It costs $10 a/per person [syntax]

[1] It costs [$10 a person].

[2] It costs [$10 per person].

The bracketed elements in [1] and [2] are noun phrases functioning as direct object of "costs". The only difference is that in [1] "a person" is a noun phrase and in [2] "per person" is a preposition phrase. In both cases they are postmodifiers of "$10", and they combine with it to yield the larger bracketed noun phrases.

We know that "$10 a person" and "$10 per person" are constituents since they are portable, cf. "[$10 a person] is what it cost" / "[$10 per person] is what it costs", where the bracketed noun phrases function as subject of the sentence.