Fallacious arguments springing from careless reading of Thomas Piketty’s *Capital in the Twenty-First Century* continue to abound. This note is aimed at clearing up one especially tricky and seductive type of fallacy.

If you have been reading Piketty carefully, you know that two fundamental concepts lie at the logical foundation of his study of the dynamics of inequality in capitalist systems: wealth and income. You know that he defines the term “capital” in such a way that it can be used interchangeably with the term “wealth.” You also know that Piketty divides all income into two types: income from capital and income from labor.

There is a simple rule which describes the growth of wealth in a society from any year i to the next year i+1:

- W
_{i+1 }= W_{i} + S_{i}

where S_{i} is that society’s total savings in year i. The savings in year i can always be expressed as some percentage s_{i }of the total national income for that year. That latter number is the savings *rate*. So we can rewrite equation 1 as:

- W
_{i+1} = W_{i} + s_{i}Y_{i}

In fact, we could treat equation 2 as an implicit definition of the savings rate:

- s
_{i} = (W_{i+1 }– W_{i})/Y_{i}

The rate at which a society saves in any given year is just the change in its wealth from that year to the next year, expressed as a proportion of national income for the first year. We also know there is always a rate at which national *income* changes from year i to year i+1. Like any rate of annual change, we can define it in this familiar way:

- g
_{i} = Y_{i+1}/Y_{i} – 1

For example, if national income grows from $1 trillion to $1.02 trillion from one year to the next, then income has grown at a rate of 0.02 or 2%. The quantity g_{i} is usually called the national income *growth* rate. But, of course, it is possible for g_{i} to be negative, in which case the income in year i+1 is smaller than the income in year i, and the economy is not growing, but shrinking.

Another important quantity that can be defined in terms of the two fundamental concepts of wealth and income is the capital-to-income ratio β:

- β
_{i} = W_{i}/Y_{i}

As we said, national income is the sum of income from capital or wealth and income from labor:

- Y
_{ i }= YW_{i} + YL_{i}

(Note that ‘YW’ and ‘YL’ are not multiplications, but single variables refering to income from wealth and income from capital respectively.) From the values of income from capital and total capital in year i, we can define the *rate of return to capital* in that year like this:

- r
_{i} = YW_{i}/W_{i}

And from the values of income from capital and total national income in year i, we can also define the capital share of national income for that year:

- α
_{i} = YW_{i}/Y_{i}

From equations 5, 7 and 8, the following identity immediately follows:

- α
_{i }= r_{i }β_{i}

This is the law Piketty calls the *First Fundamental Law of Capitalism*. Some have wondered what this law has to do with capitalism specifically, since it is an identity that is true of any economic system. That’s a fair enough criticism. But notice that the law only has important application to any system for which there is a kind of income that can be called “return to capital”. These are economic systems in which there is wealth that is privately owned, where some of that wealth has an economic use that goes beyond personal consumption, and where there are market exchanges that provide the owners of the wealth with a flow of income in exchange for the use of the capital. If a system lacks these features, then equation 9 will only be vacuously true, since α and r will both be zero.

Continue reading →