Cooper on r > g and the Return to Capital

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:

• In any society in which some people derive incomes from their ownership of the stuff that can be owned, and not just from their labor, there is a return to capital. That’s just a conceptual point. There are only two kinds of income: income derived from the ownership of wealth and income earned by labor. The sum of the first kind of income is the total return to capital; the sum of the latter kind of income is the total return to labor.

• If we can in any way measure the value of that total income from wealth ownership, and also measure the value of all of the stuff that is owned, and compare those values, then we can speak meaningfully of a rate of return to capital.

• Some of the capital income generated in any given year, beyond what is needed to replace any existing wealth that has been lost due to depreciation, might be saved and added to the stock of existing wealth; also, some of that saved income might be invested in the production of new components of the wealth stock or even in altogether new forms of wealth. But neither of these things have to happen. It is easy to imagine stagnant societies in which the owners of existing wealth receive a steady annual income from that ownership, out of which they continually replace that portion of the existing stock of wealth that is lost due to depreciation, and then consume all of the rest.

• Even if a portion of the return to capital is routinely reinvested in the creation of additional wealth – as it is in all modern developed economies – so that national income and national wealth both grow continually, there is no logical reason whatsoever why the rate at which either wealth is growing or income is growing must be equal to or greater than the average annual rate of return to capital.

• The rate of return to capital is not the rate at which something is increasing: it’s not the rate at which income is increasing and not the rate at which wealth is increasing. It’s also not the rate at which the capital-to-income ratio is increasing or the capital share of income is increasing. Nor can one derive any of these latter rates from knowledge of the rate of return to capital alone. The rate of return to capital in any given year is simply the level at which wealth ownership is rewarded with income in that year, measured in proportion to the total value of owned wealth.

Cooper isn’t persuaded of the truth of Piketty’s inequality by the thought experiment I presented of a zero growth society in which r is consistently 5% of owned wealth and g = 0%. But, Cooper’s reservations about the thought experiment aside, the point of that thought experiment was not to defend the fact that r is typically greater than g. My argument was instead aimed at people who have continually confused the rate of return to capital with some kind of growth rate, and was focused on showing why they are mistaken. The defense of r > g simply lies in the fact that the best available historical measurements of r and g for modern societies over the past two centuries show r clocking in as consistently higher than g. I take it that the empirical, historical evidence Piketty presents in his book successfully makes the case that r is significantly and persistently greater than g, and that his claim needs no further defense from me or anybody else. This really shouldn’t be that controversial at all. In fact, most of the responses to Piketty that I have read, whether supportive and critical, don’t challenge this assertion. Most of the critics have instead argued that either (i) the difference r – g will not be quite as large in the future as Piketty thinks it will be or (ii) the arguments Piketty presents about the structure of inequality based on his considerations about r and g fail for some other reason. And of course, there are some people who think Piketty might very well be right about the direction of inequality, but who just don’t care all that much about inequality.

Cooper also suggests my thought experiment runs afoul of Piketty’s Second Fundamental Law of Capitalism:

The problem arises because of the chosen the special case of an economy with no growth and no savings. Using Piketty’s second ‘law’ which gives the capital income ratio, β, as the savings rate, s, divided by the growth rate g, β = s/g, therefore we get β = 0/0.

In my view, Cooper commits a serious fallacy here in interpreting and applying Piketty’s 2nd law. I have already addressed this fallacy several times, including in these posts:

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

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

Nothing Magical about Piketty’s Mathematics

And I addressed it again today in a long piece that derives and clarifies the second fundamental law. So it is not necessary to cover this well-worn ground again. But the fundamental point is that Piketty simply does not equate the capital-to-income ratio with s/g. Most of the examples he discusses of plausible real world economies exhibit capital-to-income ratios significantly less than s/g: for example, we can imagine a society much like our own in which β is 4 and s/g is 10. All the second law says is that if s and g are positive and remain constant over time, then the capital-to-income ratio β will converge toward s/g. But in the degenerate case where s is any fixed positive number and g = 0, then the capital-to-income ratio does not converge to either s/g or any other number. The year-by-year sequence of capital-to-income ratios instead diverges, and will increase without bound by incremental additions of s. By definition, the capital to income ratio for year i+1 is always given by:

