Of course it depends on the price of getting slaves, if they can be captured cheap enough (or sell themselves as indentured servants) it will almost always be economic (because they can literally be worked to death), but if slaves has to be given a subsistence-level of food, clothing and housing, their conditions has to be compared to poor free people.

As I understand it most agriculture is most efficient in family-sized holdings (actual size dependent on technology) because at least long-term managing soils and animals is a lot about judgement and care. Taylorism seems to work only with certain crops like cotton where you had fairly uniform movements. So in most agriculture it would not be economical to keep slaves if they are not captured cheap enough.

Note that the north-south divide starts at the time when the cotton gin has recently been invented, Britain has destroyed Indian production of textiles and moved it to Britain. Britain has also lost the US colonies, banned slave trade, and enforcing that ban on others. When the divide reaches conflict the South has 75% of the worlds cotton production.

On the other hand institutions has a strong path-dependency so slavery in the south can very well be explained with it already existing, going back to the triangle trade. If slavery had not existed, poor labourers would probably had sufficed just as well. As they did after the abolition.

So I would postulate that slavery is instituted on a large scale when there is an abundance of slaves and stays until it is uneconomical and some shock causes the institutions to change.

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