Question regarding the ideal target mROI for budget optimization

[tomoyahiroe]

Hi Meridian Team,

I'm moving this question from Discord as suggested. I have a question regarding the theoretical target for mROI when calculating flexible budget scenarios.

I’ve outlined my current understanding below and would appreciate any feedback or corrections.

My Understanding

Note on Definitions:
Please note that throughout this discussion, ROI and mROI refer specifically to the incremental parameters estimated by the Meridian model (causal effects), rather than the general accounting definitions.

Initially, I assumed the ideal target would be $mROI = 1$. The logic was simple: if $mROI > 1$, we should increase spend, and if $mROI < 1$, we should reduce it until it converges to 1.
However, considering how Meridian defines ROI and mROI (and to avoid ambiguity), I decided to frame this as a profit maximization problem:

$$ \begin{equation}\max_{Ad, \mathbf{V}}\ Y - C -Ad\end{equation} $$

• $Y$: Sales (revenue)
• $Ad$: Advertising spend
• $\mathbf{V}$: A set of non-advertising decision variables
• $C$: Other costs

If we assume a variable cost rate $r$ (where $C = rY$), the profit function becomes:

$$ \begin{equation}\max_{Ad, \mathbf{V}}\ (1-r)f(Ad, \mathbf{V}) -Ad\end{equation} $$

Taking the First Order Condition with respect to $Ad$:

$$ (1 - r) \frac{\partial f}{\partial Ad} - 1 = 0 \implies \frac{\partial f}{\partial Ad} = \frac{1}{1 - r} $$

Since the left-hand side $\frac{\partial f}{\partial Ad}$ represents the mROI, it suggests that the optimal target mROI is actually $\frac{1}{1 - r}$ , rather than a flat $1$.

To clarify why this partial derivative represents mROI in this context:
We treat the MMM as a functional form within a Structural Causal Model (SCM). By estimating the model, we are essentially solving the identification problem for the causal effect of advertising on sales. Once the model is identified, the partial derivative $\frac{\partial f}{\partial Ad}$ directly yields the marginal effect, which is equivalent to the mROI (the expected additional revenue for a marginal unit of spend).

My questions

1. Is this interpretation of the "ideal" mROI correct within the context of Meridian’s optimization logic?
2. Are there any specific nuances in Meridian (e.g., how it handles lagged effects or diminishing returns) that might invalidate this simple FOC approach?

I would love to hear your thoughts. Thanks!

[dattatreyam23]

Hi @tomoyahiroe,

Thank you for contacting us!

Yes, your logic is overall correct. Your interpretation of targeting an "ideal" marginal ROI (mROI) to maximize overall ROI theoretically aligns with the First-Order Condition (FOC) in a fully unconstrained scenario. However, there are a few important nuances to how this mROI target is defined:

• Optimizing to an mROI of 1 assumes that your return is revenue and producing sales has no cost.
• Optimizing to an mROI of 1/(1-r) is optimizing for a profit mROI of zero—meaning the next dollar spent yields no incremental profit. This holds because (1-r) is your gross margin percent.

Furthermore, applying a simple mathematical approach is complicated by specific nuances in how Meridian calculates media effects and real-world limits:

• Saturation (Diminishing Returns via Hill Function): As spending on a channel increases, marginal returns eventually diminish. Meridian captures this using the Hill function. Because this function can sometimes form an S-curve, evaluating mROI at very low spend levels might mathematically imply you should reduce spend. In reality, you actually need to increase spend to get past the first half of the "S" to reach optimal efficiency.

• Lagged Effects (Adstock Function): Meridian uses the Adstock function to model carryover effects over time. To be direct: if your optimization windows are shorter than the lagged effects of your media, channels with lagged effects will be penalized in the calculation.

• Budgetary Constraints: Real-world optimizations rarely allow for perfect mROI equalization. Flexible and fixed budget scenarios in Meridian often include upper and lower boundaries on channel spend. A highly efficient channel might hit its maximum budget constraint while its mROI is still significantly higher than other channels, effectively overriding the simple FOC rule.

In short: The logic of equalizing mROI is correct, but these built-in Hill and Adstock functions mean Meridian must use advanced algorithms to find that optimal balance, as simple formulas cannot account for the complex time delays and non-linear saturation effects.

Hope this helps. Feel free to reach out to us with any further queries regarding the same.

Thank you,

Google Meridian Support Team

[tomoyahiroe]

Hi @dattatreyam23

Thank you for the clarification. It is very helpful to know that my theoretical approach is overall correct, and I now have a much clearer understanding of the practical factors—such as the Hill and Adstock functions—that I need to keep in mind.

Since my primary goal was to validate the mROI logic and its interpretation within Meridian, I believe my questions have been fully addressed. I will take these insights into account as I move forward with my analysis.

I really appreciate your time and the deep expertise you’ve shared!!