“This is the best deal, this is the only possible deal,” the Prime Minister told the public a few weeks ago. By the time of the parliamentary debate on the deal, that has become “No one likes this deal, but this is the only possible deal.” In other words, the choices, as many have said, are this deal, no deal or no Brexit.

Yet only yesterday the European Commission’s chief negotiator, Michel Barnier, put a rather different face on it, saying: “If the UK chooses to shift its red lines in the future, and it makes that choice, a choice to be ambitious, and go beyond a simple free trade agreement, which would be quite something, then the European Union will be immediately ready.. to give a favourable response.”

So which is it? The legal position is clear. Michel Barnier was distinguishing between the withdrawal agreement, the long legal text that includes the “backstop”, and the political declaration, which sets out the framework for the future trading relationship between the UK and the EU. It is the latter, not the former, which (according to him) could still be renegotiated.

The political declaration will not be legally binding; whereas the legal commitments in the withdrawal agreement, including the financial settlement and the “backstop”, will – a fundamental weakness of the UK’s position I pointed out well over a year ago. But that doesn’t mean it is just empty words, on either the UK side or that of the EU27. As the government has always said – and as a letter from EU Commission President Jean-Claude Juncker and EU Council President Donald Tusk on Monday reemphasised – they are a package, and both sides are committed to negotiating about the future relationship in good faith on the basis of the political declaration.

We don’t know where those negotiations will end up. But the political declaration sets out where they will start and in which direction they will set off. Yes, another parliament could in principle change course – but that will be far from easy, either domestically or with the EU, however UK politics evolves.

So the political declaration does matter – in economic terms, far more than the withdrawal agreement. And it is here that the Prime Minister’s claim that this is the best or only deal falls down. What the political declaration says now is that the UK will leave the single market and the customs union, and that there will therefore be new regulatory barriers, customs processes and checks. The phrase “frictionless” trade – or even “as near to frictionless as possible” – appears nowhere.

In other words, voting for and signing this deal – including this version of the political declaration – is a mandate from parliament to leave the single market and to introduce major new barriers to trade, in both goods and services, between the EU27 and the UK. And every credible independent analysis shows that this will result in significant economic damage to the UK (and indeed to a lesser extent the EU27). There is no getting around the fact that this is the implication of the deal on the table – and to the credit of government economists, they have said precisely this.

But Barnier’s point is also clear. That this is our choice, not that of the EU. We are choosing to leave the single market – we are not being kicked out of it – because of the red lines that the Prime Minister has chosen. Foremost among those is, of course, her determination to end free movement. As if to emphasise that this is our choice, not the EU’s, the political declaration says: “The United Kingdom has decided that the principle of free movement of persons between the Union and the United Kingdom will no longer apply” – the only passage in the document where it makes clear one side has made a specific choice.

So is there an alternative political declaration that the EU would accept and that would point towards a radically different future relationship – a very different Brexit? Absolutely. It would take officials just a few days to construct a political declaration that dropped the UK red line on freedom of movement and, instead, pointed towards a UK that remained, de facto or de jure, within the single market, with freedom of movement continuing – although perhaps with some new modifications or controls, of the sort that already exist for non-EU participants in the single market. Whether this would look like “Norway”, “Switzerland”, or, more likely, some specifically British solution would take years of negotiation to work out – but the direction of travel would be very clear and very different from that implied by the current deal. And so would the destination.

Would this really be a “better” Brexit? Economically, there is little doubt. From a political perspective, this question is far more complex – we would, at least in some respects, become a “rule-taker” on lots of regulatory issues. But that is likely to be true in practice with any form of Brexit, since the UK just isn’t large enough to go its own way.

But what is clearly false is to claim that it would not “respect the referendum result”, either in a legal and technical sense. We would indeed have left the EU. And not only is the UK public far more relaxed about immigration than it was two years ago, but, more importantly, it has been clear for some time, and across different polls, that if forced to choose the electorate regard staying in the single market as a higher priority than ending free movement.

What about the backstop? Whatever its pros and cons, at this point it is indeed too late for the UK to change it in any significant respect. But equally there is no doubt that if both sides make a political commitment now to “frictionless trade”, meaning a single market and a permanent customs union, that the chance of the backstop becoming operational – and, even it did in a legal sense, the chances of it resulting in any new customs or regulatory barriers between Great Britain and Northern Ireland – are far less.

So if we want a “better” Brexit deal – at least as far as jobs, growth and trade are concerned – it is not the EU that is standing in the way, but the Prime Minister, and her determination to make ending free movement the top UK policy objective – overriding any and all other economic and political considerations – in any deal. If that changes, a better deal – a better Brexit – is there for the taking.

Jonathan Portes is professor of economics and public policy at King’s College London.