Many budget allocation strategies assume that every channel follows the same pattern: the first dollar is the most productive, and each additional dollar yields a slightly lower return.

The charts below show what that pattern looks like.

The log shape means that the first dollar is the most productive, and each subsequent dollar is worth a little less. When every channel looks like that, the game plan is to spread the budget to as many channels as possible and equalize the marginal CPAs to maximize profit.

But not every channel looks like that. Some have a warm-up region where the early spend is the least efficient, not the most. On those channels, the logic above breaks, and so does the “test small, scale the winners” playbook that most of the industry runs on autopilot. 

The difference comes down to one question about the channel: Is the response curve C-shaped or S-shaped?

The answer can change how you approach channel testing and channel measurement, including any MMM analysis. Moreover, Google has been incorporating more S-shaped campaign types, and after its Google Marketing Live announcements, this trend seems set to continue.

The two shapes — and the only part that matters

The response curve plots output (conversions, revenue) against input (spend). This generally results in two types of curves in marketing.

• C-shaped (concave): Diminishing returns from the very first dollar. A log or power curve. Picture the top-left quarter of a circle: steep at the start, flattening as you go.
• S-shaped (sigmoid): A slow, inefficient start, then an inflection point where it gets steep, followed by a flattening into saturation. A logistic curve.

The response curve itself isn’t what you allocate against. You allocate against the marginal curve, the derivative, which answers the question: “What did the next dollar buy me?” That’s where the shapes diverge in a way that matters.

• For a C-curve, marginal return is highest at the first dollar and falls in only one direction. Marginal CPA rises from the first dollar onward. If conversions are a*ln(s), marginal conversions per dollar are a/s, so marginal CPA is s/a, climbing in a straight line as you scale. There’s no warm-up. The cheapest conversion you’ll ever buy is the first one.
• For an S-curve, marginal return starts low, rises to a peak at the inflection point, then falls. Marginal CPA is U-shaped. It’s expensive at the start, bottoms out around the inflection point, then climbs into saturation.

That region of increasing marginal returns is the whole story. It’s the difference between a channel where small budgets are productive and one where they are wasted.

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How this looks in a marketing campaign

Say your CPA goal is $50. Here is an S-shaped channel, modeled as Conversions = 1000 / (1 + e^(-0.25(s – 20))), with spend in the thousands and the inflection at $20,000/month:

Run the $10,000 test that a sane person runs before committing real budget. Average CPA comes back at $132, marginal around $94. If those two metrics are all you look at, you conclude that this channel can’t hit $50, so let’s kill it.

That verdict is wrong. At $20,000 to $25,000, the channel is running at an average of $32 to $40, and the marginal dollar in the $15,000 to $25,000 band costs $18. That’s not “barely viable.” In that band, it’s the best marginal buy you have. The small test fell within the warm-up and reversed the conclusion.

In a C-shaped channel, the small test would have shown you the best the channel can do. On an S-shaped channel, it shows you the worst.

This is the trap. The standard playbook is “test small, scale what works.” On S-curves, small tests systematically condemn channels that would’ve worked at scale because the test is structurally stuck in the inefficient region.

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The allocation logic, restated

C-shaped channels, go wide

The optimization is convex. There’s one global optimum, the equimarginal rule from the marginal-CPA post applies cleanly, and the solution is usually interior, meaning lots of channels get funded.

Even a small allocation is productive because the first dollar is the best dollar. Run many channels lean, reallocate continuously at the margin, and pull back the instant marginal CPA crosses your goal.

S-shaped channels, go deep or skip

The optimization is non-convex. A small allocation can be strictly worse than zero because below the inflection your marginal return sits under your target, and you’ve sunk money to get nowhere.

The decision isn’t “how much.” It is binary: commit past the threshold, or don’t fund it at all. There’s a real minimum viable budget, and it’s often above normal test budgets. You can’t sprinkle an S-curve and expect efficiency, and you can’t evaluate one on an underfunded test.

Those two rules can look like they fight each other, but that’s only true to a certain point. Past the inflection, an S-curve is concave, so the equimarginal rule governs it exactly as it governs a true C. The S-specific instruction — commit a block instead of sprinkling — is only about the trip from zero to past the inflection.

Shape is therefore mostly a launch-and-evaluation problem. Getting a new prospecting channel into its efficient range requires a committed block and patience with ugly early numbers. Once it clears the inflection, you manage it at the margin like everything else, right up until you consider cutting it hard, where shape matters again because the downside is a cliff, not a ramp.

This is the part that’s genuinely counterintuitive, and it echoes the original marginal-return point: The right move isn’t always the one that looks most efficient at a small scale.

Which channels are which?

The historical default was concave. Simon and Arndt reviewed more than 100 studies and concluded that advertising follows the law of diminishing returns, a concave response. 

