{"id":38827,"date":"2020-09-16T08:29:56","date_gmt":"2020-09-16T13:29:56","guid":{"rendered":"https:\/\/fourpillarfreedom.com\/?p=38827"},"modified":"2020-09-16T08:29:56","modified_gmt":"2020-09-16T13:29:56","slug":"how-to-apply-zipfs-law-to-your-finances","status":"publish","type":"post","link":"https:\/\/fourpillarfreedom.com\/how-to-apply-zipfs-law-to-your-finances\/","title":{"rendered":"How to Apply Zipf’s Law to Your Finances"},"content":{"rendered":"

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\n5 min read<\/span><\/p>\n

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.<\/span><\/p>\n

\"Zipf's<\/p>\n

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<\/a> in the graph below:<\/span><\/p>\n

<\/p>\n

This phenomenon is often referred to as\u00a0Zipf’s Law<\/strong>, named after linguist George Zipf, who, in the 1940s, discovered a similar pattern for word frequency in several different languages.<\/span><\/p>\n

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%.<\/span><\/p>\n

\"Zipf's<\/p>\n

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).<\/span><\/p>\n

More generally speaking, Zipf’s law is just an example of a power law<\/a>, which is a type of distribution that looks like the following:<\/span><\/p>\n

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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.<\/span><\/p>\n

To me, some of the most interesting (and useful) power laws show up in personal finance.<\/span><\/p>\n

The Power Laws of Personal Finance<\/strong><\/span><\/h2>\n

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.<\/span><\/p>\n

Average Household Expenditure<\/strong><\/span><\/h3>\n

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

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When you graph these numbers, you can see a clear power law distribution:<\/span><\/p>\n

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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.<\/span><\/p>\n

Net Worth Growth<\/strong><\/span><\/h3>\n

Here’s another no-brainer example: compound interest<\/strong>.<\/span><\/p>\n

Consider the scenario where an individual invests $10k per year and earns 7% annual returns:<\/span><\/p>\n

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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.\u00a0<\/span><\/p>\n

This means the first $100k represents about 26% of the entire journey to $1 million in terms of\u00a0time<\/em>:<\/span><\/p>\n

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The next $100k only represents 17% of the total time:<\/span><\/p>\n

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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:<\/span><\/p>\n

\"Zipf's<\/p>\n

Entrepreneur Income Growth<\/strong><\/span><\/h3>\n

Anyone who has started an online business and stuck with it for several years has likely experienced this type of income growth:<\/span><\/p>\n

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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.<\/span><\/p>\n

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.<\/span><\/p>\n

Money and Happiness<\/strong><\/span><\/h3>\n

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<\/a>:<\/span><\/p>\n

\"graph<\/p>\n

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.\u00a0<\/span><\/p>\n

If we graphed how much additional happiness you gained by each income increase, it might look something like this:<\/span><\/p>\n

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Yet another power law distribution.<\/span><\/p>\n

Use Zipf’s Law to Your Advantage<\/strong><\/span><\/h2>\n

Physics professor Albert Allen Bartlett once said:<\/span><\/p>\n

“The greatest shortcoming of the human race is our inability to understand the exponential function.”<\/span><\/p><\/blockquote>\n

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.<\/span><\/p>\n

Household Spending:<\/strong><\/span><\/p>\n

Earlier we saw that housing represents up a huge\u00a0<\/em>percentage of average household spending:<\/span><\/p>\n

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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.<\/span><\/p>\n

The exception to this rule are decisions that you can make once\u00a0<\/em>that will benefit you over and over again financially. Some simple examples include:<\/span><\/p>\n