Better schools must mean better social mobility. That is the common sense idea widely shared across political parties. The good news is that school standards are rising and the gap in educational performance between advantaged and disadvantaged pupils is narrowing. But the uncomfortable truth is that there is no straightforward link between this better education and better social mobility. Here are three possible reasons.

First, even though state schools are free, they operate within a wider market economy. Even if parents cannot pay to get their children into what they see as a good school directly, they can do so indirectly by buying houses in the neighbourhood. So if a school improves, house prices in its area rise and, other things being equal, it becomes more socially selective.

According to the latest research from the Department for Education, there is a 7 per cent increase in house prices around the top decile of secondaries and 8 per cent for the top decile of primary schools. This is before some schools add subtle forms of selection to ensure they boost their performance, admitting the children most likely to perform well. The pressures to be a “good” school exacerbate this behaviour and can have the opposite effect to what is intended.

John Goldthorpe, the doyen of British social mobility researchers, adds a second challenge to the conventional wisdom. The good news is the slight decline in the importance of social class in determining if a child gets to a “good” school. But there has also been a weakening of the link between an individual’s educational performance at school and the quality of the job they get afterwards. Employment has become more socially selective and this has more than offset the weakening of social selectivity of schooling.

I wrestled with a version of this problem when university minister. We worked hard to get universities to recruit more widely and we made some progress. But even when a disadvantaged student ended up with the same degree as a more advantaged student, their job prospects then diverged.

My successor, Jo Johnson, has rightly broadened the role of the Office for Fair Access, so that universities can spend access funding on programmes helping disadvantaged students into the jobs market, such as helping with the costs of unpaid internships during the summer vacation.

The third issue is that England has an intense educational arms race, which I investigate in my forthcoming book, A University Education. Our system of nationwide competitive entry to university is unusual. In many countries there is a right to go to your local university if a student gets above a certain mark in a school-leaving exam.

But in England teenagers have to apply individually to specific universities that operate competitive entry, with nobody guaranteed a place. It makes competition to boost school exam performance intense.

This arms race extends further up the educational scale. Jobs that used to require A levels now require a degree and even a master’s qualification, as jobs become more complex and demanding.

Looking beyond schools

We need to extend public funding for education out later and later. For example, the government is now introducing loans for master’s students - an excellent initiative to remove a financial barrier to social mobility.

This is different from the conventional view that educational resources should be shifted to early years. It is also different from the fashionable idea that there should be a fixed pot of loan funding for every 18-year-old to spend on tertiary education, which would be a barrier to funding master’s students and mature students, who may already have done an apprenticeships, or to graduates going back to university for a second chance. We also need to do more to deliver education and training for the 50 per cent not currently going to university.

Behind all this, there is a broader question I hope we will address at the forthcoming discussion at the Institute of Education (see box, below). Imagine you are an education minister who secures an extra billion pounds a year to spend on education to boost social mobility. What would you spend it on? The evidence to answer this question is not robust. I would be tempted by investing more in 16-18 year olds who were protected neither by the school spending pledge nor benefited from the extra resources for universities when we moved to £9,000 fees and loans. Or I would put more into adult literacy and numeracy programmes.

Good schools are, of course, better than bad schools. But beyond that, we have probably too much hope in that stage of education as a means of boosting social mobility.

Lord Willetts is executive chair of the Resolution Foundation. His book, A University Education, is published by Oxford University Press on 23 November