[Update: I'm getting an awful feeling this post may be totally wrong. Does (1/b)/[1+Var(V(i))/Var(Y(i))] = b if the Lamarck model is true? If so, my proposal won't work. Wish I could do math.] [Update2. Nope, I think it will work. But Majromax in comments has come up with a simpler way to do the same thing. See my 01:19 pm comment.]
Economists normally run a regression of son's income on father's income, and interpret the estimate beta as a measure of "intergenerational mobility". (High beta means low "mobility"). I think we could learn something interesting by reversing the regression -- run a regression of father's income on son's income -- and comparing the results of the two regressions.
This post is about: econometrics (errors in variables bias); genetics (regression toward the mean); and mobility/inequality. I don't know much about any of those three areas, so I will probably get things wrong. I am not 100% confident my proposal will work, but I would like people with better technical skills than me to take a look at it.
Ideally, we should all agree on how we would interpret the results before we run any regressions.