This claim stems from theoretical physicist and urban theorist Geoffrey West, portrayed in this NYT article. The paper is called Growth, innovation, scaling, and the pace of life in cities (PNAS 2007, doi: 10.1073/pnas.0610172104) and available online.
The indicators are listed in table 1 and clarified in the supporting information (extracts below).
It seems that the phrase every time the size of a city doubles is misleading: they found a correlation between city population and various metrics (number of patents, wages, GDP). They are not tracking the same city over a period of time. Thus the claim that is indeed supported by the study is: "If a city has twice the population of another city, it has also around 15 percent higher wages, GDP and number of patents granted per capita (if both cities are in the US, or both in Europe or both in China)."
- New patents refers to number of new patents granted by the U.S. Patent and Trademark Office over the period of 1 year to authors residing in a given metropolitan statistical area. Inventors refers to number of patent authors over a given year, inferred from the same source.
- Private R&D employment (U.S.) and private R&D establishments (U.S.) data are taken from the U.S. Economic Census, which provides information on the employment levels and number of establishments engaged in conducting original investigation undertaken on a systematic basis to gain new knowledge (research) and/or the application of research findings or other scientific knowledge for the creation of new or significantly improved products or processes (experimental development).
- Total wages
- “supercreative” professions are computer and mathematical; architecture and engineering; life, physical, and social science occupations; education; training and library; arts, design, entertainment, sports, and media occupations.