May 13, 2018

a*(x^b)

My entire career is based on a simple equation:

  • a*(x^b)
What the heck is Kevin talking about?

Let's think about this for a moment.
  • You spend $100,000 on paid search.
  • The average of your five attribution vendors say that paid search delivered $350,000 in sales.
  • You convert 40% of sales to profit.
  • Profit = $350,000 * 0.40 - $100,000 = $40,000.
  • Somebody smart in your company says "HOW MUCH SHOULD WE BE SPENDING ON PAID SEARCH?"
Somebody smart in your marketing department knows LTV and knows that you can actually afford to lose $25,000 instead of making $40,000 because the customer will pay you back within eighteen months.

How much should you spend so that you will lose $25,000?

That's where a*(x^b) becomes pretty darn important.

The simplest version of the equation is this:
  • 1*(x^0.5) ... better known as the SQUARE ROOT RULE.
  • You take the square root of what you previously spent and what you want to spend, and apply that to sales.
Here's what the table looks like in our example:


The equation suggests that we could spend $240,000.

Unless we have valid test results, we don't "know" that $240,000 is optimal. So we test our way north toward $240,000 until we find the right answer. And on the way, we actually learn what our version of "a" and "b" are in a*(x^b) ... we fit an equation and we know the answer.

Say we test spending $140,000 instead of $100,000 ... and after adjusting for seasonal differences we learn that we generate $391,000. We now have three data points that can be used to identify "a" and "b".
  • We know if we spend $0 we get $0, so that is point one (0,0).
  • We know that if we spend $100,000 we get $350,000 ... so this is our second point ... if we spend a 100% of our old budget we get 100% of old budget net sales (1,1).
  • We know that if we spend $140,000 we get $391,000 ... so this is our third point ... we spend (140,000 / 100,000) = 1.4 to get (391,000 / 350,000) = 1.117.
Three data points.
  • 0 , 0.
  • 1 , 1.
  • 1.4 , 1.117.
I plug the three data points into my "CurveExpert" software which I've been using since the mid 1990s. The fitted equation looks like this:


And the actual equation looks like this:
  • a*(x^b).
  • a = 0.985.
  • b = 0.407.
  • (0.985)*((new spend / old spend)^0.407).
The actual equation allows us to "optimize" paid search spend, after adjusting for seasonality.


This is the table we use to determine the "optimal" level of paid search spend.

Notice that our original guess ... using the "square root rule" ... well, that guess wasn't a bad guess at all, was it? All we knew was that $100,000 wasn't enough to spend, and we guessed that $240,000 was the "optimal" level. After testing a spend level of $140,000, we learned that $220,000 was the "optimal" level.

In other words, the square root rule was a wonderful starting point, wasn't it?

That's the power of a*(x^b).

It turns out that a*(x^b) is everywhere we look in e-commerce, retail, and old-school catalog marketing.

Copy every single line of this blog post ... print it and put it in your cubicle or your Executive Board Room. When somebody has a question about something, go back to this blog post and run your own analysis and answer hypothetical questions quickly and reasonably accurately.

Have fun, too!!

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