  Publicly traded companies are fundamental units of contemporary economies and
markets and are important mechanisms through which humans interact with their
environments. Understanding the general properties that underlie the processes
of their growth has long been of interest, yet fundamental debates about the
effects of firm size on growth have persisted. Here we develop a scaling
framework that focuses on company size as the critical feature determining a
variety of tradeoffs, and use this to reveal novel systematic behavior across
the diversity of publicly-traded companies. Using a large database of 31,553 US
companies over nearly 70 years, and 3,160 Chinese companies over 24 year, we
show how the dynamics of companies expressed as scaling relationships leads to
a quantitative, analytic theory for their growth. This theory produces several
predictions that are in good agreement with data for both the US and China,
whose markets have strikingly different histories and underlying structures. In
both cases sales scale sublinearly with assets and exhibit nearly identical
exponents leading, surprisingly and nontrivially, to assets that grow as a
power law in time rather than exponentially, as often assumed. On the other
hand, liabilities scale linearly in the US (exponent of $1.0$) but
superlinearly in China (exponent of $1.09$). We show that such small
differences in scaling exponents can have a significant impact on the character
and long-term evolution of growth trajectories. These results illustrate that
while companies are part of a larger class of growth phenomena driven by
incomes and costs that scale with size, they are unique in that they grow
following a temporal power-function which sets them apart from organisms,
cities, nations, and markets, whose growth over time is often exponential.
