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The grammatical subject is "relationship."

"According to the law of demand" is a sentence modifier and should have a comma after it.

The main verb is "exists," and its grammatical subject is "relationship." "Negative" is an adjective that modifies "relationship."

The rest of the sentence, from "between" to the end, is a prepositional phrase with two objects--"price" and "quantity"--and several modifiers. The whole prepositional phrase is adverbial and modifies "exists."

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What is the difference between a demand curve and a demand schedule, and how do they each represent the relationship between price and quantity demanded in economics?

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And quantity demanded is shown on?

And quantity demanded is shown on?