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 out 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.

In my post “Piketty on the Dynamics of Inequality: Four Useful Theorems”, I tried to focus attention on the most important phenomena involved in Piketty’s account of increasing inequality by focusing on an imagined class of *pure rentiers*: people who perform no paid labor, and all of whose income is therefore income from capital. I then proved some theorems about the rates at which rentier income and wealth grow over time, and the conditions under which the rentier share of income and rentier share of wealth will be increasing. I want to focus here on the rentier share of income.

I computed the rate of growth of the rentier *share* of national income to be:

- (1+ρr
_{R})/(1+g) – 1,

where ρ is the rate at which rentiers save, r_{R} is the rate of return rentiers receive on their wealth and g is the rate of growth of national income. Since the rentier share of national income will be growing whenever the value of this formula is greater than 0, it can easily be seen that the rentier share of income is increasing if

- ρr
_{R}> g.

I also pointed out that rentier wealth grows at the same rate as rentier income, so that if rentier income is growing faster than national income and the rentier share of income is thus increasing, then rentier wealth is also growing faster than national income. Now let’s suppose our pure rentiers receive the same average return on their capital as the population as a whole, so that r_{R} = r. Then the rentier share of income is increasing just in case:

- ρr > g

Now suppose that r ≤ g. Then since ρ is always a number between 0 and 1, it follows that no matter what proportion of their income they save each year, the rentiers’ share of income can’t go up. In order for the rentiers’ income share to increase, at least under the simplifying assumption that r_{R} = r, r has to be *greater* than g.

And notice, by the way, that this formula allows us to see why the *ratio* of r to g is hugely important in Piketty’s framework. If r/g = 5, for example, then rentiers need only save some amount in excess of 20% of their income to get a rising income share. If r/g = 3, on the other hand, then the rentier share of income can increase if the rentiers save more than 1/3^{rd} of their income. But if r = g or r is less than g, then even if the rentiers’ save 100% of their income, and receive the full average return on their savings, their share of income cannot increase.

*That’s* why r > g is so important in Piketty’s analysis. Given a uniform return on capital r, the truth of r > g is a *necessary condition* for rentier income share growth, and the greater the *difference* between r and g, the more propitious are the circumstances for this kind of growing inequality.

Piketty emphasizes these points repeatedly in his book. He does not employ my simplifying tool of looking at pure rentiers. Rather, in bringing out the importance of r > g for the structure of inequality, he analyzes growing inequality in terms of the dynamics of inherited wealth, i.e. he looks at the role played by flows of inherited wealth and flows of income from inherited wealth. But I believe the frameworks point in fundamentally the same direction. For if you divide each capital owner into two parts, so to speak, the part that is a pure rentier and receives only income from capital, and the part that is a pure laborer and receives only labor income, then the dynamics of income from inherited wealth will map onto the dynamics of pure rentier income. (In a future post, I will present Piketty’s inherited wealth framework in more detail.)

So we see that r > g is, *by itself*, important as a necessary condition on the growth of rentier income shares under conditions of roughly equal returns on capital. But while it is a *necessary* condition, it is not a *sufficient* condition. The role of savings rates among capital owners is extremely important to Piketty’s account. We can only understand Piketty’s full argument by returning to the actual text of *Capital in the Twenty-First Century*, because that argument has been much compressed and severely distorted by many of the most popular accounts and reviews of the book. Here is what Piketty says when he first introduces the role of r > g as part of his discussion of the inequality of capital ownership:

*Several mechanisms may be at work here, and to my knowledge there is no evidence that would allow us to determine the precise share of each in the overall movement. We can, however, try to hierarchize the different mechanisms with the help of the available data and analyses. Here is the main conclusion that I believe we can draw from what we know.*

*The primary reason for the hyperconcentration of wealth in traditional agrarian societies, and to a large extent in all societies prior to World War I (with the exception of the pioneer societies of the New World, which are for obvious reasons not representative of the rest of the world or the long run) is that these were low growth societies in which the rate of return on capital was markedly and durably higher than the rate of growth.* (pp. 350-51)

He then goes on to explain how the “fundamental force for divergence” functions. He first presents a hypothetical example of a society in which g = 1% and r is 5%, and then says that “saving one-fifth of the income from capital (while consuming the other four-fifth) is enough to insure that capital inherited from the previous generation grows at the same rate as the economy.” Note the perfect consistency of this statement with the formula ρr > g that I presented above, if we assume the mapping between the model based on pure rentier wealth and the model based on inherited capital.

