OK, let me try to explain it again.

Last week, City AM carried the usual blah blah saying that high house prices are the result of a lack of supply:

The UK, and particularly London, is in the midst of what should be seen as a housing crisis. According to LSE professor Paul Cheshire, new build houses are about 40 per cent bigger in the Netherlands and 38 per cent bigger in Germany than they are in England. And yet housing goes for 45 per cent less per square metre in the Netherlands and, in Germany, prices did not rise throughout the entire 1971 to 2002 period.

A new paper from the Adam Smith Institute lays the blame at the door of the Green Belt. Despite the fact that 90 per cent of the UK is undeveloped, with half of the remainder gardens, Britain is hamstrung by rules hindering development in the places where people most want to live – around successful cities, particularly in the South East.

A reader's letter a couple of days ago referred to an "iron law of economics: if you increase the supply of something, the price goes down".

It's kindergarten Faux Lib economics though. An article in the same paper a couple of days ago looked at actual hard facts. Always start with facts and work backwards to the conclusion. Or establish that there isn't a conclusion, they are just random facts:

London’s population is forecast to hit its highest-ever level this year, surpassing its pre-war peak of approximately 8.6m. Yet it remains one of the least densely populated major cities in the world, according to research by LSE Cities, based at the London School of Economics...

By Burdett’s calculations, London is 30 per cent bigger than it was when its population last peaked in 1939. Even if London’s population density grew, it would bring with it a raft of benefits. If everything is nearby, cars are used less, for example.

LSE Cities’ research finds that holding family income and size constant, petrol consumption per family per year declines by 106 gallons (401 litres) as the number of residents per square mile doubles.

A higher population density can also facilitate a more diverse range of businesses and services than lesser ones can – such as hospitals, airports, theatres, and museums.

So that is the reason why housing in London is so expensive.

It has five time the population of other conurbations in the UK, as well as having most airports and relatively easy rail or car travel to the continent. So there is "a more diverse range of businesses and services etc".

So, glossing over the debate whether we should build outwards at a low density or increase density in the middle (they have much the same effect), what happens if we build more housing in London?

1. People who are currently 'priced out' will move there.

2. So the supply of housing goes up, but the number of people and hence demand for housing goes up pro rata and the effect cancels out.

3. There will be an even more diverse range of businesses and services.

4. So the cost of renting housing and commercial premises will go up even more.

If this were all not true, then the highest house prices would be in the Scottish Highlands and Islands and houses in London would cost tuppence ha'penny.

Whether this is all A Good Thing or A Bad Thing, and if so, what we ought to do about it (liberalise construction and introduce Land Value Tax, please!), or whether it is all irrelevant is an entirely separate debate.
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@ Jim in the comments, strictly speaking, the best kind of LVT is based on "site premium" (as explained on KAALVTN) i.e. that part of the total rental value of a home or office or anything else which relates purely to the location.

But most people are unfamiliar with this concept, and simply taking a small percentage of current house prices is a good enough approximation, i.e. for most homes, the site-only rental value is around 3.5% of its current selling price.

It is not a straight line, unfortunately, it's clearly zero % at the very bottom (homes selling for £50,000 and under), rising to about 4% in the upper middle (homes selling for £400,000 - £600,000), then falling again to 2% or lower for multi-million pound homes at the very top. But hey.