  Our goal in this paper is to study the market impact in a market in which the
order flow is autocorrelated. We build a model which explains qualitatively and
quantitatively the empirical facts observed so far concerning market impact. We
define different notions of market impact, and show how they lead to the
different price paths observed in the literature. For each one, under the
assumption of perfect competition and information, we derive and explain the
relationships between the correlations in the order flow, the shape of the
market impact function while a meta-order is being executed, and the expected
price after the completion. We also derive an expression for the decay of
market impact after a trade, and show how it can result in a better liquidation
strategy for an informed trader. We show how, in spite of auto-correlation in
order-flow, prices can be martingales, and how price manipulation is ruled out
even though the bare impact function is concave. We finally assess the cost of
market impact and try to make a step towards optimal strategies.
