As I see it, there are a couple of different reasons to fit hyperbolic growth models — or, rather, models of form (dY/dt)/Y = aY^b + c — to historical growth data.
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I think the distinction between testing a theory and testing a mathematical model makes sense, but the two are intertwined. A theory will tend naturally to to imply a mathematical model, but perhaps less so the other way around. So I would say Kremer is testing both a theory and and model—not confined to just one side of that dichotomy. Whereas as far as I can see the sum-of-exponentials model is, while intuitive, not so theoretically grounded. Taken literally, it says the seeds of every economic revolution that has occurred and will occur were present 12,000 years ago (or in Hanson (2000), 2 million years ago), and it’s just taking them a while to become measurable. I see no framework behind it that predicts how the system will evolve as a function of its current state rather than as a function of time. Ideally, the second would emerge from the first.
Note that what you call Kremer’s “Two Heads” model predates him. It’s in the endogenous growth theory of Romer (1986, 1990), which is an essential foundation for Kremer. And Romer is very much focused on the modern era, so it’s not clear to me that “For the purposes of testing Kremer’s theory, the pre-industrial (or perhaps even pre-1500) data is nearly all that matters.” Kuznets (1957) wrote about the contribution of “geniuses”—more people, more geniuses, faster progress. Julian Simon built on that idea in books and articles.
A lot of the reason I’m skeptical of Kremer’s model is that it doesn’t seem to fit very well with the accounts of economic historians and their descriptions of growth dynamics....it seems suspicious that the model leaves out all of the other salient differences that typically draw economic historians’ attention. Are changes in institutions, culture, modes of production, and energetic constraints really all secondary enough to be slipped into the error term?
Actually, I believe the standard understanding of “technology” in economics includes institutions, culture, etc.—whatever affects how much output a society wrings from a given amount of inputs. So all of those are by default in Kremer’s symbol for technology, A. And a lot of those things plausibly could improve faster, in the narrow sense of increasing productivity, if there are more people, if more people also means more societies (accidentally) experimenting with different arrangements and then setting examples for others; or if such institutional innovations are prodded along by innovations in technology in the narrower sense, such as the printing press.
Actually, I believe the standard understanding of “technology” in economics includes institutions, culture, etc.--whatever affects how much output a society wrings from a given input. So all of those are by default in Kremer’s symbol for technology, A. And a lot of those things plausibly could improve faster, in the narrow sense of increasing productivity, if there are more people, if more people also means more societies (accidentally) experimenting with different arrangements and then setting examples for others; or if such institutional innovations are prodded along by innovations in technology in the narrower sense, such as the printing press.
Just on this point:
For the general Kremer model, where the idea production function is dA/dt = a(P^b)(A^c), higher levels of technology do support faster technological progress if c > 0. So you’re right to note that, for Kremer’s chosen parameter values, the higher level of technology in the present day is part of the story for why growth is faster today.
Although it’s not an essential part of the story: If c = 0, then the growth is still hyperbolic, with the growth rate being proportional to P^(2/3) during the Malthusian period. I suppose I’m also skeptical that at least institutional and cultural change are well-modeled as resulting from the accumulation of new ideas: beneath the randomness, the forces shaping them typically strike me as much more structural.
I agree with much of this. A few responses.
I think the distinction between testing a theory and testing a mathematical model makes sense, but the two are intertwined. A theory will tend naturally to to imply a mathematical model, but perhaps less so the other way around. So I would say Kremer is testing both a theory and and model—not confined to just one side of that dichotomy. Whereas as far as I can see the sum-of-exponentials model is, while intuitive, not so theoretically grounded. Taken literally, it says the seeds of every economic revolution that has occurred and will occur were present 12,000 years ago (or in Hanson (2000), 2 million years ago), and it’s just taking them a while to become measurable. I see no framework behind it that predicts how the system will evolve as a function of its current state rather than as a function of time. Ideally, the second would emerge from the first.
Note that what you call Kremer’s “Two Heads” model predates him. It’s in the endogenous growth theory of Romer (1986, 1990), which is an essential foundation for Kremer. And Romer is very much focused on the modern era, so it’s not clear to me that “For the purposes of testing Kremer’s theory, the pre-industrial (or perhaps even pre-1500) data is nearly all that matters.” Kuznets (1957) wrote about the contribution of “geniuses”—more people, more geniuses, faster progress. Julian Simon built on that idea in books and articles.
Actually, I believe the standard understanding of “technology” in economics includes institutions, culture, etc.—whatever affects how much output a society wrings from a given amount of inputs. So all of those are by default in Kremer’s symbol for technology, A. And a lot of those things plausibly could improve faster, in the narrow sense of increasing productivity, if there are more people, if more people also means more societies (accidentally) experimenting with different arrangements and then setting examples for others; or if such institutional innovations are prodded along by innovations in technology in the narrower sense, such as the printing press.
Just on this point:
For the general Kremer model, where the idea production function is dA/dt = a(P^b)(A^c), higher levels of technology do support faster technological progress if c > 0. So you’re right to note that, for Kremer’s chosen parameter values, the higher level of technology in the present day is part of the story for why growth is faster today.
Although it’s not an essential part of the story: If c = 0, then the growth is still hyperbolic, with the growth rate being proportional to P^(2/3) during the Malthusian period. I suppose I’m also skeptical that at least institutional and cultural change are well-modeled as resulting from the accumulation of new ideas: beneath the randomness, the forces shaping them typically strike me as much more structural.