Response curves: breadth vs. budget commitment

Many budget allocation strategies in online marketing silently assume that every channel follows the same pattern: the first dollar invested delivers the highest return, and each additional dollar yields slightly less. This assumption matches a logarithmic or power curve with diminishing marginal returns. As long as every channel looks like that, the rule is simple: spread budget widely, equalize marginal cost-per-acquisition (CPA), and maximize total profit.

But not every channel behaves that way. Some have a warm-up phase where early spend is the least efficient—not the most productive. On those channels, the classic logic breaks down, along with the widespread playbook of "test small, scale the winners." The decisive question is: Is the response curve C-shaped or S-shaped?

The answer changes channel testing, measurement models, and any marketing mix modeling (MMM) analysis. Google is increasingly integrating campaign types with S-shaped behavior; after Google Marketing Live announcements, this trend is likely to continue.

Two curve shapes and what matters at the margin

The response curve plots output—conversions or revenue—against input, meaning media spend. In marketing, this generally produces two curve types.

• C-shaped (concave): Diminishing returns from the first dollar. A steep start, then flattening—like the top-left quarter of a circle.
• S-shaped (sigmoid): A slow, inefficient start, then an inflection point with steep growth, followed by saturation. A logistic curve.

What matters is not the total curve itself, but the marginal curve—the derivative. It answers: What did the next dollar buy? On C-curves, marginal return is highest at the first dollar and only falls from there. Marginal CPA rises from dollar one. There is no warm-up; the cheapest conversion is the first one.

On S-curves, marginal return starts low, rises to the inflection point, then falls again. Marginal CPA is U-shaped: expensive at the start, cheapest around the inflection, expensive again in saturation. This region of increasing marginal returns separates channels where small budgets make sense from those where they burn money.

How this plays out in campaigns

Take a CPA goal of $50 and an S-shaped channel with an inflection at $20,000 monthly spend. A cautious $10,000 test might return an average CPA of $132 and a marginal value around $94. Teams that stop at those metrics kill the channel—incorrectly.

At $20,000 to $25,000 spend, average CPA sits at $32 to $40; the marginal dollar in the $15,000 to $25,000 band costs about $18. The small test stayed in the warm-up and reversed the verdict. On C-channels, a mini-test shows the best possible outcome; on S-channels, the worst. The standard playbook systematically condemns channels that would have worked at scale.

Allocation logic: go wide or go deep

C-shaped channels: use breadth

Optimization is convex. The equimarginal rule applies cleanly: many channels receive small but productive shares. Every dollar counts because the first is the best. Teams run many channels lean, reallocate continuously at the margin, and pull budget back as soon as marginal CPA crosses the target.

S-shaped channels: go deep or skip

Here optimization is non-convex. A small allocation can be worse than investing nothing, because below the inflection marginal return sits under target. The decision is binary: commit a budget block past the threshold or do not start. S-curves cannot be sprinkled and should not be judged on underfunded tests.

Beyond the inflection, an S-curve behaves concavely—then the equimarginal rule applies like on C-channels. The S-specific instruction mainly concerns the path from zero past the inflection.

Which channels are which?

For a long time, advertising was treated as globally concave. Later studies showed threshold effects: mature accounts often already operate in the efficient range, so warm-up phases disappear from data. Platform shifts make thresholds visible again.

AI Max is migrating from C toward S: broad and keywordless matching needs conversion volume to learn; below a data threshold, systems explore inefficiently. Performance Max is especially tricky because it blends a harvesting layer (brand, retargeting, shopping) with a prospecting S-layer. Early efficiency looks great because existing demand is harvested—the prospecting warm-up stays hidden in averages.

Measurement and operating rules

What teams allocate against is marginal incremental return—the slope at the current operating point. Holdout tests, time-sliced marginal CPAs, and multi-cell scaling tests capture different slices of the same reality. MMM estimates the full curve from aggregate data but pays with modeling assumptions.

• Separating a true S-curve from a concave curve with a high saturation point is hard—dashboards can hide inflection points.
• Learning phases may be one-time training costs; afterward a channel may behave concavely at the margin. Every shape call stays provisional.
• Established accounts often sit above the inflection and wrongly believe in C-behavior—until a hard cut reveals the S-shape.

Harvesting channels are C-curves: fund the first dollars, then cap early because the tail is rarely incremental. Prospecting channels are the S-curves and the real growth source: commit past warm-up or do not start—and judge on incremental lift rather than attributed CPA. Classic search rewards breadth; PMax, AI Max, and Meta prospecting reward depth on fewer, sufficiently funded bets.