Cultural transmission between individuals can take various forms. One-to-many transmission refers to the case when each individual in a population socially acquires cultural traits from one particular individual who occupies a special social status, such as teacher or powerful authority. Researchers have argued that one-to-many transmission accelerates cultural change compared with one-to-one transmission, which occurs between a pair of individuals. In contrast, a recent mathematical analysis has demonstrated that the rate of cultural change is not necessarily higher with one-to-many transmission under the assumption that cultural variants are selectively neutral. Here we analyze models of one-to-one and one-to-many transmission in a situation where cultural variants are not selectively neutral. Our analysis suggests that one-to-many transmission tends to show higher rate of cultural change than one-to-one transmission when cultural variants are selectively disfavored.