To crush your enemies, and steal their cattle for your sons!

As I gave a nod to statistical tricks and subtle shell games very recently, the material I review subsequently should be viewed with skepticism and caution. A few days ago I also pointed to a paper which describes and models intergenerational transfers of wealth across various societies. In other words, what parents transmit to children. From the perspective of someone who reads this blog, obviously parents transmit genes to their offspring.

To the left is an old scatterplot from Francis Galton which shows the dependence of the height of children upon the average height of parents. Obviously you have to take into account sex, but if you use standard deviation units and control for gender it works fine. There is a close relationship here between the axes, though it is not perfect. In modern developed societies the heritability of height is ~0.8-0.9. That means that 80-90% of the variance of the trait within the population can be explained by the variance in genes within the population. On an individual level this would result a relatively good correspondence between parental and offspring heights. In terms of the scatterplot, as heritability approaches 1, the slope converges upon 1. As heritability approaches 0, the slope approaches zero. This makes graphical sense, insofar as slope of 0 indicates no ability to predict the value of Y (children) from X. It's just random noise.

Of course there is the problem of confounds in these sorts of studies. One has to control for environmental effects, interactions and correlations. The paper that I would like to put a spotlight on in fact focuses on these issues, and not so much on genetics, except in passing. But the general formalisms resemble those of quantitative genetics, and the issue under examination is heritability of wealth, and therefore the variance in well-being across generations which persists.

To the paper, Intergenerational Wealth Transmission and the Dynamics of Inequality in Small-Scale Societies:

Small-scale human societies range from foraging bands with a strong egalitarian ethos to more economically stratified agrarian and pastoral societies. We explain this variation in inequality using a dynamic model in which a population's long-run steady-state level of inequality depends on the extent to which its most important forms of wealth are transmitted within families across generations. We estimate the degree of intergenerational transmission of three different types of wealth (material, embodied, and relational), as well as the extent of wealth inequality in 21 historical and contemporary populations. We show that intergenerational transmission of wealth and wealth inequality are substantial among pastoral and small-scale agricultural societies (on a par with or even exceeding the most unequal modern industrial economies) but are limited among horticultural and foraging peoples (equivalent to the most egalitarian of modern industrial populations). Differences in the technology by which a people derive their livelihood and in the institutions and norms making up the economic system jointly contribute to this pattern.

As a rule of thumb I generally am skeptical of the findings of social science which utilizes econometrics which are totally counter-intuitive. And of course findings which confirm what we know are not of great interest. To me this paper has wended its way into the Golden Mean, mostly plausible in light of what is known, but with a few kinks that make it non-trivial in its results. To me the primary one is the parity, and to some extent superiority, of heritability and value placed on material wealth among pastoralists vis-a-vis agriculturalists. Many would be surprised, though it is notable that one of the most powerful pastoralists known to history likely conferred upon his descendants and relatives incredible reproductive fitness just by their relation to him.

The heart of this paper is a simple (relatively) model that offers up some parameters, and the data sets which they crunch to see what the real empirical statistics in the world are. So consider:

w_i = βw_ip + ( 1 - βw ) + λ_i

w = wealth (or, more accurately, the log of wealth)

β = heritability of the wealth, it is basically h², the slope of the offspring-on-parent regression line

λ = "exogenous shock", random outside factors which might increase or decrease wealth

The subscripts "i" and "p" refer to the individual in the current generation and the parental generation.

In terms of insight, what the equation above tells you is that when β = 1 there is no regression back to the mean of the population in wealth. Differences between parent and child are simply a function of the random outside forces. As β approaches zero the power of noise to disrupt parent-child correlations diminishes, and regression back to the mean becomes a more noticeable phenomenon.

They also focused on the variation in wealth within a given generation. This sort of inequality is often measured by the Gini coefficient. To model the dispersion of wealth, they used this equation:

μ = σ²_λ/(1 - β²)

This a steady-state variance, the state when the population is at equilibrium. As β approaches 1 the variance within population of wealth starts to get really large. This is because of the long term multiplier effects which amplify the importance of exogenous shocks, the variance of which is defined by σ²_λ. The rich get richer, and the poor get poorer, and down unto the generations are boons and curses passed. Naturally the bigger the range of shocks, the numerator, the more variance in wealth there is in a society as well.

