As far as I am aware, slavery was not officially outlawed in Europe until after the Renaissance; rather, it was either officially legal or had no official status throughout the Middle Ages, and that between the Roman Empire and the Middle Ages, the practice of slavery simply melted away.
On the flip side, slavery of any variety is extremely illegal in most modern countries, and yet there is a trade in "human trafficking." This practice is not historical but I'm giving it as an example of how illegal slave trade can exist.
My point is that official legality is clearly not the be all end all of slavery, and there are other factors--economic? Social? Race-related? National identity related?--that impact the actual practice of slavery. Are there any historical examples of slavery practice ramping up or melting down despite constant legal status that could help clarify this situation?
Couple of things, I'll just list it off and give sources.
The first, and most important, is that a human being is an efficient retainer for value and equity. If I own someone and train them, I can roughly guess how much money they will produce barring no accidents. They do not spoil like produce, deteriorate like metal, etc. Any civilization that does not possess a government backed currency or other method for retaining value and equity is going to have some elements of slavery present.
Source: Debt: The First 5000 Years
Second, and this ties into the first point, just because they don't call it slavery doesn't mean it isn't slavery. European serfdom was essentially slavery. It's not until after the bubonic plague that serfs began to assert their right to work for different landowners, ask for competitive wages, and begin to negotiate the terms of their labor.
Source: English Law in the Age of the Black Death
Finally, while slaves are certainly more productive than free labor, it's not technically free. You can either feed them, clothe them, and shelter them out of your pocket OR let them do this themselves. Giving them that freedom gives them bargaining power, which they will use to push for better hours. It wasn't really efficient to have large numbers of slaves in the northern portion of the United States because you only need them at certain times of the year. Cotton, which created a massive boom in the South in the 1850's, is a year round crop. If you aren't planting and picking, you're ginning it or turning the soil. Without some kind of crop or resource that is profitable when slaves are harvesting it, the practice is going to fade.
Source: Many Thousands Gone: The First Two Centuries of Slavery in North America
A major topic in economic history, this: I can't do justice to the whole literature, but hope someone else chimes in.
A classic work is Evsey Domar's (1970) "The Causes of Slavery and Serfdom" which links the rise of slavery/serfdom to the land-labor ratio. Boiled down, his argument is that where the land-labor ratio is high (i.e. labor is scarce relative to land, so wages would be high), landowners have an incentive to collude politically to maintain institutions that deny labor freedom (to move, contract freely etc.) that would allow those high wages to come about. Conversely, where the land-labor ratio is low (i.e. labor is abundant, so wages would be cheap), free labor rules.
Domar uses this to explain the emergency of serfdom in Russia, the rise of slavery in the southern US (both places with high land-labor ratios), as well as the decline of serfdom in Western Europe following increasing population density in the Middle Ages.
[Note that the legal institutions are endogenous in this type of analysis - laws change as a function of economic fundamentals, and perhaps provide "friction," but are not ultimately determinants of the viability of slavery or not.]
Have a look at Galenson's work (e.g. here) on indentured servitude, where there is a real interplay between economic forces and legal changes (the abolition of slavery in the British Empire).