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Thread: Integration problem with maximum function inside integral

  1. #1
    Feb 2010

    Integration problem with maximum function inside integral

    Hello. I have two questions, I'll be extremely cheerful if anyone can provide an answer.

    I have got two equations as follows

    $\displaystyle r*U=b+\alpha _{0}*\int_{w_{R}}^{\infty}[W(w)-U]dF(w)$ (1)

    $\displaystyle r*W(w)=w+\alpha _{1}*\int_{0}^{\infty}\max (0,W(w^{'})-W(w)))dF(w')+\lambda [U-W(w)]$ (2)

    where w' is not the derivative of w, but just describing another value of w.

    He says: Let $\displaystyle W(w_{R})=U$, then he evaluates the equation (2) at $\displaystyle w=w_{R}$ (so he gets r*U=....)

    Then combines this with (1), (Because (1) and the (2) evaluated at $\displaystyle w=w_{R}$ are equal) to solve for $\displaystyle w_{R}$:

    to get: $\displaystyle w_{R}=b+(\alpha _{0}-\alpha _{1})\int_{w_{R}}^{\infty }[W(w')-U]dF(w')$

    Now my problem is: how does he actually gets that final equation? Because I cannot manage to evaluate (2) at $\displaystyle w=w_{R}$ because that maximum function in the integral really confuses me. Moreover, I don't understand how he merges two integrals with different intervals.

    Okay this was the first question.

    Second one is, he differentiates the equation (2) to get

    $\displaystyle W'(w)=(r+\lambda +\alpha _{1}[1-F(w)])^{-1}$

    I can't manage to get this either. I get all my problems come down to that integral with a maximum function in it.

    Okay, I hope I could describe it well enough.


    Last edited by ichoosetonotchoosetochoos; Feb 12th 2010 at 06:53 PM.
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