# How to determine statistical 'sweet spot' for impressions

I have 1000s of advertising campaigns in which I have recorded hourly impressions, clicks, conversions and revenue for.

eCPM = revenue/impressions * 1000


eCPM is generally a figure used to gauge the performance of a campaign.

However, eCPM is greatly affected by impression amount. The higher the impressions, generally the lower the eCPM...in most cases. This is also the case with CTR, generally. It's an extremely rare occasion that eCPM isn't affected by an increase of traffic.

I started thinking that maybe there was a way to calculate a 'sweet spot' in which the perfect amount of impressions is delivered so the optimal eCPM is reached for the campaign. Likewise would occur with other campaigns.

This would then allow me to know how to distribute my impressions in the most optimised way.

Does anyone know a formula or process in which I can calculate the optimal traffic amount for a campaign?

Calling $n_i$ the number of impressions, $r$ the revenue, $r_{max}$ a hypothetical maximum revenue (if you spent an infinite amount of money, you would reach everyone with your adds) and $\alpha$ a parameter of your model.
Intuitively, this could look like $r=f(n_i)=r_{max}(1-\exp(\alpha n_i))$ and you would have to find $\alpha$ and $r_{max}$ (which would be easy given the particular form of the model).
Then you would maximize $f(x)/x$ over $x$ to achieve the best eCPM.
You could go futher by specifying less restrective assumptions about $f$.