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.

To see this, imagine a simple society of barons and serfs. Let’s say there are 100 barons and 1000 serfs. The barons own all of the land, and the serfs farm that land. The barons also own all the tools, guns, whips, stocks, pikes, etc., which they use to control the serfs, and also to put implements in the serfs’ hands for more efficient farming. In addition to farming the land, some serfs are put to work manufacturing new tools, guns, whips, stocks and pikes to replace the old stock of this equipment as it depreciates. However, the barons see to it that these industrial serfs always manufacture only just enough new stuff to cover the depreciation of the old stuff. As a result the stock of capital equipment never changes; it stays exactly the same over time.

The barons do no work to speak of, except perhaps for the occasional whipping of a serf when that serf gets out of line; so virtually all labor income goes to the serfs. The serfs produce 500 pounds of grain per worker per year, for a total of 500,000 pounds per year. That’s the net annual income in this society. Of this amount, 100,000 pounds go each year to the barons for their consumption. The serfs are permitted to divide the remaining 400,000 pounds. So the annual return to capital is 100,000 pounds of grain, and the annual capital share of income is 100,000/500,000; that is, 20%. The labor share of income is thus 80%.

Due to rigid regulation of reproduction, the population is absolutely stable: the serf population remains at more or less 1000 souls, year after year, and the size of the barony remains fixed at more or less 100 barons, year after year. And the society’s income never grows on a per capita basis either. So the annual income is 500,000 pounds of grain each year – year after year after year.

Thus, the growth rate in this society is 0%. That’s Piketty’s g.

On average, all of the food that is produced on the land is consumed in the year it is produced. So there is no net saving of food over time. And the barons never expand their net holdings of land. And we have already seen that the stock of capital equipment never shrinks or grows. So the total stock of wealth is constant over time. Just as there is no growth in income, there is also no growth in wealth.

Sometimes the barons trade a little land or grain among themselves, or some of the guns and such, and based on the prices that prevail when these exchanges are made they have computed that the total value of all of the owned land and all of the owned capital equipment stock combined is equal in value to 2 million pounds of grain. Thus the ratio of total owned wealth to annual income is 2 million divided by 500,000 or 4/1. That’s Piketty’s capital-to-income ratio β. The annual rate of return to capital can be computed by dividing the total annual income from capital by the total stock of capital. That’s 100,000/2,000,000, which is 5%. The rate of return to capital can also be computed in an equivalent way by dividing the total annual capital *share* of income by the capital-to-income ratio, which in this case is 20% divided by 4, and is again equal to 5%.

So the rate of return to capital is 5%. That’s Piketty’s r.

So in this case, since 5% > 0%, r is clearly greater than g. And notice that everything in this society is completely stable: the population, the total annual income, the total annual income going to labor, the total annual income going to capital, the total level of wealth. These facts remain the same year after year after year. Total income doesn’t grow; nor does capital income. And total wealth never grows either.

Note that in this particular case, none of the return to capital is reinvested in ways by which it could gain a return. All net income from capital comes in the form of grain, and each year more or less 100% of the capital income is consumed. If that wealth were saved in some durable form – for example, a portion of surplus grain might be stocked each year in some permanently icy cave – then total wealth would increase, and the rate of return on capital might decline, since there is probably no way to generate an additional income from the stored food. (Note that, for Piketty, capital and wealth are the same things, so the stored hoards of food count as part of the society’s capital.) The total annual return to capital would stay constant, yet the value of the capital stock would increase; so the rate of return to capital would fall. One thing Piketty does in his book, however, is document the return to capital over time, and he finds that it has remained remarkably consistent despite periods of sustained growth in the capital stock, generally around 4% to 5%, and almost never going below 3% or above 6%. So, in general, people manage to find ways of generating a solid return on the wealth they choose to accumulate.

There is also no *growing* inequality in our imagined society, although things are clearly very unequal since the serfs own nothing and the barons own everything, and because the per capita income of the barons is 1000 pounds and the per capita income of the serfs is 400 pounds. On Piketty’s framework, the fact that r is greater than g does not *in itself* suffice to explain growing inequality. Growing inequality in the real world is determined (in part) by the fact that capital is unevenly distributed in the first place, and capital owners often save very significant portions of their income and increase their wealth faster than the rest of us do; and they generally manage to earn a rate of return on their additional wealth that is roughly consistent with the rate they were earning before, or that at least declines more slowly than the capital stock grows. Also, the wealthiest among them often manage to earn rates of return on their wealth significantly higher than the rates of return earned by less wealthy owners of capital. This latter phenomenon is, Piketty says, a major factor in the spectacular growth of the highest incomes around the world in recent history. In my previous post, “Why Is r > g So Significant for Piketty?”, I attempted to explain the precise role played by r > g in combination with other factors is fostering inequality.

