The École d'économie de Paris (the Paris School of Economics) is actually situated in the most un-Parisian part of the city. It is on the boulevard Jourdan in the lower end of the 14th arrondissement, bordered on one side by the Parc Montsouris. Unlike most French parks, there is a distinct lack of Gallic order here; in fact, with lakes, open spaces, and its greedy and inquisitive ducks, you could very easily be in a park in any British city. The campus of the Paris School of Economics, however, looks unmistakably and reassuringly like nearly all French university campuses. That is to say, it is grey, dull and broken down, the corridors smelling vaguely of cabbage. This is where I have arranged an interview with Professor Thomas Piketty, a modest young Frenchman (he is in his early 40s), who has spent most of his career in archives and collecting data, but is just about to emerge as the most important thinker of his generation – as the Yale academic Jacob Hacker put it, a free thinker and a democrat who is no less than "an Alexis de Tocqueville for the 21st century".

This is on account of his latest work, which is called Capital in the Twenty-First Century. This is a huge book, more than 700 pages long, dense with footnotes, graphs and mathematical formulae. At first sight it is unashamedly an academic tome and seems both daunting and incomprehensible. In recent weeks and months the book has however set off fierce debates in the United States about the dynamics of capitalism, and especially the apparently unstoppable rise of the tiny elite that controls more and more of the world's wealth. In non-specialist blogs and websites across America, it has ignited arguments about power and money, questioning the myth at the very heart of American life – that capitalism improves the quality of life for everyone. This is just not so, says Piketty, and he makes his case in a clear and rigorous manner that debunks everything that capitalists believe about the ethical status of making money.

The groundbreaking status of the book was recognised by a recent long essay in the New Yorker in which Branko Milanovic, a former senior economist at the World Bank, was quoted as describing Piketty's volume as "one of the watershed books in economic thinking". In the same vein, a writer in the Economist reported that Piketty's work fundamentally rewrote 200 years of economic thinking on inequality. In short, the arguments have centred on two poles: the first is a tradition that begins with Karl Marx, who believed that capitalism would self-destruct in the endless pursuit of diminishing profit returns. At the opposite end of the spectrum is the work of Simon Kuznets, who won a Nobel prize in 1971 and who made the case that the inequality gap inevitably grows smaller as economies develop and become sophisticated.

Piketty says that neither of these arguments stand up to the evidence he has accumulated. More to the point, he demonstrates that there is no reason to believe that capitalism can ever solve the problem of inequality, which he insists is getting worse rather than better. From the banking crisis of 2008 to the Occupy movement of 2011, this much has been intuited by ordinary people. The singular significance of his book is that it proves "scientifically" that this intuition is correct. This is why his book has crossed over into the mainstream – it says what many people have already been thinking.

"I did deliberately aim the book at the general reader," says Piketty as we begin our conversation, "and although it is obviously a book which can be read by specialists too, I wanted the information here to be made clear to everyone who wants to read it.' And indeed it has to said that Capital in the Twenty-First Century is surprisingly readable. It is packed with anecdotes and literary references that illuminate the narrative. It also helps that it is fluently translated by Arthur Goldhammer, a literary stylist who has tackled the work of the likes of Albert Camus. But even so, as I note that Piketty's bookshelves are lined with such headache-inducing titles as The Principles of Microeconomics and The Political Influence of Keynesianism, simple folk like me still need some help here. So I asked him the most obvious question I could: what is the big idea behind this book?

"I began with a straightforward research problematic," he says in elegant French-accented English. "I began to wonder a few years ago where was the hard data behind all the theories about inequality, from Marx to David Ricardo (the 19th-century English economist and advocate of free trade) and more contemporary thinkers. I started with Britain and America and I discovered that there wasn't much at all. And then I discovered that the data that did exist contradicted nearly all of the theories including Marx and Ricardo. And then I started to look at other countries and I saw a pattern beginning to emerge, which is that capital, and the money that it produces, accumulates faster than growth in capital societies. And this pattern, which we last saw in the 19th century, has become even more predominant since the 1980s when controls on capital were lifted in many rich countries."

So, Piketty's thesis, supported by his extensive research, is that financial inequality in the 21st century is on the rise, and accelerating at a very dangerous pace. For one thing, this changes the way we look at the past. We already knew that the end of capitalism predicted by Marx never happened – and that even by the time of the Russian revolution of 1917, wages across the rest of Europe were already on the rise. We also knew that Russia was anyway the most undeveloped country in Europe and it was for this reason that communism took root there. Piketty goes on to point out, however, that only the varying crises of the 20th century – mainly two world wars – prevented the steady growth of wealth by temporarily and artificially levelling out inequality. Contrary to our perceived perception of the 20th century as an age in which inequality was eroded, in real terms it was always on the rise.