Now we move to variables that they purport to measure more directly in the paper.

W = λE^eM^mR^r

W is well being, which is contingent upon wealth. &lambda you met above. E is "embodied" wealth. Like the genetic endowment of size. M is material wealth. I hope you are familiar with that. And finally, R is "relational" wealth. Basically the social networks which you use to get by in life, more or less (more in Italy, less in Finland). What are the superscripts? They measure the weight of each wealth component within a society, so that:

e + m + r = 1

In a society where all that matters for wealth is material accumulation, m = 1, and e & r = 0. Additionally, the authors use a multiplicative as opposed to additive model to generate well being because they believe that different types of wealth can complement and presumably lead to a greater fullness of life. Throughout the paper they refer to the lettered weights above more generally as α.

In terms of their formalism and how they calculate β, α, etc. for each society they're rather tight-lipped in the text of the paper. It's all in the supplements, which are free for anyone to read (as well as checks for robustness).

Let's move to some data. Here are the societies which they surveyed:

And now their β values (I've removed references to the primary literature which was used to calculate the β):

Remember, β is basically the heritability of wealth. In some of these societies it seems that wealth is basically not heritable, while in others it is significantly heritable. In qantitative genetics heritabilities north of 0.5 are not shabby, and some of these are well north of 0.6. The letters again signify the wealth class. It is clear from visual inspection that material wealth, M, is more heritable than relational wealth, R, and this is more heritable than embodied wealth, E.

Table 2 combines the different societies and puts them into 4 categories, hunter-gatherer, horticulturalist, agriculturalist and pastoralist. The β values for the three different wealth classes are shown for each lifestyle category, as well as the α values, which indicate the weight within that category of that class of wealth. How did they calculate the α's? They relied on ethnographers. For a longer answer, they put it in the supplements. Not only does this table show heritability, it also shows the inequality, the Gini coefficient, weighted by the value of that wealth class in a given society. Obviously if material wealth is distributed unequally, but few put much stock in material wealth, then it is of less consequence than would otherwise be the case.

The interesting result here is that pastoralists and agriculturalists are in one cluster, and hunter-gatherers and horticulturalists are in another cluster. Horticulturalists seems to mean slash & burn agriculturalists, so that productivity is not land limited, but labor limited. As for the nomadic pastoralist, these data suggest that they were subject to the same forces of inequality which were the burden of civilized peasant societies. Another thing to observe is that heritability of wealth is correlated with more inequality. Expected, but nice to see it born out by these data (though if it wasn't born out by these data, a cynic might ask whether this paper would have been submitted!). Here is a table from the supplements which shows the Gini coefficients in more detail:

What sort of factors are driving the heritability of wealth and inequality? To analyze this they threw the hunter-gatherers & horticulturalists into one category, and the farmers & pastoralists in another. They then use M, R and E as proxies to infer that technology is responsible for 45% of the difference, and institutions 55%. In the former case, the capital which more advanced societies require can often be transmitted, and also used to amplify wealth and generate economies of scale. In the latter case there are cultures which enforce gift-giving, lavish funerals and feasts, and so forth on wealthy individuals (the Athenians demanded litguries from their elites, the Romans "bread & circuses").

At this point one might want something more concrete, and there is something to that effect in the supplements. Specifically, a table which shows the ratio of the probability that someone born in the top 10th and 20th percentiles will retain that status to the probability that someone in the bottom 10th and 20th percentiles will obtain that status. So for example if the value is 10 for the decile at a given β (remember, β is measuring parent-child transmission of wealth), that means there is a 10 times greater odds that a person born in the top decile will remain in that decile than that someone in the bottom decile will ascend to the top decile. Here is the table:

It is important to note that even small values of β make a large difference in terms of odds when it comes to movement between the top and bottom of society. Interestingly, they also found that both wealth class (M, R and E) and lifestyle (agricultural, etc.) had an independent impact on wealth inequality. In other words, lifestyle's effect can not be reduced to the weights of influence of wealth class. They claim this is a robust finding, but I don't know what to make of it. Perhaps there are institutional effects which are systematic among agriculturalists and pastoralists generating this relationship.

