Let N = { 1 , … , n }. For a given ( r...

gaiaecologicaq2

gaiaecologicaq2

Answered

2022-07-07

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.

Answer & Explanation

Marisol Morton

Marisol Morton

Expert

2022-07-08Added 13 answers

ln is a strictly increasing monotonic function then with x k > 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 ) > 0
the 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

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