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.

The most important rates in Piketty’s analytic framework are:

r – the average rate of return to capital

s – the savings rate

g – the rate of growth of national income

β – the ratio of national wealth to national income

Piketty demonstrates the following results: For any given current values of s and g, the capital-to-income ratio will be rising so long as s/g is greater than the current value β, and it will be falling if s/g is less than the current value of β. We can also very easily compute the annual rate of wealth increase as s/β. (For one example of this computation, see my piece on Piketty’s second fundamental law, cited above.) We can see quite evidently from this result that wealth will be growing faster than national income whenever β is less that s/g, and will grow more slowly than national income whenever β is greater than s/g. If β happens to be exactly equal to s/g, then wealth will grow at the same rate as national income.

But, contra Wolfers and Summers, the rate of wealth growth can obviously be quite far from r, and doesn’t even have anything directly to do with r. For example, if in some given year we have s = 10%, β = 4 and g = 2%, then wealth will growing at a rate of 2.5% annually, which is certainly greater than g, but much less than the typical historical values Piketty finds for the rate of return on capital of between 4% to 5%. And, as Bruenig notes, β can be increasing even when r < g, so long as the value of β remains below s/g. So wealth can grow faster than income even if r < g.

A question then arises as to the circumstances under which an increasing β will drive an increase in inequality. Of course, since capital is always unequally distributed, then capital income is also unequally distributed; and any increase in the capital-to-income ratio will likely result in an increase in capital income flowing to the owners of wealth in a pattern that will tend to reflect the prevailing inequalities in capital ownership. But the question is whether this will result in greater or less inequality in the share of wealth and income going to various classes of people in the economy.

Piketty’s analytic machinery for addressing this question is complex and nuanced, and depends on a number of independently moving parts. For one thing, as he emphasizes, inequality occurs along several different dimensions, and there is no one single parameter that can capture all of these dimensions equally well.  Also, the dynamics of inequality depend not just on the rates discussed above, but also on further inequalities in the rates at which different kinds of income recipients are able to save their incomes and on differences in the average rates of return captured by various classes of capital owners. But one convenient framework for getting at some of Piketty’s central points is to focus on the class of pure rentiers: those people whose entire income is capital income. I introduced this framework, and developed it a bit, in “Piketty on the Dynamics of Inequality: Four Useful Theorems.” One thing I did in that post was calculate the conditions under which the pure rentiers’ wealth share and income shares would be increasing. It turns out that rentiers’ income share will increase just in case ρr_R > g, where ρ is the rate at which rentiers save and r_R is the rate of return rentiers receive on their wealth. Note that it follows from this formula that if r_R = r, that is, if rentiers receive the same average rate of return as all capital owners in the economy, and if r < g, then no matter how much rentiers save their income share will decrease, since ρ is always a value between zero and one. For the rentier income share to grow under conditions of r < g, rentiers must earn returns on capital at a higher rate than the average capital owner.

I hasten to add at this point that Piketty’s analysis of the structure of inequality, especially his analysis in Chapter 12 of the growth of top incomes in the global economy, does make use of the fact the wealthiest owners of capital tend to achieve rates of return on their wealth significantly higher than average.

But returning to the main subject of this post, I have to say that the numerous fallacies surrounding Piketty’s views on the role of r – the rate of return to capital – in determining the structure of inequality have become quite annoying. Most of these fallacies are easy to avoid and the relevant computations involve only elementary algebra. Why are so many smart people making them? By perpetuating these elementary fallacies and attributing them to Piketty, prominent authors are needlessly damaging Piketty’s intellectual reputation and public understanding of Piketty’s important work.