<p>Before going streight to your question, let me stress an important point.
The utility function you have written is the utility function for all family members of the household. Indeed, you are implicitily assuming that the household size  <span class="math-container">$L(t)=L(0)e^{nt}$</span> follows this normalization <span class="math-container">$L(0)\equiv1$</span>. Eventually, <span class="math-container">$L(t)=e^{nt}$</span>.</p>
<p>Then, <span class="math-container">$\rho -n$</span> is the effective discount rate because the household derives utility from the consumption per capita of its additional members in the future as well.</p>
<p>Clearly, you need to add a discounting term otherwise the utility function might get an infinite value (not an interesting case to analyze).</p>
<p>Finally, <span class="math-container">$\rho -n &gt;0$</span> ensures finite discounted utility. Take the limit of the discounting term and see what happens for <span class="math-container">$\rho -n &lt;0$</span></p>

CC BY-SA 4.0
2023-05-09T18:30:59.617Z
2023-05-09T18:30:59.617Z
Tony
<p>Consider the following utility function in an infinite time-horizon set up:</p>
<p><span class="math-container">\begin{equation}
U= \int_{0}^{\infty} e^{-(\rho-n)t}u(c(t)) dt
\end{equation}</span></p>
<p>Why we assume that <span class="math-container">$\rho &gt;n$</span>, where <span class="math-container">$\rho $</span> is the pure rate of time preferences and <span class="math-container">$n$</span> is the population growth rate?</p>

<macroeconomics><microeconomics>
Subjective discounting factor in the utility function
Dimitru
# Subjective discounting factor in the utility function

(Asked by: username_0 on 2023-05-09)

Consider the following utility function in an infinite time-horizon set up:

\begin{equation}
U= \int\_{0}^{\infty} e^{-(\rho-n)t}u(c(t)) dt
\end{equation}

Why we assume that $\rho >n$, where $\rho $ is the pure rate of time preferences and $n$ is the population growth rate?

**Tags:** macroeconomics, microeconomics

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## 1 answers

### Answer by username_1 on 2023-05-09. Score: 4 | Accepted answer

Before going streight to your question, let me stress an important point.
The utility function you have written is the utility function for all family members of the household. Indeed, you are implicitily assuming that the household size $L(t)=L(0)e^{nt}$ follows this normalization $L(0)\equiv1$. Eventually, $L(t)=e^{nt}$.

Then, $\rho -n$ is the effective discount rate because the household derives utility from the consumption per capita of its additional members in the future as well.

Clearly, you need to add a discounting term otherwise the utility function might get an infinite value (not an interesting case to analyze).

Finally, $\rho -n >0$ ensures finite discounted utility. Take the limit of the discounting term and see what happens for $\rho -n <0$


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