Apply Zipf’s Law to Your Finances


5 min read

If you take a look at the most populous cities in the United States, you’ll notice an interesting trend: The most populous city, New York City, has a population about twice the size of the second largest city, Los Angeles, three times the size of the third largest city, Chicago, four times the size of the fourth largest city, Houston, and five times the size of the fifth largest city, Phoenix.

Zipf's law for city populations

It turns out that this trend in city rank vs. population holds for nearly every city that has a population of at least 100,000 in the U.S., as was once illustrated by economist Xavier Gabaix in the graph below:

This phenomenon is often referred to as Zipf’s Law, named after linguist George Zipf, who, in the 1940s, discovered a similar pattern for word frequency in several different languages.

For example, in English, “the” is the most frequently used word at 7%, which is used twice as often as the next most common word “of” at 3.5%, and three times as often as the next most common word “and” at 2.3%.

Zipf's law for word frequency

Interestingly, Zipf’s Law also applies to urban population sizes in nearly every developed country across the world and it works well when used for metropolitan areas, which are areas defined by the natural distribution and connectivity of populations rather than arbitrary political boundaries (e.g. counting Oakland and San Francisco as one metro area as opposed to two different cities).

More generally speaking, Zipf’s law is just an example of a power law, which is a type of distribution that looks like the following:

Power laws are fascinating because they show up all over the place, from the size of craters on the moon to the frequency of family names to the size of power outages to volcanic eruptions, and more.

To me, some of the most interesting (and useful) power laws show up in personal finance.

The Power Laws of Personal Finance

When I first read about Zipf’s law, I immediately thought of several instances where it plays out in personal finance. Here are a few.

Average Household Expenditure

Once classic example of a power law distribution in finance relates to the data behind average household expenditure, which describes how the typical household in the U.S. spends their income:

When you graph these numbers, you can see a clear power law distribution:

Housing accounts for one-third of all spending, which is about twice as much as transportation costs and roughly three times as much as food costs. A classic example of Zipf’s Law.

Net Worth Growth

Here’s another no-brainer example: compound interest.

Consider the scenario where an individual invests $10k per year and earns 7% annual returns:

It will take them nearly 8 years to accumulate their first $100k, but only about 5 years to get the next $100k, then another 3.78 years to get the next $100k, and so on. 

This means the first $100k represents about 26% of the entire journey to $1 million in terms of time:

The next $100k only represents 17% of the total time:

Each $100k comes faster and faster, thanks to the nature of compound interest, and when we graph the time it takes to accumulate each $100k we can see a clear power law distribution:

Zipf's law in investing

Entrepreneur Income Growth

Anyone who has started an online business and stuck with it for several years has likely experienced this type of income growth:

The early years are a pure grind. But as you gain more experience, more knowledge, and acquire more assets, your income begins to grow at a much quicker pace than you could anticipate.

I’ve experienced this trajectory myself. I made about $5,000 in online income during my first year, $10,000 in my second year, $25,000 in my third year, and I anticipate that I’ll make about $50,000 during my fourth year.

Money and Happiness

Countless studies have shown that more income is associated with more happiness, but only to a certain point, as illustrated by this simple chart from 8000 Hours:

graph of happiness and income

It’s pretty easy to understand why. Moving from an income of $0 to $30,000 will pull you out of poverty. Moving from $30,000 to $60,000 might make you less stressed out about money in general and allow you to live more comfortably. Moving from $60,000 to $100,000 allows you to buy more luxury goods, but it has a relatively small impact on overall happiness. And anything beyond that is negligible. 

If we graphed how much additional happiness you gained by each income increase, it might look something like this:

Yet another power law distribution.

Use Zipf’s Law to Your Advantage

Physics professor Albert Allen Bartlett once said:

“The greatest shortcoming of the human race is our inability to understand the exponential function.”

Understanding power law distributions and the nature of exponential functions gives you a massive advantage if you know how to apply it to your finances. Here are a few ways that you can do so.

Household Spending:

Earlier we saw that housing represents up a huge percentage of average household spending:

This means that you should spend a significant amount of time deciding where to live and ensure that you don’t overspend on a house that’s unnecessarily large for your family’s needs. Saving tens of thousands of dollars with one housing decision is worth more than tens of thousands of tiny decisions made about saving on coffee, Netflix, and other relatively tiny expenses.

The exception to this rule are decisions that you can make once that will benefit you over and over again financially. Some simple examples include:

  • Using a free financial app like Trim to negotiate lower cable, internet, and phone bills with any provider on your behalf.
  • Using free software like Personal Capital to track your cash flow and net worth so you don’t have to waste time manually entering numbers in spreadsheets.
  • Using a free chrome extension like Wikibuy that automatically tells you if an item that you’re about to buy on Amazon can be bought somewhere else (like eBay or some other site) for a lower price.
  • Setting up automatic retirement account contributions so you don’t have to remember to manually invest money each month.

Net Worth Growth:

Earlier we saw that net worth growth is annoying slow in early years and shockingly fast in later years:

 

This means you should avoid getting discouraged early on with slow growth because getting that first $100k is truly a big deal. It might not feel like a lot, but it represents a huge chunk of the journey to $1 million. Recognize that most of that $100k will come purely from savings, not investment returns. Focus more on generating an above-average income and less on generating above-average investment returns. 

Entrepreneur Income Growth:

For anyone who wants to start an online business, it’s also important to remember that the first few months (and even years) can be a real drag. Most online businesses – whether it’s informational websites, affiliate websites, or ecommerce stores – fail because the owner simply quits after less than one year.

If you choose to pursue online income, you need to have short-term impatience and long-term patience. Take action every day, but also recognize that your actions won’t be rewarded until months or years later.

Money and Happiness:

Lastly, we saw that more income equates to more happiness, but only to a certain point. This means that yes, you should prioritize increasing your income so that you can live a more stress-free life and control more of your time, but keep in mind that entering the top 1% of income-earners will require a massive time and energy commitment without rewarding you with a massive boost in happiness.

Always remember that money can’t directly buy happiness, but it can buy you the free time needed to find happiness.

Zach

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