In the article [How SPF Changes with How Much Sunscreen You Use](https://labmuffin.com/spf-changes-how-much-sunscreen-use/) (last updated Dec 29, 2017), the author says that insufficient sunscreen disturbingly decreases protection *exponentially*, in other words, that sunscreen application has *increasing* marginal utility:

>The amount used in the sunscreen studies to determine SPF is $2$ milligrams of sunscreen per centimetre of skin.
>
>On average, **people apply only a quarter to a half of the recommended amount**, and in most studies, it’s closer to $0.5$ mg/cm².
>
>In physics, you’d expect **protection** to **decrease exponentially with less sunscreen**. This is based on a relationship called the Beer-Lambert law, and is illustrated [as](https://www.ncbi.nlm.nih.gov/pubmed/9453080)
>
>[![enter image description here][1]][1]
>
> it’s the middle of the graph that’s disturbing.
>
>**Applying half the recommended amount of sunscreen cannot give more than SPF $\mathbf{10}$**, even with SPF $50$ sunscreen!
>
>by $0.5$ mg/cm², we’re below SPF $3$ for all products.

Now, intuitively, the initial layer of sunscreen makes a greater difference than any subsequent layer.

So, does the theory *really* contradict our intuition that each layer of the recommended amount of sunscreen is less beneficial than the previous layer?

  [1]: https://i.sstatic.net/WMpct.png