Direct expense - Atai

…It can also be read as “Atae.” It is also written as “Hane” or “Hane-nao.” It was mainly given to local clans called “Kuni no Miyatsuko” (provincial governors) before the Taika Reforms, and after the reforms, this surname was often used by the gunji (district governors) and their families, who were the descendants of the Kuni no Miyatsuko. There are various theories about the meaning of “Nao,” but it is said to be derived from the fact that it is of the same root as “Atahi” (value), meaning something comparable, and that the Kuni no Miyatsuko, who once had the same power as the Emperor and ran local politics, were called “Atahi.” …

*Some of the terminology explanations that mention "cost direct" are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

…〈あたえ〉ともよむ。費，費直とも表記され，主として大化改新以前の地方豪族である国造（くにのみやつこ）に授けられ，改新以後は，その国造の後裔である郡司とその一族とが，この姓を有している場合が多い。直の語義は諸説あるが，アタヒ(値)と同根で，匹敵するものの意で，かつて天皇と同等の権力を持って地方の政治を行っていた国造を〈あたひ〉と呼んだことに由来するといわれている。…

※「費直」について言及している用語解説の一部を掲載しています。

出典｜株式会社平凡社世界大百科事典 第２版について | 情報