  In a recent Nature paper, Gabaix et al. \cite{Gabaix03} presented a theory to
explain the power law tail of price fluctuations. The main points of their
theory are that volume fluctuations, which have a power law tail with exponent
roughly -1.5, are modulated by the average market impact function, which
describes the response of prices to transactions. They argue that the average
market impact function follows a square root law, which gives power law tails
for prices with exponent roughly -3. We demonstrate that the long-memory nature
of order flow invalidates their statistical analysis of market impact, and
present a more careful analysis that properly takes this into account. This
makes it clear that the functional form of the average market impact function
varies from market to market, and in some cases from stock to stock. In fact,
for both the London Stock Exchange and the New York Stock Exchange the average
market impact function grows much slower than a square root law; this implies
that the exponent for price fluctuations predicted by modulations of volume
fluctuations is much too big. We find that for LSE stocks the distribution of
transaction volumes does not even have a power law tail. This makes it clear
that volume fluctuations do not determine the power law tail of price returns.
