Marketing Science
Hill Function Calculator
Explore how the Hill coefficient transforms marketing response — from concave curves with immediate diminishing returns to S-curves with threshold effects. A core saturation model in Marketing Mix Modeling.
Hill Function
f(x) = xⁿ / (Kⁿ + xⁿ)
x = media execution (spend, impressions, GRPs) — n = shape — K = half-saturation point
Channel Presets
Curve Shape
Response Curve
Interpretation
S-curve (n = 1.50). This channel requires threshold investment before becoming efficient. The inflection point at $17,100 marks where returns accelerate.
Common for awareness channels like TV or OOH that need frequency to be effective. Spending below the threshold produces minimal returns — either commit to the required level or reallocate.
Scope
The Hill function models saturation — how response diminishes as execution increases. In practice, K and n are not set manually. They are estimated by the model using Bayesian inference with informative priors, often calibrated from industry benchmarks or experimental data. This calculator lets you build intuition for how shape parameters affect budget allocation decisions before the model estimates them from data.
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