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The goal of this paper is to disentangle the roles of volume and ofparticipation rate in the price response of the market to a sequence oftransactions. To do so, we are inspired the methodology introduced inarXiv:1402.1288, arXiv:1805.07134 where price dynamics are derived from orderflow dynamics using no arbitrage assumptions. We extend this approach by takinginto account a sophisticated market participant having superior abilities toanalyse market dynamics. Our results lead to the recovery of two square rootlaws: (i) For a given participation rate, during the execution of a metaorder,the market impact evolves in a square root manner with respect to the cumulatedtraded volume. (ii) For a given executed volume $Q$, the market impact isproportional to $\sqrt{\gamma}$, where $\gamma$ denotes the participation rate,for $\gamma$ large enough. Smaller participation rates induce a more lineardependence of the market impact in the participation rate.

The two square root laws of market impact and the role of sophisticated market participants