cut price vs cut rate

cut price and cut rate both are nouns.

Word	Noun	Adjective	Verb	Adverb
cut price	Yes	No	No	No
cut rate	Yes	No	No	No

As nouns, cut price and cut rate are synonyms defined as:

Other synonyms of cut price include bargain rate, cheapness.

cut price (noun)	cut rate (noun)
a price below the standard price	a price below the standard price