static/dynamic optimization
That's a coincidence, because they assume nondepreciating capital. If $\delta&gt;0$ was positive we'd have
\begin{align}
\dot K = Y - C - \delta K
\end{align}
which gives
\begin{align}
\dot K = 0  \quad \Rightarrow\quad C = Y - \delta K.
\end{align}

But the static optimization would still read
\begin{align}
&amp;U(C(t)):=\max_{c(t-n,n)}\int^\infty_0 u(c(t-n,n))dn \\
&amp;s.t.\quad \int^\infty_0 u(c(t-n,n))dn \leq C(t) 
\end{align} 
Actually they define an indirect utility function with subject to a resource constraint. In every period (thus static) the sum of individual consumption cannot exceed aggregate consumption.