I know this is a trivial thing to point out in the context of such a discussion but the expression you want is "carrot and stick". A cane is a kind of stick that you can also hit with, the verb "to cane" means to hit someone with a stick.
Thanks for the correction. English is not my first language, in German the equivalent is "Zuckerbrot und Peitsche" (sugar-bread and whip) I must have somehow made the mental leap from "sugar" via "sugarcane" to "cane" and completely forgot about the carrot.
