  We develop a theory for the market impact of large trading orders, which we
call metaorders because they are typically split into small pieces and executed
incrementally. Market impact is empirically observed to be a concave function
of metaorder size, i.e., the impact per share of large metaorders is smaller
than that of small metaorders. We formulate a stylized model of an algorithmic
execution service and derive a fair pricing condition, which says that the
average transaction price of the metaorder is equal to the price after trading
is completed. We show that at equilibrium the distribution of trading volume
adjusts to reflect information, and dictates the shape of the impact function.
The resulting theory makes empirically testable predictions for the functional
form of both the temporary and permanent components of market impact. Based on
the commonly observed asymptotic distribution for the volume of large trades,
it says that market impact should increase asymptotically roughly as the square
root of metaorder size, with average permanent impact relaxing to about two
thirds of peak impact.
