Income inequality and the Rule of 72 – part 1

  

Income inequality is an issue fraught with political stake-in-the-ground positioning, but there is some very basic math that, if understood, deflates a lot of the political grandstanding from either side. If we get the math out of the way then perhaps the policy implications and choices become clearer.

The first principle applied here is that nature itself does not usually count linearly in nice base-ten addition. Most natural growth is instead exponential and logarithmic in form. In other words, populations and economies grow in “compounding multiples” rather than in fixed increments.

In a related manner, don’t think of “the economy” in political terms, rather view it as the sum total of a growing natural population of humans creating wealth among itself in an always-innovating “beehive” of mutual support. That growth is naturally exponential, not linear. You probably did not enjoy learning about exponents and logarithms in school, but a simple “math hack” called the Rule of 72 will make this concept easier to apply.

The second principle I like to summarize using an old camp song that goes, “When two, and two, and fifty make a million…” In an economy the size of the United States, tiny incremental changes don’t just add up, as per the principle above. Rather they multiply up to very significant end results over time. The growth rates “compound” exponentially.

The Rule of 72

An old accounting trick called the “Rule of 72” is a quick way of estimating how long it takes any amount of money to double in this “compounding” economy. Let’s say that you are one of the millions of Americans stuck in a non-managerial or first-tier managerial job where pay increases have been limited to 3% per year or less for a lot of years now. The Rule of 72 says to divide that “3” into 72, and result is that it will take 24 years for your salary to double at that “compound growth rate” of 3%.

In short, you are stuck in a slow-growth segment of the natural economy. If you were making $30,000 twenty-four years ago at this writing (or back in 1994) and your salary grew fairly constantly at a 3% rate, you would have only passed $60,000 in salary this past year. [1] If you only earned an average 2% raise from that same starting point, it would take a whopping 36 years (72 divided by 2, or the year 2030) to see $60,000.

But let’s say that your boss gets a 7% per year average raise over that same period of time in salary and bonuses. Per the Rule of 72, divide 72 by 7 and you get about 10.3. Thus, it only takes about ten years for his salary to double (yes, I should say “his or her,” but part of the societal problem is that it is still mostly “his”), and then it will double again in the next ten years. In that 24 years in which your salary has only doubled, your boss’s salary may have gone up five times! [2]

And if the average raise is 10% per year, it only takes seven years for that salary to double, and another seven to double again, with yet a third doubling in another seven years. (totaling eight times – 2 times 2 times 2 – over 21 years). Now, clearly not every corporate executive has received raises like this every year, but neither have non-managerial workers seen consistent raises even at the rate of 3% over the last two decades.

The point of my second principle is that pay raises happen one individual pay-raise at a time, but then are multiplied by many millions of wages earners in the U.S. If there is any differential between different levels of workers and management, the effect over time will be an ever-widening income differential. No grand conspiracy is required here, just lots of businesses making lots of individual pay decisions, all thinking that their pay-raise schedules are justified. And yet this consistent differential causes significant and fast-increasing income inequality over time.

The chart below shows how each $30,000 in salary grows at different rates of average pay increase over twenty years. Which colored line represents your slot in the economic beehive?

20-year growth

Growth of $30K in salary over 20 years

You can clearly see the effect that time has here, which is why this is called “the time value of money.” Now, what would happen if we do the same calculations over, say, thirty years instead of twenty, but we reduce the differences between the wage increases? We’ll set the bottom group of workers at a very meager 1.5% increase per year, the second group remaining as in our earlier example at 3%, but the last two groups reduced to just 6% and 8% respectively, much smaller increments than before. Note in the chart below below that the extra ten years of compounding time more than makes up for the tighter range of pay increases. Small differences mean a lot going out 30 years. That top salary is almost 10 times where it started out!

30-year growth

Growth of $30K in salary over 30 years

So, how does this second chart match the historical record in recent years? (Spoiler: All too well.) Stay tuned for Part Two to look at recent history, or wait for Part Three for my take on what needs to change.

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And Part Two is now posted.


Notes:

  1. The actual computation is simply to take the annual interest rate, add 1 (making 1.03 in this case), and then use the number of years as the exponent (or the 24th power in this case), giving you about 2.03. Multiply that by the $30,000 salary and you get $60,900. In shorter terms, the factor is (1 + i) n, where i is the interest rate per period and n is the number of interest-compounding periods.
  2. 1.07 to the 24th power equals 5.07.