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This question has a specific context.
While reading Adam Smith's The Wealth of Nations, edited, with notes and marginal summary by Edwin Cannan in 1904, one comes across the following lines on the second page:

The causes of this improvement, in productive powers of labour, and the order, according to which is produce is naturally distributed among the different ranks and conditions of men in the society, make the subject of the First Book of Inquiry.

Against "the causes" written at the outset in first line, there is a note, I presume by Edwin Cannan, which reads:

Only one cause, the division of labour, is actually treated.

Here my aim is to know what other causes of increase in productivity of labour are known today (or were known to Edwin Cannan, back in 1904) that Adam Smith missed in his classic text.

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In one word: Technology.

It's a bit abstract of course, but it summarises a lot of things very well. It's usually the A parameter in the models we have in growth theory (we augment e.g. Labour by Technology). In empirical studies it has been tried to asses productivity by coming up with a measure called TFP (Total Factor Productivity).

A lot of specific things could potentially be subsumed under this broad term. Most importantantly it is the stock of knowledge (e.g. books, blueprints,...) that we improve and build upon. The growth of technology usually depends on the actual stock of knowledge in a certain economy. It can be achieved by Research & Development (in case the economy is on the technology frontier) or simply by technology adoption (e.g. in the case of catch-up economies).

Example: Knowing how to build a tractor, we only need one unit of labour (the guy who drives the tractor) to harvest, say, 100 kilos of grain per day instead of 50 workers that have to cut the same amount by hand. This would be a tremendous increase in productivity because the other 49 people can now do another job in which they potentially add to the economy's output. For building a better tractor (that drives with less fuel, can harvest more, ...) the economy needs the blueprint of the first tractor to build upon, and so on and so forth...

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Labor productivity, as measured by output per worker or output per worker hour, is a function of many factors. Let me mention a few in the framework of growth theory. Many economists consider at least three factors that determine economic growth: human capital, physical capital, and the “efficiency” with which human capital and physical capital are used to produce output (the jargon is “Total Factor Productivity”, or TFP). “Labor productivity” in the question is affected by all these three factors.

First, labor productivity is clearly affected by human capital, the skills and capabilities of workers, which in turn is determined by many factors, including education, training, and health. So anything that improves education, training, and health is likely to increase labor productivity.

Second, labor productivity also depends on stock and quality of physical capital. Physical capital includes durable production inputs such as infrastructure, factory buildings, machines, and computers. Tractors and airplanes have vastly improved the labor productivity of U.S. farmers. But not only new generations of physical capital improves labor productivity, the quantity of physical capital also matters for productivity, at least before saturation. More assembly lines in a factory usually means less downtime of production. More computers in the firm means fewer workers have to share computers with others.

Third, more efficient ways to combine human capital and physical capital also contribute to increasing labor productivity. Technology and organizational advances play important roles in this regard. Assembly lines, telecommunications, and the invention of modern firms fall into this category. Note that some technological progress is embodied in the physical capital, e.g. cellphones and computers make workers more efficient in coordinating production.

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When there is a decrease in the amount of hours worked and an increase in output (production/GDP) this will increase productivity of labour.

According to some sources, the growth in labor productivity depends on three main factors: investment and saving in physical capital, new technology and human capital.

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