<p>The reason why the <span class="math-container">$e^{nt}$</span> term is there is because you want to multiply the whole utility by the number of people. You are actually not substituting consumption into the utility but multiplying the utility of an individual by the number of individuals. So the problem would look like:</p>
<p><span class="math-container">$$\int_{0}^{\infty}\frac{c^{1-\sigma}}{1-\sigma}e^{nt} e^{-\rho t}dt$$</span></p>
<p>At least this is how it is explained by Barro &amp; Sala-i-Martin in their textbook Economic Growth 2nd ed. pp 87 where they say [emphasis mine]:</p>
<blockquote>
<p>The multiplication of <strong><span class="math-container">$u(c)$</span></strong> in equation ...  by family size, <span class="math-container">$L = e^{nt}$</span>, represents the adding up of <strong>utils</strong> for all family members alive at time <span class="math-container">$t$</span>.</p>
</blockquote>
<p>So the way how I understand it you are multiplying all the individual utilities by the population.</p>
<p>In fact I am quite sure of it since in Romer Advanced Economics 4th ed in chapter 2 pp 54 there is Ramsey model where the objective function is given as:</p>
<p><span class="math-container">$$\int_{t=0}^{\infty} e^{-\rho t}\frac{c^{1-\sigma}}{1-\sigma} \frac{L(t)}{H} dt$$</span></p>
<p>which is essentially exactly the case you are looking for where the only difference is that Romer also adds the <span class="math-container">$H$</span> parameter which is the number of households so <span class="math-container">$L(t)/H$</span> would be population per household. Romer also later substitutes <span class="math-container">$L(t)=L(0)e^{nt}$</span> which for <span class="math-container">$L(0)$</span> normalized to 1 gives almost the same function (with that extra <span class="math-container">$H$</span> parameter - but for <span class="math-container">$H=1$</span> it would give you exactly the same result).</p>

CC BY-SA 4.0
2020-12-02T21:16:49.183Z
2020-12-02T21:26:07.660Z
2020-12-02T21:26:07.660Z
1muflon1
<p>Let's say that we have the sum of the utility of a social planner</p>
<p><span class="math-container">$$\int_{0}^{\infty}U\left(C\right)e^{-\rho t}dt$$</span></p>
<p>where <span class="math-container">$C$</span> is the total consumption. If we want to write this by a per capita variable <span class="math-container">$c=\frac{C}{L}$</span> where <span class="math-container">$L=e^{nt}$</span> is the total number of population that grows exogenously at rate <span class="math-container">$n$</span>. We can reformulate this such as</p>
<p><span class="math-container">$$\int_{0}^{\infty}U\left(c\right)e^{nt}e^{-\rho t}dt$$</span></p>
<p>Until now, everything is straightforward. However, if we have started by a specified functional form such as a CRRA utility</p>
<p><span class="math-container">$$\int_{0}^{\infty}\frac{C^{1-\sigma}}{1-\sigma}e^{-\rho t}dt$$</span></p>
<p>We would not have reached the result above and should have some exponent term with <span class="math-container">$\sigma$</span>.  What is the  logic behind? There is something I miss.</p>

<economic-growth><dynamic-optimization>
Resolution - Ramsey growth model with per capita variables
optimal control
# Resolution - Ramsey growth model with per capita variables

(Asked by: username_0 on 2020-12-02)

Let's say that we have the sum of the utility of a social planner

$$\int\_{0}^{\infty}U\left(C\right)e^{-\rho t}dt$$

where $C$ is the total consumption. If we want to write this by a per capita variable $c=\frac{C}{L}$ where $L=e^{nt}$ is the total number of population that grows exogenously at rate $n$. We can reformulate this such as

$$\int\_{0}^{\infty}U\left(c\right)e^{nt}e^{-\rho t}dt$$

Until now, everything is straightforward. However, if we have started by a specified functional form such as a CRRA utility

$$\int\_{0}^{\infty}\frac{C^{1-\sigma}}{1-\sigma}e^{-\rho t}dt$$

We would not have reached the result above and should have some exponent term with $\sigma$. What is the logic behind? There is something I miss.

**Tags:** economic-growth, dynamic-optimization

---

## 1 answers

### Answer by username_1 on 2020-12-02. Score: 2 | Accepted answer

The reason why the $e^{nt}$ term is there is because you want to multiply the whole utility by the number of people. You are actually not substituting consumption into the utility but multiplying the utility of an individual by the number of individuals. So the problem would look like:

$$\int\_{0}^{\infty}\frac{c^{1-\sigma}}{1-\sigma}e^{nt} e^{-\rho t}dt$$

At least this is how it is explained by Barro & Sala-i-Martin in their textbook Economic Growth 2nd ed. pp 87 where they say [emphasis mine]:

> The multiplication of **$u(c)$** in equation ... by family size, $L = e^{nt}$, represents the adding up of **utils** for all family members alive at time $t$.

So the way how I understand it you are multiplying all the individual utilities by the population.

In fact I am quite sure of it since in Romer Advanced Economics 4th ed in chapter 2 pp 54 there is Ramsey model where the objective function is given as:

$$\int\_{t=0}^{\infty} e^{-\rho t}\frac{c^{1-\sigma}}{1-\sigma} \frac{L(t)}{H} dt$$

which is essentially exactly the case you are looking for where the only difference is that Romer also adds the $H$ parameter which is the number of households so $L(t)/H$ would be population per household. Romer also later substitutes $L(t)=L(0)e^{nt}$ which for $L(0)$ normalized to 1 gives almost the same function (with that extra $H$ parameter - but for $H=1$ it would give you exactly the same result).


---

