<blockquote>
  <p><em>...find economic theories that support the exponential hypothesis, but, as I've said, I haven't found any.</em></p>
</blockquote>

<p>I don't think you will. For purposes of economic argument, the characteristic of the Exponential distribution that stands out and must be defended, is that it has its maximum at zero.</p>

<p>Then, arguing that an Income distribution follows the Exponential, means that we accept that the most probable situation to be observed is zero income. Is it? I don't think so.</p>

<p>As regards the study you link to, the main piece of evidence for the fit of the Exponential to the data is Fig 1 page 2. The authors write that they consider it a very good fit. I most certainly don't: the histogram in Fig 1. has clearly a mode at strictly positive values and not at zero. Βut this is the most critical piece of evidence as to whether the data follows an Exponential or not. </p>

<p>At page 4, the authors try to deal with this critical deviation of the data from the Exponential graph, by mentioning that there may be under-reporting at low values, but that's a very vague argument, unsupported and unsubstantiated.</p>

<p>a) From a theoretical perspective, it is not realistic to argue that the most probable income observed is zero
b) The data support this by providing a strictly positive mode for the frequency.</p>

<p>Therefore I don't see a case for the Exponential.</p>

CC BY-SA 3.0
2016-10-07T22:42:33.127Z
Thanks for your answer, dear Alecos. But, could we say, then, that this is a sort of exponential-ish distribution or something? You see, they're quite emotional about this, and I have to come up with something.
numberfive
CC BY-SA 3.0
2016-10-08T10:43:13.730Z
@NormanSimon Fitting distribution to data is a challenging problem, exactly because in many cases it is _not_ difficult to find a distribution that fits "a large part" of the data -and then be tempted to argue "well, it's good enough", while it isn't really. How good is the fit at the extreme parts of the distribution is usually critical, for example to conduct statistical tests. There are formal tests to judge the quality of a fitting exercise, and I don't see them in the study. "Having to come up with something" begs "no matter what", which doesn't provide for a valid scientific argument.
Alecos Papadopoulos
CC BY-SA 3.0
2016-10-08T12:02:53.297Z
In the UK, incomes of zero are the most common (when rounded to £1 a week): this includes those dependent on others and also self-employed individuals who failed to make a profit in a particular year; a consequence is that people with zero income on average have a higher standard of living than those with a fairly low positive income.  What fails the exponential distribution is the low number of adults with small but positive incomes.
Henry
CC BY-SA 3.0
2016-10-08T18:04:38.603Z
Actually the paper tells a story of "exponential above a certain threshold" which is not technically correct because the exponential should have a peak at 0. My main question, however, is if there's a recognised economic theory for this kind of results, even if that economic model claims that the exponential is "truncated".
numberfive
CC BY-SA 3.0
2016-10-07T21:20:29.860Z
2016-10-07T21:20:29.860Z
Alecos Papadopoulos
<p>Something to consider is that the log-normal distribution can look like an exponential distribution for high $\sigma$.</p>

<p>For example, here's a histogram over market capitalization for firms in Compustat.<a href="https://i.sstatic.net/HvLWn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/HvLWn.png" alt="enter image description here"></a>.</p>

<p>Kinda looks exponentially, but here's a QQ-plot checking the distribution over the log of market capitalization for normality. <a href="https://i.sstatic.net/bfxYd.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bfxYd.png" alt="enter image description here"></a></p>

<p>Not perfect, but not bad! And here's a histogram of log10 marketcap:
<a href="https://i.sstatic.net/zKSkd.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/zKSkd.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2016-10-08T18:02:25.713Z
Thanks Matthew. Actually I showed them a similar exercise with lognormals instead of exponentials, and they conceded that they can be sometimes a better fit. They however, want to know if there is an economic theory that supports the idea of an exponential distribution in the lower tail. I have read Acemoglu's work, but he's more concerned with Pareto than exponentials.
numberfive
CC BY-SA 3.0
2016-10-08T05:58:55.423Z
2016-10-08T06:05:29.573Z
2016-10-08T06:05:29.573Z
Matthew Gunn
<p>I have been working on income distributions for a while, and I've been asked to do a theoretical justification for the idea that income distribution follows an exponential function for low incomes. There's a lot of literature about Pareto distributions in the upper tail of the distribution, but I haven't been able to find literature for the lower tail and the exponential.</p>

<p>There's empirical work on this topic, mainly done by physicists: <a href="https://arxiv.org/pdf/cond-mat/0008305.pdf" rel="nofollow">https://arxiv.org/pdf/cond-mat/0008305.pdf</a></p>

<p>One of those physicists asked me to find economic theories that support the exponential hypothesis, but, as I've said, I haven't found any.</p>

<p>Could you point some pieces of economic literature to get me on the right way? Thank you.</p>

<reference-request><theory>
How to theoretically explain an exponential income distribution?
numberfive
# How to theoretically explain an exponential income distribution?

