The phrase carrot and stick denotes a promised reward coupled with a threatened punishment as a method of persuasion or coercion. (The French phrase la carotte et le bâton is a loan translation from English.)

This phrase alludes to the method of tempting a donkey to move forward by dangling a carrot before it, and beating it with a stick if it refuses.

It is recent. The earliest figurative allusion to dangled carrots as an inducement that I have found is from the Coventry Standard (Coventry, Warwickshire) of 23^rd March 1867:

Mr. Gladstone […] again resumes the old tactics of the Whig-radical cabal, whose grand object has ever been to make use of the Reform question, not to settle it, but to keep it ever dangling before the public eye with the pretence that Whigs and Radicals only can settle it, though long experience has proved that they never mean to do so. The leader of the opposition on Monday night would be well represented by the old tableau of an artful jockey riding a donkey and urging it on by the persuasive means of a bunch of carrots before its nose at the end of a long stick, which carrots the donkey is permitted sometimes to smell but never to taste or touch. Just so it is with the party of which Mr. Gladstone assumes to be the leader. They ride the popular hobby; show the Reform carrot from time to time; but never did, and never mean to let their deluded victims taste the food.

The earliest mention of the carrot-and-stick method that I have found is in The Yorkshire Post (Leeds, Yorkshire) of 18^th November 1920, which reported that at the House of Commons, during the discussion of the Agriculture Bill,

Mr. Acland (Lab., Camborne) said what was being dangled before the farmer were two carrots and a stick. The two carrots were guaranteed prices and further security of tenure, and the stick was Government control. This clause was necessary to promote security of tenure.

On 21^st July 1938, The Sydney Morning Herald (Sydney, New South Wales, Australia) published Nazification of Austria. The Pangs of Indigestion, an article about the consequences of the Anschluss (the annexation of Austria by Germany), by the British statesman Winston Churchill (1874-1965); about “the mass of the Austrian wage-earners and peasants”, Churchill wrote:

STICK AND CARROT.

Meanwhile, their masters proceed with high-handed authority to mete out rewards and tasks at their good pleasure. Two opposite kinds of immigration into Germany are in progress. Numerous trains carry droves of many scores of thousands of Austrian wage-earners to labour under stern rule in the old Reich. At a time when all Germans are under industrial or military conscription, the life of these deportees must tend in many ways to resemble the older forms of servitude. Far from home, under hard taskmasters, they labour to build up that martial strength, the final purposes of which Europe has yet to learn.
Side by side with these unfortunates run the excursion trains of those who are chosen by favour for the “Strength through Joy” organisation. These elect are permitted to gape in awe-struck admiration at the spectacle of Nazi power. Thus, by every device from the stick to the carrot, the emaciated Austrian donkey is made to pull the Nazi barrow up the ever steepening hill.