"Bought" the League????? So, are you saying Man Utd and Liverpool won 18 apiece just through homegrown talent and hard graft? So noble of them not to "buy" the League, eh?
No, not really. If you know your football history Man Utd, Arsenal and Liverpool have always been built on success, they're more famous for what they've won than who they bought.
With success brings financial reward. It just so happens that in the modern game where winnings have increased those 3 have won silverware over the years that have added to their wealth so they're able to afford excessive transfer fees/wages.
Liverpool, Man Utd, and Arsenal are where they are today from foundations built from decades of being successful. Not adding to the fact that during the modern game football clubs have been branded and are run and mass global businesses where the more success you have, the more revenue comes in.
The reason why Chelsea are where they are now is mainly down to how they spent a [Poor language removed] load and reaped a reward. They are labelled a "£ig Clu£" in the Sky Sports monopoly "big 4" due the main factors of Champions League exposure (its all about Champions League and nothing else nowdays), and winning the odd few trophies in the modern game.
Liverpool, Arsenal, Man Utd are the most successful clubs in English football and you could say their financial success has been earned.
Chelsea's, and what is going to be Citeh's eventually, success will always be seen as being bought rather than earned.