static/dynamic optimization The interesting paper Calvo and Obstfeld (1988) uses two-stage optimization on an OLG model which then reduces to a standard representative agent framework. 

First stage optimization consists on a static optimization which makes the optimal allocation between different cohorts vertically(c.f equation (9) in the paper.) Authors solve this first stage problem as:

$$\mathcal{L}=u\left[c\left(t-n,t\right)\right]\Delta\left(n\right)P\left(n\right)e^{\rho n}+\lambda\left[C\left(t\right)-\int_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)d\left(n\right)\right]$$

where $C\left(t\right)=\int_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)dn$. 

$n, P(n), \Delta\left(n\right)$ are given in the paper and not relevant for my question at this moment. 

Normally, in this paper, the dynamics of capital accumulation are given as follows:

$$\dot{K}\left(t\right)=Y\left[K\left(t\right)\right]-C\left(t\right)$$

In fact, my question is trivial but I could not be sure.

Is the part with bracket in Lagrangian comes from $\dot{K}=0$ which gives us the equality $Y\left[K\left(t\right)\right]=C\left(t\right)=\int_{0}^{\infty}c\left(t-n,t\right)P\left(n\right)dn$ ?

As the Lagrangian is for a "static" problem, I think it makes sense but I can not be sure if this is the case.

Any suggestion or hints are welcome.
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.