Actuall, not so. See what "white privilege" is REALLY ABOUT, in the words of the cultural theorist who came up with the phrase in 1989 Peggy McIntosh.
There is no question that some white people suffer from classism, but class is something you can get out of, whereas race is not.
It is DEFINITELY not just about economics. It's about living black in a white world that you did not create and over which you've exercised little to no control for centuries. Where you are ALWAYS a 2nd class citizen--a class from which you cannot escape because no matter how rich you are, you still have do deal with a whole culture (with a minority of exceptions) that, whether they admit it or not, will always define you as "less than" no matter what you achieve.