This question might be more difficult to answer than it first seems. On one hand, it's easy to just look at the numbers and see if the big number in the end turns out positive or negative. After all, surly we can see how much money was spent on the empire, vs how much money was derived from it.

However, looking more deeply into the matter, things are not so simple. We have to consider non-quantifiable matters, such as international prestige into account, as well as security concerns. Was holding Gibraltar good for the British exchequer? Probably not, but it helped secure Mediterranean trade. Did France benefit from holding kilometers of dead Sahara land? Probably not, but they stopped the Brits from holding it, tell ya what. So when asking if the empire was good for European countries, you have to look beyond money and raw resources.

Take Britain, for example. In 1988, Patrick K. O'Brien suggested in The Costs and Benefits of British Imperialism, 1846-1914 that the empire was a waste of money: " Only conquests of loot and pillage of the kind maintained by King Leopold in the Congo seem capable of providing metropolitan traders and investors with supernormal profits. In this sense the British empire can be plausibly represented, in Adam Smith's words, as "a sort of splendid and showy equipage, not an empire but the project of an empire, not a gold mine but the project of a gold mine'".

Meaning, while the cultural urge to conquer was framed as beneficial for the empire and its citizens, in the end, the costs outweigh the benefits. This attitude is repeated in Davis and Huttenback's Mammon and the persuit of Empire that came out the same year.

However, Avner Offer raised a counterpoint in 1993's The British Empire, 1870-1914: A Waste of Money?. Offer criticized O'Brian, Davis and Huttenback and claimed that one must look at the empire up to the second world war to see its benefits. He argues that while Britain did spend a considerable amount of money defending certain colonies like India and the Dominions, it got paid back by having these places open for British investments and trade and, maybe more importantly, kept them further away from Britain's rivals. Furthermore, since the whole structure of European society, culture and wealth at the time relied on the existance of Empire and European supremacy, it is almost impossible to ask "what if". He concludes: "More than two generations of well-informed investors chose to send their funds overseas. Their assets and income were safe, they felt, under British naval protection. To imagine a better return to these assets at home, we need to visualize a different society with a different distribtuion of wealth, or at least a different set of financial institutions....The Empire was an adjunct of British wealth. It provided protection for that wealth, but also the Temptation to fight for it... the empire extended Britain's capacity to wage war, by adding to its military, demographic and economic resources"

I tend to go with Offer's POV myself. Basically, as the public came to see the empire as a safe place to send and receive materials and wealth from, the empire itself became a part of commerce and defined it. It was profitable because of its mere existence as a part of the 19th century financial, economic and cultural landscape.

Did those analysis consider the contributions to wars, especially WWI and WWII?