<blockquote>
  <p>A random variable <span class="math-container">$X$</span> has a Pareto distribution with Pareto exponent
  <span class="math-container">$\theta$</span> if  <span class="math-container">$$\text{P}(X&gt;x)=\begin{cases} 
 \left(\frac{x}{x_m}\right)^\theta \quad \text{ if } x\geq x_{min}\\  \
 \ 1 \quad \quad \quad  \text{ if } x&lt;x_{min} \end{cases}$$</span></p>
</blockquote>

<p>In this case, the Pareto exponent is <span class="math-container">$\theta = \alpha - 1$</span>. Remember that <span class="math-container">$P(X&gt;x)=\int_x^\infty p(x')\mathrm{d}x'$</span>.</p>

CC BY-SA 4.0
2019-09-01T16:55:02.173Z
2019-09-01T16:55:02.173Z
Angela Pretorius
<p>I use the <a href="https://cran.r-project.org/web/packages/poweRlaw/poweRlaw.pdf" rel="nofollow noreferrer">poweRlaw</a> package in R to fit a power law to my data. I am trying to figure out what is the value of the Pareto exponent. </p>

<p>Assume the probability mass function is defined by:</p>

<p><span class="math-container">$$
p(x) = \frac{\alpha-1}{x_{min}} \left(\frac{x}{x_{min}} \right)^{-\alpha}
$$</span></p>

<p>and the complementary cumulative density function is defined by:</p>

<p><span class="math-container">$$
P(x) = \int_x ^\infty p(x’) dx’ = \left(\frac{x}{x_{min}}\right)^{-\alpha + 1}
$$</span></p>

<p>Is the Pareto exponent <span class="math-container">$\alpha$</span> or <span class="math-container">$- \alpha + 1$</span> or  <span class="math-container">$\alpha - 1$</span>?
In most literature, the CCDF is used to describe the income/wealth distribution, and <span class="math-container">$-(\alpha-1)$</span> is the slope of the CCDF on a log-log plot, so <span class="math-container">$\alpha-1$</span> seems the most intuitive. I'm pretty sure the R library <a href="https://cran.r-project.org/web/packages/poweRlaw/poweRlaw.pdf" rel="nofollow noreferrer">poweRlaw</a> returns <span class="math-container">$\alpha$</span> as defined above.
I am using <a href="https://arxiv.org/abs/cond-mat/0412004" rel="nofollow noreferrer">Newman</a> as a reference.</p>

<macroeconomics><pareto-efficiency><academic-graduate><wealth><r>
What is the difference between a power law exponent and the Pareto exponent?
0x90
# What is the difference between a power law exponent and the Pareto exponent?

(Asked by: username_0 on 2019-08-20)

I use the [poweRlaw](https://cran.r-project.org/web/packages/poweRlaw/poweRlaw.pdf) package in R to fit a power law to my data. I am trying to figure out what is the value of the Pareto exponent.

Assume the probability mass function is defined by:

$$
p(x) = \frac{\alpha-1}{x\_{min}} \left(\frac{x}{x\_{min}} \right)^{-\alpha}
$$

and the complementary cumulative density function is defined by:

$$
P(x) = \int\_x ^\infty p(x’) dx’ = \left(\frac{x}{x\_{min}}\right)^{-\alpha + 1}
$$

Is the Pareto exponent $\alpha$ or $- \alpha + 1$ or $\alpha - 1$?
In most literature, the CCDF is used to describe the income/wealth distribution, and $-(\alpha-1)$ is the slope of the CCDF on a log-log plot, so $\alpha-1$ seems the most intuitive. I'm pretty sure the R library [poweRlaw](https://cran.r-project.org/web/packages/poweRlaw/poweRlaw.pdf) returns $\alpha$ as defined above.
I am using [Newman](https://arxiv.org/abs/cond-mat/0412004) as a reference.

**Tags:** macroeconomics, pareto-efficiency, academic-graduate, wealth, r

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## 1 answers

### Answer by username_1 on 2019-09-01. Score: 1 | Accepted answer

> A random variable $X$ has a Pareto distribution with Pareto exponent
> $\theta$ if $$\text{P}(X>x)=\begin{cases}
> \left(\frac{x}{x\_m}\right)^\theta \quad \text{ if } x\geq x\_{min}\\ \
> \ 1 \quad \quad \quad \text{ if } x

In this case, the Pareto exponent is $\theta = \alpha - 1$. Remember that $P(X>x)=\int\_x^\infty p(x')\mathrm{d}x'$.


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