The revenues taken from the masters tournament are no doubt the highest of any tournament, and perhaps by multiples. They are a non profit and the membership doesn't take distributions from any profits, those get utilized to further the course and the worldwide efforts to promote amateur golf. I'm sure they have a large amount of revenue that they keep back for purchasing half of the city of Augusta and paying off Augusta CC for land, which until utilized is invested and they have retained earnings year over year.
I heard through the grapevine that members at ANGC pay no annual or monthly dues and that's because The Master's itself makes so much money that it dwarfs what the club can take in annually from a paying membership. This in of itself gives the ANGC board free reign to pick and choose who they want and don't want as members of their club, thus its exclusivity.