The most common and most likely answer to this crossword clue is the 7 letter word SCRIMPS
I've seen this crossword clue in The Times
S
C
R
I
M
P
S
Buddy explains!

The straight part
We know that "Lives frugally" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lives frugally = scrimps.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SCRIMPS mean?
SCRIMPS noun- A pinching miser; a niggard.
- To make too small or short.
- To limit or straiten; to put on short allowance.
- To be frugal.