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I have the following problem. Given this function

$E[\pi] = (1-r)[\alpha b- (1-p)C-K]+T $

I would like to find the maximum w.r.t. $r$ given this constraint:

$U = (1-r)b-T \geq 0$.

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, $\alpha,b,C,K$ and $T$ are all positive constants. If it is useful for the resolution of the problem, we can also assume that $\alpha$ is between $0$ and $1$. An important assumption (namely, assumption $\&$) is that $(1-p)C+K>\alpha b$. 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

$T \frac{\alpha b - (1-p)C-K}{b} + T$.

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?

For the admin: This is a new post since the old one was blocked because it lacked many information. I hope that in its current form it is acceptable to start a discussion. Let me know if I need to add anything else. Thank you.

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Hint.

Calling $\phi = (1-p)C+K-\alpha b\gt 0$ and considering

$$ U = (1-r)b-T \geq 0\Rightarrow r \le 1-\frac Tb $$

we can formulate the problem as

$$ \min_r (r-1)\phi+T,\ \ \text{s. t.}\ \ \cases{r\ge 0\\ r\le1-\frac Tb} $$

now as $(r-1)\phi+T$ is linear, the solution is one of the set

$$ \left\{T-\phi,T-\phi\frac Tb\right\} $$

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  • $\begingroup$ Using your hint here's my attempt (I stack with the max problem). $\max_r -(1-r) \phi+T$, in order to maximize, I have to minimize $(1-r)$. Given the constraint, $r=1-T/b$ at least. If $T/b \leq 1$, then $r=1-T/b$ is the solution. If $T/b>1$, then the problem has no solution because I cannot respect the constraint given that $r \in [0,1]$. Am I right? Say that $T/b \leq 1$, can I say something on the sign of $E[\pi]$ evaluated on $r=1-T/b$ with these data? $\endgroup$ Jun 10 '21 at 15:02

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