Extrakt04

2022-06-22

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:

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

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 $\mathrm{\&}$) 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 $\mathrm{\&}$ 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?

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

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

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

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 $\mathrm{\&}$) 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 $\mathrm{\&}$ 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?

Brendon Fernandez

Beginner2022-06-23Added 14 answers

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

$U=(1-r)b-T\ge 0\Rightarrow r\le 1-\frac{T}{b}$

we can formulate the problem as

$\underset{r}{min}(r-1)\varphi +T,\text{}\text{}\text{s. t.}\text{}\text{}\{\begin{array}{l}r\ge 0\\ r\le 1-\frac{T}{b}\end{array}$

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

$\{T-\varphi ,T-\varphi \frac{T}{b}\}$

$U=(1-r)b-T\ge 0\Rightarrow r\le 1-\frac{T}{b}$

we can formulate the problem as

$\underset{r}{min}(r-1)\varphi +T,\text{}\text{}\text{s. t.}\text{}\text{}\{\begin{array}{l}r\ge 0\\ r\le 1-\frac{T}{b}\end{array}$

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

$\{T-\varphi ,T-\varphi \frac{T}{b}\}$