Learn English – Difference between “infirmary” and “sick-bay”

It costs $50 needs an adverbial modifier if any is considered appropriate.

It usually costs $50.

It costs $50 Thursdays.

It costs $50 a bottle.

Adverbs, and the rare adverbial objective (noun used adverbially), fit the bill.

More commonly, to-infinitive clauses and especially prepositional phrases will be used:

It costs $50 to get there.

It costs $50 on the tram.

It costs $50 for a bottle.

It costs $50 per person.

It costs $50 for each person.

But

*/?It costs $50 each person.

would probably not usually be considered acceptable; it would be an adverbial objective usage, with the preposition 'for' dropped from the prepositional phrase, and this often does not give an acceptable-sounding variant. (This particular example may sound more acceptable in the States than in the UK. 'It costs $50 a person' would however be quite acceptable.)

abate is related to bate etymologically.

Bate

Origin

late Middle English: from Old French batre 'to beat' (see also batter1).

Abate

Origin

Middle English (in the legal sense): from Old French abatre 'to fell', from a- (from Latin ad 'to, at') + batre 'to beat' (from Latin battere, battuere 'to beat').

The word picture for bate is a hawk beating its wings against the air to fly. This word picture extends metaphorically to its noun definition: "foul mood".

Bate

VERB

Falconry (Of a hawk) beat the wings in agitation and flutter off the perch:

the hawks bated and immediately the breeze got in their feathers

NOUN

British informal (dated) An angry mood:

he got into a stinking bate

"Bate" obtains the meaning of "reducing the intensity of a force" as Shakespeare used it in Merchant of Venice. In each of the uses below, bate can reflect the meaning "subdue", although in some of them it could also reflect the original word picture of "beat" or "flutter".

"With bated breath and whispering humbleness,"

"His tedious measures with the unbated fire"

"These griefs and losses have so bated me,"

"And bid the main flood bate his usual height;"

By adding the prefix "a" (a phonetic variant of "ad") to the base of bate we get the picture of beating "at" something or someone.

Abate

VERB

Make (something) less intense:

nothing abated his crusading zeal

Again, abate obtains the meaning of "reducing the intensity of a harmful force" from the outcome of beating that harmful force back. In fact, "subdue" is the primary connotation of abate.

This is the clear meaning in Shakespeare's Merchant of Venice, when BASSANIO says,

"You would abate the strength of your displeasure."

So in the end, the common etymology creates common meaning across the board, although bate is rarely used now, except for the phrase "bated breath."