When car crashes occur in Formula 1, cars suffer obvious damage.
Are there any economic "fines" to pilots who are recognized to be the cause of accidents?
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Crashes are a part of the sport as they are often either caused by mechanical issue with the car or by driver errors, and there is no economic penalty/sanction mentioned anywhere in the official FIA Sporting Regulations.
Here is a list of the penalties that can be carried out by the stewards (regulatory body in Formula One), according to article 38.3 of the latest FIA Sporting Regulations (last updated on 07/12/2017):
38.3 The stewards may impose any one of the penalties below on any driver involved in an Incident:
a) A five second time penalty. The driver must enter the pit lane, stop in his pit stop position for at least five seconds and then re-join the race. The relevant driver may however elect not to stop, provided he carries out no further pit stop before the end of the race. In such cases five seconds will be added to the elapsed race time of the driver concerned.
b) A ten second time penalty. The driver must enter the pit lane, stop in his pit stop position for at least ten seconds and then re-join the race. The relevant driver may however elect not to stop, provided he carries out no further pit stop before the end of the race. In such cases ten seconds will be added to the elapsed race time of the driver concerned. In both of the above cases the driver concerned must carry out the penalty the next time he enters the pit lane and, for the avoidance of doubt, this includes any stop the driver makes whilst a VSC or safety car procedure is in use.
c) A drive-through penalty. The driver must enter the pit lane and re-join the race without stopping.
d) A ten second stop-and-go time penalty. The driver must enter the pit lane, stop in his pit stop position for at least ten seconds and then re-join the race. If any of the four penalties above are imposed upon a driver, and that driver is unable to serve the penalty due to retirement from the race, the stewards may impose a grid place penalty on the driver at his next Event. If any of the four penalties above are imposed during the last three laps, or after the end of a race, Article 38.4(b) below will not apply and five seconds will be added to the elapsed race time of the driver concerned in the case of (a) above, 10 seconds in the case of (b), 20 seconds in the case of (c) and 30 seconds in the case of (d).
e) A time penalty.
f) A reprimand.
g) A drop of any number of grid positions at the driver’s next Event. If any of the seven penalties above are imposed they shall not be subject to appeal.
h) Disqualification from the results.
i) Suspension from the driver’s next Event.
As you can see, economic fines are not included in this list. Therefore to answer your question: no, there are officially no economic fines carried out to drivers even if the accident caused is 100% their fault. The worst that can happen to a driver is disqualification from the current event and future events, which are far worse than a small fine to them, especially if they are fighting for the championship.
However, this isn't always consistent as the stewards sometimes carried out worse punishments than the ones stated in the article that I just quoted. I have two examples for you:
an extremely serious breach of the regulations, which had the potential to cause injury to others
Schumacher turned in on Villeneuve at the hairpin at Jerez in a bid to take them both out of the race, what would have handed the German what would then have been his third world championship title.
The move failed and, not only did Villeneuve go on to take the crown, but Schumacher was stripped of his runner-up position in the title chase because of his actions.
If you have more example, please add them to my answer.
There are no such fines for drivers who commits accidents, as these would just go against the sport, and the Entertainment that comes with it. If you start fining drivers for each damage on the car, they'll just stop taking risks. Cars are made to break, the way they are built keeping in mind that anything can break on it so it's usually not a problem. Also, a team could greatly benefit from a risky overtake that enventually ends up in a contact, so that would be a shame if the team asked the drive for a Financial compensation.
You'll find some info in the following NYT article.