What is it called when a company allows another to use its brand for a fee or percentage of profits?

The scenario described is accurately characterized by a licensing agreement. In this arrangement, a company, known as the licensor, permits another business, the licensee, to use its brand, trademarks, or intellectual property in exchange for a fee or a percentage of the profits generated from that usage. This practice is common across various industries, such as fashion and entertainment, where established brands can extend their reach and profitability through collaborations with other firms.

The other terms do not accurately describe this arrangement: brand extension refers to launching a new product using an existing brand name, which does not involve another company using the brand. Merchandising generally pertains to the promotion and sell of goods associated with a brand but is not about allowing another party to use the brand itself. Matching funds typically relate to financial contributions in some collaborative funding context but do not involve brand usage. Thus, the concept of a licensing agreement thoroughly encompasses the use of a brand by another entity for a specified financial return.