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You have a right to a livable, or habitable, rental. That means your landlord must ensure that conditions are safe. A broken window is not safe for anyone, and could lead to more serious damage and crime.
Generally speaking, landlords are responsible for fixing a broken window belonging to a rental property. But of course there are exceptions. If you broke a window and the landlord can prove it, then the landlord may make the repair but you will have to pay. Probably, the money will be deducted from your security deposit.
Blowing in the Wind
But what about the opposite situation? Assuming you did not break the window, but you still want it fixed fast, what do you do? Calling in a repair man quickly is probably in the landlord's best interest whoever broke the window.
Upkeep of the property maintains its value. Yet you are the one living with the wind blowing through your studio, so you probably feel more urgency about the matter than the landlord does. Do you have to wait?
Deduct Repairs From Your Rent
The answer is maybe. You have to try exchanging with your landlord first. Then if you do find yourself paying out of your own pocket for a broken window, deduct that payment from your rent. Be clear and explicit about what happened, what you paid for, and why you are reducing your rent payment.
You must include a receipt for the repair along with your reduced rent. It is wise to keep a copy of the receipt for your own records, and write a note explaining the calculations you made.
If your decision to call in a repair man was made after exchanges with the landlord, mention those in your note. To the extent you recall, mention dates and times, whether you left a message, or who you spoke to.
Basically, you are creating a record of events that led to your decision to make the repair yourself. A landlord with multiple properties may have preferred repair people - to the extent that you make an independent decision to repair, be prepared to explain.
Get It Done
Regardless of who broke the window and who will repair it, you have the right to working doors and windows in your home and the landlord has an interest in maintaining the property. Talk and together you can work it out so that window gets fixed.