Institutions could create huge carrots for members with one child families. For example. Annual rewards for delaying the birth of the first child (e.g. providing an annual payment on a mortgage forgiven if the couple eventually has no more than one child,) no additional rewards after that birth, and if a second child arrives, a take back of rewards (the institutional contribution with interest) is added to the remaining principle of the mortgage.)
Alternatively, one child per family behaviors could be implemented by an anti fertility vaccination at birth with antidotes issued by the institution when the one child is desired or a child is lost.