How do AIs gift gold city-states?

I see that Pedro of Brazil has easily 30K gold. We have already had two votes for world leader... why he hasn't just allied every city-state with his gold, I have no idea.

Is there some formula for how much gold they will give to city-states to keep it about constant regardless of how much money they have?

If AIs actually spent their gold, you'd never win on higher difficulties.
They've just been given some "generous" gold generating algorhythm to make sure they don't cripple themselves(.)