How to Allocate Budget Across Media Channels

There is no single formula for the ideal budget allocation across media channels. There is something better: a mathematical method that replaces guesswork and maximizes return on investment.

In practice, many managers go by trial and error: “move R$ 1,000 from Google to Meta and see what happens”. That tends to reinforce biases and lead to inefficient decisions, especially when each channel's performance is not well measured or testing is not systematic. The choice between Facebook Ads and Google Ads, for example, is rarely “one or the other”: it is almost always “how much in each”. In this article, we show the system we use at Marktech to answer that question with rigor.

• Spend and return: the curve is not a straight line
• How to estimate the curve for each channel
• The optimization problem
• The solution: equal marginal returns
• Cautions before applying it

Spend and return: the curve is not a straight line

It is reasonable to expect more spend to bring more results. But the rate of return tends to fall as we invest more, whether because the auction gets more expensive or because the audience saturates. In other words, the marginal return of each real is not constant, but decreasing. The behavior looks more like logarithmic or inverted exponential functions than straight lines.

The intuition is simple. The first reais in the budget reach the cheapest audiences, the ones most likely to convert. The last ones fight for expensive impressions against every competitor. The last real tends to yield less than the first.

These curves have two characteristics:

• Monotonically increasing: more spend, more results;
• Concave: the rate of return decreases as spend grows.

How to estimate the curve for each channel

With historical data and reasonable tracking, we can extract (spend, leads) pairs per channel: each week, month or campaign becomes a point on the chart. With those points, we fit representative functions using techniques such as the least squares method. The result is an estimated return curve for each channel, as in the example below.

The quality of the fit depends on the quality of the measurement. When part of the conversions happens far from the click, as in top-of-funnel channels, aggregate attribution models help complete the picture: we cover that in the article on Marketing Mix Modeling and Meridian. For the method in this article, the essential thing is to have, per channel, a reliable series of spend and results.

The optimization problem

With the curves modeled, the problem becomes: how to distribute a total budget I across channels x₁, x₂, ..., x_n, maximizing total leads?

We assume functions of the form:

y_i(x_i) = a_i · (1 − e^−b_i·x_i)

Here, a_i is the ceiling of channel i (the maximum number of leads it can deliver, no matter how much you invest) and b_i controls how fast the channel saturates. Our objective is:

Maximize: Σ y_i(x_i)

Subject to: Σ x_i = I, with x_i ≥ 0

The solution: equal marginal returns

Since the curves are smooth and concave, this is a convex optimization problem, with an optimal solution. That is a strong guarantee: there are no local optima to get in the way, and the point found is the best possible one.

At the ideal allocation point, the marginal return per real invested must be the same across all channels. That is, the derivative of each return function y_i(x_i) must be equal:

dy_i/dx_i = a_i·b_i·e^−b_i·x_i

At the optimum, we have:

a₁·b₁·e^{−b₁·x₁} = a₂·b₂·e^{−b₂·x₂} = ... = a_n·b_n·e^−b_n·x_n

This condition guarantees there is no “gain” left in moving resources from one channel to another. If some channel yielded more at the margin, it would suffice to take one real from another and put it there, and the previous allocation would not be optimal. There is still the constraint that the sum of the investments equals the total available:

Σ x_i = I

With that, we solve the problem by finding the x_i that equalize the derivatives and respect the budget constraint. In practice, we use simple numerical methods to compute this solution efficiently. Below, an example with three channels.

Cautions before applying it

Before you rush off to optimize, a few cautions:

• The inverted exponential is an approximation, not a law of nature;
• The model depends on historical data and good tracking;
• There are practical limits: minimum budgets per channel, decision granularity, among others.

Even so, the approach provides a structured, defensible and adaptable foundation. Ideal for anyone who wants to move past “let's see how it goes” and make decisions with more rigor.

If today your budget split is decided by eye, start small: build the historical series of spend and results per channel, fit the first curves and compare the suggested allocation with the current one. The difference between the two is the price of improvisation.