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Well, you've inadvertently waded into what is probably the biggest ongoing debate in the historiography of slavery in North America right now! There is a huge debate raging at the moment regarding the exact role slavery has had to play in the development of modern capitalism, and the industrial revolution specifically. It's a debate that has been raging since at least 1944 with the development of what historians of slavery call the 'Williams thesis', named for Eric Williams and his book Capitalism and Slavery, in which he argued for an intrinsic link between the two. There is today a broad consensus that there is something of an essential relationship between the two and that slavery had a significant role to play in the industrial revolution and the development of capitalist economy; the debate largely focused on just how significant that role is, and whether or not it was definitive (in other words, was slavery a necessary prerequisite for industrialisation, or did it simply give it a helping hand?). For my part, I am unconvinced by the argument that there exists a definitive causal relationship between slavery and the industrial revolution - it has a role to play, absolutely, but I am not satisfied with the arguments that have been advanced to suggest it is the driving force behind western industrialisation.
The Williams thesis essentially holds that slavery's contribution to industrialisation is one of material investment. According to Williams, in the 17th and 18th centuries slavery in the New World provided the ingredients necessary for industrialisation. Extremely profitable farming operations allowed for the accumulation of vast wealth surpluses that provided the capital to finance the industrial revolution; the transatlantic slave trade created complex international markets, with ports and shipping lines that also carried goods and messaged, along which industrial supplies and consumer goods could later be ferried. As industrialisation gets underway in earnest, slavery begins to go into decline as it becomes apparent that industrial wage labour is more profitable and socially agreeable. It is, according to Williams, abolished in the 19th century as it becomes more profitable to 'proletarianise' slaves and turn them into sharecroppers or wage labourers, aided by improvements in agricultural efficiency made possible by the explosion in engineering creativity brought on by industrialisation. In this conceptualisation, slavery is essentially a feudalistic and pre-capitalist enterprise that is fundamentally incompatible with but necessary for the development of industrial capitalism.
Historians of slavery now widely recognise that the Williams thesis is fundamentally wrong in this regard. Slavery was not in decline in the 19th century - on the contrary, particularly in the United States, it has been shown to be a thriving and enormously profitable enterprise that did not appear to be dying out of its own accord. The picture in the British Caribbean is a little more complicated - there is still some disagreement over whether or not the region was in economic decline by the 19th century. Generally speaking, there is agreement that Britain's colonies were troubled but certainly not in any kind of terminal danger or decline that meant slavery would inevitably die out (and in some parts, like Barbados, slavery was very much a healthy, expanding institution). Nor are most historians convinced that it is a wholly pre-capitalist enterprise, either. Eugene Genovese took up that mantle most notably after Williams, and that argument has been thoroughly picked apart over the years (Walter Johnson's Soul by Soul: Life Inside the Antebellum Slave Market is a solid critical response to the way in which Genovese has misinterpreted the historical record in this regard). I'm going to talk about this in more detail towards the end - and you'll see why I'm leaving it for the end - but suffice to say, New World slavery was not incompatible with capitalism.
So what about Williams' contention that slavery was a necessary prerequisite to industrialisation? Well, this is where the debate gets rather more heated and complicated. There are many historians today still arguing in favour of that component of the Williams thesis, and the most notable lately would probably be Edward Baptist in his book The Half Never Told: Slavery and the Making of American Capitalism. Baptist alleges that the enormous profits being made from cotton and sugar helped to finance the industrial revolution in the United States and Britain, respectively.
What has also not been explained satisfactorily, in my view, is at what point slavery becomes uncoupled from the wider capitalist economy. Slavery is not, as I go into more detail later, a pre-capitalist phenomenon; it is not a relic carried over from some distant feudal past. It is thoroughly and wholly compatible with capitalist economic practices. Yet scholarly studies consistently fail to find that slavery was particularly and uniquely significant to the ongoing economic prosperity of the United States or Great Britain in the 19th century. The abolition of slavery gives rise to only a small economic shock in the United States, one which is perhaps also partly explained by the end of the Civil War dragging down growth as well - economic well-being recovers very quickly. Likewise, assessments of cotton's contribution to Gross Domestic Product (the value of everything the economy produces in a year) are modest. Baptist's claims in his book depend partly on the immense value of cotton to the US economy which other scholars have shown are based on a fundamental misunderstanding of economic theory: due to what is essentially an accounting error in how he tabulated the worth of cotton production in the 19th century, he has accidentally doubled it, whilst at the same time he miscalculates GDP in such a way that makes his estimates of cotton's contribution to the economy worthless (he excludes asset sales for instance but includes slave sales, even though slave sales are essentially asset sales in a slave economy). Though their 1974 work is rightfully deeply criticised, one of the more positive contributions Fogel and Engerman have made to the scholarship on slavery is demonstrating that the Southwest was just as wealthy as the Northeast in 1860 if you include wealth caught up in slaves (Gavin Wright has shown without slaves the North was 64% richer), only concentrated in the hands of a minority. It seems illogical that there should be such a huge dependency on that kind of wealth that somehow disappears to the point where slavery is abolished with a negligible impact on the wider economy.
Likewise, Baptist's thesis argues that cotton drove industrialisation in Britain, but he does a very bad job at proving it - and indeed, the advent of meaningful cotton mill operations in northern England significantly predates the explosion in cotton production in the Southern US, rather than being driven by it. It is also worth emphasising that Jamaica is particularly suited to growing cotton, and particularly to hand-picking cotton. To this day Jamaican cotton is worth four to five times Egyptian cotton. Yet we do not see any meaningful effort on the part of British planters to tap into this lucrative resource even in Jamaica, which was not a sugar monopoly colony; whilst this may simply reflect poor economic planning, it seems to be rather odd that if these planters were providing the finance and capital to drive industrialisation in northern England, that they would not be logically trying to also tap into the emerging market for cotton goods. Sugar remained king in the Caribbean, even though it was a much less stable market than cotton. And in any event, I would also stress that, although the Civil War in the United States did have an impact on the British economy, there is evidence that Britain was able to heavily supplement its shortfall in cotton imports from both India and Egypt. Britain's own abolition of slavery and the slave trade has been estimated to have cost just 2% of national income, suggesting it was far from the driver of economic growth. The profits from Britain’s colonies are just not enough to finance industry’s wholesale development and growth.
So what role did slavery play in the industrial revolution? Certainly, it was a source of finance, and the plantations of the South and the Caribbean did produce raw materials that helped to fuel industrialisation - though the advent that made this possible, the invention of the cotton gin, comes twenty to thirty years into industrialisation and coincides with the emergence of cotton mills. But the evidence for both the United States and Britain is that the vast majority of the wealth created by slavery remained trapped within the slave system, disappearing upon abolition. The end of slavery is devastating to the elite of both the South and the British Caribbean, wiping out vast quantities of wealth over night and bringing eventual ruin to many estates. Slave owners were, largely speaking, investing in more slaves rather than in industry - though some certainly did invest in industry (as did some slave traders who were not necessarily slave owners). And we know that in Britain, about half of all planters were not actually resident in the Caribbean and had other economic interests. But the evidence for slavery being the driving and causal factor behind industrialisation is, in my view, rather weak and the argument has yet to be made convincingly. There are too many inconsistencies and problems with the developments in the Williams thesis since 1944. It has a role to play but it is fundamentally wrong, in my view, to attribute industrialisation to slavery. It's just not that simple or straight-forward, and that argument rests on a very simplified, inaccurate view of economic development in the 18th and 19th centuries.