A Hill curve is the mathematical shape a marketing mix model uses to describe what happens to a channel as spending on it rises. It maps media execution - impressions, gross rating points, euros of spend - onto a number between zero and one representing how much of that channel's maximum achievable effect has been extracted. The curve climbs steeply, then flattens. That flattening carries the commercial argument: the tenth thousand euros spent on display in a week is assumed to do less work than the first.

The function takes its name from Archibald Vivian Hill, a British physiologist who wrote it down in 1910 to describe how oxygen binds to haemoglobin. Nothing in the equation is biological. Marketing science borrowed it because advertising exposure and receptor binding share a structural feature, a finite pool of available responses that gets progressively used up. Inside a modern mix model the curve does one of two jobs. Adstock handles time. Hill handles volume.

The two parameters

Google's Meridian defines the function as Hill(x; ec, slope) equal to 1 divided by one plus (x divided by ec) raised to the power of negative slope, according to the framework's saturation documentation. Two numbers control everything.

The half-saturation point, written ec, is the level of media execution at which the curve reaches exactly 0.5, meaning half the channel's maximum effect. Meridian scales input data so that ec equal to one places half-saturation at the median non-zero media units per capita across geographies and time periods. An ec of 3 puts it at three times that median, a channel with substantial headroom. The default prior is a normal distribution centred on 0.8 with a scale of 0.8, truncated to the range 0.1 to 10. Outside those bounds, the documentation notes, the data contains no real information about where half-saturation falls.

The slope governs curvature. At one or below the curve is concave, with returns diminishing from the first unit. Above one it is S-shaped, convex below ec and concave above it, implying a threshold where small amounts of spending achieve almost nothing. Meridian's default prior for slope on paid and organic media is Deterministic(1), fixing the parameter rather than estimating it, for two stated reasons: the value is difficult to learn, and the budget optimiser only guarantees a global optimum on concave curves.

Meta's Robyn uses the same function with different labels. Its saturation_hill transformation takes alpha, controlling the shape between C-shape and S-shape, and gamma, setting the inflexion point, with recommended bounds of 0.5 to 3 and 0.3 to 1 respectively. Because Robyn transforms each media variable individually, parameter counts climb fast: ten media variables with geometric adstock produce 32 hyperparameters, of which twenty are alphas and gammas.

Where the curve sits in the transaction

Mix models do not observe individual impressions. They regress an aggregate outcome, usually weekly revenue by geography, on aggregate spending, with the transformations sitting between the raw media series and the coefficient.

Order matters. In Meridian the boolean hill_before_adstock argument defaults to false, so spend is adstocked first and the Hill function applied to the accumulated, decayed quantity rather than the week's raw execution. The media effect for a channel in a geography and week is then a geo-level coefficient multiplied by Hill applied to Adstock applied to the spend series.

The curve appears in a second place. For channels supplying reach and frequency data rather than impressions alone, Meridian fits a Hill function over frequency, with priors on ec and slope set jointly so that implied optimal frequency has a prior mean of 2.1 and a 90 percent interval of 1.0 to 4.4.

What planners see is the downstream output. Fitted parameters generate a response curve per channel, and the gradient at current spending is marginal return on investment. Meridian's September 2025 release added marginal ROI priors, on the reasoning that a channel deep in saturation can show a strong average return and a poor return on the next unit of budget.

From haemoglobin to media plans

Hill presented the equation to the Physiological Society at King's College London on January 22, 1910, in a communication on the aggregation of haemoglobin molecules and its dissociation curves. He was 23. The same form surfaced independently as the Langmuir adsorption isotherm.

Advertising response modelling arrived from the other direction. Practitioners had argued about concave against S-shaped sales response since the 1950s, and the ADBUDG function introduced in 1970 produced a bounded curve with an exponent controlling which of the two applied. Dew, Padilla and Shchetkina note in their 2024 paper that the Hill function is equivalent both to ADBUDG and to the log-logistic cumulative distribution function.

The bridge into current software was a 2017 Google Research paper by Yuxue Jin, Yueqing Wang, Yunting Sun, David Chan and Jim Koehler, Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects. It paired an adstock decay function for carryover with a Hill function for what it called shape effects, and derived return on ad spend and marginal return from posterior samples. That specification became the template. Meta Marketing Science released Robyn as an open-source R package in 2021 with a Hill saturation transform at its core. Google unveiled Meridian in March 2024 on a limited basis and opened it globally on January 29, 2025.

Why the shape decides budgets

Mix modelling regained commercial weight as user-level attribution degraded, a shift documented in 2024. The Hill parameters convert a fitted model from a report into a plan.

In an Ads DevCast episode published on May 28, 2026, Google developer relations engineer Jeff Li described Hill curves and adstock as "universally accepted in the world of advertising", illustrating the output with a display channel where the next $100,000 of spend returns half what the previous $100,000 returned. The episode framed the pairing as the engine behind Meridian's budget headroom analysis, separating channels with room to absorb money from channels already flat.

That machinery has been packaged for non-modellers. The Scenario Planner launched on February 19, 2026 as a code-free interface plotting optimised against non-optimised budget by channel, then opened in beta with per-channel spend constraints, and Meridian reached Google Analytics 360 on May 20, 2026. Commercial vendors compete on the same curve: IAB Australia profiled twelve providers in September 2025.

Limitations and disputes

The most damaging criticism is internal to the function. Different combinations of ec and slope produce almost identical curves over any finite range of observed spending, a weak-identifiability problem established in the 2017 Google paper. Fixing the slope at one, as Meridian does, is a response to that rather than a claim about consumer behaviour.

