Cities are positive feedback loops in space. Cities exist only because it is more important for people and organizations to be near each other than far from each other.
There has been recent chatter (Ryan Avent again) about agglomeration making cities wealthier, and only if densities were higher would more agglomeration benefits be achieved.
Certainly evidence shows cities with greater density produce greater "wealth". Glaeser argues it is about the speed of the spread of ideas, in addition to the classical reasons about transportation costs for people and goods. However cause and effect are not clear. I will pose a contrary hypothesis:
Cities with firms that are more agglomeration-benefitting produce higher densities; cities with firms that are less agglomeration-benefitting produce lower densities.
In other words, density is the effect of agglomeration economies, rather than the cause of agglomeration benefits.
The agglomeration-benefits of an industry change over time, and are more important when an industry is young, and in the growth phase than when it is old, mature, and locked-in.
This hypothesis imples, e.g., increasing densities in Phoenix will not suddenly make Phoenix more productive because the firms in Phoenix don't benefit much from the additional clustering (and disbenefit from the negative externalities of density such as crowding, pollution, congestion, and the higher costs of services and land that accompany high density). A city like New York or London, in which the Finance Industry (among others) locates, or Washington in which the Government Industry locates, benefits more significantly from the daily walking distance, face-to-face interactions possible by agglomeration. However even in those capital cities there may be limits to agglomeration such that the marginal benefits of an additional person or job may not outweigh the marginal cost.
Since there can only be one national capital per nation, and perhaps one financial capital per nation, the opportunities for creating financial mega-cities of the order of New York, London, Paris, Tokyo, etc. are limited, just as there can only be one effective political capital; second order cities will not magically become first order just by increasing their density.
To take a case I am familiar with, Minneapolis, downtown has not seen much new commercial growth in over a decade, though there has been a significant (though regionally relatively small) increase in residential density. Minneapolis might be termed a provincial capital. It has a large hinterland, a Federal Reserve Bank, many regional and national bank operations (Wells Fargo, US Bank, etc.), and a large number of headquarters of large firms (regionally especially, though many are not downtown, Target, Best Buy, General Mills, 3M, Medtronic, etc.), as well as emerging clusters in a few economic sectors.
In Downtown there are plenty of vacant lots (i.e. parking lots) for development to occur, and no real constraints on new office (or residential) construction downtown. Clearly the private benefits of building downtown are not as great to the firms making location decisions as locating that new building in suburban areas. Unlike New York, the zoning downtown is not a binding constraint. In brief, the private share that firm will attain from agglomeration benefits of the CBD do not outweigh the costs, (including the opportunity cost of building in some other locale). Those suburban places too have some (weaker) agglomeration benefits, but those may be sufficient. The location within a metro-area still produces benefits (which are weaker than the CBD benefits on daily walking distance face-to-face metric, but still enable daily or weekly driving distance face-to-face), such as shared labor pools, and all that.
If there are agglomeration benefits of locating downtown, they are not sufficient that the local firms (acting through local government) bribe these non-CBD locating firms to move downtown to enable those spillovers. Only a few cases where the spillovers are believed strong result in sufficient side payments (bribes, tax increment financing, etc.) (the Minnesota Twins and the Target HQ building come to mind).
The Twin Cities has some tax-based sharing, so may differ from other regions in the local willingness to bribe, but it also suggests that if agglomeration benefits exist for the marginal new business choosing a location, they are relatively weak. The mental model of agglomeration producing huge benefits if only density limits (as in New York and Washington) were lifted may not apply in many other cities.