The “carrot-and-stick” approach was popularized by the article “The Carrot and the Stick” in The Economist of June 29, 1946. In the 19th century, several written sources mentioned that a donkey could be made to do work by carrots (reward) or sticks (punishment).
   
“Carrots and sticks” have been mentioned especially in economics and foreign policy discussions. The reverse of the term—“sticks and carrots” or “stick-and-carrot”—is used much less frequently.
   
   
Wikipedia: Carrot and stick
Carrot and stick (also “carrot or stick”) is an idiom that refers to a policy of offering a combination of rewards and punishment to induce behavior. Some claim that this usage of phrase is erroneous, and that in fact comes from the figure of a carrot on a stick. In this case, the driver would tie a carrot on a string to a long stick and dangle it in front of the donkey, just out of its reach. As the donkey moved forward to get the carrot, it pulled the cart and the driver so that the carrot would always remain out of reach.
 
The earliest citation of this expression recorded by the Supplement to the Oxford English Dictionary is to The Economist magazine in the December 11, 1948, issue.
 
World Wide Words
Q&A: Carrot and stick
(...)
The combination of carrot and stick, with the image of an animal being offered a tasty encouragement at one end while being thumped with a stick at the other, is of the nineteenth century:
 
It was this carrot and stick discipline to which Mr. John Mill was subjected, and which he accepted dutifully as flowing from that perfect wisdom of which up to this time his father had been the representative.
[The Reality of Duty: As Illustrated by the Autobiography of Mr John Stuart Mill, by Lord Blatchford; Contemporary Review, August 1876.]
 
The next example I can find, however, is much more recent, in an article on the problems of post-war Britain in The Economist in July 1946, which said that a healthy economic system required both the carrot (the incentives of reward) and the stick (the threat of poverty if a man was unwilling to work) and that the trouble with the country was that in recent years both the carrot and the stick had been whittled away until there was little of either left. The editor of the Winnipeg Free Press, which reproduced part of the article the following month, was struck by its novelty, calling it “an arresting metaphor”.
 
Had it been lurking in the language for nearly a century, or did The Economist writer reinvent it? What seems more than probable from the written evidence is that the modern metaphor of carrot and stick as encouragement combined with punishment derives from this article or another in the same journal two years later.
   
Merriam-Webster Dictionary
Main Entry: car·rot–and–stick
Pronunciation: \ˌker-ət-ən-ˈstik, ˌka-rət-\
Function: adjective
Etymology: from the traditional alternatives of driving a donkey on by either holding out a carrot or whipping it with a stick
Date: 1876
: characterized by the use of both reward and punishment to induce cooperation “carrot–and–stick foreign policy”
     
(Oxford English Dictionary)
carrot, n.
Usually, the edible root itself. 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.
1948 Economist 11 Dec. 957/2 The material shrinkage of rewards and the lightening of penalties, the whittling away of stick and carrot.
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.
 
stick-and-carrot adj. phr. [see CARROT n. 2a], characterized by both the threat of punishment and the offer of reward;
1963 Times 4 Mar. 11/7 President Ayub has..given himself *stick-and-carrot powers to deal with the ‘Ebdonians’.
1977 ‘J. LE CARRÉ’ Hon. Schoolboy vi. 136 It’s a stick and carrot job. If you don’t play, the comic will blow the whistle on you… That’s the bad news… The good news is five hundred US into your hot little hand.
     
29 June 1946, New York (NY) Times, “Output in Britain Shows Steady Rise,” pg. 28:
Not everyone would agree with the moral The Economist draws, which is that incentives have been dulled and goads to greater effort removed by stifling of competition at all levels of the economic system. “The human donkey requires either a carrot in front or a stick behind to goad it into activity,” is the way The Economist puts it, adding that the tendency has been, even before the war, to whittle away at both carrot and stick.
   
28 July 1946, New York (NY) Times, “New Terms Seen Needed For Business-Labor Parley” by Russell Porter, pg. 2:
In this connection, American business men have been reading with much interest an article that appeared recently in The Economist of London under the heading of “The Carrot and the Stick.” This leading British journal held that the decline in British enterprise, competition and production as compared to the United States and Russia was due to neglect of the age-old lesson that “the human donkey requires either a carrot in front or a stick behind to goad it into activity.”
   
18 January 1948, New York (NY) Times, “Stalin—History’s Top Capitalist?” by Geoffrey Crowther (editor of The Economist of London), pg. SM45:
The one is a carrot-and-stick economy; the other is a featherbed economy.
 
OCLC WorldCat record
Carrot and stick in rural China
Author: Wolf Isaac Ladejinsky; Council on Foreign Relations.
Publisher: New York : Council on Foreign Relations, ©1957.
Edition/Format: Book : English
 
OCLC WorldCat record
Comment On . . . “Continuing Education: Carrot or Stick?”
Author: Barbara Conroy
Edition/Format: Article : No language available
Publication: Catholic Library World, 49, 8, 343-5, Mar 78
Database: ERIC The ERIC database is an initiative of the U.S. Department of Education.
Other Databases: WorldCat
Summary: Discussion of the pros and cons of continuing education—mandatory and otherwise—for librarians.
 
OCLC WorldCat record
Carrots and sticks : pay, supervision and turnover
Author: Jonathan S Leonard; National Bureau of Economic Research.
Publisher: Cambridge (1050 Massachusetts Avenue, Cambridge, Mass. 02138) : National Bureau of Economic Research, 1987.
Series: NBER working paper series ; working paper, no. 2176; NBER working paper series, working paper no. 2176
Edition/Format: Book : English
   
OCLC WorldCat record
Carrots and sticks : the TRC and the South African amnesty process
Author: Jeremy Sarkin-Hughes
Publisher: Antwerp : Intersentia, ©2004.
Edition/Format: Book : English
 
OCLC WorldCat record
Carrots and sticks principles of animal training
Author: Paul McGreevy; Robert A Boakes
Publisher: Cambridge Univ. Press 2006
Edition/Format: Book : English