Takes in better size of paper (8)
I believe the answer is:
foolscap
'size of paper' is the definition.
(I've seen this before)
'takes in better' is the wordplay.
'takes in' becomes 'fools' ('fool' can be a synonym of 'take in').
'better' becomes 'cap' (both can mean to beat somebody else's attempt).
'fools'+'cap'='FOOLSCAP'
(Other definitions for foolscap that I've seen before include "something near A4" , "About 330 x 200 (or 400) mm size of paper" , "sheet" , "Cop's foal (anag)" , "type of paper" .)