Over time, Piketty acknowledges, as wealth builds and the capital-to-income ratio increases, the price of capital may eventually decline to such a degree that the gap between r and g becomes small and inequality is no longer increasing. But between now and then, economic inequality might grow to horrible proportions, and then stay stuck there for a long time afterward.

Another query; this about the second fundamental law. Piketty seems to suggest that are restrictions to the use of it. On page 168, third paragraph from below, he says ‘ unless we assume that the growth rate is actually zero’. I wonder whether this is one of the conditions. Thanks.

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Because the second fundamental law is a limit theorem for a ratio of two functions, it must contain an implicit restriction for the case where the denominator is zero. But I don’t think the passage you cite is pointing to that implicit restriction. In that passage he’s just pointing out that the denominator in W/Y won’t be growing if the growth rate is zero, and so it’s just an exception to the case he just described.

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I boggle a bit at the feudal tale, though it makes the points. Actual distribution systems are very complex. Piketty seems to have done enough to get us to that point, in say chemical engineering, where knowing the general flow equations, we can get into a systems build to get desired outcomes. At some point, we need better measures that will not be a quantification. That wasn’t his job. One wonders what the feudal tale would expand into in a similar description of the present.

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It seems to me that your comment Brad DeLong’s recent post about the second law clarifies some confusions for

me (as well as Seth Ackerman’s comments about some of these models ‘not even wrong’). I wonder whether you can make another post clarifying the intent of the second law. There may be one already that I missed. Thanks.

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I am working on a new one based on that DeLong blog comment. But for now, have you seen this one:

http://ruggedegalitarianism.wordpress.com/2014/05/28/lets-end-the-confusion-over-pikettys-second-fundamental-law/

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I found this post very instructive, and I thank and congratulate you, Dan Kervick, for it. Together with your previous posts on this subject, it helps clarify this discussion on Piketty.

I am not sure you are fully aware, but there is a clear parallel between Marx’s tendency of the profit **rate** to fall and your discussion about how the **rate** of return on capital might fall, even though the return on capital (measured both in physical units or as a percentage of total net output) is not falling. For some reason, this is a topic non-Marxists often seem incapable or unwilling to understand.

I do have a quibble, though. Strictly speaking, just because the fraction of net output accruing to each class remains unchanged (out of the 500K pounds of grain, the 400K going to serfs and the 100K going to barons) one cannot conclude that “there is also no growing inequality in our imagined society”.

The internal distribution of grain among serfs or barons could be changing, increasing or decreasing inequality; we just have no way of telling from the information provided. Say, imagine for instance that one of the serfs gets the full 400K, while the others get nothing: there doesn’t seem to be anything in the scenario to preclude this outcome.

Anyway, thanks again and keep up the good work!

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Thanks Magpie. Yes, I agree that deeper analysis is necessary to get at the underlying dynamics of inequality, including the relationships between inequality of labor income and inequalities of capital income. In our world, we have capitalist owners paying top managers massive salaries to engineer the highest possible returns to ownership, and the mangers are able to convert much of that income into capital and further capital income.

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Hello Dan, good post.

On the other hand, you must realize that your illustration describes a feudal economy (and literally at that) with no capital growth whatsoever, and where the real growth of the economy is zero. And yes, the example of the barons and the serfs make perfect sense, but makes perfect sense in a feudal world.

But then, we live in a capitalist economy, in which capital grows, and moreover, its sole purpose is growth, i.e., being a capitalist means that I am trying to make money out of money as my first and foremost aim; and capital is merely the resources in the very process of doing that.

So, your illustration does not apply. And I understand that you are trying to tell a story through a simple enough example, but I want to remind you that this is exactly the point current economists fail: Simple illustrations (of Robinson Crusoes or feudal lords and whatnot) do not illustrate capitalism. I am curious what would happen if you get to a more realistic illustration: Replace the baron with a capitalist who wants his capital to grow… then this is a radically different situation.

Cheers, and keep the posts coming,

Burak

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