# On Piketty's Capital: r, g, and s

If you've read any of the reviews of Piketty's book, including the reviews of it from the many who have not read it, you are probably familiar now with "r > g". Many of the reviews do an iffy job of explaining what that means though, and so you may not however fully understand what is going on with it. Here I attempt my own explanation that incorporates savings.

To start, r refers to the rate of return on capital. If you have $100 of capital and it brings back $5 of capital income each year, r is 5%. Now g refers to the growth in the national income. If the national income is $100 one year and then $101 the next, g is 1%. Thus "r > g" means that the rate of return on capital is greater than the national income growth rate. Piketty does not say that this makes it totally impossible to reach any other outcome than extreme wealth concentration. He says that it is a *powerful force for divergence*. That is, it pushes hard in the direction of severe wealth inequality.

The mechanics of how this work don't just involve r and g though. They also involve, as few reviewers except Robert Solow have pointed out, savings, or s. This should be obvious. After all, if wealthy people consumed all their r-derived income each year and laborers saved at least some of their wages, eventually laborers would pile up more wealth than the super-capitalists have.

The longevity of the the super-capitalists' advantage is not determined solely by r. It is determined by r multipled by s, i.e. the rate of return on capital multiplied by the savings rate. What you find, in fact, is that someone who is living solely off capital income sees their 1) wealth, 2) capital income, and 3) annual savings grow by r*s each year. If they start out with $100 wealth with a rate of return of 10% and a savings rate of 50%, then their capital income in the first year will be $10 and their savings will be $5. The next year, the wealth will be $105, the capital income will be $10.5, and the savings will be $5.25. All of those numbers are up 5% (which is the same as 10% (r) multiplied by 50% (s)). So again, r*s is driving the show for the income and wealth pile of the super-capitalist who lives solely off of capital income.

If a super-capitalist has a savings rate that is the same as the savings rate of a laborer, eventually the laborer will overtake them in wealth. This is because, after the first year, the laborer will be adding savings from both capital and labor income to their wealth pile each year, causing it to grow at a faster rate than the super-capitalist's pile.

But this can take some time. If we assume r = 5, s = 10, and g = 1 (g here being per-capital income growth), and assume further that the super-capitalist starts out with a wealth pile big enough to give them a capital income equal to the laborer's labor income (who starts out with $0 wealth), then it would take 140 years for the laborer to overtake the super-capitalist. Good luck with that.

That's a very conservative estimate though because, in fact, it is the case that the wealthier a person is, the easier it is for them to have a higher savings rate. If we assume in the prior example that the laborer's savings rate stays at 10% permanently (including on capital income), but the super-capitalist's savings rate is 30%, the super-capitalist will never be overtaken in wealth and will actually pull further ahead each year.

At the extreme ends of wealth, the amount of capital income that can be saved each year should approach 100%. In those cases, r will actually run the show because r*s will approximately equal r. A super-capitalist of that sort will actually see their income and wealth increase by approximately r each year. People without such wealth could not possibly keep up with that savings rate and, since r > g, they cannot expect to overtake them through growth in labor earnings either.

So ultimately, the extent to which r > g pushes extreme wealth inequality depends on savings, both overall and the savings of particular types of people. But Piketty does not say otherwise on this point and, in fact, differential savings rates between wealthy and non-wealthy will tend to push wealth inequality even harder. What's important to note though is that r > g is not meant to suggest, as a purely technical matter, that you will definitely see wealth inequality and concentration. Piketty is careful to say it is simply a *powerful force of divergence*. And on that somewhat more narrow point, it is hard to imagine how he could be wrong.

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