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I have

$$\int_{-\bar{\lambda}}^{\lambda^{*}}\left(\frac{D-\rho_{F}\ell M-\rho_{A}\ell A}{\left(1-\rho_{F}\ell\right)D}-\tilde{\lambda}\right)dG(\tilde{\lambda})$$

where G is a continously increasing CDF on the interval $-\bar{\lambda},\bar{\lambda}$ and $\tilde{\lambda}$ is an IID random variable drawn from corresponding PDF to $G$ with mean 0. Finally, $\lambda^*=\frac{D-\rho_{F}\ell M-\rho_{A}\ell A}{\left(1-\rho_{F}\ell\right)D}$. I need to take FOC wrt the variables $M$, and $A$.

I am guessing to use the leibniz formula but am struggling with getting something "nice"

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  • 1
    $\begingroup$ Behind the curtain is the Riemann-Stieltjes integral $\endgroup$
    – Jean Marie
    Apr 21, 2020 at 14:37
  • $\begingroup$ Indeed, but I'm not sure it changes much here wrt. to the final derivation? $\endgroup$
    – user469216
    Apr 21, 2020 at 14:40
  • 1
    $\begingroup$ OK. I though it was one of your concerns. A little advice : simplify ! For example, I don't see the interest to keep the front fraction for the explanation of your issue... $\endgroup$
    – Jean Marie
    Apr 21, 2020 at 14:49
  • $\begingroup$ Youre right. I will delete the fron constant :-)! $\endgroup$
    – user469216
    Apr 21, 2020 at 14:51
  • $\begingroup$ The role of $\lambda^*$ in $\int_{-\bar{\lambda}}^{\lambda^{*}}\left(\lambda^*-\tilde{\lambda}\right)dG(\tilde{\lambda})$ which is present at two places (upper bound and inside the parenthesis) isn't clear : is it a constant or a parameter that, sooner or later, will vary ? I think that, if you want to have answers, to have to make things clearer. $\endgroup$
    – Jean Marie
    Apr 21, 2020 at 17:00

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