Response curves: a library of testable, comparable shapes.
Answer:
A curve library does not supply ready-to-use coefficients. It makes the behaviour imposed by each form explicit: constancy, diminishing returns, an acceleration threshold, an asymptote or local flexibility. Each profile must state its equation, identifiable parameters, observed spending range, marginal slope, half-saturation point, uncertainty and extrapolation risks. The selected form is an economic hypothesis to confront with data and experimentation.
Method and decision:
First choose whether the mechanism requires monotonicity, saturation, a threshold or locally changing behaviour. Fit comparable economic units and align the horizon with adstock; control price, promotion, distribution, trend and seasonality. Then compare out-of-sample error, parameter stability, residuals, derivative and half-saturation point under several windows and priors. Publish the observed and tested minimum and maximum spend, current budget, usable range, response interval and update rule. Use a curve only within its documented range and select the simplest form whose marginal slope remains stable under validation and sensitivity analysis.
Worked example:
Five observations between €0k and €200k are fitted with exponential saturation and a Hill curve. Their validation error differs by only 1.6 points. At €200k, published responses are 108.08 and 103.22 units; at €300k, exponential predicts 118.78 and Hill 112.47. With €8k contribution per unit, the extrapolation difference is (118.78 − 112.47) × 8 = €50.48k. Published slopes at €200k are 0.1692 and 0.1362 unit per €k, so their uncertainty must be estimated before ranking channels. The €300k budget lies outside the documented domain; no spend above €200k is recommended without a test. A €20k increment can be funded as sequential learning.
Limits:
A teaching library does not replace market-specific calibration. Adstock, targeting and creative quality may shift a curve over time. Single-channel curves describe synergies and substitutions poorly.