The Usenet Newsgroup alt.usage.english has debated this expression several times, most recently in spring 1998. No one there presented definitive evidence, but dictionaries agree that the proper expression is “the carrot or the stick”.
One person on the Web mentions an old “Little Rascals” short in which an animal was tempted to forward motion by a carrot dangling from a stick. I think the image is much older than that, going back to old magazine cartoons (certainly older than the animated cartoons referred to by correspondents on alt.usage.english); but I’ll bet that the cartoon idea stemmed from loose association with the original phrase “the carrot or the stick” rather than the other way around. An odd variant is the claim broadcast on National Public Radio March 21, 1999 that one Zebediah Smith originated this technique of motivating stubborn animals. This is almost certainly an urban legend.
Note that the people who argue for “carrot on a stick” never cite any documentable early use of the supposed “correct” expression. For the record, here’s what the Supplement to the Oxford English Dictionary has to say on the subject: “carrot, sb. Add: 1. a. fig. [With allusion to the proverbial method of tempting a donkey to move by dangling a carrot before it.] An enticement, a promised or expected reward; freq. contrasted with “stick” (=punishment) as the alternative.”
[Skipping references to uses as early as 1895 which refer only to the carrot so don’t clear up the issue.]
“1948 Economist 11 Dec. 957/2 The material shrinking of rewards and lightening of penalties, the whittling away of stick and carrot. [Too bad the Economist’s writer switched the order in the second part of this example, but the distinction is clear.]
“1954 J. A. C. Brown Social Psychol. of Industry i. 15 The tacit implication that . . . most men . . . are . . . solely motivated by fear or greed (a motive now described as “the carrot or the stick”).
“1963 Listener 21 Feb. 321/2 Once Gomulka had thrown away the stick of collectivization, he was compelled to rely on the carrot of a price system favourable to the peasant.”
The debate has been confused from time to time by imagining one stick from which the carrot is dangled and another kept in reserve as a whip; but I imagine that the original image in the minds of those who developed this expression was a donkey or mule laden with cargo rather than being ridden, with its master alternately holding a carrot in front of the animal’s nose (by hand, not on a stick) and threatening it with a switch. Two sticks are too many to make for a neat expression.
For me, the clincher is that no one actually cites the form of the “original expression.” In what imaginable context would it possibly be witty or memorable to say that someone or something had been motivated by a carrot on a stick? Why not an apple on a stick, or a bag of oats? Boring, right? Not something likely to pass into popular usage.
This saying belongs to the same general family as “you can draw more flies with honey than with vinegar.” It is never used except when such contrast is implied.