George Cooper continues to have doubts about Thomas Piketty’s famous inequality r > g, which says that the rate of return to capital is greater than the rate of growth of national income. Cooper raises those doubts in connection with a recent post of mine in which I attempted to dispel some of the confusion surrounding Piketty’s inequality by making it clear that the rate of return to capital is not any kind of growth rate. But Cooper doesn’t raise any serious objections to Piketty’s inequality, at least so far as I can tell. In fact, Piketty’s inequality ought to be regarded as among the least controversial points Piketty makes in *Capital in the Twenty-First Century*.

The thought experiment I introduced was designed to clarify certain fundamental points about the conceptual relationship between wealth, income, the rates at which wealth and income grow, and the rate at which the ownership of capital is rewarded by income. Without going back over the whole thought experiment, let me summarize what I take to be the essential takeaway points:

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.

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.

Neil Irwin, looking at the latest US economic data, finds an ugly fly in the ointment of our economic recovery – such as it is – that lends a stinking savour to the odors wafting from the labor scene:

* The latest economic data out Tuesday morning was generally good. Home building activity rebounded nicely in May after weak results in April. Consumer prices rose 0.4 percent in May, such that inflation over the last year is now 2.1 percent, about in line with what the Federal Reserve aims for.*

*But that inflation news carried with it a depressing side note. Now that the May Consumer Price Index has been published, it is possible to determine inflation-adjusted hourly earnings for the month. And the number isn’t good.*

*Average hourly earnings for private-sector American workers rose about 49 cents an hour over the last year, to $24.38 in May. But that wasn’t enough to cover inflation over the year, so in “real” or inflation adjusted terms, hourly worker pay fell 0.1 percent over the last 12 months. Weekly pay shows the same story, also falling 0.1 percent in the year ended in May.*

*Pause for just a second to consider that. Five years after the economic recovery began, American workers have gone the last 12 months without any real increase in what they are paid.*

Strangely, faced with these indications that our *soi-disant* recovery is a completely bogus affair as far as American workers are concerned, Irwin’s mind immediately turns, as so many mainstream minds seem to turn in these days of social and political paralysis, to yet more neurotic worrying about deliberations inside the conclave of monetary alchemists who run the Fed:

J. D. Alt writes:

*How would Thomas Piketty propose to save the city of Norfolk, Virginia? He teaches us, ad-nauseum, that what the U.S. collective state has to spend on such things as sea walls, flood gates, elevating infrastructure and roadways, buying-out property owners so they can relocate to higher ground, etc., etc., is limited to the number of tax dollars that can be collected from U.S. citizens—as if the collective state itself were like a club, and if the clubhouse needs repairing, the club members must first pay a special assessment of dues—or, alternatively, the club can borrow dollars from the supply of Capital owned by the wealthiest 1% of its membership, or (as a creative alternative) the rebuilding effort could be structured in such a way that the newly elevated Norfolk would pay rent to the one percent in perpetuity for the privilege of living above sea-level.*

I find this characterization puzzling, and do not understand where these teachings are to be found in Piketty. Piketty has very little to say about the factors that both determine and limit the capacities of government in carrying out ambitious projects. Far from issuing teachings on this subject “ad nauseum”, as Alt puts it, reflections of these kinds are simply not a significant component of Piketty’s work on distribution and economic inequality.

Alt continues:

*For Mr. Piketty, the calculation is: what is the availability of “Capital”? His entire book is an exercise in quantifying and compartmentalizing in various ways the distribution of “Wealth” throughout the world—and projecting how its historical 5% growth rate might be distributed in the future. Presumably, he would use these calculations to arrive at some number of Dollars that theoretically could be made available to the effort of saving Norfolk.*

But at this point, I have little idea what Alt is talking about. First of all, I don’t know where Alt’s figure for a 5% historical growth rate for wealth comes from. For Piketty, wealth grows at the annual rate s/β, where s is the national savings rate for that year and β is the capital-to-income ratio in the same year. A more typical rate of wealth growth would be something like 2%, which is the rate we would get with a national savings rate of 10% and a capital-to-income ratio of 500%. Perhaps Alt is confusing the rate of wealth growth with the rate of return on capital. The rate of return on capital, for which Piketty does indeed find historical values of around 5% in many periods, is a very different animal, and is not any kind of growth rate at all.

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.

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.

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 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.

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:

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

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: