There were time and motion studies in the early 20th century which found that if a city could have double the population then productivity would increase and per person income would increase by 7%.
Currently the world is about 56% urbanized. Only 10% are in megacities with population over 10 million.
100% urbanization would double world GDP with increased productivity.
Another 30% boost in productivity would come from having everyone in megacities.
China is integrating many cities into megaregions and the five largest will have 100 to 150 million and by 2030-2040 they will be fairly well integrated. This should boost per capita income by about 20% over a 10 million person city.
Having a 10-25 billion person global city would add another 70% per person productivity boost. Global population in 2100 will be about 13-25 billion.
A city is generally defined as requiring one to two-hour transportation between any point within the city.
This is a huge challenge of future infrastructure.
SpaceX BFRs flying ten times per day could get any spaceport on Earth to another Spaceport within about 30-40 minutes.
High-speed ET3 rail could also eventually achieve such speeds and to more points with more energy efficiency.
There will be pan-Asian high-speed rail network built by China with speeds in the 250-400 mile per hour range. However, for the global city you would not want to travel more than 10 minutes on each end of a trip on such rail. This would be 25-40 miles especially considering a short waiting for a connection.
Vacuum tubes for trains could eventually boost the speed of trains up to 4000 mph. ET3 or Space Travel On Earth could eventually reach speeds of 400/4000mph local/int’l while being 50x energy efficient than electric car or train. Conventual infrastructure transporting freight, food, water, energy, data and people globally today can easily quadruple between 2020-2050 through the construction of ET3. Cities such as London to Beijing, Beijing to New York City, NYC to Los Angeles will respectively require 75, 120, and 45 minutes of travel time with ET3.
Full speed ET3 could transport people 1000 miles in just over 15 minutes.
Then buildings would need multi-elevator connections and self-driving cars with higher speed and multi-level capability.