What is the difference between a power law exponent and the Pareto exponent? I use the poweRlaw 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 returns $\alpha$ as defined above.
I am using Newman as a reference.
A random variable $X$ has a Pareto distribution with Pareto exponent
  $\theta$ if  $$\text{P}(X&gt;x)=\begin{cases} 
 \left(\frac{x}{x_m}\right)^\theta \quad \text{ if } x\geq x_{min}\\  \
 \ 1 \quad \quad \quad  \text{ if } x&lt;x_{min} \end{cases}$$


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