Something curious happened when I tried to potty train my two-year-old recently. To begin with, he was very keen on the idea. I'd read that the trick was to reward him with a chocolate button every time he used the potty, and for the first day or two it went like a breeze – until he cottoned on that the buttons were basically a bribe, and began to smell a rat. By day three he refused point-blank to go anywhere near the potty, and invoking the chocolate button prize only seemed to make him all the more implacable. Even to a toddler's mind, the logic of the transaction was evidently clear – if he had to be bribed, then the potty couldn't be a good idea – and within a week he had grown so suspicious and upset that we had to abandon the whole enterprise.

It's a pity I hadn't read What Money Can't Buy before embarking, because the folly of the chocolate button policy lies at the heart of Michael Sandel's new book. "We live at a time when almost everything can be bought and sold," the Harvard philosopher writes. "We have drifted from having a market economy, to being a market society," in which the solution to all manner of social and civic challenges is not a moral debate but the law of the market, on the assumption that cash incentives are always the appropriate mechanism by which good choices are made. Every application of human activity is priced and commodified, and all value judgments are replaced by the simple question: "How much?"

Sandel leads us through a dizzying array of examples, from schools paying children to read – $2 (£1.20) a book in Dallas – to commuters buying the right to drive solo in car pool lanes ($10 in many US cities), to lobbyists in Washington paying line-standers to hold their place in the queue for Congressional hearings; in effect, queue-jumping members of the public. Drug addicts in North Carolina can be paid $300 to be sterilised, immigrants can buy a green card for $500,000, best man's speeches are for sale on the internet, and even body parts are openly traded in a financial market for kidneys, blood and surrogate wombs. Even the space on your forehead can be up for sale. Air New Zealand has paid people to shave their heads and walk around wearing temporary tattoos advertising the airline.

According to the logic of the market, the matter of whether these transactions are right or wrong is literally meaningless. They simply represent efficient arrangements, incentivising desirable behaviour and "improving social utility by making underpriced goods available to those most willing to pay for them". To Sandel, however, the two important questions we should be asking in every instance are: Is it fair to buy and sell this activity or product? And does doing so degrade it? Almost invariably, his answers are no, and yes.

Sandel, 59, has been teaching political philosophy at Harvard for more than 30 years, and is often described as a rock star professor, such is the excitement his lectures command. In person there is nothing terribly rock star about him; he grew up in a middle-class Jewish family in Minneapolis, studied for his doctorate at Balliol college in Oxford as a Rhodes Scholar, and has been married for decades to a social scientist with whom he has two adult sons. His career, on the other hand, is stratospheric.

Sandel's justice course is said to be the single most popular university class on the planet, taken by more than 15,000 students to date and televised for a worldwide audience that runs into millions. His 2009 book Justice, based upon the course, became a global bestseller, sparking a craze for moral philosophy in Japan and earning him the accolade "most influential foreign figure" from China Newsweek. If you heard a series of his lectures broadcast on Radio 4 in the spring you would have glimpsed a flavour of his wonderfully discursive approach to lecturing, which is not unlike an Oxbridge tutorial, only conducted with an auditorium full of students, whom he invites to think aloud.

In keeping with his rock star status, Sandel is currently embarked upon a mammoth world tour to promote his new book, and when we meet in London he has almost lost his voice. His next sleep, he croaks, half smiling, isn't scheduled for another fortnight, and he looks quite weak with jetlag. Understandably, then, he isn't quite as commanding as I had expected. But although I found his book fascinating – and in parts both confronting and deeply moving – in truth, until the very last pages I didn't find it quite as persuasive as I had hoped.

This may, as we'll come on to, have something to do with the fact that its central argument is harder to make in the US than it would be here. "It is a harder sell in America than in Europe," he agrees. "It cuts against the grain in America." This is truer today than ever before, he adds, for since he began teaching Sandel has observed in his students "a gradual shift over time, from the 80s to the present, in the direction of individualistic free-market assumptions". The book's rather detached, dispassionate line of inquiry into each instance of marketisation – is it fair, and does it degrade? – was devised as a deliberate strategy to "win over the very pro-market American audience" – and it certainly makes for a coolly elegant read, forgoing rhetoric for forensic examination in order to engage with free market economics in terms the discipline understands. But I'm just not entirely sure it works.

If, like me, you share Sandel's view that moral values should not be replaced by market prices, the interesting way to read What Money Can't Buy is through the eyes of a pro-market fundamentalist who regards such a notion as sentimental nonsense. Does he win you over then?

He certainly provides some fascinating examples of the market failing to do a better job than social norms or civic values, when it comes to making us do the right thing. For example, economists carried out a survey of villagers in Switzerland to see if they would accept a nuclear waste site in their community. While the site was obviously unwelcome, the villagers recognised its importance to their country, and voted 51% in favour. The economists then asked how they would vote if the government compensated them for accepting the site with an annual payment. Support promptly dropped to 25%. It was the potty-and-chocolate-buttons syndrome all over again. Likewise, a study comparing the British practice of blood donation with the American system whereby the poor can sell their blood found the voluntary approach worked far more effectively. Once again, civic duty turned out to be more powerful than money.

However, a true believer in the law of the market would surely argue that all this proves is that sometimes a particular marketisation device doesn't work. For them it remains not a moral debate but simply one of efficacy. Sandel writes about the wrongness of a medical system in which the rich can pay for "concierge doctors" who will prioritise wealthy patients – but to anyone who believes in markets, Sandel's objection would surely cut little ice. They would say it's a question of whether or not the system is fulfilling its purpose. If the primary purpose of a particular hospital is to save lives, then if it treats a millionaire's bruised toe while a poorer patient dies of a heart attack in the waiting room, the marketisation has clearly not worked. But if the function of the hospital is to maximise profits, then treating the millionaire's sore toe first makes perfect sense, doesn't it?

