Posted on July 23, 2020

Regular readers will know I’m a big fan of the “square root rule” which is a quick rule of thumb to calculate the diminishing returns from increasing media spend.

The square root rule says

$$ Y \propto \sqrt{X} $$

where *Y* is the output (e.g. conversions) and *X* is the input (e.g. media spend).

This tells you that, as a rough rule of thumb, doubling your media spend will increase conversions by a factor of

$$\sqrt{2} \approx 1.4$$

or 40%.

The rule is never perfectly accurate but it is a pretty good approximation when you don’t know very much about a channel.

I first heard about it from Kevin Hillstrom and the PPC nerds at the Rimm Kaufmann group (RIP to what was once a great blog - I can’t find the old posts since they got taken over).

It is also a special case of a Cobb Douglas production function because it uses a power law to model diminishing returns. In this post I want to expand on other aspects of Cobb-Douglas with a potential application to media/agency life.

The Cobb Douglas model is *Y* = *K**L*^{α}*C*^{β}

where *Y* is the total output, *L* is the labour input, *C* is the capital input and *α*, *β* are the “output elasticities”. *K* is a constant that is sometimes referred to as “productivity”.

This is more complicated than the square root rule because it has two inputs (labour and capital) rather than just one (media spend).

The obvious way (to me!) to extend this approach to media buying it to look at the inputs in terms of media spend (capital) and agency fees (labour).

Then the profitability of a campaign is *P* = *K**L*^{α}*C*^{β} − *L* − *C*

When is profitability maximised?

Let’s make things slightly simpler by looking for the optimum fee as a *percentage* of the total input. i.e. *L* + *C* = 1. Then the equation simplifies to *P* = *K*(1 − *C*)^{α}*C*^{β} − 1

Differentiating

$$ \frac{dP}{dC} = -K(1-C)^{\alpha-1}C^{\beta-1}(\alpha C + \beta (C-1)) $$

(thanks Wolfram Alpha!)

There are three zeros for this function:

*C*= 1 i.e. all input is spend on media, zero on fees and the output is (of course) 0.*C*= 0 i.e. all input is spent on fees and zero on media. The output is (of course) 0.*α**C*+*β*(*C*− 1) = 0. This is the interesting one where profit is maximised.

A quick rearrangement of the third equation shows that profit is maximised when

$$C = \frac{\beta}{\alpha+\beta}$$

Remember that *β* is the elasticity of media spend which, in my experience, is usually around 0.5. Plugging that into the above equation and simplifying gives

$$ C = \frac{1}{2 \alpha + 1} $$

I have **no idea** what the value of *α* should be. If we assume that a 10% media fee is optimal (BIG if) then this implies that *α* = 0.05

In practical terms, this predicts that doubling an agency fee without any change in media spend would increase output by a factor of

$$\sqrt[20]{2} \approx 1.04$$

or 4%.

This seems low to me.

https://www.tandfonline.com/doi/abs/10.1080/135048595357249?journalCode=rael20 mentions the following:

Output-labour elasticities varied from a low of 0.18 for tobacco to a high of 0.70 for apparel

This would imply that the optimum agency fee percentage would be between 35% and 72% assuming the output elasticity is somewhere between the two extremes of apparel and tobacco.

This seems high to me.