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Urbanisation is not intended to be UKIP argument, this is also migration from inside the UK to London.
House prices in large German cities have also gone up while they remain low in rural areas, although not as much as in the UK. And financing for housing in Germany is cheaper than the UK (I know people on 10-20 year fixed at less than 2%, with high LTVs). I think your points are correct, but I am not sure they are the main contributors.
The whole ‘immigrants cause rising house prices’ argument is just reheated UKIP bollocks, I am afraid.
UK house prices are driven by the massive influx of capital into the asset class, leveraged with cheap credit. Add into that a bit of top end demand for money laundering and safety deposit purposes and you have an affordability crisis. There are countries all over the world with high levels of immigration that don’t have our house price crisis (ie Germany), just as there are areas of the U.K. with low immigration but high house prices (ie Edinburgh).