Channel Physics

Channels do not return what you put in, in proportion. Treating spend as a straight line is the field’s most expensive habit.

A team I worked near found a channel that seemed too good to argue with. At a modest spend, somewhere around ten thousand a month, it returned customers at a cost that made every other channel look wasteful. The logic that followed felt airtight. If ten thousand produced this, then a hundred thousand should produce roughly ten times as much, so they raised the budget by a factor of ten and waited for the corresponding return. What they got was closer to three times the customers, at more than three times the cost per customer, and the channel that had been the obvious winner was now an expensive disappointment. Nobody could quite say what had gone wrong, because nothing had gone wrong in the way they were looking for. No campaign had failed. The channel had simply been pushed up a curve that bends, and they had been planning as though it were a straight line.

For a while the disappointment was read as bad luck or weak execution, a creative that had gone stale, an audience that had been used up, a competitor who had moved in. Some of those things may even have been happening at the edges. None of them was the cause. The cause was structural, and it would have shown up no matter how fresh the creative stayed, because the channel had been carried into a region of its own response where money stops turning into customers, and no amount of skill applied to the spending can change the shape of the curve the spending is climbing.

This is the most expensive habit in go-to-market, the assumption that a channel returns output in proportion to input, that if a little works then a lot works proportionally more. Channels do not behave that way, almost none of them do, and the way they actually behave has a clean shape that has been understood for a long time in any field that buys results with money. Until you can reason about that shape, you will keep making the same mistake the team made, pouring budget into a winner until it stops winning and never quite understanding why the arithmetic that looked so solid at the small scale fell apart at the large one.

The shape of a channel

The single most important fact about a channel is that its response to spend is concave, which is a precise way of saying that each additional dollar buys less than the dollar before it. The first dollars into a channel reach the people who are easiest to reach and most ready to act, and they return a great deal. As you spend more, you reach further into the population, toward people who are harder to move, less ready, more expensive to convince, and the return on each new dollar shrinks. Eventually you are spending real money to reach people who were never going to respond much, and the curve flattens toward a ceiling. This is the basic physics of buying attention from a finite population with varying willingness to respond, not a flaw in any particular channel.

It is worth seeing why the concavity is close to inevitable, because it makes the shape feel less like an empirical accident and more like something you could have predicted from first principles. Any channel reaches a population, and within that population people differ in how ready they are to respond and how expensive they are to reach. A channel that optimizes at all reaches the cheap, ready people first, because that is what optimization does, so the early spend skims the most responsive layer and returns brilliantly. Each later increment has to reach a less responsive layer, since the responsive ones are already taken, so each increment costs more and returns less, until the channel is spending heavily to reach people who will barely respond at all. The curve bends because the population is finite and ordered by willingness, and that ordering is a feature of essentially every channel, which is why essentially every channel saturates. The same logic shows up in several functional forms, and another common one, sometimes called a Hill curve, adds a slow start before the channel finds its audience, then a steep middle, then the same flattening. The flattening is the universal part.

Let me write the response of a channel as a function, $r(x)$, the amount of demand a channel produces at an input of $x$, where $x$ is the spend or effort put into it. The claim that the channel is concave and saturating is the claim that this function rises quickly at first, then bends, then flattens toward a maximum. One simple form that captures this, and that I will use because it is easy to reason with, is

$$r(x) = r_{\max}\left(1 - e^{-x/k}\right),$$

where $r_{\max}$ is the ceiling the channel approaches at very high spend, the most demand it can ever produce no matter how much you pour in, and $k$ is a scale that sets how quickly the channel approaches that ceiling. The particular formula matters less than its shape. At low spend the response climbs nearly in proportion to what you put in, which is the regime where the channel feels linear and the early arithmetic seems to work. As spend rises, the response bends away from that early line and begins to flatten, and past a certain point almost the entire budget is buying almost nothing, because the channel is close to its ceiling and there is little demand left in it to capture.