In the 21st century, this is not only the case in the so-called "rich" countries – the US, the UK and western Europe – but also in Russia, China and other countries which are emerging from a phase of development. The real danger is that if this process is not arrested, poverty will increase at the same rate and, Piketty argues, we may well find that the 21st century will be a century of greater inequality, and therefore greater social discord, than the 19th century.

As he explains his ideas to me with formulae and theorems, it still sounds a little too technical (I am someone who struggled with O-level maths). But by listening carefully to Piketty (he is clearly a good and patient teacher) and by breaking it down into bite-sized chunks it does all start to make sense. For this beginner he explains that income is a flow – it moves and can grow and change according to output. Capital is a stock – its wealth comes from what has been accumulated "in all prior years combined". It's a bit like the difference between an overdraft and a mortgage, and if you don't ever get to own your house you'll never have any stock and always be poor.

Student protests against tuition fee increases in 2010. Piketty says: 'It is a perfect example of how to inflict debt on the public sector.' Photograph: Dominic Lipinski/PA

In other words, in global terms what he is saying is that those who have capital and assets that generate wealth (such as a Saudi prince) will always be richer than entrepreneurs who are trying to make capital. The tendency of capitalism in this model is to concentrate more and more wealth in the hands of fewer and fewer people. But didn't we already know this? The rich get rich and the poorer get poorer? And didn't the Clash and others sing about it in the 1970s?

"Well actually, we didn't know this, although we might have guessed at it," says Piketty, warming to his theme. "For one thing this is the first time we have accumulated the data which proves that this is the case. Second, although I am not a politician, it is obvious that this movement, which is speeding up, will have political implications – we will all be poorer in the future in every way and that creates crisis. I have proved that under the present circumstances capitalism simply cannot work."

Interestingly, Piketty says that he is an anglophile and indeed began his research career with a study of the English system of income tax ("one of the most important political devices in history"). But he also says that the English have too much blind faith in markets which they do not always understand. We discuss the current crisis in British universities, which having imposed fees now find that they are short of cash because the government miscalculated what students would have to pay and is now unable to ensure that the loans handed out to cover the fees will ever be repaid. In other words, the government thought it was on to a sure money-maker by introducing fees; in fact, because it could not control all the variables of the market, it was gambling with the nation's money and looks set to lose spectacularly. He chuckles: "This is a perfect example of how to inflict debt on to the public sector. Quite extraordinary and quite impossible to imagine in France."

For all that he is keen on Britain and the United States, Piketty says that he only really feels at home in France. Capital in the Twenty-First Century is constructed out of a plethora of French references (the historian François Furet is key), and Piketty declares that he understands the French political landscape best of all. He was brought up in Clichy in a mainly working-class district and his parents were both militant members of Lutte Ouvrière (Workers' Struggle) – a hardcore Trotskyist party which still has a significant following in France. Like many of their generation, disappointed by the failure of near-revolution of May '68, they dropped out to raise goats in the Aude (this was a classic trajectory for many babacools – leftist hippies – of that generation). The young Piketty worked hard at school, however, studying in Paris and finishing up with a PhD from the London School of Economics at the age of 22. He then moved on to Massachusetts Institute of Technology, where he was a noted prodigy, before moving back to Paris to finally become director of the school where we are now sitting.

Piketty advocates a wealth tax, but admits that making the rich pay more will be difficult, as François Hollande has found. Photograph: Philippe Wojazer/Reuters

His own political itinerary began, he tells me, with the fall of the Berlin Wall in 1989. He set out to travel across eastern Europe and was fascinated by the wreckage of communism. It was this initial fascination that led him towards a career as an economist. The gulf war of 1991 also influenced him. "I could see then that so many bad decisions were taken by politicians because they did not understand economics. But I am not political. It is not my job. But I would be happy if politicians could read my work and draw some conclusions from it."

This is slightly disingenuous as Piketty did actually work as an adviser to Ségolène Royal in 2007, when she was the socialist candidate in the presidential elections. This was not a happy period for him – his love affair with the politician and novelist Aurélie Filipetti, another Royal acolyte, ended around then with acrimonious accusations on both sides. Fair enough, after this murky business, that Piketty might want to distance himself from the everyday rough and tumble of real politics.