βi+1 = (Wi + sYi)/(1 + g)Yi

So if β0 is the capital to income ratio for year 0, and if g = 0, then we obtain the following capital to income ratios for succeeding years, for as long as s remains constant:

β1 = β0 + s

β2 = β0 + 2s

β3 = β0 + 3s

In the hypothetical example I used in my thought experiment about the zero growth society, I tried to use numbers for the relative sizes of the laboring class of serfs and the capital class of barons that would seem plausible given our intuitions about political reality and human nature. And yes, I agree with Cooper that in a society with only one super-baron, that baron would have limited consumption potential and would likely not command an income from capital as great as that commanded by all 100 barons in the original example. The overall return to capital would be lower. Of course, if g = 0, then no matter how low the return to capital goes, r would still be greater than g so long as there is any capital income at all. It should also be noted that we probably have to posit some pretty weird Dr. Evil circumstances to imagine a society in which a single baron can persistently enforce ownership claims and command a capital income from 1000 serfs without being toppled by an easy rebellion.

We could also imagine a socialist society in which all land, resources and implements are collectively owned and in which everyone does a fairly equal share of work and receives a fairly equal share of annual income. Would this be a society in which there is no capital income and only labor income? Or would it be a society in which there is only capital income and no labor income? Or would it be a society in which there is both labor income and capital income? I think the best thing to say is that under such conditions, the distinction between capital income and labor income collapses. Capital income as such comes from ownership, and ownership comes from power and the ability to employ that power to command an income from the work of others that does not depend on any work one does oneself to participate in generating that income. In a society of democratically shared and collectively distributed power, nobody has the ability to extract an income from the work of others unless those others acquiesce in the bestowal of that income.

Finally, Cooper makes this comment with regard to capital returns in my thought experiment:

what I think Dan has described is not a model of workers and  capitalists but rather a model with workers (serfs) and supervisors (barons)  who between them take all of the output leaving nothing left over for the return on capital.

But I think this confuses the return on capital with savings, investment, or both. As I noted above, a society can have a steady return to capital even if there is no net savings or net investment of that capital. In fact, many traditional feudal societies operated in something approaching this fashion for centuries. Cooper then asks:

Can you have a return on capital without a mechanism for reinvesting that capital?

Given the above, I still think the answer to this question is clearly “yes”. Of course, in actually existing modern economies, a substantial portion of capital is reinvested, and there is ongoing growth of both wealth and income. The point of my thought experiment was just to show that there is no determinate relationship between the amount of income derived from the ownership of wealth and the amount of income that might be invested in the creation of new wealth.

9 thoughts on “Cooper on r > g and the Return to Capital

  1. Hi, Dan,
    I was mainly going to offer (yet another) comment on your math in the “Derivation”” post, but your increasingly effervescent communist ranting in a couple others has me so appreciative that I have to begin by commending you! “Class Warfare is Naked Aggression, not Misguided Policy”, and “In a society of democratically shared and collectively distributed power, nobody has the ability to extract an income from the work of others unless those others acquiesce in the bestowal of that income”–keep up that good work and you’ll soon have ISR begging to cross-post your stuff! 🙂 You know the old saying, “A socialist is a social democrat who’s been mugged by reality”…

    Re the (very nice but very lengthy) math in the “Derivation” post, I suspect you know the following and have your own reasons for avoiding it, but on the off chance that’s not the case: (and I’m going to use B for “Beta” here)

    When you want to find the Limit as i –> oo of B-sub i , and show that it equals s/g, you can save oodles of time (and teach or remind readers of an important notion associated with infinite sequences) by using the ploy generally applied to quickly finding the limit of a recursively-defined sequence. Just set the i’th term equal to the (i + 1)st, dropping the i’s, and solve. At the limit they must “essentially” be equal–not rigorous, of course, but wherever there actually is a limit it must be findable by this method (if the equation can be solved, that is, but this one is trivial.)

