Britain's housing woes might seem like a relatively recent predicament, but Kandinsky's super-smart devised piece ties them to the past. Offering a potted history of post-war property, Trap Street finds the foundations of today's crisis in the fabric of the welfare state itself.

Skipping back and forth in time, Trap Street follows a single-parent, working class family over half a century. In 1967, Valerie Welch moves her two young children into the new, prettily-named Austen Estate. In 2017, her daughter returns from her Spanish home – bought with the rent that followed right-to-buy – to find it all but run-down. It's managed by property guardians, crowded with immigrants and due for demolition to clear space for luxury flats.

Writers James Yeatman and Lauren Mooney splice that story with social context – news reports and census stats, names, dates and facts – without ever turning Trap Street into a documentary. Its argument is that the social idealism that made space for the poor in the mid-Sixties ultimately ends up partitioning them off. For all their utopian hope, the brutalist inner-city estates and the concrete new towns were ghettos-in-waiting. In empowering the poor they enabled individualism and, with it, the breakdown of the very communities that were forged. A motif of cleaning makes the case for constant care.

Kandinsky's title comes from cartography. Trap streets were fake roads inserted by early mapmakers to throw plagiarists off. These streets both existed and didn't; there, but not there. Yeatman and Mooney thread that image throughout, framing gentrification as a process of social erasure. It pops up all over the place – in virtual reality tours designed for overseas investors and in affordable housing that gets dropped when profits dip. The absences find echoes in the skips and scratches of Zac Gvirtzman's live vinyl and oboe score.

There are two other invisible, ever-presents: class and the patriarchy. While Valerie constantly favours her son, Graham, over her daughter Andrea, a Jane Austen-style estate sits as the backdrop to Joshua Gadsby and Naomi Kuyck-Cohen's design. For all that 50 years seems to eliminate class and encourage gender equality – names change, so that Terrys and Vals become Ruperts and Esmes, then Saamirs and Thandis – the two social structures are always lurking in the background; there, but not there. Just as trap streets find their way from map to map – several still exist in today's A to Zs – structural errors ripple through the years as well.

It is, perhaps, too catch-all a cause. The idea that architects and policymakers are more likely to be men, that women are shut out, is fine and true – only one in ten senior partners at major architectural firms are female – but ‘It's the patriarchy, stupid' feels like no answer at all. It's almost so over-riding as to seem like a shrug.

Nonetheless, from a collage of images and narrative fragments, Trap Street makes an intricate, rigorous case. Kandinsky's one of the brightest young companies around, but their theatre is almost entirely cerebral – more essayistic than it is fully embodied. The Welches exist to illustrate a thesis, but they never really take on lives of their own and, really, only one moment asks us to care – the sight of a woman being prized from her home. Andrea Welch (Danusia Samal) brandishes a bedsheet with a painted protest: I LIVE HERE. She's asserting her very existence in a world that ignores her – there, but not there to the end.

Trap Street runs at the New Diorama until 31 March.