my understanding is that the colonies were not extremely profitable in the way a sugar island could be counted as profitable, but they were of a net positive economic value to Britain. It is fashionable to say that Britain gained immensely post revolution by being able to trade while not incurring the military costs. The trade was there prior, while the colonies were forced to pay the military costs. I think it fair to say that Britain did not lose the trade advantage they had prior. Circumstances (ie an Industrial Revolution and increased trade demand from the colonies) may have contributed to increased revenue. I think the profitability was sabotaged by the costs of the French and Indian War (an action dictated by Britain in its desire for expansion and desire to hurt France, not by the 13 colonies) and the costs of attempting to put down the revolution. Takeaway those two things, and the colonies were a net positive economic value (I think)