All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
This matter depends on the tradition, and this tradition differs from one society to another. However, in principle it is the husband who should furnish his house with the regular requirements, but if the spouses agree to share the expenses of furnishing the house, half-half, or any other agreement, then there is no harm in this, and in this case each one of them is obliged to fulfil what he/she agreed upon. Similarly, if the husband conditions on the wife to furnish the house and she agrees on this condition, then she is obliged to fulfil this condition, as Muslims should fulfil their conditions. The same thing applies if the tradition is so to an extent that this is known in the society, or that the husband paid her more money in addition to her dowry in return for furnishing the house.
Allaah Knows best.