Power indices suggest that adding new members to a voting body may increase the power of an existing member, even if the number of votes of all existing members and the decision rule remain constant. This phenomenon is known as the paradox of new members. This paper uses the leading model of majoritarian bargaining and shows that the paradox is predicted in equilibrium for past EU enlargements. Furthermore, a majority of members would have been in favor of the 1981 enlargement even if members were bargaining over a fixed budget.