In a paper, I see a way of Pareto distribution's parametrization:
" Specifically, I assume that varieties can be indexed from highest to lowest benefit on a continuum from $[0, n]$, with
$$ z_i p_i^{1-\sigma} \equiv[1+(1-\sigma) / \theta]\left(b i^{1 / \theta}\right)^{1-\sigma} $$
Recall that $p_i$ is the price of a variety and $z_i$ captures anything else (taste, quality, etc.) that shifts demand up or down given price. This saysanything else (taste, quality, etc.) that shifts demand up or down given price. This says that varieties should be ranked in terms of their relative expenditure shares rather than prices. Relative expenditure shares for varieties $i$ and $j$, given by
$$ z_i p_i^{1-\sigma} / z_j p_j^{1-\sigma} $$
(which I hereafter call the "relative intensive margin"), are inversely proportional to their relative rank $(i / j)^{\frac{1-\sigma}{\theta}}$. The parameter $\theta$ captures asymmetry across varieties (i.e. the extent to which marginal varieties are less valuable due to price, quality or taste) and as $\theta \rightarrow \infty$ varieties become symmetric and have identical expenditure shares. "
A similar question is this
I wonder if I want to change Pareto distribution to Lognormal distribution, whether there can be a similar way like this.
In fact equations above gives a function that describe the second plot in the link. I wonder how to get the first plot.