There is a point on this curve worth naming, the place where the bending becomes severe, which I will call the saturation knee and write $x^\star$. Below the knee the channel is still returning well and more spend is reasonable. Above the knee the channel has given most of what it has and additional spend is increasingly wasted. The team that scaled from ten thousand to a hundred thousand had found a channel that was performing beautifully because it was sitting well below its knee, and then they spent it straight through the knee and out into the flat region beyond, where the marginal dollars bought almost nothing, and the average performance collapsed because it was now dominated by all that wasted spend at the top.

Average and margin, the confusion that costs the most

The team’s mistake has a precise name, and it is the confusion between the average return and the marginal return, which are different quantities that the field routinely treats as one.

The average return is the total demand a channel produces divided by the total spend, the number people usually mean when they say a channel’s cost per customer or its return on spend. The marginal return is something else, the demand produced by the next dollar, the slope of the response curve at the current level of spend, which I will write as

$$r'(x) = \frac{dr}{dx}.$$

For a concave channel these two quantities diverge, and the divergence is the whole problem. The average return stays relatively high even as you move up the curve, because it includes all those productive early dollars in the denominator, while the marginal return falls steadily and, past the knee, falls toward nothing. A channel can have an attractive average return and a marginal return close to zero at the same time, which means it looks like a good place to spend more precisely when it has almost nothing left to give. The team looked at the average, saw a winner, and poured budget into a channel whose marginal return was already collapsing, because the number they were watching, the average, was the wrong number for the decision they were making.

This error is neither small nor occasional, and it is built into the metrics the field watches, which are almost all averages, cost per acquisition and return on ad spend and the rest, none of which tell you the one thing that should govern the decision to spend more, which is what the next dollar will do rather than what the average dollar has done. A discipline that thought clearly about channels would watch the margin, because the margin is what every spending decision actually turns on, and the average is a summary of the past that is actively misleading about the next move.

A set of numbers makes the divergence concrete, using the curve above with a ceiling of a thousand customers and a scale $k$ of twenty thousand dollars. At a spend of ten thousand, the channel produces about three hundred and ninety customers, an average cost near twenty-five dollars each, and the next dollar at that point is still buying customers at around thirty-three dollars each, so average and margin sit close together and the channel really is a good place to spend. Now move to a spend of a hundred thousand. The channel produces about nine hundred and ninety customers, only two and a half times as many for ten times the money, the average cost has risen to roughly a hundred dollars, and the marginal cost, the cost of the next customer at that level of spend, has exploded to nearly three thousand dollars. The average has quadrupled and still looks survivable. The margin has risen by a factor of about ninety and is screaming that the channel is finished. The team watched the average, which lied gently, and never computed the margin, which would have told them the truth in time to stop.

Where the money should go

Once you are thinking in terms of marginal return, the question of how to split a budget across several channels has a clean and slightly surprising answer, and it is the same answer that shows up wherever a limited resource is allocated across competing uses.

Suppose you have a fixed budget to divide among several channels, each with its own concave response curve. The allocation that produces the most total demand is the one where the marginal return is equal across all the channels you are funding. The argument is simple once you see it. If one channel were returning more on its next dollar than another, you could move a dollar from the lower-marginal channel to the higher-marginal one and gain more than you lost, so the allocation could be improved, which means it was not optimal. The only arrangement that cannot be improved this way is the one where every funded channel has the same marginal return, a common value I will call $\lambda$, set so that the spends add up to the budget,

$$r'_1(x_1) = r'_2(x_2) = \cdots = r'_n(x_n) = \lambda, \qquad \sum_i x_i = B.$$

This is the equimarginal principle, and it is one of those results that sounds abstract and then quietly reorganizes how you think about everything. It says the goal of allocation is to push every channel to the same marginal return and no further, to fund each one up to the point where its next dollar matches the next dollar everywhere else. A channel sitting below that common marginal return is being starved and deserves more. A channel pushed past it, with a marginal return below the common level, is being overfed and should be cut back, no matter how good its average still looks. And a channel whose very first dollar already returns less than the common level belongs at zero, unfunded, because no amount you could spend on it would beat spending the same money where the slope is steeper. The winner the team loved was, by this logic, dramatically overfunded, its marginal return driven far below what other channels would have returned on the same dollars, and the budget that went into its flat region would have produced far more demand spread across channels that still had slope left.

