# Market impact in optimization objective function

Can anyone help to explain why when the square root market impact model is used in the standard mean-variance optimization, the exponent becomes $$\frac{5}{3}$$ in the objective function? I suspect this has to do with expressing market impact in different units but I could not find a clear explanation.

EDIT: Near the top of page 8, the authors state:

... show that the square root function is more appropriate for modelling market price impact, thus suggesting market impact costs grow at a rate slower than quadratic. Therefore in this section we consider the case with p ∈ (1, 2) in objective function (1).

https://pdfs.semanticscholar.org/b96d/c0273a25d27ab1aef9f8e300701f1689d738.pdf

• Can you link to a textbook or lecture slides that show this? – chrisaycock Jul 4 '20 at 1:15
• Can you show where they find $p=5/3$? I see where they consider $p\in(1,2)$, but I am not seeing any statement that $p=5/3$. Furthermore, cumulating impact with a square root impact function usually leads to something on the order of $x^{3/2}$, not $x^{5/3}$. – kurtosis Jul 28 '20 at 16:13