Sublime

a more complex accounting of the resource base for an urban system, its transportation, and its functions is required to understand more realistic spatial patterns.

does not account for exponentially growing historic processes of human development and economic expansion over time,

and strategic investment underpinning economic growth. We used deductive reasoning to derive general properties of cities, such as scaling relations and the laws of geography, as averages of processes over populations and time. We have seen that such “laws” describe the system as a whole but, like any average, do not actually apply to any specific

for increases in productivity.

In turn, we will see how the city as a whole—conceived as a social and economic network of these agents—derives its own balance of incomes and costs from those of its primary agents as an emergent phenomenon.

to the corresponding population share,

growth and those experiencing population

we have found new routes for understanding cities as (approximate) spatial equilibria, for deriving their structural complexity in terms of professions and neighborhoods, for understanding the first cities and their “urban package” of technologies and institutions, and for demystifying the microscopic foundations of learning

For each neighborhood j, log wj is the amount of information needed to describe its statistical pattern of income, given that we start by knowing the aggregate income distribution across the city.