[Question] How can labour productivity growth be an indicator of automation?

I was listening to an interview with the economist Paul Krugman the other day about whether robots will be taking our jobs. He pointed to data showing that the growth in productivity has declined in recent years, citing this as evidence that AI and robotics are not (yet at least) going to be automating away many jobs.

This puzzled me a little and led me to do a bit of research into the economists definition of productivity (specifically what they call labour productivity). I’m no expert on economics so this is just my best attempt.

Labour productivity is calculated by a simple formula:

Market Value Produced /​ Hours worked

So if your economy produces $100 in 10 hours the labour productivity is 10 $/​h.

What struck me as odd was that Krugman’s analysis didn’t seem to take into account the interaction between this productivity measure and reduced prices due to automation. Suppose a given market is producing $100 by selling 10 units at $10 per unit for 10 hours of labour. If automation allows us to produce 100 times more units with fewer hours of labour at first glance it seems like a no-brainer that his should increase labour productivity. But suppose that demand is not elastic, then the ability to produce 100 times more units with similar variable costs means that the price will fall dramatically. So now, the value of the market produced is much less (i.e., 10 units times a much lower price). Thus even though it takes many fewer working hours to produce the goods, the value of the goods has also fallen.

Since both the denominator and the numerator would fall as a result of large scale automation, why does Krugman suggest that rising labour productivity would be the main indicator of whether large scale automation is happening in the economy? Is there some additional assumption economists are making when they make such claims? (e.g., that demand for goods is infinitely elastic)

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