A Derivation of Piketty’s Second Fundamental Law of Capitalism

Several people have asked me for a more mathematically precise formulation of Thomas Piketty’s Second Fundamental Law of Capitalism, as well as a more careful statement of its proof and of the boundary conditions over which the proof applies, with some attention to the unusual and exceptional cases. I have written up some notes that are an attempt to comply with those requests. While I have attempted to be as precise as possible in laying out the proof, I have tried to avoid introducing any more rigor than is absolutely necessary for an informed reader to understand how the proof goes through. I have assumed a familiarity with reasoning about limits at the level of elementary calculus.

The notes can be downloaded here.

The Growth of Wealth and the Rate of Return on Capital

Justin Wolfers has posted some slides purporting to deal with the arguments of Thomas Piketty’s Capital in the Twenty-First Century. Unfortunately, the discussion outlined in Wolfers’s slides suggests that while he has read some of the more prominent recent responses to Piketty –  including Lawrence Summers’s review of Piketty in Democracy: A Journal; a recent paper by Per Krussel and Tony Smith on Piketty’s second fundamental law of capitalism; and some posted comments on Piketty by Debraj Ray –  he doesn’t seem to have read much of Piketty himself. I say this because Wolfers repeats some of the same interpretive errors that appear in those other works, despite the fact that the errors are quite easy to avoid, and even obvious, to anyone who has worked directly with Piketty’s text.

I commented on some of Debraj Ray’s criticisms of Piketty in my post “Why Is r > g So Significant for Piketty?” And I dealt obliquely with some of the Krussel and Smith arguments in “Piketty’s Second Fundamental Law and Some Fallacious Reasoning about Savings.” I will likely return again to these critics’ arguments again in future posts. I also dealt with the interpretive errors springing from Summers’s review in my post “Summers’s Review of Piketty: Underestimating the Argument for the Forces Driving Inequality.” Summers errs in attributing to Piketty (i) the view that wealth grows at the rate of return to capital, (ii) the view that as long as the return to capital exceeds an economy’s growth rate, wealth-to-income ratios will tend to rise and (iii) the tacit presupposition that returns to capital are 100% reinvested. Matt Bruenig has posted two fine new pieces this week, here and here, that challenge Wolfers on errors that seem to have their source in Summers, and has already dealt with most of the key points I would make on that score. But I would like to add just a bit to his discussion of the relationship (or rather lack of relationship) between the growth of wealth and the rate of return to capital.

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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.

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The Rate of Return on Capital Is Not a Growth Rate

This post is simpler than many of my more recent posts, and is aimed only at clearing up a single popular confusion I have run across from time to time concerning the rate of return on capital, and the role that rate plays in Thomas Piketty’s arguments in Capital in the Twenty-First Century.

Some people, when they first come across Piketty’s claim that the rate of return on capital has been significantly and persistently higher than the rate of growth throughout history, will respond in one of these two fashions:

• That’s crazy! If the rate of return on capital were higher than the growth rate, then capital would inevitably expand faster than the rest of the economy, and would eventually absorb everything.

• That’s horrible! But it explains everything. If the rate of return on capital exceeds the growth rate, that means capitalists are getting rich faster than the rest of us, and the gap between wealthy capitalists and everyone else is expanding.

I suspect both of these responses often spring from a common confusion. The idea seems to be that the rate of return on capital is the rate at which something is growing: perhaps it’s the rate at which either capital itself is growing or the rate at which the income from capital is growing. But the rate of return on capital is not a growth rate.

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Why Is r > g So Significant for Piketty?

In the course of several posts, I have attempted to clarify various aspects of Thomas Piketty’s framework for thinking about wealth and income dynamics, and to lay bare his analysis of the forces of divergence that lead to greater inequalities in income and wealth. One of the points I have made several times is that it is not the bare truth of Piketty’s famous inequality r > g alone that suffices to generate sharply greater economic inequalities, but the way in which significant gaps between r and g combine with other factors – including the rates at which different classes of capital owners save and the varying rates of return these capital owners obtain from their savings – that drives the growth in economic inequality. Piketty is quite clear about all of this, and examines these interacting phenomena in some detail in chapters 10, 11 and 12 of Capital in the Twenty-First Century.

But it occurs to me that in doing all of this exegesis, I have run the risk of losing track of why the basic inequality r > g is important to Piketty in the first place. After all, this is the formula whose truth Piketty singles out for special attention, and tends to invoke in summing up his central results in abbreviated form. So in this post I would like to rectify that omission, and explain the role of r > g.

