I have the following problem. Given this function E [ π ] = ( 1...



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I have the following problem. Given this function
E [ π ] = ( 1 r ) [ α b ( 1 p ) C K ] + T
I would like to find the maximum w.r.t. r given this constraint:
U = ( 1 r ) b T 0
It is an economic problem that I am formalizing, but this is not relevant to its solution, I am only interested in the mathematical resolution of the problem. I provide some background. Our variable r is a number between 0 and 1, p is some probability, α , b , C , K and T are all positive constants. If it is useful for the resolution of the problem, we can also assume that α is between 0 and 1. An important assumption (namely, assumption &) is that ( 1 p ) C + K > α b. Readers used to economics may recognize that the objective function is a sort of expected profit and the constraint is a sort of expected utility. I tried with Kuhn-Tucker, but with miserable results since deriving w.r.t. r does not yield any expression with r.

Now I'm following a more intuitive approach. I start by assuming that U = 0 is the constraint, then I can find an expression for r from the constraint and I substitute it in the target function. After easy steps, I get this
T α b ( 1 p ) C K b + T
At this point, I can use assumption & to conclude that the first T above will be negative, but I'm not able to conclude whether it will be lower, equal or higher to/than the second T because it is multiplied by some constant.

At this point I'm stuck. I do not know if my approach can work. Could you please suggest a nicer way to proceed? I am on the right track or it's a dead end?

Answer & Explanation

Brendon Fernandez

Brendon Fernandez

Beginner2022-06-23Added 14 answers

Calling ϕ = ( 1 p ) C + K α b > 0 and considering
U = ( 1 r ) b T 0 r 1 T b
we can formulate the problem as
min r ( r 1 ) ϕ + T ,     s. t.     { r 0 r 1 T b
now as ( r 1 ) ϕ + T is linear, the solution is one of the set
{ T ϕ , T ϕ T b }

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