It bothers me that the metaphor of the carrot and the stick has become perverted. To me it describes the holding of a stick with a carrot on a string at the end of it and dangling it in front of a donkey to motivate it to move. Any other interpretation does not hold up under simple analysis.
Current thinking regards the carrot as a reward and the stick as punishment. This implies that the driver had two sticks—one to dangle the carrot and one to strike the donkey. Or, he could have turned the single stick around as appropriate, either to punish the donkey or motivate it.
Does this make sense to you?