Dividing an amount proportionately among unequal amounts

Question

I have \$100000. I want to divide the amount proportionately to 300 employees. I want the money divided inversely from the top down. So, the top earner get \$0 and the lowest paid employee(s) get the most. If there are 10 employees at the lowest pay rate, they all get the same amount. Is there a formula or program I can create to create this?

Answer 1

According to conversation in the comment session, employees are ranked by the salary group and the distribution of the \$100,000 bonus should be distributed inversely. (If it is proportionally inversely, then top earner will get slightly more than \$0.)

So, let $a_i$ for $i=1,2,\dots,300$ be the salary of each employee, and $A=\max a_i$ be the salary of the top earner. One posibility is to have an inverse linear distribution using the variable $x_i=A-a_i$, and have each bonus $b_i$ proportional to $x_i$.

In this case we know that $b_i=\alpha x_i$, and that $$\sum_{i=1}^{300}b_i=100000.$$ Replacing $b_i$: \begin{align} \sum_{i=1}^{300}\alpha x_i&=100000 \\ \alpha\sum_{i=1}^{300} x_i&=100000 \\ \alpha\sum_{i=1}^{300} A-a_i&=100000 \\ \alpha\left(\sum_{i=1}^{300} A-\sum_{i=1}^{300}a_i\right)&=100000 \\ \alpha\left(300A-\sum_{i=1}^{300}a_i\right)&=100000 \end{align} Where $\sum a_i$ is sum of all salaries: the total wage package $W$. So, from hear we have: \begin{align} \alpha(300A-W)&=100000 \\ \alpha&=\frac{100000}{300A-W} \end{align} So the bonification of each employee will be: \begin{align} b_i &= \alpha x_i\\ b_i &= \frac{100000}{300A-W} x_i\\ b_i &= \frac{100000}{300A-W}(A-a_i)\\ \end{align}

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If you prefer an inverse proportional distribution, the distribution variable would be $x_i=\frac1{a_i}$. However in this distribution the top earner will have a part of the bonus (although a smaller part than the other employees).

A fix to this distribution is to make $x_i=\frac1{a_i}-\frac1A$.

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I have treated each salary independently, however if you have only a few possible salaries you can adopt a different strategy:

Let $A_1,A_2,\dots,A_n$ with $n\ll 300$ be the possible salaries and let's assume that $A_1>A_2>\dots>A_n$ (salaries are ordered).

Let $k_i$ be the number of employees that earn $A_i$.

So the total number of employees are $\sum_{i=1}^{n}k_i=300$ and the total wage package is $\sum_{i=1}^{n}k_iA_i=W$.

We define the distribution variable $X_i$ with some inverse criteria, v.g. $X_i = A_1-A_i$ (for the lineal distribution), $X_i=\frac1{A_i}$ (for the inverse proportional distribution), $X_i=\frac1{A_i}-\frac1{A_1}$ (for the shifted inverse proportional distribution), $X_i=i-1$ for a ranked distribution, etc.

Let $B_i$ be the bonus for each salary rank, and have it proportional to the distribution variable $B_i=\alpha X_i$

Now, you have: $$\sum_{i=1}^{n}k_iB_i=\alpha\sum_{i=1}^{n}k_iX_i=100000,$$ so $$B_i=\alpha X_i=X_i\frac{10000}{\sum_{i=1}^{n}k_iX_i}.$$

The result of $\sum_{i=1}^{n}k_iX_i$ (or $\sum_{i=1}^{200}x_i$) depends on your criteria for choosing $X_i$.