Well, I don't know what 'the poor's' problems are, probably varies, and I've heard that the obesity epidemic isn't confined to poor folk.
I can tell you that a healthy diet of whole foods is cheaper, though. It's not a difficult thing to prove, really. It may involve eating less calories, though. Since obesity, not hunger, is the problem, studies that focus on cost per calorie are, of course, bullshit. (LARD!) Wrong focus. Cost per healthy, appropriate diet should be the measure...of course.