# Solve Joint Distribution From An Equation

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.

• Is your curly bracket $\{ \}$ means anything? or it has the same use as regular bracket $()$? Jul 13, 2022 at 13:19
• @Redsbefall Just a regular one. Jul 13, 2022 at 13:33
• Can you share the link to the e-book? Jul 13, 2022 at 13:37
• @Redsbefall It's not a book it's a paper on econometrica. onlinelibrary.wiley.com/doi/10.1111/j.1468-0262.2002.00441.x Jul 13, 2022 at 13:48
• 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. Jul 19, 2022 at 12:23