The practical shape of this is a rule the field almost never follows. Do not fund channels to the hilt by their average performance. Fund each one until its marginal return falls to the level of the others, then stop, and put the rest where the slope is still steep. Following that rule would change most budgets I have seen, usually by taking money out of a small number of beloved, saturated channels and spreading it into channels that looked worse on average and had far more left to give on the margin.

The same numbers show why concentrating a budget in one beloved channel is usually wrong. Suppose the hundred thousand had instead been split, and a second channel was available whose responsive early layer was still entirely untouched. At the point where the first channel has swallowed the whole hundred thousand, its next dollar buys a customer for nearly three thousand dollars. The second channel, sitting at zero spend, would buy its first customers for around twenty dollars each, which is the same low cost the first channel once offered before it was driven up its own curve. Moving a dollar from the first channel to the second turns a three-thousand-dollar customer into a twenty-dollar one, a gain so large it is almost embarrassing, and the gains keep coming until the two channels have been pushed to the same marginal return. The beloved channel was a saturated channel being force-fed while a better use of the money sat idle, and the only way to see that was to compare margins rather than averages.

The complication of time

There is one more feature of channels that the static curve leaves out, and it matters enough to name, which is that a channel’s response is not instantaneous. Spend today produces some demand today and more demand later, trailing off over time, because attention and memory and consideration all have duration. A burst of spend casts a shadow forward, an effect that decays gradually rather than ending when the spend ends.

The standard way to capture this is to say that the effective input of a channel carries over from one period to the next, decaying by some factor each period. If $\alpha$ is the fraction of effect that persists, then the effective accumulated input behaves like

$$A_t = x_t + \alpha A_{t-1},$$

so that today’s effective input is today’s fresh spend plus a decayed remainder of everything that came before. The larger $\alpha$ is, the longer a channel’s memory, the more a past burst keeps working after it ends. This carryover is why turning a channel off does not stop its returns immediately, and why turning it on does not produce its full effect at once, and why measuring a channel over too short a window badly misjudges it, attributing to this week’s spend what last month’s spend is still delivering, or crediting a channel with nothing in the weeks before its delayed effect arrives. The decay is rarely measured, and so the timing of channel effects is rarely understood, and decisions get made on windows too short to see what the channel is actually doing.

A concrete version of this measurement trap is common enough to be worth picturing. A team runs a burst of spend in a channel with a long memory, a high $\alpha$, and measures the result in the two weeks that follow. Much of the channel’s effect has not arrived yet, because it is still decaying out into the population over the following months, so the two-week measurement badly understates the channel and the team cuts it. The cut looks costless at first, because the decaying tail of the earlier spend keeps delivering customers for a while, and the team now credits those customers to whatever channel it moved the money into. The channel that actually worked is punished and the channel that happened to be running when the tail landed is rewarded, and the whole misjudgment follows from measuring a delayed effect in a window too short to contain it.

The curve is not given

All of this assumes you know the response curve, and you do not, at least not at first, because the curve is never handed to you and has to be discovered. This is worth dwelling on, because it is where the engineering actually lives. A channel does not announce its ceiling or its knee. You learn the shape the way engineers learn any response, by varying the input on purpose and watching the output, spending at different levels deliberately and recording what comes back, until the curve emerges from the points. This is parameter variation, the patient method of changing one thing at a time to map a relationship, and it is the only honest way to know where a channel’s knee sits before you spend straight through it.

