Derive the Demographic Structure in the Steady State I am reading a paper with following description on the demographics in their model: &quot;... each (representative) agent lives for $T$ periods ... We assume that each individual has $e^{f}$ children at age $B$. Since we consider only steady states, we need to derive the stationary age distribution of this economy associated with this fertility rate. Our assumptions imply  $N(a, t)=e^{f} N(B, t-a)$ and $N\left(t^{\prime}, t\right)=0, t^{\prime}&gt;T$. It is easy to check that in the steady state $N(a, t)=\phi(a) e^{\eta t}$, where $\phi(a)=\eta \frac{e^{-\eta a}}{1-e^{-\eta T}}$ and $\eta=f / B$ is the growth rate of population.&quot;
I have no idea how is this steady state calculated?
I don't know the paper nor the notation, so I am just guessing here. I gues $N(a,t)$ is the number of agents of age $a$ at time period $t$.
Let's follow the number of age $B$ accross generations:
$$
\begin{align*}
N(B,t) &amp;= e^f N(B, t- B),\\
&amp;= e^{2f} N(B, t - 2 B),\\
&amp;= \ldots,\\
&amp;= e^{f t/B} N(B,0),\\
&amp;= e^{\eta t} N(B, 0).
\end{align*}
$$
Then using this in the definition of $N(a,t)$, we have:
$$
N(a,t) = e^f N(B, t- a) = e^f e^{\eta(t - a)}N(B,0).
$$
Next, I assume that at period $0$ there is a mass of size 1 but nobody lives longer than $T$ periods, so integrating across all ages:
$$
\begin{align*}
&amp;\int_0^T N(a,0) da = 1,\\
\to &amp;\int_0^T e^f e^{-\eta\, a}N(B,0) da = 1,\\
\to &amp;e^f N(B,0) \left[-\frac{e^{-\eta a}}{\eta}\right]^T_0 = 1,\\
\to &amp;e^f N(B, 0) \left[1 -e^{-\eta T}\right] = \eta,\\
\to &amp;e^f N(B, 0) = \frac{\eta }{1 - e^{-\eta T}}
\end{align*}
$$
So subsituting into the expression for $N(a,t)$ gives:
$$
N(a,t) = \eta\frac{e^{-\eta a}}{1 - e^{-\eta T}} e^{\eta t}
$$