I am currently studing economics and I encountered this problem that is driving me crazy. I need to solve for C1 from this set of equations:
$\frac{1}{C_1}= \frac{p}{C_2^h}+\frac{q}{C_2^m}+\frac{z}{C_2^l}$
$C_2^h=Y_2^h+(1+r)Y_1-(1+r)C_1$
$C_2^m=Y_2^m+(1+r)Y_1-(1+r)C_1$
$C_2^l=Y_2^l+(1+r)Y_1-(1+r)C_1$
The first one is the Euler equation from an optimization problem with unincertainty on the output of the second period. Then, $p+q+z=1$ because they are the probabilities assigned to each state. The heading also mentionates that:
$Y_1=E \left\{ Y_2 \right\}$
I managed to get to:
$C_2^hC_2^mC_2^l=pC_1C_2^mC_2^l+qC_1C_2^hC_2^l+zC_1C_2^hC_2^m$
Also I tried to open that equation and replace the C2 but it was effort wasted.
Hope someone out there has an answer to this puzzle (or maybe has already faced this problem in the past and knows the key to solve it)
Thx in advance. Greetings from Argentina.