The field mostly skips this. It finds a spend level that works, settles there, and never maps the curve around it, so it knows a single point and almost nothing about the slope or the ceiling, which is exactly the knowledge it lacks when the moment comes to scale. Mapping the curve costs something, because some of the spend at the different levels is deliberately not optimized, it is spent to learn rather than to return, and that feels wasteful to a field trained to optimize every dollar. The waste is tuition. A channel whose curve you have mapped is one you can scale with some confidence, knowing where the knee is and what the next dollar will do, and a channel known only as a single point is one you will eventually scale into the flat, because you could not see the bend coming.

The hypothesis

Let me state the claims of this essay as a hypothesis sharp enough to be tested and, if it is false, to fail.

The hypothesis has three parts. The first is structural, that channel response is concave and saturating, well described by a curve that rises, bends at a knee, and flattens toward a ceiling, so that marginal return falls as spend rises and approaches zero past the knee. The second is normative, that the allocation of a budget which produces the most demand is the one that equalizes marginal return across funded channels, the equimarginal condition. The third is empirical and is the provocative one, that observed go-to-market budgets rarely equalize marginal return, that they systematically overfund a few channels with attractive averages and low margins while starving channels with worse averages and higher margins, so that most budgets are misallocated in a measurable and correctable direction.

All three are testable. Estimate the response curve of each channel, which is itself real work but well-understood work, then compute the marginal return of each channel at its current level of spend, and check whether they are equal. The hypothesis predicts they will not be, and it predicts the direction of the error, too much in the saturated favorites and too little in the channels with slope to spare. If you ran this and found marginal returns already equalized across channels, the budget would be optimal and the hypothesis wrong, and that would be a genuine and useful surprise. I have never seen it, because the field allocates by average and the average points the wrong way, but the test is available to anyone willing to estimate the curves.

What it changes

The deepest change here is in what the field treats as the object of attention. The team that scaled its winner was reasoning about a point, the channel’s performance at the spend level where they had found it, and they assumed the point would travel with them as they scaled. The thing that actually governs a channel is a curve rather than a point, the whole response function with its slope and its knee and its ceiling, and almost every channel mistake comes from reasoning about a point on a curve as though the curve were a line through that point. A discipline would estimate the curve, watch the margin rather than the average, allocate to equalize marginal return, and account for the carryover that smears a channel’s effect across time. None of this is exotic. It is the ordinary economics of spending a limited budget on resources with diminishing returns, and it has been understood for a century in other domains, and go-to-market has mostly not imported it, which is why it keeps scaling winners into the ground.

It would also change the emotional relationship the field has with its winners. A channel that returns brilliantly at small spend produces a kind of attachment, a sense that this is the thing that works and deserves more, and the attachment is exactly what drives the overfunding. Thinking in curves cools that attachment in a useful way, because it reframes the beloved channel as a curve with a knee rather than a winner without a ceiling, and it turns the question from how much do we love this channel into where is its marginal return relative to everything else. That is a less romantic way to run a budget, and a far more productive one.

There is a thread that runs forward from here, because a curve is not a fixed thing. Channels saturate faster as you push them, competitors crowd in, audiences fatigue, and the response function you estimated last quarter is not the one you face today, so the knee moves and the ceiling shifts and yesterday’s optimal allocation drifts out of true. A system whose curves are always moving is a system you cannot set and forget, one that has to be sensed and corrected continuously as its own behavior changes, which is no longer a question of allocation at a moment but of control over time, and control is its own body of knowledge that go-to-market has barely begun to touch.

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References

• On concave response curves, diminishing returns, and saturation in marketing spend, see the literature on advertising response functions and marketing-mix modeling.
• On the equimarginal principle for allocating a constrained budget across activities with diminishing returns, see standard treatments in microeconomics and optimization (the first-order conditions for constrained maximization).
• On carryover and decay (adstock) in advertising effect over time, see the modeling literature originating with the adstock formulation.