You would be liable to pay for these costs, since they are costs arising naturally and reasonably foreseeably out of the accident for which you are at fault. If you do not pay, they could sue you for the money and if they did, would likely be awarded it by a court; on the other hand, they may not choose to sue, since it may not be worth the time, effort, etc. of a lawsuit for the amount at stake. Only if they did sue and win would you be compelled or required to pay.