It really is an interesting question to ask what "the economy" actually even is. You start with the 15th century, so let's start there. The Caribbean economy in the 15th century was the Arawak (Taino) economy seems to have been stable for centuries, punctuated only by hurricanes and raids from Carib war canoes (buccaneers first leaned their island hopping raising ways from the Carib tribesmen but I digress). They fished, grew cocoa, pineapple, guava, cassava, tobacco, and maize, and organized themselves democratically and communally. So depending on how you view economics, they weren't doing too bad.
Then of course they were attacked, sickened, and enslaved by the Spanish. Those strong enough resisted enslavement to the death or fled to the mountainous interiors but needless to say the Spanish arrival wrecked the existing economy. Some Arawaks kept on scratching out a living hunting and smoking wild boar (on the "boucan" - strips of wood, also something they taught buccaneers whose trade was boat hunting as they escaped indentured servitude before it was piracy, but I digress again).
Anyway the new economy was basically if the place had gold, like Hispaniola, concentrate on it and make slaves, Taino or imports from the Spanish Main or Africa dig for it, and if it didn't have gold, fruitlessly dig for it anyway and finally give up and leave behind a few poor neglected ranchers along with runaway African slaves (Maroons) and surviving Tainos. Limited agriculture outside of that, much of which was poorly suited for the land. Not a great economic start for the Caribbean at the end of the 15th century.
It wasn't till the 17th century that sugar production really started to take off, though it was implemented in a limited way, first on the Canaries, then São Tomé and the Azores, and finally in the Caribbean, but the Spanish didn't focus on it. The English did after their takeover of Jamaica (ill-defended and ignored because of there being no gold), and so did the French on Saint-Domingue, in the 1600s. During this time in the region you also have near constant warfare between the powers (Google "no peace beyond the line) which often threatened to embroil Europe in war, along with constant raiding buccaneers and Carib war parties. They mined plenty of gold though, I guess, but it was really not very important compared to the gold from the mainland.
In the 1700s the slavocracy got into full swing as I mentioned, which also produced wars between the maroons and the plantation owners, hindering development of the interior, and also tying up British troops to put down revolts, maintain an uneasy order, and unsuccessfully traipse into the mountains to hunt maroons. You could say it was a good business though, if all you are analyzing is the plantocracy. I don't know how good it was for the slave peasants on the islands or the peasants back in the mother countries, though. And by the end Of the century you see revolution and the collapse of the plantocracy on the horizon. The American Revolution then the French gave the slaves the language of the Rights of Man, and so you had the Haitian Revolution with it's devastating effects and a Second Maroon War on Jamaica.
So yeah honestly I think it would be tough to argue that the economy was good in the Caribbean at any point after 1492. Maybe in the independence era with tourism things are a bit better, but up till then it was exploitation by the sugar estates and the United Fruit Company and foreign domination from the USA, Spain, UK, etc.
But the 19th and 20th centuries aren't under study, so