Analyst Phil Hayward of Wellington, New Zealand provides  a provocative perspective on why urban intensification (densification in  the urban cores) is incapable of compensating for the huge house price  increases attributable to urban containment boundaries. Writing on Making New Zealand for Urban Planning that  Works, he notes that “planners and advocates and politicians and even economists,  are making an assumption that urban intensification is a potential route to  housing affordability.” 
The  assumption involves changing zoning so that “X number of housing units” can be  constructed in existing urban locations “instead of X number of housing units”  on pristine ex-urban land. The latter is assumed to be an evil to be avoided,  and that the former is a perfect substitute in terms of “sufficient housing  supply to enable affordability.
Hayward continues:
Common sense tells us that there are  quite a few potential problems with this assumption. For example, NIMBYs will  obstruct the intensification and reduce the rate of housing supply so the  policy will fail. Therefore, what we need is the removal of NIMBY rights of  protest and appeal, and the policy will then work. 
Hayward’s analysis suggests that: 
And  generally, the data runs in that direction - not only does intensification  within a regulatory boundary "not restore affordability", it seems  that the more density you “allow”, the higher your average housing unit price  gets. The correlation runs the opposite way to the assumption.
Indeed, “Paul Cheshire and his colleagues at the London School of Economics believe this is due to the  ‘bidding war’ at the margins of each income-level cohort of society, for  ‘slightly more space,’" according to Hayward. “But when a market is  allowing people to consume "as much space as they want", which has  only really occurred in the automobile era, the “bidding war” effect is absent.”
Boston and Atlanta provide powerful examples.
…(The)  difference is that Boston has de facto growth boundaries / green belts while  Atlanta does not. The ironic implication is that fringe growth containment  pushes median multiples up less, when there are severe restrictions against  density – otherwise Boston should be the most expensive city in the data, not  Hong Kong. The evidence suggests that this is because there is a total absence  of “bidding war for slightly more space” - everyone has "more than they  want" already. The median multiple of 6 rather than 3, represents the  effect of demand for "living in Boston", period, and they simply  don't provide enough houses to keep the median multiple down like Atlanta does  (in the face of staggering population growth in Atlanta, by the way).
Perhaps the most important conclusion is that “there is no  evidence that any city anywhere in the world has ‘freed up intensification  processes’ enough to result in floor space being built faster than site values  inflate. 
The bottom line is a mistaken impression that high density  housing “will remain available as a substitutable option to suburban family  housing even if the latter is forced up in price deliberately by central  planner's policies. The lesson that needs to be learned urgently, is that this  is impossible; the two things are inter-related.”    
But when a market is allowing people to consume "as  much space as they want", which has only really occurred in the automobile  era, the “bidding war” effect is absent. The evidence supports this, with most  median-multiple-3 cities being from 600 to 2500 people per square km. Another  interesting case study would be Liverpool; it lost approximately 50% of its  population from the 1950's to the 2000's (similar to Detroit) - yet its median  multiple is over 7. And its density is still 4,400 per square km (presumably it  would have been double this, or more, in 1950). This is prima facie evidence  that 4,400 people per square km within a growth boundary, are still going to be  dissatisfied with their living space, to the extent that they will be engaging  in an unwitting bidding war against each other for a little more of it. Of  course under these conditions, the lowest socio-economic cohort is denied all  options other than crowding tighter and tighter in rented accommodation or even  illegal “living space”. In UK cities, rental advertisements include options  like a ¼ share in 2 rooms, with communal access to kitchen and bathroom shared  by even more tenants in further rooms. In median-multiple-3 housing cities, the  same real rent would apply to a whole house of reasonable size and standard. 
There might be other policy mixes by  which housing supply within a growth boundary could be made the means of  keeping housing affordable, but publicly and politically, the debate is nowhere  near tackling the complexities involved.