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Taking Stock at My Cottage Piketty Industry

Well, I never intended to begin a mini-industry of Piketty interpretation and exegesis. I just set out to read Capital in the Twenty-First Century carefully so that I could assemble it’s full argument in my head and on paper, and then critically evaluate it. But in the process of working up my interpretation, I ran into a profusion of reviews, journalistic responses, and blog comments, many from reputable sources, that were so muddled and full of interpretive errors that I felt increasingly compelled to correct them. I’ve been trying to do my small bit to prevent the public record from becoming so polluted with erroneus representations of what Piketty says and doesn’t say that the work itself would end up buried under a mountain of confusion.

Here are my most recent Piketty posts, each of which appeared over the past two weeks, listed from the most recent to least recent. I believe the third and fourth posts in the list are probably the most important:

What Is Piketty’s Model?

Emergency: The World Needs Much Better Piketty Reviews!

Piketty on the Dynamics of Inequality: Four Useful Theorems

Let’s End the Confusion over Piketty’s “Second Fundamental Law”

The Financial Times’s Lazy Reading of Piketty

Naked Piketty – with a Bonus Spreadsheet!

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What Is Piketty's Model?

Alex Tabarrok interprets Thomas Piketty’s model of wealth and income dynamics in terms of a “super-simple” Solow growth model with the steady state requirement that investment is equal to depreciation. And he concludes on that basis that Piketty is committed to the idea that the economies he is modeling always invest in sufficient amounts to offset any depreciation, prior to any additional consumption and any additional saving. This is what he says.

Capital depreciates–machines break, tools rust, roads develop potholes. We write D(epreciation)=dK where d is the rate of depreciation and K is the capital stock.

Now the model is very simple. If I>D then capital accumulates and the economy grows. If I<D then the economy shrinks. Steady state is when I=D, i.e. when we are investing just enough each period to repair and maintain the existing capital stock.

Steady state is thus when sY=dK so we can solve for the steady state ratio of capital to output as K/Y=s/d. I told you it was simple.

Now let’s go to Piketty’s model which defines output and savings in a non-standard way (net of depreciation) but when written in the standard way Piketty’s saving assumption is that I=dK + s(Y-dK). What this means is that people look around and they see a bunch of potholes and before consuming or doing anything else they fill the potholes, that’s dK. (If you have driven around the United States recently you may already be questioning Piketty’s assumption.) After the potholes have been filled people save in addition a constant proportion of the remaining output, s(Y-dk), where s is now the Piketty savings rate.

As I have indicated previously, my strong suspicion is that this is a quite wrong-headed approach to understanding what Piketty is up to, and that any attempt to read Piketty as putting forward some kind of steady state equilibrium growth model distorts his argument.

Here’s how I interpret the rudiments of Piketty’s inherently dynamic model of wealth and income growth:

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Emergency: The World Needs Much Better Piketty Reviews!

Reviews of Thomas Piketty’s Capital in the Twenty-First Century continue to roll out, many by professional economists. Yet I continue to be frustrated by the fact that almost none of the  economists’ reactions to Piketty that I have read display any close familiarity with chapters 7 through 12 of the book, where all of the actual analysis of the structure of inequality is contained. Most of these reviews seem to go no further than Chapter 6, with Piketty’s now-famous inequality r > g then tossed into the salad for good measure, on the basis of which the reviewer then attributes to Piketty large claims about the dynamics of inequality based entirely on r > g and the contents of those introductory chapters. But everything in Chapters 1 through 6 is prefatory to the analysis of the structure and dynamics of inequality that follows.

The economists’ reviews and responses are also riddled with confusion over Piketty’s Second Fundamental Law of Capitalism. I have dealt with most of those confusions already in three posts:  Nothing Magical about Piketty’s MathematicsSummers’s Review of Piketty: Underestimating the Argument for the Forces Driving Inequality; and Let’s End the Confusion over Piketty’s “Second Fundamental Law”. But the confusions continue to appear in the work of very competent economists.

