  Zipf's law states that the number of firms with size greater than S is
inversely proportional to S. Most explanations start with Gibrat's rule of
proportional growth but require additional constraints. We show that Gibrat's
rule, at all firm levels, yields Zipf's law under a balance condition between
the effective growth rate of incumbent firms (which includes their possible
demise) and the growth rate of investments in entrant firms. Remarkably, Zipf's
law is the signature of the long-term optimal allocation of resources that
ensures the maximum sustainable growth rate of an economy.