But no matter. What have we learned? Capitalism is bad. Hooray! What's the answer? Socialism? Hope so. "It is not quite so simple," he says, disappointing this former teenage Marxist. "What I argue for is a progressive tax, a global tax, based on the taxation of private property. This is the only civilised solution. The other solutions are, I think, much more barbaric – by that I mean the oligarch system of Russia, which I don't believe in, and inflation, which is really just a tax on the poor." He explains that oligarchy, particularly in the present Russian model, is quite simply the rule of the very rich over the majority. This is both tyrannical and not much more than a form of gangsterism. He adds that the very rich are not usually hurt by inflation – their wealth increases anyway – but the poor suffer worst of all with a rising cost of living. A progressive tax on wealth is the only sane solution.

But for all that he is talking sense, much of it common sense, I put to him that no political party in Britain or the United States, of left or right, would dare to go to the polls with such idealistic ideas. The present government of François Hollande is widely despised not because of the president's sexual peccadilloes (in contrast, these are pretty much widely admired) but because of the punitive tax regime he has been seeking to impose.

"This is true," he says. "Of course it is true. But it is also true, as I and my colleagues have demonstrated in this book, that the present situation cannot be sustained for much longer. This is not necessarily an apocalyptic vision. I have made a diagnosis of the past and present situations and I do think that there are solutions. But before we come to them we must understand the situation. When I began, simply collecting data, I was genuinely surprised by what I found, which was that inequality is growing so fast and that capitalism cannot apparently solve it. Many economists begin the other way around, by asking questions about poverty, but I wanted to understand how wealth, or super-wealth, is working to increase the inequality gap. And what I found, as I said before, is that the speed at which the inequality gap is growing is getting faster and faster. You have to ask what does this mean for ordinary people, who are not billionaires and who will never will be billionaires. Well, I think it means a deterioration in the first instance of the economic wellbeing of the collective, in other words the degradation of the public sector. You only have to look at what Obama's administration wants to do – which is to erode inequality in healthcare and so on – and how difficult it is to achieve that, to understand how important this is. There is a fundamentalist belief by capitalists that capital will save the world, and it just isn't so. Not because of what Marx said about the contradictions of capitalism, because, as I discovered, capital is an end in itself and no more."

Piketty delivers this speech, erudite and powerful, with a quiet passion. He is, one would guess, a relatively modest and self-effacing character, but he loves his subject and it is indeed a delight to find oneself in the midst of a private seminar on money and how it works. His book is indeed long and complicated but anyone who lives in the capitalist world, which is all of us, can understand the arguments he makes about the way it works. One of the most penetrating of these is what he has to say about the rise of managers, or "super-managers", who do not produce wealth but who derive a salary from it. This, he argues, is effectively a form of theft – but this is not the worst crime of the super-managers. Most damaging is the way that they have set themselves in competition with the billionaires whose wealth, accelerating beyond the economy, is always going to be out of reach. This creates a permanent game of catch-up, whose victims are the "losers", that is to say ordinary people who do not aspire to such status or riches but must be despised nonetheless by the chief executives, vice-presidents and other wolves of Wall Street. In this section, Piketty effectively rips apart one of the great lies of the 21st century – that super-managers deserve their money because, like footballers, they have specialised skills which belong to an almost superhuman elite.

"One of the great divisive forces at work today," he says, "is what I call meritocratic extremism. This is the conflict between billionaires, whose income comes from property and assets, such as a Saudi prince, and super-managers. Neither of these categories makes or produces anything but their wealth, which is really a super-wealth that has broken away from the everyday reality of the market, which determines how most ordinary people live. Worse still, they are competing with each other to increase their wealth, and the worst of all case scenarios is how super-managers, whose income is based effectively on greed, keep driving up their salaries regardless of the reality of the market. This is what happened to the banks in 2008, for example."

It is this kind of thinking that makes Piketty's work so attractive and so compelling. Unlike many economists he insists that economic thinking cannot be separated from history or politics; this is what gives his book the range the American Nobel laureate Paul Krugman described as "epic" and a "sweeping vision". Piketty's influence indeed is growing well beyond the small enclosed micro-society of academic economists. In France he is becoming widely known as a commentator on public affairs, writing mainly for Le Monde and Libération, and his ideas are frequently discussed by politicians of all hues on current affairs programmes such as Soir 3. Perhaps most importantly, and unusually, his influence is growing in the world of mainstream Anglo-American politics (his book is apparently a favourite in the Miliband inner circle) – a place traditionally indifferent to French professors of economics. As poverty increases across the globe, everyone is being forced to listen to Piketty with great attention. But although his diagnosis is accurate and compelling, it is hard, almost impossible, to imagine that the cure he proposes – tax and more tax – will ever be implemented in a world where, from Beijing to Moscow to Washington, money, and those who have more of it than anyone else, still calls the shots.

Thomas Piketty will be talking about Capital at the IPPR, London WC2, on 30 April, and at the LSE, London WC2, on 16 June