How to vary the thickness of a Bezier curve?

I have made a bezier tube and I want some parts of it to be large and others to be small.

When I try to set the depth for one of its vertices, it just changes the depth for the whole object. Is there a way to change the depth for separate parts of the bezier curve?

• @Zachooz It sounds like the shortcut wasn't working correctly (what you described sounds like the normal behavior scaling with S). Does it work if you press Shink Fatten in 3D view > Tool shelf (T) > Tool tab > Transform? – gandalf3 Apr 18 '14 at 20:37