# Piketty’s Second Fundamental Law and Some Fallacious Reasoning about Savings

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. Wi+1 = Wi + Si

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

1. Wi+1 = Wi + siYi

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

1. si = (Wi+1 – Wi)/Yi

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:

1. gi = Yi+1/Yi – 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 gi is usually called the national income growth rate. But, of course, it is possible for gi 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:

1. βi = Wi/Yi

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

1. Y i = YWi + YLi

(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:

1. ri = YWi/Wi

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:

1. αi = YWi/Yi

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

1. αi = ri βi

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:

1. βi +1 = Wi+1/Yi+1

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

1. βi +1 = (Wi + siYi)/(1+gi)Yi
1. βi +1 = (Wi/Yi + si)/(1+gi)
1. βi +1 = (βi + si)/(1+gi)

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 Xi defined for a given year i, let’s use the expression °Xi to define the rate at which X is changing in that year. So we can first define the rate of change in wealth as:

1. °Wi = Wi+1/Wi – 1

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

1. °Wi = (Wi + siYi)/Wi – 1
1. °Wi = [1 + si(Yi/Wi)] – 1
1. °Wi = si(1/βi)
1. °Wi = sii

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:

1. °βi = βi+1i – 1

And using equation 13, we get:

1. °βi = [(βi + si)/(1+gi)]/βi – 1

Some simple further manipulation gives us:

1. °βi = (1+ sii)/(1+gi) – 1
1. °βi = (1+ si/βi)/(1+gi) – (1 + gi)/(1 + gi)
1. °βi = (sii – gi)/(1 +g i)
1. °βi = (°Wi – gi)/(1 + gi)

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:

1. For any numbers x and y, if s(i) is constantly equal to x and g(i) is constantly equal to y, then limi →∞ 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:

1. For constant s and g, limi →∞ β(i) = s/g.

Or like this:

1. 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 2nd 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 2nd 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 2nd 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%

βi 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 βi changes along with them. The actual value of βi changes too over time, but those changes depend on changes in the values of Wi and Yi over time, not on the values of si and gi.

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.

## 12 thoughts on “Piketty’s Second Fundamental Law and Some Fallacious Reasoning about Savings”

1. Andrew Edwards says:

Thanks for the great rebuttal to some of fallacious reasoning deployed against Piketty. I have two key take aways. First, economists will have to engage with Piketty’s definitions if they want to rebut him. This is obviously a powerful move on Piketty’s part, and a valid one. Piketty’s definitions are *almost my favorite thing about the book. Two (and Piketty himself could have said this better) that Piketty’s laws only count as fundamental laws of capitalism because they assume the historical institutional configuration that produces recognizably capitalist societies. As a historian, I’m particularly interested in the latter. Any more thoughts on it?

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2. Paul Boisvert says:

H, Dan,
Nice work on the math, as always. But a couple points: first, near the end, I think you have a typo. In the example about 5 paragraphs from the end, you write “current growth rate is 2%”, but many of the rest of the calculations proceed as if you had meant 1%. For you write “B-sub i is converging towards 10%”, and “s/g in the long run, which in this case is s/0.01”. This error persists throughout the remaining paragraphs, with some using implications from g = 2% while others use those from g = 1%.

Secondly, and more importantly, I’ve been commenting mathwise over at NC on Yves’s continuing (perhaps diminishing, I hope) confusion of r (the return to W) with the growth rate of W, and a question occurred to me, which should have been obvious sooner, and which I hoped might be addressed in your post above, though I don’t see it. Perhaps you can clarify, since you’ve read the book whereas I only understand the math:

We know that “Beta”, the long-run convergence value for W/Y is determined by s and g, not r. So the “destination” towards which W/Y is trending has nothing to do with r. Hence, why Piketty’s emphasis on the issue of whether r > g? Who cares?

It is true that the convergence value of “Alpha”, the capital share of national income, is partially determined by r, since it’s just r times Beta. But why is it important whether r > g? A larger r determines a larger Alpha, but whether r happens to be greater than g or less than g seems irrelevant. If r was 3%, one would get the same limiting Alpha whether it were the case that s = 7.5% and g = 2.5%, or s = 10.5% and g = 3.5%. How does the fact that r > g in one case but not the other lead to important conclusions? I assume this involves real (political, societal) economic issues, not just the math itself(?)

I note that Piketty’s (and/or the media reviewers’) focus on r > g may be one reason why there is so much confusion, like Yves’s, about r vs. the growth rate of W. “The Capital/Income ratio has nothing to do with r at all, let alone with whether r > g!”, might be the best way to start off explications in the future! 🙂

Finally, a trivial piece of mathematical (notational) advice, since you’ve done so much of it above so well: I would avoid naming variables with two letters, such as your

Y i = YWi + YLi

The YWi looks like your previously named “Y” TIMES your previously named Wi, and I assumed it was at first. I then realized you had defined a new variable, “YW”, then subscripted it. It’s almost impossible even for mathematically adept readers to follow algebraic work with “double-letter” variables. Better to invent a new single-variable name, I find!

