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:

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:

2. W_i+1 = W_i + s_iY_i

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

3. 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:

4. 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:

5. β_i = W_i/Y_i

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

6. 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:

7. 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:

8. α_i = YW_i/Y_i

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

9. 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.

All of the above should seem relatively uncontroversial as an interpretation of Piketty’s conceptual scheme. These are the most basic parts of his analytic framework. But starting from these basic elements we can deduce other identities and laws. For example, we might be interested in understanding how the capital-to-income ratio for any year, along with the savings rate and growth rate for that year, determines the capital-to-income ratio for the succeeding year. By equation 5, the definition of the capital-to-income ratio, we have:

10. β_{i +1} = W_i+1/Y_i+1

And by applying equations 1, 4 and a little algebra we get:

11. β_{i +1} = (W_i + s_iY_i)/(1+g_i)Y_i

12. β_{i +1} = (W_i/Y_i + s_i)/(1+g_i)

13. β_{i +1} = (β_i + s_i)/(1+g_i)

We can also define the rates at which both wealth and the capital-to-income ratio grow between any year i and the succeeding year i + 1.  For any variable quantity X_i defined for a given year i, let’s use the expression °X_i to define the rate at which X is changing in that year. So we can first define the rate of change in wealth as:

14. °W_i = W_i+1/W_i – 1

And by employing algebra and some of the equations above, we get:

15. °W_i = (W_i + s_iY_i)/W_i – 1

16. °W_i = [1 + s_i(Y_i/W_i)] – 1

17. °W_i = s_i(1/β_i)

18. °W_i = s_i/β_i

So that’s the rate at which wealth is changing in any given year. If the savings rate is 10% and the capital-to-income ratio is 4, then wealth is increasing at a rate of 2.5% per year. We can perform a similar calculation to find the rate at which the capital to income ratio is changing. By definition, that rate of change is:

19. °β_i = β_i+1/β_i – 1

And using equation 13, we get:

20. °β_i = [(β_i + s_i)/(1+g_i)]/β_i – 1

Some simple further manipulation gives us:

21. °β_i = (1+ s_i/β_i)/(1+g_i) – 1

22. °β_i = (1+ si/βi)/(1+g_i) – (1 + g_i)/(1 + g_i)

23. °β_i = (s_i/β_i – g_i)/(1 +g _i)

24. °β_i = (°W_i – g_i)/(1 + g_i)

So, if wealth is growing at 2.5%, as in the example above, and the rate of growth is 2%, then the rate of change in the capital-to-income ratio is 0.5% divided by 1.02, which approximately 0.49%. Note that for a given rate of wealth increase, then as g grows, the numerator of this expression shrinks and the denominator increases, so the capital-to-income level grows at a slower rate. Similarly, the capital-to-income ratio grows more rapidly when the growth rate of income is small. That’s important, because as Piketty argues in the book, the forces for wealth and income divergence operate more strongly the faster the capital-to-income ratio is growing.

We now come to the mathematical law Piketty calls the Second Fundamental Law of Capitalism. When Piketty first introduces the law, he abbreviates it as “β = s/g” and also expresses it with the statement “β = s/g in the long run.” But in the text, both from the examples he uses and his more careful descriptions of the law, and in his online technical appendix where he sketches the proof of the law, it is clear the second fundamental law is a “long-term asymptotic law” or a convergence theorem that depends on the computation of limits in connection with the analysis of infinite sequences. We can express it more carefully like this:

25. For any numbers x and y, if s(i) is constantly equal to x and g(i) is constantly equal to y, then lim_{i →∞} β(i) = x/y.

