<p>The paper is assuming that some form of "law of large numbers" (LLN) applies for the continuum. The expected value of capital for an individual agent is
$$\mathbb{E}[R\cdot 1_{R\geq R^*}]=\int_{R^*}^1 R\,dR\tag{1}$$
The LLN assumption says that when we have a mass-1 continuum of agents, their actual total will equal their expectation in (1).</p>

<p>What justifies this assumption? Imagine that rather than having a continuum of agents, we simply had a very large number $N$, with each of these agents still having $R$ drawn from the same uniform distribution on $[0,1]$, resulting in an expected value of capital given by (1). Then as $N\rightarrow \infty$, the <a href="http://en.wikipedia.org/wiki/Law_of_large_numbers" rel="nofollow">standard law of large numbers</a> implies that the sample mean of capital will approach the value in (1). If, as we take this limit, we normalize each agent to have a size of $1/N$, then the sample mean will equal the total amount of capital - so in the limit, the amount of capital will always be (1).</p>

<p>Here comes the weird and not-entirely-correct thing that economists do. We often say that having a mass-1 continuum of agents is <em>like</em> taking this $N\rightarrow \infty$ limit (after all, there is an infinite number of agents in the continuum...). We therefore infer that the total value of capital in the continuum will equal the expected value given in (1).</p>

<p>Like I said, this is not entirely correct or rigorous. <a href="http://www.sciencedirect.com/science/article/pii/0022053185900596" rel="nofollow">Judd famously complained</a> about it, pointing out that if we take a continuum of agents each with independent random values, then the resulting function will usually not even be Lebesgue integrable, so we can't even define the aggregate value in a rigorous way. <a href="http://www.sciencedirect.com/science/article/pii/S0899825603001751" rel="nofollow">Al-Najjar recently suggested</a> that we replace the continuum assumption with a discrete construction that actually implements the $N\rightarrow \infty$ limit. There are some other suggestions out there too. </p>

<p>Most economists, though, just ignore these technical concerns and continue to assume that when you have a continuum with independently distributed values, you get the expected value when you integrate. I don't mind this, since it's just a convention, and they are right in spirit, since a continuum realistically is intended to model a very large number of agents. (This issue came up once with a coauthor, and after reading Judd and Al-Najjar and others for a few hours, we just threw our hands up and decided to ignore the problem! My personal, technically acceptable way to resolve it is to say that agents - rather than living on a continuum - live in the probability space itself, and define aggregation over agents as probability-weighted integration over this probability space. But this is pretty technical and obscure, and the basic point is that 95% of economists ignore these issues.)</p>

CC BY-SA 3.0
2015-05-10T09:47:18.443Z
(+1) for bringing up and discussing the "obscure technical issue".
Alecos Papadopoulos
CC BY-SA 3.0
2015-05-09T16:52:38.917Z
2015-05-09T16:52:38.917Z
nominally rigid
<p>I won't go into detail of this model because it's really just one point that i'm confused about. This question is based on a model by Matsuyama (2007).</p>

<ul>
<li>Agents are deciding whether to invest in a project or not</li>
<li>The project gives a payoff of R units of capital</li>
<li>R is a random, uniformly distributed variable over the interval [0,1]</li>
<li>Agents only decide to
invest if return on investment > return on saving; let's call this
cutoff level of capital R*</li>
<li>Question: what is the total level of
capital in this economy?</li>
<li>Answer: capital = the integral of R from R*
to 1.  [1]: <a href="https://i.sstatic.net/O7gIj.png" rel="nofollow noreferrer">https://i.sstatic.net/O7gIj.png</a></li>
</ul>

<p>I understand that an integral is like a continuous sum... I just don't quite understand how we can just sum of all the R's to obtain the total level of capital? What if all the R's don't actually occur?</p>

<p>Thank you!</p>

CC BY-SA 3.0
2015-05-10T15:30:48.947Z
Clarifications please: 1) How much do agents invest, if they decide to invest? Is it normalized in all cases to unity? 2) You write "R is return on investment". If it is return on investment, then what happens to it? Is it consumed, or does it augment the capital base? Also, how many agents are they? We learn that the _payoff_ is continuously distributed in $(0,1)$ not that there exists a "continuum of agents", So is their number discrete and equal to say, $n$, or indeed we have a "continuum of agents"? Finally, provide also a link for the paper please.
Alecos Papadopoulos
<macroeconomics><financial-economics><investment><capital-returns><credit-friction>
Capital in a credit frictions model
Natasha
# Capital in a credit frictions model

(Asked by: username_0 on 2015-05-09)

I won't go into detail of this model because it's really just one point that i'm confused about. This question is based on a model by Matsuyama (2007).

