How Do We Measure Productivity (and Technology)?

Productivity of people is hugely important to human society, and a major focus in macroeconomics.
But it can’t be measured directly.
When we measure it indirectly it’s called total factor productivity. It gets this name because it’s from all factors of productivity totaled together: we can’t really break down which factors are getting more productive.
We measure total factor productivity as a residual. This means that we net out other stuff that we understand, and what’s left over must be productivity (or something close to it).
So, we start with real GDP. Then we subtract out the effect of labor. We can measure people working really well, and the hours they work pretty well. Measuring the differences between individuals is really hard.
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DIGRESSION: What’s critical for a follow-up post is that we’re dealing with aggregates. This means that we can get totals, and from those we can get per capitas. That’s a form of average productivity. But we can’t get the marginal productivity that we really need to know to apply our theories properly. And we can’t get productivity on an individual basis to assess how accurate the average productivity is as a summary of what individual people do.
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Then we subtract out capital, which is a little harder. Counting physical capital is pretty easy, and assessing the utilization rate of it is a bit harder. Again, we run into the problem of measuring differences in productivity between different models, makes, and vintages of machinery. We run into the same problem with human capital, like education and learning-by-doing. And it gets worse with social capital, like legal and regulatory enforcement.
When we’re all done with that, what we’re left with is a residual called total factor productivity. Roughly, this is a measure of technology, or at least the effects and usage of technology.
Do note that we can’t separate out types and vintages of technology either. So this is a mix of high tech, like iPhones, and low tech, like freedom of association.