As a general rule, unless your lease or rental agreement states otherwise, your tenant may not withhold the cost of repairs from rent. However, if there is a violation that may endanger or materially impair the health, safety,or well-being of a tenant, the tenant may make the repairs himself and withhold the cost of the repairs from rent up to an amount equal to four months of the rent. To do so, he must first have the violation certified by a housing inspector or court and provide you with written notice of the violation, and you will then have up to 14 days to complete the work. You should consult with an attorney to determine whether this repair may constitute a health or safety violation.
Depends on who is responsible for its replacement under your lease.
To questioners from West Virginia & New York. Although I am licensed to practice in your state, I practice on a day-to-day basis in Massachusetts. I answer questions in your state in areas of the law in which I practice, and in which I feel comfortable trying to offer you assistance based on my knowledge of specific statutes in your state and/or general principles applicable in all states. It is always best, however, to work with attorneys and court personnel in your own area to deal with specific problems and factual situations.
Tenants are generally obligated to report issues to the landlord (check your lease.) Moreover, tenants cannot generally perform repairs on their own unless the landlord refuses to repair and/or unless the condition is emergent.
In other words: although you have an obligation to maintain a habitable apartment, you also should be entitled to try to fix things on your own, before the tenant does it.
A broken A/C unit is not generally an emergency. Moreover, it is something which can often be repaired (instead of replaced.) In many cases that would suggest that you would not have to pay for the A/C unit. That said, I am confused by your claim that the tenant broke it: why and how do you assume that to be true?
You should really consult an attorney.
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I agree with the others that you must have been notified of the problem and then given a reasonable time to repair or replace the appliance. Because the air conditioner does not affect the habitability of the leased premises, whether you have an obligation to fix the air conditioner at all will depend somewhat on the terms of your lease agreement.
In order for a tenant in Massachusetts to withhold rent, the landlord must first be on notice of a defect in the property which substantially impairs the habitability of the property. Could a broken A/C substantially impair the value of a unit? I guess that depends on the unit and how dire the need is in the summertime.
The part of the fact pattern that stands out here, however, is the fact that they broke the A/C to begin with. Did they admit that to you? That's a bad fact for them, and bolsters your claim that not only are you owed full rent, but maybe even some recompense from your tenants.
That having been said, understand this: Massachusetts is a very tenant-oriented jurisdiction. Going to battle with your tenant in district court is a tough road to hoe. You may be better off simply looking for new tenants at the expiration of their lease.