Neal Hockley has criticised me for eliding the distinction between social mobility, equality of opportunity and meritocracy.

He's right. They are different things. For example, if social positions were allocated at random we'd have social mobility and equality of opportunity but not meritocracy. If there were different chances of success, we'd have social mobility but not equality of opportunity and maybe meritocracy or not depending upon how jobs were allocated. And if jobs were allocated by merit, and merit were 100% heritable, we'd have meritocracy but not social mobility.

This last possibility is no mere thought experiment. Some research has found a significant correlation between parental income and hippocampal gray matter density, which in turn suggests that children from poorer backgrounds are, on average, less able to form memories and to learn than their richer counterparts. Poor kids, then, are at a biological disadvantage.

This has important implications for equality of opportunity. If you think that equal opportunity should mean that everyone has an equal chance to become qualified (as distinct from the non-discrimination conception of equal opportunity), then it implies that kids from poor backgrounds should have better education to offset their biological disadvantage.

This is the argument for a pupil premium - extra funding for schools in poor areas.

But this doesn't go far enough. For one thing, it's not obvious that it's sufficient to offset the tendency for better schools to be located in more expensive (pdf) areas, or for increases in the quanity of education to give people from rich families an increasing advantage - two ways in which state-provided education has increased inequalities of opportunity.

And for another thing, the disadvantages associated with a poor background begin before children get to school and - given the weak link (pdf) between school funding and quality - are thus only weakly remediable by education funding.For this reason Julian Betts and John Roemer have estimated that to create substantive equality of opportunity in the US would require massive inequalities in educational funding.

All of this suggests that substantive equality of opportunity is not "equality-lite". it requires very radical changes - and perhaps technically and politically infeasible ones.It might be that policies to achieve greater equality of incomes are more achievable than equality of opportunity.