The Zohar explains that the thirteen parts of Arich Anpin's beard correspond to the Thirteen Attributes of Mercy.... The Biblical source for the number of parts of Arich Anpin's beard appears in the book of Exodus. After the sin of the golden calf, G-d proclaimed to Moses the Thirteen Attributes of His Mercy: "G-d (1), merciful(2), and gracious(3), slow (4) to anger (5) and abundant in love (6) and truth (7). Keeping mercy (8) to the thousandth generation (9) forgiving sin (10), rebellion (11), and error (12) and cleansing (13)" [Exodus 34:6-7]
Inner Space by Rabbi Aryeh Kaplan pg. 101-102
The Attributes of Mercy are mentioned again in the Torah in the book of Numbers. After the incident of the spies who return to the wilderness with a bad report about the land of Israel, G-d is ready to annihilate the Jewish people because of their lack of faith even after repeated miracles. Moses prays to G-d in order to arouse His mercies and he quotes the words which G-d once declared listing the Thirteen Attributes. This time, however, Moses only metions nine of these Attributes: "Slow (1) to anger (2), abundant (3) in love (4), forgiving sin (5) and rebellion (6); cleansing those who repent (7), and not cleansing those who do not repent (8) but keeping in mind the sins of the fathers for their children, their grandchildren and their great grandchildren (9)." [Numbers 14:18]. The verses in the book of Exodus listing the Thirteen Attributes represents Arich Anpin, and the verse in the book of Numbers listing Nine Attributes represents Zer Anpin which is "smaller," i.e. programmed to transmit G-d's mercies only in proportion to humanity's ability to receive and emulate them. In addition, the hair of both Atik Yomin and Arikh Anpin is depicted as the white hair of a grandfather, while the hair of Zer Anpin is depicted as the black hair of a young man.
Thus the "size" and "color" of their respective beards indicate that at the level of Arikh Anpin, G-d's love and mercy are far greater than that of Zer Anpin. This is readily understandable when we remember the Partzufim were created as intermediaries which would allow man to receive G-d's ultimate good through his own efforts. On the level of Keter, G-d's mercy is unmitigated and pure. Like a grandfather, His desire is to bestow the greatest good irrespective of how deserving His "grandchildren" are. For their own good, however, He constricts His mercy through the smaller Partzuf of Zer Anpin and thereby, like a parent, makes His giving dependent on His children's merit. These two modes of giving do not contradict each other. They both emanate from G-d's love for all His creatures.