The Anatomy of Adstock & Saturation: Modeling Diminishing Returns and Memory Effects

🎯 Executive Takeaway

Every marketing channel operates under two universal constraints: adstock decay (the lingering psychological memory of past impressions) and saturation (diminishing marginal returns as you exhaust high-intent audiences). Modeling both correctly is the difference between blindly over-spending on exhausted channels and capturing massive upside through optimal cross-channel capital reallocation.

The Two Laws of Marketing Physics

Imagine running a high-impact billboard or a Super Bowl commercial. If you measure revenue strictly within the 60-second broadcast window, the campaign would look like an abysmal disaster.

Humans do not see an advertisement and immediately drop their fork to place an order. They remember the brand tomorrow while brewing coffee, discuss it with a coworker on Thursday, and finally complete checkout on Sunday afternoon. Conversely, if an e-commerce brand spends $100,000 per month on Google Search, jumping to $1,000,000 per month will not generate 10x the revenue; you will quickly run out of in-market searchers and begin paying exorbitant bids for barely-relevant queries.

In Marketing Mix Modeling, these two phenomena are modeled through Adstock Transformation and Saturation Curves.

Figure 4.1: Mathematical representation of geometric adstock carryover (left) and Hill saturation S-curve dynamics (right).

Part 1: Adstock Transformations (Memory & Lag)

First coined by British advertising researcher Simon Broadbent in 1979, adstock mathematically mimics how human memory retains an advertising message over time.

Geometric Adstock Decay

The simplest and most common formulation is recursive geometric decay:

Geometric Adstock Formula Adstock_t = Spend_t + \lambda \cdot Adstock_{t-1} \text{Half-Life} = \frac{\ln(0.5)}{\ln(\lambda)}

Here, $\lambda \in [0, 1)$ represents the retention rate.

If $\lambda = 0.2$, the ad has a rapid burnout: 80% of its effect evaporates after the first day. This is typical for direct-response flash sales or SMS promo blasts.
If $\lambda = 0.8$, the half-life is over 3 weeks. A single impression echoes deep into the quarter. This is typical for high-consideration B2B software, luxury goods, and automotive branding.

Weibull Adstock (Delayed Peak)

Geometric decay assumes that the maximum impact happens immediately on day zero. But what about channels like podcasts, YouTube sponsorships, or catalog mailers? A podcast listener downloads an episode on Monday, listens during their gym workout on Wednesday, and visits the URL on Saturday.

Social Vriddhi MMM implements Weibull PDF and CDF adstock distributions. By adding a shape parameter, our engine can model a delayed peak where the peak impact occurs 3 to 7 days after the media dollar is spent, capturing true real-world consumption patterns.

Part 2: Saturation Curves (Diminishing Marginal Returns)

Every media buyer knows the painful sensation of fatigue: you double the budget, but CPAs double right along with it.

Why Logarithmic Functions Are Inadequate

Early econometric models often used simple $\log(x)$ or $\sqrt{x}$ transformations. But log transformations suffer from severe flaws:

They grow indefinitely towards infinity as spend increases (violating physical reality).
They assume that the very first dollar spent has infinite marginal return, failing to capture the minimum threshold required to cut through consumer clutter.

The Superiority of the Hill Function

Borrowed from biochemistry (where Archibald Hill formulated it to model oxygen binding to hemoglobin), the Hill saturation function is the gold standard of modern MMM:

Hill Saturation Equation Hill(x) = \frac{x^S}{K^S + x^S}

Where:

$x$: The adstocked media spend.
$K$ (Half-Saturation Point): The spend level at which the channel achieves 50% of its maximum possible outcome. It defines where diminishing returns truly start biting.
$S$ (Shape Parameter): Controls the slope and S-curve nature. When $S > 1$, the function exhibits an "S-curve" behavior, reflecting that you need a minimum volume of impressions before consumer awareness kicks in.

The Crucial Difference: Average ROAS vs. Marginal ROAS

The single most expensive mistake in media allocation is confusing Average ROAS (aROAS) with Marginal ROAS (mROAS).

If you spent $50,000 on Google Search last month and generated $200,000 in revenue, your Average ROAS is 4.0x. Your agency will enthusiastically recommend spending another $20,000 next month.

However, because Google Search is operating at the saturated top of its Hill curve, the Marginal ROAS on that additional $20,000 might only be 0.85x! In other words, you will spend $20,000 to generate only $17,000 in new sales.

💡 The Golden Rule of Budget Optimization

Optimal capital allocation across channels is achieved not when Average ROAS is equal, but when the Marginal ROAS of the next dollar is equal across every single channel in your media mix.

This is precisely what Social Vriddhi MMM's Scenario Planner and Optimizer solves continuously. By calculating the real-time derivative (slope) of each channel's Hill curve, our system dynamically directs your next dollar into whichever channel has the highest available marginal efficiency.