Finally, they found that the more important a wealth class was (the α) the higher its heritability was (the β). This makes sense; if you value a particular type of wealth you will take greater pains to pass it on.

I'll let them conclude:

Our principal conclusion is that there exist substantial differences among economic systems in the intergenerational transmission of wealth and that these arise because material wealth is more important in agricultural and pastoral societies and because, in these systems, material wealth is substantially more heritable than embodied and relational wealth. By way of comparison, the degree of intergenerational transmission of wealth in hunter-gatherer and horticultural populations is comparable to the intergenerational transmission of earnings in the Nordic social democratic countries—the average β for earnings in Denmark, Sweden, and Norway is 0.18— whereas the agricultural and pastoral societies in our data set are comparable to economies in which inequalities are inherited most strongly across generations, the United States and Italy, where the average β for earnings is 0.43. Concerning wealth inequality, the Gini measure in the hunter-gatherer and horticultural populations is almost exactly the average of the Gini measure of disposable income for Denmark, Norway, and Finland (0.24); the pastoral and agricultural populations are substantially more unequal than the most unequal of the high-income nations, the United States, whose Gini coefficient is 0.37....

The part that caught my attention obviously is that the most egalitarian advanced countries exhibit levels of wealth heritability on par with hunter-gatherers. I'm struck by this because I've been mooting the idea for years that modern Western societies have seen somewhat of a shift back toward values of hunter-gatherers. Specifically, the monopolies of power, influence and status which accrued through coordinated actions of male lineage groups as well as socially dominant institutions seems to be decreasing. People from more "traditional" countries are wont to characterize the West as "decadent," commenting in particular upon women who are given the license of freedoms which would bring dishonor upon the lineage of men of power in many "traditional" societies. I put the quotes there because I believe that these traditional societies are transients, they emerged after the rise of peasant polities which slid along the Malthusian boundary. The elites of these societies had to control their populations, while the common people had to make decisions which maximized their survival to the next day. Some uncertainty and volatility might have been mitigated by grain storage, but the day to day reality of life was miserable. In these mass societies a small elite was able to control and extract surplus rents to fuel a life of relative ease and affluence.

Peter Turchin, a quantitative ecologist, has constructed a model which is predicated on demographic cycles based on these extraction patterns by elites. Turchin's model takes for granted the values of β and α which are specified here. He illustrates how historical events may hinge on the flux of Gini coefficients, roughly conceived. Hunter-gatherer and pastoralist societies were no utopias, and in fact they may have remained further below the Malthusian limit because of heightened mortality from war and conflict. But, their relatively dispersed and decentralized nature, along with the low βs reported here, made it difficult for powerful rent-seeking lineages and classes to emerge. Without these rent-seeking lineages humans lived on a more animal level of individual choice, and personal satisfaction. Though I have come to doubt the utility of the "Stone Age Mind" model of classical evolutionary psychology, I do think there is some truth or likelihood that aspects of our societies or cultures are at sharp variance with human nature. Traditional societies which arrange marriages, or strictly control the affairs of women, still produce literature or oral narratives which valorize forbidden love (see the tale of Diarmuid and Gráinne). To my knowledge there are few ballads singing the praises of arranged matches.

Fleshing out the connection between this theory and description with what is being reported in this paper is for another time. In particular, I need to think about the implication in regards to the high heritability of wealth among pastoralists. If you don't have access to the paper, the free supplements are sufficient (I've shown you the tables within the text anyhow).

Citation: Intergenerational Wealth Transmission and the Dynamics of Inequality in Small-Scale Societies, Monique Borgerhoff Mulder, Samuel Bowles, Tom Hertz, Adrian Bell, Jan Beise, Greg Clark, Ila Fazzio, Michael Gurven, Kim Hill, Paul L. Hooper, William Irons, Hillard Kaplan, Donna Leonetti, Bobbi Low, Frank Marlowe, Richard McElreath, Suresh Naidu, David Nolin, Patrizio Piraino, Rob Quinlan, Eric Schniter, Rebecca Sear, Mary Shenk, Eric Alden Smith, Christopher von Rueden, and Polly Wiessner (30 October 2009), Science 326 (5953), 682. [DOI: 10.1126/science.1178336]