I, the King, wish to share the Kingdom's wealth with the People. If the Kingdom's population keeps growing, how long before they collectively are richer than the Royal Family?
An estimate would say 'Probably 100 years'. An exact formula says never.
Stick with me here.
Let's say this is a verrrryy nice king. What goes around comes around- he shares his wealth with his people. For every sum of cash(for ease, we're going to call this amount $2C$) that comes into the kingdom, he takes $\frac{1}{2}C$ for himself. He then decides to group the kingdom by population and closeness to himself: every group has one more person than the one above it. He's on the top by himself, his Queen and heir are one below, his three knights below that, four nobles, etc., all the way down to his thousand peasant farmers. Everyone in each tier gets the cash divided up like this:
So, the fraction of C you get is one over two to the power of the number of people in your tier.
The rich stay rich and the poor stay poor, but everyone's grateful to the king because he gives each of them enough to sustain their own lifestyles, but critically not enough for anyone to move up or down. When you're born, you're shoved into the 'lowest bottom pile'. 130
If someone dies higher up and you have a right to that space, it's yours and you get the cash. For this reason, this system can scale towards infinity.
His Royal Highness is also very intelligent. He knows money talks, and is aware of the meltdown that could occur if he suddenly became very unpopular and the people had more wealth than he did. Or he's an egomaniac. Either way, he wants his family to be #therichest.
With this system, what's the maximum population size you can have before the people have more wealth than the king?
An estimate would say "Well, given the current population growth, I'd guess 20 years your Highness?". However, a Medieval mathematician (<PERSON>) proved that this sequence converges to 2. Hence why I used $2C$ at the start. Read that proof. It's truly brilliant.
Finally, some simple math:
$Wealth_{King} = \frac{1}{2}$
$Wealth_{Queen} + Wealth_{Heir} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$
$Wealth_{Royal\,Family} = 1$
$Wealth_{Infinite\,population} = 2-1 = 1$
$Wealth_{Infinite\,population} = Wealth_{Royal\,Family}$
So, as long as the population is less than infinite, the King can share the Kingdom's wealth with his people and his family will always be richer than everyone else put together. Neat, right?
I'm aware this is such a botched explaination of pretty much everything, so let me know if something's unclear and I'll try to clean it up in an edit.
Hope it helps! 587