At the peak of the American whaling industry, what share of the workforce were involved in it?
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I've done some research into this, but I can offer only a partial answer at present.
The graph in the article you linked shows that the national output from whaling peaked in the 1850s. The population of the United States in that year (based on the 1850 US Census) was 23,191,876.
At its peak, the United States whaling fleet consisted of a total of 735 ships (out of 900 in the world). The Whaleman’s Shipping List has listings for 20 ports in 1855, the largest of which was New Bedford, Massachusetts (also where the newspaper was published). The New Bedford whaling fleet peaked at 329 vessels (valued at more than $12 million), and more than 10,000 men were employed on those vessels.
By 1860, vessels sailing from Bristol County Massacheussets (New Bedford is the largest town in Bristol County) produced 80% of the output of the United States whaling industry. 10,458 hands were employed on these vessels, out of a total of 12,301 employed on board the entire US whaling fleet. By contrast, the number employed in "refining sperm oil and whale oil" in Bristol County in 1860 was just 117 (out of a total of 337 employed in those industries in the whole United States).
[source - Lance E. Davis, Robert E. Gallman, Karin Gleiter: In Pursuit of Leviathan: Technology, Institutions, Productivity, and Profits in American Whaling, 1816-1906, University of Chicago Press, 2007]
Clearly the numbers employed in refining the oil were tiny compared to the numbers actually employed on board the whaling fleet. Based on these numbers, a total of 12,638 people were directly employed in producing the whale oil in 1860. This number is presumably slightly down from the peak that occurred a few years earlier, but probably not by much.
The total workforce in the United States has been estimated to be only about 11,110,000 people. I have to say that I am not entirely convinced by this number, given that we know the total United States population in 1860 was 31,443,321 (based on the 1860 US Census), however it does give us a starting point to work from.
Based on the figure of 11,110,000, we can say that just 0.11% of the workforce was directly employed at the sharp end of whale oil production in 1860.
Just for comparison, the estimated workforce in 1850 was 8,250,000. If a similar number were employed in whaling in 1850 (which, based on the graph in your article, this seems a reasonable estimate), then the percentage of the workforce rises to 0.15%.
Now, obviously these figures don't include all the "downstream activities" that you mentioned in the question. Clearly there were a large number of jobs that supported the whaling fleet, from boat-builders and sail-makers, to those that sold the victuals and fitted out the vessels before they sailed. In addition, a large number of people would have been required to market and distribute the finished product. Unfortunately, I haven't yet been able to locate any sources to quantify the exact numbers of people employed in those industries (which I why I said this is just a partial answer).
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