The problem being solved is

max_{\{C_{s},K_{s}\}} E [∑_{t=0}^{∞} β^{t} \frac{C_{t}^{1−γ}}{1 − γ} ]

subject to

C_{t} + K_{t} = A_{t}K_{t−1}^{α} + δK_{t−1} .

The Euler equations then emerge as

C_{t}^{−γ} = λ_{t}
λ_{t} = βE_{t} [λ_{t+1}(αA_{t+1}K_{t}^{α−1} + δ)] .

These can be solved to eliminate λ, producing

C_{t}^{−γ} = βE_{t} [C_{t+1}^{−γ}(αA_{t+1}K_{t}^{α−1} + δ)]

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.