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"Irrational exuberance" is a phrase used by the then-Federal Reserve Board Chairman, Alan Greenspan, in a speech given at the American Enterprise Institute during the Dot-com bubble of the 1990s. The phrase was interpreted as a warning that the market might be somewhat overvalued.
Greenspan's comment was made on December 5, 1996 (emphasis added in excerpt):
[...] Clearly, sustained low inflation implies less uncertainty about the future, and lower risk premiums imply higher prices of stocks and other earning assets. We can see that in the inverse relationship exhibited by price/earnings ratios and the rate of inflation in the past. But how do we know when irrational exuberance has unduly escalated asset values, which then become subject to unexpected and prolonged contractions as they have in Japan over the past decade? [...]
The prescience of the short comment within a rather dry and complex speech would not normally have been so memorable; however, it was followed by immediate slumps in stock markets worldwide, particularly the Nasdaq Composite, provoking a strong reaction in financial circles and making its way into colloquial speech. Greenspan's comment was well remembered, although few heeded the "warning." The phrase was picked up by Yale professor Robert Shiller (or was it the other way around?), who used it as the title of his book, Irrational Exuberance, in 2000.
By the mid-to-late 2000s the losses were recouped and eclipsed by a combination of events, including the 2000s commodities boom and the United States housing bubble. However, the late-2000s recession of 2007 onwards wiped out these gains. The second market slump brought the phrase back into the public eye, where it was much used in hindsight, to characterize the excesses of the bygone era. In 2006, upon Greenspan's retirement from the Federal Reserve Board, The Daily Show with Jon Stewart held a full-length farewell show in his honor, named "An Irrationally Exuberant Tribute to Alan Greenspan."
It had become a catchphrase of the boom to such an extent that, during the economic recession that followed the stock market collapse of 2000, bumper stickers reading "I want to be irrationally exuberant again" were sighted in Silicon Valley and elsewhere.
See also 
- Did Greenspan Steal the Phrase Irrational Exuberance?, http://www.ritholtz.com/blog/2013/01/did-greenspan-steal-the-phrase-irrational-exuberance/