Hello and thank you for allowing me the opportunity to assist you.
Question: “What happens if I do not pay my rent on a leaseed apartment?”
Answer: If you vacate the apartment and stop paying rent then the landlord can seek a monetary judgment against you. That said, the landlord would need to make a “reasonable effort” to find a new tenant so that his damages are limited.
704.29 Recovery of rent and damages by landlord; mitigation.
(1) Scope of section. If a tenant unjustifiably removes from the premises prior to the effective date for termination of the tenant's tenancy and defaults in payment of rent, or if the tenant is removed for failure to pay rent or any other breach of a lease, the landlord can recover rent and damages except amounts which the landlord could mitigate in accordance with this section, unless the landlord has expressly agreed to accept a surrender of the premises and end the tenant's liability. Except as the context may indicate otherwise, this section applies to the liability of a tenant under a lease, a periodic tenant, or an assignee of either.
In plain English, you are liable for rent for the entire rental term until the landlord is able to rerent it. If he makes a “reasonable effort” to rerent it but cannot, then you will owe rent until your lease expires by its terms. If the landlord does not make a “reasonable effort” to rerent it, then you would be relieved of your liability under the lease (though the burden is on you to prove the effort was lacking).
If the landlord gets a judgment against you, he will be able to go after your assets, including garnishment of your wages, levy on your bank accounts, and possibly a lien on your new house.
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