In Postel-Vinay & Robin 2002 they show an equation: $$\left\{\delta+\mu+\lambda_{1} \bar{F}(p)\right\} \ell(\varepsilon, p)=\left\{(\delta+\mu) h(\varepsilon)+\lambda_{1} \int_{p_{\min }}^{p} \ell(\varepsilon, x) d x\right\} f(p)$$ where $h$ is the pdf of $ε$, and $\bar{F}(p)=1-F(p)$ is the inverse cdf of $p$.

They then say that this solves as $$\ell(\varepsilon, p)=\frac{1+\kappa_{1}}{\left[1+\kappa_{1} \bar{F}(p)\right]^{2}} h(\varepsilon) f(p)$$ where $\kappa_{1}=\lambda_{1} /(\delta+\mu)$, but I can't see how.

Also note that at this time we don't know that if the joint distribution is independent or not because the authors integrate this result to get $\ell(p)=\frac{1+\kappa_{1}}{\left[1+\kappa_{1} \bar{F}(p)\right]^{2}} f(p)$ and thus $\ell(\varepsilon, p)=h(\varepsilon) \ell(p)$ and finally argue that $ε$ and $p$ are uncorrelated.

  • $\begingroup$ Is your curly bracket $\{ \}$ means anything? or it has the same use as regular bracket $()$? $\endgroup$
    – Redsbefall
    Jul 13, 2022 at 13:19
  • $\begingroup$ @Redsbefall Just a regular one. $\endgroup$ Jul 13, 2022 at 13:33
  • $\begingroup$ Can you share the link to the e-book? $\endgroup$
    – Redsbefall
    Jul 13, 2022 at 13:37
  • $\begingroup$ @Redsbefall It's not a book it's a paper on econometrica. onlinelibrary.wiley.com/doi/10.1111/j.1468-0262.2002.00441.x $\endgroup$ Jul 13, 2022 at 13:48
  • $\begingroup$ Why dont you ask the author? (e-mail etc.) If it is from paper it is usually not elementary, also by looking at the equation. $\endgroup$
    – Redsbefall
    Jul 19, 2022 at 12:23


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