How to deal with Prescott's formulation of time to build in his original RBC model?
By re-indexing all investment projects in the form of $s_{J,t}$, I get the following two equivalent constraints
\begin{align}
 c_t+y_{t+1}-y_t&amp;\leq f(k_t)-\sum_{i=1}^\infty\varphi_j s_{J,t-(J-j)} % Resource Constraint
 \\ 
 k_{t+1}&amp;=(1-\delta)k_t+S_{J,t-(J-1)} %Law of motion for capital formation
\end{align}
Hence, the Lagrangian for the maximization problem is
\begin{multline}
\mathcal{L}
 = \sum_{t=0}^\infty\beta^tu\left(c_t,1- \alpha_0n_t - \eta(1-\alpha_0) a_t\right) %Objective Function
 - \sum_{t=0}^\infty\lambda_{1t}\left(c_t+y_{t+1}-y_t+\sum_{i=1}^\infty\varphi_j s_{J,t-(J-j)}-f(k_t)\right) % Resource Constraint
\\ - \sum_{t=0}^\infty\lambda_{2t}\left(k_{t+1}-(1-\delta)k_t-S_{J,t-(J-1)}\right) %Law of motion for capital formation
 - \sum_{t=0}^\infty\lambda_{3t} \left(a_{t+1}-(1-\eta)a_t-n_{t}\right) % Law of motion for past leisure
\end{multline}
where the state variables at time $t$ are $a_t, k_t, y_t$ and the decision variables at time $t$ are $c_t, n_t, s_{Jt}, y_{t+1}$. It is easy to see that all constraints must bind. 
The necessary conditions are
\begin{align}
 &amp;\frac{\partial \mathcal{L}}{\partial c_t} =0\implies \beta^t \frac{\partial u}{\partial c_t}=\lambda_{1t}
\\ &amp;\frac{\partial \mathcal{L}}{\partial s_{J,t}} =0\implies \sum_{i=1}^J\lambda_{1,t+J-i}\varphi_i=\lambda_{2,t+J-1}
\\ &amp;\frac{\partial \mathcal{L}}{\partial k_t}=0\implies \lambda_{1t}\frac{\partial f}{\partial k_t}+\lambda_{2t}(1-\delta)=\lambda_{2,t-1},
\end{align}
which holds for all $t$.
Combining 14,15,16 and simplifying, we have
\begin{equation}
 \frac{\partial u}{\partial c_t}\frac{\partial f}{\partial k_t}
 + (1-\delta)\sum_{i=1}^J\beta^{1-i} \frac{\partial u}{\partial c_{t+1-i}}\varphi_i
 = \beta^{-1}\sum_{i=1}^J\beta^{1-i} \frac{\partial u}{\partial c_{t-i}}\varphi_i
\end{equation}
In steady state, $\frac{\partial u}{\partial c_t}=\frac{\partial u}{\partial c_{t'}}$, so we have 
\begin{equation}
\frac{\partial f}{\partial k_t}
= \sum_{i=0}^{J-1}\beta^{J-i-1} \varphi_{J-i} \left(\frac{1-\beta}{\beta}+\delta\right)
\end{equation}
One observation is that since in steady state, no borrowing can occur between household and firm, the real interest rate $r=\frac{1-\beta}{\beta}$. We see that $\sum_{i=0}^{J-1}\beta^{J-i-1} \varphi_{J-i} $ is the price of the capital in steady state.