I have the following problem. Given this function
I would like to find the maximum w.r.t. given this constraint:
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 is a number between 0 and 1, is some probability, and are all positive constants. If it is useful for the resolution of the problem, we can also assume that is between 0 and 1. An important assumption (namely, assumption ) is that . 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. does not yield any expression with .
Now I'm following a more intuitive approach. I start by assuming that is the constraint, then I can find an expression for from the constraint and I substitute it in the target function. After easy steps, I get this
At this point, I can use assumption to conclude that the first above will be negative, but I'm not able to conclude whether it will be lower, equal or higher to/than the second 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?