8. What if
A2 improves the quality of his ad copy increasing his CTR by 100%
Let’s now relax Assumption 1) above
where we assumed that the CTR of an ad was dependent only on its position
and not on the ad copy. In particular, let’s say that A2’s ad
copy is superior and gets twice the number of clicks compared to A1 in
any given position.
In tabular form, again, here are the parameters
and (new) values

Conversion Rate 
Margin per Sale 
Position 1 CTR 
Position 2 CTR 
Impressions per day 
Cost per Click in Pos. 1 
Cost per Click in Pos. 2 
A1 
5% 
$100 
10% 
6% 
100 
X 
$1.00 
A2 
4% 
$75 
20% 
12% 
100 
Y 
$1.00 
With this, let’s again solve for the “Point
of Position Indifference” for both A1 and A2.
For A1, nothing changes including the CTRs.
Thus X is still $2.60; i.e. A1 will continue
to increase her bid for the number 1 position until she is paying $2.60;
after that she would prefer the number 2 position.
For A2, the CTRs have changed, doubled to
be precise. So let’s recalculate this “Point of Position Indifference”
which occurs when A2’s daily Expected Net Profits from the number 1 position
= daily Expected Net Profits from the number 2 position.
Daily Expected Net Profits daily from the number
1 position
Clicks per Day * Expected Net Profits per
Click =
Impressions per Day * Click Through Rate
* (Expected Gross Profits per Click – Cost per Click) =
100 * 0.20 * (0.04*$75 – Y) where Y is
the click cost that A2 pays for the number 1 position (1c + A1’s bid)
Daily Expected Net Profits daily from the number
2 position =
Clicks per Day * Expected Net Profits per
Click =
Impressions per Day * Click Through Rate
* (Expected Gross Profits per Click – Cost per Click) =
100 * 0.12 * (0.04*$75  $1.00)
“Point of Position Indifference” occurs
when we equate these:
100 * 0.20 * (0.04*$75 – Y) = 100 * 0.12
* (0.04*$75  $1.00)
Solving for Y; Y = 1.80 (i.e. Y did not
change either)
So the surprising result is that
“Point of Position Indifference” for A2 does not change even when A2’s
ad copy is so much better that it is now getting twice as many clicks as
A1 for any given position.
Thus, even with a superior ad copy, A2 must,
like before, concede the number 1 position to A1 who has the higher expected
profits per click.
The reason that this somewhat counter intuitive
result is true is that the CTR from A2’s ad copy is uniformly better than A1 (twice as much in both position 1 and position 2).
So even though A2’s absolute daily profits can increase substantially
as the CTR increases due to superior ad copy, the point of position indifference
between position 2 and position 1 does not change since the ad’s 100%
improvement in CTR applies equally to both position 2 and position 1.
(The result could be different if, for instance, the CTR on the number
1 position increases much more as a percentage than the CTR on the number
2 position).