(Asked by: username_0 on 2016-10-07)

I have been working on income distributions for a while, and I've been asked to do a theoretical justification for the idea that income distribution follows an exponential function for low incomes. There's a lot of literature about Pareto distributions in the upper tail of the distribution, but I haven't been able to find literature for the lower tail and the exponential.

There's empirical work on this topic, mainly done by physicists: <https://arxiv.org/pdf/cond-mat/0008305.pdf>

One of those physicists asked me to find economic theories that support the exponential hypothesis, but, as I've said, I haven't found any.

Could you point some pieces of economic literature to get me on the right way? Thank you.

**Tags:** reference-request, theory

---

## 2 answers

### Answer by username_1 on 2016-10-07. Score: 4

> *...find economic theories that support the exponential hypothesis, but, as I've said, I haven't found any.*

I don't think you will. For purposes of economic argument, the characteristic of the Exponential distribution that stands out and must be defended, is that it has its maximum at zero.

Then, arguing that an Income distribution follows the Exponential, means that we accept that the most probable situation to be observed is zero income. Is it? I don't think so.

As regards the study you link to, the main piece of evidence for the fit of the Exponential to the data is Fig 1 page 2. The authors write that they consider it a very good fit. I most certainly don't: the histogram in Fig 1. has clearly a mode at strictly positive values and not at zero. Βut this is the most critical piece of evidence as to whether the data follows an Exponential or not.

At page 4, the authors try to deal with this critical deviation of the data from the Exponential graph, by mentioning that there may be under-reporting at low values, but that's a very vague argument, unsupported and unsubstantiated.

a) From a theoretical perspective, it is not realistic to argue that the most probable income observed is zero
b) The data support this by providing a strictly positive mode for the frequency.

Therefore I don't see a case for the Exponential.

> **Comments:**
> **username_0**: Thanks for your answer, dear Alecos. But, could we say, then, that this is a sort of exponential-ish distribution or something? You see, they're quite emotional about this, and I have to come up with something.  
> **username_1**: @NormanSimon Fitting distribution to data is a challenging problem, exactly because in many cases it is _not_ difficult to find a distribution that fits "a large part" of the data -and then be tempted to argue "well, it's good enough", while it isn't really. How good is the fit at the extreme parts of the distribution is usually critical, for example to conduct statistical tests. There are formal tests to judge the quality of a fitting exercise, and I don't see them in the study. "Having to come up with something" begs "no matter what", which doesn't provide for a valid scientific argument.  
> **username_2**: In the UK, incomes of zero are the most common (when rounded to £1 a week): this includes those dependent on others and also self-employed individuals who failed to make a profit in a particular year; a consequence is that people with zero income on average have a higher standard of living than those with a fairly low positive income.  What fails the exponential distribution is the low number of adults with small but positive incomes.  
> **username_0**: Actually the paper tells a story of "exponential above a certain threshold" which is not technically correct because the exponential should have a peak at 0. My main question, however, is if there's a recognised economic theory for this kind of results, even if that economic model claims that the exponential is "truncated".  

---

### Answer by username_3 on 2016-10-08. Score: 2

Something to consider is that the log-normal distribution can look like an exponential distribution for high $\sigma$.

For example, here's a histogram over market capitalization for firms in Compustat.[![enter image description here](https://i.sstatic.net/HvLWn.png)](https://i.sstatic.net/HvLWn.png).

Kinda looks exponentially, but here's a QQ-plot checking the distribution over the log of market capitalization for normality. [![enter image description here](https://i.sstatic.net/bfxYd.png)](https://i.sstatic.net/bfxYd.png)

Not perfect, but not bad! And here's a histogram of log10 marketcap:
[![enter image description here](https://i.sstatic.net/zKSkd.png)](https://i.sstatic.net/zKSkd.png)

> **Comments:**
> **username_0**: Thanks Matthew. Actually I showed them a similar exercise with lognormals instead of exponentials, and they conceded that they can be sometimes a better fit. They however, want to know if there is an economic theory that supports the idea of an exponential distribution in the lower tail. I have read Acemoglu's work, but he's more concerned with Pareto than exponentials.  

---

