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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I'm not an economist, but here's an answer based on my understanding.

Suppose the market produces 10 widgets with 10 hours of labor.  Those widgets are worth $1 each.  Now, an innovation comes along that allows twice as many units to be produced with the same amount of labor.

The company can now produce 10 units with only 5 hours of labor.  It then reallocates the five hours, either by assigning the workers to other products, dismissing them to find other work, or whatever.

Clearly, the economy is no less productive at this point.  When the now surplus labor moves to another use, the economy is strictly more productive, producing more than before for the same amount of labor.

As you point out, if production is twice as efficient, the price will drop, and quantity demanded will increase at the new, lower, price.  So, most likely, the company will wind up keeping the same amount of labor, but producing 20 units instead of 10.

It's true that all things being equal, the value of the 20 units will not be twice the value of the 10 units from before, since the price will drop substantially.  So it's true that twice the productivity will not measure twice the monetary value.  

However: the lower price produces deflation.  That product deflates in price by around 50 percent, but in the overall US (say) economy, it amounts to a very small amount of deflation. Seeing that deflation, the Fed realizes it should increase the money supply (print more money) to keep the overall price level the same (or to keep it at its target 2% inflation, or whatever).

What happens, then, is: the price of widgets is lower, both before and after inflation, but the price of everything else is slightly higher to compensate.

If you add up the economy using all the old quantities but the new prices, they have to stay exactly the same, because inflation is zero.  But: adding in the additional 10 widgets means that GDP (after inflation) has increased by their value, which means GDP is higher, with inflation at zero, and the same amount of labor.  


In summary:

In terms of goods produced, the country is obviously more productive after the innovation, because it has 10 more widgets with the same amount of labor.  The monetary value might not show that -- it could indeed go down if the price of widgets falls enough.  However, if you choose to measure in dollars instead of widgets, you have to adjust for inflation to keep the dollars constant.  If you do that, you can prove mathematically that the overall value of everything produced must be higher.  That's because the more the price of widgets drops, the more deflation you have, and the two cancel out, leaving only the value of the extra widgets.

Thanks for your comment Phil. That's helpful, I hadn't considered the question of where labour shifts after less of it is needed to produce an existing good. 

I understand you as saying that as productivity increases in a field and market demand becomes saturated then the workers move elsewhere. This shift of labour to new sectors could (and historically did) lead to more overall productivity, but I think this trend may not continue with the current waves of automation. It seems possible that now areas of the economy where workers move to are those les... (read more)

But even if workers move to less productive industries, productivity must still go up, adjusted for inflation. Suppose 5 workers lose their jobs because it takes 5 fewer workers than before to make 10 widgets.  The country is now making the same as before, but with 5 fewer workers.  So productivity is higher than before, if the 5 workers remain unemployed.  (Same output, less labor). If the 5 workers get jobs elsewhere, even if they are almost completely unproductive and make only 1 grommet combined, the country is still more productive than before -- more output (1 extra grommet), same labor. If productivity is output/labor, it must always be true, mathematically, that even if the (now) surplus labor is even minimally productive, average productivity rises.   For the case where the workers stay put making widgets and it's just that more widgets get made, that's just a special case where the surplus labor stays in the same industry, and the "proof" is the same as before.
Ah, thanks for clarifying. So the key issue is really the adjusted for inflation/deflation part. You are saying even if previously expensive goods become very cheap due to automation, they will still be valued in "real dollars" the same for the productivity calculation.  Does this mean that a lot rides on how economists determine comparable baskets of goods at different times and also on how far back they look for a historical reference frame?
I'm saying that if previously expensive goods become very cheap due to automation, the total for all goods will be valued higher in "real dollars".  For that one good, the total dollar value could indeed be lower, even after overall inflation (such as, for instance, if the price drops by a factor of 20, but only 10 times as many items are produced). But for the economy as a whole, the value in "real dollars" will always at least stay the same after productivity improvements that lower some prices relative to the status quo.  That's because even though that one good may be lower in value even after adjusting for deflation caused by the lower price, the other goods in the economy will make up the difference and more by being higher in value after adjusting for deflation.