I can summarize the problem briefly. The reviewers have persistently read the second law as an outright equation, some kind of approximate identity or long-term equilibrium condition of the form β  = s/g (or equivalently K/Y = s/g) which they then fold into an equilibrium model framework, and from which they then proceed to deduce bizarre results that they attribute to Pikeety. For example, Per Krussell and Tony Smith argue that the 2nd Fundamental Law is implausible because “implies saving behavior that, as the growth rate approaches zero, requires the aggregate economy to save 100% of GDP each year.”  They also say that, together with the 1st Fundamental Law α = rβ, which is a genuine identity, the 2nd fundamental law:

delivers the central relationship of Piketty’s book: capital’s share of income is r x s/g. This formula is alarming because it suggests that were the economy’s growth rate to decline towards zero, as Piketty argues it will, capital’s share of income could increase explosively.

But these readings, I believe, are quite incorrect. To be fair, Piketty does abbreviate the law in the form of a strict equation when he introduces it. But in his ample discussion in the text of the book right after he introduces the 2nd law, and in his online technical appeandix, he makes it clear that the law is only an asymptotic and inherently dynamic law that could just as well have been abbreviated in the form “β → s/g”, and does not license to importation of k/y = s/g as a constraint on the computation of some kind of equilibrium condition.  Indeed, the very idea of equilibria seems foreign to Piketty’s open-ended, empirical and irreducibly dynamic perspective which emphasizes historical contingency and volatility.

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Piketty on the Dynamics of Inequality: Four Useful Theorems

Matt Breunig has posted three excellent new pieces on Thomas Piketty’s analysis of the dynamics of inequality. They’re at the Demos Policyshop blog, and can be found here, here and here. Since Breunig comes to many of the same interpretive conclusions I have reached myself, I will just refer the reader to these pieces without much further comment on them. But I do want to call special attention to one thing Breunig says in response to a recent critical essay on Piketty by John Aziz. In characterizing Piketty’s account of inequality, Aziz says that according to Piketty inequality will tend to increase when the rate of economic growth for the entire economy is less than the average return on capital. But Breunig notes in response:

This is fine enough as a gloss of an explanation, but is it not strictly true. Piketty’s actual point [is] that the larger the spread between r and g, the more forcefully the dynamics of capital income pushes in the direction of increasing wealth inequality.

Breunig raises an important point here, and it bears emphasis. Piketty couches most of his arguments about equality and inequality in terms of forces of divergence and forces of convergence. His approach is to identify those conditions under which the forces of divergence will predominate, and if so, how strongly they will predominate. His view is that at the beginning of the 21st century, the conditions appear to be in place for the forces of divergence to acquire renewed strength, although he also stresses that nothing is certain, and the exact course of 21st century inequality depends on a host of political, demographic, technological and economic factors. One thing Piketty routinely stresses, however, is that the forces for divergence operate very strongly when r is “significantly and durably” higher than g, and automatically lead to a very high concentration of wealth.

I want to add a bit of precision to these initial statements, and in the process shed some light on why Piketty lays very significant stress on the fact that different kinds of wealth owners earn different rates of return on their wealth, and also on the fact that the wealthy save their incomes at higher rates than those who are less wealthy. Both of these phenomena play an important role in Piketty’s analysis of the forces for divergence and the structure of inequality in Chapters 10, 11 and 12 of Capital in the Twenty-First Century.

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Let's End the Confusion over Piketty's "Second Fundamental Law"

A great deal of unnecessary confusion continues to surround Thomas Piketty’s Second Fundamental Law of Capitalism. I have already dealt with the most common source of confusion here, and refer the reader to that post. But the same misreading has reappeared frequently in the ongoing debate over Piketty’s Capital in the Twenty-First Century, so it is worth returning to the topic briefly.

James Hamilton writes:

On page 168 of Piketty’s book the reader is introduced to “the second fundamental law of capitalism” according to which β = s/g, where β denotes the capital/income ratio, s the economy’s saving rate, and g the overall economic growth rate. Note that a curious corollary of this “law” is the claim that if the economy is not growing (g = 0), the capital/income ratio β has to be infinite.

But that is a misreading. Piketty’s Second Fundamental Law is not an identity or an approximate identity. It is, as Piketty makes clear at some length on pages 166-170 of the book, a long-term asymptotic law. For the benefit of the general reader, Piketty abbreviates the statement of the law in the form  “β  = s/g”. But it might be stated more carefully this way: “For a fixed savings rate s and growth rate g, the capital-to-income ratio β converges over time to s/g.” He might have abbreviated it in more conventional fashion in the form β → s/g. Piketty sketches an elementary proof of this convergence theorem in his online technical appendix. Like any limit theorem proved over the real numbers, it requires implicit restrictions on the range of the variables to avoid singularities.

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