    That is, set B = (B + s)/(1 + g), cross-multiply, and solve for B = s/g in 2 seconds. In any case where there actually is a finite limit (rather than divergence to oo or by oscillation), that must be it. For those who want rigor, all you need to show is that, assuming w.l.o.g. that B-sub 0 is s/g as well, of course.)

    And in any case where the answer obviously doesn’t make sense, hence you suspect there is no such limit, which here (it is evident from inspection of the result) is for non-positive g, you can then, if you like, work through the longer series approach to see what is happening. Or perhaps more simply just calculate a few terms and see if there is an obviously divergent pattern, which there will be for simple sequences, like this one.

    But your very rigorous series derivation is of course nicely done as well, and you did say you wanted the rigor–and are obviously enjoying the math! Thanks for all the enjoyable reading!


  2. Sorry, one part got messed up: In above comment, replace sentence beginning “For those who want rigor…” with following:

    For those who want rigor, all you need to show is that, assuming w.l.o.g. that B-sub 0 is s/g, heading downwards.


  3. I didn’t know about that trick Paul. But as you say, it makes perfect sense. Since convergence entail Cauchy convergence, then at the limit the difference between successive terms goes to zero, and the relation between successive returns approaches identity.

    In this case, though, I’m glad I did the careful consideration by cases, because some people had specifically asked me about the case of negative growth, and others had gotten entangled in fallacies about the case of g=0, and in both cases beta diverges.


  4. Hi, Dan,
    Yes, your series approach was clearly more useful for many of your purposes, especially since it gave you an explicit formula for B in the nth year, hence allowing you to calculate just how far B still might be from the limit value in future years (reinforcing your case against the confusion of current values of B with s/g.) The “trick” is useful only for finding the limit, not for studying behavior along the way.

    Since you know enough math to get Cauchy convergence (!), I’ll try one last time to post the “rigor” sentence correctly. For 0 < B-sub 0 < s/g, a few seconds of trivial algebra show that B-sub i < B-sub(i + 1) s/g > 0, one gets a decreasing sequence bounded below, which must converge as well, as long as g is again > 0.

    One unrelated comment on your “MMT” posts, including the old ones at NC re Wray’s take. While you are convincing that Piketty doesn’t seem to warrant criticism on this issue, the point of Wray’s post was tactical. His original post was a bit breathless(!) and not well-argued, but the notion was that focusing on taxation is a worse tactic (now, given current politics) than simply saying “there are non-tax, MMT-oriented ways to substantially improve growth and lower unemployment”.

    There is no doubt this is an extremely plausible position. You can’t watch political campaigning in this country without realizing that “taxation” is by a huge margin the main touchstone issue in American politics. The fact that the right deliberately conflates who’s being taxed and how much, to confuse voters into supporting them, is a given, but correcting that will take a vast political educational effort–especially since Democrats effectively support the rich as well, just a tad less than the right does (and “Kansas” voters are well aware of this). Wray says “save that for the long run, just spend without taxing now, which MMT shows is fine.”

    Wray is open to the counterargument that convincing people that MMT is fine (i.e., won’t lead to Zimbabwe) will also take a vast political educational effort, so why not keep the “tax fairness” issue as part of the effort. It’s kind of a chicken and egg thing–which set of issues will more effectively “carry the raised consciousness of the people” along with it, and hence should be the primary focus. But there’s a pretty compelling case that that right now (which is when people are suffering, as usual!) opposition to taxation is the harder issue to overcome tactically. Imagine a Democrat trying to explain why the rich should be taxed more, rather than just saying “printing money doesn’t lead to inflation as long as there are unused societal resources that it will mobilize.” The latter is just Econ 101, is obviously true, and doesn’t challenge anything about capitalism itself.