* Agents are deciding whether to invest in a project or not
* The project gives a payoff of R units of capital
* R is a random, uniformly distributed variable over the interval [0,1]
* Agents only decide to
  invest if return on investment > return on saving; let's call this
  cutoff level of capital R\*
* Question: what is the total level of
  capital in this economy?
* Answer: capital = the integral of R from R\*
  to 1. [1]: <https://i.sstatic.net/O7gIj.png>

I understand that an integral is like a continuous sum... I just don't quite understand how we can just sum of all the R's to obtain the total level of capital? What if all the R's don't actually occur?

Thank you!

**Tags:** macroeconomics, financial-economics, investment, capital-returns, credit-friction
> **Comments:**
> **username_1**: Clarifications please: 1) How much do agents invest, if they decide to invest? Is it normalized in all cases to unity? 2) You write "R is return on investment". If it is return on investment, then what happens to it? Is it consumed, or does it augment the capital base? Also, how many agents are they? We learn that the _payoff_ is continuously distributed in $(0,1)$ not that there exists a "continuum of agents", So is their number discrete and equal to say, $n$, or indeed we have a "continuum of agents"? Finally, provide also a link for the paper please.  

---

## 1 answers

### Answer by username_2 on 2015-05-09. Score: 4

The paper is assuming that some form of "law of large numbers" (LLN) applies for the continuum. The expected value of capital for an individual agent is
$$\mathbb{E}[R\cdot 1\_{R\geq R^\*}]=\int\_{R^\*}^1 R\,dR\tag{1}$$
The LLN assumption says that when we have a mass-1 continuum of agents, their actual total will equal their expectation in (1).

What justifies this assumption? Imagine that rather than having a continuum of agents, we simply had a very large number $N$, with each of these agents still having $R$ drawn from the same uniform distribution on $[0,1]$, resulting in an expected value of capital given by (1). Then as $N\rightarrow \infty$, the [standard law of large numbers](http://en.wikipedia.org/wiki/Law_of_large_numbers) implies that the sample mean of capital will approach the value in (1). If, as we take this limit, we normalize each agent to have a size of $1/N$, then the sample mean will equal the total amount of capital - so in the limit, the amount of capital will always be (1).

Here comes the weird and not-entirely-correct thing that economists do. We often say that having a mass-1 continuum of agents is *like* taking this $N\rightarrow \infty$ limit (after all, there is an infinite number of agents in the continuum...). We therefore infer that the total value of capital in the continuum will equal the expected value given in (1).

Like I said, this is not entirely correct or rigorous. [Judd famously complained](http://www.sciencedirect.com/science/article/pii/0022053185900596) about it, pointing out that if we take a continuum of agents each with independent random values, then the resulting function will usually not even be Lebesgue integrable, so we can't even define the aggregate value in a rigorous way. [Al-Najjar recently suggested](http://www.sciencedirect.com/science/article/pii/S0899825603001751) that we replace the continuum assumption with a discrete construction that actually implements the $N\rightarrow \infty$ limit. There are some other suggestions out there too.

Most economists, though, just ignore these technical concerns and continue to assume that when you have a continuum with independently distributed values, you get the expected value when you integrate. I don't mind this, since it's just a convention, and they are right in spirit, since a continuum realistically is intended to model a very large number of agents. (This issue came up once with a coauthor, and after reading Judd and Al-Najjar and others for a few hours, we just threw our hands up and decided to ignore the problem! My personal, technically acceptable way to resolve it is to say that agents - rather than living on a continuum - live in the probability space itself, and define aggregation over agents as probability-weighted integration over this probability space. But this is pretty technical and obscure, and the basic point is that 95% of economists ignore these issues.)

> **Comments:**
> **username_1**: (+1) for bringing up and discussing the "obscure technical issue".  

---

