We know that the Gemara darshens the laws of Purim from Yom Kippur. If so why are we not obligated to fast on Purim?
יליף פורים פורים מיום הכפורים
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We do not decree what the people won't follow, which is why we do not have two days of Yom Kippur in the diaspora even though we do for Rosh Hashana a few days earlier and Sukkot a few days later. It would be too dangerous to fast for 49 hours and the people wouldn't do it.
Fasting, we know, is something people can do, so you would think it would be required on Purim which is like (ki) Yom Kippur(im). However, as anybody who learned this lesson the hard way knows, drinking to the point of drunkenness on an empty stomach is a terrible idea. You thought you were afflicting yourself on Yom Kippur? That would be nothing compared to the morning after Purim.
If the rabbis required the fast then either people would ignore the fast or they would not drink. Drinking is more important, so we opt for that.
The question is answered in reverse by the achrei achronim. They point out that the gezeirah shava could work in the other direction -- why don't we feast on Y"K? The answer is in the name of the day, it is only "k"purim, like Purim. In what way is it different? Vnahafoch hu, in that everything is different!
There's another g'zeira shava comparing "בתשעה לחדש בערב" (Vayikra 23:32, "on the ninth of the month in the evening") to "בערב היא באה" (Esther 2:14, "in the evening she would come") which compares Purim to Erev Yom Kippur. Just as eating on Erev Yom Kippur is a mitzva, so too is eating on Purim a mitzva.
Furthermore, just as eating on Erev Yom Kippur is equivalent to fasting on both Yom Kippur and Erev Yom Kippur (B'rachos 8b)1, so is Purim equivalent to fasting on Yom Kippur, Erev Yom Kippur, and Purim itself. Thus, by eating and feasting on Purim, we commemorate the three day fast of the Jews in the time of Esther (Esther 4:16).
1 כל האוכל ושותה בתשיעי מעלה עליו הכתוב כאילו מתענה תשיעי ועשירי