Does a person's statement that the product is cheaper elsewhere constitute cutting off the livelihood of the seller of the advertised product?

There is no harm in guiding someone to a seller who offers a product at a cheaper price, unless it is confirmed that the advertiser has already sold to someone or has been bargained with by someone, and they have agreed upon a specific price. In that case, guiding others is forbidden, due to the Prophet's (peace and blessings be upon him) saying: "Let no one sell over the sale of his brother," and "Let no one bid over the bid of his brother."

Considered within the meaning of this prohibition is guiding to the availability of a product at a cheaper price elsewhere if that displeases the seller to whom the buyer had initially committed, due to the strife and dispute that such an action might cause.

However, if the buyer has not committed to the advertiser's product, there is no harm in guiding to another seller at a cheaper price, because the default rule is permissibility, and this is not considered cutting off someone's livelihood, but rather an act of kindness and benefit to people.