It won't hurt. Whether it will help depends on the magistrate. At least it shows that you aren't willfully just blowing off your obligation.
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It may, it may not. You can absolutely request to the court to come to terms on letting you pay a partial amount and then agreed upon amounts every month but ultimately it will be up to the court to decide.
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You need to pay the purge amount to stay out of jail. It may help to pay partial, but if you are in jail you will not be able to do anything. Contempt of court can carry a 180 days in jail punishment, and you still owe the money, and probably lose your job. Is that what you want? You can be placed in the custody of the sheriff for a considerable amount of time, and you say you have no one you can borrow from. Well, this is what happens when child support is ignored. Best of luck to you and be sure to have some sort of evidence, if nothing more than your own testimony, about what you did to try to come up with the purge.
Each judge is different. I practice in Indiana, and I always tell clients that if they have an extra $5 in their pocket and they are behind on child support they need take that $5 and pay it towards their support. Most judges look at it as you should feed your kids before you feed yourself, so every little bit usually helps to show that you are trying.
How about a loan from your employer? Sometimes that will help. Stress to the court that you will lose your job and then you will not be able to pay anything at all. In these tough economic time the judge should consider that.
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If what you say is accurate, and the proof was presented at the contempt hearing, it is difficult to understand how a purge was set that you cannot meet. In any event, pay what you can. At the very least, it will show good faith when you appear at the commitment hearing, and perhaps result in an amendment to the purge in the nature of a payment plan. Bring your bank statements, income statements, and employer to the commitment hearing.