Why did slavery pay? In what circumstances? In a Malthusian world, a laborer’s pay is just enough to allow him to raise two kids to adulthood. That’s all the average peasant makes, so that’s all an employer has to pay. Someone who raised slaves for labor would have to spend just as much or go out of business, in the long run.
So, in the Classical world, slaves mainly made economic sense (for the masters, natch) only if you could steal them, generally through wars of conquest. If Rome enslaved a 20-year Thracian farmer, they could get decades of work out of him without having to pay for his upbringing. All of this assumes that the Romans had special expertise in conquest and social control, and for the most part they did. Of course, if they picked the wrong Thracian, they could be sorry.
As the Roman Empire cut back on conquest, slaves cost more. Since the Empire had a fairly long period of internal peace and good government, living standards naturally dropped. Population growth far outpaced technical innovation, which was never a Roman strong point. Slavery seems to have become less and less important. Certainly there was no long-term, self-sustaining population of slaves.
Slavery in the American colonial era is a very different story. Although many slaves were imported from Africa, the slave population reproduced quite successfully. As C. Vann Woodward wrote: “So far as history reveals, no other slave society, whether of antiquity or modern times, has so much as sustained, much less greatly multiplied, its slave population by relying on natural increase.” In a mostly-Malthusian world, that is exactly what you would expect.
Before the War of the Rebellion, the natural growth of the slave population was about 25% per decade, greater than any country in Europe. That doesn’t necessarily mean that the slave health and nutrition was equivalent to that of white Americans, or that their lives were easy: but they had a higher standard of living than the typical peasant in a country with a stable population. The main underlying reason was that the US, in those days, was one of the least Malthusian place that has ever existed. The Amerindian population had collapsed (not before transmitting highly productive crops such as maize) and land was abundant. Food was abundant and labor was not.
You can see why slavery might pay, but why African slavery? There is reason to think that they were more productive farm workers than Europeans in the American South , largely because they were significantly more resistant to a number of tropical diseases. Almost all people from central and west Africa are Duffy negative and are highly resistant to vivax malaria. A number of different mutations common in those same populations, such as sickle-cell, Hemoglobin C, G6PD deficiency,etc. etc. give substantial protection against falciparum malaria. Although the genetic details are at this point not understood, people from sub-Saharan Africa are also resistant to hookworm and are more likely to survive yellow fever. One would suspect that they also have greater heat tolerance.