Milk and economic development: No use crying

http://www.economist.com/news/finance-and-economics/21647352-ability-digest-milk-may-explain-how-europe-got-rich-no-use-crying

Humans can digest lactose with the help of an enzyme called lactase.  Two-thirds of people stop producing it after they have been weaned.  However, one-third of people have "lactase persistence" which allows them to continue to produce lactase into adulthood.  The article discusses a recent paper that argues that this genetic quirk helps to explain why some countries are rich and others are poor.

Justin Cook, the author of the paper, uses data on historical migration flows to estimate the ethnic composition of 108 countries in Africa, Asia, and Europe in 1500.  He then estimates what proportion of the population would have been able to digest milk based off data on the lactose tolerance of different ethnic groups.  Cook estimates that pre-colonial countries in western Europe tended to have the highest rates of lactase persistence.  The lowest levels were in Sub-Saharan Africa and South-East Asia.

One-standard-deviation increase in the incidence of lactase persistence was associated with a 40% rise in population density.  People who could digest milk used resources more efficiently than those who couldn't.  They could extract milk, or liquid energy, from livestock.  The nutritional value of milk also could have balanced the pre-colonial diet to reduce the incidence of disease.  If animal milk was used as a substitute for breast-feeding, it could reduce weaning time and thus, the time between mothers' pregnancies.  These ideas support the theory that societies with higher lactase persistence could support higher population densities.

As common theory goes, the higher the population density, the faster the growth.  Areas with higher population densities see an increase in infrastructure, an increase in the development of cities, and an increase in technological innovation.  The main point of the paper is that areas with higher population densities in pre-colonial times tend to be relatively rich today.

Do you think it is a coincidence that areas with higher population densities in pre-colonial times are relatively richer today than areas with lower population densities in pre-colonial times?  Do you think that Cook's theory makes sense?  Are there weaknesses in his theory?  What are the strengths of his theory?