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:
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.