Saturation describes diminishing returns: an additional spend increment produces less response as exposure rises.
Saturation describes diminishing returns: an additional spend increment produces less response as exposure rises. Yet the selected shape determines the recommended spend. A logarithmic, Michaelis–Menten or Hill function may fit the same central observations and extrapolate very differently beyond the historical range.
Decision and method.
Optimise only within a range where saturation shape is identified, stable and compatible with observed or experimental spending. Document minimum, median, maximum, periods near each level and correlation with other channels. Fit several families with comparable priors or bounds; compare validation, parameters, monotonicity and extrapolation. Publish the derivative—the response to the next equal budget increment—not only total attributed contribution. When plausible curves diverge outside history, bound the recommendation or require a test.
Worked example.
Maximum response is €500k, half-saturation is €50k, and spend rises from €40k to €80k. Michaelis–Menten gives 500 × 40 ÷ (50 + 40) = €222k at €40k; at €80k it gives 500 × 80 ÷ 130 = €308k. Doubling spend adds €86k, not €222k. The increment is acceptable only if €86k covers its cost and remains robust across other credible functions.
Checks and limits.
Keep candidate spend in a covered or explicitly tested range; compare several functions on the same data; publish marginal return and its interval at the candidate point; stress-test against still-plausible curves. Adstock and saturation are hard to separate when spend barely varies; spend saturation may instead reflect media-price or inventory-quality changes; asymptotes are rarely identified without material variation or external information.
Resources and sources.
Download saturation curves. Related: read response curves, separate memory and saturation, compare the next increment. Sources: Hill (1910); Michaelis and Menten (1913); Jin et al. (2017).