Finding a closed-form solution for the following fixed point algorithm,

Question

\begin{equation} X^{(t)}=\frac{SK}{\bigg(\frac{ZK}{(1+X^{(t-1)})} +\frac{Y(RK+Y)}{R^{2}}\bigg)}\ \end{equation} For $ \text{t}=1,2,...$,

where the initial value $X^{(t-1)}={R}$

Please, any help or any hint to find the solution closed-form.

Answer 1

Rewrite $$ X^{(t)} = \frac{SK}{\frac{ZK}{1 + X^{(t-1)}} +\frac{Y(RK+Y)}{R^2}} = \frac{A}{\frac{B}{1 + X^{(t-1)}} + C}. $$ Take limit $$X = \frac{A}{\frac{B}{1 + X} + C}.$$ This is equivalent to $$CX^2 + (B + C - A)X - A = 0.$$ with solutions $$X = {A-B-C\pm{\sqrt{C^2+\left(2\,B+2\,A\right)C+B^2-2AB+A^2}}\over{2C}}.$$