    But the former is actually a critique of capitalism as an entire system–the rich must earn their money unjustly, if it’s ok to take more of it away from them than others; hence capitalism, which allows (encourages!) the rich to (unjustly) earn money, is an unjust system. People who hear that then may begin to wonder, why have an unjust system and regulate it? Why not just change to a new system? Things that are unjust in this country (assaulting people, robbing them, defrauding them, etc.) are, um, illegal. If capitalism is unjust, why shouldn’t it be illegal too?

    Suddenly the utterly dominant, most firmly entrenched ideology in the history of the planet is under attack… And how’s that been working out these last 40 years… 🙂

    As a socialist I of course believe both propositions–capitalism is inherently unjust, but you can easily and quickly have somewhat less injustice in it under MMT. Both will indeed need a massive effort to convince others of their truth. But MMT itself has little ideological content (it would apply just as well to a socialist society as a capitalist one, which is why I’m so enamored of it.) Overcoming ideology is always the hard(est) task, and Wray/MMT’s tactic of avoiding it has a very strong case behind it, if your goal is to start reforming and improving capitalism as soon as possible.

    Lastly, your call for MMT to provide more math models I don’t see as necessary. We can easily tell if the economy is “inflating” too rapidly, and just reduce spending. It’s like driving a car–you don’t need a math model to tell if you should be accelerating or decelerating to meet current driving conditions. I also note that inflation is always accompanied by more total, societal income with which to pay the inflated prices, hence is a wash other than in distributional terms. It’s true that to solve such a distributional issue one would have to look at tax or income support measures, thus supporting your side of the argument that MMT alone isn’t sufficient. But inflation also reduces real debt, largely held by the un-wealthy, so even without further distribution fixes it’s still not much of a price to pay overall for sustained growth and low unemployment–especially since MMT’s JG would further ameliorate inflation’s distributional effects. It’s almost totally the wealthy (the owners of inflating money) who oppose inflation, so why would we care if it temporarily runs a bit too high because we “decelerate” it a bit too slowly?


  5. and the “rigor” sentences STILL don’t post correctly!! must be an html symbol issue…. maybe the greater than and less than signs?? How about this, no symbols: in the most common case,the B’s form an increasing sequence bounded above by s divided by g, so they must converge. That’s all I’m sayin’ 🙂


  6. This comment is strictly off topic but the link below may explain some of the really extreme reactions Piketty’s work has generated. And, considering the website where the post was published, that seems something really telling.

    The Summer’s Most Unread Book Is…
    By Jordan Ellenberg

    “A simple index drawn from e-books shows which best sellers are going unread (we’re looking at you, Piketty).”


  7. I found Piketty’s book dull and am obliged by Dan’s careful reading. We were reading neoclassicism as a dud back in the 80s. Standard reading back then included this paper by Philip Mirowski –

    Deeper in than Dan’s good explanations of just what the puzzles ‘solved’ by Piketty actually are, are questions of how we get into ‘discussion’ without much consideration of how (and if) concepts like ‘energetics’ and maths and physics more generally transfer to the social sphere. Mirowski explains this well. To me as a scientist, Piketty reads as ‘unempirical’ as the early marginalists. Social mice, even in times of plenty, produce an inequality very similar to the human plight, presumably without any of the nine or so versions of economics. What economics concepts are, really, is missing and we have ‘conventionalism with sums’ rather than empirical adequacy.

    Missing from Dan’s thought experiment is any conception of what work produces surplus (as opposed to, say, ‘glut of unwanted’) and the potential to live differently and better other than through differentiation as an owner or through neurotic consumption. One wonders if we can create one that challenges just what economic concepts are? The easy one is full robot heaven, which challenges what work motivation is (as does winning the lottery). Others include what the few left when the planet has burned might think of an economics in which we use our expertise to frack, produce another thousand channels playing soap re-runs and called idiot savants differentiating and integrating stochastic assumptions economists. The Domesday Book actually compares feudal structures for efficiency and Dan would lose ‘his serfs’ to ‘me’ if I could squeeze more out of them in my outfit. What might more complex arguments on economics be – ones not relying on assumptions (often just ignored) like comparative advantage and fictions that economics produces the right goods?


  8. Pingback: My Piketty Series Resurfaces | Samma Vaca

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