Ever ask, "**How much more or how much less should I be spending on advertising?**" Here's a simple model based on the square root rule to help answer that question, along with an Excel spreadsheet to help with the arithmetic. (To skip equations, click here for the applied section.)

The square root "rule" says sales, **S**, increases with the square root of advertising, **A**, like so:

In symbols:

where **k** is a calibration constant, determined from an actual observation of sales and advertising **(S0, A0)**.

If your effective profit margin (defined as 1 minus cost-of-goods percentage minus variable selling expense percentage) is **m**, we get this formula for profit **P** as a function of advertising spend **A** :

Of course, this toy model is not correct.

The only way to determine the actual relationship between sales and advertising is empirically, through testing.

Nonetheless, assuming the relationship follows these equations provides some insight.

Strengths of this model:

**Decreasing returns to scale.**The slope of the curve get less steep (formally, its second derivative is negative). This means the more you advertise, the more you suffer decreasing returns to scale. Because you buy the best advertising first (or at least try your hardest to do so), the more advertising you buy, the less effective each incremental advertising chunk is at producing sales. Real life works this way, and the model captures that.**Self-consistency.**The model's predictions are internally consistent for any**A_0**.**Theoretical basis.**This equation is a specific case of the well-studied Cobb-Douglas production function from economics.**Simplicity.**Because of the model's simple form, we can solve for the optimal advertising level which maximizes profit.

Weaknesses of this model:

**Smoothness.**The model says the relationship between sales and advertising is smooth. Not so in real life -- you buy advertising in chunks (channels, media, spots, insertions, campaigns, keywords) with different inherent degrees of "chunkiness" and differing performance. For example, you can't buy 4/5ths of a Superbowl ad.**Scope.**The model says you can always buy more sales with more advertising, on up to infinity. Not so in real life -- at some point, you've tapped out the market of potential buyers, and additional advertising returns nada.**Objective.**Optimizing**P(A)**will maximize operating profits. While most catalogers and direct marketers run their advertising so as to generate maximum profit, many other marketers do not, instead intentionally choosing to "over-advertise" to increase brand awareness or drive top line sales. All valid and good, just be aware this model and spreadsheet adopts the direct marketing perspective and aims to maximize bottom line.

We can maximize profits, **P(A)**, by differentiating and setting the derivitive equal to zero. This gives up the optimal level of advertising, **Amax**, which coincidentally is also the optimal profit, **Pmax**.

Under this model, the optimal A/S is

To offer some intuition for this, if your effective profit margin is **m** cents per sales dollar, you maximize your operating profit (in dollar terms, not percentage terms) by sinking half of your effective profit margin (in percentage terms) in marketing, keeping the other half as contribution towards profit.

Again, this is a *highly *simplified model for advertising returns to scale. Use this only as starting point.

The only certain way to determine your optimal advertising level is through careful testing of the A vs. S elasticity curve for your business.

Enough Math! How Much Should I Advertise?

We've prepared an Excel spreadsheet that handles all this arithmetic.

To use this model, change the gray shaded cells. Enter your ad cost for a recent campaign in cell C8 and the corresponding tracked sales in C9. Enter your average cost-of-goods sold, expressed as a percentage of net sales, in C10. Enter your other variable selling expenses (credit card discount, shipping subsidy, pick and pack cost, dunnage, etc), again as a percentage of net sales, in C11.

The model will give you a base P&L for your campaign. It will also estimate what would have occurred had your advertising been 30% higher or 30% lower. And it will estimate the advertising level that would have maximized your operating profit.

For an example, I loaded the model with the numbers from Kevin Hillstrom's December post on the square root rule. Kevin gave a (made-up?) example about about a CFO who stormed in on the online marketing team demanding they cut their budget by 10%. With the P&L Kevin provided ($10mil sales, $2mil advertising, 60% COGS, 13% variable), Kevin notes the CFO was likely right -- the last 10% of budget wasn't pulling in enough sales to justify it. Cutting $200k from the ad budget might bump the bottom line up by $60k.

The spreadsheet goes further, suggesting that, in this made-up scenario, profits would be maximized by cutting the budget in half. *(Slow down there, cowboy!)*

Take some care before slashing (or, alternatively, before heavily increasing) an ad budget by such a large factor based on a simple model. The model could be wrong several ways. The assumptions of the model might be incorrect: your actual sales vs. advertising curve might not be a smooth square root curve. Your advertising team might not be able to identify the worst performing corners of the advertising budget to cut. (If that's the case, you may have other problems.) And your goal might not be maximum online sales: you could be advertising to drive your top line, to build brand, or to increase store traffic.

On the other hand, suppose the online marketing team from Kevin's example was generating the same $10mil in sales (still with 60% COGS and 13% other variable) but was achieving that with a $1.2mil budget (12% A/S), rather than the $2mil budget described (20% A/S). In that case, the model suggests the team is underadvertising to the tune of $300k, and that increasing the ad spend to $1.5mil would increase operating profits by $20k and sales by $1.25 mil. In this case, the CFO could rant that the marketing team was being too conservative.

Before making significant changes to your ad spend, use careful testing to determine your true elasticity between advertising and sales.

This simple model can start that ball rolling by giving you some sense of the direction and scale to test.

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*Thanks to Kevin Hillstrom for posts on the "square root rule" in December and in January, to Roger Cortesi for his LaTex to image rendering page, and for Peter Newbury and Joost Winne for LaTex cheat sheets.*

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