As property prices go up, lower earners will be squeezed out. In the absence of a massive house-building programme in central London, the capital will become to feel more like Paris, with a rich centre and a poorer outer-ring. Rent policy can slow or hasten this process but not reverse it.
Should we care? Instinctively I want to answer "yes". I look on with misgivings as the Highbury street on which I live becomes steadily fancier - even though I know that I have contributed to that process and stand to gain from it financially.
It has been an article of faith among socially-minded reformers since the days of Joseph Rowntree and Ebenezer Howard that "mixed communities" are a good thing and income segregation bad. Yet the evidence in favour of mixed-income neighbourhoods is weak. Poor children from rich neighbourhoods do not seem to do any better in life than those from poor neighbourhoods. LSE economists Paul Cheshire and Henry Overman argue that there might even be benefits for poor people living in poor neighbourhoods: shops are cheaper and public services tailored to them.
The evidence on the impacts of mixed communities is indeed fairly mixed, and the point about costs is an important one, but I'm not sure it's the whole story. If the cheaper areas of the future are going to be in the suburbs then the cost of transport for the poor to get to city centre jobs is going to be higher. School quality and environmental amenity are also likely to be lower in cheaper areas - that's part of the reason why they're cheaper, after all.
So I think the benefits to the poor of being 'squeezed out' of affluent city centres are still fairly ambiguous, even leaving aside the very large transitional costs facing anyone who does make such a move (as demonstrated by the fact that the people affected by the cuts to housing benefit generally seem pretty unhappy about it).
More broadly I think these dilemmas highlight a very important shift in how our cities function. To simplify massively, in the past when cities had lots of dirty industry they had dirty environments as a result, particularly towards the centre. That encouraged richer people to move out of inner cities as soon as they could afford to and transport allowed. On the other hand, poorer people could save on transport costs by living in the centre, close to the jobs but also close to the pollution.
But over time, as incomes rose and as environmental regulation strengthened, cities lost most of their dirty industry (and associated crime, perhaps). Inner city environments improved drastically, which meant that the rich have started to come back in, lowering their transport costs at the same time. That pushes up housing costs in the centre. So the poor have to choose between staying put and paying higher housing costs, or moving out and paying higher transport costs. That's if the poor rent in the private sector, anyway. If they own their own place or live in social housing, they get to benefit from an improved inner city environment without higher housing costs. So there's an argument that inner city social housing is of increasing benefit to the poor as city centre environments improve. You can also see why it's of increasing interest to those who think we should sell it off.
Obviously that's a very broad sketch with quite a few simplifications and assumptions thrown in. But I think the link between 'greener cities' and displacement of the poor is real enough.
 E.g. it assumes that the centre still has the lion's share of the jobs, which is the case in many European cities but not in some US ones.