<p>I don't know the paper nor the notation, so I am just guessing here. I gues <span class="math-container">$N(a,t)$</span> is the number of agents of age <span class="math-container">$a$</span> at time period <span class="math-container">$t$</span>.</p>
<p>Let's follow the number of age <span class="math-container">$B$</span> accross generations:
<span class="math-container">$$
\begin{align*}
N(B,t) &amp;= e^f N(B, t- B),\\
&amp;= e^{2f} N(B, t - 2 B),\\
&amp;= \ldots,\\
&amp;= e^{f t/B} N(B,0),\\
&amp;= e^{\eta t} N(B, 0).
\end{align*}
$$</span>
Then using this in the definition of <span class="math-container">$N(a,t)$</span>, we have:
<span class="math-container">$$
N(a,t) = e^f N(B, t- a) = e^f e^{\eta(t - a)}N(B,0).
$$</span>
Next, I assume that at period <span class="math-container">$0$</span> there is a mass of size 1 but nobody lives longer than <span class="math-container">$T$</span> periods, so integrating across all ages:
<span class="math-container">$$
\begin{align*}
&amp;\int_0^T N(a,0) da = 1,\\
\to &amp;\int_0^T e^f e^{-\eta\, a}N(B,0) da = 1,\\
\to &amp;e^f N(B,0) \left[-\frac{e^{-\eta a}}{\eta}\right]^T_0 = 1,\\
\to &amp;e^f N(B, 0) \left[1 -e^{-\eta T}\right] = \eta,\\
\to &amp;e^f N(B, 0) = \frac{\eta }{1 - e^{-\eta T}}
\end{align*}
$$</span>
So subsituting into the expression for <span class="math-container">$N(a,t)$</span> gives:
<span class="math-container">$$
N(a,t) = \eta\frac{e^{-\eta a}}{1 - e^{-\eta T}} e^{\eta t}
$$</span></p>

CC BY-SA 4.0
2021-05-15T12:51:44.733Z
2021-05-15T13:21:59.207Z
2021-05-15T13:21:59.207Z
tdm
<p>I am reading <a href="https://www.aeaweb.org/articles?id=10.1257/aer.104.9.2736" rel="nofollow noreferrer">a paper</a> with following description on the demographics in their model: &quot;... each (representative) agent lives for <span class="math-container">$T$</span> periods ... We assume that each individual has <span class="math-container">$e^{f}$</span> children at age <span class="math-container">$B$</span>. 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  <span class="math-container">$N(a, t)=e^{f} N(B, t-a)$</span> and <span class="math-container">$N\left(t^{\prime}, t\right)=0, t^{\prime}&gt;T$</span>. It is easy to check that in the steady state <span class="math-container">$N(a, t)=\phi(a) e^{\eta t}$</span>, where <span class="math-container">$\phi(a)=\eta \frac{e^{-\eta a}}{1-e^{-\eta T}}$</span> and <span class="math-container">$\eta=f / B$</span> is the growth rate of population.&quot;</p>
<p>I have no idea how is this steady state calculated?</p>

CC BY-SA 4.0
2021-05-14T21:58:21.310Z
@Giskard Yes, I update the link. Please note that this demographic setting is rather a separate part and has nothing to do with the main part of the model.
Alalalalaki
<macroeconomics><steady-state><demographics>
Derive the Demographic Structure in the Steady State
Alalalalaki
# Derive the Demographic Structure in the Steady State

(Asked by: username_0 on 2021-05-14)

I am reading [a paper](https://www.aeaweb.org/articles?id=10.1257/aer.104.9.2736) with following description on the demographics in their model: "... 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 $N\left(t^{\prime}, t\right)=0, t^{\prime}>T$. 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."

I have no idea how is this steady state calculated?

**Tags:** macroeconomics, steady-state, demographics

> **Comments:**
> **username_0**: @Giskard Yes, I update the link. Please note that this demographic setting is rather a separate part and has nothing to do with the main part of the model.  

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## 1 answers

### Answer by username_1 on 2021-05-15. Score: 3 | Accepted answer

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:
$$
\begin{align\*}
N(B,t) &= e^f N(B, t- B),\\
&= e^{2f} N(B, t - 2 B),\\
&= \ldots,\\
&= e^{f t/B} N(B,0),\\
&= e^{\eta t} N(B, 0).
\end{align\*}
$$
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:
$$
\begin{align\*}
&\int\_0^T N(a,0) da = 1,\\
\to &\int\_0^T e^f e^{-\eta\, a}N(B,0) da = 1,\\
\to &e^f N(B,0) \left[-\frac{e^{-\eta a}}{\eta}\right]^T\_0 = 1,\\
\to &e^f N(B, 0) \left[1 -e^{-\eta T}\right] = \eta,\\
\to &e^f N(B, 0) = \frac{\eta }{1 - e^{-\eta T}}
\end{align\*}
$$
So subsituting into the expression for $N(a,t)$ gives:
$$
N(a,t) = \eta\frac{e^{-\eta a}}{1 - e^{-\eta T}} e^{\eta t}
$$


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