<p>As a mathematical fact, <span class="math-container">$\int_0^1 K~\mathrm di =K$</span> is perfectly fine. The way I would read this would be that every agent owns the average amount; the integral gives a population average. This works equally well if you have <span class="math-container">$n$</span> households, and each has a weight of <span class="math-container">$1/n$</span>. Then, we have <span class="math-container">$\sum_{i=1}^n 1/n~K=K$</span>. So, you might think of the integral representing that each individual has infinitesimal weight and owns an infinitesimal share of the total capital.</p>

CC BY-SA 4.0
2024-10-08T23:03:46.750Z
Hi, thanks for the answer. About your answer, if K means "average amount" of capital, then how do you calculate the "total amount" of capital in the whole economy? Does that mean the "total amount" is infinity?
HumbleHongbo
CC BY-SA 4.0
2024-10-08T23:51:26.597Z
Do you have a specific context? Subtleties come in when one looks at population growth, otherwise you can simply treat everything as "per capita."
Michael Greinecker
CC BY-SA 4.0
2024-10-08T22:53:27.770Z
2024-10-08T22:53:27.770Z
Michael Greinecker
<p>When I was taking a first-year Macroeconomics class, the Professor always mentioned that &quot;The economy has identical households with a measure of 1&quot;. I don't really how to aggregate all the households as a representative.</p>
<p>For example, suppose there are uncountably infinite many households in the economy:</p>
<p>Label <span class="math-container">$i\in [0,1]$</span>, the individual capital level is <span class="math-container">$k$</span> and total capital in the economy is <span class="math-container">$K$</span>.</p>
<p>Then we have:
<span class="math-container">$\int_{0}^{1} k(i)\ di = K$</span>, by identity of all the households, then we have <span class="math-container">$\int_{0}^{1} k\ di = K$</span>, which implies <span class="math-container">$k(i) = k = K, \forall i \in [0,1]$</span>.</p>
<p>The economy meaning is that individual capital alway equal to the total number of capital, which is not realistic.</p>
<p>I wonder does this mean that the aggregation I learnt is mathematically wrong?</p>
<p>(I asked the professor this question several times and this guy just dodged everytime :(</p>
<p>Thanks so much in advance for any suggestions!</p>
<p>EDIT: I just realized the meaning of &quot;integral&quot; is &quot;average&quot; rather than &quot;summation&quot;, so now the question is: is that true for this economy, the total amount of capital is always +<span class="math-container">$\infty$</span>?</p>

<macroeconomics>
General question about aggregation of identical households as a representative one
HumbleHongbo
# General question about aggregation of identical households as a representative one

(Asked by: username_0 on 2024-10-08)

When I was taking a first-year Macroeconomics class, the Professor always mentioned that "The economy has identical households with a measure of 1". I don't really how to aggregate all the households as a representative.

For example, suppose there are uncountably infinite many households in the economy:

Label $i\in [0,1]$, the individual capital level is $k$ and total capital in the economy is $K$.

Then we have:
$\int\_{0}^{1} k(i)\ di = K$, by identity of all the households, then we have $\int\_{0}^{1} k\ di = K$, which implies $k(i) = k = K, \forall i \in [0,1]$.

The economy meaning is that individual capital alway equal to the total number of capital, which is not realistic.

I wonder does this mean that the aggregation I learnt is mathematically wrong?

(I asked the professor this question several times and this guy just dodged everytime :(

Thanks so much in advance for any suggestions!

EDIT: I just realized the meaning of "integral" is "average" rather than "summation", so now the question is: is that true for this economy, the total amount of capital is always +$\infty$?

**Tags:** macroeconomics

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## 1 answers

### Answer by username_1 on 2024-10-08. Score: 2 | Accepted answer

As a mathematical fact, $\int\_0^1 K~\mathrm di =K$ is perfectly fine. The way I would read this would be that every agent owns the average amount; the integral gives a population average. This works equally well if you have $n$ households, and each has a weight of $1/n$. Then, we have $\sum\_{i=1}^n 1/n~K=K$. So, you might think of the integral representing that each individual has infinitesimal weight and owns an infinitesimal share of the total capital.

> **Comments:**
> **username_0**: Hi, thanks for the answer. About your answer, if K means "average amount" of capital, then how do you calculate the "total amount" of capital in the whole economy? Does that mean the "total amount" is infinity?  
> **username_1**: Do you have a specific context? Subtleties come in when one looks at population growth, otherwise you can simply treat everything as "per capita."  

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