The Denominator Disposition
Making do often doesn't make sense
My first project at Scale damn near killed me.
Our goal seemed easy enough: correctly sort grocery-ish items into the proper category for a large grocery delivery company. The scale of the task was the first challenge, with about three million items to sort through and five thousand categories to place them in. The items themselves were often challenging as well: duplicate, sparsely described, poorly named, missing an image. Finally, many items defied easy categorization, forcing us to split hairs and flesh out absurdly detailed definitions. Is fruit salad in “Fruit” or “Prepared Food”? Would a stuffed plush toy of Gritty be in “Stuffed Animals” or “Plush Toys”?
This was back in 2021, before the age of ChatGPT and LLMs that can use the internet. Ours was a human solution: piece together the spotty information on each item with the help of the internet and put it in the right category.
I worked 100 hours a week for 2.5 months, neglecting my pregnant wife and spending every waking moment in front of the computer in order to deliver the project and make my reputation at Scale. The long hours mostly went into the cycle of optimization: observe, tweak, observe, tweak, over and over again. I felt I had all the pieces of the answer in front of me, and that I only had to assemble them in just the right way to complete the project, no extra resources required.
My feeling seemed true for most of the project, but as the deadline approached, my methods were falling short. My optimizations to the existing system were hitting diminishing returns. I felt I now had an impossible task, that extra resources were required, but it was too late to ask for them. The project ended as a string of hit milestones (i.e. number of items categorized), then one massive miss on the final delivery.
(Don’t worry, the customer gave us a redo on the final delivery and we were able to turn it around. They’re still a customer to this day.)
A Box of My Own Making
On the project, my problem was this: I did not ask for more resources. With more humans working on labeling, we could have hit our quality bar on the final delivery. Yes, it would have increased costs, but over the long term the customer ended up being worth so much more than any extra spend I ever would have needed. I was too focused on making do with what I had.
After we failed our final delivery, my boss basically did the thing I should have done on a different project: he threw a bunch of costly-but-quality taskers at it after he got good foundations set up. It was then I learned to always consider asking for more on my work projects, even if ultimately I chose not to make the ask.
In the intervening years, I have wondered what it was about me that stopped me from asking for more. I started seeing parallels in my personal life too. Pay someone to mow the lawn? I have a lawnmower, I can make do—forget that I don’t have time with raising two kids to do it nearly often enough and that it’s a poor use of my time even when I do get around to it. Get a new charging cable long enough to use while I’m on the couch? No, I can charge it in the other room, even though I’ll probably get up two or three times to check it after I hear a buzz.
I’m not lazy. I’m not especially frugal. I do these things because by nature I’m an optimizer, a muddler, a lemonade maker from life’s lemons—even when life’s oranges are on sale just around the corner. My disposition is to work with what I have.
Top and Bottom
I have come to call this denominator thinking. It’s a problem-solving frame that focuses on reducing the denominator to make your metric go up.
Contrast with numerator thinking, where the focus is on growing the numerator to achieve the same effect.
There are many expressions of the numerator vs denominator dichotomy:
Growth vs efficiency
Abundance vs scarcity
Non-zero sum vs zero sum
I think a lot of smart people are by nature denominator thinkers because optimization is a problem crying out to be solved. With enough thought and work, a system can become more efficient/profitable/fair etc.2 They feel simply adding more resources is wasteful, greedy, or lazy.
Critically, I think more people are denominator thinkers than they need to be because denominator thinking is primarily a disposition. A disposition I have! Even with clear proof in my past that I can grow the numerator, I still naturally take a denominator lens.
(I also believe numerator thinking is a disposition, based on observation of others, but I’m going to stick to what I know.)
This denominator disposition is limiting as we’ve seen. Now let’s see how to drill numerator thinking into that denominator disposition and lift that limit.
Growth Blindset
“Growth mindset” is an overused and annoying term, but it is probably the popular concept closest to numerator thinking. Where growth mindset is a way of life, though, numerator thinking is a tool.
