Two parts of the campaign perform at certain ROI. We want to change the bids and maximally increase the conversion value, without changing the average ROI.

The growth (G) of the total conversion value after the change is:

– where V is the value of click and Clk is the number of clicks.

Before the changes all parts of the campaign have the same ROI. Therefore:

After the changes, the average ROI remains unchanged i.e. the total value of clicks is equal to the total cost multiplied by (ROI+1):

At the maximum growth (G):

 

The discriminant of the quadratic equation is:

and the solution:

 

Changing CPC by ΔCPC will maximise the income without changing the total ROI:

The elasticity data are available in AdWords interface (Bid Simulator) or via AdWords API (Bid Landscapes). The “1” and “2” may be any part/segment of the campaign: keyword, adgroup, segment. For example, “1” can be “mobile” and “2” – “desktop”, or “1” can be a keyword and “2” – all the other keywords in the campaign (in this case we should calculate total elasticity of all the other keywords).

How to calculate total elasticity of two campaigns?

From definition:

If in all campaigns the relative change of CPC is the same:

The formula for total elasticity is:

 

This model assumes that price elasticity does not change within the range of changes in campaigns. However, in reality, price elasticity decreases as CPC increase. Therefore significant changes of bids may be inaccurate and sometimes may have no meaning (e.g. change of bid below minus 100%). In practice, if the formula implies very significant changes of bids, you should not modify bids more than by 20-25% and after this modification, measure the elasticity again and continue optimisation at the next iteration.