If I am right, should I explain to him that scarcity means not enough so the answer shortage is more appropriated than bare.
As well as meaning, it's about parts of speech. Scarcity, Shortage, Abundance, Plenty can all be nouns ("there is a scarcity of affordable housing"; "there is a shortage of affordable housing"; "For whosoever hath, to him shall be given, and he shall have more abundance: but whosoever hath not, from him shall be taken away even that he hath" (Matthew 13:12), "So shall thy barns be filled with plenty, and thy presses shall burst out with new wine." (Proverbs 3:10) - these are actually quite archaic uses). Shortage is occasionally an adjective ("physics is a shortage subject") but the others rarely, if ever. None of them are verbs in common usage.
By contrast, Bare is rarely (never?) a noun; it is almost always an adjective ("he had bare feet") or a verb ("It's sunny, so I am going to bare my legs today").
So there are plenty of sentences in which you can replace "scarcity" with "shortage" and still have it both grammatically correct and meaning roughly the same, and likewise "abundance" and "plenty". And you can replace any of them with any other of them and still have something grammatically correct, although the meaning will be inverted. You can't replace any of the first four with bare and have it be grammatically correct. Just reading it aloud should tell you that, even if you can't put names to the reasons. "There is a bare of housing".