Should-Read: An entertaining rant from Doug Campbell. I wonder what triggered this? It is certainly true that the abstract of Acemoglu, Johnson and Robinson (2001), "Reversal of Fortune: Geography and the Making of the Modern Income Distribution" is simply wrong where it says "countries... that were relatively rich in 1500 are now relatively poor": they should have written "countries... that were densely populated and relatively urbanized in 1500—and thus had done well either because of geographic advantages or because of successful matching of technology to environment—are now relatively poor". "Rich" is the wrong word for, as Doug correctly points out, Malthusian model reasons. It is marriage patterns (and, to some degree, vulnerability to epidemics and violence) that determines whether those people still alive to consume in the old days were relatively rich or poor: Doug Campbell: On the Uses (and Abuses) of Economath: The Malthusian Models: "The first year of an Econ Ph.D. feels much more like entering a Ph.D. in solving mathematical models by hand than it does with learning economics...

...very little reading or writing... loads and loads of fast algebra.... Why?... The first reason is that mathematical models are useful! Take the Malthusian Model... the birth rate... increasing in income... the death rate... decreasing... income per person... negatively related to population, and... technological growth... slow relative to population growth, and you can explain a lot of world history.... Income in a Malthusian economy is determined solely by birth and death rate schedules, and is uncorrelated with technology. Using this model, you can explain, for example, why incomes before 1800 were roughly stagnant for centuries despite improving technology.... It is a very simple, and yet powerful, model. And it makes (correct) predictions that many historians (e.g., Kenneth Pomeranz), scientists (e.g., Jared Diamond), and John Bates Clark-caliber economists (see below) get wrong.

Douglas L Campbell On the Uses and Abuses of Economath The Malthusian Models

A second beneficial reason is signalling.... I gratuitously put a version of the Melitz model in my job market paper, and when I interviewed, someone remarked that I was "really teched up!"... versions of peacock feathers....

A third reason to use math is that it is easy to use math to trick people.... If you make your assumptions in plain English, they will sound ridiculous.... A particularly informative example is the Malthusian model proposed by Acemoglu, Johnson, and Robinson in the 2001 version of their "Reversal of Fortune" paper.... They take the same basic assumptions, assign a particular functional form to how population growth is influenced by income, and arrive at the conclusion that population density (which is proportional to technology) will be proportional to income! They use the model:

$ P_{t+1} = {\rho}\left(P_{t}\right) + \lambda\left(y - \bar{y}\right) + {\epsilon}_t $

where $ P_{t+1} $ is population density at time t+1, $ P_{t} $ is population at time t, $ {\rho} $ is a parameter (perhaps just less than 1), $ \lambda $ is a parameter, $ y $ is income, $ \bar{y} $ is the level of Malthusian subsistence income, and $ {\epsilon}_t $ is an error term. If you impose a steady state ($ P^* $ and $ y^* $) and solve for p*, you get:

$ P^* = \frac{\lambda(y - \bar{y})}{1 - \rho} $

Population density is increasing in income, and thus that income per person should have been increasing throughout history.... What exactly was the source of the difference in the classical Malthusian model and the "MIT" malthusian model? The crucial assumption, unstated in words but there in greek letters for anyone to see, was that income affects the level of population, but not the growth rate in population... that a handful of individuals could and would out-reproduce the whole of China and India combined if they had the same level of income.... What this model does successfully is reveal how cloaking an unrealistic assumption in terms of mathematics can make said assumption very hard to detect, even by tenured economics professors at places like MIT. Math in this case is used as little more than a literary device designed to fool the feebleminded. Fortunately, someone caught the flaw, and this model didn't make the published version in the QJE. Unfortunately, the published version still included the view that population density is a reasonable proxy for income in a Malthusian economy, which of course it is not. And the insight that Malthusian forces led to high incomes in the Neo-Europes was also lost.... People are very susceptible to bullshit when written in equations.... This episode shows some truth to Bryan Caplan's view that "The main intellectual benefit of studying economath ... is that it allows you to detect the abuse of economath"...

You can argue that in a preindustrial society a high population density makes you more vulnerable to epidemics—and thus that there is some drag on population from a high population density. But you can equally well argue that in a preindustrial society a low population density keeps you from developing immunity and leaves you a greenfield devastatingly vulnerable to epidemics.