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Specific proof of quasiconcavity for the sum of quasiconcave functions

  1. Sep 14, 2012 #1
    I want to prove that a function ψ(a1,a0), which is additively decomposed in two quasiconcave functions, is also quasiconcave. My numerical simulations seem to suggest it is possible, but I cannot get around it. I have tried using the definition of quasiconcavity (i.e. plugging ψ(λa + (1-λ)b) >...etc), but I can not prove the inequality. I also tried using the border Hessian, but i cannot sign the determinant. Any suggestions??

    Here are the details of the problem:

    ψ(a1,a0)=R(a1)+R(a0)

    Where a1 [itex]\in[0,1][/itex] and a0 [itex]\in[0,1][/itex], and

    R(aj)=F(Bl(aj)) ∫[itex]^{1}_{aj}[/itex] t*P(t)dt + F(Bh(aj)) ∫[itex]^{1}_{aj}[/itex] t*(1-P(t))dt for j[itex]\in{1,0}[/itex]

    and the functions F(.), Bl(.), Bh(.) and P(.) have the following properties:

    F',Bl',Bh',P'>0
    Bl''>0
    Bh''<0
    F'' and P'' can be signed as required for the proof

    Thanks a lot for any hints!
     
  2. jcsd
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