Either the observations are wrong or the theory is nonsense.

The observations

1. The larger the conurbation-the denser the population-the better the links to other places, the higher the average productivity-wages-profits. This is partly because of synergies-specialisation-agglomeration benefits; and partly because higher wages at the top of a pyramid depend on how wide the base of the pyramid is. This pulls up the overall average while widening the difference between top and bottom. The fag packet calculation is double the size of a conurbation and average wages go up by 5%.

2. Within any country, there is free movement of people and workers. Between leaving home and 'settling down', plenty of people are willing move from lower wage to higher wage areas. Once all housing in the higher wage area is occupied, this pushes rents up to the point where a new equilibrium is reached and the rent differential is equal to the wage differential. At this stage, there is little financial advantage in moving and net internal migration falls to a trickle.

3. This is easily measured by looking at average wages minus average rents in different regions of any country, this 'basic minimum' is much the same anywhere.

See for example here

(Clearly, rent and wage differentials between countries if there is no free movement of people and workers between them can be sustained almost indefinitely.)

Anybody who disputes this can leave the conversation right now.

The theory

1. Supply of housing is kept below its optimum level by one or more of the following:
i.  Councils' planning departments (for reasons unspecified) not handing out enough planning permissions.
ii. NIMBYs putting political pressure on councils not to hand out planning permissions.
iii. Land owners holding on to land as long as possible to bank the biggest gains.
iv. Home builders restricting new supply to keep prices up.

Choose your favourite according to political bias!

2. Housing is a perfectly ordinary good, like coffee beans, cars or carpets. If there is an increase in supply, prices will fall, and vice versa. This is easily observable in the short term, like if there is a coffee glut, or in Germany post-unification when East Germans sick and tired of Trabis, were itching to buy decent second hand VWs and second hand car prices doubled for a year or two. This levels off very quickly, as manufacturers make a certain minimum profit margin. Any less than that and some go out of business, any more than that and they manufacture more, or new players enter the market.

3. Therefore, if somehow home builders/land bankers could be persuaded to increase output (despite it being against their interest to do so), prices and rents would fall. Simple, 'housing crisis' solved! Duh, why did nobody think of that before?

(I always ask these people, OK, if it's all about supply, and there are more homes within the M25 than in the whole of Scotland, why are prices so much higher within the M25? None has ever answered that, the closest they ever get is 'it's different this time'.)

Either the observations are wrong or the theory is nonsense

OK, let's measure 'affordability' in terms of how much disposable income working tenants (actual or potential internal migrants) have left over after paying rent ('the basic minimum'). We know from observation that within any country or area with freedom of movement, this is pretty much the same everywhere and however many homes there are in any area, the basic minimum will still be the same all over the country.

(This is why people at the bottom of the wage pyramid in a wealthy country are negatively impacted by freedom of movement - they are competing for jobs with people with a much lower basic minimum expectation, who are happy to accept lower wages or pay higher rents).

This is where is becomes nigh impossible to reconcile the theory to the observed facts:

Probability A. We increase supply of homes in high wage-immigration areas, there will be more agglomeration benefits = higher average wages, so the trajectory is that rents there will increase rather than fall. Assuming that in emigration areas there is the opposite effect and wages fall, then the basic minimum must fall. This actually exacerbates the whole thing.

Possibility B.We increase supply of homes in high wage-immigration areas, people move there but there are no agglomeration benefits. Wages, and hence rents, and hence affordability stay the same.

Possibility C. A greater mind than mine can actually reconcile the theory with the observations.