White is better after 7.b3.
7...Bxf3 8.Qxf3 cxb3?? 9.Qxa8.
7...cxb3 8.Bxb5+ seems pretty bad for black.
Besides, who cares about doubled pawns when they mean a stronger center, a half open g file and the bishop pair. Don't sweat the really minor stuff...
Consider the following opening moves:
1. d4 d5 2. c4 c6 3. Nf3 dxc4 4. e3 b5 5. a4 Bb7 6.axb5 cxb5 7. b3
From this point, it appears that white can regain the missing pawn. But white's queen is guarding both b3 and f3. So black can capture the knight on f3 with the bishop on b7, and then if white wants to keep the queen guarding b3, the pawn on g2 must recapture the bishop, leaving doubled f pawns.
A few possible continuations:
7. ...Bxf3 8. Qxf3 cxb3
7. ...Bxf3 8. gxf3 cxb3 9. Qxb3
7. ...cxb3 8. Qxb3 Bxf3 9. gxf3
What is white's best plan here? Should white have played differently around moves 5-7?