Growth mindset asks people to change how they think about everything. It is explicitly a way of understanding and interpreting everything that happens to the subject. Implicitly, if you can’t achieve a growth mindset, you are living a worse life. If you’re not in the growth mindset, you’re wrong.
By contrast, the numerator/denominator dichotomy accepts natural dispositions and focuses on categories of solutions to concrete problems. The only merit is in the products of numerator thinking; the subject is incidental to the outcome. While numerator thinking is the general recommendation, denominator thinking has its place depending on circumstance.
Go back to my personal example at the top. Solving the problem of my project wasn’t about me believing in myself or something - it was about seeing a solution based on growing the top. I had a blindspot for numerator thinking, based on my disposition rather than a rational consideration of all options.
I think a lot of problems have numerator solutions, some more achievable than others depending on circumstance but all worth considering. Examples:
Personal finance: budgeting/cutting expenses vs making more money
Writing: tailoring to your existing audience vs publishing what you want and attracting a variety of readers
Dieting: cutting calories vs exercising more
Digital ads: crafting some choice copy vs trying tons of different variations
Machine learning: designing techniques based on limited human knowledge vs throwing increasingly cheap compute at the problem
A friend of mine is a great numerator thinker. I particularly notice it for day-to-day problems or little inconveniences. When he has a problem in his life that can be solved with a cheap purchase, he makes it. For a long time I thought it was impulsivity, but now I understand the wisdom of spending a little bit once to make an annoyance disappear forever, even if that annoyance is small; the ROI of the freed mental cycles is almost always there.
I have learned to copy this tendency, but my natural inclination is to hack a solution with what’s available or just deal with it. I have to remind myself that the money I’m paying for this or that doodad is buying me free mental time, which I am comfortable paying dearly for. Nowadays I just pull the trigger and don’t worry about the occasional times the purchase was worthless or the problem ended up being transient.
Again, numerator thinking is not always the best option. If you are in a situation that truly cannot be grown out of, then go ahead economizing. But I think it’s an underused option. It is a tool in the toolbox. My point here is to shine a light on this blindspot and ask you to more regularly take a look.
But when to use this new tool? And why might so many of us have so little innate inclination for it?
All Things in Time
I contend numerator thinking is the right default when you can take a long-term view.
Growth often takes time and stability. Yes, obviously I’m tempted to bring development economics in here, but to make it more plain let’s consider food. Back in the day when we didn’t know where our next meal was coming from, humans valued calories above all else. The goal was maximum caloric density, from a constrained supply of food. Optimizing is classic denominator thinking.
Now that food is abundant, we don’t have to optimize our calories. We can balance calories with nutrients like vitamins and minerals for better health, plus taste and variety for more enjoyment. By growing the numerator on the supply of food over decades in the modern food system - arguably over centuries or millennia if you include the development of staple crops - we unlocked greater wellbeing.
Similarly, recall any post-apocalyptic movie you’ve ever watched and you’ll envision a world returned to the denominator. The constraints of a broken world are immovable, reinforced with the weight of civilizational rubble, so the characters have to do the best with what they’ve got. No wonder the food usually sucks.
If you exist in a time and place where long-term planning is impossible, you can’t make any sort of implicit (or explicit) ROI calculation. You have to play it safe. Most of human existence has been like this. But recently it hasn’t been, for many people at least. Probably that includes you.
So next time you need to take a crack at a problem, see if you can frame it in numerator and denominator language. Feel which set draws you, but think which set will solve your problem better.
I’m tempted to call these right- and left-coded, respectively, but it’s not quite true anymore. Center-left people and organizations have lurched towards an agenda of abundance (e.g. Kamala’s embrace of YIMBY at the DNC), but haven’t gone full Cowen to assert growth primacy. On the reverse, Trump’s love of tariffs is all about the denominator - there’s only so much industry to go around and we have to protect what’s ours, that kind of thing.
I think school trains this view as well. Homework problems and tests are puzzles designed to be put together with existing pieces in just the right way, with clear boundaries about what is and isn’t allowed or what you do and do not need to know. Problem solving in the real world is not like this.