The function s(i) is the function that maps each year i to a savings rate for year i, over some domain of years running from an initial year 0 out to infinity. So to say “s(i) is constantly equal to x” is just an abbreviated way of saying that s(i) = x for every value of i for which s(i) is defined. We can express the law more concisely and informally like this:

26. For constant s and g, lim_{i →∞} β(i) = s/g.

Or like this:

27. For constant s and g, β(i) converges to s/g.

Now this might all seem a bit pedantic. But it is useful going through all of these equations in this careful way to help avoid falling victim to some seductive fallacies that involve Piketty’s 2^nd fundamental law. The law is designed to give a snapshot of how the capital-to-income ratio is changing at any given time, based on the current savings rate and the current growth rate at that time. It tells you what would happen to the capital-to-income ratio over time if the current rates of g and s were held constant and extrapolated out to infinity. It does not help you estimate the current capital-to-income ratio by giving an approximate value for that ratio. The current capital-to-income ratio could be quite far from s/g, where s is the current savings rate and g is the current growth rate. What you learn when you learn the current values of s and g is the number to which the capital-to-income ratio is converging, not anything about what the current capital-to-income ratio is.

Think of it in terms of this analogy: At any given moment in the Earth’s elliptic revolution around the sun, our planet has an instantaneous velocity: an instantaneous speed at which it is moving in addition to a direction in which it is heading. Suppose there is always some star at which the instantaneous motion of the Earth is pointing at any moment in its orbit. Now if, for a given moment of time, and instead of continuing along its orbit, the Earth were to begin moving at its current instantaneous rate of speed in its current instantaneous direction indefinitely, then you could calculate at which star the Earth would end up and the time it would take to get there. The only difference is that the 2^nd fundamental law, is an asymptotic convergence law, which means that it would continually get nearer to s/g over infinite time, but never reach that value.

But the 2^nd fundamental law is more useful that the analogy I just presented, because the economy does not follow an orbit with continuously cycling growth rates and savings rates, but tends to persist with savings rates and growth rates that move within narrow bounds for extended periods of time. So, s/g gives you a value toward which the economy would actually head over time, if the current rates of growth and savings didn’t change. Call the value of s/g, the present convergence value for the capital-to-income ratio. The convergence value might be nowhere near the actual present value of the capital to income ratio β = W/Y.

Now, suppose you learn that the current stock of wealth in 2014 is $4 trillion and the current annual national income is $1 trillion; and suppose you also learn that the current savings rate is 10% and the current growth rate is 1%. Then you can conclude these facts:

β₂₀₁₄ = 4

°W₂₀₁₄ = 2.5%

°β₂₀₁₄  ≈ 1.49%

β_i is converging toward 10.

Now here are the fallacies you need to avoid. Suppose you reason like this: “Well the capital-to income ratio is 4, and 4 is equal to s/g, which in this case is s/0.01, so s must now be 4 x (0.01) or 4%.” If you reasoned that way, you would be committing a very serious fallacy. The capital-to-income ratio is not equal to s/g. It is only converging toward s/g. Nor need it even be approximately equal to s/g. You cannot use what you know about the current capital-to-income ratio to get an approximate current value for s/g, and then use that estimate to infer some conclusion from the current growth rate about the current – or near term or short term – value for s. The present convergence value for the capital-to-income ratio depends on whatever are the current values for s and g, and if those latter values change over time, the convergence value for β_i changes along with them. The actual value of β_i changes too over time, but those changes depend on changes in the values of W_i and Y_i over time, not on the values of s_i and g_i.

Here is a slightly more subtle fallacy. Suppose you say “Well the capital-to income ratio is 4, and 4 is equal to s/g in the long run, which in this case is s/0.01, so s must over the long run converge to  4 times (0.01) or 4%.”

That is also fallacious. If the level of wealth is $4 trillion, and the current national income is $1 trillion, then the current capital-to-income ratio β is 4. And if the current rate of growth is 2% while the current savings rate is 10%, then the current convergence value for β is 10. There is nothing in this that allows one to assume the current value of β constant over time for the long run, and then to use that long run constant value to compute long-run values for s from g or g from s.

Here is the upshot: One can deduce no information about savings behavior whatsoever from knowledge of the current capital-to-income ratio – neither the current, near-term or long-term rate of savings, nor the long-run path of the rate of savings – even if you know the current rate of growth, or the path of the rate of growth over time.

 

Addendum: Thanks to Paul Boisvert for spotting a mistaken substitution of “2%” for “1%” in the latter part of the essay.