Marginal return: placing the next euro, not rewarding the past.
Evidence status
What was reproduced, and what it does not prove.
The fixture and the Python run verify the response formula, the robust marginal return, the bounds and the rescaling of the educational allocator. They verify neither an empirical curve nor a market return.
- 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.
Local derivative
For a curve R(s), the marginal return is dR/ds at level s. A curve can have an excellent average return while already being almost saturated.
Common economic unit
Channels must be compared in incremental contribution after costs and over a coherent horizon. Comparing attributed revenue, leads and product margin creates a meaningless ranking.
Risk and constraints
Minimum presence, inventory capacity, buying step, delay of effect and confidence in the curve limit the theoretical equalisation.
Method
Building an allocation through small decisions
- 01Estimate the curves
Connect spend and contribution with deferred effects, saturation and context factors; publish the observed zone and the uncertainty.
- 02Compute the next tranche
Compare R(s + Δ) − R(s) for a step compatible with media buying and with the precision of the model.
- 03Apply the constraints
Discard the moves that violate minimum, maximum, available reach, contracts, creative quality or commercial capacity.
- 04Re-estimate after the move
An allocation changes the point of the slope. The calculation must be repeated after each significant tranche or step.
Four slopes at the current spend point
The curves show the marginal return of the next tranche of €10,000. The cumulative height of the channel is not enough to decide.
Moving €50,000 to gain €40,000 of contribution
At the current level, channel A returns €0.60 per additional euro and channel B €1.40. A tranche of €50,000 can be removed from A without falling below its minimum.
Projected loss on A: 50,000 × 0.60 = €30,000. Gain on B: 50,000 × 1.40 = €70,000. Net contribution gain: €40,000, before transition cost and interval of uncertainty.
Make a first step if the prudent bound stays positive, then recompute both slopes; do not extrapolate €1.40 to all future tranches.
Acceptance conditions
What must be true to act.
- 01
All channels are compared in the same unit of contribution and over the same horizon.
- 02
The slope is computed at the current spend level and not at zero.
- 03
Intervals of uncertainty and extrapolated zones are visible.
- 04
Reallocation progresses in steps with re-estimation.
Limits
What this analysis does not prove.
- Curves estimated on little variation can give unstable slopes.
- Interactions between channels make separate optimisation incomplete.
- A mathematical optimum can be unexecutable because of contracts, creative or buying capacity.
References
Works cited.
- Little, J. D. C. (1970), “Models and Managers: The Concept of a Decision Calculus”, Management Science.
- Jin, Y. et al. (2017), “Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects”.
- Sethi, S. P. (1977), “Dynamic Optimal Control Models in Advertising”, SIAM Review.