A simple quantitative genetic model must rely on the assumption of an underlying normal distribution of something, EQ we can call it, that is the additive genetic part of whatever determines income. We can impute this since, now, we have no way to measure it. Like IQ originally, and like our construction of AQ,’amish quotient’, here, we could come up with an estimator if we could measure what we think we ought to measure. For the moment we assume that there is such a direction in character space. An immediate problem is that income is far from normally distributed but we can impute a mapping from the observed income distribution to EQ. Income percentiles are well known and published for many countries. The figure is derived from Swedish data given by Björklund and Jäntti .
The the top panel shows the conventional Lorenz curve for the Swedish data: the horizontal axis is income rank and the vertical axis is the percentile of national income at that rank, i.e. it is a conventional cumulative distribution of income. The circles show data points along a green line which is a spline through the data points.
The bottom panel has the same vertical axis but the distribution along the horizontal axis is the imputed normal distribution of EQ. For example the 50th percentile of income maps to the mean of the imputed distribution, the 84th percentile maps to +1 standard deviation of a standard normal, the 16th percentile maps to –1 standard deviation, and so on. This figure gives us estimate of the income of a person given his value of an underlying normally distributed EQ, shown as superimposed on the figure. The virtue of this is that we can instantly apply a century’s worth of quantitative genetic theory and knowledge.
In each panel a computed offspring distribution is shown as a bar two standard deviations wide with a line to the parents’ EQ at each mid-quintile, i.e. at percentiles 10,30,50,70,and 90. The lines are not vertical because of regression to the mean.
Given a quantitative genetic model we know, for example, that offspring of a couple should be distributed symmetrically around the mid-parent value, regressed toward the mean.
A clear exposition of this is in a post in Steve Hsu’s blog , with a contribution from James Lee, here. This particular figure is computed with an additive heritability of 0.9 and a spousal correlation of 0.9. These are both at the high end of plausibility, as suggested by Clark’s data. We also should remember that family (cultural) transmission, of wealth or values or whatever is likely indistinguishable from genetic transmission and will increase the heritability.
From this we can compute, and perhaps derive explicit expressions, for the long term movement, i.e. EQ, of one’s descendants. It should not be difficult to derive longer term expectations from a model like that in the bottom panel. For example starting at some initial EQ the distibution of descendants’ EQ should be distributed in subsequent generations along the EQ distribution, the X axis of the bottom panel, according to some reaction-diffusion process similar to the Fisher-Komogorov equation where the diffusion is given by the the incomplete assortative mating and the heritability less than unity. The deterministic part would describe the greater number of surviving offspring with high EQ as documented in Clark’s Farewell to Alms .
To paraphrase what we hear often in the social sciences, there is no need to invoke any social or cultural transmission at all.
- Clark, G., 2014. The Son Also Rises: Surnames and the History of Social Mobility. Princeton University Press. ↩
- Intergenerational income mobility in Sweden compared to the United States. A. Björklund and M. Jäntti.
The American Economic Review (1997):1009–1018. ↩
- Clark, G., 2007. A Farewell to Alms. Princeton: Princeton University Press. ↩