Looking for the integral:
With $b>0, L>0, \alpha_0>b,\sigma>0, x>L$ ,
$$\phi(x;\alpha_0,\sigma_)=\int_b^\infty\alpha L^{\alpha } x^{-\alpha -1} \frac{e^{-\frac{\left(\log (\alpha -b)-\log (\alpha_0-b)+\frac{\sigma ^2}{2}\right)^2}{2 \sigma ^2}}}{\sqrt{2 \pi } \sigma (\alpha -b)}d\alpha$$
$\textbf{Background}$: This is the density of a Pareto distribution $\alpha L^{\alpha } x^{-\alpha -1} $ with its tail exponent $\alpha$ distributed according to a shifted lognormal with mean $\alpha_0$ and lower bound $b$. In other words $\alpha + b$ follows a $\text{LogNormal}\left(\log (\alpha_0-b)-\frac{\sigma ^2}{2},\sigma \right)$.