A response curve converts spend into expected incremental effect.
A response curve converts spend into expected incremental effect. Its form directly drives allocation: linear means the next euro is always worth as much, exponential saturation imposes immediate diminishing returns, and a Hill curve permits acceleration before saturation. Best historical fit is not necessarily best decision function. Compare plausibility, stability, out-of-sample error and marginal behaviour over the actionable range.
Decision and method.
Retain the simplest function that reproduces observations, respects expected mechanisms and yields stable marginal returns in the allocation range. Assemble spend–effect pairs from credible historical variation, experiments or calibrated model, retaining dispersion and context. Fit at least a linear reference and two saturating forms with parameters consistent with expected sign, asymptote and response speed. Compare predictive error, parameter stability and interval coverage on held-out periods, separately for high and low spend. Calculate level, average return and local marginal slope at current and proposed budgets; reject allocation driven only by unstable extrapolation.
Worked example.
Observed effects in thousands of euros are 0, 49, 79, 97, 108 for spend 0, 50, 100, 150, 200 k€. A line gives 124 k€ at 200 k€; exponential saturation with ceiling 125 and speed 0.01 gives 108.1 k€. Line error at final point: 124 − 108 = 16 k€, or 14.8%. Exponential slope at 200 k€: 125 × 0.01 × exp(−0.01 × 200) = 0.169 k€ effect per k€ spent. Retain exponential saturation in this range; Hill remains a sensitivity scenario until acceleration is observed.
Checks and limits.
Fitted points represent incremental or calibrated effect; recommended range remains near observed spend; out-of-period error, parameters and marginal slope are published together; every important extrapolation includes a second plausible function. An aggregated function hides creative, zone and segment differences. Parameters change with price, competition, media quality and season. Without spending variation, curvature cannot be identified.
Resources and sources.
Download curve parameters. Related: explore saturation, compare saturation functions, calculate marginal slope, bound reallocation. Sources: Vidale and Wolfe (1957); Little (1970); Chan and Perry (2017).