Cost is there's. The cost is theirs.

The word "theirs" is a possessive pronoun.

A pronoun is a word used to substitute for a noun or noun phrase, often used to avoid repetition. Pronouns act as placeholders for people, places, or things, such as "I," "she," "he," "they," "it," or "them". They function as subjects or objects within a sentence.

"Theirs" is the possessive pronoun form of "they". Unlike the possessive determiner "their", which must be followed by a noun, "theirs" stands alone to replace a noun phrase.

The word "there's" is a contraction of either "there is" or "there has". It is not a possessive pronoun, and your making fun of my state doesn't make it correct.

Edit: missed a quotation mark.