The above advice is trivial mainly because once you begin using subscripts at all, you instantly lose 99.9% of your readers… 🙂 So it’s not like it will make a difference to anyone but a math professor like me…

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• Paul Boisvert says:

Hi, Dan,

It seems to me that fruitful mathematical discussion of the book should thus take a different tack, and START with the growth rate of “rentier” share of total income. It isn’t r, but rather “rho”(r), that is crucial, as you point out. In fact, this is obviously precisely the confusion that Yves and others have made–they think r > g means “rho(r) > g”.

In the future, if I comment on such confusion out in the blogosphere, I will rename–consistent with my take on using one-letter variables–that rate rho(r) as, say, “Q”, and make the headline “Is Q > g?”! Once one has started with this issue, then asking “what are the limiting values of alpha and beta” can be done with less confusion, I think.

Also, it seems to me that Piketty didn’t think through his own PR. The fact that a “necessary” condition for rentier share of income increasing is r > g seems not to justify it being the primary focus, since it ignores rho. Who cares whether it is necessary, the question is “Is rentier/capital share of income actually (or likely to be) increasing?”–which it may not be, even if r > g!

At any rate, thanks again for all the skillful explication–I owe many hours of my life (freed from wading through the book) to you!

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• Dan Kervick says:

“and START with the growth rate of “rentier” share of total income. It isn’t r, but rather “rho”(r), that is crucial, as you point out.”

Rho also isn’t a growth rata, Paul. It’s the savings rate for the rentiers – so parallel to s, but for the class of rentiers rather than the population as a whole.

What I’ve learned is that almost every time Piketty says that the key fact is r > g, he is using the formula “r > g” as a kind of shorthnad for “r is significantly and persistently greater than g”. So the point is that the greater the difference between r and g, and the longer that difference is maintained, the easier it is for capital owners to increase their share of wealth and income.And that’s definitely true.

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• Paul Boisvert says:

Hi, Dan,
Sorry, I wasn’t clear in my phrasing. The two sentences you quote above aren’t related in the way you (understandably) took them to be. The “growth” rate I was referring to in the first was the “rho times r” that I mentioned in the second. I understand well that neither rho nor r individually are growth rates. But “rho times r” (using r in place of “r-subR” ) is a “pure rentier income” growth rate, right, as you point out in your “4 useful equations” post:?

However, having now gone through that post on 4 useful equations thoroughly, and though admiring the math therein, I guess I have to retract the whole comment anyway–I think that the simplest way to conceptualize the whole deal quantitatively remains to just focus on s and g after all, not on the relationship between r, or r(rho), and g.

Very simply, if W/Y is currently below s/g, then current values of Beta and Alpha must rise until that is no longer true. Obviously, if we dislike that rise, we need to lower s/g, i.e., raise g and/or decrease s. The role of r is best considered in terms of (undesirably) raising s, rather than in comparison to g, it seems to me: since the wealthy save more of their income than others, and bigger r gives them more income, it will increase s, whether or not r happens to be currently greater than g.

Moreover, since bigger s allows greater growth in Alpha and Beta (towards higher s/g), the wealthy will then have yet more income and wealth, on which to further increase s, and so on upwards. I don’t see much need here for detailed math–other than on Beta approaching s/g, which is very easy to show. I suppose it might also be useful to point out that the rate of increase of Alpha is (s/Beta – g)/(1 + g), so that bigger r, leading to bigger s, creates immediately bigger growth in Alpha than otherwise, as well as increasing the long run limits on Alpha and Beta.

And all that is just for constant r–the (political, not mathematical) fact that the very wealthy get higher r than others obviously just further increases their ability to increase s, by the previous mechanism. And of course raising g has the (wonderful) opposite effect from raising r, but that is again true whether or not r > g.

So upon reflection I don’t see much need for the detour into the “pure rentier” scenario, and especially not to focus on how that math supports r > g, particularly since r does not actually need to be greater than g for Alpha or Beta to increase in the real world (rather than the “pure rentier” world.) While I agree that in general a lower r implies that more new capital is formed by those who earn income, rather than those who “purely” collect it as rents, many wealthy people also earn a lot of income, so this may not broaden the distribution of capital too much–but at least that does raise a distributional issue, which none of the rest of the math does. Still, I don’t see that r > g is the most salient (mathematical) focus.

However, I certainly agree with your second paragraph above in a political (not so much mathematical) sense. Pointing out the political (distributional) consequences of r > > g is of course a good thing, though I don’t think we need the detailed math to appreciate them. But of course, capitalism in general has long been open to a devastating political critique, so Piketty isn’t treading new ground there, though every little bit helps!

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3. This is very nice. But doesn’t this reinforce the notion that in Piketty’s treatment, s and g are exogenous? If we think capital leads to growth, shouldn’t g = f(s)? And wouldn’t that change the dynamics?

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