Relationship between order size and spread for direct market order

Question

Suppose that I am placing a market order directly in the order book of an exchange. For market orders, it seems quite clear that larger orders obtain larger spreads due to the fact that - without loss of generality - a large market buy order must reach further into the limit sell orders of the order book than would a smaller market order.

Assuming that this initial premise is true (and if it's not, please tell me!), what would be the nature of this relationship? Would it be linear? Quadratic? Logarithmic? Any insights would be appreciated, including links to existing research on the subject.

Answer 1

In theory or when doing simulation with static orderbook your assumption is right. In general it is correct that large size orders have larger market impact. However, in real life there are other things you should take into account. Exchanges have market makers or some sort of participants who have higher priority and your market order does not necessarily reach orderbook.
As for relationship, it is arguably square-root. If you look market impact you will find enough literature you decide for yourself whether that it true or not. You can refer to this link for more info.