Piketty then describes the historical example of France during the period 1820 to 1913, where r was in reality around 5% on average and g was approximately 1%, while capital income was around 40% of all income. He then points out that “it was enough to save one-quarter of this” capital income to generate an overall national savings rate of 10%, and this was “sufficient to allow wealth to grow slightly more rapidly than income, so that the concentration of wealth trended upwards.” Again this is precisely as we would expect given what has been discussed above.

Also, note that in developing his detailed analyses, Piketty rarely refers to the bare fact that r > g, but always stresses the size of the gap, using terms like “significantly”, “markedly” or “distinctly”. For example, in addition to the passage quoted above, we have:

*The fact that the return on capital is distinctly and persistently greater than the growth rate is a powerful force for a more unequal distribution of wealth.* (p. 361)

*… once the rate of return on capital significantly and durably exceeds the growth rate, the dynamics of the accumulation and transmission of wealth automatically lead to a very highly concentrated distribution, and egalitarian sharing among siblings does not make much of a difference*. (p. 364)

*Whenever the rate of return on capital is significantly and durably higher than the growth rate of the economy, it is all but inevitable that inheritance (of fortunes accumulated in the past) predominates over saving (wealth accumulated in the present.) *(pp. 377-8)

Note the similarity of the expressions:

“markedly and durably,”

“distinctly and persistently,”

“significantly and durably.”

So in addition to the fact that the literal inequality r > g is in itself a necessary condition for certain kinds of capitalist income divergence, I think it is fair to say that Piketty also uses the expression “r > g” as a kind of convenient verbal shorthand for the more complex idea that *r is significantly and persistently greater than g*.

One final topic is worth mentioning in this context. There is a debate going on between Debraj Ray and Branko Milanovic over Piketty’s book. The debate contains several threads, but one key point of contention is what appear to be Ray’s claim that Piketty is simply wrong in holding that r > g has anything at all to do with growing inequality, and his further claim that Piketty fails to recognize the role of the size of capitalist savings *net of consumption* out of their total income. But both claims seem incorrect to me. The first claim is incorrect for the reasons discussed above. If we look at societies in which rates of return to capital are relatively consistent across all capital owners, then r > g is a necessary condition on the growth of the rentiers’ income share.

And the second claim is incorrect because, as we have seen, Piketty is fully aware of the important role of capitalists’ savings rates in generating increased inequality. The fact that increasing inequality depends not just on r and g, but on the rate at which inheritors of capital save their capital income is quite an important part of Piketty’s own analysis, not an oversight. We have already cited the passage from page 351 where he employs these considerations. But we should also call attention to Piketty’s projections for 21^{st} century inheritance flows that are discussed on pages 398-401. In this passage, Piketty makes it clear that he assumes savings behavior similar to that observed in the past, which includes increased savings rates correlated with both higher incomes and the size of the initial endowment. And the savings rates he is talking about are savings of *total* income; he is not assuming there is something special about capital income.

So I don’t think Professor Ray is detecting a lacuna in Piketty’s argument. He’s just pointing to an incompleteness in the popular capsule summaries of Piketty’s book that have been circulated in the Republic of Punditry. One such capsule summary goes something like this:

“r > g, therefore disaster!”

But Piketty’s actual argument in the text of *Capital in the Twenty-First Century* is much more complex and nuanced.

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The way I would look at this is through the eyes of ‘Harrod neutral growth’. If labour saving inventions and capital saving inventions are balanced then the capitalists to maintain their rate of return will increase capital in line with output. The capital/output ratio remains the same. Thus if growth is 2% capital will also increase by 2% and the residue of the capitalists’ income will be consumed. The ratio of the capital owned to total income will remain the same. They won’t get relatively richer.

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