[...] I'm a parent and [...] I use this punishment [...]
I'm looking for any criticisms you might have, so feel free to shout at me in the comments.
Let me preface my answer by stating that I do not condemn or criticize you or your behaviour at all. I will try to explain why punishing them by confiscating small amounts of money might just not work and/or not give the results you wanted.
I am not proposing to be weak and indifferent towards your children; only pointing out why punishment oftentimes simply does not work and is a waste of your and their time and energy.
Punishment is a difficult topic for two reasons, in my opinion and experience:
- It is an inhibitor, not a motivator.
- Children may grow immune to the current level of punishment and there is a natural limit to the severity of punishment you can dole out.
Inhibitor instead of motivator
Punishing creates mind states that are all-round negative. The culprit will never have the opinion that it was justified to punish them, and be happy about the fact that their actions were regulated. Instead, they will view you as evil, inappropriate, unjust, mean. It will breed resentment and if it happens often it will likely turn happy, joyful children into moody unhappy beings.
It does nothing to achieve better behaviour. Yes, it might have some effect in the regard that you hear less swearing from your children. But you can be pretty sure that they will swear just as before when you are not around to hear it. If you punish them for hitting each other, they will find ways to hurt each other without you noticing. Or one will come and insist that the other hit them; you will be hard pressed to judge whether the statement is true or not.
Speaking of which: it makes you a judge/sheriff instead of a benevolent parent. A judge is far removed from a trusted entity. And the job is very hard, as you also are your own jury, prosecution and detective... trust me: you would not rather not be in this place.
Most importantly: It does not teach them how to behave instead. It only teaches them to hide things from you and distrust you. It certainly does not earn you their respect (which all of this is about, in the end).
Immunity to punishment
What happens if their money is gone? No more punishment; they are now free to do whatever they want. What will you do then?
Do they need to buy essential things (food) from their money? I guess not. So money is probably not that important to them anyways.
The same goes for other kinds of punishment. House arrest? What happens if they actually like sitting in their room alone? Withdrawing favourite junk food? What happens if they don't care so much? What if you simply cannot find another punishment?
I can't, at the top of my head, think of any kind of punishment which does not suffer this problem, except those kinds that are generally not so much accepted by society these days (i.e., beating them, severe hunger, spending a night alone in the woods amongst wild animals, etc.). If you do even the worst imaginable kinds of punishment long enough, they will get used to it, rendering it even more useless as it was before, and what then?
Try "action and reaction" instead
Punishment is arbitrary. Try to work with action and reaction (a.k.a. cause and effect) instead.
For anything they do wrong, there needs to be a negative effect that lets them know that they did something wrong. This is not punishment, but the cold hard facts of life. It is an immediate effect that is directly related to the cause. If you cannot find a negative effect to some action then, well, the action was fine in the first place!
If there is any conceivable way that the reaction to their wrongdoing comes from someone else than yourself (or from the Universe itself), then you do not need (and should not) do anything at all but let things play out, making sure that you do not give them an easy way out of the stuff they brought upon themselves (as appropriate, of course; if they ignore red lights, you are allowed to pull them back before the car hits them...).
For example, if your 15 year old child deigns to go outside in winter in slippers and T-Shirt, they will get ill. Assuming they know the relationship between body temperature and illness, this is enough of a cause+effect. You do not need to chew them out for getting ill, and you actually do not need to hinder them going out. Inform them about the outside temperature (in a friendly manner), just to give them a chance in case they just forgot to pick up a coat, and be done with it. You never know, they might be immune to severe cold, and never get ill! You can be sure that a sincere "wow, you are tough, this would be much too cold for me and I'd certainly get ill" will make them very happy, instead of "you will stay inside until you pick warmer clothes" (which is 100% useless in the big picture).
Example: They break something that belonged to you? Now is the time to confiscate their money and buy a new something. Don't take a symbolic dollar, but make it count. If they break some expensive porcelain they might be unhappy to find out that these things might cost as much as their half-yearly allowance (feel free to lessen the effect a little bit, of course, you do not need to be 100% draconic here). The important thing is that at this point you make it only about the replacement. Not about emotions. Stay calm and composed. If they have run out of money, then another effect would be that you cannot buy them XYZ which you would normally have bought... things like that.
Example: Your 5-course Sunday meals are very important to you and a cornerstone of your week, nay, of your whole life? The children are rude and spoiling it? Then simply do not let them attend (not as punishment, but as matter of fact - be friendly with them, let them do whatever they want, give them something else to eat). They can sit in their room or go play with their friends while you merrily wine and dine. If they are OK with that (because they don't much like these kinds of events), then that is perfectly fine and you enjoy your free time with adult talk. If you want them to sit with you and pretend to be happy while you know that they hate it, then you are causing the trouble.
Example: they have been rude to you for the last 30 minutes when you were driving in your car with them. You pass your favourite ice cream parlour; they want ice. It is perfectly valid (and "enough") to calmly tell them that due to all the recent bickering you are not really in the mood for ice right now. You do not make it about how respectless they are, or that you don't want to spend the money, or whatever. Just make sure they know that you are disappointed and not in the mood; they cannot argue with that. It is not punishment "you did X to me, so I do Y to you", but it teaches them that if they drive someone mad, that has a direct effect in itself.
Try to apply this kind of thinking to anything where it is even remotely possible. Yes, getting a correct reaction for swearing and rudeness might be quite difficult, but most other things work pretty well. This also goes hand in hand with putting more responsibility on their shoulders. Let them figure stuff out, you don't need to hand-hold them that much.
A different, almost unrelated spin on the money issue: 13 year olds can easily start to earn money instead of living on the dole (heck, 7 year olds can figure it out...). Cut their allowance. They get money for doing good things/chores. They are 100% free not to do them, but then they do not get money. Heck, give them an extra dollar for every day they did not cuss at you. Make it as objective and clinical as in the real world (no arbitrary giving or taking, 100% clear upfront rules). Done.
There is a book "The Entitlement Trap" which has a very elaborate scheme to get rid of all possible reasons for fighting, if you wish to go this way. Some people may think all of this is too much "capitalism", but if you are not against that out of principle, their "game rules" certainly are logically watertight. They also manage to put a lot of positive connotations into the matter - i.e., it is a family activity, not just a payment scheme.
Acceptance, motivation, good example
Relax, accept them for what they are, try to understand why they behave like they do. Make sure you understand that there is a separation between you and them, you are not one entity. Be patient. Be very strict with yourself to present good examples at any time. Think a lot. Motivate them in some manner. Not "if you do X, you will get Y", but by finding activities that you can do together where they have plenty of positive emotions and really enjoy the time together with you. Sports, play, rituals...
Remember that it is your job to love your children, it is not the job of your children to love you. You had a choice of putting children into the world; they did not have the choice of their parents. It does happen that even the best of the best parents get "bad" children, and it does happen that you cannot do anything about it. Arbitrary people do not necessarily like each other, and there is no reason to believe that it is different between children and parents. This might sound cruel, but for all intents and purposes it is the truth. Accepting that might help when the going gets rough. Them swearing at you and being rude is a phase that is very likely to occur multiple times, and there is very little indeed you can do about it.