How can I solve $\phi_1$ and $\phi_2$?

Question

Let be

$$\rho(1)=\frac{\phi_1}{1-\phi_2}, \rho(2)=\frac{\phi_1^2+\phi_2(1-\phi_1)}{1-\phi_2}$$

How can I solve $\phi_1,\phi_2$?

My idea:

1. $\rho(1)=\frac{\phi_1}{1-\phi_2}\Leftrightarrow \phi_1=\rho(1)(1-\phi_2)$.
2. Then, I put $\phi_1$ in $\rho(2):$ $$\rho(2)=\frac{(\rho(1)(1-\phi_2))^2+\phi_2(1-\rho(1)(1-\phi_2))}{1-\phi_2}=\rho(1)[\rho(1)(1-\phi_2)-1]+\frac{\phi_2}{1-\phi_2}.$$

For the next step, I don't know.

Answer 1

Your idea is correct, but you have some typo and some mistake. From the first equation we have: $$ \phi_1=\rho(1)(1-\phi_2) $$ and, substituting, the second equation becomes: $$ \rho(2)=\frac{\rho^2(1)(1-\phi_2)^2+\phi_2[1-\rho(1)(1-\phi_2)]}{1-\phi_2} $$ that, for $1-\phi_2 \ne 0$, gives: $$ \rho(2)(1-\phi_2)=\rho^2(1)(1-\phi_2)^2+\phi_2[1-\rho(1)(1-\phi_2)] $$ and this is a second degree equation in $\phi_2$ that you can solve with the quadratic formula.