precise but not all are linearly independent) and I think by solving them you will get $\pi_s$.
Giskard
CC BY-SA 3.0
2015-05-05T10:20:16.620Z
Sorry would you mind expanding? I though of writing out all of the equations to solve for pi, but we're not given any endowment in the first period which somewhat confused me in terms of writing a budget constraint for that..
mathfinalshelp
CC BY-SA 3.0
2015-05-05T10:34:07.557Z
Yes, you are correct, the individual endowments will also be variables.
I was wrong yesterday, there are more then six equations.
Let $e_{s,i}^A$ denote the endowment of agent $A$ in state $s$ of good $i$.
Then in equilibrium
$$
\forall s,i: e_{s,i}^A + e_{s,i}^B = x_{s,i}^A + x_{s,i}^B.
$$
Using these, the optimality conditions (given by MRS) and one of the budget constraints you ought to get your answer.
Giskard
CC BY-SA 3.0
2015-05-05T10:38:48.337Z
A trick you might want to use is that since
$$
MRS(x_{1,i}^A,x_{2,i}^A) = \frac{x_{2,i}^A}{x_{1,i}^A} = \frac{p_{1,i}}{p_{2,i}}
$$
and
$$
MRS(x_{1,i}^B,x_{2,i}^B) = \frac{x_{2,i}^B}{x_{1,i}^B} = \frac{p_{1,i}}{p_{2,i}},
$$
you will also have
$$
\frac{x_{2,i}^A + x_{2,i}^B}{x_{1,i}^A + x_{1,i}^B} = \frac{p_{1,i}}{p_{2,i}}.
$$
This works very well with the equilibrium constraints.
Giskard
<consumer-theory><risk><financial-markets><uncertainty>
Probability of states of nature
mathfinalshelp
# Probability of states of nature

(Asked by: username_0 on 2015-05-04)

I've been given the following question and would really appreciate any help on part a. I've looked over all of my resources for this course and we have always been given the probability of the realisation of each state, so any hints on where to start would be great. I understand that they are not the same as the risk neutral probabilities, but that's about all we've been given on the topic!

![enter image description here](https://i.sstatic.net/cvHkN.png)

**Tags:** consumer-theory, risk, financial-markets, uncertainty

> **Comments:**
> **username_1**: I believe that the asset prices are equilibrium prices. In equilibrium both agents make optimal choices given the prices and their budget constraints. Also, in equilibrium (with insatiable preferences) the aggregate demand in any state will match the aggregate supply. From this, you get several equations (six to be precise but not all are linearly independent) and I think by solving them you will get $\pi_s$.  
> **username_0**: Sorry would you mind expanding? I though of writing out all of the equations to solve for pi, but we're not given any endowment in the first period which somewhat confused me in terms of writing a budget constraint for that..  
> **username_1**: Yes, you are correct, the individual endowments will also be variables. I was wrong yesterday, there are more then six equations. Let $e_{s,i}^A$ denote the endowment of agent $A$ in state $s$ of good $i$. Then in equilibrium $$ \forall s,i: e_{s,i}^A + e_{s,i}^B = x_{s,i}^A + x_{s,i}^B. $$ Using these, the optimality conditions (given by MRS) and one of the budget constraints you ought to get your answer.  
> **username_1**: A trick you might want to use is that since $$ MRS(x_{1,i}^A,x_{2,i}^A) = \frac{x_{2,i}^A}{x_{1,i}^A} = \frac{p_{1,i}}{p_{2,i}} $$ and $$ MRS(x_{1,i}^B,x_{2,i}^B) = \frac{x_{2,i}^B}{x_{1,i}^B} = \frac{p_{1,i}}{p_{2,i}}, $$ you will also have $$ \frac{x_{2,i}^A + x_{2,i}^B}{x_{1,i}^A + x_{1,i}^B} = \frac{p_{1,i}}{p_{2,i}}. $$ This works very well with the equilibrium constraints.  

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## 0 answers

