I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywords to find it. Please help.Say there is a function G ( k , x ) = ∫ a x f ( k , t ) d t, and I wish to maximize G ( k , x ) w.r.t. k. Under what conditions is this maximization problem equivalent to maximizing f ( k , t ) w.r.t. k?In short, when is the following true: max k { ∫ a x f ( k , t ) } d t ⇔ max k { f ( k , t ) }

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I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywords to find it. Please help.Say there is a function   G  (  k  ,  x  )  =      ∫    a    x    f  (  k  ,  t  )  d  t, and I wish to maximize   G  (  k  ,  x  ) w.r.t.   k. Under what conditions is this maximization problem equivalent to maximizing   f  (  k  ,  t  ) w.r.t.   k?In short, when is the following true:      max          k            {          ∫      a      x        f    (    k    ,    t    )    }    d  t  ⇔      max          k        {  f  (  k  ,  t  )  }

amanhantmk · Accepted Answer

Do you mean to find the argmax of   k? Assuming nice differentiability and solution is interior, the solution is given by             d                      d            k        G  (  k  ,  x  )  =      ∫    a    x              d                      d            k        f  (  k  ,  t  )      d    t  =  0. You see immediately that if   k is the   arg  ⁡  max  f  (  k  ,  t  ) for all   t  ∈  (  a  ,  x  ) then your claim is true.

I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywo

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