I'm having some errors while copmiling the following:
\begin{equation}
k_{t+1}=\left\{
\begin{array}{l}
$A.s.k_{t}^{\alpha}+(1-\delta).b_{t}^{P}-\delta.b_{t}$ \quad \text{if $\frac{b_{t}+b_{t}^{P}}{(1-\epsilon).s.k_{t}^{\alpha}} < 1$} \\
$s.k_{t}^{\alpha}-b_{t}$ \quad \text{if $\frac{b_{t}+b_{t}^{P}}{(1-\epsilon).s.k_{t}^{\alpha}} \geq 1$}
\end{array}
\right.
\end{equation}
It looks as I want in pdf but I'm getting some missing $ inserted.
Thanks for your help
The array environment is already in math mode, so you don't need the $ there. Following does not produce the Missing $ inserted.
Not sure what the . means in your equations, but did you perhaps intend to use a \cdot?
Your cases example in the comments had an extra & at the end. I have added that example in the MWE.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
k_{t+1}=\left\{
\begin{array}{l}
A.s.k_{t}^{\alpha}+(1-\delta).b_{t}^{P}-\delta.b_{t} \quad \text{if $\frac{b_{t}+b_{t}^{P}}{(1-\epsilon).s.k_{t}^{\alpha}} < 1$} \\
s.k_{t}^{\alpha}-b_{t} \quad \text{if $\frac{b_{t}+b_{t}^{P}}{(1-\epsilon).s.k_{t}^{\alpha}} \geq 1$}
\end{array}
\right.
\end{equation}
\begin{equation}
k_{t+1}=
\begin{cases}
A.s.k_{t}^{\alpha}+(1-\delta).b_{t}^{P}-\delta.b_{t}
& \quad \text{if } \frac{b_{t}+b_{t}^{P}}{(1-\epsilon).s.k_{t}^{\alpha}} < 1
\\
s.k_{t}^{\alpha}-b_{t}
& \quad \text{if } \frac{b_{t}+b_{t}^{P}}{(1-\epsilon).s.k_{t}^{\alpha}} \geq 1
\end{cases}
\end{equation}
\end{document}
casescomment in the answer.