Is it permissible for an employee to keep the remaining amount from the housing allowance if he finds housing for less than the allocated value (2900), knowing that the responsible authority mandates spending the full amount on rent?

Contracts between two parties are binding on everyone as long as their terms are within the permissible bounds of Islamic law; as Allah Almighty says: (O you who have believed, fulfill [your] contracts), and as the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions, except for a condition that makes a haram thing halal, or a halal thing haram."

So, if the institution determined the amount as a reserve to provide housing, then the employee must return any excess over the rent to the institution. However, if the agreement was to pay the amount as compensation for housing, and no increase or decrease is demanded, then the amount belongs to the employee, and any surplus from it is his.