We all know the usual story about social mobility. It's been falling steadily for ages and continued to fall during the Labour years. Plenty of politicians, journalists and pundits will line up to tell anyone willing to listen that things have got worse.

It sounds like a compelling story. The problem is, it's not clear it's true. To understand why, we need to differentiate between two types of social mobility. The most common, and the one that gets by far the most attention, concerns the extent to which your parents determine your life chances (termed intergenerational mobility by the wonks). By this measure, it's true that studies published during the 2000s showed a fall in social mobility – but those studies compared a cohort born in 1958 with one born in 1970.

Now there are quite a few possible explanations for the fall in mobility between those two groups. Most concern the nature of British society in the 1960s and 1970s compared to the 1970s and 1980s. Many experts highlight the impact of the rapid expansion of university education for the middle classes, while many pundits point to the decline of grammar schools (a view contradicted by recent research). What all these explanations have in common is that they don't have much to do with Tony Blair or Gordon Brown, or anything else going on in Britain in the 2000s.

The truth is, we know much less about what has happened to social mobility of those born since the 1970s, not least because the key study that would have helped shed light on this was cancelled by Margaret Thatcher in 1980 (a mistake that David Willetts has ensured this government won't repeat). We won't get definitive evidence of what really happened to mobility in the Labour years until 2020, when the real "Blair babe" generation enters adulthood.

For now, the best we can do on intergenerational mobility is try to discern likely future trends by looking at the link between parental background and the performance of children around the millennium (using early test results at school). All in all, we can be pretty confident that mobility between generations in the UK – even if it has stopped falling – is still depressingly low, as it tends to be in highly unequal societies.

But that all brings us on to the second definition of social mobility, and here there's more light to shed. It considers the extent to which people can climb the earnings ladder within their own lifetime (it's termed intragenerational mobility). It asks whether people get stuck at the same point in the earnings distribution throughout their career, or whether they can earn their way up relative to their peers.

This type of mobility is almost entirely ignored in the political debate. Only a handful of academics have looked into it. Yet it, too, is fundamental to the character of our society. For many people – those living on low to middle incomes in particular – being able to work your way up in society is of great economic, social and psychological importance. And an economy in which those who start their careers on a high wage always stay at the top – regardless of performance – isn't going to be either fair or efficient.

So, what has happened to this type of mobility? A major new study, published today by Lee Savage at the Resolution Foundation (PDF), tells us what changed in intragenerational mobility during the 2000s compared to the 1990s. It tracks a large number of people in their thirties through the 1990s and compares how socially mobile this group was compared to another group in their thirties during the 2000s.

The results are fascinating. The good news – and there is more of it than you might expect – is that the chances of someone moving a long way up the earnings distribution – enough to really change a person's standard of living – increased by over 20 per cent in the 2000s compared to the 1990s. More interesting still is the change in mobility across the earnings distribution.

As the first chart below shows, when we look at the position of the lowest earners in society, we see a small fall in the proportion who stayed at the bottom of the wage pile throughout the 2000s; a sizeable increase (31 per cent) in those who moved up from the bottom to the middle; and a doubling in the proportion who leaped right up to the top.

Of course, upward mobility requires downward mobility. So when we look at the richest 20 per cent of wage earners (see chart below), it's significant that we see a small fall between the 1990s and the 2000s in the proportion who started the decade at the top and stayed there.

So much for the good news. Most people looking at these charts will, of course, notice something rather bleaker. The overriding story remains that, regardless of whether you were in the 1990s or 2000s generation, if you started off at the top of the earnings distribution you were much more likely to stay there than move somewhere else. And if you started off at the bottom you were likely to stay there, too.

Mobility may have picked up, but from a very low base. The doubling of the chance of moving from the bottom to the top in the 2000s loses much of its gloss when you realise that the absolute increase was from a measly 3 per cent to 6 per cent. So, all in all, some important if modest gains – certainly enough to confound the story of the social mobility pessimists who say things only ever get worse – but not exactly a revolution in opportunity.

Gavin Kelly is the chief executive of the Resolution Foundation.