Extrapolation is the second problem, and Google's documentation states it directly: estimation is based on the observed range of media data, and curves extended beyond that range should be treated with caution. A large budget expansion rests on the assumed functional form rather than on evidence. Niklas Heusch's August 21, 2026 paper made the same point from experimental data, noting that posterior response curves track the true curve across the tested range and fan out beyond it. That paper reported a conventionally specified model returning 10.61 times return on ad spend for a paid search channel whose true value was 4.20.

Whether saturation and time-varying effectiveness can be told apart is contested. The 2024 Wharton and London Business School paper found the two conflated in roughly 90 percent of simulation settings under high carryover, and priced one resulting misallocation at $227,000 across 14 weeks.

The S-shape itself is disputed inside Google. Marketing researchers Pablo Perez and Jesus Martin Calvo argued in February 2026 that S-shaped response curves contradict consumer behaviour models and brand lift data, reflecting the difficulty aggregate models have with low execution levels rather than a real threshold. Governance concerns run alongside. CIMM warned on July 31, 2026 that default priors in open-source frameworks can reconfigure data asymmetry rather than remove it, an argument that lands on a deterministic slope prior shipped by a media seller, and PPC Land examined the same tension through Meta's Robyn in April 2026.

What Hill curves are not

Adstock is the other half of the pair. It distributes the effect of spending across subsequent weeks through a decay parameter; Hill compresses the effect of volume within a period. A model can misestimate one while getting the other right.

A response curve is the output, not the transformation. It plots incremental outcome against spend in currency, combining the Hill shape with the fitted coefficient and any scaling. Two different Hill parameter sets can produce near-identical response curves.

Logistic saturation is an alternative form, the default in pymc-marketing and in the Heusch paper. It also produces diminishing returns and is not interchangeable with Hill. The reach transformation is the special case with the shape parameter fixed at one, named in the 2017 Google paper for relating television reach to gross rating points.

Recent developments

Attention has shifted from the curve to how it is calibrated. Meridian GeoX, announced on May 5, 2026 in Google's pre-Google Marketing Live measurement push, runs geographic experiments and feeds the results back as priors. Heusch's counter-argument is that collapsing an experiment into one lift figure discards the information that identifies curvature, and that tests at different spending levels can estimate the curve directly. Google cut its minimum incrementality experiment budget to $5,000 on November 11, 2025, lowering the entry price without changing the design requirement.

Confidence in the outputs remains uneven. IAB's State of Data 2026, published February 7, 2026, found up to 75 percent of buy-side decision-makers rating attribution, incrementality and mix modelling as underperforming.

Timeline

  • January 22, 1910: Archibald Vivian Hill presents his equation on haemoglobin dissociation curves to the Physiological Society in London
  • 1970: the ADBUDG response function, mathematically equivalent to the Hill form, enters advertising budget modelling
  • April 2017: Google Research publishes "Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects", pairing adstock with a Hill saturation function
  • 2021: Meta Marketing Science releases Robyn as an open-source package with a two-parameter Hill saturation transform
  • August 14, 2024: "Your MMM is Broken" is posted to arXiv, documenting conflation between nonlinear and time-varying specifications
  • January 29, 2025: Google opens Meridian globally after testing with hundreds of brands
  • September 30, 2025: Meridian adds binomial adstock decay, non-media variables and channel-level contribution priors
  • November 11, 2025: Google lowers its minimum incrementality experiment budget to $5,000
  • February 7, 2026: IAB State of Data 2026 records buy-side dissatisfaction with mix modelling and incrementality
  • February 19, 2026: Meridian Scenario Planner launches as a code-free budget interface
  • May 5, 2026: Meridian GeoX and Meridian Studio are announced
  • May 20, 2026: Meridian is integrated into Google Analytics 360
  • May 28, 2026: Google's Ads DevCast publishes a technical walkthrough of Meridian's Hill curve and adstock mathematics
  • July 31, 2026: CIMM warns that default priors in open-source frameworks carry allocation consequences
  • August 21, 2026: "Structural Estimation of Marketing Mix Model Parameters from Geo-Experiments" is posted to arXiv

Summary

Who: Data scientists and measurement teams at advertisers, agencies and modelling vendors fit Hill curves; Google maintains the parameterisation used in Meridian, Meta Marketing Science maintains Robyn's alpha and gamma version, and the pymc-marketing project maintains an open-source alternative. The functional form originates with the physiologist Archibald Vivian Hill and entered advertising practice through a 2017 Google Research paper by Yuxue Jin, Yueqing Wang, Yunting Sun, David Chan and Jim Koehler.

What: A two-parameter saturation function that converts media execution into a bounded measure of how much of a channel's achievable effect has been captured, with a half-saturation point setting where the curve reaches half its maximum and a slope or alpha parameter setting whether returns diminish immediately or only after a threshold.

When: Published in 1910, formalised for advertising through the ADBUDG lineage from 1970 and the 2017 Bayesian mix modelling paper, and embedded in open-source frameworks from 2021 onward. Contested in published research through 2024 and 2026.

Where: Inside marketing mix models operating on aggregated weekly or geo-level data across search, social, retail media, display, audio, connected television and offline channels, and exposed to planners through response curves, budget optimisers and scenario planning interfaces.

Why: Budget allocation across channels requires an estimate of what the next unit of spend will return, not what the last one did on average. The Hill curve is the standard device for producing that estimate. Its parameters are also weakly identified, its extrapolation beyond observed spending is unsupported by data, and independent research has shown it can be confused with time-varying effectiveness, which means the curve determines allocation decisions on evidence that is thinner than its smooth rendering in a dashboard suggests.