What is a Hill function?
The Hill function originated in biochemistry, where it described how binding sites become progressively occupied until they are full. In media planning it has been adopted for exactly the same shape of behaviour: a marketing channel that responds eagerly to early spend, then increasingly resists further investment as the most responsive audience is exhausted. Plotted with spend on the horizontal axis and response - revenue, leads, or conversions - on the vertical, a Hill curve traces a characteristic S: a slow start, a steep middle, and a long flattening tail that bends toward a ceiling but never quite reaches it.
Two numbers define the shape. The half-saturation point marks the spend level at which the channel delivers half of its maximum achievable response, and the slope (or shape) parameter governs how abruptly the curve turns from steep growth into a plateau. Because the function is grounded in this small set of interpretable parameters, it can be fitted to historic campaign data and then used to describe where a channel sits today and how it is likely to behave as you push more budget through it. The concept sits alongside adstock, which models how advertising impact carries over in time rather than across spend.
How do Hill functions model saturation?
Saturation is the point at which a channel can no longer convert extra spend into proportional results. Every channel has finite responsive audiences, finite quality inventory, and finite attention - so as budget rises, you increasingly buy impressions from people who were already going to convert, or from audiences who never will. The Hill function encodes this directly: in the steep region of the curve, an extra pound of spend produces a large jump in response; near the top, the same pound barely moves the line at all.
That bending behaviour is precisely what makes a Hill function useful for forecasting. The slope of the curve at any spend level is the marginal return - the productivity of the next pound - and as the curve flattens, marginal return falls even while total response keeps inching upward. This is why a channel can look healthy on blended averages while quietly destroying the efficiency of incremental spend, a distinction explored in marginal ROAS versus average ROAS. Reading the curve, not the average, is how you find the budget ceiling - the spend level beyond which additional investment stops paying for itself, a dynamic explained in detail in how saturation curves predict paid-media budget ceilings.
Hill function vs linear assumptions
Most spreadsheet-based planning quietly assumes linearity. If a channel returned a 4.0 ROAS last quarter, the plan implicitly expects it to return 4.0 again at double the budget - a straight line through the origin. Real media performance almost never behaves this way. A linear assumption has no ceiling, so it always rewards moving more money into whatever channel showed the best historic average, and it consistently overstates how much a larger budget will actually deliver.
A Hill function corrects this by building the ceiling into the model. Where the linear view sees limitless upside, the Hill curve shows the upside tapering as spend grows, which changes the recommendation entirely: the optimal allocation is not “everything into the best average” but a balance that keeps each channel on the productive part of its own curve. This is the core argument for media planning without spreadsheets - once non-linear saturation is in play, a linear model is not a simplification, it is a source of systematic error.
How ElenIQ uses Hill saturation curves
ElenIQ’s forecasting engine fits saturation curves to your historic channel data, combining a Hill-style response shape with adstock to capture both diminishing returns and carryover. Rather than reporting a single backward-looking average, it produces a response curve per channel and reads the marginal return at your current and proposed spend levels - so the recommendation is always about the next pound, not the last campaign. That is what “forecast before you spend” means in practice: you can see the budget ceiling before you hit it.
From there, planning becomes a question of moving budget along each curve toward the point of equal marginal return. You can pressure-test allocations with the budget allocation simulator to see how shifting spend changes forecast output, then commit with confidence. This forecast-led, marginal-thinking approach is built into Dex, the ElenIQ forecasting assistant, so performance teams and agencies plan around saturation rather than being surprised by it.
Related terms
- Adstock - how advertising impact carries over in time after exposure.
- iROAS - the incremental return on ad spend that saturation modelling protects.
- Marginal ROAS vs average ROAS - why the slope of the curve matters more than the blended average.
Frequently asked questions
What is a Hill function?
A Hill function is an S-shaped (sigmoidal) curve borrowed from biochemistry and applied in marketing to model how channel response flattens as spend rises. It captures diminishing returns: output climbs steeply at first, then bends toward a ceiling as the channel approaches saturation.
How do Hill functions model saturation?
A Hill function is shaped by two parameters - a half-saturation point, where the channel reaches half its maximum response, and a slope (or shape) parameter that controls how sharply the curve bends. Together they describe the point at which extra spend stops producing proportional returns, which is the essence of saturation modelling.
How is a Hill function different from a linear assumption?
A linear assumption treats every pound of spend as equally productive, so doubling the budget is expected to double the result. A Hill function instead bends toward a ceiling, recognising that audiences and inventory are finite. Linear models systematically overstate the return on incremental spend at higher budgets.
Why do Hill functions matter for budget planning?
Because they reveal the budget ceiling where additional spend stops paying for itself. By fitting a Hill curve to historic data, you can forecast where marginal return falls below your target and allocate budget to where the next pound works hardest, rather than over-investing in an already saturated channel.