The dissent came later: Vakratsas, Feinberg, Bass, and Kalyanaram found that threshold effects do exist and that response is not necessarily globally concave. Their explanation for why thresholds were so hard to find is the useful part. Mature accounts already operate inside the effective range, so the warm-up never shows up in the data, and most studies fit a concave model (the double-log) that can’t reject an S-curve even when one is present.

The platform shift has made the threshold visible again. Here is a fuller map, ordered roughly from C to S. The shape column is an inference from how each system targets and learns, not a measured constant, and the right shape for your account still has to be measured.

Two rows do most of the work.

AI Max is the live example of a channel migrating from C toward S. Swapping explicit keywords for broad and keywordless matching means it needs conversion volume to learn which queries convert, so below a data threshold, it explores badly.

The mixed independent results fit that: Google reports about 14% more conversions on average and up to 27% for exact-match-heavy campaigns, while independent testing reports 84% of advertisers seeing neutral or negative results. Much of that spread is accounts that turned it on without the conversion volume to clear the learning region.

Performance Max is the trap, because its curve is a composite. It blends a harvesting layer (branded, retargeting, Shopping against existing intent) with a prospecting layer (keywordless expansion across surfaces). The harvesting layer is a cheap C that pays off on the first dollar. The prospecting layer is the S underneath.

Blended, the early efficiency looks great, because you are mostly skimming demand you already had, and the average hides the prospecting warm-up entirely. That is also why the platform is glad to optimize it for you: the blend flatters the headline number. You can’t read PMax or run the shape analysis on it until you split the harvesting from the prospecting.

The throughline runs in two layers. Rules-based auctions capture the best inventory first, which yields concavity; machine-learning systems must be fed before they are efficient, which introduces a threshold. Underneath both, harvesting existing demand is concave and mostly non-incremental, while creating new demand is the S-shaped part where the real growth and the real warm-up cost both sit.

Average versus marginal: total over spend, or the slope where you stand.

What you allocate against is marginal incremental return, the slope of the incremental curve at your operating point. A holdout fixes the first axis only. Time-sliced marginal CPA on attributed data fixes the second only. A multi-cell scaling test gets both, at a cost. 

MMM (method 1) estimates the whole curve from aggregate data and sidesteps click attribution entirely, but pays in identifiability and modeling assumptions instead. Most arguments about ‘what is working’ are two people standing on different axes.

There are two major cautions, and I would flag both as genuinely unsettled rather than settled facts. 

• Separating a true S-curve from “concave with a high half-saturation point” is hard, because a concave model will fit S-shaped data well enough to hide the inflection (this is the Vakratsas point, and it applies to your own dashboards as much as to academic studies). 
• The learning phase may be a one-time fixed cost to train the model rather than a permanent feature of the steady-state curve. If it is transient, the channel may behave concavely at the margin once it is trained, and the S you measured was a startup artifact. The truth is probably a mix: a one-time training cost, plus an ongoing minimum-volume requirement to stay efficient. Treat every shape call as provisional and re-check it.

One more failure mode, and this one is not unsettled science but a matter of where you are standing on the curve. An S only looks like an S if your data spans the inflection. 

Above the inflection, an S is concave, mathematically identical to a C. Look at only the $20,000-and-up rows of the table above: marginal CPA rises monotonically from $18, a textbook C-curve, and the convex warm-up is invisible because you are no longer operating in it. 

Established accounts usually sit past the inflection, which is exactly why Vakratsas found thresholds so hard to detect, and why you can run an S-shaped channel for years, correctly, while believing it is concave. The tell arrives the day you cut hard and fall off the inflection instead of easing down a slope.

When to go wide and when to go deep

The marginal-return post told you to equalize marginal CPAs across the program. That rule is still correct, but the shape of the curve tells you how you’re allowed to get there. 

• On C-shaped channels, you can get there by sprinkling, because every dollar is productive and breadth is the natural answer. 
• On S-shaped channels, you have to commit a block of budget past the inflection before the channel earns its place, and then concentrate rather than spread.

Lay the harvest-versus-create cut on top. Harvesting channels (branded, retargeting, non-brand search) are your C-curves: fund the first dollars, then cap them early, because they saturate fast and most of the tail isn’t incremental, no matter how strong the attributed ROAS looks. 

Prospecting channels (Meta, YouTube, LinkedIn, the expansion half of PMax) are your S-curves and your only real source of incremental growth: commit past the warm-up or don’t start, and judge them on incremental lift rather than attributed CPA, or you’ll kill the thing that was working.

Classic search rewards going wide. PMax, AI Max, and Meta prospecting reward going deep on fewer bets and giving each enough volume to clear the warm-up. Run an S-curve like a C-curve and you’ll starve it, read the underfunded result, and kill a channel that would’ve been one of your best.