You should never use $$
in LaTeX, see Why is \[ ... \] preferable to $$ ... $$?
I suggest using multline*
, but with three lines, which makes a bit clearer the structure of the equation.
\documentclass[xcolor=table]{beamer}
\begin{document}
\begin{frame}
\begin{multline*}
\Lambda_{t}\biggl[1+\phi\biggl(\frac{K_{t+1}}{K_{t}}-g\biggr)\biggr]=\\
\beta E_{t}\Lambda_{t+1}\biggl[1-\delta+\alpha a_{t+1}
\biggl(\frac{K_{t+1}}{X_{t+1}h_{t+1}}\biggr)^{\!\alpha-1}+{}\\
\phi\frac{K_{t+2}}{K_{t+1}}\biggl(\frac{K_{t+2}}{K_{t+1}}-g\biggr)-
\frac{\phi}{2}\biggl(\frac{K_{t+2}}{K_{t+1}}-g\biggr)^{\!2}\,\biggr]
\end{multline*}
\end{frame}
\end{document}
I prefer to size manually the fences; note also \!
in the exponents near to large closing parentheses and \,
before the last ]
, so it will be slightly moved farther from the exponent.

You may want to have the relation and operation signs in the continuation lines; just be consistent across your document. In this case the input should be
\begin{multline*}
\Lambda_{t}\biggl[1+\phi\biggl(\frac{K_{t+1}}{K_{t}}-g\biggr)\biggr]\\
=\beta E_{t}\Lambda_{t+1}\biggl[1-\delta+\alpha a_{t+1}
\biggl(\frac{K_{t+1}}{X_{t+1}h_{t+1}}\biggr)^{\!\alpha-1}\\
{}+\phi\frac{K_{t+2}}{K_{t+1}}\biggl(\frac{K_{t+2}}{K_{t+1}}-g\biggr)-
\frac{\phi}{2}\biggl(\frac{K_{t+2}}{K_{t+1}}-g\biggr)^{\!2}\,\biggr]
\end{multline*}
that will produce

Note the {}
preceding +
in the continuation line (or +{}
if you prefer to have it at the end of a line), which ensures correct spacing of the symbol.
Don't worry if this occupies more space than a two line approach; cramming as much as possible in a slide is wrong to begin with.
amsmath
provides many environments for splitting equations, e.g.split
inside the equation ormultline
as standalone equation environment.\[
and\]
than the deprecated$$