"I suspect that you have – we have – a certain idea of what a hospital is for, such that a purely profit-driven one misses the mark; it's deficient in some way; it falls short of what hospitals are properly for. You would say, wouldn't you, that that hospital – that market-driven one – is not a proper hospital. They've misidentified, really, what a hospital is for. Just as if they were a school that said: 'Our purpose isn't, really, primarily, to educate students, but to maximise revenue – and we maximise revenue by offering certain credentials, and so on,' you'd say: 'Well, that's not a proper school; they're deficient in some way.'"

I would, I agree. But a rabid rightwinger wouldn't. They would say the profit motive is in itself blameless, and pursuing it by mending people's bodies or expanding their minds is no different to making motor cars, as long as it works.

"My point is that the debate, or the argument, with someone who held that view of the purpose of the hospital would be a moral argument about how properly to understand the purpose of a hospital or a school. And, yes, there would be disagreement – but that disagreement, about purpose, would be, at the same time, a moral disagreement. I'd say 'moral disagreement', because it's not just an empirical question: How did this hospital define its mission? It's: What are hospitals properly for? What is a good hospital?"

I don't think that would convince a hardliner at all. Similarly, I imagine a hardline rightwinger might read Sandel's chapter about the practice in the US of corporations taking life insurance policies out on their staff, often unbeknown to the employees, and think: what's the problem? Sandel writes about the "moral tawdriness" of companies having a financial interest in the death of an employee, but as he doesn't suggest it would tempt them to start killing their staff, these policies would strike many on the right as a rational financial investment.

At this point Sandel begins to peer at me across the table with an expression of mild disgust and disbelief. Is this woman really, I think I can see him wondering, from the Guardian? So I explain hastily that I tried very hard to read his book wearing Thatcherite glasses.

"You tried a bit too hard," he says wryly. "You shouldn't have tried so hard. You should have gone with the flow a bit more." Which feels like a disappointing answer.

The irony is that I think Sandel would have written a more powerful book had he not tried to argue the case on free-market economists' own dry, dispassionate terms. It is, as he rightly points out, the language in which most modern political debate is conducted: "Between those who favour unfettered markets and those who maintain that market choices are free only when they're made on a level playing field." But it feels as if by engaging on their terms, he's forcing himself to make an argument with one hand tied behind his back. Only in the final chapter does he throw caution to the wind, and make the case in the language of poetry.

"Consider the language employed by the critics of commercialisation," he writes. "'Debasement', 'defilement', 'coarsening', 'pollution', the loss of the 'sacred'. This is a spiritually charged language that gestures toward higher ways of living and being." And it works, for the book suddenly makes sense to me. His closing elegy to what is lost by a society that surrenders all decisions to the market almost moved me to tears.

"Does that mean I should have just started and ended with the poetry, and forgotten about the argumentative and analytical part?" he asks. "I want to address people who are coming to this from different ideological directions." But funnily enough, I think the poetry might well do a better job of persuading those very sceptics he's trying to convert.

A fascinating question he addresses is why the financial crisis appears to have scarcely put a dent in public faith in market solutions. "One would have thought that this would be an occasion for critical reflection on the role of markets in our lives. I think the persistent hold of markets and market values – even in the face of the financial crisis – suggests that the source of that faith runs very deep; deeper than the conviction that markets deliver the goods. I don't think that's the most powerful allure of markets. One of the appeals of markets, as a public philosophy, is they seem to spare us the need to engage in public arguments about the meaning of goods. So markets seem to enable us to be non-judgmental about values. But I think that's a mistake."

Putting a price on a flat-screen TV or a toaster is, he says, quite sensible. "But how to value pregnancy, procreation, our bodies, human dignity, the value and meaning of teaching and learning – we do need to reason about the value of goods. The markets give us no framework for having that conversation. And we're tempted to avoid that conversation, because we know we will disagree about how to value bodies, or pregnancy, or sex, or education, or military service; we know we will disagree. So letting markets decide seems to be a non-judgmental, neutral way. And that's the deepest part of the allure; that it seems to provide a value-neutral, non-judgmental way of determining the value of all goods. But the folly of that promise is – though it may be true enough for toasters and flat-screen televisions – it's not true for kidneys."

Sandel makes the illuminating observation that what he calls the "market triumphalism" in western politics over the past 30 years has coincided with a "moral vacancy" at the heart of public discourse, which has been reduced in the media to meaningless shouting matches on cable TV – what might be called the Foxification of debate – and among elected politicians to disagreements so technocratic and timid that citizens despair of politics ever addressing the questions that matter most.

"There is an internal connection between the two, and the internal connection has to do with this flight from judgment in public discourse, or the aspiration to value neutrality in public discourse. And it's connected to the way economics has cast itself as a value-neutral science when, in fact, it should probably be seen – as it once was – as a branch of moral and political philosophy."

Sandel's popularity would certainly indicate a public appetite for something more robust and enriching. I ask if he thinks academia could do with a few more professors with rock star status and he pauses for a polite while before smiling. "That's a question I would rather have you answer than me, I would say." That someone as unflashy and mild-mannered as Sandel can command more attention in the US than even a rightwing poster boy academic such as Niall Ferguson must, I would say, be some grounds for optimism. On a purely personal level, I ask, is there any downside to engaging with the world through the eyes of moral philosophy, rather than simple market logic?

"None but the burden of reflection and moral seriousness."