Let N = { 1 , … , n }. For a given ( r 1 , … , r n ) ∈ R + + n . I need to solve max ( k 1 , … , k n ) ∈ R + + n ∏ i ∈ N [ ∑ j ∈ N ( r j − k j ) + r i ln ⁡ ( k i r i ) ] . I could not come up with a closed form solution of the maximizers by using first order conditions, because of the log. Is it possible to tell, that there does not exist a closed form solution and we thus have to maximize the product numerically.

Question

Let   N  =  {  1  ,  …  ,  n  }. For a given   (      r    1    ,  …  ,      r    n    )  ∈            R              +      +        n  . I need to solve                              max                      (                          k              1                        ,            …            ,                          k              n                        )            ∈                                          R                                            +                +                            n                                                ∏                      i            ∈            N                                                [                          ∑                              j                ∈                N                                                    (                              r                j                            −                              k                j                            )                        +                          r              i                        ln            ⁡                          (                                                k                  i                                                  r                  i                                            )                        ]                          .            I could not come up with a closed form solution of the maximizers by using first order conditions, because of the log. Is it possible to tell, that there does not exist a closed form solution and we thus have to maximize the product numerically.

Marisol Morton · Accepted Answer

ln is a strictly increasing monotonic function then with       x    k    &amp;gt;  0  max      ∏    k        x    x    ≡  max      ∑    k    ln  ⁡  (      x    k    )                              max                      (                          k              1                        ,            …            ,                          k              n                        )            ∈                                          R                                            +                +                            n                                      f        (        k        )        =                  ∏                      i            ∈            N                                                [                          ∑                              j                ∈                N                                                    (                              r                j                            −                              k                j                            )                        +                          r              i                        ln            ⁡                          (                                                k                  i                                                  r                  i                                            )                        ]                          .            so under the hypothesis that      ∑          j      ∈      N            (          r      j        −          k      j        )    +      r    i    ln  ⁡      (                  k        i                    r        i              )    &amp;gt;  0the problem presents a more amenable formulation as  max  F  (  k  )  =  ln  ⁡  (  f  (  k  )  )  =      ∑          i      ∈      N        ln  ⁡      (          ∑              j        ∈        N                    (              r        j            −              k        j            )        +          r      i        ln    ⁡          (                        k          i                          r          i                    )        )  with            ∂      F              ∂              k                  ν                      =                    r                  ν                            k                  ν                    −      1                      ∑                  j          ∈          N                            (                  r          j                −                  k          j                )            +              r        i            ln      ⁡              (                              k            i                                r            i                          )              =  0  ⇒      r          ν        =      k          ν      and the stationary global point is attained at      r          ν        =      k          ν        ,  ∀  ν  ∈  N