In MA how much will your auto insurance increase after an at fault accident?

It depends upon State law and the way in which the insurer has "filed rates" with the regulator. That is, some states have prohibitions against insurers increasing rates when a claim does not exceed a stated amount, even if the insured was At Fault for the collision. I cannot comment on Mass. law in that regard; your insurance agent would have that information.

Naturally, if a claim would have an adverse impact upon your rate, you can choose to pay the claim without the involvement of the insurer. This would be a business decision for you to make. Among the factors to consider are the amount of the claim, whether it is for property damage only or bodily injury damages, the amount, if any, of the immediate rate increase, and for how long the higher rate would be applied/collected. The premium that you pay is a function of the rate. The rate is a dollar amount per $100 worth of insurance. Therefore the premium is calculated by multiplying the amount of coverage that you have (in $100 increments) by the rate.

If the claim is for bodily injury, where the amount of damages cannot be immediately determined, you may wish to have the insurer handle it regardless of the impact upon the insurance premium that you pay in the future. This is because your liability insurance not only pays sums for which you may be found liable, but it also provides a defense if you are sued. That is, it pays defense counsel to defend you and related court costs. In all events, if you are going to have the insurer handle the claim for you, it is important that you report the claim to it promptly and cooperate in all respects with its investigation of the claim.