If your agency has ever shown you a smooth response curve with a red dot marked “current spend”, you have already met the Hill curve. It is the piece of math inside almost every marketing mix model that decides which channel gets more money next quarter.
PPC Land has published a deep explainer on the function. Here is what performance teams actually need to take from it.
What a Hill curve does
A Hill curve maps media execution — impressions, GRPs, euros of spend — onto a number between 0 and 1 representing how much of a channel’s maximum achievable effect has been extracted. It rises steeply, then flattens. That flattening is the whole commercial argument: the tenth thousand euros of display in a week is assumed to work less hard than the first.
Inside a mix model there is a clean division of labour. Adstock handles time — how spend echoes into later weeks. Hill handles volume — how much of this week’s push actually lands.
The name comes from Archibald Vivian Hill, a British physiologist who wrote the equation down in 1910 to describe oxygen binding to haemoglobin. He was 23. Nothing in it is biological; marketing borrowed it because ad exposure and receptor binding share one structural feature — a finite pool of responses that gets used up.
Two parameters run everything
Google’s Meridian defines it as Hill(x; ec, slope). Two numbers control the shape:
- ec, the half-saturation point — the spend level where the channel delivers half its maximum effect. Meridian scales data so ec = 1 sits at the median non-zero media units per capita. An ec of 3 means a channel with real headroom. Default prior: normal, centred on 0.8, scale 0.8, truncated to 0.1–10.
- slope — at 1 or below the curve is concave (diminishing returns from unit one). Above 1 it turns S-shaped, implying a threshold below which small budgets do almost nothing.
Meridian’s default prior for slope is Deterministic(1) — fixed, not estimated. Two stated reasons: it is hard to learn from data, and the budget optimiser only guarantees a global optimum on concave curves.
Meta’s Robyn uses the same function with different labels: alpha for shape (bounds 0.5–3) and gamma for the inflexion point (0.3–1). Because Robyn transforms each media variable individually, ten media variables with geometric adstock produce 32 hyperparameters — twenty of them alphas and gammas.
Why marketers should care
Because average ROI is a lousy allocation signal. A channel deep in saturation can post a strong average return and a terrible return on the next dollar. Meridian’s September 2025 release added marginal ROI priors for exactly that reason.
In a Google Ads DevCast episode published on 28 May 2026, developer relations engineer Jeff Li called Hill curves and adstock “universally accepted in the world of advertising”, illustrating with a display channel where the next $100,000 returns half what the previous $100,000 did.
The cracks in the curve
This is the part dashboards hide.
Different combinations of ec and slope produce nearly identical curves over any observed spend range — a weak-identifiability problem documented in Google’s own 2017 paper. Fixing slope at 1 is a workaround, not a behavioural claim.
Extrapolation is worse. Google’s docs say estimation is based on the observed range and curves beyond it should be treated with caution. Niklas Heusch’s 21 August 2026 paper reported a conventionally specified model returning 10.61x ROAS on paid search whose true value was 4.20.
A 2024 Wharton and London Business School paper found saturation and time-varying effectiveness conflated in roughly 90% of simulation settings under high carryover, pricing one misallocation at $227,000 across 14 weeks. And IAB’s State of Data 2026 found up to 75% of buy-side decision-makers rate attribution, incrementality and mix modelling as underperforming.
What to do on Monday
Ask three questions of any MMM output. What spend range was actually observed? Was the slope estimated or fixed? Have the curves been calibrated with a geo experiment — Google cut its minimum incrementality test budget to $5,000 on 11 November 2025, so the excuse of cost is thinner than it was.
Trust the curve inside your tested range. Test before you trust it outside.
Source: PPC Land
