# Calculus and Marketing Spend

Posted on March 19, 2020

Before I went on holiday this post by Mack Grenfell was doing the rounds about how to use calculus to pick the optimum value of marketing spend to maximise profit.

As is often the case with online writing that I think is mostly good but which misses some important details I am not the target audience for that post but I still feel a response is worse writing.

First of all, I think Mack does a very good job (better than I have ever done) of explaining why this is an important problem; there is a relationship between the amount spent on marketing and the return from it and this relationship has diminishing returns (i.e. you spend money on the best stuff first so extra money you add in after that won’t quite perform so well).

If you can model this relationship between the level of spend and the number of conversions then you can use calculus to figure out at which point on the curve your maximum profit lies.

One interesting question is what kind of model should you use for this relationship?

Switch the line of best fit’s setting to logarithmic, it looks like this line fits the data much more closely

Mack Grenfell uses a logarithmic curve for this which has a few interesting properties. Let’s follow the assumptions of the model for a bit:

1. The formula is α + βln(spend + γ) where α, β and γ are constants. (Mack omits the constant γ but I have included it because you can use it to make sure the predicted number of conversions is never negative even for low spends).
2. Ignoring the α and γ terms, this tells us that doubling spend will increase conversions by βln(2) because βln(2spend) = βln(2) + βln(spend) (this is part of how the ln function is defined)
3. Which means that doubling spend will always increase conversions by the same amount no matter what spend you start with.

Now we have a vague idea of what Mark’s model says about how revenue/conversions will change as spend increases we can ask ourselves “is this realistic?” or “is this true?”. This helps decide if it is a good model or not.

For me, this does not smell right. It says that the increase in conversions going from \$1000 to \$2000 is the same amount as going from \$10,000 to \$20,000 which doesn’t match up to my experience.

### The Square Root Rule

Instead of a logarithmic function I prefer to use the square root function. I first heard about the “square root rule” from Kevin Hillstrom; it simply says that revenue/conversion is equal to αspend0.5. One advantage of this is that there are fewer parameters so it is dead easy to fit but that isn’t really the main point here.

A more general form is αspendβ. I just find that 0.5 is a good starting value for estimates when you haven’t done any testing yet.

For the square root rule a doubling of spend will always lead to a ~40% increase in revenue/conversions. In the more general case, revenue/conversion will increase by a factor of 2β after doubling the spend.

This model is a simplification of the Cobb Douglas production function which is used to predict economic output given inputs of capital and labour. It has just occurred to me that this could also be used to balance the split between media spend (capital) and management fees (labour) but I guess I will leave that for another post!

### Bid Simulator

I use the square root rule when I’m making proposals or for non Google Ads channels. Within Google Ads you are better looking at the data from bid simulator directly.