You've inherited a marketing mix model, but how do you make decisions from it? You have a simple channel mix, but it's not clear how you're supposed to divide the budget between Facebook and TikTok.

GoolyBib Office – Second to Last Week of the Quarter

You're used to forecasting the diminishing returns of Facebook ads each quarter for finance, but now you've launched TikTok ads you have to do it for two.

Budget Proposal
Hey,
Can you get me the latest numbers for the budget proposal?
How will do we need to divide the budget now we've got TikTok running?
Ty,

Allocating your media spend is one of the most important jobs in marketing. Having some justification for how you make spending decisions is key if you want to win the respect of finance and senior leadership.
The motivating factor for allocating budget is determining diminishing returns. One channel might have a higher ROI than another, but that doesn't mean you can put 100% of your budget there and expect the same results. Effective budget allocation is about finding the tipping point where you maximize ROI across channels, by taking account each channels unique response curve.

Budget Optimization

Alibaba Founder

Spending money is much more difficult than making money.

The trick to optimizing budget is to understand when a channel gets more or less efficient. Modeling the number of incremental conversions you get for each level of daily spend will give you that channel's response curve. Plotting this curve will help you make the tradeoff between volume and efficiency, and give you hard numbers to back up your spending decisions.

Splitting spend across channels

Based on the diminishing returns in your model, there should be an optimal choice of where to allocate spend... but how to find it? Let's start by plotting the response curves for Facebook and TikTok.

You settle down with your latest marketing mix model to forecast how much to spend. This time you have to adapt your process to account for two channels, allocating budget between them.

To determine how much you should spend on a channel, you need to forecast the amount of conversions you'll get at each spend level. If there's a linear relationship between spend and conversions, then your response curve will be a straight line: if you get 100 conversions for $400 you'll get 200 conversions for $8.
Marketing channels rarely work this way in practice, so it makes sense to incorporate diminishing returns: this shows up as a curve downwards, where you get less incremental conversions at each spend increment. So for example going from $400 to $800 might get you 150 more conversions, not the 200 you'd expect from a linear relationship.
To plot these response curves, you just need to map out the spend levels you want to calculate, then apply the function you get from your model. Apply the diminishing returns function to the spend level, then multiply the result by the coefficient in your model to get the estimated conversions. Do this across multiple spend levels and you should get a smooth curve for that channel, and you can choose your point along it.
Do you want more conversions at a higher CPA? Or less conversions at a cheaper CPA? Once you have your response curve estimated, you have the power to make that tradeoff! This can lead to more realistic goal-setting discussions with leadership and management. It can also help you cut the noise, and identify periods when you've geniunely improved performance, which would show as the whole curve moving downwards (a lower CPA for the same volume of conversions).

Response Curves

For diminishing returns you can use the formula x^B, or your spend variable to the power of B, where B is the saturation point. For example if the channel was 50% saturated and you spent $100 that day, you'd transform the variable with the formula $100 ^ 0.5, then multiply the result by the coefficient in your model. Finding the saturation point is a case of trial and error: testing different saturation points and seeing what improves the model accuracy, which in this scenario has already been done for you, and is supplied in the model.

Calculate the response curves for each channel and plot the results. What do the relative curves tell you about the returns from each channel?

Add up the contribution to conversions for Facebook and TikTok at the current points they're at now, which is $7k daily on Facebook and $2.5k on TikTok. What's the expected blended CPA at these spend levels?

What would the CPA be if you split a $10k budget evenly across both channels ($5k each)?

Choosing your place on the curve

You may not know the math, but thankfully someone else does, and you can use their solution. Use the SciPy Python library to optimize your media spend based on the diminishing returns and coefficients in your model.

Stuck in the office late working on this, you can't quite get the right split, and you don't remember the math to calculate it properly. You reach out to a friend for help. 

Hey,
Of course, happy to help
We have a budget optimizer script we use
Let me share it with you
Just modify it to your model and run

The math for optimization under contraints can be complicated, but thankfully someone has done it for you! The SciPy library gives you a ready-made optimizer to use, you just have to pass it an objective function that represents your model. This is the function you'd use to calculate conversions given the inputs of facebook and tiktok spend, as well as the constant (thankfully your model is quite simple!).
As well as an objective function, you need to pass the optimizer some sensible constraints, bounds and initial guesses. In our case the constraint is that spend must equal the total marketing budget and that no channel can spend negative or more than the budget.
We pass it an initial guess of just dividing the budget equally across channels, and from there it iterates to find the optimal solution. Once it's finished running, it prints out the results of the model, so we can make our decision.
To modify this script we need to adjust the model coefficients, the diminishing returns, the marketing budget and the constant, so that we get results relevant to our business.

Run the optimizer with no diminishing returns (dr = 1 for each variable). What happens to the spend allocation?

In the absence of diminishing returns, budget optimization becomes a linear programming exercise. The solution is always to just spend 100% of your budget on whichever channel has the highest ROI. It's often the case in practice that you don't have any diminishing returns in your model, but you should assume that you will encounter this effect as you increase spend. So use common sense when interpreting the results of the media optimizer, and only increase spend gradually to check if performance matches what the model predicted. 

Adjust the budget to 10k, and update the model coefficients, variable names, diminishing returns and the constant value to match our model. What split does the optimizer recommend?

Because you have an objective function that calculates the number of conversions in your model given the facebook and tiktok spend, you can use it to simulate different scenarios. For example you can see how many conversions you'd get if you spent nothing on facebook, or if you increased or decreased your spend. You just pass the right values into the objective function and it tells you how many conversions you'd get, which you can divide into the total budget to get an estimated CPA. This is exactly what the optimizer is doing, but with multiple iterations until it finds the best fit, but you can use it for manual scenario planning as well.

Add two new scenarios: one that calculates what the CPA would be if you spent everything on facebook ($10k), nothing on tiktok, the other if you split the budget evenly. Would spending everything on Facebook be better or worse than splitting the budget evenly?

RE: Budget Proposal
This is really cool! Advanced stuff.
You'll have to explain to me how it works, but for now I'll get this sent off to finance.
Appreciate it,

Automated media budget optimizer

Gustav Gieger

Head of Performance

Dushyant Dixit

Consultant

Charlotte Cook

Sam Smirnov

Brand Strategist

William Winters

Data Scientist

Alexandra Anderson

Owner

Danielle Oscar

Ashton Donaghy

How do you allocate marketing budget across channels? You can't just guess: leadership won't be happy without seeing justification for the numbers you send them. You need a budget optimization script.

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