Response curves: choosing a shape that stays credible outside the sample.
Evidence status
What was reproduced, and what it does not prove.
The fixture and the Python run verify the saturating response curve, its weighted marginal derivative and the sealed outputs of the educational allocator. They do not select a curve shape from market observations.
- Technical specification of the budget allocator, specification in French (opens in a new tab)
v1, 31 August 2026 · CC BY 4.0 · specification in French · Method document written by marketing-mix.net. The figures are illustrative: they are not market measurements.
Key distinctions
Three conditions before calculating.
Incremental effect
The explained variable must represent a causal or calibrated effect, not an attributed sale. A very precise curve on a biased signal industrialises the bias instead of securing the allocation.
Observed support
The function is more credible within the spend range genuinely observed. A recommendation far from the support requires an explicit assumption, a range of uncertainty and preferably a test.
Decision slope
The allocation depends on the local derivative: the expected contribution of the next euro. Two functions close in level can produce opposite decisions if their slopes diverge at the current budget.
Method
Comparing the functions on fit and on decision
- 01Prepare the identified points
Assemble the spend–effect pairs coming from credible historical variations, from experiments or from a calibrated model; keep the dispersion and the context of each point.
- 02Fit several families
Estimate at minimum a linear reference and two saturating shapes; impose parameters compatible with the expected sign, asymptote and speed of response.
- 03Validate out of period
Compare predictive error, parameter stability and interval coverage on withheld periods; examine high and low spends separately.
- 04Compare the decisions
Compute level, average return and marginal slope at the current budget and at the proposed budgets; refuse an allocation that depends solely on an unstable extrapolation.
Three functions, three marginal decisions
The same observations are fitted by a straight line, an exponential saturation and a Hill curve; their extrapolations and their slopes are compared.
The straight line overestimates the effect at €200k by 15%
The observed effects, in thousands of euros, are 0, 49, 79, 97 and 108 for spends of €0, €50k, €100k, €150k and €200k. A straight line gives €124k at €200k, while an exponential saturation with a ceiling of 125 and a speed of 0.01 gives €108.1k.
Error of the straight line at the last point: 124 − 108 = €16k, that is 14.8%. Exponential slope at €200k: 125 × 0.01 × exp(−0.01 × 200) = €0.169k of effect per €k spent.
The exponential saturation is retained within this range. The straight line would keep funding the lever for too long; the Hill curve remains a sensitivity scenario as long as the acceleration zone is not observed.
Acceptance conditions
What must be true to act.
- 01
The fitting points represent an incremental or calibrated effect.
- 02
The recommended range stays close to the observed spends.
- 03
Out-of-period error, parameters and marginal slope are published together.
- 04
A second plausible function accompanies any material extrapolation.
Limits
What this analysis does not prove.
- An aggregate function hides the differences of creative, area or segment.
- The parameters change with price, competition, media quality and season.
- The absence of spend variation prevents the curvature from being identified.
References
Works cited.
- Vidale, M. L. & Wolfe, H. B. (1957), “An Operations-Research Study of Sales Response to Advertising”, Operations Research, 5(3), 370–381.
- Little, J. D. C. (1970), “Models and Managers: The Concept of a Decision Calculus”, Management Science, 16(8), B466–B485.
- Chan, D. & Perry, M. (2017), “Challenges and Opportunities in Media Mix Modeling”, Google Research.