At PMG, data and technology are at the heart of everything we do. So, when our clients ask us “How much we should be spending?”, we typically start with a data-driven approach. After some research, we decided to pursue the idea of calculating the point of diminishing returns to answer such a question.

## When will the next dollar we invest make us less than the dollar we spent?

Most of us look at ROI, return on investment, and think if we are operating better than 1:1 we are making money as opposed to losing money. However, it’s possible to be operating at a good overall ROI while wasting dollars unnecessarily.

Diminishing returns is that point where our last dollar brings us back exactly what we invested, one dollar. The next dollar we spend will most likely generate less than a dollar in return, causing us to lose money. The goal of this analysis is to find that point.

## Theory

The strategy was to plot revenue and spend combinations, fit a curve, and finally find the point of diminishing returns. We had to determine what type of curve we would use as well as what span of data would make up our points. Once that was determined, we had what we needed to find the point of diminishing returns.

For one particular client, we looked at monthly and weekly aggregations of the data, but our best results came from using daily data to make up our points. It allowed us to shorten the time frame to ensure a homogeneous dataset while giving us a sufficient number of data points to fit a model. We normalized by day of week to minimize that effect.

Spend against revenue is typically modeled with a log curve. As we start spending money, we see a quick rise in revenue, followed by a gradual tapering off until we reach the point where the curve becomes flat, indicating that spending more will not increase revenue at all. Mathematically, there are some nice interpretations and properties that will come in handy when performing our final calculations.

Recall, the point of diminishing returns is the point where it’s no longer advantageous to spend more money. Numerically, this is where spending one more dollar will get you exactly one more dollar in revenue. Given that our curve is looking at revenue as a function of spend, this point is where the slope of the curve is equal to 1. Therefore, we can find the point where the derivative of our model is equal to 1 to find the point of diminishing returns. It’s important to reiterate that the point of diminishing returns is not the point where your overall ROI is 1:1. It is where that last dollar is 1:1. For example, your point of diminishing returns could come at a spend level that leads to an overall ROI of 8:1.

## Math Deep Dive

Let’s take a closer look at how we calculate the point of diminishing returns. Once we have our data, we fit a linear model on the log of cost.

### Revenue = f(Spend) = m * log(Spend) + b

Recall we said the point of diminishing returns is where the slope of the curve is equal to 1. We simply take the derivative of our new model. One of the advantages to fitting our model using the natural log of spend is the straightforward derivative.

Now set the derivative equal to 1 and solve for spend:

The point of diminishing returns turns out to be the coefficient for the log of spend from our linear model.

We can generalize this to work for any last dollar ROI target:

We simply divide the coefficient for the log of spend from the linear model by your target last dollar ROI.

## Learnings and Next Steps

We’ve done a couple of diminishing returns analyses for clients to this point. The biggest learning so far has been the importance of your revenue attribution model. One case saw the point of diminishing returns change 4x based on the percentage of post view revenue that was credited. Revenue must be measured accurately for this methodology to work.

Immediate next steps are implementing the results and tracking overall performance to verify the theory. We also need to work on a solution for revenue spikes caused by holidays and promotions.

The potential for this is huge. The end game would be full-on media mix modeling. This would require a cross-channel attribution model to ensure revenue was being measured consistently across channels and tactics. If that is in place and we can find the right solution, this could be used to automatically assign budgets across the board.