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\begin{document}
The problem being solved is
\[ \max_{\set{C_s,K_s}} E\left[\sum_{t=0}^\infty \beta^t \frac{C_t^{1-\gamma}}{1-\gamma}\right]\]
subject to
\[
   C_t + K_t = A_tK_{t-1}^\alpha +\delta K_{t-1}\:.
\]
The Euler equations then emerge as
\begin{align*}
   C_t^{-\gamma}&=\lambda_t \\
   \lambda_t&=\beta E_t\left[\lambda_{t+1}(\alpha
   A_{t+1}K_t^{\alpha-1}+\delta)\right]\:.
\end{align*}
These can be solved to eliminate $\lambda$, producing
\[
   C_t^{-\gamma}=\beta E_t\left[C_{t+1}^{-\gamma}(\alpha
   A_{t+1}K_t^{\alpha-1}+\delta)\right]
\]
Note that I am following a convention that is uniform in my own writing,
but adhered to in only a minority of the economics literature, that one
can determine from a variable's time subscript whether it is known at $t$.
Most of the macro growth literature dates what I call $K_t$ as $K_{t+1}$,
even though under the usual interpretation of the model that would put $K_{t+1}$ in
the information set at $t$.
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