There was one post on Monday: “Robot Geometry (Very Wonkish):”

And now for something completely different. Ryan Avent has a nice summary of the argument in his recent book, trying to explain how dramatic technological change can go along with stagnant real wages and slowish productivity growth. As I understand it, he’s arguing that the big tech changes are happening in a limited sector of the economy, and are driving workers into lower-wage and lower-productivity occupations.

But I have to admit that I was having a bit of a hard time wrapping my mind around exactly what he’s saying, or how to picture this in terms of standard economic frameworks. So I found myself wanting to see how much of his story could be captured in a small general equilibrium model — basically the kind of model I learned many years ago when studying the old trade theory.

Actually, my sense is that this kind of analysis is a bit of a lost art. There was a time when most of trade theory revolved around diagrams illustrating two-country, two-good, two-factor models; these days, not so much. And it’s true that little models can be misleading, and geometric reasoning can suck you in way too much. It’s also true, however, that this style of modeling can help a lot in thinking through how the pieces of an economy fit together, in ways that algebra or verbal storytelling can’t.

So, an exercise in either clarification or nostalgia — not sure which — using a framework that is basically the Lerner diagram, adapted to a different issue.

Imagine an economy that produces only one good, but can do so using two techniques, A and B, one capital-intensive, one labor-intensive. I represent these techniques in Figure 1 by showing their unit input coefficients:

Here AB is the economy’s unit isoquant, the various combinations of K and L it can use to produce one unit of output. E is the economy’s factor endowment; as long as the aggregate ratio of K to L is between the factor intensities of the two techniques, both will be used. In that case, the wage-rental ratio will be the slope of the line AB.

Wait, there’s more. Since any point on the line passing through A and B has the same value, the place where it hits the horizontal axis is the amount of labor it takes to buy one unit of output, the inverse of the real wage rate. And total output is the ratio of the distance along the ray to E divided by the distance to AB, so that distance is 1/GDP.

You can also derive the allocation of resources between A and B; not to clutter up the diagram even further, I show this in Figure 2, which uses the K/L ratios of the two techniques and the overall endowment E:

Now, Avent’s story. I think it can be represented as technical progress in A, perhaps also making A even more capital-intensive. So this would amount to a movement southwest to a point like A’ in Figure 3:

We can see right away that this will lead to a fall in the real wage, because 1/w must rise. GDP and hence productivity does rise, but maybe not by much if the economy was mostly using the labor-intensive technique.

And what about allocation of labor between sectors? We can see this in Figure 4, where capital-using technical progress in A actually leads to a higher share of the work force being employed in labor-intensive B:

So yes, it is possible for a simple general equilibrium analysis to capture a lot of what Avent is saying. That does not, of course, mean that he’s empirically right. And there are other things in his argument, such as hypothesized effects on the direction of innovation, that aren’t in here.

But I, at least, find this way of looking at it somewhat clarifying — which, to be honest, may say more about my weirdness and intellectual age than it does about the subject.